domain: chim sdv

text:
Aurore Van De Walle

Jelena Kolosnjaj-Tabi

Yoann Lalatonne ⊥∥

Claire Wilhelm
email: claire.wilhelm@univ-paris-diderot.fr

Ever-Evolving Identity of Magnetic Nanoparticles within Human Cells, the Interplay of Endosomal Confinement, Degradation, Storage, and Neo-Crystallization

Considerable knowledge has been acquired in inorganic nanoparticles synthesis and nanoparticles potential use in biomedical applications. Among different materials, iron oxide nanoparticles remain unrivaled for several reasons. Not only they respond to multiple physical stimuli (e.g. magnetism, light) and they exert multifunctional therapeutic and diagnostic actions, but also they are biocompatible, and they integrate endogenous iron-related metabolic pathways. With the aim to optimize the use of (magnetic) iron oxide nanoparticles in biomedicine, different bio-physical phenomena have been recently identified and studied.

Among them, the concept of "nanoparticle's identity" is of particular importance. Nanoparticles identities evolve in distinct biological environments and over different periods of time. In this account, we focus on the remodeling of magnetic nanoparticles identity following nanoparticles journey inside the cells. For instance, nanoparticles functions, such as heat generation or magnetic resonance imaging, can be highly impacted by endosomal confinement. Structural degradation of nanoparticles was also evidenced and quantified in cellulo and correlates with the loss of nanoparticles magnetic properties. Remarkably, in human stem cells, the nonmagnetic products of nanoparticles degradation could be subsequently reassembled into neosynthetized, endogenous magnetic nanoparticles. This stunning occurrence might account for magnetic particles naturally found in human organs, especially the brain. However, mechanistic details and the implication of such phenomena in homeostasis and disease have yet to be completely unraveled. This Account aims to assess the short and the long-term transformations of magnetic iron oxide nanoparticles in living cells, particularly focusing on human stem cells. Precisely, we herein overview the multiple and ever-evolving chemical, physical and biological magnetic nanoparticles identities and emphasize the remarkable intracellular fate of these nanoparticles.

INTRODUCTION

Nanomedicine emerged in an attempt to overcome the limitations of conventional therapeutic and diagnostic approaches. Nanoparticles might, for instance, serve as carriers to deliver active ingredients to the target site, which could result in therapy improvement. In terms of diagnosis, nanoparticles might not only detect anomalous tissues, but also discerern abnormal cells or precancerous lesions. In addition, inorganic nanoparticles could be used as physical mediators to treat malignancies upon remote activation with light, magnetism or electric field.

These remote triggers induce in turn a physical effect (e.g. heat, movement, force) via nanoparticles located in situ, resulting in a precise temporal and spatial control of the treatment.

Such physical approaches are currently being tested in clinical trials (such as Aurolase® therapy, mediated by AuroShell® nanoparticles made of a gold nanoshell) or received European market approval as a medical device (such as NBTXR3®, hafnium oxide nanoparticles, which locally increase the radiation dose). Notwithstanding these novel developments, magnetic iron oxide nanoparticles were the first and the only inorganic nanoparticle-based drugs (marketed as Feridex® (Endorem®), and Resovist® (Cliavist®)) that obtained the marketing authorization as medicinal products and became largely available over the globe.

The ease to synthesize and functionalize iron oxide nanoparticles, their potential as multifunctional therapeutics (e.g. hyperthermia or drug delivery agents) or diagnostic tools (e.g. magnetic resonance contrast agents), the possibility to magnetically guide them, as well as their advantage of being biocompatible, make them particularly attractive for nanomedicine. [START_REF] Xie | Surface-Engineered Magnetic Nanoparticle Platforms for Cancer Imaging and Therapy[END_REF][START_REF] Yoo | Theranostic Magnetic Nanoparticles[END_REF][START_REF] Mai | Nanosystems Based on Magnetic Nanoparticles and Thermo-or Ph-Responsive Polymers: An Update and Future Perspectives[END_REF][START_REF] Zhu | Magnetic Nanoparticles for Precision Oncology: Theranostic Magnetic Iron Oxide Nanoparticles for Image-Guided and Targeted Cancer Therapy[END_REF][START_REF] Carregal-Romero | Magnetically Triggered Release of Molecular Cargo from Iron Oxide Nanoparticle Loaded Microcapsules[END_REF] Moreover, iron oxides are being increasingly envisaged as cues to regulate intracellular biological processes, or tools for intracellular cargo delivery. [START_REF] Kim | Magnetic Nanotweezers for Interrogating Biological Processes in Space and Time[END_REF][START_REF] Tay | Nanostructured Materials for Intracellular Cargo Delivery[END_REF] In regenerative medicine, nanotechnology deployed additional therapeutic assets to repair damaged tissues. Owing to the potential of magnetic nanoparticles to trace, spatially organize, and stimulate stem cells, such versatile nanoparticles portend a paradigm shift to tissue regeneration procedures and are tested in numerous preclinical studies. [START_REF] Li | Chemical Engineering of Cell Therapy for Heart Diseases[END_REF][START_REF] Cheng | Magnetic Antibody-Linked Nanomatchmakers for Therapeutic Cell Targeting[END_REF][START_REF] Vandergriff | Magnetic Targeting of Cardiosphere-Derived Stem Cells with Ferumoxytol Nanoparticles for Treating Rats with Myocardial Infarction[END_REF][START_REF] Wang | Externally Applied Static Magnetic Field Enhances Cardiac Retention and Functional Benefit of Magnetically Iron-Labeled Adipose-Derived Stem Cells in Infarcted Hearts[END_REF] However, one major obstacle engineered nanoparticles (including the magnetic ones) still face in their therapeutic mission, concerns a decreased stability and loss of targeting potential in the complex biological environment. [START_REF] Mulens-Arias | The Intrinsic Biological Identities of Iron Oxide Nanoparticles and Their Coatings: Unexplored Territory for Combinatorial Therapies[END_REF][START_REF] Kolosnjaj-Tabi | Biotransformations of Magnetic Nanoparticles in the Body[END_REF][START_REF] Feliu | In Vivo Degeneration and the Fate of Inorganic Nanoparticles[END_REF][START_REF] Roy | Assembly and Degradation of Inorganic Nanoparticles in Biological Environments[END_REF][START_REF] Soenen | The Effect of Nanoparticle Degradation on Poly (Methacrylic Acid)-Coated Quantum Dot Toxicity: The Importance of Particle Functionality Assessment in Toxicology[END_REF][START_REF] Soenen | Intra) Cellular Stability of Inorganic Nanoparticles: Effects on Cytotoxicity, Particle Functionality, and Biomedical Applications[END_REF] Indeed, despite efforts to customize nanoparticles surface, made with the ambition to deliver nanoparticles to a specific target, the most probable nanoparticles' fate is to end up within endosomes of hepatic and splenic macrophages, where nanoparticles undergo gradual "digestion". Therefore, the topical nano-bio-interface query of the current and following decade relates to processes occurring within the (intra)cellular environment, which profoundly alters the physical and chemical properties of nanoparticles.

Because of the difficulty to perform qualitative and quantitative metrics on nanoparticles in situ, within concerned subcellular environments, this issue is still overlooked. Moreover, analyses are even harder over long periods of time. Nevertheless, the early studies evaluating nanoparticles long-term intracellular transformations unanimously lead to the premise that intracellular (endosomal) environment transforms nanoparticles identity. [START_REF] Roy | Assembly and Degradation of Inorganic Nanoparticles in Biological Environments[END_REF][START_REF] Soenen | Intra) Cellular Stability of Inorganic Nanoparticles: Effects on Cytotoxicity, Particle Functionality, and Biomedical Applications[END_REF][START_REF] Goodman | The Surprising in Vivo Instability of near-Ir-Absorbing Hollow Au-Ag Nanoshells[END_REF][START_REF] Kreyling | In Vivo Integrity of Polymer-Coated Gold Nanoparticles[END_REF][START_REF] Vlasova | Enzymatic Oxidative Biodegradation of Nanoparticles: Mechanisms, Significance and Applications[END_REF][START_REF] Ivask | Complete Transformation of Zno and Cuo Nanoparticles in Culture Medium and Lymphocyte Cells During Toxicity Testing[END_REF][START_REF] Liu | Biodegradation of Bi-Labelled Polymer-Coated Rare-Earth Nanoparticles in Adherent Cell Cultures[END_REF] Finally, while nanotechnology emerged rather recently, magnetic nanoparticles could be traced as far back as Archean, [START_REF] Fayol | Managing Magnetic Nanoparticle Aggregation and Cellular Uptake: A Precondition for Efficient Stem-Cell Differentiation and Mri Tracking[END_REF] when magnetic particles allowed organisms magnetotaxis along the geomagnetic field. [START_REF] Faivre | Magnetotactic Bacteria and Magnetosomes[END_REF] And surprisingly, bacterial cells [START_REF] Kolinko | Biosynthesis of Magnetic Nanostructures in a Foreign Organism by Transfer of Bacterial Magnetosome Gene Clusters[END_REF] are not the only ones capable of magnetic bio-mineralization. Magnetite nanoparticles were also found in humans, [START_REF] Kirschvink | Magnetite Biomineralization in the Human Brain[END_REF][START_REF] Brem | Magnetic Iron Compounds in the Human Brain: A Comparison of Tumour and Hippocampal Tissue[END_REF][START_REF] Sant'ovaia | Magnetic Susceptibility and Isothermal Remanent Magnetization in Human Tissues: A Study Case[END_REF] but their origin (environmental versus biosynthesized) is still subject to discussions. [START_REF] Maher | Magnetite Pollution Nanoparticles in the Human Brain[END_REF][START_REF] Gieré | Magnetite in the Human Body: Biogenic Vs[END_REF] We just demonstrated that magnetic nanoparticles are no stranger to mammalian cells, as pristine human mesenchymal stem cells (MSCs) can also generate magnetic nanoparticles anew, from non-magnetic precursors. [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] In this Account, we focus on magnetic nanoparticles, which have an unrivaled advantage for therapy, diagnosis, and tissue engineering approaches. In order to apprehend nanoparticles full potential, we first describe how the structure and functionality of nanoparticles evolve upon cellular processing. In order to provide specific answers that link nanoparticles fate to the therapeutic outcome, we describe how endosomal compartmentalization alters both nanoparticles theranostics properties and nanoparticles integrity. The latter is lost over time, when the iron sheds away from nanoparticles and integrates the endogenous mechanisms involved in iron homeostasis. This gradual breakdown of magnetic nanoparticles can be monitored and quantified, as the loss of nanoparticles integrity correlates with the loss of nanoparticles magnetic properties. Yet, strikingly, the loss of magnetic properties is, in some cases, only transient. Within the cells, an intriguing phenomenon can occur, where exogenous nanoparticles completely degrade, to endogenously reform anew.

