Thomas Burger
email: thomas.burger@cea.fr

Can omics biology go subjective because of artificial intelligence? A comment on "Challenges and Opportunities for Bayesian Statistics in Proteomics" by Crook et al

Keywords: Bayesian modelling, uncertainty representation, proteomics, mass spectrometry, statistical significance, artificial intelligence

In their recent review, 1 Crook et al. diligently discuss the basics (and less basics) of Bayesian modeling, survey its various applications to proteomics and highlight its potential for the improvement of computational proteomic tools. Despite its interest and comprehensiveness on these aspects, the pitfalls and risks of Bayesian approaches are hardly introduced to proteomic investigators. Among them, one is sufficiently important to be brought to attention; namely, the possibility that priors introduced at an early stage of the computational investigations detrimentally influence the final statistical significance.

Let us start with a toy example: As any p-value, the one associated to a protein differentially expressed (referred to as 𝑃𝑟𝑜𝑡) is counter-intuitive to interpret. [START_REF] Cohen | The earth is round (p<. 05)[END_REF] Thus, one may prefer the associated posterior error probability (PEP), reading:

PEP 𝑃𝑟𝑜𝑡 = 𝑃(𝐻0|𝑃𝑟𝑜𝑡) = 𝑃(𝐻0) 𝑃(𝑃𝑟𝑜𝑡)
⋅ 𝑃(𝑃𝑟𝑜𝑡|𝐻0) = 𝑃(𝐻0) 𝑃(𝑃𝑟𝑜𝑡) ⋅ pvalue 𝑃𝑟𝑜𝑡 PEP 𝑃𝑟𝑜𝑡 quantifies the probability that 𝑃𝑟𝑜𝑡 is not differentially expressed, which directly interprets in terms of risk for the confirmatory wet-lab experiments. Unfortunately, to compute it from the p-value, one needs the probability of the null hypothesis 𝑃(𝐻0), i.e., the prior probability that a random protein is stable. If based on his faith in the project, an investigator (called Archibald) assumes 10% of the proteins to be changers while only 1% of them truly are, their PEPs are altered. Therefore, in Archibald's publication, the PEPs will not reflect the risk that confirmatory experiments discard the putative biomarkers. Naturally, here the error lies in Archibald's overconfidence, so that he should be the one to blame instead of the statistical theory he inappropriately used. Now, let us consider another investigator (Bianca) who finely estimates the prior to best guide her future confirmatory targeted proteomics experiments. She decides to rely on her knowledge of the pathways involved in the disease studied: Proteins of those have a lower probability of being stable than others. Doing so leads her to isolate a handful of putative biomarkers (which unsurprisingly, belongs to the pathways promoted by the prior). She will investigate them thanks to future grants, and meanwhile she prepares her manuscript about the discovery experiments. To compute the statistical significance values demanded by the journal, she naturally relies on the PEPs she computed and used. In other words, she does the same mistake as Archibald, which is to confuse how she deals with the risk/benefit balance along her investigations and the statistical significance of the very experiment she publishes. However, Archibald's mistake is easier to spot and arguably imputable to a data-dredging attempt, while on the contrary, Bianca's one is probably involuntary and certainly more difficult to identify. Finally, despite Bianca being apparently more scrupulous than Archibald, her approach is not immune to selfconfirming biases (the pathways she promoted in the prior will provide significance values that will attract subsequent investigations).

While any statistical model can be used more or less correctly, we defend that Bayesian modeling, despite its power and versatility, can be a fertile ground for mistakes like Bianca's one. To understand why, a small historical detour about the interactions between biology, statistics and artificial intelligence (AI) is necessary.

In life sciences, statistics have long occupied a post-hoc quality control role, i.e., the objectivation that the discoveries claimed were not the result of luck. Because of the cost of wet-lab experiments or of clinical assays, preliminary statistical design has also thrived, but its role has mainly been to limit the risk of insufficiently significant statistics at the final stage. Therefore, until recently, biostatistics have essentially followed a goal of objectivity, in sharp contrast with the wet-lab investigations where the scientist's intuition, knowledge and skills, intrinsically subjective and personal, could act out. However, with the advent of omics biology, massive data are daily produced by high-throughput technologies, and computational biologists rely on multiple tools based on AI or statistics to correct batch effects, to profile patients or proteins (possibly undergoing imputations due to missing measurements), to visualize interactions with networks, and to build predictive models. In other words, this knowledge extraction process shares many formulae with significance computations, yet it is conceptually closer to the wet-lab investigation process: the way one makes the data speak along the course of a complex and iterative AI-aided process is partly subjective and does not share the objectivity aim of statistical significance.

The reason why statistical tools can be used both for significance assessment and for "subjective" inferences lies in the underlying probability theory. In the everyday language, a 0.5 probability has two meanings. It can either tell that success and failure, despite random, are equally frequent (e.g., head or tails game); or that in absence of sufficient information, the decision is random from the decision maker viewpoint, despite a possibly deterministic outcome (e.g., in which of my hands is the coin hidden?). This distinction between frequentist 3 and subjective [START_REF] De Laplace | Théorie analytique des probabilités[END_REF] views has been almost as old as probability theory itself, [START_REF] De Moivre | The Doctrine of Chances: or, a method for calculating the probabilities of events in play[END_REF] and while its implications have long been discussed in economy and game theory [START_REF] Jeffreys | The theory of probability[END_REF][START_REF] Walley | Statistical reasoning with imprecise probabilities[END_REF] or in AI, [START_REF] Pearl | Probabilistic reasoning in intelligent systems: networks of plausible inference[END_REF][START_REF] Fagin | Reasoning about knowledge[END_REF] they are scarcely so in omics biology.

As the natural tool to model the apparent random nature of the physical world, the probability of a given outcome should amount to its frequency of observation on the long run. [START_REF] Mises | Probability, statistics, and truth[END_REF] Such "frequency probabilities" have been instrumental to the development of statistical analysis, starting from demography [START_REF] Poisson | Recherches sur la probabilité des jugements en matière criminelle et en matière civile: précédées des règles générales du calcul des probabilités[END_REF] and progressively extending to a variety of application domains such as quality control, [START_REF] Student | The probable error of a mean[END_REF] insurance, etc. and of course, evidence-based medicine. Nowadays, a clinical trial is essentially a way to refine the frequencies of false positives and negatives associated to a given biomarker or drug. Indeed, both patients and policy makers would not accept that undesirable effects or false diagnostics occur at a higher frequency than originally claimed.

Another early application of probability theory has been gamble. However, gambling can also target unique events, for which frequencies of occurrences are either unavailable or unreliable (e.g., stock exchange crashes, football competitions, elections). To do so, partial information, intuition, and possibly emotions can be leveraged (e.g., accounting for the popularity of a football player, who could attract confident bets despite recent deceiving results). To this end, other theories have blossomed, where a probability is essentially a degree of belief in the decision maker's mind. [START_REF] Cox | Probability, frequency and reasonable expectation[END_REF][START_REF] De Finetti | La prévision: ses lois logiques, ses sources subjectives[END_REF] Such probabilities are called "subjective", "epistemic" or "Bayesian", as they are specific to a given state of knowledge (possibly incomplete or partially wrong).

