Thermodynamic phase-field modelling predicts non-linear evolution of tumour spheroid dynamics
Patient-derived tumour spheroids provide experimentally tractable models of tumour growth and treatment response, but standard radius-based measurements do not directly identify the tissue-scale mechanisms that shape internal structure. This study develops and calibrates a thermodynamic continuum model of untreated spheroid internal structural growth, in which viable and necrotic cell populations, nutrient transport, mechanical relaxation, and cell-fate transitions are represented as coupled spatial fields. Because the model does not prescribe internal concentric spheroid compartments like traditional spheroid models, the experimentally reported intermediate (inhibited) radius was treated as an exploratory model-to-data mapping and optimised over different candidate field-derived radii. The calibrated model reproduced the experimental trajectories with aggregate uncertainty-weighted root-mean-square error (WRMSE) values ranging from 0.49 to 1.24 on the metric defined in the manuscript, and improved average fit quality by approximately 40% relative to an existing Greenspan-type compartment model. The model recovered the growth-inhibited and necrotic structure observed in spheroids as an emergent consequence of spatial cell-state dynamics rather than prescribing concentric compartments. Biological and nutrient-associated model parameters were locally constrained to the experimental data, although compensatory relationships were observed among nutrient-response and death parameters. The framework was further extended by introducing a drug transport field as a proof-of-concept demonstration of spatial treatment-response modelling. These results show that a continuum cell-growth model can recover classical spheroid structure while preserving flexibility for exploratory field-based analysis of drug response.