Centre for Multidisciplinary Mathematical Modelling Seminar Series
June 17 @ 10:30 am - 11:30 am
Professor Graeme Wake, Massey University, New Zealand
Beating the big C: Cell population growth models
Graeme Wake is Professor Emeritus of Industrial Mathematics at Massey University (New Zealand). Professor Wake has made significant contributions to applied mathematics in the areas of combustion, mathematical biology (particularly to agricultural problems) and industrial mathematics.
Links to the Cancer Society research group in NZ, led to this long-term project in living cell population dynamics. A model for cell populations, which are structured by size and are undergoing growth and division simultaneously, leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term.
This is akin to the famous pantograph equation, with advanced multiplicative functionality but here in size. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic, as time goes to infinity, to an attracting certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems.
In this presentation, we derive a novel solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solutions are known explicitly, higher-order terms in the asymptotic form can be readily obtained. These low parameter solutions are used to underpin experimental work in developing new anti-cancer drugs.