Breast cancer and non-malignant breast data: Quantification by FPT - Signal Processing in MRS with Biomedical Applications - page 312

herein taken to be approximately 1 Hz, allowing slight variations, and also
assuming that the peaks were Lorentzian. The line widths are proportional to
Im(ν k ). It is emphasized that the mentioned smallest chemical shift difference
of≈2×10 −4 ppm is 4 times less than the typical line width of 8×10 −4 ppm.
The phases φ k (1≤k≤9) from complexvalued d k were all set to zero, so
that every d k becomes real, d k =|d k |.

Table 10.1 displays the input data for the normal breast tissue (upper panel
(i)), fibroadenoma (middle panel (ii)) and for breast cancer (bottom panel
(iii)). The diagonal FPT (−) was used to analyze the FIDs with the coe

cients{p − r ,q − s }of the polynomials P − K and Q − K computed by solving a system

of linear equations from chapter 4 (section 4.10) by treating the product in
G N (z −1 )∗Q − K (z −1 ) = P − K (z −1 ) as a convolution. We solved the character
istic equation Q − K (z −1 ) = 0 to extract the peak parameters. As discussed,

this leads to K unique roots z − k (1≤k≤K), so that the sought ω − k is

deduced via ω − k = (i/τ) ln (z − k ). The k th metabolite concentration C − k

was

computed from the absolute value|d − k |of the reconstructed amplitude d − k

as

C − k = 25|d − k |C ref M/g ww.

We systematically increased the signal length for the same bandwidth (i.e.,
3 acquisition times) to demonstrate the constancy of the spectral parameters
for all three signals. The spectral parameters were examined at total orders
K = 500, 750 and 1000, where 2K = N P and N P denotes partial signal
length. Convergence occurred at K = 750 for all three FIDs, and remained
stable thereafter. We determined whether a given reconstructed resonance
was true or spurious by using the concept of Froissart doublets throughout
the whole Nyquist interval with a particular focus on the frequency range of
interest from 1.3 ppm - 3.3 ppm. For all three examined FIDs, of the 750
total resonances, 741 were identified as spurious Froissart doublets by their
zero amplitudes and the polezero coincidences; the nine remaining resonances
were genuine.

10.4.1 Pade-reconstruction of MRS data for normal breast
tissue
Table 10.2 shows the data reconstructed by the FPT (−) for the normal breast
tissue. Three illustrative partial signal lengths (N P = 1000, 1500 and 2000)
are shown therein. At N P = 1000 (upper panel (i)), only eight of the nine
resonances were identified. In the interval where there should be two peaks,
PCho (k = 4) at 3.220012 ppm and PE (k = 5) at 3.220215 ppm, only a single
resonance was identified at 3.220189 ppm. Since this peak was closer to PE,
we gave it that assignment.
At N P = 1000, the reconstructed amplitudes and, therefore, the concen
trations for that single peak were approximately the sum of (PCho + PE).
At N P = 1000 all the reconstructed spectral parameters were fully exact for
lactate (Lac, k = 1) at 1.330413 ppm, alanine (Ala, k = 2) at 1.470313 ppm,