  Bayesian optimal design is considered for experiments where the response
distribution depends on the solution to a system of non-linear ordinary
differential equations. The motivation is an experiment to estimate parameters
in the equations governing the transport of amino acids through cell membranes
in human placentas. Decision-theoretic Bayesian design of experiments for such
nonlinear models is conceptually very attractive, allowing the formal
incorporation of prior knowledge to overcome the parameter dependence of
frequentist design and being less reliant on asymptotic approximations.
However, the necessary approximation and maximization of the, typically
analytically intractable, expected utility results in a computationally
challenging problem. These issues are further exacerbated if the solution to
the differential equations is not available in closed-form. This paper proposes
a new combination of a probabilistic solution to the equations embedded within
a Monte Carlo approximation to the expected utility with cyclic descent of a
smooth approximation to find the optimal design. A novel precomputation
algorithm reduces the computational burden, making the search for an optimal
design feasible for bigger problems. The methods are demonstrated by finding
new designs for a number of common models derived from differential equations,
and by providing optimal designs for the placenta experiment.
