Least-squares chemical shift separation for 13C metabolic imaging

Authors

  • Scott B. Reeder MD, PhD,

    Corresponding author
    1. Department of Radiology, University of Wisconsin, Madison, Wisconsin
    2. Department of Medical Physics, University of Wisconsin, Madison, Wisconsin
    3. Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin
    4. Department of Medicine, University of Wisconsin, Madison, Wisconsin
    • Department of Radiology, E3/311 CSC, University of Wisconsin, 600 Highland Ave., Madison, WI 53792-3252
    Search for more papers by this author
  • Jean H. Brittain PhD,

    1. Global Applied Science Lab, GE Healthcare, Madison, Wisconsin
    Search for more papers by this author
  • Thomas M. Grist MD,

    1. Department of Radiology, University of Wisconsin, Madison, Wisconsin
    2. Department of Medical Physics, University of Wisconsin, Madison, Wisconsin
    3. Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin
    Search for more papers by this author
  • Yi-Fen Yen PhD

    1. Global Applied Science Lab, GE Healthcare, Menlo Park, California
    Search for more papers by this author

Abstract

Purpose

To describe a new least-squares chemical shift (LSCSI) method for separation of chemical species with widely spaced peaks in a sparse spectrum. The ability to account for species with multiple peaks is addressed.

Materials and Methods

This method is applied to imaging of 13C-labeled pyruvate and its metabolites alanine, pyruvate, and lactate. The method relies on a priori knowledge of the resonant frequencies of the different chemical species, as well as the relative signal from the two pyruvate peaks, one of which lies near the alanine peak. With this information a least-squares method was utilized for separation of signal from the three metabolites, facilitating tremendous reductions in the amount of data required to decompose the different chemical species. Optimization of echo spacing for maximum noise performance of the signal separation is also described.

Results

Imaging an enriched 13C phantom at 3.0T, the LSCSI method demonstrates excellent metabolite separation, very similar to echo planar spectroscopic imaging (EPSI), while only using 1/16th as much data.

Conclusion

This approach may be advantageous for in vivo hyperpolarized 13C metabolic applications for reduced scan time compared with EPSI. J. Magn. Reson. Imaging 2007;26:1145–1152. © 2007 Wiley-Liss, Inc.

THE ADVENT OF HYPERPOLARIZED 13C (1) has renewed interest in imaging with this isotope for a variety of applications, including vascular and metabolic flux imaging. 13C-labeled pyruvate and its metabolites, lactate and alanine, have great potential for in vivo metabolic imaging, particularly for applications in oncology (2), cardiac perfusion (3, 4), as well as angiography and pulmonary perfusion (5). 13C-labeled pyruvate (pyr), with a labeled carbon at the C1 position may provide an excellent means of probing important metabolic pathways within cells. Under aerobic conditions, pyruvate is converted to alanine through transamination by alanine aminotransferase, rapidly incorporating the labeled carbon at the C1 position into alanine (ala), as shown in Fig. 1a. However, under anaerobic conditions pyruvate is converted to lactate by lactate dehydrogenase via the anaerobic glycolytic pathways, incorporating the labeled carbon at the C1 position into lactate (lac). Finally, a C1-C2 pyruvate ester (PE) may also form. A schematic of the 13C spectrum at 3.0T centered at 32 MHz is shown in Fig. 1b, with the relative chemical shifts noted. As shown in this figure, the NMR spectrum of pyruvate and its two metabolites lactate and alanine is relatively sparse, with single peaks for lactate and alanine and two peaks for pyruvate (6). Sparse datasets such as these are very well suited for chemical shift-based imaging methods (7–11).

Figure 1.

a: Schematic of biochemical pathways of labeled pyruvate in both aerobic and anaerobic respiration. b: Acquired 13C spectrum for lactate (dashed), alanine (dotted), and pyruvate (solid). Frequency shifts are relative to lactate at 3.0T. The smaller of the two pyruvate peaks (242 Hz downfield from lactate) is from the pyruvate ester, which has a similar chemical shift as pyruvate hydrate, which is present in vivo.

