The development of techniques to hyperpolarize 13C in metabolically active substrates presents significant opportunities and challenges. With signal-to-noise ratio (SNR) enhancements on the order of the 10,000-fold (1), the prospect of observing in vivo dynamic tissue metabolism may provide hitherto inaccessible diagnostic information (2, 3).
Golman et al. (2) recently demonstrated the metabolic imaging of 13C1 pyruvate and its downstream metabolites lactate, alanine, and bicarbonate in cross-sections of normal pigs and rats, in a rat tumor model, and in a pig model of myocardial ischemia (4). A phase-encoded chemical shift imaging (FIDCSI) sequence was used to generate images with a 5 mm × 5 mm in-plane resolution and 10 mm slice thickness in a scan time of 13 s (2). A multiecho steady-state free precession (SSFP) pulse sequence was also used to demonstrate the feasibility of imaging with 7 mm × 7 mm resolution in a scan time of 400 ms in a pig model of renal disease (4).
The 13-s imaging time was shown to be adequate to satisfy T1 constraints, but the shorter scan time of the SSFP sequence allows for imaging of processes with high temporal variation. Metabolic imaging of the myocardium, shown by Golman and Petersson (4) to be sensitive to changes in the tricarboxylic acid cycle function that result from ischemia, is a prime example. Ideally, multislice imaging within a single diastole would produce high-quality images by minimizing motion artifacts. Another application requiring high temporal resolution scanning is imaging of metabolism in flowing blood. This is of interest in order to distinguish metabolism occurring in the blood prior to arrival in the organ of interest because there is rapid conversion of pyruvate to lactate and to alanine in blood (5). Metabolic modeling is a third application. As noted by Golman et al. (2), studies of metabolism using thermally polarized 13C-enriched substrates require long scan times due to the low sensitivity of the 13C nucleus. Therefore, steady-state conditions are created via infusion of substrate over a relatively long period of time. In the case of hyperpolarized substrates, the substrates are not injected in trace amounts but rather in concentrations that are significantly higher than those that are present naturally. Although significant dilution occurs in the blood before the substrate reaches the target tissue, the concentration of the substrate is higher than its normal physiologic level. Reported doses of hyperpolarized pyruvate, 0.79 mmol/kg (6), exceed reported physiologic levels of pyruvate, which range from 0.022 to 0.094 mmol/kg (7). As a result, the metabolism that is observed in imaging experiments in the minutes following injection is not steady-state metabolism. Little is known about reaction rates in the nonequilibrium state. Because these rates may vary in time, high temporal resolution is necessary to adequately study them with in vivo MRS.
When using hyperpolarized samples, the magnetization decays toward its thermal equilibrium value and is not recoverable. As a result, SNR in metabolic images is not proportional to the number of averages performed, but rather to the fractional polarization of the injected substrate and to the concentration of the molecule(s) at the imaging site (8). There is therefore no SNR penalty in performing a single-interleaf spiral imaging experiment rather than a multishot experiment on a sample containing a single resonance (9). Spiral CSI, which uses a multiecho spiral acquisition to acquire both spatial and chemical shift information in the presence of multiple resonances, is an attractive option for performing rapid chemical shift imaging and seems well-suited to the fast metabolic imaging of hyperpolarized metabolites (10). A distinction between imaging of a sample containing a single hyperpolarized resonance and imaging of a sample containing multiple metabolites is that in the presence of multiple metabolites with different chemical shifts, image quality is also affected by how well the peaks are resolved. Here, we propose a general formulation to optimize acquisition parameters and to perform image reconstruction, and we provide a detailed analysis for the spectrum that appears following injection of hyperpolarized 13C1 pyruvate. This work is to demonstrate the feasibility of exploiting the sparsity of the spectrum (11) to obtain metabolic imaging in approximately 100 ms for a field of view (FOV) of 8 cm and 0.5 cm isotropic resolution.
