Investigating the regional effect of the chemical shift displacement artefact on the J‐modulated lactate signal at ultra high‐field

The present work aims to show the applicability of an analytical model for the optimisation of the STEAM sequence timing parameters for lactate detection at ultra high‐field. The effects of the chemical shift displacement artefact on the J‐modulated signal for a weakly‐coupled spin system were considered in the three applied directions of field gradients and the product operator formalism was used to obtain expressions for the signal modulation in each compartment of the excited volume. The validity of this model was demonstrated experimentally at 7 T in a phantom and acquisitions with optimised parameters were performed on a healthy volunteer. The spectra acquired with TE = 144 ms with the optimised mixing time and TE = 288 ms showed easily detectable lactate peaks in the normal human brain. Additionally, the acquisition with the longer TE resulted in a spectrum with less lipid/macromolecular contamination. The simulations shown here demonstrated that the proposed analytical model is suitable for correctly predicting the resulting lactate signal. With the optimised parameters, it was possible to use a simple sequence with sufficient signal‐to‐noise ratio to reliably distinguish lactate from overlapping resonances in a healthy brain.

spins interactions between the methyl and methine groups. To resolve the lactate methyl signal at 1.31 ppm from its overlapping resonances, some MRS methods exploit J-modulation effects in combination with the use of a long TE to reduce the intensity of the MM and lipid signals. The PRESS 7 sequence has been widely used for lactate detection in a clinical setting (field strengths of 3 T and lower). In this spin-echo based experiment, J-evolution operates on the methyl group, which is refocused by the 180 pulses, to obtain an inverted doublet peak at a TE of 144 ms (1/J, J = 6.933 Hz).
However, ultra high-field, which offers increased SNR and spectral resolution, imposes power and hence bandwidth restrictions due to specific absorption rate (SAR) limitations. Therefore, the PRESS sequence is associated with a large chemical shift displacement artefact (CSDA), as a result of the limited bandwidth of the slice-selective refocusing pulses. On the other hand, the STEAM 8,9 sequence minimises the CSDA effects in comparison to the PRESS sequence, due to the larger bandwidth of the 90 pulses relative to the 180 pulses, which makes the former a popular sequence for use at higher field strengths. To further minimise the CSDA effects, the semi-LASER 10 sequence, which uses adiabatic refocusing pulses to address the bandwidth limitation at ultra high-field, has been employed in several studies for lactate measurement. 5,11 However, this sequence might only be available on certain scanners as part of research packages.
The modulation in the STEAM sequence arising from scalar coupling is governed by both TE and mixing time (TM). The optimisation of the STEAM sequence timing parameters has been investigated for the improved acquisition of scalar-coupled metabolites at ultra high-field, such as glutamate. [12][13][14] For lactate measurement, previous work reported an analytical expression derived for the STEAM signal modulation in a weaklycoupled spin system of the lactate form, 15 AX 3 , where A corresponds to the proton in the methine group and X to those in the methyl group.
However, the effects of the CSDA were not included in the latter, it being assumed that both chemical groups experience all 90 pulses.
Although STEAM simulations performed at 4 T with a TE of 70 ms have not shown substantial localisation artefacts for weakly-coupled spin systems, 16 the effects of the CSDA should be taken into account when using longer TEs (144 ms) at higher field strengths. Analytical solutions for the response of the STEAM signal have been obtained considering the effect of the CSDA in two dimensions. 17 Numerical solutions incorporating two 18 and three-dimensional 16 localisation have also been implemented.
However, there is no complete three-dimensional analytical model with the consideration of the CSDA for assisting the optimisation of the STEAM sequence in the detection of weakly-coupled spin systems (AX n ). The model derived in this work enables the accurate determination of the signal as a function of TE, TM and pulse bandwidth for any AX n metabolite, including lactate (n=3), as well as the localisation of the signal throughout the excited region. This model was validated with a phantom experiment and was also applied to the optimisation of the STEAM sequence timing parameters for lactate detection in the human brain.

