³¹P MRSI at 7 T enables high-resolution volumetric mapping of the intracellular magnesium ion content in human lower leg muscles

2.1 ³¹P MRSI measurements

The lower legs of three healthy volunteers (one female/two male; age: 25–30 y) were examined on a 7T whole-body MR scanner (Magnetom 7T, Siemens Healthineers, Erlangen, Germany) using a circularly polarized, double-resonant ³¹P-¹H birdcage coil (RAPID Biomedical, Rimpar, Germany; 26.5 cm inner diameter and 22 cm length). All examinations were approved by the local ethics committee of the Medical Faculty of the University of Heidelberg, and written informed consent was received from all subjects. One of the subject's lower legs was positioned centrally in the coil by elevating it marginally from the resting position by using a cushion. To reduce movements during the measurement, the leg was slightly fixed with cushions while taking care to avoid strong compression of the leg. A comfortable position for the volunteer was ensured, but no special efforts were made during the measurement to further control the subject's state.

³¹P MRSI data of the lower leg were acquired using an acquisition-weighted 3D CSI sequence with the following parameters: matrix size = (24 × 24 × 16), nominal spatial resolution = (8 × 8 × 16) mm³, T_R = 240 ms, α = 20°, Δf = 5000 Hz, 1024 time points; Hamming-weighted k-space acquisition with 16 averages in k-space center. The total acquisition time was 56 min. The entire protocol, i.e. positioning of the subject, B₀ shimming, acquisition of morphological ¹H images and ³¹P MRSI, took approximately 75 min.

2.2 Data processing

Reconstruction, post processing and evaluation of the acquired ³¹P MRSI datasets were performed using MATLAB 2020a (The MathWorks, Natick, MA, USA).

To assess the potential for reducing the measurement time of such acquisitions, the acquired 56-min scans from all volunteers were additionally reconstructed using only six central k-space averages while maintaining the shape of the Hamming-weighting. These datasets simulate scans with a reduced measurement time of 28 min and are in the following referred to as the 28-min scans. Hence, for each volunteer two MRSI datasets were processed and investigated: first, the complete 56-min scan, and second, the retrospectively time-reduced 28-min scan.

In a supplementary analysis, an even further reduction to 21 min (reconstruction with only four central k-space averages) in combination with the application of the “moderate low-rank denoising” approach²⁰ to partly compensate for the SNR loss, was investigated (see Supporting Information S2, which is available online). Note, that no denoising was applied for the 56-min and 28-min scans.

All MRSI datasets were processed by one-fold spatial zero-filling resulting in a matrix size of (48 × 48 × 32), and application of a 10-Hz Gaussian filter in the time domain. Evaluation of the ³¹P spectra was performed in two steps using the home-built MATLAB implementation of the AMARES algorithm²¹ as described in section 2.3 of Korzowski et al.²⁰ Amplitudes, frequencies, and linewidths were quantified for the following six metabolites: phosphocreatine (PCr), the three resonances of ATP, inorganic phosphate (P_i) and glycerophosphocholine (GPC). All metabolites were modeled as singlets with Gaussian lineshape.

2.3 Calculation of free magnesium ion concentration [Mg²⁺_free] maps

The local chemical shift differences δ_Pi-PCr and δ_αβ were obtained from the quantification results of the individual localized spectra, and applied voxel wise to Equations (1) and (2) to yield volumetric pH and [Mg²⁺_free] maps for each dataset and volunteer. For visualization purposes, all maps were masked to show only voxels in muscle tissue. As criterion, a minimal PCr amplitude was chosen.

2.4 ROI analysis

To compare pH and [Mg²⁺_free] values from different muscle groups, 3D ROIs were defined for the TA, Sol, and GM in the Medical Imaging Interaction Toolkit²⁵ using morphological ¹H images. Mean values and standard deviations for pH and [Mg²⁺_free] were calculated for each dataset and volunteer individually (subject mean values), as well as across all volunteers (study population mean).

3.1 Spectral quality

In all volunteers, the 56-min scan yielded localized ³¹P MR spectra of high quality, enabling the robust quantification of all detectable metabolites across the whole measured volume (representatively shown for three voxels from the dataset of volunteer 2; Figure 1A). The localized spectra from the retrospectively time-reduced 28-min scan (Figure 1B) show an approximately 1.8-fold lower SNR compared to the 56-min scan. However, also in the 28-min scans from all volunteers the spectral quality was sufficient for the quantification of all metabolites.

3.2 3D [Mg²⁺_free] mapping

The quantified spectra revealed spatial variations in the B₀-shift corrected chemical shifts of β-ATP and P_i. For all volunteers, the largest difference is observed between the muscle groups TA and Sol (indicated as blue-shaded area in Figure 1A). The generated δ_αβ- and δ_Pi-PCr-maps (Figure 2) show local differences of these chemical shifts of up to 0.1 ppm. The large variation in δ_Pi-PCr across the lower leg is assumed to be mainly attributed to differences in local pH values and demonstrates the necessity to incorporate the local pH value into the calculation of [Mg²⁺_free] (see Equation 1).

