Derivative shape estimations with resolved overlapped peaks and reduced noise for time signals encoded by NMR spectroscopy with and without water suppression

Abstract

Derivative estimation in magnetic resonance spectroscopy (MRS) possesses several attractive features. It has the ability to enhance the inaccessible spectral details when time signals encoded by MRS are analyzed by nonderivative shape estimators. These unfolded subtle spectral features can be diagnostically relevant in differentiating between healthy and diseased tissues. Within the realm of shape estimators, the prerequisite for the success of MRS in the clinic is reliance upon accurate derivative signal processing. However, derivative processing of encoded time signals can be very challenging. The reason is that such spectra may suffer from severe numerical instabilities since even small perturbations (noise) in the input data could produce large errors in the predicted output data. Nevertheless, it is presently demonstrated that this obstacle can be surmounted by an adaptive optimization. The benefit is simultaneously increased resolution and reduced noise in quantitatively interpretable lineshapes. The illustrative spectra are reconstructed from time signals encoded by proton MRS with and without water suppression.

1 Introduction

This study is on signal processing with applications to nuclear magnetic resonance (NMR) spectroscopy, which is in medicine renamed as magnetic resonance spectroscopy (MRS). For conventional, nonderivative data analysis, shape and parameter estimations are employed, as qualitative and quantitative methods, respectively [1, 2]. The fast Fourier transform (FFT) is the most frequently used linear shape estimator mainly due to its automatic application of the Cooley-Tukey algorithm [3]. However, the FFT has a low resolving power and poor signal-to-noise ratio (SNR). The situation is even worse in the derivative fast Fourier transform (dFFT) for noise-corrupted time signals, or equivalently, free induction decay (FID) curves. This has been shown using FIDs from simulations with noise [4,5,6] and from encodings [7,8,9,10,11,12,13,14,15,16,17].

The dFFT works excellently for idealized noiseless synthesized time signals, but fails flagrantly for realistic data such as encoded FIDs. The reason for this failure of the dFFT is in strongly amplified instability of envelopes (i.e. total shape spectra) when the frequency-dependent derivative operator is repeatedly applied to the FFT. For encoded FIDs, the action of this operator on the FFT deteriorates both SNR and resolution of the ensuing dFFT. Therefore, processing encoded FIDs by the dFFT is ill-conditioned, despite the fact that this estimator actually tackles a direct, well-posed problem. Ill-conditioning is most frequently encountered in inverse problems [18].

The cause-effect relationship is at the core of distinguishing between direct and inverse problems. Finding an effect from the given cause is a direct problem (e.g. determining the outcome of propagation of waves through a medium of known characteristics). An inverse problem is finding the cause from the observed effects (e.g. inferring the unknown properties of the medium by analyzing the output data). Almost all of medicine is within the realm of inverse problems.

Both direct and inverse problems can be ill-conditioned. According to Hadamard [19], a problem is said to be ill-conditioned if its solutions are either (i) nonexistent or (ii) nonunique or (iii) do not depend continuously on the input data (or on the initial conditions). The latter means that solutions may have large errors even for small perturbations (noise) in the input data (or from round-off errors in finite-precision numerical computations). Such solutions are unstable and, therefore, unphysical.

However, despite its severity, the ill-conditioning difficulty is not insurmountable and the solutions may turn out to be physical. In other words, ill-conditioned problems can be made tractable by some suitable regularizations leading to stabilized solutions. The present regularization for linear signal processing is implemented by optimizing derivative Fourier shape estimations.

This methodology is applied to three entirely different types of FIDs encoded by proton MRS. Two sets of FIDs of short length (0.5 KB) were encoded with and without water suppression at a 1.5T General Electric (GE) clinical scanner from the standard Philips phantom, which contains a mixture of ethanol, methanol and acetate dissolved in demineralized water. The third FID transients of long length (16 KB) have been encoded in vitro with water suppression by a 600 MHz (14.1T) Bruker spectrometer from human cancerous ovarian cyst fluid.

For these encoded FIDs, the optimized derivative spectra are of high quality with simultaneously improved resolution and SNR. All the reconstructed resonances are separated from the overlapped spectral structures by gradually increasing the derivative order. The obtained bell-shaped resonance profiles are amenable to quantitative interpretations. In other words, despite belonging to linear shape estimators, the optimized derivative processor from this study crossed the barrier and ended up as a quantification-equipped, parameter estimator.

2 Theory

The features of the standard, unoptimized dFFT have been elaborated in Refs. [4,5,6,7,8,9,10,11,12,13,14,15,16,17] and need not be restated here. The present optimized dFFT has two objective functions, decaying exponentials for water-suppressed and decaying Gaussians for water-unsuppressed encoded FIDs. For the given FID, a spectrum $\textrm{S}(\nu )$ is a complex-valued function of linear frequency $\nu$. The mth derivative $(m=1,2,...)$ of this spectrum is the result $\textrm{S}^{(m)}(\nu )$ of the application of the operator $\textrm{D}_m=(\text{ d }/\text{d }\nu )^m$ to $\textrm{S}(\nu )$.

Spectra $\textrm{S}$ and $\textrm{S}^{(m)}=\textrm{D}_m\textrm{S}$ will be presented through graphs plotted in the magnitude mode as $\left| \textrm{S}\right|$ and $\left| \textrm{D}_m\textrm{S}\right|$. This is most convenient for interpretation because, by definition, magnitude envelope spectra are positive-definite and this obviates the need for phase corrections. Moreover, as opposed to the derivative absorptive lineshapes, magnitude profiles of complex derivative spectra have the possibility to significantly suppress the sidelobe artifacts around resonances by increasing the derivative order m [4,5,6].

With the augmented nonnegative integer m, a derivative spectrum has the peak widths and heights decreased and increased, respectively. Therefore, for a direct comparison of derivative spectra for different m, it is useful to normalize such lineshapes. The normalized magnitude spectra are defined as follows. From the results $\left| \textrm{D}_{m}\textrm{S}\right|$ $(m=0,1,2,...)$, where $\left| \textrm{D}_{0}\textrm{S}\right| \equiv \left| \textrm{S}\right|$, the first extracted are the maximal values $\textrm{max}\left| \textrm{D}_{m}\textrm{S}\right|$ for $m\ge 0$ in a small chemical shift band around a selected peak (the same for the FIDs with and without water suppression):

$$\begin{aligned} \textrm{max}\left| \textrm{D}_{m}\textrm{S}\right| \equiv \left| \textrm{D}_{m}\textrm{S}\right| ^\textrm{max}\quad (m=0,1,2,...). \end{aligned}$$

(1)

Subsequently, the scaling factors $\textrm{R}_{m}>0$ are deduced for the varying values of m as:

$$\begin{aligned} \textrm{R}_{m}= {\displaystyle {\frac{ \left| \textrm{D}_{m}\textrm{S}\right| ^\textrm{max} }{ \left| \textrm{S}\right| ^\textrm{max} } }}\quad (m=1,2,...). \end{aligned}$$

(2)

Finally, the normalized derivative magnitude spectrum $\left| \textrm{D}_{m}\textrm{S}\right| _\textrm{N}$ is obtained by means of the scaling:

$$\begin{aligned} \left| \textrm{D}_{m}\textrm{S}\right| _\textrm{N}={\displaystyle { \frac{\left| \textrm{D}_{m}\textrm{S}\right| }{\textrm{R}_{m}} }}\quad (m=1,2,...), \end{aligned}$$

(3)

or equivalently,

$$\begin{aligned} {\displaystyle { \left| \text {D}_{m}\text {S}\right| = \frac{\left| \text {S}\right| ^\text {max}}{\left| \text {D}_{m}\text {S}\right| ^\text {max}} }} \quad (m=1,2,...). \end{aligned}$$

(4)

In other words, the normalized derivative spectrum $\left| \textrm{D}_{m}\textrm{S}\right| _\textrm{N}$ is the product of the corresponding unnormalized spectrum $\left| \textrm{D}_{m}\textrm{S}\right|$ and the quotient of the two maximal values (nonderivative/derivative) as $\left| \textrm{S}\right| ^\textrm{max}/\left| \textrm{D}_{m}\textrm{S}\right| ^\textrm{max}\, (m=1,2,...)$. The nonderivative spectrum $\left| \textrm{S}\right|$ is unnormalized. For brevity, both the optimized and unoptimized derivative spectra will be denoted by the same symbol $\left| \textrm{D}_{m}\textrm{S}\right|$. It will be clear from the context to which of the two variants of the spectra a particular reference is made in the presentation of the results. In the figures, no confusion should arise since each sub-figure (panel) will be marked by “Optimized” or “Unoptimized”.

