## Section: New Results

### Computational Diffusion MRI

#### Reducing the Number of Samples in Spatio-Temporal dMRI Acquisition Design

Participants : Patryk Filipiak, Rutger Fick [TheraPanacea, Paris] , Alexandra Petiet [ICM, CENIR, Paris] , Mathieu Santin [ICM, CENIR, Paris] , Anne-Charlotte Philippe [ICM, CENIR, Paris] , Stéphane Lehericy [ICM, CENIR, Paris] , Philippe Ciuciu [CEA, Université Paris-Saclay] , Demian Wassermann [Inria Parietal] , Rachid Deriche.

Acquisition time is a major limitation in recovering brain white matter microstructure with diffusion magnetic resonance imaging. The aim of this work is to bridge the gap between growing demands on spatiotemporal resolution of diffusion signal and the real-world time limitations. We introduce an acquisition scheme that reduces the number of samples under adjustable quality loss. Finding a sampling scheme that maximizes signal quality and satisfies given time constraints is NP-hard. Therefore, a heuristic method based on a genetic algorithm is proposed in order to find suboptimal solutions in acceptable time. The analyzed diffusion signal representation is defined in the q$\tau $ space, so that it captures both spatial and temporal phenomena. The experiments on synthetic data and in vivo diffusion images of the C57Bl6 wild-type mouse corpus callosum reveal superiority of the proposed approach over random sampling and even distribution in the q$\tau $ space.

This work has been published in [12].

#### Dmipy, a Diffusion Microstructure Imaging toolbox in Python to improve research reproducibility

Participants : Abib Olushola Yessouffou Alimi, Rutger Fick [TheraPanacea, Paris] , Demian Wassermann [Inria Parietal] , Rachid Deriche.

The recovery of microstructure-related features of the brain's white matter is a current challenge in diffusion MRI (dMRI). In particular, multi-compartment (MC)-based models have been a popular approach to estimate these features. However, the usage of MC-models is often limited to those hard-coded in publicly available toolboxes.

In this work, we present Diffusion Microstructure Imaging in
Python (Dmipy), a diffusion MRI toolbox which allows accessing any
multi-compartment-based model and robustly estimates these
important features from single-shell, multi-shell, and
multi-diffusion time, and multi-TE data. Dmipy follows a *building block*-based philosophy to microstructure imaging,
meaning an MC-model can be constructed and fitted to dMRI data
using any combination of underlying tissue models, axon dispersion
or diameter distributions, and optimization algorithms.using less
than 10 lines of code, thus helps improve research
reproducibility. In describing the toolbox, we show how Dmipy
enables to easily design microstructure models and offers to the
users the freedom to choose among different optimization
strategies.We finally present three advanced examples of highly
complex modeling approaches which are made easy using Dmipy.

This work has been published in [21], [30].

#### Non-parametric graphnet-regularized representation of dMRI in space and time

Participants : Rutger Fick [TheraPanacea, Paris] , Alexandra Petiet [ICM, CENIR, Paris] , Mathieu Santin [ICM, CENIR, Paris] , Anne-Charlotte Philippe [ICM, CENIR, Paris] , Stéphane Lehericy [ICM, CENIR, Paris] , Demian Wassermann [Inria Parietal] , Rachid Deriche.

Effective representation of the four-dimensional diffusion MRI
signal–varying over three-dimensional q-space and diffusion
time $\tau $ – is a sought-after and still unsolved challenge in
diffusion MRI (dMRI). We propose a functional basis approach that
is specifically designed to represent the dMRI signal in this
q$\tau $-space. Following recent terminology, we refer to our
q$\tau $-functional basis as *q$\tau $-dMRI*. q$\tau $-dMRI can be seen as a
time-dependent realization of q-space imaging by Paul Callaghan
and colleagues. We use GraphNet regularization - imposing both
signal smoothness and sparsity – to drastically reduce the number
of diffusion-weighted images (DWIs) that is needed to represent
the dMRI signal in the q$\tau $-space. As the main contribution, q$\tau $-dMRI
provides the framework to – without making biophysical assumptions
- represent the q$\tau $-space signal and estimate time-dependent
q-space indices (q$\tau $-indices), providing a new means for studying
diffusion in nervous tissue. We validate our method on both
in-silico generated data using Monte–Carlo simulations and an
in-vivo test-retest study of two C57Bl6 wild-type mice, where we
found good reproducibility of estimated q$\tau $-index values and
trends. In the hope of opening up new $\tau $-dependent venues of
studying nervous tissues, q$\tau $-dMRI is the first of its kind in
being specifically designed to provide open interpretation of the
q$\tau $-diffusion signal.

