T L Peck

Al J E Strocker

Al J Dechow

Al L C Renaud

Al C Massin

Massin

Keywords: NMR micro-antenna, Micro-probe, Sensitivity processing, Artificial Neural Network Table I. 2

well as Mr. Christian GONTRAND, from INL laboratory, for all their support and wise advice, otherwise my thesis work would not be achieved with a great quality.

I warmly thank Mr. Lionel ROUSEAU for his availability, his advice and the interest showed in my thesis subject as well as for the achievements made at Cleanroom of l'ESIEE Paris, during the prototype manufacturing. I would also like to thank Sir. Stephane ROQUES and all the members of the cleanroom, at Icube laboratory, Strasbourg for their help in the characterization of microprobe prototypes. La maladie d'Alzheimer (MA) est l'une des maladies neurodégénératives les plus fréquentes chez les personnes âgées. Selon le dernier rapport mondial sur la MA, près de 55 millions de personnes vivent avec la démence de la MA. En France, plus d'un million de personnes souffrent de cette démence, et cela arrive parfois, avant l'âge de 65 ans. La perte de mémoire est souvent le premier symptôme de la MA qui permet d'orienter le diagnostic ; il s'agit donc d'un enjeu de santé publique.

Cette maladie neurodégénérative résulte de la progression de deux principales lésions cérébrales : l'accumulation anormale de protéines appelées peptide amyloïdes (plaques séniles) et de protéines conduisant à la dégénérescence des neurones. L'évolution progressive de la MA peut donc être détectée 20 ans avant l'apparition des symptômes de démence (phase prodromique), par le biais des certains marqueurs fonctionnels et/ou métaboliques.

Malheureusement, le diagnostic formel de la MA ne peut être obtenu que par autopsie du cerveau post-mortem. Il convient de noter que les marqueurs qui reflètent l'évolution de la maladie peuvent être biochimiques ou anatomiques.

L'objectif majeur de la recherche clinique actuelle sur la MA est d'améliorer la détection précoce et présymptomatique du dysfonctionnement des neurones. Ainsi, le domaine de la recherche sur la MA est très actif, et plusieurs essais de traitements ont déjà été effectués. Par exemple, les essais visant à nettoyer les plaques séniles ont été mis en place, mais les résultats n'ont pas été concluants jusque-là. Une raison possible serait que les patients qui ont participé à l'essai étaient à un stade trop avancé de la maladie. L'origine de la MA et sa progression ne sont donc pas encore très bien comprises, un diagnostic précoce devrait permettre de mieux comprendre cette pathologie et de mettre en place les traitements pharmacologiques actifs nécessaires pour ralentir la maladie, si possible avant qu'elle n'ait un impact sur la qualité de vie du patient.

Actuellement, malgré les efforts pour diagnostiquer la MA grâce aux techniques de neuroimagerie par les méthodes de suivi de l'inflammation (atrophie) de l'hippocampe par Imagerie par Résonance Magnétique (IRM) fonctionnelle ou structurelle, ou encore le suivi des biomarqueurs du liquide céphalo-rachidien ou le suivi des biomarqueurs sanguins, toutefois le diagnostic de la MA est actuellement de façon tardive. Il faut noter que certaines de ces méthodes de diagnostic ne fournissent pas d'informations reproductibles.

Pour notre étude, nous avons choisi la technique de Spectroscopie RMN (SRM) pour suivre l'évolution des biomarqueurs (métabolites cérébraux) qui pourraient éventuellement être représentatives de la MA, grâce à un modèle animal (la souris ou le rat). La Résonance Magnétique Nucléaire (RMN) est une méthode d'analyse très puissante pour identifier des échantillons chimiques et biologiques. Au cours des 20 dernières années, elle est devenue un outil incontournable pour le diagnostic des maladies ; ainsi la SRM semble être un bon candidat pour notre recherche en raison de la capacité de la technique à quantifier la concentration de certains métabolites dans une zone d'intérêt très précise. Néanmoins, l'obtention de spectres RMN in vivo de haute résolution constitue un véritable enjeu méthodologique. En effet, la principale limitation de la RMN est sa sensibilité en termes du rapport signal sur bruit (SNR)

qui décroît lorsque l'on cherche à augmenter la résolution spatiale. Pour tenter de s'affranchir de cette limitation du SNR, sans une trop forte pénalisation en termes de résolution spatiale ou de temps d'acquisition, il est judicieux d'améliorer l'aimantation transversale moyenne mesurée pendant la réception du signal et également la sensibilité de détection.

Plusieurs stratégies ont été proposées dans la littérature, visant à améliorer la sensibilité RMN:

1. L'augmentation de l'intensité du champ statique B0, puisque la sensibilité RMN en spectroscopie varie comme / . Des aimants supraconducteurs avec des champs statiques de 4,7 (200 ) jusqu'à 23,3 (1 ) sont désormais utilisés en spectroscopie « conventionnelle ».

2. L'amélioration de la sensibilité de détection par :

L'amélioration du couplage magnétique entre l'antenne assurant la réception du signal RMN et l'échantillon par l'adaptation de la taille du capteur avec celle de l'échantillon. Le facteur de remplissage est ainsi augmenté (ƞ ≈ 1) et par conséquent, le SNR est augmenté et la sensibilité de mesure sera améliorée, grâce à une sensibilité optimal du capteur RF.

La réduction des sources de bruit thermique par le refroidissement de l'antenne de détection RMN.

3. La réduction de la largeur de raie : la mise en rotation rapide de l'échantillon autour d'un axe à 54,7 ° de la direction du champ (Magic Angle Spinning) est utilisée depuis longtemps pour réduire l'élargissement dipolaire dans les milieux solides.

4. L'utilisation de réseaux d'antennes : ici, on ne joue plus sur des facteurs physiques, mais sur des éléments techniques et instrumentaux puisqu'il s'agit de superposer les signaux de capteurs indépendants.

Dans ce travail de thèse, notre stratégie portrait sur l'amélioration de l'antenne de détection en termes de rapport signal sur bruit (SNR). Comme on le sait, le SNR est lié à deux principaux paramètres qui peuvent être améliorés : d'une part, le champ magnétique en adaptant la taille de la micro-bobine à celle de l'échantillon, et d'autre part, la résistance équivalente R de l'antenne qui peut être minimisée. Par ailleurs, nous pouvons également apporter un soin particulier au processus de microfabrication des antennes.

Pour cette étude, nous avons appelé micro-antenne RMN, la micro-probe associée à son circuit Par ailleurs, le principal inconvénient de la micro-probe réside dans sa faible sensibilité en termes de Rapport Signal sur Bruit ( ) due à la résistance globale de la micro-bobine et à celle de la ligne de transmission et de la partie de connectique (wire-bonding or underpass & vias), y compris la résistance supplémentaire correspondante au bruit de couplage élevé entre la parie active de la micro-probe (micro-bobine) avec l'échantillon. Notre première approche pour la réalisation de la micro-probe pour des applications in vitro a été présentée dans les thèses de N. [START_REF] Baxan | Mise En Oeuvre de Microantennes RMN En Perspective d'étude in Vivo de Metabolites Par Spectroscopie[END_REF] et A. [START_REF] Kadjo | Micro-Capteur Implantables : Etude Des Criteres de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie RMN in Vivo[END_REF] Pour la réalisation de ce nouveau prototype de micro-probe, nous avons reparti notre approche en deux parties principales :

1. Simulation de performance en termes de facteur de qualité et SNR Nos approches de simulations nous ont permis de déterminer les critères de performance optimal pour les prototypes de micro-probes (voir Figure . 2) et de le valider par la modélisation à l'aide du logiciel ADS. Cependant, ce travail de simulation est très chronophage et il reste difficile de déterminer l'impact de chaque partie qui constitue le dispositif (micro-probe). Le machine Learning (ML) c'est un champ de l'IA qui permets aux programmes des ordinateurs d'exécuter des taches sans être explicitement programmé pour la réaliser. Cela est possible par l'apprentissage de l'algorithme en utilisant une expérience de l'ensemble de données.

Un modèle ANN qui est une des branches de ML nécessite un grand jeu de données correspondant aux entrées et à leur cible (sorties). Dans le jeu de données, pour notre étude, nous pouvons noter les prédicteurs et les cibles comme suit : Les variables indépendantes peuvent être liées à la cause des états d'un système ou à la caractéristique du système pour prédire l'évolution (conventionnellement dénotées par X : Prédicteurs). La variable dépendante peut être considérée comme l'état final du système ou la cible, comme le but final de l'étude ou la tentative de prédiction (conventionnellement dénotée par Y : Target). Certes, pour notre jeu de données, les cibles de la simulation sont bien connues, qui correspondent aux performances de la microsonde en termes de facteur Q et de résistance R, ainsi que les prédicteurs du système, qui sont les paramètres géométriques, les propriétés électriques du substrat, le domaine de fréquence.

Prédicteurs (X)

Dans cette section, nous présenterons sur la Figure. Le principe de la méthodologie de notre approche consiste en plusieurs étapes : Génération de données à partir des simulations de performance de la microsonde.

Pré-traitement des données pour voir la distribution des données, sélectionner les paramètres optimaux et nettoyer les données si nécessaire.

Entraîner le modèle d'IA, en sélectionnant l'algorithme de d'optimisation des paramètres et hyperparamètres optimaux pour atteindre une bonne précision pour la généralisation du modèle.

Tester le modèle avec de nouvelles données et évaluer sa précision.

Grâce aux données de simulations réalisées à l'aide de la plateforme 3D-TLE, nous avons développé un modèle par l'approche IA, basé sur une architecture de machine Learning, Réseau Artificiel de Nuerons (RAN) pour démontrer la possibilité de prédire les critères de performance de la micro-probe (facteur Q et résistance ) à partir de sa géométrie, de la nature Pour trouver le meilleur modèle pour entraîner les données, nous devions définir la forme du modèle (architecture du réseau de neurones), sélectionner les meilleures caractéristiques de nos données, plus précisément identifier quels prédicteurs sont importants, et régler plusieurs paramètres de l'architecture (nombre de neurones dans les couches cachées, la profondeur du réseau, définir les fonctions d'activation pour chaque couche) afin de trouver les hyperparamètres optimaux. En général, c'est un travail d'itération qui est effectué pour trouver le meilleur ajustement pour nos données par rapport au modèle (hyperparamètres optimaux).

Suite à l'essai de plusieurs architectures, avec un réseau peu et très profond, notre modèle choisi est l'architecture Shallow qui correspond à une architecture ayant une seule couche cachée. Cependant, intuitivement, elle ne semblait pas être le meilleur choix au vue de sa simplicité en comparaison avec d'autres l'architectures et de la complexité du problème. L'optimisation du nombre de neurones dans la couche cachée a été réalisée grâce à un script développé sur MATLAB. Nous avons fait 1000 itérations pour trouver le nombre optimal de neurones dans la couche cachée (hyperparamètre), et 500 itérations pour trouver les paramètres optimaux pour le réseau.

Pour évaluer la qualité de notre modèle, il était important de diviser nos données (jeu des données). Nous les avons donc répartis aléatoirement de la manière suivante : 70 % de nos données ont été utilisés pour l'entraînement, 15 % pour la validation et 15 % pour le test.

Artificial Neuronal architecture

Model training parameters

Model performances

En général, il est souhaitable qu'un modèle puisse tenir compte de la tendance des données, sans s'adapter au bruit dans les données. L'ajustement du bruit est appelé « surajustement » (overfitting). L'utilisation de données de validation pendant le processus de training nous a permis d'éviter l'overfitting. Pour les données de validation, il existe deux approches : la validation croisée (cross validation) et la méthode de validation hold-out, celle qui a été retenue pour l'entraînement de notre modèle. Notre modèle entraîné nous a permis de prédire le facteur de qualité et la résistance, pour 60 observations. Sur les figures qui suivent, nous avons l'allure du factor de Qualité et l'allure de résistance de la micro-probe, dans les cas de données de tests (Figure . 12). L'approche IA, plus particulièrement ANN associée à 3D-TLE, est une méthode que nous avons développée pour généraliser l'optimisation de notre système multiparamétrique très complexe (micro-probe). Notre modèle ANN a été entraîné pour prédire le facteur Q de la micro-probe avec une précision de test de 99,67 % ( = 0,032), la prédiction de la The range that we choose for the process optimization Alzheimer disease (AD) is one of the most frequent neurodegenerative diseases in older people. According to the latest world AD report, nearly 55 million people live with this dementia. AD results from the progression of two brain injuries: Abnormal accumulation of proteins called peptide amyloid (senile plaques) and tau proteins (neurons degeneration). However, AD progressive evolution can be detected 20 years before the onset of dementia symptoms (prodromic phase), by certain functional and metabolic markers. Unfortunately, AD is still facing a late diagnosis. NMR spectroscopy (MRS) appears to be a good candidate for AD biomarkers assessment due to its ability to quantify cerebral metabolites concentration in a very precise area of interest (ROI). Unfortunately, the NMR main disadvantage is its sensitivity. Several works from the literature report some approaches to improve this sensitivity. In this thesis work, our strategy is to improve the NMR receiver (RF antenna) in terms of Signal to Noise Ratio (

). As known, the is related to the two main parameters that can be enhanced; on the one hand, the magnetic field by adapting the micro-probe size to the sample, and on the other hand, the equivalent Resistance which can be minimized. In addition, we can also improve the antenna microfabrication process.

For this study, we have named NMR micro-antenna, the micro-probe associated with its matching and tuning circuit. The micro-probe is the micro-coil part associated with a micro-Transmission Line (TL) and the different connecting types. Usually, the micro-probe manufacturing task is time-consuming and expensive if we have to explore all prototypes dimensions and compare them experimentally. So, simulation makes possible the examination of different scenarios without experiments to then select the main parameters for the required performances. It can also provide meaningful data trends and guide the manufacturing process of the micro-device. In this work, we propose a new approach to optimize an implantable NMR micro-probe. We developed a platform (3D-TLE: Transmission Line Extractor) associated with Advanced Design System (ADS) software to extract the parameters of the microprobe design. Thus, we obtained a complete electrical circuit to model any micro-probe shape using any type of substrate. The simulations were carried out for four types of micro-coil (square, circular, rectangular and ellipsoidal), with two types of substrates (glass and silicon respectively), at the frequency range of 200 to 900 . In the case of rectangular micro-coil (1000 × 500 μ ²), the global resistive loss was around = (3.897 ± 0.193) Ω, taking into the account the contribution of the different parts of the micro-probe. Furthermore, we also developed an AI model, in particular an Artificial Neural Network (ANN), which we trained with the dataset generated by micro-probe simulations (6621 observations). Our trained ANN predicted the micro-probe factor with an accuracy of 99.67 % ( = 0.032), and the Resistance with an accuracy of 91.34 % ( = 0.042).

Based on previous results, we fabricated the different micro-probe prototypes on a glass and silicon substrate. Thanks to the also optimized fabrication process, we reached very small micro-coil dimensions up to 250 × 250 μ ², never achieved before. Our work shows that we can accurately design and fabricate any optimal micro-probe or probe through the required performances, working frequency and substrate type with less computing time (2 minutes instead of more than 15 hours). Thus, this new optimization approach using AI tools combined with simulation tools (multi-parameter optimization work) implies a breakthrough in biomedical research towards NMR micro-detector used for brain exploration, early diagnosis, and treatment follow up.

Résumé

Mots-clefs : Micro-antenne RMN, Micro-sonde, Sensibilité, Réseau Neuronal Artificiel, Spectroscopie RMN, Maladie d'Alzheimer, Biomarqueurs, Microfabrication.

La maladie d'Alzheimer (MA) est l'une des maladies neurodégénératives les plus fréquentes chez les personnes âgées. Selon le dernier rapport mondial sur la MA, près de 55 millions de personnes vivent avec cette démence. La MA résulte de la progression de deux principales lésions cérébrales : L'accumulation anormale de protéines appelées peptide β amyloïde (plaques séniles) et de protéines tau (dégénérescence des neurones). Cependant, l'évolution progressive de la MA peut être détectée jusqu'à 20 ans avant l'apparition des symptômes de démence (phase prodromique), par le biais de certains marqueurs fonctionnels et/ou métaboliques. Malheureusement, le diagnostic formel de la MA ne peut être obtenu que par autopsie du cerveau post-mortem. La Spectroscopie RMN (SRM) semble être un bon candidat pour l'évaluation des biomarqueurs de la MA en raison de sa capacité à quantifier la concentration des métabolites cérébraux dans une zone d'intérêt (ROI) très précise. Cependant, le principal inconvénient de la RMN est sa sensibilité. Plusieurs approches d'amélioration de cette sensibilité ont été proposées dans la littérature. Dans ce travail de thèse, notre stratégie consiste à améliorer le récepteur RMN (antenne RF) en termes de rapport signal sur bruit (

). Comme on le sait, le est lié à deux principaux paramètres qui peuvent être améliorés : d'une part, le champ magnétique en adaptant la taille de la micro-sonde à celle de l'échantillon, et d'autre part, la résistance équivalente de l'antenne qui peut être minimisée. Par ailleurs, nous pouvons également apporter un soin particulier au processus de microfabrication des antennes.

Pour cette étude, nous avons appelé micro-antenne RMN, la micro-sonde associée à son circuit d'adaptation et d'accord. La micro-sonde est la partie de la micro-bobine associée à la micro-ligne de transmission (TL) et aux différents types de connectiques. En général, la fabrication de la micro-sonde est longue et coûteuse, quand il s'agît d'explorer plusieurs prototypes de dimensions et matériaux différents et les comparer expérimentalement. La simulation permet d'en affranchir et examiner tous les scénarios possibles sans expérimentations, puis de sélectionner les principaux paramètres pour obtenir les performances requises. Elle peut également fournir des tendances de données significatives et guider le processus de fabrication. Dans ce travail, nous avons proposé une nouvelle approche pour optimiser une micro-sonde RMN implantable. Nous avons développé une plateforme (3D-TLE : Transmission Line Extractor) associée au logiciel Advanced Design System (ADS) pour extraire les paramètres de la microsonde. Nous avons ainsi proposé un circuit électrique complet permettant de modéliser n'importe quelle forme de micro-sonde et son type de substrat. Nous avons effectué des simulations pour quatre types de micro-bobine (carrée, circulaire, rectangulaire et ellipsoïdale), avec deux types de substrat (verre et silicium), dans une gamme de fréquences de 200 à 900 . Dans le cas de la micro-bobine rectangulaire (1000 × 500 μ ²), la perte résistive globale, en tenant compte la contribution des différentes parties de la micro-sonde a été estimée à = (3,897 ± 0,193) . En outre, nous avons également développé un modèle

Introduction

Alzheimer disease (AD) is one of the most frequent neurodegenerative diseases in older people. According to the latest world AD report, nearly 55 million people live with this dementia [START_REF] International | Journey through the Diagnosis of Dementia[END_REF]. AD results from the progression of two brain injuries: Abnormal accumulation of proteins called peptide amyloid (senile plaques) and tau proteins (neurons degeneration).

However, AD progressive evolution can be detected 20 years before the onset of dementia symptoms (prodromic phase) by certain functional and metabolic markers [START_REF] Frisoni | The Clinical Use of Structural MRI in Alzheimer Disease[END_REF]. Unfortunately, AD is still facing a late diagnosis.

For our study, we chose the NMR Spectroscopy (MRS) technique to follow the evolution of biomarkers (brain metabolites) that can possibly be representative of AD, using an animal model (mouse or rat). Nuclear Magnetic Resonance (NMR) is a very powerful analytical method to identify chemical and biological samples. Over the last 20 years, it has become an essential tool for the diagnosis of diseases. MRS seems to be a good candidate for our research because of the technique's ability to quantify the concentration of certain metabolites in a very precise area of interest (ROI). Nevertheless, obtaining high resolution in vivo NMR spectra is a real methodological challenge. Unfortunately, the NMR main disadvantage is its sensitivity.

Several strategies have been proposed in the literature, to improve the NMR sensitivity:

1. Increasing the static field strength , as the NMR sensitivity in spectroscopy varies as [START_REF] Hoult | The Signal-to-Noise Ratio of the Nuclear Magnetic Resonance Experiment[END_REF]. Superconducting magnets with static fields from 4. 7 (200 ) to 23.3 (1 ) are now used in "conventional" spectroscopy.

2. The improvement of the detection sensitivity by: Improving the magnetic coupling between the antenna receiving the NMR signal and the sample by adapting the size of the sensor to that of the sample [START_REF] Hoult | The NMR Receiver: A Description and Analysis of Design[END_REF]. The filling factor is thus increased (ƞ ≈ 1) and consequently, the SNR is also increased and the measurement sensitivity will be improved, thanks to an optimal sensitivity of the RF sensor [START_REF] Griffin | Metabolic Profiling of Rodent Biological Fluids via 1H NMR Spectroscopy Using a 1 Mm Microlitre Probe[END_REF].

The reduction of thermal noise sources by cooling the NMR detection antenna.

3. Reduction of the linewidth: the rapid rotation of the sample around an axis at 54.7° from the field direction (Magic Angle Spinning) [START_REF] Tsang | Metabolic Characterization of Distinct Neuroanatomical Regions in Rats by Magic Angle Spinning1H Nuclear Magnetic Resonance Spectroscopy[END_REF] has been used for a long time to reduce the dipole broadening in solid media [START_REF] Rabeson | Quantitation with QUEST of Brain HRMAS-NMR Signals: Application to Metabolic Disorders in Experimental Epileptic Seizures[END_REF].

4. The use of antenna arrays: here, we no longer play on physical factors, but on technical and instrumental elements since it is a question of superimposing the signals of independent sensors [8] [9].

In this thesis work, our strategy is to improve the NMR receiver (RF antenna) in terms of Signal to Noise Ratio ( ). As known, the SNR is related to the two main parameters that can be enhanced; on the one hand, the magnetic field by adapting the micro-probe size to the sample, and on the other hand, the equivalent Resistance which can be minimized. In addition, we can also improve the antenna microfabrication process.

For this study, we called NMR micro-antenna, the microprobe associated with its matching and tuning circuit. The microprobe is the part of the micro-coil associated with the microtransmission line (TL) and the different types of connectors. The connecting wire allow linking the center of the micro-coil with the transmission line, and another one that will connect the transmission line with the tuning and adaptation circuit. Our implantable NMR microprobe (also called needle coil) should lead to an analysis tool that ensures high-resolution spectroscopic measurements reproducibility and encourages an in vivo application. More specifically, to monitor the variation of cerebral metabolic markers in small volume with low concentration by NMR spectroscopy using animal models (rat or mouse).

Our first approach performed in vitro, presented in the thesis of N. [START_REF] Baxan | Mise En Oeuvre de Microantennes RMN En Perspective d'étude in Vivo de Metabolites Par Spectroscopie[END_REF] and A. [START_REF] Kadjo | Micro-Capteur Implantables : Etude Des Criteres de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie RMN in Vivo[END_REF], could not be achieved in vivo without a complementary work (design and simulations) for the optimization of the microprobe sensitivity ( ). This is of crucial importance because the source signals are so small that they are comparable to thermal noise (sensitivity of the microprobe), which can be linked to its resistance. To overcome these resistive losses of the micro-antenna proposed by N. developed a method for optimizing NMR micro-coils using a physical approach. Despite these improvements, the quality factor remained low (e.g., = 1.4) and insufficient for the expected in vivo applications.

We aim to develop an NMR implantable microprobe, leading to an accurate analysis of metabolites biomarkers for in vivo applications (in particular, early diagnosis of Alzheimer's disease). Therefore, the objective of my thesis is to propose a new approach to optimize an implantable, biocompatible NMR microprobe specially designed for the target application which requires a high resolution and highest sensitivity.

In the following of my thesis document, I have divided my work in four chapters:

Chapter 1 presents the general context of the project (an introductory part to Alzheimer disease earlier diagnostic), the problematic and the thesis objectives. We briefly introduced the main NMR principles, the NMR signal detection, the state-of-the-art of NMR planar antennas, and the micro-antennas, the definition of microprobe performance parameters, and then we finalized this chapter with a review from previous work carried out by our team, based on the optimization of such microprobes and its prospect for in vivo applications.

In chapter 2, we presented the numerical modelling of the microprobe in terms of the This chapter aims to present an overview of the project thesis; we begin by introducing Alzheimer's disease, the markers and diagnosis methods, and the present study hypothesis for earlier detection of AD biomarkers. We also report a brief introduction to the NMR principle and its sensitivity.

We report the antenna criteria of performance, the state-of-the-art of planar NMR antennas.

We reviewed the work about the NMR micro-antenna optimization previously performed by our team to enhance its performances in terms of SNR and Q factor, for in vivo applications.

Moreover, according to our team previous results, we presented This neurodegenerative disease results in the progression of two brain injuries: Abnormal accumulation of proteins called peptide amyloids (senile plaques) and tau protein (neurons degeneration). Therefore, the disease's progressive evolution can be detected 20 years before the dementia symptoms (prodromic phase) appear. Note that biomarkers reflecting the state of the disease can be biochemical or anatomical.

The major goal of current clinical research in AD is to improve early detection and presymptomatic detection of neuron dysfunction [START_REF] González | Atlas of Biomarkers for Alzheimer's Disease[END_REF]. So, the AD research field is very active, and several treatments test are currently underway. Trials aimed at cleaning the main lesions observed in the brains of people with AD (the amyloid plaques) have been set up, with an unsuccessful result so far. No improvement in symptoms has been observed even though the amount of plaques is decreasing. A possible reason for this failure is that the patients who participated in the trial were too advanced in their disease. Today, more and more prevention trials are underway in which these plaques are targeted at very early stages of the disease when symptoms are very mild or even non-existent [START_REF]Les traitements de la maladie d'Alzheimer -ICM[END_REF].

The A very common approach used by the scientific community is the MCI as a prodromal phase, defined as a traditional stage between normal ageing and dementia, and grouping together subjects suffering from mild cognitive disorders. Subjects with these characteristics have a higher risk of developing AD later [START_REF] Petersen | Mild Cognitive Impairment: Clinical Characterization and Outcome[END_REF]. Therefore, in Fig. [START_REF] Popuri | Using Machine Learning to Quantify Structural MRI Neurodegeneration Patterns of Alzheimer's Disease into Dementia Score: Independent Validation on 8,834 Images from ADNI, AIBL, OASIS, and MIRIAD Databases[END_REF]. Others confirm with a similar precision of about 75 % [START_REF] Basaia | Automated Classification of Alzheimer's Disease and Mild Cognitive Impairment Using a Single MRI and Deep Neural Networks[END_REF].

As we can see in the previous paragraph, the field of neuroimaging markers has made revolutionary advances, and we have different neuroimaging modalities being applied to AD detection. Moreover, the authors are interested in combining different markers and test neuroimaging techniques to predict the possibility of AD conversion in the MCI area. However, neuroimaging is expensive and technically challenging [START_REF] Khan | Biomarkers in Alzheimer's Disease (Academic Press is an imprint of[END_REF].

I.1.3. The objectives of the presents study and hypothesis

Despite the efforts to diagnose AD thanks to neuroimaging techniques (structural, functional and molecular) by monitoring hippocampus inflammation (atrophy) using functional or structural Magnetic Resonance Imaging (MRI) [START_REF] Klunk | Imaging Brain Amyloid in Alzheimer's Disease with Pittsburgh Compound-B: Imaging Amyloid in AD with PIB[END_REF] [24], cerebrospinal fluid biomarkers [START_REF] Hulstaert | Improved Discrimination of AD Patients Using -Amyloid(1-42) and Tau Levels in CSF[END_REF] or monitoring blood biomarkers [START_REF] Mapstone | Plasma Phospholipids Identify Antecedent Memory Impairment in Older Adults[END_REF], but the AD is still facing a late diagnosis. Note that some of these methods do not provide reproducible information [START_REF] Snyder | Developing Novel Blood-Based Biomarkers for Alzheimer's Disease[END_REF]. The area under each NMR peak corresponds to the concentration of that particular metabolite. Differences in the peak area of metabolites in the rat brain provide a measure of the level of neurochemical processing, which is reflective of the pathophysiologic state. Moreover, the cerebral metabolites and also their concentration in the specific ROI are very well known [29] [28]. The point is to quantify the concentration of these metabolites and lead to a possibility of monitoring their variation, distinguishing the abnormal changes corresponding to neurodegenerative diseases such as Alzheimer's disease.