In the calculated 3D maps of pH and [Mg²⁺_free] from the 56-min scan (Figures 3 and 4), differences of up to 0.1 pH units and 0.55 mM [Mg²⁺_free] can be observed between individual muscle groups. Due to the high resolution, locations of larger pH gradients match well the edges of muscle groups visible on the morphological images (i.e., in the coronal view of the pH maps; Figure 3). The same applies to the [Mg²⁺_free] maps (Figure 4), but with a slightly different pattern compared to the pH maps.

The pH and [Mg²⁺_free] maps resulting from the 28-min scan (Figures 5 and 6) show overall comparable patterns as the maps from the 56-min scan. For pH (cf. Figures 3 and 5), only minor differences between the 28-min and 56-min scan are observable. In the [Mg²⁺_free] maps from the 28-min scan (Figure 6), some areas show more extreme values compared to the maps from the 56-min scan (Figure 4). However, the trend of higher and lower values in the individual muscle groups is the same for both scans.

The patterns in the maps from the supplementary analysis of the 21-min scan with additional denoising (Supporting Information Figures S1 and S2) also match well the patterns of the maps from the complete 56-min scan, and compensate well for outliers. However, the contrast is partly lost (i.e. in the TA, see Supporting Information Figures S1 and S2).

3.3 ROI analysis

For all volunteers, the ROI analysis of the complete 56-min scan (Figure 7, blue color) shows differences in pH (left column) and [Mg²⁺_free] values (right column) between the investigated muscle groups. In the TA, the pH value is clearly lower than in the Sol and the GM, whereas the [Mg²⁺_free] value is elevated. When comparing the Sol and the GM, a slight trend towards higher pH and lower [Mg²⁺_free] in the Sol can be observed for all three volunteers. These differences are also found in the study population mean values (Table 1). The TA shows the highest [Mg²⁺_free] value, followed by the GM and the Sol with the lowest value.

The described trends for the 56-min scan are also preserved in the ROI analysis of the retrospectively time-reduced 28-min scan (Figure 7, red color). For each volunteer, the pH and [Mg²⁺_free] subject mean value from the 28-min scan (Figure 7, red color) agrees with the corresponding subject mean value from the 56-min scan (Figure 7, blue color). The calculated SDs (black whiskers) from the 28-min datasets are clearly higher than the ones from the 56-min scan, due to the larger fluctuations of values, already observed in the qualitative investigation of the maps (see section 3.2). The study population mean values for pH and [Mg²⁺_free] from the 28-min scan agree with the study population mean values from the 56-min scan (Table 1).

Together with the qualitative investigation of the generated maps, the ROI analysis demonstrates the potential of reducing the measurement time of the ³¹P MRSI acquisition by a factor of 2 while preserving the overall trend of results (i.e. different pH and [Mg²⁺_free] values in the different muscle groups).

The ROI analysis of the denoised 21-min scan (Supporting Information Figure S4) showed a good agreement of mean pH and [Mg²⁺_free] values between 21-min and 56-min scan for the Sol and the GM. However, in the TA, a bias of higher pH and lower [Mg²⁺_free] values was found for the denoised 21-min scans.

4.1 Observed local chemical shift differences

The observed local differences of the calculated [Mg²⁺_free] values result from the local differences of δ_αβ (Figure 2), which are prone to quantification errors. Considering the raw precision of the measurement itself, i.e. estimated as halve the spectral resolution in the noiseless case, being approximately 0.02 ppm, concentration differences of larger than ± 0.15 mM can be reliably distinguished (calculated for [Mg²⁺_free] = 1.4 mM, which is the worst case, as the error increases for larger [Mg²⁺_free]). Given the noise level in the 56-min scan, Cramér-Rao lower bounds (CRLBs) of ∼0.01 ppm were found, making it reasonable to use the raw measurement precision for a coarse uncertainty estimation. For the 1.8 times higher noise level in the 28-min scans, the CRLBs are still in the region of the measurement precision (CRLBs increase linearly with noise level), such that it is again plausible to apply the upper limit of ± 0.15 mM for the quantification of [Mg²⁺_free] in all measurements.

Concerning the influence of quantification errors of the chemical shifts on the pH value, errors of 0.02 ppm on δ_Pi-PCr result in an error of ± 0.03 pH units. However, this propagates only slightly to the calculation of [Mg²⁺_free] (± 0.015 mM, one magnitude smaller than the error due to an error on δ_αβ) and can be assumed to be neglectable.