To re-emphasize, throughout the text, the terms ‘optimized’ and ‘unoptimized’ refer to the dFFT with and without the mentioned attenuating or tempering objective functions, respectively. Moreover, whenever the ‘bare’ acronym dFFT is used with no adjective, this will be reserved exclusively for the usual, unoptimized dFFT [4,5,6,7,8,9,10,11,12,13,14,15,16,17]. In the present optimization, the attenuation parameters in the mentioned objective functions are simultaneously adapted or tailored to the encoded FID (through its total acquisition time T) and to each derivative order m separately. The universal analytical form of this explicit twofold dependence will be elaborated in a subsequent related article to be published very soon.

3 Results and discussion

The results to be reported in this Section are the magnitude spectra $\left| \textrm{S}\right|$ for $m=0$ and $\left| \textrm{D}_m\textrm{S}\right|$ for $m\ge 1$, obtained using the encoded FIDs, zero-filled once. The presented reconstructions stem from three entirely different kinds of FIDs encoded by proton MRS. Two of these FIDs encoded with and without water suppression refer to a standard phantom provided by a manufacturer of clinical scanners based on NMR and the associated spectra will be analyzed first (Figs. 1–8). Subsequently, analyzed will be the spectra from the FIDs encoded in vitro from human cancerous ovarian cyst fluid (Figs. 9 and 10).

3.1 Phantom data

For quality assurance of clinical scanner functioning and for assessing the quantification capabilities of signal processors, the most practical are the standard test samples that contain various known substances of fixed concentrations. The main manufacturers of clinical scanners for MRS provide such test samples that are commonly referred to as phantoms. An example of these test samples is the polyethylene Philips phantom for proton MRS [20, 21]. This is a plastic sphere of 10 cm diameter. It is filled with the three main substances, ethyl alcohol (ethanol, EtOH, ${\mathrm{CH_3CH_2OH}},$ 80%, 10 ml), acetic acid (acetate, Ace, ${\mathrm{CH_3COOH}},$ 98%, 5 ml) and methyl alcohol (methanol, MeOH, ${\mathrm{CH_3OH}}$). The solvent is demineralized water ($\mathrm{H_2O}$). For some technical purposes, also added to this mixture are phosphoric acid (${\mathrm{H_3PO_4}},$ 98%, 8 ml), 1 ml 1% arquad solution and copper sulfate (${\mathrm{CuSO_4}},$ 98%, 8 ml). The Philips Manuals [20, 21] do not give the volumes of methanol and water.

Using a 1.5T GE clinical scanner (the Larmor frequency $\nu _{\textrm{L}}=63.863375$ MHz) [22] at the Astrid Lindgren Children Hospital (Stockholm), encoded by proton MRS were two sequences of FIDs, one with and the other without water suppression. Water suppression from the encoded FIDs was made during the measurements using the customary inversion recovery procedure. Encoding was performed by employing the conventional single-voxel proton spectroscopy with the point-resolved spectroscopy sequence (PRESS) [22].

The acquisition parameters were: the short total signal length of $N=$512 data points (0.5 KB), the echo time TE=272 ms, the repetition time TR=2000 ms, the bandwidth BW=1000 Hz, the sampling time $\tau =1/\textrm{BW}=$1 ms, the total duration of each encoded time signal $T=N\tau$=512 ms (the total acquisition time) and the number of excitations NEX=128 (i.e. the number of the encoded transients FIDs). These 128 FIDs were automatically averaged in the scanner to improve SNR. The same specifics apply to encodings with and without water suppression.

With the known content of the Philips phantom for proton MRS, the anticipation is that a spectrum should have ten peaks altogether: one giant resonance for water, seven for ethanol (a triplet and a quartet), one for methanol and one for acetate. In a lower-resolution case, regarding ethanol from the mixture in the phantom, only three singlet peaks would appear. This would be reminiscent of a seminal study of Arnold et al. [23] with pure ethanol using a laboratory-built NMR spectrometer (0.76T). These three peaks are assigned to three different arrangements of ethanol known as the methyl (${\mathrm{CH_3}}$), methylene (${\mathrm{CH_2}}$) and hydroxyl (${\textrm{OH}}$) groups.

The quotients of their peak intensities for the same peak widths should give the ratios of the number of protons (from hydrogen atoms) corresponding to each resonance. Thus, for ethyl alcohol ${\mathrm{CH_3CH_2OH}}$, the sought ratios should be 3:2:1, associated with the molecular groups ${\mathrm{CH_3}}$, ${\mathrm{CH_2}}$ and ${\textrm{OH}}$, respectively. In Ref. [23], the detected peaks were of unequal widths, so that the expected ratios 3:2:1 have been found to hold approximately using the peak areas instead of the peak heights. For the Philips phantom, the OH peak is invisible in the spectrum since it is swamped by the dominant ${\mathrm{H_2O}}$ resonance, which is presently taken to be located at 4.87 ppm (parts per million), corresponding to a temperature of $\mathrm{20^oC}$ according to Ref. [22].

In a higher-resolution case, as in a remarkable, combined experimental-theoretical work of Arnold [24], the ${\mathrm{CH_3}}$ and ${\mathrm{CH_2}}$ groups of ethanol should undergo J-splitting due to the nuclear spin-spin interactions of nonequivalent protons. The protons are identical, of course, but they become ’nonequivalent’ when embedded in different molecules or in different groups of the same molecule. As a result of J-splitting, the spectrum should display the multiplets at the pertinent chemical shifts. Thus, the methyl ${\mathrm{CH_3}}$ and methylene ${\mathrm{CH_2}}$ groups of ethanol ${\mathrm{CH_3CH_2OH}}$ should have two different multiplicities, a triplet (t) and a quartet (q), respectively.

Generally, for two groups of molecules with multiplets, the number of the peaks in one fixed group is predicted by the “$n+1$” rule, where n is the number of hydrogen protons in the other group involved in J-splitting. Moreover, the binomial distribution within the Pascal triangle can predict the ratios of the relative intensities of the individual peaks in the multiplet from the given group. This is possible under the assumption that these peaks have the same widths. For unequal widths, the peak areas should be taken as the relative intensities [23].

