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RSL/MRSL Journal Club 8.27.2010 VIPR/Radial MRI Kitty Moran

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Outline

- VIPR
- 2. Aliasing Artifact in Radial MRI

History of Radial MRI

Projection Acquisition was the first MRI k-space trajectory (Lauterbur, 1973)

Bernstein,et al. Handbook of MRI Pulse Sequences2004

History of Radial MRI

- In initial MRI scanners rectilinear sampling was utilized because of B0 inhomogeneity and gradient non-linearity
- With rectilinear k-space trajectories, the effect of these errors is a shift in the location and amplitude of the point spread function, resulting in geometric and intensity distortion
- With PA the effect is not only a shift in location and amplitude but the shape of the point spread function is effected resulting in a blurring artifact

Bernstein,et al. Handbook of MRI Pulse Sequences2004

Undersampling Cartesian K-space

- Aliasing results in wrap around artifacts
- Overlap in image space in the direction in which the undersampling occurs
- Replications occur at frequency of 1/Δkr where Δkr is the sampling frequency in k-space
- To avoid aliasing you want to meet the criteria 1/Δkr = FOV

Sampling in Polar Coordinates

- Data sampled in the radial (r) and angular (θ) direction
- In MRI the image is the inverse Fourier transform of the acquired signal data:

- Translating to polar coordinates

Scheffler, Hennig, Reduced Circular Field-of-View Imaging, 2005

- Scheffler and Hennig considered 5 different sampling patterns in their investigation of aliasing artifacts in polar sampling
- Sampling in either the kr and/or kθ direction
- In the first two examples we consider continuous sampling in one direction and discrete sampling in the other

Discrete sampling in kr, continuous sampling in kθ

Discrete concentric rings in k-space

We assume S(kr,kθ) is circularly symmetric then S(kr,kθ) S(kr) then

Discrete sampling in kr, continuous sampling in kθ

Sum over zero order Bessel functions

Sum over zero order Bessel functions

- Concentric ring lobes surrounding the main peak result in radial aliasing artifacts
- Artifacts occur when object size exceeds the circular field of view diameter of 1/Δkr
- Object smears circularly
- The effective field of view (unaliased region) in polar sampling is a circle of radius 0.5/Δkr

Discrete sampling in kθ, continuous sampling in kr

- Fourier-Bessel integral becomes independent of kr
- Corresponding integrals diverge

- In the case of continuous radial sampling, the extent of sampling in the radial direction must be limited to a finite value otherwise total measuring time and signal power become infinite

Discrete sampling in kθ, continuous sampling in kr

Where anθ = 2πrcos(nθΔkθ – θ)

PSF consists of radial stripes in the sampling direction that produce streak artifacts

- Intensity plot along the angular coordinate θ for a fixed distance r is similar to the Dirichlet kernel for Cartesian sampling

For angular undersampling the radial increment Δkr is much smaller than the maximal angular increment Δkθ

- Similar to case of continuous radial sampling leads to a star-like PSF
- However, small circular flat region at the center
- Angular undersampling allows for artifact free imaging as long as the radius of the object does not extend past the radius of the reduced field of view

Similar to the first type of continuous angular sampling

- PSF shows equidistant ring lobes surrounding the main center peak
- Amplitude of successive rings decreases as a function of the radial distance kr

“optimal sampling”, increased angular sampling

- Circular flat region of the PSF linearly expands to the first ring lobe
- Nθ = 2πNr
- So intensity profile in the radial direction is composed of two parts
- Within the interval (-rΔkr,rΔkr) the profile is approximately given by the Bessel function and therefore is similar to continuous angular sampling
- Beyond the first ring lobe, wild oscillations occur

VIPR

- Vastly Undersampled Isotropic Projection Reconstruction
- Contrast magnetic resonance angiography
- Capture first-pass arterial enhancement
- Separate arterial from venous enhancement
- Time-resolved methods are desirable to eliminate dependence on specific bolus timing
- Trade-off in MRA between spatial resolution and coverage can be greatly reduced by acquiring data with an undersampled 3D projection trajectory in which all three dimensions are symmetrically undersampled

Barger, et al.

VIPR

- Fully sampled 3D PR trajectories are not frequently used
- Np = πNr2
- Aliased energy from undersampling the 3D projection trajectory resembles noise more than coherent streaks
- Oversampling at the center of k-space and interleaving the 3D projections so that the spatial frequency directions are coarsely sampled every few seconds, allows for the possibility of utilizing a ‘sliding window’ reconstruction
- By using a temporal aperture that widens with increasing spatial frequencies, temporal resolution with fast, flexible frame rates can be obtained without any loss in resolution or volume coverage

VIPR

- 3D spherical coordinate system
- Readout direction defined by angle θ from the kz-axis and angle Φ from the ky-axis
- Resolution in all three dimensions determined by the maximum k-space radius value (kmax)
- Diameter of the full field of view is determined
- by the radial sample spacing (Δkr)

VIPR

- Stenotic carotid phantom
- Separate random noise from undersampled artifact
- equal number of excitations and scan time

1,500 projections with 16 signal averages

6,000 projections with 4 signal averages

1,5000 projections with 16 signal averages

12,000 projections with 2 signal averages

24,000 projections with no signal averages

VIPR

- Plot of relative noise due to increased aliased energy from undersampling
- Fewer than 15% of number of projections required for fully sampled k-space leads to significant degradation of image quality
- Two carotid phantoms separated by 10 cm
- Two fully sampled datasets were subtracted to determine noise not due to undersampling artifact
- Noise ratio is noise measured in undersampled datasets relative to noise from subtracted fully sampled datasets