| Theory
The STEAM sequence ( Figure 1) is composed of three slice-selective 90 pulses, each of which is subject to a CSDA. The excited volume, defined as the region in which at least one of the A or X spins is excited by at least one of the RF pulses, can be segmented into partial volumes, according to the excitation pattern. For each direction there are three possible patterns of excitation, two in which only one of the spins experiences the pulse (A or X spins) and one in which both spins experience the pulse. Therefore, the excited volume comprises a total of 3x3x3 = 27 compartments ( Figure 2). F I G U R E 1 Schematic diagram of a standard STEAM sequence. Three 90 pulses are applied in conjunction to generate a stimulated echo. Each of the RF pulses is applied concurrently with slice selection gradients mutually orthogonal to each other (G s ). Dephasing/rephasing gradients G 1 and G 3 are introduced during TE/2 intervals, while G 2 gradients, which crush unwanted coherences, are added during the TM interval The product operator formalism was used to derive expressions for the X spin signal from each compartment as a function of TE and TM, following the convention defined in. 19 Signal modulation in the central compartment (number 14 in Figure 2b (ii) and shown in darker blue in Figure 2b (i) has been previously described in. 15 It is referred to as the 'ideal condition', in which both chemical groups experience all three RF pulses. This work extends the analysis to include the remaining compartments, which is required to fully describe the three-dimensional model. A similar approach to 15 was adopted here: all 90 pulses were assumed to be ideal selective, applied with a phase of 0 . This means that their bandwidth profile has a sharp cut-off, i.e., all spins within the bandwidth experience the same flip angle, whereas those outside the bandwidth are unaffected.
After the application of the first pulse, which transforms longitudinal (L) into transverse magnetisation (single-quantum coherence, SQ), each of the spins will undergo chemical shift and J-coupling evolution during TE/2. The objective is to determine the X spin signal: some of this arises directly from excitation of X spins, some is transferred to the X spins from the excited A spins. Thus, A spins terms that do not contribute to the F I G U R E 2 Representation of the CSDA effects in three dimensions on the weakly-coupled spin system signal during a STEAM sequence. The excited volume can be decomposed into three regions in each slice selection (SS) direction, two where only one of the spins experiences the pulse and one in which both do. The location of the central plane, in which A and X spins are affected by the first pulse, is illustrated in b (i) and the signal from each compartment (labelled 10-18) is shown in b (ii). The location of the planes in which only the A or X spins experience the first pulse is shown in a (i) and c (i), respectively. The central compartment, shown in darker blue in a-c (i) and identified by the number 14 in b (ii), is referred to as the 'ideal condition' in which both A and X spins experience all three pulses. The signal contributions yielding from each chemical group are shown for each compartment in a-c (ii) together with the weightings for the calculation of the partial volumes, where k is the ratio between the chemical shift difference and RF bandwidth transfer of transverse magnetisation to the X spins or will generate higher-order coherences in the next interval are excluded. After the second pulse, the transverse magnetisation is converted to longitudinal, single-and multiple-quantum coherences terms. Only those terms insensitive to the gradient G 2 ( Figure 1) applied in the TM period will survive, i.e., zero-quantum coherences (ZQ) and longitudinal magnetisation; transverse and higher-order coherences are dephased by the gradient. The final pulse is used to convert the longitudinal and zero-quantum coherences back to single-quantum terms, which can be detected in the acquisition period. The dephasing created by the gradient G 1 in the first TE/2 interval is recovered by the gradient G 3 present in the last TE/2 interval ( Figure 1). In addition, only in-phase signal at the X spins resonance and detected in the y-axis was considered. The possible coherence pathways contributing to the X spin resonance (top) and the respective signals (bottom) are: and where ΔΩ is the chemical shift difference (in Hz) between A and X spins and n is the number of X spins. S A1 , S A2 and S A3 (Equation (1) (i) -(iii)) represent the signal contribution arising from A spins, while S X1 , S X2 and S X3 (Equation (2) (i) -(iii)) arise from X spins.
