Semidefinite relaxations in optimal experiment design with application to substrate injection for hyperpolarized MRI

TL;DR

This paper develops semidefinite relaxation methods to solve optimal input design problems in MRI, enabling efficient computation of injection profiles and assessing practical solutions' optimality.

Contribution

It introduces semidefinite relaxation techniques for optimal experiment design with amplitude and norm constraints, applied to hyperpolarized MRI substrate injection.

Findings

Relaxation is tight for l2-norm constraints, enabling global optimality.

Relaxation provides a near-optimality guarantee for current boxcar injection profiles.

Method efficiently computes optimal injection profiles in MRI applications.

Abstract

We consider the problem of optimal input design for estimating uncertain parameters in a discrete-time linear state space model, subject to simultaneous amplitude and l1/l2-norm constraints on the admissible inputs. We formulate this problem as the maximization of a (non-concave) quadratic function over the space of inputs, and use semidefinite relaxation techniques to efficiently find the global solution or to provide an upper bound. This investigation is motivated by a problem in medical imaging, specifically designing a substrate injection profile for in vivo metabolic parameter mapping using magnetic resonance imaging (MRI) with hyperpolarized carbon-13 pyruvate. In the l2-norm-constrained case, we show that the relaxation is tight, allowing us to efficiently compute a globally optimal injection profile. In the l1-norm-constrained case the relaxation is no longer tight, but can be used to prove that the boxcar injection currently used in practice achieves at least 98.7% of the global optimum.