Finally, probabilities have supported the development of two approaches: First, the observation-based modelling of a world submitted to randomness; and second, the digital encoding and combination of multiple state of knowledge for automated reasoning and AI. With Crook and his coauthors, we likely agree on several points: First, their review clearly makes the distinction between frequentist and Bayesian probabilities. Second, frequentist modelling, like the Bayesian one or any other one, comes with limitations, so that hierarchizing them is pointless. Third, epistemic probabilities are more general than their frequentist counterpart, as it is always possible to build one's "subjective" viewpoint thanks to preliminary frequency estimations. Consequently, any epistemic framework is especially interesting for computational biologists, who rely on an expertise that is missing to the general-purpose data scientist to blend the results of an "objective" experiment (e.g., differential expression analysis) with more "subjective" background knowledge (e.g., the pathways most likely involved) as to propose new hypotheses, as ambitioned by Bianca. In this context, Bayesian methods are powerful and they deserve a better exposure in the proteomics community, a feature accomplished by Crook et al. Nevertheless, casting everything in a subjective formalism because of its larger conceptual genericity can be a double-edged sword: using a same language for AI-aided reasoning and for statistical significance analysis increases the risk of mating the latter with the former, as exemplified with Bianca's mistake. More precisely, when a protein is deemed as a putative biomarker because of its significant differential abundance, the associated significance value is classically understood in a frequentist context, for cultural reasons: Biologists have learnt to split the "subjective" investigation part from the "objective" (and essentially technical) significance computation one. Contrarily, for IA (including Bayesian) experts, there is a continuum from data modelling to decision making and their skill should express along this entire process, possibly using advanced tools (which description here would be off-topic) that make it possible to finely calibrate their priors as well as the subsequent significance values, to finally exhibit the best knowledge inference model possible. However, for computational biologist, it is probably more customary and safer to keep a clear albeit artificial distinction between the tools used for subjective AI-aided reasoning and those for statistical significance analysis.

To do so, specific tools are already available, as many researchers in AI have long identified that one weakness of Bayesian modelling lies in its single way of handling the various facets of uncertainty. [START_REF] Dubois | Representing partial ignorance[END_REF] Although it is probably the reason of its success (any combination of subjective and frequentist probabilities yields a subjective probability, which can be subsequently combined, and so on) it is also its Achille's heel in an applicative context that facilitates confusions. To cope for this, multiple epistemic probability-like theories have developed in the last decades, where the [0;1] interval is used to encode different type of knowledge: Fuzzy set theory, [START_REF] Zadeh | Fuzzy sets[END_REF] Possibility theory, [START_REF] Dubois | Possibility Theory[END_REF] Dempster-Shafer theory, [START_REF] Shafer | A mathematical theory of evidence[END_REF] etc. Essentially, they propose to make an explicit difference between the uncertainty resulting from randomness (classically captured by frequencies) and the one resulting from the investigator's subjectivity (possibly imprecise, partial, doubtful, etc.). In practice, the knowledge of equally frequent events in a head and tails game and the ignorance about which of the two hands holds the hidden coin are no longer encoded in the same way. Instead of focusing on the same gamble they lead to, one focuses on the different states of knowledge they correspond to, as to better incorporate them in an automated line of reasoning. In theory, relying on these frameworks should authorized more refined modeling. However, from the computational biologist's practical viewpoint, their application is hurdled by their complexity.

As relying on advanced Bayesian tools or on alternative frameworks for subjective inference are equally demanding in terms of theoretical skills, a third path has witnessed a growing popularity in computational biology, namely, empirical Bayes approaches. [START_REF] Carlin | Bayesian methods for data analysis (3 rd Edition)[END_REF] Their central dogma is that priors should not be elicited by experts, as doing so would lead to too subjective biases that are incompatible with sound statistical significance. On the contrary, if the prior distributions are trained from the data, the final statistics can be expected to better generalize to new data. This is practically confirmed by the multiple empirical Bayes methods that are acknowledged in the state-of-the-art (e.g., Limma). [START_REF] Ritchie | limma powers differential expression analyses for RNA-sequencing and microarray studies[END_REF] Despite, their incorrect use is not immune to opening a breach for subjective spoiling of statistical significance. Notably, if the prior parameters are estimated on the very same data as those subsequently processed, one increases the risk of reinforcing some data specificities (a.k.a., overfitting, another form of self-confirming bias). Let us illustrate this on the same PEP example, yet applied to Peptide-Spectrum Match (PSM) validation. Assuming a classical PSM score reading 𝑆 𝑃𝑆𝑀 = -10 ⋅ log 10 (pvalue 𝑃𝑆𝑀 ), where pvalue 𝑃𝑆𝑀 is the p-value resulting from testing the match between the theoretical and empirical peak values in fragmentation spectra, it is possible to define the following PEP, which accounts for the peptide's length 𝐿:

PEP 𝑃𝑆𝑀 = 𝑃(𝑚𝑖𝑠𝑚𝑎𝑡𝑐ℎ|𝑃𝑆𝑀, 𝐿) = 𝑃(𝑚𝑖𝑠𝑚𝑎𝑡𝑐ℎ|𝐿) 𝑃(𝑃𝑆𝑀|𝐿) ⋅ pvalue 𝑃𝑆𝑀 Doing so is insightful for two reasons: First, as the previous PEP, it easily interprets in terms of mismatch probability. Second, it purposely offers to account for the fact that random matches are not equally probable on shorter and longer amino acid sequences, for combinatory reasons. In an empirical Bayes setting, the ratio 𝑃(𝑚𝑖𝑠𝑠𝑚𝑎𝑡𝑐ℎ|𝐿)/𝑃(𝑃𝑆𝑀|𝐿) can easily be "trained" from the data at hand: One essentially has to bin the amino acid sequences of the searched database according to their length. However, doing so can lead to unintentional significance dredging. Notably, if the searched database is too small, it may only contain a handful of very long peptides, with pairwise distinct lengths. Then, as each length 𝐿 is specific to a peptide, 𝑃(𝑚𝑖𝑠𝑚𝑎𝑡𝑐ℎ|𝐿) = 0, so that the PEP will be nil too, regardless of pvalue 𝑃𝑆𝑀 , as sometimes reported by MaxQuant's users. [START_REF] Couté | Beyond Target-Decoy Competition: Stable Validation of Peptide and Protein Identifications in Mass Spectrometry-Based Discovery Proteomics[END_REF] In other words, whatever the quality of the match between the fragmentation spectrum and its theoretical counterpart, the PSM will be validated because the empirical prior had overfitted the data.

Finally, whatever the AI framework (i.e., classical Bayesian, empirical Bayesian, or even other subjective ones), the risk of mating the statistical significance with more subjective information exist and will increase, as inherent to the computational biology approach, and as a consequence of the ever-growing workflow complexity. Naturally, as in any scientific field, the evil is in the details, and the main safeguards we can think of are the scientists' rigor and integrity. Moreover, as sketched by Crook et al. in their review, Bayesian theory is rich and many advanced tools are available to refine the modelling. As for non-intentional dredging (i.e., Bianca's error), we propose two guidelines: First, whenever possible, keep separated computational workflows for biological knowledge inference and for statistical significance computation. Second, if it is not possible to separate them, provide an explicit formulation (both in mathematical and in everyday languages terms) of the prior knowledge incorporated in the AI-aided reasoning, as to subsequently foster a critic eye on possibly too optimistic biological conclusions.

Acknowledgement:

The author is grateful to Oliver Crook and to the anonymous reviewers for their constructive comments.

Funding: This work was supported by grants from the French National Research Agency: ProFI project (ANR-10-INBS-08), GRAL project (ANR-10-LABX49-01) and MIAI @ Grenoble Alpes (ANR-19-P3IA-0003).


Krzysztof Talaśka
email: krzysztof.talaska@put.poznan.pl

Dominik Wojtkowiak

Wilczy Dominik

Antoine Ński

Ferreira

Dominik Wilczyński

Antoine Ferreira
email: antoine.ferreira@insa-cvl.fr

Computational Methodology for Drug Delivery to the Inner Ear using Magnetic Nanoparticle Aggregates

Keywords: Magnetic nanoparticles, aggregation design, drug delivery, inner ear, targeted therapy, Lagrange -Euler FEM analysis

Computational methodology using Finite Element Modeling of the process of magnetic nanoparticles (MNPs) transport through the biological barrier (RWM)of the round windows membrane of inner ear. these investigations. Presented research confirms the definite usability of the Lagrange-Euler method for the precise determination of the required magnetic force value FM to control the accelerated motion of MNP aggregates of complex shapes through RWM. It is possible to determine the predominant parameters with a precision of less than 5% for single-layer aggregates and spatial aggregates crossing the RWM.

