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

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...

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Vydáno v:	NMR in biomedicine Ročník 33; číslo 10; s. e4353 - n/a
Hlavní autoři:	Li, Jing‐Rebecca, Tran, Try Nguyen, Nguyen, Van‐Dang
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Oxford Wiley Subscription Services, Inc 01.10.2020 Wiley
Témata:	acceleration Bioengineering Biological products Bloch-Torrey equation Closed-form representations Complicated geometry Computation Computer Science Computer simulation controlled study Diffusion diffusion coefficient diffusion MRI diffusion weighted imaging Eigen decomposition Eigenvalues Eigenvalues and eigenfunctions Eigenvectors Encoding (symbols) extracellular matrix finite element analysis Finite element method finite elements Formalism Geometry Heterogeneous medium Imaging inverse Laplace transform Laplace eigenfunctions Laplace transforms Life Sciences magnetic field Magnetic field gradient Magnetic fields Magnetic resonance imaging Mathematical analysis Mathematical models MATLAB Matrix algebra matrix formalism Matrix methods Modeling and Simulation Neurons Numerical methods Operators (mathematics) oscillation Oscillation frequency Partial differential equations priority journal pyramidal nerve cell Representations Sequences Signal encoding Signal systems simulation Simulation and modeling Simulation framework surface property Tissue diffusion MRI Bloch-Torrey equation Laplace eigenfunctions simulation Matrix Formalism finite elements
ISSN:	0952-3480, 1099-1492, 1099-1492
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Shrnutí:	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. 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. The matrix formalism representation requires around 100 eigenmodes to match the reference signal when the cell geometry originates from realistic neurons. We 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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	0952-3480 1099-1492 1099-1492
DOI:	10.1002/nbm.4353