Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces

Abstract

The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b‐values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch‐Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

A simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI

“…To meet such a huge computing demand, we leverage the numerical matrix formalism based on finite element discretization implemented in the SpinDoctor toolbox [34]. The theory of this method is explained in section 2, and the convergence properties have been studied by Li et al [52,53]. Nonetheless, the previous implementation is not fast enough.…”

“…First of all, we could remove the impermeability assumption. Agdestein et al [53] have extended the numerical matrix formalism to include permeable compartments. We can solve the complete BT equation system, with permeable membranes using numerical matrix formalism.…”

“…For complex geometries, it is nearly impossible to find their analytical eigenfunctions. Li et al [52,53] proposed a way to numerically compute the eigenstates for complex geometries to make the matrix formalism useful for practical simulation. The idea is to discretize the complex geometries and calculate the eigenstates in the FE basis.…”

“…For the dMRI simulations, we adopt the numerical matrix formalism (NMF) method [50][51][52][53] based on a finite-element (FE) discretization and implemented in the SpinDoctor Toolbox [34,46]. Integrating matrix formalism with a finite element method (FEM) brings significant advantages in terms of computational efficiency.…”

Incorporating interface permeability into the diffusion MRI signal representation while using impermeable Laplace eigenfunctions

“…In a previous work, we presented a numerical implementation of the matrix formalism for permeable interfaces (Agdestein et al 2021), called the numerical matrix formalism method, where the permeability interface conditions are incorporated in the Laplace eigendecomposition step. In this paper, we present a new method, where the diffusion MRI signal of a permeable medium is computed using only impermeable Laplace eigenfunctions.…”