Magnetic

Paul Francis and co-workers examined the original cholinergic hypothesis of AD in a review [START_REF] Francis | The Cholinergic Hypothesis of Alzheimer's Disease: A Review of Progress[END_REF]. Their study describes the biochemical changes of neurotransmitter markers occurring in the patient's brain with AD. Although, several studies focused on neurodegenerative diseases that result in disruptions of the cholinergic innervation network. Cholinergic system deficiency is found mainly in patients with Alzheimer's disease [START_REF] Francis | The Cholinergic Hypothesis of Alzheimer's Disease: A Review of Progress[END_REF] [31], with neocortical electroencephalogram (EEG) activity [START_REF] Puoliväli | Moderate Cortical EEG Changes in Apolipoprotein E-Deficient Mice during Ageing and Scopolamine Treatment but Not after Nucleus Basalis Lesion[END_REF], Parkinson [START_REF] Pahapill | The Pedunculopontine Nucleus and Parkinson's Disease[END_REF] and with epilepsy [START_REF] Steriade | Bases Cellulaires Des Transitions de l'état de Sommeil Aux Paroxysmes Épileptiformes[END_REF].

∆E = γ ħB (I.3)
According to the electromagnetic principle, we know that, ∆ = ħ ; So, we deduce:

ω = γ B (I.4)
The equation (I.4) by a classical approach is associated with the Larmor frequency. Thus, some values of the gyromagnetic ratio for attractive atomic nuclei are given in Table I For our application, we are effectively interested in the Proton, a nucleus that is naturally about 99.98% abundant and mainly present in brain tissue. According to the Fermi distribution, applicable to a particle with a half odd integer spin [START_REF] Eisberg | [END_REF], the spins per volume concentration of an energetic state as a function of the probability P and the total number of spins per volume is :

n = N P = ∈ ħ (I.5)
Moreover, according to Bloembergen [START_REF] Bloembergen | Relaxation Effects in Nuclear Magnetic Resonance Absorption[END_REF], the ratio of Boltzman distributions is a good approximation of the ratio of the concentration of the two equilibrium states is:

↓ ↑ = ↓ ↑ = e ∆ = e ħ (I.6)
The ratio ħ ≈ 6.81 × 10 , if the previous ratio is ≈ 0 we can make a Tylor secondorder series approximation of equation (I.6) as follows:

↓ ↑ ≅ 1 - ħ = 1 -ε B (I.7)
Besides, the summation of both energy state probability must be one ( ↓ + ↑ = 1), then we might deduce:

↑ ↓ ≈ 1 -ε B (I.8)
With:

P ↓ = (I.9)

P ↑ = (I.10)
Note that the difference between equation (I.9) and (I.10) is given as follows:

P ↑ -P ↓ = ε B (I.11)
From equation (I.11), we have defined the difference between both energetic states (level).

Thus, in terms of concentration is defined in the following equation:

∆n = N (P ↑ -P ↓ ) = N ħ (I.12)
These few differences between both energetic states justify the source of the Magnetization per volume vector [START_REF] Abraham | Introduction to NMR Spectroscopy[END_REF] [41] as the defined equation :

M = N ² ħ² ( ) (I.13)
The number of spins per volume unit is denied by the molecular concentration (mol/kg) the number of Avogadro and the number of nuclei participating in magnetic resonance into the molecule , weighted by the average percentage of the resonant isotropic . This percentage is representative of the abundance of the resonant isotope relative to the stable element [START_REF] Rosman | Isotopic Compositions of the Elements 1997 (Technical Report[END_REF].

N = n × C N (I.14)
Through the magnetization vector, we can measure NMR signals. We can also see that is directly proportional to the intensity and the magnitude . The signal is minimal (weak); that's why the NMR experiments require strong static magnetic fields, especially when analysing small samples.

I.2.3. The Radiofrequency (RF) magnetic field

To measure the Magnetization of a sample on the axis ( ), a transversal direction generally is required to apply an RF pulse. That signal corresponds to an RF magnetic field which is applied in a perpendicular axis's direction, compared to the the magnetic field axis. Therefore, the RF pulse causes the magnetization vector to tilt from its equilibrium position to a direction predefined by the chosen NMR sequence. The pulse sequence is transmitted using a planar or volumetric NMR antenna, depending on the application.

Two types of RF pulse are frequently used to switch the magnetization into a suitable plan for NMR measurements of the signal:

90 -RF Pulse switches the Magnetization from its equilibrium direction to the transverse plane.

180 -RF Pulse shifts the Magnetization from its equilibrium position to an antiparallel direction.

We can demonstrate by quantum mechanics theories; there is a transition of protons from a low energy level to a higher level (parallel protons are oriented in anti-parallel) when we apply an RF pulse. It occurred when the RF pulse achieved the same frequency as the Lamor frequency (equation (I.4)). It means that the system attends to the resonance condition = .

Relaxation is the phenomenon that makes possible the proton analysis, so there exist two types of relaxation.

Relaxation (longitudinal): Magnetization returns to the equilibrium position (spinnetwork).

Relaxation (transversal): Magnetization returns out the ions of states in phase (spinspin).

I.2.4. The detection of signals

The NMR signal can be detected using an RF receptor antenna after the sample ⃗ excitation.

The sample (atomic nuclei) magnetization ⃗ get precession around the static magnetic field ⃗ axis (generally ) having an angular velocity as defined in equation (I.4). There is a direct interaction between the Electromagnetic Force (EMF) induced in the S section by a rotating magnetic moment element in the sample volume V at point P;

meanwhile, the field ⃗ at the same point P produced by the current in the S section [START_REF] Hoult | The Principle of Reciprocity[END_REF]; consequently, EMF can be calculated as follows:

δξ(t) = -( ⃗ M ⃗ (t))dv (I.15)
Where:

( ) is the electromotive force induced into the turn S by an elementary volume , ⃗ is the magnetic field created in S section by the current .

To calculate the signal induced by the whole sample, we have to integrate the equation (I.15) into the complete volume , which correspond to the examined volume:

ξ(t) = -∭ ( ⃗ M ⃗ (t))dv (I.16)
Electromagnetic Field (EMF) signal can be calculated as the following equation (I.17), by Solving the equation (I. [START_REF] Sörensen | and for the Alzheimer Disease Neuroimaging Initiative, Amyloid Biomarkers as Predictors of Conversion from Mild Cognitive Impairment to Alzheimer's Dementia: A Comparison of Methods[END_REF]), considering that is an alternative sinusoidal signal and ⃗ ( )

Magnetization.
The macroscopic magnetization ⃗ under a static magnetic field, varies over time according to a law given by the Bloch equation:

⃗ ( ) = γ M ⃗ B + e ⃗ - e ⃗ - ⃗ ⃗ (I.16.1)
More details of the calculation of the ( ), even how to define ⃗ ( ) which corresponds to the macroscopic magnetization after the RF impulsion is applied, are given in [44] [4].

ξ(t) = S e * sin (ω t) (I.17)
With: There is an apparent decrease in FID because of the relaxation time and also to the field. The static magnetic field generally is never uniform at the microscopic scale, implying no synchronization of the elementary transverse magnetizations, which is faster than the relaxation due only to . Therefore, considering the inhomogeneity of the magnetic field , the true characteristic of relaxation time is represented by * called the true transverse relaxation time , as defined in the equation (I.17

S = ω M V (I.17

I.2.5. The NMR instrumentation

An MRI or MRS environment is defined by an NMR system that is composed largely of an intense static magnetic field , a variable RF magnetic field source , magnetic field gradient coils and antennas (transmitter and receiver). The functionality of the NMR instrument can be briefly described as follows: in the presence of the static magnetic field , the magnetic moments of the nuclei produce a nuclear paramagnetic polarization at equilibrium states. The application of the RF magnetic field , perpendicular to the constant magnetic field , generates a forced precession of the whole nuclei polarization around . This results in a component of the nuclear polarization, perpendicular to both the static magnetic field and the RF magnetic field , as it was recalled in the section.I.2.4 (chapter I), under standard laboratory conditions, this component can induce a measurable voltage signal (FID). This voltage is the NMR signal that the receiver antenna can then detect. This phenomenon will be followed by the progressive return of the magnetization to the equilibrium state.

Despite the technological progress of the NMR instrumentation, the main limitation of NMR technique is the sensitivity. So, several strategies to improve that sensitivity have been employed:

1. Increasing the static field , knowing that the NMR sensitivity varies as [START_REF] Hoult | The Signal-to-Noise Ratio of the Nuclear Magnetic Resonance Experiment[END_REF]. For Superconducting magnets with static fields starting with 4.7 ( 200) up to

(1

) that are now used in "conventional" spectroscopy.

The improvement of the detection sensitivity by:

Improving the magnetic coupling between the antenna receiving the NMR signal and the sample by reducing the size of the sensor [START_REF] Hoult | The NMR Receiver: A Description and Analysis of Design[END_REF]. The filling factor is thus increased, and consequently, the signal to noise ratio (SNR) of the receiver increase and the antenna will optimize the measurement sensitivity.

Reduction of thermal noise sources by cooling the NMR detection antenna [START_REF] Griffin | Metabolic Profiling of Rodent Biological Fluids via 1H NMR Spectroscopy Using a 1 Mm Microlitre Probe[END_REF].

3. Reduction of the width of spectrum ray: the rapid rotation of the sample around an axis at 54.7 ° from the magnetic field direction (Magic Angle Spinning) [START_REF] Tsang | Metabolic Characterization of Distinct Neuroanatomical Regions in Rats by Magic Angle Spinning1H Nuclear Magnetic Resonance Spectroscopy[END_REF] has long been used to reduce dipolar broadening in a solid environment. This technique has recently been developed for the analysis of tissue samples [START_REF] Rabeson | Quantitation with QUEST of Brain HRMAS-NMR Signals: Application to Metabolic Disorders in Experimental Epileptic Seizures[END_REF].

4. The use of capillary tubes for the analysis of very small solution quantities -devices proposed by manufacturers or suppliers of accessories. Recently it was used for a quantitative evaluation of NMR spectrometer performance [START_REF] Chi | Capillary-Tube Package Devices for the Quantitative Performance Evaluation of Nuclear Magnetic Resonance Spectrometers and Pulse Sequences[END_REF].

5. The use of an antenna array: here, we no longer play on physical factors but on technical and instrumental elements since it is a question of superimposing the signals of each sensor [8] [9].

For our case, we focused on the approach which consists of enhancing the sensitivity of the RF receiver coil. According to Sweedler [START_REF] Olson | High-Resolution Microcoil 1H-NMR for Mass-Limited, Nanoliter-Volume Samples[END_REF], reducing the antenna size, to adapt with the sample size should allow a considerable increase in insensitivity. Thus, the NMR detection idea was developed using microscopic devices and tools. This is how the notion of planar "microprobe"

or "micro-coil" became popular. Its optimization has been a subject of several theses in our team. In this study, we focus mainly on the RF receiver part, particularly the microprobe. We discussed its review and sensitivity criteria in the following Section I.3.

I.3. NMR antennas and their criteria of performance

Our study mainly focuses on the planar antennas, which we will demonstrate to be suitable for analysing the small volume later in this work.

Note that the planar coil has several advantages, e.g., a high magnetic field close to the conductor wire surface, the possibility of its implantation, close to the sample (in vivo applications) and adapting its dimension to one of the samples, facility to microfabrication, comparing with the volumic antenna. Therefore, the planar described features represent a significant asset for NMR spectroscopy's small volume and concentrations detection.

I.3.1. Stat-of-the-art of the surface antennas

One of the first proton NMR spectroscopy studies using planar micro-antennas was carried out by Peck et al. [START_REF] Peck | NMR Microspectroscopy Using 100 Μm Planar RF Coils Fabricated on Gallium Arsenide Substrates[END_REF]. He used a micro-coil with 4.5 spiral turns, a thickness of 2 μ , an inner diameter of 97.5 μ . It is a gold electroplated layer onto a GaAs (Arsenide Gallium) substrate. The antenna was suitable for NMR application, particularly in a static magnetic field of 7.05 (300 for the proton resonance frequency). Then, Stocker et al. [START_REF] Stocker | Nanoliter Volume, High-Resolution NMR Microspectroscopy Using a 60-Μm Planar Microcoil[END_REF] had analysed a volume of 880 , from which they could observe sucrose peaks. The planar microantenna with 3.5 , 60 μ inner diameter, 3 μ track thickness and also etched on a GaAs substrate, and they placed the sample above 50 μ from the micro-antenna. Their analysis of water in a 5.9 field (250 frequency) gave a Signal-to-Noise Ratio of about = 25

per acquisition and a ray width less than 2 . Therefore, by this study, they demonstrated the usefulness of planar antenna for high-resolution NMR spectroscopy experiments and how it may also be suitable for localized NMR studies [START_REF] Stocker | Nanoliter Volume, High-Resolution NMR Microspectroscopy Using a 60-Μm Planar Microcoil[END_REF].

The first planar micro-antenna integrated separately into a micro-fluid system for the spectroscopic study was performed by Trumbull et al. [START_REF] Trumbull | Integrating Microfabricated Fluidic Systems and NMR Spectroscopy[END_REF]. They analysed 30 of water sample within 5.9 magnetic field ( 250), getting a proton spectrum with a of 23,5

per acquisition and about 1.4 for bandwidth of spectrum resolution.

Based on the planar micro-antenna developed by the team of T. L. Peck, many studies about planar micro-antenna have been published, showing the growth improving the low sensitivity of NMR measurements. Massin et al. [START_REF] Massin | Planar Microcoil-Based Microfluidic NMR Probes[END_REF] compare planar micro-antennas from different inner diameters (2000, 1000 and 500 μ ), corresponding to detection volumes of about 470 , 120 , 30 . They demonstrated that the sensitivity increased, reducing the planar coil diameter. Through that, they get a 1 spectrum for 160 μ sucrose diluted in 470 of sample (for magnetic field of 7 corresponding to 300 for the resonance frequency).

Eroglu et al. [51] have developed a micro-antenna with an internal radius of about 750 μ , 3 spiral turns, 100 μ track width, 20 μ thickness and 100 μ for the gap distance between the conductor tracks. They reached a quality factor of about 39, focusing on a comparison between theory and experience. They studied the field of views in terms of parameters. Furthermore, they performed micro-imaging experiments on two concentric capillaries: an outer capillary with an internal diameter of 1 and an inner capillary with an internal diameter of 500 μ containing demineralized water and a fat, respectively. They achieved a spatial resolution of 30 μ within a static magnetic field of 11.74 . The microantenna allows acquiring micro-images of living cells: Frog ovocytes and pancreatic cells isolated from rats [START_REF] Gimi | NMR Spiral Surface Microcoils: Applications[END_REF].

Many authors have also studied the potential of micro-antennas for MRS analysis of nanoliter volumes [53] [54]. A first study focusing on the analysis of nano-volumes in highresolution of MRS was presented by Olson et al. [START_REF] Olson | High-Resolution Microcoil 1H-NMR for Mass-Limited, Nanoliter-Volume Samples[END_REF]. An improvement in mass sensitivity by a factor of about 140 was reported using a solenoid micro-antenna with an active volume of

5

(relative to a conventional antenna). These authors are the first to obtain a 1 spectrum from a sample of α-sacciform peptide cells ( -), Alypsia California.

Ehrmann et al. [START_REF] Ehrmann | Sample Patterning on NMR Surface Microcoils[END_REF] proposed the designing of a planar NMR microprobe designed for cells analysis and characterization using lipid vesicles as cell substitutes. He was able to align the sample in areas of homogeneous RF-field on the highly sensitive surface of the planar NMR microprobe using microcontact printing. So, he successfully recorded a H NMR spectrum of a sucrose solution confined within the lipid vesicles. They achieved a total detection volume between 1 to 2ƞ from 3D Finite elements simulations. After weighting the = 68 by the number of acquisitions, the detection limit of this experiment, encourage the use of this micro-antenna a great as asset to measure metabolites in rodents models [START_REF] Baxan | Limit of Detection of Cerebral Metabolites by Localized NMR Spectroscopy Using Microcoils[END_REF] for in vivo applications; however, an optimization work is still necessary. He also performed a biocompatibility study. Furthermore, he demonstrated that a rectangular antenna with at least a surface area of 500 × 400 μ ² causes pressure and stress in the brain tissue, followed by relatively large scar necrosis after the implanting. Moreover, this inflammation can cause a reduction in the field of view of the micro-antenna and, therefore, may affect the NMR signal measurements. He also demonstrates that micro-coil positioning in intracerebral structures is well reproducible under stereotaxic conditions. The rat kept the micro-antenna for four months before it died [START_REF] Baxan | Vivo Animal NMR Studies Using Implantable Micro Coil[END_REF]. He highlights that some improvements are still necessary to get a metabolic response for in vivo detection of small concentrations and volumes.

A new planar micro-antenna design was presented by Woytasik et al. [START_REF] Woytasik | Characterization of Flexible RF Microcoils Dedicated to Local MRI[END_REF]. That consist of a planar micro-coil manufactured on both sides of a flexible substrate, which allows the microantenna to take the form of the study object (sample). Different external diameters (3 , 5 and 15

) were compared with another flat antenna. They reach an gain factor of 6 and 2 at a distance of 1 μ and 4 μ respectively inside the phantom.

A high spatial resolution antenna to study the internal structure of certain biological cells by micro imaging was developed by Weiger et al. [START_REF] Weiger | NMR Microscopy with Isotropic Resolution of 3.0 Μm Using Dedicated Hardware and Optimized Methods[END_REF]. The optimization of their equipment made it possible to attend 3 μ for isotropic resolution. A planar micro-antenna, with 20 μ for inner diameter, placed within a strong gradient, 6500 / for the three directions, and a static field of 18.8 , was used for experiments on a phantom composed of glass fibres immersed in water doped with copper sulphate ( ).

A portable system for molecular cell analysis was developed by Lee et al. [START_REF] Lee | Chip-NMR Biosensor for Detection and Molecular Analysis of Cells[END_REF]. That NMR miniaturized diagnostic system consists of a microfluidic microarray with several planar microcoil, a printed, electronic circuit (tuning, matching, amplification) and a small magnet that generate a magnetic field from 0.1 0.5 . The compact system allows bacteria and cancer cell profile detection. Biomarkers of the protein ( -fetoprotein) could also be identified for a sample volume of about 5 10 μ .

A comparative study between planar, solenoidal and Helmholtz antennas was carried out by Ehrmann et al. [START_REF] Ehrmann | Microfabricated Solenoids and Helmholtz Coils for NMR Spectroscopy of Mammalian Cells[END_REF]. He noticed the variation of the quality factor Q and the electrical resistance as a function of the working frequency. A comparison in terms of the SNR, the homogeneity of the field and the width at mid-height for each antenna was performed. Continuing the comparison of micro-antennas, the 3D micro-coils (solenoidal, helmholtz) offer great sensitivity, magnetic field much more homogeneous than the planar micro-coil, but their disadvantages come from the micro-manufacturing difficulty, which can harm the electronic characteristics. For example, the Helmholtz micro-antenna manufactured on two silicon substrates lead to poor circuit quality because of the eddy currents in the semiconductor substrate [START_REF] Massin | Planar Microcoil-Based Microfluidic NMR Probes[END_REF]. The other drawback of a 3D micro-coil is: the sample must be inside the microantenna, which is not suitable for in vivo application: monitoring of the AD biomarkers variation using an animal model. What is crucial to highlight here is the homogeneity of spin excitation as a function of sample diameters of living biological cells placed on the surface of the planar micro-antenna with an inner diameter of 1 , two turns copper line, 15 μ for the thickness.

When comparing planar antennas with those of the solenoidal type from the literature, we can notice that planar antennas micro-manufacturing is less complicated than others (volumic antennas) presented in the literature on the small size. Planar micro-antenna can provide a good and also a high spatial resolution for a very small sample volume placed in the vicinity of the antenna [START_REF] Rivera | MRI Visualization of Small Structures Using Improved Surface Coils[END_REF]. Finally, they can be integrated with less difficulty than solenoids into microanalysis systems and easily used to build array antennas. On the other hand, the disadvantage is the inhomogeneity of the field the antenna compared to the volumic antennas, especially when the working frequency domain is very high i.e. a very intense magnetic field [START_REF] Bosch | Surface Coil Spectroscopy[END_REF].

The choice among different types of micro-antennas depends strongly on the field of application, the nature of the sample to be analysed, the conditions and constraints of the work to be performed [START_REF] Kadjo | Micro-Capteur Implantables : Etude Des Criteres de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie RMN in Vivo[END_REF]. Our attention was focused on such implantable planar antennas for in situ or in vivo applications, minimally invasive, and that would allow the detection of a small amount of solutions. Our team could not yet use this original solution to detect cerebral metabolites by MRS; at least, we have demonstrated its potential for in-vivo applications.

Ginefri et al. [START_REF] Ginefri | Implanted, Inductively-Coupled, Radiofrequency Coils Fabricated on Flexible Polymeric Material: Application to in Vivo Rat Brain MRI at 7T[END_REF] developed an implanted, inductively-coupled RF coil, fabricated on a flexible material, for in vivo rat brain MRI analysis at 7 T. The system allows acquiring the rat brain with a good resolution (150μ × 150μ ), reaching a visualization of a fine cerebral structure. The implantable coil was fabricated in a PDMS substrate on both faces and reached a Q factor of 114.

Deborne et al. [START_REF] Deborne | Implantable NMR Microcoils in Rats: A New Tool for Exploring Tumor Metabolism at Sub-Microliter Scale?[END_REF] evaluated the potential of the miniaturized implantable coil to acquire in vivo proton NMR spectrum in sub-microliter ROI and get metabolites information. For his study, a rat model was used, and the micro-coil was implanted in the right cortex for glioma diagnosis. The implanted micro-coil had about 450 μ × 3 for width and length, and its detection volume was about 450 , with a sensitivity of 76. It was demonstrated that the microprobe is promising to investigate brain tumors.

My thesis aims to develop an optimized RF microprobe (microsensors) to analyse microliter volumes located within a larger sample through the MRS. The microprobe should be implanted in situ for a sufficient period (several weeks to several months) to monitor the concentration variation of the biomarkers through an animal model of AD. This development could lead us to an earlier diagnosis of AD. Knowing that the microprobe development has been performed by our team for years, my main role consists of proposing a new approach to optimize an implantable NMR microprobe. In the next section, we are going to describe planar antenna miniaturization, which is one of the strategies already applied by our team to increase the NMR sensitivity.

I.3.2. NMR antenna miniaturization

During the last decade, research showed the possibility of implementing NMR instrumentation at a small scale, especially with RF resonators, which can be used as NMR signal detectors. The actual NMR field trend is the analysis of very small mass and concentration samples; the main limitation is the low representing the experiment performance.

Numerous studies using micro-antennas show the growing interest in improving the low sensitivity of NMR measurements by reducing the size of the receiver sensor for the NMR signal. Whatever some reasons are imposing this reduction in the sample volume: difficulties in producing the reagent in large quantities or miniaturization of the analysis system, it leads to questioning the design of the receiving RF coils.

Conventional NMR instruments used for small volume analysis give a small because of their large dimensions compared to the sample size. When the detection RF coil size matches the sample's size, the MR signal amplitude is maximized, which means that the filling factor is close to one. Therefore, an NMR micro-device specially designed for micro-samples is required to improve the parameter and increase the sensitivity [START_REF] Gimi | NMR Spiral Surface Microcoils: Applications[END_REF]. Considering the need for small quantities of tissue, examining by NMR spectroscopy or micro-imaging, it is possible to create probes with such limitations, i.e. compatible working volume [START_REF] Schoeniger | The Design and Construction of a NMR Microscopy Probe[END_REF]. Thus, to analyze samples having volume close to the nanoliter, it is necessary to fabricate, by microelectronics technology, antennas having dimensions of several micrometers [START_REF] Peck | Design and Analysis of Microcoils for NMR Microscopy[END_REF]. This way requires investigations essentially based on the present state-of-the-art of microelectronics, the optimization of the manufacturing process, as we are going to present in chapter IV, before the microfabrication of the final prototype of the microprobes.

On the biomedical scope, accurate quantification of metabolite variations is associated with spatial localization of small resonators (microprobe) placed directly into a well-defined region of interest (ROI) appears essential, as we will present in the next section. Before we have summarized in several planar antenna works performed from 1994 until now (2022), studied for different teams in the whole world (Table I. 2). Furthermore, we have summarized in Table I. [START_REF] Hoult | The Signal-to-Noise Ratio of the Nuclear Magnetic Resonance Experiment[END_REF] , the works about the planar micro-antennas already performed by our team. In these tables, we recall all pertinent parameters of the microprobe, performances criteria value, fabrication methods and application. All summarized information, give an overview of works already performed on planar antenna and a basis for our project startling point.

I.3.3. The performance criteria of the NMR antennas

The reciprocity principle [START_REF] Hoult | The Principle of Reciprocity[END_REF] is usually used as a basis for optimizing the sensitivity of NMR antennas. In the following section, we will present the criteria for antenna performance, particularly defining the sensitivity of NMR antennas. We first introduce the noise, then the parameter, finalize the Q factor and the last Signal to Noise Ratio (SNR) parameter.

I.3.3.1. Noises

Several sources of noise damage the signal's spectral domain. We are interested mainly in noise from the internal NMR system (receiver antenna). Noises sources coming from numerical acquisition, from other passives and actives electronics components we generally neglect in the first instance.

The previous section showed that the NMR signal is detected through an antenna. Thus, the NMR amplitude signal is too much weak. Hence, we have to determine the noise signal generally coming from the charges (electrons, ions) thermal agitation, the electronic circuits used in reception and the conduction behaviour of the sample.

Therefore, we can list the principal sources of noises for NMR systems:

Noise due to the electrical resistance of the RF coil in reception.

Noises due to capacitive (dielectric) and inductive loss on the sample.

Noise due to electronic circuits at the acquisition chain.

Imperfections of the Faraday cage of the NMR dispositive. It isolates no more from the disturbance in external electromagnetic fields.

Generally, the NMR system principal noise source comes from the sample. Noises come from the detection when the one induces more significant noise than the sample noise.

Concretely, it happens when we adapt the NMR antenna dimension with the sample size of around 2 μ to 3 μ for our application case. It can be emphasized that the sample noise is neglected when decreasing the sample dimension [START_REF] Peck | Design and Analysis of Microcoils for NMR Microscopy[END_REF]. Therefore, in our case, NMR experiment noise will come mainly from the micro-antenna [START_REF] Hoult | The Signal-to-Noise Ratio of the Nuclear Magnetic Resonance Experiment[END_REF].

A material or system with some charges (ions and electrons) does not have a perfect rest state for particle charges, even if no electric current is applied. If the temperature is not equal to zero, the system becomes a seat of thermal agitations. The Brownian motion principle can often model the random movement of charges. We can demonstrate by Langevin stochastic equation applied to a charge moving that we can explore two crucial phenomena which are connected: The fluctuation (Langevin force) and dissipation (friction force). These results stipulate that when a system gets the equilibrium state, there is no fluctuation without dissipation [START_REF] Kittel | Introduction to Solids Physics Book[END_REF]. Hence, it was demonstrated to fluctuate voltage magnitude and current intensity inside a conductor in a quasi-stationary regime [START_REF] Nyquist | Thermal Agitation of Electric Charge in Conductors[END_REF]. So, the conductor voltage fluctuation depends on the resistance and the temperature T as defined in the formula:

= 4 ∆ (I.19)
Where: is the Boltzmann constant, the resistance of the conductor, represents the system's temperature and ∆ the width band of the system.

I.3.3.2. Parameter of NMR antenna

Scattering parameters or S-parameters describe the electrical behaviour of a linear electrical system. This parameter is useful for several branches of electrical engineering, including electronics, communication systems, and more particularly for MW engineering.

Basically, many electrical properties of circuit components (inductor, capacitors, resistors) can be expressed using S-parameters, such as gain, return loss, voltage standing wave ratio (VSWR), reflection coefficient and amplifying stability.

