The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas

Abstract

This paper is a natural three-dimensional extension of a simple two-dimensional mathematical model of glioma growth and diffusion. The model was originally constructed to simulate a case of recurrent anaplastic astrocytoma treated with chemotherapy, and then modified to allow estimation of the effects of the extent of surgical resection and of variations in growth and diffusion to cover the whole range of glioma behavior. Growth is considered to be constant and exponential (analogous to continuously compounding interest) and is expressed as a decimal fraction per day; the diffusion coefficient is expressed as cm2 per day. Model predictions suggest that diffusion, practically ignored until the present, is a more important component of glioma growth than the growth rate. Even with very early diagnosis, only those tumors with a low diffusion coefficient and a rapid growth rate benefit from a wide resection. Surgical resections generally fail, just as dropping fire-fighters into the burned out center of a forest fire fails, the action being on the periphery as the tumor cells or fires spread out from the center.