Modeling the Effect of Binding Kinetics in Spatial Drug Distribution in the Brain

Abstract

A 3-dimensional mathematical model is developed to determine the effect of drug binding kinetics on the spatial distribution of a drug within the brain. The key components, namely, transport across the blood-brain barrier (BBB), drug distribution in the brain extracellular fluid (ECF), and drug binding kinetics are coupled with the bidirectional bulk flow of the brain ECF to enhance the visualization of drug concentration in the brain. The model is developed based on the cubical volume of a brain unit, which is a union of three subdomains: the brain ECF, the BBB, and the blood plasma. The model is a set of partial differential equations and the associated initial and boundary conditions through which the drug distribution process in the mentioned subdomains is described. Effects of drug binding kinetics are investigated by varying the binding parameter values for both nonspecific and specific binding sites. All variations of binding parameter values are discussed, and the results show the improved visualization of the effect of binding kinetics in the drug distribution within the brain. For more realistic visualization, we suggest incorporating more brain components that make up the large volume of the brain tissue.

Figure 1

3D brain unit domain, W_in, W_out domains, and bulk flow directions.

Figure 2

Drug exchange between W_pl and W_ECF.

Figure 3

Drug transport in brain capillaries and active transport across the BBB into the brain ECF (a). A cubical lattice (blue) represents a piece of 3-dimensional brain tissue. The cubical lattice is formed of the network of smaller cubic lattices (red). The arrows indicate the bidirectional bulk flow of the brain ECF (b).

Figure 4

Drug distribution in blood plasma (μ) for P = 0.1 × 10⁻⁷ms⁻¹ and v_blood=1 × 10⁻⁶ms⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the blood plasma (μ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 5

Drug distribution in blood plasma (μ) for P = 0.5 × 10⁻⁷ms⁻¹ and v_blood=1.5 × 10⁻⁶ms⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the blood plasma (μ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 6

Drug distribution in blood plasma (μ) for P = 2 × 10^−6.9ms⁻¹ and v_blood=1 × 10⁻⁶ms⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the blood plasma (μ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 7

Drug distribution in blood plasma (μ) for P = 0.1 × 10⁻⁷ms⁻¹ and v_blood=1.1 × 10^−5.95ms⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the blood plasma (μ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 8

Drug distribution in brain ECF (ρ) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the brain ECF (ρ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 9

Drug distribution in brain ECF (ρ) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 5 × 10⁻¹s⁻¹, and k_2off = 3 × 10^−0.2s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the brain ECF (ρ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 10

Drug distribution in the brain ECF (ρ) for k_1on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻³(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the brain ECF (ρ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 11

Drug distribution in the brain ECF (ρ) for k_1on = 0.5(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 2 × 10⁻¹s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the brain ECF (ρ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 12

Drug distribution in the brain ECF (ρ) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 2.5 × 10⁻⁴(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1.5 × 10⁻³s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in the brain ECF (ρ) for t = 2s, t = 8s, and t = 14s, respectively.

Figure 13

Drug distribution in specific binding sites (B₁) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₁) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 14

Drug distribution in non-specific binding sites (B₂) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in non-specific binding sites (B₂) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 15

Drug distribution in non-specific binding sites (B₂) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 5 × 10⁻¹s⁻¹, and k_2off = 3 × 10^−0.2s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in non-specific binding sites (B₂) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 16

Drug distribution in specific binding sites (B₁) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 5 × 10⁻¹s⁻¹, and k_2off = 3 × 10^−0.2s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₁) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 17

Drug distribution in non-specific binding sites (B₂) for k_1on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻³(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in non-specific binding sites (B₂) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 18

Drug distribution in specific binding sites (B₁) for k_1on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻³(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₁) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 19

Drug distribution in specific binding sites (B₁) for k_1on = 0.5(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 2 × 10⁻¹s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₁) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 20

Drug distribution in non-specific binding sites (B₂) for k_1on = 0.5(μmolL⁻¹s)⁻¹, k_2on = 1 × 10⁻²(μmolL⁻¹s)⁻¹, k_1off = 2 × 10⁻¹s⁻¹, and k_2off = 1 × 10⁻¹s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in non-specific binding sites (B₂) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 21

Drug distribution in specific binding sites (B₁) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 2.5 × 10⁻⁴(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1.5 × 10⁻³s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₁) for t = 2 s, t = 8 s, and t = 14 s, respectively.

Figure 22

Drug distribution in nonspecific binding sites (B₂) for k_1on = 1 × 10⁻¹(μmolL⁻¹s)⁻¹, k_2on = 12.5 × 10⁻⁴(μmolL⁻¹s)⁻¹, k_1off = 1 × 10⁻²s⁻¹, and k_2off = 1.5 × 10⁻³s⁻¹. The plots (a), (b), and (c) indicate the distribution of drug in specific binding sites (B₂) for t = 2 s, t = 8 s, and t = 14 s, respectively.