THE FIRST STEPS OF NANOPARTICLES INTRACELLULAR JOURNEY: INTERNALIZATION AND STORAGE IN ENDOSOMES.

Many magnetic nanoparticle synthesis methods were developed for prospective medical applications, each chemical approach providing different levels of control over nanoparticles shape, size, composition, and biocompatibility. The most conventional, easy, and inexpensive synthesis method is the co-precipitation, [START_REF] Massart | Preparation and Properties of Monodisperse Magnetic Fluids[END_REF] which allows obtaining aqueous dispersions of iron oxide nanoparticles. The high temperature thermal decomposition process using surfactants leads to an excellent control over particle size and monodispersity. [START_REF] Park | Ultra-Large-Scale Syntheses of Monodisperse Nanocrystals[END_REF][START_REF] Avugadda | Esterase-Cleavable 2d Assemblies of Magnetic Iron Oxide Nanocubes: Exploiting Enzymatic Polymer Disassembling to Improve Magnetic Hyperthermia Heat Losses[END_REF] The resulting nanoparticles are hydrophobic and a post-functionalization is necessary for water transfer.

Aqueous sol-gel methods are used as well, but require the use of heat treatments to obtain high crystallinity. The polyol sol-gel process however takes advantage of higher boiling points to avoid further heat treatments. [START_REF] Cai | Facile Synthesis of Superparamagnetic Magnetite Nanoparticles in Liquid Polyols[END_REF] The resulting nanoparticles have intermediate size distribution in between co-precipitation and hydrothermal routes and can be directly dispersed in water due to the presence of polyols at nanoparticles surface. Finally, non-aqueous sol gel approach, such as the benzyl alcohol route, 39 also provides high crystallinity, purity, and reproducibility, without the need of surfactants or heat treatment. Subsequently, the extraction from benzyl alcohol is required to obtain water dispersion. All these chemical synthesis approaches provide a magnetic core, and they all result in the covering of the core by a chosen coating. The latter governs nanoparticles stability in aqueous dispersions and also influences nanoparticles interactions with cells. However, as soon as the nanoparticles are placed in biological environments, such as in cell cultures or inside the body, their surface is readily changed by adsorption of adjacent ions and proteins.

The nanoparticles and protein complexes are then systematically internalized by cells via endocytosis (Figure 1a) and stored within the endosomes (Figure 1b-e). At a given incubation time, nanoparticles concentration, and nanoparticles features (e.g. coating, shape and size) will result in varying quantities of nanoparticles per cell. Magnetic nanoparticles taken up by cells can be quantified by the determination of total iron mass or by magnetometry and magnetophoretic approaches. The latter correlates the magnetization of the cells to the amount of intact nanoparticles. To totally discriminate endogenous iron from nanoparticles-provided exogenous iron, iron isotopes with low natural abundance can be used as building blocks for nanoparticles. [START_REF] Masthoff | Introducing Specificity to Iron Oxide Nanoparticle Imaging by Combining 57fe-Based Mri and Mass Spectrometry[END_REF][START_REF] Weissleder | Superparamagnetic Iron Oxide: Pharmacokinetics and Toxicity[END_REF] Coating can drive differential capture rates, particularly in vitro, mainly due to the variation of the nanoparticle charge and the aggregation status, [START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF] and nanoparticle size can control cellular responses. [START_REF] Jiang | Nanoparticle-Mediated Cellular Response Is Size-Dependent[END_REF] The internalization of magnetic nanoparticles can also be enhanced by the application of a remote magnet. [START_REF] Martina | The Effect of Magnetic Targeting on the Uptake of Magnetic-Fluid-Loaded Liposomes by Human Prostatic Adenocarcinoma Cells[END_REF][START_REF] Espinosa | Janus Magnetic-Plasmonic Nanoparticles for Magnetically Guided and Thermally Activated Cancer Therapy[END_REF] Importantly, the cell-loading procedure has to be tightly monitored, as not only the nanoparticles concentration, but also the decrease of nanoparticles suspension stability (e.g. aggregate formation) may lead to increased cell toxicity. [START_REF] Fayol | Managing Magnetic Nanoparticle Aggregation and Cellular Uptake: A Precondition for Efficient Stem-Cell Differentiation and Mri Tracking[END_REF][START_REF] Van De Walle | Role of Growth Factors and Oxygen to Limit Hypertrophy and Impact of High Magnetic Nanoparticles Dose During Stem Cell Chondrogenesis[END_REF]

ENDOSOMAL CONFINEMENT IMPACTS THERAPEUTIC AND IMAGING FUNCTIONS.

Cellular internalization of magnetic nanoparticles modifies their therapeutic functions. For magnetic hyperthermia (Figure 1g), heating was evaluated for nanoparticles in water or in cells (Figure 1f) [START_REF] Espinosa | Magnetic (Hyper) Thermia or Photothermia? Progressive Comparison of Iron Oxide and Gold Nanoparticles Heating in Water, in Cells, and in Vivo[END_REF] . Regardless of the type of nanoparticles we tested, it appeared that cellular internalization decreased the heating efficiency. Nevertheless, two different behaviors were evidenced. When the magnetic core is superparamagnetic and the magnetic relaxation is governed by Néel mechanism, the decrease in heat generation (about 2-fold) upon cellular internalization is not very pronounced. By contrast, when the magnetic core is ferromagnetic, the relaxation is governed by Brownian motion. As the latter is inhibited by the strong intracellular confinement of the nanoparticles in the endosomes, the heating is dramatically affected (> 10-fold decrease in generated heat). This finding recently led to chemical strategies to enable nanoparticle motion in endosomes and restore intracellular magnetic hyperthermia efficiency. [START_REF] Zyuzin | Confining Iron Oxide Nanocubes inside Submicrometric Cavities as a Key Strategy to Preserve Magnetic Heat Losses in an Intracellular Environment[END_REF] Worthy of note, we just mention here that confinement may affect other nanomedicine modalities, such as laser-mediated photothermia. The latter mostly relies on the use of plasmonic gold nanoparticles, and an opposite effect could be observed, with an increased heat generation in the intracellular environment. [START_REF] Espinosa | Cancer Cell Internalization of Gold Nanostars Impacts Their Photothermal Efficiency in Vitro and in Vivo: Toward a Plasmonic Thermal Fingerprint in Tumoral Environment[END_REF][START_REF] Sangnier | Endosomal Confinement of Gold Nanospheres, Nanorods and Nanoraspberries Governs Their Photothermal Identity and Is Beneficial for Cancer Cells Therapy[END_REF] In a similar way, the intracellular confinement of magnetic nanoparticles, which are T2-agents (negative contrast agents) due to their high r2 values (transverse relaxivity), is beneficial for magnetic resonance imaging (MRI). When confined in cells, the magnetic nanoparticles r1 value (longitudinal relaxivity) decreases massively (100-fold or more), while r2 only falls by a factor of 2.5, resulting in a remarkable increase in the r2/r1 ratio (almost 10-fold), governing the MRI negative contrast (Figure 1h). [START_REF] Di Corato | High-Resolution Cellular Mri: Gadolinium and Iron Oxide Nanoparticles for in-Depth Dual-Cell Imaging of Engineered Tissue Constructs[END_REF] Single cells can thus be tracked by MRI, [START_REF] Sharifi | Superparamagnetic Iron Oxide Nanoparticles for in Vivo Molecular and Cellular Imaging[END_REF] such as lymphocytes (Figure 1i), labeled with magnetic nanoparticles, which could be detected and distinguished while infiltrating a tumor. [START_REF] Smirnov | In Vivo Single Cell Detection of Tumor-Infiltrating Lymphocytes with a Clinical 1.5 Tesla Mri System[END_REF] Cells loaded with magnetic nanoparticles can also be guided using external magnets. In this case, the cells magnetization is directly proportional to the amount of intact nanoparticles, and is therefore not impacted by the endosomal confinement. We have previously described the feasibility of magnetic guiding and showed that nanoparticle-loaded cells preferentially migrate to the limb on which an external magnet was placed. 54 [START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF][START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] . (f-g) Effect of intracellular confinement on magnetic hyperthermia potential assessed by calorimetry. The heating efficiency of nanoparticles compared in water and in cells (f), after exposure to magnetic hyperthermia (g) for iron oxide nanoparticles. Data from ref [START_REF] Espinosa | Magnetic (Hyper) Thermia or Photothermia? Progressive Comparison of Iron Oxide and Gold Nanoparticles Heating in Water, in Cells, and in Vivo[END_REF] . (h) Intracellular confinement of the magnetic nanoparticles impacts their relaxivity, remarkably increasing their r2/r1 ratio, favorable for their use as T2 contrast agents for MRI. Data from ref [START_REF] Di Corato | High-Resolution Cellular Mri: Gadolinium and Iron Oxide Nanoparticles for in-Depth Dual-Cell Imaging of Engineered Tissue Constructs[END_REF] . (i) MRI detection of individual lymphocytes (white arrows) labeled with magnetic nanoparticles in vivo, in a murine tumor, detected 48h after intravenous injection. Data from ref [START_REF] Smirnov | In Vivo Single Cell Detection of Tumor-Infiltrating Lymphocytes with a Clinical 1.5 Tesla Mri System[END_REF] .

