Signal-to-Noise Ratio Comparison of Encoding Methods for Hyperpolarized Noble Gas MRI Lei Zhao, Arvind K. Venkatesh, Mitchell S. Albert, and Lawrence P. Panych Department of Radiology, Harvard Medical School and Brigham and Women’s Hospital, 75 Francis Street, Boston, Massachusetts 02115 Received June 26, 2000; revised November 1, 2000 Some non-Fourier encoding methods such as wavelet and direct encoding use spatially localized bases. The spatial localization feature of these methods enables optimized encoding for improved spatial and temporal resolution during dynamically adaptive MR imaging. These spatially localized bases, however, have inherently reduced image signal-to-noise ratio compared with Fourier or Hadamad encoding for proton imaging. Hyperpolarized noble gases, on the other hand, have quite different MR properties compared to proton, primarily the nonrenewability of the signal. It could be expected, therefore, that the characteristics of image SNR with respect to encoding method will also be very different from hyperpolarized noble gas MRI compared to proton MRI. In this article, hyperpolarized noble gas image SNRs of different encod- ing methods are compared theoretically using a matrix description of the encoding process. It is shown that image SNR for hyperpo- larized noble gas imaging is maximized for any orthonormal en- coding method. Methods are then proposed for designing RF pulses to achieve normalized encoding profiles using Fourier, Had- amard, wavelet, and direct encoding methods for hyperpolarized noble gases. Theoretical results are confirmed with hyperpolarized noble gas MRI experiments. © 2001 Academic Press Key Words: non-Fourier encoded MRI; signal-to-noise ratio; hyperpolarized noble gas MRI; spatially selective RF excitation; wavelet encoding. 1. INTRODUCTION With spatially selective RF excitation, non-Fourier encoding methods such as Hadamard, wavelet, and direct encoding can be implemented for magnetic resonance imaging (MRI). These non-Fourier encoding methods, especially those with spatially localized basis functions, can be used for adaptive imaging where the data acquisition strategy is modified according to information obtained during imaging (1). With adaptively op- timized encoding bases, data acquisition redundancy can be reduced, thus improving temporal and spatial resolution during dynamic imaging. Unfortunately, in proton MRI, the spatially localized bases that are especially useful for adaptive imaging, such as wavelet and direct encoding, give a significantly reduced image signal- to-noise ratio (SNR) compared with Fourier or Hadamard encoding. For example, the SNRs of wavelet and direct encod- ing were shown (2) to be  N/3 and  N , respectively, relative to the SNR of Fourier or Hadamard encoding, for an equal number of encoding steps, N. In Fourier or Hadamard encod- ing, all spins within the field-of-view (FOV) participate in each of the encoding steps, whereas in wavelet and direct encoding, not all of the spins contribute, resulting in a lower SNR. This reduced image SNR greatly limits the applications of these spatially localized non-Fourier encoding bases in proton MRI. The SNR situation for hyperpolarized noble gas MRI (3) is quite different than for proton MRI. Because each RF excita- tion depletes some of the nonrenewable hyperpolarized mag- netization, spatially localized encoding methods which use significantly fewer excitations within a given volume element, such as wavelet and direct encoding, cause much less depletion of the hyperpolarized magnetization. As a result, larger flip angles can be used on each RF excitation, thereby boosting image SNR. Thus, relative image SNRs for spatially localized encoding methods with hyperpolarized noble gas imaging dif- fer from those for proton imaging. In this paper, hyperpolarized noble gas image SNRs are analyzed theoretically using different orthogonal encoding bases. From this analysis, different encoding basis sets are optimized to produce maximal image SNR. The experimental results with optimized encoding bases are then compared with theoretical predictions. 2. THEORY 2.1. A Matrix Representation for MRI Encoding In order to analyze image SNR and to optimize encoding bases, we adopt a matrix description of the encoding process, described elsewhere in detail (4, 5) and summarized here in brief. For simplicity, we will consider a 1-dimensional encod- ing model. The results for multidimensional magnetic reso- nance (MR) encoding techniques can be represented as sepa- rable 1D operations in multiple dimensions. Let s ( x ) represent a 1D MR signal density to be “imaged.” Define ( x ) as a function that is centered at x = 0 and has a spread of x , which will serve as a sampling or point-spread function of a pixel. We then define a spatially localized set of Journal of Magnetic Resonance 148, 314 –326 (2001) doi:10.1006/jmre.2000.2253, available online at http://www.idealibrary.com on 314 1090-7807/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.