Considering a two-port system (Fig. The relationship between the reflected and incident signal, on port 1, and 2 can be defined by the S-parameter matrix.

= (I.20)

The matrix coefficient is defined as follows: In my thesis, we work with RF antennas receivers. So, we aim to perform its behaviours simulation at reception, specifically to detect the NMR signal from a sample source. Hence, we focus on the reflection parameter ( ), which represent the return loss of the signal whose the definition in dB is given in the following equation:

= (I.
= -20 log ( ) (I.23)
Where, represents the incident wave in the antenna system, and the reflected wave from the antenna system.

If is zero ( = 0 ), then all wave power is reflected from the antenna system, and nothing can be converted into radiation, thus by reciprocity principle can be translated by no signal converted to an EMF for the RF receiver antennas. The receiver antenna ideally starts resonating when < -10 .

Since antenna resonating at a frequency ω , is able to create a signal overvoltage during the resonance regime. The characteristic of a resonant circuit is its ability to strongly amplify the signal at a given frequency without amplifying it at other frequencies. The amplification is done in a range of frequencies, the smallest possible around the resonance frequency, called bandwidth ∆ . The parameter representing the resonant circuit's performance is the quality factor, which is introduced in the following section I.3.3.3.

I.3.3.3. Q -Factor of the NMR antenna

The quality factor is a pertinent parameter that can correspond to the width at half-height of the resonance peak of the antenna, given by the measurement or simulation of the parameter of a receiver antenna (defined in session I.3.3.2).

Q = ∆ (I.24)
With: ∆ω that represent the width at half-height of the resonance peak of the antenna and ω is the resonance frequency of the antenna.

∆ω = ω -ω (I.24.1)
From the energetic point of view, the Q factor of a system is also defined as follows the formula:

Q = 2π (I.25)
The stocked magnetic energy is available by the inductive effect of the micro-antenna for an injected current intensity. Then the other part of the energy is dissipated by the Joule effect.

It is also a standard criterion to determine the electrical circuit's performance modelling the NMR micro-antenna.

Q = { } { } = (I.26)
Where Z corresponds to the system (resonator) impedance measured in [Ω]

We need a system with the smallest bandwidth to have a very selective resonator, leading to the highest quality factor possible. It helps to ensure that the system (resonator) be more performant and selective (sensitive), avoiding noise from the measurements.

I.3.3.4. Signal Noise Ratio (SNR)

The SNR is one of the most critical factors to assess an instrument's performance in the sample measurements over a reasonable period. We can define SNR as the ratio between the amplitude of the free precession signal (FID amplitude) and the average quadratic value of the noise. This value is associated with the efficiency of the NMR antenna. When we include the acquisition number [START_REF] Baxan | Limit of Detection of Cerebral Metabolites by Localized NMR Spectroscopy Using Microcoils[END_REF] the is defined as :

= × * × √∆ (I.30)
Where √∆ is the width of the spectra [Hz]

The Q factor of the microprobe is an easily measurable parameter that can also define de RF sensitivity of the antenna, as in the given formula [START_REF] Darrasse | Perspectives with Cryogenic RF Probes in Biomedical MRI[END_REF].

= ƞ ∆ (I.31) S ∝ ƞQ (I.32)
In the optimal case of the filling factor ɳ ≈ 1, the RF probe sensitivity is quickly evaluated by the Q factor that is easy to calculate from the circuit's frequency response.

From the previous sensitivity equations, in terms of we can note that the NMR sensitivity experiments also depend on the micro-probe geometrical parameters. More precisely, we can see in equation (I.28) that the depends on some experimental parameters as:

: is the resonant angular frequency, is the static Magnetization, Vs is the sample volume, kB is the Boltzmann constant, and ∆ is the spectrum bandwidth.

When we fix it, we can optimize the sensitivity by tunning the micro-antenna geometrical parameters, which are directly linked with the ratio and the resistance .

So, the RF sensitivity is one of the criteria we use to optimize the NMR antenna, easily deduced as follows:

S ∝ SNR ∝ √ (I.33)
For whatever optimized microprobe, we can fix the √ and the experimental parameters.

They are essential to optimize the experimental SNR and the acquisition time.

I.4. Sensitivity and Limit of detection of the NMR experiments

The investigation of mass-limited and concentration limited samples by MRS requires several performance criteria to validate the feasibility of the experiment using NMR microprobes.

These criteria allow computing the approximate mass or concentration of sample needed to acquire the desired Signal to Noise Ratio (SNR) for a specific peak achieved during a particular scan time. These figures of merit are going to be explained more in detail in the following sections as the concentration and mass sensitivity ( and ), also the concentration and mass limit of detection (LODc and LODm).

I.4.1. The sensitivity of the NMR experiment

In reviews dedicated to NMR microprobe analytical investigation [START_REF] Lacey | High-Resolution NMR Spectroscopy of Sample Volumes from 1 NL to 10 ΜL[END_REF] [74], the sensitivity of a microprobe might define the performance. This indicator of sensitivity in terms of sample concentration and the amount of material (molar sensitivity), respectively as:

S = [ ] (I.34) S = (I.35)
where [ ] is the sample concentration ( / ), and mol is the solute number of moles.

These sensitivity definitions were taken up by C. Massin [START_REF] Massin | Planar Microcoil-Based Microfluidic NMR Probes[END_REF], relating them to a single acquisition, thus defining them as follows:

I.4.2. Limit of Detection (LOD)

The Limit of Detection (LOD) or Lower Limit of Detection (LLD) is a concept that helps to determine the lowest amount of substance (mass or concentration) that can be detected in comparison with a measurement made in the absence of the sample within a confidence interval.

These parameters were very well introduced in the thesis of N. Baxan [START_REF] Baxan | Mise En Oeuvre de Microantennes RMN En Perspective d'étude in Vivo de Metabolites Par Spectroscopie[END_REF]; They represent useful parameters for NMR experiments.

When we have to analyse a chemical substance, assuming that the white field does not exist or is at least constant, the LOD shall be determined from the highest peak of the SNR on the baseline. Different teams have used NMR detection limit calculations for many applications to determine the performance of micro-coils of solenoidal geometry since 1995 [START_REF] Olson | High-Resolution Microcoil 1H-NMR for Mass-Limited, Nanoliter-Volume Samples[END_REF] [74] [73] [START_REF] Lacey | 1H NMR Characterization of the Product from Single Solid-Phase Resin Beads Using Capillary NMR Flow Probes[END_REF]. It was demonstrated that the noise of the NMR spectrum increases as the root mean square value of the number of acquisitions while the signal remains proportional to the .

That is also proportional to the time experiment . On the other hand, it is possible to standardize the detection limit taking into account the number of acquisitions and the duration time of experience as follows: The minimum of the spin number quantity that we can detect represents the micro-antenna sensitivity for catching a few spin amounts; that is defined as follows [START_REF] Boero | Integrated NMR Probe for Magnetometry[END_REF]:

N = √∆F (I.40)
Where: is the number of spins per unit of volume and corresponds to the sample volume.

I.5. In-vivo implantation made by our team

I.5.1

. Advantages of the Implanted micro-antennas

In practice, the study of cerebral metabolism in small animals can be carried out either by using a "macro-coil" or external commercial antenna such as a volumetric antenna or through an implantable micro-antenna, which is our aim work. For that, a comparative study between a commercial antenna (Rapid Biomedical 3.2 diameter surface) and a planar micro-coil (racetrack) was made by our team, the thesis of A. Kadjo [START_REF] Kadjo | Micro-Capteur Implantables : Etude Des Criteres de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie RMN in Vivo[END_REF]. A series of measurements, in vitro from 10 brain metabolites solution of known concentration in distilled water, allowed evaluating its performance using the photon detection limit ( ) criterion. demonstrates how useful the implantable micro-antennas should be for in vivo detection of metabolites by NMR [START_REF] Kadjo | Micro-Capteur Implantables : Etude Des Criteres de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie RMN in Vivo[END_REF]. Some demonstrative examples of brain metabolites spectra from volumes solution with 8 64 acquired using the micro-antenna, at 4.7 can be seen more in detail [START_REF] Baxan | Vivo Animal NMR Studies Using Implantable Micro Coil[END_REF].

An advantage of micro-antennas compared with the commercials macro antennas is that the magnetic losses in low conductivity samples are neglected [START_REF] Peck | Design and Analysis of Microcoils for NMR Microscopy[END_REF] because the main source of noise is essentially from the microprobe (micro-coil + micro path and micro connection wire)

resistance. This can be explained by the small volume of samples of the order of microliters, even about some nanoliters. The micro-antenna reduction dimension also favours the filling factor, allowing the microprobe sensitivity enhancement. So principal challenge rest to optimize the microprobe. All these results represent an asset for my thesis project and justify the use of a micro-antenna completely designed for our application instead of a commercial antenna. However, detecting the brain metabolites in small volume brain areas of a few meters cubic in LDT & PPT tissue presents a considerable difficulty in MRS. This difficulty is due to the low SNR achieved using conventional NMR sensors. Thus, in the last ten years, our research topic has focused on developing a new generation of optimal NMR sensors with a filling factor close to one (ƞ ≈ 1). They are designed to be closer as possible to the region of interest, allow to obtain the best possible SNR, and consequently lead to biomedical observations by NMR spectroscopy.

To address the issue of SNR of the micro-antenna and enhance it for our application, we first have to review the micro-antenna, last optimization work performed by our team, validate the last pertinent results and propose some enhancement on the optimization methodology

proposing a new approach to model the implantable microprobe as we are going to see in the next chapter.

Conclusion

In the actual chapter, we have presented an overview of the project thesis: the introduction of Alzheimer's disease, the markers and diagnosis methods, and the present study's hypothesis for earlier detection of AD biomarkers. We also reported a brief introduction to the NMR principle and its sensitivity.

Moreover, we presented the review of our planar micro-antennas, from the conception to the in vitro and in vivo applications, highlighting their criteria of performance, the manufacturing methods and applications. We also reported in this chapter a summary of work previously done to optimize a racetrack micro-antenna and its advantage in comparison with a commercial one.

Meanwhile, according to our team previous results, we presented and validated the in vivo implanting pertinence for Magnetic Resonance Spectroscopy (MRS), representing a crucial starting point (hypothesis) for this project. Our team's previous works and literature propositions represent promising results for the NMR domain and have raised several questions.

It demonstrated the possibility of optimising the micro-coil performance, proposing new methods/approaches for microprobe modelling and simulations. So far, the feasibility of their use in vitro and in vivo applications.

In the next chapter, we will model the microprobe antenna in terms of the magnetic field representing its sensitivity and in terms of the resistance value calculation, which represent the resistive losses of the micro-antenna.

Chapter II

Modelling of magnetic field and resistive loss of the micro antenna Currently, many commercial software provides tools to design and simulate the performance of a 3D system (i.e, to calculate electrical parameters from the geometry and simulate the magnetic field). In the case of microprobe, the designing and performances simulation in terms of Signal to Noise Ratio (SNR) and Q factor, of such 3D system becomes too complex.

Therefore, simulation work must be performed to design and to improve the performances II.1. Review of the previous micro antennas: 1 st and 2 nd prototypes Before the microprobe fabrication, there are two crucial tasks, (i) the optimization study, which helps us predict the device performance and (ii) the design of the optimal prototype according to the proposed specifications.

Before presenting the approach I chose for my work, I reviewed the previous works and prototypes -coil and transmission line with optimal dimensions, developed by our team. These prototypes were the starting point of my whole thesis.

In The preliminary studies about its simulations were reported in the thesis of J. Rosillo Trejo [START_REF] Rosillo | Contribution à l'amélioration de la sensibilité d'un micro-récepteur RMN implantable[END_REF] and T. Cong Troung [START_REF] Truong-Cong | Optimisation Par Approche Physique Des Micro-Antennes RMN Fabriquees Par Techniques Microelectroniques : Etude Theorique et Experimentale[END_REF]. They have modelled the micro antennas by an electrical circuit and calculated by a physical approach the B magnetic field. In their approach, they considered only the micro-coil without the transmission line (TL), which they modelled separately. It was assumed that the sensitivity of the micro-coil (active part) could lead them to deduce the sensitivity of the microprobe, and then of the micro-antenna. Thus, they selected the best

Micro-coil

Micro connecting wire :

wire-bonding

Micro Transmission Line

Tunig and matching circuit geometry and proposed the optimal configuration for our application. Table II. 1, we have summarized the four optimized micro-coil geometries with their different dimensions and parameters. All the parameters representing the microprobe prototype must be taken into account, and be varied judiciously. Thus, we will use them to validate our own methodology which could be generalized to other types of microprobe modelling for other and other applications cases. Transmission lines with two ground planes have the advantage of low attenuation and their wideband application in inhomogeneous environments [START_REF] Bhattacharya | Analysis of Broadside Offset Coupled Strips for Broadband Filter Applications[END_REF]. For our simulation work, we keep the previous optimized layout shape for the micro-tracks, ground /signal/ground, with the same for the track line, and added some improvements in the new design.

Configuration

II.1.2. Limitations of the previous designs

In this section, we have pointed out the limitations of previous design prototypes, highlight the parts that require to be improved. The underpass & vias solution represents an interesting solution, instead of wire-bonding to improve the microprobe performance. Its weak influence on the NMR signal was already reported [START_REF] Mendes | Wafer-Level Integration of on-Chip Antennas and RF Passives Using High-Resistivity Polysilicon Substrate Technology[END_REF]. However, according to the post-layout simulations of the prototype fabrication [START_REF] Rosillo | Contribution à l'amélioration de la sensibilité d'un micro-récepteur RMN implantable[END_REF], the underpass & vias increases the total resistance to about 1.5 Ω [START_REF] Rosillo | Contribution à l'amélioration de la sensibilité d'un micro-récepteur RMN implantable[END_REF].

The impact of Vias (TSV) incorporated on a silicon substrate was also already reported in several studies for the case of high frequency. The TSV technology has been considered as a solution for 3D interconnections' problems. The main advantages of the TSV are the higher interconnection density, less parasitic effects and less power consumption [83] [84]. We do not forget that a good characterization of TSV electrical parameters and modelling of coupling effects are highly needed. However, this is not the scope of this chapter but will be discussed in detail in the chapter III. According to all these facts, the underpass & vias are still a good solution, so we proposed this connecting solution for the 3 rd prototype of the microprobe.

Nevertheless, it still requires some enhancements in the fabrication process to improve the probe performance. For the 3 rd prototype fabrication, we also chose the transmission line proposed on the 2 nd prototype with some upgrades. We have also proposed another micro-wire connecting solution (air-bridge).

II.2. The design and modelling of the microprobe.

As we previously emphasized in chapter I, our team has been working on design and optimization of implantable NMR micro-coils for many years. Several proposals were made in terms of design and of performance (Q factor, SNR). In previous modelling work, the micro transmission line (TL) and wire-bonding have not been taken into account to estimate resistive losses. In this section, we will present the methods used to validate the previous works done by our team.

The aim is to model the microprobe completely, predict its performance very accurately and then propose an enhancement for the next prototype fabrication. First, we need to validate our methodology with the previous results of the micro-coil simulations, assuming that the TL is the same for all geometry and will not impact the performance of the micro-coil.

To develop a microprobe prototype, we have followed the methodology summarized in This chapter mainly focuses on the conception of the microprobe and its performance simulation. We performed the simulation of the magnetic field generated by the micro-coil and the whole micro antenna, and then we calculated the RLC of the micro-coil by an analytical approach.

The magnetic field generated by the micro-coil was simulated using CST MWS (Computational System Technology Microwave Studio) for the micro-coil part, according to different geometries (given in Table I.1) and COMSOL Multiphysics to simulate the impact of the substrate to the magnetic field and then simulate the magnetic field from whole microprobe prototype.

We have also estimated the RLC (Resistance -Inductance -Capacitance) parameters of the micro-coil by an analytical approach, of which the formulas were presented in section II.2.2.

This approach is widely inspired by formulas in literature and previous works done by our team (For example, the calculation of self and mutual inductance of the micro-coil). To do this, we coded a MATLAB script to calculate all parameters. We were able to validate our results with those of COMSOL Multiphysics and ADS (Advanced Design Software).

The approach to calculating the RLC parameters takes into account several physical phenomena that we will describe in the subsequent sections.

II.2.1. Simulation of the magnetic field

We have designed the four geometries as defined above, and the dimensions characteristics are detailed in the following paragraph. The comparative study of the magnetic field generated for four micro-coil geometries was performed using CST MICROWAVE Studio®, a software-based on Finite Integration Technique (FIT), where Maxwell's equations is written on a grid space both in time and frequency domain. The magnetic field analysis was set to 300 working frequency domain, which corresponds to 7 and 500 for 11.4 . Notice that in our case, we have a static approach, and the magnetic field will not depend on the frequency. We also assumed that the current through the micro-coil is unitary current intensity (i.e., = 1 ). Note that even if the rectangular design appears to be the most sensitive at the centre; its sensitivity decreases rapidly compared to others less sensitive in the centre (e.g., Square and circular).

II.2.2. Electrical parameters of the micro-coil (RLC)

This section focuses on the main approach and formulas used to determine the RLC parameters thanks to analytical calculations done by a developed MATLAB script.

We present an AC resistance calculation approach to determine the different micro-coils geometries resistances. That approach aims to take into account the skin and proximity effect as will be modelled into the later part of this work, with the complete microprobe through the 3D -TLE platform.

For the Inductance calculation, we used the approach previously proposed in our team which is simplest to model the analytical calculations compared to the approach proposed in the literature.

We also present the formula to model the capacitive effect in section II.3.2.4. All formulas modelling capacitive effect were extracted from the literature, and it represents the basis for the capacitive modelling of the microprobe;

The formulas we defined to estimate the RLC parameters of the micro-coil need, analytical calculation of the length of the micro-coil wire. That is why, in the section. II.

l = ∫ r + ( )²dθ (II.5)
Where is the number of spires and r represents the polar coordinate of the circular spiral.

II.2.2.1.3. Ellipsoidal micro-coil shape

We assume the ellipsoidal planar micro-coil is a composition of a half circle on each extreme side with a length size corresponding to the outer circle diameter.

l = 2nd + ∫ r + ( )²dθ (II.6)
The analytical modelling seems to agree with direct measurements. It can be useful to calculate the ellipsoidal planar coil's resistance, inductor, and capacitance like the previously related shape. In this approach, the current passes through all conductor surface cross-sections. We consider that the current density is uniform across the entire surface S, whereas it happens differently.

II.2.2.3. Skin effect and proximity effect approach

When we inject an alternating current through a metallic wire, we create an alternating magnetic field having flux lines concentric with the wire axis. The magnetic field generated induces eddy currents throughout its wire as well as inside another nearby conductor. Its The proximity effect also appears when we place a fed wire nearby another one (complicated phenomena to model analytically) [START_REF] Dwight | Proximity Effect in Wires and Thin Tubes[END_REF]. Hence, we neglected its influence, even though it was shown that it is essential to estimate microprobe electrical losses [START_REF] Eroglu | Estimate of Losses and Signal-to-Noise Ratio in Planar Inductive Micro-Coil Detectors Used for NMR[END_REF]. The proximity effect modelling is so complex to calculate analytically [START_REF] Truong-Cong | Optimisation Par Approche Physique Des Micro-Antennes RMN Fabriquees Par Techniques Microelectroniques : Etude Theorique et Experimentale[END_REF].

II.2.2.4. Skin effect for a micro-conductor

Let's consider a micro-wire fabricated by copper electro-plating (σ = 5.96 × 10 , = 1 and = 4 × 10 ) we want to determine whether the skin effect might be taken into account for a specific conductor radius. We have to compare the skin thickness given in equation II.8 against the wire conductor thickness.

To illustrate our example, we studied the optimized geometrical parameters of the micro-coil from previous thesis work carried out by our team (i.e., micro-coil with = 40 μ and = 20 μ ), which correspond to an average radius = 28.284 μ . So, we can deduce the threshold frequency, = 5.3127 , keeping in mind that our frequency domain is 300 or 500 .

Through the equations (II.8), we can calculate the skin effect thickness for 3.764 μ and 2.915 μ wire radius. Therefore, the skin effect must be taken into account to determine the accurate microprobe resistance for both working frequency cases, knowing that the working frequency domain defined for our application is higher compared to the threshold frequency previously determined for the geometry of the micro-coil ( = 5.3127

). Thanks to the same process, we can easily calculate the threshold frequency for the underpass & via. For its diameter 60 μ and thickness = 150 , we have a threshold frequency = 1.42 , which is also higher than our frequency domain. This allows us to conclude that we must include the skin effect calculation to estimate accurately the resistive losses from the NMR microprobe.

II.2.2.5. Resistance modelling with skin and proximity effect

We defined the AC resistance, the addition of DC resistance plus skin and proximity effect resistances. That definition includes the frequency impact, knowing that the skin and proximity effect appears when working at the high-frequency domain or simply when the threshold condition is verified. We previously demonstrated in section II. The two cases model the skin effect phenomena. Fig. 9 (a) represents the ideal case for a wire with a rectangular surface section, where the rectangular corners lack any fabrications residues or some shape defaults, whereas that is more realistic for a conductor wire with an almost ellipsoidal surface cross-section (b) because we consider the microfabrication defaults

= ( -2 ). ( -2 ) = ( -2 ). ( -2 )
and neglected the proximity effect. From all previous considerations, we can calculate AC resistance using the following equation: have proposed different formulas to determine the self-inductor and mutual inductor for the recline rectangular cross-section conductor [87]. Nonetheless, that approach is still complex for the conductor inductance calculation.

R
In radio-frequency (RF) microsystems, the sizing study to determine the inductance of the system is necessary because the performance is strongly influenced by high resonance selffrequency. Practically, the geometry of our microprobe is quite complex; the discretization (segmentation)

of the conductor into many small straight elements is decisive for calculating the inductance.

However, given the various existing formulas, the theoretical analysis of the inductance of a coil with a rectangular cross-section remains challenging and often inaccurate.

The method to calculate a conductor's inductance was demonstrated in the thesis of T. Cong

Troung [START_REF] Truong-Cong | Optimisation Par Approche Physique Des Micro-Antennes RMN Fabriquees Par Techniques Microelectroniques : Etude Theorique et Experimentale[END_REF]. His approach considers the interaction between self-inductance and mutual inductance for several geometries and conductor shapes.

The calculation of conductors' self-inductance and mutual inductance with a rectangular cross-section is identical to a thin strip-line. However, the discretization of the conductor into many filaments will be performed in both directions of the conductor cross-section [91] [89].

The self-inductance from a conducting wire with a rectangular cross-section surface can be calculated through the following formula [START_REF] Truong-Cong | Optimisation Par Approche Physique Des Micro-Antennes RMN Fabriquees Par Techniques Microelectroniques : Etude Theorique et Experimentale[END_REF]:

L = μ.
[ln + ] (II.17)

Following some enhancement on the previous formula of the self and mutual inductance, the planar conductor is calculated depending on analytical expressions [START_REF] Rosa | The Self and Mutual Inductances of Linear Conductors[END_REF]:

L = μ.
[ln + + .

( ) ] (II.18)

M = M = M = μ.
[ln

-1 + ] (II .19)
Where = with = 4 × 10 /

II.2.2.7. The capacitor of the microprobe model

A capacitor is the capacity of a device to store charges, depending on geometrical parameters.

Generally, it is defined by the following equation:

C = ε . (II.20)
These formulas represent the capacitive coupling of the micro-device according to different parts. We will see more in detail in the following sections; There are several parasitic capacities from the microprobe due to the coupling between the loops and the one with the substrate and oxide layer, in the case of the silicon substrate.

We have an interline coupling capacitance C , its value is calculated according to the equation defined by Nurmi Osborn [START_REF] Osborn | All-Copper Chip-to-Substrate Interconnects for High Performance Integrated Circuit Devices[END_REF]. From the concerned lines, surfaces and fringe capacities (capacities between the coper layer and the substrate: and '). This capacity is taken at the line extremities as well as the The coupling capacity between the ground plan and the concerned line surface can be referred to in the literature as line self-capacity.

II.2.3. Summary of electrical and magnetic parameters of the micro-coils

To validate our studies, we have modelled four micro-coils, simulated its magnetic field and then calculated analytically its RLC (resistance , inductance , capacitance ) parameters using a MATLAB script that we developed. In Table II. 2, we summarized the parameters of planar simulated micro-coils (Fig.II.11) We can confirm that the rectangular micro-coil is the best geometry due to its optimal sensitivity (greater magnetic field at micro-coil centre) and good quality factor (Table . II.2).

Geometry [ / ] [Ω] [ƞ ] [ ] Q [ √ ] Rectangular
We also observed from these results that the rectangular and the optimized ellipsoidal shapes, with the same characteristics (i.e., , , , and outer diameter, see section.II.

II.2.4. The impact of the geometrical parameters

After we validated the methodology for magnetic field simulation and the resistive losses estimation, we simulated the impact of the dimensions of the micro-coil on its performances.

First of all, we selected the rectangular geometry; to balance the micro-coil parameters, we fixed its thickness to = 40 μ , the value of turn number to = 4. For the following simulation, we adjust the , and external diameter (dimension).

In In the following Table II. 3, we summarized the magnetic (B (z = 0)) and electrical parameters ( ) and its deduced performance parameters ( and ) for each dimension of a rectangular micro-coil with n = 4, = 40 μ . Where w is the width of the conductor, is the thickness of the conductor, s is the distance between the turns, the number of spiral turns, is the resistance of the micro-coil, is the inductance, is the quality factor and is the component of the field generated by the micro-coil at its centre.

Dimensions [μ ²] [μ ] [μ ] [Ω] [ ] [ ] Q [ / ] [ √
[μ ²] [μ ] [μ ] [ / ] [ƞ ] [ ] [ ] [ √ ] 1000× 1000

II.2.5. Design limitation of the smaller micro-coils

When we reduce the dimension of the micro-coils, it is essential to define the minimum outer diameter, that can be reached for the micro-coil layout design and its manufacturing. We consider the geometrical parameters such as , , (width of the central track); the external diameters will have a threshold for the respective geometrical parameters mentioned above.

In this section, we will define the limit of design of a micro-coil, considering the fabrication constraints. It requires considering the minimal distance between microwires and the space necessary to the centre pad of the micro-coil.

The centre track links the micro-coil part with the connector track (Transmission line) through the underpass & vias connecting wire solution, air bridge or wire-bonding when we consider the central micro-track having the corresponding dimension (60 × 60 μ ²) defined previously by our team in the context of the 1 st prototype fabrication (pad for wire-bonding connecting solution). In our case, we can achieve a minimum dimension of 40 × 40 μ ², knowing that the smallest wire-bonding diameters can have 15 to 25 μ , for an aluminium bond wire [START_REF] Schuettler | Microassembly and Micropackaging of Implantable Systems[END_REF], usually ranging from 18 μ to 33 μ in diameter for a gold wire [START_REF] Ducata | Chip Bonding Tools, Fine Ceramic & Machining Parts[END_REF]).

Through the following formula, we can define the threshold outer diameter of the microcoil:

D _ = 2n × [w + s] + p (II.24)
Where: is the number of turns and s the width of the conductor and the distance between turns of the spires, the width of the central track, is the micro-coil wire width. Since the number of turns has already been optimized to = 4, we proposed to keep the number of spires turns and the dimension of the track as optimal for the issue of its manufacture.

Therefore, we obtained suitable and for the manufacturing of micro-coils smaller than 400 × 400 μ ². For micro-coils dimensions of 500 × 250 μ ², 600 × 300 μ , 1000 × 250 μ ², we proposed the value of the following size for the , parameters suitable for our limited manufacturing feasibility. Table II. 5. Geometric parameters with the limiting diameter for the feasibility Here, we keep as limit dimensions 400 × 400 μ ² being the optimal one, but we propose also to manufacture a square of 250 × 250 μ ² with the parameters and predefined in Table II.

5. In this way, we are going to confirm it feasibility for prototype microfabrication with wirebonding connecting solution. According to our results presented in the Table II. 6, we can notice that the square geometries (500 × 500 μ ²) present the highest ratio than the rectangular (1000 × 500 μ ²). Therefore, it would be wise to say that the best configuration of micro-coil is the square geometry for the dimensions given in Table II As previously mentioned, we use this software to simulate the magnetic field generated by a 3D model of the microprobe. So, we briefly introduced the main electromagnetics principle, particularly Maxwell's equations which is a fundamental base of the software RF module to calculate the magnetic field in a 3D system through the Finite Element Method (FEM).

