Optimal experiment design for physiological system identification
Hyperpolarized carbon-13 magnetic resonance imaging is a new medical imaging method that has enabled the real-time observation of perfusion and metabolism in vivo. To generate an image, the user must choose a flip angle at which to perturb the magnetic spins associated with each of the compounds to be imaged. We consider the problem of optimally choosing a time-varying sequence of flip angles in order to achieve the best estimates of rate parameters in a physiological model.
We have developed a model of the observed image data as a function of the chosen flip angles. This allows us to formulate the choice of flip angles as a nonlinear optimization problem, using the Fisher information about the unknown parameters as the objective function. We have found that the resulting optimized flip angle schemes provide more reliable estimates of the model's rate parameters than the constant flip angle schemes currently used in practice.
Trajectory-based computation of reachable sets
Reachability analysis is a method of studying the behaviour of a dynamical system under the effect of inputs, disturbances or uncertainties. It provides us with the ability to simultaneously study all possible trajectories of the system's state, allowing us to make definitive assertions about the safety and reliability of perturbed or uncertain systems. However, the reachability analysis of state space models can become computationally demanding as the number of states in the model increases.
We consider a trajectory-based approach to the reachability problem where we first numerically simulate a number of sample trajectories of the system and next establish a bound on the divergence between the samples and neighbouring trajectories using the measure (or logarithmic norm) of the Jacobian. Trajectory-based approaches have the advantage that numerical simulation is a relatively inexpensive operation, even for systems with a large number of states. Thus, they can scale to a large number of state dimensions.