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Portable simulation framework for diffusion MRI

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A FEniCS-based framework brings portable, parallel diffusion MRI simulation to the cloud.

desk verdict A useful, reproducible cloud/HPC FEM framework for diffusion MRI, but the multi-compartment interface method only handles bipartite compartment topologies, so the advertised generality is narrower than claimed. read the letter →

arxiv 1908.01719 v1 pith:L6U2WWYI submitted 2019-08-05 cs.CE physics.comp-ph

Van-Dang Nguyen , Massimiliano Leoni , Tamara Dancheva , Johan Jansson , Johan Hoffman , Demian Wassermann , Jing-Rebecca Li This is my paper · ORCID
classification cs.CEphysics.comp-ph MSC 65M6092C55
keywords diffusionMRIsimulationBloch-TorreyequationfiniteelementmethodFEniCSpartitionofunitypermeableinterfacespseudo-periodicboundaryconditionscloudcomputing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that finite-element simulation of the diffusion MRI signal no longer has to be a specialist, installation-heavy activity. It presents an open-source Python framework, built on FEniCS and packaged in Docker and Singularity containers, that solves the Bloch-Torrey equation for single- and multi-compartment geometries with permeable membranes, T2 relaxation, arbitrary diffusion-encoding sequences, and periodic boundary conditions. The authors validate the computed signals against the matrix-formalism reference for multilayered disks, spheres, cylinders, and tori, and they demonstrate that the same solver runs in Google Colaboratory notebooks and with MPI on cloud or HPC clusters. If the framework works as described, it gives the community a free, reproducible, parallel alternative to the Matlab-based SpinDoctor toolbox for FEM-based diffusion MRI simulation.

What carries the argument

The load-bearing object is the PUFEM interface treatment: an element-wise constant phase function $\Phi_h$ taking only the values 0 and 1 distinguishes the two sides of every interface, so the permeability jump condition $\{\mathbf{D}\nabla U \cdot \mathbf{n}\} = -\kappa[U]$ can be written into the weak form as a single bilinear term. Because $\Phi_h$ is two-valued, the framework requires every multi-compartment geometry to be sorted into two groups, odd and even, with no interior interfaces inside either group. The other key device is the transformation $u = U \exp(i\gamma F(t) \mathbf{g} \cdot \mathbf{x})$, which converts the pseudo-periodic boundary conditions of a periodically repeated box into ordinary periodic conditions and makes the time-dependent phase factor part of the PDE rather than the boundary. Containers, Docker for notebooks and Singularity for HPC, carry the whole FEniCS stack, and the $\theta$-method with a fixed time step is the time discretization.

What would settle it

Take a three-compartment domain in which compartment A touches both B and C, with B and C not touching each other. Running the framework's CreatePhaseFunc on this arrangement should reveal that the two-group odd-even requirement cannot encode A-B and A-C simultaneously; if it nevertheless produces a signal, comparing that signal to the matrix-formalism reference for the same geometry would settle whether the interface treatment remains valid outside its stated assumption.

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Extended reading notes

Core claim

The central claim is that a portable, containerized FEniCS-based implementation of the Bloch-Torrey equation can serve as a practical simulation tool for diffusion MRI. The method solves the complex transverse magnetization in a finite-element discretization, imposes permeable interface conditions through a partition-of-unity finite element method with a two-valued phase function, and handles periodic domains by transforming the pseudo-periodic boundary conditions into ordinary periodic ones through $u = U \exp(i\gamma F(t) \mathbf{g} \cdot \mathbf{x})$. Numerical experiments show that simulated signals match reference matrix-formalism signals for three-layered disks, spheres, tori and cylinders, that arbitrary gradient waveforms such as PGSE, double-PGSE and OGSE can be specified symbolically, and that the same code runs on a free cloud notebook, a local container, and a multi-node HPC cluster with MPI. The paper presents this as a free, reproducible, Python-based alternative to existing finite-element toolboxes.

Load-bearing premise

The framework's multi-compartment interface conditions rest on the assumption that every compartment can be split into two groups with no interfaces inside either group; a tissue with connected compartments within a group cannot be represented by the two-valued phase function.

