Design and application of prefocused pulses by simulated annealing

Abstract

A method is proposed for producing inherently refocused selective excitation radio frequency pulses using the numerical technique of simulated annealing (Z-3). The control parameters of the method are discussed and preliminary experimental results are shown, including application to a suitably modified volume-selective STEAM sequence (4). A prefocused pulse is one which provides uniform excitation of nuclear magnetic resonance responses across a selected region of the frequency domain, with minimal phase dispersion. Consequently, in an NMR imaging experiment which uses prefocused pulses to select spatial slices there is no requirement for additional refocusing lobes to the gradient pulses. The advantages of this are twofold. Primarily, the switching demands imposed on the gradient amplifier are reduced, but second, the overall sequence length may be considerably shortened. Figure la shows a conventional sliceselecting pulse and gradient combination and is compared in Fig. 1 b to a method based on prefocused pulses. In this work, existing numerical methods (5, 6) aimed at generating such pulses have been repeated. However, we have incorporated the method of simulated annealing since that algorithm can “escape” local minima in the convergence pathway and thereby arrive at the global minimum. Furthermore, simulated annealing also permits the inclusion into the optimization procedure of constraints based on hardware considerations. Conceptually, the optimization method of simulated annealing is analogous to the thermodynamic process of a molten crystal cooling into a stable, low-energy configuration. The essence of the method is that during the cooling process the system is constantly changing its structural configuration and that, at a given thermal energy, there is a finite chance that a less stable configuration will be adopted (based on Boltzmann statistics). This might be a vital intermediate state allowing a route to a more desirable lower-energy configuration. As the temperature is lowered, the probability of making such upward changes decreases and eventually a stable arrangement is achieved. If the cooling has been sufficiently slow, this final configuration will have the lowest possible associated structural energy. By analogy, these considerations can be applied to the optimization of the pulse shape. It is important to consider the implementation of the optimized pulse within the context of an experimental system. Hence, constraints representing hardware limitations may need to be incorporated.