Hyperpolarized 13C metabolic imaging applications that require short scan times, such as cardiac imaging with breath-holding, will require significant reductions in scan time from current spectroscopic imaging methods, as well as the ability to acquire all required data during the equilibration of the polarized nuclei. We have recently reported the development of a chemical shift-based method that requires the acquisition of only a few images at different echo times (TE). This method is an iterative least-squares chemical shift-based method, used primarily for the decomposition of water from fat, but also capable of decomposing multiple chemical species (10, 12). Known as IDEAL (iterative decomposition of water and fat with echo asymmetry and least-squares estimation), this method allows the use of a flexible number of echoes acquired at arbitrary echo spacings in order to separate different chemical species. Flexibility in the number of echoes facilitates noise optimization in the estimation of signal from different species (9), as well as the ability to apply IDEAL to various pulse sequences such as fast spin-echo (12), spoiled gradient echo (13), and steady-state free precession (SSFP) (14–17). Recently, several authors have reported the use of IDEAL for separation of 13C metabolites, in combination with SSFP (18, 19), gradient echo-spin echo (GRASE) (20), and spiral imaging (21), while others have reported the use of spiral chemical shift imaging in combination with 13C metabolite separation (22). All such chemical shift-based methods require a priori knowledge of the resonant frequencies of the species. With this knowledge, separation of species into different images can be achieved with relatively few images acquired at a few echo times. Unfortunately, the conventional IDEAL algorithm is applicable for multiple species consisting of a single peak, well separated from all peaks of the other chemical species. Some species, such as 13C-labeled pyruvate, however, may have more than one peak, often with minimal or no separation from the peaks of other species. The IDEAL algorithm, which models spectra of different species with a single peak, will fail in these circumstances because there may be insufficient spectral resolution to separate signal from these different peaks.

In this work we present a novel strategy for the least-squares chemical shift imaging (LSCSI) method based on the IDEAL algorithm, tailored for separation of pyruvate and its metabolites lactate and alanine, labeled with 13C at the C1 position. This method accounts for a multiplicity of spectral peaks, even if some peaks have little separation from those of other species.

THEORY

Chemical Shift Separation

13C-labeled pyruvate is an important probe to interrogate dynamic metabolic processes in vivo. The schematic of the relevant 3.0T spectrum shown in Fig. 1b shows that relative to lactate, alanine has a single peak at −242 Hz, pyruvate has a main peak at −602 Hz, and the pyruvate ester lies at −242 Hz. Knowing these relative shifts, and assuming that the relative area of the two pyruvate peaks is known, the signal from a voxel containing these three species measured at N arbitrary echo times, tn (n = 1, …, N), can be modeled as:

equation image(1)

where ρL, ρA, and ρP are the total signal contributions from lactate, alanine, and pyruvate, respectively, ΔfL, ΔfA, ΔfPyr, and ΔfPE are the resonant frequencies of lactate, alanine, the main pyruvate peak, and the pyruvate ester peak relative to the receiver frequency, and rPE and rPyr are the relative fractions of total signal from the two pyruvate peaks, such that rPE + rPyr = 1.0. The outer phase term, results from the local field inhomogeneity (θ), which can be estimated using an iterative algorithm (10) or measured from separate calibration 1H images. For this discussion we will assume θ is known and has been demodulated from Eq. 1. Details of an iterative solution for the field map have been described elsewhere (10). After demodulation of the field map, Eq. 1 can be written in matrix format, ie:,

equation image(2)

where

equation image

and

equation image

Using the Penrose-Moore pseudoinverse (23), least-squares estimates of the signal vector comprising signals from alanine, lactate, and pyruvate can be made:

equation image(3)

noting that the superscript “H” denotes the Hermitian transpose. The signal estimation shown in Eq. 3 is a maximum likelihood estimator and will provide the maximum possible noise performance for a given combination of images acquired at different echo shifts. It is important to note that all terms in the coefficient matrix, A, are known, and that at least three images with different echo times (tn) are required to decompose the signals from the three species. It is also important to note that additional species with different chemical shifts can be easily added to the above formulation. In general, the LSCSI method requires at least N images with different echo times in order to separate N species with unique chemical shifts. If more than N images are used, the additional images will improve the noise performance of the species signal estimation, although at the cost of longer scan time.

Echo Timing Optimization

Although the separation of chemical species can be performed with an arbitrary combination of echo time shifts, careful selection of the echo shifts is important to optimize the noise performance of the estimation process. The effect of echo shifts can be analyzed using the condition number of the A matrix in Eq. 2 (21). The major disadvantage of using the condition number for noise optimization is that it does not permit direct optimization of noise performance for the estimation of individual species. For this reason, we prefer to calculate the effective number of signal averages (NSA), which is calculated from the inverse of each diagonal component of the covariance matrix of A, ie:

equation image(4)

where m = 1 (alanine), 2 (lactate), 3 (pyruvate). Unlike the condition number, the NSA can be calculated for each of the three chemical species, permitting a more comprehensive understanding of the overall noise performance. Direct optimization of the noise performance for the estimation of the individual species may be important for applications where the estimation of one specific species (eg, lactate) may be more important that other species (eg, alanine). Analytical solutions for NSA are derived in the Appendix.

