Last modified: 2014-03-31

#### 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, in an unperturbed situation, is observed experimentally to exhibit the attribute of that of an asymptotic ”Steady-Size-Distribution” (SSD). i.e.,

n(x, t) ∼ scaled (by time) multiple of a constant shape y(x) as t → ∞.

By developing a simple evolutionary model to describe this outcome, criteria for the growth/decline of the cohort can be determined. This enables precise estimates to be made on the characteristics of the growth/division process. By incorporating induced cell apoptosis (death) we can then provide a diagnostic tool for clinicians.

The calculus involved is different, in fact, it is “non-local” where cause and effect are separated by time, space, or in this case, by size. This involves little-developed-techniques often included in the undergraduate curriculum.

This work is relevant to the underlying understanding of cell tumor growth. It is believed that the symmetric division of stem cells increases the risk of cancer. Alternatively, asymmetric division of stem cells provides a degree of protection against cancer. Therefore, a new model is needed of cell-growth with asymmetrical division (two or more daughter cells of different sizes from a single “division-event”). This model must capture the key features from earlier models with symmetrical cell division, where the cell-size distribution tends asymptotically to one of constant shape when the cohort is not disturbed; this being a well known observation. This is also still called a steady-size-distribution (SSD).

A model is proposed which does this for different types of cellular evolution and amounts to a hyperbolic integer-differential equation. Separated solutions again answer the question of SSD behavior and the time constant can be a principal eigenvalue of a singular first-order integro- ODE. More general questions arise as to whether these solutions span the space of all solutions.