MRI measurements from a decade-old study of the physical properties of brain tissue observed a dynamic, pulsating fluid flow in the interstitial spaces of the brain attributed to the cardiac cycle. The effects of this cyclic flow pattern on the spatial distribution of molecules in the brain are modeled in this paper. The effects of oscillatory flow on the dispersion or volumetric transmission of a molecule that is advected by this flow is modeled by a mechanism hitherto neglected in the literature. An oscillatory random walk model is used to estimate the spread or effective diffusivity due to the oscillatory advection. Then, respiration effects are also estimated and the additional dispersion of molecules due to this are calculated in our model. Our model indicates that the observed oscillatory flow in the interstitial spaces due to cardiac as well as respiratory pulsatility can induce an effective diffusivity when the spread of the molecule is observed over times long compared with a cycle of the oscillation. This would help explain the high-volume transmission within the interstitium or brain parenchyma found in MRI measurements of a marker infused into the cerebrospinal fluid in human subjects that is well above what would be expected. Interstitial spaces should be viewed as a region of dynamic oscillatory flow driven by cardiac and respiratory cycles. This oscillatory flow could result in a significant dispersion of molecules and explain the higher-than-expected effective diffusion suggested in human studies. It may be possible to augment or slow this flow and concomitant spread by applying external forces.
The principal result in this paper is an endogenous mechanism, overlooked hitherto as far as we are aware, that increases the spread of a molecule within the interstitial spaces of the brain.
The extent or effectiveness of the spread is co-determined by the cardiac and respiratory pulsations which result in oscillatory fluid flow in the interstitium and the spatial randomness (and spacing) of the branchings in the interstitial pathways. The motivation for this paper was a remark in a study of the spread of an MRI contrast reagent in (Vinje, Zapf et al. 2023) which serves as a surrogate tracer for some therapeutic molecules of interest in the treatment of brain disease cannot be explained by diffusion alone.
The plan of the paper is as follows: The body of the paper (Section 2, Methods) begins by developing a result in random walk. A standard elementary result in random walk in three dimensions is that if a particle takes a step of length L in time , randomly oriented in space with respect to the previous step, then at the end of N steps, its mean distance (over samples of such walks) from the starting point is 0, and its mean squared distance is L 2 N. As is well known, this is classical diffusion when taken to the continuum limit. It is shown first from numerical study that if each step is random, but constrained to have a negative projection onto the previous step’s direction, then diffusion still occurs though with a reduced diffusivity, 1/3 rd of the conventional random walk’s. For the purposes of this paper this is referred to as an “oscillatory random walk”. Section 2.2 then argues that the interstitial pathways are sufficiently randomly branched to constitute pathways for a random walk. Then, provided a certain constraint is met, oscillatory fluid flows in the brain that can advect particles will result in the oscillatory random walk referred to. This is the proposed mechanism that enhances particle spread. The final piece of the argument of course is the existence of such fluid flows. The evidence for this from an experiment followed by a model calculation is presented in Section 2.3. The Results Section 3 ties together the calculations from the previous section to make the case for the mechanism and its magnitude in the human brain. An important aspect of this mechanism is that it is independent both of the magnitude of the particle diffusivity and of particle size, within limits that will be made clear in the section.
There is a vast literature attempting to understand perivascular and interstitial fluid flows in the brain along with implications for drug delivery. In order not to impede the development, we a discussion of these is in the final Section 4, Discussion, which begins with the motivating remark of (Vinje, Zapf et al. 2023) mentioned earlier and then proceeds to discuss other literature which is complementary to our work. In conclusion, some speculations on the possibility of enhancing this mechanism to distribute drugs in the brain with artificial means is offered.
The central result of this paper and its implications, developed in subsequent sections to apply to the brain interstitium, is illustrated with some simplified models. In Figure 1, a twodimensional square lattice (which has the virtue of allowing an exact calculation, in fact in any dimension) is shown, with the interstitial pathways in white being arranged in a “Manhattan"type grid with square blocks representing the cellular obstructions. Figure 1a shows a particle advected, beginning with the black dot. In the first half-cycle (Figure 1a) the particle is advected to a (taxicab) distance of 2. The key assumption made is that at any branch, the fluid flow and therefore the advected particle can choose any of the branches as long as it is not oppositely directed to the pressure driving the flow. The particular assumptions for the walk on this lattice is clearly shown in Figure 1a: the particle will arrive at any of the positions marked in red with equal probability. Following the same assumption, mutatis mutandis, for the second half-cycle results in the particle arriving at the green dots: each dot representing a contribution of equal probability. It is obvious that the particle has not returned to the origin: a cloud of particles will spread. The mean distance is of course zero, the center of mass of the dots is the origin. The picture can be converted into an exact, if elementary, result: The mean square distance traversed in one cycle, with obstacles of unit length, and distances traversed twice that, is 16/9 in the twodimensional model and 48/25 in the three-dimensional, both are close to 2. If one cycle is about one second (as for the cardiac cycle) and the distance traversed is 20 microns this result in a mean square distance in one second of 100 m 2 . The effective diffusivity of the particle is therefore augmented by 100/6 ~ 16 m 2 /second using the result that the mean square distance is 6 D t for a Wiener o
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