Synopsis

The statistical distribution of a number of measurement parameter values in heterogeneous tissues can be analyzed based on MR spectra, using a paradigm recently introduced by us. However, the MR resonances in question are affected by influences on line shape other than those exerted by the measurement parameter to be analyzed, e.g., by magnetic-field inhomogeneity. We demonstrate here that desired and spurious line shape contributions can be efficiently disentangled by judiciously employing the convolution theorem. This method should facilitate future applications of our method of tissue heterogeneity assessment.

Introduction

Signal filtering techniques have been established practice in high-resolution 1D and 2D NMR spectroscopy for many decades ¹, and are mostly aimed at improving signal-to-noise ratio and/or resolution of MR spectra. One frequently used procedure is the Lorentzian-Gaussian lineshape transformation, consisting of (i) deconvolution with a Lorentzian function, followed by (ii) convolution with a Gaussian of the same linewidth. Morris et al. have suggested to replace the Lorentzian in this scheme by an empirical reference lineshape to compensate for lineshape contributions that are not typically Lorentzian ².

We present here a new deconvolution strategy for cases with significant lineshape broadening due to variations, across an observed volume, of a particular parameter of interest, x, with known influence on chemical shift, δ, i.e., δ = f(x). The goal of our algorithm is to compensate for all lineshape contributions that are not due to x, without otherwise affecting the experimentally obtained lineshape. The parameter x in question, indirectly observable by MRS, may be pH (³¹P) ³; Mg²⁺ (³¹P) ⁴ or Ca²⁺ concentration (¹⁹F) ⁵; temperature (¹H) ⁶, and others. Here, we discuss a computer simulation for x = temperature, as significant thermal heterogeneity occurs, e.g., in tissues under hyperthermia treatment. Simulations and experiments for further parameters, x, were also performed (data not shown).

Methods

Results and Discussion

The basic assumption of our novel approach is that any x-related contribution to the line shape in question is independent of x-unrelated contributions. Under this condition, the x-related contribution can be obtained from an experimentally obtained line shape by removing all x-unrelated (spurious) contributions through deconvolution (Fig. 1, red arrow). The chemical shift of a water ¹H MRS resonance is a linear function of temperature. We simulated an experiment where the water ¹H MRS spectrum of a given biological tissue is first measured at thermal equilibrium (i.e., before application of hyperthermia) to obtain the water line shape in the absence of temperature gradients (Fig. 1, bottom center; Lorentzian reference spectrum, S_ref, representing spurious lineshape contributions only). Then, raw spectra obtained during hyperthermia (S_raw in Fig. 1, bottom left; Lorentzian) are deconvolved with the reference spectrum to yield line shapes defined by temperature gradients only (S_decon in Fig. 1, bottom right; chemical-shift converted to temperature).

In practice, deconvolution is executed using the deconvolution theorem (blue arrows) ⁶ as follows. First, iFT of S_raw and S_ref yield FID_raw and FID_ref (Fig. 1, top left and top center), respectively; however, in practical experiments FID_raw and FID_ref are usually acquired directly. Then, complex division of FID_raw with FID_ref yields FID_decon (Fig. 1, top right) which is Fourier transformed to give S_decon. Note that in this example, S_ref has been set on resonance (0 Hz), corresponding to an unmodulated exponential function (FID_ref). If S_ref is not on resonance, FID_decon needs to be complex multiplied with the iFT of a synthetic spectrum representing a delta function at the chemical shift of S_ref. The same algorithm can be applied to cases where x represents a parameter other than temperature, as mentioned in Introduction.

A comparative summary of the statistical descriptor values obtained for the simulated temperature distributions in Fig. 1, with and without deconvolution, is given in Table 1. Since in this basic simulated example all resonances are symmetric, there is virtually no difference between means, medians and maxima for raw vs. deconvolved distributions, within the precision of the simulation (at 200 digital points or histogram bins for each distribution). However, as deconvolution results in a narrowing of the distribution curve, standard deviation is necessarily decreased and kurtosis increased for S_decon vs. S_raw, whereas skewness and entropy are influenced by the cutoff at high and low temperatures (ranges judiciously chosen; identical for S_decon and S_raw). Further results confirmed our algorithm for a range of bi and trimodal distributions as well (data not shown). The basic equations, generalized for any applicable parameter x, are given in Fig. 2.

Conclusion

Acknowledgements

References