Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 14

nance lineshapes such as Lorentzians, Gaussians or their sum. Similar LS
techniques are also employed in the time domain by fitting a sum of damped
complex exponentials or Gaussians or their product to a given FID. Examples
of such fittings are the “variable projection” (VARPRO), the socalled “ad
vanced method for accurate, robust and e cient spectral fitting” (AMARES),
etc. [73]-[75]. As another option to these fittings of every individual peak,
similar LS adjustments of the given in vivo FFT spectrum as a whole are also
frequently practiced with MRS data. This can be done via, e.g., the “linear
combination of model” (LCModel) in vitro spectra [76]. The LCModel uses
a separately encoded FID on a phantom to set up a basis of “model in vitro
spectra” that are subsequently fitted to the studied in vivo spectrum which is
subjected to quantification. Such model metabolites from in vitro spectra can
alternatively be preselected from the corresponding data banks, if available,
for every concrete case. In practice, no su ciently matching data bases exist
for patients, and preencoding with phantoms is invariably not close enough
to in vivo FIDs. These fittings can use some prior information to constrain
the variations of the adjusted parameters during nonlinear minimization of
the LS residuals. Such residuals are defined as the difference between the
observed envelopes of spectra or FIDs and the corresponding modeled data.
In practice, these fittings are set up by means of iterations. In particular, the
LCModel uses the LevenbergMarquardt nonlinear fitting algorithm [77] and
gives some error assessments of the KramerRao type or the like.
The most serious fault of all the existing fitting recipes is nonuniqueness,
as the same envelope spectrum can be freely adjusted to have any subjectively
chosen number of resonances. In practice, all curve fitting algorithms are sen
sitive to the choice of the number of components and extremely sensitive to
even small errors in the input data. Moreover, in every LS minimization based
upon nonlinear fitting techniques, such as VARPRO, AMARES, LCModel,
etc., the found minimum is, in fact, local, rather than global. This means
that there is more than one minimum having statistically the same χ 2 and LS
residuals, within the prescribed threshold for the sought accuracy. In other
words, different minimae can give nearly the same residual or error spectra
(defined as the model function minus the input data), but the ensuing collec
tions for predictions of spectral parameters could be very different and there
is no criterion to state which of the sets of the estimates for the sought param
eters is correct (if any). In order to patch such fundamental inconsistencies,
some prior information is customarily used in LS fittings in MRS, but this
could easily lead to biased estimates in quantification even when the assessed
KramerRao bounds decrease, as pointed out in Refs. [8]-[11]. This is par
tially due to the usage of nonorthogonal expansion sets. With such sets,
any change of the weighted sum of the squares of the LS residuals, caused by
alterations of one or more adjustable parameters, could largely be compen
sated by independent variations of all the remaining free parameters [78]. To
recapitulate, all fitting algorithms are sensitive to the choice of the number of
components to be fitted and are also extremely sensitive to even small errors