INTRACELLULAR MAGNETIC NANOPARTICLES USE IN TISSUE ENGINEERING

The recent use of magnetic nanoparticles for tissue engineering, as for instance to enhance stem cells differentiation, largely motivates the study of their long-term fate within these stem cells, whose ultimate vocation is to repair permanently damaged tissue. For these bioengineering applications, the magnetic nanoparticle-loaded cells become magnetoresponsive and can be manipulated at distance using magnetic forces (Figure 2a), a powerful tool for functional tissue engineering. Magnetic forces were for instance used to compact cells and create macro-scale cellular assemblies, without the use of scaffolds (Figure 2b-c). The tissues formed can be magneto-mechanically stimulated at a distance using either static or dynamic magnetic fields applied via permanent magnets [START_REF] Du | A 3d Magnetic Tissue Stretcher for Remote Mechanical Control of Embryonic Stem Cell Differentiation[END_REF] or electromagnets, 4 respectively (Figure 2d). Magnetic stimuli were used to orient the differentiation of stem cells toward specific tissue types, [START_REF] Fayol | Use of Magnetic Forces to Promote Stem Cell Aggregation During Differentiation, and Cartilage Tissue Modeling[END_REF] showing for instance that a cyclic magnetic stretching of embryonic stem cells largely increases the generation of cardiomyocytes. [START_REF] Mary | High-Throughput Differentiation of Embryonic Stem Cells into Cardiomyocytes with a Microfabricated Magnetic Pattern and Cyclic Stimulation[END_REF][START_REF] Du | A 3d Magnetic Tissue Stretcher for Remote Mechanical Control of Embryonic Stem Cell Differentiation[END_REF] The magnetic arrangement of cells has also been exploited for the development of various tissue structures such as cell spheroids, sheets, or rings (Figure 2e-g). [START_REF] Fayol | Use of Magnetic Forces to Promote Stem Cell Aggregation During Differentiation, and Cartilage Tissue Modeling[END_REF][START_REF] Shimizu | Construction of Multi-Layered Cardiomyocyte Sheets Using Magnetite Nanoparticles and Magnetic Force[END_REF][START_REF] Zwi-Dantsis | Remote Magnetic Nanoparticle Manipulation Enables the Dynamic Patterning of Cardiac Tissues[END_REF][START_REF] Frasca | Formation of a Three-Dimensional Multicellular Assembly Using Magnetic Patterning[END_REF] Generally, magnetically labeled cells are attracted toward the bottom of a culture dish with a magnet being placed underneath, but cell spheroids could also be formed via magnetic levitation, with cells being in suspension at the liquid-air interface (Figure 2h). 61 (a) After a simple incubation, magnetic nanoparticles are internalized within the endosomes of cells. These magnetized cells can be directed by remote magnetic forces, using permanent or electromagnets. (b) Homogenous size cell aggregates can be formed magnetically using a patterns of magnetic micro-tips. (c) Cell aggregates can be forced to fuse using remote magnetic attractors, gradually combining two-by-two, then four-by-four, etc. to form a millimeter-scale tissue, here for cartilage tissue engineering. Data from ref [START_REF] Fayol | Use of Magnetic Forces to Promote Stem Cell Aggregation During Differentiation, and Cartilage Tissue Modeling[END_REF] (d) Cell aggregates can also be mechanically stimulated via stretching using movable magnetic micro-tips. Data from ref [START_REF] Du | A 3d Magnetic Tissue Stretcher for Remote Mechanical Control of Embryonic Stem Cell Differentiation[END_REF] (e) Magnetic cell patterning can be achieved on arrays, with cells immediately attracted to the surface where the magnetic micro-tips are positioned. Data from ref [START_REF] Mary | High-Throughput Differentiation of Embryonic Stem Cells into Cardiomyocytes with a Microfabricated Magnetic Pattern and Cyclic Stimulation[END_REF] (f-g) Different tissue shapes can be achieved such as cell-sheets (f), or rings (g), reproduced with permission from refs [START_REF] Shimizu | Construction of Multi-Layered Cardiomyocyte Sheets Using Magnetite Nanoparticles and Magnetic Force[END_REF] and [START_REF] Zwi-Dantsis | Remote Magnetic Nanoparticle Manipulation Enables the Dynamic Patterning of Cardiac Tissues[END_REF] , respectively, copyright 2007 and 2020 John Wiley & Sons. (h) Cell assembly formed after magnetic levitation. Reproduced with permission from ref [START_REF] Souza | Three-Dimensional Tissue Culture Based on Magnetic Cell Levitation[END_REF] , copyright 2010 Nature publishing group.

CAN MAGNETIC NANOPARTICLES DEGRADE IN VIVO?

For any use of magnetic nanoparticles for therapeutic or diagnostic purpose, the main issue and challenge is to determine nanoparticles long-term fate after achieving their therapeutic mission. The initial steps towards the understanding of magnetic nanoparticles fate in the biological environment have been to study their transformation in vivo, in rodents. For this, magnetic nanoparticles were injected intravenously, and the location and repartition of iron were observed in organs, by distinguishing the magnetic from the non-magnetic form. [START_REF] Kolosnjaj-Tabi | Biotransformations of Magnetic Nanoparticles in the Body[END_REF][START_REF] Mejías | Long Term Biotransformation and Toxicity of Dimercaptosuccinic Acid-Coated Magnetic Nanoparticles Support Their Use in Biomedical Applications[END_REF][START_REF] Rojas | Time-Course Assessment of the Aggregation and Metabolization of Magnetic Nanoparticles[END_REF][START_REF] Stepien | Effect of Surface Chemistry and Associated Protein Corona on the Long-Term Biodegradation of Iron Oxide Nanoparticles in Vivo[END_REF] Magnetism measurements were performed together with histological and TEM imaging, showing the course of nanoparticles degradation within diverse organs. Most of magnetic nanoparticles were found within the liver and the spleen. Inside these organs, the nanoparticles were located intracellularly, within endosomes. The latter gradually fuse with lysosomes, and an acid-induced degradation of the nanoparticles takes place. [START_REF] Kolosnjaj-Tabi | Biotransformations of Magnetic Nanoparticles in the Body[END_REF] In order to demonstrate how exogenous iron integrated endogenous iron metabolism and reuse pathways, iron isotopes (e.g. [START_REF] Zwi-Dantsis | Remote Magnetic Nanoparticle Manipulation Enables the Dynamic Patterning of Cardiac Tissues[END_REF] Fe or [START_REF] Fayol | Use of Magnetic Forces to Promote Stem Cell Aggregation During Differentiation, and Cartilage Tissue Modeling[END_REF] Fe) containing nanoparticles were used. Nanoparticle-constituting iron could thus be differentiated from endogenous iron ( 56 Fe), with the isotopes being incorporated within hemoglobin. [START_REF] Masthoff | Introducing Specificity to Iron Oxide Nanoparticle Imaging by Combining 57fe-Based Mri and Mass Spectrometry[END_REF][START_REF] Weissleder | Superparamagnetic Iron Oxide: Pharmacokinetics and Toxicity[END_REF] These in vivo experiments brought a general understanding of iron oxide nanoparticles fate within the organism, and more importantly evidenced that magnetic nanoparticles degrade. This generated nanosafety interrogations, [START_REF] Liu | Applications and Potential Toxicity of Magnetic Iron Oxide Nanoparticles[END_REF] as released iron ions could lead to metal toxicity [START_REF] Sabella | A General Mechanism for Intracellular Toxicity of Metal-Containing Nanoparticles[END_REF] as well as Fenton-derived cytotoxicity, which we will discuss further in sections 7 and 8 of this Account.

CAN BIODEGRADATION BE QUANTIFIED IN CELLULO?

For regenerative medicine applications involving the labeling of stem cells with magnetic nanoparticles, assessments at the entire organism level were insufficient to conclude on the intracellular fate of the magnetic nanoparticles, defining the need for in vitro "close-circuit" analyses allowing precise quantifications.

