Improved component characterization of multi

Improved component characterization of multi-exponential NMR relaxation data through two-dimensional
stabilization of the inverse Laplace transform
Hasan Celik, Mustapha Bouhrara, David A. Reiter, Kenneth W. Fishbein, and Richard G. Spencer
Laboratory of Clinical Investigation, National Institute on Aging, National Institutes of Health, Baltimore, MD 21224, USA
Nuclear magnetic resonance (NMR) relaxation properties of tissue, such as brain, cartilage and muscle, both in vivo and
ex vivo, are of great interest. NMR relaxation data permits examination of tissue water mobility, providing important insight
into macromolecular composition. Therefore, multi-exponential analysis of NMR transverse relaxation data has become a
widespread technique to determine relative weights and time constants (T 2) of underlying exponentially-decaying
components. This analysis requires application of a numerical inverse Laplace transform (ILT) to the time-domain decay
data. However, the ILT is well-known to be ill-conditioned; even with substantial signal-to-noise, there can be a number of
T2 distributions that fit the data equally well. In addition, small perturbations of the data due to noise or other experimental
factors can result in large changes in the corresponding T 2 histogram. Finally, the results can be very dependent upon
the degree of regularization applied in an effort to obtain more meaningful results. For these reasons, for cases in which
underlying exponential components are not widely separated, interpretation of ILT results can be highly problematic.
An important extension of the multi-exponential analyses described above utilizes two-dimensional relaxation
experiments. In these experiments, one dimension of data, the direct dimension, is collected as a function of time, as for
the case of conventional T2 measurements. A second dimension, the indirect dimension, is collected as a function of an
appropriate incremented delay in the pulse sequence. The indirect dimension can be selected to reflect a variety of tissue
biophysical properties, e.g. longitudinal relaxation constant (T 1), apparent diffusion constant (ADC) or magnetization
transfer rate (MTR). This permits tissue components to be distinguished based on two NMR parameters. Construction of
this 2D distribution of NMR parameters (e.g. T2-T1) from the 2D NMR data requires a 2D ILT.
We have previously demonstrated a substantially greater degree of stability and markedly decreased sensitivity to the
extent of regularization in parameter distributions obtained using the 2D ILT as compared to the 1D ILT under certain
conditions [1]. This surprising effect was explored using extensive simulations and phantom experiments, indicating that
the 2D ILT could be used to determine T 2 distributions with increased accuracy as compared to distributions obtained
using the 1D ILT, even on an equal experimental time basis. Here, we will present the theoretical underpinnings of the
increased robustness of the 2D inversion results as compared to 1D inversion, based on the discrete Picard condition. In
addition, analytic expressions for the data and model resolution matrices will be presented for the 2D ILT. Using these
expressions, the behavior of the 2D transform is directly compared to that of the 1D transform using resolution tests
applied to underlying spike models.
[1] Celik et. al,J. Magn. Res.,236:134-139, 2013