  Joint modelling of longitudinal and survival data is increasingly used in
clinical trials on cancer. In prostate cancer for example, these models permit
to account for the link between longitudinal measures of prostate-specific
antigen (PSA) and the time of clinical recurrence when studying the risk of
relapse. In practice, multiple types of relapse may occur successively.
Distinguishing these transitions between health states would allow to evaluate,
for example, how PSA trajectory and classical covariates impact the risk of
dying after a distant recurrence post-radiotherapy, or to predict the risk of
one specific type of clinical recurrence post-radiotherapy, from the PSA
history. In this context, we present a joint model for a longitudinal process
and a multi-state process which is divided into two sub-models: a linear mixed
sub-model for longitudinal data, and a multi-state sub-model with proportional
hazards for transition times, both linked by shared random effects. Parameters
of this joint multi-state model are estimated within the maximum likelihood
framework using an EM algorithm coupled to a quasi-Newton algorithm in case of
slow convergence. It is implemented under R, by combining and extending the
mstate and JM packages. The estimation program is validated by simulations and
applied on pooled data from two cohorts of men with localized prostate cancer
and treated by radiotherapy. Thanks to the classical covariates available at
baseline and the PSA measurements collected repeatedly during the follow-up, we
are able to assess the biomarker's trajectory, define the risks of transitions
between health states, and quantify the impact of the PSA dynamics on each
transition intensity.
