Norbert W Lutz
email: nlutz@umars.fr

Monique Bernard
email: monique.bernard@univ-amu.fr

Ph.D William E Hull

Multiparametric quantification of the heterogeneity of free Ca 2+ concentration by 19 F MR spectroscopy †

Keywords: calcium quantitation, tissue heterogeneity, 19 F MRS, lineshape analysis, heterogeneity descriptors, statistical distributions, histograms, free intracellular Ca 2+ Graphical

For biological tissue that is heterogeneous with respect to free intracellular Ca 2+ concentration ([Ca 2+ ] i ), the lineshape of the 19 F MRS resonance of injected [Ca 2+ ]-sensitive 4-FBAPTA or BAPTA-FF reflects the statistical distribution of [Ca 2+ ] i values. While conventional 19 F MRS of these fluorinated Ca 2+ reporter molecules only provides one [Ca 2+ ] i value per spectrum, our specially designed lineshape analysis reveals at least eight quantitative statistical parameters (descriptors) characterizing the [Ca 2+ ] i distribution within the observed tissue volume. To this end, the [Ca 2+ ]-sensitive 19 F MRS resonance is transformed into a [Ca 2+ ] i curve. Subsequently, the digital points of this [Ca 2+ ] i profile are used to build a histogram using dedicated algorithms. The following statistical descriptors are computed from this histogram: weighted mean and median, standard deviation, range, mode(s), kurtosis, skewness, and entropy. Our new method is thoroughly validated through in-silico and experimental models. The potential of combining statistical [Ca 2+ ] information with spatial resolution is demonstrated by simulated statistical CSI maps. This proof of principle should form the basis of future in-vivo studies in physiology and medicine, notably in heart and muscle research.

Introduction

Metal ions play fundamental roles in biological organisms. Calcium (Ca 2+ ) is an important second messenger in signal transduction, and is involved in neurotransmitter release from neurons and in muscle contraction [START_REF] Brini | Calcium in health and disease[END_REF]. Ca 2+ is also required as a cofactor by many enzymes. While bone is the principal mineral storage site for calcium, organelles such as mitochondria and the endoplasmatic reticulum provide intracellular calcium storage [START_REF] Wilson | Steatosis inhibits liver cell store-operated Ca(2)(+) entry and reduces ER Ca(2)(+) through a protein kinase C-dependent mechanism[END_REF]. From these organelles, Ca 2+ is released during particular cellular events. Thus, the resting Ca 2+ concentration in the cytosol is only on the order of several 100 nmol, but specific signals can trigger an increase by at least two orders of magnitude. Ca 2+ is actively pumped from the intracellular to the extracellular space to maintain a [Ca 2+ ] gradient of a factor of 20,000 to 100,000, as the extracellular Ca 2+ concentration may easily be in the millimolar range. There is growing evidence that calcium/sodium exchange has particular pathophysiological implications, notably in neurological diseases [START_REF] Annunciato | Sodium calcium exchange: a growing spectrum of pathophysiological implications[END_REF]. Furthermore, calcium metabolism disorders have been associated with gut, kidney and bone [START_REF] Peacock | Calcium metabolism in health and disease[END_REF].

The mechanisms behind these disorders are complex, but are critically dependent on the intracellular presence of free Ca 2+ . Therefore, methods for in-vivo measurement of free-Ca 2+ concentrations in biological tissue have been proposed, e.g., 19 F MR spectroscopy ( 19 F MRS) of fluorinated intracellular Ca 2+ reporters such as 4F-BAPTA (4,4'-difluoro-BAPTA) or 5F-BAPTA (5,5'-difluoro-BAPTA) [START_REF] Smith | Intracellular calcium measurements by 19F NMR of fluorine-labeled chelators[END_REF][START_REF] Metcalfe | Free cytosolic Ca2+ measurements with fluorine labelled indicators using 19FNMR[END_REF]. [In the remainder of this work, "Ca 2+ " exclusively refers to free Ca 2+ , unless specified otherwise]. Although both 4F-BAPTA and 5F-BAPTA specifically form chelates with Ca 2+ , 4F-BAPTA is characterized by fast exchange (on the NMR time scale) with Ca 2+ , but 5F-BAPTA by slow exchange. These chelators were employed to determine intracellular [Ca 2+ ] in thymocytes from BALB/c mice [START_REF] Smith | Intracellular calcium measurements by 19F NMR of fluorine-labeled chelators[END_REF][START_REF] Metcalfe | Free cytosolic Ca2+ measurements with fluorine labelled indicators using 19FNMR[END_REF], in pig lymphocytes, Ehrlich ascites tumor cells, rat basophil leukemic cells and perfused rat hearts [START_REF] Metcalfe | Free cytosolic Ca2+ measurements with fluorine labelled indicators using 19FNMR[END_REF], but also in intact ferret hearts (notably by Marban et al. [START_REF] Marban | Ca2+ transients in perfused hearts revealed by gated 19F NMR spectroscopy[END_REF][START_REF] Marban | Quantification of [Ca2+]i in perfused hearts. Critical evaluation of the 5F-BAPTA and nuclear magnetic resonance method as applied to the study of ischemia and reperfusion[END_REF]), and in rabbit hearts [START_REF] Corretti | Glycolytic inhibition and calcium overload as consequences of exogenously generated free radicals in rabbit hearts[END_REF]. Another fast-exchange Ca 2+ chelator, BAPTA-FF (5,5',6,6'-tetrafluoro-BAPTA), has also been used to study intracellular free Ca 2+ in rabbit and rat heart by 19 F MRS [START_REF] Murphy | Measurement of cytosolic free calcium in perfused rat heart using TF-BAPTA[END_REF][START_REF] Chen | Measurement of free Ca2+ in sarcoplasmic reticulum in perfused rabbit heart loaded with 1,2-bis(2-amino-5,6difluorophenoxy)[END_REF][START_REF] Chen | Gender differences in sarcoplasmic reticulum calcium loading after isoproterenol[END_REF]. Note that all cardiac experiments mentioned here have been done successfully on the beating heart. Unsurprisingly, applications in vivo and in situ soon followed, e.g., intracellular [Ca 2+ ] measurements in kidney, spleen, and brain of male Sprague-Dawley rats by using 19 F MRS of 5F-BAPTA [START_REF] Song | Determination of intracellular calcium in vivo via fluorine-19 nuclear magnetic resonance spectroscopy[END_REF]. Thus, feasibility and sufficient sensitivity of the F-BAPTA 19 F MR spectroscopic technique in general has been unequivocally proven for intact, living mammals. Newer developments were directed at yielding information on spatial Ca 2+ distribution within the studied organ or tissue region. For instance, 19 F CEST MRI of 5F-BAPTA has been suggested for mapping Ca 2+ concentration [START_REF] Bar-Shir | Metal ion sensing using ion chemical exchange saturation transfer 19F magnetic resonance imaging[END_REF], benefiting from signal amplification through saturation transfer. Alternatively, Angelovski et al. have developed a range of smart Ca 2+ -sensitive T 1 relaxation agents designed to enable (mostly extracellular) Ca 2+ imaging based on the water 1 H signal [START_REF] Moussaron | Ultrasmall Nanoplatforms as Calcium-Responsive Contrast Agents for Magnetic Resonance Imaging[END_REF][START_REF] Dhingra | Towards extracellular Ca2+ sensing by MRI: synthesis and calcium-dependent 1H and 17O relaxation studies of two novel bismacrocyclic Gd3+ complexes[END_REF]. While these recently presented methods are expected to provide some information on the physical location of tissue regions characterized by differing Ca 2+ concentrations, they only partially reflect the [Ca 2+ ] variations actually existing within the object studied. This intrinsic shortcoming is due to the fact that methods based on digital images only yield one single concentration value for each image pixel, the latter representing a voxel from within the imaged object. Thus, any heterogeneity with respect to Ca 2+ concentration within such a voxel is necessarily ignored, and is replaced by a single concentration value that may or may not be close to the average value for that voxel. The relatively large voxel sizes typically used in such mapping techniques (several microliters) aggravate the problem, as tissue voxels of this order of magnitude are biologically heterogeneous and comprise different microenvironments.