Editorial extensions

If this is right

  • Diffusion MRI researchers can run finite-element simulations from a web browser through Google Colaboratory, with no local FEniCS installation.
  • The same Python code scales to MPI-parallel execution on cloud platforms and HPC clusters, with reported timings of a few minutes to about an hour per b-value depending on mesh size.
  • Users can specify arbitrary gradient waveforms symbolically, including PGSE, double-PGSE, and cosine or sine OGSE, with automatic conversion between b-value and gradient strength.
  • Tissue models can include cell-wise discontinuous diffusion tensors and T2 relaxation, and thin or tubular structures can be simulated on 1D manifolds instead of full 3D meshes.
  • Because the framework is open source and containerized, published notebooks can be re-run directly, which supports reproducible comparison of new diffusion MRI methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The two-group restriction of the phase function means the framework, as presented, cannot handle geometries where one compartment connects to two others that are themselves in the same group; extending $\Phi_h$ to a genuinely multi-valued indicator would be the natural generalization.
  • Adopting Python ODE solvers that handle mass matrices and sparse Jacobians could close the gap with SpinDoctor's adaptive time-stepping and likely reduce the fixed-step $\theta$-method cost.
  • The validation set is concentric layered geometries against matrix formalism; applying the framework to non-concentric or branching connected compartments would be a testable next step before relying on it for realistic neuron networks.
  • If the claimed portability holds, the framework lowers the barrier to reproducing published diffusion MRI simulations, which may shift more of the field's method-development workflow onto cloud notebooks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a portable, open-source, Python-based simulation framework for diffusion MRI built on FEniCS, containerized with Docker/Singularity, and deployable on Google Colaboratory and HPC clusters. The framework solves the Bloch-Torrey equation with PUFEM-based interface conditions, pseudo-periodic boundary conditions, arbitrary gradient waveforms, manifold discretizations, and MPI parallelism. Numerical validation compares simulated signals against matrix-formalism references for multilayered disks, spheres, tori, and cylinders, and the paper reports timings and parallel scaling. The central claim is that the framework provides a general, reproducible, multi-compartment simulation tool.

Significance. If the claims are properly qualified, this is a valuable software contribution to computational diffusion MRI: the code is open source, containerized, and accompanied by ready-to-run Colab notebooks, which directly supports reproducibility. The validation against matrix formalism uses an independent external reference with no fitted parameters, and the timing and parallel-scaling experiments are concrete. The main value is as a Python-based alternative to SpinDoctor with cloud/HPC portability. However, the demonstrated generality is narrower than advertised because of the two-valued phase-function restriction and the incomplete validation of arbitrary gradient sequences.

major comments (3)
  1. [Section 3.2, Eq. (A.6)] The multi-compartment interface treatment is restricted to compartment topologies whose adjacency graph is bipartite. The phase function Φh takes only two values, so all compartments must be partitioned into an even group and an odd group with no interfaces inside either group; the paper states this explicitly and Fig. 2 illustrates it. The abstract and introduction, however, promise simulation of "complex tissue micro-structures" and general "multi-compartment domains" without this qualification. All numerical examples (multilayered disk, sphere, torus, cylinder, neuron-in-box) satisfy this restriction, so the demonstrated results remain valid, but the scope of the advertised generality must be narrowed or the method must be extended to handle non-bipartite compartment adjacency graphs.
  2. [Section 4, Fig. 8] Validation of the "arbitrary temporal profiles" feature is incomplete. The text states that simulated signals match the matrix-formalism reference for PGSE and cos-OGSE only, but the figure also displays sin-OGSE, Trapezoidal PGSE, Double PGSE, and Double Trapezoidal PGSE without reference comparisons. Since general-sequence support is a central advertised capability, the authors should either report quantitative errors against the matrix-formalism reference for all f(t) shown, or state explicitly that those sequences are demonstrated without reference validation.
  3. [Section 4, Figs. 4–7] The accuracy claim "match very well" is never supported by quantitative error metrics. The paper would be strengthened by reporting a relative error norm (e.g., max or L2 relative difference between the FEM signal and the matrix-formalism signal) for each geometry and b-value range; without such numbers, the reader cannot judge the accuracy across mesh resolutions and time steps, and the stated Δt choices are not justified.
minor comments (5)
  1. [Abstract] "Completely portable across multiple platforms" is stronger than what is demonstrated; the workflow is tested on Ubuntu-based Docker/Singularity and Colab, not across the full range of platforms implied by "completely."
  2. [Introduction and Conclusion] The introduction reads "benefits from from long-established" with a duplicated word, and the conclusion spells "muti-compartment" instead of "multi-compartment."
  3. [Section 4] The phrase "time step size of Δt = 200µm" should use microseconds (200 µs) rather than micrometers; the same unit error appears in the MultilayeredStructures paragraph.
  4. [Section 4, Fig. 5] The sentence "ten times as larger" should read "ten times larger," and the comparison between the two timesteps should be rewritten to clarify which method requires the smaller step.
  5. [Section 5.3 vs. Section 3.3] The script-based parallel interface is stated not to support the artificial-permeability implementation; this limitation should be noted in Section 3.3, where both periodic options are described, rather than appearing only in the parallelization example.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the numerical claims are validated against an external matrix-formalism benchmark, and no fitted parameter is relabeled as a prediction.

