Eric M. Baker^{1}, Brendan Moloney^{1}, Xin Li^{1}, Erin W. Gilbert^{2}, and Charles S. Springer^{1}

^{1}Advanced Imaging Research Center, Oregon Health & Science University, Portland, OR, United States, ^{2}Surgery, Oregon Health & Science University, Portland, OR, United States

### Synopsis

**Monte Carlo random walk simulations of water
molecule displacements in realistic cell ensembles are presented. Within an ensemble, the cells have stochastic
distributions of sizes, shapes, and interstitial spacings. The probability of molecules permeating the
cell membrane is varied. The
irreducible, fundamental system parameters are the cell density, ρ, the mean cell volume, <V>, and the steady-state cellular water efflux rate constant, <k**_{io}>. Even
though the self-diffusion coefficient is that of pure water, non-Gaussian
displacements are observed.

### Introduction

There
is growing evidence the pseudo-first order, homeostatic
cellular water efflux rate constant [k_{io}] has a significant
contribution from the membrane Na^{+},K^{+}-ATPase metabolic
rate [MR_{NKA}].^{1} This
vital enzyme activity has not been accessible *in vivo*. The rate constant k_{io}
can be precisely measured with ^{1}H_{2}O MR when the
longitudinal shutter-speed [к_{1}] is sufficiently large: only with
very high extracellular contrast agent (CA) concentration^{1} or very
low magnetic field strength.^{2}
Neither circumstance is achievable in clinical MRI. A new diffusion-weighted imaging [DWI]
analysis requires neither CA nor к_{1}, and yields the irreducible,
fundamental cell biology parameters, cell density [ρ], and mean
cell volume <V>, as well as <k_{io}>.^{1} Monte Carlo random walks [RWs] in parsimonious,
ordered ensembles of identical spheres simulate experimental
DWI data almost absolutely.^{1}
Here, we use arrays of randomly sized/shaped [“Voronoi”] cells.^{3} **Figure
1** is a 3D rendering of an ensemble with properties given in the
legend. ### Methods

Ensemble partitioning begins with ρ stipulation, specified by a random
Poisson distribution of cell centers. From
no extracellular space, <V>_{max}
= 1/ρ [*i.e.*; intracellular volume
fraction, v_{i} = 1], we shrink the cells almost uniformly. Each interstitial space [“gap”] is less than a
target, δ*,
or a given fraction, κ,
of the center-to-center distance.
Experimental k_{io} values contain an active term proportional to MR_{NKA}, but we can simulate
RW displacements with only the passive term [k_{io}(p) = <A/V>P_{W}(p)],
where <A/V>
is the mean cell <area/volume> ratio and P_{W}(p) is the passive water permeability coefficient.^{1} Theory^{4} shows P_{W}(p) is proportional
to p_{p}, the permeation probability.
After evaluating <A/V>
and
<V>
geometrically, we vary the RW ensemble-average <k_{io}>
value by varying p_{p}. Thus, specification
of δ*,
κ, and p_{p}
achieves a stipulated <V> and <k_{io}>.

**Figure 2** is a 2D cartoon where a RW
starts at r_{0}. Steps inside and
outside cells are light blue and dark blue, respectively. After the RW, the water molecule has reached
r: the displacement [d] from r_{0} is colored red. The direction of a potential
diffusion-encoding pulsed field gradient [**G**]
is shown, and the d projection on **G**
is (r – **G**_{0})**G**: **G**_{0}
is the r_{0} projection on **G**. The self-diffusion coefficient, D = D_{0}
(3.0 μm^{2}/ms; 37^{o}C pure water) obtains inside and outside
cells: steps of 1 μs move 0.13 μm if unhindered.

### Results

If
we set k_{io} = 0 s^{-1} [*i.e.*,
p_{p} = 0], we can separately monitor the <d^{2}> time‑courses for intra- and extracellular
molecules [**G** = 0, here]. **Figure 3**
shows such for 100 *different*
ensembles each of size (502 μm)^{3}, <V> = 9.2 pL, and ρ = 80,400 cells/μL; thus <<d^{2}>> *vs.*
RW time. There were 100,000 [80 ms]
walks per ensemble. The intra- and extra-cellular
<<d^{2}>> traces [H_{2}O_{in}, H_{2}O_{out}]
are blue and green, respectively. Even
though D = D_{0} in each space, neither trace exhibits Gaussian
behavior – a straight line with slope 6D, often assumed
for extracellular water.^{5}
Though the Fig. 3 green curve is almost linear, its general slope
is clearly not that [6D_{0}] of the black dashed straight
line for pure water [ρ = 0], H_{2}O_{free}. **Figure
4** shows simulations for different <k_{io}> values. When
k_{io} is non-zero, we no longer distinguish intra- and extra‑cellular
molecules because they spend time in each space [Fig. 2]. The single Fig. 4 k_{io} = 0
curve is the volume fraction-weighted average of the Fig. 3 curves. For a given RW time, <d^{2}> increases with <k_{io}>. Pure water behavior [Fig. 3] is
approached when <k_{io}> is sufficiently large, no matter the ρ and <V> values. ### Discussion

Analogous
simulations with isotropic, avascular identical sphere ensembles match experimental
DWI data almost absolutely^{1} and simulations as here are even better
[*separately submitted*]. Since the displacement
curve tangential slope continuously varies, there is no overall D defined for any
time-course. Intracellular diffusion can
be characterized as “restricted,” and extracellular diffusion as “hindered:” therefore,
no overall displacement represents a single Gaussian process. One cannot designate discrete D values
for each space [D_{o} > D_{i}], as often done.^{5} Significantly, just the presence
of semipermeable membranes is sufficient to impart non-Gaussian
character. ### Acknowledgements

**Grant Support:** NIH:
R44 CA180425, Brenden-Colsen Center for Pancreatic Care. ### References

**1.** Springer, *JMR*,
**291:**110-126 (2018). **2.** Ruggiero,
Baroni, Pezzana, Ferrante, Crich, Aime, *Angew.
Chem.* *Int. Ed. Engl.*, **57:**7468-7472
(2018). **3.** Van Nguyen,
Li, Grebenkov, Le Bihan, *J. Comput. Phys.*, **263:**283-302
(2014). **4.** Regan, Kuchel, *Eur. Biophys. J.*, **29:**221-227 (2000). 5. Benveniste,
Hedlund, Johnson, *Stroke*, **23:**746-754 (1992).