We suggest here a new approach taking into account the entire (microscopic as well as macroscopic) heterogeneity of tissue with respect to intracellular Ca 2+ concentrations. Our concept is based on the paradigm that the overall line shape of an MRS resonance whose chemical shift is a function of Ca 2+ concentration, directly reflects the statistical distribution of the Ca 2+ concentrations within the probed tissue volume. For [Ca 2+ ] analysis by 19 F MRS of fluorinated chelators, this implies that fast exchange between metal ion and ligand is required. At present, our algorithms provide at least eight quantitative statistical descriptors (in addition to one or several modes, i.e., maxima of the distribution curve) characterizing the distribution of Ca 2+ concentration values: mean, median, standard deviation, range, skewness (asymmetry), kurtosis (pointedness), standard and normalized entropy (smoothness). In many cases, the relative sizes of sample regions representing characteristic concentration ranges may also be calculated. The underlying formal scheme is similar to the algorithms previously introduced by us to analyze heterogeneity in Mg 2+ concentration [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF], pH [START_REF] Lutz | Quantitative in-vivo characterization of intracellular and extracellular pH profiles in heterogeneous tumors: a novel method enabling multiparametric pH analysis[END_REF] and temperature [START_REF] Lutz | Multiparametric quantification of thermal heterogeneity within aqueous materials by water 1H NMR spectroscopy: paradigms and algorithms[END_REF][START_REF] Lutz | Thermal heterogeneity within aqueous materials quantified by 1H NMR spectroscopy: Multiparametric validation in silico and in vitro[END_REF] by 31 P or 1 H MRS, respectively; these papers [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF][START_REF] Lutz | Quantitative in-vivo characterization of intracellular and extracellular pH profiles in heterogeneous tumors: a novel method enabling multiparametric pH analysis[END_REF][START_REF] Lutz | Multiparametric quantification of thermal heterogeneity within aqueous materials by water 1H NMR spectroscopy: paradigms and algorithms[END_REF][START_REF] Lutz | Thermal heterogeneity within aqueous materials quantified by 1H NMR spectroscopy: Multiparametric validation in silico and in vitro[END_REF] discuss mathematical details of general importance to the quantification of the heterogeneity of physicochemical parameters by way of statistical MRS line shape analysis.

In addition to presenting our algorithm, we provide the proof of principle by transforming experimental BAPTA-FF 19 F MR spectral lines into statistical distribution curves for Ca 2+ concentrations, i.e., into the corresponding statistical concentration profiles. Although both BAPTA-FF and 4F-BAPTA form fast exchanging complexes with Ca 2+ , 4F-BAPTA is not currently commercially available. Nonetheless, we assess the potential of 4F-BAPTA by numerically modeling 19 F MR spectra as a function of Ca 2+ concentration. These are not ab-initio, but empirical simulations; both chemical shift and linewidth values are taken from actual 19 F MR spectroscopy experiments previously conducted on 4F-BAPTA solutions in the presence of varying Ca 2+ concentrations [START_REF] Smith | Intracellular calcium measurements by 19F NMR of fluorine-labeled chelators[END_REF][START_REF] Yanagida | Simultaneous multicompartment intracellular Ca2+ measurements in the perfused heart using 19F NMR spectroscopy[END_REF], at a time when 4F-BAPTA was still available commercially. This bestows high practical relevance upon our calculated models.

The mathematical and statistical equations given here are formally analogous to the equations presented previously for studying Mg 2+ concentrations [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF]. Therefore, the basic mathematical validity of the transformation of MR line shapes to ion concentration profiles, and the justification for deriving statistical parameters from these profiles, are already documented [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF], and will not be repeated here. The purpose of our current validation is to examine the following questions: (i) How do the resulting quantitative descriptors of statistical [Ca 2+ ] distribution have to be interpreted in terms of distribution curve properties? (ii) What are the ranges of the statistical descriptor values for [Ca 2+ ] distribution curves to be expected from actual in-vivo experiments? (iii) How sensitive are the resulting [Ca 2+ ] distribution curves, and their associated descriptor values, to actual [Ca 2+ ] gradients vs. spurious effects, i.e., curve shape contributions unrelated to [Ca 2+ ] distribution? (iv) What are the effects of 19 F MR line shape correction by our empirical deconvolution method on the resulting distribution curve shapes and, consequently, on the derived descriptors of [Ca 2+ ] concentration profiles? Moreover, we discuss potential and limitations of our method. Based on our results, we also suggest the development of further [Ca 2+ ] reporter molecules optimized for use in our analysis of statistical distributions. When used with monovoxel 19 F MR acquisition, our analysis is fast and will provide valuable statistical information that is complementary to the spatially resolved data obtainable by the Ca 2+ mapping techniques mentioned above. When used in multivoxel 19 F MR experiments such as chemicalshift imaging (CSI), our new method will be able to generate a combination of full statistical and some spatial information on the distribution of these metal ion concentrations in one single experiment; this is demonstrated by parameterized CSI maps modeled with calculated [Ca 2+ ] profiles. Once translated to the field of biomedical research, our multiparametric analysis is expected to significantly enhance the analysis of intracellular free Ca 2+ concentrations in vivo by offering detailed quantitative characterization of tissue heterogeneity with respect to [Ca 2+ ], thus complementing fast [Ca 2+ ] mapping by 19 F-CEST or other MRI methods. We also briefly discuss the potential of our approach in the context of the increasingly important data analysis by "machine learning" procedures.

Theory

The use of 19 F MRS for measuring Ca 2+ concentrations is based on the dependence of the chemical shift, , of a 19 F resonance of the reporter molecule (ligand) in question (4F/4'F of 4F-BAPTA, or 5F/5'F of BAPTA-FF), on the concentration of free Ca 2+ to be determined. The relationship between the measured chemical shift,  mea , and [Ca 2+ ] is given by the following general equations [START_REF] Gupta | NMR studies of intracellular metal ions in intact cells and tissues[END_REF] ] values closer to basal levels; further advantages of BAPTA-FF have been described elsewhere [START_REF] London | NMR-sensitive fluorinated and fluorescent intracellular calcium ion indicators with high dissociation constants[END_REF]. For 5F/5'F of BAPTA-FF, δ BAP and δ CaBAP values are confirmed experimentally.

