[1] John Marriott. The round-trip contrast problem. In 53rd IEEE Conference on Decision and Control, pages 5788-5793, Dec 2014. [ bib | DOI ]
The contrast problem in nuclear magnetic resonance imaging has been treated by a variety of recent works. In this process, the image is typically captured multiple times to filter noise from the image, and to do so the system must return to its initial state before repeating the experiment. A natural motivation is to perform this return trip as quickly as possible, in particular with live subjects. In this article we introduce and discuss preliminary results of this so-called return trip of the contrast problem. The tools of geometric optimal control theory have been effectively applied to the contrast problem, and they are similarly employed to this problem which shares many characteristics. The time-minimal transfer in the single-spin case and preliminary results in the two-spin case are presented.

[2] Monique Chyba, Sergio Grammatico, Van T. Huynh, John Marriott, Benedetto Piccoli, and Ryan N. Smith. Reducing actuator switchings for motion control of autonomous underwater vehicles. In American Control Conference (ACC), 2013, pages 1406-1411, 2013. [ bib ]
A priority when designing control strategies for autonomous underwater vehicles is to emphasize their cost of implementation on a real vehicle. Indeed, the major issue is that due to the vehicles' design and actuation modes usually under consideration for underwater platforms, the number of actuator switchings must be kept to a small value to ensure feasibility and precision. This constraint is typically not satisfied by optimal trajectories, for instance. Our goal is to provide a trajectory which preserves with great accuracy some of the properties of a desired trajectory that reduces the implementation cost. We first introduce the theoretical framework and illustrate our algorithm on two AUV applications. In both cases, we can achieve similar localization results in the same fixed time with respect to the reference trajectory, but with significantly fewer actuator switchings.

[3] Bernard Bonnard, Monique Chyba, and John Marriott. Feedback equivalence and the contrast problem in nuclear magnetic resonance imaging. Pacific Journal of Optimization, 9(4):635-650, 2013. [ bib ]
The theoretical analysis of the contrast problem in NMR imaging is mainly reduced, thanks to the Maximum Principle, to the analysis of the so-called singular trajectories of the control system modeling the problem: a coupling of two Bloch equations representing the evolution of the magnetization vector of each spin particle. They are solutions of a constrained Hamiltonian equation. In this article we describe feedback invariants related to the singular flow to distinguish the different cases occurring in physical experiments.

[4] Bernard Bonnard, Alain Jacquemard, Monique Chyba, and John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields, 3(4):397-432, 2013. [ bib ]
The analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on B1 ×B2, where B1 and B2 are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points.

[5] Bernard Bonnard, Monique Chyba, and John Marriott. Singular trajectories and the contrast imaging problem in nuclear magnetic resonance. SIAM Journal on Control and Optimization, 51(2):1325-1349, 2013. [ bib | DOI | arXiv | http ]
In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control whose solutions can be parameterized using Pontryagin's maximum principle and analyzed using geometric optimal control. In particular, the optimal problem can be mainly reduced to the analysis of the Hamiltonian dynamics describing the singular trajectories, and encoding their optimality status.

[6] Monique Chyba and John Marriott. Exceptional trajectories in contrast problem in nuclear magnetic resonance. Cybernetics and Physics, 1(4):243-251, 2012. [ bib | http | http ]
In this paper we focus on the contrast problem in medical imaging. It consists of using a single magnetic field to control a pair of non-interacting spins, each representing a specific substance, with the goal of maximizing the difference of the moduli of the magnetization vectors of the two substances. Prior work analyzed the saturation contrast problem which brings one of the spins to magnetization zero while maximizing the modulus of the other. Here we relax the saturation constraint that one of the spins must reach magnetization zero, providing more flexibility to obtain a higher contrast. We focus on the study of exceptional arcs and construct bang-exceptional extremals based on a methodology that searches for the highest possible contrast. Numerical calculations are provided, and we graphically illustrate the results.

[7] Bernard Bonnard, Monique Chyba, Steffen J. Glaser, John Marriott, and Dominique Sugny. Nuclear magnetic resonance: The contrast imaging problem. In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 5559-5564, December 2011. [ bib | DOI | .pdf ]
Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal problem and the Pontryagin Maximum Principle is used to select the optimal trajectories among the extremal solutions. Correlation between the contrast problem and the optimal transfer time problem is demonstrated. Further, we develop some analysis of the singular extremals and apply the results to examples of cerebrospinal fluid/water and grey/white matter of the cerebrum.

[8] Urszula Ledzewicz, John Marriott, Helmut Maurer, and Heinz Schättler. Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment. Mathematical Medicine and Biology, 27(2):157-179, 2010. [ bib | DOI | http | .pdf ]
Two mathematical models for tumor anti-angiogenesis, one originally formu- lated by Hahnfeldt, Panigrahy, Folkman and Hlatky (1999) and a modifica- tion of this model by Ergun, Camphausen and Wein (2003), are considered as optimal control problem with the aim of maximizing the tumor reduc- tion achievable with an a priori given amount of angiogenic agents. For both models, depending on the initial conditions, optimal controls may con- tain a segment along which the dosage follows a so-called singular control, a time-varying feedback control. In this paper, for these cases the efficiency of piecewise constant protocols with a small number of switchings is investi- gated. Through comparison with the theoretically optimal solutions, it will be shown that these protocols provide generally excellent suboptimal strate- gies that for many initial conditions come within a fraction of 1% of the theoretically optimal values. When the duration of the dosages are a priori restricted to a daily or semi-daily regimen, still very good approximations of the theoretically optimal solution can be achieved.

[9] Urszula Ledzewicz, Heinz Schättler, and John Marriott. Piecewise constant suboptimal controls for a system describing tumor growth under angiogenic treatment. Proceedings of the 3rd IEEE Multi-Conference on Systems and Control, pages 77-82, July 2009. [ bib | .pdf ]
A mathematical model for tumor anti-angiogenesis formulated by Ergun et al. [9] is considered as an optimal con- trol problem with the aim of maximizing the tumor reduction achievable with an a priori given amount of anti-angiogenic agents. Optimal controls contain a segment along which the dosage follows a time-varying feedback control. With current medical technologies such a design is not realistic. In this paper the efficiency of piecewise constant, open-loop protocols with a small number of switchings is compared with the theoretically optimal solution derived earlier. It is shown that these protocols generally provide excellent suboptimal strategies, even when the times of applications are restricted to follow daily patterns.

[10] Urszula Ledzewicz, John Marriott, Helmut Maurer, and Heinz Schättler. The scheduling of angiogenic inhibitors minimizing tumor volume. Journal of Medical Informatics & Technologies, 12:23-28, 2008. [ bib | .pdf ]
The efficiency of piecewise constant protocols with a small number of switchings is in- vestigated for a mathematical model of tumor anti-angiogenesis formulated originally by Hahnfeldt et al. in [8]. By comparing with the theoretically optimal solution derived ear- lier [12] it will be seen that for the problem of minimizing the primary cancer volume with a given amount of angiogenic inhibitors to be administered constant protocols already provide very good suboptimal strategies.


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