Thus, the “$n+1$” rule applied to the two ethanol groups, $\mathrm{CH_3}$ ($n_\mathrm{CH_3}=3$) and $\mathrm{CH_2}$ ($n_\mathrm{CH_2}=2$), implies that the J-splitting pattern for the lineshapes related to the former and the latter molecular group will yield 3 ($n_\mathrm{CH_2}+1=3$) and 4 ($n_\mathrm{CH_3}+1=4$) peaks with the intensity ratios 1:2:1 and 1:3:3:1, respectively. Overall, even with a relatively low static magnetic field (1.5T) in the GE clinical scanner, the spectrum of the Philips phantom should show ten peaks, excluding the ethanol OH peak, which is hidden beneath the large water resonance. In the Philips Manual [20] (p. 51), in Refs. [9, 10] as well as in the present illustrations, these peaks are numbered as follows:

# 1 (singlet, s), water ($\mathrm{H_2O}$),

# 2-5 (quartet, q), methylene protons from the $\mathrm{CH_2}$ group of ethanol (EtOH, $\mathrm{CH_3CH_2OH}$),

# 6 (singlet, s), methanol (MeOH, $\mathrm{CH_3OH}$),

# 7 (singlet, s), acetate or acetic acid (Ace, $\mathrm{CH_3COOH}$) and

# 8-10 (triplet, t), methyl protons from the $\mathrm{CH_3}$ group of ethanol (EtOH, $\mathrm{CH_3CH_2OH}$).

The derivative spectra $(m\ge 1$) reconstructed without and with optimization are shown in Figs. 1–8. Therein, also shown are the nonderivative spectra $(m=0$). Following the procedure outlined in Sect. 2, the derivative spectra are normalized to the maximal values of the magnitude lineshapes within a narrow band around a selected peak. The latter peak is chosen to be the Ace resonance (near 2 ppm) and the maximum of the magnitude lineshape is extracted from the narrow band 1.875$-$2.125 ppm. For consistency in comparisons, the Ace peak height is used for normalization of spectra reconstructed from both the water-suppressed and water-unsuppressed FIDs.

Specifically, Figs. 1–3 and 4–6 are for the FIDs encoded with and without water suppression, respectively. Furthermore, on each of the remaining two illustrations (Figs. 7 and 8) from this sub-Section, juxtaposed are the optimized derivative spectra alone for the FIDs with and without water suppression. Each of these 8 figures contains two columns and four rows. Depicted in different columns of the given figure, derivative spectra are drawn as the full red and blue curves for the results without and with optimization, respectively. Figures 7 and 8 for the optimized derivative spectra on the left and right columns refer to the FIDs without and with water suppression, respectively.

The nonderivative spectra $(m=0)$ are shown as full black curves. They are given merely for referencing, which is useful when monitoring the additional information brought by derivative spectra (unoptimized and optimized alike) for varying order m. The waveforms of the FIDs are not plotted as they can be found in Refs. [9, 10].

The Fourier-based nonderivative and derivative processings by the FFT and dFFT, respectively, belong to shape estimations. For encoded FIDs, the nonderivative estimation $(m=0)$ by the FFT cannot generally reduce an envelope to a set of isolated peaks (components) because there will always be some overlapped resonances. The past experience within MRS demonstrated that the derivative processing by the dFFT failed completely for the realistic FIDs (synthesized with noise [4,5,6] as well as encoded [7,8,9,10,11,12,13,14,15,16,17]). This occurred at $m\ge 3$ with worsening the already low resolution and poor SNR of the FFT.

Our goal with Figs. 1–8 in this sub-Section as well as with Figs. 9 and 10 in the next sub-Sect. 3.2 is to find out whether this troublesome situation can be rescued by the optimized derivative estimation. Furthermore, such a possibility is examined in Figs. 1–3 and 4–6 for the FIDs encoded with and without water suppression, respectively. It is also of interest to inter-relate the reconstructions utilizing only the optimized derivative processing of the FIDs with and without water suppression to verify whether the latter results could compare favorably to the former predictions.

As stated earlier, regarding the ethanol multiplets, a spectrum for a low magnetic field strength cannot resolve the compound structures in $\mathrm{CH_2}$ and $\mathrm{CH_3}$. Therefore, in such a case, each of the lineshapes for the $\mathrm{CH_2}$ and $\mathrm{CH_3}$ molecules would appear as a single broadened peak. Although the magnetic field strength of 1.5T is also relatively low, the FFT spectrum for the Philips phantom has been found in Ref. [20] to be structured (albeit roughly). Therein, the three tallest peaks were seen and assigned to $\mathrm{CH_3}$, Ace and $\mathrm{CH_2}$ near 1, 2 and 3.6 ppm, respectively. Moreover, some hints of the shouldered sub-structures emerged around the chemical shifts of $\mathrm{CH_2}$ (peaks # 2-5) and $\mathrm{CH_3}$ (peaks # 8-10) as being superimposed on an elevated background baseline. This has also been encountered in Refs. [9, 10] and will presently be illustrated, as well.

3.1.1 Optimized and unoptimized derivative spectra from FIDs encoded with water suppression

This sub-Section deals with spectra reconstructed from FIDs encoded by proton MRS from Philips phantom [20] with water suppression. In Fig. 1, the nonderivative envelopes are shown with (a) and without (e) the presence of the residual water resonance. On (a), the latter peak is still more than about 4 times stronger than the Ace resonance, which is the most intense structure in the remaining spectrum. On (e), the resonances #2-10 appear more or less clearly, but resolution and SNR are low so that this envelope cannot be quantified.

Panels (b-d, f-h) in Fig. 1 are on derivative estimations $(m=1-3)$ without (b-d) and with (f-h) optimization. Relative to the nonderivative spectrum (e, $m=0$), both resolution and SNR are greatly improved in the first derivative spectra $(m=1)$ on (b, f). For instance, the Ace peaks are narrower on (b, f) than on (e). Resolution enhancement on (b, f) is most evident in the separation of the individual peaks within each of the two multiplets, $\mathrm{CH_2}$ and $\mathrm{CH_3}$. Thus, the peaks # 2-5 in $\mathrm{CH_2}$ and # 8-10 in $\mathrm{CH_3}$ are so distinctly split apart that the dips between them descend very close to background baseline.

On (b, f), the peak intensity ratios 1:3:3:1 in $\mathrm{CH_2}$ and 1:2:1 in $\mathrm{CH_3}$, predicted by the first-order quantum-mechanical perturbation theory [24], are satisfied only crudely. The actual inequalities seen in the peak heights from $\mathrm{CH_2}$ (# 3, 4) and from $\mathrm{CH_3}$ (# 8, 10) are partially caused by the fact that the bottoms of these resonances are bumpy as they lie on top of the uneven baseline. The other reason is in the slight inequality of the peak widths so that the peak areas should be used to check more accurately the ratios 1:3:3:1 in $\mathrm{CH_2}$ and 1:2:1 in $\mathrm{CH_3}$ as has been done in Ref. [24].

It is then evident that already the spectra for $m=1$ (b, f) exhibit a net superiority of derivative relative to nonderivative (e, $m=0$) estimations. The real issue is, however, whether this encouraging result at the onset of derivative processing $(m=1)$ would persist for $m\ge 2$. This is important in practice given that the overall quality of either of the two reconstructions for $m=1$ (b, f) is insufficient to finalize signal processing at this early stage of derivative estimations.

The first derivative spectra without (b) and with (f) optimization are, in fact, very similar, although the former is slightly noisier than the latter. This is obvious from comparing the left hand sides of the bottom parts of e.g. the Ace peaks. Furthermore, noise on (b) distorts and incompletely masks the peak # 2 in $\mathrm{CH_2}$. The baseline on the far left on (b) toward 4.25 ppm is about twice higher than that on (f). Still, the improvement by optimization on the level of merely the first derivative spectra (f, $m=1$) is only minor.

The second derivatives ($m=2$) paint quite a different picture when it comes to discriminating between estimations without (c) and with (g) optimization. In the unoptimized estimation, the noisy pattern, hinted on (b) for $m=1$, becomes notably expressed on (c). After the Ace peak, the baseline is considerably more lifted on (c) compared to (b). For example, at 4.25 ppm, the end of the baseline is about three times higher on (c) than on (b). Further, to the right of $\mathrm{CH_3}$, toward 0.5 ppm, the baseline on (c) is slightly higher than on (b).