Special calibration techniques may also take into account pH and other effects on the 19 F MRS resonances of the reporter molecules, depending on tissue type and physiological conditions. However, since our current report is strictly focused on the proof of principle of our statistical approach, we do not currently include fine-tuning through tissue-specific chemical shift-to-[Ca 2+ ] conversion algorithms. A practical example of our calculations is presented in the template provided as an EXCEL spreadsheet in Supplementary Information (S1 Supplemental Software). Further explanations pertaining to the general paradigm underlying our method, as well as to the statistical parameters that can be determined, are contained in Supplementary Information (S1 Supplemental Materials) . distributions [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF]. As in the latter case, most quantitative parameters suggested in this report are universally applicable to any given distribution of concentration values. Taken together, the parameter values given in our tables and diagrams explore the basic behavior to be expected for statistical descriptors in in-vivo distribution curves. While inspection of Figure 1 allows one to qualitatively assess the distinctness of adjacent modes, quantitative evaluation is made possible by comparing statistical descriptor values as a function of mode intensity ratios. In Figure 2, select descriptor values are plotted against intensity ratios for rapid identification of dominating trends. Since the changes in some descriptor values are rather subtle, we also provide the complete set of numerical data in Table 1 that additionally includes results for 4.7-T simulations. The improved distinction of two modes differing by 1.0 M for 7.0-T vs. 4.7-T simulations mentioned at the end of the preceding paragraph, is reflected in larger high-field changes (with resonance amplitudes) for those descriptors that characterize profile shapes, i.e., skewness, kurtosis, standard and normalized entropy (Table 1). For instance, normalized entropy, H n , decreases only by about 0.05 units for the low-field profiles, but by about 0.07 units for the high-field profiles when going from the nearly symmetric distribution (height ratio 1:1) to increasingly asymmetric distributions (down to 1:8). In addition, skewness varies by no more than 0.4 units for the low-field profiles, but by 0.7 units for the highfield profiles. The same trends hold for max. kurtosis increase (by 0.37 and 1.26, respectively), and for max. standard entropy decrease (by 0.33 and 0.47, respectively). shows that mode area ratios, a 1 :a 2 , are proportional to intensity ratios, i 1 :i 2 . Note that the units of the parameters shown in the middle and bottom panels are dimensionless.

Results

Studies of multimodal [Ca

Validation of statistical algorithms by way of simulated bimodal [Ca

Owing to the high resolution of the underlying 19 F spectra, the resulting [Ca 2+ ] 1 and [Ca 2+ ] 2 modes are perfectly invariant, which is not the case for low-field simulations (4.7 T, Table 1) that produce largely overlapping [Ca 2+ ] profiles (Fig. 1). As expected, the relatively narrow high-field spectral lines (in ppm) resulted in smaller [Ca 2+ ] standard deviations and ranges than the relatively broad low-field lines (Table 1).

As was to be anticipated, weighted [Ca 2+ ] means and medians were closest for [Ca 2+ ] distribution profiles that deviated the least from a unimodal symmetric curve (for i 1 :i 2 = 0 in Fig. 2, as well as for i 1 :i 2 = 0.125 and 0.25, corresponding to ratios 1:8 and 1:4 in Table 1). While kurtosis was near zero for the unimodal [Ca 2+ ] profile (i 1 :i 2 = 0), decreasing values with increasing i 1 :i 2 accurately reflect the progressive replacement of one dominating peak by two peaks (Fig. 2). Furthermore, the transition from a left-skewed to a right-skewed profile from i 1 :i 2 = 0.125 to i 1 :i 2 = 1.0 is truthfully represented by increasing skewness values. Another expected feature of our bimodal curves is that both area ratios (a 1 :a 2 ) and peak height ratios (h 1 :h 2 ) increase nearly proportionally with i 1 :i 2 . In summary, statistical descriptors of [Ca 2+ ] distributions calculated by our algorithms for a series of in-silico 4-FBAPTA 19 F MRS models, straightforwardly reflect the behavior of the underlying [Ca 2+ ] heterogeneity. Analogous data for 4-FBAPTA 19 F spectra based on Lorentzians are presented in Supplementary Information, including Figure S1 and Table S1 (S1 Supplemental Materials).

In practice, 19 Indices a -f, top, refer to diagrams shown in Fig. 3. Indices a -e, bottom, refer to diagrams shown in Fig. S2. As expected, the apparent [Ca 2+ ] profile based on a single high-concentration 5F/5'F resonance from BAPTA-FF ('e' in Fig. 3 ] descriptors that are consistent with the respective lineshape characteristics.

Measured profiles of [Ca

Experimental [Ca 2+ ] profiles indicated a very large influence of spurious effects on MRS lineshapes, in agreement with previously published BAPTA-FF 19 F spectra from homogeneous solutions; these spectra have been shown to exhibit broad and [Ca 2+ ]-dependent linewidths for the 5F/5'F peak.

Quantitative [Ca 2+ ] heterogeneity analysis with 19 F CSI -evaluation of numerical simulations.

Next, we demonstrated the feasibility of combining statistical [Ca 2+ ] information with spatial resolution by simulating statistical CSI maps. These numerical simulations serve to demonstrate the proof of principle for statistical [Ca 2+ ] distribution maps based on 19 F 4F-BAPTA MRSI, with the underlying [Ca 2+ ] profiles being evaluated with our algorithms presented above; they are not intended to mimic any particular biological tissue. This goal was achieved by (i) simulating 23 [Ca 2+ ] distribution profiles with varying but well-defined characteristics, and (ii) assigning these profiles to 8x8 cells of a grid (Fig. 5)

underlying the 8x8 pixels of derived CSI maps, followed by interpolation (to 256x256 pixels) as is general practice in CSI maps (Fig. 6). In addition, we simulated a "morphological" high-resolution (256x256) MR image as a reference for localization (arbitrary pattern, Fig. S3).

The rationale behind the construction of our exemplary [Ca 2+ ] profiles and their distribution over the grid was to create readily verifiable patterns as follows: in each CSI map (i) the simulated highconcentration peak should increase from 4.5 to 7. The CSI maps clearly reflect the statistical properties of the CSI pixels. In Fig. 6, lowest mean [Ca 2+ ] values are seen in the bottom right corner of the relevant CSI map ("mean"), representing the distributions with the most intense low-concentration mode; and highest values are present near the top left corner where a rather broad high-concentration mode dominates the distribution (Fig. 5). Standard deviations were highest in the top left corner of the pertinent map because of the particularly broad overall distribution, whereas [Ca 2+ ] values are concentrated within a small range for CSI pixels at and close to the right of the center (Fig. 5). Highest kurtosis values were generally found in the right half of the kurtosis map, with a prevalence of profiles dominated by narrow single peaks. By contrast, the presence of multiple and/or broad peaks lead to low kurtosis values. The transition from left to right skew is best seen in the bottom half of the skewness map, whereas the broadening of the distributions at the top of the map tended to wash out any major left or right skew (predominantly slight right skew; Fig. 5).

Discussion

Proof of principle.

