Least Squares Reconstruction for Sparse Spectra: Application to Hyperpolarized 13 C Imaging (2006)

Y. S. Levin, D. Mayer, Y-F. Yen, R. Hurd, D. M. Spielman Department of Radiology, Stanford University School of Medicine, Stanford, CA, United States, Department of Electrical Engineering, Stanford University School of Engineering, Stanford, CA, United States, Advanced Sciences Laboratory, GEHealthcare, Menlo Park, CA, United States Introduction Hyperpolarization of C has made possible the imaging of metabolism in vivo; an injected compound and its downstream metabolites can be distinguished on the basis of chemical shift (1,2) . Rapid imaging is necessary because the hyperpolarized signal decays with T1. Although the bandwidth of the carbon spectrum is large, it is also sparse. Spectral sampling can therefore be performed below the Nyquist rate, allowing spectral peaks to alias into regions of the spectrum that contain no metabolites. Spiral CSI (spcsi) (3), which maximizes efficiency of data acquisition by varying both x and y gradients simultaneously is a technique that allows the user to take advantage of this aliasing. A conventional reconstruction (CR) of the acquired data would involve performing a 1D FFT in kf to separate the metabolites. The resulting data, a 3D data set in kx, ky and frequency, are demodulated with the known frequency of each of the metabolites and then gridded. This work proposes an alternate reconstruction technique based on a least squares (LS) algorithm. The technique allows the user to include additional a priori knowledge in the reconstruction, to better choose acquisition parameters, and to minimize acquisition time. Theory For a spectrum containing known peaks, the signal as a function of time and k-space location can be expressed as y(kx,ky,t) = Σj M( kx,ky,νj ) exp( -i2πνjt ), where νi are the known frequencies present in the spectrum. For spiral CSI in which a single k-space trajectory is repeated for a total of M echoes, the signal for each kx, ky point can be written in matrix form as y = Am, where y is a vector corresponding to the M observations of the kx, ky point, m is a vector describing the spectral components of the point, and the elements of A are Amn = exp(-i2πνn (TEm + τ(kx,ky)). τ(kx,ky) is the time relative to the k-space origin. Any known information about the peaks, such as J-coupling and T2 values, can be included in A. LS methods can be used to solve for y for each k-space point, and gridding can be performed to obtain metabolic images. The user is free to choose the echo times TEm subject to the FOV and resolution constraints of the desired images. In a technique similar to that proposed by Glover et al for fat/water separation (4) and further developed by Reeder et al for IDEAL imaging (5), A can be optimized. Here, condition number is used to find optimal TE values and to determine the relative benefit of acquiring additional echoes. Procedure The reconstruction and optimization technique was investigated through simulation and phantom experiments. Simulations were performed for a set of peaks at 0,-212,-363.5,-693.7 Hz, corresponding to the resonance frequencies of C1 lactate, C1 alanine, C1 pyruvate and C bicarbonate, respectively. This spectrum would be expected to appear after administration of C1 alanine (ala), lactate (lac), or pyruvate (pyr). This is true assuming that investigation is performed on an organ that exhibits little or no anaplerosis, in which pyr is incorporated into oxaloacetate. In practice, optimization of each TE independently is a computationally complex problem. Therefore, the TEs were constrained to be equally spaced, with the spacing ∆TE = 1/SW. Because the condition number of A varies with τ, condition number was computed as a function of SW and time (9.1 ms for the FOV = 8 cm, in-plane resolution = 0.5 X 0.5 cm, single-interleaf acquisition considered here). After optimization, simulated data sets were created and images were reconstructed using least squares estimation for each k-space point. The phantom contained three tubes of 1.5M 99% enriched C1 ala, lac, and a pyr-pyruvate hydrate C1-C2 ester (PPE). For lac on resonance at 3T, ala appears at -212 Hz, and the two PPE peaks are at -243 and -594 Hz. J-coupling patterns/constants were also measured in the phantom and incorporated into matrix A for improved reconstruction. Phantom measurements were performed on a GE 3T MR scanner equipped with self-shielded gradients (40 mT/m, 150 mT/m/ms). A doubly tuned (H-C) birdcage coil (diameter = 44mm) was used for both RF excitation and signal reception. The implemented sequence consisted of a slice(z)-selective excitation (5.4 mm) and a single-interleaf spiral readout gradient. A total of 32 echoes were acquired at SW = 109.7 Hz. 64 averages were acquired to obtain sufficient SNR in the absence of hyperpolarization. Images were reconstructed according to the LS method described above using 21 echoes and 8 echoes. The reconstructions were compared to the CR of the same data and to a separately acquired fidcsi (16 X 16 phase encodes, SW = 2000 Hz, npoints = 2048, TR = 2s) data set. Figure 1