Enhanced spectral resolution for correlated spectroscopic imaging using inner-product and covariance transform: a pilot analysis of metabolites and lipids in breast cancer in vivo

Acquisition duration of correlated spectroscopy in vivo can be longer due to a large number of t1 increments along the indirect (F1) dimension. Limited number of t1 increments on the other hand leads to poor spectral resolution along F1. Covariance transformation (CT) instead of Fourier transform along t1 is an alternative way of increasing the resolution of the 2D COSY spectrum. Prospectively undersampled five-dimensional echo-planar correlated spectroscopic imaging (EP-COSI) data from ten malignant patients and ten healthy women were acquired and reconstructed using compressed sensing. The COSY spectrum at each voxel location was then generated using FFT, CT and a variant of CT called Inner Product (IP). Metabolite and lipid ratios were computed with respect to water from unsuppressed one-dimensional spectrum. The effects of t1-ridging artifacts commonly seen with FFT were not observed with CT/IP. Statistically significant differences were observed in the fat cross peaks measured with CT/IP/FFT. Spectral resolution was increased ~ 8.5 times (~ 19.53 Hz in FFT, ~ 2.32 Hz in CT/IP) without affecting the spectral width along F1 was possible with CT/IP. CT and IP enabled substantially increased F1 resolution effectively with significant gain in scan time and reliable measure of unsaturation index as a biomarker for malignant breast cancer.

www.nature.com/scientificreports/While the multi-voxel spectroscopic imaging covers a larger area of the breast, the spectral resolution along indirect F 1 (t 1 ) spectral dimension is generally low due to the scan time limitations.Combined with artifacts due to t 1 -ridging caused by factors like subject motion and instrumental fluctuations 17 , this can lead to loss of/ corrupted cross peaks, limiting the full potential of the technique.While zero-filling can improve the resolution by interpolation to some extent 18 , it doesn't improve the separation between different resonant frequencies.It can also introduce ringing artifacts along the F 1 domain and does not mitigate t 1 -ridging as well.
Recently, covariance NMR 19,20 has been applied to J-resolved spectroscopic imaging in vivo for increased F 1 spectral resolution without introducing ringing artifacts 21 .It replaced the second Fourier transformation applied to the t 1 dimension with a covariance transformation (CT).The resultant spectrum has a spectral resolution in the indirect dimension equal to that in the direct dimension.Therefore, fewer number of t 1 increments were required to extract the spin correlations than what is required in a conventional FFT based spectral analysis.This facilitates reducing the scan time while achieving higher spectral resolution along F 1 22 .Although this approach has been used in various NMR experiments like total correlation spectroscopy (TOCSY) and nuclear Overhauser effect spectroscopy (NOESY) [23][24][25] , the adaptation to in vivo has been limited.A variant of the covariance NMR spectroscopy called inner-product (IP) NMR spectroscopy, is yet another approach which further improves the covariance NMR by making it robust against changes in the carrier frequency 26 .In this work, we applied the CT and IP approaches to 5D EP-COSI in-vivo to show its advantages over the conventional FFT based COSY spectrum in terms of both improved spectral resolution and minimal influence of t 1 -ridging, while exploring the possibility of further acceleration in scan time.It is shown the biomarkers such as unsaturation index (UI) quantified from the COSY cross peaks may be unambiguously determined from a CT/IP spectrum in the presence of t 1 -ridging.

Data acquisition
The 5D EP-COSI data was acquired on a Siemens 3 T Skyra scanner (Siemens Healthineer, Erlangen, Germany) with a dedicated "receive" 24-channel phase-array breast coil and a body "transmit" coil (FOV: 160 × 160 × 120mm 3 , 1.5 mL voxel volume, TR/TE were 1500/35 ms), running on VE11C software platform.64 t 1 points sampled were used along F 1 15 with a spectral bandwidth (SW) of 1250 Hz and 512 complex t 2 points, and a SW of 1190 Hz along F 2 .A three-pulse sequence 27 was employed before the global water suppression.A nonwater suppressed scan with one t 1 point was acquired for eddy current phase correction and coil combination 28 .Two spatial and one spectral dimensions (k y ,k z ,t 1 ) were non-uniformly sampled with an exponentially-weighted sampling density along t 1 and gaussian sampling density along the k y -k z plane for an acceleration factor of 8.The total scan time was 28 min and 48 s.

