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Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo
Lecture aims to explain: 1. Applications of Fourier transform in frequency analysis
2. The coverage of k-space in an imaging experiment
3. The Gradient echo and k-space diagrams
4. General spin echo imaging
Applications of Fourier transform to frequency analysis
t)πfes(t) 0t/T2 2cos(−=
Calculation of NMR line broadening Task of Fourier transform is to find amplitudes of different harmonics constituting the signal, i.e. reconstruct the spectrum. This can be used to convert a time dependence in a frequency dependence
πft-i2ˆ g(t)edt(f)g ∫=
For an FID signal with decay given by:
This gives:
20
222
2
)f(f4π1/T2/T(f)s
−+=ˆ
0 100 200 300 400 500-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FID
sig
nal (
arb.
uni
ts)
Time (microseconds)
With full width at half maximum (FWHM):
2
1πT
=Δf0 2 4 6 8 10
x 104
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Frequency (Hz)
Four
ier
tran
sfor
m o
f FID
sig
nal
Homogeneous broadening (FWHM):
2
1πT
=Δf
Homogeneous vs inhomogeneous broadening
In the case when inhomogeneities are present, inhomogeneous broadening (FWHM): *
1
2πT=Δf
The latter expression is an approximation as an inhomogeneously broadened lineshape is generally described with a Gaussian giving non-exponential decay in the time domain
The coverage of k-space in an imaging experiment
Typical parameters in an MR imaging experiment
The goal of MR imaging: to determine the spatial distribution of a given species within the sample
In experiment this can be done by determining the “frequency content” of the resulting MR signal, provided a well-defined spatial field gradient is superimposed on the homogeneous static field: Larmor precession of the spins (once flipped in the xy-plane) will depend on their spatial coordinate – frequency encoding
In a typical clinical MR imaging a static field of 1.5 T can be used and a linear gradient G=10 mT/m
1D imaging equation (reminder)
Demodulated signal s(t) is rewritten in a form where the spatial frequency k=k(t) given by:
kz-i2ρ(z)e dzs(k) π∫=
If the gradient G is constant over time τ=(0,t), k(t) is simply:
∫=t
0
Gdτ2πγk(t)
The spin density ρ(z) of the sample is found by taking the inverse Fourier transform of the signal:
kzi2 s(k)edk(z) πρ +∫=
Gt2πγk(t) =
In order to calculate this integral precisely, ‘good coverage’ of k-space is required
Limiting factors for (1D) imaging Need to collect a uniform distribution of points in k-space – only necessary to sample the signal at constant rate in the presence of a constant gradient
Main factors preventing collection of continuous data over all k-space are (i) finite time of the imaging experiment; (ii) relaxation wiping out the signal within a finite period of time
Sampling both negative and positive values of k can be achieved by changing the sign of the gradient
Note: gradients themselves cause spin dephasing as they are effectively magnetic field inhomogeneities
0 100 200 300 400 500-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FID
sig
nal (
arb.
uni
ts)
Time (microseconds)
The gradient echo and k-space diagrams
1D MRI experiment
(c) Gradient is applied (‘dephasing lobe’) (d) Second refocusing gradient is applied (‘rephasing gradient lobe’) The condition for the gradient echo in (d):
123E tttT −+=Important: decay due to single gradient is reversed which allows extension of the k-space
k-space diagram
As followed from the imaging equation, ‘good coverage’ of k-space is required
In an FID experiment with a single gradient the following k-space coverage is possible:
k-space diagram in a gradient echo experiment
Both positive and negative k can be accessed in a single experiment
General spin echo imaging
Sequence diagrams employing spin and gradient echo - 1
Sequence diagrams employing spin and gradient echo - 2
SUMMARY Task of Fourier transform is to find amplitudes of different harmonics constituting the signal, i.e. reconstruct the spectrum. This can be used to convert a time dependence in a frequency dependence (or in spatial coordinate if frequency encoding is used) .
t)πfes(t) 0t/T2 2cos(−=Example for an FID signal with decay:
20
222
2
)f(f4π1/T2/T(f)s
−+=ˆ Giving full width at half
maximum (FWHM): 2
1πT
=Δf
Gradients, necessary for imaging, introduce strong inhomogeneity. Gradient echo is used to compensate for this and extend the range in the k-space where the signal is collected.
Combination of gradient and spin echo are used to serve this purpose, which will also remove inhomogeneities of the static field.
k-space diagrams are employed to demonstrate the range of k-space covered in an experiment