Synopsis

Sodium(I) (Na⁺) plays a key role in basic cell function. Noninvasive methods for measuring intracellular Na⁺ concentrations ([Na⁺]_i) in biological tissue have been developed on the basis of ¹⁹F NMR spectroscopy. By contrast, ²³Na MRI at most provides approximate (relative) contributions of intracellular Na⁺ to total ²³Na by relying on a number of theoretical assumptions, as it does not detect Na⁺_i selectively. Since [Na⁺]_i values are often not uniform throughout a given tissue volume (or voxel), we have designed an approach for quantifying total [Na⁺]_i heterogeneity by exploiting the lineshape of the ¹⁹F resonance of an appropriate [Na⁺]_i reporter molecule.

Introduction

The role of Na⁺ in cells ranges from the maintenance of electrolyte balance and the regulation of osmotic pressure to the generation of nerve impulses in neurons and cell signaling ¹. Na⁺ is also crucial in metabolic cell functions, e.g., in Na⁺/K⁺-ATPase and many other enzymes. The mechanisms involved are complex, but are critically dependent on the intracellular concentration of free Na⁺ ([Na⁺]_i). Therefore, methods for in-vivo measurement of [Na⁺]_i in biological tissue have been proposed, e.g., MRS techniques based on chemical-shift effects of [Na⁺]_i on the ¹⁹F MRS resonance of FCrown-1 ². However, these methods have not taken into consideration [Na⁺]_i heterogeneity within a measured volume or voxel, nor have MRS methods based on chemical-shift reagents such as dysprosium ³ or thulium ⁴ complexes. ²³Na MRI has been used to map Na⁺ concentration in vivo ⁵, but also neglects intravoxel [Na⁺] variations. Moreover, ²³Na MRI does not detect intracellular ²³Na selectively.

We suggest a new approach taking into account the total (microscopic plus macroscopic) [Na⁺]_i heterogeneity of a given tissue volume. Our concept is based on the paradigm that the shape of a ¹⁹F MRS resonance whose chemical shift is determined by the concentration of free Na⁺ ions, directly reflects the statistical distribution of Na⁺ concentrations within the probed volume. Currently, our algorithms provide ≥ 8 quantitative statistical descriptors characterizing a measured distribution of Na⁺ concentration values: mean, median, standard deviation, range, skewness (asymmetry), kurtosis (pointedness), standard entropy and normalized entropy (smoothness). The algorithms employed are analogous to those previously introduced by us to analyze heterogeneity in Ca²⁺ concentration by ¹⁹F MRS ⁶. We provide here the proof of principle for our method by investigating the statistical properties of [Na⁺] distribution curves derived from modeled ¹⁹F MRS spectra of FCrown-1.

Methods

Results and Discussion

First, the algorithms of our method were validated for a series of modeled bimodal [Na⁺] distributions (Fig. 1). Expectedly, the increasing weight of the high-concentration mode in Figure 1 is reflected in increasing mean and median values, but also in decreasing skewness values, as the distribution increasingly turned from right-skewed to left-skewed (Table 1 top, from left to right; see also Figure 2 for selected descriptors). The transition toward a more unimodal distribution is also reflected in progressively decreasing range and standard deviation. As anticipated, kurtosis increased as the overall [Na⁺] distribution curve approached a more unimodal shape. The trend toward increasing dominance of a single mode caused entropy to decrease accordingly.

Since the two modes were rather well resolved despite their partial overlap, modes 1 and 2 remained almost completely constant. For the same reason, peak area ratios were close to theoretical values derived from the underlying peak intensities, i₁:i₂ (straight line in second panel of Fig. 2). However, peak height ratios necessarily deviated from this straight line, due to the nonlinearity of the relationship between the chemical-shift and concentration scales.

Quantitative effects of deconvolution on statistical [Na⁺] distribution descriptors can best be analyzed by comparing descriptor values of a well-defined distribution curve before (moderate) deconvolution (Fig. 1 c; Table 1 top, column "Gauss 1:4") with the same distribution curve after deconvolution (Fig. 3 a; Table 1 bottom, column "Gauss 1:4 decon"). Range and standard deviation decreased due to the line narrowing effect of the deconvolution procedure. However, since the two modes were relatively well resolved even before deconvolution, deconvolution only generated minor changes for the other distribution descriptor values.

Nonetheless, skewness values clearly became more negative upon deconvolution, which is a consequence of the more marked left skew. In Figure 3 b, mode areas are plotted for the calculation of height and area ratios for our trimodal Lorentzian-based [Na⁺] distribution (Fig. 1 d; Table 1 bottom, column "Lorentz 1:1:1").

Conclusion

Acknowledgements

References