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A partition of unity finite element method for computational diffusion MRI. (English) Zbl 1416.65356
Summary: 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 approaches have been taken to solve the Bloch-Torrey equation but an efficient high performance computing framework is still missing. In this paper, 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 are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests 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 and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.
MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
92C55 Biomedical imaging and signal processing
65-04 Software, source code, etc. for problems pertaining to numerical analysis
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[1] Torrey, H. C., Bloch equations with diffusion terms, Phys. Rev., 104, 563-565, (1956)
[2] Topgaard, D., Multidimensional diffusion MRI, J. Magn. Reson., 275, 98-113, (2017)
[3] Stejskal, E. O.; Tanner, J. E., Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys., 42, 1, 288-292, (1965)
[4] Tanner, J. E., Transient diffusion in a system partitioned by permeable barriers. Application to NMR measurements with a pulsed field gradient, J. Chem. Phys., 69, 4, 1748-1754, (1978)
[5] Xu, J.; Does, M.; Gore, J., Numerical study of water diffusion in biological tissues using an improved finite difference method, Phys. Med. Biol., 52, 7, (2007)
[6] Yuan, Z.; Fish, J., Toward realization of computational homogenization in practice, Int. J. Numer. Methods Eng., 73, 361-380, (2008) · Zbl 1159.74044
[7] Hagslatt, H.; Jonsson, B.; Nyden, M.; Soderman, O., Predictions of pulsed field gradient NMR echo-decays for molecules diffusing in various restrictive geometries. Simulations of diffusion propagators based on a finite element method, J. Magn. Reson., 161, 2, 138-147, (2003)
[8] Loren, N.; Hagslatt, H.; Nyden, M.; Hermansson, A.-M., Water mobility in heterogeneous emulsions determined by a new combination of confocal laser scanning microscopy, image analysis, nuclear magnetic resonance diffusometry, and finite element method simulation, J. Chem. Phys., 122, 2, (2005)
[9] Moroney, B. F.; Stait-Gardner, T.; Ghadirian, B.; Yadav, N. N.; Price, W. S., Numerical analysis of NMR diffusion measurements in the short gradient pulse limit, J. Magn. Reson., 234, 165-175, (2013)
[10] Li, J.-R.; Calhoun, D.; Poupon, C.; Bihan, D. L., Numerical simulation of diffusion MRI signals using an adaptive time-stepping method, Phys. Med. Biol., 59, 2, 441, (2014)
[11] Nguyen, D. V.; Li, J.-R.; Grebenkov, D.; Bihan, D. L., A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging, J. Comput. Phys., 263, Supplement C, 283-302, (2014) · Zbl 1349.78070
[12] Beltrachini, L.; Taylor, Z. A.; Frangi, A. F., A parametric finite element solution of the generalised Bloch-Torrey equation for arbitrary domains, J. Magn. Reson., 259, Supplement C, 126-134, (2015)
[13] Russell, G.; Harkins, K. D.; Secomb, T. W.; Galons, J.-P.; Trouard, T. P., A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue, Phys. Med. Biol., 57, 4, N35, (2012)
[14] Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. Appl. Mech., 60, 2, 371-375, (1993) · Zbl 0775.73337
[15] Verwer, J. G.; Hundsdorfer, W. H.; Sommeijer, B. P., Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math., 57, 1, 157-178, (1990) · Zbl 0697.65072
[16] Larsson, F.; Runesson, K.; Saroukhani, S.; Vafadari, R., Computational homogenization based on a weak format of micro-periodicity for rve-problems, Comput. Methods Appl. Mech. Eng., 200, 1, 11-26, (2011) · Zbl 1225.74069