full rationale

The paper's central numerical claim—that the simulated diffusion MRI signals match reference signals—is checked against an independent external benchmark, the matrix formalism of Grebenkov [45], not against the framework's own equations or fitted values. The simulation parameters are fixed inputs, not calibrated to the reference signals: 'Unless stated otherwise, the simulations were performed for a PGSE with Δ = 43100 µs, δ = 10600 µs, b−values between 0 and 10000 s/mm², and the diffusion coefficient of D = 3×10⁻³ mm²/s... The simulated signals are compared to the reference ones computed by the matrix formalism (MF) method [45].' The PUFEM interface treatment is inherited from the authors' prior Journal of Computational Physics paper [16], and the pseudo-periodic BCs are based on prior work [10, 12, 15, 16], but this is ordinary reuse of published methods; the load-bearing accuracy evidence in this paper is the comparison against [45], which is external to the fitted values and to the authors' own formulas. The two-valued phase function restriction stated in Section 3.2 ('the compartments need to be sorted into two groups oddgroup and evengroup... in each group, however, there is no interface between two compartments') limits the framework's multi-compartment generality to bipartite compartment topologies, but this is an explicit scope limitation, not a circular reduction. No equation in the paper defines a predicted quantity in terms of the very quantity it is supposed to predict, and no fitted parameter is renamed as a prediction. Therefore no circular step is present, and the self-citations are not load-bearing in the sense required to raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The framework introduces no new physical entities and fits no free parameters. Its central claim rests on the Bloch-Torrey model, the matrix formalism benchmark, and inherited PUFEM/periodic-BC numerical machinery from the authors' prior papers. The most paper-specific axiom is the two-group partition requirement for multi-compartment interfaces, which is a stated limitation of the phase-function approach.

assumptions (6)
  • domain assumption Bloch-Torrey equation with permeability interface conditions models the diffusion MRI signal (Eqs. 1-2).
    Standard physical model in dMRI simulation, cited to Torrey [22] and Tanner [23].
  • standard math Matrix formalism provides exact reference signals for multilayered structures.
    Used as ground truth in Section 4; independent benchmark from Grebenkov [45].
  • standard math The transformation u = U e^{i γ F(t) g·x} converts pseudo-periodic BCs to periodic BCs (Appendix A).
    Derivation from [10,12]; the paper applies it without re-proving.
  • standard math PUFEM with element-wise constant phase function Φh accurately imposes discontinuous magnetization and flux interface conditions.
    Method adopted from authors' prior JCP paper [16]; treated as established.
  • ad hoc to paper Multi-compartment domains can be partitioned into two groups with no intra-group interfaces.
    Stated in Section 3.2 and Fig. 2; restricts applicability to bipartite-like compartment arrangements.
  • standard math Artificial permeability κe = max{D/h} with operator splitting gives stable approximation of pseudo-periodic BCs.
    Based on Nitsche's method [49,50] and authors' prior work [15,16].

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Cite this review

Pith. "Pith review of Portable simulation framework for diffusion MRI." pith.science (2026). https://pith.science/paper/L6U2WWYI

@misc{pith2026190801719,
  author       = {Pith},
  title        = {Pith review of: Portable simulation framework for diffusion MRI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6U2WWYI}},
  note         = {Machine review of arXiv:1908.01719}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 1908.01719 by the authors.

