John N. Maidens

Reachability Analysis of Nonlinear Systems Using Matrix Measures

Matrix measures, also known as logarithmic norms, have historically been used to provide bounds on the divergence of trajectories of a system of ordinary differential equations. In this technical note we use them to compute guaranteed overapproximations of reachable sets for nonlinear continuous-time systems using numerically simulated trajectories and to bound the accumulation of numerical simulation errors along simulation traces. Our method employs a user-supplied bound on the matrix measure of the systems Jacobian matrix to compute bounds on the behavior of nearby trajectories, leading to efficient computation of reachable sets when such bounds are available. We demonstrate that the proposed technique scales well to systems with a large number of states.

Optimizing Flip Angles for Metabolic Rate Estimation in Hyperpolarized Carbon-13 MRI

Hyperpolarized carbon-13 magnetic resonance imaging has enabled the real-time observation of perfusion and metabolism in vivo. These experiments typically aim to distinguish between healthy and diseased tissues based on the rate at which they metabolize an injected substrate. However, existing approaches to optimizing flip angle sequences for these experiments have focused on indirect metrics of the reliability of metabolic rate estimates, such as signal variation and signal-to-noise ratio. In this paper we present an optimization procedure that focuses on maximizing the Fisher information about the metabolic rate. We demonstrate through numerical simulation experiments that flip angles optimized based on the Fisher information lead to lower variance in metabolic rate estimates than previous flip angle sequences. In particular, we demonstrate a 20% decrease in metabolic rate uncertainty when compared with the best competing sequence. We then demonstrate appropriateness of the mathematical model used in the simulation experiments with in vivo experiments in a prostate cancer mouse model. While there is no ground truth against which to compare the parameter estimates generated in the in vivo experiments, we demonstrate that our model used can reproduce consistent parameter estimates for a number of flip angle sequences.

Optimal experiment design for physiological parameter estimation using hyperpolarized carbon-13 magnetic resonance imaging

Hyperpolarized carbon-13 magnetic resonance imaging is a new medical imaging modality that has enabled the real-time observation of perfusion and metabolism in vivo. The rates at which perfusion and metabolism occur are important for disease diagnosis and treatment monitoring. 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 first formulate a discrete-time model describing perfusion, exchange, relaxation and measurement error. We then show how to compute the Fisher information for the unknown parameters of this model and present time-varying flip angle schemes that maximize the Fisher information. Through numerical studies, we demonstrate that the optimal flip angle scheme provides better estimates of the models rate parameters than a constant flip angle scheme.

Parallel dynamic programming for optimal experiment design in nonlinear systems

We present a method of computing optimal input trajectories for parameter estimation in nonlinear dynamical systems using dynamic programming. In contrast with previously published dynamic programming formulations, we avoid adding an equation for the dispersion to the system state, allowing for more efficient solutions. This method is applicable whenever the design metric is linear in the Fisher information and is applicable to a general class of noise models. We implement this algorithm in the Julia programming language, and exploit parallelism to increase computation speed. A motivating application for this investigation is the design of dynamic acquisition sequences for magnetic resonance imaging (MRI). We also benchmark the performance of our parallel implementation on a low-dimensional population dynamics model.

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

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.

Symmetry reduction for dynamic programming and application to MRI

We present a method of exploiting symmetries of discrete-time optimal control problems to reduce the dimensionality of dynamic programming iterations. The results are derived for systems with continuous state variables, and can be applied to systems with continuous or discrete symmetry groups. We prove that symmetries of the state update equation and stage costs induce corresponding symmetries of the optimal cost function and the optimal policies. Thus symmetries can be exploited to allow dynamic programming iterations to be performed in a reduced state space. The application of these results is illustrated using a model of spin dynamics for magnetic resonance imaging (MRI). For this application problem, the symmetry reduction introduced leads to a significant speedup, reducing computation time by a factor of 75×.

Trajectory-based reachability analysis of switched nonlinear systems using matrix measures

Matrix measures, or logarithmic norms, have historically been used to provide bounds on the divergence of trajectories of a system of ordinary differential equations (ODEs). In this paper we use them to compute guaranteed overapproximations of reachable sets for switched nonlinear dynamical systems using numerically simulated trajectories, and to bound the accumulation of numerical errors along simulation traces. To improve the tightness of the computed approximations, we connect these classical tools for ODE analysis with modern techniques for optimization and demonstrate that minimizing the volume of the computed reachable set enclosure can be formulated as a convex problem. Using a benchmark problem for the verification of hybrid systems, we show that this technique enables the efficient computation of reachable sets for systems with over 100 continuous state variables.

An Improvement to Test Case Failure Prediction in the Context of Test Case Prioritization

Aim: In this study, we aim to re-evaluate research questions on the ability of a logistic regression model proposed in a previous work to predict and prioritize the failing test cases based on some test quality metrics. Background: The process of prioritizing test cases aims to come up with a ranked test suite where test cases meeting certain criteria are prioritized. One criterion may be the ability of test cases to find faults that can be predicted a priori. Ranking test cases and executing the top-ranked test cases is particularly beneficial when projects have tight schedules and budgets. Method: We performed the comparison by first rebuilding the predictive models using the features from the original study and then we extended the original work to improve the predictive models using new features by combining with the existing ones. Results: The results of our study, using a dataset of five open-source systems, confirm that the findings from the original study hold and that our predictive models with new features outperform the original models in predicting and prioritizing the failing test cases. Conclusions: We plan to apply this method to a large-scale dataset from a large commercial enterprise project, to better demonstrate the improvement that our modified features provide and to explore the models performance at scale.

Simulation-Based Reachability Analysis for Nonlinear Systems Using Componentwise Contraction Properties.

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set.

Control and Optimization Problems in Hyperpolarized Carbon-13 MRI

Hyperpolarized carbon-13 magnetic resonance imaging (MRI) is an emerging technology for probing metabolic activity in living subjects, which promises to provide clinicians new insights into diseases such as cancer and heart failure. These experiments involve an injection of a hyperpolarized substrate, often pyruvate labeled with carbon-13, which is imaged over time as it spreads throughout the subject’s body and is transformed into various metabolic products. Designing these dynamic experiments and processing the resulting data requires the integration of noisy information across temporal, spatial, and chemical dimensions, and thus provides a wealth of interesting problems from an optimization and control perspective. In this work, we provide an introduction to the field of hyperpolarized carbon-13 MRI targeted toward researchers in control and optimization theory. We then describe three challenge problems that arise in metabolic imaging with hyperpolarized substrates: the design of optimal substrate injection profiles, the design of optimal flip angle sequences, and the constrained estimation of metabolism maps from experimental data. We describe the current state of research on each of these problems, and comment on aspects that remain open. We hope that these challenge problems will serve to direct future research in control.