Estimating Unknown Transition Times Using a Piecewise Nonlinear Mixed-Effects Model in Men with Prostate Cancer

Abstract

It may be clinically useful to know when prostate-specific antigen (PSA) levels first begin to rise rapidly and to determine if the natural history of PSA progression is different in men with locally confined prostate cancers compared to men with metastatic tumors. This article uses a nonlinear mixed-effects model to describe longitudinal changes in PSA in men before their prostate cancers were detected clinically. Repeated measurements of PSA are available for 18 subjects with a diagnosis of prostate cancer based on prostate biopsy. PSA measurements were determined on repeated frozen serum samples collected from subjects with at least 10.0 years and up to 25.6 years of observation before the cancer was detected. A piecewise model is used to describe this data. The model is linear long before the cancer was detected and exponential nearer the time the cancer was detected. The time at which the PSA levels change from linear to exponential PSA progression is unknown but can be estimated by including random terms that allow each subject to have his own transition time. The model also accounts for two groups of patients—those with local or regional cancer and those with advanced cancer or whose cancer has metastasized. Various parameters are allowed to differ between these two groups. By backward elimination of statistically nonsignificant parameters, a model is found that adequately describes the data. The model represents a situation where local/regional and advanced/metastatic cancers have similar rates of PSA progression, but advanced/metastatic cancers are diagnosed later. Piecewise mixed-effects models may be useful in a variety of research settings where it is necessary to estimate the unknown time of an event.