The monitoring of nanoparticles fate in vitro, inside living cells, and on the long term, is not simple, due to constitutive cell division and cell death. Chemical models were thus developed, consisting of media mimicking the acidic environment of lysosomes. [START_REF] Levy | Degradability of Superparamagnetic Nanoparticles in a Model of Intracellular Environment: Follow-up of Magnetic, Structural and Chemical Properties[END_REF] TEM imaging was subsequently used to visualize in situ the gradual morphological alterations on edges and vertexes of magnetic nanocubes, which could be correlated to degradation. [START_REF] Lartigue | Biodegradation of Iron Oxide Nanocubes: High-Resolution in Situ Monitoring[END_REF] More recently, an in vitro tissue-mimicking model, made of human stem cells, was developed to quantitatively assess bio-transformations in the quiescent biological environment. [START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] This model consists in gathering stem cells by centrifugation, after which the cells form a cohesive 3D spheroid that can remain viable for months (Figure 3a). The stem cell-spheroids mature as tissues and produce a self-secreted extracellular matrix. As cell division stops, nanoparticle dilution is avoided. In such models, TEM imaging revealed a loss of structural integrity of internalized nanoparticles (Figure 3b-e). Furthermore, the intracellular biodegradation of iron oxide nanoparticles was quantitatively monitored using magnetometry measurements (Figure 3f-g).

The cellular magnetism reflected nanoparticles integrity, and a decrease in magnetism indicated nanoparticles degradation. Analyses were made with magnetic nanoparticles produced by different synthesis protocols: co-precipitation (rock-like spherical nanoparticles), [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] benzyl alcohol synthesis route (nanospheres), [START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF][START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF] high temperature polyol synthesis (flowerlike multicores), [START_REF] Mazuel | Magneto-Thermal Metrics Can Mirror the Long-Term Intracellular Fate of Magneto-Plasmonic Nanohybrids and Reveal the Remarkable Shielding Effect of Gold[END_REF] thermal decomposition (nanocubes or iron oxide gold dimers), [START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] or bacterial synthesis (biogenic magnetosomes). [START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] Altogether, the results obtained in the stem cell-spheroid model confirmed a progressive decrease in magnetism, indicating a degradation of all nanoparticles. The latter was almost complete one month following nanoparticles exposure, for low doses of internalized nanoparticles per cell (in the 1 pg of iron per cell range), without impacting cell differentiation. Besides, the degradation kinetics could be correlated to nanoparticles intracellular content (slower degradation at a higher dose) [START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF] and to nanoparticle coating (a polymer coating, PAA [START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF] or molecular imprinted polymer (MIP)) [START_REF] Boitard | Biological Fate of Magnetic Protein-Specific Molecularly Imprinted Polymers: Toxicity and Degradation[END_REF] , where polymers resulted in a slower degradation of nanoparticles in comparison to citrate coated ones).

Importantly, for all tested iron oxide nanoparticles, nanomagnetism brings substantial advantage to quantitatively track nano-transformations within cells.

For these standard magnetometer analyses, cells and tissues have to be fixed to be processed. By contrast, a bench-sized device was recently tested for longitudinal real-time monitoring of the degradation of nanoparticles within the same living cells, over several days, without affecting either cellular viability or tissue formation (Figure 3g). [START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF] This approach confirmed a progressive degradation of the nanoparticles over time and delineated small fluctuations in the degradation process, probably linked to biological events. [START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] The spheroids remain viable along extensive culture periods allowing long-term monitoring of nanoparticles biotransformations. (b-e) TEM images of intact nanoparticles in suspension (framed panel) and same nanoparticles, internalized and metabolized by cells, visualized after 21 to 25 days of culture within the tissue model. Nanoparticles biodegradation in cellulo is demonstrated for a panel of nanostructures: (b) iron oxide nanospheres, (c) gold/iron oxide nanodimers, (d) iron oxide nanocubes, (e) magnetosomes. Data from refs [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF][START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] . (f) The magnetic moment of the spheroids is obtained via vibrating sample magnetometry (VSM) over time. Results evidence a progressive decrease in magnetic iron while total iron remains constant (right panel), demonstrating a progressive degradation of the magnetic nanoparticles in the tissue model over time. Data from refs [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] (g) Magnetometry measurements on the tissue model, in realtime, while keeping the cells viable were also achieved using a small bench-top magnetic sensor device. Data from ref [START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF] . Measurements were either obtained punctually (left panel) or continuously on a same spheroid (right panel).

IRON METABOLISM AS AN ESSENTIAL STEP IN NANOPARTICLES DEGRADATION

PROCESSES

Iron is essential in many biological functions. [START_REF] Aisen | Iron Metabolism[END_REF] The natural human iron metabolism confers a particular advantage to magnetic nanoparticles in nanomedicine, as their iron oxide core might be bio-assimilated. However, the iron redox properties, which are intimately associated to its activity with proteins, also imply a potential Fenton-derived toxicity. To minimize toxicity, specific protein systems operate within the body and 'chaperone' the metal, allowing its uptake, transport, storage, and reuse.

Upon nanoparticles internalization and concentration within endosomes, others [START_REF] López-Castro | From Synthetic to Natural Nanoparticles: Monitoring the Biodegradation of Spio (P904) into Ferritin by Electron Microscopy[END_REF][START_REF] Hohnholt | Handling of Iron Oxide and Silver Nanoparticles by Astrocytes[END_REF][START_REF] Tamion | Ferritin Surplus in Mouse Spleen 14 Months after Intravenous Injection of Iron Oxide Nanoparticles at Clinical Dose[END_REF][START_REF] Gutiérrez | Bioinorganic Transformations of Liver Iron Deposits Observed by Tissue Magnetic Characterisation in a Rat Model[END_REF][START_REF] Ruiz | Biotransformation of Magnetic Nanoparticles as a Function of Coating in a Rat Model[END_REF] and us 1-3, 42, 55, 70 not only observed the specific synthetic nanoparticles, but also distinct, more or less electron-dense structures, with sizes ranging from 5 to 7 nm. The number of these structures, which were identified as ferritin proteins, increased over time, as the number of endosomal nanoparticles was decreasing. These structures were present both inside the endosomes and inside the cytosol. The role of the ferritin proteins is to intercept the iron ions that are released from synthetic nanoparticles, and their structure is the same, regardless of the initial shape of cell-internalized nanoparticles, as evidenced in Figure 4.

A method to address the integration of iron released over nanoparticles degradation, consists in assessing the impact of nanoparticles degradation on iron binding proteins expression by quantitative PCR. The genes related to iron metabolism include the ones associated to iron storage (ferritin/FerrH & FerrL), transport (transferrin/TFR1; dimetal transporters/DMT1), and export (ferroportin /SLC40A1). Ionic species released upon the degradation of chemically synthesized iron oxide nanoparticles (Figure 4a, 4b, 4d), [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] or magnetosomes made by magnetotactic bacteria (Figure 4e), [START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] can all be stored within the ferritin protein core. As a consequence, the degradation systematically correlates with an up-regulation of ferritin genes expression (Figure 4c,4f). [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF][START_REF] Mazuel | Magneto-Thermal Metrics Can Mirror the Long-Term Intracellular Fate of Magneto-Plasmonic Nanohybrids and Reveal the Remarkable Shielding Effect of Gold[END_REF] A homeostatic balance in gene expression of transport and export proteins is also generally observed, with a decreased expression of TFR1, to limit iron uptake by cells, and increased ferroportin, allowing the export of iron excess.

WHAT HAPPENS NEXT? RECRYSTALLIZATION OF BIOGENIC MAGNETIC

NANOPARTICLES

It is now obvious that in the cellular environment, magnetic nanoparticles evolve over time and start rapidly degrading. As magnetic nanoparticles mainly consist of an iron oxide core, the metal (iron) ions stemming from degraded nanoparticles subsequently integrate the physiological iron metabolism pathway and transition into iron depots, the ferritins. Within ferritin proteins, the iron is stored in the non-magnetic ferrihydrite form.

Phenomenally, we just demonstrated that under specific conditions, stem cells can also exploit ionic species of iron, released during the process of degradation of nanoparticles made by coprecipitation [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF] or biological magnetosomes [START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] . The ionic species were then used to resynthesize novel, biogenic, magnetic nanoparticles (Figure 5). These de novo synthesized nanoparticles a priori exhibit a magnetite structure. In our experiment, the internalized magnetic nanoparticles were first degraded (within the first 3 days), this degradation correlated with a decrease in cellular magnetization. Subsequently, the cells almost completely remagnetized, reflecting the re-synthesis of magnetic nanoparticles (Figure 5a-d). These recrystallized magnetic nanoparticles made by human stem cells are around 8 nm in size (Figure 5e-h).

The neo-synthesis of biogenic nanoparticles most probably involves the ferritin protein, which commonly stores the iron in its non-magnetic ferrihydrite form, but can also contain magnetic iron phases such as magnetite or maghemite. [START_REF] Quintana | Electron Nanodiffraction and High-Resolution Electron Microscopy Studies of the Structure and Composition of Physiological and Pathological Ferritin[END_REF] While the biomineralization within the ferritin cage naturally occurs in bacteria, the production of magnetoferritin [START_REF] Wong | Biomimetic Synthesis and Characterization of Magnetic Proteins (Magnetoferritin)[END_REF] from the apoferritin protein has been mostly reported in reconstituted in vitro suspension. Nevertheless, the presence of magnetism sometimes observed in mammals has recently been attributed to iron-rich ferritins. [START_REF] Tamion | Ferritin Surplus in Mouse Spleen 14 Months after Intravenous Injection of Iron Oxide Nanoparticles at Clinical Dose[END_REF] Interestingly, the nanoparticles neo-synthetized by human stem cells described above, [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] have the size of the cavity of the ferritin protein, potentially indicating that magnetoferritin could be formed in human cells in case of iron excess. However, detailed processes of this biomineralization in mammalian cells have yet to be unraveled.