This work has been published in [11].

#### Resolving the crossing/kissing fiber ambiguity using functionallyCOMMIT ( Convex Optimization Modeling for Microstructure Informed Tractography)

Participants : Matteo Frigo, Isa Costantini, Samuel Deslauriers-Gauthier, Rachid Deriche.

The architecture of the white matter is endowed with kissing and crossing bundles configurations. When these white matter tracts are reconstructed using diffusion MRI tractography, this systematically induces the reconstruction of many fiber tracts that are not coherent with the structure of the brain. The question on how to discriminate between true positive connections and false positive connections is the one addressed in this work. State-of-the-art techniques provide a partial solution to this problem by considering anatomical priors in the false positives detection process. We propose a novel model that tackles the same issue but takes into account both structural and functional information by combining them in a convex optimization problem. We validate it on two toy phantoms that reproduce the kissing and the crossing bundles configurations, showing that, through this approach, we are able to correctly distinguish true positives and false positives.

This work has been published in [25].

#### Reducing false positive connection in tractograms using joint structure-function filtering

Participants : Matteo Frigo, Guillermo Gallardo Diez, Isa Costantini, Alessandro Daducci [EPFL, Lausanne] , Demian Wassermann [Inria Parietal] , Samuel Deslauriers-Gauthier, Rachid Deriche.

Due to its ill-posed nature, tractography generates a significant number of false positive connections between brain regions. To reduce the number of false positives, Daducci et al. proposed the COMMIT framework, which has the goal of re-establishing the link between tractography and tissue microstructure. In this framework, the diffusion MRI signal is modeled as a linear combination of local models associated with streamlines where the weights are identified by solving a convex optimization problem. Streamlines with a weight of zero do not contribute to the diffusion MRI data and are assumed to be false positives. Removing these false positives yields a subset of streamlines supporting the anatomical data. However, COMMIT does not make use of the link between structure and function and thus weights all bundles equally. In this work, we propose a new strategy that enhances the COMMIT framework by injecting the functional information provided by functional MRI. The result is an enhanced tractogram filtering strategy that considers both functional and structural data.

This work has been published in [31].

#### Combining Improved Euler and Runge-Kutta 4th order for Tractography in Diffusion-Weighted MRI

Participants : Cherifi Dalila [IEEE University of Boumerdes, Algeria] , Boudjada Messaoud [IEEE University of Boumerdes, Algeria] , Morsli Abdelatif [IEEE University of Boumerdes, Algeria] , Girard Gabriel [EPFL, Lausanne] , Rachid Deriche.

In this work, we develop a general, deterministic tractography algorithm (CIERTE), which is a combination of Improved Euler and Range-Kutta fourth-order algorithm and test it on synthetic and real data. The proposed tractography method is validated using seven metrics of the tractometer evaluation system and positively compared to state-of-the-art tractography algorithms.

This work has been published in [9].

#### Fiber orientation distribution function from non-negative sparse recovery with quantitative analysis of local fiber orientations and tractography using DW-MRI datasets

Participants : Thinhinane Megherbi [USTHB, Algiers] , Gabriel Girard [EPFL, Lausanne] , Ghosh Aurobrata [AI Innovation Lab, Verisk Analytics] , Fatima Oulebsir-Boumghar [USTHB, Algiers] , Rachid Deriche.

In this work, we propose, evaluate and validate a new Diffusion Weighted MRI method to model and recover high quality tractograms even with multiple fiber populations in a voxel and from a limited number of acquisitions.

Our method relies on the estimation of the Fiber Orientation Distribution (FOD) function, parameterized as a non-negative sum of rank-1 tensors and the use of a non-negative sparse recovery scheme to efficiently recover the tensors, and their number. Each fiber population of a voxel is characterized by the orientation and the weight of a rank-1 tensor.