Figure 1
Figure 1. A composed domain Ω = Ω0 ∪ Ω1 (a), and a PGSE sequence (b). Oscillating Gradient Spin Echo (OGSE) [25], nonstandard diffusion sequences such as double diffusion encoding [26, 27, 28, 29] and multidimensional diffusion encoding [30] can be modelled. Concerning the boundary conditions (BCs) on the exterior boundaries ∂Ω, there are two options that are very often employed. One is placing the spins to be simulated suffi… view at source ↗
Figure 2
Figure 2. For multi-compartment domains, the compartments need to be sorted into two [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. A typical Google Colaboratory notebook for diffusion MRI simulation. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Simulations of diffusion inside a three-layered disk of [5 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The artificial permeable method approaches the pseudo-periodic BCs. The time [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Three-layered structures and their signals for a PGSE with ∆ = 43100 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: T2−effects of diffusion inside a three-layered cylinder for a PGSE with ∆ = δ = 10ms, permeability κ = 10−5m/s to the magnetization at b = 4000s/mm2 (a, b, c), and to the signals for b between 0 and 4000s/mm2 (d). (a) Signals (b) Temporal profiles [PITH_FULL_IMAGE:fig…
Figure 8
Figure 8. Figure 8: Simulated signals inside a disk of radius 5 [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: A comparison between signals inside a neuron from the drosophila melanogaster for [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: The mesh of an extracellular space with 462 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]
Figure 11
Figure 11. Figure 11: The simulation performance is verified for a single computational node on the [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

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Works this paper leans on

51 extracted references · 35 canonical work pages

  1. [1]

    B. D. Hughes, Random walks and random environments / Barry D. Hughes, Clarendon Press Oxford ; New York, 1995

    work page 1995

  2. [2]

    C.-H. Yeh, B. Schmitt, D. Le Bihan, J.-R. Li-Schlittgen, C.-P. Lin, C. Poupon, Diffusion microscopist simulator: A general monte carlo simulation system for diffusion magnetic resonance imaging, PLoS One 8 (10) (2013) e76626, pONE-D-13-18755[PII]. doi:10.1371/journal. pone.0076626. URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3794953/

    doi:10.1371/journal 2013

  3. [3]

    M. G. Hall, D. C. Alexander, Convergence and parameter choice for monte- carlo simulations of diffusion mri, IEEE Transactions on Medical Imaging 28 (9) (2009) 1354–1364. doi:10.1109/TMI.2009.2015756

    arXiv 2009

  4. [4]

    Palombo, C

    M. Palombo, C. Ligneul, C. Najac, J. Le Douce, J. Flament, C. Escartin, P. Hantraye, E. Brouillet, G. Bonvento, J. Valette, New paradigm to assess brain cell morphology by diffusion-weighted mr spectroscopy in vivo, Proc Natl Acad Sci U S A 113 (24) (2016) 6671–6676, 201504327[PII]. doi: 10.1073/pnas.1504327113. URL http://www.ncbi.nlm.nih.gov/pmc/articles...

    work page doi:10.1073/pnas.1504327113 2016

  5. [5]

    K. V. Nguyen, E. H. Garzon, J. Valette, Efficient gpu-based monte-carlo simulation of diffusion in real astrocytes recon- structed from confocal microscopy, Journal of Magnetic Reso- nancedoi:https://doi.org/10.1016/j.jmr.2018.09.013. URL http://www.sciencedirect.com/science/article/pii/ S1090780718302386

    work page doi:10.1016/j.jmr.2018.09.013 2018

  6. [6]

    P. A. Cook, Y. Bai, S. Nedjati-Gilani, K. K. Seunarine, M. G. Hall, G. J. M. Parker, D. C. Alexander, Camino: Open-source diffusion-mri reconstruc- tion and processing, 2006

    work page 2006

  7. [7]

    S. N. Hwang, C.-L. Chin, F. W. Wehrli, D. B. Hackney, An image-based finite difference model for simulating restricted diffusion, Magnetic Reso- nance in Medicine 50 (2) (2003) 373–382. doi:10.1002/mrm.10536. URL http://dx.doi.org/10.1002/mrm.10536

    work page doi:10.1002/mrm.10536 2003

  8. [8]