Notable lifting of the baseline on (c) is observed at the higher frequencies from the band 0.5$-$4.25 ppm because these are located closer to the water peak at 4.87 ppm. Although the spectral wiggles (the remnants of noise) are somewhat reduced on (c) relative to (b), the peak separations in the multiplets of either $\mathrm{CH_2}$ or $\mathrm{CH_3}$ are deteriorated when comparing $m=2$ with $m=1$ on the left column of Fig. 1. Further, on (c), the elevated baseline fills up the lower parts of the dips in $\mathrm{CH_2}$ (between # 2 and 3 as well as # 4 and 5) and in $\mathrm{CH_3}$ (between # 8 and 9 as well as # 9 and 10).

However, none of these disadvantages exist in the optimized spectrum (g, $m=2$). Quite the contrary, passing from $m=1$ (f) to $m=2$ (g), the baseline becomes flatter throughout the analyzed band (0.5$-$4.25 ppm) on account of the squeezed and lowered tails outside the peaks # 2, 6-8 and 10. This results in an enhanced resolution on (g) with respect to (f). Thus, for the second derivatives ($m=2$) between the unoptimized (c) and optimized (g) estimations, the latter processing emerges as more adequate.

As to the spectra from the third derivatives, they are shown on (d, h). The unoptimized spectrum is worsened for $m=3$ (d) compared to $m=2$ (c). The background baseline is higher on (d) than on (c), as seen by looking at the beginning (0.5 ppm) as well as at the end (4.25 ppm) of the displayed chemical shift range. The formerly observed peaks # 2 and 5 in $\mathrm{CH_2}$ are now degraded to the mere shoulders leaning to the sides of their taller neighbors (# 3 and 4, respectively). In $\mathrm{CH_2}$, lifted to the higher levels are the dips between the peaks # 2 and 3 (as well as # 4 and 5). A similar observation also holds true in $\mathrm{CH_3}$ for the dips between the peaks # 8 and 9 (as well as # 9 and 10). With such a trend, the gap between the heights of e.g. the peaks # 9 and 10 is increased when juxtaposing $m=3$ (d) to $m=2$ (c).

Fig. 1

Spectra in the magnitude mode. Nonderivative spectrum ($m=0$) in the wider (a) and zoomed (e) chemical shift bands 0.5$-$5.5 ppm and 0.5$-$4.25 ppm with the water peak included and excluded, respectively. Normalized derivative spectra ($1\le m\le 3$) without (b-d) and with (f-h) optimization. Usage of the water suppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Fig. 2

Spectra in the magnitude mode. Nonderivative spectrum ($m=0$) in the wider (a) and zoomed (e) chemical shift bands 0.5$-$5.5 ppm and 0.5$-$4.25 ppm with the water peak included and excluded, respectively. Normalized derivative spectra ($4\le m\le 6$) without (b-d) and with (f-h) optimization. Usage of the water suppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Fig. 3

Spectra in the magnitude mode at 0.5$-$4.25 ppm with the water peak excluded. Nonderivative spectrum (a, $m=0$). Normalized derivative spectra (b-h, $1\le m\le 6$) with optimization. The red curve on (h) is the derivative spectrum $(m=6)$ without optimization. Usage of the water suppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

In contrast, the optimized spectrum for $m=3$ (h) is further improved relative to its predecessor for $m=2$ (g). On (h), the whole baseline is flatter and lower than on (g). This is reflected particularly on the sides of the Ace and $\mathrm{CH_3}$ resonances. From a fuller inspection of the juxtaposed lineshapes on (g) and (h), it follows that the optimized processing gradually enhances its performance with the increased derivative order m. Moreover, on (h, $m=3$), there is a huge improvement in resolution and SNR over (e, $m=0$). Most importantly, a comparison between the unoptimized (d) and optimized (h) estimations for $m=3$ shows the superiority of the latter over the former processing.

The just performed analysis for $0\le m\le 3$ is extended to $4\le n\le 6$ in Fig. 2. The same nonderivative spectrum ($m=0$) from Fig. 1 is replotted in Fig. 2 on (a, e). Scrolling down through (b-d) in Fig. 2, it appears that the deteriorating process in the unoptimized estimation, initiated already for $m=2$ in Fig. 1, is getting even worse. For example, the visibility of the shoulders # 2 and 5 in $\mathrm{CH_2}$ for $m=3$ in Fig. 1 is now diminished for $m=4$ (b) in Fig. 2.

Eventually, these shoulders are lost for $m=5$ (c) and $m=6$ (d) so that from the sought four resonances in $\mathrm{CH_2}$ only two rough peaks are observed. A similar spectral impoverishing also occurs in $\mathrm{CH_3}$, where the still visible tops of the peaks # 8 and 10 for $m=4$ (b) merge to a shoulder and a bulge for $m=5$ (c), respectively, only to be washed out from the spectrum for $m=6$ (d). Hence, the original triplet in $\mathrm{CH_3}$ (e) is reduced to a singlet for $m=6$ (d).

The peak widths on (b-d) in Fig. 2 broaden gradually when stepping up with the derivative order from $m=4$ to 6. Therein, the higher derivatives in the unoptimized processing increase the overlaps in the adjacent resonances and this significantly decreases resolution. Also the SNR is dramatically lowered on (b-d). Herein, the baseline becomes more elevated with augmentation of m. Thus, at e.g. 4.25 ppm, the baseline surpasses the tallest peak height in $\mathrm{CH_2}$ on (d). Moreover, also at 4.25 ppm, the baseline for $m=6$ (d) is about 2.5 higher than for $m=0$ (e). This finding and the loss of four resonances out of the sought nine peaks demonstrate that the performance of the unoptimized derivative estimation (d) is worse than in the nonderivative processing (e).

On the other hand, the optimized derivative processing in Fig. 2 met with success. Herein, panels (f-h) testify to the accuracy and reliability of the results from the optimization. Observe the steadily maintained flatness of the baseline at $4\le m\le 6$ and the improving SNR. Moreover, high resolution is achieved for $m=6$ (h), where the formerly overlapped peaks on (e) become decoupled. The overall outcome is an excellent visualization and delineation of the individual resonances particularly on (h) with $m=6$.

This enables a reasonably accurate quantification since the approximate values of the resonant chemical shifts and the stabilized peak heights can be read off from the lineshape on (h) or printed as an output from the same computer program. The peak areas too can approximately be estimated on (h) by integration. The peak area is proportional to the abundance (concentration) of the given molecule in the chemical compound. The peak widths (proportional to the reciprocal to the spin-spin relaxation times) are deduced from the peak areas and peak heights of the reconstructed bell-shaped spectral profiles. It then follows from Fig. 2 that the optimized derivative processing is capable of yielding the adequate estimates of the sought peak parameters (peak positions, widths, heights, areas) for a relatively low derivative order ($m=6$) on (h). Thus, despite starting as a shape estimator, the optimized derivative processor ends up by becoming a parameter estimator.

In Figs. 1 and 2, we focused on comparing the unoptimized and optimized spectra. Now, when the evidence-based superiority of the latter to the former processing is established, Fig. 3 is offered to summarize the optimized spectra alone for $0\le m\le 6.$ This permits realizing in full how actually good the overall merit of optimization is by viewing its spectra in a single figure. The nonderivative source spectrum (a, $m=0$) shows e.g. the Ace peak as a wide asymmetric resonance with an elevated bulky bottom portion and the extended tails on each side of the center. Compare this pattern to the thin, well-delineated, symmetric Ace resonance with basically no tails for $m=6$ (g). A similar progress is also evident in the $\mathrm{CH_3}$ triplet and the $\mathrm{CH_2}$ quartet when proceeding from $m=0$ (a) to $m=6$ (g). On (a) for $m=0$, both of these multiplets are very crude, showing only some glimpses of what are supposed to be the peaks # 2 and 5 in $\mathrm{CH_2}$ as well as # 8 and 10 in $\mathrm{CH_3}$. Such an obscurity went through a systematic sharpening for the increased m on (b-g).

