Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

Moutal, Nicolas

doi:10.1007/s10915-019-01055-5

Cited by 29 publications

(31 citation statements)

References 96 publications

(166 reference statements)

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“…In order to compare the performance of both transit models we have the necessity to obtain an analytical solution. A recent study 26 presented an elegant transcendental equation F(λ ) := 0 that is to be solved in order to obtain the eigenvalues λ of the diffusion operator. The eigenvalues are the roots of F(λ ) where λ is an auxiliary variable considered when F is evaluated on a continuous range.…”

Section: Analytical Solutionmentioning

confidence: 99%

“…that link compartments together 26 . The problem of finding eigenvalues is therefore reduced to that of finding the roots of that transcendental equation…”

Section: Analytical Solutionmentioning

confidence: 99%

“…Here, ı denotes the imaginary unit √ −1. The analytical diffusion signal is calculated using an expression derived for uniform initial seeding 26 . In equation (19), we approximate the integral numerically using trapezoidal integration over the finely-discretised domain.…”

Section: Dwi and Other Potential Applicationsmentioning

confidence: 99%

“…This was first suggested by Tanner et al 25 to estimate the DWI signal in a system of equi-spaced parallel plates. Very recently, Moutal et al 26 presented a semi-analytical method to obtain the particle density distribution anywhere in a domain with arbitrary barriers.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Random walk diffusion simulations in semi-permeable layered media with varying diffusivity

Alemany¹,

Rose²,

Garnier-Brun³

et al. 2022

Preprint

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In this paper we present analytical and random walk based solutions to diffusion in semi-permeable layered media with varying diffusivity. We propose a new random walk transit model (hybrid model) based on treating the membrane permeability and the change in diffusion as two infinitesimal separate phenomena. By conducting an extensive analytical flux analysis, the performance of our hybrid model is compared with a commonly used membrane model (reference model). We numerically demonstrate the limitations of the reference model and show the capability of our new model to overcome these restrictions. The suitability of both random walk transit models for the application to simulations of the diffusion tensor cardiovascular magnetic resonance (DT-CMR) is assessed in a histology-based domain. We consider a larger range of permeabilities to show the potential of our model to other possible applications beyond biological tissue.

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Section: Analytical Solutionmentioning

confidence: 99%

“…that link compartments together 26 . The problem of finding eigenvalues is therefore reduced to that of finding the roots of that transcendental equation…”

Section: Analytical Solutionmentioning

confidence: 99%

Section: Dwi and Other Potential Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Random walk diffusion simulations in semi-permeable layered media with varying diffusivity

Alemany¹,

Rose²,

Garnier-Brun³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. The matrix formalism has been previously applied with permeable membranes for simple geometries such as multilayered plates, spheres, or cylinders, 34,38 with sometimes thousands of permeable cell membranes in the one‐dimensional case 39 . The formulation we propose is valid for arbitrary multi‐compartment geometries with three‐dimensional shapes.…”

Section: Introductionmentioning

confidence: 99%

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

Agdestein

Tran²,

2021

NMR in Biomedicine

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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.

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Homogenization of a Multivariate Diffusion with Semipermeable Interfaces

Aryasova,

Pavlyukevich,

Pilipenko

2024

J Theor Probab

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We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

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Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

Cited by 29 publications

References 96 publications

Random walk diffusion simulations in semi-permeable layered media with varying diffusivity

Random walk diffusion simulations in semi-permeable layered media with varying diffusivity

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

Homogenization of a Multivariate Diffusion with Semipermeable Interfaces

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