Data reconstruction
The undersampled 5D EP-COSI data was reconstructed using a Group Sparsity (GS)-based compressed sensing (CS) algorithm 15,29 to estimate the unacquired samples along the k y -k z -t 1 dimensions.The dominant lipid peak around 1.3 ppm was zeroed in the Fourier transform of the non-water suppressed signal to obtain a waterdominant time domain signal for the eddy current phase correction and coil combination.The spectral peak volume integrals were computed as described in 16 .The quantified proton resonances along the diagonal (F 1 -F 2 ), and off-diagonal are listed in Table 1.

Covariance and inner-product COSY processing
After the GS-CS reconstruction, a hybrid spectral-spatial data matrix D(x, y, z, t 2 , t 1 ) was outputted where x, y and z were the Fourier transform of k x , k y and k z dimensions in k-space.After Fourier transforming the direct spectral dimension (t 2 ), we get the mixed time-frequency matrix A(F 2 , t 1 ) for every special location (x = 1, 2, 3…, 16; y = 1, 2, 3…, 16; z = 1, 2, 3…, 8), with a stack of 1D spectra.Fourier transforming the indirect dimension t 1 of A yielded the conventional Fourier transformed COSY spectrum S(F 2 , F 1 ) of size (512 × 64).
A covariance transform was instead obtained from A in the following manner 25 .
Step 1 Make matrix A offset free by subtracting the average 1D spectrum ( A avg ) from it.A = A − A avg .This 1D spectrum is formed by averaging over the t 1 dimension in A.
Step 2 Apply a singular value decomposition (SVD) 30 to the transposed mixed time-frequency matrix A T = U .W. V T , where U and V are the singular vectors and W is the diagonal matrix with singular values as its diagonal elements.
Step 3 The final covariance transformed spectrum is then calculated as, S cov = ÃT .Ã 1/2 = Ũ.W. ṼT , which gives a high-resolution spectrum of size (512 × 512), with the spectral width along indirect spectral dimension same as that of the direct spectral dimension.
The similarity between covariance and Fourier-transformed spectra is based on the Parseval's theorem as described in 20 .It follows that the covariance spectrum C = ÃT .A correspond to the 2D spectrum squared, i.e., S 2 , provided that the A avg in the calculation of covariance vanishes.However, it is shown later that the condition of vanishing average might not hold near the spectral center due to the relatively slow spin precision, as well as

Ethical approval and informed consent
The study was approved by the institutional review board of the University of California, Los Angeles.The subjects in the study provided written informed consent.

FFT versus CT versus IP
A 2D spectrum from a healthy subject generated using FFT, FFT with zero-filling along t 1 , covariance transform and inner product are compared in Fig. 1 as intensity plots and contour plots.FFT based spectrum shown in Fig. 1a without zero-filling had a matrix size of 512 × 64 while zero-filled FFT, CT and IP in Fig. 1b-d had matrix size of 512 × 512.Only 0-6 ppm range is shown in the figure.While FFT and zero-filled FFT had a SW of 1250 Hz along F 1 , both CT and IP had a SW of 1190 Hz.All four spectra had a SW of 1190 Hz along F 2 .Arrow 1 indicates the ringing caused by zero-padding in zero-filled FFT.Arrow 2 is pointed at the false peaks in FFT and zero-filled FFT which is in fact due to a mild t 1 -ridging.Arrow 3 shows the region where IP based spectra differed from the CT.
Panels (e-h) show enlarged regions from figures in panels (a-d) comprising 0.5-3.5 ppm along both axes.Apart from the ringing indicated by arrow 1 in zero-filled FFT, this figure shows the effect of t 1 -ridging on the cross-peaks between 0.9 and 1.3 ppm, as pointed out by arrow 1 in (e, f) and the lack of this degradation in (g, h).
Another 2D spectrum from a malignant lesion identified in a 45-year-old patient is shown in Fig. 2. Sections (a)-(d) shows the 2D spectra generated using FFT, FFT with zero-filling along t 1 , CT and IP.T 2 -weighted MR image with the white box representing the VOI placement is shown in (e).Red square shows the location of the extracted 2D spectrum.Arrows 1 and 2 points out the stronger effect of t 1 -ridging in (a) and (b) as compared to Fig. 1.Examples of spurious correlation appearing in CT and IP are pointed out by arrows 3 and 4.
Table 1.Metabolites and lipids identified in the 2D COSY spectra of breast tissues.

