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3. EXPERIMENTAL VALIDATION 3.1. Experimental setup
3.1.1. Crossing angle resolution of the **model**
The crossing angle resolution (CAR) is the minimum de- tectable angle between two different FOs. We performed its evaluation on the 2-fiber **model** (M = 2) using synthetic data on a single voxel. We generated the simulated data sets ac- cording to the spherical deconvolution method proposed in [12]. They simulate multiple FOs assuming that the fiber ori- entation distribution function is a sum of equally weighted delta functions. They perform its convolution with the kernel proposed in [13], which models restricted **diffusion** within a cylindrical fiber of radius ρ = 5µm and length L = 5mm. We chose this method because (i) they implemented it on-line, as part of the fanDTasia toolbox 3 , (ii) this method has been used in more than 20 papers and (iii) the spherical deconvolution in its discrete version is close to multi-compartment models.

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(a) Gamma Distributions (b) **Signal** Optic Nerve (c) **Signal** Sciatic Nerve Fig. 1: Signals generated using the Callaghan **model**.
3.2 **Signal** fitting and Effect of Regularization
In our first experiment we test how many spatial or temporal basis functions we need to fit a 3D+t **diffusion** **signal**. We choose to study in the case of restricted **diffusion** in a cylindrical compartment, since this is a good **model** **for** white mat- ter tissue in highly organized areas. We generate the noiseless **signal** as described in Section 2.3 with the sampling scheme we described in Section 3.1 . We then fit the **signal** with increasing maximum order **for** the spatial and temporal basis. We then compute the mean squared error (MSE) of the fitted **signal** compared to the ground truth. We show a heat map of the results in Figure 2a where we see that the **signal** fitting in this specific **signal** **model** only improves very little after a spatial order of 6 and a temporal order of 5. Using Eq. ( 5 ) this means we fit 300 coefficients to accurately represent the 3D+t **signal**.

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ing GFA [ Cohen-Adad et al. , 2011 ] or PA [ Fick et al. , 2016a ], while large-scale comparisons like those **for** FA are missing.
Moreover, the typical criterium **for** being a biomarker is that the measure of inter- est should provide a statistically significant difference between healthy and diseased populations. However, care should be taken in prematurely calling a non-specific marker such as **diffusion** anisotropy a biomarker. As an illustration, in the particular case of Parkinson’s disease, after many studies had claimed that FA could be used as a diagnostic biomarker, a systematic review of these studies actually showed that on its own, it cannot [ Hirata et al. , 2016 ]. It is likely that the non-specificity of **diffusion** anisotropy will continue to confound its interpretation as a biomarker **for** pathology. On multi-compartment-based anisotropy: To overcome this lack of specificity, multi-compartment approaches strive to separate the **signal** contributions of different tissue compartments using biophysical models. However, it is important to realize that these models still describe **diffusion** anisotropy in some reparameterized way. **For** example, while NODDI is a multi-compartment **model** that separates the **signal** contributions of CSD, intra- and extra-axonal compartments, it can only describe one axon bundle using a Watson distribution with a single ODI, which is a function of concentration parameter κ. Illustrating the Watson ODFs together with the **signal**- based ODFs in Fig. 7 , we indeed find its similarity to others, in particular DTI. Of course, ODI has a different interpretation than FA, but it is important to see how they are related. Furthermore, µFA describes the per-axon micro-environment and is theoretically insensitive to crossing or dispersed axon configurations.

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We present the ﬁrst 4D basis and reconstruction algorithm **for** representing and reconstructing the axisymmetric b-tensor encoded diﬀusion **signal**. We study the properties of the **signal** to inform how many and which b-tensors should be used, thereby providing sampling recommendations. Although preliminary, the high reconstruction accuracy achieved on Magic DIAMOND **model** is promising **for** interpolation of b-tensors encoded measurements to enable algorithms **for** recovering the DTD to achieve higher accuracy and/or reduce the number of measurements.

Figure 1
The **signal** S is defined as the linear combination of the IC, EC and CSF compartments, where the anisotropic components IC and EC are convolved with a Watson distribution W of parameters d and k. The **signal** shape each compartment Ex depends on the characterising diffusivities and the S0 responses are designed **for** defining the 3-S0 **model**. Each parameter in the table was drawn from a normal distribution of mean and standard deviation equal to the ones listed. The S 0 ic and S

values, considering b 0 = 0 and E(0) = 1 **for** any u ∈ S 2 , which makes our estimation
more reasonable and accurate. Otherwise, there is no warranty **for** the estimated **signal** ˜
E(0) = 1. Another advantage is that **for** the single shell HARDI data, considering b = 0 can let us have 2 shells, which will largely improve the results. The second one is how to determine the parameter ζ in basis. The authors in [8] proposed an experience strategy **for** it, which is dependent on the radial truncation order N. However, we think the parameter should be just dependent on the **signal**, not on the basis order. Considering E (q) = exp(−4π 2 τq 2 D), b = 4π 2 τq 2 , and a typical **diffusion** coefficient of D = 0.7 × 10 −3 mm 2 /s, a typical b-value b = 3000s/mm 2 , we set ζ = 1

