A partition of unity finite element method for computational diffusion MRI

Tran

et al. 2019

NeuroImage

Self Cite

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation. Under the assumption of negligible water exchange between compartments, the time-dependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a time-dependent Neumann boundary condition.

Section: Discussionmentioning

confidence: 99%

SpinDoctor: A MATLAB toolbox for diffusion MRI simulation

Tran

et al. 2019

NeuroImage

Self Cite

“…(A.3) shows the flux is also discontinuous. Following the same PUFEM approach proposed in [16], we obtain the following weak form…”

Section: Discussionmentioning

confidence: 99%

“…The pseudo-periodic BCs (Eq. 4) can be implemented weakly through the use of an artificial permeability coefficient, κ e [15,16]. The artificial permeability condition at the external boundaries take two equations for the master side and the slave side of the mesh.…”

Section: Discussionmentioning

confidence: 99%

“…(a) Neuron 04b_pyramidal7aACC (b) The mouse neuron embedded in a box (c) Strong parallel scaling of one-node and multi-node simulations Figure 11: The simulation performance is verified for a single computational node on the neuron 04b_pyramidal7aACC and for multiples nodes on the mouse neuron embedded in a box presented in Section 7.6 in [16]. The strong scaling on Tegner is good both on one node (32 CPUs) and multi-node (scaling up to 500 CPUs).…”

Section: (B3)mentioning

confidence: 94%

“…The methods imposed in our framework are based on the partition of unity finite element method (PUFEM) to manage the interface conditions [16]. Both weak and strong implementation of the periodicity with some advantages and disadvantages are included.…”

Section: Appendicesmentioning

confidence: 99%

See 2 more Smart Citations

Portable simulation framework for diffusion MRI

Journal of Magnetic Resonance

Leoni

Dancheva

et al. 2019

Self Cite

The numerical simulation of the diffusion MRI signal arising from complex tissue micro-structures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the Bloch-Torrey 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 elements 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 elements 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 elements 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 open-source software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations. * I am corresponding author Email addresses: vdnguyen@kth.se (Van-Dang Nguyen ), jjan@kth.se (Johan Jansson), jhoffman@kth.se (Johan Hoffman), demian.wassermann@inria.fr (Demian Wassermann), jingrebecca.li@inria.fr (Jing-Rebecca Li)

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

Tran

2020

NMR in Biomedicine

Self 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.