It can be concluded that the MNPs and their aggregates should not be larger than 500 nm -750 nm to cross the RWM with high velocities of penetration close to 800 nm/s for magnetic forces of hundreds 10 -14 Newtons.

Conclusions:

The proposed Lagrangian-Eulerian approach is capable of accurately predicting the movement parameters of MNP aggregates of irregular shape that are close to the experimental test cases. The presented method can serve as a supplementary tool for the design of drug delivery systems to the inner ear using MNPs.

HIGHLIGHTS

 The main goal of the proposed study is to improve the efficiency of the ear treatment by targeting drug delivery to the inner ear, namely the cochlea.

Although drugs have been proposed to prevent damage or restore functionality to hair cells, a difficulty with such treatments is ensuring adequate drug delivery to the cells. To do so, we present a methodology for the evaluation of the magnetic forces needed to move magnetic particle nanorobots (termed MNPS) and their aggregates through the cochlea round window membrane (RWM). The presented finite element method can be an additional tool during the design of drug delivery systems to the inner ear using MNPs.

Abstract

Background and Objective: The main goal of the proposed study is to improve the efficiency of the ear treatment via targeted drug delivery to the inner ear, i.e. the cochlea. Although pharmacotherapy has been proposed as a solution to prevent damage or restore functionality to hair cells, the main challenge in such treatments is ensuring adequate drug delivery to the cells. To this end, we present a methodology for the evaluation of the magnetic forces needed to move magnetic particle nanorobots (abbreviated as MNP) and their aggregates through the cochlea round window membrane (RWM).

Methods:

The FEM -Lagrangian-Eulerian approach (Abaqus software) was used to determine the specific parameters of movement of the nanoparticles crossing the RWM. This method results in a high consistency of FEM simulations and in-vivo experimental results in regards to the required magnetic force during the movement of spherical nanoparticles with a given viscosity ηave. Based on the analysis of the experimental studies found in subject literature, the sizes of the MNPs and their aggregates able to cross RWM with and without the application of magnetic force FM have been determined.

Results:

The present work accounts for both the experimental and theoretical aspects of

Introduction

Treating diseases of the inner ear is of particular interest for researchers, mainly due to the difficulty in delivering systemic drugs to the targeted location [START_REF] Guigou | Use of Super Paramagnetic Iron Oxide Nanoparticles as Drug Carriers in Brain and Ear: State of the Art and Challenges[END_REF]. These problems derive from the presence of natural anatomical and physiological barriers [START_REF] Shapiro | Shaping magnetic fields to direct therapy to ears and eyes[END_REF]. This manifests itself particularly through the limited exchange of substances across the blood-perilymph barrier as well as the exchange between the plasma and the fluids of the inner ear [START_REF] Mittal | Nanoparticle-based drug delivery in the inner ear: current challenges, limitations and opportunities[END_REF]. Modeling techniques can lead to an improved design of fine, magnetic swimmers able cross the biological barriers and exert active control over the delivery of drugs to the inner ear. An analysis of the inner ear structure, in particular the cochlea, shows the following physical barriers separating the fluid in the cochlea from the inner ear and the circulatory system are: the oval window (OW), the blood inner ear barrier, and the round window membrane (RWM) [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF]. What follows from such naturally occurring limitations for the delivery of pharmaceuticals to the cochlea, it is important to explore intelligent and non-invasive systems of targeted drug delivery to the desired location and in the exact amounts [START_REF] Fisher | Micro and nanoparticle drug delivery systems for preventing allotransplant rejection[END_REF]- [START_REF] Komaee | Feedback Control for Transportation of Magnetic Fluids With Minimal Dispersion: A First Step Toward Targeted Magnetic Drug Delivery[END_REF].

The current repertoire of drug delivery techniques include application either through the middle ear, to be deposited near the round window (intratympanic delivery), reliance of diffusion through the round window to reach the inner ear, or direct injection via the round window into the cochlea (intracochlear delivery), or via propulsion utilizing ultrasonic waves [START_REF] Sumner | Steady streaming as a method for drug delivery to the inner ear[END_REF][START_REF] Liao | Ultrasound-induced microbubble cavitation via a transcanal or transcranial approach facilitates inner ear drug delivery[END_REF]. In the case of intratympanic delivery, a variety of methods are used in order to maintain contact with the membrane.

Hydrogels can be used as carriers for drugs or micro/nanoparticles in order to extend contact time with the oval or round windows, as well as combinations thereof in the so-called ferrogels which can be manipulated through the use of magnetic fields. The most popular techniques employ nanoparticles as drug carriers [START_REF]Modeling of Optimal Targeted Therapies using Drug-Loaded Magnetic Nanoparticles for the Liver Cancer[END_REF]. Nanoparticles ranging from 10 nm to 200 nm in diameter can serve as an effective drug carrier [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF], [START_REF] Chen | Disposition of nanoparticle-based delivery system via inner ear administration[END_REF]- [START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF]. Placed near or directly on the surface of the physical barrier: RWM or OW, they are able to penetrate the barrier via the phenomenon of diffusion [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF], [START_REF] Chen | Disposition of nanoparticle-based delivery system via inner ear administration[END_REF]- [START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF]. RWM is often used for analyzing the feasibility and potential of drugs delivery [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF]. For the most part, this is the result of the manifested permeability of RWM [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF], [START_REF] Chen | Disposition of nanoparticle-based delivery system via inner ear administration[END_REF]- [START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF]. Out of all types of nanoparticles used to carry drugs to the cochlea, the methods employing magnetic nanoparticles (MNPs), also known as supermagnetic iron oxide NPs (SPIONs) show excellent potential for further development and application [START_REF] Shapiro | Shaping magnetic fields to direct therapy to ears and eyes[END_REF][START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF]. An external magnetic field enables to control the direction and velocity of MNPs in order to guide them to the desired location [START_REF] Mahmood | A Novel Design of an MPI-Based Guidance System for Simultaneous Actuation and Monitoring of Magnetic Nanoparticles[END_REF]. Depending on the MNP size, they can be channeled through blood vessels [START_REF] Tehrani | A Novel Scheme for Nanoparticle Steering in Blood Vessels Using a Functionalized Magnetic Field[END_REF][START_REF] Forbes | Validation of High Gradient Magnetic Field Based Drug Delivery to Magnetizable Implants Under Flow[END_REF] or through selected body tissues [START_REF] Cui | Enhanced intracellular delivery and controlled drug release of magnetic PLGA nanoparticles modified with transferrin[END_REF]. It is important that the interaction of the MNPs and the external magnetic field can greatly accelerate the movement of the MNPs in its current medium [START_REF] Du | Magnetic Targeted Delivery of Dexamethasone Acetate Across the Round Window Membrane in Guinea Pigs[END_REF]. Test results demonstrate a decided increase in the concentration of MNPs inside the cochlea (approximately 330%) under an active magnetic field in comparison to experiments carried out without the presence of an external magnetic field [START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF]. Scientific research of the last years confirms that this method is potentially useful for drug delivery to the cochlea [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF][START_REF] Thaler | Visualization and analysis of superparamagnetic iron oxide nanoparticles in the inner ear by light microscopy and energy filtered TEM[END_REF]. An increasing number of articles present the results of experimental in vivo and in vitro studies in this area. They all clearly emphasize the advantage of magnetically active drug carriers (MNPs) in comparison to other types of NPs [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF][START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF][START_REF] Du | Magnetic Targeted Delivery of Dexamethasone Acetate Across the Round Window Membrane in Guinea Pigs[END_REF]]- [START_REF] Thaler | Visualization and analysis of superparamagnetic iron oxide nanoparticles in the inner ear by light microscopy and energy filtered TEM[END_REF]. In addition to the experimental studies involving the methods of measuring movement time of MNPs through RWM, the visualization of this process, the determination of the magnetic field strength, the estimation of the magnetic push or pull force, a continuous pursuit of methods of designing systems enabling the delivery of MNPs to the cochlea is called for [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF][START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF][START_REF] Du | Magnetic Targeted Delivery of Dexamethasone Acetate Across the Round Window Membrane in Guinea Pigs[END_REF]]- [START_REF] Thaler | Visualization and analysis of superparamagnetic iron oxide nanoparticles in the inner ear by light microscopy and energy filtered TEM[END_REF].