Dimensions [μ ²] Geometry [μ ] [μ ] R [ ] L [ƞ ] [ ] [ √ ] × Rectangular 22 20

II.3.1. Electromagnetic principle of the approach

To build a micro-coil model and simulate its electromagnetic behaviour as an RF receiver in MRI, we need to deal with a set of equations named after Maxwell equations (see appendix A.1) and solve them accurately to better approach the model to the actual system. In electromagnetism, modelling is naturally complicated, especially in a three-dimensional representation, vector character of the manipulated magnitudes and the relative complexity of the associated mathematical equations. Above all, the modelling strategy is rarely apparent, given the diversity of the different possible formulations (e.g., the possibility of using Maxwell's equations directly or of utilizing the propagation equations, of constructing an equivalent problem or not, of working in the harmonic regime or not, of choosing one coordinate system or another). For our case, we used the linear solver and a physics-controlled mesh. We carried out a stationary and frequency domain study (quasi-stationary approach).

II.3.2. Simulation of substrate impact to the magnetic field

The microprobe prototype is fabricated on a silicon or glass substrate. On the previous approach modelled in section II.2.1.1, we modelled the copper micro-coil, and the substrate part was neglected. In this section, we first design the complete micro-coil, which is the design of a micro-coil in a substrate. Then, we simulate the magnetic field generated in the case of different substrates (Glass, Silicon, PEEK, PDMS, Quartz); at the end, we will perform a simulation showing the impact of the frequency.

The table II.7 list the physical properties of substrate used for our simulations [95] [96].Where is the electrical conductivity, is the thermal conductivity, and are the relative permeability and permittivity, is the heat capacity by mass and is mass density of the material.

Table II. 7. The physical properties of the substrate material used for our simulations, at room temperature.

We begin by modelling the rectangular micro-coil (the best geometry), including the substrate part and then inserting it inside a system (airbox) to propagate the magnetic field. This model was conceived using COMSOL Multiphysics, more precisely the RF module. We needed to ensure that the electromagnetics simulations are reliable and robust. For our study, it is pertinent to see how the frequency and substrate can impact the magnetic field generated by the micro-coil, a simulation of the micro-coil designed on a silicon substrate.

We plotted the magnetic field varies with the perpendicular direction (z-direction). Thus, in Fig. The implantable microprobe consists of a copper layer deposited onto a wafer (Glass or silicon substrate). The copper is either directly deposited at a glass substrate or a Silicon dioxide layer for the silicon substrate. For this simulation, we have used the rectangular micro-coil as active part with thickness = 40 μ , inter-spiral gap = 20 μ , the wire width = 22 μ , with 1000 × 500 μ ² surface section occupied.

Our complete simulated system is represented by the microprobe inside a spherical volume of 0.6 in diameter, centred on the surface of the active part of the microprobe (rectangular micro-coil).

Fig. II

. 20 shows the magnetic field lines generated around the micro-coil and the microtransmission line. We note that the magnetic field is higher at the centre of the micro-coil.

Nonetheless, the micro-transmission line also irradiates a magnetic field that is not negligible. Table II. 8 shows the magnetic field on the centre of the micro-coil for two frequency cases.

Working frequency

Micro-coil MF : We compared the magnetic field generated by the simplest model (micro-coil) and the complete one (microprobe). The results show a magnetic field changing when we change the model, as well as the working frequency.

( = 0 μ ) [ ] 10
z-coordinate (μm)

Conclusion

In this chapter, we reported a new methodology to model the implantable NRM microprobe in terms of the magnetic field and resistive loss. In section II.1.1, we reviewed the optimization work made previously by our team, and we confirmed that from four geometries (square, circular, rectangular and ellipsoidal), the rectangular one is the best regarding its higher magnetic field at its centre ( ( = 0) = 10.1

). Even if its magnetic field variation decreases faster than the one from the other geometry (Fig. II. 5).

We also extrapolated the conception and 3D modelling of the complete microprobe, thanks to COMSOL Multiphysics, simulating the magnetic field in the case of 300 and 500

working frequency. We noted that the magnetic field changes according to the model, and a significant magnetic field change from the coil part is indicated when we change the working frequency. Moreover, we simulate the impact of the substrate to the magnetic field on the simple design of the micro antenna since one of the complete models rests complex to perform due to the long computational time.

Our simulations lead to the prediction and validation of an optimal prototype design and validating using commercial software and analytical calculation to estimate the losses. As mentioned, the simulation task takes considerable time, and it is pretty challenging to deduce the impact of each part that constitutes the microprobe device. Work will be done, intending to enhance our methodology and get a complete scientific explanation of the results from our simulations work. Using this methodology, we proposed an electrical circuit that models the complete Microprobe (micro-coil, connecting micro-wire and TL) in chapter III.

Chapter III

3D -Electrical model of the Microprobes and prediction of their performances criteria (Q -factor and SNR)

One of the NMR technique's main challenges is to improve the sensitivity in terms of Signal to Noise Ratio (SNR) and spectral resolution to detect weak metabolites concentration in a small region of interest (ROI ~ 2 μ -3 μ ). An approach that increases sensitivity consists in miniaturizing of the NMR receiver. Thus, the RF coil size will be adapted to the sample dimension that minimizes its equivalent resistance , enhances the magnetic field , and increases the . Another way is to improve its performance by optimizing the microfabrication process [57] [63].

However, microfabrication processes involve long and costly steps. Simulation might be a way to shorten this time and save enough effort for experimentation. Simulations can also provide significant data trends, a comparison between prototypes and process options. So, we developed a platform called 3D Transmission Line Extractor (3D-TLE), in collaboration with INL (Institut des Nanotechnology de Lyon). This tool was used to design planar NMR micro antennas and calculate their equivalent RLC parameters analytically.

In this chapter, we also presented the AI approach, based on the Artificial Neuronal The data saved in a .txt file contain the geometrical parameters and technical description (material electrical properties and structure coupling). It also allows the design of the 3D structure of the modelled devise (microprobe).

As mentioned above, all parameters are defined in a .txt file (For more detail, see appendix A. 3 and4). The file is composed of four classes, namely:

Layers: represents the basic information about the layers;

Components: defines every single component;

Paths: represent the interconnections between components;

Couplings: defines the coupling relationship between components;

After running the platform, the netlist of a 3D structure that corresponds to its equivalent RLCG circuit can be exportable under any CAD (Computer Aided Design) software; for our case, we used ADS® (Advanced Design System) through S-parameters simulation and Q factor.

The previously presented syntax (Fig. III.2) represents the base to build a 3D model of the devices and extract their electrical parameters. Among structures that compose a 3D device, we can list Underpass, Vias or TSV (Through Silicon Vias), metallic strip lines on a substrate, substrates, oxides layer, etc.

For the following study, we will present how to model the underpass & vias part, the approach to model the losses on the substrate in the case of high frequency, which are the most complicated parts to consider when we design the complete electrical model of the microprobe.

III.1.2. The model of Underpass & Vias (TSV -Through Silicon Vias)

3D interconnecting requires the consideration of the global electrical model to evaluate complete system performance correctly.

The TSV can be defined as a metal conic structure surrounded by an oxide layer and filled With:

R = R _ -R _ (III.3) R _ = R _ + T _ (III.4) R _ = R _ + T _ (III.5)
Self and mutual inductances for cylindrical conductors are considered, and we have defined them as the following (III.6 and III.7) equations.

L = μ sinh ⁄ + ⁄ - ⁄ + 1 + (III.6) H M = μ sinh + - + 1 (III.7)
In our case, underpass & vias altogether were modelled as a U structure, where both TSV (vias) are connected toward a signal Transmission Line, which can be easily modelled rectangular micro-transmission Line.

Series elements (partial inductances and resistors) model the signal propagation through the TSV or transmission Line connector. Parallel components are used to describe the interactions between the TSV and its environment, for example, the substrate or another compound close to the TSV in a high-frequency case. The currents propagate vertically from the oxide layer to the ground lines. A resistor models this path in parallel with a capacitance plus the capacitance of the thin oxide layer ( ) separating the TSV from the substrate [START_REF] Ma | Modèles compacts électroniques du premier ordre et considération de bruit pour les circuits 3D[END_REF]. Finally, to estimate all the interactions, it is required to extract the substrate losses in terms de resistance and capacitance value to quantify its contribution to the electrical loss of our NMR implantable device.

III.1.3. Substrate losses in the high-frequency domain

In our working frequency domain (200 to 900

), the effects of the substrate must be added to the electrical circuit to model our device. Generally, we can represent the substrate as an network to estimate the system's loss more accurately. The modelling becomes quickly complex when the coupling effects are considered between all the components of the system (TSV, underpass, signal and ground transmission line, micro-coil and oxides layer in the case of the silicon substrate). In addition, the substrate can be anisotropic with different doping values, i.e., different resistivity and permittivity throughout a specific volume. The substrate is therefore modelled as a stack of parallel heterogeneous dielectric layers because of the doping [START_REF]An Efficient and Simple Compact Modeling Approach for 3-D Interconnects with IC׳s Stack Global Electrical Context Consideration[END_REF].

In this perspective, an extraction method was proposed relying on the Transmission Line method applied to a multi-layered substrate and the Green electrostatic functions [101] [102].

The method is incorporated in 3D-TLE platform, and we used it for our global modelling of the microprobe system [START_REF] Gontrand | 3D Substrate Modeling; from a First Order Electrical Analysis, towards Some Possible Signal Fluctuations Consideration, for Radio Frequency Circuits[END_REF]. As in the case of the microprobe model, a current is applied on a contact pad located on the top or inside the substrate. A voltage is then measured at the level of another contact, enabling to deduce impedance matrix [ T Z ] whose dimensions correspond to the number of contacts. Resistance generally models the substrate coupling between two elements in parallel with a capacitance [START_REF] Gontrand | Towards a Modeling Synthesis of Two or Three-Dimensional Circuits Through Substrate Coupling and Interconnections[END_REF].

The approach was already integrated into 3D -TLE to model the compound of the microprobe, such as: underpass & vias and the transmission line (TL) of the microprobe embedded on the substrate. Note that TL corresponds to a copper layer embedded on a substrate surface. The transmission line was designed as a coplanar waveguide at the top of a high resistive substrate with a layer. The coplanar lines (TL) were previously reviewed in chapter II. It can be assimilated into three lines (groundsignal -ground), and the vias with substrate coupling have been modelled.

III.2. The electrical circuit of the implantable NMR microprobe

We proposed an equivalent 3D electric circuit that models the microprobe on a glass and a silicon substrate, respectively. To start, we presented a brief review of the electrical circuits (Previous electrical models) to study the NMR receiver performance in terms of electrical losses. Then an analytical study is done on the impedance circuit's equations allowing us to calculate the performance parameters like parameters and deduce the Q factor.

This Section presents advancements in 3D-TLE development and illustrates its efficiency and accuracy in our application case: Transmission Line (TL) modelling of an implantable NMR microprobe and the underpass & vias within a substrate.

III.2.1. The RLC electrical circuit and PI model

The simplest representation of an NMR microprobe or any passive RF receiver is an RLC circuit resonating at the Lamor frequency of the sample being studied. This representation is only valid as long as the wavelengths involved are much greater than the length of the microprobe conductor. Otherwise, the propagation phenomena are no longer negligible.

Therefore, more complex models must be considered to represent the different parts of the RF receiver correctly with an equivalent circuit and to take into account the effective addition of extra components that induce non-negligible capacitive and inductive effects (Substrate,

underpass & vias, micro-transmission line and oxide layer in the case of silicon substrate).

For the RLC circuit, the capacitor can be placed in series or parallel with the Resistance and inductor giving resonant circuits. Some proprieties of the RLC parallel and series model of our planar micro-coil was studied by simulation in the thesis of J. Trejo Rosillo [START_REF] Rosillo | Contribution à l'amélioration de la sensibilité d'un micro-récepteur RMN implantable[END_REF]. Note that the Q factor of our complete model was deduced by calculating the complete impedance of the electrical circuit. The impedance expression can ever be written as a composition of a real impedance part and an imaginary part.

From the complete electrical circuit of the microprobe on the glass substrate, we determined the Q-factor analytically as detailed in the following formulas:

In the first place, we define: = + with the notation: i∈ {sign,gnd,und,vias} .

Z μ & = μ _ _ μ _ _ + Z + (III.17) Z μ _ _ = Z + + Z & + Z (III.18) Z & = 2. R + ( ) (III.19)
The Q factor is given by the equations following:

Q μ _ _ = { μ _ _ } { μ _ _ } (III.20)
Where: Following the same procedure, we could carry out the analytical calculations of the equivalent impedance of the microprobe model on the silicon substrate. The Q factor is defined as the equations following:

R μ _ _ = R {Z μ _ _ } (III.21) L μ _ _ = μ _ _ (III.
Q μ _ _ = { μ _ _ } { μ _ _ } (III.31)
Where:

R μ _ _ = R {Z μ _ _ } (III.32) L μ _ _ = μ _ _ (III.33)
Note that all these analytical formulas are too complex to evaluate the value of each parameter manually, so a MATLAB script was developed to calculate the value of each parameter ( , Table III. 1 summarizes the extracted value of different parts of the microprobe. The parameter represent the length of each part of the microprobe. The micro-coil length is calculated using the analytical formula in section II.2.2.1.

and
With the proposed equivalent electrical circuit of the complete microprobe, we could simulate the performance of the electric model in terms S parameters and deduce the Q factor, using ADS. We also used the analytical formulas defined in the Section III.3.1 and section III.3.2 to deduce the value of the microprobe more quickly.

The S parameter criteria were already defined in chapter I (For more details, see section I.3.3.2). Note that from a plot, we can deduce the Q factor of the resonator (Microprobe).

Because of that, we were able to add a circuit for the tunning and impedance matching of microprobe in ADS software to achieve the required resonance frequency for our application to accurately measure the Q factor.

In our case, ADS software provides variable capacitors that we combine in serial and parallel associated with the microprobe. We variated their value until the resonance peak was placed at the frequency of 300 .

Measuring the Q factor of the model and then comparing it with the previous value from our team, we evaluated the contribution of each part of the microprobe. This process is necessary to validate the proposed electrical models for the microprobe on the glass and silicon substrate. When all this is validated, we compared the value of the model (an electrical circuit) with one of the analytical formulas. In this way, we could generalize the process for other geometries and dimensions.

In Section III.2.4, we will present the results from the simulations of the frequency response of parameters of different models in the glass and silicon substrate previously introduced (RLC, PI and complete model).

Then in Section III. We have summarized in Table III.2 the results from ADS simulations to illustrate the impact of each part of the microprobe on glass and silicon substrate cases, specifically their performance at 300 and 500 of (Resistance , self-inductance and Quality factor of each electrical circuit model). Then we compared the results with the analytical results. When we add the TL to the model to complete it, we should increase its global

Resistance. But note that even it happens, the Inductance increase of almost 53 % is still proportionally higher than Resistance increasing (27 %). This explains the increase of the Q factor even when we add the TL. Then, when we add the underpass & vias, we note that the Resistance increases considerably (16 % on glass substrate and 20 % on silicon substrate) and the Inductance not so much (less than 1 % for both substrates cases).

The same analyse was performed in the case of 500 . In the Note that the percentage variation for the case of 500 , in both substrates is quite similar to the one of 300 MHz. The difference is about the percentage intensity due to the frequency that is much higher than 300 . when we move the frequency from 300 to 500 , in the glass substrate and secondly, the Q factor growth is 18 % in the silicon substrate.

Therefore, we can now conclude that each part of the microprobe (micro-coil, TL, underpass & vias) and the substrate make an influence on the variation of Resistance, self -Inductance and Q-factor parameter. The influence is mostly understood through the several models we described in the previous sections (Section III.2.1 and Section III.2.2), more particularly thanks to our original model representation of the electrical circuit from the complete microprobe. The TL contributed to the increase of the Q-factor, while the underpass & vias can deteriorate the Q-factor of the microprobe when it is not optimized.

At 300 , the fabrication changes of microprobe substrate from glass to silicon provided an impact on the performance of the Q-factor (decreasing by 7 %). At 500 , the Q-factor of the microprobe degraded by 27 % if used with silicon substrate instead of the glass substrate.

Thus, the best working frequency for our application could be 500 , a microprobe on a glass substrate. We were sure that we could optimize considerable resistive loss due to the underpass & vias embedded on a silicon substrate or simply change the connecting wire type to wire-bonding, airbridge, which require an adaptation on the manufacturing process microprobe prototype.

From the results of our simulations, we could affirm that the prediction of the performance parameters of the microprobe with an optimized underpass & vias with a thickness ( = 10 μ instead of 0.5 μ ), is much better, as we can summarize in the widely used in several fields of applications, particularly in IMR for image processing and analysis [START_REF] Aguilera | A Guide for On-Chip Inductor Design in a Conventional CMOS Process for RF Applications[END_REF], brain image segmentation [START_REF] Okada | Modeling of Spiral Inductors[END_REF], image classification and segmentation of the human brain for AD diagnostics [START_REF] Yue | On-Chip Spiral Inductors with Patterned Ground Shields for Si-Based RF ICs[END_REF], spectra reconstruction from under-sampled data [START_REF] Wang | A Sparse Model-Inspired Deep Thresholding Network for Exponential Signal Reconstruction --Application in Fast Biological Spectroscopy[END_REF],

very fast reconstruction of NMR spectra from limited experimental data [START_REF] Akkus | Deep Learning for Brain MRI Segmentation: State of the Art and Future Directions[END_REF] For more than one decade, AI has been improving its development in many application domains. Very recently, AI technology has been used to solve complex problems such as antenna parameters extractions, structural design and data interpretation for radar and Multiple

Input and Multiple Output (MIMO) systems [113] [114]. Artificial Neural Networks (ANN) is one of the AI branches beneficial for solving problems of modelling where the relationship between system inputs and outputs are relatively too complex. The advantages of the Artificial Neural Network (ANN) to model system are: reduce computational complexity and time.

Moreover, decrease some cost compared to traditional computational modelling and leads to highly accurate data set analysis [START_REF] Qu | Accelerated Nuclear Magnetic Resonance Spectroscopy with Deep Learning[END_REF]. These facts represent a great asset to perform the design and optimization of our microprobe structure.

In the following section III.3.2, we have briefly recalled the principle of the classical methodology for optimal microprobe design. Then in section III.3.3 and section III.3.4, we have presented a new approach and some basic AI concepts that were useful for this work.

III.3.2. Traditional simulation method for microprobe model

Experiments in real life are often limited, either by the execution time, financial cost, or simply by the impossibility of performing them. For example, observing and analyzing deformation mechanisms of tissue in vivo at the nanoscopic scale is still tricky, even impossible in many cases, mainly due to the limitations of microscopy techniques. Computer simulations can overcome these constraints either by replacing observations and analyses that are difficult to measure. Simulation can also be complementary, completing experiments that have already been made. Furthermore, numerical simulations can also reveal previously unsuspected mechanisms, or still largely misunderstood, in many disciplinary fields.

In our case, we have been carried out a simulation of an implantable microprobe designed for in vivo applications. Usually, microprobe manufacturing is time-consuming and expensive.

So, simulation makes possible the examination of different scenarios without experimentations and then select the main parameters for the required performances. It can also provide meaningful data trends and guide the manufacturing process. Until now, our team has applied a classical methodology for microprobe design simulation and performance study [37] [43].

The used optimization processes of microprobe design are reminded on a block diagram ( Fig. We need an approach that can generalize the design and optimization of multi-parameters systems. Thus, in the Fig. III. 20, we shows the principles of classical methods (a), comparing to an AI approach (b). Using the classical method, usually we might know the model that will be used to simulate the performance of the system. In the Machine Learning approach, we do not need to know the model, we can train an AI model (black-box) directly with the expected input and output, and then we can predict the required performances according to the characteristics of the system. A supervised ML algorithm experience a dataset containing features, but each example is also associated with a label or target. This requires training the model with a known set of input data (Predictors, conventionally designated ) and known responses (output or target, conventionally designated or ). Supervised learning can be used since the data you must predict has outcomes that are very well known. Regression methods of ML are used to predict the continuous values of a magnitude using an experience of the dataset. The dataset is characterized by two types of variables: dependents ( in my thesis designated by ) and independent variables ( ).

Independent variables can be related to the cause of the states or the feature of the system to predict the evolution (Predictors). The dependent variable can be seen as the state or target, as the final goal of the study or trying to predict (Target). Certainly, for our dataset, the simulation results are well-known, which correspond to the performance of the microprobe in terms of the Q factor and Resistance, as well as the system predictor, which are the geometrical parameters, substrate electrical properties, frequency domain.

To manage our datasets, we developed our predictive model using the Artificial Neural Network (ANN), a computer system inspired by the biological neural network, as we will see later in the following section. Some neurons may influence the environment by triggering actions [START_REF] Schmidhuber | Deep Learning in Neural Networks: An Overview[END_REF]. Learning (training) is about finding weights that make the NN show the desired behaviour of the model.

The training of different architectures is very well understood by mathematical models [122] [120]. For our study, we trained the AI model based on the family of feedforward Neural Networks via the Back Propagation technique, which we will describe its principle in section (III.3.3.6).

III.3.3.2. The principle of an Artificial Neuron

The human brain is very efficient for solving tasks such as object recognition by the human eye or object segmentation in images [START_REF] Dicarlo | How Does the Brain Solve Visual Object Recognition?[END_REF]. This reason has motivated the research community to understand the human brain by proposing models representing its behaviours In the following equation, we defined the mathematical model of an artificial neuron (perceptron) without activation function ( ):

= ∑ + (III.35)
123

The previous regression model can be represented in the matrix form as the following equation:

= + (III.35)
The goal here is to build a system that can take a vector ∈ as input and predict the value of a scalar ∈ as an output before we apply the activation function. Where ∈ is the vector of the parameters.

The output of the neuron is defined as the following equation (III.36). The activation function, we denoted by which allow the evaluation and transformation of the linear output of the neuron:

= → = ( + ) (III.36)
The activation functions add non-linearity to the output, which enables Neural Network (NN)

to solve nonlinear problems. The standard activation functions include Linear, sigmoid, Tanh, and ReLu, but many others also exist [127] [128]. These are the most currently used for regression and classification problems.

One of the most frequent used activation functions is the sigmoid. This function is also used to model a logistic regression algorithm which is often associated with the single perceptron.

Logistic regression is also used to illustrate the idea behind a simple neuron network approach.

Thus, in the next section, we will introduce the loss functions, more precisely, the log loss function that is commonly used for classification problems and associated with a logistic regression algorithm [START_REF] Yegnanarayana | Artificial Neural Networks[END_REF]. Note that, logistic regression we used here to illustrate the different steps used for a standard neuron network which are at the base of several neuronal network architectures;

III.3.3.3. The loss function and descendent gradient of a Logistic regression

The loss or cost function is a function that allows evaluating the error made by an AI model. If the model prediction is completely wrong, the loss function will be the highest value, and if not, the loss function is a lower value. So that, give information about how well the model is learning from the dataset.

The cost function is defined as follows:

= = -∑ ( ) log ( ) + (1 -( ) ) log(1 -( ) ) (III.37)
Where:

represents the number of observations : is the model ouput (predicted value)

: the true value of the output.

The challenge of this approach is to find how to minimize the log loss function, which means minimizing the error of the AI model to train from the dataset. For this, one of the most used methods is the gradient descendent.

Gradient descend is a first-order iterative algorithm for the optimization of most of ML methods. For example, in deep learning is used to find a local minimum of the loss function, which is a differentiable function.

The formula used for the gradient descendent algorithm is the following equation: The idea is to make repeated steps in the opposite direction to the gradient of the loss function at the current point. The second condition has to be repeated several times until we find the minimum of the log loss function.

We can easily demonstrate that, for a logistic regression the and are calculated by the following formula:

= ∑ ( ( ) -( ) ) (III .39)
And:

= ∑ ( ( ) -( ) ) (III.40)
Equations (III.39) and (III.40) can easily be implemented with the aim of the weight and bias of the neuronal are adjusted on presentation of each sample containing the information about the data tendency. Note that the cost function of the logistic regression is a continuous and differentiable function with a local minimum, as we can determine using the previously shown equation. But for the more complicated case, where an analytical function does not define the gradient, other more complex gradients algorithms can help find an approach suitable for the application case [START_REF] Ding | Activation Functions and Their Characteristics in Deep Neural Networks[END_REF].

III.3.3.4. Vectorization of the mathematical model of a neuron

In computer programming, vectorization consists of transforming data in the form of vectors, matrices or N-dimensional arrays in order to perform mathematical operations on the dataset in the simplest and fastest way. Vectorization is a valuable technique in ML and DL because we deal with a huge amount of data in these domains of AI.

The vectorization technique reduces the computation time of the model by avoiding computing

the data one by one. In contrast, in the neural network, the same computation must be done several times for different data. So, it is more practical to vectorize the data to process all data at once and help to avoid a consuming time for computation.

Therefore, to illustrate the technique in the following section, we will vectorize the equation of the artificial neuron model for logistic regression that is the algorithm previously described in section III.3.3.3.

a) The ANN model trained for several observations can be defined as follows:

( ) = ∑ ( ) + (III.41)
Where we define as the number of the dataset (observations), is the number of neuron inputs (predictors), e.g., in our case, we have = 6621 observations, and = 12 predictors.

= ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) ( ) ⋮ ( ) ( ) ( ) ( ) ⋮ ( ) ⋱ ( ) ( ) ( )
⋮ ( ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ∈ × (III .42)
And:

= ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) ( ) ⋮ ( ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ∈ × → = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) ( ) ⋮ ( ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ∈ × (III.43) Thus: = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) ( )
⋮ ( ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) ( ) ⋮ ( ) ( ) ( ) ( ) ⋮ ( ) ⋱ ( ) ( ) ( )
⋮ ( ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ × ⎝ ⎜ ⎜ ⎜ ⎛ ⋮ ⎠ ⎟ ⎟ ⎟ ⎞ + ⎣ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎤ (III .44)
With:

= ⎝ ⎜ ⎜ ⎜ ⎛ ⋮ ⎠ ⎟ ⎟ ⎟ ⎞ ∈ × = ⎣ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎤ ∈ × (III.45)
Therefore, we can define the net model for observation in matrix form as the equations:

= × + (III.46) = → = ( × + ) (III.47) b)
We can demonstrate that vectorization of the log loss function is given by the formula:

= -∑ ( ) log + (1 -( ) ) log(1 -) (III.48)
c) The vectorization of the descendent gradient we can define as:

⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ - ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (III.49)
We deduce the vectorization of the descendent gradient as defined:

= - (III.50)
Note that the matrix notation does not consider the index; moreover, it looks like the previous formula without vectorization and the bias coefficient calculation. When we replace the equation (III.39) in the Jacobian, we can deduce the following formula for the gradient vectorization:

= ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ( , ) = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ∑ ( ) ( ( ) -( ) ) ⋮ ⋮ ⋮ ∑ ( ) ( ( ) -( ) ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (III.51) = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ( ) ( ) - ( ) + ⋯ + ( ) ( ( ) -( ) ) ⋮ ⋮ ⋮ ( ) ( ) - ( ) + ⋯ + ( ) ( ( ) -( ) ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ×( ) (III.52)
So, we can deduce the gradient matrix of the weights parameters as:

= × ( -) (III.53)
Following the same procedure as in the equation (III.51), we can deduce the gradient for the bias as follows:

= × ( -) (III.54)
So, the general formulation of the vectorization of the model, the loss function and descendent gradient avoid the utilization of several loop "for" during the model coding for computation of the model. We can also generalize the calculation of the amount of the dataset and variable (predictor). The only parameter we will have to update is the size of the matrix according to the case. All these concepts are already embedded in the MATLAB GUI application for Deep learning, more precisely Neural Network Toolbox; we performed the optimization of the parameters and hyperparameters, as well as the data pre-processing before the model training.