The proof of principle for the first method enabling quantitative characterization of heterogeneity in Ca 2+ concentration was provided in a basic feasibility study. The underlying paradigms and algorithms were presented and validated for 19 F MRS of two Ca 2+ reporter molecules, 4-FBAPTA and BAPTA-FF. Calculated statistical descriptors of [Ca 2+ ] distribution profiles were evaluated based on in silico and experimental in vitro approaches, for different magnetic field strengths and [Ca 2+ ] distribution models. Thus, the fundamental suitability of this method for the determination of quantitative parameters describing the statistical distribution of intracellular [Ca 2+ ] has been established. profile shapes that no longer truthfully reflect the underlying [Ca 2+ ] distribution. There are several ways of dealing with spurious effects. Prior to signal acquisition, the shim on the very volume to be used for 19 F MRS should be optimized. Modern MR systems provide shim sequences that automatically adjust the shim gradients such that the field is optimally homogenized for a user-defined volume. This procedure can minimize line shape distortions due to magnetic field inhomogeneity. Despite these precautions, resulting line shapes will still be affected by spurious effects. Therefore, experimentally observed line shapes represent a combination of [Ca 2+ ] heterogeneity and spurious effects, although the latter can be largely compensated for by the reference deconvolution technique mentioned above and in the following subsection, and first introduced and discussed in much more detail in previous papers [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF][START_REF] Lutz | Multiparametric quantification of thermal heterogeneity within aqueous materials by water 1H NMR spectroscopy: paradigms and algorithms[END_REF][START_REF] Lutz | Thermal heterogeneity within aqueous materials quantified by 1H NMR spectroscopy: Multiparametric validation in silico and in vitro[END_REF] (see also S1 Supplemental Materials). However, even the apparent descriptor values obtainable without reference deconvolution (subsection 3.2) are useful if they are employed as relative (as opposed to absolute) values.

The influence of spurious effects. Current limitations of our

For instance, differences in apparent descriptor values for a particular group of experimental animals vs.

those obtained for a control group provide information on the difference in [Ca 2+ ] heterogeneity between these groups. Of course, in this case all individual measurements involved should be performed under identical experimental conditions.

Our validation results closely parallel previously obtained findings for [Mg 2+ ] distributions (obtained from analogous simulated ATP- 31 P MR spectra) that had also been studied for reference-deconvolved spectra [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF]. Therefore, it was not necessary to repeat the [Ca 2+ ] distribution analysis of subsection 3.2. for the corresponding deconvolved spectra; instead, we simply point here to the results contained in that previous report [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF] indicating increased sensitivity to differences in metal ion concentration upon deconvolution. In future in-vivo applications, reference functions to be applied in deconvolution will be empirical, i.e., they will involve the corresponding experimentally measured Ca 2+ -sensitive F-BAPTA resonances from appropriate biological tissues with homogeneous [Ca 2+ ], in analogy with the previously described application of 31 P MRS to ATP-. This will lead to significantly decreased values for evaluable [Ca 2+ ] differences, again in analogy with results previously obtained in [Mg 2+ ] distribution analysis [START_REF] Lutz | Multiparametric quantification of heterogeneity of metal ion concentrations, as demonstrated for [Mg2+] by way of 31P MRS[END_REF].

Moreover, ongoing developments geared toward [Ca 2+ ]-dependent deconvolution functions should further increase the sensitivity of our method to small [Ca 2+ ] differences, as they will take into account F-BAPTA linewidth variations with [Ca 2+ ] due to varying F-BAPTA/Ca 2+ exchange rates.

Sensitivity of F-BAPTA 19 F MRS and MRSI. Conventional [Ca 2+

] analysis by 19 F MRS of BAPTA-FF in biological tissues has demonstrated that a high-quality 1D spectrum can be acquired over a period of 5 minutes for a single-volume acquisition, yielding a SNR of about 10 [START_REF] Murphy | Measurement of cytosolic free calcium in perfused rat heart using TF-BAPTA[END_REF][START_REF] Chen | Measurement of free Ca2+ in sarcoplasmic reticulum in perfused rabbit heart loaded with 1,2-bis(2-amino-5,6difluorophenoxy)[END_REF]. This is also the SNR we consider to be the minimum for our new evaluation strategy. Note that no matter how low the Ca 2+ concentration, we always measure the full intensity of the BAPTA 19 F MR signal. This is due to the fact that variations in Ca 2+ concentration under conditions of fast exchange (on the NMR scale) between Ca 2+

and the fluorinated BAPTA compound only change the chemical shift of the [Ca 2+ ]-sensitive 19 F-BAPTA peak, but not its intensity. Thus, it is only the fluorinated BAPTA concentration that determines sensitivity in terms of SNR, not the Ca 2+ concentration of the sample. The feasibility of F-BAPTA MRS in terms of SNR has also been confirmed for actual in vivo and in situ measurements with a related F-BAPTA compound [START_REF] Song | Determination of intracellular calcium in vivo via fluorine-19 nuclear magnetic resonance spectroscopy[END_REF], as mentioned in the Introduction section. In all cases, tissue concentrations of the Ca 2+ reporter molecules in question typically are in the millimolar range [START_REF] Murphy | Measurement of cytosolic free calcium in perfused rat heart using TF-BAPTA[END_REF][START_REF] Chen | Measurement of free Ca2+ in sarcoplasmic reticulum in perfused rabbit heart loaded with 1,2-bis(2-amino-5,6difluorophenoxy)[END_REF], i.e., in the same concentration range as many metabolites that are comfortably measured by in vivo 1 H monovoxel MRS and MRSI (CSI) methods. Note that the 19 F nucleus (i) is only slightly less NMR sensitive than 1 H, (ii) occurs at 100% natural abundance, and (iii) has the added advantage of no background signal when compared to 1 H MRS and MRSI, As was to be expected on the basis of these considerations, the practical feasibility of in vivo F MRSI has been repeatedly demonstrated for fluorinated drug metabolites [START_REF] Gonen | Simultaneous 3D NMR spectroscopy of proton-decoupled fluorine and phosphorus in human liver during 5-fluorouracil chemotherapy[END_REF][START_REF] Dzik-Jurasz | Gallbladder localization of (19)F MRS catabolite signals in patients receiving bolus and protracted venous infusional 5-fluorouracil[END_REF][START_REF] Klomp | Optimization of localized 19F magnetic resonance spectroscopy for the detection of fluorinated drugs in the human liver[END_REF], but also for genereporter molecules [START_REF] Kodibagkar | Imaging beta-galactosidase activity using 19F chemical shift imaging of LacZ gene-reporter molecule 2-fluoro-4-nitrophenol-beta-Dgalactopyranoside[END_REF]. Owing to the absence of particular sensitivity issues, 19 F MRSI would also be feasible for BAPTA-FF, 4-FBAPTA and other Ca 2+ -sensitive fluorinated chelates. In these cases, one 19 [START_REF] London | NMR-sensitive fluorinated and fluorescent intracellular calcium ion indicators with high dissociation constants[END_REF]. Since our new approach was experimentally investigated using the same fluorinated [Ca 2+ ] chelator, we directly refer to their results in the following. As for pH, significant effects were only found for rather acidic milieus, i.e., below pH 6.25 to 6.50 (Fig. 6 in [START_REF] London | NMR-sensitive fluorinated and fluorescent intracellular calcium ion indicators with high dissociation constants[END_REF]). For instance, from the upper (alkaline) limit down to pH 7.1, the chemical shift of the ] without correction for Mg 2+ and pH [START_REF] Smith | Intracellular calcium measurements by 19F NMR of fluorine-labeled chelators[END_REF]. The same authors have demonstrated that other ions (Zn 2+ , Fe 2+ , Mn 2+ , etc.) are in slow exchange with 4F-BAPTA, and therefore will not affect the 4F-BAPTA/Ca 2+ resonance [START_REF] Smith | Intracellular calcium measurements by 19F NMR of fluorine-labeled chelators[END_REF].

