M. A. J. Chaplain, L. Graziano, L. Preziosi; Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math Med Biol 2006; 23 (3): 197-229. doi: 10.1093/imammb/dql009
This paper presents a mathematical model of normal and abnormal tissue growth. The modelling focuses on the potential role that stress responsiveness may play in causing proliferative disorders which are at the basis of the development of avascular tumours. In particular, we study how an incorrect sensing of its compression state by a cell population can represent a clonal advantage and can generate hyperplasia and tumour growth with well-known characteristics such as compression of the tissue, structural changes in the extracellular matrix, change in the percentage of cell type (normal or abnormal), extracellular matrix and extracellular liquid. A spatially independent description of the phenomenon is given initially by a system of non-linear ordinary differential equations which is explicitly solved in some cases of biological interest showing a first phase in which some abnormal cells simply replace the normal ones, a second phase in which the hyper-proliferation of the abnormal cells causes a progressive compression within the tissue itself and a third phase in which the tissue reaches a compressed state, which presses on the surrounding environment. A travelling wave analysis is also performed which gives an estimate of the velocity of the growing mass.