    J. Xu, M. Does, J. Gore, Numerical study of water diffusion in biological tissues using an improved finite difference method, Physics in Medicine and Biology 52 (7). URL http://view.ncbi.nlm.nih.gov/pubmed/17374905

    work page arXiv

Show all 51 references
  1. [9]

    K. D. Harkins, J.-P. Galons, T. W. Secomb, T. P. Trouard, Assessment of the effects of cellular tissue properties on ADC measurements by numerical simulation of water diffusion, Magn. Reson. Med. 62 (6) (2009) 1414–1422. URL http://dx.doi.org/10.1002/mrm.22155 25

    2009 doi

  2. [10]

    Russell, K

    G. Russell, K. D. Harkins, T. W. Secomb, J.-P. Galons, T. P. Trouard, A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue, Physics in Medicine and Biology 57 (4) (2012) N35. URL http://stacks....

    2012

  3. [11]

    B. F. Moroney, T. Stait-Gardner, B. Ghadirian, N. N. Yadav, W. S. Price, Numerical analysis of NMR diffusion measurements in the short gradient pulse limit, Journal of Magnetic Resonance 234 (0) (2013) 165–175. URL http://www.sciencedirect.com/science/article/pii/ S1090780713001572

    2013

  4. [12]

    D. V. Nguyen, J.-R. Li, D. Grebenkov, D. L. Bihan, A finite elements method to solve the blochtorrey equation applied to diffusion magnetic resonance imaging, Journal of Computational Physics 263 (Supplement C) (2014) 283 – 302. doi:https://doi.org/10.1016/j.jcp.2014.01.009. URL...

    2014 doi

  5. [13]

    Beltrachini, Z

    L. Beltrachini, Z. A. Taylor, A. F. Frangi, A parametric finite element solution of the generalised blochtorrey equation for arbitrary domains, Journal of Magnetic Resonance 259 (Supplement C) (2015) 126 – 134. doi:https://doi.org/10.1016/j.jmr.2015.08.008. URL http://www.scien...

    2015 doi

  6. [14]

    D. V. Nguyen, J. R. Li, D. S. Grebenkov, D. L. Bihan, Modeling the diffu- sion magnetic resonance imaging signal inside neurons, Journal of Physics: Conference Series 490 (1) (2014) 012013. URL http://stacks.iop.org/1742-6596/490/i=1/a=012013

    2014

  7. [15]

    V. D. Nguyen, A FEniCS-HPC framework for multi-compartment Bloch- Torrey models, Vol. 1, 2016, pp. 105–119, QC 20170509. URL https://www.eccomas2016.org/

    2016

  8. [16]

    Nguyen, J

    V.-D. Nguyen, J. Jansson, J. Hoffman, J.-R. Li, A partition of unity finite element method for computational diffusion mri, Jour- nal of Computational Physics 375 (2018) 271 – 290. doi:https: //doi.org/10.1016/j.jcp.2018.08.039. URL http://www.sciencedirect.com/science/article/pi...

    2018 doi

  9. [17]

    Nguyen, J

    V.-D. Nguyen, J. Jansson, H. T. A. Tran, J. Hoffman, J.-R. Li, Diffusion mri simulation in thin-layer and thin-tube media using a discretization on manifolds, Journal of Magnetic Resonance 299 (2019) 176 – 187. doi:https://doi.org/10.1016/j.jmr.2019.01.002. URL http://www.scienc...

    2019 doi

  10. [18]

    Li, V.-D

    J.-R. Li, V.-D. Nguyen, T. Nguyen Tran, J. Valdman, B. Cong Trang, K. Van Nguyen, V. D. Thach Son, H. A. Tran, H. T. A. Tran, T. M. Phuong Nguyen, SpinDoctor: a Matlab toolbox for diffusion MRI simulation, arXiv e-prints (2019) arXiv:1902.01025arXiv:1902.01025

    2019 arXiv

  11. [19]

    Logg, K.-A

    A. Logg, K.-A. Mardal, G. N. Wells, Automated solution of differential equations by the finite element method : the FEniCS book, Springer Verlag, 2012, xIII, 723 s. : ill

    2012

  12. [20]

    FEniCS, Fenics project, http://www.fenicsproject.org

  13. [21]

