Unified model of brain tissue microstructure dynamically binds diffusion and osmosis with extracellular space geometry

We present a universal model of brain tissue microstructure that dynamically links osmosis and diffusion with geometrical parameters of brain extracellular space (ECS). Our model robustly describes and predicts the nonlinear time dependency of tortuosity ($\lambda =\sqrt{D/D^{*}}$) changes with very high precision in various media with uniform and nonuniform osmolarity distribution, as demonstrated by previously published experimental data ($D$ = free diffusion coefficient, $D^{*}$ = effective diffusion coefficient). To construct this model, we first developed a multiscale technique for computationally effective modeling of osmolarity in the brain tissue. Osmolarity differences across cell membranes lead to changes in the ECS dynamics. The evolution of the underlying dynamics is then captured by a level set method. Subsequently, using a homogenization technique, we derived a coarse-grained model with parameters that are explicitly related to the geometry of cells and their associated ECS. Our modeling results in very accurate analytical approximation of tortuosity based on time, space, osmolarity differences across cell membranes, and water permeability of cell membranes. Our model provides a unique platform for studying ECS dynamics not only in physiologic conditions such as sleep-wake cycles and aging but also in pathologic conditions such as stroke, seizure, and neoplasia, as well as in predictive pharmacokinetic modeling such as predicting medication biodistribution and efficacy and novel biomolecule development and testing.

Figure 1

Schematic illustration of the geometrical setting used in the multiscale mathematical model under homogenization procedure.

Figure 2

Geometric model of the brain tissue. Depictive example showing that the method of homogenization effectively smooths substructure variations caused by brain spatial heterogeneties. A cross section in the medium alternately intersects the two phases (a). A cross section of the solution at a given time in ECS (b) is compared to the solution of the homogenized equation (c).

Figure 3

The initial shape of the cell is represented by the dash-dotted line. Using this initial shape, the solution of our level set equation (32) is represented by the dashed line, which is approximated by the square shape that is represented by the solid line.

Figure 4

Tortuosity plotted as a function of volume fraction. A comparison is made between our model prediction and the formula $\lambda =\sqrt{2-\alpha }$.

Figure 5

(a) Tortuosity ($\lambda$) plotted as a function of time for multiple values of initial osmolarity difference across the cellular membrane ($\sigma$). (b) Volume fraction $\alpha$ plotted as a function of time for multiple values of initial osmolarity difference across the cellular membrane ($\sigma$). Values and units of these parameters are given in Table 1.

Figure 6

(a) Tortuosity ($\lambda$) plotted as a function of time for multiple values of water permeability of the membrane ($p_f$). (b) Volume fraction ($\alpha$) plotted as a function of time for multiple values of membrane water permeability ($p_f$). Values and units of these parameters are given in Table 1.

Figure 7

(a) Nonuniform initial distribution of osmolarities in the selected tissue. Initial ECS osmolarity in the small circular region in the tissue is 150 $\mathrm{mol}/{\mathrm{m}}^3$ and in other regions of the tissue is 300 $\mathrm{mol}/{\mathrm{m}}^3$. Initial ICS osmolarity is 300 $\mathrm{mol}/{\mathrm{m}}^3$ throughout. (b) Tortuosity plotted as a function of time for the points A, B, C, and D. Values and units of these parameters are given in Table 1.

Figure 8

Comparison of concentration profiles as a function of time between the microscopic model and the homogenized model in the case of nonuniformly distributed initial osmolarity. (a) The circles are the concentration of ${\varphi }_{\epsilon }$ in the microscopic model and the dashed line is the the concentration of ${\varphi }_0$ in the homogenized model. (b) The dashed line is the concentration of ${\psi }_{\epsilon }$ in the microscopic model and the circles are the concentration of ${\psi }_0$ in the homogenized model.

Figure 9

Comparison of concentration profiles as a function of time between the microscopic model and the homogenized model in the case of nonuniformly distributed initial osmolarity. (a) The circles are the concentration of ${\varphi }_{\epsilon }$ in the microscopic model and the dashed line is the concentration of ${\varphi }_0$ in the homogenized model. (b) The circles are the concentration of ${\psi }_{\epsilon }$ in the microscopic model and the dashed line is the concentration of ${\psi }_0$ in the homogenized model.

Figure 10

Impression of a typical cell and the selected square region in it.