Nadia A S Smith^{1}, Jessica E Talbott^{1}, Chris A Clark^{2}, and Matt G Hall^{1,2}

We investigate the effect of the sub-voxel patterning of permeability in muscle tissue on the diffusion signal via a finite element simulation of diffusion MRI on a model of muscle tissue. We observe that permeability with a disordered pattern leads to statistically significant differences in diffusion signal at high b and longer diffusion times.

Diffusion MRI measurements are known to contain information about the
micro-environment experienced by diffusing spins. There has been considerable
interest in using diffusion as a probe of microstructure in the brain, particularly
in white matter,^{1} and in muscle.^{2} Muscle tissue differs
markedly from white matter – fibres are larger, and have a hierarchical structure ^{3}
which means their diffusion properties are different. In particular, diffusion
in muscle tissue is known to exhibit a strong time dependence.^{4}

Duchenne muscular dystrophy (DMD) is a genetic muscle wasting condition
affecting approximately 1 in 3600 boys.^{5} DMD pathology causes
progressive muscle loss and is invariably fatal.^{6} DMD tissue is often
imaged for fat-fraction, but replacement of muscle with fat is the end stage of
the pathology, and this has driven interest in new methods which are sensitive
to earlier stage microstructural changes. Diffusion MRI is one such approach.

Recent simulation-based investigation of DMD pathology revealed that
changes in permeability have a large effect on the diffusion signal.^{7}
DMD is known to cause increased permeability of muscle fibres. Changes
in permeability are not uniform across space, however. Permeable fibres are spatially randomly distributed and histological evidence for permeability
changes reminiscent of a percolation cluster has been reported.^{8}

The effect of spatial distribution of permeability on the diffusion signal has not been extensively studied. Previous simulation-based investigations typically considered permeability as a uniform change across tissue. This work calculates the diffusion signal as a function of diffusion time and b-value in models of permeable muscle tissue in two scenarios: one where permeability is changed uniformly across the tissue, and another in which barriers are permeable or impermeable randomly across the tissue. In both cases the total expected flux across all barriers is the same. We find that permeability pattern does affect diffusion signal, although in a b-value regime which is currently only accessible on preclinical hardware.

We generate
synthetic pulsed gradient spin-echo (PGSE) diffusion-weighted signal curves over
a wide range of scan parameters using a Finite Element Method (FEM) to numerically
solve the Bloch-Torrey equations for spins in muscle tissue excited via a
synthetic PGSE sequence, and integrate over the sample volume to compute the diffusion-weighted
signal.^{9} A hexagonal lattice with a fixed unit cell size was used as
a tissue model. Hexagonal cells are separated by a narrow interstitial space
with a lower diffusivity. Exchange between the cells and interstitial space was
via boundaries with a controlled permeability (fig. 1).

FEM allows efficient handling of different permeability configurations and different diffusivities
in the tissue vs. interstitial space. The disordered permeability pattern was
created by randomly selecting a fixed fraction of boundaries as permeable (fig.
1d). The probability that a particular boundary is permeable is the percolation
probability, *p _{perc}*. The permeability value of a boundary is

FEM
simulations were performed using COMSOL Multiphysics™ 5.3a (COMSOL Group,
Stockholm, Sweden). Synthetic diffusion-weighted signals were obtained from
simulations of both uniform and non-uniform patterns for the gradient strengths
of 0.02–0.10 T/m, diffusion times from 50–250 ms, and muscle diffusivity in
{1E-9, 1.7E-9 and 2.3E-9} m^{2}/s. Gradient duration was 20 ms,
gradients were perpendicular to fibres. Interstitial diffusivity was 2.3E-9 m^{2}/s.
Dimensionless reduced permeability was 3.125E-5 and percolation probability was
30%.

[1] Tsien C, Cao Y, Chenevert T. Clinical applications for diffusion magnetic resonance imaging in radiotherapy. Semin Radiat Oncol. 2014;24(3):218-26.