Interestingly, in the optimized spectra from Fig. 3, between the triplet and quartet, the latter is stabilized first. The stabilization of the $\mathrm{CH_2}$ quartet is hinted already for $m=2$ (c) and is maintained steadily thereafter with practically no change on (e-g) for $m=4-6$. On the other hand, the $\mathrm{CH_3}$ triplet has a slower convergence and attains its fully stabilized lineshape at a higher derivative order, i.e. $m=6$ (g). Overall, it is gratifying to record the robustness and steadiness of the optimized estimation in Fig. 3 with the concluding reconstruction for $m=6$ (g).

The spectrum for $m=6$ (g) is identical to its counterpart for $m=7$ (not shown to avoid clutter). Instead, the spectra from the unoptimized and optimized estimations for $m=6$ are plotted together on (h). We have already encountered this comparison on panels (d) and (h) in Fig. 2. However, on (h) in Fig. 3, the repeated comparison on the same panel is also deemed instructive as it displays a dramatic improvement in resolution and SNR with the optimization. On (h), observe especially the $\mathrm{CH_2}$ and $\mathrm{CH_3}$ structures that are clearly separated by the optimized estimation, but destructured by the unoptimized processor.

Fig. 4

Spectra in the magnitude mode. Nonderivative spectrum ($m=0$) in the wider (a) and zoomed (e) chemical shift bands 0.5$-$5.5 ppm and 0.5$-$4.25 ppm with the water peak included and excluded, respectively. Normalized derivative spectra ($1\le m\le 3$) without (b-d) and with (f-h) optimization. Usage of the water unsuppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Fig. 5

Spectra in the magnitude mode. Nonderivative spectrum ($m=0$) in the wider (a) and zoomed (e) chemical shift bands 0.5$-$5.5 ppm and 0.5$-$4.25 ppm with the water peak included and excluded, respectively. Normalized derivative spectra ($4\le m\le 6$) without (b-d) and with (f-h) optimization. Usage of the water unsuppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Fig. 6

Spectra in the magnitude mode. Nonderivative spectrum (a, $m=0$). Normalized derivative spectra (b-h, $1\le m\le 6$) with optimization. The red curve on (h) is the derivative spectrum $(m=6)$ without optimization. Usage of the water unsuppressed FIDs encoded by proton MRS (1.5T). Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

The achievement of the optimized processing in Fig. 3 cannot be overstated, especially given that the nonderivative estimation is uninterpretable on (a, $m=0$). For instance, panel (g, $m=6$) gives not only the well-resolved spectrum, useful for visualizations, but also readily interpretable in a quantitative manner. It is noteworthy that the optimized processor is capable of making such a critical advance consisting of a switch from a qualitative shape estimation to a quantitative parameter estimation, while advantageously preserving the computational expedience of the Cooley and Tukey fast algorithm [3].

3.1.2 Optimized and unoptimized derivative spectra from FIDs encoded without water suppression

Spectra for the averaged FID generated from the 128 time signals encoded without water suppression are displayed in Figs. 4–6. These figures are made in the same way as Figs. 1–3. The first striking aspect of Fig. 4 is seen in the nonderivative spectrum ($m=0$) with (a) and without (e) inclusion of the intact water peak. On (a), the height of the water resonance is about 525 au, relative to about 4.5 au for the residual water peak discussed in the part 3.1.1 of sub-Sect. 3.1. On (e), the almost everywhere smooth spectral lineshapes fall off steadily with decreasing frequencies. Such a spectrally opaque pattern is dictated by the large water peak (a) whose far-extending tail distorts all the remaining structures throughout the entire Nyquist interval, including the band of interest, 0.5$-$4.25 ppm.

The closer the water peak (4.87 ppm) to a resonance, the stronger the distortion of that resonance. This explains why the lineshape on (e) incessantly rises when going from 0.5 to 4.25 ppm. For the same reason, $\mathrm{CH_2}$ near 3.6 ppm is completely invisible. Although, Ace and $\mathrm{CH_3}$ resonances are positioned further away from the water peak, their presence is nevertheless manifested merely through some slight deviations from the overall smooth lineshapes on (e). The numbers of the invisible peaks are given on (a, e) around their expected locations, just for an orientation.

The derivative spectra for $m=1$ (b, f) in Fig. 4 begin to uncover some of the structures formerly hidden on (e). For instance, the unoptimized estimator (b) senses that there might be a spectral content in the proximity of $\mathrm{CH_2}$, Ace and $\mathrm{CH_3}$. Nevertheless, this processor is unable to make such a sensing transparent, other than producing some wiggles, ripples and spikes. By comparison, the optimized estimator $m=1$ (f) clearly propels $\mathrm{CH_3}$, Ace and MeOH into prominence, despite the presence of a still high baseline. Moreover, peaks # 3-5 in $\mathrm{CH_2}$ are also detected.

Peak # 2 is the closest structure to the water resonance (4.87 ppm) and that is why it persistently rides high as a mere shoulder on the ever climbing lineshape with increasing frequencies. The optimized processor (f) is capable of extracting eight (# 3-10) out of the sought nine peaks. The reason is that the employed optimization, in a wave-packet type fashion, shrinks the formerly wide base of the water peak by appreciably lowering its tail. The difference between (b) and (f) is considerable. Thus, already this very onset of derivative processing $(m=1)$ paves the directionality of the relative performance of the unoptimized and optimized estimators.

The trend initiated on (b, f) for $m=1$ is continued for $m=2$ on (c, g) in Fig. 4. On (c), all that the unoptimized processor does is to somewhat reduce the ripples near the chemical shifts 1.1, 2.1 and 3.6 ppm around $\mathrm{CH_3}$, Ace and $\mathrm{CH_2}$, respectively. The Ace peak appears as a small asymmetrical structure lying on the baseline, which is higher for $m=2$ (c) than for $m=1$ (b). However, the optimized estimator for $m=2$ (g) progressed immensely by reference to $m=1$ (f). On (g), the baseline is flattened at 0.5$-$3.25 ppm, albeit with some tiny superimposed undulations. The only residual portion of the high baseline formerly present on (f) is a low-lying rippled strip in a small sub-band 3.75$-$4.25 ppm on (g) within the whole band $\nu \in [0.5,4.25]$ppm.

The implication is that the optimized processor for $m=2$ on (g) is effective in localizing the water peak to a narrow range around 4.78 ppm with its tail being mostly cut-off. Consequently, all the peaks # 2-10 are nicely delineated on (g). The largest improvement, when relating $m=2$ (g) to $m=1$ (f), is within $\mathrm{CH_2}$ whose four peaks descended close to the chemical shift axis in the former panel. In particular, the former (f) highly elevated shoulder # 2 developed into a distinct peak.

Concentrating on $m=3$ (d, h) in Fig. 4 leaves little doubt as to where the predictions by the unoptimized and optimized estimations would be heading. Thus, the unoptimized processor (d) brings no benefit regarding resolution. Nothing else is improved either on (d) with respect to (c). In particular, the SNR is decreased since the baseline is higher for $m=3$ (d) than for $m=2$ (c). In fact, the background baseline keeps on rising when traversing from (b) to (d). For example, at 4.25 ppm, it is about 11 au (b), 14 au (c) and 16 au (d). This trend smears down and diminishes the weaker spikes and wiggles at the locations of $\mathrm{CH_2}$ and $\mathrm{CH_3}$.