Advanced simulation methods based on the finite element method (FEM) can be of definite use in this matter, which already has been recognized in the research area covering MNPs delivery systems as well as the modelling of the inner ear, the cochlea and RWM [START_REF] Talaśka | An Approach to Identifying Phenomena Accompanying Micro and Nanoparticles in Contact With Irregular Vessel Walls[END_REF]- [START_REF] Gylienė | Vibrational and numerical evaluation of human incus mechanical properties[END_REF]. We would like to propose a new and previously unused approach to determining the design parameters of the delivery system of the MNPs to the cochlea based on the numerical simulation analysis. The proposed FEM simulations are based on the Lagrangian-Eulerian approach, verified with selected results of experimental studies, which is useful for determining the value of the magnetic force FM needed to accelerate the movement of MNPs in a location such as the RWM tissue. Thus it becomes possible to determine the time necessary to move individual MNPs and their aggregates as well as to determine the relationship between the magnetic force FM and the velocity v of the MNPs in the design process. This approach can be a successful complement of the research toolkit at the stage of designing and planning the application of MNPs in a new operating area.

Materials and Methods

When designing the delivery system for MNPs for transportation through RWM, several aspects call for a thorough analysis. Primarily, it is necessary to consider the complex structure of the RWM, the factors which affect its permeability and the complex phenomenon of aggregating individual MNPs in larger concentrations of particles for efficient drug delivery.

The research method presented in the present work is based on the use of numerical simulation methods. The simulation analyses conduced in this work employ the commercial software ABAQUS/Explicit. In order to develop the model for RWM, MNPs and their aggregates, the Lagrangian-Eulerian method was used. This method is used predominantly in the various analyses of the interaction between structural objects and liquids [START_REF] Talaśka | An Approach to Identifying Phenomena Accompanying Micro and Nanoparticles in Contact With Irregular Vessel Walls[END_REF]. Its distinctive characteristics include the possibility to precisely calculate the forces acting on an object through a surrounding body as well as the ability to determine the object movement parameters within another body, making this method particularly usable for the analysis considering the interests presented in this work.

For detailed information about the presented method, please refer to [START_REF]Abaqus Documentation[END_REF].

The anatomy of RWM

From the structural point of view, RWM is a three-layered membrane. From the side of the middle ear, one can distinguish the epithelium cell layer, the inner middle connective tissue, and from the side of perilymph -the inner layer of the epithelial cell layer [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF][START_REF] Pyykkö | Nanoparticle based inner ear therapy[END_REF]. Figure 1 shows a schematic view of the RWM structure. The middle layer is the most complex one. Structurally, it comprises fibrocytes, collagens, microfibrils, elastin and nerve ends [START_REF] Pyykkö | Nanoparticle based inner ear therapy[END_REF]. The two outer layers consist of epithelial cells. The RWM thickness in mammals examined in the course of the experimental studies: a guinea pig, a rat, and a chinchilla varies from a few to dozens of μm [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Kechai | Recent advances in local drug delivery to the inner ear[END_REF]. In humans, the RWM thickness varies from about 40 μm in the vicinity of its center, to about 70 μm on the fringes [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Kechai | Recent advances in local drug delivery to the inner ear[END_REF]. Importantly, the structure and the anatomy of the RWM in animals is similar to humans [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF]. This makes it possible to transfer the results of animal research to applications involving the human inner ear. The shape of the human RWM is close to circular, slightly elliptic: with the longer diameter equal to about 2.3 mm (1.8-3.2 mm) and the shorter diameter approximately 1.9 mm (1.4-2.4 mm) [START_REF] Nomura | Otological significance of the round window[END_REF].

Steering MNPs across the RWM

RWM is characterized by semi-permeability [START_REF] Pyykkö | Nanoparticle based inner ear therapy[END_REF][START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. It has been demonstrated that a number of substances as well as miniature objects are able to permeate RWM [START_REF] Kechai | Recent advances in local drug delivery to the inner ear[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. These include: sodium ions, antiseptics, antibiotics, toxins, and, interestingly, even latex spheres with a diameter of about 1 μm [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. This is of key importance considering the drug delivery strategies employing MNPs. It has been confirmed that NPs placed on the outer surface of the RWM are able to permeate it crosswise in a specific time [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF][START_REF] Pyykkö | Nanoparticle based inner ear therapy[END_REF][START_REF] Kechai | Recent advances in local drug delivery to the inner ear[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. The effectiveness of RWM permeability by different substances as well as NPs is influenced by many factors including size, configuration, concentration and electrical charge of the substance and the NPs. In addition, the thickness and the condition of RWM should be taken into account [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF][START_REF] Zou | Efficient penetration of ceric ammonium nitrate oxidant-stabilized gammamaghemite nanoparticles through the oval and round windows into the rat inner ear as demonstrated by MRI[END_REF]. Considering the three-layered nature of RWM, the outer layers will provide the greatest resistance to the NPs permeation through RW [START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. There are two ways of movement of NPs through the layers of epithelium cells: transcellular and paracellular transportation [START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF][START_REF] Zou | Efficient penetration of ceric ammonium nitrate oxidant-stabilized gammamaghemite nanoparticles through the oval and round windows into the rat inner ear as demonstrated by MRI[END_REF]. Figure 2 shows a schematic illustration of the two ways of NPs transportation through the layer of epithelium cells. When it comes to the middle layer, the NPs will move between the component structures (e.g. collagens, microfibrils, elastin). Figure 3 shows the schematic of the NPs movement process through the middle connective layer. NPs placed on the outer surface of the RWM will permeate it over time, primarily via the phenomenon of diffusion [START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. At such a small object scale (20-100 nm) Brownian motion will also be significant [START_REF] Uma | Nanoparticle Brownian motion and hydrodynamic interactions in the presence of flow fields[END_REF], naturally assisting in the general phenomenon of diffusion. MNPs and their aggregates will move through RWM via paths offering the lowest mechanical resistance. Particular tissues can be penetrated by single MNPs and their aggregates up to specific sizes. Table 1 collects the information about the size (hydrodynamic diameter) of nanoparticles and their aggregates, which are able to cross the RWM barrier after being applied to its surface.

