IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 6, JUNE 2012 1967 Gradient-Coil Design: A Multi-Objective Problem Clemente Cobos Sánchez , Mario Fernández Pantoja , Michael Poole , and Amelia Rubio Bretones Dept. of Ingeniería de sistemas y Electrónica, University of Cádiz, E. Superior de Ingeniería, 11002 Cádiz, Spain Dept. of Electromagnetism, University of Granada, Fuentenueva s/n, 18071 Granada, Spain School of Information Technology and Electrical Engineering, University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia In this work, the design of gradient coils for magnetic resonance imaging (MRI) is studied as a multi-objective optimization (MOP) problem, which is successfully solved by using Pareto optimality formalism. The proposed approach is illustrated using a stream function inverse boundary element method (IBEM), as the coil design paradigm that is capable of including numerous design requirements or objectives. These are frequently in conflict, which stresses the need to deal efficiently with the tradeoff between different coil properties. It is shown that the inclusion of many of the most commonly used coil design requirements (such as field homogeneity, uniformity, magnetic stored energy, power dissipated, torque balanced ) reduces the problem to a convex MOP, where Pareto optimal solutions can be efficiently found by using suitable convex optimization procedures. Pertinent examples are studied to illustrate the versatility of the proposed MOP approach, which can be used to obtain a comprehensive understanding of the coil design problem, as well as to handle the different coil requirements efficiently and how they should be combined to yield the best solution for a given problem. Index Terms—Magnetic resonance imaging, optimization methods. I. INTRODUCTION O VER the last decades, magnetic resonance imaging (MRI) has become an invaluable tool for diagnostic medicine. This noninvasive technique is based on the nuclear magnetic resonance (NMR) phenomenon, and relies on the use of well-defined and controlled magnetic fields, such as the linear magnetic field gradients which are used to spatially encode the signals from the sample. These field gradients are generated by coils of wire, usually placed on cylindrical sur- faces [1], although other geometries can be employed [2]–[5]. In order to provide optimal performance and patient comfort, an ideal gradient coil should have several properties, such as: minimal stored magnetic energy, high gradient-to-current ratio, minimal resistance, good field gradient linearity, and minimal interaction with the rest of the MRI system and patient. Gra- dient coil design therefore seeks to find optimal positions for the multiple windings of the coil so as to produce fields with the desired spatial dependence and properties [1]. This can be seen as an inverse problem in which several conflicting per- formance attributes or objectives need to be optimized simul- taneously. Unfortunately, there does not exist a single solution for the coil design problem that optimizes all the objectives at the same time. Rather, there exist a range of solutions that op- timize each objective to varying degrees; which represents a tradeoff between the various coil properties. Typically, a gra- dient coil designer will manually explore this range of solutions by designing many coils until a satisfactory result is obtained. In this work, a method of analyzing the tradeoffs in coil design is presented with the aim of simplifying this somewhat tedious process. Manuscript received August 25, 2011; revised November 04, 2011; accepted December 06, 2011. Date of publication December 15, 2011; date of current ver- sion May 18, 2012. Corresponding author: C. Cobos Sánchez (e-mail: clemente. cobos@uca.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2179943 Du and Parker stressed the importance of obtaining an overall view about all aspects of the achievable coil performance [6]. Their work was restricted to the design of -gradient coils com- posed by multiple modified Maxwell pairs. While this is a valu- able approach, their exploration of the solution space includes all intermediate solutions obtained in the metaheuristic opti- mization process and therefore lacked an appropriate concept of optimality to keep only the most optimal solutions. The problem of dealing with multiple and conflicting objec- tives in gradient-coil design is demonstrated in approaches such as stream function inverse boundary element method (IBEM) [7], [8]; a flexible group of coil-design techniques that has al- lowed the inclusion of many coil features in the design process. For example, minimal power dissipation [9], minimal electric field induced in a prescribed conductor [10], or minimized max- imum current density [11], [12]. Problems of multi-criteria nature, like coil design, character- ized by the nonexistence of a single optimal solution capable of minimizing all objectives, are well known in fields such as economics or engineering [13]–[15], and more precisely they are especially frequent problems in electromagnetic industrial design [16], [17], which are often referred to as multi-objec- tive optimization (MOP) [18]–[20]. MOP problems have been successfully tackled by introducing new concept of optimality of a solution based on Pareto dominance principles [21], [22]. The application of Pareto optimality provides a set of nondom- inated and equally valid solutions (the Pareto front), so that for each of Pareto optimal solution there exists no other solution that is better in all objectives. The set of the Pareto optimal so- lutions represents the tradeoff between the objectives in a MOP problem. For any solution on the Pareto front it is impossible to achieve a better value in one objective without at least one of the other objectives deteriorating. Furthermore, Pareto optima theory has been proven to be a highly suitable approach for the solution of MOP electromag- netic design problems [23], [24]. There are several methods that find the solutions on the Pareto front that can be classified into two main groups [20]: determin- 0018-9464/$26.00 © 2011 IEEE