VIPR – Time Resolved

- Coronal MIPs of time resolved images volumes
- Times are time after contrast injection

VIPR – Time Resolved

- Effect of increasing the width of the top of the temporal filter
- Maximum width of the temporal aperture varied from 4 sec to 40 sec
- CNR calculated between artery and background

VIPR – Isotropic Resolution

- Arterial image volume shown 10 seconds after injection
- Isotropic resolution allows for reformat in any plane

VIPR

- Advantages:
- No penalty in terms of coverage and spatial resolution
- Resolution limits are constrained by noise and artifact
- Isotropic resolution provides advantageous in terms of reformatted images
- Oversampling center of k-space so more robust to motion
- Improved coverage
- Limitations:
- Artifacts from undersampling decreases CNR

HYPR

HighlY constrained back Projection

Data are acquired in time frames comprised of interleaved and equally spaced k-space projections

All of the acquired data is combined and used to form a composite image (either through filtered back projection or gridding and inverse fourier transform)

Each projection in a time frame is inverse fourier transformed and back projected in image space

Each back projected time point is weighted by the respective projection of the composite image

The contributions from all projections in a time frame are summed

In both the polar and Cartesian cases, we have replicates at multiples of 1/Δkr

- For Cartesian sampling, the replicates are exact duplicates of the main lobe
- For Polar sampling the replications differ from the main lobe in shape and amplitude
- In Cartesian sampling, exactly M oscillation periods occur between successive replicates
- In polar sampling approximately M oscillation periods occur between successive replicates

Differences between polar and Cartesian undersampling

Polar sampling generally produces streaking and radial aliasing arifacts

These artifacts appear either inside or outside the circular field of view depending on the size of the object

Even for optimal polar sampling, streaks and rings can be observed outside the displayed window

In Cartesian sampling, the Dirichlet-shaped PSF of Cartesian sampling just generates an infinite amount of identical copies of the object separated by the dimensions of the rectangular field of view

In polar sampling the discretization occurs in both the radial and azimuthal directions

We can focus on the radial effects of polar sampling because we effectively have a circularly symmetric ring sampling distribution

The Jacobian is the given area that a discrete sample subtends

In Cartesian sampling, each sample subtends the same constant area (ΔxΔy), however, in equidistant polar sampling, the sample size changes as a function of radius

To normalize the area for each sampling in polar sampling, each sample is weighted by its areal extent which is equivalent to correcting for the sampling density

The Fourier-Bessel (Hankel) transform is for circularly symmetric objects

Main advantages of Projection Acquisition:

Shorter echo times

Reduced sensitivity to motion

Improved temporal resolution for certain applications

Bernstein, pg 898

Trajectory:

3D spherical coordinate system

Readout direction defined by angle θ from the kz-axis and angle Φ from the ky-axis

Resolution in all three dimensions determined by the maximum k-space radius value (kmax)

Diameter of the full field of view is determined

by the radial sample spacing (Δkr)

PA motion artifacts consist of streaks propagating perpendicular to the direction of object motion and are displaced from the image of the moving object itself

Aliasing in Projection Acquisitions:

Since the first lobe ring occurs at 1/Δkr and the object smears circularly, the inner edge of the smeared replicate occurs at (1/Δkr – rmax) where rmax is the maximal radial extent of the object

Thus if rmax is greater than 0.5/Δkr the the replication will overlap the original object

The effective field of view (unaliased region) in polar sampling is a circle of radius 0.5/Δkr

Object size 0.375/Δkr, 3*1/Δkr shown

Object size 0.554/Δkr

In both the polar and Cartesian cases, we have replicates at multiples of 1/Δkr

For Cartesian sampling, the replicates are exact duplicates of the main lobe

For Polar sampling the replications differ from the main lobe in shape and amplitude

Oscillations occur due to the finite sampling extent and are thus often referred to as truncation artifact

In Cartesian sampling, exactly M oscillation periods occur between successive replicates

In polar sampling approximately M oscillation periods occur between successive replicates

In the Cartesian case the complex exponentials are purely periodic while the Bessel function’s non-purely periodic nature leads to destructive interference on summation

In polar sampling, then the radial sampling requirement is the same as in Cartesian sampling, sampling must be above the Nyquist rate. In other words if the radial spacing in k-space is Δkr, the object must be limited to within a circle of diameter 1/Δkr in the image domain to avoid polar aliasing

However, if we assume S(kr,kθ) is circularly symmetric then S(kr,kθ) S(kr) so

Now assume that we sample at the center of k-space (kr = 0) as well as at M equally spaced concentric rings, for radial spacing of Δkr, the ring distribution is given by

The the Fourier-Bessel transform of the sampled pattern is

Sum over zero order Bessel functions weighted by Δkr2

The Cartesian and principal polar PSFs are expressed as summations of complex exponentials and weighted zero-order Bessel functions of the first kind, respectively:

In the Cartesian case the complex exponentials are purely periodic while the Bessel function’s non-purely periodic nature leads to destructive interference on summation

So the FT of S(kr) is

For M = 9, s(r) looks like

VIPR

- Gradient waveforms Gx, Gy and Gz are modulated to trace out radial lines at different θ and Φ angles
- The endpoints of the projections are ordered to smoothly revolve about the kz-axis from the upper pole to the equator
- For N total projections, the equations for the gradient amplitude as a function of projection number n are

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