Tumour Cell Biology and Some New Non-local Calculus

Abstract

Living cell populations which are simultaneously growing and dividing are usually structured by size, which can be, for example, mass, volume, or DNA content. The evolution of the number density $n(x,t)$ of cells by size $x$, in an unperturbed situation, is observed experimentally to exhibit the attribute of that of an asymptotic “Steady-Size-Distribution” (SSD). That is, $n(x,t) \sim$ scaled (by time $t$) multiple of a constant shape $y(x)$ as $t \rightarrow \infty$, and $y(x)$ is then the SSD distribution, with constant shape for large time. A model describing this is given, enabling parameters to be evaluated. The model involves a linear non-local partial differential equation. Similar to the well-known pantograph equation, the solution gives rise to an unusual first order singular eigenvalue problem. Some results and conjectures are given on the spectrum of this problem. The principal eigenfunction gives the steady-size distribution and serves to provide verification of the observation about the asymptotic growth of the size-distribution.