Anthony G. Christodoulou^{1,2}, Jaime L. Shaw^{1,3}, Xiaoming Bi^{4}, Behzad Sharif^{1,5}, and Debiao Li^{1,3}

Quantitative first-pass myocardial perfusion imaging is a potentially powerful tool for diagnosing coronary artery disease. However, quantification is complicated by ECG misfires and the nonlinear response of signal intensity to contrast agent concentration. Here we propose a method overcoming the curse of dimensionality to simultaneously image cardiac motion, contrast dynamics, and T1 relaxation in 2D and 3D, using a low-rank tensor imaging framework for cardiovascular MR multitasking. This non-ECG, first-pass myocardial perfusion T1 mapping method accounts for the signal intensity nonlinearity, allowing direct quantification of contrast agent concentration at any cardiac phase in any cardiac cycle.

For ECG-free, time-resolved T1 mapping, the desired multidimensional image $$$I(\mathbf{x},c,\tau,t)$$$ is a function of spatial location $$$\mathbf{x}$$$, cardiac phase $$$c$$$, saturation recovery (SR) time $$$\tau$$$, and heartbeat index $$$t$$$. Representing this image as a 4-way tensor $$$\mathcal{A}$$$ with elements $$$A_{ijk\ell}=I(\mathbf{x}_i,c_j,\tau_k,t_\ell)$$$, signal correlation can be exploited by modeling $$$\mathcal{A}$$$ as low-rank^{4,5}, i.e., as the outer product of a core tensor $$$\mathcal{C}$$$ and basis matrices^{6}:$$\mathcal{A}=\mathcal{C}\times_1\mathbf{U_x}\times_2\mathbf{U}_\mathrm{c}\times_3\mathbf{U}_τ\times_4\mathbf{U}_\mathrm{t},\qquad(1)$$or in collapsed form,$$\mathbf{A}_{(1)}=\mathbf{U_xC}_{(1)}(\mathbf{U}_\mathrm{t}\otimes\mathbf{U}_τ\otimes\mathbf{U}_\mathrm{c})^T,\qquad(2)$$where each $$$\mathbf{U}$$$ contains a limited number of basis functions for the corresponding
dimension. $$$\mathcal{A}$$$ can then be reconstructed by estimating $$$\mathbf{\Phi}=\mathbf{C}_{(1)}(\mathbf{U}_\mathrm{t}\otimes\mathbf{U}_τ\otimes\mathbf{U}_\mathrm{c})^T$$$ from subspace training data^{4,5} and
then fitting $$$\mathbf{\Phi}$$$ to the remainder of the sparsely sampled data
to recover $$$\mathbf{U_x}$$$:$$\hat{\mathbf{U}}_\mathbf{x}=\arg\min_\mathbf{U_x}\|\mathbf{d}-E(\mathbf{U_x\Phi})\|_2^2+R(\mathbf{U_x}),\qquad(3)$$where $$$\mathbf{d}$$$ is the measured data, $$$E(\cdot)$$$ describes multichannel MRI encoding and sampling, and $$$R(\cdot)$$$ is a regularization functional.

The proposed method employed an ECG-free continuous-acquisition SR-FLASH prototype pulse sequence with readouts collected throughout the entire SR period. For 2D, radial acquisition was performed using a golden-angle ordering scheme, interleaved with 0° radial spoke acquisition every other readout as subspace training data. For 3D, stack-of-stars acquisition was performed with golden-angle ordering for the polar coordinates and variable-density Gaussian random sampling for $$$k_z$$$, interleaved with 0° spoke acquisition at $$$k_z=0$$$ every other readout.

Explicit-subspace low-rank matrix imaging^{7}
was first used to obtain an image $$$I(\mathbf{x},t')$$$ with a
single “real-time” dimension $$$t'$$$. $$$I(\mathbf{x},t')$$$ depicts the overlapping effects of
cardiac motion, T1 recovery, and contrast agent dynamics, allowing image-based
cardiac phase identification. The matrix $$$\mathbf{\Phi}$$$ was
estimated after LRT completion of the subspace training data^{4}. $$$\hat{\mathbf{U}}_\mathbf{x}$$$ was
reconstructed according to Eq. 3, using spatial total variation as the
regularization functional $$$R(\cdot)$$$.