In contradistinction, for $m=3$ (h), the optimized estimator has gone through some notable advances. This is most obvious for SNR in the vicinity of $\mathrm{CH_2}$. Herein, at e.g. 4.25 ppm, the baseline reaches the height of about 0.2 au (h) compared to 1 au (g). Nevertheless, further improvements are needed, especially for resolution to attain a deeper splitting, particularly in $\mathrm{CH_3}$ (between # 8 and 9 as well as # 9 and 10).

Figure 5 for $4\le m\le 6$ finalizes the presentation of the reconstructions by the unoptimized and optimized estimations using the water-unsuppressed FID. In the unoptimized processing, the mentioned tendency of a near-disappearance of all the weaker structures around 1.1 in $\mathrm{CH_3}$ and near 3.6 ppm in $\mathrm{CH_2}$ is intensified on (b-d). By the time the derivative order $m=6$ (d) is reached, almost nothing is left from the ripples in $\mathrm{CH_2}$ and, moreover, $\mathrm{CH_3}$ is also practically gone.

This occurs despite the relatively small changes of the maximal values (between 17 and 18 au) of the baseline at 4.25 ppm on (b-d). The height of the delineated Ace peak on (b-d) is maintained at about 5 au because the whole lineshape is normalized to this resonance, as stated earlier. Compared to the totally uninformative nonderivative spectrum for $m=0$ (e) it is conclusive that the unoptimized estimator even for $m=6$ (d) has nothing meaningful to give, except to smooth out the artifacts (ripples, wiggles, spikes).

This is in stark contrast to the outcome secured by the optimized estimations (f-h). In particular, the nonderivative (e, $m=0$) and the optimized derivative (h, $m=6$) spectra bring in bold relief the full effect of the advantage of the latter with respect to the former reconstruction. On (f-h), the baseline steadily remains at the very low level, almost everywhere close to the chemical shift axis. Herein, all the peaks are systematically refined. For $m=6$ (h), both resolution and SNR achieved their sufficiently good quality to facilitate quantitative interpretations in the same manner as discussed with Fig. 2h for the water-suppressed FID.

Utilizing the water-unsuppressed FID, the nonderivative (a, $m=0$), the optimized derivative (b-g, $1\le m\le 6$ and the unoptimized derivative (h, $m=6$) spectra are summarized in Fig. 6. It makes an interesting visualization to depict in one graph all the results of the optimized processing. The nonderivative spectrum (a, $m=0$) has no usable content, while the finalization (g, $m=6$) of the optimized estimation provides the quantitatively interpretable information. This latter finding is especially significant when contrasted to the corresponding unoptimized outcome on (h, $m=6$).

The unoptimized spectrum for $m=6$ is divided by a factor of 4 to scale down the whole lineshape so as to encapsulate it into the frame of panel (h). This amply informs about the high level of the baseline in the unoptimized envelope. The most striking feature for $m=6$ (h) is the proper full component content provided by the optimized processor compared to the spectral paucity of the envelope in the unoptimized estimation. Specifically, on (h) at 0.5$-$4.25 ppm, in the optimized spectrum, nine well separated resonances are detected relative to only a minuscule Ace peak in the unoptimized lineshape.

3.1.3 Optimized derivative spectra from FIDs encoded with and without water suppression

Both of the remaining plots (Figs. 7 and 8) from this sub-Section compare the optimized spectra for the FIDs without (a-d) and with (e-h) water suppression. While the nonderivative spectrum on (a) in Fig. 7 is structureless, its counterpart on (e) is structured (albeit roughly). The disparity between (a) and (e) is so huge that, at first, it would seem hopeless to try to extract some meaningful information from any spectra using the raw, water-unsuppressed FID.

However, such a skepticism is eliminated by the optimized derivatives (b-d). This is seen in an enormous jump from (a) for $m=0$ to (b) for $m=1$. Namely, already the optimization for $m=1$ (b) is able to pull out a substantial amount of information from the water-unsuppressed FID. This would not be apparent in the time domain of encodings. It, however, becomes visible in the complementary, frequency domain, where several recognizable peaks appear to emerge on (b) from their former (a) hidings.

Especially in the proximity of $\mathrm{CH_2}$ and MeOH, there is practically nothing superimposed on the smooth baseline on (a, $m=0$). Yet, the optimization for $m=1$ (b) successfully segments out that part of the baseline by sub-structuring it into one shoulder (# 2) and four peaks (# 3-6). These peaks may look rough and deformed on (b), but they quickly get straightened up for $m=2$ (c) and $m=3$ (d) to become the well-shaped resonances. The same goes for the other resonances (# 7-10).

Reconstruction of spectra with the water-suppressed FID should be more manageable. This is evidenced already in the nonderivative envelope (e, $m=0$), which is reasonable. Therein, a bulky background lifts and distorts all the resonances, but at least some of them are still visible. Therefore, as we pass to derivative estimations, the first task for the optimization is to diminish the background so as to allow all the obscure resonances to pop up in a more distinct way, especially around $\mathrm{CH_2}$ and MeOH. That is precisely what is accomplished on (f, $m=1$). From then and on, the further improvements are seen on (g, $m=2$) as well as on (h, $m=3$).

Fig. 7

Spectra in the magnitude mode at 0.5$-$4.25 ppm with the water peak excluded. Nonderivative spectra ($m=0$) without (a) and with (e) water suppression in the FIDs. Normalized, optimized derivative spectra ($1\le m\le 3$) without (b-d) and with (f-h) water suppression in the FIDs. Usage of the FIDs encoded by proton MRS (1.5T) with and without water suppression. Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Fig. 8

Spectra in the magnitude mode at 0.5$-$4.25 ppm with the water peak excluded. Nonderivative spectra ($m=0$) without (a) and with (e) water suppression in the FIDs. Normalized, optimized derivative spectra ($4\le m\le 6$) without (b-d) and with (f-h) water suppression in the FIDs. Usage of the FIDs encoded by proton MRS (1.5T) with and without water suppression. Ordinates are in arbitrary units (au), abscissae are in parts per million (ppm). For details, see the main text (Color online)

Comparing the two columns in Fig. 7, the most striking are the panels (b) and (f), particularly within the band 3.5$-$4.25 ppm, which contains $\mathrm{CH_2}$ and MeOH. It is at these chemical shifts that the large unsuppressed water peak at 4.87 ppm, through its wide tail, exerts the strongest distortions, as observed on (b). Such severe spectral deformations are absent from (f) due to processing the water-suppressed FID. It then comes as no surprise that resolution and SNR are better on (g, h) than on (c, d), respectively. However, what is surprising is that the spectrum on (d) with the water-unsuppressed FID could be of a qualitatively comparable value to its counterpart on (h) with the water-suppressed FID.

Still, as to the finer details, some significant quantitative differences between (d) and (h) are transparent. Thus, e.g. the MeOH peak is wider on (d) than on (h). Also, the dips between the peak pairs # (2, 3) as well as # (3, 4) in $\mathrm{CH_2}$ are more elevated on (d) than on (h). The same remark applies to the peak pairs # (8, 9) as well as # (9, 10) in $\mathrm{CH_3}$. Note also that on the left and right columns, the heights and widths of the peaks are unequal because of employing two different objective functions in the optimization for the FIDs encoded without and with water suppression.

Figure 8 ($4\le m\le 6$) on the optimized derivative spectra furthers the comparisons between the reconstructions based of the FIDs without and with water suppression. The gain is minimal for $m=4$ (b, f) compared to the associated finding in Fig. 7 for $m=3$ (d, h), respectively. Eventually, the definite improvements in Fig. 8 are made for $m=5$ (c, g) and most notably for $m=6$ (d, h). Therein, in spite of employing the two drastically different FIDs, without (d) and with (h) water suppression, the obtained resolution and SNR in the optimized derivative spectra for $m=6$ are very similar for both kinds of encoded time signals.

