  We consider Bayesian inference for stochastic differential equation mixed
effects models (SDEMEMs) exemplifying tumor response to treatment and regrowth
in mice. We produce an extensive study on how a SDEMEM can be fitted using both
exact inference based on pseudo-marginal MCMC and approximate inference via
Bayesian synthetic likelihoods (BSL). We investigate a two-compartments SDEMEM,
these corresponding to the fractions of tumor cells killed by and survived to a
treatment, respectively. Case study data considers a tumor xenography study
with two treatment groups and one control, each containing 5-8 mice. Results
from the case study and from simulations indicate that the SDEMEM is able to
reproduce the observed growth patterns and that BSL is a robust tool for
inference in SDEMEMs. Finally, we compare the fit of the SDEMEM to a similar
ordinary differential equation model. Due to small sample sizes, strong prior
information is needed to identify all model parameters in the SDEMEM and it
cannot be determined which of the two models is the better in terms of
predicting tumor growth curves. In a simulation study we find that with a
sample of 17 mice per group BSL is able to identify all model parameters and
distinguish treatment groups.
