A simple growth model for tumor spheroids

Mathematical models of tumor spheroid growth provide essential insights into avascular tumor dynamics, hypoxia development, and treatment response. Here, we present a simple biophysically motivated growth model that integrates energy conservation, oxygen diffusion, and necrosis formation into a unified framework. The model captures the key features of spheroid development, including initial exponential growth, deceleration due to diffusion limitation, and growth inhibition mediated by necrotic tissue. Using published growth data from human colon carcinoma (HCC1), mouse colon carcinoma (MCC26), V79 fibroblasts, and EMT6-Ro spheroids, we demonstrate that the model accurately reproduces observed growth kinetics across multiple phases. Importantly, the model predicts viable rim thickness and oxygen distribution over time, in agreement with theoretical expectations. Fitted parameters further enable estimation of single-cell mass, oxygen consumption rate, and the energetic cost of cell production. By linking geometry, metabolism, and growth regulation, this minimal model offers a transparent yet quantitative description of tumor spheroid behavior, with potential applications in radiobiology and therapy modeling.