John T. Lundstrom^{1,2}, Megan E. Poorman^{3}, Andrew Dienstfrey^{4}, and Kathryn E. Keenan^{3}

^{1}Department of Physics, University of Colorado, Boulder, CO, United States, ^{2}Associate of the National Institute of Standards and Technology, Boulder, CO, United States, ^{3}Physical Measurement Laboratory, National Institute of Standards and Technology, Boulder, CO, United States, ^{4}Information Technology Laboratory, National Institute of Standards and Technology, Boulder, CO, United States

We investigate non-linear optimization to produce quantitative parameter maps in MR Fingerprinting. Our Monte Carlo analysis shows non-linear optimization to be robust with respect to noise. We also find non-linear optimization to yield consistent results when initialized by dictionary matching using either sparse or densely populated dictionaries. Our research outcomes suggest non-linear optimization is an effective enhancement to the MR Fingerprinting pipeline, allowing for smaller dictionary sizes and more accurate parameter retrieval. Future work will include enhancements to, and analysis of, the computational efficiency of non-linear optimization.

We minimize $$$T_1$$$ and $$$T_2$$$ in a least squares sense

$$\min_{T_1, T_2} || \mathbf{v} - \rho \mathbf{b}(T_1, T_2)||^2$$

Where $$$\mathbf{v}$$$ is the measured signal, $$$\mathbf{b}$$$ is the modeled signal as determined by the Bloch equation, and $$$\rho$$$ is the proton density. For a given $$$T_1$$$ and $$$T_2$$$, there exists an optimal value for $$$\rho$$$ given by a linear analytical solution.

$$\rho_{opt} = \frac{1}{||\mathbf{b}||^2} \left(\mathbf{b_r}\cdot\mathbf{v_r} + \mathbf{b_i}\cdot\mathbf{v_i} + i\left(\mathbf{b_r}\cdot\mathbf{v_i} - \mathbf{b_i}\cdot\mathbf{v_r}\right)\right).$$

We use the above equations along with the MATLAB built-in function, fminsearch, to evaluate the first equation.

In the Monte Carlo simulation, we imitate a measured signal by inserting Gaussian noise into a signal generated from a Bloch solver; the width of the Gaussian noise is determined by the SNR level, $$$ SNR = 10\log_{10}\left(\frac{||\mathbf{v}||^2}{n\sigma^2}\right)$$$, where $$$\mathbf{v}$$$ is the measured signal, $$$n$$$ is the number of time points, and $$$\sigma$$$ is the width of the Gaussian noise. We choose arbitrary nominal values for $$$T_1$$$, $$$T_2$$$, and proton density of 500ms, 50ms, and $$$2 + 3i$$$ respectively. In the experiment, we select 10 levels of SNR and perform 720 Monte Carlo realizations at each level. In a single realization, we generate and insert complex Gaussian noise into the nominal signal, initialize $$$T_1$$$ and $$$T_2$$$ to $$$\pm 10$$$ms from their nominal value, perform non-linear optimization, and record the parameter values after optimization.

To assess the opportunity to use smaller dictionary sizes, we simulate MRF using a numerical phantom (Figure 2). We use an SSFP sequence with undersampled spiral readout (Figure 3A-B). We run the MRF simulation using dictionary matching alone and using dictionary matching with optimization; for each of these, we use both densely populated and sparse dictionaries (Figure 3C). We plot the distributions of the residuals and compare results.

Figure 5 shows the distribution of the residuals resulting from four methods of simulating MRF on the numerical phantom. From the distributions that include optimization, we notice they are centered around zero and are normally distributed. This suggests optimization is well-behaved and produces accurate results. We also notice that when using optimization, the distributions are almost identical, independent of the dictionary type used. This suggests non-linear optimization converges to a global minimum over a wide range of initial conditions and is robust with respect to initial conditions. In turn, with optimization, smaller dictionary sizes could be used, reducing the computer memory and computation time required for dictionary matching. The time savings in dictionary matching would then be used in non-linear optimization where more accurate parameter measurements may be made, as we are no longer confined to the discrete grid of dictionary entries. Further research will include enhancements to the non-linear optimization algorithm to increase the computational efficiency.

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Figure 1: Simulated MRF pipeline with the proposed non-linear optimization enhancement after dictionary matching.

Figure 2: Numerical phantoms used in simulated MRF.

Figure 3: Pulse sequence and dictionary used for data acquisition. (A) Flip angle and repetition time (TR) pattern for the SSFP sequence used to acquire data. (B) Normalized, single arm k-space trajectory used to acquire data in k-space. For each subsequent RF pulse, the trajectory was rotated by 127.5 degrees. (C) Sparse and dense dictionary entries such that T_{1} is greater than T_{2}.

Figure 4: (A) The ten SNR levels spaced logarithmically from 1-14.8 dB. (B) Monte Carlo distributions with varying SNR. In all parameters, T_{1} (blue), T_{2} (orange), Re{ρ} (yellow), and Im{ρ} (purple) distributions are unbiased, centered around the nominal value, and variance decreases as SNR increases.

Figure 5: Residual distributions after simulating MRF on a numerical phantom using dictionary matching alone (red), dictionary matching with optimization (blue), a sparse dictionary (light), and a dense dictionary (dark).

DOI: https://doi.org/10.58530/2022/2596