Importantly, the biosynthesis of magnetic nanoparticles correlated with no cellular toxicity. This innocuousness of in cellulo-synthesized nanoparticles was observed even at high doses of intracellular iron (over 10 pg per cell). On the contrary, when biosynthesis did not occur, cellular toxicity was detected at these doses. This indicates that biosynthesis might act as a potential protective mechanism. c-d), the cells magnetism (a,c) evidences an important decrease from day 0 to 3 (degradation) followed by a clear increase back between day 3 and 9, indicating the synthesis of magnetic nanoparticles anew produced by the cells themselves. The magnetization curves obtained at day 3 and day 9 (b,d) clearly demonstrate the emergence of a magnetic cellular signal. (e-h) Electron microscopy imaging and analysis of the biosynthesized nanoparticles. TEM images show the biosynthesized magnetic nanoparticles (8 nm in diameter) located in the endosomes of the cells at day 21 for both the synthetic nanoparticles (e) and for the magnetosomes (g). Selected Area Electron Diffraction (SAED) pattern of endosomes filled with biosynthesized nanoparticles (targeted endosome is shown in the boxed images) display diffraction rings that can be indexed to lattice planes of magnetite, again for both nanoparticles (f) and magnetosomes (h).

MAGNETISM IN HUMANS

The biosynthesis of magnetic nanocrystals in microorganisms is vastly documented and exploited; yet, the studies describing the genesis of magnetic crystalline materials in mammalian cells are rather scarce. To the best of our knowledge, the studies described above were the first to report the possibility for an in situ synthesis of magnetic crystals in human cells (precisely, mesenchymal stem cells MSCs). [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] Nevertheless, the existence of magnetic crystals in human brain samples was evidenced almost three decades ago. [START_REF] Kirschvink | Magnetite Biomineralization in the Human Brain[END_REF] The spatial distribution of magnetite was studied on dissected whole human brains and showed that magnetite is preferentially partitioned in the cerebellum and brain stem. [START_REF] Gilder | Distribution of Magnetic Remanence Carriers in the Human Brain[END_REF] Individuals (healthy subjects and patients suffering from neurodegenerative diseases) were monitored by magnetic resonance imaging (MRI) or underwent magnetoencephalography, which allowed tracking and quantifying magnetite (Figure 6a-b). [START_REF] Khan | Using the Magnetoencephalogram to Noninvasively Measure Magnetite in the Living Human Brain[END_REF] These analyses revealed the presence of magnetite in both populations, but larger amounts were detected in the brain of aging male subjects and in patients suffering from neurodegenerative diseases such as Alzheimer. There is now general consensus that Alzheimer disease is associated with a strong deregulation of the iron metabolism, leading to iron excess. [START_REF] Liu | Iron and Alzheimer's Disease: From Pathogenesis to Therapeutic Implications[END_REF] It could be thus hypothesized that magnetic nanoparticles form due to iron overload and minimize oxidative stress, which would be otherwise massively engendered by iron ions.

After the pioneering works on the brain, magnetic crystals were also found in other human tissues, such as the heart, spleen, liver, ethmoid bone, cervical skin, and tumors. [START_REF] Kirschvink | Magnetite Biomineralization in the Human Brain[END_REF][START_REF] Brem | Magnetic Iron Compounds in the Human Brain: A Comparison of Tumour and Hippocampal Tissue[END_REF][START_REF] Sant'ovaia | Magnetic Susceptibility and Isothermal Remanent Magnetization in Human Tissues: A Study Case[END_REF][START_REF] Calderón-Garcidueñas | Combustion-and Friction-Derived Magnetic Air Pollution Nanoparticles in Human Hearts[END_REF][START_REF] Lu | Chemical Multi-Fingerprinting of Exogenous Ultrafine Particles in Human Serum and Pleural Effusion[END_REF][START_REF] Murros | Magnetic Nanoparticles in Human Cervical Skin[END_REF] The source of magnetic crystals has not been clearly ascertained. A large body of literature suggests an exogenous origin of magnetic crystals. Exogenous sources mainly include atmospheric pollution (particularly diesel exhaust and particulates derived from brake abrasion). [START_REF] Gieré | Magnetite in the Human Body: Biogenic Vs[END_REF] Airborne magnetite can thus penetrate the brain, probably via the olfactory nerve (Figure 6c). However, some previous studies also suggested a potential formation of magnetic crystals within the human body. This assumption was only recently empirically evidenced in vitro in human MSC, either transfected with the magnetotactic bacterial gene mms6, [START_REF] Elfick | Biosynthesis of Magnetic Nanoparticles by Human Mesenchymal Stem Cells Following Transfection with the Magnetotactic Bacterial Gene Mms6[END_REF] or by us in pristine human MSC after exposure to synthetic nanoparticles. [START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF] Interestingly, it has been indicated that exogenous magnetic nanocrystals can be distinguished from endogenous ones by their shape and their crystal structure. [START_REF] Maher | Magnetite Pollution Nanoparticles in the Human Brain[END_REF] Magnetite crystals might also be the most plausible cause allowing some humans to transduce the changes in magnetic fields strength into an active neural response. This conclusion was drawn after exposing subjects to very weak magnetic fields (35 μT) oriented along different directions, while monitoring the electrical activity of participant's brains by electroencephalography (EEG) (Figure 6d-f). [START_REF] Wang | Transduction of the Geomagnetic Field as Evidenced from Alpha-Band Activity in the Human Brain[END_REF] When the magnetic field was shifted, the participants did not experience any obvious feelings. However, in some participants, the EEG revealed a drop in amplitude of their EEG alpha waves. Such responses typically occur when persons detect and process a sensory stimulus and could reflect some kind of individual differences in navigational ability. The importance of weak magnetic fields was also emphasized in orientation studies conducted on animals. [START_REF] Wiltschko | Magnetic Orientation in Animals[END_REF] Migratory birds can for instance filter magnetic signals and only respond to those that were environmentally relevant.

CONCLUSION AND PERSPECTIVES

Magnetic nanoparticles have an undisputed potential in the medical field. However, the scientific community obviously did not gather enough data to obtain a full picture of the behavior of magnetic nanoparticles in the biological environment. And yet, such research should be performed to vulgarize theranostics applications. The overall identity of a nanoparticle is a result of the interplay of its physical, synthetic, and biological identities. For magnetic nanoparticles, the magnetic core defines their physical identity and is responsible for their multiple functionalities (imaging, heat generation, magneto-responsiveness). Conversely, nanoparticles chemical design (size, shape, surface coating) defines the synthetic identity. The interactions with biological fluids affect the surface chemistry (and alter nanoparticles synthetic identity). Subsequently, the nanoparticles sustain modifications in physiological fluids (undergo aggregation, opsonization). These modifications occur under different circumstances, in different media (within the blood or inside intracellular compartments) and at different points of time, and inflict an ever-evolving biological identity. This has been widely acknowledged, but, remarkably, the investigations rarely went beyond the surface -into nanoparticles core. In this Account, we sought to consider the evolution of magnetic nanoparticle biological identity from nanoparticles surface throughout the core. The biotransformations that occur all through the core, and inevitably occur inside the cells, directly affect nanoparticles physical identity. Thanks to the development of a stem cells-spheroid model, these transformations could be monitored over one month. During this period, the magnetism of the spheroid was systematically used as the direct signature of the nanoparticle physical identity. Biotransformations first reshape nanoparticles synthetic identity (size, shape, surface moieties/coating) and can subsequently lead to nanoparticles total degradation. When this total degradation takes place, ionic species are released, which in some cases triggers an ultimate transformation: the intracellular magnetic recrystallization, as we observed in pristine stem cells. This empirical evidence of a biogenic magnetic synthesis by human cells recalls the natural presence of magnetite crystals in humans, particularly in the brain. Studies indicate that a strong dysregulation of the iron metabolism can be associated with Alzheimer disease and leads to iron excess. This could lead to the hypothesis that magnetic biosynthesis could take place in these conditions and act as a protective mechanism to avoid oxidative stress engendered by free iron ions via the formation of magnetic iron oxides, such as magnetite, potentially in the core of the ferritin protein. Interestingly, this is also consistent with the observation that magnetic biosynthesis occurs in stem cells and enables stem cells to resist high doses of iron (delivered by synthetic nanoparticles). This iron resistance correlates with both, low toxicity and preservation of stem cell differentiation potential. It should also be noted that magnetic particles in humans could also be of environmental, and not of biogenic origin. However, the magnetic nanoparticles evidenced in the human brain appear to differ in shape, when compared to typical environmental nanoparticle pollutants. Nevertheless, it is possible that the fate of chemically synthesized magnetic nanoparticles and the fate of environmental nanoparticles might converge, meaning that regardless of their origin, nanoparticles might degrade and recrystallize anew. In the future, the understanding of the aspects of such intracellular bioinorganic synthesis might provide clues explaining the yet unknown origin of naturally occurring magnetic nanoparticles in humans, with a potential connection with neurodegenerative diseases. Moreover, it might open the gate to novel bioinspired synthesis methodologies in the field of bioinorganic chemistry and materials science.

