Dan Ma^{1}, Stephen Jordan^{2}, Rasim Boyacioglu^{1}, Michael Beverland^{2}, Yun Jiang^{1}, Darryl Jacob^{3}, Sherry Huang^{4}, Helmut G Katzgraber^{2}, Julie Love^{2}, Mark A Griswold^{5}, Matthias Troyer^{2}, and Debra F McGivney^{1}

^{1}Radiology, Case Western Reserve University, School of Medicine, Cleveland, OH, United States, ^{2}Microsoft, Seattle, WA, United States, ^{3}Physics and Astronomy, Texas A&M University, College Station, TX, United States, ^{4}Biomedical Engineering, Case Western Reserve University, Cleveland, OH, United States, ^{5}Case Western Reserve University, School of Medicine, Cleveland, OH, United States

### Synopsis

MR Fingerprinting (MRF) is a fast quantitative MR imaging technique
that simultaneously quantifies multiple tissue properties. We propose to use
quantum-inspired optimization to characterize the optimization landscape by
using an appropriate cost function to account for signal features and create an
optimization frontier. The simulation results from the optimized MRF sequences
showed reduced bias and variance as compared to those from the original
empirical design. The in vivo maps from the optimized sequences showed improved
image quality as well.

### Introduction

MR Fingerprinting^{1} (MRF) is a fast quantitative MR technique
that simultaneously quantifies multiple tissue properties. Given the large
number of degrees of freedom in designing the flip angle and TR patterns,
optimization of the MRF scan is challenging. Traditional optimization methods
are not well-equipped to handle this non-convex and high-dimensional problem,
and will likely converge to a local minimum. Here, we propose to use
quantum-inspired optimization^{2} (QIO) to better understand the
optimization landscape by using an appropriate cost function to create optimal
FA and TR patterns. The QIO methods applied here mimic the effects of quantum
mechanics which has been reported to drastically outperform previous classical
methods^{3}. They also allow for a flexible cost function that can
include any number of additional constraints. Using these algorithms across
many different sequence patterns, we have identified an optimization frontier,
which appears to define a limits of MRF sequence performance.
### Method

Beginning with an MRF-FISP
sequence^{4}, we formulated a cost function to balance four features
that are desirable in the MRF signal evolutions, namely, signal magnitude,
orthogonality between signals from different tissue types, smoothness, and scan
duration. We adapted substochastic Monte Carlo, a quantum-inspired optimization
algorithm introduced in [2], to continuous variable problems and applied it to
optimize FA and TR patterns to maximize distinguishability between two
representative tissue types, white matter (T1 = 800 ms, T2 = 40 ms) and gray
matter (T1 = 1400 ms, T2 = 60 ms). Different weights were assigned to each
feature in the cost function to generate a frontier representing the trade-off
between the inner product of WM and GM and the average signal intensity.
The quantitative accuracy,
precision and image quality from the optimized and conventional MRF-FISP
sequence (MRF0) were evaluated using Monte Carlo simulations and from in vivo
scans. For each sequence, signals from GM and WM were simulated with SNR
ranging from 0 to 20 dB (10log_{10}(S_signal/S_noise)) with a total of 100 noise
levels, where S_signal is the time averaged signal intensity. At each noise
level, 2000 repetitions were simulated by adding complex random Gaussian noise.
The resulting signal evolutions were matched to the MRF dictionary separately
to obtain T1, T2 and M0 values. The mean and standard deviation of the T1 and
T2 results at each noise level were calculated and compared to the ground
truth. Separately, an in vivo scan was performed in compliance with the IRB and
was performed in a Siemens 3T Skyra scanner. Both the optimized and original MRF
sequences were applied to the same volunteer with an FOV of 300x300 mm2, image
resolution of 1.2x1.2 mm^{2} using a single shot spiral acquisition,
resulting in an acceleration factor of 48. ### Results

The weights in the cost function
were varied to obtain an optimization frontier, which provides an empirical
limit for the performance of MRF sequences that are specific to the cost
function and constraints used in the optimization. Figure 1 shows one example
of the frontier between signal magnitude and orthogonality of the cost function
based on gray matter and white matter signals with 1000 TRs, along with the
estimate from the conventional empirical design (MRF0). The optimized sequences
show better performance on joint magnitude and orthogonality estimation, as
demonstrated by a lower inner product between gray and white matter at the same
level of average magnetization. Based on the frontiers generated using 1000 TRs
and 500 TRs, one sequence optimized for 1000 TR and another optimized for 500
TRs were created by adjusting the weights between signal magnitudes, orthogonality
and smoothness. Figure 2 and 3 compare the percentage of mean error and
standard deviation of the T1 and T2 results from the optimized and MRF0
sequences from Monte Carlo simulations. At lower SNR, the T1 and T2 results
from the optimized sequences have lower bias and variance than those from the
MRF0. Figure 4 compares the in vivo T1 and
T2 maps from the optimized and MRF0 sequences with single shot spiral
acquisition with 1000 TRs. The T2 map from the optimized sequence shows no
shading artifacts between frontal and parietal lobes, and thus provides better
brain structure contrast in the parietal lobe. ### Conclusion

Using quantum inspired
algorithms, an optimization frontier can be mapped out by accounting for signals’
magnitude, orthogonality and smoothness, which indicate the optimal acquisition
parameters of the MRF scans. By taking advantages of this powerful optimization
framework, more factors in terms of signal characteristics and actual scan
conditions could be incorporated in order to generate the optimal MRF scans
under different circumstances. ### Acknowledgements

The authors would like to acknowledge fundings from Siemens Healthineers, Microsoft and NIH grant NIH 1R01EB016728### References

1: Ma et al, "Magnetic Resonance Fingerprinting", Nature, 2013, 495(7440):187-192

2: Jarret, Jordan, and Lackey, "Adiabetic Optimization versus Diffusion Monte Carlo Methods", Physical Review 94(4), 042318

3. Max-SAT
2016 - Eleventh Max-SAT Evaluation. Available at: http://maxsat.ia.udl.cat/results-incomplete/. (Accessed: 5th
November 2018)

4. Jiang
et al., “MR Fingerprinting Using Fast Imaging with Steady State Precession
(FISP) with Spiral Readout.” MRM, 2015,74(6):1621-1631