The works [START_REF] Hoshiar | Studies of aggregated nanoparticles steering during magnetic-guided drug delivery in the blood vessels[END_REF][START_REF] Georgieva | Surface characteristics of nanoparticles determine their intracellular fate in and processing by human blood-brain barrier endothelial cells in vitro[END_REF] demonstrate that MNPs with a hydrodynamic diameter of up to 200 nm are able to cross e.g. epithelium cells. The study [START_REF] Zou | Size-dependent passage of liposome nanocarriers with preserved post transport integrity across the middle-inner ear barriers in rats[END_REF] provides results which indicate the clear influence of the NPs size of and the ability to penetrate RWM (diameter of NPs -95 nm to 240 nm). Studies determining the size of the individual NPs can be used as grounds for arguing the size of the aggregates able to cross the RWM. In the conclusion to the work [START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF], it has been shown that NPs measuring from 10 to 640 nm are able to cross RWM. The work [START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF] presents the results of the experimental studies of movement of MNPs 300 nm in diameter through RWM. It has been clearly stated that there is an increase in the velocity of crossing RWM when using the external magnetic field. The study [START_REF] Konstantinova | Nano-TiO2 penetration of oral mucosa: in vitro analysis using 3D organotypic human buccal mucosa models[END_REF] demonstrates that epithelium cells can be penetrated by NP aggregates sized 200 -400 nm. Additionally, it has been noted that aggregates of this size, most likely are coming from the layers of epithelium cells via paracellular transport. The work [START_REF] Thaler | Visualization and analysis of superparamagnetic iron oxide nanoparticles in the inner ear by light microscopy and energy filtered TEM[END_REF] demonstrates a clear influence of an external magnetic field acting on the MNPs sized 482.8 ± 158 nm, the MNPs crossed RWM within an hour, much faster than without the magnetic field. Furthermore, the works [START_REF] Hoshiar | Studies of aggregated nanoparticles steering during magnetic-guided drug delivery in the blood vessels[END_REF][START_REF] Do | Guidance of magnetic nanocontainers for treating alzheimer's disease using an electromagnetic, targeted drug delivery actuator[END_REF] state that MNP aggregates with the average diameter of 770 nm were able to cross the barrier of epithelium cells assisted by the external magnetic field. The abovementioned research results serve to confirm that objects less than 1000 nm in diameter, in particular below 750 nm, are able to cross RWM [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF][START_REF] Hoshiar | Studies of aggregated nanoparticles steering during magnetic-guided drug delivery in the blood vessels[END_REF][START_REF] Do | Guidance of magnetic nanocontainers for treating alzheimer's disease using an electromagnetic, targeted drug delivery actuator[END_REF]. Crossing RWM naturally (without additional stimuli to expedite the process) is time-consuming. It can last anywhere from a few to dozens of hours. Using MNPs as micro-swarm drug carriers has the potential benefit of accelerating this process [START_REF] Dormer | Magnetically-Targeted, Technetium 99m-Labeled Nanoparticles to the Inner Ear[END_REF][START_REF] Du | Magnetic Targeted Delivery of Dexamethasone Acetate Across the Round Window Membrane in Guinea Pigs[END_REF]- [START_REF] Thaler | Visualization and analysis of superparamagnetic iron oxide nanoparticles in the inner ear by light microscopy and energy filtered TEM[END_REF]. In essence, the operating mechanism is the external magnetic field generating an extra pushing or pulling force to move the MNPs.

Forces acting on individual MNPs and their aggregates

The force necessary to displace a solid within a liquid of specific viscosity (drag force) may be calculated from the following formula (Stokes equation) [START_REF] Gao | Magnetic Assisted Transport of PGA Nanoparticles Through a Human Round Window Membrane Model[END_REF]:

𝐹 𝑑 = 6𝜋𝑅 𝑀𝑁𝑃 η u (1)
where: Fd -drag force; RMNP -radius of MNP (or aggregate); 𝜂 -liquid viscosity; 𝑢 -velocity. The pushing or pulling force for a single MNP (magnetic nanoparticle) or their aggregate generated by magnetic field should be equal to the drag force if achieving a specific velocity of the MNPs is desired. The magnetic force can be calculated from the equation below [START_REF] Gao | Magnetic Assisted Transport of PGA Nanoparticles Through a Human Round Window Membrane Model[END_REF][START_REF] Shapiro | Control to Concentrate Drug-Coated Magnetic Particles, to Deep Tissue Tumors for Targeted Cancer Chemotherapy[END_REF]:

𝐹 𝑚 = 2𝜋𝑅 𝑀𝑁𝑃𝜇 0 3 3 • 𝜒 1+𝜒 3 ∇𝐻 2 (2)
where: Fm -magnetic force; 𝜇0 -permeability in free space; 𝜒 -volume susceptibility of the nanoparticle; H -magnetic field strength. As they move, nanoparticles will form aggregates in time. One of the main forces responsible for this phenomenon is the electrostatic force. The equation to calculate this force is provided, among others, in the work [START_REF] Gao | Magnetic Assisted Transport of PGA Nanoparticles Through a Human Round Window Membrane Model[END_REF]:

𝐹 𝑒 = 2𝜋𝜀 𝑟 𝜀 0 𝑅 𝑀𝑁𝑃 𝑠 2 κ𝑒 -κs 1+𝑒 -κs (3)
where: Fe -electrostatic force; 𝜅 -reciprocal double layer thickness; 𝜀𝑟, 𝜀0 -relative and static permittivity of water; s -particle-particle separation distance. In addition, one can define a separate magnetic force of attraction of two adjacent particles. The formula to determine the value of this force is provided, among others, in the works [START_REF] Gao | Magnetic Assisted Transport of PGA Nanoparticles Through a Human Round Window Membrane Model[END_REF][START_REF] Wassel | Dispersion of Super Paramagnetic Iron Oxide Nanoparticles in Poly(D,L-Lactide-co Glycolide) Microparticles[END_REF]:

𝐹 𝑚𝑑 = 24𝜋𝜇 0 𝑅 𝑀𝑁𝑃 6
where: Ms -saturation magnetization.

The resultant of these forces together with the actual viscosity of the RWM material affects the condition of movement of MNP aggregates; in particular this applies to the average velocity of their motion.

The processes of MNPs aggregation

The process of aggregation of MNPs, along with the shape and size of the individual MNPs has a decisive influence on their magnetic properties [START_REF] Rojas | Time-course assessment of the aggregation and metabolization of magnetic nanoparticles[END_REF]. These processes are very complicated at the level of detail. This is connected with the structure of MNPs with a magnetic core (e.g. Fe304) and its surrounding hydration layer [START_REF] Wang | Effects of monovalent and divalent metal cations on the aggregation and suspension of Fe3O4 magnetic nanoparticles in aqueous solution[END_REF]. Therefore, the process of aggregation is affected by two main groups of factors: chemical and physical (mechanical). The group of chemical factors includes the following features and phenomena: pH, valences, hydrated ionic radii, hydration layer and hydration forces [START_REF] Wang | Effects of monovalent and divalent metal cations on the aggregation and suspension of Fe3O4 magnetic nanoparticles in aqueous solution[END_REF]. On the other hand, the physical factors can include: magnetic attraction, electrostatic forces and specific ion effect of metal cations [START_REF] Wang | Effects of monovalent and divalent metal cations on the aggregation and suspension of Fe3O4 magnetic nanoparticles in aqueous solution[END_REF]. Drug delivery system designers employing MNPs try to eliminate as many factors affecting uncontrolled aggregation as possible, which may hinder MNP delivery. An example of such an action is coating the MNPs core with different substances. Such an approach can potentially eliminate the electrostatic attraction of MNPs, as was the case with dextran, for example [START_REF] Ch | Aggregation of dextran coated magnetic nanoparticles in aqueous medium: Experiments and Monte Carlo simulation[END_REF][START_REF] Jun | Chemical design of nanoparticle probes for high performance magnetic resonance imaging[END_REF]. Due to the fact that dextran is non-ionic, the determined zetapotential of such MNPs in a wide range of pH values is close to zero [START_REF] Ch | Aggregation of dextran coated magnetic nanoparticles in aqueous medium: Experiments and Monte Carlo simulation[END_REF].

