Computational and Mathematical Methods in Medicine
Volume 10 (2009), Issue 2, Pages 117-138
doi:10.1080/17486700802200784
Original Article

A Stochastic and State Space Model for Tumour Growth and Applications

Wai-Yuan Tan,1 Weiming Ke,2 and G. Webb3

1Department of Mathematical Sciences, The University of Memphis, Memphis, TN, USA
2Department of Mathematics and Statistics, South Dakota State University, Brookings, SD, USA
3Department of Mathematics, Vanderbilt University, Nashville, TN, USA

Received 18 June 2007; Accepted 1 May 2008

Copyright © 2009 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We develop a state space model documenting Gompertz behaviour of tumour growth. The state space model consists of two sub-models: a stochastic system model that is an extension of the deterministic model proposed by Gyllenberg and Webb (1991), and an observation model that is a statistical model based on data for the total number of tumour cells over time. In the stochastic system model we derive through stochastic equations the probability distributions of the numbers of different types of tumour cells. Combining with the statistic model, we use these distribution results to develop a generalized Bayesian method and a Gibbs sampling procedure to estimate the unknown parameters and to predict the state variables (number of tumour cells). We apply these models and methods to real data and to computer simulated data to illustrate the usefulness of the models, the methods, and the procedures.