Details of the mathematical model can be found in the appendix. Briefly, a Krogh cylinder geometry was used where a cylindrical blood vessel segment is surrounded by tissue with a radius approximately equal to half the local inter-capillary distance (
Figure 1B). Blood flows from the arterial end, where a systemic two-compartment model with biexponential decay defines the concentration. The local blood concentration is determined by the blood velocity, permeability of the vessel wall, and fraction of free drug (not bound to blood cells or plasma proteins). Cellular uptake in the blood was ignored (
File S1) given the slower kinetics relative to blood flow [22] (
link). A mixed boundary condition is used at the capillary interface, where the flux at the capillary wall determined by the permeability is equal to the diffusive flux into the tissue. In the tissue, the free drug undergoes radial and axial diffusion along with agent specific reaction terms. For small molecules, this involves cellular uptake and metabolism (e.g. oxygen utilization, irreversible trapping over short time scales by FDG phosphorylation, reversible uptake for doxorubicin). For antibodies, this involves reversible binding and dissociation with irreversible internalization. Due to the lack of functional lymphatics in tumors, lymphatic drainage was ignored [23] (
link). The following equations defined the plasma concentration, plasma tissue interface, and tissue concentration (for first order kinetics):
where [C]
plasma is the total concentration of drug in the plasma, t is time, v is the local blood velocity, L is the length along the vessel segment, R
cap is the capillary radius, H is the hematocrit, P is the vessel wall permeability, f
free is the fraction of drug that is unbound, [C]
tissue,free is the unbound concentration in the tissue (overall/pseudohomogenous concentration), and epsilon the void fraction. D is the effective diffusion coefficient in tissue, r is the radial distance from a vessel, and k
rxn defines the local reaction rate (which is first order in this example equation).
The method of lines was used with axial and radial variations and solved with a stiff solver using Matlab (The Mathworks; Natick, MA). A sparse Jacobian was defined to decrease simulation times.
Thurber G.M, & Weissleder R. (2011). A Systems Approach for Tumor Pharmacokinetics. PLoS ONE, 6(9), e24696.