1. Gradients (Continued), Signal Acquisition and K-Space Sampling
G Practical MRI 1
Gradients (Continued), Signal Acquisition and K-Space Sampling

2. Spin Echo and Gradient Echo
When the prephasing gradient is applied the spins accumulate phase (differently with location). After the 180° pulse, they will continue to accumulate phase under the influence of the readout gradient and will refocus at the echo time.
The echo is maximum when the area of
the readout gradient is equal to the area
of the prephasing lobe
If the echo coincide with the RF echo,
then off-resonance effects are minimized
In the gradient echo sequence we don’t have the refocusing pulse, so the prephasing lobe has the opposite polarity. What does this tell you about an important difference with spin-echo?
Bernstein et al. (2004) Handbook of MRI Pulse Sequences

3. Readout Gradient Design
The duration of data acquisition Tacq is determined by the receiver bandwidth ±BW and the number of k-space data points along the readout direction nx ( ∆t = sampling time)
The amplitude of the readout gradient plateau can be derived from the FOV along the readout direction Lx
Which for a constant readout gradient has a simple k-space expression
The higher the readout gradient amplitude, the smaller the FOV that can be achieved.

4. Phase Encoding Gradients
Phase encoding creates a linear spatial variation of the phase of the magnetization
It is implemented by applying a gradient lobe while the magnetization is in the transverse plane, but before the readout
By varying the area under the phase encoding gradient, different amounts of linear phase variation are introduced
The resulting signal can be reconstructed with Fourier transforms to recover spatial information about the objects
It is typically used to encode information orthogonal to the frequency-encoded direction

5. Qualitative Description
Bernstein et al. (2004) Handbook of MRI Pulse Sequences
a) At the end of the RF excitation pulse, the transverse magnetization has the same phase (direction) in each pixel.
b) After the phase-encoding gradient is applied, the phase of the transverse magnetization varies at each location along the phase-encoded direction

6. Spin Echo and Gradient Echo
Bernstein et al. (2004) Handbook of MRI Pulse Sequences
In the spin-echo pulse sequence, the phase encoding gradient lobe can occur either before or after the RF refocusing pulse. In both pulse sequences they usually occur approximately at the same time as the prephasing gradient lobe.

7. Implementation
The phase-encoding gradient waveform can overlap with other gradient lobes (not with the frequency-encoded readout)
Usually has the same shape (typically a trapezoid) and time duration for each phase-encoding step and the amplitude is scaled to give the desired ky
Some pulse sequences collect a single line of k-space for each excitation, starting at one edge of k-space and moving continuously to the other edge. Echo train pulse sequences (e.g. EPI) that collect multiple ky lines per excitation, may collect lines in a different order (if the central lines of k-space are acquired first  centric)

8. Phase-Encoding Gradient Design
For N phase-encoding steps:
To satisfy the Nyquist criterion the phase-encoding step size must be chosen so that
The area under the largest phase-encoding lobe can be calculated from:
To minimize TR, the phase-encoding lobes are made as short as possible. What does this tell you about the phase-encoding steps at the edge of k-space?
To minimize TR, the phase-encoding lobes are made as short as possible.

9. Phase-Encoding Gradient Design
For N phase-encoding steps:
To satisfy the Nyquist criterion the phase-encoding step size must be chosen so that
The area under the largest phase-encoding lobe can be calculated from:
To minimize TR, the phase-encoding lobes are made as short as possible. Answer: therefore the phase-encoding steps at the edge of k-space use the maximum gradient amplitude and maximum slew rate
In full Fourier encoding, lines are collected symmetrically around the ky = 0 line. In partial Fourier encoding, one half of k-space is partially filled. The missing data are either zero-filled or restored exploiting some consistency criterion (e.g. Hermitian conjugate symmetry)

10. Slice Selection Gradients
Each application that uses spatially selective RF pulses requires a slice-selection gradient to achieve the desired spatial localization
It is typically a constant gradient that is played concurrently with the selective RF pulse
The RF envelope is modulated with a predetermined shape (e.g. a SINC waveform)
The RF bandwidth ∆f of the RF pulse and the amplitude of the slice-selection gradient determine the location and thickness of the imaging slice
The gradient direction (any combination of the three gradients) determines the normal to the slice plane

11. Qualitative Description
Slice Thickness =
∆f = RF bandwidth
Gz = magnitude of the gradient
Bernstein et al. (2004) Handbook of MRI Pulse Sequences
Note: the slice direction z is not necessarily the z-axis

12. Carrier Frequency Offset
For a general slice that does not pass through the gradient isocenter, the RF carrier frequency must be changed. The proper offset can be calculated from:
Bernstein et al. (2004) Handbook of MRI Pulse Sequences or
What happens if the slice-selection gradient is not spatially uniform?