For the plane in which both spins experience the first 90 pulse, nine compartments (3x3) can be identified (compartments 10 to 18, as depicted in Figure 2b (ii)). The first 9 compartments represent the plane in which only A spins experience the first pulse (Figure 2a (ii)) and in compartments 19 to 27 only X spins experience the pulse (Figure 2c (ii)). The signal in these two planes is equivalent to that in the central plane, excluding the contributions related to the spin that is not affected by the first pulse. If the X spins are not affected by the first pulse, the chemical shift dephasing in the last interval is not refocused. With respect to the A spins, their magnetisation would remain oriented along the longitudinal axis after the first pulse and it would not therefore be possible to transfer any magnetisation to the X spins.
Expressions for the signal from each compartment are given in Figure 2 (see Equations (1) and (2) Provided the metabolite of interest is uniformly distributed throughout the excited region (homogeneous case), the net signal S is given by: Note that the net signal is the same as would be obtained from a model that just considers the central plane (two-dimensional model), and thus the resulting modulation pattern is also equivalent. However, this simplified approach does not fully reflect the spatial origin of the signal contributions.
[Corrections added on 11 November 2020, after first online publication: The degree sign had been mistakenly replaced by the letter r throughout equations 1 and 2 and was reinstated.] Note also that when TE is very short (TE 0) or is set to a multiple of 2/J, e.g., TE = 288ms, for the methyl group of lactate, all terms vanish due to the sin 2 (πJTE/2) term, except one and Equation (3)

| RESULTS
Expressions for the signal from each compartment are given in Figure 2. Equations (1) and (2)  For a TE value of 288ms, sin 2 πJTE=2 ð Þ= 0, so all signal components in Equations (1) and (2) vanish apart from S X1 and there is no dependency on TM. For a TE of 144ms, cos πJTE=2 ð Þ= 0, and so S A1 , S A2 , S A3 and S X1 terms all vanish, leaving just S X2 and S X3 terms non-zero. The total signal, as given by Equation (3), is S = − S X2 ð1 −k 2 Þ + S X3 k 2 , and only the S X2 term has a dependence on TM. For TE = 72ms, most compartments contribute to the total signal.
The total signal (Equation (3)) is plotted in Figure 4 as a function of TM for the same three TE values and three k values: k = 0 (no CSDA), k = 0.27 (as used in our experimental work) and k = 0.5. Note again the lack of dependence on TM and the CSDA for TE = 288ms, and that this yields the maximum signal (in the absence of relaxation effects). When TE = 144ms, an inversion of the lactate peak is obtained, reaching its maximum at TM = 16.9ms. The efficiency of the inversion, i.e., the maximum obtainable negative peak amplitude, is reduced when the CSDA is taken into account (increasing k values). For TE = 72ms, the signal is never maximised nor fully inverted, thus diminishing the suitability for in vivoapplications. Additionally, when TE is set to values in between multiples of 1/J, as in the case of TE = 72ms, the choice of k not only affects the amplitude of the TM-modulated signal, but also the position of its maxima/minima. Figure 5 shows the variation of the amplitude of the lactate doublet with TM and k for three TE values (0, 10 and 20ms) that represent the typical range of short TE values used at higher field strengths. For TE = 0ms, the maximum amplitude (0.5) is observed irrespective of TM or k. For TE = 10ms, variation with TM is observed that is dependent on the amount of CSDA, but the reduction in signal is never more than 1.1%. At TE = 20ms, more significant variation is observed, with a potential loss in signal of about 6.4%.
The experimental TM dependence of the total signal for TE = 144ms is shown in Figure 6   A 40 x 30 x 25mm 3 voxel was placed in the brain of a healthy volunteer and 264 averages were used to obtain the spectra. The red box represents the voxel at the lactate frequency and blue box the voxel at the water frequency (pulse bandwidth = 2799 Hz). The spectra were phase corrected and displayed in MATLAB with 2Hz of line broadening. Glx represents the combined peaks of glutamate and glutamine and mI refers to myoinositol and 11.58 Hz, respectively. The SNR of the same peak was 105 (TE = 144 ms) and 42 (TE = 288 ms). Clear signals from the methyl group of lactate are seen in both cases (inverted in the case of TE = 144 ms).