In the next section, we will present the generalization of the artificial neuron for several neurons and layers, which is the base for the architecture chosen for the training of our model. Feedforward networks are of extreme importance to machine learning practitioners. They represent the basis of many important commercial AI applications [START_REF] Jurafsky | Speech and Language Processing: An Introduction to Natural Language Processing[END_REF].

Specifically, we will introduce a feedforward computation for the network shown in For an illustration of this model. This feedforward network has an activation function for all layers that is computed by a function [ ] ( [ ] ). The values of these hidden units are then used as the input for a second layer, and so on, until the last layer, which corresponds to the output of the network architecture.

= 1 [ ] = [ ] + [ ] [ ] = [ ] ( [ ] ) (III.55) = 2 [ ] = [ ] [ ] + [ ] [ ] = [ ] [ ] (III.56) ⋮ = [ ] = [ ] [ ] + [ ] [ ] = [ ] ( [ ] ) (III.57)
With:

= [ ] = [ [ ] [ ] …. [ ] ] (III.58) = ⎣ ⎢ ⎢ ⎡ ⋯ ⋮ ⋱ … . . ⋱ . . … ⋱ ⋮ ⋯ ⎦ ⎥ ⎥ ⎤ ∈ × (III.59)
Basically, each neuron of the input layer distributes its value to all the neurons in the middle layer. Along with each connection between input and middle neurons, there is a connection weight so that the middle neuron receives the product of the value from the input neuron and the connection weight; thus, neurons in the middle layer take the sum of its weighted inputs and then applies a nonlinear function (activation function) to the sum. The result of the function then becomes the output for the entire architecture [START_REF] Tangri | Predicting Technique Survival in Peritoneal Dialysis Patients: Comparing Artificial Neural Networks and Logistic Regression[END_REF]. All this process represents the first Feedforward propagation of the network computation before evaluating the training performance (by backpropagation).

Theoretically, PB provided with a simple layer of hidden units is sufficient to map any mathematical problem. If provided with an appropriate number of hidden units (neurons), they will also be able to minimize the error of nonlinear functions of high complexity [START_REF] Pasupa | A Comparison between Shallow and Deep Architecture Classifiers on Small Dataset[END_REF].

Back-propagation refers only to the method for computing the gradient, while another algorithm, such as stochastic gradient descent, is used to learn using this gradient. Furthermore, back-propagation is often misunderstood as being specific to multilayer neural networks, but in principle, it can compute derivatives of any function (for some functions, the correct response is to report that the derivative of the function is undefined) [START_REF] Ding | Activation Functions and Their Characteristics in Deep Neural Networks[END_REF]. Specifically, we will describe how to compute the gradient for our neural architecture with an arbitrary loss function.

For a regression model, the cost function is defined as the mean square error of the true value of the system output and the prediction from the feedforward propagation process.

= ( [ ] -) (III.60)
The aim is to minimize the cost function; thus, the gradient of the cost function is defined according to the matrices of weights and bias parameters using the chain rule.

[ ] = [ ] [ ] [ ] [ ] [ ] = [ ] ′ [ ] ( [ ] ) [ ] (III.61) [ ] = 2( [ ] -) [ ] ′ [ ] ( [ ] ) (III.62) = 2( [ ] -) ′ [ ] ( [ ] ) (III.63)
The following formula gives the overage of the cost function for all training:

〈 [ ] 〉 = ∑ [ ]
(III.63)

〈 [ ] 〉 = ∑ 2( [ ]( ) -( ) ) [ ]( ) ′ [ ] ( [ ]( ) ) (III.64)
With:

[ ] = [ ] [ ] + [ ] [ ] + ⋯ + [ ] [ ] (III.65) [ ] = [ ] ( [ ] ) (III.66)
The most effective modern optimization algorithms are based on gradient descent, but many useful loss functions have no useful derivatives (the derivative is either zero or undefined everywhere). These two problems mean that we rarely use empirical risk minimisation in the context of deep learning. Instead, we must use a slightly different approach, in which the quantity that we actually optimize is even more different from the quantity that we truly want to optimize. For our approach, we used a GUI from MATLAB Toolbox that takes into account all these details and provides different types of training functions and algorithms for optimization of the model dataset training.

III.3.4. The evaluation metrics of a trained model

One way to evaluate the performance in terms of the model's accuracy to predict the data correctly is when we test with new data. For the regression problem, a quantitative way is using some statistical metrics that we will describe in the following sections. We defined as the amount of test data K. In MSE metric, we calculate the square of our error, and then we take its mean. This is a quadratic scoring method, which means that the penalty is not proportional to the error like in MAE metric but to the square of the error. It gives relatively higher weight (penalty) to large errors or outliers while smoothening the gradient for smaller errors.

= ∑ ( ) -

(III.70)
The advantages of this metric are that we obtain extreme losses for a larger value. It is more sensitive to the outlier than the MAE. The error value from the calculation has the same magnitude scale as the predicted magnitude, while the MSE does not have.

III.3.4.5. Relative Absolute Error (RAE)

RAE is also known as the residual sum of a square, where we take the total absolute error and normalize it by dividing by the total absolute error of the simple predictor.

134

Test the model with new data and evaluate the accuracy of the model The tensor analogy matches very well with the representation of the parameters defining our system (Microprobe designing features and performances parameters). All these assumptions were required for the microprobe performance prediction. Note that the SNR parameter (defined in chapter I, section (I.3.3.4)) is a function of the targets parameters of the system (Q factor and Resistance). Therefore, we trained an ANN model to predict all these parameters, which represent the target of the system.

Our dataset used to train the model was generated from numerical simulation of the microprobe Resistance and Q factor using the 3D-TLE (Transmission Line Extractor) platform, associated Those uniformly distributed features represent good predictors whose we train with the AI model, and those that do not variate should not impact the learning process of the model, so we did not select them as the model predictors. To find the best model training, we need to define which shape of the model we should use, select the best feature, know what features are important, and tune several parameters to find the optimal parameters. We needed to iterate many times to find the best fit for our data.

In Fig. III. 34, we show the ANN shallow architecture, which is the architecture that we chose for our study. Shallow is a neural network that provides only a single hidden layer. We provided an architecture with 14 neurons in the input layer (corresponding to the number of data predictors) and two neurons in the output layer (corresponding to the data targets). The number of neurons for the hidden layer will be optimized by a script, which calculates the optimal number of neurons according to the RMSE criterion. Therefore, when the model fits the noise, we call it overfitting. So, using validation data during the training process helps prevent overfitting. For validation data, there are two approaches:

cross-validation and holdout validation methods. In our work, we have chosen the holdout method.

To evaluate the quality of the chosen model, it was important to divide our data (data sets). We The trainrp function is the fastest algorithm (5 minutes for computation time). However, it did not perform well on function approximation problems. The memory requirements for this algorithm are relatively small in comparison to the other algorithms considered.

The conjugate gradient algorithms, in particular trainscg, seems to perform well over a wide variety of problems, particularly for our case, which seems to be similar to the trainlm. The SCG algorithm is almost as fast as the LM algorithm on function approximation problems. The conjugate gradient algorithms have relatively modest memory requirements. Furthermore, the SCG performance seems to be similar to the LM algorithms performance. Now, we have the performance of the model to generalize the prediction of our data. Thus, we used completely new data to predict the Resistance and Q factor of the microprobe. In Trial l number r of f optimal l hidden n layer r neuronal l

Conclusion

In this chapter, we proposed an electrical circuit that model the microprobe on a glass and silicon substrate. For that, we developed a 3D Transmission Line Extractor (TLE) platform; a homemade software dedicated to model the 3D transmission line (TL). Our goal was to model an NMR microprobe with a micro-coil, micro-TL, underpass, and vias structures. Thus, the effort was drawn on modelling and analysing the micro transmission lines and the TSV, simulated on a frequency range of 200 MHz to 900 MHz and then proposed the optimization in the designing to obtain a better-extracted value enhance the microprobe performance parameters (Q factor and Resistance value). The result obtained (Q factor of 110 for microprobe on a glass substrate, with a connecting wire solution -airbridge and a Q factor of 71 for a microprobe on a silicon substrate, with a connecting wire solution -underpass & vias with an optimized thickness t = 10 μm instead of 0.5 μm), We have validated these outcomes with ADS simulations; they were confirmed to be in good agreement (Table . III.2).

We also reported the training of an ANN model to predict the microprobe performances.

Our study demonstrates how well ANN can help us to predict the microprobe performance based on its geometrical parameters, materials electrical properties and its own working frequency. The Q factor was accurately predicted (99.79% of accuracy which correspond to = 0.0322); while, the prediction accuracy was of 91% ( = 0.042) which still need some improvements.

However, predicting the Q-factor that is correlated with the microprobe sensitivity remains a good opportunity to model the 3D Microprobe for NMR implanting before the microfabrication process and allow us to solve a complex problem that has been challenging for years.

IV.1. Microprobe cleanroom manufacturing process

There is almost three methods for the fabrication of micro-systems [START_REF] Tangri | Predicting Technique Survival in Peritoneal Dialysis Patients: Comparing Artificial Neural Networks and Logistic Regression[END_REF] [137] such as : (i)

Bulk micromachining, (ii) Surface micromachining and (iii) LIGA process, that is the Germanic abbreviation of Röntgenlithographie, Galvanoformung, Abformung representing different steps of the process [START_REF] Park | Time-Sensitive Multi-Dimensional Recommender in Database System[END_REF]. The first two methods are the most widely used in the microsystem fabrication in silicon and glass substrates [START_REF] Varadan | RF MEMS and Their Applications[END_REF]. However, the fabrication of our microprobe prototypes was performed using the surface micromachine method, more precisely the copper electroplating technique.

It is known that, the implementation of the micro-probe for our application lead the choice of the manufacturing process, from the connecting wire solution until the choice of the substrate to be used in order to have the desired performances for our application. From our team, a state of the art of different fabrication processes of the microprobe on silicon and glass substrate was reported in the Theis of J. Trejo Rosillo [START_REF] Rosillo | Contribution à l'amélioration de la sensibilité d'un micro-récepteur RMN implantable[END_REF]. The most important points to highlight are: (i)

Micro-coil fabricated by electrodeposition on glass silicon can achieve a good quality factor.

Though, a glass substrate does not allow the embedding of an active or passive circuit as in the case of a silicon substrate. (ii) Micro-coil manufactured on the silicon substrate reach a poor Q factor when we work on the high-frequency domain. Several works from the literature reported different MEMS coil designs, optimizing their performance, mainly investigating their performance for different coil materials, substrates and different fabrications techniques to maximize the Q factor [START_REF] Choudhary | MEMS: Packaging and Technology[END_REF]. The glass substrate has been preferred compared to a silicon substrate for the fabrication of the NMR microprobe. The Glass substrate has a low conductivity (about 10 15 / ) and is biocompatible for our application. For our study, we performed the fabrication of the microprobe prototype on both substrates: silicon and glass substrate.

IV.1.1. Specifications of the microprobe prototype fabrication

According to the optimal microprobe specification project, the following Tables summarises the provided microprobe geometries, dimensions, (number of spires turns) and the quantity in the designed layout for wire-bonding in Table IV The vacuum deposition method is used to produce a thin film and coatings. It is characterized by a process in which the material goes from a condensed phase vapour phase to a thin film condensed phase. Due to a high electric field effect, the argon-ion is accelerated toward the material to deposit and then spray on the wafer surface. For our case, we used sputtering and evaporation, which are the most common. The thickness of the spread photoresist depends mainly on its viscosity and the coater centrifuge rotation speed, which can be calculated by the equation below:

t = √ (IV.1)
Where is the centrifuge constant (between 80 and 100), is the percentage of the photoresist viscosity, and is the speed in revolutions per minute divided by 1000.

To optimize the microfabrication process, we spread a positive photoresist (AZ4562) on the prepared wafer surface. We were able to reach a mould thickness of 50 μ maximum with a positive photoresist and 40 μ for a negative photoresist. Positive photoresist gradually is more widely used since they offer better process mastering for small geometry features [START_REF] Luo | Review of Recent Advances in Inorganic Photoresists[END_REF].

Moreover, positive photoresist have the advantage of producing thick and resistant layers which can be used as a mould for the electrodeposition of metal [START_REF] Sotnik | Nano Devices and Microsystem Technologies: Brief Overview[END_REF].

Baking and relaxing: after the spin coating step, we kept the wafer relaxing for a few minutes to place and sequentially apply heat. We put on the electric heating plate of 64. This step consisted of removing the insoluble photoresist and the unprotected seed layer (copper and Titanium layer) after the growth of the copper layer. To achieve this, we immersed the wafer into an acetone bath. The photoresist dissolution was also accelerated using an ultrasound tray.

So then, the seed layer removing; First, the copper layer was removed by dipping the wafer in a solution of 60 from and 10 oxide water ( ) during one minute. Secondly, the / layer was also removed by using the solvent, for one minute.

The schematic representation of the Lift-Off step, before and after the process is shown in Fig.

IV. 13. This step was carefully carried out, and the time ultrasound tray was critical to ensure that the seed layer was completely removed while ensuring that all microstructures designed on the wafer did not get out to the wafer.

IV.1.2.5. Stage 5: Electroless gold plating

Electroless plating is defined as a coating process without applying an external current. It allows a uniform metallic coating regardless of the geometric shape of the surface after the sample is immersed in the solution. It is as if a chemical reaction occurs between the copper layer and the gold.

After removing the seed layer, the copper-plated on the wafer was exposed to an oxidation.

To avoid this, we had to growth a gold layer on top of the copper layer by electroless gold plating; So, the wafer was immersed in a gold solution; In the To examine and validate the fabrication process of the microprobe's prototypes, we performed their characterizations (morphological and electrical).

The morphological characterizations allowed us to check the shape of the prototypes, the impurities, the manufacturing defects (residuals resin and short-circuits) and this characterization allowed also examining the surface roughness and verify the thickness of the deposited copper layer. For this, we used two principal equipment: a microscope and a profilometer.

The electrical characterization allowed estimating the resistance and verifying the shortcircuits. The measure of the microprobe resistance was carried out using a conventional multimeter equipped with two micro-tips. The Table IV. 3 shows the results of the measurement of parameters of the three selected microprobes. All measurements were reported at frequency of 300 . One can notice that our results are in good agreement with those of the simulations. What is interesting to emphasize in this section is that all these results show that the first trial of microprobe manufacturing still needs to be improved. Therefore, in the next section, we will show the steps that we improved and then carry out for the fabrication of the final prototype according to the required specifications. IV.1.4. Optimization of Micro-manufacturing process

After the first trial of the microprobe prototype fabrication, all the obtained prototypes were characterized. We noticed some major defects on certain shapes of the microprobe, and sometimes the required thickness was not reached.

Based on these results, from the previous microprobe fabrication trial (1 st trial), we have a focus on the optimization of the following process steps:

1. Photolithography (UV lighting time, Spin coating process)

2. Developing (timing, photoresist and baking time) We used a positive photoresist (AZ4562) on the previous trial fabrication, but we could not reach the required thickness (40 μ ). Thus, we changed to a negative photoresist, and we also changed the mask. Note that the negative photoresist we get using for this experiment is suitable for spreading a thick mould layer.

Therefore, we aimed to define the optimal parameters for the spin coating equipment in the first place, knowing that it is needed to deposit at least 40 μ thickness of the photoresist. Then we optimized the baking time adapted to the resist and the UV insolation time to achieve the expected shape during the etching section.

For this experiment, we used a positive mask with some enhancement even for the optimization of the space on the wafer. In the following The electroplating was performed in two steps since it was unknown how to evaluate exactly the time required to reach the standard thickness with the new photoresist.

We first deposited electroplating only 4 μ for 30 to determine the rate of plating on the wafer. So, to attend at least 30 μ (like 28 μ ), we restarted the electroplating Therefore, the pictures show how well we could optimize the microprobe fabrication process since the images show the micro-coil part without any defect compared to the previous one (1 st trial of microfabrication). Thanks to this, we could fabricate microprobes whose characterization results will be presented in the next session.

IV.1.6. Characterization of the optimized prototype.

After the manufacturing process, the obtained microprobes were covered with a resin layer to protect and prevent them from peeling off the wafer support.

This photoresist layer needed to be removed before characterization of the microprobe.

The removal process of the protective resin layer also needed to be optimized. So, first of all, we selected seven microprobes. The aim here was to find and validate the optimal time for the removal process of the resin.

The experiment was carried out by plonging the microprobe into the recipient with an acetone solution. 14 30 for microprobe ( ° 1) and 10 for microprobe ( ° 3 to ° 7).

Microprobe ( ° 1) resin removing without ultrasound, °6 complete 10 minutes on the ultrasound, ° 4 was 5 minutes with and 5 minutes without and the rest of microprobe was 7 minutes without ultrasound and 3 minutes with ( ° 2, 3, 5, and 7).

In For this, we have chosen seven microprobes, selecting four on all corners (one for each one), and then 3 for both extreme middle lines and one on the centre area of the wafer (Fig. IV. 33).

We remove the resist of 7 microprobes, one by one for 5 minutes on the ultrasonic cleaning (acetone) + 2 minutes in isopropyl alcohol. Table (IV. 9.1 to IV. 9.7) show the measurements of the cooper thickness from different areas of the wafer (microprobe ° to ° ). From our thickness measurements of the micro-coil part and transmission line on different positions of the wafer, we estimated that the average thickness of the microprobe was about = (28.18 ± 1.

Measurement of the microprobes parameters

The parameters of the four microprobes were measured by a NANO VNA ZVL Rhode & Schwarz. We measured four micro-coil geometries such the square, circular, rectangular and ellipsoidal). The measurements were performed in the frequency range of 10 to 6 , calibrated for 4000 points of frequency to measure. In the following table we summarized the value of parameters of four geometries at 300 and 500 . In the Table IV.12, we compared the parameters of the ellipsoidal geometry determined by different approach. Note that the microprobe parameters are different according to the approach used, and mainly each is complementary to determine the parameters of the microprobe accurately.

Coil n°Geometry

Dimensions (μ ²) R [Ω] [ ] R [Ω] [ ] 1

Conclusion

In this chapter, we have described the processes of micro-antenna prototypes fabrication on glass or silicon substrates. Due to the inconvenience of previous connection points (wirebonding or underpass & vias) between the centre of the micro-coil and the signal transmission tracks on the previous prototype (1 st and 2 nd ) fabricated by our team, we have proposed a design of the microprobe with a connecting wire solution such airbridge. Thus, we presented the specifications of the microprobe characteristics for our fabrication.

Based on the defined specifications, we optimized the fabrication process, intending to From the point of view of the microprobe fabrication step, we did two trials. The prototype was characterized by morphological information. The first trial was not suitable for our applications; that is why we have optimized the fabrication process in the second trial (final optimized prototype of the micro-antenna). So, the morphological characterization shows satisfactory results for our application (Fig. IV. 33). We enhance in the first place the thickness of the prototype from trial 1 to the second one (1 st trial prototype thickness: (15.71 ± 3.12) μ and 2 nd trial prototype thickness: (27.10 ± 1.37) μ .

The measurement of the electrical characteristics of the prototypes fabricated during the first and second trial were carried out on the glass wafer. The results obtained show that the resistive and inductive losses measured match very well with the simulated results (Table IV.

3
and Table IV. 11). Furthermore, the final prototype after optimization of the process (second trial) should be suitable in terms of electrical performance due to great results of morphological characterization (no short-circuits, less residuals resin or impurities compared to the first trial, and the highest thickness, as well as the electrical characterization.

General Conclusions and Discussions

Based on the previous studies conducted by our team to develop implantable NMR microantenna, and by analysing the points not yet explored, this theoretical and experimental work comes as a complement to optimize the performances of the expected micro-antennas for our application.

In this work, we have proposed new methods to estimate the additional losses of all parts of the microprobe at the frequency range, 200 to 900 . Indeed, the analytical calculation of the resistance ( ), inductance ( ) and Q factor of the complete microprobe had been performed through an electrical circuit proposed in this work to model the microprobe in a silicon and glass substrate. Our approach takes into account all the constituents and characteristics of the systems: the skin, and proximity effect, the substrate, the intermediates layers ( ) for the prototypes manufactured on the silicon substrate.

From our results, we could note that, at 300 , the changes of microprobe substrate from glass to silicon provided an impact on the performance of the Q-factor (decreasing by 7 %). At 500 , the Q-factor of the microprobe degraded by 27 % if used with silicon substrate instead of the glass substrate. Thus, the best working frequency for our application could be 500 , for a microprobe on a glass substrate. We were sure that we could optimize An AI approach, specifically ANN combined with 3D-TLE, is a method we developed to generalize the optimization of our highly complex multi-parametric system (micro-probe). Our ANN model was trained using the dataset generated by the simulation work done beforehand.

Our last optimized ANN model allowed us to predict the Q-factor of the micro-probe with a test accuracy of 99.67% ( = 0.032), the strength prediction with an accuracy of 91.34 % ( = 0.042),

Considering the geometrical parameters of the μ-antennas that was defined from simulation work, the micro-moulding fabrication technique with electroplating was performed to obtain 40 μ . We had carried out a UV photolithography optimization choosing 226 for the UV insolation with a new photoresist (AZ152NXT -Negative photo-resin). We also optimized the second trial of the microfabrication process, the electroplating step (5 ℎ for copper electroplating, copper sulphate electrolyte bath ( ) with a speed flow of about 20 /ℎ and applied current density of 200 / ²).

We have optimized the fabrication process of the microprobe in order to respect the specifications defined in the project (see section IV. Thanks to all these results, we strongly believe that 3D-TLE associated with the ANN approach can help establish a relationship between several microprobe geometric parameters, material properties, and the working frequency with the microprobe performances parameters (Resistance and Q factor).

Actually, our AI model can perform the prediction of microprobe performance in 2 minutes instead of 15 hours of computing time. Thus, this new optimization approach using AI tools Finally, this thesis work also allowed us to demonstrate that we can design and fabricate optimized prototypes of different probes dedicated to the applications by knowing the desired performances, the working frequency and the materials needed for its fabrication.

= σ [START_REF] Rabeson | Quantitation with QUEST of Brain HRMAS-NMR Signals: Application to Metabolic Disorders in Experimental Epileptic Seizures[END_REF] where ε is the permittivity of vacuum, μ is the permeability of the vacuum, and σ is the electrical conductivity. In the SI system, the permeability of vacuum is chosen to be μ = 4π × 10 H/m. The velocity of an electromagnetic wave in a vacuum is given as c and the permittivity of a vacuum is derived from the relation:

ε = = 8.854 × 10 = × 10 (8)
The electromagnetic constants ε , μ , and c are available in COMSOL Multiphysics as predefined physical constants.

A consequence of Maxwell's equations is that changes in time of currents and charges are not synchronized with changes of the electromagnetic fields. The changes of the fields are always delayed relative to the changes of the sources, reflecting the finite speed of propagation of electromagnetic waves. Under the assumption that this effect can be ignored, it is possible to obtain the electromagnetic fields by considering stationary currents at every instant. This is called the quasi-stationary approximation. The approximation is valid because the magnetic field variations in time are small, and the studied geometries ( ≈ 1 ) are considerably smaller than the wavelength ( = 1 ) [START_REF] Ahamed | Field and Wave Electromagnetics -David K Cheng[END_REF] [147].

A.2. Mesh

In finite element analysis, the resolution domain is discretized into subdomains called mesh elements. These elements are the elementary bricks whose mesh will represent the geometric system to be simulated. The elements are geometrical primitives composed of several nodes. In COMSOL we have two types of mesh, physics-controlled mesh and user-controlled mesh.

Meshing considerations in COMSOL Multiphysics:

1. How to divide the geometry: sequence type

× N = N × n (8)
How many epochs a model should run to train is based on many parameters related to the dataset and the goal of the mode. So, while there have been efforts to turn this process into an algorithm, often a deep understanding of the data itself is indispensable, that why the preprocessing stage is crucial before the model training.

A.10. Algorithm selected for the optimization of the final model.

I

  would like to thank, Mr. Youssef ZAIM WADGHIRI, from NYU school of Medicine, (University of New York), and Mr. Lionel ROUSEAU from l'ESIEE, Paris for accepting to examine my thesis manuscript. Thanks to Mr. Luc HEBRARD from University of Strasbourg, Mrs. Alexandra PETIET from ICM Paris, and Mrs. Carole CHAIX, from CNRS, ISA Lyon for accepting to be jury on my thesis work. I would like to thank of course all the professionals of ISA, all my doctoral colleagues more particularly those who shared this work with me: Abel RANGEL TREJO, Adam ADEWOLU and Leticia KONADU. The scientific exchanges with my team, the project partners and collaborators during the years of my thesis contributed to my personal training, enriched my scientific and techniques knowledge which really helped me in the achievement of this work. Finally, I dedicate this work to all my family. I express my gratitude to my mother Rebeca Paulo PEDRO, my lovely girlfriend Marion DEROIN, my Brother Manuel Pedro BERNARDO and friends. During the last three years, you were always in my heart, and some were available to support me and help me to overcome my difficulties. "Ose savoir!" -By : Emmanuel Kant List of Acronyms AD : Alzheimer's disease AC : Alternative Current AI : Artificial Intelligence ADS : Advanced Design Systems ANN : Artificial Neuronal network BP : Basal Prosencephalon BP : Back Propagation CAD : Computer Aided Design CST MWS : Computational Simulation technology Microwave Studio CSF : Cerebral Spinal Fluid DUT : Device Under test DC : Direct Current DL : deep Learning Ex-Vivo: outside of a living body EM : Electromagnetics EMF : Electromotive Force EMF: Electromagnetic Field EEG : Electroencephalogram FDG : Fluorodeoxyglucose FP : Forward Propagation FFP : Feed Forward Propagation FID : Free Induction Decay FIT : Finite Integration Technique FWHM : Full Width at Half Maximum GABA : Gamma-AminoButyric Acid GUI : Graphical User Interface In-Vivo : inside of a living body In-Vitro : in glass testing LTD : Latero Dorsal Tegmental LW : Lorentzian width LOD : Limit Of detection LIGA : Rontgenlithographie Galvanoformung Abformung NMR : Nuclear Magnetic Resonance NAA : N-Acetyl aspartic acid NN : Neural network MEMS : Micro-Electro-Mechanical System MEK : Methyl Ethyl Ketone MIBK : Methyl Isobutyl Ketone MRS : Magnetic Resonance Spectroscopy MRI : Magnetic resonance Imaging MCI : Mild Cognitive Impairment MW : Microwave MIMO : Multiple Input and Multiple Output ML : Machine Learning MAE : Mean Absolute Error MAPE : Mean Absolute Percentage Error MSE : Mean Square Error RMSE : Root Mean Square Error RAE : Relative Absolute Error RSE : Relative Square Error RH : Relative humidity PET : Positron Emission Tomography PPT : PedunculoPontine Tegmental PCB : Printed Circuit Board PVD : Physical Vapour Deposition PEEK : Polyether Ether Ketone PDMS : Poly Dimethyl Siloxane ROI : Region Of Interest RF : Radio Frequency RMSE : Root Mean Square Error SNR : Signal to Noise Ratio SPICE : Simulation Program with Integrated Circuit Emphasis TL : Transmission Line TSV : through Silicon Vias VSWR : Voltage Standing Wave Ratio 3D -TLE : 3 Dimensions Transmission Line Extractor RLC : Resistance Inductance capacitance UV : Ultraviolet VNA : Vector Network Analyser THESIS SUMMARY FRENCH VERSION 1 Résumé de thèse Mots clefs : Micro-probe, Micro-bobine, Spectroscopie RMN, Maladie d'Alzheimer, Biomarqueurs, Réseau de Neurones Artificiels (RNA), Facteur de Q, Rapport Signal sur Bruit (RSB ou SNR), Intelligence Artificiel (IA).

  d'adaptation et d'accord. La micro-probe est la partie micro-bobine associée à la micro-ligne de transmission (TL) et aux différents types de connectiques. Les connectiques permettent de relier le centre de la micro-bobine à la ligne de transmission, et également la ligne de transmission au circuit d'accord et adaptation. La Figure. 1 montre le premier prototype de micro-antenne RMN implantable, fabriquée par notre équipe. Cette micro-antenne a été conçu, pour être un outil d'analyse implantable de manière chronique pour des applications in vivo assurant la reproductibilité des mesures par spectroscopie RMN haute résolution. Plus précisément, pour suivre la variation des métabolites cérébraux (biomarqueurs) dans des petits volumes à faible concentration en utilisant des modèles animaux (rat ou souris).