The latter findings are of significant practical importance. Namely, we likewise found that the adopted optimization is also equally successful for the FIDs encoded with and without water suppression by in vivo proton MRS at 3T from a borderline ovarian tumor in a patient. These two types of FIDs have been kindly provided to us by Professor Eva Kolwijck (from the group of Professor Ron Wevers, Radboud University, Nijmegen, The Netherlands) and the corresponding optimized derivative spectra will be reported shortly.

There are two principal reasons to operationalize the presently employed optimization in the clinic with MRS tumor diagnostics. First, encoding FIDs from patients with partial water suppression extends the long examination time which, in turn, impacts adversely on the cost-effectiveness of MRS in the clinic. This hampers the wider routine usage of MRS in hospitals, not only for diagnostics, but also for follow-up and screening of patients. Second, the existing water-suppression techniques applied during the FID encodings unavoidably distort all the remaining peaks in the spectrum and lead to some unknown errors on the diagnostically relevant quantitative information (e.g. metabolite concentrations).

These spectral distortions are not evenly distributed throughout the given frequency band of diagnostic interest. The closer the given metabolite to the residual water peak, the more severe the distortion. As a result, the metabolite concentration ratios too might be prone to errors. Such ratios with certain cut-points are frequently used to differentiate between the healthy and diseased tissues. This could compromise the diagnostic accuracy and clinical reliability of MRS in the health care systems. Therefore, there is every interest to pursue further investigations aimed at wide-spread applications of MRS with water-unsuppressed FIDs and to analyze the results by the concept of the optimized derivative estimation.

3.2 Patient data

The just performed benchmarking of the optimized derivative processing on the standard Philips phantom for FIDs encoded by proton MRS is essential because the content of the examined specimen is known. A further valuable test of the same methodology would be to show the patient data associated with the FIDs encoded by proton MRS from some specimens of unknown content. To that end, this sub-Section is concerned with processing patient data using the FIDs encoded by in vitro MRS from the excized samples bathed in the $\mathrm{D_2O}$ solvent [25]. While the analyzed phantom data correspond to 1.5T, the patient data encodings were at a much stronger magnetic field (14.1T). In the latter case, only the selected representative illustrations will be given here, and the more detailed results will be published soon in a separate report.

Encodings of the FIDs by in vitro proton MRS in the quadrature mode were made by the authors of Ref. [25] who kindly provided to us their measured data. They employed a Bruker 600 MHz spectrometer to encode the FIDs from the samples of human ovarian cyst fluid. We were given the FIDs related to two patients diagnosed histopathologically to have benign (serous cystadenoma) and malignant (serous cystadenocarcinoma) ovarian cysts. For brevity, only the malignant ovarian cyst fluid data will presently be analyzed.

Fig. 9

Spectra in the magnitude mode at chemical shifts 4.25$-$5.1 ppm (a) and 0.975$-$1.6 ppm (b) with the water peak included and excluded, respectively. Normalized fourth derivative spectral representations (a,b,d-f, $m=4$) with optimization. Nonderivative spectrum (c, $m=0$). In the narrow band around 1.0 ppm, the two smaller peaks are assigned to the doublet of isoleucine (Iso). Usage of the water suppressed FIDs encoded by proton MRS (14.1T). The spectral intensities on the ordinates in au and abscissae as chemical shifts in ppm. For details, see the text (Color online)

The encoding parameters of the received FIDs were: the Larmor frequency $\nu _\textrm{L}=600\,\textrm{MHz}$ $(B_0\approx 14.1$T), the long total signal length of $N=16384$ (16 KB), the echo time TE=30 ms, the repetition time TR=1200 ms, the bandwidth BW=6667 Hz, the sampling time $\tau =1/\textrm{BW}\approx$0.15 ms, the total acquisition time $T=N\tau =2.46\textrm{s}$ (the total duration of each encoded FID transient) and the number of excitations NEX=128. Encoding was done with water suppression and the residual $\mathrm{H_2O}$ resonance was presaturated during the relaxation delay of 1200 ms.

For a partial noise suppression in the time domain, the encoded FIDs were averaged. Before processing, we zero-filled once this averaged FID. Similarly to the phantom data in Sect. 3.1, the reconstructed magnitude spectra in the unoptimized and optimized derivatives for the patient data were normalized also to the maximal values of the magnitude lineshapes within the small band 1.875$-$2.125 ppm around the Ace peak (for its prominence in this band of the whole spectrum, see Ref. [17]).

3.2.1 Optimized derivative spectra in sequential estimation and visualized quantification

Figure 9 shows the optimized derivative spectra (a, b, d-f, $m=4$) in the magnitude mode reconstructed using the just specified FIDs encoded from samples with ovarian cancerous cyst fluid. Shown also is the nonderivative envelope (c, $m=0$). Among the derivative lineshapes, the first shown are the lactate quartet (a) and doublet (b). The subsequent panels focus on the chemical shifts around the lactate doublet, including several other doublets such as alanine (Ala), threonine (Thr), three hydroxybutyric acid (3-HB) or $\beta -$hydroxybutyric acid ($\beta -$HB), valine (Val) and isoleucine (Iso). The quartet (a) and doublet (b) of lactate are fully resolved down to the chemical shift axis. Hence, these multiplet resonances are quantifiable e.g. by a numerical quadrature due to the well defined integration boundaries. Because of the lactate doublet dominance, there is a huge dynamic span of the intensities on the ordinate preventing to discern any other resonance on (b).

However, once the lactate doublet from (b) has been quantified, it is possible to extract its lineshape (together with a small, immediate surrounding) from the full spectrum. This would permit visualizing and quantifying the remaining peaks. A procedure of this kind goes under the name of ’sequential quantification’, which has recently been put forward [13]. With such a goal, three different versions of sequential quantification are systematically illustrated on (d-f).

For the nonderivative envelope ($m=0$), the ordinate scale is reduced from 4500 au (b) to 420 au (c) by truncating most of the upper part of the lactate doublet. This is of no further concern since the two lactate peaks have already been quantified on (b). In this way, the other resonances can emerge more visibly from their former embedding in the background baseline.

However, they are noted to ride on the high tails of the lactate doublet. Thus, no quantification therein can be made because the integration limits are undefinable for a numerical quadrature to unequivocally determine the peak areas. Various adjustments, performed usually by employing some fitting techniques with e.g. a linear combination of Lorentzians (or Gaussians or Voigtians), would not help either as no reliable results could be obtained due to the unknown contributions from the elevated background baseline. The customary attempts to reduce the baselines producing an artificial profile from fitting by a spline polynomial of a given degree (usually 3 or 4) are prone to uncontrollable errors since the subsequent subtraction of spectra would inevitably wash out some of the true resonances.

Fig. 10

Spectra in the magnitude mode at chemical shifts 0.975$-$1.6 ppm outside the water peak. Normalized derivative spectra with (a-d, $m=1-4$) and without (e, $m=4$) optimization. In the narrow band around 1.0 ppm, the two smaller peaks are assigned to the doublet of isoleucine (Iso). Usage of the water suppressed FIDs encoded by proton MRS (14.1T). The spectral intensities on the ordinates in au and abscissae as chemical shifts in ppm. For details, see the text (Color online)

By contrast, these problems are simultaneously solved in the optimized fourth derivative spectra (d-f, $m=4$). Herein, all the resonances are fully resolved down to the chemical shift axis. It is seen that the individual metabolite lineshape profiles are sharp, isolated and situated on the completely flattened background baseline. Therefore, every such resonance (left and right of the lactate doublet) is amenable to accurate quantification on (d-f). There are three equivalent illustrations of sequential quantification with different displays of the lactate doublet intensities:

• (d) mostly cut off from its top,

• (e) fully shown by reference to the right-hand ordinate with the common abscissa, and

• (f) divided by a factor of 25 to be shown as intact on the same ordinate alongside all the other resonances.