Using both deterministic and probabilistic tractography algorithms, we show that our method is able to accurately reconstruct narrow crossing fibers and obtain a high quality connectivity reconstruction even from a limited number of acquisitions. To this end, a validation scheme based on the connectivity recovered from tractography is developed to quantitatively evaluate and analyze the performance of our method. The tractometer tool is used to quantify the tractography obtained from a simulated DW-MRI dataset including a high angular resolution dataset of 60 gradient directions and a dataset of 30 gradient directions, each of them corrupted with Rician noise of SNR 10 and 20. The performance of our FOD model and its impact on the tractography results are also demonstrated and illustrated on in vivo DW-MRI datasets with high and low angular resolutions.

This work has been published in [15].

#### Solving the Cross-Subject Parcel Matching Problem Using Optimal Transport

Participants : Guillermo Gallardo Diez, Nathalie Gayraud, Maureen Clerc, Demian Wassermann [Inria Parietal] , Samuel Deslauriers-Gauthier, Rachid Deriche.

Matching structural parcels across different subjects is an open problem in neuroscience. Even when produced by the same technique, parcellations tend to differ in the number, shape, and spatial localization of parcels across subjects. In this work, we propose a parcel matching method based on Optimal Transport. We test its performance by matching parcels of the Desikan atlas, parcels based on a functional criteria and structural parcels. We compare our technique against three other ways to match parcels which are based on the Euclidean distance, the cosine similarity, and the Kullback-Leibler divergence. Our results show that our method achieves the highest number of correct matches.

This work has been published in [32], [26].

#### A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI

Participants : Mauro Zucchelli, Samuel Deslauriers-Gauthier, Rachid Deriche.

Rotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any spherical function. Specifically, our method is based on the spherical harmonics series expansion of a given function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features.

This work has been published in [29].

#### Rational invariants of ternary forms under the orthogonal group

Participants : Paul Görlach [MPI for Mathematics in the Sciences] , Evelyne Hubert [Inria, AROMATH] , Théodore Papadopoulo, Rachid Deriche.

In [68], [69], [81] we started to explore the theory of tensor invariants as a mathematical framework for computing new biomarkers for HARDI. We pursued this work and, in collaboration with the project-team GALAAD/AROMATH , we succeeded to develop a complete set of rational invariants for ternary quartics [39]. Being rational, they are very close to the polynomial invariants developed in [69] but they constitute a complete set of invariants. They are also good tools to understand better the algebraic invariants of [81] and some others based on spherical harmonics decomposition [55]. We determined a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $O\left(3\right)$ on the space $R{[x,y,z]}_{2d}$ of ternary forms of even degree 2d. The construction relies on two key ingredients. On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $B\left(3\right)$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $B\left(3\right)$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this work. The expression of the $B\left(3\right)$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $O\left(3\right)$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $B\left(3\right)$-invariants to determine the $O\left(3\right)$-orbit locus and provide an algorithm for the inverse problem of finding an element in $R{[x,y,z]}_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.

This work has been sumitted and is currently under review. A preprint is available in [39].

#### Edema-informed anatomically constrained particle filter tractography

Participants : Samuel Deslauriers-Gauthier, Drew Parker [UPenn, USA] , François Rheault [SCIL, Sherbrooke University, CA] , Steven Brem [UPenn, USA] , Maxime Descoteaux [SCIL, Sherbrooke University, CA] , Ragini Verma [UPenn, USA] , Rachid Deriche.

In this work, we propose an edema-informed anatomically constrained tractography paradigm that enables reconstructing larger spatial extent of white matter bundles as well as increased cortical coverage in the presence of edema. These improvements will help surgeons to maximize the extent of the resection while minimizing the risk of cognitive deficits. The new paradigm is based on a segmentation of the brain into gray matter, white matter, corticospinal fluid, edema and tumor regions which utilizes a tumor growth model. Using this segmentation, a valid tracking domain is generated and, in combination with anatomically constrained particle filter tractography, allows streamlines to cross the edema region and reach the cortex. Using subjects with brain tumors, we show that our edema-informed anatomically constrained tractography paradigm increases the cortico-cortical connections that cross edema-contaminated regions when compared to traditional fractional anisotropy thresholded tracking.