In general, this method permits the acquisition of echoes that are arbitrarily and unequally spaced, and an overall optimization requires careful selection of the spacing between the N acquired echoes. For multiecho acquisitions, such as the implementation we describe below, however, it is usually impractical and inefficient for pulse sequences to acquire unequally spaced. Therefore, we limit our current optimization to equally spaced echoes.

Figure 2 plots the NSA for three and four equally spaced echoes, based on 3.0T chemical shifts of 13C-labeled pyruvate, lactate, and alanine using Eq. A.2. From these figures numerous local maxima and minima are noted, providing a guide for echo spacing selection in order to maximize the signal-to-noise ratio (SNR) performance in the estimation of the different chemical species.

Figure 2.

Noise performance of the LSCSI method measured using the effective number of signal averages (NSA) plotted for the three species at 3.0T for (a) three and (b) four equally spaced echoes. In addition, the condition number, a measure of how well Eq. 2 is posed numerically, is plotted for (c) three and (d) four equally spaced echoes. The smaller the condition number, the better posed the estimation problem. Good agreement between the location of the minima of the condition number and the maxima of the corresponding NSA plots is seen. Although a very useful means of optimizing the choice of echoes, the condition number is limited when compared to the NSA approach, which provides a direct measure of the SNR performance for all species.

It is important to note that the effects of Nyquist aliasing are accounted for in the NSA plots, and Nyquist sampling criteria do not need to be strictly obeyed with this least-squares approach. Fourier approaches, which have fixed frequency bins for the reconstructed signal, but no a priori information on the location of spectral peaks, require Nyquist sampling to resolve different species. With the LSCSI method, however, the spectral locations are known and, therefore, all effective aliasing is also known. Aliasing will not affect the decomposition of spectral peaks, so long as aliasing does not superimpose two peaks to the same effective frequency. Peaks that are superimposed on one another from aliasing cannot be resolved, and this effect is reflected in the NSA plots, as shown in Fig. 2. For example, the relative frequency between lactate and alanine is 210 Hz at 3.0T. For an equally spaced, four echo acquisition, the optimal sampling rate to distinguish between lactate and alanine using four equally spaced echoes is 1/(4*210) = 1.2 msec, which provides uniform sampling around the unit circle generated in 2π of phase developed between lactate and alanine. This echo spacing corresponds to the local maxima shown at 1.2 msec spacing in the plot in Fig. 2b. However, we would expect aliasing of the lactate and alanine peaks on one another if the echo spacing equals a full period, ie, 1/210 = 4.8 msec, ie, when lactate and alanine are in-phase for each echo time. This situation corresponds to a broad minimum of the plot in Fig. 2b, reflecting the inability of the LSCSI method to resolve lactate from alanine at this echo spacing. There are numerous other local minima and maxima described by the NSA plot, which account for the effects of temporal undersampling in the Fourier sense and the resulting aliasing that occurs.

An alternative method for echo spacing optimization is the use of the condition number, a commonly used tool for assessing how well a numerical estimation problem is posed for linear systems of equations (24). The lower the condition number, the better the problem is posed. Figure 2c,d plots the condition number for the species estimation problem for the matrix A in Eq. 2 for three and four echoes, respectively. The local minima of the condition number represent optimal echo spacings, which agree closely with the corresponding local maxima in the NSA plots of Fig. 2a,b. The main advantage of using the condition number is its relative simplicity (single scalar value) and its computational efficiency, since the condition number function is widely available in most numerical libraries, including MatLab (MathWorks, Natick, MA). However, a major disadvantage of the condition number, and the reason why we prefer the use of NSA, is that the NSA directly reflects the SNR performance of the estimation problem for each of the species.

It is important to note that the echo spacing optimization with both the NSA and condition number was performed for equally spaced echoes, which is the most practical situation for multiecho acquisitions. However, for applications that acquire a single echo per TR, more flexible, unequally spaced echoes can be used, which has the potential to improve the noise performance further. This requires a more involved optimization that is beyond the scope of this work, but would apply the same optimization principles illustrated above.

