Revisiting the Bloch Equation through Averaging Bahman Tahayori, Leigh A. Johnston, Iven M.Y. Mareels, and Peter M. Farrell Abstract—A novel approach to finding an approximate analytic solution to the Bloch equation is developed in this paper. The method is based on time scaling and averaging of the Bloch equation after transformation to a rotating frame of reference. In order to accomplish the scaling, a novel time scaled magnetisation vector is introduced. The resultant time scaled system is subsequently approximated through averaging, a technique that to the best of our knowledge, has not previously been applied in the nuclear magnetic resonance context. Our proposed method of approximating the solution to the Bloch equation is valid for continuous wave excitation as well as the traditional pulse excitation with an arbitrary envelope, making this a widely applicable technique unlike previously proposed methods. Comparison of the approximate analytic solution and simulation results clearly indicates that the error is negligible when the field inhomogeneities are small compared to the excitation field amplitude. Extremum seeking techniques may be applied to determine the optimal excitation, given the form of the approximate solution. This result is applicable to a range of research areas including nuclear magnetic resonance, magnetic resonance imaging and optical resonance problems. I. INTRODUCTION The Bloch equation, developed in 1946, describes the empirical behaviour of an ensemble of spins in the presence of an external magnetic field [1]. This equation is applicable to nuclear magnetic resonance, magnetic resonance imaging (MRI) and optical resonance problems. In 1949 Torrey presented an analytic solution for a long- lived constant pulse excitation by adopting the Laplace trans- form [2]. Madhu and Kumar provided an analytic solution for the Bloch equation in response to the application of a con- stant radio frequency field [3]. In their approach, the Bloch equation corresponding to each magnetisation component is written as a third order differential equation, with the solution following from these equations. Solving the Bloch equation during the period in which a time-varying external magnetic field is applied, termed the excitation period, is of crucial importance for slice selection in MRI. The bilinear form of the Bloch equation makes it very hard to find a closed form solution for an arbitrary excitation pattern. Several approximate solutions to the Bloch equation under restrictive limitations have been proposed [4], [5]. These approaches are limited by the excitation pattern they consider. In [6] an approximate solution is proposed for a rectangular pulse excitation when the relaxation terms are ignored. It has been generally accepted that an analytic solution does not exist for an arbitrary pulse excitation [6]. This work is supported by NICTA Victorian Research Laboratory Life Sciences Program. The authors are with The Department of Electrical and Electronics Engineering, The University of Melbourne, VIC 3010, Australia. {b.tahayori,l.johnston,i.mareels,p.farrell} @ee.unimelb.edu.au. Fig. 1. The main steps of the approach presented in this paper to find an approximate analytic solution to the Bloch equation. Several different numerical techniques have been used to solve the Bloch equation [7], [8]. Most MRI simulators implement approximate numerical solutions to the Bloch equation based on rotation matrices [9], [10]. The Shinnar- Le Roux method, used universally in MRI machines to selectively excite a slice, is based on a discrete approximation to the Bloch equation which simplifies the solution of the optimal slice selective pulse to the design of two polynomials [11]. We present a novel technique for finding an approximate analytic solution to the Bloch equation that retains impor- tant features of the Bloch equation, and can therefore be applied to the design of improved MRI pulse sequences. Our approach is based on a combination of time scaling and averaging methods from dynamical systems theory [12], [13]. We verify the success of the averaging method in simulations without formally establishing its validity. The steps of the method to find an approximate solution in the laboratory frame of reference are shown in Fig. 1. Since the magnetic resonance signal is demodulated after being received, it is sufficient to determine the solution in the rotating frame of reference. In this paper, all analytic solutions and simulation results represent the spin system response as observed from a frame of reference rotating at the Larmor frequency of the static magnetic field. In Section II we present an overview of the proposed approach including the transferral of the Bloch equation to the rotating frame of reference, and the novel application of time scaling and averaging to the resultant system. Section Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 ThTA10.6 978-1-4244-3124-3/08/$25.00 ©2008 IEEE 4121