Realistic Microstructure Simulator (RMS): Monte Carlo simulations of diffusion in three-dimensional cell segmentations of microscopy images

doi:10.1016/j.jneumeth.2020.109018

Journal of Neuroscience Methods

Volume 350, 15 February 2021, 109018

https://doi.org/10.1016/j.jneumeth.2020.109018 Get rights and content

Highlights

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RMS is an open-sourced Monte Carlo pipeline of diffusion simulations.
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RMS focuses on diffusion, exchange and other contrasts in realistic permeable cells.
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The implementation of corner reflections largely reduces the computational load.
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Accurate simulation of exchange requires additional correction terms.
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Precision is further achieved by GPU-based parallel computations.

Abstract

Background

Monte Carlo simulations of diffusion are commonly used as a model validation tool as they are especially suitable for generating the diffusion MRI signal in complicated tissue microgeometries.

New method

Here we describe the details of implementing Monte Carlo simulations in three-dimensional (3d) voxelized segmentations of cells in microscopy images. Using the concept of the corner reflector, we largely reduce the computational load of simulating diffusion within and exchange between multiple cells. Precision is further achieved by GPU-based parallel computations.

Results

Our simulation of diffusion in white matter axons segmented from a mouse brain demonstrates its value in validating biophysical models. Furthermore, we provide the theoretical background for implementing a discretized diffusion process, and consider the finite-step effects of the particle-membrane reflection and permeation events, needed for efficient simulation of interactions with irregular boundaries, spatially variable diffusion coefficient, and exchange.

Comparison with existing methods

To our knowledge, this is the first Monte Carlo pipeline for MR signal simulations in a substrate composed of numerous realistic cells, accounting for their permeable and irregularly-shaped membranes.

Conclusions

The proposed RMS pipeline makes it possible to achieve fast and accurate simulations of diffusion in realistic tissue microgeometry, as well as the interplay with other MR contrasts. Presently, RMS focuses on simulations of diffusion, exchange, and T₁ and T₂ NMR relaxation in static tissues, with a possibility to straightforwardly account for susceptibility-induced $T_{2}^{*}$ effects and flow.

Introduction

The MRI measurements of self-diffusion of water molecules in biological tissues provide the sensitivity to the diffusion length scales ranging from microns to tens of microns at clinically feasible diffusion times. As the feasible range of diffusion lengths is commensurate with the sizes of cells, diffusion MRI allows one to evaluate pathological changes in tissue microstructure in vivo. To balance between accuracy and precision in estimation of tissue parameters through biophysical modeling of diffusion MR signal, assumptions are inevitably made to simplify tissue microgeometry (Grebenkov, 2007, Jones, 2010, Kiselev, 2017, Jelescu and Budde, 2017, Alexander et al., 2018, Novikov et al., 2019). It is necessary to validate the assumptions of models before use, either through experiments in physical phantoms (Fieremans and Lee, 2018), or testing the model functional forms in animals and human subjects (Novikov et al., 2018a), or numerical simulations (Fieremans and Lee, 2018).

So far, numerical simulation is the most flexible and economic choice among all kinds of validation. Benefiting from the recent advances in microscopy, realistic cell geometries for simulations have been directly reconstructed from the microscopy data of neuronal tissues in 2 dimension (2d) (Chin et al., 2002, Xu et al., 2018) and 3d (Nguyen et al., 2018, Palombo et al., 2019, Lee et al., 2020b, Lee et al., 2020a), as shown in Fig. 1. The emerging need for simulations in realistic substrates prompts the development of open-source software congenial to physicists, biologists and clinicians.

Here, we describe our implementation of Monte Carlo (MC)-based diffusion simulations: the Realistic Microstructure Simulator (RMS), which entails a fast and accurate model validation pipeline. While our pipeline has been recently announced and applied to simulate diffusion MRI in axonal microstructure (Lee et al., 2020b, Lee et al., 2020a, Lee et al., 2020d), these publications are mainly focused on the physics of diffusion and model validation. In this work, we describe the methodology in detail, building on the algorithms introduced by our team over the past decade (Fieremans et al., 2008, Fieremans et al., 2010, Novikov et al., 2011, Novikov et al., 2014, Burcaw et al., 2015, Fieremans and Lee, 2018, Lee et al., 2020c), and in particular, derive the finite MC-step effects relevant for the interactions (reflection and permeation) of random walkers and membranes.

