Revisiting excitation pattern design for magnetic resonance imaging through optimisation of the signal contrast efficiency

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Abstract

The design of excitation signals for Magnetic Resonance Imaging (MRI) is cast as an optimal control problem. Here, we demonstrate that signals other than pulse excitations, which are ubiquitous in MRI, can provide adequate excitation, thus challenging the optimality and ubiquity of pulsed signals. A class of on-resonance piecewise continuous amplitude modulated signals is introduced. It is shown that despite the bilinear nature of the Bloch equations, the spins system response is largely analytically tractable for this class of signals, using Galerkin approximation methods. To challenge the optimality of the pulse excitation, an appropriate cost criterion, the Signal Contrast Efficiency (SCE), is developed. It is to be optimised subject to dynamics expressed by the Bloch equations. To solve the problem the Bloch equation is transferred to the excitation dependent rotating frame of reference. The numerical solutions to the problem for different tissue types show that for a short period of time, pulse excitations provide the maximum signal contrast. However, the problem should be solved for longer periods of time which may result in a different answer than a pulse. For this purpose, the approximate analytic solution which is derived based on averaging the Bloch equation in the excitation dependent rotating frame of reference will be used to find the optimal excitation pattern. The solution to the optimisation problem is potentially useful for all forms of MRI including structural and functional imaging. The objective of this paper is to show that while classically transient response of pulses have been monitored so far, the optimal excitation pattern may be the steady state response of a non-pulse excitation.

Introduction

Magnetic Resonance Imaging (MRI) is one of the major tomographic imaging modalities. A Magnetic Resonance (MR) signal is generated by recording the current induced in a receive coil by fluctuations in nuclear magnetisation produced by a time-varying (Radio Frequency (RF)) externally applied magnetic field. The behaviour of the (proton) spin system at a classical level in the presence of an external field is completely described by the Bloch equation [1],M˙(t)=γM(t)×Bext(t)+1T1(M0Mz(t))ez1T2Mxy(t).Here, γ is the gyromagnetic ratio, and ez is the unit vector in the z-direction. T1 and T2 are longitudinal and transverse relaxation time constants, respectively. M is the bulk magnetisation vector (to be measured), dependent on both position and time. M0ez is the thermal equilibrium created by an ideally uniform static field oriented in the z-direction (aligned with the static external field).

It is worthwhile to mention that the Bloch equation is a model to describe the Nuclear Magnetic Resonance (NMR) phenomenon, thus it does not represent the quantum behaviour of spins. However, experimental results have shown that the Bloch equation describes spin dynamics very well at a classical level [2], hence it is universally used in finding excitation patterns for NMR as well as MRI.

In the above equation the external field is the superposition of a static and uniform magnetic field in the z-direction (with amplitude B0), and a time-varying, spatially dependent, excitation field, Bxy(t), expressed asBext(t)=Bxy(t)+B0ez.

In order to identify and spatially localise the MR signal from the induced current, which is a function only of time, gradient fields are carefully designed to map spatial dependency to frequency dependency in the received signal, a property that is enabled by the fact that proton spin frequency at a particular location scales linearly with the magnitude of the external field at that location.

The first step in two dimensional MR imaging is typically to excite a thin slice of the object by applying a selective RF pulsed magnetic field. To reduce partial volume effects, the design of a RF pulse with good frequency selectivity is of crucial importance. Slice selection is based on the Bloch equation, which is nonlinear in nature.

From the early days of MRI, several methods have been proposed to solve the selective excitation problem [3]. These methods have been derived through approximate solutions to the Bloch equation [4], [5], [6], or are based on computer simulations that predict the magnetisation response of the spin system to various excitatory inputs [7], [8]. For small flip angles (the amount of perturbation of the local thermal equilibrium magnetisation from its initial orientation), the design of the selective pulse is based on the Fourier analysis of the Bloch equation [4], [9], valid for flip angles smaller than π/2.1 The design of better pulses requires application of optimal control theory [10], [11], [12], [13].

