Modeling of drug diffusion in a solid tumor leading to tumor cell death

Abstract

It has been shown recently that changing the fluidic properties of a drug can improve its efficacy in ablating solid tumors. We develop a modeling framework for tumor ablation, and present the simplest possible model for drug diffusion in a spherical tumor with leaky boundaries and assuming cell death eventually leads to ablation of that cell effectively making the two quantities numerically equivalent. The death of a cell after a given exposure time depends on both the concentration of the drug and the amount of oxygen available to the cell. Higher oxygen availability leads to cell death at lower drug concentrations. It can be assumed that a minimum concentration is required for a cell to die, effectively connecting diffusion with efficacy. The concentration threshold decreases as exposure time increases, which allows us to compute dose-response curves. Furthermore, these curves can be plotted at much finer time intervals compared to that of experiments, which is used to produce a dose-threshold-response surface giving an observer a complete picture of the drug's efficacy for an individual. In addition, since the diffusion, leak coefficients, and the availability of oxygen is different for different individuals and tumors, we produce artificial replication data through bootstrapping to simulate error. While the usual data-driven model with Sigmoidal curves use 12 free parameters, our mechanistic model only has two free parameters, allowing it to be open to scrutiny rather than forcing agreement with data. Even so, the simplest model in our framework, derived here, shows close agreement with the bootstrapped curves, and reproduces well established relations, such as Haber's rule.

Tumor ablation due to inhomogeneous -- anisotropic diffusion in generic 3-dimensional topologies

“…We now impose a specific initial condition. In [17] a bump function (compact Gaussian) decaying to zero just within the domain was used. During inhomogeneous -anisotropic diffusion, the drug does not diffuse evenly, and hence a bump function that extends to the endpoints would not capture the irregularities expected in such a problem.…”

“…As done in [17], we use a binary population model: the tumor cell is dead after some exposure time τ (which is much larger than the diffusive time scale), if at any time during the diffusion process the drug concentration is above some given threshold value u T (τ ), otherwise it is alive. To simplify computation, the model assumes natural cell death rate is equivalent to the cell population growth rate, and effectively negate each other.…”

“…From the representative sample we calculate what u T (τ = 24 hrs), u T (τ = 48 hrs), u T (τ = 72 hrs) must be in order to produce the empirically observed response. Then we have three equations with three unknowns, which is solved explicitly in [17]. In this paper, we use the following threshold values.…”

“…In [17], Rahman et al present a simple drug diffusion -binary population model. It is assumed that a drug is being injected directly into the center of a homogeneous -isotropic spherical tumor, and hence the diffusion is radial with constant diffusivity.…”

“…While the model of Rahman et al [17] performed well against artificial replication data, the simplicity is burdened by the baggage of assumptions. Transport of drugs in the brain is a complex process due to the highly inhomogeneous and anisotropic structure of brain tissue, local pressure differences, and chemcial interactions of the drug with the surrounding tissue.…”