Nonlinear joint models for individual dynamic prediction of risk of death using Hamiltonian Monte Carlo: application to metastatic prostate cancer

BMC Med Res Methodol. 2017 Jul 17;17(1):105. doi: 10.1186/s12874-017-0382-9.


Background: Joint models of longitudinal and time-to-event data are increasingly used to perform individual dynamic prediction of a risk of event. However the difficulty to perform inference in nonlinear models and to calculate the distribution of individual parameters has long limited this approach to linear mixed-effect models for the longitudinal part. Here we use a Bayesian algorithm and a nonlinear joint model to calculate individual dynamic predictions. We apply this approach to predict the risk of death in metastatic castration-resistant prostate cancer (mCRPC) patients with frequent Prostate-Specific Antigen (PSA) measurements.

Methods: A joint model is built using a large population of 400 mCRPC patients where PSA kinetics is described by a biexponential function and the hazard function is a PSA-dependent function. Using Hamiltonian Monte Carlo algorithm implemented in Stan software and the estimated population parameters in this population as priors, the a posteriori distribution of the hazard function is computed for a new patient knowing his PSA measurements until a given landmark time. Time-dependent area under the ROC curve (AUC) and Brier score are derived to assess discrimination and calibration of the model predictions, first on 200 simulated patients and then on 196 real patients that are not included to build the model.

Results: Satisfying coverage probabilities of Monte Carlo prediction intervals are obtained for longitudinal and hazard functions. Individual dynamic predictions provide good predictive performances for landmark times larger than 12 months and horizon time of up to 18 months for both simulated and real data.

Conclusions: As nonlinear joint models can characterize the kinetics of biomarkers and their link with a time-to-event, this approach could be useful to improve patient's follow-up and the early detection of most at risk patients.

Keywords: Calibration; Discrimination; Hamiltonian Monte Carlo; Individual dynamic prediction; Nonlinear joint model.

MeSH terms

  • Algorithms*
  • Bayes Theorem*
  • Biomarkers, Tumor / analysis
  • Humans
  • Kinetics
  • Male
  • Models, Biological
  • Monte Carlo Method*
  • Neoplasm Metastasis
  • Nonlinear Dynamics*
  • Prostate-Specific Antigen / analysis
  • Prostatic Neoplasms / metabolism
  • Prostatic Neoplasms / pathology
  • Prostatic Neoplasms, Castration-Resistant / metabolism
  • Prostatic Neoplasms, Castration-Resistant / pathology
  • Risk Factors


  • Biomarkers, Tumor
  • Prostate-Specific Antigen