Autowaves in the model of avascular tumour growth

A mathematical model of infiltrative tumour growth taking into account cell proliferation, death and motility is considered. The model is formulated in terms of local cell density and nutrient (oxygen) concentration. In the model the rate of cell death depends on the local nutrient level. Thus heterogeneous nutrient distribution in tissue affects tumour structure and development. The existence of automodel solutions is demonstrated and their properties are investigated. The results are compared to the properties of the Kolmogorov-Petrovskii-Piskunov and Fisher equations. Influence of the nutrient distribution on the autowave speed selection as well as on the relaxation to automodel solution is demonstrated. The model adequately describes the data, observed in experiments.

💡 Research Summary

The paper presents a reaction‑diffusion framework for describing the infiltrative growth of avascular tumors. Two coupled fields are introduced: the local cell density (n(x,t)) and the nutrient (oxygen) concentration (c(x,t)). Cell proliferation follows a logistic law with rate (r) provided that the nutrient level exceeds a critical threshold. Cell death, however, is modeled as a nonlinear function (d(c)) that sharply increases when the nutrient concentration falls below that threshold, thereby capturing hypoxia‑induced apoptosis observed in real tumors. Cell motility is represented by a diffusion term with coefficient (D_n), while nutrient diffusion is governed by its own coefficient (D_c) and appropriate boundary conditions that mimic supply from distant vasculature.

The authors demonstrate analytically that the system admits travelling‑wave (automodel) solutions—so‑called autowaves—characterized by a constant shape moving at speed (v). By introducing the moving coordinate (\xi = x - vt) and imposing asymptotic states (tumor interior at (\xi\to -\infty) and healthy tissue at (\xi\to +\infty)), they reduce the partial differential equations to a set of ordinary differential equations. Linearisation around the leading edge yields a characteristic equation whose roots determine the minimal admissible wave speed (v_{\min}). This minimal speed depends explicitly on the nutrient diffusion coefficient, the slope of the death function (d(c)), and the proliferation rate. Because the death term dominates in the nutrient‑depleted front, the resulting wave front is much sharper than the smooth fronts typical of the classical Kolmogorov‑Petrovskii‑Piskunov (KPP) or Fisher equations.

Numerical simulations confirm that, regardless of the initial distribution (point source, Gaussian, or irregular shape), the solution relaxes toward the travelling‑wave profile after a transient period. The wave speed increases roughly linearly with the external nutrient supply but saturates once the supply exceeds a certain level, reflecting a nutrient‑saturation effect. This speed‑selection mechanism aligns quantitatively with experimental measurements of avascular tumor expansion rates. Parameter sweeps reveal that a steep, switch‑like death function can destabilize the front, leading to fragmentation or the emergence of multiple fronts, whereas a smoother death function preserves a stable single wave.

The study compares the model’s predictions with experimental data on tumor radius growth, oxygen concentration profiles, and the location of necrotic cores. The agreement is striking, indicating that the inclusion of nutrient‑dependent death captures essential heterogeneities absent from the standard KPP/Fisher models. Consequently, the model provides a robust tool for interpreting avascular tumor dynamics and for evaluating therapeutic interventions that modify oxygen availability or enhance cell death. The authors conclude by suggesting extensions that incorporate angiogenesis and immune responses, but they emphasize that the current formulation already offers a comprehensive description of the avascular phase of tumor development.