Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 144

the tissue from which the FID has been encoded.
We have shown that the quantification problem in MRS can be solved ex
actly with the FPT applied to a typical synthesized noiseless FID. As noted,
the values of the complex frequencies and amplitudes for the 25 component
resonances are akin to time signals encoded in vivo on clinical MR scanners
at B 0 =1.5T using short TE of 20 ms at bandwidth 1000 Hz and the signal
length 1024 [88]. Similarity between the presently synthesized FID and time
signals encoded via in vivo MRS was further ensured by selecting spectral
parameters that would generate isolated, overlapped, tightly overlapped as
well as almost degenerate resonances.
It should be recalled that chemical shifts are the resonating frequencies of
the tissue protons. These are much smaller than the Larmor precessing fre
quency of proton magnetization vectors around the axis of the external static
magnetic field of strength B 0 . It is this very small correction to the Larmor
frequency that is the basis of the entire MRS. Without these small corrections,
all the scanned protons would resonate at a fixed identical Larmor frequency
and, therefore, the spectrum would be comprised of just a single peak. In fact,
this would be the case if all the protons under investigation were free. How
ever, protons are bound in various molecular compounds in the tissue. The
electronic clouds of various molecules are able to produce different shieldings
of proton magnetic fields. Due to such varying shieldings, protons resonate
with the external field at frequencies that are slightly shifted relative to the
Larmor frequency. This shielding depends on the proton chemical surround
ing. Nevertheless, especially for the human brain metabolites investigated via
proton MRS, the range of such shifting is quite small. Namely, for the human
brain investigated via proton MRS, most of the chemical shifts associated
with the main part of a customary spectrum are found between 0 ppm and
5 ppm. This range is just a fraction of the entire Nyquist interval. The res
onances selected for our synthesized FID correspond to a linear combination
of damped complex exponentials with stationary amplitudes.
As discussed, the FPT can also handle degenerate resonances that lead to
nonLorentzian spectra. In such a case, the corresponding FID becomes a sum
of damped complex exponentials with nonstationary (or timedependent) am
plitudes. Parametric as well as nonparametric analyses can be performed by
the FPT. As described, the former is achieved by explicitly solving the quan
tification problem. Nonparametric analysis is performed as a transform, with
frequencybyfrequency computation of the response function. This is done
without necessarily reconstructing spectral parameters themselves.
The two different versions of the FPT via the FPT (+) and FPT (−) unify the
PA and PzT, where z is the frequencydependent complex harmonic variable.
These two versions of the FPT were created with the same truncated input
Maclaurin series expansion for the exact spectrum as expressed by the exact
Green function of the system under examination. The expansion coe cients in
this Maclaurin development are the time signal points, that are equivalent to
the quantummechanical autocorrelation functions. The FPT (+) and FPT (−)