    J. S. Hale, L. Li, C. N. Richardson, G. N. Wells, Containers for portable, productive, and performant scientific computing, Computing in Science Engineering 19 (6) (2017) 40–50. doi:10.1109/MCSE.2017.2421459

    2017

  14. [22]

    H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev. 104 (1956) 563–565. doi:10.1103/PhysRev.104.563. URL https://link.aps.org/doi/10.1103/PhysRev.104.563

    1956 doi

  15. [23]

    J. E. Tanner, Transient diffusion in a system partitioned by permeable barriers. application to nmr measurements with a pulsed field gradient, The Journal of Chemical Physics 69 (4) (1978) 1748–1754. arXiv:https: //doi.org/10.1063/1.436751, doi:10.1063/1.436751. URL https://doi....

    1978 doi

  16. [24]

    E. O. Stejskal, J. E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, The Journal of Chemical Physics 42 (1) (1965) 288–292. URL http://dx.doi.org/10.1063/1.1695690

    1965 doi

  17. [25]

    M. D. Does, E. C. Parsons, J. C. Gore, Oscillating gradient measurements of water diffusion in normal and globally ischemic rat brain, Magn. Reson. Med. 49 (2) (2003) 206–215. doi:10.1002/mrm.10385

    2003 doi

  18. [26]

    Shemesh, S

    N. Shemesh, S. N. Jespersen, D. C. Alexander, Y. Cohen, I. Drobn- jak, T. B. Dyrby, J. Finsterbusch, M. A. Koch, T. Kuder, F. Laun, M. Lawrenz, H. Lundell, P. P. Mitra, M. Nilsson, E. zarslan, D. Top- gaard, C.-F. Westin, Conventions and nomenclature for double diffusion encodi...

    2016

  19. [27]

    Dhital, M

    B. Dhital, M. Reisert, E. Kellner, V. G. Kiselev, Intra-axonal dif- fusivity in brain white matter, NeuroImage 189 (2019) 543 – 550. doi:https://doi.org/10.1016/j.neuroimage.2019.01.015. URL http://www.sciencedirect.com/science/article/pii/ S1053811919300151 27

    2019 doi

  20. [28]

    D. S. Novikov, E. Fieremans, S. N. Jespersen, V. G. Kiselev, Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation, NMR in Biomedicine 32 (4) (2019) e3998. doi:10.1002/nbm.3998. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/nbm.3998

    2019 doi

  21. [29]

    R. N. Henriques, S. N. Jespersen, N. Shemesh, Microscopic anisotropy misestimation in spherical-mean single diffusion encoding mri, Mag- netic Resonance in Medicine 81 (5) (2019) 3245–3261. arXiv: https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.27606, doi:10.1002/mrm.27606....

    2019 doi

  22. [30]

    Topgaard, Multidimensional diffusion mri, Journal of Magnetic Reso- nance 275 (2017) 98 – 113

    D. Topgaard, Multidimensional diffusion mri, Journal of Magnetic Reso- nance 275 (2017) 98 – 113. doi:https://doi.org/10.1016/j.jmr.2016. 12.007. URL http://www.sciencedirect.com/science/article/pii/ S1090780716302701

    2017 doi

  23. [31]

    Z. Yuan, J. Fish, Toward realization of computational homogenization in practice, International Journal for Numerical Methods in Engineering 73 (2008) 361–380. doi:10.1002/nme.2074

    2008 doi

  24. [32]

    Bashari Rad, H

    B. Bashari Rad, H. Bhatti, M. Ahmadi, An introduction to docker and analysis of its performance, IJCSNS International Journal of Computer Science and Network Security 173 (2017) 8

    2017

  25. [33]

    G. M. Kurtzer, V. Sochat, M. W. Bauer, Singularity: Scientific containers for mobility of compute, PLOS ONE 12 (5) (2017) 1–20. doi:10.1371/ journal.pone.0177459. URL https://doi.org/10.1371/journal.pone.0177459

    2017 doi

  26. [34]

    National Academies of Sciences, Medicine, Reproducibility and Repli- cability in Science, The National Academies Press, Washington, DC, 2019

    E. National Academies of Sciences, Medicine, Reproducibility and Repli- cability in Science, The National Academies Press, Washington, DC, 2019. doi:10.17226/25303. URL https://www.nap.edu/catalog/25303/ reproducibility-and-replicability-in-science

    2019 doi

  27. [35]

    Melenk, I

    J. Melenk, I. Babuka, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1) (1996) 289 – 314. doi:https://doi.org/10.1016/S0045-7825(96)01087-0. URL http://www.sciencedirect.com/science/article...

