An alternative approach to modelling relapse in cancer with an application to adencarcinoma of the prostate

This paper proposes an alternative approach to modelling relapse in cancer. In particular, the dynamic model for the tumor or biomarker will be subjected to a lower elastic boundary at which the process either will be absorbed or reflected. The likelihood of reflection then can be interpreted as the probability of relapse. This framework will be exemplified for prostatic cancer by extending the recently proposed stochastic model of Dayananda et al. [P.W.A. Dayananda, J.T. Kemper, M.M. Shvartsman, A stochastic model for prostate-specific antigen levels, Math. Biosci. 190 (2004) 113] that focussed on the dynamics of the prostate-specific antigen (PSA) biomarker. Analytical results for the conditional density function, given a non-negative lower boundary, are obtained for the extreme cases of certain cure and of certain relapse. Simulations illustrate the relevance of the relapse probability and of the normal value of the biomarker for the design of treatment strategies. The paper thus points to two additional (patient-specific) characteristics that might enter treatment design and monitoring of progress in therapy.

Keywords: Absorbing boundary; Cancer; Conditional probability density function; Elastic boundary; Prostate cancer; Prostate-specific antigen (PSA); Reflecting boundary; Relapse; Stochastic differential equations

MSC: 60J35; 60J70; 60K30; 92B15