13. Carrier Frequency Offset
For a general slice that does not pass through the gradient isocenter, the RF carrier frequency must be changed. The proper offset can be calculated from:
Bernstein et al. (2004) Handbook of MRI Pulse Sequences or
Also subject to water/fat offset due to chemical shift
Answer: If the slice-selection gradient is not spatially uniform, the offset δz will also vary and the selected slice will not be planar (e.g. potato-chip-shaped slice)
This often occurs for large value of δz due to gradient non linearity and in the
presence of perturbations of Gz due to local gradient induced by magnetic
susceptibility variations

14. Slice-Rephasing Gradient
Bernstein et al. (2004) Handbook of MRI Pulse Sequences
The slice-selection gradient results in some phase dispersion of transverse magnetization across the slice that causes signal loss
A slice-rephasing or -refocusing lobe is associated with the
slice-selection gradient

15. Any questions?

16. Signal Acquisition and k-Space Sampling

17. Bandwidth and Sampling
The readout (or receive) bandwidth is the range of spin precession frequencies across the FOV
This range depends on the FOV and the amplitude of the frequency encoding gradients
Half-bandwidth = BW (±BW at the scanner)
For a readout gradient Gx, the full range of precession frequencies across an object of length D is equal to

18. Bandlimiting Filter
If an FOV Lx smaller than D is desired, the signal bandwidth must be reduced by applying a band-limiting filter (sometimes also
called an analog anti-alias or hardware filter) prior to the sampling step. After applying the anti-alias filter, the bandwidth is:
Bernstein et al. (2004) Handbook of MRI Pulse Sequences
The A/D converter then samples the signal at intervals Δt = 1/2BW
The Nyquist sampling requirements apply both to the k-space and spatial domains

19. Nyquist Theorem
“If a function x(t) contains no frequencies higher than B Hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.”
x(t) +B -B
Harry Nyquist
February 7, 1889
April 4, 1976

20. MRI Receiver
An MRI receiver removes the Larmor precession frequency of the transverse magnetization
Same effect as if the received data were acquired in the rotating frame
The signal induced in the coil by the precessing magnetization is:
The term ωt in the sine function is removed by multiplying the signal by a sine or cosine oscillating at or near ω, followed by low-pass filtering  demodulation
The real signal induced in the receive coil is then converted into a complex signal suitable for Fourier transform  quadrature
phase of the transverse magnetization
phase of the
coil sensitivity

21. Demodulation
Consider multiplying the function sin[(ω + Δω)t] by sin(ωt):
can be eliminated by a low-pass filter with the appropriate bandwidth
Similarly, multiplying by cos(ωt):
can also be eliminated by a low-pass filter

22. Quadrature Detection
Demodulating the signal by multiplying by sin(ωt) and cos(ωt) followed by low-pass filtering results in two separate signals:
The two signals can be combined in quadrature:
Using complex notation:

23. K-Space
After demodulation to remove the rapid signal oscillation caused by the B0 field, the time-domain signal created by transverse magnetization is:
Defining:
The signal becomes:
Transverse magnetization
Receive coil sensitivity
(accumulated phase)
As time evolves, S(t) traces a path
k(t) in k-space
The gradient amplitude and γ
determine the speed of k-space
traversal
The total distance is determined by
the area under G(t)

24. K-Space Trajectory The k-space trajectory is the path traced by k(t)
Illustrates the acquisition strategy
Influences which type of artifacts can result
Determines the image reconstruction algorithm
The most popular trajectory is a Cartesian raster in which each line of k-space corresponds to the frequency encoding readout at each value of the phase-encoding gradient
Image can be reconstructed using FFT
What is main drawback of Cartesian trajectories?

25. K-Space Trajectory The k-space trajectory is the path traced by k(t)
Illustrates the acquisition strategy
Influences which type of artifacts can result
Determines the image reconstruction algorithm
The most popular trajectory is a Cartesian raster in which each line of k-space corresponds to the frequency encoding readout at each value of the phase-encoding gradient
Image can be reconstructed using FFT
Answer: Long scans, because (except for echo train) each line requires a separate RF excitation pulse

26. Examples of K-Space Trajectory
Cartesian raster
(no echo-train)
Radial
projections
Echo-Planar Imaging (EPI)
Spiral acquisition
Bernstein et al. (2004) Handbook of MRI Pulse Sequences

27. 2D Acquisitions
2D imaging involves slice selection and spatial encoding within the selected slice
Slice selection is accomplished by a gradient played concurrently with a selective RF pulse
Occasionally by saturating signal outside the slice
To cover an imaging volume with a 2D acquisition, multiple sections must be acquired
Sequential or Interleaved acquisition

28. Sequential vs. Interleaved Acquisition
Bernstein et al. (2004) Handbook of MRI Pulse Sequences

29. Data Acquisition Efficiency
In sequential acquisition, the magnetization within a slice is repeatedly excited every TR
If TR is longer than the actual length of the pulse sequence waveforms (Tseq), the scanner become inactive for a period:
Idle time = TR – Tseq
Data acquisition efficiency is the scanner-active time divided by the total scan time
Sequential acquisition have TR ≈ Tseq

30. 3D Acquisitions
A 3D or Volume MR acquisition simultaneously excites an entire set of contiguous slices per TR
The set of slices if called a “slab”
Rectilinear sampling is the most common strategy
Additional “phase encoding 2” or “slice encoding”
Reconstructed by a 3D Fourier transform
Non-rectilinear sampling also possible
Radial/spiral sampling in-plane and phase encoding along slice direction (stack of projections or spirals)
3D-projection acquisition (only frequency encoding)

31. 3D Rectilinear Sampling
Bernstein et al. (2004) Handbook of MRI Pulse Sequences

32. Minimum Slice Thickness
Thin slices reduce partial volume averaging and reduce intra-voxel phase dispersion
In 2D imaging:
In 3D imaging Δz is inversely proportional to the area under the largest phase encoding gradient:
G can’t be arbitrarily increased (hardware limits) nor
Δf reduced (worse slice profile and chemical shift)
Samples along
slice direction
(slice-encoding area ranges from
+Amax to –Amax in Nphase2 steps)
Step size in k-space
from each slice
encoding

33. Any questions?

34. See you next week!