Digital Signal Processing Reference
In-Depth Information
Therefore, S
k
=|d
k
|/2, which states the wellknown fact that the Lorentzian
peak area S
k
is independent of the resonance width. This reduces (3.14) to
the following simple expression for obtaining metabolite concentrations
|d
k
|
≡C
met
k
C
k
=
C
ref
C
k
.
(3.15)
2
The physical meaning of the concentration C
me
k
is in representing the abun
dance of the k th metabolite in the scanned tissue. On the other hand,
encoding in proton MRS proceeds via detection of time signals emanating
from protons that are bound to various chemical compounds/molecules in the
examined tissue. Therefore, the concentration C
me
k
is proportional to the
number of protons underneath the k th metabolite peak. In various imple
mentations of the existing sequence encodings in clinical scanners for MRS,
due to several realistic experimental limitations (echo time delay, TE = 0,
finite repetition times, TR =∞, etc.), certain empirical modifications to the
metabolite concentrations are usually introduced via division of the peak ar
eas S
k
by a product of exponentials of the magnetization relaxation type,
S
k
/{exp (−TE/T
2k
)[1−exp (−TR/T
1k
)]}. Here, T
1k
and T
2k
are the k th
metabolite relaxation times [143]. In our computations of metabolite con
centrations, we shall use (3.15) as it stands, with no recourse to the latter
empirical corrections.
The presentation of the results and their analysis is facilitated by referring
to resonances according to their numbers k (also denoted by n
◦
k
in the tables)
where 1≤k≤K (K = 25). These numbers are ordered so that the 1st
resonance is a lipid and the 25th is water. In proton MRS, the following
abbreviations are used for some of the major MRdetectable metabolites, e.g.,
H
2
O, Cho (choline), Cr (creatine), Glu (glutamine), Gln (glutamate), mIns
High quality in vivo time signals encoded via proton MRS are obtained with
water suppression. In such FIDs, the residual water content does not seriously
disturb the determination of concentrations of the neighboring metabolites.
In order to closely mimic this situation, we set up the presently synthesized
FID to contain the residual water metabolite. This resonance will have a small
and broad shape determined by the chosen values of the parameters|d
25
|and
Im(ν
25
) seen at the chemical shift 4.68 ppm on the 25th line in Table 3.1.
Upon inspection of the values of all the other fundamental parameters for
the synthesized time signal from Table 3.1, it can be seen that the corre
sponding exact absorption spectrum, defined as the real part of the associated
complexvalued spectrum, possesses a variety of structures, including isolated,
overlapped, tightly overlapped and nearly degenerate resonances. The exact
absorption component shape spectra for the clinically most informative fre
quencies (0 - 5 ppm), will have isolated, but closelylying resonances, that are
associated with the following 10 peaks: k = 8, 9, 10, 13, 18, 19, 20, 21, 24, 25 lo
cated at chemical shifts 2.261 ppm, 2.411 ppm, 2.519 ppm, 2.855 ppm, 3.481
ppm, 3.584 ppm, 3.694 ppm, 3.803 ppm, 4.271 ppm, 4.680 ppm, respectively.
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