Multiplet resonances in MRS data from normal and cancerous prostate - Signal Processing in MRS with Biomedical Applications - page 343

The absolute values{|d k |}of the input amplitudes{d k }were deduced

from the reported metabolite concentrations{C k | =

2C k /C ref . Here, the reference material was taken to be the TSP molecule
(3(trimethylsilyl) 3,3,2,2tetradeuteropropionic acid). The reference con
centration of TSP, C ref was computed from the data reported in Ref. [443].
Specifically, in [443] there was a total of (TSP + D 2 O) = 3.79 mg, where TSP
comprised 0.75% of that weight. There were altogether 0.02845 mg of TSP.
Because the molecular weight of TSP = 172.23 g, there were 0.1652 M of
TSP and the mass of wet tissue = 15.11 mg, such that the concentration of
TSP = 0.1652 M/15.11 mg ww = 10.93 M/g ww.
We set all the phases φ k (1≤k≤27) from complexvalued d k to zero, and
thereby every d k becomes real d k =|d k |. Tables 11.1 , 11.2 and 11.3 display
the input data for the normal glandular, stromal prostate and for prostate
cancer, respectively.
The diagonal FPT (−)

}using (3.15) via |d k

was employed to analyze the FIDs. The coe cients

{p − r ,q − s }of the polynomials P − K and Q − K were computed by solving the sys

tems 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. To extract the peak
parameters, we solved the characteristic equation Q − K (z −1 ) = 0. 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 FPT (−)

extracts the spectral parameters{ω − k ,d − k

}(1≤k≤K) of
every physical resonance using only the input raw FID without any editing or
modification. The k th metabolite concentration C − k of the tissue wet weight

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

as C − k =|d − k |C ref /2 = 5.465|d − k

|M/g ww.
In order to confirm the constancy of the spectral parameters for all three
signals, we progressively increased the signal length for the same bandwidth.
We examined the spectral parameters at total orders K = 250, 300 and 350
where 2K = N P and N P denotes partial signal length. Convergence occurred
at K = 350, 300 and 300 for all the FIDs in the respective cases of normal
glandular and normal stromal prostate and for prostate cancer. All continued
to be stable thereafter. In the parametric signal processing, we determined
whether a given reconstructed resonance was true or spurious by computing
signal poles and zeros, as prescribed by the concept of Froissart doublets
within the fast Pade transform. Spurious resonances have zero amplitudes
due to the coincident poles and zeros of the Pade complexvalued spectrum.
The constancy in nonparametric signal processing was also checked by
computing a sequence of the Pade shape spectra{P − m /Q − m }(m = 1, 2, 3,...)

in the whole Nyquist range, which includes the frequency interval of interest
from 1.3 ppm to 4.2 ppm.
For normal glandular prostate, of the 350 total resonances, 323 were found
to be spurious due to zero amplitudes and the polezero coincidences, thus
yielding the 27 genuine resonances. For normal stromal tissue and prostate