  Predicting cancer dynamics under treatment is challenging due to high
inter-patient heterogeneity, lack of predictive biomarkers, and sparse and
noisy longitudinal data. Mathematical models can summarize cancer dynamics by a
few interpretable parameters per patient. Machine learning methods can then be
trained to predict the model parameters from baseline covariates, but do not
account for uncertainty in the parameter estimates. Instead, hierarchical
Bayesian modeling can model the relationship between baseline covariates to
longitudinal measurements via mechanistic parameters while accounting for
uncertainty in every part of the model.
  The mapping from baseline covariates to model parameters can be modeled in
several ways. A linear mapping simplifies inference but fails to capture
nonlinear covariate effects and scale poorly for interaction modeling when the
number of covariates is large. In contrast, Bayesian neural networks can
potentially discover interactions between covariates automatically, but at a
substantial cost in computational complexity.
  In this work, we develop a hierarchical Bayesian model of subpopulation
dynamics that uses baseline covariate information to predict cancer dynamics
under treatment, inspired by cancer dynamics in multiple myeloma (MM), where
serum M protein is a well-known proxy of tumor burden. As a working example, we
apply the model to a simulated dataset and compare its ability to predict M
protein trajectories to a model with linear covariate effects. Our results show
that the Bayesian neural network covariate effect model predicts cancer
dynamics more accurately than a linear covariate effect model when covariate
interactions are present. The framework can also be applied to other types of
cancer or other time series prediction problems that can be described with a
parametric model.