The noise-free MR signal received from a sample with N resonances, as a function of time, can be expressed as
where νn are the known frequencies of the metabolites present in the spectrum, M(x,y,νn) is the magnetization arising from the nth resonance in the isochromat denoted by the (x,y) coordinate, and P(νn,t) is a term that accounts for signal modulation such as that associated with J-coupling and T2*. For CSI, the goal is to choose a sampling scheme in kx, ky and t that optimizes the metabolic images subject to specified FOV, spatial resolution, and imaging time requirements.
This general multidimensional optimization problem is more readily solved with the inclusion of additional constraints. First, we note that Eq.  can be simplified by expressing it in terms of kx, ky.
Constraining our consideration of pulse sequences to those using a kx-ky spiral trajectory repeated several times (echoes) to obtain spectral information, Eq.  can be expressed as follows:
where the number of elements in the vector s is equal to the number of echoes acquired, and
τ(kx,ky) is the time of acquisition of the kx-ky point relative to the acquisition of the k-space origin, kx = ky = 0, and is referred to henceforth as τ. TEm is the time of acquisition of the k-space origin for the mth echo.
Known information about the peaks, such as J-coupling and T2* values, can be included in P. Least-squares methods then can be used to solve for m at each k-space point:
Two-dimensional Fourier transformation (including gridding, if necessary) can be performed to obtain metabolic images. The method yields estimates of metabolite amplitudes as a function of position. Optimization can then be performed by selecting the best echo times TEm subject to the FOV and resolution constraints.
In order to maintain image quality while minimizing scan time, root mean square error (rMSE) was chosen as the metric for optimization:
where i is an index referring to the ith resonance, mkx,ky,i represents the true amplitude of the ith resonance at the kx, ky point, m̂kx,ky,i represents the estimated value, σ2 represents the variance of the estimated value, and E() refers to the expected value. The first term in the numerator represents bias. The second term, which represents variance of the estimate, can be expressed in terms of the underlying noise variance in the collected data, σ2, as
(Akx,kyTAkx,ky)ii−1, often referred to as 1/NSA (number of signal averages) (12, 13), describes the propagation of noise into each of the metabolic images, and is minimal when the peaks are well-separated. T refers to the amount of time that data acquisition occurs. η is the SNR efficiency of the trajectory that is, by definition, equal to one for uniform-density k-space sampling, and is less than one for nonuniform sampling (14).
The choice of metric differs from that traditionally used in fat-water imaging, where the NSA alone is used to optimize echo times (12, 15–17). For a particular pulse sequence and signal model, this is sufficient. However, when comparing various pulse sequences and signal models, the rMSE, which includes estimator bias, is the more appropriate metric. For MRSI, there is a separate rMSE value for each metabolite, and in this work we have chosen to minimize the maximum rMSE over the set of all metabolites:
For a given acquisition technique, the goal is to choose imaging and reconstruction parameters to minimize the rMSE in the metabolic images while minimizing imaging time. Two factors should be considered.
Choice of Signal Model
In contrast to traditional Dixon imaging, where the fat and water peaks are modeled as delta functions (12, 15–17), it may be beneficial to include additional parameters in the signal model for the 13C metabolites, such as J-coupling constants and T2* relaxation times. The J-coupling constants of the metabolites are typically known a priori. Knowledge of T2* can be obtained with any of several mapping techniques using standard 1H-MRI before injecting the hyperpolarized 13C compound. While knowledge of the resonance frequencies of the metabolites must be included in the matrix A, whether to include prior knowledge of J-coupling patterns of the metabolites and knowledge of T2* results in several possibilities:
The first possibility models the peaks as delta functions; the second includes known 1H-13C J-coupling patterns, where Kn is the total number of protons coupled to the 13C nucleus with the resonance frequency νn. The third includes both J-coupling and T2*. Assuming no other contributions to the spectrum, the option of Eq. [9c] is an unbiased estimator.
