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 {Ivm} as the contrast-weighting images associated with the simultaneously-excited Ns imaging slices at the mth repetition time, the data model for SMS-MRF can be written as
dm,c=FmNs∑v=1SvcIvm+nm,c,
where v and c respectively denote the slice and coil index, Fm the undersampled Fourier encoding matrix, dm,c the measured k-space data, Sc the sensitivity maps, and nm,c the measurement noise. Furthermore, the contrast-weighting function Ivm for each slice can be expressed as Ivm=Φvm(Tv1,Tv2)ρv, where Φvm(Tv1,Tv2) denotes the contrast-weighting function, and ρv the spin density. Given that nm,c is complex Gaussian noise, the maximum likelihood reconstruction can be formulated as
{ˆTv1,ˆTv2,ˆρv}=argmin
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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