MATERIALS AND METHODS

A 13C phantom intended to mimic the in vivo 13C spectrum was constructed from three spheres doped separately with 13C-labeled alanine, lactate, and pyruvate. Specifically, each sphere contained 1.125 M 13C-1-pyruvate hydrate ester, 1.375 M 13C-1-alanine, and 20% sodium 13C-1-lactate (1.77 M). The sodium 13C-1-pyruvate, sodium 13C-1-alanine, and 20% sodium 13C-1-lactate were purchased from Cambridge Isotope Labs (Andover, MA). For long-term stability, the sodium 13C-1-pyruvate was converted to 13C-1-pyruvate pyruvate-hydrate ester by the addition of a stoichiometric concentration of HCl. The three spheres were arranged in triangular configuration and placed inside a 250-mL polyethylene bottle filled with water.

Figure 1b plots a spectrum of the phantom obtained with conventional 2D-chemical shift imaging, obtained to measure the relative frequencies of the three chemical species, as well as the relative signal contributions from the two pyruvate peaks. Specific parameters of the free induction decay acquisition included: TR = 90 msec, field of view (FOV) = 6 cm, 10° flip, 256 spectral readout points, BW = ±4000 Hz, and a 12 × 12 matrix.

The chemical shift of lactate and alanine is similar to that in vivo, but the pyruvate-pyruvate-hydrate ester peaks are shifted relative to 13C-1-pyruvate and its hydrate. The pyruvate peak in the phantom appears at about −602 Hz relative to lactate (Fig. 1). The two labeled carbons of the pyruvate/pyruvate-ester are contained within a single molecule are present in a 1:1 stoichiometric ratio. Therefore, the expected signal from the two carbons should be similar, rather than at ≈8% as in the free pyruvate:pyruvate-hydrate equilibrium (25). The ratio of the pyruvate moieties was adjusted to reflect those in the phantom, measured with conventional spectroscopic methods, and may be different than those expected in vivo. The decomposition method can easily be adjusted to account for such variations in the composition of the relative peak amplitudes of the different moieties.

MR Imaging was performed using a GE Signa Excite 3T clinical scanner (VH/i, GE Healthcare, Waukesha, WI). A custom-built 4-cm diameter dual tuned birdcage 1H/13C coil was used for this study. The transmit gain was adjusted manually using an external reference (bottle of corn oil).

Chemical shift imaging was performed using a 2D echo planar spectroscopic imaging (EPSI) sequence with flyback readout gradients (26). The EPSI waveform permitted simultaneous sampling in the time domain and one spatial frequency domain, with phase encoding gradients used to sample the other spatial frequency axis. Although all data acquired during the EPSI-flyback gradient could, in principle, be used to reconstruct the spectroscopic image, only the data acquired during positive EPSI gradient plateau were used for reconstruction in this work to avoid complicating phase shifts resulting from filter and gradient timing delays (27, 28). The number of positive-and-flyback cycles (or echoes) in the EPSI train determines the number of timepoints of spectral sampling, with the time period between two acquired echoes equal to the inverse of spectral bandwidth. In this work, 64 echoes separated by 2.028 msec were acquired for a spectral bandwidth of 493 Hz. Other image parameters included: FOV = 6 cm, slice = 20 mm, TR = 1 second, flip = 10°, 12 × 12 matrix, and receive bandwidth was ±6.4 kHz. The resulting spatial resolution was 5 × 5 × 20 mm3. EPSI data were reconstructed using conventional 3D Fourier transformation, with 64 separate spectral bins. For the LSCSI method, a local NSA maximum for all three species occurs at ≈2.0 msec (Fig. 2b), corresponding closely to the echo spacing of the EPSI data (2.028 msec). Therefore, the first four echoes of the EPSI data were used for the LSCSI reconstruction. Based on the areas under the peaks from the plot in Fig. 1b, the relative proportion of signal of the pyruvate peaks was 61% for the main peak and 39% for the ester peak. The relative proportions are different than 50:50, most likely due to differences in J-coupling of the two peaks, which are below the spectral resolution of the CSI data shown in Fig. 1b. Images were Fourier-transformed in the two spatial dimensions and pixels processed individually using Eq. 3. Chemical shift correction in the readout direction was performed for both methods (29). Because the pyruvate signal is comprised of two peaks, the average of the two pyruvate frequencies was used when corrected for chemical shift with the LSCSI method. As a result, the chemical shift in the readout direction was reduced but not completely corrected. However, for this experiment the bandwidth per pixel is 1068 Hz, and the relative chemical shift between the two pyruvate peaks of 360 Hz translating to a spatial shift of 0.35 pixels.