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In silico data: As first data set, we generated time- dependent **diffusion** data using our Python-based [9] im- plementation of the two-compartment **model** with the intra- cellular fraction modeled with a set of Watson dispersed crossing cylinders [10] and the extra-cellular one modeled with a temporal zeppelin [1]. The angles of crossings were controlled by a random variable with a distribution centered at π/2 and a dispersion π/4. Following the study of Zhang et al. [10], we used the concentration parameters of the Watson distribution κ ∈ {0.25, 1, 4, 16}, where κ = 16 means highly concentrated and κ = 0.25 means highly dispersed cylinders. Additionally, we considered two variants of our data set per- turbed with Rician noise having **Signal**-to-Noise Ratio (SNR) 20 and 10, respectively.

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However, because of the low spatial resolution in **diffusion** **MRI**, it has been shown that the ST **model** is an average of several **diffusion** processes arising from multiple populations of tissues in the voxel, ultimately leading to an inaccurate description of the microstructure in most parts of the brain white matter. In addition, even within a single homogeneous tissue population, non-Gaussian dif- fusion has been observed [13]. This has led to the extension of the ST **model** to mixture models, often called multi-compartment models (MCM), where the voxelwise **diffusion** **signal** is modeled as a linear combination of compartmental **diffusion** signals arising from underlying homogeneous **diffusion** processes [8]. To the best of our knowledge, the only mathematical framework available **for** multi- compartment image processing [12] is however limited to multi-tensor images (mixture of single-tensor signals, see [10]) as it relies on log-Euclidean geometry. In this work, we present an alternative to the log-Euclidean space on tensors called the Bayes Hilbert space [3] **for** processing compartmental **diffusion** signals instead of tensors. It provides a unified framework that can accommodate an- alytic compartment **diffusion** signals of any type (Gaussian and non-Gaussian). Section 2 gives a general introduction to Bayes Hilbert spaces with motivation and setup in **diffusion** **MRI**. Section 3 describes a simulation study and a tractog- raphy application on real data to compare Euclidean, log-Euclidean and Bayes interpolation. Results commented in Section 4 show that Bayes interpolation fea- tures improved robustness to noise and yields better pyramidal tract reconstruc- tions when combined with MCM-based deterministic tractography to account **for** streamline atlas priors when **diffusion** models cannot be locally trusted.

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We show that by exploiting the DW signal’s phase it is possible to character- ize axonal tortuosity, which can be linked to elongation/compression phenomena in pathological scenarios. Compressed axons are partially convoluted and show irregular undulation with a non-uniform tortuosity along the longitudinal direc- tion (see fig. 1 right). Hence, we **model** the compressed axon as a sinusoid with varying tortuosity, inducing asymmetry in its shape. Moreover, as the degree of compression increases, the tortuosity variation rate along the axon also increases. Spins diffusing within these axons undergo an asymmetric **diffusion** process re- sulting in an informative DW signal’s phase, sensible to increasing tortuosity rates. The amount of **diffusion** asymmetry thus gives important insights into the underlying degree of compression.

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test the accuracy of the results and to provide criteria **for** improving the modeling and the methodology.
A significant aspect of simulation is that the features (e.g. cell membrane’s shape and prop- erty) are completely under the control of users, so that the role of specific property can be examined by altering its attribute. In dMRI, tissue modeling is the task of intensively research in order to clarify the relationship between **MRI** **signal** and tissue microscopic characteris- tics. Budde and Frank provide an excellent example of how simulations and experiments can be complementary tools to study the impact of tissue property [ Budde and Frank ( 2010 )]. They constructed a biophysical **model** **for** beading axonal membranes, which was validated by the histological evidence of biological nerve fibres, thus highlighting the significant impact of morphological variations of axons on ADC. In their study, the role of simulation offers the flexibility to adjust the magnitude of morphological variations of the axonal membranes, while the role of dMRI experiment verifies the phenomenon in realistic biological conditions. In the future, DMS will be further applied to investigate different tissue characteristics under healthy and pathological states via cross validation between simulations and experiments. This also reflects the benefit and the need of 3D rendering of histological images **for** the reconstruction of tissue configuration, as described in the previous section (Chapter 7.2.1 ). Although DMS simulations were mostly applied to simulate the tissues in the CNS system, DMS has the potential to be broadly applied to different **MRI** research fields such as **diffusion** in the porous media [ Shemesh et al. ( 2010a )], the acinar tree of lung [ Perez-Sanchez et al.