    1996 doi

  28. [36]

    J. R. G. N. W. Benjamin Kehlet, Anders Logg, Fenics project, https: //bitbucket.org/fenics-project/mshr/ (2019)

    2019

  29. [37]

    A. Logg, G. N. Wells, Dolfin: Automated finite element computing, ACM Trans. Math. Softw. 37 (2) (2010) 20:1–20:28. doi:10.1145/1731022. 28 1731030. URL http://doi.acm.org/10.1145/1731022.1731030

    2010

  30. [38]

    Geuzaine, J

    C. Geuzaine, J. F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Interna- tional Journal for Numerical Methods in Engineering

  31. [39]

    Ribes, C

    A. Ribes, C. Caremoli, Salom platform component model for numeri- cal simulation, in: 31st Annual International Computer Software and Applications Conference (COMPSAC 2007), Vol. 2, 2007, pp. 553–564. doi:10.1109/COMPSAC.2007.185

    2007 doi

  32. [40]

    Beta cae systems, ansa pre-processor: The advanced cae pre-processing software for complete model build up., https://www.beta-cae.com

  33. [41]

    Schl¨ omer, meshio,https://github.com/nschloe/meshio (2019)

    N. Schl¨ omer, meshio,https://github.com/nschloe/meshio (2019)

    2019

  34. [42]

    Inc., Google colaboratory, https://github.com/jupyter/ colaboratory (2014)

    G. Inc., Google colaboratory, https://github.com/jupyter/ colaboratory (2014)

    2014

  35. [43]

    J. H. et al., Matplotlib

  36. [44]

    K. I. Sandia Corporation, Paraview

  37. [45]

    D. S. Grebenkov, Pulsed-gradient spin-echo monitoring of restricted dif- fusion in multilayered structures, Journal of Magnetic Resonance 205 (2) (2010) 181 – 195. doi:https://doi.org/10.1016/j.jmr.2010.04.017. URL http://www.sciencedirect.com/science/article/pii/ S1090780710001199

    2010 doi

  38. [46]

    J. P. Kinney, J. Spacek, T. M. Bartol, C. L. Bajaj, K. M. Harris, T. J. Sejnowski, Extracellular sheets and tunnels modulate glutamate diffusion in hippocampal neuropil, Journal of Comparative Neurology 521 (2) (2013) 448–464. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.10...

    2013 doi

  39. [47]

    Carim-Todd, K

    L. Carim-Todd, K. G. Bath, G. Fulgenzi, S. Yanpallewar, D. Jing, C. A. Barrick, J. Becker, H. Buckley, S. G. Dorsey, F. S. Lee, L. Tessarollo, Endogenous truncated trkb.t1 receptor regulates neuronal complexity and trkb kinase receptor function in vivo, Journal of Neuroscience...

    2009 doi

  40. [48]

    Developers, Scipy, https://www.scipy.org (2001)

    S. Developers, Scipy, https://www.scipy.org (2001)

    2001

  41. [49]

    J. Nitsche, ¨Uber ein variationsprinzip zur l¨ osung von dirichlet-problemen bei verwendung von teilr¨ aumen, die keinen randbedingungen unterwor- fen sind, Abhandlungen aus dem Mathematischen Seminar der Universit¨ at 29 Hamburg 36 (1) (1971) 9–15. doi:10.1007/BF02995904. URL...

    1971 doi

  42. [50]

    Hansbo, Nitsche’s method for interface problems in computational me- chanics, GAMM-Mitteilungen 28 (2) (2005) 183–206

    P. Hansbo, Nitsche’s method for interface problems in computational me- chanics, GAMM-Mitteilungen 28 (2) (2005) 183–206. doi:10.1002/gamm. 201490018. URL http://dx.doi.org/10.1002/gamm.201490018 30 (a) Three-layered disk (b) Three-layered sphere (c) Three-layered torus (d) Si...

    2005 doi

  43. [87]

    25901, doi:10.1002/mrm.25901

    arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm. 25901, doi:10.1002/mrm.25901. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/mrm.25901

    doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.