The Klein–Gordon equation in mixture models of tumour growth

A mixture model of tumour microenvironment is considered, which consists of a solid phase for the tumour cells, a liquid phase for the interstitial fluid, and a nutrient phase. The balance equations for the three phases take into account exchange of mass between tumour and nutrients, and exchange of drag forces between the constituents. Under rather natural assumptions, the determination of the nutrient density is reduced to the solution of a Klein–Gordon equation, with source term depending on mass injection from outside. A chain of decoupled equations for the remaining unknowns is then determined in terms of the nutrient density. Finally, the growth of tumour volume is investigated under the assumption of spherical symmetry.