Figure 1 :

 1 Figure 1: Endosomal internalization of magnetic nanoparticles and effect of intracellular confinement on hyperthermia and imaging applicability. (a) Nanoparticles are internalized within cellular endosomes by a simple incubation procedure, where cultured cells are coexposed to nanoparticles. (b-e) Nanoparticles confinement in cells' endosomes shown by transmission electron microscopy (TEM); (b) magnetic nanoparticles obtained by coprecipitation, (c) Iron (magnetic)/gold nanodimers and (d) magnetic nanocubes, both obtained by thermal decomposition, (e) biological magnetosomes extracted from magnetotactic bacteria. Data from refs[START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF][START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] . (f-g) Effect of intracellular confinement on magnetic hyperthermia potential assessed by calorimetry. The heating efficiency of nanoparticles compared in water and in cells (f), after exposure to magnetic hyperthermia (g)

Figure 2 :

 2 Figure2: Magnetic tissue engineering using magnetically labeled stem cells. (a) After a simple incubation, magnetic nanoparticles are internalized within the endosomes of cells. These magnetized cells can be directed by remote magnetic forces, using permanent or electromagnets. (b) Homogenous size cell aggregates can be formed magnetically using a patterns of magnetic micro-tips. (c) Cell aggregates can be forced to fuse using remote magnetic attractors, gradually combining two-by-two, then four-by-four, etc. to form a millimeter-scale tissue, here for cartilage tissue engineering. Data from ref57 (d) Cell aggregates can also be mechanically stimulated via stretching using movable magnetic micro-tips. Data from ref[START_REF] Du | A 3d Magnetic Tissue Stretcher for Remote Mechanical Control of Embryonic Stem Cell Differentiation[END_REF] (e) Magnetic cell patterning can be achieved on arrays, with cells immediately attracted to the surface where the magnetic micro-tips are positioned. Data from ref[START_REF] Mary | High-Throughput Differentiation of Embryonic Stem Cells into Cardiomyocytes with a Microfabricated Magnetic Pattern and Cyclic Stimulation[END_REF] (f-g) Different tissue

Figure 3 .

 3 Figure3. Biodegradation of magnetic nanoparticles over time evidenced in a 3D stem cellspheroid model via magnetic measurements. (a) Upon centrifugation, stem cells are cultured in 3D, forming a cohesive structure. Data from ref[START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] The spheroids remain viable along extensive culture periods allowing long-term monitoring of nanoparticles biotransformations. (b-e) TEM images of intact nanoparticles in suspension (framed panel) and same nanoparticles, internalized and metabolized by cells, visualized after 21 to 25 days of culture within the tissue model. Nanoparticles biodegradation in cellulo is demonstrated for a panel of nanostructures: (b) iron oxide nanospheres, (c) gold/iron oxide nanodimers, (d) iron oxide nanocubes, (e) magnetosomes. Data from refs[START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Curcio | Transformation Cycle of Magnetosomes in Human Stem Cells: From Degradation to Biosynthesis of Magnetic Nanoparticles Anew[END_REF][START_REF] Sangnier | Impact of Magnetic Nanoparticle Surface Coating on Their Long-Term Intracellular Biodegradation in Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] . (f) The magnetic moment of the spheroids is obtained via vibrating sample magnetometry (VSM) over time. Results evidence a progressive decrease in magnetic iron while total iron remains constant (right panel), demonstrating a progressive degradation of the magnetic nanoparticles in the tissue model over time. Data from refs[START_REF] Van De Walle | Biosynthesis of Magnetic Nanoparticles from Nano-Degradation Products Revealed in Human Stem Cells[END_REF][START_REF] Mazuel | Massive Intracellular Biodegradation of Iron Oxide Nanoparticles Evidenced Magnetically at Single-Endosome and Tissue Levels[END_REF] (g) Magnetometry measurements on the tissue model, in realtime, while keeping the cells viable were also achieved using a small bench-top magnetic sensor device. Data from ref[START_REF] Van De Walle | Real-Time in Situ Magnetic Measurement of the Intracellular Biodegradation of Iron Oxide Nanoparticles in a Stem Cell-Spheroid Tissue Model[END_REF] . Measurements were either obtained punctually (left panel) or continuously on a same spheroid (right panel).

Figure 4 :

 4 Figure 4: Storage of iron in ferritin proteins, upon ionic release following nanoparticles degradation. TEM micrographs of (a) rock-like nanoparticles made by coprecipitation, (b) spherical nanoparticles made by benzyl alcohol synthesis, (c) nanocubes made by thermal decomposition, and (d) magnetosomes made by bacterial synthesis, internalized in stem cells and cultured in cellular spheroids for 21-25 days, showing little if any intact nanoparticles and evidencing the approximatively 6-nm electron-dense dots within the endosomes and within the cytoplasm of cells. These dots correspond to the ferritin protein, loaded with iron. Increased gene expression of the L-subunit of ferritin for cells labeled with rock-like nanoparticles (e) and the H-subunit of ferritin for cells labeled with magnetosomes (f) at days 21-25 supports the transfer of the released iron into the protein cage. Data from refs 3, 55 .

Figure 5 :

 5 Figure 5: Biomineralization of magnetic nanoparticles anew by human cells following degradation of synthetic (8-nm made by co-precipitation) or biogenic (magnetosomes made by bacteria) nanoparticles. Data from refs 2, 3 . (a-d) Magnetometry assessment of the recrystallisation process. After internalization in human stem cells of the nanoparticles (a-b) or the magnetosomes (c-d), the cells magnetism (a,c) evidences an important decrease from day 0 to 3 (degradation) followed by a clear increase back between day 3 and 9, indicating the synthesis of magnetic nanoparticles anew produced by the cells themselves. The magnetization curves obtained at day 3 and day 9 (b,d) clearly demonstrate the emergence of a magnetic cellular signal. (e-h) Electron microscopy imaging and analysis of the biosynthesized nanoparticles. TEM images show the biosynthesized magnetic nanoparticles (8 nm in diameter) located in the endosomes of the cells at day 21 for both the synthetic nanoparticles (e) and for the magnetosomes (g). Selected Area Electron Diffraction (SAED) pattern of endosomes filled with biosynthesized nanoparticles (targeted endosome is shown in the boxed images) display diffraction rings that can be indexed to lattice planes of magnetite, again for both nanoparticles (f) and magnetosomes (h).

Figure 6 :

 6 Figure 6: Magnetic nanocrystals in the human brain. (a) Higher magnetization intensities are detected post-mortem in the brain stem and cerebellum than the cerebral cortex. Reproduced with permission from ref 79 , copyright 2018 Nature publishing group. (b) Magnetoencephalogram of the brain of healthy human subjects indicates source localization and mass. Reproduced with permission from ref 80 , copyright 2019 John Wiley & Sons. (c) Presence of pollution-derived magnetite nanoparticles in the human brain. Reproduced with permission from ref 33 , copyright 2016 National Academy of Sciences. (d-f) Geomagnetic sensing in humans. Reproduced with permission from ref 86 , copyright 2019 Society for Neuroscience. (d) Schematic drawing of the Faraday cage was specifically designed for the testing of human magnetoreception. (e) Schematic showing the magnetic field rotations used in the experiment. The participant sits facing North, while the downwards-pointing field rotates CW (blue arrow) from Northwest to Northeast or CCW (red arrow) from Northeast to Northwest. (f) EEG from a single channel (Fz) centered over the top of the head. Pink lines show the 100 millisecond

ACKNOWLEDGMENTS

The authors acknowledge financial support provided by the European Research Council (consolidator grant MaTissE #648779).

AUTHOR INFORMATION

Corresponding Author E-mail: claire.wilhelm@univ-paris-diderot.fr ORCID Claire Wilhelm: 0000-0001-7024-9627

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.

Biographies


Lucas José Da

Silva Moreira

Mirko Fiacchini
email: mirko.fiacchini@grenoble-inp.fr

Gildas Besancon
email: gildas.besancon@grenoble-inp.fr

Francesco Ferrante
email: francesco.ferrante@grenoble-inp.fr

Herve Roustan
email: herve.roustan@riotinto.com

Herve Roustan State

Gildas Besançon

Hervé Roustan

State Affine Modeling and Observer Design for Hall-Héroult Process

Keywords: Aluminum electrolysis, modeling, state estimation, Kalman filter

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Hall-Héroult process is the electrolysis method to produce aluminum at industrial scale [START_REF] Grjotheim | Aluminium Smelter Technology: A Pure and Applied Approach[END_REF]. In short, it is based on an electrochemical reaction that requires the dissolution of alumina (Al 2 O 3 ) in molten cryolite bath under a high-temperature condition. During the electrolysis process, carbon anodes are dipped in the solution, while the produced liquid aluminum is accumulated at the bottom of the cell and carbon dioxide gas is expelled out of the system. The process is summarized by the following chemical equation:

Al 2 O 3(diss) + 3 2 C (anode) → 2Al (l) + 3 2 CO 2(g) (1)
In large industrial plants, each cell has a large number of anodes connected in parallel and supported by a common busbar. The height of this bar can be set during the operation to adjust the anode-cathode distance (ACD).

High DC current, in the order of kilo-amps, is applied to each anode. This large intensity is required to increase the amount of aluminum produced. The chemical reagent, alumina, is periodically injected into the system as powder by feeders along with the pot cell. A simple schematic view of a Hall-Héroult cell is shown in Figure 1.

The hazardous conditions inside the pot such as high temperature, magnetic field and the presence of corrosive elements, make it difficult to develop sensors for continuous measurement. Moreover, some important state variables of the process are not measured at any time during the operation. This inconvenient can lead to inaccurate results [START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF] and it restrains the process production efficiency [START_REF] Yao | State and parameter estimation in Hall-Héroult cells using iterated extended Kalman filter[END_REF].

⋆ This research was partially supported by project PIANO.