Thus, one can assume that local differences larger than 0.03 pH units, and 0.15 mM [Mg²⁺_free] can be distinguished in our measurements. In the 3D ROI analysis of the individual volunteer measurements (Figure 7) and of the study population mean values (Table 1), differences between the TA and the Sol of ∼0.05 pH units and ∼0.17 mM [Mg²⁺_free] were observed. Given the considerations above and the fact that the 3D ROIs were quite large (300–1200 voxels), one can assume these differences between different muscle groups to be real and not due to quantification uncertainties.

However, intra-muscular differences, i.e. voxelwise differences, as partly seen in the calculated maps (Figures 3-6), cannot be reliably approved in this manner, but the patterns in the pH and [Mg²⁺_free] maps show the same trends in all volunteers, as well as across all subjects. This hints toward the assumption that also slight intra-muscular differences are present, e.g. in the Sol, which is a large muscle, or the TA, where differences in the oxidative capacity in proximo-distal direction were found previously.²⁶ In a supplementary analysis of potential differences of pH and [Mg²⁺_free] values in the TA in proximo-distal direction (see Supporting Information S3) some differences can be seen, but no clear trend, i.e. a gradient in one direction, was observed.

The conclusion that the observed local differences are real, is further supported by similar observed structures in pH and [Mg²⁺_free] maps. A mutual effect on the pH and [Mg²⁺_free] calculation is unlikely, as the shift differences of δ_αβ and δ_Pi-PCr are opposing. The downfield shift of β-ATP in the TA (compared to Sol and GM) could be explained by either an increased [Mg²⁺_free] value or an increased pH value (increasing pH results in a downfield shift of ATP-resonances²²). An increased pH value would however be reflected in a downfield shift of δ_Pi, which is contradicted by the observation (see Figure 1, δ_Pi is shifted upfield in the TA compared to Sol and GM). Consequently, the herein observed local differences in calculated [Mg²⁺_free] values are unlikely to be explained by pH differences, which is further supported by findings in Reyngoudt et al.⁴ comparing [Mg²⁺_free] values calculated with pH values determined by ³¹P and ¹H MRS. Moreover, the opposing chemical shift differences of δ_αβ and δ_Pi-PCr would also explain the nearly constant chemical shift difference of γ-ATP in all muscle groups. As the resonance of γ-ATP is known to be stronger influenced by pH than the resonance of α and β,²² the opposing effects on the chemical shift of γ-ATP due to pH and [Mg²⁺_free] could possibly balance each other out, resulting in the same δ_γ, although the physiological conditions in the individual muscle groups are presumably different.

These considerations strongly support the idea that the observed local differences of δ_Pi and δ_αβ are resulting from different physiological conditions, for example, different muscle fiber compositions (oxidative vs. glycolytic fibers). Note, however, that different blood perfusion or oxygenation states due to compression of the leg in some parts cannot be excluded and might contribute to the observed differences, although care was taken during positioning and fixation of the subject's leg in order to avoid compression.

4.2 Comparison with other studies

To the best of our knowledge, this is the first study reporting volumetric, high-resolution [Mg²⁺_free] maps from the lower leg of healthy volunteers, enabling the determination of [Mg²⁺_free] values in various muscle groups within the same measurement. Compared to previous studies, this work was performed at a magnetic field strength of 7 T using a volume resonator and a 3D MRSI acquisition, thus allowing for a whole volume coverage and high spatial resolution. Due to the achieved small effective voxel size of about 3 mL, the obtained ³¹P MR spectra are less affected by partial volume effects, allowing for a reliable assignments of spectra to individual muscle groups, and reducing effects of tissue mixing on the [Mg²⁺_free] quantification.

The herein determined [Mg²⁺_free] values for the muscles TA, Sol, and GM in healthy volunteers show clear differences. The found difference between the Sol and GM is in accordance with the results from Widmaier et al.,¹⁰ who reported an 11 % lower value for [Mg²⁺_free] in the Sol compared to the GM. However, the use of a surface coil at only one position in the study of Widmaier prevented the acquisition of [Mg²⁺_free] values in the TA. [Mg²⁺_free] values for the TA and the Sol and GM were reported in Reyngoudt et al.,⁴ however from multiple non-localized measurements using a surface coil enabling only a coarse localization of the acquired signal. In this study from Reyngoudt et al., no differences in [Mg²⁺_free] values between different measured muscle groups were found, which is in contrast to our results, showing clear differences for the [Mg²⁺_free] values in the TA, Sol, and GM. These finding might emphasize a potential benefit for performing localized acquisitions, also in terms of application to pathologies.