Diagonal peaks
Cross-peaks

Choice of t 1 points and acceleration feasibility
Figure 5 shows the effect of different schemes of t 1 sampling while using FFT and CT.The 2D spectrum here is the same spectrum as shown in Fig. 1, retrospectively undersampled along t 1 to study feasibility of further acceleration.Figure 5a shows the conventional FFT spectrum using the full 64 points along t 1 ranging TEs from 35 to 85.4 ms at 800 μs intervals, giving 1250 Hz SW.Choosing either first 32 or last 32 points as shown in (b) and (c) reduces the spectral resolution along F 1 by half and shows heavy t 1 -ridging effects (arrows 1 and 2).Sampling every other t 1 points for 2 × acceleration as shown in (d) and (e) on the other hand halves the SW along F 1 while retaining the same spectral resolution causing folding artifacts indicated by arrow 3.
Panels (f)-(j) in Fig. 5 show the results of same cases as in panels (a)-(e) when CT is used instead of FFT.Arrows 1 and 2 in (g) and (h) points out the spurious correlations appearing in the spectrum.The results for IP are shown in Supplementary Figure 1 were the panels (a)-(e) correspond to the same cases as in panels (f)-(j) of Fig. 5.

Quantitation comparison
Figure 6 shows bar graphs comparing the mean (95% CI) of different metabolite and lipid ratios with respect to 1D water peak area in both malignant ((a), (c) and (e)) and healthy ((b), (d) and (f)) breasts.Plots in (a) to (d) show the ratios for different cross peaks while (e) and (f) shows the ratios for diagonal peaks in the spectrum.A statistically significant difference between the estimation of ratios from IP, CT and FFT was determined by Figure 7 shows bar graphs comparing the UI between IP, CT and FFT as well as across healthy and malignant groups.(a)-(b) show the results for malignant and healthy groups compared between IP, CT and FFT when the UI is computed from cross peaks above and below the diagonal.(c)-(d) show UI from cross peaks above and below the diagonal compared between healthy and malignant groups.Statistically significant differences (p < 0.05) were observed between healthy and malignant groups for UI computed from cross peaks both above and below diagonal for IP and CT, and above the diagonal for FFT.The measured values of UI from the cross peaks above and below the diagonal were very close for IP and CT with difference being < 2% for malignant and 13% for healthy, whereas they were much larger for FFT with a 25% difference for malignant and 38% difference for healthy as shown in (a) and (b).

Discussion
Due to the scan time limitations, conventional 2D L-COSY spectrum is acquired with limited number of t 1 points.Even with non-uniform sampling and CS based reconstruction, 64 t 1 points are usually used to achieve reasonable scan time.This results in poor spectral resolution along F 1 .In this work, prospectively undersampled 5D EP-COSI data were reconstructed using GS-CS and 2D COSY spectra from multiple locations in malignant and healthy breast masses were analyzed using FFT, CT and IP for enhanced F 1 spectral resolution, reduced t 1 -ridging, and acceleration feasibility for faster scan times.

Limitations of FFT based spectrum
One the main requirements of FFT based analysis is that the time increments (∆t 1 ) for t 1 points should fulfill Nyquist theorem (∆t 1,Nyq = 1/SW) to avoid aliasing artifacts.Doubling the duration of t 1 increment for example, results in halving the SW along F 1 (625 Hz instead of 1250 Hz) while retaining the same spectral resolution (~ 19.53 Hz) if 32 t 1 points are collected for 2 × acceleration.This results in aliasing artifacts along F 1 due to the lack of sufficient SW (see Fig. 5d, e).
On the other hand, if SW is retained by keeping ∆t 1 = 800 μs while reducing the number of t 1 increments to 32, it results in poorer spectral resolution (~ 39.1 Hz) (see Fig. 5b, c).Common approach of zero-padding before FFT achieves interpolation to a larger matrix size, increasing the digital resolution, without actually increasing the spectral resolution.It can also introduce ringing in the spectrum in case of discontinuity in the time domain (see Fig. 1).The ringing can however be minimized by applying appropriate filters before zero-padding.

