Pad´e quantification of malignant and benign ovarian MRS data - Signal Processing in MRS with Biomedical Applications - page 285

to converge to all the physical resonances.
However, the situation improves dramatically with N/16 = 64 signal points,
shown on the middle panel in Table 9.2 . By comparing Tables 9.1 and 9.2,
we demonstrate that the FPT succeeded in reconstructing exactly to all six
decimal places all the spectral parameters of each of the twelve peaks with
N/16 = 64 signal points for data derived from benign ovarian cysts. In other
words, all the parameters are identical to the input data at the signal length
N/16 = 64. From the reconstructed spectral parameters at N/32 = 32 and
N/16 = 64, the metabolite concentrations were computed. It can be seen
from the comparison of Tables 9.1 and 9.2 that the concentrations retrieved
by the FPT for N/16 = 64 are exactly equal to the input concentrations.
The bottom panel in Table 9.2 reveals that the convergence is stable with an
increased number of signal points. Namely, at N/8 = 128, all of the spectral
parameters remain identical to those at N/16 = 64. At even longer fractions
N/M (M < 8) of the full FID including N = 1024 (M = 1), we verified that
all the peak parameters reconstructed by the FPT remained unchanged.
Figure 9.1 compares the convergence performance of the FFT and the FPT
for the absorption total shape spectra at three different signal lengths for
the FID corresponding to the benign ovarian cyst data. The three panels
on the left present the absorption spectra of the FFT at N/32 = 32 (top,
(i)), N/16 = 64 (middle, (ii)) and N/8 = 128 (bottom, (iii)). These spectra
generated via the FFT are obviously rough and yield no interpretable infor
mation, whatsoever. The panels on the right show the Padegenerated spectra
at these same three signal lengths. Concordant with Table 9.2, at N/32 = 32,
nine of the twelve metabolites are detected and identified via the FPT. To
be quantitatively identified, the other three resonances: isoleucine, threonine
and choline require 64 signal points. At N/16 = 64, all the peak heights are
correct, and, in agreement with Table 9.2, the total absorption shape spec
trum is fully converged in the FPT at that signal length. At the longer signal
length of N/8 = 128, the convergence is fully maintained, i.e., it is stable.
The convergence patterns of the FFT and FPT are further compared in
Fig. 9.2 . The top two panels recapitulate the rapid convergence of the FPT
attained at N/16 = 64 signal points (right upper panel). At signal lengths
N/32 = 32 and N/16 = 64 the FFT yielded rough and uninformative spec
tra (middle panels in Fig. 9.2). The bottom panels in Fig. 9.2 depict the
convergence pattern of the absorption spectra in the FFT at two large signal
lengths (N = 8K = 8192,N = 32K = 32768) where K denotes the kilobyte (K
= 1024). The first FID length for which the positivedefinite Fourier absorp
tion spectra are obtained is very high, (N = 8K). All the twelve resonances
are seen to be resolved in the FFT at N = 8K on the lower left panel in
Fig. 9.2, but several peak heights are incorrect. This implies that some of
the metabolite concentrations estimated from the Fourier spectra by either
fitting or peak integrations will be insu ciently accurate even at N = 8K.
Moreover, at N = 8K there are significant baseline distortions which would
further compromise both fitting and numerical peak integrations. Eventually,