Mathematical Modeling of Multicellular Tumor Spheroids Quantifies Inter-Patient and Inter-Tumor Heterogeneity

Abstract

In the study of brain tumors, patient-derived three-dimensional sphere cultures provide an important tool for studying emerging treatments. The growth of such spheroids depends on the combined effects of proliferation and migration of cells, but it is often challenging to make accurate distinctions between increase in cell number versus the radial movement of cells. To address this, we formulate a novel model (which we refer to as the reaction-diffusion, advection-reaction-diffusion model) in the form of a system of two partial differential equations (PDEs) and show that it more accurately fits our data as compared to simpler PDE models, including the reaction-diffusion model that has been found to be clinically relevant. We propose a numerical approach for calculating travelling-wave speeds, which are shown to be strongly associated with population heterogeneity. Having fitted the model to our dataset we show that a subset of the cell lines are best described by a “Go-or-Grow”-type model, which constitutes a special case of our model. Finally, we investigate whether our fitted model parameters are correlated with patient age at diagnosis and survival. Multiple model parameters are correlated with patient age, and a single model parameter with patient survival. Our work improves on current modelling approaches, and provides a novel step towards understanding associations between mathematical model parameters and clinical data.