Choice of Echo Times
In order to allow for robust reconstruction, the echo times must be carefully chosen. Each of the TEm in Eq.  can be chosen independently, subject to the FOV and image resolution, as well as the maximum strength and slew rate constraints of the gradient system. For uniform sampling, in which the echo spacing is a constant ΔTE, only two parameters, TE1 and ΔTE, need to be specified. For nonuniform sampling, the complexity of the optimization problem increases with each additional echo. For a case in which nonuniform sampling is deemed important, any of a number of nonlinear optimization techniques can be used to optimize the echo times (18, 19).
The choice of echo times is complicated by chemical shift evolution during each echo. In general, because the matrix A varies with τ, noise propagation into the metabolic images varies with spatial frequency. It is not obvious how to weight the spatial frequencies in coming up with a composite rMSE because the power spectral density of the metabolic images is not known a priori. Two possible choices are to include only the k-space origin in the calculation of rMSE or to compute the rMSE for each point in k-space and then use an unweighted average. For these two cases, Eq.  can be expressed as
The theory described above was applied to the single-interleaf spiral CSI (spCSI) pulse sequence described by Mayer et al. (11). The goal was to analyze the ability to perform fast metabolic imaging, and to compare the rMSE obtainable from such rapid imaging to that obtained with a 13-s rapid FIDCSI acquisition analogous to that described by Golman. In the description below, the spectral width (SW) is the reciprocal of the sampling interval in the FIDCSI simulation, and is the reciprocal of the echo spacing (SW = 1/ΔTE) for the uniformly sampled spCSI simulation. Spectral resolution is the reciprocal of the duration of a single readout.
Simulations were performed for a set of peaks at 0, −212, −363.5, −693.7 Hz, corresponding to the resonance frequencies of 13C1 lactate (lac), 13C1 alanine (ala), 13C1 pyruvate (pyr), and 13C bicarbonate, respectively, at 3T (20). This spectrum would be expected to appear after administration of 13C1 ala, lac, or pyr (3, 4, 21). This is true assuming that investigation is performed on an organ that exhibits little or no anaplerosis, in which 13C label from pyr is incorporated into oxaloacetate and subsequently into other molecules. In order to incorporate J-coupling information into the simulations, J-coupling patterns were measured on a phantom containing three vials, one each of 13C1 lac, 13C1 ala, and 13C1 pyr, demonstrating that lactate is approximately a quintet with J = 3.75 Hz, and alanine a quintet with J = 4.5 Hz. Because of the instability of the pyruvate solution, pyr actually formed a pyruvate-pyruvate ester with two resonances, one at −243 Hz and one at −565 Hz. Figure 1 shows an image of a cross-section through the three phantoms, with each vial labeled with its corresponding spectrum. For the simulations, the J-coupling pattern for the −243 Hz resonance, a quartet with J = 4 Hz, was used for modeling pyr, and the true resonance frequency of pyr, −363.5 Hz, was used. Thus, for the simulation of pyr, the phantom provided only a J-coupling pattern. The resonance frequency was obtained from the literature. Bicarbonate was modeled as a singlet because all of its protons are in fast exchange with surrounding water. A single T2* value was assumed for all species under the assumption that T2* is dominated by local magnetic field variations and not by spin-spin relaxation. This is a valid assumption because the inherent T2 values for these metabolites is on the order of 2–5 s (8, 22). Linewidth was simulated at both 1 Hz and 5 Hz, corresponding to T2* of 318 ms and 63 ms, respectively. This is not unreasonable for a well-shimmed volume considering that the gyromagnetic ratio of the 13C nucleus is approximately one-quarter that of the 1H nucleus.