RESULTS

Figure 3 shows separate pyruvate, lactate, and alanine images generated using EPSI (top row) and the LSCSI method (bottom row). Separate EPSI images were obtained from the sum of three frequency bins centered on the corresponding spectral frequency for each chemical species. The pyruvate ester and main pyruvate peaks were added together to form the EPSI pyruvate image. Excellent qualitative separation of the three metabolites can be seen with both the EPSI and LSCSI methods.

Figure 3.

Metabolic 13C images acquired using only four echoes with LSCSI (top), and all 64 images with EPSI (bottom). Separate pyruvate, lactate, and alanine images show excellent qualitative agreement between EPSI and LSCSI.

Finally, Fig. 4 plots profiles through the three vials containing the different metabolites. Normalization of signal levels to the maximum pyruvate signal was performed to account for the differences in absolute signal from the fact that LSCSI only used 4 of the 64 echoes. The same scaling factor was subsequently used for lactate and alanine peaks. Very good agreement in the profiles measured with all three methods was found, including good agreement of the relative amplitudes of the three species.

Figure 4.

Profiles plotted through the different metabolite vials, (a) pyruvate, (b) lactate, (c) alanine. show good agreement between LSCSI and EPSI. Relative scaling of the signal from the two methods was achieved by normalizing the maximum values of pyruvate signal for both methods. The same scaling factor was then used for alanine and lactate, indicating that the relative signal values from the two methods are in very good agreement. Note that the scale for each graph, which is shown in this figure, was adjusted to maximize the signal from each profile to best display the good agreement between the two methods.

DISCUSSION

This study demonstrates the feasibility of a least-squares chemical shift imaging method for separation of 13C metabolites. This method achieves similar separation as EPSI that uses the Fourier transform in the spectral dimension for decomposition of different frequency bins. In addition, only 1/16th the number of images was required, which may accelerate the overall acquisition time, depending on the pulse sequence acquisition method. The LSCSI method is able to distinguish alanine from the pyruvate ester through a priori knowledge of the relative portions of the two pyruvate peaks. Without this knowledge or knowledge of the metabolite resonant frequencies, separation of metabolites is not possible with chemical shift approaches such as this. The use of a priori information (resonant frequencies of the species and the relative amplitudes) is what distinguishes LSCSI from EPSI, which assumes fixed, equally spaced frequencies that are unrelated to the resonant frequencies of the different species.

The major disadvantage of this approach is that a priori knowledge of the chemical species is required. For specific applications such as 13C imaging with sparse datasets and well-defined spectral peaks, this is a minor limitation. Perhaps a more important consideration is requirement that the relative proportion of the two pyruvate peaks must be known in vivo. Unlike our phantom, in vivo an equilibrium between pyruvate and pyruvate-hydrate occurs, and ≈8% of pyruvate will be in the form of pyruvate-hydrate (25). In vivo, the time constant for conversion of pyruvate to pyruvate-hydrate is not well known. If the time constant of this equilibrium is much shorter than that of the metabolic flux of pyruvate into alanine and/or lactate, the relative amounts of pyruvate-hydrate to pyruvate should be constant. Otherwise, the assumption that the relative amounts of pyruvate and pyruvate-hydrate are fixed may not be valid. In addition, the noise performance NSA calculations are heavily influenced by the relative proportions of the pyruvate and should be calculated for different proportions of signal from the two peaks, and the noise performance for pyruvate should improve as the resonant peak near alanine becomes smaller relative to the main pyruvate peak. Further in vivo work will be required to explore these issues.

The LSCSI method described in this work may be well suited for metabolic imaging with hyperpolarized 13C, using SSFP, and should be easily adaptable to this pulse sequence. Balanced SSFP methods hold great promise for in vivo 13C-labeled pyruvate metabolic imaging (18, 19). The slow decay of transverse magnetization of the 13C signal is ideal for SSFP, which maintains coherence of long T2 species such as hyperpolarized 13C-labeled metabolites. Multiecho chemical shift methods using SSFP have been described for water-fat separation (10, 14, 15) as well as water/fat/acetone/silicone separation (16), and may be well suited for hyperpolarized 13C applications. In order to maintain coherent signal that is relatively insensitive to field inhomogeneities, the TR of SSFP imaging must be kept short to avoid banding artifacts (30), limiting acquisition to short echo trains, such as three to four echoes. Thus, methods such as LSCSI may be well suited for metabolic imaging with 13C-labeled pyruvate. Fortunately, the gyromagnetic ratio of 13C is relatively low (10.71 MHz/T compared to 42.58 MHz/T for 1H), such that banding artifacts caused magnetic field inhomogeneities from tissue susceptibility differences will have a smaller impact, and TRs of 8–10 msec should be easily tolerated with minimal artifact. This is tempered in part by imaging at higher field strengths, which increases susceptibility differences, as well as the need for longer readout gradients to achieve adequate resolution due to the lower gyromagnetic ratio, an effect that tends in increase the TR. Echo spacings of 1.1 msec with a three echo acquisition or 1.3 msec with a four echo acquisition provide optimal noise performance (Fig. 2), and it should be feasible to acquire such images in a reasonable TR, relatively free of banding artifacts. If high spatial resolution or low bandwidth imaging is desired, then a single readout per TR can be performed, which will help shorten TR, although this will substantially increase the acquisition time.

