A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging

Nguyen, Van Dang; Li, Jing-Rebecca; Grebenkov, Denis S.; Bihan, Denis Le

doi:10.1016/j.jcp.2014.01.009

Cited by 62 publications

(69 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There, the authors present a FE approach using first-order basis functions in space and an explicit second-order approximation in time, which does not make such constraining approximations. To the best of our knowledge, this latter paper by Nguyen et al [14] is the first to do so. Even though their approach allows to consider arbitrary geometries inside the volume of interest, it still has limitations in the way the meshes have to be generated, imposing a hard constraint as the symmetry of the nodal positions on the outermost faces.…”

Section: Introductionmentioning

confidence: 82%

“…Finally, to compute (9f), a discontinuous FE approach was considered. Under this method, the solution is allowed to be discontinuous at the compartment interfaces but not inside each region [14]. The discretisation was then obtained by doubling the nodes at the interfaces, each of them belonging to each region.…”

Section: Formulation Of Element Matrices For Polynomial Basis Functionsmentioning

confidence: 99%

“…Recently, a flexible FE formulation of the standard BT equation (i.e. without flow and transverse relaxation terms) has been proposed [14]. There, the authors present a FE approach using first-order basis functions in space and an explicit second-order approximation in time, which does not make such constraining approximations.…”

Section: Introductionmentioning

confidence: 99%

“…We extend the formulation given in [14] by considering the flow and relaxation terms, allowing a better representation of the phenomena under scope. We obtain parametric expressions of the corresponding matrices considering basis functions of arbitrary order.…”

Section: Introductionmentioning

confidence: 99%

“…This is crucial for achieving reliable solutions with a minimum number of iterations. We introduce an implicit scheme to solve the BT equation with similar computational load as the explicit method used in [14], which makes it highly competitive in the field. The presented framework is built on the basis of arbitrary discretisations without imposing special constraints to the geometrical meshes to be used.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains

Beltrachini

Taylor

Frangi

2015

Journal of Magnetic Resonance

View full text Add to dashboard Cite

Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request.

show abstract

Section: Introductionmentioning

confidence: 82%

Section: Formulation Of Element Matrices For Polynomial Basis Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains

Beltrachini

Taylor

Frangi

2015

Journal of Magnetic Resonance

View full text Add to dashboard Cite

show abstract

Large‐scale magnetic resonance simulations: A tutorial

Concilio

2020

Magnetic Reson in Chemistry

View full text Add to dashboard Cite

Computational modeling is becoming an essential tool in magnetic resonance to design and optimize experiments, test the performance of theoretical models, and interpret experimental data. Recent theoretical research and software development made possible simulations of large spin systems, for example, proteins with thousands of spins, in reasonable time. In the last few years, the Fokker–Planck formalism also re‐emerged due to its ability to handle spatial dynamics. The purpose of this tutorial is to describe advantages and disadvantages of the most common formalisms, the latest developments and strategies to improve the computational efficiency, and to guide users in the setting up of a simulation using theSpinachsoftware.

show abstract

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

Tran

Nguyen

2020

NMR in Biomedicine

Self Cite

View full text Add to dashboard Cite

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 biological cell 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. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

show abstract

A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging

Cited by 62 publications

References 32 publications

A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains

A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains

Large‐scale magnetic resonance simulations: A tutorial

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

Contact Info

Product

Resources

About