This work has been published in [24].

#### Towards the assessment of myelination using time-dependent diffusion MRI indices

Participants : Abib Olushola Yessouffou Alimi, Alexandra Petiet [ICM, CENIR, Paris] , Mathieu Santin [ICM, CENIR, Paris] , Anne-Charlotte Philippe [ICM, CENIR, Paris] , Stéphane Lehericy [ICM, CENIR, Paris] , Demian Wassermann [Inria Parietal] , Rachid Deriche.

In this work, we study the sensitivity of time-dependent diffusion MRI indices or q$\tau $-indices to demyelination in the mouse brain. For this, we acquire in vivo four-dimentional diffusion-weighted images-varying over gradient strength, direction and diffusion time-and estimate the q$\tau $-indices from the corpus callosum. First order Taylor approximation of each index gives fitting coefficients $\alpha $ and $\beta $ whose variance we investigate. Results indicate that, cuprizone intoxication affects mainly index coefficient $\beta $ by introducing inequality of variances between the two mice groups, most significantly in the splenium and that MSD increases and RTOP decreases over diffusion time $\tau $.

This work has been published in [35].

#### An Analytical Fiber ODF Reconstruction in 3D Polarized Light Imaging

Participants : Abib Olushola Yessouffou Alimi, Yves Usson [UMR5525 TIMC-IMAG CNRS] , Pierre-Simon Jouk [CHU Grenoble-Alpes] , Gabrielle Michalowicz [CHU Grenoble-Alpes] , Rachid Deriche.

Three dimensional polarized light imaging (3D-PLI) utilizes the birefringence in postmortem tissue to map its spatial fiber structure at a submillimeter resolution. In this work, we propose an analytical method to compute the fiber orientation distribution function (ODF) from high-resolution vector data provided by 3D-PLI. This strategy enables the bridging of high resolution 3D-PLI to diffusion magnetic resonance imaging with relatively low spatial resolution. First, the fiber ODF is modeled as a sum of $K$ orientations on the unit sphere and expanded with a high order spherical harmonics series. Then, the coefficients of the spherical harmonics are derived directly with the spherical Fourier transform. We quantitatively validate the accuracy of the reconstruction against synthetic data and show that we can recover complex fiber configurations in the human heart at different scales.

This work has been published in [22].

#### fMRI Deconvolution via Temporal Regularization using a LASSO model and the LARS algorithm

Participants : Isa Costantini, Patryk Filipiak, Kostiantyn Maksymenko, Samuel Deslauriers-Gauthier, Rachid Deriche.

In the context of functional MRI (fMRI), methods based on the deconvolution of the blood oxygenated level dependent (BOLD) signal have been developed to investigate the brain activity, without a need of a priori knowledge about activations occurrence. In this work, we propose a novel temporal regularized deconvolution of the BOLD signal using the Least Absolute Shrinkage and Selection Operator (LASSO) model, solved by means of the Least-Angle Regression (LARS) algorithm. In this way, we were able to recover the underlying neurons activations and their dynamics.

This work has been published in [23], [37].

#### A Second Order Multi-Stencil Fast Marching Method with a Non-Constant Local Cost Model

Participants : Susana Merino-Caviedes [Universidad de Valladolid] , Lucilio Cordero-Grande [King's College London] , Maria Tereza Perez [Universidad de Valladolid] , Pablo Casaseca-de-La-Higuera [Universidad de Valladolid] , Marcos Martín-Fernández [Universidad de Valladolid] , Carlos Alberola-Lopez [Universidad de Valladolid] , Rachid Deriche.

The Fast Marching method is widely employed in several fields of image processing. Some years ago a Multi-Stencil version (MSFM) was introduced to improve its accuracy by solving the Eikonal equation for a set of stencils and choosing the best solution at each considered node. The following work proposes a modified numerical scheme for MSFM to take into account the variation of the local cost, which has proven to be second order. The influence of the stencil set choice on the algorithm outcome with respect to stencil orthogonality and axis swapping is also explored, where stencils are taken from neighborhoods of varying radius. The experimental results show that the proposed schemes improve the accuracy of their original counterparts, and that the use of permutation-invariant stencil sets provides robustness against shifted vector coordinates in the stencil set.