Date of Award

Summer 1989

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematics and Statistics


Computational and Applied Mathematics

Committee Director

John A. Adam

Committee Member

John H. Heinbockel

Committee Member

Philip Wohl

Committee Member

Lloyd Wolfinbarger, Jr.


A study of several complementary mathematical models that describe the early, prevascular stages of solid tumor growth by diffusion under various simplifying assumptions is presented. The advantage of these models is that their degree of complexity is relatively low, which ensures fairly straightforward comparisons with experimental or clinical data (as it becomes available), yet they are mathematically sophisticated enough to capture the main biological phenomena of interest.

The tumor growth and cell proliferation rate are assumed to depend on the local concentrations of nutrients and inhibitory factors. The effects of geometry and spatially non-uniform inhibitor production and non-uniform nutrient consumption on the prevascular tissue growth are examined. The concentrations of nutrients and growth inhibitor are governed by diffusion processes, and thus the equations are of diffusion type in spherically symmetric geometries. Since a key characteristic of cancerous diseases is uncontrolled growth, the sensitivity of a model to the nature of different mitotic control functions is examined and the stability of subsequent tissue growth is discussed. A limiting size for the stable tissue growth is provided, and in related models the time-evolution of the tissue prior to that limiting state is described via a growth (integro-differential) equation for the different phases of tumor growth; the kernel of which depends on the solutions of the spherically symmetric diffusion equations for the concentration of nutrient and growth inhibitor within the tumor. Conditions on the existence and uniqueness of solutions to two classes of non-linear time-independent diffusion equations, which arise in tumor growth models, are also examined.

A detailed study of theoretical models of the type constructed here provides useful insight into the basic biological mechanisms of tumor growth, and therefore may offer possibilities for optimization of cancer therapy (e.g. chemo- or radio-therapy).