Single MNPs tend to quickly form larger aggregations. First, combinations of two MNPs (dimers) and three MNPs (trimers) are formed. Subsequently, chains of MNPs are formed, eventually leading to the formation of complex structures of different shape and varying size [START_REF] Hoshiar | Studies of aggregated nanoparticles steering during magnetic-guided drug delivery in the blood vessels[END_REF][START_REF] Heinrich | Effects of magnetic field gradients on the aggregation dynamics of colloidal magnetic nanoparticles[END_REF][START_REF] Faraudo | Cooperative magnetophoresis of superparamagnetic colloids: theoretical aspects[END_REF]. The study [START_REF] Faraudo | Cooperative magnetophoresis of superparamagnetic colloids: theoretical aspects[END_REF] provides conclusions from an experimental observation showing that the length of the aggregate chain is limited. It was noted that when the chain length exceeds 14 particles, the lateral aggregation accelerates. On the basis of this conclusion, it can be assumed that this process leads to a more rounded, ellipsoidal, in the ideal case -spherical aggregate shape. 5). These analyses can be useful in determining the MES model parameters for the aggregates of relatively large size (less than 750 nm) which comprise MNPs of smaller sizes (e.g. 20 nm). It is therefore conclusive that controlling the parameters of IRWM and ID leads to satisfactory results. It is important insomuch as larger objects moving in areas with lower resistance to movement will utilize channels with a diameter not exceeding 1000 nm [START_REF] Mao-Li | Permeability of round window membrane and its role for drug delivery: our own findings and literature review[END_REF][START_REF] Goycoolea | Round Window Membrane. Structure Function and Permeability: A Review[END_REF]. Analyses employing the Lagrangian-Eulerian approach are very time consuming. Using them to calculate the total time required for the movement of MNPs through RWM would be impossible. However, such method can still be of use in the current approach. It is sufficient to determine the values of the specified parameters of analysis (force, velocity, viscosity, etc.) in a sufficiently short time, enabling the stabilization of these parameters during calculations. The larger the object (this applies to both individual MNPs and their aggregates), the longer it takes to stabilize the parameters. For MNPs with the diameter of DFE = 50 nm, the total time of analysis was tA = 5 ns, whereas the time needed for the stabilization of the force value FM-FEM was tPS = 2 ns. MNPs with the diameter DFE = 750 nm, required a total analysis time of tA = 150 ns, whereas the time required for the stabilization of the force value FM-FEM was tPS = 60 ns.

Development of the numerical FEM model

Results

The present work analyzes the formation of aggregates which can be divided into two groups. The first group includes single-layer aggregates consisting of between a few and a few dozen MNPs. The second group applies to aggregates with a large number of MNPs, forming complex, solid shapes. In the first section two approaches to modelling the aggregate shapes have been proposed. In the second section the concept of average mock viscosity ηave of RWM has been proposed together with determination of its value and demonstration of its utility.

Development of the MNP aggregate models

MNPs begin to form aggregates within a short time. The possible range of shapes is extremely varied. Considering their transportation through RWM, their maximum size will be limited by the size of the natural passageways through RWM. In accordance with the information in section 2.2, the maximum dimension does not exceed 1000 nm and the majority of the literature sources confirms that MNPs and their aggregates with dimensions not exceeding 750 nm facilitate the most efficient transportation.

Figure 5 shows the aggregate formation diagram. Assuming that the diameters of MNPs correspond to the hydrodynamic diameters of these particles, forming the aggregate entails them coming into contact. First, clusters of 2 and 3 particles are formed, and later they create more complex forms by merging together. First, one layer is formed (Figure 5e) and then more complex, solid structures are formed (Figure 5f). In the case of single-layer aggregates, it is possible to propose an algorithm for describing its shape. This is justified due to the simplification of assigning particular shapes of aggregates consisting of MNPs coated with specific substances which intend to arrest the process of aggregation. Consequently, small aggregates with simple shapes are formed, comprising between a few and a few dozen individual MNPs. The results of using such a substance has been presented in works [START_REF] Ch | Aggregation of dextran coated magnetic nanoparticles in aqueous medium: Experiments and Monte Carlo simulation[END_REF][START_REF] Jun | Chemical design of nanoparticle probes for high performance magnetic resonance imaging[END_REF]. Figure 6 presents a way of describing the shape of single-layer aggregates, which can be useful when modelling them for the purpose of the FEM model construction. For example, the designation For analyzing solid aggregates with complex shapes, the best solution is to build a three dimensional model that corresponds to the shape of the aggregate. Figure 7 shows the result of development of such a model based on microphotography [START_REF] Ge | Distribution of PLGA nanoparticles in chinchilla cochleae[END_REF].

There are therefore two scenarios for researching MNPs and their aggregates. In the case of simple single-layer structures, it is possible to use the proposed shape description algorithm. Whereas for more complex forms of solid aggregates, it is possible to develop their three dimensional models.

Determination of average mock viscosity ηave

Viscosity is the primary parameter of liquids. Its exhibits a very broad range of possible values, eg. The viscosity of water at 25°C is ηw = 0.89 mPa•s, and the viscosity of tar (25°C) is ηt =1•10 7 Pa•s. From the biological perspective, RWM is an object composed of materials with different characteristics. It is assumed that intracellular viscosity is approximately 1 mPa•s [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF][START_REF] Fushimi | Low viscosity in the aqueous domain of cell cytoplasm measured by picosecond polarization microfluorimetry[END_REF]. However, the viscosity of the other constituents of the connective tissue, e.g. loose collagen matrix, is much higher and ranges from 8-13 Pa•s [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF][START_REF] Chan | Viscosities of implantable biomaterials in vocal fold augmentation surgery[END_REF]. MNPs and their aggregates, provided their size is appropriate (diameter less than 750 nm), move through RWM along the path of the lowest mechanical resistance (lowest viscosity). Based on the results of experimental studies contained in the works [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF][START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF][START_REF] Rojas | Time-course assessment of the aggregation and metabolization of magnetic nanoparticles[END_REF] it is possible to determine the average mock viscosity of RWM as ηave. The [START_REF] Rojas | Time-course assessment of the aggregation and metabolization of magnetic nanoparticles[END_REF] concludes that the viscosity of RWM material should be at least 1000 times greater than the viscosity of water. The exact value range can be determined on the basis of the information about the size of MNPs and the time of movement through RWM. The [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF] determines the magnetic force value FM enabling MNPs of the specified size to travel through the model RWM of the guinea pig in the course of one hour. In [START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF], the required magnetic force FM value required for MNPs 300 nm in diameter to move through the RWM of the rat within one hour was determined. Table 6 summarizes the thickness values of RWM of the guinea pig and the rat available in various literature sources. The average thickness of the rat's RWM is approximately 13.75 μm, whereas for the guinea pig it is approx. 20 μm. According to [START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF], the magnetic force FM with a value in the range from 0.3 fN to 1.2 fN facilitates an effective acceleration of MNP movement for particle diameters of 300 nm. After one hour, a large number of MNPs which have successfully crossed the RWM was observed. Of course, the transportation time by different particles was not the same, but it can be assumed that 1h constitutes the average time that is required. The results of the work [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF] apply to MNP aggregates with a diameter from 50 nm to 130 nm crossing within 1h. The force FM needed to facilitate this process was 0.651 fN.

Assuming the thickness of the model RWM equal to the thickness of the guinea pig RWM, it is possible to determine the average mock viscosity of RWM as ηave. These results can then be verified using the results provided in [START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF]. Table 7 contains the predicted values of average mock viscosity ηave on the basis of the results of experimental studies presented in the works [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF][START_REF] Sarwar | Magnetic Injection of Nanoparticles Into Rat Inner Ears at a Human Head Working Distance[END_REF]. The viscosity values were derived using the Stokes law. After accounting for the extreme RWM thickness values, it can be assumed that the average mock viscosity ηave ranges between 0.0239 Pa•s and 0.1913 Pa•s.

. FEM model verification using the results of experimental studies

The work [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF] provides very interesting and valuable research results on the estimation of the magnetic force values FM generated by the proposed arrangement of magnets. The values of these forces were used to estimate the velocity of the vagr aggregates with a diameter of DA-MNP = 50-130 nm, derived from MNPs with the diameter of 10 nm. The velocity value was estimated for the movement of aggregates in perilymphatic fluid with assumed viscosity value of ηpf = 0.855 mPa•s. Table 8 shows the value of the magnetic force FM generated by individual magnets. It also provides the predicted velocities vagr used as a basis for the FEM analysis carried out for the aggregate with a diameter of DA-MNP = 130 nm. The work [START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF] assumes a close to spherical shape of these aggregates.