[2] Oudeman J, Mazzoli V, Marra MA, et al. A novel diffusion-tensor MRI approach for skeletal muscle fascicle length measurements. Physiol Rep. 2016;4(24):e13012.

[3] Saladin KS. Anatomy and Physiology, 3rd Ed. New York: Watnik, 2010. p 405–406.

[4] Porcari P, Hall MG, Clark CA, Greally E, Straub V, Blamire AM, The effects of ageing on mouse muscle microstructure: a comparative study of time‐dependent diffusion MRI and histological assessment, NMR in Biomed 2018:13(3):e3881

[5] Bushby K, Finkel R, Birnkrant DJ, Case LE, Clemens PR, Cripe L, Kaul A, Kinnett K, McDonald C, Pandya S, Poysky J, Shapiro F, Tomezsko J, Constantin C. Diagnosis and management of Duchenne muscular dystrophy. Part 1: diagnosis, and pharmacological and psychosocial management. Lancet Neurol 2010;9:77–93.

[6] Emery AEH & Muntoni F, Duchenne Muscular Dystrophy ISBN-10: 9780198515319, ISBN-13: 978-0198515319, OUP Oxford; 3 edition (4 Sept. 2003)

[7] Hall M, Clark C. Diffusion in Hierarchical Systems: A Simulation Study in Models of Healthy and Diseased Muscle Tissue. Magn Reson Med. 2017 Sep; 78(3): 1187-1198.

[8] Straub V, Rafael JA, Chamberlain JS, Campbell KP. Animal models for muscular dystrophy show different patterns of sarcolemmal disruption. J Cell Biol. 1997;139(2):375-85.

[9] Moroney B, Stait-Gardner T, et al. Numerical analysis of NMR diffusion measurements in the short gradient pulse limit. J Magnetic Resonance. 2013; 234:165-175.

[10] Sprinthall, R. C. (2011). Basic Statistical Analysis (9th ed.). Pearson Education. ISBN 978-0-205-05217-2.

Figure 1: Computational
domain: a) blue region corresponds to muscle tissue and red to interstitial space. The same equations are applied in both domains, but the diffusivity in the muscle *D*_{holes} can be different to that of the interstitial *D*_{walls}; b) the dimensions of unit cells and gaps
between them are shown; c) uniform case – all boundaries are flux boundaries
and rate of flux across boundaries is *h*_{perm}**p*_{perc}; d)
non-uniform case – only *p*_{perc} (here set to 30%) of boundaries are open (shown
in blue) with rate of flux across boundaries set to *h*_{perm}.

Figure 2:
Comparison of signals, *log(S/S*_{0}) [dimensionless], against diffusion time, Δ [s], for the uniform and mean of the non-uniform boundaries patterns. Each plot corresponds to a different muscle diffusivity, *D*_{holes} [m^{2}/s],
and each coloured line within the plots corresponds to a different gradient
strength, *g* [T/m].
The non-uniform curve has been calculated as the mean of the different *N* samples of random configurations of non-uniform
open boundaries, and the error bars are ±*std dev/√N*.

Figure 3: Comparison of signals, *log(S/S*_{0}) [dimensionless], against b-value [mm^{2}/s], for the uniform and mean of the non-uniform boundaries patterns. Each plot corresponds to a different diffusion time, Δ [ms], and each coloured line within the plots corresponds to a muscle diffusivity, *D*_{holes} [m^{2}/s]. The non-uniform curve has been calculated as the mean of the different *N* samples of random configurations of non-uniform open boundaries, and the error bars are *±std dev/√N*.

Figure 4: Heat maps showing the normalised difference (in %) of the
uniform vs. non-uniform signals as a function of gradient strength and
diffusion time, for each muscle permittivity. To ensure
the differences are statistically significant, a two-sided Z-test ^{10} was carried out and showed most p-values were < 10^{-5}
over the parameter range. The few
p-values that were greater than the chosen confidence level of 0.05 are shown in
white. These correspond to those combinations of parameters that have the
smallest percentage differences between uniform and non-uniform, thus making it
harder to show if the differences are statistically significant.