To assess repeatability of resting myocardial blood flow (MBF)
measurements, *n*=8 healthy volunteers were imaged on a 3 T Siemens Verio. Pulse
sequence parameters were FA=10°, TR/TE=3.6/1.6 ms, FOV=270$$$\times$$$270 mm^{2}, matrix size=160$$$\times$$$160, spatial resolution=1.7$$$\times$$$1.7 mm^{2}, and slice thickness=8 mm. Image reconstruction was
performed for 15 cardiac bins and 42 saturation
times. Two 0.1 mmol/kg doses of Gadovist were administered 20 to 30 minutes
apart. Subjects were instructed to hold their breath for as much of the 45 s
scan duration as possible, followed by shallow breathing. To demonstrate the
feasibility of 3D imaging, the same process was performed for a
healthy volunteer using FA=10°, TR/TE=5.9/2.7 ms,
FOV=256$$$\times$$$256$$$\times$$$96 mm^{3}, matrix size=128$$$\times$$$128$$$\times$$$12, spatial resolution=2.0$$$\times$$$2.0$$$\times$$$8.0 mm^{3}.

For quantification, $$$T_1(t)$$$ was calculated for the left
ventricular (LV) blood pool and six myocardial segments in the 2D images at
end-diastole. Contrast agent concentration was calculated as$$Gd(t)={\Delta}R_1(t)/\gamma=\left(\tfrac{1}{T_1(t)}-\tfrac{1}{T_1(0)}\right)/\gamma,\qquad(4)$$where $$$\gamma$$$ is the T1 relaxivity of the contrast agent. Fermi deconvolution^{8}
of each myocardial $$$Gd(t)$$$ by the LV $$$Gd(t)$$$ yielded MBF for each myocardial segment.

1. DiBella EV, Chen L, Schabel MC, et al. Myocardial perfusion acquisition without magnetization preparation or gating. Magn Reson Med. 2012 Mar;67(3):609-13.

2. Sharif B, Dharmakumar R, Arsanjani R, et al. Non-ECG-gated myocardial perfusion MRI using continuous magnetization-driven radial sampling. Magn Reson Med. 2014 Dec;72(6):1620-8.

3. Chen D, Sharif B, Dharmakumar R, et al. Quantification of myocardial blood flow using non-ECG-triggered MR imaging. Magn Reson Med. 2015 Sep;74(3):765-71.

4. Christodoulou AG, Shaw JL, Sharif B, Li D. Proc ISMRM. 2016:867.

5. Liang ZP. Spatiotemporal imaging with partially separable functions. Proc IEEE ISBI. 2007:988-991.

6. Kolda TG, Bader BW. Tensor decompositions and applications. SIAM Rev. 2009 Aug;51(3):455-500.

7. Christodoulou AG, Zhang H, Zhao B, et al. High-resolution cardiovascular MRI by integrating parallel imaging with low-rank and sparse modeling. IEEE Trans Biomed Eng. 2013 Nov;60(11):3083-92.

8. Jerosch-Herold M, Wilke N, Stillman AE, Wilson RF. Magnetic resonance quantification of the myocardial perfusion reserve with a Fermi function model for constrained deconvolution. Med Phys. 1998 Jan;25(1):73-84.

9. Muehling OM, Jerosch-Herold M, Panse P, et al. Regional heterogeneity of myocardial perfusion in healthy human myocardium: assessment with magnetic resonance perfusion imaging. J Cardiovasc Magn Reson. 2004 Jan;6(2):499-507.

10. Likhite D, Suksaranjit P, Adluru G, et al. Interstudy repeatability of self-gated quantitative myocardial perfusion MRI. J Magn Reson Imaging. 2016 Jun;43(6):1369-78.

Fig. 1: Example results showing contrast agent
dynamics for both systole and diastole, pictured at one saturation recovery time.

Fig. 2: Example 3D results pre–myocardial enhancement
and at peak myocardial enhancement, pictured at diastole for one saturation recovery time. This demonstrates the ability to scale to
whole-heart imaging.

Fig. 3: Signal intensity curves take the form of 2D
surfaces when including saturation recovery. $$${\Delta}R_1(t)$$$ can be mapped from these surfaces.

Fig. 4: Time-resolved T1 mapping allows direct
calculation of Gd concentration as $$${\Delta}R_1/\gamma$$$, where $$$\gamma$$$ is the relaxivity of
the contrast agent.

Fig. 5: (a) Two-way ANOVA table, and (b) repeatability
statistics for *n*=8 healthy subjects.
There is a nonsignificant difference (*p*=0.44)
between repetitions and a nonsignificant difference (*p*=0.47) between segments (as expected for healthy volunteers).