Gradients (Continued), Signal Acquisition and K-Space Sampling G16.4427 Practical MRI 1 Gradients (Continued), Signal Acquisition and K-Space Sampling

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

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.

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

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

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.

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)

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.

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)

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

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

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?

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

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

Any questions?

Signal Acquisition and k-Space Sampling

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

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

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

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

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

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:

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)

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?

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

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

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

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

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

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)

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

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

Any questions?

See you next week!