Advantages of CT and IP
An advantage of CT and IP over traditional 2D FFT is that the indirect dimension of S is not required to be sampled with a time increment ∆t 1 that fulfills the Nyquist theorem.Furthermore, if number of t 1 point needs to be reduced to accelerate the acquisition, a wider range of t 1 evolution times can be used with CT and IP without sacrificing the SW, since the SW along F 1 in CT and IP will be equal to that of F 2 dimension.It is also reported that probing wider range of t 1 evolution times allows better discrimination between true and spurious spin correlations 31 .Unlike zero-filled FFT, CT and IP facilitate true spectral resolution enhancement along F 1 as clearly shown in the results section.Even tough uniform increments along t 1 are shown in results section to demonstrate acceleration feasibility of CT and IP in comparison with FFT, non-uniform sampling along t 1 is also feasible with both IP and CT.The choice of specific set of increments needs further investigation and is the subject of future work.One approach would be to use the prior knowledge simulations to identify the set of t 1 increments that will maximize the sensitivity of metabolites or lipids of low concentrations.Another advantage of the CT and IP is the lack of t 1 -ridging artefacts.This results in cross peaks that are much better defined in the ppm range where t 1 -ridges are present in the FFT spectrum, for example, the lipid crosspeaks near 1.3 ppm along F 2 (see Figs. 1, 2, 3, 4).Consequently, the quantitation of these cross-peaks improves substantially.It was observed that there is a larger difference between the ratios of cross peaks on either side of the diagonal in FFT as opposed to CT and IP.This resulted in a larger difference between these cross-peaks on either side of the diagonal with FFT as opposed to CT and IP, despite the symmetric property of the COSY spectra (see Figs. 6c, d, 7).This can cause variation in the measures of UI which is one of the important potential biomarkers available in the COSY spectra depending on whether upper cross peaks or lower cross peaks are used for its computation.Ideally, the ratios should be similar on either side of the diagonal, but the lower resolution along F 1 , and t 1 ridging effects could influence these measures.Earlier studies have reported this measure from the cross peaks either above or below the diagonal or an average of the peaks on either side 16,32 .Statistically significant difference among IP, CT and FFT were observed in the cross peaks CP2, CP3, CP4 and CP5, especially in healthy controls.In the ratios from malignant tissues, only the difference in CP2 was statistically significant.This may be because of the fact that the healthy breast tissues generally have dominant fat peaks compared to malignant tissues.As a result, the chances of t 1 ridging is higher near the aforementioned cross peaks in healthy tissues.
Furthermore, CT and IP based approaches give substantial gain in actual spectral resolution.The CT and IP based spectra gives the same bandwidth along F 1 as that of F 2 .Therefore, these methods gave a spectral resolution of ~ 2.32 Hz along F 1 , while FFT based spectra had a spectral resolution of ~ 19.53 Hz in the experiments shown in results section.The metabolite peaks between 3 and 4 ppm were therefore much better resolved with these (c) FFT spectrum using last 32 t 1 points.Both (b) and (c) reduce the spectral resolution along t 1 by half and show heavy t 1 -ridging effects (arrows 1 and 2).Sampling only (d) odd t 1 points or (e) even t 1 points for 2× acceleration halves the SW along t 1 while retaining the same spectral resolution causing folding artifacts indicated by arrow 3. (f) CT spectrum using 64 points along t 1 ranging TEs from 35 to 85.4 ms at 800 μs intervals, giving 1250 Hz SW.(g) CT spectrum using first 32 t 1 points (t 1 SW = 1250 Hz).(h) CT spectrum using last 32 t 1 points (t 1 SW = 1250 Hz).Arrows 1 and 2 in (g) and (h) points out the spurious correlations appearing in the spectrum.(i) Sampling every other t 1 points starting from t 1 = 1.Effective t 1 SW = 1250 Hz. (j) Sampling every other t 1 points starting from t 1 = 2. Effective t 1 SW = 1250 Hz.
techniques compared to FFT in spectra from malignant tissues (see Figs. 2, 3, 4).This is especially important considering the ability of 5D EP-COSI in vivo detection and quantitation of metabolites like mI and Gly, in addition to measuring lipid-based biomarkers 16 .Reports from ex vivo breast cancer tissues have also shown the role of these metabolites including mI, Gly and Cho in identifying malignancy 5,33 .Even when not as resolved as that of CT and IP, the FFT based spectra also showed higher intensities in 3 ppm-4 ppm range in malignant tissues.Hence, the results of quantitation showed that these metabolites were elevated in malignant breast tissues compared to healthy ones with all three methods (see Fig. 6e, f).Since mI and Gly are separated by only 0.006 ppm, it requires < 1 Hz spectral resolution to disentangle them at 3 T.Despite this factor, it can be argued that the combined effect of high spectral resolution (~ 2.32 Hz) and absence of t 1 -ridging made the identification and analysis of these potential bio markers for breast cancer in CT and IP much less ambiguous as compared to FFT.

