Section: New Results
Diffusion MRI
J.R. Li, K. V. Nguyen and T. N. Tran
Diffusion Magnetic Resonance Imaging (DMRI) is a promising tool to obtain useful information on microscopic structure and has been extensively applied to biological tissues.
We obtained the following results.

The BlochTorrey partial differential equation can be used to describe the evolution of the transverse magnetization of the imaged sample under the influence of diffusionencoding magnetic field gradients inside the MRI scanner. The integral of the magnetization inside a voxel gives the simulated diffusion MRI signal. This work proposes a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The variable thickness of the threedimensional domains is included in the weak formulation established on the manifolds. We conducted a numerical study of the proposed approach by simulating the diffusion MRI signals from the extracellular space (a thin layer medium) and from neurons (a thin tube medium), comparing the results with the reference signals obtained using a standard threedimensional finite element discretization. We show good agreements between the simulated signals using our proposed method and the reference signals for a wide range of diffusion MRI parameters. The approximation becomes better as the diffusion time increases. The method helps to significantly reduce the required simulation time, computational memory, and difficulties associated with mesh generation, thus opening the possibilities to simulating complicated structures at low cost for a better understanding of diffusion MRI in the brain [12].

The nerve cells of the Aplysia are much larger than mammalian neurons. Using the Aplysia ganglia to study the relationship between the cellular structure and the diffusion MRI signal can potentially shed light on this relationship for more complex organisms. We measured the dMRI signal of chemicallyfixed abdominal ganglia of the Aplysia at several diffusion times. At the diffusion times measured and observed at low bvalues, the dMRI signal is monoexponential and can be accurately represented by the parameter ADC (Apparent Diffusion Coefficient). We performed numerical simulations of water diffusion for the large cell neurons in the abdominal ganglia after creating geometrical configurations by segmenting high resolution T2weighted (T2w) images to obtain the cell outline and then incorporating a manually generated nucleus. The results of the numerical simulations validate the claim that water diffusion in the large cell neurons is in the short diffusion time regime at our experimental diffusion times. Then, using the analytical short time approximation (STA) formula for the ADC, we showed that in order to explain the experimentally observed behavior, it is necessary to consider the nucleus and the cytoplasm as two separate diffusion compartments. By using a two compartment STA model, we were able to illustrate the effect of the highly irregular shape of the cell nucleus on the ADC [13].

The complex transverse water proton magnetization subject to diffusionencoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment BlochTorrey partial differential equation. Under the assumption of negligible water exchange between compartments, the timedependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a timedependent Neumann boundary condition. This work describes a publicly available MATLAB toolbox called SpinDoctor that can be used 1) to solve the BlochTorrey partial differential equation in order to simulate the diffusion magnetic resonance imaging signal; 2) to solve a diffusion partial differential equation to obtain directly the apparent diffusion coefficient; 3) to compare the simulated apparent diffusion coefficient with a shorttime approximation formula. The partial differential equations are solved by $P1$ finite elements combined with builtin MATLAB routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen. SpinDoctor provides builtin options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extracellular space enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting; 5) permeable membranes; Builtin diffusionencoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscillating Gradient Spin Echo. We describe in detail how to use the SpinDoctor toolbox. We validate SpinDoctor simulations using reference signals computed by the Matrix Formalism method. We compare the accuracy and computational time of SpinDoctor simulations with MonteCarlo simulations and show significant speedup of SpinDoctor over MonteCarlo simulations in complex geometries. We also illustrate several extensions of SpinDoctor functionalities, including the incorporation of $T2$ relaxation, the simulation of nonstandard diffusionencoding sequences, as well as the use of externally generated geometrical meshes [10].

The numerical simulation of the diffusion MRI signal arising from complex tissue microstructures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the BlochTorrey equation by finite elements is a more recently developed approach for this purpose, in contrast to random walk simulations, which has a longer history. While finite element discretization is more difficult to implement than random walk simulations, the approach benefits from a long history of theoretical and numerical developments by the mathematical and engineering communities. In particular, software packages for the automated solutions of partial differential equations using finite element discretization, such as FEniCS, are undergoing active support and development. However, because diffusion MRI simulation is a relatively new application area, there is still a gap between the simulation needs of the MRI community and the available tools provided by finite element software packages. In this paper, we address two potential difficulties in using FEniCS for diffusion MRI simulation. First, we simplified software installation by the use of FEniCS containers that are completely portable across multiple platforms. Second, we provide a portable simulation framework based on Python and whose code is open source. This simulation framework can be seamlessly integrated with cloud computing resources such as Google Colaboratory notebooks working on a web browser or with Google Cloud Platform with MPI parallelization. We show examples illustrating the accuracy, the computational times, and parallel computing capabilities. The framework contributes to reproducible science and opensource software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations [11].

We performed simulations for a collaborative project with Demian Wassermann of the Parietal team on distinguishing between spindle and pyramidal neurons with Multishell Diffusion MRI [34].

We continued in the simulation and modeling of heart diffusion MRI with the postdoc project of Imen Mekkaoui, funded by InriaEPFL lab. The project is cosupervised with Jan Hesthaven, Chair of Computational Mathematics and Simulation Science (MCSS), EPFL. An article on this topic is under preparation.