A model of effective diffusion and tortuosity in the extracellular space of the brain

Abstract

Tortuosity of the extracellular space describes hindrance posed to the diffusion process by a geometrically complex medium in comparison to an environment free of any obstacles. Calculating tortuosity in biologically relevant geometries is difficult. Yet this parameter has proved very important for many processes in the brain, ranging from ischemia and osmotic stress to delivery of nutrients and drugs. It is also significant for interpretation of the diffusion-weighted magnetic resonance data. We use a volume-averaging procedure to obtain a general expression for tortuosity in a complex environment. A simple approximation then leads to tortuosity estimates in a number of two-dimensional (2D) and three-dimensional (3D) geometries characterized by narrow pathways between the cellular elements. It also explains the counterintuitive fact of lower diffusion hindrance in a 3D environment. Comparison with Monte Carlo numerical simulations shows that the model gives reasonable tortuosity estimates for a number of regular and randomized 2D and 3D geometries. Importantly, it is shown that addition of dead-end pores increases tortuosity in proportion to the square root of enlarged total extracellular volume fraction. This conclusion is further supported by the previously described tortuosity decrease in ischemic brain slices where dead-end pores were partially occluded by large macromolecules introduced into the extracellular space.

FIGURE 1

An example of the fitting procedure based on Eq. 26. Median effective diffusion coefficient was computed from fits corresponding to all individual counting boxes (eight in this case). This example shows fitting for effective diffusion along the x₁ axis in a 2D environment with square obstacles. To detect possible anisotropy, two sets of counting boxes were used, one with a₂ → ∞ and the second one with a₁ → ∞.

FIGURE 2

Geometrical arrangement of the 2D model composed of random polygons. Molecules are seen in black close to the release site. See text for details on modeling the 2D effective diffusion as a 3D process restricted to a thin layer.

FIGURE 3

Dead-end pore diffusion. Cellular elements were removed to reveal the distribution of the diffusing molecules. The molecules are rendered as unrealistically large spheres to aid visualization. Because the molecules readily enter the dead-end pores, the effective diffusion observed on a macroscopic scale appears to be delayed. The elements are 3 μm across.

FIGURE 4

(A) 2D environment with pores. When dead-end volume fraction is added to the initial well-connected volume fraction α₀, the total ECS volume fraction α increases but the hindrance of the environment increases as well. This prediction is contained in Eq. 19 and confirmed by numerical simulations (data points for effective diffusion along both x₁ and x₂ axes are shown). In contrast, if α is increased by adding only well-connected space to α₀, the effective diffusion approximately follows Maxwell's curve with decreasing hindrance for higher volume fractions (Eq. 27a). The 2D model used α₀ = 0.129. (B) 3D environment with pores. Dead-end pores added to the well-connected 3D environment increase the diffusion hindrance, as Eq. 25 predicts. Note that the mutual relationships are qualitatively similar to the 2D case (A) but all values are significantly shifted toward lower hindrance. This effect is characteristic of the transition to three dimensions. Addition of dead-end pores can lead to tortuosities commonly observed in the nervous tissue. The 3D model used α₀ = 0.1 and the molecules were allowed to diffuse for 2 s, divided into 4 × 10⁶ time steps.

FIGURE 5

Decreasing the ECS volume fraction α by narrowing the channels between cellular elements decreases the tissue permeability only slightly. Both a simple model composed of cubes and a more realistic one with random convex polyhedra follow fairly closely Maxwell's homogenization estimate (Eq. 27b). The lowest achievable diffusion permeability is given by the narrow channel approximation (Eqs. 21 and 22). It is clear that manipulation of the uniformly wide and well-connected channels cannot account for experimental diffusion data in nervous tissue.

FIGURE 6

Illustration of the random 3D geometry composed of convex polyhedra. The gaps between the elements are uniform. Typical size of one cellular element is 3 μm. See text for more details on generating this model.

FIGURE 7

A single element of the 3D environment with dead-end pores. Pockets were made in every face of the cube, taking care to avoid mutual intersections while achieving maximum possible dead-end volume fraction. The orientation of each element in the environment was randomly selected from the 24 possible orientations. The width of the pore channels was identical to the gaps between the elements (0.104 μm).