Mathematical and computational models of drug transport in tumours

Abstract

The ability to predict how far a drug will penetrate into the tumour microenvironment within its pharmacokinetic (PK) lifespan would provide valuable information about therapeutic response. As the PK profile is directly related to the route and schedule of drug administration, an in silico tool that can predict the drug administration schedule that results in optimal drug delivery to tumours would streamline clinical trial design. This paper investigates the application of mathematical and computational modelling techniques to help improve our understanding of the fundamental mechanisms underlying drug delivery, and compares the performance of a simple model with more complex approaches. Three models of drug transport are developed, all based on the same drug binding model and parametrized by bespoke in vitro experiments. Their predictions, compared for a 'tumour cord' geometry, are qualitatively and quantitatively similar. We assess the effect of varying the PK profile of the supplied drug, and the binding affinity of the drug to tumour cells, on the concentration of drug reaching cells and the accumulated exposure of cells to drug at arbitrary distances from a supplying blood vessel. This is a contribution towards developing a useful drug transport modelling tool for informing strategies for the treatment of tumour cells which are 'pharmacokinetically resistant' to chemotherapeutic strategies.

A general model of coupled drug release and tissue absorption for drug delivery devices

“…The above equation is well accepted to describe binding-unbinding processes in biological media, and has been used by several other authors in various contexts, including for drug release to the arterial wall [28,15] and tumour drug delivery [28,29]. Loosely speaking, binding describes a phenomenon opposite to dissolution, and Eq.…”

Delivery of cancer therapeutics to extracellular and intracellular targets: Determinants, barriers, challenges and opportunities

“…These publications are in two general categories. The first category describes the development of theoretical models to capture the known tumor pathobiological parameters and the physical processes underlying the transport processes (e.g., [250–254]). While these publications do not provide experimental verifications of the proposed models, they are excellent resources for model development.…”

Drug delivery in a tumour cord model: a computational simulation

“…Applying the principle of mass action leads to three coupled ordinary differential equations which describe the system [9]: $\begin{array}{cc}right\; left\; right\; left\; right\; left\; right\; left\; right\; left\; right\; left3pt0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em{V}_1\frac{\mathrm{d}{C}_1}{\mathrm{normald}t}& =a{k}_1(C_2-C_1),\end{array}$ $\begin{array}{cc}right\; left\; right\; left\; right\; left\; right\; left\; right\; left\; right\; left3pt0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em{V}_2\frac{\mathrm{normald}{C}_2}{\mathrm{normald}t}& =a{k}_1(C_1-C_2)-V_2{k}_2{C}_2(C_0-C_3)+V_2{k}_{-2}{C}_3\end{array}$ $\begin{array}{cc}right\; left\; right\; left\; right\; left\; right\; left\; right\; left\; right\; left3pt0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\; 2em\; 0.278em\text{and}{V}_2\frac{\mathrm{normald}{C}_3}{\mathrm{normald}t}& =V_2{k}_2{C}_2(C_0-C_3)-V_2{k}_{-2}{C}_3,\end{array}$in which k 1 is the rate constant for the transmembrane transport of drug, a is the area of the interface between the extracellular and intracellular spaces (the surface area of the cells), k 2 and k −2 are the drug binding and unbinding rates, respectively, and C 0 is the concentration of binding sites within the cell. This model is illustrated by the schematic in figure 1.…”

“…This model is illustrated by the schematic in figure 1. Values for the kinetic rate constants for the binding process, given in table 1, have been derived from a bespoke experimental binding assay, outlined in [9].

Figure 1.A three-compartment model of drug distribution in tissue.

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“…The parameter k 0 has been estimated based on the value of r 1 (transport rate between cell layers) in the multilayer model of [8]. )variablevaluedescriptionsource of estimate l 1.6×10 −5   m vessel radiushistology [12] L r 1.96×10 −4   m cord radius (vessel + 9 cells)histology [13,14] L z 5.0×10 −4   m cord length (25 cells)histology [15] r 1.0×10 −5   m cell radiushistology [16] δ 0.0625extracellular : intracellular volume ratio parameterhistology [8,17] α 1.94028×10 5   m −1 membrane surface : tissue volume ratio$\alpha =2/\left(r\sqrt{1+\delta }\right)$ k 0 2.5×10 −6   m   s −1 permeability between cellsexperiment [8] k 1 1.0×10 −6   m   s −1 permeability across cell membraneexperiment [9] k 2 0.90×10 −6  μ M −1   s −1 drug binding rateexperiment [9] k −2 14.0×10 −6   s −1 drug unbinding rateexperiment [9] β 140/9 μ M unbinding/binding ratio β = k −2 / k 2 k v 2.8×10 −6   m   s −1 permeability across vessel wallestimate [18]…”