Norbert W Lutz^{1} and Monique Bernard^{1}

Materials such as biological tissue are often characterized by considerable heterogeneity. This can manifest itself in significant variability of certain physicochemical parameter values across the measured volume. If the chemical shift of a particular MR resonance varies systematically with such a parameter, the resulting lineshape can be used to quantitatively characterize the heterogeneity with respect to this parameter. This is achieved by transforming the MRS lineshape into a curve representing the statistical distribution of the parameter values in question, followed by the derivation of a histogram. We study here two important conditions for the statistical evaluation of such spectrum-derived histograms.

Introduction

Heterogeneous materials such as biological tissue are known to significantly broaden MRS lineshapes when compared to homogeneous solutionsResults and Discussion

Each digital point of an MRS resonance was
transformed into the analog of a histogram bin. Thus, we first analyzed the
conditions for constructing correct histograms. In a conventional histogram,
the height (= weight W_{k}) of a bin is proportional to the number of
measurements that result in parameter values falling into a given interval (Fig. 1
top; arbitrary max. count = 7 in this example). For our purpose, the counts of histogram bins, b_{1},
b_{2}, ... b_{m}, are replaced with the heights of digital
points from MRS resonances. These points define the positions of the **center lines** of "surrogate
bins", x_{1}, x_{2}, ... x_{m} (Fig. 1 bottom, vertical
lines; arbitrary max. height = 2800 in this example). Although for conventional histograms, the
median is determined by identifying the *x*
value at half the total cumulative sum (*x*
= *x*_{custot/2}), our
point-based histograms require that to this value, half the difference between x_{custot/2} and the next
higher *x* value, x_{custot/2+1}, be chosen: *x* = *x*_{custot/2} +
0.5(*x*_{custot/2+1}- *x*_{custot/2}).

Evaluations of simulated temperature distribution curves (example: Fig. 2) demonstrated that correction for the center of histogram bins indeed lead to noticeable changes for medians. Based on our calculations, uncorrected medians for monomodal (mode 3), bimodal (modes 2 and 3) and trimodal (modes 1 to 3) distributions amounted to 62.19, 58.92 and 56.61 °C, respectively, whereas the corresponding corrected medians were 62.77, 59.51 and 57.19 °C, thus stressing the relevance of the corrections introduced here.

Furthermore, if the number of
bins, m, of a histogram is not optimally chosen, the
standard deviation, *s*, derived from the histogram becomes dependent on the m/n
ratio, with $$$s=\sqrt{\frac{\sum_{k=1}^m W_{k}(x_{k}-\overline{x})^{2}}{n-1}}$$$.
The mathematical expressions for several other
statistical descriptors, notably skewness and kurtosis, also contain *s*. In our
algorithms, these descriptors are rendered independent of m/n by proportionally
upscaling all W values (corresponding to raw intensities), which does not affect the statistics of the
distribution. Based on bimodal pH distributions with well-defined m values, we
determined that all three descriptors become virtually n-independent for $$$n>>(10\times m)$$$,
with $$$n=\sum_{k=1}^mW_{k}$$$ (example: m = 40 for Fig. 3, derived from a
phantom 3-APP ^{31}P MR spectrum). Thus, a scaling factor, N (Fig. 3
top left), such that $$$\sum_{k=1}^mW_{k}>>(10\times m)$$$, avoids algorithmic
artifacts. Corrections for MRS lineshape contributions unrelated to *x* will be presented separately.

Conclusion

Two conditions must be met to obtain accurate quantitative descriptors of statistical distributions of measurement parameter1. Juchem C, Merkle H, Schick F, Logothetis NK, Pfeuffer J. Region and volume dependencies in spectral line width assessed by 1H 2D MR chemical shift imaging in the monkey brain at 7 T. Magn. Reson. Imaging 2004;22:1373–1383.

2. Lutz NW, Le Fur Y, Chiche J, Pouyssegur J, Cozzone PJ. Quantitative in-vivo characterization of intracellular and extracellular pH profiles in heterogeneous tumors: a novel method enabling multiparametric pH analysis. Cancer Res 2013;73:4616–4628.

3. Lutz NW, Bernard M. Multiparametric quantification of thermal heterogeneity within aqueous materials by water 1H NMR spectroscopy: paradigms and algorithms. PLOS ONE 2017;12:e0178431.

Fig. 1. Analogy between a conventional histogram (top) and a histogram generated from the digital points of an MRS resonance (bottom). While the heights of conventional histogram bars are defined by the number of measurements of parameter values that fall into given intervals (bins b_{1}, b_{2}, ...b_{m}), the heights of lineshape-derived histogram bars are proportional to the heights (intensities) of digital points. The vertical lines (bottom) indicate
the positions
of the center lines of
histogram bars ("surrogate bins"), x_{1}, x_{2}, ... x_{m}.

Fig. 2.
Numerically simulated
trimodal temperature distribution curve, derived from an idealized simulated ^{1}H MRS resonance of water obtainable from (semi-)aqueous materials such as biological tissue. Distinct areas under the temperature distribution curve are color coded, and correspond to particular temperature ranges under characteristic curve maxima (= modes).

Fig. 3. Top left: Experimentally measured pH distribution curve from a phantom with two 3-APP solutions at different pH values. The curve was derived from the underlying ^{31}P MRS lineshape (W = raw intensity). Red dots indicate the positions of the digital points directly derived from the corresponding points of the ^{31}P MR spectrum. Large scaling factors (N) result in large W_{k} and, therefore, large n values (see text). For increasing n, the values for standard deviation, skewness and kurtosis converge toward n-independent values (diagrams top right and bottom).