JOURNAL CLUB:“General Formulation for Quantitative G-factor

Calculation in GRAPPA Reconstructions”

Breuer, Griswold, et al. Research Center Magnetic Resonance Bavaria, Wurzburg,

Germany

Mar 31, 2014Jason Su

Motivation• GRAPPA is becoming the dominant form of parallel

imaging– Creating reliable g-factor maps is an important tool to have– Allows the evaluation and optimization of different acquisition

schemes (CAIPIRINHA or even just how to choose Ry, Rz)

• In our studies, we are beginning to wonder what is an acceptable level of acceleration, esp. for visualizing thalamus– G-factor is a critical quantity for this analysis

Theory: GRAPPA• Interpolate missing data in k-

space from neighboring samples with a kernel– Other coils are considered

neighbors

• Attain the linear interpolation weights from central ACS region– Here R=3 and using a 2x3(?) kernel– Ssrc [NcNsrc x Nrep]– Strg [NcNtrg x Nrep]– w [NcNtrg x NcNsrc]

• Input the source samples from all coils

• Output the target points for all coils

Theory: GRAPPA• w can be found with pseudoinverse– ACS is our training data– Find the least squares linear regression of the source to target

points– Predict missing data by sweeping (correlating) over the data

• Convolve the flipped kernels, wkl, for all channels• Sum the contributions from all channels to produce one channel of

data– Validate against sampled data?

Theory: Image Domain Weights

• Combine kernels for different target points together into a single kernel by lining up the target points

• Get the kernel and image dimensions to match by zero-padding• Then:

– By FT and linearity– Here · is element-wise multiplication

Theory: Noise Propagation

• We are interested in how the noise is modified by the GRAPPA kernel– Replace I, the actual image, with the noise image

– The variance of the output noise is then:

• By variance of linear combinations

Theory: Noise Propagation

• I think this would be computed separately for every pixel

• Diagonal entry on quadratic form of covariance matrix with some scale factors– Familiar in form to CRLB covariance

Theory: G-factor

• The g-factor for a coil image• Computed pixel-wise to obtain the whole map

Theory: Combined Images

• For SOS set pk=Ik*/ISOS

– What is ISOS?

• SNR-optimal image combination for both nonnormalized and B1-normalized have equivalent g-factor– Requires coil sensitivities

Theory: Multiple Kernels

• Rm = reduction factor for kernel m

• fm = fraction of k-space kernel applied over

• gm = g-factor associated with that kernel• Each kernel affects the whole image, so we sum the

contributions from each

• For ACS data (R=1, g=1, f = ACS/total lines)• What about edge kernels?

Methods• Siemens 1.5T• 2D Phantom

– TE/TR = 7.1/40ms, α=30deg., bw=100Hz, 256x256– Noise only image with α=0 to measure noise correlation– R = [2, 3, 4]

• 3D In Vivo– MPRAGE: TE/TR = 4.38/1350ms, TR=800ms, α=15deg., bw=180Hz,

256x192x160– Noise only image with α=0– Rectangular and CAIPIRINHA sampling, R=2x2

• 3x3x3 kernel with 24x24x32 ACS block

• Simulated non-cartesian GRAPPA– Variable density– PROPELLER, R = [2, 3, 4]

• Validation against pseudomultiple replica

Pseudomultiple Replica

• Generate 300+ artificial images by adding bootstrapped noise– Collected noise images are randomly reordered

and added to the acquired coil data• Compare analytic g-factor to simulated g-

factor

Pseudomultiple Replica

Results: 2D Phantom

• Perfect match• Overestimation

without including noise correlation

Results

Results: In vivo and PROPELLER

Discussion

• Can be used to identify the optimal reconstruction kernel, acceleration factor, sampling scheme

• For multiple kernels:– Can treat kernels that share source points as

having uncorrelated noise– Why?