A Spatially Resolved Mechanistic Growth Law for Cancer Drug Development Predicting Tumor Growing Fractions

Abstract

Mathematical models used in preclinical drug discovery tend to be empirical growth laws. Such models are well suited to fitting the data available, mostly longitudinal studies of tumor volume; however, they typically have little connection with the underlying physiologic processes. This lack of a mechanistic underpinning restricts their flexibility and potentially inhibits their translation across studies including from animal to human. Here we present a mathematical model describing tumor growth for the evaluation of single-agent cytotoxic compounds that is based on mechanistic principles. The model can predict spatial distributions of cell subpopulations and account for spatial drug distribution effects within tumors. Importantly, we demonstrate that the model can be reduced to a growth law similar in form to the ones currently implemented in pharmaceutical drug development for preclinical trials so that it can integrated into the current workflow. We validate this approach for both cell-derived xenograft and patient-derived xenograft (PDX) data. This shows that our theoretical model fits as well as the best performing and most widely used models. However, in addition, the model is also able to accurately predict the observed growing fraction of tumours. Our work opens up current preclinical modeling studies to also incorporating spatially resolved and multimodal data without significant added complexity and creates the opportunity to improve translation and tumor response predictions.

Significance: This theoretical model has the same mathematical structure as that currently used for drug development. However, its mechanistic basis enables prediction of growing fraction and spatial variations in drug distribution.

FIGURE 1

Schematic of the modeling concept. A growing tumor is modeled as a proliferating shell encapsulating a necrotic (nonproliferative core) with the boundaries between regions determined dynamically by considering nutrient diffusion. The assumed geometry and model variables and parameters are labeled in the cross-section with and being the necrotic and total tumor radii, respectively. (Histologic image—a day-24 CDX xenograft with Ki-67 stain.)

FIGURE 2

A–C, Fits of the diffusion-limited model to CDX treatment data for protocols 1, 2, and 3, respectively. Treatment protocols are defined in the Materials and Methods. D, Fit of both the diffusion-limited and exponential-linear models to five representative datasets from protocol 1. E and F, VPCs of both the treated and control CDX data, respectively. The VPC is based on 1,000 simulations, the shaded regions represent the 95% confidence intervals of the 5th (yellow), 50th (purple), and 95th (orange) percentiles of the simulated data. The experimental data median, 5th and 95th percentiles are marked (obtained using rolling average).

FIGURE 3

A, Simulated tumor response (blue) along with the predicted growth fraction (red) for dosing strengths 25, 50, 75, and 100 mg/kg (blue: top to bottom curves, respectively, red: bottom to top curves, respectively) dosed on days 1, 8, and 15. B, and C, Simulated tumor dynamics for the CDX xenografts [SW620 (B) and Calu6 (C) cell lines, simulated curves using population parameters from Supplementary Tables S1 and S2. The endpoint box plots are derived from histologic examination of necrotic area for eight SW620 xenografts and 10 Calu6 xenografts].

FIGURE 4

A, Heatmap of the simulated internal drug distribution within the tumor immediately following dosing on day 15. The grey dashed circle shows the predicted necrotic region. The tumor edge is indicated by the black solid line. B, Percentage of drug concentration reaching inner necrotic radius (corresponding to gray circle in A) as a function of drug localization parameter . C, Tumor volume (blue) and growth fraction (red) dynamics for dosing strength fixed at 50 mg/kg dosed on days 1, 8, 15, for (blue: bottom to top curves, respectively, red: top to bottom curves, respectively). describes full drug penetration, with increasing corresponding to a reduction in drug penetration. At , the drug effect is largely restricted to the surface of the tumor.