Based on the photograph of PLGA particles as presented in Figure 8, the FEM model of the aggregates (A and B) has been developed and the analysis of their velocity has been carried out with the assumed magnetic force value FM (magnet 3). Due to their irregular shape, velocities of the aggregates vagr will be different for different directions of movement. To find the difference in their values, the analysis has been conducted along the four main directions: x, y, DS and DG. Figure 9 shows the discretized models of MNPs aggregates A and B. It also indicates the directions relative to which the velocity of the aggregate movement has been studied. Furthermore, the velocity of the aggregates in each such direction has been given. Estimation of MNPs velocity in a medium with a specific viscosity based on the Stokes law applies only to spherical shapes. If it were assumed that the aggregates A and B (Figure 8) were spheres with the diameter DSA = 93 nm and DSB = 100 nm, their velocities in perilymphatic fluid at the force 0.651 fN (magnet 3) would equal to vA= 869 nm/s and vB = 808 nm/s. When analyzing the velocity value of the aggregates in Figure 9, it is easy to notice significant differences in velocity values depending on the direction of movement. It can be clearly concluded that assuming the spherical shape for the purpose of simplification can in many cases lead to measurement errors.

Single-layer and spatial aggregates

For the purpose of this study, aggregates formed of MNPs with a diameter DMNP = 50 nm have been chosen. is the significant increase in the calculation time as the viscosity value increases.

However, for a given shape at the assumed velocity, a linear increase in the value of viscosity appears to effect a linear increase in the value of the required magnetic force FM. Based on this assumption, it is possible to perform calculations for smaller viscosity values and re-calculate the value of the magnetic force FM for higher values. The presented results confirm the validity of the above mentioned assumption, because the ratio of the calculated values of magnetic forces FM does not deviate from the ratio of the average mock viscosity ηave by more than 5.5%. In addition, the value of magnetic forces FM has been calculated for the spatial aggregates in Figure 7. The parameters of the analysis were similar to single-layer aggregates. The calculation was performed for a single designated direction of movement. The obtained results confirm the feasibility of analyzing any aggregate shape (Figures 1415). When modelling the spatial aggregate, its faithful recreation is of crucial importance, both in the stage of creating a solid model, and afterwards when breaking an object into finite elements.

Discussion

The topic of precise drug delivery to the inner ear using MNPs is extremely complex.

RWM is characterized by semi-permeability, therefore objects with a diameter of less than 1000 nm are able to cross it. Based on the results of the research presented in subject literature, it can be concluded that in order to freely cross the blood-perilymph barrier, the MNPs and their aggregates should not be larger than 500 nm -750 nm.

Most scientists estimate the movement parameters of individual MNPs using the Stokes law. The major limitation of this method is the fact that that it only allows for the analysis of spherical objects. Figure 9 illustrates the scale of possible differences in the estimated velocity value of MNPs for shapes that are actually irregular. This is despite that fact that analyzed aggregates were close to spherical. Another issue was the viscosity value assumed during the analysis. An important aid when analyzing the processes of MNPs crossing the RWM is the proposed average mock viscosity ηave. In accordance with the presented calculations based on the results of the experimental research presented in selected publications, it can be concluded that the average mock viscosity ηave value is between 28 and 222 times higher than the viscosity of perilymphatic fluid. The method presented in this work makes it possible to precisely determine the movement parameters of magnetic aggregates of any shape in the medium with a specific viscosity. In connection with different ways of reducing the aggregation process, the aggregates can be analyzed in two ways. Single-layer aggregates can be accurately described using the proposed algorithm. This can be helpful in the case of MNPs coated with substances intended to arrest the process of aggregation. In turn, large spatial aggregates can be analyzed as solid objects with external dimensions corresponding to the dimensions of the hydrodynamic aggregate.

The determined value of the average mock viscosity ηave presented in this study can also be helpful in the simplified analysis of transport of MNPs through RWM.

Conclusion

Treatment of the inner ear disorders poses a challenge both due to anatomical and physiological barriers of the ear. The anatomic inaccessibility of the inner ear coupled with poor penetration through the blood-perilymph barrier and permeation of drug molecules through RWM results in sub-therapeutic concentrations of the drug at the site of action. In this work, we present a methodology for the evaluation of the magnetic forces needed to move magnetic particles and their aggregates through the cochlea round window membrane for efficient intracochlear drug delivery. To our knowledge the employment of the Lagrangian-Eulerian method to describe the penetration of magnetic nanoparticles guided by magnetic fields through the RWM has not yet been addressed. The main advantage of the proposed FEM methodology is that it allows the simulation of aggregates of particles with different shapes and sizes instead of only spherical objects. The analysis of the predictive magnetic force results complement the magnetically guided drug delivery experiments, both in-vitro and in-vivo, which were published earlier. These simulation findings hold a great potential to improve the penetration capabilities of MNPs used for drug delivery into the cochlea [START_REF] Leterme | Superparamagnetic Nanoparticle Delivery to the Cochlea Through Round Window by External Magnetic Field: Feasibility and Toxicity[END_REF], [START_REF] Rathnam | Developments in Bio-Inspired Nanomaterials for Therapeutic Delivery to Treat Hearing Loss[END_REF]. Further experimental work using the presented methodology could make it possible to further clarify the value of the average viscosity ηave with regard to RWM.

Tables Table 1: The sizes of NPs and their aggregates that are able to cross the barrier of RWM.

Figure 4

 4 Figure4presents a schematic overview of the FEM model development process. As a result of the additional force assisting the movement in RWM material (magnetic force FM), single MNPs, and more often their aggregates, move through areas with the smallest resistance to movement (lowest viscosity). Consequently, an approach has been proposed to determine the value of the required magnetic force FM effecting the movement of single MNPs and their aggregates in the medium with the specified viscosity value (average mock viscosity ηave). For the purpose of the present paper, it has been assumed that the resistance force during motion of MNPs with a specific velocity in the medium with specific characteristics can be balanced with the magnetic force FM. Therefore, once the resistance force value is known, it will be possible to determine the necessary parameters of the magnetic field to cause MNPs movement with a given velocity. In line with the assumptions, the material in which MNPs movement takes place has been modelled as an Eulerian body (element type EC3D8R). However, due to decisively higher rigidity in relation to the RWM material, MNPs have been modelled as a rigid Lagrangian body (element type C3D10M) for the purpose of FEM model calibration. The dimensional scale of the problem under consideration proves challenging for the development of a FEM model. The first step is to determine the effective size of the finite elements D FE in the model. Afterwards, it must be determined which dimension of the RWM sample α RWM in relation to the dimensional characteristic of the MNPs D FE should be taken into account in order to obtain the calculation results within a reasonable time. For the purpose of model calibration, the movement of individual MNPs in a medium with known propertiesthe perilymphatic fluid -has been modelled. The determined viscosity value is ηpf = 0.86 mPa•s [19]. The used MNP diameter is 50 nm. The assumed movement velocity is vM = 10 nm/s. The magnetic force value FM, determined in the FEM analysis and required to effect the movement of MNPs with the specified velocity, was compared with the force value determined on the basis of the Stokes law. Calibration consisted of thickening the FEM grid in order to achieve a more accurate result and selecting an

1L( 3 )- 5 -

 35 1R(1) indicates that we are dealing with a 3-layer aggregate. The middle layer is the most numerous, comprising 5 singular MNPs; next, the 1L(3) stands for 1 MNP to the left of the middle layer is attached to 3 MNPs. Whereas 1R(1) denotes 1 MNP on the right side, attached to the first MNP of the middle layer. For larger three dimensional aggregates, the proposed formulation becomes too complicated to employ. However, aggregates consisting of dozens of MNPs form solids of irregular shapes by combining and partially permeating hydrational layers. The beginning of this process is shown in Figure 6c. Figs. 6c, 6d and 6e demonstrate examples of actual MNP aggregates.

   show the obtained value of the magnetic force FM which is required to maintain the movement velocity vagr = 10 nm/s of selected aggregates in a medium with the average mock viscosity ηave from 0.0239 Pa•s to 0.1913 Pa•s. Such assumed velocity makes it possible to cross the rat RWM in less than 23 min, and less than 34 min for a guinea pig RWM. The value of the dimension indicator IRWM fluctuated close to 2 (relative to each dimension of the aggregate). The DFE value in each case was less than 15 nm. When analyzing the results shown in Figures10-14, it can be concluded that the proposed method of calculating the magnetic force value FM is effective for a wide range of viscosity values. The negative side of the employed method

FIGURESFigure 1 :

 1 FIGURES

Figure 2 :

 2 Figure 2 : Process of MNPs movement through the layers of epithelium cells: 1 -drop of gel with MNPs, 2 -epithelium cell -EC, TJ -tight junction, MNP -magnetic nanoparticle, A-MNPs -aggregates of magnetic nanoparticles, A -paracellular transport of MNPs, B -transcellular transport of MNPs, D -dimer, T -trimer.

