Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 141

the number of the sought physical resonances. This could severely hamper
diagnostics that are supposed to be aided by signal processing. For example,
in brain tumor diagnostics via MRS, as reviewed in chapter 8, several incon
clusive findings have been reported with contradictions depending on whether
some typical metabolites were included or excluded from the basis set of in
vitro spectra used in the LCModel for fitting the corresponding in vivo data
[144, 145].
All fitting algorithms available in MRS use the customary LS adjustments.
This is accomplished in either the time or the frequency domain. In the fre
quency domain, fittings attempt to adjust the envelope spectrum by a chosen
model in order to reconstruct the most important component shape spec
tra of individual resonances and their spectral parameters. As stated earlier,
the LS technique is a minimization of the LS residual, which represents the
squared difference between the input and the modeled information. Any se
lected model in fittings is based upon variations of the free parameters until
eventually a minimum of the LS residual is found. Invariably, a detected
minimum is a local extremum rather than the global minimum. If inter
preted properly, this would have grave consequences especially for nonlinear
fitting (e.g., VARPRO, AMARES, LCModel, etc.) which yields several local
minimae with comparable χ 2 and LS residuals for the given Fourier enve
lope spectrum, thus giving different predictions of spectral parameters with
no way to tell which of the set of estimates is correct. The parameters to
be estimated are often constrained by some conditions to facilitate fittings.
Similarly, in the time domain, the studied time signal is fitted in MRS via
adjustable parameters to match the damped harmonic oscillations in the en
coded raw data. Regardless of the studied domain (time or frequency), fitting
prescriptions from MRS invariably use nonorthogonal expansion functions.
In most cases these basis sets are damped complex exponentials, Gaussians
and/or their combinations as implemented in VARPRO, AMARES, etc. Al
ternatively, rather than modeling individual resonances, fittings in MRS also
use the entire model spectra with some preselected metabolites that play the
role of a nonorthogonal basis set expansion as done in the LCModel. As men
tioned, in all such fittings, nonorthogonality of expansion sets implies that
any alteration in the weighted sum of the LS residuals caused by changes of
one or more adjustable parameters in the selected mathematical model, could
be compensated to a large extent by completely independent variations of the
other free parameters [78]. This arbitrariness becomes particularly problem
atic when the number of adjustable parameters in fittings is not small. Such
a mathematical illconditioning leads to nonuniqueness of all the LS fitting
algorithms currently used in MRS and elsewhere.
In contradistinction, the FPT yields unique reconstructions. In the FPT,
the complex frequencies are unequivocally determined by solving the char
acteristic equation, which as the socalled secular equation is equivalent to
the Schrodinger equation [13]. This leaves no room whatsoever for freedom
in 'varying' the amplitudes, since they are unambiguously fixed by their def