Fig. 1. Pot Schematic View

Some researchers have been developing nonlinear estimators to overcome this information limitation and obtain the desired process states [START_REF] Kolås | State estimation is the real challenge in NMPC[END_REF][START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF][START_REF] Hestetun | Detecting abnormal feed rate in aluminium electrolysis using extended kalman filter[END_REF]. However, those works are mostly based on the approximate Extended Kalman Filter [START_REF] Moore | Optimal filtering[END_REF], and do say not much about the underlying physics.

The present paper aims to provide a simple model for the Hall-Héroult process, yet capturing the main features of a regular operation by combining physical-chemical aspects with experimental models. In continuation of the work developed in (da Silva Moreira et al., 2020), the system is modeled in a state affine form, so as to allow an observer implementation here using exact Linear Kalman results, in the spirit of (T ¸iclea and Besançon, 2013) for instance. An identification procedure for a first model calibration is then presented. This procedure is validated with operational collected data and tested for different conditions in APX pot cell of Rio Tinto Laboratoire des Recherches de Fabrications (LRF) located in Saint Jean de Maurienne, France. For confidentiality reasons, the data in the y-axis in all plots have been removed.

The paper goes on as follows: section 2 gives an overview of the problem we consider, section 3 presents the modeling that we propose, and section 4 provides online state estimation implementation and results. Some conclusions are summarized in section 5.

PROBLEM DESCRIPTION

The process efficiency is directly related to the distance gap between the cathode and anode, (ACD) [START_REF] Côté | Predicting instability and current efficiency of industrial cells[END_REF]. This value is not constant during the operation due to the chemical reaction. The carbon anodes are consumed and they are replaced after some time. Moreover, the liquid aluminum layer increases its height because of the production. Furthermore, current perturbations and bath composition can also affect the ACD. However, the hazardous conditions inside the pot make it impossible to develop a sensor for continuous measurement. The ACD is critical since a large distance decreases the pot cell production and a small value can cause a shortcircuit between the produced aluminum and the anode [START_REF] Keniry | Digital processing of anode current signals: an opportunity for improved cell diagnosis and control[END_REF]. Unfortunately, only a few papers have modeled this dynamical behavior [START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF][START_REF] Yao | Estimation of spatial alumina concentration in an aluminum reduction cell using a multilevel state observer[END_REF].

The dissolved alumina concentration (wAl 2 O 3 ) is also an important quantity that is not continuously measured.

Commonly, just a few samples per week are manually taken, which makes it difficult to obtain an experimental model. This quantity is again very important: A low concentration triggers the so-called "anode effect" [START_REF] Bearne | The development of aluminum reduction cell process control[END_REF] which can cause interruptions on the production and produces greenhouse gases. Conversely, a high concentration makes it possible to see sludge formation.

In practice, only the line current (I), line voltage (V ), the busbar position (BM ), and the frequency of the alumina feeding (F ) are continuously collected by sensors. For the system regulation, an indirect measurement called the pseudo-resistance (R) is commonly used to adjust the ACD and wAl 2 O 3 . It is calculated according to the following formula:

R(t) = V (t) -V ext I(t) (2)
where V ext is the voltage intercept of cell voltage versus current extrapolated from a small change in current to zero current.

Based on the pseudo-resistance value, the alumina feeding frequency is modified. It alternates between two predetermined periods to have faster or slower feeding. This value is adjusted by the ACD regulation to ensure control stability and obtain a good current efficiency [START_REF] Homsi | Overview of process control in reduction cells and potlines[END_REF]. A nonlinear relation between R, ACD and wAl 2 O 3 is also known to take the form of Fig. 2 [START_REF] Haupin | Interpreting the components of cell voltage[END_REF]. From this complex relation, it is not easy to obtain some information. All this motivates the development of a model for the unknown state dynamics, and an observer to get an online estimation for it.

1 2 3 4 5 Alumina Concentration (%) Pot Resistance ACD =4.2 cm ACD =4.3 cm ACD =4.4 cm ACD =4.5 cm ACD =4.6 cm ACD =4.7 cm ACD =4.8 cm
Fig. 2. Typical pot resistance curve as function of alumina concentration and ACD

MODELING

The aim of this section is to obtain a discrete-time model that explicitly relates ACD and wAl 2 O 3 to the pot pseudo-resistance R. The model is obtained using a hybrid approach that combines physical-chemical aspects and information extracted from data.

Anode-Cathode Distance

From the problem description, it is possible to define the ACD variation ( d dt ACD) as the height variation between the aluminum ( d dt Al height ) and carbon ( d dt C height ) layers, plus the anodes busbar beam movement ( d dt BM ):

d dt ACD(t) = d dt Al height (t)- d dt C height (t)+ d dt BM (t) (3)
Each element height can be written as a function of the mass production by the density relation. Hence, the aluminum and carbon height variation become:

d dt Al height (t) = 1 ρ Al S d dt m Al (t) (4) d dt C height (t) = 1 ρ C S d dt m C (t) (5)
where ρ Al is the liquid aluminum density, m Al is the aluminum produced mass, ρ C is the carbon density, m C is the carbon produced mass, and S is the average reaction surface area.

The produced aluminum mass rate ( d dt m Al ) is given by the Faraday's law of electrolysis:

d dt m Al (t) = C e Al m 3F I(t) (6)
where C e is the current efficiency, Al m is the aluminum molar mass, I(t) is the instantaneous current applied to the pot, F is the Faraday's constant, and 3 is the valency number of ions of the aluminum given by: 4Al

3+ + 6O 2-+ 3C (anode) → 4Al (l) + 3CO 2(g) (7)
From the chemical balance of equation ( 1), it is possible to relate the aluminum mass production with the carbon mass consumption. This leads to the carbon mass consumption rate d dt m C formula:

d dt m C (t) = C m 4F I(t) (8)
where C m is the aluminum molar mass and 4 is the stoichiometric coefficient.

Hence, equation ( 3) can be rewritten using the above expressions:

d dt ACD(t) = 1 SF C e Al m 3ρ Al - C m 4ρ C I(t) + d dt BM (t) (9)
The beam movement derivative is considered as one of the system inputs (u 1 ):

u 1 (t) = d dt BM (t) (10)
Hence, by defining β as:

β = 1 SF C e Al m 3ρ Al - C m 4ρ C (11)
And using equation ( 11) in ( 9), the ACD dynamics results:

d dt ACD(t) = βI(t) + u 1 (t) (12)
Using Euler discretization in equation ( 12), it is possible to obtain a discrete-time model for the ACD:

ACD[n + 1] = ACD[n] + T s (u 1 [n] + βI[n]) (13
) where T s is the sampling time.

As it is not possible to measure the ACD during the operation of the plant, β cannot be obtained experimentally. Hence, this parameter is calculated using theoretical values.

Alumina Concentration

The dynamics of alumina concentration wAl 2 O 3 can be modeled as the difference between the quantity injected by the feeders (wAl 2 O 3in ) and the consumed by the chemical reaction (wAl 2 O 3cons ):

d dt wAl 2 O 3 (t) = wAl 2 O 3in (t) -wAl 2 O 3cons (t) (14)
The quantity injected by the feeders at time t can be represented by:

wAl 2 O 3in (t) = N m in M F (t)
where N is the number of feeders in the pot, m in is the amount of mass injected by feeders, M is the total bath mass, and F is the frequency of feeders. The alumina injected in the system does not dissolve instantaneously. This effect can be approximated by a time delay D introduced in equation ( 15):

wAl 2 O 3in (t) = N m in M F (t -D) (16)
The frequency of the feeders is considered as one of the system inputs (u 2 ):

u 2 (t) = F (t -D) (17)
Then, equation ( 16) becomes:

wAl 2 O 3in (t) = N m in M u 2 (t) (18)
The Al 2 O 3 consumption is given by the Faraday's law divided by the bath mass:

wAl 2 O 3cons (t) = Al 2 O 3m C e 6FM I(t) (19)
where Al 2 O 3m is the alumina molar mass and 6 is the number of electrons required for the electrolysis to perform.

Replacing equations ( 18) and ( 19) in ( 14):

d dt wAl 2 O 3 (t) = N m in M u 2 (t) - Al 2 O 3m C e 6FM I(t) (20)
And defining:

α 1 = N m in M , α 2 = Al 2 O 3m C e 6FM (21)
Equation ( 20) can be written as:

d dt wAl 2 O 3 (t) = α 1 u 2 (t) -α 2 I(t) (22)
Using Euler discretization in equation ( 22):

wAl 2 O 3 [n + 1] = wAl 2 O 3 [n] + T s (α 1 u 2 [n] -α 2 I[n]) (23)
Then, it is possible to obtain a discrete-time model for the alumina concentration. The next step is to identify parameters D, α 1 and α 2 .

The time-delay is estimated as a result of an unconstrained nonlinear optimization problem. The cost function is defined as the mean square error between the resistance measurements and simulated values from a model using ACD and wAl 2 O 3 values in many data sets. Then, the optimization searches the best time-delay to decrease fitting error.