[17] Nguyen, V.-D.; Béchet, E.; Geuzaine, C.; Noels, L., Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation, Comput. Mater. Sci., 55, 390-406, (2012)
[18] Sandstöm, C.; Larsson, F.; Runesson, K., Weakly periodic boundary conditions for the homogenization of flow in porous media, Adv. Model. Simul. Eng. Sci., 1, 1, 12, (2014)
[19] Nguyen, V. D., (A FEniCS-HPC Framework for Multi-Compartment Bloch-Torrey Models, vol. 1, (2016)), 105-119, QC 20170509
[20] Melenk, J.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Eng., 139, 1, 289-314, (1996) · Zbl 0881.65099
[21] Wadbro, E.; Zahedi, S.; Kreiss, G.; Berggren, M., A uniformly well-conditioned, unfitted Nitsche method for interface problems, BIT Numer. Math., 53, 3, 791-820, (2013) · Zbl 1279.65134
[22] Hansbo, P.; Larson, M. G.; Zahedi, S., A cut finite element method for a stokes interface problem, Appl. Numer. Math., 85, Supplement C, 90-114, (2014) · Zbl 1299.76136
[23] Sommeijer, B.; Shampine, L.; Verwer, J., Rkc: an explicit solver for parabolic pdes, J. Comput. Appl. Math., 88, 2, 315-326, (1998) · Zbl 0910.65067
[24] Grebenkov, D. S., Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered structures, J. Magn. Reson., 205, 2, 181-195, (2010)
[25] Walter, A.; Gutknecht, J., Permeability of small nonelectrolytes through lipid bilayer membranes, J. Membr. Biol., 90, 3, 207-217, (1986)
[26] Nitsche, J., Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamb., 36, 1, 9-15, (1971) · Zbl 0229.65079
[27] Hansbo, P., Nitsche’s method for interface problems in computational mechanics, GAMM-Mitt., 28, 2, 183-206, (2005) · Zbl 1179.65147
[28] Logg, A.; Mardal, K.-A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element MethodThe FEniCS Book, (2012), Springer Verlag, xIII, 723, s.: ill
[29] FEniCS, Fenics project
[30] Fenics-hpc; FEniCS-HPC
[31] Jansson, N.; Jansson, J.; Hoffman, J., Framework for massively parallel adaptive finite element computational fluid dynamics on tetrahedral meshes, SIAM J. Sci. Comput., 34, 1, C24-C41, (2012) · Zbl 1237.68244
[32] Hoffman, J.; Jansson, J.; de Abreu, R. V.; Degirmenci, N. C.; Jansson, N.; Müller, K.; Nazarov, M.; Spühler, J. H., Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry, Comput. Fluids, 80, 10, 310-319, (2013) · Zbl 1284.76223
[33] Hoffman, J.; Jansson, J.; Degirmenci, C.; Jansson, N.; Nazarov, M., Unicorn: A Unified Continuum Mechanics Solver, (2012), Springer, Ch. 18
[34] Jansson, N.; Hoffman, J.; Jansson, J., Framework for massively parallel adaptive finite element computational fluid dynamics on tetrahedral meshes, SIAM J. Sci. Comput., 34, 1, C24-C41, (2012) · Zbl 1237.68244
[35] Kirby, R. C., FIAT: Numerical Construction of Finite Element Basis Functions, (2012), Springer, Ch. 13
[36] Hoffman, J.; Jansson, J.; Jansson, N.; Nazarov, M., Unicorn: a unified continuum mechanics solver, (Automated Solutions of Differential Equations by the Finite Element Method, (2011), Springer)
[37] Hoffman, J.; Jansson, J.; Jansson, N.; Johnson, C.; de Abreu, R. V., Turbulent flow and fluid-structure interaction, (Automated Solutions of Differential Equations by the Finite Element Method, (2011), Springer)
[38] Carim-Todd, L.; Bath, K. G.; Fulgenzi, G.; Yanpallewar, S.; Jing, D.; Barrick, C. A.; Becker, J.; Buckley, H.; Dorsey, S. G.; Lee, F. S.; Tessarollo, L., Endogenous truncated TrkB.T1 receptor regulates neuronal complexity and TrkB kinase receptor function in vivo, J. Neurosci., 29, 3, 678-685, (2009)
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