For these reasons, we can neglect both pH and [Mg 2+ ] effects in the context of our current report.

However, further refinements such as optimization for particular tissue types should be envisaged in future research projects. Such tissue-specific work would also include (i) considering tissues with particularly high or low protein levels, and (ii) studying the influence of 19 F MR spectrum deconvolution on the practical in vivo limitations of our method. This would greatly simplify the deconvolution process. To the best of our knowledge, such chelators are not currently available. Therefore, it would be strongly desirable to see such fluorinated compounds developed in the future. Once available, these new compounds would render our new method even more powerful by facilitating work at relatively high field strengths (9.4 or 11.7 T, and above). In addition to exchange rates, attention must be paid to the dissociation constant, K d, of the complex between Ca The past few years have seen an impressive increase in publications applying "machine learning" procedures to the analysis of results from biomedical investigations (see [START_REF] Lan | A Survey of Data Mining and Deep Learning in Bioinformatics[END_REF] and [START_REF] Camacho | Next-Generation Machine Learning for Biological Networks[END_REF] for reviews). These techniques include numerous data mining algorithms used for preprocessing, classification and clustering as well as various optimized neural network architectures in "deep learning" methods. These advanced routines can be thought of as higher-level statistics or "meta-statistics", and have been successfully used in metabolomics and other omics sciences. In this context, the multiple statistical descriptors generated by our new approach may be combined with further metabolic, biochemical, biological and imaging parameters to characterize tissue in vivo by using neural network and other machine learning algorithms.

This deeper level of analysis should facilitate the discovery of complex patterns relevant to healthy and diseased tissues that may be difficult or impossible to identify in the conventional way, i.e., by operators not assisted by artificial intelligence.

Methods

A series of Tris base/HEPES-buffered BAPTA-FF solutions was prepared with varying Ca 2+ concentrations [START_REF] London | NMR-sensitive fluorinated and fluorescent intracellular calcium ion indicators with high dissociation constants[END_REF]. 19 F MR spectra were acquired at 376 MHz with a spectral width of 30 ppm, an FID length of 8 k data points (0.364 s), and a total of 2048 transients. Spectra were apodized with 20 or 40 Hz before Fourier transformation. After referencing to the 6,6'- 19 F resonance and baseline correction, spectra were transferred to our EXCEL template (see S1 Supplemental Software). Subsequently, spectra were either directly subjected to chemical shift-to-[Ca 2+ ] conversion or processed with reference deconvolution (described above) before conversion. Statistical analysis of the underlying distribution of [Ca 2+ ] values ensued, based on the equations given above and in S1 Supplemental Materials (the latter file also contains further methodological details).

Figure 1 .

 1 Figure 1. Simulated 4FBAPTA 19 F MRS resonances and their envelopes (first and third row), and [Ca 2+ ]profiles derived from the respective envelopes (second and fourth row). The spectral envelopes represent the sums of two partially overlapping Gaussians, for amplitude ratios (upfield : downfield) 1:1 (a,e), 1:2 (b,f), 1:4 (c,g) and 1:8 (d,h). The centers of the individual Gaussians underlying each envelope were chosen to be at 1.73 and 1.90 ppm in all cases; the linewidth of each individual Gaussian was 30 Hz, corresponding to 0.11 ppm at 7.0 T (a -d), and 0.16 ppm at 4.7 T (e -h). Areas that can be associated with distinct modes are color-coded. The vertical black lines represent [Ca 2+ ] distributions in the form of Dirac  functions. The right tails of the [Ca 2+ ] distributions (second and fourth row) decrease more slowly to the baseline than the left tails. The origin of this asymmetry, introduced by ppm-to-[Ca 2+ ] conversion, is the nonlinearity between the ppm and the [Ca 2+ ] scales, as mentioned in the text. This nonlinearity "stretches" the distribution curve out to the right, and "compresses" it to the left for the particular concentration ranges chosen. A well visible consequence of these effects is the difference in line width; the "[Ca 2+ ] 2 " widths are larger than the "[Ca 2+ ] 1 " widths, although in the underlying simulated spectra (first and third row), both simulated resonances have identical widths.

2+ ] profiles corresponding to 4 -FBAPTA 19 F

 419 MRS resonances. First, effects of varying [Ca 2+ ] heterogeneity on [Ca 2+ ] distribution parameters were studied for [Ca 2+ ] profiles derived from the envelopes of two simulated, partially overlapping Gaussian (Fig. 1) or Lorentzian (Fig. S1) 4-FBAPTA 19 F resonances centered at 1.73 and 1.90 ppm (30 Hz linewidth). Note that spectral envelopes change in shape when converted to [Ca 2+ ] profiles, due to the nonlinear relationship between chemical shift and [Ca 2+ ]. As the relative amplitudes of the resonances (upfield : downfield ratios) decrease from left to right (Fig. 1, first and third row), the respective derived [Ca 2+ ] profiles change shapes correspondingly (Fig. 1, second and fourth row), with [Ca 2+ ] 2 modes increasingly dominating associated [Ca 2+ ] 1 modes. The two modes differing by 1.0 M can be much better discerned for 7.0-T simulations (second row) than for 4.7-T simulations (fourth row).

Figure 2 .

 2 Figure 2. Statistical descriptors for the [Ca 2+ ] profiles presented in Figure 1 a -d, and for a [Ca 2+ ] profile obtained from a unimodal simulated Gaussian resonance (i 1 :i 2 = 0). The straight line in the center panel

  2+ ] heterogeneity. BAPTA-FF solutions have been prepared to model a bimodal[Ca 2+ ] distribution, with modes being centered at 0.18 and 0.31 mM. Since the K d CaBAP value for BAPTA-FF is much higher (65 M) than for 4-FBAPTA (2.9 M), the optimally measurable [Ca 2+ ] range is increased by one to two orders of magnitude compared to 4-FBAPTA, as pointed out in the Theory section. Our phantom [Ca 2+ ] concentrations were chosen as to be within the range characteristic of realworld tissue exhibiting elevated [Ca 2+ ] i values. For instance, Ca 2+ concentration attributed to heterogeneous sarcoplasmic reticulum populations in perfused rabbit heart has been shown to decrease from about 1.5 to 0.3 mM[START_REF] Chen | Measurement of free Ca2+ in sarcoplasmic reticulum in perfused rabbit heart loaded with 1,2-bis(2-amino-5,6difluorophenoxy)[END_REF], and even further down to < 0.1 mM[START_REF] Yanagida | Simultaneous multicompartment intracellular Ca2+ measurements in the perfused heart using 19F NMR spectroscopy[END_REF] upon perfusion with a Ca 2+ -ATPase inhibitor. Thus, our BAPTA-FF solutions prepared to model unimodal and bimodal [Ca 2+ ] distributions (0.18 and 0.31 mM Ca2+) were within the [Ca 2+ ] range given in these references[START_REF] Chen | Measurement of free Ca2+ in sarcoplasmic reticulum in perfused rabbit heart loaded with 1,2-bis(2-amino-5,6difluorophenoxy)[END_REF][START_REF] Yanagida | Simultaneous multicompartment intracellular Ca2+ measurements in the perfused heart using 19F NMR spectroscopy[END_REF].Moreover, the [Ca 2+ ]-sensitive 5F/5'F resonance of this compound is also more broadened by spurious effects (essentially due to fast-to-intermediate exchange with [Ca 2+ ]).

