Title:
Application of the Multidimensional Inverse Laplace Transform and Compressed Sensing in Magnetic Resonance Relaxometry for Tissue Characterization
Abstract:
Fourier transform nuclear magnetic resonance spectroscopy (FTNMR) has been extraordinarily
successful for characterizing samples with molecular components that resonate at different
frequencies in an external magnetic field. In FTNMR, 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 higherdimensional relaxometry can be partially ameliorated by
the use of compressed sensing.
