Breast cancer and non-malignant breast data: Quantification by FPT - Signal Processing in MRS with Biomedical Applications - page 314

β−glucose (β−Glc, k = 7) at 3.250224 ppm, taurine (k = 8) at 3.270141 ppm
and myoinositol (mIns, k = 9) at 3.280132 ppm.
Although the reconstructed amplitude|d − k |and metabolite concentration

were completely correct for choline (k = 3), the chemical shift frequency,
Re(ν − k ) and the line width, Im(ν − k ) for choline were exact for five of the six
decimal places (3.210123 ppm instead of 3.210124 ppm, 0.000830 ppm instead
of 0.000831 ppm respectively). At N P = 1000 for glycerophosphocholine
(k = 6) the line width, Im(ν − k ) was exact for five of the six digits (0.000831
ppm instead of 0.000832 ppm), whereas the reconstructed Re(ν − k ) ,|d − k

|and

concentration were all completely correct.
Full convergence was achieved at N P = 1500 in the reconstructed param
eters for all nine resonances (middle panel (ii) of Table 10.2 ) . We show that
convergence was stable at a higher signal length N P = 2000 on the bottom
panel (iii) in Table 10.2, where it can be confirmed that all the reconstructed
parameters remain fully correct. This stability continues at yet higher N P
and at the full signal length N, as theoretically predicted by the FPT.
In Fig. 10.1 the absorption component shape spectra and the total shape
spectra as reconstructed by the FPT (−) are displayed: N P = 1000, 1500 and
2000 for the normal breast data. As can be seen on the right upper panel (iv),
at N P = 1000, the absorption total shape spectrum is converged. However,
this was not so for the component shape spectrum (left upper panel (i)),
where only one peak (phosphoethanolamine, k = 5) at 3.22 ppm is seen and
is overestimated, whereas phosphocholine, (k = 4) is unresolved.
At N P = 1500 in the left middle panel (ii) of Fig. 10.1 the component shape
spectrum is converged such that peaks k = 4 and 5 are resolved and have the
true heights, as do all the other peaks. Therein, the small phosphocholine
peak completely underlies phosphoethanolamine. This can be expected since,
as noted, the difference between the chemical shifts of the peaks k = 4 and 5 is
approximately four times less than the line widths. At N P = 2000 convergence
is confirmed to be stable, as seen in the lower panels for both the absorption
component shape spectrum (iii) and the total shape spectrum (vi), and at
still higher N P including the full signal length N.
Figure 10.2 illustrates the convergence of metabolite concentrations for the
normal breast data for the same three partial signal lengths: N P = 1000, 1500
and 2000 as in Fig. 10.1 and Table 10.2. The input data are shown by the
symbol “x” and the Padereconstructed data are represented by open circles.
Before convergence, at N P = 1000 neither the concentrations of PCho nor
PE are correctly assessed in the reconstruction (top panel (i)), although the
other metabolite concentrations are exact.
At N P = 1500 (middle panel (ii)) and N P = 2000 (bottom panel (iii)), all
of the metabolite concentrations are fully correct, as seen both numerically
and by the graphic representations. This is seen by the “x” being centered
within the open circles, such that there is complete agreement between the
input and reconstructed data.