Conolly et al. [10] provide a mathematical basis for RF pulse design and an algorithm to solve for the optimal pulse, defined to be the pulse that steers the magnetisation from the initial state closest to the desired final state. It is shown that such an optimal control does exist. The objective is to find a control law, B1(t), that at the end of the pulse duration, TP, drives M(0) to M(TP)=D, where D is the desired spatial distribution of magnetisation after the excitation is turned off. The authors have formulated and solved the problem for three different optimisation criteria, including minimum distance, minimum time, and minimum energy.

The Shinnar-Le Roux method is a recursive algorithm for finding the optimal pulse for selective excitation [11]. This method is based on a discrete approximation to the Bloch equation which simplifies the solution of the selective pulse to the design of two polynomials. In this case RF pulse design is reduced to a polynomial design problem that is solved through finite impulse response (FIR) filters. The Shinner-Le Roux (SLR) technique for designing selective RF pulses is universally used in MRI machines. Several variants of the SLR method have been proposed for design of selective RF pulses [14].

Ulloa et al. [13] expand the Bloch equation and objective function in a Chebyshev series. In this way they have converted a differential equation formulation to a set of linear algebraic equations. They have solved the problem for the minimum distance, minimum energy, and minimum time criteria. Their simulations show improvements over the SLR method.

Brockett and Khaneja [12] consider stochastic models for constructing the optimal excitation, based on a conditional entropy approach. In practice, due to field inhomogeneities, application of an on-resonance pulse cannot simultaneously excite an ensemble of spins. Li [15], and Li and Khaneja [16] have formulated and solved this problem, finding a procedure to design compensating pulses that are able to simultaneously bring all the desired spins into the transverse plane.

Improvements to the received MR signal may be achieved via two principal approaches. First, it is possible to increase the received signal strength through better hardware design, including higher strength magnetic fields, more homogeneous fields, and more sensitive coils. The second approach is via design of a more efficient RF excitation signal. In this paper we focus on the latter method. We formulate the image contrast optimisation problem based on the Bloch equation, and demonstrate the feasibility of maintaining spin coherency for a novel continuous wave excitation signal, thus providing a potential alternative to the traditional pulsed MR excitation.

Section snippets

Theory

A block diagram of the MRI system is depicted in Fig. 1. The inputs to this system are the main static field and the RF excitation signal. The result of the RF excitation is an MR signal that, if encoded properly, retains spatial information about the object. No rigorous mathematical analysis exist in the MRI literature proving optimality of pulse excitation for generation of image contrast. We therefore formulate the optimisation problem in order to determine whether pulses, or some other form

Numerical solution results

To solve the problem we have used the TOMLAB optimisation environment which is a powerful MATLAB toolbox for numerical solutions to the optimisation problems [31], [32]. The numerical solution is based on time discretisation. To generate a sufficiently accurate solution we have considered 200 nodes in our simulations. The number of nodes represent the number of grid points in this discretisation process. Here, we present the results for intensity optimisation as well as the contrast

Discussion

The results of the numerical solution of the problem indicate that for short periods of time a pulse excitation generates an optimal contrast between different tissue types. However, in MRI to improve the contrast to noise ratio of the image a sequence of pulses are applied and the time of imaging is very long. Thus, the problem developed here must be solved for longer periods of time which as shown in the motivating example a periodic excitation pattern may generate a better magnetic resonance

Conclusion

In this paper we have formulated the problem of designing the optimal excitation pattern for Magnetic Resonance Imaging based on a Signal Contrast Efficiency performance measure. An excitation pattern was represented that maintains the coherency of the spins in the steady state. Moreover, it was shown that this continuous wave excitation can generate an improved signal intensity for a single tissue type, relative to the traditional pulse excitation. The numerical methods results represented in

Acknowledgment

This work is supported by NICTA Victoria research laboratory, biomedical and life science.

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      Recent work [10], inspired by quantum optics [11], has experimentally demonstrated that a spin system excited by a Rabi modulated CW achieves substantial periodic steady-state magnetisation. The frequency components of this steady-state magnetisation are restricted to harmonics of the excitation modulation frequency [9] and a maximum harmonic magnitude is achieved when a secondary resonance condition is met [10]. The Rabi resonance condition is also being investigated in CW electron paramagnetic resonance (EPR) [12].

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