Figure. 1 :

 1 Figure. 1:Le premier prototype de la micro-antenne fabriquée par notre équipe. Devant cette nécessité d'examiner par spectroscopie RMN de faibles quantités de matière, dans notre équipe, nous voulons explorer une région précise dans un ensemble beaucoup plus vaste ; ce qui nous a orientés vers le choix d'antenne implantable. L'objectif de plusieurs thèses

  Les tests in vivo n'ont pu être réalisés qu'avec un travail complémentaire d'optimisation (conception et simulations) de la sensibilité de la micro-probe, en termes de son facteur de qualité et . Ce point est d'une extrême importance, car les signaux sources sont si faibles qu'ils sont comparables au bruit thermique, qui peut être lié à la résistance de la micro-antenne. Pour améliorer ses pertes résistives et plus particulièrement celles dues à sa connectique (wire-bonding), J. Trejo Rosillo (2014) a proposé dans sa thèse une nouvelle façon de connecter le centre de la micro bobine avec la micro-probe (underpass & vias). De même, T. Cong Troung (2014) a développé une méthode d'optimisation des micro-bobines RMN par une approche physique. Malgré ces améliorations, le facteur de qualité est resté faible (par exemple, = 1,4) ; insuffisant pour les applications in vivo attendues. D'autre part, nous pouvons améliorer les performances de la micro-probe en optimisant son processus de fabrication. La tâche de fabrication nécessite plusieurs étapes coûteuses en temps et argent, surtout quand il s'agît d'explorer la fabrication des prototypes avec différentes dimensions, différents matériaux et les comparer expérimentalement. La simulation permet de s'en affranchir et d'examiner différents scénarios de systèmes sans avoir recours à l'expérimentation, puis de sélectionner les principaux paramètres pour obtenir les performances requises. Elle peut également fournir des tendances de données significatives et réaliser des comparaisons entre les prototypes possibles et guider le processus de fabrication. La micro-probe peut être considérée comme un microsystème complexe multiparamétriques. Le nombre de paramètres augmente la complexité de l'étude analytique, d'une part et rend la fabrication très sophistiquée et coûteuse, il nous a fallu recourir aux outils de simulations, puis aux outils de prédictions. Par conséquent, l'objectif de ma thèse a été de proposer une nouvelle approche pour optimiser la micro-probe NMR implantable (microbobine + micro-ligne de transmission (TL) + connectiques). La Figure. 2 montre le dessin du nouveau prototype optimisé et fabriqué. Sur la Figure. 2, nous avons également la description de chacune des parties et leurs dimensions respectives.

Figure. 2 :

 2 Figure. 2: Le prototype de la micro-probe aux dimensions optimisées : version avec une connectique type air-bridge

2 .

 2 Prédictions des performances des nouveaux prototypes de micro-probe par le biais d'un modèle de l'IA. Pour cela, nous avons développé une nouvelle plateforme « maison » 3D -TLE (Transmission Line Extractor) en collaboration avec l'INL (Institut de Nanotechnologie de Lyon). 3D-TLE est une plateforme créée sur MATLAB GUI (Graphical User Interface), qui peut générer un réseau SPICE (Simulation Program with Integrated Circuit Emphasis). Elle peut extraire automatiquement les impédances équivalentes de tout type de composant ou dispositifs de différentes géométries et de matériaux quelconques. Cet extracteur est 100 % compatible avec le simulateur de base SPICE, et avec le logiciel de modélisation ADS (Advanced design system). Sur la Figure. 3, nous avons le schéma blocs des différentes étapes pour la simulation des performances de différentes géométries et dimensions de la micro-probe.

Figure. 3 :

 3 Figure. 3: Schématisation des principales étapes des simulations de micro-probes. Le principe consiste à créer d'abord, un fichier .txt avec tous les paramètres géométriques du système, les propriétés des matériaux et le couplage entre les différentes parties constituant le système. Ensuite, un fichier spice est généré avec les paramètres électriques (RLC), avec lequel la performance en termes de facteur Q et SNR ont été simulé par ADS et calculé analytiquement.

Figure. 4 :

 4 Figure. 4: Une micro-bobine circulaire et carrée à trois spires ( : diamètre extérieur, : diamètre interne, w : largeur du conducteur, t : épaisseur du conducteur et s : distance de séparation entre deux conducteurs)

Figure. 5 :Figure. 6 :

 56 Figure. 5: Le modèle électrique de la micro-probe sur un substrat de silicium (modèle complet) Les paramètres de performances ( ), de notre modèle électrique ont été calculés analytiquement dans la gamme de fréquences de fonctionnement des micro-antenne allant de 200 MHz à 900 MHz. Les résultats analytiques ont également été validés avec des résultats de simulations sur ADS (Tableau. 1) et des mesures après fabrication des prototypes de microprobes. Notre modèle électrique nous a permis également d'étudier la contribution de chaque partie et estimer les pertes résistives introduites.

Figure. 7 :

 7 Figure. 7: Variation de la composante z du champ magnétique en fonction de la direction perpendiculaire (y) de la micro-probe.

  première était de trouver une méthode efficace, pour exploiter et traiter les ensembles de données issues des différentes simulations (Figure. 8). L'autre idée est de, construire un modèle d'optimisation qui pourrait qui pourrait être généralisé à l'optimisation de tout système complexe multiparamétrique comme le nôtre (voir Figure. 9).

Figure. 8 :

 8 Figure. 8: Le parallèle entre la modélisation Classique et l'approche Machine Learning. La Figure.8 montre le parallèle entre la méthode classique de simulation et la méthode avec une approche IA. Pour la suite de ce rapport, on s'est focalisé sur l'approche IA qui constitue la deuxième partie de l'approche utilisé pour la réalisation du nouveau prototype de la microprobe.

Figure. 9 :

 9 Figure. 9: Le modèle tenseur de notre système multiparamétrique. Les méthodes de régression de Machine Learning sont utilisées pour prédire les valeurs continues tel que le prix des appartements dans une ville, d'une grandeur physique tel que la température d'un système, le facteur de qualité du système dans notre cas. L'ensemble de données du système peut être caractérisé par deux types de variables : les variables dépendantes et les variables indépendantes.

Figure. 10 :

 10 Figure. 10: Le schéma block représentant la procédure de l'approche IA pour la perdition des performances de la microsonde.

  de son substrat, et de sa fréquence de travail. Le modèle ANN a été entraîné et testé à travers le jeu de données généré par les simulations (6621 observations). Pour ce faire, nous avons utilisé la boîte à outils de MATLAB, Application GUI Deep Learning (Figure. 11).

Figure. 11 :

 11 Figure. 11: Interface de l'outils d'entrainement du model IA.

Figure. 12 :Tableau. 2 :

 122 Figure. 12: Prédiction du facteur Q et résistance R de la micro-probe en fonction du nombre d'observations, superposée aux valeurs réelles. On peut noter sur la Figure. 12 que les prédictions de notre modèle sont en accord avec les vraies valeurs de chaque paramètres (le facteur Q et la résistance R). Les performances du test des modèles IA, en termes de la métrique RMSE et < ² > sont résumés dans le tableau suivant :

Figure. 13 :Figure. 14 :Figure. 15 :Figure. 17 :

 13141517 Figure. 13: Les mages des micro-bobines fabriquées dans le substrat de Verre. Géométries : circulaire ( × μ ²), ellipsoïdale ( × μ ²), rectangulaire ( × μ , × μ ²)

Fig. I. 1 .

 1 Fig. I. 1. Natural progression of biological markers of AD [2] ................................................ Fig. I. 2. The spectrum of cerebral metabolites of the rat [28] ................................................ Fig. I. 3. Anatomy of the main cholinergic pathways in the rat brain [35]. ............................. Fig. I. 4. Magnetic atomic nuclei behaviour, before and after setting it into the magnetic field .................................................................................................................................................. Fig. I. 5. Energy diagram describing the distribution of the nucleon quantum stats at two different energy levels, before and after setting into the static magnetic field B0 [37] ........... Fig. I. 6. The principle of emission and reception of FID signal. a) The rotating of magnetization M induces a EMF into the S loop; b) The signal at point P is proportional to the magnetic field B1 [43] ...................................................................................................................................... Fig. I. 7. The plot of ξ(t), detected NMR signal in the time domain, FID [43] .................... Fig. I. 8. An implantable planar micro-antenna (with ellipsoidal micro-coil): first prototype developed by our team [57] ...................................................................................................... Fig. I. 9. A DUT (Device Under Test) with two ports ............................................................. Fig. I. 10. Comparison of the LOD of a commercial antenna with the rack trace antenna [58] .................................................................................................................................................. Fig. I. 11. (a) Micro-antenna implantation inside the ROI, using a steréotaxy device, (b) representing amagnetic anchoring screw and the micro-antenna site (Dorsal plans of rat skull) [56]. .......................................................................................................................................... Fig. I. 12. In vitro NMR spectrum of 10 brain metabolites in aqueous solution using the planar micro-antenna (prototype developed by our team) ..................................................................

Fig. III. 15 .

 15 Fig. III. 15. The variation of Resistance a) and Inductance from complete electrical models of microprobe on a glass substrate and silicon ........................................................................... Fig. III. 16. The statistical study of RLQ parameters, comparing different models of microprobe modelled on a glass and silicon substrate, for ...................................................... Fig. III. 17. The statistical study of RLQ parameters, comparing different models of microprobe modelled on a glass and silicon substrate, for ...................................................... Fig. III. 18. A statistical study of Q-factor variation from microprobe operating at and for both substrate cases ................................................................................... Fig. III. 19. Block diagrams of the classic simulation methodology for microprobe parameters optimization [43] .................................................................................................................... Fig. III. 20. (a) Traditional method vs (b) Machine Learning approach ................................ Fig. III. 21. Machine Learning Workflow [116] .................................................................... Fig. III. 22. Two categories of machine learning techniques: Supervised and unsupervised learning [117] ......................................................................................................................... Fig. III. 23. Human neuron (a) versus a standard mathematical model of a neuron (b) [124] [125] ....................................................................................................................................... Fig. III. 23. Representation of the standard activation functions ........................................... Fig. III. 25. Artificial Neural networks are organized in layers consisting of a set of

Fig. III. 35 .

 35 Fig. III. 35. The microprobe Q factor and Resistance R prediction, using the optimal number of neurons according to the evaluation of RMS from training, validation and testing .............. Fig. III. 36. The microprobe Q factor and Resistance R prediction, using the min and max RMSE of the testing model according to the trial of training with different weights and bias parameters .............................................................................................................................. Fig. III. 37. ANN architecture for Microprobe dataset training model .................................. Fig. III. 38. The RMSE variation in terms of the number of neurons in the hidden layer ..... Fig. III. 39. (a) The performance variation as a function of epoch (train, validation and test) (b) The error histogram for the training model ............................................................. Fig. III. 40. A plot of the regression of the (a) Training dataset, (b) the validation and (c) Test dataset ..................................................................................................................................... Fig. III. 41. The prediction and the true value of the Q factor and Resistance of the microprobe ................................................................................................................................................ Fig. III. 42. RMSE performance of the optimal trial .............................................................. Fig. III. 43. The microprobe global Q factor and Resistance R prediction, after trial number optimization ............................................................................................................................

  Fig.I.1 AD occur up to 20 years before the onset of dimension symptoms through certain markers such as amyloid , functional and metabolic markers, atrophy from whole brain, hippocampe, Entorhinal cortex and temporal neocortex. We can note their evolution in different regions of AD stage, particularly in Mild Cognitive Impairment (MCI) where we can distinguish a high evolution of these markers.

Fig. I. 1 .

 1 Fig. I. 1. The representation of natural progression of biological markers of AD [2]

  Fig. I. 2. The spectrum of cerebral metabolites of the rat [28]

Fig. I. 3 .

 3 Fig. I. 3. Anatomy of the main cholinergic pathways in the rat brain [35].

Fig. I. 4 . 37 ]

 437 Fig. I. 4. Magnetic atomic nuclei behaviour, before and after setting it into the magnetic field

Fig. I. 6 . 43 ]

 643 Fig. I. 6. The principle of emission and reception of FID signal. a) The rotating of magnetization M induces a Electromotive Force (EMF) into the S loop; b) The signal at point P is proportional to the magnetic field [43] Figure I.6 illustrates the NMR signal Free Induction Decay (FID) principle for a coil section S [43]. The variable magnetization ⃗ , around an axis, can produce an alternating magnetic field in the S section of a conductor coil. Assuming that the cross-sectional S of the coil through which the current travers, the coil can generate a magnetic field ⃗ at the point P (Fig. I. 6.

. 1 )

 1 ⃗ , ⃗ , ⃗ are representing the unitary vectors for the reference plane. , are the longitudinal and transversal relaxation time. T * is representing the true relaxation phenomena of , which we take into account the constant inherent inhomogeneity of the magnetic field [3]. * = + γ ∆B (I.18)

  .2). Generally, we name the NMR signal detected by an RF antenna; Free Induction Decay (FID) is a sinusoidal signal damped by an exponential signal as shown in Fig. I. 7.

Fig. I. 7 .

 7 Fig. I. 7. The plot of ξ(t), detected NMR signal in the time domain, FID [43]

  Baxan et al.[START_REF] Baxan | Mise En Oeuvre de Microantennes RMN En Perspective d'étude in Vivo de Metabolites Par Spectroscopie[END_REF] proposed a planar micro-antenna with the following geometrical characteristics: Ellipsoidal, 1000 × 500 μ ², = 22 μ wire width, = 20 μ wire spacing and thickness = 38 μ and 4 copper line. What provides an observation volume of about 0.2 μ for In vitro measurements by proton NMR spectroscopy. It was performed under a static field of 4.7 with a sample containing five brain metabolites: taurine, lactate, NAA, choline and GABA. The taurine concentration was 50 / and the metabolites remaining was 100 / .

Fig. I. 8 .

 8 Fig. I. 8. An implantable planar micro-antenna (with ellipsoidal micro-coil): first prototype developed by our team [57] From our team, A. Kajdo [58] reports the performance criteria of the first micro-antenna prototype proposed by Baxan et al. (Fig. I. 8). He studied its optimization for MRS in vivo acquisition. He also performed a biocompatibility study. Furthermore, he demonstrated that a

  TableI. 3. Summary of our team works, geometrical dimensions, electrical parameters analysis, and materials for μmanufacturing and its application

  I. 9), we can represent different signals which correspond to the incident and reflected signal.

Fig

  Fig. I. 9. A DUT (Device Under Test) with two ports

  20.1) = (I.20.2) = (I.20.3) Linear system (DUT) (e.g : Resonator) = (I.20.4) The 2-port S-parameters have the following generic descriptions: is the input port voltage reflection coefficient is the reverse voltage gain is the forward voltage gain is the output port voltage reflection coefficient.

=:

  (I.27) = (I.27.1) Replacing the expression of the NMR signal and the noises signal in terms of voltage magnitude, we can define the SNR in the time domain as follows: is the resonant angular frequency, Corresponds the magnetic field generated by the micro-coil, M0 is the static Magnetization, Vs is the sample volume, kB is the Boltzmann constant, is the antenna resistance, and ∆ is the spectrum bandwidth. The NMR experiments are generally analysed in the frequency domain (spectra). Hence, we must define the SNR formula in the frequency domain. The experimenter has easy access to the SNR parameter in the spectral domain, usually due to the accumulation of several signal acquisitions. That means the number of acquisitions each other of duration as well as which the amplitude of the signal, we assumed the signal decreased exponentially with the time constant * . So, by Fourier Transformation (TF), we transform the in the frequency domain equation:

37 )

 37 It is also possible for the sensitivity formula in terms of acquisition time from the experiment duration. For years, the scientific community has had these sensitivity formulas to define how performant the experiment can be[START_REF]Micro-Capteur Implantables : Etude Des Critères de Performance En Vue de l'optimisation Des Acquisitions Par Spectroscopie Rmn in Vivo Kadjo -Recherche Google[END_REF] [START_REF] Baxan | Vivo Animal NMR Studies Using Implantable Micro Coil[END_REF].

Fig. I. 10 .to 4 and 30 / to 64 )

 1064 Fig. I. 10. Comparison of the LOD of a commercial antenna with the rack trace antenna [58]

I. 5 . 2 .

 52 Previous implantations of the Micro-coil in a rat model Successful micro-antenna implantation was carried out by our team in the first place [56]. As shown in Fig. I.11 (a), the micro-antenna placed inside the ROI, taking into account the perpendicularity between the magnetic field , generated by the RF micro-antenna and the statical magnetic field .

Fig. I. 11

 11 Fig. I. 11. (a) Micro-antenna implantation inside the ROI, using a steréotaxy device, (b) representing amagnetic anchoring screw and the micro-antenna site (Dorsal plans of rat skull) [56]. The experimental outcomes demonstrate that the micro-antenna is "minimally invasive" [58] [56]. The implantation was performed without significant brain lesions or postoperative complications. In the first place, the area of interest was localized by NMR imaging and correlated by histology. The measurements correspond to the coordinates of the Paxinos and Walton atlas [36]. The useful in vivo measured volume was about 1.86 , corresponding to almost the in vitro magnitude measured (2 μ ).

Fig. I. 12

 12 Fig. I. 12 shows an in vitro NMR spectrum of ten brain metabolites with a 50 / concentration, which was acquired in a magnetic field of 4.7 T by the first micro-antenna prototype developed by our team.

  of the NMR probe adapted to the target application. Several studies had implemented electromagnetic simulation tools (FEMLAB, Maxwell 2D/3D, Magnetica, CST MWS, COMSOL Multiphysics, etc.) to model micro antennas, optimize their SNR parameter and consequently their performance[78] [79].This chapter reports a new methodology of the microprobe (micro-coil + transmission line +underpass & vias or airbridge) modelling in terms of the magnetic field and resistive loss. In the first section, we reviewed the optimization work made previously by our team, and then we validated the new method, comparing our results. Magnetic field simulations were performed through CAD (Computer Aided Design), specifically for EM simulations such as CST MWS (micro-coil part) and COMSOL Multiphysics (complete microprobe). Then electrical parameters, simulations based on analytical calculations of RLC (Resistance, Inductance and Capacitance).

  the second part of this chapter, I presented my own modelling work and the main results obtained. II.1.1. The micro-coil and transmission line modelling The Fig. II. 1 shows the 1 st prototype of the Implantable NMR micro antenna fabricated by our team [56]. The micro-antenna consisted of the microprobe (i.e., micro-coil, Transmission Line (TL), and wire-bonding), associated with an electrical circuit fabricated on a PCB for signal matching and tuning.

Fig. II. 1 .

 1 Fig. II. 1. Micro antenna prototype (the 1 st fabricated by our team) [56].

1 .

 1 Fig. II. 2. Design and dimensions of the 2 nd prototype of the micro-probe [80]

  The first point of improvement concerns the wire-bonding, used in the first versions of the microprobe to connect the center of the micro-coil to the TL and from the TL to the PCB. The connecting wire (wire-bonding) presented two major disadvantages: its delicate welding and its fragility during implantation. To solve this, our team (J. Rosillo Trejo and T. Cong Truong Thesis) have proposed a new micro-connecting wire (underpass & vias) and optimization of the transmission line layout as we have previously seen in the precedent section.

Fig. II. 3 .

 3 Fig. II. 3. Each step represents a suitable phase to follow to design and develop an optimal prototype. So, the process is divided into two parts: First, the conception and simulation, and secondly, the experimental part, which corresponds to the optimization, characterization and validation of the microprobe fabrication.

Fig. II. 3 .

 3 Fig. II. 3. Block diagram of our methodology to develop the new optimized microprobe prototype

  Fig. II. 4. The four geometries of micro-coil designed in CST MWS

Fig. II. 5

 5 Fig. II. 5 shows B Magnetic field as a function of z -direction where we simulated the magnetic field around the micro-coil. We select the variation of the magnetic field into the perpendicular plan which represents the field of view of the micro-coil, and most precisely its sensitivity.

2 . 2 . 1 ,

 221 we start by presenting the modelling of the length for the four geometries of micro-coil that are useful for the analytical estimation of RLC. II.2.2.1. The length modelling of the four micro-coil geometries Before describing the formulas to analytically calculate the micro-coil length, we show the micro-coil shape geometrical parameters description. To illustrate the modelling, we have three spiral turns circular and square micro-coil (Fig. II. 6) and a four spiral turn ellipsoidal geometry (Fig. II. 7). The descriptions of all geometrical parameters of a coil are given in the Fig.II.6.

Fig. II. 6 .

 6 Fig. II. 6. Circular and square micro-coil shape with three spiral turns ( : outer diameter, : inner diameters, : width of wireline, : the thickness of wire and : separation distance between two wires) [43]

Fig. II. 7 .

 7 Fig. II. 7. Ellipsoidal micro-coil shape with four spiral turns [43]

:

  phenomena (skin effect) contribute to increasing the alternative resistance on the conductor wire. The following equation defines the skin effet thickness [ ] in a conductor: Frequency domain of the alternating current [ ] : is the electrical conductivity of the conductor wire [ ]The skin effect formula demonstrates the increase in the conductor wire resistance at the highfrequency domain qualitatively. The threshold frequency can be determined by the equation (II.10). This formula allows determining the frequency from which the skin effect appears for a conductor wire radius.

  Fig. II. 9. Skin effect phenomena illustration in the case of a rectangular (a) and ellipsoidal (b) conductor surface section. The blue area represents the surface cross-section region where the current is going to pass-through S = S -S (II.12)

Fig. II. 10 .

 10 Fig. II. 10. Coplanar line structure for two close wires with the rectangular surface [43]

Fig. II. 11 .

 11 Fig. II. 11. The four modelled geometries: = 4, = 40 μ , = 22 μ , = 20μ

  Fig. II. 12, we showed the fields variations generated by different dimensions of rectangular micro-coils.

Fig. II. 12 .

 12 Fig. II. 12. Variation of the magnetic field as a function of the z-direction for different dimensions of rectangular micro-coil

3 .

 3 The electrical and magnetic parameters of each rectangular micro-coil at , for different dimensions The same work was performed for several dimensions of square micro-coil; the results for the magnetic field value on the centre of the micro-coil are shown in Fig. II. 13. and electrical parameters () and parameters are summarized in TableII. 4.

Fig. II. 13

 13 Fig. II. 13. variation as a function of the z-direction, for different dimensions of the square micro-coil

  The smallest dimension generates the most intense field in the centre of the micro-coil geometry section, 250 × 250 μ ² for the square geometry (Fig. II. 13) and 500 × 250 μ ² for the rectangular (Fig. II. 12).

Fig. II. 14 .

 14 Fig. II. 14. Representation of the limiting external diameter as a function of the geometric parameters of the μcoil To better illustrate our approach to define the threshold dimension of a micro-coil, let's take a micro-coil, with the following geometrical parameters: the number of turns ( = 4), conductor width ( = 22 μ ), conductor thickness ( = 40 μ ), inter-turn distance ( = 20 μ ) and width of the center micro-path ( = 60 μ ). The value of the threshold diameter can be evaluated at about = 400 μ . This implies that the micro-coil must have a minimum surface of 400 × 400 μ ². Reducing the size of the surface of the micro-coil to increase its field strength means to vary the parameters , , or , which could modify the value of the resistance.

The airbox volume was 4 × 2 × 8 ,

 8 Fig. II. 15. a) where the micro-coil on substrate support is modelled, the red line help to define the direction in which the magnetic field will evaluate (field of the view of the micro antenna). In Fig. II. 15.b), we plot different plans in which the magnetic field is propagated.

Fig. II. 15

 15 Fig. II. 15. a) The micro-coil modelled on the substrate inside a rectangular system. b) 3D view of the magnetic field variation on the parallel and perpendicular plan against micro-coil Fig. II. 16 shows the field line propagated on the parallel micro-coil plan. It can be highlighted in the figure the magnetic field line symmetry, according to both perpendicular plans. The field line is more concentrated on the coil centre and at the conductor vicinity.

Fig. II. 16 .

 16 Fig. II. 16. Lines of the magnetic field generated by the micro-coil: 2D view (a) situated at = μ and 3D view (b) From plotted data in the Fig. II. 16, we could extract the 1D plot of the magnetic field to analyse how the z-component of the magnetic field change as a function of the perpendicular direction at several working frequencies.

II. 17 ,Fig. II. 17 .

 1717 Fig. II. 17. Profile of the z magnetic field component from several working frequencies. In Fig. II. 18, we have shown the impact of the substrate. For this, we fixed the frequency simulation (300); Then we plotted the magnetic field for fives substrates (Glass, Silicon, PEEK, PDMS, Quartz).

Fig. II. 18 .

 18 Fig. II. 18. Magnetic field z component as a function of the z-direction for substrates types We studied how the magnetic field variates in the z-direction (perpendicular direction) and the plane containing the microprobe. Here, we have only the micro-coil (active part) of the microprobe designed on a substrate. We change the properties of the materials which correspond to each type of substrate for our study. We noted that the magnetic field did not significantly change when we changed the substrate type (Fig.II.18). As well as, the magnetic field did not considerably change when we varied the frequency from the range of 200 900 (Fig.II.17)

Fig. II. 19 .

 19 Fig. II. 19. The 3D layout of the proposed microprobe: new prototype (Coil Needle) with rectangular coil The microprobe layout design shown in Fig. II. 19 includes different parts such as:

Fig. II. 20 .

 20 Fig. II. 20. 3D view of the microprobe magnetic field linesEven if microprobe transmission line also generates a magnetic field, it does not Impact the magnitude of the magnetic field generated by the micro-coil (active part).

Fig. II. 21 .

 21 Fig. II. 21. Magnetic field component as a function of microprobe parallel direction ( )

Fig. II. 22

 22 Fig. II. 22 shows that the magnetic field generated by the microprobe at 300 is lower than at 500 in the centre ( = 5850 μ ). That simulation was performed for a design of the microprobe on the silicon substrate.

  Network (ANN) model to demonstrate the possibility of predicting the microprobe performance criteria (Q factor and Resistance) from its geometrical dimensions, electrical properties of the used substrate, and its working frequency. Our ANN model was trained and tested through the dataset generated by simulations (6621 observations).III.1. 3D -TLE (Transmission Line Extractor)3D-TLE is a platform created in MATLAB GUI (Graphical User Interface) which can generate the netlist of a given structure. 3D TLE can automatically extract impedances of arbitrary shapes and materials. This extractor is 100% compatible with SPICE (Simulation Program with Integrated Circuit Emphasis) core simulator, like ADS (Advanced design system). Our in-lab-made software (3D-TLE) is based on the parasitic effects, and we can see in Fig. III. 1 the graphical user interface of the platform.

Fig. III. 1 .

 1 Fig. III. 1. User Interface of 3D-TLE (Transmission Line Extractor)This extractor is based on the closed-form expressions of the RLCG (Resistance, Inductance, Capacitance, Conductance) compact models that were built from the Transmission Line Method and 3D full-wave electromagnetic analyses[START_REF] Nabil | Electrical Modeling of Tapered TSV Including MOS-Field Effect and Substrate Parasitics: Analysis and Application[END_REF].

Fig. III. 2 .

 2 Fig. III. 2. 3D-TLE extraction tool environment [100]

  by anti-corrosion (Fig. III. 3).

Fig. III. 3 .

 3 Fig. III. 3. 3D Compact model of the TSV as defined in 3D-TLE [96] As shown in the Fig. III. 4, the TSV electrical model consists of serial elements (partial Resistance and Self-Inductance) representing the signal propagation and also parallel elements modelling the TSV environment coupling. In particular, when these elements are fed by a current, its vertical current path goes through the oxide layer and reaches the ground lines. Only the inductive coupling between TSV is considered for the Low Frequencies.

Fig. III. 4 .

 4 Fig. III. 4. The electrical model of the TSV orientation according to its I (1) /O (2) ports [99] [98]

Fig. III. 6 .

 6 Fig. III. 6. The electrical circuit that models the micro-coil on the silicon substrate (PI Model)

Fig. III. 7 .

