Case Studies in cancer and its treatment by radiotherapy

Mathematical modelling has played a major role in certain aspects of the wide area of study of cancer and its treatment. Relevant background material is followed by mathematical modelling case studies in tumour growth and the response of tumour cells to irradiation. These components are synthesized in Case Study 4 where the problem of the derivation of optimal radiotherapy treatment schedules is discussed. The case studies are presented in a form suitable for classroom development.     

Some mathematical aspects of wave motion

A survey is made of some mathematical techniques and theorems which can have relevance at a moderately advanced level to certain classes of wave propagation problems.     

Colour conductivity of hard spheres

We present an analytic solution for the d-dimensional (d > 1) hard-sphere free flight trajectories in a thermostatted colour field. The solution shows that particles can only reach a finite distance in the direction perpendicular to the field in the absence of collisions. Using a numerical algorithm we designed to simulate many-body hard-sphere systems with curved trajectories, we study the onset of the instability leading to phase separation in the two-dimensional case for a range of field strengths and three densities. For the two fluid densities we find that phase separation occurs for sufficiently strong fields regardless of the initial configuration, and that the phase-separated state eventually becomes a collisionless, non-ergodic steady state. For solid densities the phase-separated configuration is stable and conducting, but is not an attractor for other charge distributions because of the impossibility of particle rearrangement.