Changyu Sun^{1}, Austin Robinson^{2}, Christopher Schumann^{2}, Daniel Weller^{1,3}, Michael Salerno^{1,2,4}, and Frederick Epstein^{1,4}

^{1}Biomedical Engineering, University of Virginia, Charlottesville, VA, United States, ^{2}Medicine, University of Virginia, Charlottesville, VA, United States, ^{3}Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, United States, ^{4}Radiology, University of Virginia, Charlottesville, VA, United States

Multiband (MB) excitation and in-plane acceleration of first-pass perfusion imaging has the potential to provide a high aggregate acceleration rate. Our recent slice-SPIRiT work formulated MB reconstruction as a constrained optimization problem that jointly uses in-plane and through-plane coil information and MB data consistency. Here we extend these methods to develop k-t slice-SPARSE-SENSE and k-t slice-L+S reconstruction models. First-pass perfusion data with MB=3 and rate-2 k-t Poisson-disk undersampling were acquired in 6 patients. The slice-L+S reconstruction showed sharper borders and greater contrast than slice-SPARSE-SENSE and had better image quality scores as assessed by two cardiologists.

$$\mathop {\arg \min }\limits_{L,S} {\left\| {H\left( {\begin{array}{*{20}{c}} {{L_1} + {S_1}}\\ {{L_2} + {S_2}}\\ \vdots \\ {{L_{{N_s}}} + {S_{{N_s}}}} \end{array}} \right) - {y_{MB}}} \right\|^2} + {\lambda _L}{\left\| {\begin{array}{*{20}{c}} {{L_1}}\\ {{L_2}}\\ \vdots \\ {{L_{{N_s}}}} \end{array}} \right\|_*} + {\lambda _s}{\left\| {T\left( {\begin{array}{*{20}{c}} {{S_1}}\\ {{S_2}}\\ \vdots \\ {{S_{{N_s}}}} \end{array}} \right)} \right\|_1}, (1)$$

where the operator $$$H$$$ (Figure 1C) is defined as $$$H = PQE$$$, $$$N_s$$$ is the number of MB slices, $$$P_z$$$ is the CAIPIRINHA phase modulation matrix for the $$$z^{th}$$$ slice, $$$F$$$ is the fast Fourier transform, $$$E_z$$$ is the sensitivity map for the $$$z^{th}$$$ slice, $$$y_{MB}$$$ is the MB data, $$${\lambda _L}$$$ is the weight for the low-rank constraint, $$${\lambda _S}$$$ is the weight for the temporal frequency sparse constraint, $$$T$$$ is the temporal sparsity operator, $$$m = \left( {\begin{array}{*{20}{c}} {{m_1}}\\ {{m_2}}\\ \vdots \\ {{m_{{N_s}}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{L_1} + {S_1}}\\ {{L_2} + {S_2}}\\ \vdots \\ {{L_{{N_s}}} + {S_{{N_s}}}} \end{array}} \right)$$$ represents concatenated $$$N_s$$$ coil-combined images undergoing reconstruction as $$$L+S$$$ for all slices, $$$m_z$$$ is the coil-combined image

We use variable splitting

We define the conjugate of $$$H$$$ as $$${H^*} = {E^*}{Q^*}{P^*}$$$ (Figure 1D) and redefine $$${Q^*} = \left( {\begin{array}{*{20}{c}} {{F^{ - 1}}{K_1}}& \cdots &0\\ \vdots & \ddots & \vdots \\ 0& \cdots &{{F^{ - 1}}{K_{{N_s}}}} \end{array}} \right)$$$, where $$$K_z$$$ is a matrix that performs a convolution using the split slice-GRAPPA kernel (SP-SG)

For comparison, a k-t slice-SPARSE-SENSE method (Figure 1A) using the temporal total variation

$$\mathop {\arg \min }\limits_{{m_1},{m_2},...{m_{{N_s}}}} {\left\| {H\left( {\begin{array}{*{20}{c}} {{m_1}}\\ {{m_2}}\\ \vdots \\ {{m_{{N_s}}}} \end{array}} \right) - {y_{MB}}} \right\|^2} + {\lambda _1}{\left\| {T\left( {\begin{array}{*{20}{c}} {{m_1}}\\ {{m_2}}\\ \vdots \\ {{m_{{N_s}}}} \end{array}} \right)} \right\|_1} , (2)$$

where $$${\lambda _1}$$$ is empirically chosen as 0.01.

Figure 1: A: The solution of the k-t slice-SPARSE-SENSE
model using CG. B: The solution of the k-t slice-L+S model using CG and
soft-thresholding with variable splitting. *Λ* is the temporal sparsity
thresholding and *SVT* is the singular value
thresholding. C: the H operator is
depicted, including use of sensitivity maps (*E*), FFT (*F*), phase modulation (*P*)
and summation. D: The approximation of *H*^{*} is depicted, including
slice-separating K kernels, phase demodulation (*P*^{*}), IFFT (*F*^{-1})
and coil combination (*E*^{*}).

Figure 2: Rate-6 accelerated perfusion imaging with MB=3
and in-plane undersampling with R=2. 2DIFFT-reconstructions illustrate the artifacts associated with MB=3 and
R=2 sampling (a). Three slices separated using split slice-GRAPPA show remaining
in-plane undersampling artifacts and slice separation artifacts, and form the
initial guess for k-t slice-L+S (b). Images reconstructed by k-t slice-L+S
demonstrate background (L) and dynamic (S) components of three slices simultaneously
(c). The superposition of L and S demonstrates slice separation and artifact
removal (d).

Figure 3: A: Comparison of k-t slice-SPARSE-SENSE
using temporal TV and k-t slice-L+S for image reconstruction applied to first-pass
perfusion MRI with MB=3 and in-plane undersampling with R=2 (three short-axis
slices from a patient are shown). Slice-L+S shows sharper borders and greater
contrast compared to k-t slice-SPARSE-SENSE. B: Blinded image-quality scores
for all 6 subjects. The bar plot shows the mean score of slice-L+S is higher than
slice-SPARSE-SENSE with lower standard deviation.

Figure 4: Rate-6 accelerated perfusion images with
MB=3 and in-plane undersampling with R=2 provides 9-slice coverage using k-t slice-L+S.