Figure 3 '

 3 a' to 'e' shows plots of the two partially overlapping resonances at different intensity ratios and after 20-Hz apodization (left column), akin to the bimodal resonances simulated for 4-FBAPTA (preceding subsections). The corresponding calculated apparent [Ca 2+ ] distributions were obtained after deconvolution with a Gaussian of 20 Hz linewidth (right column). This deconvolution only partially took into account the exchange broadening of the MRS resonance, which explains the rather broad lines even after deconvolution. Consequently, the appearance of the resulting curves is far from the combinations of two Dirac  functions theoretically expected for a combination of two solutions each of which is characterized by one discrete Ca 2+ concentration. Nonetheless, the numerical results obtained for the statistical descriptors exhibited the expected trends (Table2, top), similar to those obtained for 4-FBAPTA-based simulations. With increasing weight of the high-concentration mode (i.e., from 'a' to 'd' in Fig.3and Table2, top; see also Fig.4), mean and median [Ca 2+ ] increased as the high-concentration (right) component increasingly dominated the [Ca 2+ ] profile. The latter feature also explains that both peak height and integrated peak area ratios ('h 1 :h 2 ' and 'a 1 :a 2 ') decreased, and skewness increased. The increasing peakedness of the distribution is reflected in increasing kurtosis values, but standard and normalized entropy showed little variation (slight decrease).Numerical values for the same BAPTA-FF resonance transformed upon 40 Hz apodization are given in Table2, bottom (see Fig.S2for spectra and corresponding [Ca 2+ ] profiles). While the general trends were identical for 20-Hz and 40-Hz apodizations, the broader lines (for 40 Hz) necessarily caused increased nominal values for standard deviation and range for peaks with roughly equal intensities ('a' and 'b'). The increased overlap due to increased linewidth is reflected in altered 'h 1 :h 2 ' and integrated 'a 1 :a 2 ' values.These differences, as well as increased skewness and kurtosis values, are also particularly pronounced for the curves based on peaks with roughly equal intensities. Overall, the descriptors of the broader profiles (40 Hz apodization) are less sensitive to changes in relative intensities of the low-and high-concentration components than the narrower profiles (20 Hz apodization).

Figure 4 .

 4 Figure 4. Statistical descriptors for the [Ca 2+ ] profiles presented in Figure 3. The straight line in the center panel shows that mode area ratios, a 1 :a 2 , are proportional to intensity ratios, i 1 :i 2 . Note that the units of the parameters shown in the middle and bottom panels are dimensionless.

5

 5 M (bottom to top); (ii) the overall distribution should change from left-skewed to right-skewed (from left to right); and (iii) the distance between the low and high-concentration modes underlying individual distributions should increase from the center to the edge of the CSI maps. All profiles were created based on two more or less overlapping spectral lines, except for trimodal profiles in subregions of the bottom left and top right quadrant of the CSI map (profiles 21 and22; intermediate-concentration mode inserted between low and high-concentration modes). Note that inseveral distributions, the bimodal or trimodal character is concealed by significant mode overlap. For these CSI pixels, relative heights and areas of underlying modes were unavailable (white areas in Fig.S5, bottom panels).The statistical descriptors for each [Ca 2+ ] distribution was calculated, and one CSI map for each obtained descriptor was created (Figs. 6, S4 and S5). All panels of these figures exhibit color-coded CSI maps representing different [Ca 2+ ] distribution descriptors, each overlaid on a "morphological" image, as is common for CSI presentations. Figure6, top left panel, indicates the 'true" spatial resolution of the CSI maps (8x8 pixels, broken lines). Tables S2 -S4 contain the descriptor values for all [Ca 2+ ] distributions used, according to the numbering shown in Figure5. The same descriptor values were also arranged in spreadsheets as to directly represent all 8x8 CSI maps (EXCEL file "CSI maps pixel values.xlsx", S2 Supplemental Materials).

Figure 5 .

 5 Figure 5. [Ca 2+ ] distribution profiles, based on simulated 4-FBAPTA 19 F MR spectra, arranged in a 8x8 matrix for the calculation of CSI maps shown in Figs. 6, S4 and S5. Simulations were purposefully designed such that (i) the simulated high-concentration peak increased from 4.5 to 7.5 M, from bottom to top; (ii) the overall distribution changed from left-skewed to right-skewed, from left to right; and (iii) the distance between the low and high-concentration modes underlying the overall distributions increased

Figure 6 .

 6 Figure 6. Color-coded CSI maps for mean, standard deviation, kurtosis and skewness of the [Ca 2+ ] distribution profiles shown in Figure 5. The top left CSI map indicates the real spatial resolution based on an 8x8 matrix (broken lines), before interpolation to 256x256 pixels. The black-and-white background structure in each CSI map has been generated by simulation; see reference image in Fig. S3.

4. 5 .

 5 Future developments. Depending on the concentration range to be measured, spurious line broadening may be particularly strong due to the influence of intermediate-to-fast exchange rates on T 2 .Although advanced reference deconvolution procedures currently under development should be able to tackle this issue by compensating [Ca 2+ ]-dependent line width effects, a more straightforward, powerful and elegant solution would be the use of fluorinated [Ca 2+ ] chelators producing very fast exchange rates.

  2+ and the fluorinated chelator. As outlined above, some currently available compounds such as BAPTA-FF are only useful when Ca 2+ concentrations are involved that are one to two orders of magnitude above the basal values typical of mammalian cells. Future efforts should be directed toward the development of a pool of fast-exchanging fluorinated [Ca 2+ ] chelators representing a range of K d values, from which a particular compound can be chosen as a function of the Ca 2+ concentration of interest. Once confirmed by in-vivo experiments, potential biomedical applications of our quantitative [Ca 2+ ] heterogeneity measurements may include a range of conditions affecting the concentration of free intracellular Ca 2+ , notably in heart, muscle and brain tissue. This approach should help clarify the nature and extent of pathological processes involving major perturbations of [Ca 2+ ] regulation.

2+ ] profiles -overview. Most

	tissues containing cells that exhibit a range
	of different Ca 2+ concentrations are likely characterized by complex [Ca 2+ ] distributions due to genetic,
	physiological, metabolic and other growth conditions varying throughout the studied tissue volume. The
	presence of heterogeneous cell populations may also contribute to a lack of [Ca 2+ ] homogeneity [24].
	However, for the purpose of validating our method we initially focused on numerically simulated and
	experimental unimodal and bimodal [Ca 2+ ] profiles, because the concept of quantitative parameters of
	[Ca 2+ ] heterogeneity is best tested on the basis of well-defined distribution functions. For [Ca 2+ ]
	investigations with BAPTA-FF as reporter molecule, experimental [Ca 2+ ] distribution models were
	generated, based on specially designed samples (phantoms). By contrast, we had to limit our
	Ca 2+ /4-FBAPTA studies to numerical simulations because 4-FBAPTA is not currently available as a
	commercial product, as indicated in the Introduction section. Our simulations were performed for field
	strengths of 4.7 T and 7.0 T. These values were chosen based on the following considerations. First, we
	see a significant potential for our method primarily in preclinical research; this lead us to not focus on
	field strengths more common for clinical instruments, such as 3 T. On the other hand, focusing on very
	high fields would be problematic for currently existing fluorinated Ca 2+ reporter molecules because at very
	high fields, the exchange between Ca 2+ and chelator may no longer be in the fast-exchange range on the
	NMR scale (e.g., for Ca 2+ /BAPTA-FF there is a borderline situation of "fast-intermediate" exchange
	already starting from 8.5 T [23]). Our evaluations are in keeping with our previous validations for [Mg 2+ ]

Table 1 .

