Random Walks in Stochastic Geometries
Eric M. Baker1, Brendan Moloney1, Xin Li1, Erin W. Gilbert2, and Charles S. Springer1

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


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, <kio>. Even though the self-diffusion coefficient is that of pure water, non-Gaussian displacements are observed.


There is growing evidence the pseudo-first order, homeostatic cellular water efflux rate constant [kio] has a significant contribution from the membrane Na+,K+-ATPase metabolic rate [MRNKA].1 This vital enzyme activity has not been accessible in vivo. The rate constant kio can be precisely measured with 1H2O MR when the longitudinal shutter-speed [к1] is sufficiently large: only with very high extracellular contrast agent (CA) concentration1 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 к1, and yields the irreducible, fundamental cell biology parameters, cell density [ρ], and mean cell volume <V>, as well as <kio>.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.


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, vi = 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 kio values contain an active term proportional to MRNKA, but we can simulate RW displacements with only the passive term [kio(p) = <A/V>PW(p)], where <A/V> is the mean cell <area/volume> ratio and PW(p) is the passive water permeability coefficient.1 Theory4 shows PW(p) is proportional to pp, the permeation probability. After evaluating <A/V> and <V> geometrically, we vary the RW ensemble-average <kio> value by varying pp. Thus, specification of δ*, κ, and pp achieves a stipulated <V> and <kio>.

Figure 2 is a 2D cartoon where a RW starts at r0. 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 r0 is colored red. The direction of a potential diffusion-encoding pulsed field gradient [G] is shown, and the d projection on G is (r – G0)G: G0 is the r0 projection on G. The self-diffusion coefficient, D = D0 (3.0 μm2/ms; 37oC pure water) obtains inside and outside cells: steps of 1 μs move 0.13 μm if unhindered.


If we set kio = 0 s-1 [i.e., pp = 0], we can separately monitor the <d2> 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 <<d2>> vs. RW time. There were 100,000 [80 ms] walks per ensemble. The intra- and extra-cellular <<d2>> traces [H2Oin, H2Oout] are blue and green, respectively. Even though D = D0 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 [6D0] of the black dashed straight line for pure water [ρ = 0], H2Ofree. Figure 4 shows simulations for different <kio> values. When kio 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 kio = 0 curve is the volume fraction-weighted average of the Fig. 3 curves. For a given RW time, <d2> increases with <kio>. Pure water behavior [Fig. 3] is approached when <kio> is sufficiently large, no matter the ρ and <V> values.


Analogous simulations with isotropic, avascular identical sphere ensembles match experimental DWI data almost absolutely1 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 [Do > Di], as often done.5 Significantly, just the presence of semipermeable membranes is sufficient to impart non-Gaussian character.


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


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).


Figure 1. A representative ensemble of randomly sized/shaped digital cells is shown. The array is (484)3 μm3 = 0.11 μL, ~ 1/8 of a typical 1H2O MR voxel, and contains ~ 19,000 “Voronoi” cells. The average cell volume <V> is 5.58 pL and the cell density [ρ] is 1.53 x 105 cells/μL. The intracellular volume fraction [vi] is 0.742, and the interstitial spacing “gap” is ≤ 1.9 μm.

Figure 2. A 2D cartoon shows a Monte Carlo random walk [RW] in a portion of an ensemble such as that in Fig. 1. The walk starts at position r0. Steps inside and outside cells are light and dark blue, respectively. At the end of the RW period, the water molecule had reached position r, the displacement [d] from r0 is colored red. The direction of a potential diffusion-encoding pulsed field gradient [G] is shown with converging flux lines, and the d projection on G is (r – G0)G: G0 is the r0 projection on G.

Figure 3. Intra- and extracellular molecule displacements when no permeation occurs [kio = 0 s-1]. Each of 100 different (502 μm)3, <V> 9.2 pL, and ρ 80,400 cells/μL ensembles had 100,000 [80 ms] RWs [G = 0]. The intra- and extra-cellular <<d2>> displacements are blue and green, respectively. In each space, the diffusion coefficient D = D0 [37oC pure water; 3.0 μm2/ms]. Neither time‑course exhibits Gaussian behavior – a straight line with slope 6D. Though the green curve is almost linear, its apparent slope is clearly not that [6D0] of the black dashed straight line for pure water [ρ = 0].

Figure 4. The mean square <d2> time‑courses for RWs [G = 0] when <kio> = 30 [yellow], 10 [purple], 6 [light blue], 4 [red], 2 [green], and 0 s‑1 [dark blue; the volume fraction-weighted Fig. 3 curve average]. For a given RW period, <d2> increases with increasing <kio>. There is no overall D for any time-course.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)