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

Quantifying tissue
heterogeneity is a major challenge in biomedical MRS and MRI. We present here a
review of several closely related recent methods for deriving, from MRS lineshapes *via* histograms, quantitative
statistical information on heterogeneity, and point out current
and future biomedical applications. At present, heterogeneity in parameters such
as ion concentrations (H^{+}, Ca^{2+}, Mg^{2+}, Na^{+})
are being targeted, but also thermal heterogeneity (relevant to hyperthermia) has been proven to be accessible to this
type of spectral analysis. Future *in-vivo*
studies are expected to benefit from the wider window into the regulation of
physicochemical parameters, enabled by the new underlying paradigm.

Outline of content

The rationale
for obtaining quantitative information on the statistical distribution of
parameter values from an MRS lineshape will be presented based on a numerically
simulated, idealized trimodal MRS resonance (Fig. 1). The spectral line (a) is
first converted into a curve representing the statistical distribution of
parameter *x* (b). Here, a linear
relationship between δ and *x* is assumed; nonlinear relationships would require appropriate intensity
(I) adjustments. Since the MRS resonance is digitized (c), the *x* distribution curve also consists of
digital points, *x*_{k}, from
k=1 to k=m (d). Each digital point can be replaced with a vertical line (e)
that may serve as the center of a histogram bar. The resulting histogram (f) represents
weights (W=W_{k}) of x
values, *i.e.*, their statistical
distribution. This is the basis for the calculation of statistical
descriptors, notably weighted mean and weighted median, mode(s), range, standard
deviation, kurtosis, skewness, entropy, as well as relative sizes of regions
characterized by particular *x* value
ranges; the latter are separated by red and green vertical lines in (d)). MRS
lineshape contributions from factors other than *x* (spurious effects) will be dealt with separately. Note that the *in-silico* model presented here (Fig. 1) is based on an
arbitrary δ range from 1 to 2 ppm, covering an arbitrary range
of *x* values from 0 to 100.

Equations for statistical descriptors of *x* value distributions were derived from analogous
equations available for conventional histograms. For weighted mean, skewness
and kurtosis, the final equations are ^{2}:

weighted mean: $$$\overline{x}=\frac{\sum_{k=1}^{m}(x_k\times W_k)}{\sum_{k=1}^{m}W_k}$$$

skewness: $$$G1_{x}=\frac{n}{(n-1)(n-2)}\sum_{k=1}^{m}W_k(\frac{x_{k}-\overline{x}}{s})^{4}$$$

kurtosis: $$$G2_{x}=\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{k=1}^{m}W_k(\frac{x_{k}-\overline{x}}{s})^{4}-\frac{3(n-1)^{2}}{(n-1)(n-3)}$$$;

with $$$s=\sqrt{\frac{\sum_{k=1}^{m}W_k({x_{k}-\overline{x}})^{2}}{n-1}}$$$, and $$$n=\sum_{k=1}^{m}W_k$$$

The relative flatness (by comparison to a single Gaussian) and asymmetry of the distribution shown in Fig. 1 are clearly reflected by the negative kurtosis and skewness values, respectively (Table 1). The relative heights and areas of the modes fit well with the amplitude ratios used in the design of this curve (mode1:mode2:mode3 = 1:2:4). All calculations can be performed using an EXCEL spreadsheet. Further simulations have been performed for complete validation (data not shown).

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Fig. 1. Steps involved in converting an MRS lineshape reflecting heterogeneity with respect to parameter *x*, into a histogram. The resulting histogram serves as a basis for the extraction of statistical descriptors quantitatively characterizing the underlying heterogeneity within the measured volume (Table 1).

Table 1. Statistical descriptors characterizing the heterogeneity of parameter *x* represented by the simulated MRS resonances and derived distribution curves shown in Fig. 1.