 1 Statistical descriptors for [Ca 2+ ] curves based on in-silico models of Gaussian 19 F MR resonances of 4F-BAPTA

	descriptors for 7 T	1:1 a	1:2 b	1:4 c	1:8 d
	weighted mean (M)	2.32	2.49	2.62	2.72
	weighted median (M)	2.25	2.58	2.69	2.75
	std. deviation (M)	0.59	0.58	0.53	0.47
	range (M)	3.18	3.21	3.18	3.17
	mode, mg 1 (M)	1.77	1.77	1.77	1.77
	mode, mg 2 (M)	2.76	2.76	2.76	2.76
	global max. (M)	1.77	2.76	2.76	2.76
	peak height ratio (h 1 :h 2 )	1.55	0.78	0.39	0.20
	peak area ratio integ.	0.99	0.49	0.24	0.11
	(a 1 :a 2 )
	skewness, G1	0.343	-0.051	-0.314	-0.362
	kurtosis, G2	-0.906	-0.831	-0.295	0.364
	stand. entropy, H s	3.15	3.08	2.91	2.72
	norm. entropy, H n	0.5559	0.5486	0.5126	0.4864

Indices a -h refer to diagrams shown in Fig.

1

  F MRS line shapes underlying [Ca 2+ ] distribution curves are not only influenced by Ca 2+ effects on chemical shift, but also by spurious effects, e.g., T 2 , T 2 * etc., as outlined in the Theory section above. Assuming that the individual resonances making up the spectra plotted in Figure1are entirely defined by such spurious effects, each bimodal [Ca 2+ ] distribution shown is represented by two discrete M ([Ca 2+ ] 1 and [Ca 2+ ] 2 ), can be clearly discerned at 7.0 T for all spectral amplitude ratios from 1:1 to 1:8, but are unambiguously identified at 4.7 T only for ratios 1:1 and 1:2. For other ratios, [Ca 2+ ] differences were largely submerged by spurious line broadening. This indicates that [Ca 2+ ] profiles become more and more dominated by spurious effects as [Ca 2+ ] differences decrease below 1.0 M at 4.7 T, and below several 100 nM at 7.0 T. Consequently, the statistical descriptors presented in Figure2and Table1should be seen as "effective" or "apparent" descriptors in the presence of major spurious effects because they are greatly affected by factors other than [Ca 2+ ] distribution (see also discussion below in subsection 4.2). If obtained under these conditions, Table1shows that (apparent) statistical [Ca 2+ ] distribution descriptors relating to curve shape (skewness, kurtosis, standard and normalized entropy) are much less influenced by changes in amplitude ratios for 4.7 T vs. 7.0 T, in analogy to the case described above for "pure" [Ca 2+ ] distribution descriptors. Figure 3. a -e: Left column: sums of two partially overlapping 19 F MRS resonances (two 5F/5'F peaks from BAPTA-FF), measured at 376 MHz for model solutions containing 180 and 308 M free Ca 2+ . ; one single resonance is also shown. Right column: [Ca 2+ ] profiles obtained from corresponding spectra on the right. Areas that can be associated with distinct modes are color-coded. Two modes ([Ca 2+ ] 1 and [Ca 2+ ] 2 ) differing by 128 M can be clearly discerned for spectral amplitude ratios from 1:1 to 4:1. However, broad lines (e.g., 210 Hz linewidth for the left peak after 20-Hz apodization for improved signal-to-noise ratio, SNR) indicate that [Ca 2+ ] distribution profiles are substantially influenced by spurious effects. f: Envelope (sum) of two simulated Gaussian BAPTA-FF 19 F MRS resonances with 170 Hz linewidth (amplitude ratio 1:4, left), and the [Ca 2+ ] profile derived from this envelope (right). The centers of the individual Gaussians underlying each envelope were chosen such that the difference between the [Ca 2+ ] 1 and [Ca 2+ ] 2 modes was about 5 M.

	Spectra	were	added	as	to	yield	amplitude	ratios	between	from	1:1	to
	8:1

[Ca 2+

] values, i.e., two Dirac  functions (black vertical lines in second row of Fig.

1

). Visual inspection of this figure then suggests that for 4-FBAPTA, two modes, here equivalent to two distinct [Ca 2+ ] values differing by 1.0

Table 2 .

 2 Statistical descriptors for [Ca

	descriptors for 20 Hz LB	1:1 a	1:2 b	1:4 c	1:8 d	_1 e	simul. f
	weighted mean (M)	0.26	0.29	0.32	0.32	0.32	0.021
	weighted median (M)	0.24	0.28	0.30	0.31	0.31	0.022
	std. deviation (M)	0.10	0.11	0.12	0.12	0.08	0.003
	range (M)	0.47	0.56	0.71	0.71	0.40	0.017
	mode, mg 1 (M)	0.18	0.19	0.19	0.19	0.31	0.017
	mode, mg 2 (M)	0.31	0.31	0.31	0.31	-	0.022
	global max. (M)	0.18	0.31	0.31	0.31	0.31	0.022
	peak height ratio (h 1 :h 2 )	1.51	0.83	0.60	0.45	-	0.29
	peak area ratio integ.	1.15	0.51	0.35	0.19	-	0.20
	(a 1 :a 2 )
	skewness, G1	0.587	0.725	1.156	1.175	0.479	-0.563
	kurtosis, G2	-0.235	0.579	2.229	2.407	0.201	-0.100
	stand. entropy, H s	5.88	5.79	5.72	5.61	5.08	4.52
	norm. entropy, H n	0.61	0.60	0.59	0.58	0.57	0.62

2+

] curves based on experimental models of

19

F MR resonances of BAPTA-FF, and for a [Ca 2+ ] curve based on two simulated Gaussians

  and Table2, top) yielded the same mean, median and mode values as the envelope profile with the most dominating high-concentration component ('d'), but range and standard deviations were much smaller for the single-peak profile. Another consequence of the derivation from a

	single resonance is the finding of very small skewness and kurtosis values for 'e' (which are zero for pure
	Gaussians). By contrast, the trend toward moderate decrease of standard and normalized entropy from 'a'
	to 'd' is continued to 'e', reflecting the progressive transition from bimodal to unimodal behavior from 'a' to
	'e'.
	For comparison, a [Ca