In general, the determination of [Mg²⁺_free] using the ATP chemical shifts is a complex problem, which has been addressed in numerous studies since the first approach by Gupta et al.⁶ This first approach only involves the equilibrium reaction $$$ {\mathrm{ATP}}^{4-}+{\mathrm{Mg}}^{2+}\leftrightarrow {\mathrm{ATP}\mathrm{Mg}}^{2-} $$$, characterized by the dissociation constant K_D. Several studies extended this initial approach by including additional chemical species, which are involved in reactions with Mg and ATP, for example, [H⁺] or [K⁺].^{12-16, 27, 28} For all of these approaches aiming at the absolute quantification of [Mg²⁺_free], the largest challenge is the need for accurate characterization of all equilibria involved (in terms of equilibrium constants).

Numerous studies on the lower leg muscles applying the approach by Gupta et al. used a value of 50 μM for the dissociation constant K_D (some performed corrections for different temperatures and pH values) and reported [Mg²⁺_free] values of (0.46–0.56) mM.^7-10 In this work, the formula from Barker et al.¹⁷ was used for the calculation of [Mg²⁺_free], which is based on the calculations from Golding and Golding.¹⁵ This approach takes into account four MgATP equilibria, also including the influence of the local pH value. In contrast to the cited studies above, Golding and Golding (and therewith Barker et al.) used a K_D of 90 μM, which is the main reason for the higher [Mg²⁺_free] values observed in this work compared to the cited previous studies.^7-10 Using the lower value of 50 μM^7-10 would lower the herein calculated values to 0.63 mM in the TA, 0.48 mM in the Sol and 0.53 mM in the GM (study population mean values from the 56-min scan), which would then agree with the previous reported values. The value of 90 μM from Barker et al. and Golding and Golding^{15, 17} is supported by findings in Malloy et al.²⁹ and Zhang et al.,³⁰ who reported values of 86 μM and 87 μM for K_D.

Other studies using more complex approaches involving additional equilibria^{4, 13, 27} report [Mg²⁺_free] values in the calf muscle of healthy volunteers of (0.45–0.56) mM, which are clearly lower than the results from our work with a study population mean value across all ROIs of (1.00 ± 0.14) mM. Finding a reason for this discrepancy is difficult due to the different approaches used in the latter cited approaches^{4, 13, 27} and our work, demonstrating the need for further investigation of the processes influencing the chemical shifts of ATP. In addition to pH and temperature,^{31, 32} also the influence of the ionic strength was discussed to be considered in the characterization of the involved equilibria necessary for the calculation of [Mg²⁺_free] by means of ³¹P MRS.³³

4.3 Clinical applicability

The measurement time of 56 min for the complete acquisitions in this study is not applicable for clinical studies on larger cohorts, but we showed the potential for reducing the measurement time of the ³¹P MRSI acquisition to only 28 min while maintaining the spatial resolution, and achieving comparable [Mg²⁺_free] maps. Although the [Mg²⁺_free] maps from the time-reduced 28-min scans show some differences to the maps from the 56-min scan, the overall patterns are comparable (compare Figures 3 and 5; and Figures 4 and 6), and the same results are obtained from the ROI analysis in both cases (Figure 7).

A further reduction of measurement time of the ³¹P MRSI acquisition to clinically relevant times of only 20 min is in principle possible, but requires a minimal SNR level to ensure fitting robustness. A possibility to increase the SNR is the application of denoising approaches to the acquired ³¹P MRSI datasets. With the supplementary analysis of the retrospectively time-reduced 21-min scans with additional denoising (see Supporting Information Figure S2) we showed, that it is in principle possible to achieve comparable pH and [Mg²⁺_free] maps in a measurement time of 21 min. However, in the ROI analysis a bias of the calculated mean values (i.e., a trend toward the global mean pH and [Mg²⁺_free]) is observed in some parts of the investigated volume (i.e., in the TA, see Supporting Information Figure S3), as it could be expected due to the application of the denoising approach.²⁰ Still, this approach could be of interest for certain applications or research questions, for example, when investigating extremely large pH and/or [Mg²⁺_free] differences. Nevertheless, when aiming at measurement times of 20 min, but with accurate results (i.e., no bias in the mean values in some parts of the volume), there is still the necessity for a further increase in SNR. A further possibility would be the application of nuclear Overhauser enhancement (NOE) techniques,^34-36 but was not investigated in this study. As detection via a volume resonator is not optimal, another possibility to increase the acquired signal could be the use of a receiver array. In any case, one could also work with a slightly lower spatial resolution for the benefit of measurement time. Combining these gains in SNR could enable volumetric mapping of [Mg²⁺_free] in a clinically feasible measurement time of 20 min with still reasonable spatial resolution. In addition to the application to tumors, this might be of interest also for the application to non-focal pathologies affecting the whole muscle, for example, diabetes or Duchenne muscular dystrophy,^1-5 as the acquisition of localized spectra improves the [Mg²⁺_free] quantification by reducing effects from tissue mixing or B₀ inhomogeneities.