Simulations were performed with the following parameters: FOV = 8 cm, in-plane resolution = 0.5 cm, maximum gradient amplitude constraints = 4 G/cm, and slew rate constraints = 15 G/cm/ms. For convenience, an infinitely short slice-selection pulse was assumed, with TE = 0 ms. The performance of the simulated single-interleaf spCSI was compared to a simulated 16 × 16 FIDCSI acquisition with a 13-s imaging time, similar to the one reported in Ref.2. For the FIDCSI simulation, the allotted time was divided equally among the number of excitations required to acquire an image. Thus, the parameters of the FIDCSI simulation were: TR = 51 ms, and sampling interval of 0.5 ms giving a total of 102 points, spectral resolution of 19.7 Hz and SW of 2000 Hz.
For the spCSI simulation, rMSE was minimized as a function of imaging time. Both uniform and nonuniform sampling schemes were considered. For nonuniform sampling, a Nelder-Meade algorithm was used for echo-time optimization. For uniformly spaced echoes, the spCSI simulation used a spacing of 9.1 to 20 ms, corresponding to a SW of 109.7 Hz to 50 Hz, respectively. Imaging time was varied from 50 to 500 ms, corresponding to spectral resolutions of 20 Hz to 2 Hz, respectively. For each imaging time and SW, rMSE was calculated using the maximum possible number of echoes. For example, for an imaging time of 100 ms and SW of 109.7 Hz a maximum of 11 echoes can be acquired. For a 100-ms imaging time and SW of 90 Hz, a maximum of nine echoes can be acquired. The minimum rMSE and its corresponding SW were recorded for each imaging time. The small values of SW were necessitated by the single-shot experiment. Because of slew rate limitations, the gradients are capable of spiraling out to cover the desired FOV and resolution, followed by rewinder, in 9.1 ms. Although the spectral resolution of the simulated FIDCSI was constant at 19.7 Hz while the spectral resolution of the spCSI simulation varied with imaging time, the purpose was not to provide a systematic comparison of the two sequences. Rather, the purpose of the simulations was to explore the performance of a fast acquisition scheme in comparison with a previously published 13-s FIDCSI sequence (2).
In the first simulation, only the kx = ky = 0 point was considered for optimization. Data was generated using the J-coupling and T2* values described above for each of the four metabolites assuming equal magnitude of each peak. The simulation was performed with the value of σ2 = (0.3)2 chosen so as to achieve SNR of 3.3 in an 0.5-ms acquisition of a sample containing a single on-resonance component. This corresponds to SNR of 33.3 in the 51-ms FIDCSI acquisition described by Golman et al. (2). The minimum achievable rMSE was calculated for each prescribed imaging time and for each sequence. The same procedure was performed for each of the estimators described in Eq. .
A second simulation repeated the above simulation for a case in which decoupling is performed. The unbiased estimator is therefore characterized by T2* decay only. Modeling was performed using delta functions alone (Eq. [9a] above) and with inclusion of T2* knowledge.
In a third simulation, chemical shift evolution during the echo acquisition was considered by analyzing the matrix A for all values of τ. Imaging time was taken to be 100 ms. The rMSE was calculated for each value of τ and for each value of SW. The two possibilities described by Eq.  above were considered.
The phantom that was used to measure J-coupling patterns for the simulations was also used to demonstrate the reconstruction technique by acquiring images using a single-interleaf spCSI trajectory. Images were acquired with a GE Signa 3T Signa Excite MR scanner using a dual-tuned 1H-13C coil for transmission and reception. A schematic depiction of the single-interleaf spCSI waveform is depicted in Fig. 2, where the top portion represents the gradient waveform, and subsequent rows divide the acquisition into individual echoes. While each TE can be optimized separately subject to the gradient constraints and FOV and resolution prescription, here we used uniform sampling with TE1 = 2.688 ms. SW was optimized for a 100-ms acquisition. Because the metabolites contained in the phantom were not identical to those in used in the simulations, determination of appropriate SW and number of echoes was performed for the metabolites present in the phantom. Specifically, no bicarbonate was present in the phantom, and pyruvate-pyruvate hydrate ester (PPE) was present instead of pyruvate. The two PPE peaks were used together to model the signal from the PPE vial. The measured J-coupling constants were included. A SW of 96 Hz and nine echoes were used. Because the metabolites were not hyperpolarized, 128 averages were acquired. This was done only to provide sufficient SNR for reconstruction; in the presence of sufficient signal as in the case of hyperpolarized 13C, a single one of these acquisitions would suffice to fulfill the FOV and resolution requirements. A flip angle = 45°, TR = 5 s, SW = 96 Hz. and nine echoes were used for a spectral resolution of 10.4 Hz. No assumptions were made as to the relative peak-heights of the metabolites in the phantom.
