Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of...

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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

Transcript of Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of...

Page 1: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

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

Page 2: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

Applications of Fourier transform to frequency analysis

Page 3: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

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

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With full width at half maximum (FWHM):

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1πT

=Δf0 2 4 6 8 10

x 104

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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

Page 5: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

The coverage of k-space in an imaging experiment

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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

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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

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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

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The gradient echo and k-space diagrams

Page 10: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

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

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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:

Page 12: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

k-space diagram in a gradient echo experiment

Both positive and negative k can be accessed in a single experiment

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General spin echo imaging

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Sequence diagrams employing spin and gradient echo - 1

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Sequence diagrams employing spin and gradient echo - 2

Page 16: Lecture aims to explain - University of Sheffield/file/... · 2012. 3. 28. · Lecture 8: Basics of magnetic resonance imaging (MRI): use of gradient and spin echo . Lecture aims

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