Limitations of CT and IP
The main challenge with CT and IP is the possibility of spurious cross-peaks which has intensities above the noise level despite not representing the spin correlations.While this might arise from limited number of t 1 points, we have observed that the FFT spectra with heavy t 1 -ridging also has tendency for stronger spurious correlations in CT and IP.Interestingly, the relationship between the number of t 1 increments and spurious correlation appeared to be also dependent on the increment time used (see Fig. 5).However, the deterministic nature of the resonance positions and sampling pattern makes it easy to identify these false cross peaks 31 .Furthermore, since these spurious correlations generally appeared away from the true resonant positions, it was less problematic compared to effects of t 1 -ridging in FFT based spectra.It has also been shown that the false cross peaks can be easily identified by performing the correlation experiments with various mixing times since the artifacts should not be affected by the mixing time 34 .While two fold acceleration is demonstrated in the results, further acceleration introduced more artifacts in the form of spurious correlations.Even though the CT and IP based spectra should be reliable to measure known correlations, assessment of new cross-peaks should therefore need careful analysis to discard the possibility of spurious correlations.

Difference in CT and IP
While both CT and IP based spectra appeared almost identical, a notable difference was near residual water at 4.7 ppm.This is as expected since the theory of inner-product based transformation is designed to be robust against the choice of central frequency.Since our spectra were all centered at water, the frequencies near water undergo limited oscillation with increasing the evolution time which weakens the vanishing mean assumption in CT.IP on the other hand doesn't have this requirement and hence argued to be more robust to the influence of the central frequency.However, in our application, the residual water was not important due to the water suppressed acquisition.Though the level of water itself in ratio with fat has been reported to be of importance in malignant breast cancers, this is usually calculated form the associated non-water suppressed 1D acquisition for eddy-current correction and coil combination.Therefore, both CT and IP served the purpose in a very identical manner throughout our experiments.However, it may be noted that the choice between the two should also be based on the choice of central frequency in the spectrum.

Conclusion
Reconstruction of in vivo 5D EP-COSI using CT and IP is presented in this work showing enhanced F 1 -resolution of COSY in comparison with FFT based spectral analysis.With CT and IP, we were able to achieve ~ 2.32 Hz spectral resolution along F 1 dimension as compared to ~ 19.53 Hz resolution in FFT based spectrum.The effect of t 1 -ridging artifacts commonly seen in FFT spectra was not observed in CT or IP.Consequently, both CT and IP showed well defined and symmetrical cross-peaks on either side of the diagonal as evident by the quantified ratios of the lipid cross-peaks above and below the diagonal.Furthermore, CT and IP were found to permit www.nature.com/scientificreports/wider range of t 1 increments without affecting the actual SW along F 1 .This is particularly advantageous in spectral analysis of metabolite and lipid biomarkers including unsaturation index for breast cancer while making significant gains in scan time.