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However, the past two decades have been marked by the discovery of large inter- individual variability in brain structure and function (i.e. different phenotype), creating additional challenges in furthering the understanding of the brain and a lack of models to explain its functioning 198-201 . Studies have begun to explore and describe these pathways across large numbers of individuals 24,116,202 . Just as studies have found that individual differences in behaviour correlate with variation in local white matter microstructure, so too have tractography studies found that variation in the ‘strength’ or organization of white matter fibres relate to variations in behaviour. **For** instance, language areas and their interconnections are predominantly represented in the left hemisphere **for** most but not all healthy participants 203,204 . However, individuals vary in the degree of lateralisation and people with more symmetric patterns of connection have been shown to perform better at episodic memory tasks 203 . These differences are important consideration **for** the field of clinical neuroscience. Forkel et al. 205 demonstrated that the phenotype of the structural network supporting language (i.e. arcuate fasciculus) may interact with recovery after a stroke in the left hemisphere. Additionally, Lunven et al. 206 reported that the strength of inter- hemispheric communication is important **for** the recovery of visuospatial neglect after a stroke in the right hemisphere (see 207,208 **for** extensive reviews on the role of **diffusion** **MRI** in clinical neurosciences). Hence the study of inter-individual variability is a **new** line of research directly accessible in large scale projects such as the Human Connectome Project (https://www.humanconnectome.org), where high quality **diffusion** **MRI** data have been acquired in 1200 subjects and openly shared with the community 209 . Using **diffusion** weighted imaging tractography to characterise the functional specialisation of brain areas at the individual level may therefore benefit the domain of inter-individual variability, providing a more tailored brain **model** when doing group comparisons eventually making methods like spatial registration and smoothing obsolete.

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While algorithms have been proposed **for** recovering the DTD from b-tensor encoded measurements of the **diffusion** **signal** [ 7 ]–[ 10 ], only recently some consideration has been given to designing the b-tensor sampling scheme [ 11 ], [ 12 ]. In most cases, b-tensors are empirically chosen, often from uniform sampling of b-tensor parameters [ 9 ], [ 13 ]. Due to the inverse Laplace transform being ill-conditioned, on the order of 1000 samples are required by existing algorithms to recover marginal distributions of the full DTD [ 14 ], [ 15 ]. Such large numbers of samples require long scan times and are unrealistic **for** use in a clinical setting. Two recent studies [ 11 ], [ 12 ] proposed optimized b-tensor sampling schemes **for** the estimation of the parameters of a multi-tensor **model** of white matter. In contrast, in this work we propose a versatile method to design an optimal sampling scheme based on the properties of the **signal**, irrespective of the choice of a particular biophysical **model**.

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Keywords **diffusion** **MRI**, microstructure, multi tissue, single-TE, volume fraction, **signal** fraction
1 Introduction
**Diffusion** **MRI** (dMRI) is an imaging technique that allows to inspect the brain tissue microstructure in- vivo non-invasively. One of the most commonly studied microstructural feature is the volume fraction of a tissue in a sample. In particular, the intra-axonal (i.e., intra-cellular - IC), extra-axonal (i.e., extra-cellular - EC) and cerebro-spinal fluid (CSF) volume fractions have been investigated with models like the neurite orientation dispersion and density imaging (NODDI) ( Zhang et al. , 2012 ), ActiveAx ( Alexander et al. , 2010 ), and the multi-compartment microscopic **diffusion** imaging framework ( Kaden et al. , 2016 ). The differences between these models lie on the representation employed in describing the tissue-specific **signal** and on the assumptions made on the **model** parameters. **For** example, intra-axonal **diffusion** can be modelled as the **diffusion** within a stick or a cylinder and some models fix the value of the diffusivity or tortuosity. A unifying aspect that characterizes most of the brain microstructure models is the building-blocks concept behind their formalisation. In other words, models are defined in a multi-compartment (MC) fashion, where the dMRI **signal** is described as a linear combination of single-tissue models. The resulting models are called MC models and they require the acquisition of multi-shell dMRI data in order to accurately disentangle the contribution of each compartment ( Scherrer and Warfield , 2010 ). Thorough reviews have been dedicated to the design and validation of such models ( Jelescu and Budde , 2017 ), to the sensitivity of MC models to experimental factors and microstructural properties of the described tissues ( Afzali et al. , 2020 ), and to the abstraction of these models that allows to obtain a unified theory ( Fick et al. , 2019 ).

