Current advances in mathematical modeling of anti-cancer drug penetration into tumor tissues

Abstract

Delivery of anti-cancer drugs to tumor tissues, including their interstitial transport and cellular uptake, is a complex process involving various biochemical, mechanical, and biophysical factors. Mathematical modeling provides a means through which to understand this complexity better, as well as to examine interactions between contributing components in a systematic way via computational simulations and quantitative analyses. In this review, we present the current state of mathematical modeling approaches that address phenomena related to drug delivery. We describe how various types of models were used to predict spatio-temporal distributions of drugs within the tumor tissue, to simulate different ways to overcome barriers to drug transport, or to optimize treatment schedules. Finally, we discuss how integration of mathematical modeling with experimental or clinical data can provide better tools to understand the drug delivery process, in particular to examine the specific tissue- or compound-related factors that limit drug penetration through tumors. Such tools will be important in designing new chemotherapy targets and optimal treatment strategies, as well as in developing non-invasive diagnosis to monitor treatment response and detect tumor recurrence.

Keywords: drug distribution; drug penetration; drug pharmacodynamics; mathematical modeling; solid tumor; tumor microenvironment.

Figure 1

A schematic representation of multiple physical processes involved in drug penetration into the tumor tissue. Drug molecules are supplied from the vasculature and move through the interstitial space via diffusive and advective transports, can be activated and are subject to natural decay before they are taken up by the cells.

Figure 2

Examples of typical outcomes from various mathematical models of drug penetration through the tumor tissue. (A) A hybrid model of tumor mass and discrete vasculature (left, red-tumor tissue, brown-vasculature) was used to investigate fluid and drug extravasation from the vasculature (right, color corresponds to drug concentration) [from Wu et al. (37), Figure 18]; (B) Patterns of diffusion-(top) and advection-(bottom) dominated transport of drug molecules (red dots) through the interstitial space between the cells (white circles) [from Rejniak et al. (58), Figure 6]; (C) A gradient of the interstitial fluid pressure (top) and transvascular pressure differences (bottom) in the growing tumor mass [from Stylianopoulos et al. (30), Figure 2]; (D) A drug concentration gradient from the vessel (white) outward with colors representing high (yellow) and low (brown) levels of diffusive drug [from Thurber et al. (17), Figure 8]; (E) Spatial distributions of hypoxic (yellow), necrotic (black), and apoptotic (green) tumor cells within the tumor mass treated with angiogenesis inhibitors for 37 weeks; the treatment is supplied from vasculature (red) [from Gevertz (34), Fig.3]; (F) Structural adaptation of vessel diameters (colors represent the volume of blood flow) inside the tumors [from Pries et al. (31), Figure 6]. All figures reprinted with permissions.