In order to provide a reference for comparison of the spCSI quantitation, the reconstructed data were compared to the data obtained from a high spectral resolution FIDCSI experiment (flip angle = 45°, TR = 5 s, 16 × 16 phase encodes with SW = 2000 Hz and 2048 sample points, for a spectral resolution of 1 Hz). Although this is considerably higher resolution than the spCSI experiment, the sole purpose of this comparison was to determine whether the spCSI experiment yields quantitative estimates of the metabolites similar to a reference experiment. For this purpose, it was deemed appropriate to use a FIDCSI experiment with high spectral resolution to serve as a reference standard.
The FIDCSI data was reconstructed using three-dimensional Fourier transformation, with 5-Hz Gaussian line broadening. Peak integration was performed on each of the four peaks. The two PPE peaks were added to form an image. The images for the single-interleaf spCSI acquisition were reconstructed using the k-space-based least squares algorithm described in the text. For both FIDCSI and spCSI acquisitions, the 16 × 16 data were zero-filled to 64 × 64 to form images. A region of interest (ROI) measuring 8 × 8 pixels was chosen for each of the metabolic images; lac/PPE and ala/PPE ratios were calculated, and the FIDCSI and single-interleaf spCSI results were compared.
The Nelder-Meade optimization method found that for the set of peaks described above, the rMSE performance of a uniform sampling scheme was within 95% of nonuniform sampling for a number of echoes greater than or equal to six. This number of echoes is achieved in 66 ms for the single-interleaf spCSI acquisition. Because this is at the lower end of what may be clinically necessary, the figures below Figs. 3–5 depict the rMSE for uniform sampling. Ebel et al. (18) arrived at a similar result in analyzing a sparse proton spectrum containing eight metabolites. In that study, the authors noted that nonuniform sampling using a Nelder-Meade optimization algorithm demonstrated benefit only for a number of echoes less than 16 (18).
Figure 3a depicts simulation results of the single-interleaf spCSI sequence for an imaging time of 100 ms, with 1/rMSE plotted as a function of SW for an unbiased estimator. The modulations in the rMSE curve are due to a combination of the choice of SW, which determines the locations of the aliased peaks in the spectrum, and J-coupling and T2*, which determine the effective linewidth. Figure 3b compares the performance of single-interleaf spCSI for a well-chosen value of SW (93.4 Hz) with that of a poorly chosen value (109.7 Hz). Although the larger SW allows for acquisition of 11 echoes in 100 ms while the smaller SW allows only nine echoes, the performance of the larger value of SW is poor due to aliasing.
For the simulation in which all k-space points were considered in calculating rMSE, rMSE decreased by 9%. Because finite slew rates prevent all points in k-space from being sampled instantaneously, only one k-space point can be sampled at the optimal echo times. The optimal value of SW did not change; it remained 93.4 Hz.