 7 Fig. III. 7. Model PI block diagram The equivalent impedance, from the PI model diagram, can be analytically determined [106] :

22 )

 22 Our analytical calculations were done, associating the complete electrical circuit with a block diagram (Fig. III. 9) representing different main parts of the microprobe. Therefore, the block diagram simplifies how to handle the calculation of the model impedance and to deduce the Resistance, Inductance and Q-Factor (RLQ) parameters of the system (complete microprobe).

Fig. III. 9 .

 9 Fig. III. 9. Block diagram of the electrical circuit of microprobe on a glass substrate Fig. III. 10 shows the complete electrical circuit to model the microprobe on a silicon substrate. This model is more complex compared with the microprobe model on the glass substrate (Fig.III.8) due to the consideration of the resistive loss on the substrate, induced by the micro-coil, transmission line, in addition to the underpass & vias.

Fig. III. 10 .

 10 Fig. III. 10. The electrical circuit modelling the microprobe on a silicon substrate (complete model)

  factor) from the proposed electrical model. The analytical value from the MATLAB was compared and validated with the simulations done by ADS software. III.2.3. The electrical parameters (RLC) of the microprobe extracted by 3D-TLE The electric parameters (RLC) were extracted from each part of the microprobe at an operating frequency of 300 through the 3D-TLE platform. (See Fig.II.2 for the complete design of the microprobe with a rectangular micro-coil and an underpass & vias connecting wire solution).

2 . 5

 25 Figure. III.11 shows the S parameter results of the electrical circuit such as RLC, PI model μprobe & μpath and complete model. We denoted here by μprobe & μpath the completed model with the underpass & vias replaced by a short-circuit.

Fig. III. 11 .

 11 Fig. III. 11. parameters of the microprobe electrical model from the silicon substrate (a) and the glass (b) As already said, before simulating the full electrical model of the microprobe on the glass and silicon substrate, we proposed an intermediate model called μcoil & path (i.e, microcoil plus micro-transmission), which only includes the contribution of the micro-coil part and the one of the micro-Transmission Lines (TL). This model has been helpful to deduce the contribution of the underpass & vias in the overall resistive loss of the microprobe. In Fig. III. 11b) we can note a high variation of the resonance peak of the electrical model of the microprobe without the underpass & vias, compared to the complete model, specifically to the one of the silicon substrates. In addition, we can see in Fig. III. 11 a) & b) that when the model gets more complex, the peak of the resonance curve starts to decrease. This can be explained by the energy losses of the underpass & vias on the substrate.

Fig

  Fig. III. 12.parameters of the complete model of microprobe on a silicon and glass substrate

Fig. III. 13

 13 Fig. III. 13 shows the variation of the Q-factor simulated for the complete electrical model of the microprobe on glass and silicon substrate.

Fig. III. 13

 13 Fig. III. 13. Q-factor variation of both complete electrical models of NMR micro-probe Note that the value of the Q factor from the microprobe model on the silicon substrate stop increasing when it reaches the frequency of 500 .

Fig. III. 14 .

 14 Fig. III. 14. Resistances variation from electrical models of microprobe on a glass substrate a) and silicon b)

Fig. III. 15 .

 15 Fig. III. 15. The variation of Resistance a) and Inductance from complete electrical models of microprobe on a glass substrate and silicon

  on the ADS results of the parameters summarized in Table III. 2, we normalized the highest value of each parameter ( ) to 100%, and we performed a statistical study. We applied uniform normalization in the range of 0 to 1. The impact of each component of the micro-probe (micro-coil, TL, underpass & vias, and the substrate) in the variation of , and Q-factor parameters were detected through a good understanding of the global electrical diagram. In Fig. III. 16, we show the normalized value of parameters of different electrical models for the glass and silicon substrate at 300 .

Fig. III. 16 .

 16 Fig. III. 16. The statistical study of RLQ parameters, comparing different models of microprobe modelled on a glass and silicon substrate, for

  Fig. III. 17. The statistical study of RLQ parameters, comparing different models of microprobe modelled on a glass and silicon substrate, for

Fig. III. 18

 18 Fig. III. 18 shows the variation of Q-factor of the complete electrical circuit on glass and silicon substrate and two operating frequencies (300 and 500).

Fig. III. 18 .

 18 Fig. III. 18. A statistical study of Q-factor variation from microprobe operating at and for both substrate cases In Fig. III. 18, we can highlight two points: Firstly, the quality factor increases by 38%

III. 19 .

 19 ) of the developed homemade simulation software by T. Cong Truong (Thesis work, 2014) [43].

Fig. III. 19 .

 19 Fig. III. 19. Block diagrams of the classic simulation methodology for microprobe parameters optimization [43]

Fig 3 .

 3 Fig. III. 20. (a) Traditional method vs (b) Machine Learning approach

Fig. III. 22 .

 22 Fig. III. 22. Two categories of machine learning techniques: Supervised and unsupervised learning [117] Supervised learning develops predictive models using classification and regression techniques. Common regression algorithms include linear model, nonlinear model, regularization, stepwise regression, decision tree boosting/bagging, neural networks, and adaptive neural learning.

III. 3 . 3 . 1 .

 331 Deep Learning (DL) architecture DL is a specific kind of Machine Learning branch that corresponds to an AI able to assimilate new knowledge through a Network of Artificial Neurons. The motivation behind using a Neural Network (NN) model is to organise them in several layers and different shapes. A standard neural network consists of many simple, connected neurons, each producing a sequence of realvalued activations. Input neurons are activated through sensors perceiving the environment, and other neurons are activated through weighted connections from previously active neurons.

  since 1943[START_REF] Mcculloch | A Logical Calculus of the Ideas Immanent in Nervous Activity[END_REF] [START_REF] Hochreiter | Long Short-Term Memory[END_REF]. The basic computational unit of the brain is a neuron. Approximately 86 billion neurons are inside the human nervous system. The neurons are connected with approximately 10 synapses. A neuron receives input signals from its dendrites and produces an output signal along its axon. The axon connects to the dendrites of other neurons. Fig. III. 23 a) shows images of a human neuron.

Fig. III. 23 .

 23 Fig. III. 23. Human neuron (a) versus a standard mathematical model of a neuron (b) [127] [125] In the standard mathematical model of a neuron (Fig. III. 23b), the signal travels along the axon of a neuron and interact multiplicatively with the dendrite of the current neuron based on the synaptic strength of the dendrite (weights). The dendrites carry different signals to the current neuron body where they are summated, as well as a bias coefficient for each neuron. Bias coefficient and the weights represent the learnable parameters of a Neuron [129].

Fig. III. 24 .

 24 Fig. III. 24. Representation of the standard activation functions

  to optimize at time j+1 and j : The learning step (always > 0 : is the gradient of the loss function in time j If < 0 we have a growing up the gradient. Which means > If > 0 we have a descendent gradient. Which means <

III. 3 . 3 . 5 .

 335 The principle of an Artificial Neuronal Network An Artificial Neural Network (ANN) is defined as a group of neurons that are connected in an acyclic graph. Instead of using unstructured groups of connected neurons, they are organized into layers. This way, the outputs of some neurons become inputs to other neurons. Each Artificial Neuron has a processing node (body) represented by circles, as well as connections from (dendrites) and connections (axons) to other neurons, which are defined as arrows in Fig. III. 25.

Fig. III. 25 .Forward

 25 Fig. III. 25. Artificial Neural networks are organized in layers consisting of a set of interconnected nodes

  Fig. III. 25.

Fig. III. 26 .

 26 Fig. III. 26. Schematic procedure of the AI approach to predict the microprobe performances

Fig. III. 27 .

 27 Fig. III. 27. The tensor model of our multiparameter system To illustrate the 4 th dimension, we used the tensors analogy. We see in Fig. III. 28, several representations for dimensions of a tensor. The 4 th dimension can be generated by replicating several times the 3D case. For our case, the 3D tensor corresponds to [Performance (Geometrical parameters) (Material properties)] repeated for each frequency case which represents the 4 th dimension.

Fig. III. 28 .

 28 Fig. III. 28. The analogy of tensors dimensions [138]

  ADS software (Advanced Design System). Fig. III. 29 shows the schematic block of the main step to simulate the microprobe e performance.

Fig

  Fig. III. 29. Block diagrams of the simulation methodology: TLE user interface and 3D design of the microprobe with its dimensions

Fig. III. 30 .

 30 Fig. III. 30. The histogram of the dataset predictors Note that we have two features of the system uniformly distributed (frequency and substrate type) and two parameters that do not change (TL width part gnd and underpass width).

Fig. III. 31 .

 31 Fig. III. 31. The boxplot of 10/12 predictor with the targets (Resistance and Q factor of microprobe) In the boxplot from Fig. III. 31, we can highlight three points: Firstly, the outlier of the Q factor target represents the most pertinent value of the microprobe performance. Secondly, there are more outliers on the resistance target than the Q factor, and finally, such behaviours occur for all predictors parameters.

Fig. III. 32

 32 Fig. III. 32 and Fig. III. 33 show that the Q factor and the Resistance have a poor correlation coefficient even though microprobe Resistance is much smaller than the micro-coil Resistance.

Fig. III. 34 .

 34 Fig. III. 34. ANN architecture for Microprobe dataset training model Generally, we want a model that captures the data trend without fitting the noise in the data.

  distributed it randomly as follows: 70 % of our data were used for training, 15 % for validation and 15 % for testing. So, when we find the best model options, we trained the model using the combined training and validation data. Once we get the final model, we have applied the new observations; That is why, it was crucial to provide the test data. III.3.5.4. The testing of Backpropagation algorithms and activation function Generally, each backpropagation training section starts with different initial values for the weights and bias coefficient parameters of each ANN unit neuron. We can also fix the random divisions of the dataset for training and validation or, therefore, randomly divide data for different values of training, validation, and test for each trial. We can choose different activations functions. These additional conditions can lead to very different solutions for the same problem. One of the most important parameters to define is the type of training algorithms to optimize the cost function. It is very difficult to know which training algorithm will be optimal and fastest for a given problem. It depends on many factors, including the complexity of the problem, the number of data points in the training set, the number of weights and biases in the network, the error goal, and whether the network was being used for pattern recognition (discriminant analysis) or function approximation (regression). This section compares the various training algorithms applied to our dataset. We performed the trial training for several cases of the optimal number of neurons (I) for the hidden layer; we chose the min and max value for the RMSE metric that we denoted by the best model and poorest model among several trials of the optimal (I model) training and validation. We tested the best and poorest model with new data, and we plotted the prediction of the factor and resistance , compared to the true value. Thus, we performed our test using different backpropagations algorithms for the training of the ANN model such as: Trainlm : Levenberg-Marquardt backpropagation Trainbr : Bayesian Regularization Backpropagation Trainscg : Scaled Conjugate Gradient backpropagation Trainoss : One Step Secant backpropagation Traingdx : Variable Learning Rate Backpropagation/Gradient descent with momentum and adaptative learning rate backpropagation Trainrp : Resilient Backpropagation Traingdm : Gradinet Descendent with Momentum backpropagation We also tested two cases of activation function of the output layer (linear function and ReLu function), knowing that we used as activation function in the hidden layer the hyperbolic tangent ( ℎ).

Fig. III. 35 (

 35 optimal model testing) and Fig. III. 36 (the best and poorest model testing), we can see the plot of the prediction and true value. For this, we performed a test of 25 observations corresponding to the Resistance and Q factor from a rectangular geometry on a glass and silicon substrate with = 24μ (micro-coil width completely outside the previous splitted dataset, as well as a square geometry on a glass substrat with w = 34μm both one from 200MHz to 900MHz (See appendix A11, for more details).

Fig. III. 35 .

 35 Fig. III. 35. The microprobe Q factor and Resistance R prediction, using the optimal number of neurons according to the evaluation of RMS from training, validation and testing What we called by the optimal model (I model) corresponds to the optimal model, among the trial training and testing to choose the optimal number of neurons for the hidden layer (Test of neuron number from 1 to 1000, step of 1). And the best and poorest models correspond to the

Fig. III. 36 .

 36 Fig. III. 36. The microprobe Q factor and Resistance R prediction, using the min and max RMSE of the testing model according to the trial of training with different weights and bias parameters Note that the models (I model and the best) seem to predict the Resistance and Q factor tendency for completely new data while the poorest do not predict at least the tendency.

Fig. III. 37 .

 37 Fig. III. 37. ANN architecture for Microprobe dataset training model

Fig. III. 38 .

 38 Fig. III. 38. The RMSE variation in terms of the number of neurons in the hidden layer One of the major aspects of training an ANN model is avoiding overfitting. That is a concept in data sciences, which occurs when the model fits exactly against its training data, memorizing

Fig. III. 39 .

 39 Fig. III. 39. (a) The performance variation as a function of epoch (train, validation and test) (b) The error histogram for the training modelThe following figure shows that the target data fit very well with the prediction by regression of performance close to one (< > = 0.99961) for testing, validation, and training. We can also note that the data fit well the model; furthermore, the model fits the true data tendency.

Fig

  Fig. III. 40. A plot of the regression of the (a) Training dataset, (b) the validation and (c) Test dataset

Fig. III. 41 .

 41 Fig. III. 41. The prediction and the true value of the Q factor and Resistance of the microprobe Each network starts from different initial weights and bias coefficients and with a different division of training, validation, and test sets. So, we train several times (500 trials) to get the optimal value of bias and weights for the optimal number of neurons for the hidden layer. In the next Figures, we can see the results of the enhancement of generalization for the and the Q factor prediction.

Fig. III. 42 .

 42 Fig. III. 42. RMSE performance of the optimal trialFig. III. 42 shows the fluctuation of RMSE performance of the model trials. We can see that we have a different value for the optimal neuron number from the hidden layer when we train the

Fig. IV. 2 .

 2 Fig. IV. 2. Layout designed for underpass & vias and airbridge mask fabrication

Fig. IV. 4 .IV. 1 . 2 . 1 .

 4121 Fig. IV. 4. Layout design for the glass wafer: geometries, description of the number of the microprobes for each raw

Fig. IV. 5

 5 Fig. IV. 5 shows the sputtering equipment (Cryogenic pump, SAS loading, DC spraying, RF, Bias, Cleaning by RF etching), we used for the thin film deposition.

Fig. IV. 5 .

 5 Fig. IV. 5. Sputtering equipment type SCM 600 (ALCATEL) We started the microfabrication process by preparing the wafer, depositing the seed layer through the PVD (Physical Vapour Deposition) or cathodic pulverization technique (Fig. IV. 6). This vacuum technique allowed making the deposition of an adherence and conductor material. By this, we could reach a thin layer of 600 μ .

Fig. IV. 6 .

 6 Fig. IV. 6. Seed layer preparing process a) Glass substrate and b) silicon substrate

Fig. IV. 8 .

 8 Fig. IV. 8. The spin coating, (a) before and (b) after the photoresist spreading out

3 °

 3 photoresist and improve the photoresist adhesion to the wafer surface.Light exposure and development consist of the photoresist exposing to the UV for the mask shape printing into the wafer surface and then removing the photoresist. The mask protects the photoresist against UV light at the desired location to obtain the micro-probe fingerprints on the positive photoresist surface. In this process step, the mask alignment was necessary; this task was performed by an automatized equipment (Fig. IV. 9 (a)). After the light exposure, the wafer was immersed into a PRD 238 solution for 45 seconds to remove the soluble photoresist and reveal the microprobe patterns overprinted onto the wafer surface (areas not exposed to UV light) as in Fig. IV. 9 (b).

Fig. IV. 9 .

 9 Fig. IV. 9. UV photolithography equipment (a): wafer mask alignment to the light exposure) and (b): the wafer after photoresist cleaning up In the following figure (Fig. IV. 10), we have the characterization of the wafer (a) and the results of the micro-probe shape after wet etching of the Photoresist layer on the seed layer (b).

FigIV. 1 . 2 . 4 .

 124 Fig. IV. 10. (a) The microscope to characterize the micro-coil shape, (b) Rectangular micro-coil geometry after the photoresist development and cleaning up

Fig. IV. 13 .

 13 Fig. IV. 13. Wafer before (a) and (b) after photoresist and seed layer removing

Fig. IV. 14 .

 14 Fig. IV. 14. Wafer before (a) and (b) after photoresist and seed layer removing

  Fig. IV. 15, we show a schematic illustrating the result after the electroless process. This section is often required at the end of the microprobe fabrication process.

Fig. IV. 15 .

 15 Fig. IV. 15. The prototype ended up after the electroless process IV.1.3. Characterization of the fabricated microprobes: 1 st trial

  sections, we presented the results of the morphological (section IV.1.3.1) and electrical characterization (section IV.1.3.2) of the microprobe prototypes manufactured in the first trial of the fabrication process, as well as the equipment used. IV.1.3.1. Morphological characterization In the Fig. IV. 16 a), we showed the microscope used to examine the shape of the microprobe prototypes, and the impurities from residuals resin not completely removed, short-circuits between the microprobes strip-lines. Also, we showed in the, Fig. IV. 16 b) the Mechanical profilometer (Veeco Dektak 8 model (Bruker)) used to survey the surface roughness and measure the microprobe thickness.

Fig. IV. 16 .Fig. IV. 19 .

 1619 Fig. IV. 16. (a) The microscope (b) The optic profilometer The profilometer consists of performing a displacement of a microtip along the surface in a pre-defined direction on the surface of the material ( or direction) usually with a sensitive diamond tip. The tip follows the material irregularities by varying its vertical position. These variations are converted into electrical voltage to be analyzed and processed by the interface software. It is equipped with a 12.5 long tip, with an exerting force of 0.03 to 10 which allowed us to measure the thickness in the range of 5 to 1 , in our case we chose the tip weight of 3 . The device has a Charge Coupled Device (CCD) camera of different magnifications to facilitate the visualization of the sample spacing. The Fig. IV. 17 shows the results of the microprobe characterization by microscopy (objective zoom of 10). Manufacturing defects are easily clustered. Note that the spacing between microstrip is smallest in the Fig. IV. 17. a), b) and d) ((1) (2): short-circuit), ((3) residual resin).The copper oxidation can be seen at different sides of the microprobe, more specifically on the corners.

Fig. IV. 20 . 2 .

 202 Fig. IV. 20. Profile of the microprobe surface scanningAll previous results from the characterization of the first test of fabrication represented a good asset to validate some steps of the microfabrication process and then optimize the fabrication process to obtain the optimal prototype as defined by the specification sheet of the project.

Fig. IV. 22 .Fig. IV. 23 .

 2223 Fig. IV. 22. Digital multimeter (KEITHLEY 2100, 1kV3A, 6.5 digits) used for Resistance measurementAfter the morphological and resistance measurement of the microprobe prototype, we selected ten microprobe prototypes suitable for the rest of the experiment. Note that, on a set of 96 characterized microprobes in the first trial of the process fabrication, only 10 microprobes were suitable.

Fig. IV. 24

 24 Fig. IV. 24 shows the three microprobes were suitable for the performance measurement of parameters, among the ten microprobes set up on the PCB.

Fig. IV. 24 .

 24 Fig. IV. 24. The geometries suitable for the performance measurements

3 .

 3 Copper electroplating (electroplating time and the applied current, homogeneity of the plating and the seed layer oxidation) The Fig. IV. 25 a) shows the ten wafers prepared for the optimization of the process fabrication with the new fabricated mask (Fig. IV. 25 b)) and resin. IV.1.4.1. Photolithography Optimisation

Fig 3 . 8 .

 38 Fig. IV. 25. (a) Ten prepared wafers, (b) the optimized mask

for 3ℎ 30 .

 30 After that, we could estimate the speed of the growth of the copper layer by about 8 μ /ℎ. Which in Fig. IV. 27 was denoted by the . For wafer 2, we selected the same initial parameters defined in Table IV. 6, and we selected 5 ℎ for electrodeposition to attend et least 40 μ of thickness, which we denoted by in Fig. IV. 27.IV.1.5. Manufactured microprobe after the optimization processThe optimized prototypes are shown in the following images of the micro-coil manufactured on silicon and glass substrate. These pictures were made using a microscope with an objective zoom of 10.

Fig. IV. 28 .Fig. IV. 29 .Fig

 2829 Fig. IV. 28. The geometry of some manufactured micro-coil: circular ( × μ ²), ellipsoidal ( × μ ²), rectangular ( × μ , × μ ²)

Fig. IV. 31 .

 31 Fig. IV. 31. The geometry of some manufactured micro-coil: micro-coil: square ( × μ , × μ , × μ ²).

  Fig. IV. 32, we show the schematic distribution of the microprobe selected for the experiments (geometrical characteristics and their classification according to the resin removal results).

Fig. IV. 32 .

 32 Fig. IV. 32. The seven-microprobes selected for the resin removing and their classification according to their observation using a microscope.The characterisation of the shape of all microprobes allowed us to estimate the optimal time for microprobe resin removing. Therefore, we have fixed 5 minutes to 10 minutes in the ultrasound.This result was useful to characterise the rest of microprobes on the wafer as we will show in the next section.

Rectangular ( 400 Fig. IV. 33 .Fig. IV. 34 .

 4003334 Fig. IV. 33. Microprobe thickness mapping on different parts of the wafer support

  47 ) μ corresponding to measurement uncertainty of = 5.21 %, and = (26.02 ± 1.27) μ corresponding to measurement uncertainty of = 4.87 %. IV.1.6.2. Electrical characterization of the fabricated microprobes In the Fig. IV. 35, we show a microprobe set up on a PCB support with a wire-bonding connecting to a SMA connector type-N.

Fig. IV. 35 .

 35 Fig. IV. 35. Microprobe fabricated in the glass substrate, with a connector on a PCB support, by wire-bonding connecting wire.

Fig. IV. 36 .- 10 .

 3610 Fig. IV. 36. shows the results of parameters measures that was done, for the four geometries of microprobe. We can note, that the four geometries do have a value at least close to -10 . We could confirm that the four microprobes have almost the same self-resonance frequency, 4 for many of them.

Fig. IV. 36 .

 36 Fig. IV. 36. Measure of parameter from four geometries of microprobe

Fig. IV. 37 .

 37 Fig. IV. 37. Measure of the resistance and inductance from four geometries of microprobe

  reach the microprobe thickness ( = 40 μ ). We mainly optimized the UV photolithography and the electroplating step, whose results are summarized in Fig. IV. 26 -Fig. IV. 31.

  considerable resistive loss due to the underpass & vias embedded on a silicon substrate or simply change the connecting wire type to airbridge, which requires an adaptation on the manufacturing process of the microprobe prototype. By simulations, we could confirm that we could optimize the Q factor of microprobe ( = 71 instead of = 59) by optimizing the thickness of the underpass & vias connecting wire ( = 10 μ instead of = 0.5 μ ). The Q factor was also improved ( = 110) using a connecting wire airbridge of = 10 μ (see Table.

  III.3).

  1.1). Preliminary characterization results (morphological and electrical) obtained with the 1 st prototype fabricated on glass substrates were reported. the 1 st fabrication test allowed us to reach a microprobe thickness of = (18.90 ± 5.25) μ (Fig. IV. 18 & Fig. IV. 19), the resistance of the prototype in a glass substrate without a connector, was about = (3.897 ± 0.193) , and with a connector soldered into a PCB, was = (6.12 ± 0.12) . While, from the final optimized process, we were able to fabricate micro-probes with reduced micro-coils of dimensions up to 250 × 250 μ ² (Fig. IV. 30) with a thickness of about = (27.10 ± 1.37 ) μ (Fig. IV. 33); dimensions never reached before by our team. Our characterization results are very promising for future in vivo applications.

  combined with simulation tools (multi-parameter optimization work) implies an advance in biomedical research based on the development of NMR micro-sensors used for brain exploration, early diagnosis and treatment monitoring.

Fig. 1 :

 1 Fig.1: The shape elements type to mesh a geometry.

2 . 2 BEOL 1 A. 4 . 1 BEOL 1 A. 6 . 2 .ℎ sizes 3 .

 221411623 The shape used to divide the geometry: element type 3. Interpolation between nodes: Element order A.3. Syntaxes of the .txt file to model TL part on a silicon substrate *Modele CPW pour 240um mais avec Wgnd=60um et 398e-6 W=600e-6 L=10000e-6 T=2e-6 epsr=6.T=40u rho=1.72e-8 #COMPONENTS LINE 1 BEOL 1 W=320u L=9500u ncells=10 inter OXI(1)->SUB(1) LINE 2 BEOL 1 W=60u L=9590u ncells=10 inter OXI(1)->SUB(1) Syntaxes of the .txt file to model underpass & vias within the silicon substrate .9e-6 789.9e-6 358e-6 W=60.2u L=315.2e-6 T= -5u epsr=6.T=1u rho=1.72e-8 #COMPONENTS LINE 1 RDL 1 W=60u L=315u ncells=10 inter OXI(2)->subnode(1The sequences of Microprobe Simulation for data-generating 1. Define the dimension choosing the _ size. e.g: _ = 1000μ Define the , , , , and Run the algorithms for each geometry (Circular, Ellipsoidal, Square and Rectangular) in both substrate cases (Glass and Silicon).

4 . 6 . 8 .

 468 Then we change the t value, and we repeat the 3 rd step. a. = [40 38 20 10 2]μ b. We repeat the process until the last t value for all geometry and substrate cases (e.g: = 2μ ) 5. So, we change the and ℎ_ both simultaneously, and then we repeat step four. a. = [20 10 2 0.5] μ b. ℎ = [2 1 1 1] μ c. We change the and ℎ until the end value (e.g: = 0.5μ and ℎ = 1μ ), getting back to the step .4 for each value. So then, we start the change of w, s, taking into account the step. repeat the process from step three until step six for each value until the end one. Then we change the next value a. = [1000 900 800 700 600 500 400 300] μ 9. (Add here please the next step) A.7. The Microprobe characteristics for data simulated = 1000 μ , = 6 (Circular and square geometry: = = 30 μ and = 22 μ = 20 μ ), = 5 (all geometries = 22 μ , = 20 μ ) _ = 900 μ & 500 μ , = 5 (Ellipsoidal and Rectangular geometries):

Fig. 2 :

 2 Fig.2: Graphical representation of epoch and batch in a dataset training.

  TRAINLM is a network training function that updates weight and predicts values according toLevenberg-Marquardt optimization. Levenberg Marquardt Algorithm (LMA) is a commonly used training algorithm in data classification. It is often the fastest backpropagation algorithm, although it does require more memory than other algorithms. The network performance function that is used is Mean Squared Error. It measures the network's performance according to the mean of squared errors. Mean Squared Error measures the network's performance according to the mean of squared errors. It is an average of the squares of differences between the actual observations and those predicted. The squaring of errors tends to heavily weigh statistical outliers, affecting the accuracy of the results. In mathematics and computing, the Levenberg-Marquardt algorithm (LMA or just LM), also known as the damped leastsquares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise, especially in least-squares curve fitting.

  Pour finir, notre travail montre qu'à l'avenir nous pourrons concevoir et fabriquer avec précision n'importe quel type de micro-sonde ou sonde optimisée en fonction des performances souhaitées, de la fréquence de travail et du type de substrat. Cela avec un temps de calcul de 2 minutes au lieu de 15 heures. Ainsi, cette nouvelle approche d'optimisation utilisant des outils d'IA combinés à des outils de simulation (travail d'optimisation multiparamétriques) implique un progrès dans la recherche biomédicale basée sur le développement de micro-capteurs RMN utilisés pour l'exploration du cerveau, le diagnostic précoce et le suivi du traitement.

	résistance avec une précision de 91,34 % (	= 0,042), grâce au jeu de données générées
	par le travail de simulation réalisé au préalable.
	<30.36μm>	<33.06μm>
	<30.21μm>	<32.21μm>
	<28.35μm>	<30.33μm>	<35.39μm>
	<29.67μm>	<26.11μm>	<34.58μm>
	<38.73μm>		<37.50μm>
	<29.55μm>		<35.63μm>

Table of contents

 of List of captions of the figures and tables

  Fig. III. 7. Model PI block diagram ........................................................................................ Fig. III. 8. The electrical circuit modelling the microprobe on a glass substrate (complete model) ................................................................................................................................................ Fig. III. 9. Block diagram of the electrical circuit of microprobe on a glass substrate .......... Fig. III. 10. The electrical circuit modelling the microprobe on a silicon substrate (complete model) ..................................................................................................................................... Fig. III. 11. parameters of the microprobe electrical model from the silicon substrate (a) and the glass (b) ...................................................................................................................... Fig. III. 12. parameters of the complete model of microprobe on a silicon and glass substrate ..................................................................................................................................