3.2.2 Stabilization of optimized derivative spectra

Figure 10 illustrates the stabilization of the optimization as a function of the gradually increased derivative order for $m=1$ (a), $m=2$ (b), $m=3$ (c) and $m=4$ (d). In particular, a systematic and dramatic line narrowing by the optimization is seen in the widest resonance, the lactate doublet, when passing from $m=0$ (Fig. 9c) to $m=1$ (Fig. 10a). This latter achievement is coupled with a near complete suppression of the long-extending tail of the lactate double peak by the optimized first derivative spectrum. As a result, all the other resonances are observed as lying on a flat background baseline. Some small baseline undulations are observed around the alanine doublet (1.50–1.52 ppm) and at frequencies near the doublets of isoleucine and valine (1.0–1.02 ppm), where the former refers to the small two peaks. Still, the dip between the two lactate peaks is highly elevated possibly by the presence of macromolecules (lipids, proteins).

Most of the remnants of the macromolecules are practically washed out from the optimized second derivative spectrum (b, $m=2$). Yet, a small amount of macromolecules is still present beneath the lactate doublet. Also, the bottom of the alanine doublet begins to show the incompletely resolved two small satellites. Moreover, within 1.0$-$1.02 ppm, one of the two small peaks of the isoleucine doublet is not fully resolved either.

These circumstances necessitate the third optimized derivative (c, $m=3$). This time, the two peaks in the lactate doublet are entirely resolved. On the other hand, one of the small alanine satellite eventually needs a refinement and the like comment also applies to one of the two peaks from the isoleucine doublet.

Although arguably slight, such remaining ameliorations are calling for the optimized fourth derivative (d, $m=4$). This step actually completes the lineshape stabilization process versus the increased derivative order m. The word ’stabilization’ refers to the situation in which all the physical resonances are wholly resolved very close to the chemical shift axis. The systematics from $m=1$ (a) to $m=4$ (d) help clarify the fast convergence of stabilization within the optimization concept. It demonstrates the robustness of sequential quantification by this type of data analytical methods despite the fact that the whole procedure began by a shape estimation, which is qualitative. This is to be contrasted to the unoptimized derivative estimation, which even in the fourth order (e, $m=4$) on Fig. 10 fails to stabilize and, in fact, loses all the physical information.

Overall, the illustrations in Figs. 9 and 10 are focused on the optimization for sequential processing and visualized quantification. This is deemed to be helpful to the physician’s interpretation of the MRS examination of the patient. The clinical examples of the optimized derivative spectra shown in this sub-Section are notable given the insufficiency of the nonderivative estimation (c, $m=0$) on Fig. 9 and the breakdown of the unoptimized derivative processing (e, $m=4$) on Fig. 10. Similarly, the nonderivative envelope ($m=0$) in the FFT is unreliable even for the FIDs encoded with the high-resolution magic angle spinning (HRMAS) variant of MRS. Thus, e.g. employing the FIDs encoded at a Bruker 600 MHz ($\approx$14.1T) spectrometer by HRMAS MRS from cervical cancer, the reconstructed FFT spectra were abundant with unresolved resonances, superimposed on unidentifiable macromolecular structures embedded in the rolling background baseline [26].

To re-emphasize, one of the practical benefits of the present advance in signal processing for medical diagnostics by single-voxel MRS is that the same Cooley-Tukey computations [3] from the standard FFT can also be used in the optimized derivative estimations. With this combination, the well-resolved isolated metabolite resonances of high SNR would show up on the screen. This would facilitate a visualized quantification in optimized derivative spectra (yielding e.g. metabolite concentrations from the estimated peak areas), while the patient is still in the scanner. Moreover, a multi-voxel scanning of the adjacent tissues, followed by the quickly provided high-resolution component spectra in the optimized derivatives, would be very useful in magnetic resonance spectroscopic imaging (MRSI), as well.

Marred by insufficient specificity and poor SNR due to the usage of the low-resolution FFT envelopes for FIDs encoded volumetrically, in vivo MRSI is still far from realizing its promising potential in the clinic. A distinct progress in MRSI should be possible by the optimized derivative processing which, already during examination of the patient, can give hundreds of quantitatively interpretable spectra (i.e. components) from any desired voxel to expediently evaluate their potential relevance to the diagnostic decision-making.

This would enhance the detection sensitivity and, crucially, yield the increased specificity of MRSI with the ensuing significant reduction of the percentage of false positive findings. In cancer medicine, the strength of diagnostics based on magnetic resonance methodologies is in their nonionizing nature and the potential for early tumor diagnostics. The latter should be especially helpful if MRSI could be practical for screening of patients and a motivation toward this goal is offered by the optimized derivative shape estimation.

4 Conclusions

The theme of this study falls into the category of ill-conditioned direct problems. Both direct (properly-posed) and inverse (improperly-posed) problems can be ill-conditioned. This signifies that the solutions may not exist or may be nonunique or may not depend continuously on the input data. In the latter case, the solutions may have large errors when the input data are even slightly perturbed. Such solutions are unstable and, thus, unphysical.

Ill-conditioning exists already in nonderivative parameter estimators for noisy time signals or FIDs. In this nonlinear inverse problem, an FID is given, but its structure is unknown and this makes the reconstructions ill-conditioned (unstable). By comparison, the FFT (nonderivative) deals with a well-posed, direct problem, which is a linear mapping of the given input FID from the time to the frequency domain, with preservation of full information. However, ill-conditioning may appear in derivative shape estimations (nonparametric). An example is the dFFT, which is ill-conditioned, despite the well-posedness of the processing problem (application of the Cooley-Tukey fast algorithm to the given, i.e. known input data).

All the sought full information is contained in the encoded FIDs. Since the time and frequency domains are equivalent, the whole information from encoding is also available as intact in total shape spectra from the FFT. However, the latter reconstructed envelopes are opaque, i.e. abundant with overlapped resonances. Then, the art is to make an input FFT envelope transparent by teasing out the constituents components. Such a task is beyond the reach of the dFFT, which succumbs to noise in encoded FIDs and loses all the physical information. By contrast, the presently optimized Fourier derivatives can reconstruct the components from the known input data by overcoming ill-conditioning.

The optimized derivatives surmount the major stumbling block of overlapping peaks in spectra from MRS and improve resolution as well as SNR. It is demonstrated that this advance is superior to the dFFT. This is illustrated with the three conceptually different FIDs encoded by proton MRS. Two of the FIDs were encoded with and without water suppression from the Philips phantom, which contains a mixture of ethanol, methanol and acetate, dissolved in demineralized water. The third FID transients were encoded in vitro with water suppression from human malignant ovarian cyst fluid.

Upon stabilizing for gradually increased derivative order, the resulting optimized spectra for all three signals are of a very high quality. Therein, the overlapped peaks are separated. They are clearly visualized and amenable to reliable quantification. This is anticipated to be particularly useful when dealing with patient data encoded by single-voxel MRS (based on protons or other spin-active nuclei) since it would allow the physician to directly interpret the quantitatively reconstructed information of potentially critical diagnostic relevance.

Moreover, according to the reported findings for relatively low derivative orders, the resolution and SNR achieved by optimized derivatives are similar for the FIDs encoded with and without water suppression. The latter result is of considerable practical importance. It raises the prospect of reliance upon the water-unsuppressed FIDs in medical diagnostics by in vivo single-voxel MRS and multi-voxel MRSI.