Dominik Ludwig^{1,2}, Frederik Bernd Laun^{3}, Peter Bachert^{1}, and Tristan Anselm Kuder^{1}

Apparent exchange rate (AXR) mapping might provide an insight into the exchange of water between intra- and extracellular space by using a double-diffusion encoded sequence with varying mixing times between the two gradient pairs. The connection between AXR and permeability has already been validated for simplified two-compartment models. However, multi-compartment systems and non-periodic geometries have not been evaluated so far. In this study, we were able to show that it is also possible to reliably connect permeabilities to AXR-values for these geometries. Furthermore, the AXR was only dependent on average cell size and not on the number of compartments.

Methods

Simulations
were performed using Monte Carlo Simulations with GPU acceleration implemented
in MATLAB (The MathWorks, Inc., Natick, MA). A schematic representation of the
AXR sequence can be found in Figure 1. Transition probabilities for random
walkers encountering a membrane were calculated according to [5]. Random
walkers which were not allowed to cross the membrane were elastically
reflected. For this study, we used randomly positioned 2D cylinders with one,
two, and three diameters and a packing density of 50%. The grids used for the
simulations are shown in Figure 2. The 3 grids had been precomputed and were then
used for all simulations. More detailed information about the grids can be
found in Table 1. The grid size was set to 200 µm^{2},
since larger grids had no further influence on the simulation results. To
evaluate the influence of the membrane permeability, 10 logarithmically spaced
permeabilities between$$$\,\rho=0.0005\,$$$µm/ms and$$$\,\rho=0.05\,$$$µm/ms were
simulated. Simulation of ADC-values was carried out for 10$$$\,t_m$$$ with a maximal value of$$$\,8/AXR$$$;
values larger than one second were always set to$$$\,t_{m,max}=1\,$$$s due
to the limitation of the mixing time by T1-relaxation in real world
experiments. The gradient duration for the filter block was set to$$$\,t_f=14\,$$$ms and
the echo time was set to$$$\,T_f=28\,$$$ms, while
the diffusion weighting was set to $$$t_d=16\,$$$ms and $$$T_d=32ms\,$$$. Gradient
amplitude was set to$$$\,G_f=G_d=0.8\,$$$T/m. Equilibrium
ADC-values were simulated with$$$\,G_f=0\,$$$T/m. The
step-size for all simulations was$$$\,dt=5\,$$$µs.
In order to allow for some statistical analysis, 8 averages with
75000 particles each were simulated. The particle count was chosen to adjust
fluctuations due to Monte-Carlo-noise to a range which may be similar to actual
experiments. The reference
AXR-values were calculated using:$$AXR_{reference}=\rho\cdot\frac{S_{cell}}{V_i}\cdot\frac{1}{f_e},$$where $$$S_{cell}$$$ is the surface of the cell and $$$V_i$$$ the volume and $$$f_e=(V_{total}-V_{inside})/V_{total}$$$.
Reference AXR-values were then calculated using the average cell radius
shown in table 1. AXR-values were fitted according to [1].

Discussion and Conclusion

The deviations for high permeabilities in Figure 4 are mainly governed by the effects of noise shown in Figure 3. This limitation is, of course, not influenced by the compartment count but solely dependent on signal-to-noise and thus on the number of simulated particles in this study. However, the presented data still show that, regardless of the number of compartments, the systems are well described by the theory when only using the average cell radius and the packing density for AXR calculations. This unfortunately also means that AXR measurements presumably cannot be used to distinguish between different permeabilities and cell size distributions.Financial support by the DFG (Grant No. KU 3362/1-1 and LA 2804/6-1) is gratefully acknowledged.

One of the two Titan Xp GPUs used for this study was donated by the NVIDIA Corporation.

[1] Lasič, Samo, et al. "Apparent exchange rate mapping with diffusion MRI." Magnetic resonance in medicine 66.2 (2011): 356-365.

[2] Nilsson, Markus, et al. "Noninvasive mapping of water diffusional exchange in the human brain using filter‐exchange imaging." Magnetic resonance in medicine 69.6 (2013): 1572-1580.

[3] Tian, Xin, et al. "Evaluation and comparison of diffusion MR methods for measuring apparent transcytolemmal water exchange rate constant." JMR 275 (2017): 29-37.

[4] Ludwig, Dominik et al. “Apparent exchange rate mapping: relation to membrane permeability” ISMRM Annual Meeting 2018, In proc., E-Poster #3231

[5] Fieremans, Els, et al. "Monte Carlo study of a two‐compartment exchange model of diffusion." NMR in Biomedicine 23.7 (2010): 711-724.

Figure 1: Schematic representation of a
FEXI-sequence using two PGSE blocks. The first gradient pair used as the FEXI
filter is followed by a varying mixing time
$$$t_m$$$ during
which the magnetization is longitudinally stored, assuming that transversal
components dephase during $$$t_m$$$. The second gradient pair is a standard diffusion
weighting. This block is followed by an image acquisition.

Figure 2:
Randomly distributed 2D cylinders used for the Monte Carlo Simulations with a
packaging density of 50% and one cylinder diameter (A), two diameters (B) and
three diameters (C).

Figure 3: Exemplary
ADC-curves for the three systems with $$$\rho=0.0039\,$$$µm/ms (A), $$$\rho=0.03\,$$$µm/ms
(B). For lower
permeabilities, it is possible to determine AXR-values reliably from these
ADC-curves. For higher permeabilities, the change in ADC-values depending on the
mixing times has the same order of magnitude as the noise of the Monte-Carlo
simulation making it hard to reliably determine AXR-values from these curves.

Figure 4: Fitted
AXR-values as a function of the membrane permeability for a packaging density
of 50%. For each of the three grids (two, three and four compartments), the
AXR-values meet the expectations. For the range of
$$$\rho$$$-values
marked in red, the chosen noise contribution prevented a stable AXR fit. Using
the theoretical equation relating AXR to
$$$\rho$$$
for
cylinders and inserting the average cell size (dashed lines), the simulated AXR
values are correctly predicted also for this multi-compartment system.

Table
1:
Settings used to generate the random grids in this study.