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+, Ca2+, Mg2+, 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, xk, 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=Wk) 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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