Figure 1 . 8 Figure 3
Figure 1.Reconstructed 2D COSY spectrum of healthy tissues from a 60-year-old woman.(a) Intensity and contour plots of FFT based spectra (0-6 ppm).Arrow 2 indicates false peak from t 1 -ridging.(b) Intensity and contour plots (0-6 ppm) of FFT based spectra after zero-filling t 1 dimension to 512 points.Arrow 1 shows the ringing effect.(c) Intensity and contour plots of CT based spectra (0-6 ppm).(d) Intensity and contour plots of IP based spectra (0-6 ppm).Arrow 3 points out the region where IP spectrum is different from CT spectrum.(e) Intensity and contour plots of FFT based spectra (0.5-3.5 ppm).Arrow 1 indicates altered lipid cross-peak from t 1 -ridging.(f) Intensity and contour plots (0.5-3.5 ppm) of FFT based spectra after zero-filling along the t 1 dimension to 512 points.Arrow 2 shows the ringing effect.(g) Intensity and contour plots of CT based spectra (0.5-3.5 ppm).(h) Intensity and contour plots of the IP based spectra (0.5-3.5 ppm).

Figure 2 .
Figure 2. Reconstructed 2D COSY spectra from a malignant lesion identified in 45-year-old patient (grade 3 invasive ductal carcinoma, estrogen receptor positive, progesterone receptor positive, her2 positive, ki-67 = 20% and BI-RADS 5): (a) FFT based spectrum (b) FFT based spectrum after zero-filling along t 1 , (c) CT based spectrum, (d) IP based spectrum, (e) T 2 -weighted MR image with the white box representing the VOI placement.Red square shows the location of extracted spectrum.Arrows 1 and 2 point out the effect of t 1 -ridging in (a) and (b).Examples of spurious correlation appearing in CT and IP are pointed out by arrows 3 and 4.

Figure 3 .
Figure 3. Masking spurious correlations in (a) CT and (b) IP based spectrum shown in this figure.Panels on left show contour plots with spurious correlations.Panels on the right show the spectrum after removing spurious correlations.

Figure 4 .
Figure 4. Reconstructed 2D COSY spectra from malignant lesions identified in 41-year-old patient (grade 3 invasive ductal carcinoma and ductal carcinoma in situ, estrogen receptor positive, progesterone receptor positive, her2 positive, ki-67 = 60% and BI-RADS 5): (a) FFT based spectrum, (b) FFT based spectrum after zero-filling along t 1 , (c) CT based spectrum, (d) IP based spectrum, (e) T 2 -weighted MR image with the white box representing the VOI placement.Red square shows the location of extracted spectrum.Arrows 1 and 2 show\ the effect of t 1 -ridging in (a, b).

Figure 5 .
Figure 5. Effects of t 1 sampling in FFT and CT based spectra.(a) FFT spectrum using 64 points along t 1 ranging TEs from 35 to 85.4 ms at 800 μs intervals, giving 1250 Hz SW. (b)FFT spectrum using first 32 t 1 points.(c) FFT spectrum using last 32 t 1 points.Both (b) and (c) reduce the spectral resolution along t 1 by half and show heavy t 1 -ridging effects (arrows 1 and 2).Sampling only (d) odd t 1 points or (e) even t 1 points for 2× acceleration halves the SW along t 1 while retaining the same spectral resolution causing folding artifacts indicated by arrow 3. (f) CT spectrum using 64 points along t 1 ranging TEs from 35 to 85.4 ms at 800 μs intervals, giving 1250 Hz SW.(g) CT spectrum using first 32 t 1 points (t 1 SW = 1250 Hz).(h) CT spectrum using last 32 t 1 points (t 1 SW = 1250 Hz).Arrows 1 and 2 in (g) and (h) points out the spurious correlations appearing in the spectrum.(i) Sampling every other t 1 points starting from t 1 = 1.Effective t 1 SW = 1250 Hz. (j) Sampling every other t 1 points starting from t 1 = 2. Effective t 1 SW = 1250 Hz.

Figure 6 .
Figure 6.Mean (95% CI) of different metabolite and lipid ratios with respect to 1D water in both malignant (a, c, e) and healthy (b, d, f) breasts.Plots in (a-d) show the ratios for different cross peaks while (e, f) show the ratios for diagonal peaks in the spectrum.

Figure 7 .
Figure 7. Bar graphs comparing the unsaturation index between IP, CT and FFT as well as across healthy and malignant groups.Comparison between IP, CT and FFT when the unsaturation index is computed from cross peaks above and below the diagonal in (a) malignant group, (b) healthy group.(c) Unsaturation index from cross peaks above the diagonal compared between healthy and malignant groups.(d) Unsaturation index from cross peaks below the diagonal compared between healthy and malignant groups.