Improved algorithm for noniterative time-domain model fitting to exponentially damped magnetic resonance signals

This communication is concerned with a new method of fitting a physical model function to a magnetic resonance signal, directly in the time domain. Our primary aim is analysis of the signal in quantitative terms, i.e., describing the signal in terms of physically meaningful parameters with their statistical errors. Before explaining the new method we make some remarks about the place of time-domain model fitting in spectral analysis. The notion of quantitative description just defined goes beyond constructing a spectrum from the available time-domain data. This judgment is supported by the observation that recently proposed methods for constructing a spectrum (l-3), although very useful for many purposes, have not provided values of the physical parameters involved. In fact, if the latter are wanted afterward, it is then still necessary to fit an appropriate model function to the spectrum (4, 5). Furthermore, in addition to the fitting, the degree to which the spectrum constructed approximates the “ideal” spectrum is to be assessed. On the basis of these considerations, we advocate the use of timedomain model fitting, if quantitative description of a signal is needed. On the other hand, if the primary need is to have a spectrum, one may well use methods such as proposed in Refs. (Z-3), some of which require significantly less computer time. If the signal decays exponentially, which is not uncommon in magnetic resonance, an additional advantage of remaining in the time domain emerges, namely that the fitting procedure can be made noniterative. To the best of our knowledge noniterative fitting procedures are not yet available for the frequency domain. A method of noniterative fitting has recently been devised by Kumaresan and Tufts (6) and was subsequently applied to magnetic resonance (7, 8) under the name LPSVD; see also (9) for a related method. An error analysis was given in (6, IO). We are here concerned with an alternative noniterative model fitting procedure, devised by Kung et al. (II) using the so-called state space formalism. The method can handle considerably more data points than LPSVD because polynomial rooting is avoided. At the same time, the residue of the fit is usually better than that of LPSVD. We shall indicate that the basic idea can be explained with elementary matrix algebra, without invoking the state space formalism. In addition, a formula for efficient computer implementation is given.