Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
A partition of unity finite element method for computational diffusion MRI
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).ORCID iD: 0000-0002-3213-0040
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).ORCID iD: 0000-0002-1695-8809
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).ORCID iD: 0000-0003-4256-0463
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The Bloch-Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries. Many efforts have been made to solve the equation but there is still missing an efficient high performance computing framework. In this work, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations use extended finite elements with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as operator splitting for the exterior boundary conditions. The method appears to be straightforward to implement and it is implemented in the FEniCS for moderate-scale simulations and in the FEniCS-HPC for the large-scale simulations. The accuracy of the resulting method is validated numerically and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available. Finally, we do simulations on a complex neuron to study how the signals decay under the effect of the permeable membrane and to show that the method can be used to simulate for complex geometries that we have not done before.

Highlights:

  • The discontinuity in the magnetization across the interior interfaces of the medium is weakly imposed, allowing generalization to arbitrary order finite elements.
  • Spin exchange across the external boundaries is implemented by weakly imposing an artificial, high permeability, condition, allowing generalization to non-matching meshes.
  • Thus, optimal convergence with respect to the space discretization is achieved.
  • The second-order Crank-Nicolson method is chosen for the time discretization to reduce oscillations at high gradient strengths and allows for larger time-step sizes.
  • The method is of a high level of simplicity and suitable for parallelization.
  • An efficient open-source code is implemented in the FEniCS and FEniCS-HPC platforms.
Keywords [en]
extended finite element, computational diffusion MRI, interface conditions, pseudo-periodic conditions, Bloch-Torrey equation, FEniCS, FEniCS-HPC.
National Category
Computational Mathematics Computer Sciences Bioinformatics (Computational Biology)
Research subject
Applied and Computational Mathematics; Computer Science; Biological Physics
Identifiers
URN: urn:nbn:se:kth:diva-219268OAI: oai:DiVA.org:kth-219268DiVA, id: diva2:1162071
Note

QC 20171218

Available from: 2017-12-02 Created: 2017-12-02 Last updated: 2018-04-11Bibliographically approved
In thesis
1. High-Performance Finite Element Methods: with Application to Simulation of Diffusion MRI and Vertical Axis Wind Turbines
Open this publication in new window or tab >>High-Performance Finite Element Methods: with Application to Simulation of Diffusion MRI and Vertical Axis Wind Turbines
2018 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The finite element methods (FEM) have been developed over decades, and together with the growth of computer engineering, they become more and more important in solving large-scale problems in science and industry. The objective of this thesis is to develop high-performance finite element methods (HP-FEM), with two main applications in mind: computational diffusion magnetic resonance imaging (MRI), and simulation of the turbulent flow past a vertical axis wind turbine (VAWT). In the first application, we develop an efficient high-performance finite element framework HP-PUFEM based on a partition of unity finite element method to solve the Bloch-Torrey equation in heterogeneous domains. The proposed framework overcomes the difficulties that the standard approaches have when imposing the microscopic heterogeneity of the biological tissues. We also propose artificial jump conditions at the external boundaries to approximate the pseudo-periodic boundary conditions which allows for the water exchange at the external boundaries for non-periodic meshes. The framework is of a high level simplicity and efficiency that well facilitates parallelization. It can be straightforwardly implemented in different FEM software packages and it is implemented in FEniCS for moderate-scale simulations and in FEniCS-HPC for the large-scale simulations. The framework is validated against reference solutions, and implementation shows a strong parallel scalability. Since such a high-performance simulation framework is still missing in the field, it can become a powerful tool to uncover diffusion in complex biological tissues. In the second application, we develop an ALE-DFS method which combines advanced techniques developed in recent years to simulate turbulence. We apply a General Galerkin (G2) method which is continuous piecewise linear in both time and space, to solve the Navier-Stokes equations for a rotating turbine in an Arbitrary Lagrangian-Eulerian (ALE) framework. This method is enhanced with dual-based a posterior error control and automated mesh adaptation. Turbulent boundary layers are modeled by a slip boundary condition to avoid a full resolution which is impossible even with the most powerful computers available today. The method is validated against experimental data of parked turbines with good agreements. The thesis presents contributions in the form of both numerical methods for high-performance computing frameworks and efficient, tested software, published open source as part of the FEniCS-HPC platform.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 34
Series
TRITA-EECS-AVL ; 2018:3
Keywords
High performance finite element method, computational diffusion MRI, turbulent flow, vertical axis wind turbine.
National Category
Computer and Information Sciences Mathematics
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-225952 (URN)978-91-7729-708-6 (ISBN)
Presentation
2018-05-08, Q33, Osquldas väg 6B, Stockholm, 10:15 (English)
Opponent
Supervisors
Note

QC 20180411

Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-02Bibliographically approved

Open Access in DiVA

No full text in DiVA

Authority records BETA

Nguyen, Van DangJansson, JohanHoffman, Johan

Search in DiVA

By author/editor
Nguyen, Van DangJansson, JohanHoffman, Johan
By organisation
Computational Science and Technology (CST)
Computational MathematicsComputer SciencesBioinformatics (Computational Biology)

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 238 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf