Digital Signal Processing Reference
In-Depth Information
of the listed imaginary frequency{Im(ν
k
)}via the relationship
1
Im(ν
k
)(ppm)ν
L
(Hz)
λ
k
(s) =
(3.10)
where ν
L
is the Larmor precession frequency, which is equal to 63.864 MHz
at B
0
=1.5T. As per convention in MRS, the frequencies Re(ν
k
) and Im(ν
k
)
are reported in the dimensionless units of parts per million (ppm)
1
, whereas
λ
k
is expressed in seconds (s) and|d
k
|in arbitrary units (au). Furthermore,
chemical shifts Re(ν
k
) given in ppm and hertz (Hz) are related by
Re(ν
k
)(ppm) = 4.68−
1
ν
L
Re(ν
k
)(Hz)
(3.11)
where 4.68 ppm is the resonance frequency of water. In the case of imaginary
frequencies Im(ν
k
) expressed in ppm and Hz, the following relationship exists
1
ν
L
Im(ν
k
)(Hz).
Im(ν
k
)(ppm) =
(3.12)
These fundamental spectral parameters directly yield the peak area S
k
for
the k th resonance where 1≤k≤K. Furthermore, each peak area S
k
is
proportional to the dimensionless concentration C
met
k
of the k th metabolite,
C
met
k
∝S
k
. In this relationship, the proper dimension of the concentration
C
me
k
can be introduced by fixing the constant of proportionality as an over
all multiplying factor. Such a proportionality constant is chosen to be the
concentration of a reference metabolite, C
ref
, expressed in units of mM/g per
wet weight (ww) of tissue. Various numerical values for C
ref
can be selected
for different tissues. For example, for FIDs encoded via MRS in occipital
grey matter of the healthy human brain, the reference metabolite is usually
NAA (nitrogen acetyl aspartate) or H
2
O (water), in which case the usual
concentrations are [88, 143]
C
ref
(NAA) = 6 mM/g
ww
C
ref
(H
2
O) = 85 mM/g
ww .
(3.13)
For the selected C
ref
, the concentration C
met
of the k th metabolite M
k
is
k
deduced from the expression
C
met
k
= S
k
C
ref
.
(3.14)
For the Lorentzian lineshape (2.184), the surface S
k
is given by the area of
a rectangle, S
k
= a
k
h
k
, where a
k
is proportional to the fullwidth at the
halfmaximum (FWHM) of the k th peak, FWHM
k
= Im(ω
k
)/2. Here, h
k
is
the corresponding height defined as h
k
=|d
k
|/[Im(ω
k
)] according to (2.186).
1
Frequency is expressed in dimensionless units of ppm so that a given resonance always
appears at the same position, irrespective of the value of the magnetic field strength B
0
.
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