Tianyu Han^{1}, Teresa Nolte^{1,2}, Nicolas Gross-Weege^{1}, and Volkmar Schulz^{1,3}

MR fingerprinting offers a rapid way to accurately map multiple tissue parameters. The dictionary based reconstruction under the influence of Gaussian noise is identified as a convex optimization problem and solved by a Nelder-Mead simplex algorithm. Instead of a lengthy and uniform sampling proposed by dictionay matching, the new approach using a heuristic and incoherent sampling in the $$$T_1$$$-$$$T_2$$$ space. More robust $$$T_1$$$ estimations are obtained even under severe noise environments. Thus, a robust and fast MR fingerprinting reconstruction can be made without any dictionary.

A voxel uniformly filled with a single kind of tissue was examined throughout this study. The tissue was modeled by two relaxation times ($$$T_1$$$ = 167.87 ms and $$$T_2$$$ = 104.49 ms). The noise-free fingerprint for flip angles and repetition times as depicted in figure 1 was obtained by an extended phase graph (EPG) algorithm ^{[4]}. Moreover, different SNR circumstances were simulated by adding Gaussian distributed noise onto both real and imaginary parts of the noise-free signal ^{[2]}.
For each SNR value, the dot product of the signal with all dictionary entries was calculated and represented by a correlation map.
The dictionary-free reconstruction algorithm aims to locate the global maximum of the dot-product array without computing the entire array, i.e., dictionary. In the initial step, a trial solution based on *a priori* knowledge is used to deduce two other solutions and further forming the initial simplex. The worst solution detected by calculating dot-products between trial solution signals and the target signal is replaced by a set of simplex transformation in each step. This process is repeated until the size of the simplex becomes adequately small. The optimal solution is thus the vertex holding the largest dot-product in the finial simplex. For comparison, an MRF dictionary ($$$T_1$$$ range: [10 ms, 500 ms], $$$\Delta T_1$$$ = 2 ms; $$$T_2$$$ range: [10 ms, 200 ms], $$$\Delta T_2$$$ = 2 ms) is constructed. Furthermore, the deviation of a matched $$$T_{1/2}$$$ from the nominal value was quantified by the average estimation error $$$\delta_{T_{1/2}}$$$ defined by:

$$\delta_{T_{1/2}} = \frac{1}{M}\left(\sum_{i}^{M}\frac{\left | \delta_{T_{1/2}}^{i} \right |}{T_{1/2}}\right); \delta_{T_{1/2}}^{i} = T_{1/2}^{Defined} - T_{1/2}^{Estimated}$$

Adding noise and subsequent matching was repeated M=8 times to obtain an average error. Besides, a total estimation error $$$\delta = \delta_{T_1} + \delta_{T_2}$$$ was defined to characterize the robustness of different reconstruction routines under different SNR. A gelatin phantom was measured on a 3T MRI scanner (Achieva 3T, Philips, The Netherlands). 20 constant density spiral interleaves were combined to sample the K-space. Sparse ($$$\Delta T_1$$$ = 50 ms, $$$\Delta T_2$$$ = 50 ms) and dense ($$$\Delta T_1$$$ = 5 ms, $$$\Delta T_2$$$ = 3 ms) dictionaries ($$$T_1$$$ range: [10 ms, 2500 ms], $$$T_2$$$ range: [10 ms, 600 ms]) were used for measurement image reconstruction and compared to the simplex method.

[1] Ma D, Gulani V, Seiberlich N, et al. Magnetic resonance fingerprinting. Nature. 2013; 495(7440): 187-192.

[2] Sommer K, Amthor T, Doneva M, Towards predicting the encoding capability of MR fingerprinting sequences. Magnetic Resonance Imaging. 2017; 41: 7-14.

[3] Nelder J.A, Mead R, A simplex method for function minimization. The computer journal. 1965; 7(4): 308-313.

[4] Weigel M, Extended phase graphs: dephasing, RF pulses, and echoesâpure and simple. Journal of Magnetic Resonance Imaging. 2015; 41(2): 266-295.

[5] Jiang Y, Ma D, Seiberlich N, et al. MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. Magnetic resonance in medicine. 2015; 74(6): 1621-1631.