Mathematical Modeling of Tumour Invasion of Tissue

Abstract

In this study we develop a mathematical model using a system of partial differential equation which describes tumor invasion of tissues by focusing on the role of a generic matrix degrading enzyme such as urokinase plasminogen activator. The model consists of a system of reaction-diffusion-taxis partial differential equations describing the interactions between tumor cells, the urokinase plasminogen activator components and the host tissue. We develop a method to obtain an approximate solution of this non-linear system of partial differential equation by applying Laplace-Adomian decomposition method. In the developed method, the higher order derivative is reduced using the Laplace decomposition method and the non-linear terms are handled by Adomian decomposition method. By the developed mathematical model, we examined the invasive behaviours of tumor cells by plotting the graph of the obtained approximate solution from which the biological meaning of the solution is discussed. The results obtained from numerical computations carried out on the model equations produce, after undergoing morphological changes malignant and invasive tumour cells, i.e., cancer cells, break away from the primary tumour by loss of cell-cell adhesion, degrade their basement membrane(through the urokinase plasminogen activator system) and migrate through the extracellular matrix(via taxis) by enhancement of cell-matrix adhesion. The important factors governing the final tumour cell density distribution are extracellular matrix heterogeneity and the haptotactic response of the cells to the gradients created in the degraded matrix. Thus the only gradients in the extracellular matrix are a result of protaese degradation and hence the cells at the leading edge of the tumour are mostly affected by haptotaxis even though small clusters of cells can break away from the central mass of the tumour and invade further leading to possible metastasis. These results may lead to broadening our understanding of cancer cell invasion and in the long term, contributing to methods of patient treatment.