2+ ] distribution function, 'f', based on simulated 5F/5'F resonances of BAPTA-FF, comparable to one of the experimental BAPTA-FF-derived curves (Fig. 3 c and Table 2 c, top), has been evaluated for a lower Ca 2+ concentration range. Direct quantitative comparisons of descriptor values between measured und simulated distributions are not meaningful in view of the broad and noisy experimental lines whose shapes are very different from Gaussians or Lorentzians. The simulated apparent [Ca 2+ ] profile nevertheless shows the expected negative skewness values because it is left-skewed, as opposed to the experimental profiles with pronounced right tails. Kurtosis is close to zero because the overall shape is dominated by the Gaussian of the high-concentration component, and slightly negative due to the bulging flank caused by the low-concentration component. Standard and normalized entropy values for the simulated [Ca 2+ ] profile are of the same order of magnitude as those obtained for the experimental profiles, as these descriptors do not depend on absolute concentration values or symmetry. In summary, both measured and simulated [Ca 2+ ] heterogeneity profiles based on BAPTA-FF 19 F spectra yielded statistical [Ca 2+

  novel technique are due to spurious effects on [Ca 2+ ] distribution profiles, i.e., effects that are unrelated to chemical-shift effects of [Ca 2+ ] on the Ca 2+ -sensitive 19 F resonance of the reporter molecule in question. Spurious effects may broaden 19 F resonances principally due to T 2 and T 2 * mechanisms and 19 F-1 H J coupling, and may thus lead to [Ca 2+ ]

  F spectrum would be obtained for each CSI voxel. However, simply converting "the" chemical shift of each Ca 2+ -sensitive F-BAPTA line to a [Ca 2+ ] value would lead to a [Ca 2+ ] map that ignores any intravoxel[Ca 2+ ] heterogeneity, and may not even represent an accurate average[Ca 2+ ] value for each resolved voxel, depending on the shape of the underlying statistical[Ca 2+ ] profile. This is in direct analogy to the situation described above for 1D19 F MR spectra. Combining our new method for evaluating statistical [Ca 2+ ] profiles with the spatial resolution of CSI would be a way of adding information on statistical [Ca 2+ ] distribution to spatial information. Nonetheless, spatial resolution based on MRSI will remain somewhat limited by sensitivity, even if relatively fast CSI variants such as EPSI are used. Therefore, whenever fast Ca 2+ imaging at high spatial resolution is desired, appropriate methods based on CEST/F-BAPTA or smart T 1 relaxation agents should be used, to the extent that these recently developed methods turn out to be reliable in vivo. In such cases, our current new spectroscopic approach would provide complementary multiparametric statistical information on total [Ca 2+ ] distribution that is not available through these new Ca 2+ imaging methods.Sensitivity of our [Ca 2+] distribution curves to differences in [Ca 2+ ] within a heterogeneous sample is a separate issue; the latter is totally unrelated to the SNR issue discussed in the preceding paragraph and has been thoroughly analyzed in the Results section for two different chelators (BAPTA-FF and 4F-BAPTA), for characteristic [Ca 2+ ] ranges. A given variation in [Ca 2+ ] has a relatively large effect on the chemical shift of the19 F-BAPTA signal if the actual [Ca 2+ ] values to be analyzed are close to the K d value of the Ca 2+ /F-BAPTA complex, because changes in [Ca 2+ ] close to K d have large effects on the ratio of free vs. complexed chelator concentrations. This leads to [Ca 2+ ] distribution curves and associated statistical descriptors finely mirroring even relatively small [Ca 2+ ] differences. In other words, the "[Ca 2+ ] resolution" is relatively high. By contrast, if the actual [Ca 2+ ] values are much lower or much higher than K d , [Ca 2+ ] variations have very little effect on the ratio of free vs. complexed chelator concentrations because nearly all BAPTA species are free or complexed, respectively. In this case, effects of [Ca 2+ ] variations on chemical shift are very small and hardly discernable in derived [Ca 2+ ] distribution curves and associated statistical descriptors as small [Ca 2+ ] variations are averaged out. As a consequence, the "[Ca 2+ ] resolution" is relatively low. Accordingly, for our method to be powerful, the fluorinated [Ca 2+ ] chelator should be chosen such that the order of magnitude of its K d is approximately that of the actual [Ca 2+ ] values of interest. Conversely, a given [Ca 2+ ] chelator should only be used to study[Ca 2+ ] values that are approximately on the order of magnitude of the K d value of that chelator. These requirements are in strict analogy with those for classical applications of19 F-BAPTA MRS analysis of [Ca 2+ ] where the K d value determines the [Ca 2+ ] range for which Ca 2+ concentrations can be measured with reasonable precision. As explained above in the Theory section, London et al.[START_REF] London | NMR-sensitive fluorinated and fluorescent intracellular calcium ion indicators with high dissociation constants[END_REF] and Murphy et al.[START_REF] Murphy | Measurement of cytosolic free calcium in perfused rat heart using TF-BAPTA[END_REF] have shown that the relatively high K d value of BAPTA-FF/Ca 2+ (65 μM) makes it ideal for measuring elevations in [Ca 2+ ] i occurring, e.g., in heart during ischemia (values > 2 μM), but inappropriate for basal [Ca 2+ ] i values that are much lower. By contrast, 4-FBAPTA used in our simulations has a much lower K d value with Ca 2+ , and would therefore be useful for measuring basal [Ca 2+ ] i levels (≤ 1 μM). In summary, it is neither our new approach nor the classical19 F-BAPTA MRS method per se that define the reasonable [Ca 2+ ] range, but only the K d value of the fluorinated reporter molecule employed.

	4.4. Potential confounding parameters. In the Theory section we have stated that actual K d values for
	F-BAPTA/Ca 2+ complexes in different biological tissues may vary slightly as a function of intracellular
	parameters such as pH and ion concentration. In fact, the issue of potential interference by the most
	relevant parameters, [Mg

2+

] and pH, with

[Ca 2+

] analysis by

19

F MRS of BAPTA-FF has already been studied by London et al. in the context of the classical application of this method, both theoretically and experimentally

  5,5'-F resonance only changed by a negligible 0.2 % of the difference between the shifts for pH values of fully protonated and fully deprotonated BAPTA-FF. On the other hand, mammalian cells are not usually viable at intracellular pH values significantly below 7.0; even transiently lowering intracellular pH to 6.5 or less causes necrosis, apoptosis, or both forms of cell death. (We disregard cases where pH i is diminished only for a fraction of a second; these cases are irrelevant here because a typical [Ca 2+ ] i measurement by19 F-BAPTA-FF MRS may take a few minutes). It would be of little interest to try measuring "intracellular" [Ca 2+ ] for cells that are disintegrating, i.e., where intracellular and extracellular spaces are no longer defined in the first place. As for [Mg 2+ ], the authors found that millimolar changes could indeed lead to measurable errors in the [Ca 2+ ] determination for [Ca 2+ ] values near basal values of several hundred nanomolar. As mentioned in the above, this order of magnitude for [Ca 2+ ] is certainly too low to be measured by way of BAPTA-FF with any acceptable degree of accuracy. However, for accessible [Ca 2+ ] well above baseline levels(London et al. mention heart tissue during ischemia), changes in[Mg 2+ ] have a negligible effect on [Ca 2+ ] values measured by19 F-BAPTA-FF MRS. As for 4F-BAPTA, Smith et al. have shown that the chemical shift of its19 F resonance is very insensitive to Mg 2+ and pH, and that 4F-BAPTA therefore provides a direct measurement of[Ca 2+

This work was supported by the Centre National de la Recherche Scientifique (CRMBM UMR 7339).

Supporting Information