| DISCUSSION
The quantum mechanical formulation of the coherence pathways in PRESS is relatively straightforward, as the two selective 180 pulses can be treated independently of each other, in contrast with the selective 90 pulses in the STEAM sequence which act in combination to induce a stimulated echo. 18 In this work, simulations based on the derived analytical expressions in the product operator formalism allowed a better understanding of the spatial distribution of a weakly-coupled spin system signal, as a function of the STEAM sequence parameters. The simulations reveal that for a homogeneous volume, two-and three-dimensional analyses predict the same total signal and modulation patterns. However, the threedimensional analysis indicates where within the excited volume the signal originates and this may be important in the case of an inhomogeneously distributed metabolite.
Considering the case of lactate (AX 3 ), it was found that for a TE value of 72ms, the methyl group of lactate signal is not confined to a specific plane or set of compartments, while with a TE = 144ms, there is no signal contribution from the plane where only A spins experience the first pulse; however the signal is still distributed throughout the remaining volume. On the other hand, with TE = 288ms, the net signal is restricted to the VOI of the methyl group of lactate, which demonstrates the signal independence of k and therefore of the CSDA. The fact that, with this TE value, the signal arises from a region with defined boundaries can be exploited to optimise the positioning of the spectroscopy voxel; if the focus of the measurement is the methyl group of lactate, the VOI for this chemical group should coincide with the region of interest one desires to acquire signal from.
The information on the lactate signal distribution within the excited volume was also used in the work of Edden et al., 23  The simulations carried out with the proposed model also allowed the investigation of the signal modulation as a function of TM. It was possible to verify that with TE = 288ms, there is no TM modulation, while for a TE = 144ms, the TM modulation can be used to obtain an inverted lactate doublet. The TM that led to maximal inversion of the lactate peak for this TE, which matched in both theoretical and experimental data, was 16.9ms. However, for a non-negligible pulse length, it is not possible to achieve a complete inversion, and the inversion efficiency depends on the RF bandwidth.
The performance of the STEAM sequence at TE = 144ms with the optimal TM and at TE = 288 ms was assessed with an in vivo acquisition.
Despite not being able to achieve 100% inversion efficiency (or equivalently -0.25 of the total magnetisation), due to the limited bandwidth and the RF field inhomogeneities at 7T, an inverted lactate peak was easily detectable in a healthy human brain (TE = 144ms). By visual inspection of the spectrum acquired withTE = 288ms, a positive lactate peak could be detected, comparable in magnitude with the one obtained withTE = 144ms.
Additionally, in the acquisition with TE = 288ms, the lactate signal was less affected by lipid and MM contamination, making this measurement less dependent on the individual lipid/MM content. Although it has been suggested that a general MM profile is sufficient for an accurate metabolite quantification at 7T, 24 individual variability in any of these superimposed signals could contribute to an over-or under-estimation of the lactate concentration.
As first proposed by Lange et al. 25 for 3T acquisitions with the PRESS sequence, the use of TE = 288ms can be a strategy to overcome the CSDA. Here, the non-dependence of the signal on the CSDA at TE = 288ms allowed the detection of a clear lactate peak despite the loss due to transverse relaxation decay. Therefore, in acquisitions with the STEAM sequence at 7T, it is recommended to use TE = 288ms, with as short a TM as possible to minimise longitudinal relaxation. However, at higher field strengths than 7T, the use of TE = 144ms might be more advantageous for lactate determination, due to the shorter transverse relaxation times.
Short TE experiments are commonly employed when multiple metabolites are to be measured, and lactate is often determined in this way, for example in functional MRS studies of the response to a visual stimulus. [3][4][5] Of course, such measurements are challenging in the presence of significant lipid and other macromolecular resonances in the vicinity of the 1.31ppm lactate methyl doublet. We showed that for a TE value of 10ms, there is negligible loss of lactate signal with TM and k, and even for a TE of 20ms, the loss is less than 10%. However, as TE increases, the signal variation with TM increases, and this in turn depends on the extent of the CSDA. In such circumstances, if the SNR of lactate is critical, optimisation of TM may be worthwhile. A similar argument applies to other coupled resonances, and their signals could be optimised using the theoretical expressions developed here (for example alanine, also of the AX 3 form).