Application of the Multidimensional Inverse Laplace Transform and Compressed Sensing in Magnetic Resonance Relaxometry for Tissue Characterization
Fourier transform nuclear magnetic resonance spectroscopy (FT-NMR) has been extraordinarily successful for characterizing samples with molecular components that resonate at different frequencies in an external magnetic field. In FT-NMR, molecular components can be obtained from the observed data by a Fourier transform. Unfortunately, there are many examples in which the NMR signal is comprised of components of equal frequency, so that the Fourier transform cannot directly resolve them. An important example is studies of tissue water, the largest component of biological tissues; the differences in resonance frequency of the water within different tissue compartments are small in comparison to spectral line widths, rendering these components of the water signal indistinguishable. However, when components exhibit exponential decay with different relaxation time constants, the inverse Laplace transform (ILT) may be used instead of the FT to resolve and quantify them. Unlike the FT, the ILT is an ill- posed problem so that this procedure is fraught with difficulty. We have found that the stability of the regularized ILT is improved in higher dimensions (e.g. N = 2 or 3), leading to the potential for much improved quantitative tissue analysis in certain circumstances. The lengthy experimental times required for higher-dimensional relaxometry can be partially ameliorated by the use of compressed sensing.