Figure 3 :

 3 Figure 3 : The process of MNPs movement through the middle connective layer.

Figure 4 :

 4 Figure 4 : Diagram of the numerical FEM model: A -RWM sample, B -close up of a selected MNPs aggregate, C -FEM model, 1 -Eulerian body, 2 -Lagrangian body, FM -magnetic force enforcing movement, vagr -aggregate velocity, ηave -average mock viscosity.

Figure 5 :

 5 Figure 5 : Process diagram of aggregate formation from MNPs: a -e -single-layer aggregates, f -spatial aggregate.

Figure 6 :

 6 Figure 6: The description algorithm for a single-layer MNPs aggregate: a, b -model shapes, c -actual aggregate [49], d -actual spatial aggregate [49], e -actual aggregate [49].

Figure 7 :

 7 Figure 7: Development of the models mapping the shape of the MNPs aggregate: a -MNPs aggregate [48], b -solid model, c -MNPs aggregate [48], d -solid model.

Figure 8 :

 8 Figure 8: Microphotography of the PLGA particles [17]: DG -larger diameter, DS -smaller diameter.

Figure 9 :

 9 Figure 9: Discretized models of aggregates A and B.

Figure 10 :

 10 Figure 10: Value of the magnetic force FM for the aggregate 1L(3)-5-1R(1) in the function of the average mock viscosity ηave, arrow -direction of movement.

Figure 11 :

 11 Figure 11: Value of the magnetic force FM for the aggregate 1L(3-5)-7-1R(1-4) in the function of the average mock viscosity ηave, arrow -direction of movement.

Figure 12 :

 12 Figure 12: Value of the magnetic force FM for the aggregate 2L(1-2,4)-1L(3-5)-7-1R(1-4)-2R(1) in the function of the average mock viscosity ηave, arrow -direction of movement.

Figure 13 :

 13 Figure 13: Value of the magnetic force FM for the aggregate 2L(3,4)-1L(3-5)-7-1R(1-5)-2R(1,4)-3R(3) in the function of the average mock viscosity ηave, arrow -direction of movement.

Figure 14 :

 14 Figure 14: Value of the magnetic force FM for the aggregate from Figure 7a,b in the function of the average mock viscosity ηave, arrow -direction of movement.

Figure 15 :

 15 Figure 15: Value of the magnetic force FM for the aggregate from Figure 7c,d in the function of the average mock viscosity ηave, arrow -direction of movement.

Table 2 :

 2 Comparison of the magnetic forces obtained for various values of the dimension indicator ID, FM-S -magnetic force derived from the Stokes law, FM-FEM -magnetic force derived using FEM analysis.

	Dimension	Force FM-S	Force FM-FEM	Percentage	CPU
	indicator	[fN]	[fN]	difference	time [s]
	ID			[%]
	2	0.00405	0.0052	22.1	11.2
	5	0.00405	0.00417	2.81	25
	6.66	0.00405	0.00397	2.08	53.4
	10	0.00405	0.00394	2.86	204.4
	20	0.00405	0.00395	2.6	4479

Table 3 :

 3 Comparison of magnetic force values obtained for different dimension indicators IRWM, DMNP = 50 nm, DFE = 5 nm, ID = 10.

	Dimension	Force	Force	Percentage	CPU
	indicator	FM-S	FM-FEM	difference	time [s]
	IRWM	[fN]	[fN]	[%]
	3	0.00405	0.00366	9.69	66.7
	5	0.00405	0.00394	2.78	204.4
	7	0.00405	0.00407	0.43	524.5
	9	0.00405	0.0041	1.17	1157.4

Table 4 :

 4 Comparison of magnetic force values obtained for different dimension indicators IRWM, DMNP = 50 nm, DFE = 10 nm, ID = 5.

	Dimension	Force	Force	Percentage	CPU Time
	indicator	FM-S [fN]	FM-FEM [fN]	difference	[s]
	IRWM			[%]
	3	0.00405	0.00385	5.0	23.1
	5	0.00405	0.00417	2.9	25.0
	7	0.00405	0.00429	5.86	55.5
	9	0.00405	0.00433	6.84	89.6

Table 5 :

 5 Verification model information: FM-S -value of the magnetic force derived from the Stokes law, FM-FEM -value of the magnetic force derived using FEM analysis.

	DMNP	Force	Force	Percentage	I RWM	ID	CPU
	[nm]	FM-S [fN]	F M-FEM	difference			time [s]
			[fN]	[%]
	20	0.00162	0.00153	5.6	5	10	225.3
	50	0.00405	0.00394	2.8	5	10	204.4
	100	0.0081	0.00740	8.7	3	16.67	832.8
	250	0.0203	0.0194	4.7	3	16.67	1169.4
	500	0.0405	0.0396	2.3	2	33.33	6791.4
	750	0.0608	0.0597	1.8	2	39.47	10365.3

Table 6 :

 6 Thickness of RWM in rats and guinea pigs.

	Rat RWM	Source	Guinea pig RWM	Source
	thickness TR		thickness TGP [µm]
	[µm]
	12	[53,54]	10-30	[53,55]
	11.5	[56]	10	[25,58]
	15	[25,57]	25-30	[42]
	16	[28]	~20	[59]

Table 7 :

 7 Average mock viscosity values.

	MNP	Magnetic	Average	Viscosity
	diameter -	force FM [fN]	thickness of	ηave
	DMNP [nm] 130	0.651	RWM [µm] 20 ave	[Pa•s] 0.0956
	130	0.651	10	0.1913
	130	0.651	30	0.0638
	300	0.3	13.75 ave	0.0278
	300	0.3	11.5	0.0332
	300 300	0.3 1.2	16 13.75 ave	0.0239 0.111
	300 300	1.2 1.2	11.5 16	0.1329 0.0955

Table 8 :

 8 Values of magnetic forces generated by the magnets[START_REF] Barnes | Magnetic characterization of superparamagnetic nanoparticles pulled through model membranes[END_REF].

	Mag net	Force value	Predicted	Force	Percenta ge
		FM [fN]	velocity	FM-FEM [fN]	differenc e
			vagr [nm/s]		[%]
	1	0.572	546	0.556	2.8
	2	0.554	529	0.539	2.71
	3	0.651	622	0.634	2.61
	4	0.504	481	0.49	2.78

9(𝑠+2𝑅 𝑀𝑁𝑃 )[START_REF] Valente | Nanoparticle drug delivery systems for inner ear therapy: An overview[END_REF] (4)

Funding

This work was supported by the French Programme: Plan Cancer 2014-2019, Inserm, Project: Microrobots Targeting Glioblastoma. With financial support received from ITMO Cancer AVIESAN (Alliance Nationale pour les Sciences de la Vie et de la Santé, National Alliance for Life Science & Health) within the framework of the Plan Cancer initiative.

Conflict of Interest

The authors declare no conflict of interest was present. The funders had no influence over the preparation of the study, the collection, analyses, or interpretation of data herein as well as on in the writing of the present manuscript as well as on the decision to publish the results.