Bayesian optimal design for ordinary differential equation models with application in biological science

TL;DR

This paper develops a Bayesian optimal design framework for experiments involving nonlinear ODE models, addressing computational challenges with a probabilistic solution and Monte Carlo methods, demonstrated on biological models.

Contribution

It introduces a novel probabilistic approach combined with Monte Carlo approximation and a precomputation algorithm for efficient Bayesian optimal design in nonlinear ODE models.

Findings

Successfully designed experiments for biological ODE models

Reduced computational complexity with a new precomputation algorithm

Provided optimal designs for placenta transport models

Abstract

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.