g16.4427 practical mri 1 – 24 th february 2015 g16.4427 practical mri 1 gradients (continued),...

G16.4427 Practical MRI 1 – 24th February 2015

G16.4427 Practical MRI 1

Gradients (Continued), Signal Acquisition and K-Space Sampling


Spin Echo and Gradient EchoWhen 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 DesignThe 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

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

Bernstein et al. (2004) Handbook of MRI Pulse Sequences


Spin Echo and Gradient Echo

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)


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

The area under the largest phase-encoding lobe can be calculated from:

For N phase-encoding steps:

To satisfy the Nyquist criterion the phase-encoding step size must be chosen so that


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

Phase-Encoding Gradient Design

The area under the largest phase-encoding lobe can be calculated from:

For N phase-encoding steps:

To satisfy the Nyquist criterion the phase-encoding step size must be chosen so that

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

Note: the slice direction z is not necessarily the z-axis

Slice Thickness = ∆f = RF bandwidth

Gz = magnitude of the gradient



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:

Carrier Frequency Offset

What happens if the slice-selection gradient is not spatially uniform?

or



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:

Carrier Frequency Offset

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

or



Slice-Rephasing Gradient

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


If an FOV Lx smaller than D is desired, the signal bandwidth must be reduced by applying a band-limiting filter (sometimes alsocalled an analog anti-alias or hardware filter) prior to the sampling step. After applying the anti-alias filter, the bandwidth is:

Bandlimiting Filter

• 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



“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.”

Nyquist Theorem

Harry Nyquist

February 7, 1889April 4, 1976

x(t)

+B-B


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 magnetizationphase 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



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



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



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:

Samples alongslice direction (slice-encoding area ranges from

+Amax to –Amax in Nphase2 steps)

G can’t be arbitrarily increased (hardware limits) norΔf reduced (worse slice profile and chemical shift)

Step size in k-spacefrom each slice encoding


Any questions?


See you next week!

g16.4427 practical mri 1 – 24 th february 2015 g16.4427 practical mri 1 gradients (continued),...

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