Figure 4 depicts the best achievable rMSE of the single-interleaf spCSI sequence as a function of imaging time for each estimator. To calculate best rMSE, the optimal value of SW for a given imaging time was chosen. The gray curves represent the three possible signal models when no decoupling is performed. The black curves represent the two possible signal models when decoupling is performed. The solid gray and black lines demonstrate that for an unbiased signal model, decoupling provides lower rMSE. This is expected because decoupling effectively increases SNR. When infinite T2* is assumed, as in the dashed lines, decoupling does not provide better rMSE than that achieved without decoupling when the signal model includes known J-coupling constants. In this case, the SNR benefit of decoupling is balanced by increased bias in the signal model. For imaging times up to 150 ms, the single-shot spCSI acquisition achieves rMSE of less than 0.1 (10%) without any T2* estimation. For imaging times between 80 and 100 ms, minimum rMSE is 0.05. For comparison, the FIDCSI simulation without decoupling yielded rMSE values of 0.3052, 0.0886, and 0.0685, respectively, for the three possible signal models.
Figure 5 displays the rMSE performance of the single-shot spCSI technique without decoupling, assuming a T2* of 63 ms, corresponding to 5 Hz linewidth. Like Fig. 4, Fig. 5 demonstrates that including J-coupling in the signal model provides better performance than a simple delta-function model. However, because of the broader linewidth, incorporation of J-coupling alone into the signal model results in rMSE of greater than 10% even for the shortest acquisition times. The figure demonstrates, however, that precise knowledge of T2* is not necessary in order to achieve rMSE values less than 10%. Specifically, assuming T2* of 4 Hz or 6 Hz allow for all imaging times under consideration. This represents a ±20% error in estimation of T2*, which can readily be achieved by performing T2* mapping using proton imaging (23) before injection of the hyperpolarized pyruvate bolus.
Figure 6 depicts the results of the phantom experiments, which used a 100-ms acquisition. Individual grayscale images are shown, with a composite color overlay at the right. The result of the spCSI acquisition is on the bottom row, while the result of a FIDCSI data set is included above it for comparison. Metabolite ratios of lac/pyr and ala/pyr inside the chosen ROI were 0.71 and 0.68, respectively, for the FIDCSI experiment. For the single-interleaf spCSI experiment, the ratios were 0.75 and 0.67, evidence of good quantitative agreement between the two acquisitions.
DISCUSSION AND CONCLUSION
The ability to efficiently utilize hyperpolarized 13C signal will be crucial in any metabolic imaging experiment in which temporal resolution is important. This work provides a framework that can be used to optimize acquisition parameters and to reconstruct the data, and suggests that imaging with very high temporal resolution (imaging time of approximately 100 ms for an 8-cm FOV and 0.5 cm isotropic resolution) is feasible. Using the parameters investigated here, multislice cardiac imaging in a single diastole using a phased-array coil could be considered.
The same analysis applied here to the single-interleaf spCSI acquisition can be applied to other types of pulse sequences. For multishot sequences, such as the three-interleaf spCSI discussed by Mayer et al. (11) or the 16-shot echo planar spectroscopic imaging (ESPE) sequence discussed by Chen et al. (21), rMSE will be less sensitive to inclusion of J-coupling in the signal model because the individual readouts are shorter. Furthermore, higher values of SW are achievable. However, the single-interleaf technique is capable of higher temporal resolution imaging and does not require FA optimization While the three-shot spCSI technique can achieve imaging time of 100 ms, its minimum rMSE at that temporal resolution is slightly higher than that of the single-interleaf spCSI technique. Specifically, the minimum rMSE for the three-shot spCSI at 100 ms temporal resolution is 8.7%, while it is 7.9% for the single-shot spCSI experiment. The EPSI sequence cannot achieve an imaging time of 100 ms. Furthermore, in the absence of ramp sampling, the EPSI sequence suffers from SNR loss due to the significant fraction of time in which no data acquisition occurs. The single-interleaf technique can also be used for longer imaging times in instances where this is desirable.