Just a few alumina concentration measurements are recorded during an operational day without constant sampling time. Using the data collected at times

[τ 0 τ 1 • • • τ N ],
it is possible to organize equation ( 23) in a matrix form as follows:

(24)

    wAl 2 O 3 (τ 1 ) -wAl 2 O 3 (τ 0 ) wAl 2 O 3 (τ 2 ) -wAl 2 O 3 (τ 1 ) . . . wAl 2 O 3 (τ N ) -wAl 2 O 3 (τ N -1 )     = T s               τ1 k=τ0 u 2 (k) - τ1 k=τ0 I(k) τ2 k=τ1 u 2 (k) - τ2 k=τ1 I(k) . . . . . . τN k=τN-1 u 2 (k) - τN k=τN-1 I(k)               α 1 α 2
Based on this structure, it is possible to identify the parameters α 1 and α 2 by least square estimation. Fig. 3 shows a comparison between the model simulated with estimated parameters α 1 and α 2 and the alumina concentration values collected on the plant. The model is initialized with measurement and the simulation is started using the signals of u 2 and I. Every time a new alumina concentration measurement is available, the model is reinitialized to improve the accuracy of the prediction provided by the model. Following this approach, the mean absolute relative error between the model outputs and the measurement is 3.52 %.

Pseudo-Resistance

As for the pot pseudo-resistance, an approximate polynomial can be obtained from Fig. 2. Considering regular process operation in which alumina concentration varies between 2% and 4%, this model takes the following form:

(25) R(t) = cwAl 2 O 2 3 (t) + (d + eACD(t))wAl 2 O 3 (t) + (f + gACD(t))
where c, d, e, f , and g are some parameters to be determined. By differing the equation ( 25), it is possible to obtain a dynamic behavior model for the pseudoresistance:

d dt R(t) = eACD(t) d dt wAl 2 O 3 (t) + 2c d dt wAl 2 O 3 (t) + e d dt ACD(t) wAl 2 O 3 (t) + d d dt wAl 2 O 3 (t) + g d dt ACD(t) (26)
Using Euler discretization in equation ( 26), it is possible to obtain the resistance variation model:

R[n + 1] = R[n] + T s (e∆wAl 2 O 3 [n]ACD[n] + (2c∆wAl 2 O 3 [n] + e∆ACD[n])wAl 2 O 3 [n] + d∆wAl 2 O 3 [n] + g∆ACD[n]) (27)
where:

∆ACD[n] := ACD[n] -ACD[n -1] T s (28) ∆wAl 2 O 3 [n] := wAl 2 O 3 [n] -wAl 2 O 3 [n -1] T s (29)
Equations ( 28) and ( 29) can be respectively approximated by: ∆ACD

[n] ≈ u 1 [n] + βI[n] (30) ∆wAl 2 O 3 [n] ≈ α 1 u 2 [n] -α 2 I[n] (31)
Notice that equation ( 27) needs an initial ACD value and this values is not measurable. Indeed, only the ACD variation is calculated by equation ( 13). This problem can be solved by considering the initial ACD as an additional parameter to be identified.

For t = N T s , equation ( 27) turns into:

∆R(N T s ) = 2c∆wAl 2 O 3 (N T s )wAl 2 O 3 (N T s ) + d∆wAl 2 O 3 (N T s ) + e{∆wAl 2 O 3 (N T s )[ACD(0) + N k=1 T s [u 1 ((k -1)T s ) + βI((k -1)T s ))] + ∆ACD(N T s )wAl 2 O 3 (N T s )} + g∆ACD(N T s ) (32)
where:

∆R(N T s ) := R(N T s ) -R((N -1)T s ) T s (33)
Defining:

γ 1N := N k=1 T s [u 1 ((k -1)T s ) + βI((k -1)T s )) (34)
Equation ( 32) becomes:

(35) ∆R(N T s ) = 2c∆wAl 2 O 3 (N T s )wAl 2 O 3 (N T s ) + d∆wAl 2 O 3 (N T s ) + e{∆wAl 2 O 3 (N T s )[ACD(0) + γ 1N ] + ∆ACD(N T s )wAl 2 O 3 (N T s )} + g∆ACD(N T s ) Then, using N samples at [T s , 2T s , • • • N T s ]
, it is possible to organize equation (32) in a matrix form:

   ∆R(T s ) . . . ∆R(N T s )    =   
a 11 a 12 a 13 a 14 . . . . . . . . . . . .

a N 1 a N 2 a N 3 a N 4       c d + eACD(0) e g    (36)
where:

a 11 := 2∆wAl 2 O 3 (T s )wAl 2 O 3 (T s ) a 12 := ∆wAl 2 O 3 (T s ) a 13 := ∆wAl 2 O 3 (T s )γ 11 + ∆ACD(T s )wAl 2 O 3 (T s ) a 14 := ∆ACD(T s ) a N 1 := 2∆wAl 2 O 3 (N T s )wAl 2 O 3 (N T s ) a N 2 := ∆wAl 2 O 3 (N T s ) a N 3 := ∆wAl 2 O 3 (N T s )γ 1N + ∆ACD(N T s )wAl 2 O 3 (N T s ) a N 4 := ∆ACD(N T s )
Then, the parameters can be identified using a least square estimation algorithm. In Fig. 4, a comparison between the model output and actual measurements is shown. The mean absolute error computed for this data set is 2.21%. However, a small drift can be observed. This happens because the model structure from equation ( 27) reproduces the resistance variation and needs an initial value.

This difference between the model and the measurements accumulates because of the integrator dynamics inside of the model. Nevertheless, the parameter estimation can be considered valid since the mean absolute error is small. Based on the obtained models for ACD, wAl 2 O 3 , and R in equations ( 13), ( 23), and ( 27), respectively, and considering the current intensity as a measured disturbance in the system, it is possible to define a discrete-time state-space model of the plant:

(37)

       x[n + 1] =   1 a 12 [n] a 13 [n] 0 1 0 0 0 1   x[n] +   b 1 [n] b 2 [n] b 3 [n]   y[n] = 1 0 0 x[n]
where:

a 12 [n] = e(α 1 u 2 [n] -α 2 I[n]) (38) a 13 [n] = (2c(α 1 u 2 [n] -α 2 I[n]) + e(u 1 [n] + βI[n]) (39) b 1 [n] = (d(α 1 u 2 [n] -α 2 I[n]) + g(u 1 [n] + βI[n])) (40) b 2 [n] = (u 1 [n] + βI[n]) (41) b 3 [n] = (α 1 u 2 [n] -α 2 I[n]) (42
) which all the parameters known and the state vector is:

x[n] = x 1 [n] x 2 [n] x 3 [n] = R[n] ACD[n] wAl 2 O 3 [n] (43)
Notice that model ( 37) is affine in the state. This makes it possible to easily design an observer providing an estimate of the state x by using measurements of the resistance. To perform the state estimation, it is used a Time-Varying Linear Kalman Filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] as a state observer. The Kalman filter equations are given as follows: Prediction:

x

[n + 1] -= A[n]x[n] + B[n] P [n + 1|n] = A[n]P [n|n]A[n] T + Q noise Update: K[n] = P [n|n -1]C T (CP [n|n -1]C T + R noise ) -1 x[n] = x[n] -+ K[n](y[k] -C x[n] -) P [n|n] = (I -K[n]C)P [n|n -1]
where x is the state estimate vector, P is the covariance, Q noise is the process noise and R noise is the measurement noise matrices. Moreover, the matrices A, B and C come from the state affine system (37):

A[n] = 1 a 12 [n] a 13 [n] 0 1 0 0 0 1 (44) B[n] = [b 1 [n] b 2 [n] b 3 [n]] T (45) C = [1 0 0] (46)
To implement the observer, it is necessary to tune convariance and noise matrices. The initial covariance matrix P [0|0] is defined as:

P [0|0] =   10 -1 0 0 0 10 3 0 0 0 10 -1   (47)
The largest entry in the P matrix is related to the ACD.

As it is not possible to measure this state, we tune the covariance matrix to somehow assign more priority to the estimate of this state. The process and measurement noise matrices, Q noise and R noise respectively are chosen as:

Q noise =   10 -5 0 0 0 10 -5 0 0 0 10 -5   , R noise = 10 -5 (48)
The observer efficiency was tested in two data sets with different initial conditions in order to predict the next states measurements. For each test, it was used a different combination of ACD and wAl 2 O 3 initial values in a certain operational range, while the R values were initialized using the actual measurement. The resulting estimates are shown in Fig. 5 and 6, the solid line is the pot resistance measurement, the dashed lines are the states estimations and the "x" markers are the alumina concentration measurements.

In Fig. 5, it is possible to notice a fast convergence, for all estimate independently of the chosen initial condition.

Convergence can be verified from the final standard deviation indexes listed in Table 1. The standard deviation of the states R and wAl 2 O 3 are smaller than that of the ACD since they are experimentally validated. Fig. 6 shows the estimates provided by the observer for a long period and compare those with the measurements of wAl 2 O 3 . The values of the mean absolute error reported in Table 1 confirm the effectiveness of the proposed estimation strategy.

CONCLUSIONS

In this paper, the Hall-Héroult process is modeled by a state affine dynamic system that combines physicalchemical aspects and experimental data. Following this approach, we provide a Linear Kalman filter to estimate the states for the system.

Based on the obtained experimental results, it was possible to validate a state affine model for a complex chemical process. Moreover, the chosen observer strategy proved to be reliable by estimating with accuracy the desired states for unknown initial conditions.

  Fig. 3. Alumina Concentration Model Validation

Fig

  Fig. 4. Resistance Model Validation

  Fig. 5. States Estimations -Data Set 1

Table 1 .

 1 Estimations Final Values Comparison

		Standard deviation	Mean Absolute Relative Error
	R	2.91e-09 µΩ	0.09%
	ACD	0.0387 cm	-
	wAl 2 O 3	2.29e-06 %	6.75 %

ACKNOWLEDGEMENTS

The authors wish to acknowledge project PIANO funded by the French Fonds Unique Interministériel (FUI).