RMS is introduced as follows. In Section 2, we provide an overview of RMS implementation. Theoretical results and implementation details of particle-membrane collisions and exchange are presented in Appendices A and B; the first order correction of membrane permeability due to a discretized diffusion process is derived in Appendix C. In Section 3, we demonstrate the application of RMS to diffusion simulations in realistic axonal shapes reconstructed from electron microscopy data of a mouse brain (Lee et al., 2019). Simulated diffusion MR signals are shown to be closely related to features of cell shape, facilitating the interpretation of diffusion measurements in biological tissues. Finally, in Section 4 we provide an outlook for microstructure simulation tools in general, and RMS in particular, as a platform for MR-relevant simulations of diffusion and relaxation in microscopy-based realistic geometries.

Section snippets

Realistic Microstructure Simulator: an overview

The goal of our RMS implementation is to provide a universal platform of MC simulations of diffusion in any realistic microgeometry based on microscopy data. Therefore, the RMS has the following properties:

(i)
The simulation is performed in 3d continuous space with voxelized microgeometry. We will introduce this main feature of RMS in Section 2.1.1.
(ii)
The particle-membrane interaction of impermeable membrane is modeled as classic elastic collision. The reason for this choice will be explained in

RMS applied to intra-axonal microstructure

In this section, we describe an RMS-compatible example of a realistic electron microscopy (EM) tissue segmentation (Section 3.1), give an overview of the related biophysical models (Section 3.2), describe the RMS settings for MC in axonal geometry (Section 3.3), and outline our results for the diffusion along (Section 3.4) and transvere (Section 3.5) to the axons.

All procedures performed in studies involving animals were in accordance with the ethical standards of New York University School of

Outlook

Performing MC simulations in realistic cell geometries using the proposed RMS helps to test the sensitivity of diffusion MRI to tissue features and validate the biophysical models. RMS is an open-source platform for the Monte Carlo simulations of diffusion in realistic tissue microstructure. In addition to the examples of diffusion within intra-cellular space, it is also possible to perform simulations of diffusion in the extra-cellular space, as well as of the exchange between intra- and

Conclusions

Numerical simulations in realistic 3d microgeometry based on microscopy data serve as a critical validation step for biophysical models, in order to obtain quantitative biomarkers, e.g., axonal diameter, the degree of caliber variations and axonal undulations, for potential clinical applications. With the help of the proposed RMS pipeline, it is possible to achieve fast and accurate simulations of diffusion in realistic tissue microstructure, as well as the interplay with other MR contrasts.

Data and code availability statement

The SEM data and axon segmentation can be downloaded on our web page (www.cai2r.net/resources/software).

The source codes of Monte Carlo simulations can be downloaded on our Github page (github.com/NYU-DiffusionMRI/monte-carlo-simulation-3D-RMS). The first release of RMS supports simulations of multiple MR contrasts (diffusion, T₂ relaxation, water exchange) and the pulsed-gradient spin-echo sequence. Simulations of other MR contrasts (MT and T₁ relaxation), sequences (stimulated-echo),

Author contribution statement

Hong-Hsi Lee: Software, Validation, Formal analysis, Methodology, Investigation, Writing – Original Draft, Visualization.

Els Fieremans: Conceptualization, Methodology, Investigation, Resources, Writing – Review & Editing, Supervision, Project administration, Funding acquisition.

Dmitry S. Novikov: Conceptualization, Methodology, Investigation, Resources, Writing – Review & Editing, Supervision, Project administration, Funding acquisition.

Declarations of interest

None declared.

Acknowledgements

We would like to thank Sune Jespersen for fruitful discussions about theory, and the BigPurple High Performance Computing Center of New York University Langone Health for numerical computations on the cluster. Research was supported by the National Institute of Neurological Disorders and Stroke of the NIH under awards R01 NS088040 and R21 NS081230, by the National Institute of Biomedical Imaging and Bioengineering (NIBIB) of the NIH under award number U01 EB026996, and by the Irma T. Hirschl

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