A novel three-phase model of brain tissue microstructure

Abstract

We propose a novel biologically constrained three-phase model of the brain microstructure. Designing a realistic model is tantamount to a packing problem, and for this reason, a number of techniques from the theory of random heterogeneous materials can be brought to bear on this problem. Our analysis strongly suggests that previously developed two-phase models in which cells are packed in the extracellular space are insufficient representations of the brain microstructure. These models either do not preserve realistic geometric and topological features of brain tissue or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. In light of the highly connected nature of three-dimensional space, which limits the minimum diffusivity of biologically constrained two-phase models, we explore the previously proposed hypothesis that the extracellular matrix is an important factor that contributes to the diffusivity of brain tissue. Using accurate first-passage-time techniques, we support this hypothesis by showing that the incorporation of the extracellular matrix as the third phase of a biologically constrained model gives the reduction in the diffusion coefficient necessary for the three-phase model to be a valid representation of the brain microstructure.

Figure 1. Proposed Two-Phase Model.

Representative region of proposed microstructural model. (A) 2D two-phase model with ICS in red and ECS in blue. (B) 3D two-phase model (nonstaggered case) with ICS in red.

Figure 2. ECM Incorporation into Model.

(A) Schematic representation of a cell and some of the associated ECM components. Note how the ECM forms in close proximity to the cell that produces it. Image is adapted from:http://courses.cm.utexas.edu/jrobertus/ch339k/overheads-2/figure-07-30.jpg. (B) Our representation of the ECM in the proposed three-phase model. The square represents a convex cell body and the “x-ed” network surrounding the cell represents the ECM.

Figure 3. Properties of Three-Phase Model.

(A) The left y-axis (dashed blue line with circles) gives the average gap width in the model and the right y-axis (solid green line with squares) gives the fraction of concave cells in the model. Both plots are given as a function of particle radius (in µm). B) Effective diffusivity of 3D three-phase media at ECS volume fraction ϕ ₁ = 0.2 as a function of the particle radius. The results of the simulation are compared to the maximum two-point three-phase upper bound (Equation 3 with a = 1; solid red line) and the target diffusivity (dotted black line).

Figure 4. First-Passage-Time Algorithm.

Example of random walk in 2D using first-passage squares.