A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues Dang Van Nguyen To cite this version: Dang Van Nguyen. A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues. Mathematical Physics [math-ph]. Ecole Polytech- nique X, 2014. English. pastel-00957750 HAL Id: pastel-00957750 https://pastel.archives-ouvertes.fr/pastel-00957750 Submitted on 11 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse Présentée pour obtenir le grade de DOCTEUR DE L’ÉCOLE POLYTECHNIQUE Spécialité : Mathématiques Appliquées par Dang Van NGUYEN A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues Soutenue le 7 Mars 2014 devant le jury composé de : M. Denis GREBENKOV Co-Directeur de Thèse M. Houssem HADDAR Examinateur M. Johan JANSSON Rapporteur M. Denis LE BIHAN Rapporteur Mme. Jing-Rebecca LI Directeur de Thèse M. Rachid TOUZANI Rapporteur Acknowledgment First and foremost I would like to express my deepest gratitude to my supervisors, Dr. Jing-Rebecca Li and Dr. Denis Grebenkov, who have supported me during my thesis with their enthusiasm and patience. Knowing and working with them is a great chance for me to enrich my knowledge not only about my current work but also about my own life. I would like to thank Dr. Denis Le Bihan and Dr. Cyril Poupon who have facilitated good conditions for my work during the time I worked at Neurospin, CEA Saclay. I am very thankful for all helpful discussions in the launchpad with Fenics supporters during the course of our code development. I am grateful to J. Jansson and J. Hoffman at the KTH, Sweden for their welcome and help during and after our visit. Related to the use of the Salome platform, we would like to thank Christophe Bourcier, Nathalie Gore and Serge Rehbinder in the Laboratoire de Genie Logiciel et de Simulation at the CEA, Saclay, France. In current work related to the formulation and simulation on the new interface conditions, I also wish to thank Dr. Houssem Haddar and Caubet Fabien for their help. Finally, I would like to thank all my friends and my family, especially my wife Bich Loan for their encouragement to help me overcome some difficult periods. This work is funded in part by the ANR (Agence Nationale de la Recherche), project ‘SIMUDMRI’. Contents Thesis summary 1 List of publications 5 1 Introduction 7 1.1 Background ................................ 7 1.2 Equations of diffusion MRI ....................... 8 1.3 The apparent diffusion coefficient .................... 11 1.4 Multiple compartments Bloch-Torrey PDE . 13 1.5 Laplace PDE for the homogenized diffusion tensor . 15 1.6 Explicit solutions and approximations . 16 1.6.1 Explicit solution ......................... 16 1.6.2 Narrow pulse approximation . 17 1.6.3 Gaussian phase approximation . 17 2 A finite elements method to solve the multi-compartment Bloch- Torrey equation 21 2.1 Introduction ................................ 21 2.2 Method .................................. 23 2.2.1 Interface conditions: allowing jumps in the finite elements solution .............................. 23 2.2.2 Boundary conditions: transformation of the Bloch-Torrey PDE 25 2.2.3 When the cell interface touches the computational boundary 28 2.2.4 Time stepping for the Bloch-Torrey PDE using explicit RKC 29 2.3 Implementation on FEniCS Finite Elements platform and mesh gen- erator Salome ............................... 30 2.4 Numerical results ............................. 31 2.4.1 Local stiffness matrix for the interface jump condition . 31 2.4.2 Different positions of computational domain in periodic struc- tures ................................ 33 2.4.3 Spatial discretization ....................... 34 2.4.4 Computational efficiency ..................... 39 2.4.5 Timing on heterogeneous domain . 41 2.5 Conclusion ................................ 42 3 Applications of FEM-RKC code to study diffusion MRI 43 3.1 Effective diffusion tensor from homogenization . 43 3.2 Layer compartments and interface conditions . 47 3.2.1 Myelin sheath and slow diffusion layer around cell membranes 47 3.2.2 Three-compartment model of biological tissue . 48 3.2.3 The relationship between the two- and three-compartment models ............................... 49 3.2.4 Interface conditions for anisotropic diffusion inside the mem- brane layer ............................ 50 3.2.5 Simulations ............................ 53 3.3 Validation of Kärger model of the dMRI signal . 56 3.3.1 The Kärger model ........................ 56 3.3.2 Simulations ............................ 58 3.4 Conclusion ................................ 60 4 Modeling the diffusion magnetic resonance imaging signal inside neurons 61 4.1 Introduction ................................ 61 4.2 The Bloch-Torrey PDE in a neuron ................... 64 4.3 1D model of neurites ........................... 65 4.3.1 Problem formulation ....................... 65 4.3.2 Linear basis functions for 3D line segments . 68 4.3.3 Simulations ............................ 69 4.4 The “3D+1D” model ........................... 70 4.5 The disconnected “3D | 1D” model ................... 72 4.5.1 Exchange between neurites and soma . 72 4.5.2 The disconnected “3D | 1D” model . 74 4.5.3 Timing .............................. 77 4.6 The ADC inside neurites trees ..................... 77 4.6.1 Generation of random neurites trees . 77 4.6.2 Least squares problem ...................... 80 4.6.3 Effective length of neurites trees with variable lengths . 82 4.6.4 Fitting ADCN by two DN s ................... 85 4.6.5 A relationship between neurites trees with two lengths and single length ........................... 86 4.7 Analytical upper and lower bounds of ADC and signal inside neurites trees .................................... 87 4.8 Conclusion ................................ 89 5 Conclusion and Future work 91 5.1 Conclusion ................................ 91 5.2 Future work on mesh generation .................... 92 5.3 Future work on optimizing the FEM-RKC code . 93 5.4 Ongoing applications of FEM-RKC to dMRI . 94 List of Figures 95 List of algorithms 101 Bibliography 103 Thesis summary Diffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, no- tably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal al- lows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We formulate and implement such a numerical approach based on the linear finite elements method (FEM) for the multi-compartment Bloch-Torrey partial differ- ential equation (PDE) that allows us to work with general gradient pulses and relatively complex geometries. To compute the long-time effective diffusion tensor predicted by homogenization theory, we also considered the solution of the related steady-state Laplace equations. Compatible with the linear finite elements discretization, we used linear approxima- tion of the cellular interfaces. The semi-permeable biological cell membranes were modeled by zero-volume elements to allow jumps in the finite elements solution on the membranes. The computational domain was periodically extended by applying the appropriate artificial boundary conditions. This allows us to reduce the edge effects caused by artificial boundaries. This finite elements discretization is second order accurate in space. The spatial discretization was then coupled to the adaptive explicit Runge-Kutta Chebyshev (RKC) time stepping method that is second order accurate in time. Being an explicit time stepping method, this choice has the advantage that the linear system to be solved at each time step involves only the mass matrix. Furthermore, using a transformation of the Bloch-Torrey PDE, we reduced the oscillations in the sought-after solution and simplified the implementation of the artificial boundary conditions. In this way, we were able to keep the mass matrix of our FEM real-valued and time-independent after applying the boundary conditions. Because it is real-valued, the mass matrix is only half the size compared to a naive implementation without the transformation. Due to the time independence, the mass matrix only needs to be assembled and the artificial boundary conditions 2 applied once before the time integration. In short, our method can bring improvements in the computational time, memory usage, and the accuracy of dMRI signal simulations. Our method was implemented on the FeniCS C++