A dynamic model for tumour growth and metastasis formation

In the mathematically oriented medical literature different models are applied to describe the process of tumour growth and metastasis formation. Most of these models fall in one of the three following categories: The first ones are discrete models on the basis of single cell interactions which are then described by the aid of Mte Carlo simulations. The second ones are complex mathematical analyses of continuum models on the base of differential equations. A good overview of these approaches can be found in the articles of Ward and King [1,2] and Roose, Chapman and Maini [3]. A third interesting alternate ansatz was developed by Iwata, Kawasaki and Shigesada [4,5] which is in the following referred to as the IKS-model. They model metastasis formation from the primary tumour and from metastases from metastases and give complex analytical solutions for the density respective the abundance of metastatic colonies depending on different growth functions of the primary tumour.All the abovementioned methods have the disadvantage of complex re-analysis or the need for time consuming numerical re-calculations when input functions or constraints are to be varied. Systematic investigations and the analysis of metastasis modulating events or treatment effects upon metastasis formation are limited due to the complexity or the computing power required.In the following a mathematical model is presented which is based upon a series of successive generations of tumour development. This model enables a fast calculation of macroscopic relevant entities of the metastatic cascade. The entire programming was carried out in the C language using the graphical analysis package root, developed at CERN [6].Metastasis formation is a complex process often referred to as a cascade as each step has to be performed in a certain order. It is initiated, when the first primary malignant cell starts to proliferate. If the developing primary tumour has reached a certain size, it sends out angiogenetic