For a given voxel the effects of field inhomogeneities will shift all peaks by an equal amount. This effect is described as the outer phase term in Eq. 1. For the purposes of this discussion, this phase shift has been ignored, but may become important, particularly for in vivo applications where imperfect shimming and susceptibility effects from the body will create an inhomogeneous perturbation from the main magnetic field. There are two possible solutions to this effect. The first approach would be to apply a previously described iterative method that uses the same images acquired at different echo times, in order to measure the field inhomogeneity, and subsequently demodulate these phase shifts. The second approach would be to use a dual tuned 1H/13C coil in order to measure the field inhomogeneity from 1H images, and subsequently demodulate the resulting phase shifts from the 13C source images. Further work in vivo is needed to fully explore the effects of field inhomogeneities with 13C chemical shift imaging using the LSCSI method.

The precision of the LSCSI method to quantify metabolites is an area that requires further investigation. There are two main factors that will affect the ability to quantify metabolites: SNR performance and spectral resolution. In comparison to EPSI, LSCSI has lower SNR performance, owing to the fact that significantly fewer images are used in the reconstruction. NSA calculations can be used to determine the relative SNR loss compared to EPSI. Ignoring the effects of T2 signal loss during the EPSI acquisition, we would expect the effective signal averaging of EPSI to be ≈64 for all spectral bins, compared to NSA of 1.5, 3.0, and 4.0 for pyruvate, alanine, and lactate, respectively, using four echoes only. It is important to realize that, using LSCSI, the SNR performance varies with the different metabolites, unlike EPSI, which has constant noise performance across all spectral bins.

The effects of spectral resolution present a more complex issue. If the line width of the different metabolites is narrow relative to the spacing between the different metabolites, the LSCSI method should accurately separate the different species similar to EPSI. The precise effect from line width broadening from T2 or T2* effects requires additional analysis that is beyond the scope of this work. Based on the close agreement of the two methods in the 13C phantom, this effect appears to be minor in this situation, although its importance in vivo remains to be determined.

Finally, a complete optimization of the noise performance of this method is also required. Although uniform echo spacing is more practical for most multiecho pulse sequences, alternative acquisition schemes with nonuniformly spaced echoes may provide improved noise performance of the estimation of the different species.

In conclusion, we have described a least-squares chemical shift imaging method well suited for resolving groups of chemical species with sparse spectral data, such as 13C-labeled pyruvate and its metabolites lactate and alanine. This approach requires a priori knowledge of the frequency shifts of the different species, as well as the relative area of the two pyruvate peaks. Noise analysis of least-squares estimation method permits optimizing of echo spacing in order to maximize the noise performance of metabolite signal decomposition.

Acknowledgements

The authors thank Ralph Hurd, PhD, for assistance with the 13C phantom and image acquisition, and Sarah Nelson, PhD, and Susan Kohler, PhD, for helpful discussions.

APPENDIX

Analytical expressions of the NSA of each species can be determined from Eq. 4. First, it can be shown that

equation image(A1)

The diagonal terms of the inverse of the matrix in Eq. A1 can then be calculated, the reciprocal of which determines the NSA for each of the three chemical species, ie:

equation image(A2)

where

equation image

and

equation image

is defined to organize and simplify the mathematics. From Eq. A2, it can be seen that the maximum NSA values are achieved when aA, aL, and aP are minimized by optimized values of tn. It can also be shown that the calculated NSA values are independent of the absolute echo time; by setting tn = to + Δtn, and showing that Eq. A2 is independent of to. Therefore, the noise performance of the echo estimation is only dependent on the relative spacing between the different echoes, not the absolute echo times. Details of this analysis are omitted for brevity.

Ancillary