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4.7. Conclusions and further improvements
4.7 Conclusions and further improvements
Our analysis of the parameters estimation problem in the previous sections has re- vealed advantages and limitations of this approach. On the one hand, it seems to be possible to estimate a number of physiologically relevant parameters **for** “simple” tissues composed of not too elongated constituents. Among these parameters are the volume fractions, the apparent **diffusion** coefficient in the extra-cellular space, and the residence time which is related to the membrane permeability and surface- to-volume ratio. Clearly, these parameters can only represent averaged properties of the tissue. Nevertheless, a reliable experimental determination of these param- eters has always presented a great challenge. We showed on simplified examples a possibility of solving the parameters estimation problem. Most importantly, we analyzed the stability features of this problem. In particular, we studied the ro- bustness of the estimates against various biological and physical parameters (e.g., permeability, sequence timing, the range and number of b-values), as well as the robustness against noise. The **ODE** **model** always provided more accurate and sta- ble estimates than the Kärger **model**. This study suggests a promising perspective **for** estimating many important parameters from dMRI signals.

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acteristic time to traverse the correlation length of the cell packing.
These statements are quite difficult to interpret and verify, **for** example, even when considering a reasonable geometrical **model** of brain tissue such as a voxel-sized domain containing densely packed dendrite trees and cylindrical axons with some orientation distribution and spherical cells modeling soma and glia cells. Hence we consider numerical simulations on voxel-sized (three dimensional) domains that contain densely packed, randomly placed and oriented cells of spherical and cylindrical geometries (and tree structures) to verify the above statemetns an extremely useful future direction. We illustrate this by showing preliminary results of applying the FPK **model** to a two dimensional voxel-sized non-periodic domain in Section 5. In this paper, we study the FPK **model** **for** periodic domains because the macroscopic **model** parameters can be defined rigorously **for** periodic domains, and we are able to show the convergence of the FPK **model** to the microscopic **model** in the homogenization parameter ε in a clear way. We hope that these results can then guide us improving and generalizing macroscopic models to brain tissue dMRI.

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moi toutes les joies et les douleurs de ce parcours (votre soutenance va arriver bientôt, courage !) ; Kevish, même si tu viens d’arriver, tu t’es déjà fait une belle place dans la meilleure équipe de foot (allez l’A.S. Saint Hessienne) ; Hang Tuan et Van Dang pour m’avoir introduit l’IRMD et le modèle FPK ; Tobias pour ses photos de l’équipe de foot ; Federico perché non so come avrei fatto i primi mesi senza il tuo supporto; Irene **for** showing me that you have always a reason to be in a good mood; Shixu **for** showing me that there exist people who have never seen a musical; and finally Jake who is a very good friend (I am sorry I have no better words to describe you but I have really enjoyed the time we spent together and I am really glad that I had the opportunity to meet you!). Even if she is not really a member of DeFI group, I would like to thank Prof. Fioralba Cakoni **for** bringing always a good spirit when she is around. Au sein de l’INRIA je tiens aussi à remercier vivement Valérie Berthou pour toute l’aide qu’elle m’a apportée avant mon arrivée en France puis pendant mes trois ans ici !

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Index Terms— **Diffusion** **MRI**, White Matter, Microstruc- ture, Generalized Multi Tissue Modelling
1. INTRODUCTION
**Diffusion** **MRI** (dMRI) is able to probe brain tissue mi- crostructure in-vivo non-invasively. Two of the most com- monly studied microstructural features are the intra-axonal and cerebro-spinal fluid (CSF) volume fractions and sev- eral microstructural models have been proposed to retrieve them [1, 2, 3]. These models differ in the type of **signal** **model** used to represent a given tissue type (e.g. **diffusion** within a stick or cylinder **for** modeling the intra-axonal **diffusion**)

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Perspectives
The simulated noise-free and non-artifact images can be used **for** evaluating **diffusion** image processing algorithms, such as denoising, k-space reconstruction, and motion correction. In addition, the simulation can help optimize the imaging parameters **for** high-b value imaging and q-space imaging. Furthermore, in vivo cardiac fiber models can be constructed based on PLI data and motion information from **MRI** cine sequences and as a result in vivo DW images can be simulated. As detailed in [12], load-balancing among computing resources is close to optimal **for** the simulation phase. However, there is still room **for** improving performance, especially in the merging phase: (i) the merging phase could be improved by using multiple parallel mergers as done in [13], (ii) the influence of the checkpointing frequency on the merging phase could be studied, (iii) the placement of partial simulation results on different storage locations could be investigated to reduce the cost of the merging phase. These could reduce the amount of manual intervention by grid experts which, to date, is still required to conduct such experiments.

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