North British Differential Equations Seminar

Talks by Professor Friedman

Mathematical Analysis of Cancer Models

Evolution of free boundaries and their asymptotic behavior

We consider a tumor with cells in one of three phases: proliferating, quiescent, or necrotic, with densities P, Q, and N respectively. Living cells can go from one phase to another at a rate that depends on the nutrient concentration C. Due to proliferation of cells and to removal of necrotic cells, the cells undergo motion with velocity field V. Assuming that the tissue is a porous medium, Darcy's Law says that V is the gradient of internal pressure H. Assuming also that P+Q+N=constant we derive an elliptic equation for H and hyperbolic equations for P and Q (based on conservation of mass). Finally, C satisfies a diffusion equation. On the surface of the tumor, which is a free boundary, we impose natural kinematic conditions. We shall describe existence and uniqueness theorems for solutions of the evolution of the tumor, and for stationary solutions (dormant tumors). We shall also address the question of asymptotic stability of radially symmetric stationary solutions.

Symmetry breaking bifurcations

Most papers dealing with mathematical models of tumors consider only the case of radially symmetric solutions. In this talk we shall consider the special case where all the cells in the tumor are proliferating so that, in the notation of (I), P=constant, Q=N=0. The free boundary problem is reduced to two PDEs for the nutrient concentration C and the pressure H. We further consider only stationary solutions. In this case there exists a family of radially symmetric solutions depending on a parameter e. We shall state results regarding the existence of infinite sequence of branches of symmetry-breaking stationary solutions bifurcating from radially symmetric solutions. We shall also consider the question of asymptotic stability of the stationary solutions.


Some of the experimental drugs for cancer therapy are genetically engineered viruses. The virus adsorbs to the surface of cancer cells, penetrates and infects them, and, as these infected cells die, many virus particles burst out and infect adjacent cells. The mathematical model we consider assumes conservation laws for the densities of uninfected tumor cells, P, infected tumor cells, Q, and dead cells, N. The density of the virus particles V satisfies a diffusion-transport equation. The velocity field is the gradient of the internal pressure H; H satisfies an elliptic equation. The mathematical problem is somewhat similar to the free boundary problem in lecture (I); however the coupling terms in the present system and some of the conditions at the free boundary are different. The main question here is how to treat the tumor, with initial injection of V, or with a sequence of injections, in an optimal way. We shall report on ongoing work. As a tumor grows, it requires more nutrients. To address this need, it stimulates the formation of new blood vessels in its vicinity - this is the process of angiogenesis. One of the experimental methods to control tumor growth is based on trying to block the angiogenesis process. If time allows, I shall discuss mathematical models of angiogenesis.
GDL Last revised: