Jan N Rose^{1}, Lukasz Sliwinski^{1}, Sonia Nielles-Vallespin^{2,3}, Andrew D Scott^{2,3,4}, and Denis J Doorly^{1}

To better understand diffusion MRI in biological tissues, numerical simulations are commonly used to model the MR signal. Realistic simulation substrates built directly from histology images help to reduce the model error, but intrinsic parameters other than the microstructure have an effect too. In this study, we investigate the relationship between diffusion tensor fractional anisotropy and membrane permeability. Using a GPU-accelerated Bloch–Torrey solver, we observe a significant difference from the impermeable case for long diffusion times on the order of 1s.

Biological tissue of interest for diffusion MRI investigations is non-homogeneous. This may be modelled using compartments with varying prescribed diffusivities. To study the effects of confounding factors such as perfusion and permeability, the use of numerical simulations is increasingly common. This allows for in-silico investigation of model parameters and can even serve to validate MR measurements [1].

Recent work simulating diffusion in the myocardium [2] has demonstrated the need for the simulation substrate to replicate the large-scale aggregates of cells found in (cardiac) muscle tissue. For this, numerical models need to be built directly from histology. While the microstructure in the brain is often appropriately modelled as having impermeable membranes, long mixing times common in e.g. cardiac applications show significant discrepancies between simulated and experimentally-measured fractional anisotropy (FA) values.

In this work we investigate how a non-zero membrane permeability manifests itself in the synthesised MR signal. We performed numerical simulations of the Bloch–Torrey equation, using a finite difference-based diffusion solver. The code is written to execute efficiently on graphical processing units (GPUs).

We obtained wide-field microscopy images of 10µm Masson-stained sections from Yorkshire pigs [3]. The image resolution was 0.5µm
and simulations were performed with this pixel size. A
1x1mm^{2} region in the mesocardium, where cells are cut
nearly perpendicular to the imaging slice, was chosen for
simulations. The input image was processed in MATLAB to automatically
label intra-cellular space by
thresholding. This mask was then smoothed. We applied a
watershed algorithm to separate individual touching or overlapping
cells.

Intra- and extra-cellular
diffusivities were set to *D*_{IC}=1µm^{2}/ms
and *D*_{EC}=2µm^{2}/ms respectively [4]. The
outer pixel boundaries of cells were identified and assigned a
reduced membrane diffusivity *D*_{m}. This allows for
the inclusion of permeability according to *P*=*D*_{m}/*L*
[5],
where *L* is the histology pixel size.

We simulated a stimulated echo pulse
sequence with diffusion time Δ=1s. We calculated the 2D diffusion
tensor and mean diffusivity (MD) and FA from three gradient
directions. The membrane permeability *P* was varied from 0 (impermeable) to 2µm/ms (*D*_{m}=*D*_{IC}).
We solved the Bloch–Torrey equations for the transverse
magnetisation dynamics *M _{x}*+i

The simulation substrate is shown in Figure 1. This region was chosen as representative of the surrounding tissue in the mesocardium. The extra-cellular volume fraction is 28%. The large extra-cellular gaps are shear layers separating the aggregates of cardiomyocytes known as sheetlets [3].

To visualise the effect of membrane
permeability we consider an initial spike of concentration *ρ*(*x*_{0},
*t*_{0})=δ(*x*-*x*_{0}).
Figure 2 shows the resulting distribution at *t*=Δ for
four values of permeability. This approximates the probability
density function (PDF) from *x*_{0}. The mean distance diffused depends on the diffusivity in
the separate compartments, but visibly increases with increases in
permeability.

Figure 3 shows the magnitude of
transverse magnetisation at the echo time for three different
diffusion gradients * G*. The reduction in

The effect of varying permeability is quantified
through MD and FA (Figure 4). We observe an increase in MD and a
reduction in FA with increasing permeability, as expected. In recent work [2] we
reported a higher-than-expected FA for impermeable membranes compared
with in-vivo experiments. If we assume a primary
eigenvalue λ_{1}
normal to the image (along the long-axis of cells), we can
estimate the three-dimensional FA. We set λ_{1}=1.2µm^{2}/ms,
equal to the volume-weighted free diffusivity, in Figure 4.

- Fieremans E, Lee H-H. Physical and numerical phantoms for the validation of brain microstructural MRI: A cookbook. NeuroImage. 2018;182:39–61.
- Rose JN, Nielles-Vallespin S, Ferreira PF, Firmin DN, Scott AD, Doorly DJ. Novel insights into in-vivo diffusion tensor cardiovascular magnetic resonance using computational modeling and a histology-based virtual microstructure. Magn Reson Med. 2018;00:1–15.
- Nielles-Vallespin S, Khalique Z, Ferreira PF, et al. Assessment of Myocardial Microstructural Dynamics by In Vivo Diffusion Tensor Cardiac Magnetic Resonance. J Am Coll Cardiol. 2017;69(6):661–676.
- Seland JG, Bruvold M, Brurok H, Jynge P, Krane J. Analyzing Equilibrium Water Exchange Between Myocardial Tissue Compartments Using Dynamical Two-Dimensional Correlation Experiments Combined With Manganese-Enhanced Relaxography. Magn Reson Med. 2007;58:674–686.
- Powles JG, Mallett JD, Rickayzen G, Evans WAB. Exact analytic solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc R Soc Lond A. 1992;436:391–403.
- Hwang SN, Chin CL, Wehrli FW, Hackney DB. An image-based finite difference model for simulating restricted diffusion. Magn Reson Med. 2003;50(2):373–382.

Original histology section (left) and the resulting simulation
substrate (right) obtained from automatically processing the histology image.

Spread of concentration after an initial
spike of concentration inside a single cell for various values of membrane permeability. The
four frames from left to right correspond to permeabilities 0, 0.002µm/ms, 0.02µm/ms,
and 0.2µm/ms. A diffusion time of 1s was simulated. It is clear that
larger values of permeability allow the concentration to spread
further, which increases mean diffusivity.

Local NMR signal at the end of the simulated pulse sequences for the
three gradient directions **G** (green arrows). The synthesised signal is the
volume-averaged magnitude of transverse magnetisation *M*=||*M*_{x}+i*M*_{y}||.
Lower values of magnetisation correspond to greater signal loss due
to less restricted diffusion along **G**. For the given simulation
substrate with gaps in extra-cellular space oriented nearly at 45deg
one can appreciate the increased signal loss along the diagonal *xy*
gradient.

Plots showing the increase in MD (left) and decrease in FA
(right) with increasing membrane permeability. The left-most point
corresponds to impermeable membranes with *P*=0, which cannot otherwise be
displayed on the logarithmic axis.
2D (blue) values correspond directly to the simulation result,
whereas the 3D (red) values are based on an estimated primary
eigenvalue λ_{1}=1.2µm^{2}/ms along the cells (normal to the image).