Bo Zhao^{1,2}, Berkin Bilgic^{1,2}, Jason Stockmann^{1,2}, Lawrence L. Wald^{1,2}, and Kawin Setsompop^{1,2}

Magnetic resonance fingerprinting is an efficient quantitative MRI paradigm, which simultaneously acquires multiple MR tissue parameters. Recently, simultaneous multislice (SMS) acquisition has been used to further speed up MRF experiments. In this abstract, we present a maximum likelihood formulation to enable improved SMS-MRF reconstruction. We further describe an algorithm based on variable splitting, the alternating direction method of multipliers, and the variable projection method to solve the resulting nonlinear and nonconvex optimization problem. Representative results are shown to demonstrate that the proposed method enables more accurate MR tissue parameter maps compared to the recent SMS-MRF approach utilizing direct pattern matching.

Denoting $$${\left\{ {I_m^v} \right\}^{}}$$$ as the contrast-weighting images associated with the simultaneously-excited $$$N_s$$$ imaging slices at the $$$m$$$th repetition time, the data model for SMS-MRF can be written as

$${{\bf{d}}_{m,c}} = {{\bf{F}}_m}\sum\limits_{v = 1}^{{N_s}} {{\bf{S}}_c^v} I_m^v + {n_{m,c}},$$

where $$$v$$$ and $$$c$$$ respectively denote the slice and coil index, $$$\mathbf{F}_m$$$ the undersampled Fourier encoding matrix, $$$\mathbf{d}_{m, c}$$$ the measured k-space data, $$$\mathbf{S}_c$$$ the sensitivity maps, and $$$\mathbf{n}_{m, c}$$$ the measurement noise. Furthermore, the contrast-weighting function $$$I_m^v $$$ for each slice can be expressed as $$$I_m^v = \Phi _m^v(T_1^v,T_2^v){\rho ^v}$$$, where $$$ \Phi _m^v(T_1^v,T_2^v)$$$ denotes the contrast-weighting function, and $$$\rho ^v$$$ the spin density. Given that $$$n_{m, c}$$$ is complex Gaussian noise, the maximum likelihood reconstruction can be formulated as

$$ \left\{ {\hat T_1^v,\hat T_2^v,{{\hat \rho }^v}} \right\} = \arg \min \sum\limits_{m = 1}^M {\sum\limits_{c = 1}^{{N_c}} {\left\| {{\bf{d}}_{m,c} - {{\bf{F}}_m}\sum\limits_{v = 1}^{{N_s}} {{\bf{S}}_c^v} \Phi _m^v(T_1^v,T_2^v){\rho ^v}} \right\|_2^2} } .$$

The above formulation leads to a challenging nonlinear and nonconvex optimization problem. Here we extend the algorithm in [7] to solve this problem. More specifically, we use the following variable splitting: $$$g_m^v = \Phi _m^v(T_1^v,T_2^v){\rho ^v}$$$, and $$${h_{m,c}} = \sum\limits_{v = 1}^{{N_s}} {{\bf{S}}_c^vg_m^v} $$$ to form the following augmented Lagrangian functional:

$$ \sum\limits_{m = 1}^M {\sum\limits_{c = 1}^{{N_c}} {\left\| {{{\bf{d}}_{m,c}} - {{\bf{F}}_m}{{\bf{h}}_{m,c}}} \right\|_2^2} } + \frac{{{\mu _h}}}{2}\sum\limits_{m = 1}^M {\sum\limits_{c = 1}^{{N_c}} {\left\| {{{\bf{h}}_{m,c}} - \sum\limits_{v = 1}^{{N_s}} {{\bf{S}}_c^vg_m^v + X_m/{{{\mu _h}}}} } \right\|} } _2^2 + \frac{{{\mu _g}}}{2}\sum\limits_{m = 1}^M {\sum\limits_{v = 1}^{{N_s}} {\left\| {g_m^v - \Phi _m^m\left( {T_1^v,T_2^v} \right){\rho ^v} + Y_m/\mu _g} \right\|_2^2} },$$

where $$$\mathbf{X}_m$$$ and $$$\mathbf{Y}_m$$$ are the Lagrangian multipliers, and $$$\mu_h$$$ and $$$\mu_g$$$ are the penalty parameters. We then use the
alternating minimization scheme to minimize the above functional. In particular, note that the sub-problem with respect to $$$\left\{T_1^v,T_2^v, \rho ^v\right\} $$$ can be solved by the variable projection algorithm as in [7].

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[2] F. A. Breuer, M. Blaimer, R. M. Heiemann, M. F. Mueller, M. A. Griswold, and P. M. Jakob, “Controlled alising in parallel imaging results in higher acceleration (CAIPIRINHA) for multi-slice imaging,” Magn. Reson. Med., vol. 53, pp. 684-691, 2005.

[3] K. Setsompop, B. A. Gagoski, J. R. Polimeni, T. Witzel, V. J. Wedeen, L. L. Wald, “Blipped-controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g-factor penalty,” Magn. Reson. Med., vol. 67, pp. 1210-1224, 2012.

[4] D. A. Feiberg, and K. Setsompop, “Ultra-fast MRI of the human brain with simultaneous multi-slice imaging,” J. Magn. Reson. Med., vol. 229, pp. 90-100, 2013.

[5] H. Ye, D. Ma, Y. Jiang, S. F. Cauley, Y. Du, L. L. Wald, M. A. Griswold, and K. Setsompop, “Accelerating magnetic resonance fingerprinting (MRF) using t-Blipped simultaneous multislice acquisition,” Magn. Reson. Med. vol. 75, pp. 2078-2085, 2016.

[6] H. Ye, S. F. Cauley, B. A. Gagoski, B. Bilgic, D. Ma, Y. Jiang, M. A. Griswold, L. L. Wald, and K. Setsompop, “Simultaneous multislice magnetic resonance fingerprinting (SMS-MRF) with direct spiral slice-GRAPPA (ds-SG) reconstruction,” Magn. Reson. Med., in press. doi: 10.1002/mrm.26271

[7] B. Zhao, K. Setsompop, H. Ye, S. F. Cauley, and L. L. Wald, “Maximum likelihood reconstruction for magnetic resonance fingerprinting,” IEEE Trans. Med. Imag. vol. 35, pp. 1812-1823, 2016.

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Figure 1:
Reconstructed T1 maps using the conventional approach and the ML
reconstruction. Normalized error map associated with each reconstruction is
also included.

Figure 2:
Reconstructed T2 maps using the conventional approach and the ML reconstruction.
Normalized error map associated with each reconstruction is also included.

Figure 3:
Reconstructed spin density maps with the conventional approach and the ML
reconstruction. Normalized error map associated with each reconstruction is
also included.