"On growth of ellipsoidal tumours"

The existing mathematical models for tumour growth, to a large extent, are new and not well established as of today. This is mainly due to the fact that there are many known and unknown factors that enter the process of malignant tumour development, and no convincing arguments about their relative importance are generally established. As a consequence of this search for a credible model, almost every tumour model that has been investigated so far refers to the highly symmetric case of the spherical geometry, where the curvature is a global invariant over its outer surface. Hence, no information about the effects of the local curvature upon the shape of the exterior proliferating boundary is available. In this presentation, we discuss first the standard Greenspan model for a spherical tumour, where the basic ideas are presented, and then we extend the model to that of triaxial ellipsoidal geometry. In this way, we elevate fundamental qualitative characteristics of the growth process that are invisible in spherical geometry. One such thing is the effect of the local mean curvature on the development of the outer boundary of the tumour, as it is governed by the Young-Laplace law, which controls the interface between two non-mixing fluids. A second advantage of the ellipsoidal model is due to the way the confocal system is generated. Indeed, in contrast to the spherical system which springs out of a central point, the confocal ellipsoidal system starts out as an inflated focal ellipse which, if it is interpreted as a biological membrane, provides a much more realistic candidate for tumour genesis. Nevertheless, the investigation of the ellipsoidal model of a tumour growth is by no means completed, and a lot of further study needs to be done before final conclusions on the effects of curvature variations are drawn.