6. The electrical circuit that models the micro-coil on the silicon substrate (PI Model) ................................................................................................................................................

Fig. III. 13. Q-factor variation of both complete electrical models of NMR micro-probe .... Fig. III. 14. Resistances variation from electrical models of microprobe on a glass substrate a) and silicon b) ..........................................................................................................................

  Fig. III. 27. The tensor model of our multiparameter system ................................................ Fig. III. 27. The analogy of tensors dimensions [135] ........................................................... Fig. III. 29. Block diagrams of the simulation methodology: TLE user interface and 3D design of the microprobe with its dimensions ................................................................................... Fig. III. 30. The histogram of the dataset predictors .............................................................. Fig. III. 31. The boxplot of the predictor with the targets (Resistance and Q factor of microprobe) ............................................................................................................................ Fig. III. 32. The plot of the targets (Q -factor and micro-coil Resistance): four types of geometry .................................................................................................................................

25. Artificial Neural networks are organized in layers consisting of a set of interconnected nodes .............................................................................................................. Fig. III. 26. Schematic procedure of the AI approach to predict the microprobe performances ................................................................................................................................................

Fig. III. 33. The plot of the targets (Q -factor and Microprobe Resistance): four types of geometry ................................................................................................................................. Fig. III. 34. ANN architecture for Microprobe dataset training model ..................................

Table II .

 II ........................................ 172 micro-transmission line. ....................................................... 172 a microscope. ............................................................... 174 Fig. IV. 33. Microprobe thickness mapping on different parts of the wafer support ............. 175 Fig. IV. 34. Confocal Raman Microscopy images of the microprobes selected for the wafer mapping .................................................................................................................................. 176 Fig. IV. 35. Microprobe fabricated in the glass substrate, with a connector on a PCB support, by wire-bonding connecting wire. .......................................................................................... 180 Fig. IV. 36. Measure of 11 parameters from four geometries of microprobe ..................... 180 Fig. IV. 37. Measure of the resistance and inductance from four geometries of microprobe 181 Table I. 1. Value of gyromagnetic ratios of specific atomic nuclei of interest ........................ 2. Summary of electrical and magnetic parameters from each micro-coil geometry at ................................................................................................................................. Table II. 3. The electrical and magnetic parameters of each rectangular micro-coil at , for different dimensions ........................................................................................ Table II. 4. The electrical and magnetic parameters of each square micro-coil at , for different dimensions ...........................................................................................................

	Fig. IV. 29. The geometry of some manufactured micro-coil: rectangular (	×	μ	,
	× μ ²) Fig. IV. 30. The geometry of some manufactured micro-coil: square ( μ , × μ ²), square ( ×	×	μ	,	×
	μ μ ²), Fig. IV. 31. The geometry of some manufactured micro-coil: micro-coil: square ( , ×	×
	μ	,	×	μ	,	×	μ ²). ............................................................... 173
	Fig. IV. 32. The seven-microprobes selected for the resin removing and their classification
	according to their observation using Table I. 2. Review of all planar microprobe for NMR applications ........................................
	Table I. 3. Summary of our team works, geometrical dimensions, electrical parameters analysis,
	and materials for μmanufacturing and its application ..............................................................
	Table II. 1. Micro-coil layout and dimension of the optimal geometries [37]……………… 72
	Table II. 5. Geometric parameters with the limiting diameter for the feasibility.....................
	Table II. 6. Geometrical configurations to be manufactured, = and =	μ . ............

[START_REF] Cudalbu | Handling Macromolecule Signals in the Quantification of the Neurochemical Profile[END_REF]

. The geometry of some manufactured micro-coil: circular (

× μ ²), ellipsoidal ( × μ ²), rectangular ( × μ , ×

μ

²) ........

Table II .

 II 7. The physical properties of the substrate material used for our simulations, at room temperature. .............................................................................................................................. Table II. 8. Magnetic field z component as a function of microprobe perpendicular direction ( ) .......................................................................................................................................... Table III. 1. The electrical parameters (RLC) from different parts and according to their dimensions. ............................................................................................................................. List of the algorithms and models performance obtained during the testing of algorithms ............................................................................................................................... Table III. 5. The performance from different ANN architecture for three target cases ......... Table IV. 1. The microprobe geometry, dimensions and characteristics: wire-bonding ....... Table IV. 2. The microprobe geometry, dimensions and characteristics: underpass & vias and airbridge ................................................................................................................................. ................................................................................................................... Table IV. 4. Fundamental parameters of the photolithography process optimization ...........

	Table III. 2. The	parameters from different models of microprobe, performed at
	300	, although 500	.................................................................................................
	Table III. 3. The	electrical parameters from the best model of micro-probe, all of them
	tuned at 300	and 500	...........................................................................................
	Table III. 4. Table IV. 3. The measurement of	parameters of microprobe on	(1 st trial of
	microfabrication) .Table IV. 5. Photoresist planting (AZ 125 NXT -10A) ........................................................
	Table IV. 6. Standards parameters specifications of AZ 125 NXT -10A photoresist ..........

Table IV .

 IV 7. Parameters of the Oxygen plasma section to clean up the wafer before the copper electroplating .......................................................................................................................... Table IV. 8. The setting parameters for the electroplating of the copper layer...................... ). ...................................................................... .............................................................................................................................................

	(ANN), NMR Spectroscopy, Alzheimer Disease, Biomarkers, Micromanufacturing.
	Table (IV. 9.1 to IV. 9.7 ) show the measurements of the cooper thickness from different
	areas of the wafer (microprobe ° to ° Table IV. 11. The measurement of the microprobes	parameters at 300	and 500
	...Table IV. 12. The measurement of the microprobes	parameters at 300	and 500
	................................................................................................................................................

  variation is mainly detected in the dementia stage, although amyloid β and tau protein markers have the advantage of occurring earlier than any metabolic, structural and obviously before clinical changes. So, the biomarkers aim to provide quantitative information about the metabolites (molecular) and morphological changes occurring in the subject's brain during these transformations.There are several tools and methods available to measure the Alzheimer disease markers, such

	diagnosis and others to disease progression (dementia state) to surrogate markers in a clinical
	trial.
	AD is characterized by an accumulation of amyloid β peptides and some transformation in
	the cerebral region through the structural abnormal tau protein; these transformations in the
	long-term causes hypometabolism and cortical atrophy. Cortical atrophy represents a
	morphological biomarker; its
	I. 1, we have a theoretical
	model of the natural progression of cognitive and biological markers of AD. So, we can
	highlight two points: Some markers are sensitive to disease in MCI and useful for early

as: Amyloid Positron Emission Tomography (PET): This imaging technique quantifies amyloid plaque deposition in the brain. From the literature, there are several studies of deposit of amyloid to measure the AD progression with PET technique [14] [15] [16]. A study to predict and relate the possibility of a patient in MCI that will be in AD dementia stage. The study was performed under 19 months of monitoring, with a sensitivity of 96 % and achieved specificity 42 % [17]. Fluorodeoxyglucose (FDG) Positron Emission Tomography (PET): This PET modality allows mapping cerebral hypometabolism due to pathological neuronal loss. The results reporting the performance of this technique varies very often according to the application, but most recent work has reported the capacity of the FDG PET technique to predict. They had a sensitivity variation from 56 % to 100 %, with a specificity of 24 % to 100 % [18]. Cerebral Spinal Fluid (CSF): Analysis of CSF collected by lumbar puncture allows quantification of the concentration of amyloid-beta peptides 40 and 42 and of the Tau protein, these concentrations being respectively lower and higher in patients with AD. The best performance achieved with this technique for six months of study is a sensitivity of 81 % for predicting the AD conversion [19]. Magnetic Resonance Imaging (MRI): This structural imaging technique aims to measure focal brain atrophy due to pathological neuronal loss. A Structural MRI study carried out on six months was performed with 81% accuracy for the AD conversion prediction

Table I .

 I .1. 1. Value of gyromagnetic ratios of specific atomic nuclei of interest

	Atomic nuclei
	Spin	1	1	1	1	3
		2	2	2	2	2
	(MHz/T)	42.58	10.71	40.05	17.24	11.26

  = ρ

				(	)	(II.15)
	With:
		R = (	(	(	) ) )ρ. l	(II.16)
	Where:	represents the conductor wire length. The formula corresponding to different
	micro-coil geometry is given in section II.2.2.1.

II.2.2.6. The self and mutual inductance

The analytical calculation of the inductance of a conductor has been reported in numerous articles but, these formulas are only applicable for long conductors or conductors of specific cross-sections

[87]

[88] [89] [90]. Based on Neumann's formulations work, several other teams

  2.1 and Fig.II.6) have a very little difference in the value (Table II.2). Hence, we use the rectangular microcoil to model the complete microprobe through the 3D electrical circuit proposed in chapter III.

Table II .

 II

									]
	1000 × 500	22	20	0.518	12.468	0.185	45.319	10.68	14.839
	1000 × 400	22	20	0.476	11.312	0.170	44.476	17.80	25.826
	1000 × 350	18	18	0.508	11.28	0.155	41.55	18.96	26.601
	1000 × 350	12	20	0.589	11.76	0.163	37.38	29.80	38.829
	1000 × 250	12	20	0.538	10.59	0.149	36.902	38.72	52.789
	800 × 400	22	20	0.393	9.042	0.133	43.189	17.84	28.458
	600 × 300	12	20	0.359	6.620	0.100	34.74	22.91	38.236
	500 × 250	12	20	0.283	4.998	0.078	33.273	47.02	88.387

  III.[START_REF] Sörensen | and for the Alzheimer Disease Neuroimaging Initiative, Amyloid Biomarkers as Predictors of Conversion from Mild Cognitive Impairment to Alzheimer's Dementia: A Comparison of Methods[END_REF], we noticed two points: First, the transmission line contributes to the increase of the Q-factor on two substrate cases (32 % for a glass and 21 % for the silicon substrate). Secondly, underpass & vias decreases the value of the Q factor (18 % in the glass and 19 % for the silicon substrate).

	Working frequency In Fig.
	Electrical						Q						Q
	parameters		[ ]		[nH]			[ ]		[nH]
	Model Type	Calc.	ADS	Calc.	ADS	Calc.	ADS	Calc.	ADS	Calc.	ADS	Calc.	ADS
		RLC	0.53	0.53	12.56	12.57	45.02	44.85	0.68	0.53	12.74	12.79	59.10	75.14
	Micro-coil	PI	0.53	0.53	12.59	12.57	44.78	44.85	0.69	0.54	12.82	12.79	58.32	74.57
	Micro-coil &	Glass	0.77	0.77	25.77	26.61	65.43	65.45	1.03	0.77	27.37	27.37	83.22	111.77
	micro-path
	Silicon	0.78	0.89	26.71	27.63	65.02	58.59	1.10	0.84	27.49	30.42	81.96	80.96
	Complete	Glass	0.98	0.95	25.98	26.83	51.43	53.50	1.26	1.01	27.59	27.59	68.63	86.99
	model
	Silicon	0.94	1.10	27.99	28.01	51.20	47.89	1.30	1.77	27.72	33.24	67.83	58.85
	Table III. 2. The		parameters from different models of microprobe, performed at		, although

  Note that, the model of an airbridge is equivalent to the underpass & vias if we assume that there is no substrate loss for the connecting wire. Optimizing the transmission line, underpass & vias part associated with different geometries required several simulations and different cases with our validated model. Thanks to our MATLAB script, we automatically generated data from the simulation of four geometries, different dimensions of the micro-coil, underpass & vias dimensions, and two substrate types in the frequency range of 200 to 900 (more details about the procedure, see the appendix, A.6).

	Working frequency III.3. Prediction of the microprobe performances: Artificial
	Electrical parameters Intelligence approach [ ]	[nH]		Q		[ ]		[nH]		Q
	Model Type III.3.1. Introduction Calc.	Sim.	Calc.	Sim.	Calc.	Sim.	Calc.	Sim.	Calc.	Sim.	Calc.	Sim.
	Complete	Glass	0.77	0.73	26.80		26.80	65.87	69.10	1.03	0.79	27.57	27.55	86.76	110.20
	model Artificial intelligence (AI), one of the developments in computer technology, has been 0.78 0.91 26.88 28.01 65.31 Silicon 1.06 1.48 27.77 33.24 81.97 70.69 59.06
		Table III. 3. The		electrical parameters from the best model of micro-probe, all of them tuned at
								and

Table. III.3. The complete model on a glass substrate corresponds to a microprobe with an airbridge connecting wire ( = 10 μ ).

  Due to the simplicity of the method, it takes less computation time compared with the other methods. This method emphasizes that all errors are weighted equally on the same scale.III.3.4.2. Mean Absolute Percentage Error (MAPE)MAPE is quite similar to MAE, while MAPE is the calculated error in percentage.

			( )	( )
	= ∑		( )		× 100%	(III.68)
	Note that we have the actual value of the observation in the denominator, which can not be
	equal to zero. The advantage of this metric is that the error is normalized on a common scall,
	according to the amount of dataset.
	III.3.4.3. Mean Square Error (MSE)
	MSE, as its name suggests, is the average of the squared difference between predicted and
	actual values.
	III.3.4.1. Mean Absolute Error (MAE)
	MAE is the simplest method to calculate the error function for a regression problem. It is
	calculated as the absolute difference between the true and prediction values.
	= ∑	|	( )	-	( ) |	(III.67)

Table III .

 III Training parameters: epoch =2000, plot interval =53 epoch (see appendix, A.9, for more details about epoch), variation of number of neurons for the hidden layer (1 to 1000), number of trial training for parameters optimization 500 trials.The following table list the results of algorithms that were tested for our problem with the model performance obtained for training, validation and test of the models.[START_REF] Hoult | The NMR Receiver: A Description and Analysis of Design[END_REF]. List of the algorithms and models performance obtained during the testing of algorithmsThere are several algorithm characteristics that can be deduced from the table of experiments described (TableIII. 4). In general, on function approximation problems, for networks that contain down to a few twenty weights in the input layer, as our case, 12 weights,

	activation function (		= 1.084,		= 1.173 and		= 3.717). The
	memory requirements of trainlm are larger than the other algorithms tested.
	Train	NN	RMSE	RMSE	RMSE	RMSE	RMSE	RMSE	RMSE	Comput.
	algorithm		optimal	train	validation	test	min	max	mean	time
	Lm (Linear)	12:125:2	0.7379	0.8354	0.8995	0.8665	0.7379	2.8207	1.3247	8 h 30 min
	Lm (ReLu)	12:505:2	16.967	1.426	1.848	1.923	16.968	18.820	17.922	0h 30 min
	Scg (Linear)	12:125:2	0.7379	0.8354	0.8995	0.8665	0.7379	2.8207	1.432	7 h 30 min
	Scg (ReLu)	12:505:2	16.968	1.427	1.848	1.923	16.968	18.820	17.922	1h 30 min
	Gdm(Linear)	12:4:2	23.102	56.0913	56.6369	56.513	13.893	115.42	44.501	0h 10 min
	Gdm (ReLu)	12:163:2	17.142	298.669	292.927	288.537 17.078	1.02e6	2.99e3	0h 20 min
	Oss (Linear)	12:3:2	1.0840	3.0711	3.072	3.315	1.0477	4.795	2.2547	5h 30 min
	Oss (ReLu)	12:505:2	16.819	3.7624	4.599	4.593	16.819 224.614	19.612	2h 00 min
	Rp (Linear)	12:8:2	1.173	1.704	1.639	1.756	0.908	3.349	1.688	5h 30 min
	Rp (ReLu)	12:505:2	16.819	4.199	5.1489	5.395	16.819	26.915	17.986	0h 15 min
	Br (Linear)	12:4:2	23.102	56.187	NaN	56.5935 13.893	115.42	44.5010	0h 5 min
	Br (ReLu)	12:202:2	17.874	317.25	NaN	310.86	17.546	606.45	171.91	0h 5 min
	Gdx (Linear)	12 :7 :2	3.717	4.479	4.723	4.861	3.229	79.006	5.211	1h 30 min
	Gdx (ReLU)	12 :717 :2 17.0718	40.3672	42.3451	41.019	17.059 974.244	28.944	0h 30 min
	the Levenberg-Marquardt (LM) algorithm had the latest convergence (8 ℎ 30	for
	computation). With trainlm we obtained the lower RMSE than any of the other algorithms
	tested (	= 0.7379, for a linear activation function). However, as the activation function
	of the trainlm change to ReLu, the advantage of trainlm decreased (	= 16.967), we
	could confirm that the trainoss, trainrp and traingdx have shown lower		with a linear

  Table IV.9.1: Table IV.9.2: Table IV.9.4: Table IV.9.5:

	1	Mean Type of geometry	2 Micro-coil thickness 33.06	3 31.78 TL thickness
		std (ID)			1.33 (μ )	1.86 (μ )
		Error %		3.12 57.21	4.90 37.43
						29.76	37.29
						31.71	26.86
		Type of geometry Rectangular	Micro-coil thickness 32,71	TL thickness 23.84
		×	(ID)	μ ²		(μ ) 36.52	(μ ) 27.40
	4				5	29.47 50.51	6 31.55 26.09
		Mean			29.19 39.73	31.46 29.82
			2 std			31.39 11.35	28.27 5.97
		Circular Error %		30.28 23.69	31.55 16.86
		×	μ ²		29.77	31.46
		Type of geometry	29.92 Micro-coil thickness	28.27 TL thickness
		(ID)			32.13 (μ )	31.55 (μ )
	7					31.99 35.95	31.46 32.47
						29.99 35.38	28.27 35.00
						29.43 33.69	28.27 33.75
		Mean Ellipsoidal		30.36 37.62	30.21 36.65
		×	μ ²
			std			1.03 35.36	1.59 33.75
		Error %		2.66 35.45	4.67 34.71
		Mean			35.39	34.58
			std			1.31	1.70
		Type of geometry	Micro-coil thickness	TL thickness
		(ID) Error %		(μ ) 2.52	(μ ) 3.99
						38.82	36.43
		Type of geometry	37.09 Micro-coil thickness	35.94 TL thickness
		(ID)			(μ )	(μ )
		Type of geometry (ID) Rectangular × μ ²	Micro-coil thickness 35.62 28.10 (μ ) 39.56 28.44	36.93 TL thickness 29,90 (μ ) 33.20 29.60
						32.25 36.80 28.55	33.79 35.94 29.50
		1 Rectangular		32.12 37.66 28.22	30.09 36.93 29.90
		×	μ ²
		Mean Square		31.48 37.50 28.55 28.22	34.79 29,60 35.63 29.50
		× std	μ ²		32.76 1.30	30.53 1.28
		Mean			28.35	29.67
		Error % std		34.71 2.51 0.19	30.53 2.30 0.19
	Table IV.9.3:	Error %		35.01 0.59	30.53 0.52

Table IV

 IV

	Type of geometry	Micro-coil thickness	TL thickness
	(ID)		(μ )	(μ )
			29.70	26.33
	4		30.80	25.89
			30.50	26.33
	Ellipsoidal	29.70	25.89
	×	μ ²	30.80	26.33
			30.50	25.89
	Mean		30.33	26.11
	std		0.51	0.02
	Error %	1.39	0.84
	Table IV.9.7:
	5
	.9.6:

Table IV .

 IV [START_REF] González | Atlas of Biomarkers for Alzheimer's Disease[END_REF]. The measurement of the microprobes parameters at and

		Square	500 × 500	2.2	16.0	4.0	16.2
	2	Circular	500 × 500	2.3	16.6	4.1	16.6
	6	Rectangular	1000 × 250	4.7	19.6	8.8	19.5
	7	Ellipsoidal	1000 × 500	1.1	9.4	2.3	9.3

Table IV .

 IV [START_REF]Les traitements de la maladie d'Alzheimer -ICM[END_REF]. The measurement of the microprobes parameters at and

d'IA, en particulier un réseau neuronal artificiel (RNA), que nous avons entraîné avec le jeu de données généré à partir des simulations réalisées au préalable (6621 observations). Notre RNA a pu prédire le facteur Q de la micro-sonde avec une précision de 99,67 % ( = 0,032), et la Résistance avec une précision de 91,34 % ( = 0,042).En se basant sur ces résultats, nous avons fabriqué les différents prototypes de micro-sondes sur substrat de verre et de silicium. Grâce à l'optimisation du processus de fabrication, nous avons pu fabriquer des microbobines de très petites dimensions de l'ordre de 250 × 250 μ ², dimensions jamais atteintes auparavant.Notre travail montre qu'à l'avenir nous pourrons concevoir et fabriquer avec précision n'importe quel type de micro-sonde ou sonde optimisée en fonction des performances souhaitées, de la fréquence de travail et du type de substrat. Cela avec un temps de calcul de 2 minutes au lieu de 15 heures. Ainsi, cette nouvelle approche d'optimisation utilisant des outils d'IA combinés à des outils de simulation (travail d'optimisation multi-paramètriques) implique un progrès dans la recherche biomédicale basée sur le développement de micro-capteurs RMN utilisés pour l'exploration du cerveau, le diagnostic précoce et le suivi du traitement.

A.11. Prediction of Resistance and Q factor using completely different dataset for three trained models.

Acknowledgements

My thesis was carried out at ISA laboratory of Lyon, with Claude Bernard University. First, I would like to thank the director of the ISA, Mr. Christophe MORELL, for all supports during the three years of my thesis.

I express all my gratitude to my research work, supervisor Mrs. Latifa FAKRI-BOUCHET, as

Acknowledgements List of Acronyms

The RLC electrical circuit in Fig. III. 5 models a planar micro-coil fabricated on a glass substrate; this model does not take into account the loss on the substrate. The RLC circuit characteristics correspond to a passband filter. It describes the resonance behaviour of an NMR coil widely at reception. We can deduce the previous impedance formula as follows:

(III.8.1)

When the system resonates, the frequency of the system is defined :

We can deduce the quality factor of the RLC circuit as follows when the receiver achieves the resonance condition.

With: the micro-coil résistor, self-inductor and is the capacitor from the close wires coupling.

When the micro-coil is fabricated on the silicon substrate, the receiver can be modelled by PI model [103] [104]. This model (Fig. III.6) considers the main physical phenomena, particularly the substrate effect, because, in the case of the silicon substrate, there is a high resistive loss while flowing a current throughout the micro-coil. From the Z _ equation, we can calculate the Q factor analytically as follows:

From the literature, it was already demonstrated that the Q factor of the PI model could be calculated by the equation [START_REF] Long | The Modeling, Characterization, and Design of Monolithic Inductors for Silicon RF IC's[END_REF]: (III. [START_REF] Sörensen | and for the Alzheimer Disease Neuroimaging Initiative, Amyloid Biomarkers as Predictors of Conversion from Mild Cognitive Impairment to Alzheimer's Dementia: A Comparison of Methods[END_REF])

The equation given by Yue and Wong [START_REF] Long | The Modeling, Characterization, and Design of Monolithic Inductors for Silicon RF IC's[END_REF] illustrates the complexity of the electrical circuit to model the micro-coil in a silicon substrate. In equation (III. [START_REF] Visser | Amyloid Imaging in the Prediction of Alzheimer-Type Dementia in Subjects with Amnestic MCI[END_REF], we can see that the quality factor also depends on the substrate losses and the self-resonance of the model.

Note that the previous models (RLC and PI) represent the electrical circuit of micro-coil, considering all physical phenomena but still needing some enhancement to model the NMR implantable microprobe. In this perspective, we proposed in Section III.2.2 a complete model that includes all parts of the microprobe that considers all the capacitive and inductive coupling and the losses from the substrate part. Note that we define here as microprobe, the composition of the micro-coil, in addition to the micro transmission line (micro-path) and a connecting wire (underpass & vias, wire-bonding or air-bridge).

III.2.2. Our proposed microprobe electrical circuit

In Fig. III. 8, we show the complete electrical circuit that models the microprobe on glass.

(III.71) III.3.4.6. Relative Squared Error (RSE)

RSE is very similar to RAE, and it is widely adopted by the data science community to calculate the R square metric.

(III.72)

And:

R square is not an error percentage but is a popular metric for the accuracy of the model. It represents how close the data values are to the fitted regression line. The higher is R², better the model fits our data. Each of these metrics can be used to quantify the model prediction using the test dataset.

To sum up, MAE is the easiest of the metrics to understand since it is just the average error, while the MSE metric is more popular than MAE because the focus is geared more towards large errors. This is due to the squared term exponentially increasing larger errors in comparison to smaller ones. RMSE is one of the most popular evaluation metrics because root mean squared error is interpretable in the same units as the response vector or ( ) units, making it easy to relate its information.

III.3.5. Model training for the prediction of the micropore performance

In this section, we will present the methodology we used to train the model for microprobe performance prediction. In addition, we explored Deep Learning Tool from the Matlab GUI APP.

The methodology of our approach consists of several steps (Fig. Generating data from the microprobe performance simulations Data pre-processing to study the distribution, select the optimal parameters and clean the data if necessary.

Train the AI model, selecting the optimal parameters and hyperparameters to reach a good accuracy for the data generalization model several times. That helps us choose the best one of the optimal models as we will show the performance prediction for the first optimization and then the poorest model.

We can note in the The microfabrication requires implementing a relatively complex manufacturing process and some optimizations. It is therefore essential to present the main steps of the manufacturing process; described in detail in this chapter.

Firstly, we presented the specification required for the optimal microprobe, for our application.

Then we recalled the main stages of microprobe fabrication. Furthermore, we reported the optimization steps that were performed to fabricate the final prototype. In the end, we performed some characterizations and we validated their suitability for our in vivo applications. We provided two types of substrates (silicon and glass) and two mask layout designs with the following characteristics:

Micro-coil type

1. A glass substrate with 400 μ thickness (mask 1).

Wire-bonding for connecting micro-wire, and 40 μ for copper thickness Four our experiments, we prepared six glass wafers, knowing that most were used to optimize the fabrication process in the first trial.

First, we plated a layer of titanium/tungsten ( / ) of 100 thickness to provide the cohesion of the next layer on the glass wafer.

We could then deposit a 500 copper ( ) thickness over the entire / layer by the same technique we used to deposit the 1 st layer and pulverize using an argon plasm.

IV.1.2.

Photoresis

Based on the photoresist datasheet information that we have reminded in the following Table IV. 6, for a standard thickness, with its corresponding UV insulation time. Thus, we could determine the tendency curve and get the optimal insolation time for our expected thickness. Here, the equations have been formulated in differential form because it leads to differential equations that the finite element method can handle. For general time-varying fields, Maxwell's equations can be written as:

AZ 152NXT, photoresist specifications

The first two equations are also referred to as Maxwell-Ampère's and Faraday's, respectively.

∇ × = +

(1)

The next equations are two forms of Gauss' law: the electric and magnetic form, respectively.

∇ ⋅ =

(3)

Out of the five equations mentioned, only three are independent. The both altogether, with either the electric form of Gauss' law or the equation of continuity form such a separate system.

The equations include constitutive relations that describe the macroscopic properties of the medium to obtain a closed system. They are given as:

A.

A.8. Data regularization methods for generalization of the model

There are several techniques that can be used to avoid overfitting and improve the training model. In this section, we explain one of the more used ways very briefly when we have a small amount of dataset. Regularization is defined as a process of adding data with the aim to avoid a model overfitting. Here we are going to describe two standards methods of data regularization: augmentation and Dropout [START_REF] Brunet | Thick Photoresist Development for the Fabrication of High Aspect Ratio Magnetic Coils[END_REF].

Having more data (dataset/samples) is the best way to get better consistent estimators

206

The LMA is used in many software applications for solving generic curve-fitting problems.

However, as with many fitting algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum. The LMA interpolates between the Gauss-Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases, it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA. LMA can also be viewed as Gauss-Newton using a trustregion approach.

APPENDIX B: List of personal publications B.1. Scientific papers

B.2. International Congress papers with published proceedings: [O]

[eP]: