Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology

Numerical analysis and computational simulation of partial differential equation models in mathematical biology are now an integral part of the research in this ¯eld. Increasingly we are seeing the development of partial di®erential equation models in more than one space dimension, and it is therefore necessary to generate a clear and effective visualisation platform between the mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualisation of the results is more complicated. The scope of this paper is to apply the established marching cubes volume rendering technique to the study of solid tumour growth and invasion, and present an adaptation of the algorithm to speed up the surface rendering from numerical simulation data. As a specific example, in this paper we examine the computational solutions arising from numerical simulation results of a mathematical model of malignant solid tumour growth and invasion in an irregular heterogeneous three-dimensional domain, i.e., the female breast. Due to the di®erent variables that interact with each other, more than one data set may have to be displayed simultaneously, which can be realized through trans parency blending. The usefulness of the proposed method for visualisation in a more general context will also be discussed.