Dominik Ludwig^{1}, Frederik Bernd Laun^{2}, 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 time between the two gradient pairs. To investigate the connection between AXR and membrane permeability and to test the assumptions of the underlying theory, Monte Carlo simulations using simplified tissue models were performed. Simulations covered a broad range of membrane permeabilities to determine limits of the applicability of this technique. For the considered simplified tissue model, AXR-values could be reliably related to membrane permeabilities typically occurring in vivo.

The exchange of water between intra- and extracellular space is an important biological process in human physiology. It is therefore of high interest to noninvasively measure this parameter in clinics, potentially leading to new diagnostic possibilities. For this purpose, the filter-exchange imaging (FEXI) technique, which is also called apparent exchange rate (AXR) mapping, was proposed by Lasič et al. [1,2]. The aim of this work was to investigate if the assumptions underlying this approach are valid for a range of permeabilities by using Monte Carlo simulations and whether membrane permeabilities can thus be determined from AXR data.

A simplified biological cell-system can be described by an exchanging two compartment model. The exchange between the compartments is characterized by the intra- and extracellular life-times $$$\tau_i$$$ and $$$\tau_e$$$. The signal equation for such a system was first calculated by Kärger in 1969 [3]. This was later expanded by Lasič [1] to FEXI. The sequence used for these experiments is basically a double-diffusion encoded sequence (see Fig.1) with varying mixing times between the two gradient pairs. Full description of the AXR-theory can be found in [1]. It was shown that the Apparent Diffusion Coefficient (ADC) for such a system can be described as:

$$ADC(t_m)=ADC_{eq}+(ADC(0)-ADC_{eq})e^{-AXR\cdot t_m}=ADC_{eq}(1-\sigma\cdot e^{-AXR\cdot t_m})\, (1),$$

with
the filter effectivity $$$\sigma=1-ADC(0)/ADC_{eq}$$$.
The theory assumes no exchange during the
diffusion weightings, as well as transfer of magnetization between the
compartments only during the mixing time. In order to investigate the appropriateness of
these assumptions, a large range of cell permeabilities was simulated. Random walks were performed using a Monte Carlo
simulation implemented in MATLAB (The MathWorks, Inc., Natick, MA)
for both CPU and GPU (CUDA) computation. Transition probabilities for the semi-permeable
membranes were calculated according to [4]. If a particle is not allowed to cross
the membrane, it is reflected elastically. For representation of an in vivo-like
system, a two-dimensional rectangular ordered cylinder-array covering a start
area of about $$$6\cdot$$$$$$10^4\mu m^2$$$ was
used. Beyond this area the two compartments were endlessly repeated. As cell-like
structures, cylinders with a diameter of $$$6\mu m$$$ were
used. The fraction of extracellular space was set to
$$$f_e=0.2762$$$, which
is a typical order of magnitude for biological tissues. See Fig.2 for a
schematic representation. Permeabilities
were evaluated between $$$\rho=0.0001\mu m/ms$$$ and $$$0.1\mu m/ms$$$. In
total $$$2\cdot$$$$$$10^6$$$ particles were simulated for each mixing time split up into 4
packets of $$$5\cdot$$$$$$10^5$$$ particles for statistical analysis. The step size was $$$dt=5\cdot$$$$$$10^{-7}s$$$. Echo-time
of the filter $$$T_f$$$ and
of the diffusion weighting $$$T$$$ was
set to $$$T_f=T=50ms$$$ ($$$T_f=0.1ms$$$ for high permeabilities) with the filter strength $$$b_f=1.5ms/\mu m^2$$$. Intra-
and extracellular diffusion constants were set to $$$D_i=1\mu m^2/ms$$$ and $$$D_e=2\mu m^2/ms$$$. Mixing
times were evaluated between $$$t_m=2ms$$$ and $$$t_m=2048ms$$$, with $$$t_{m,i}=2^i[ms]$$$ and $$$i=1...11$$$. For
high permeabilities, mixing times between $$$0.1ms$$$ and $$$50ms$$$ were
evaluated in addition. AXR values were fitted using equation (1) in MATLAB taking
the mean of the 4 particle packets.

[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] Kärger, Jörg. "Zur Bestimmung der Diffusion in einem Zweibereichsystem mit Hilfe von gepulsten Feldgradienten." Annalen der Physik 479.1‐2 (1969): 1-4.

[4] 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:
2D cell-array used for simulations. The cylinders
have a diameter of 6µm. Neighboring cylinders are separated by 0.25µm in the x-
and y-direction. The geometry and structure is endlessly repeated outside the start
area of 6*10^{4}µm^{2} . The fraction of extracellular space is f_{e}=0.2762 in this
case.

Figure 3: Simulated
ADC-values as a function of the mixing time t_{m}, with their corresponding
AXR-fits. Errors are too small to be displayed. The filter was set to b_{f}=1.5ms/µm^{2} and T_{f}=50ms. The fits show
strong correlation between the simulated data and the theory for this range of
permeabilities.

Figure 4:
Simulated
ADC-values as a function of the mixing time t_{m}, with their corresponding
AXR-fits for high permeabilities. Filter
was set to b_{f}=1.5ms/µm^{2} and T_{f}=50ms (yellow and pale blue dots) and T_{f}=0.1ms (blue and purple dots). When
using the long filters, the system is already in its long-time limit, leading
to an underestimation of the AXR-value. When going to short filters and short
mixing-times, it is again possible to obtain a reasonable fit-result.

Figure 5:
Fitted AXR-values as a function of membrane-permeability ρ. T_{f}=50ms (red
dots), and T_{f}=0.1ms (black crosses). For the long filter times, the errors
are too small to be displayed. Except for the two high permeabilities simulated
with long filters, the simulated datasets correspond well with the theory.