The multiecho SSFP technique that has been proposed for metabolic imaging of sparse spectra (4, 24) can also be optimized using a least-squares analysis. With a single excitation, followed by a train of refocusing pulses, the SSFP technique rapidly samples the spatial and temporal dimensions. It differs from the single-shot spCSI technique in that larger SW and lower spectral resolution are used. Because k-space is sampled in a rectilinear fashion and because multiple refocusing pulses are needed, the technique is not capable of reaching a 100-ms imaging time even when implemented in a single shot. Like the EPSI technique, the relatively short duration of the individual readouts make reconstruction less sensitive to T2* and to J-coupling than the single-shot spiral method.
In contrast to the rectilinear sampling trajectories mentioned above, the single-interleaf spCSI acquisition uses a value of SW that is much lower than the bandwidth of the spectrum, which results in significant aliasing. The quality of peak separation is determined by the ability to differentiate the metabolites in the presence of noise. Practically, this means choosing SW so as to minimize overlap among the aliased peaks. For poorly separated peaks, least-squares fitting of the peaks results in energy from one resonance appearing as noise in the metabolic image representing another resonance. This work demonstrates that for well-chosen acquisition parameters and signal models, a very short acquisition is possible.
An additional benefit of the k-space based analysis is that it allows the user to consider evolution of chemical shift during the readout and its effect on each k-space point. This contrasts with techniques that perform peak separation and deblurring in distinct steps (11, 16). The choice of optimal SW did not change when averaging all points in k-space together rather than using only the first point. However, in instances in which higher resolution is desired, or if specific information about the power spectral density of the object is known a priori, rMSE can be used to optimize the echo times by weighting the k-space points appropriately.
One issue not addressed explicitly in this work is incorporation of field inhomogeneity into the reconstruction. This issue has been explored by other authors in the context of imaging of systems of sparse spectra (12, 16–18, 25, 26). Most of these have focused on fat-water separation, although Reeder et al. (17) considered a three-component spectrum in one instance and Ebel et al. (18) considered eight components using a spectroscopic ultra-fast low-angle rapid acquisition and relaxation enhancement (U-FLARE) sequence. Dixon-like techniques for fat-water separation must resolve only two peaks, but must include field-map estimation in their reconstructions because of the inherent ambiguity of the two-component spectrum. Field-map estimation has itself become an important subject of research (13, 27).
The situation is somewhat easier when considering metabolite imaging in the context of hyperpolarized 13C. It is most advantageous to acquire a B0 field-map using proton imaging (corrected by the γ13C/γ1H ratio) before administering the hyperpolarized agent. The simulations in this work assumed a uniform B0 field map, while the phantom experiment necessarily involved some B0 nonuniformity. Because of the small gyromagnetic ratio of 13C, field inhomogeneity is expected to be less problematic than in proton imaging. However, in cases where it is an issue, methods such as multifrequency reconstruction (16, 28) could be considered. The k-space based reconstruction would be performed multiple times, shifting the resonance frequencies of the metabolites by a range of frequencies characterized by the estimated off-resonance in the sample. The metabolic images would then be pieced together pixel-by-pixel from the multiple reconstructed images. Alternatively, a full spectroscopic reconstruction (11) could be performed first in order to align the peaks, followed by two-dimensional Fourier transformation back to the k-space domain and least-squares reconstruction.
A second aspect not considered in this work is how to deal with remaining transverse magnetization. Because fast imaging can yield images on a time-scale much shorter than the T2* of the metabolites, there can be remaining transverse magnetization after the readout is complete. Future work will focus on how best to manage this signal. Several options may be considered: 1) Perform a small tip excitation for each readout, spoiling the remaining magnetization at the end of the readout. Most of the magnetization remains along the z axis for later imaging. 2) Use a large-tip excitation for the first readout, and return remaining magnetization to the z axis via driven equilibrium (29). 3) Use refocusing pulses such as spin echo to maintain the magnetization in the transverse plane for additional imaging. Future work will include in vivo acquisitions and considerations of other hyperpolarized 13C spectra.
Y.S.L. thanks Angel Pineda and John Pauly for helpful discussions.