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Experimental Validation of Augmented Fractional MR Fingerprinting
Lixian Zou1,2, Haifeng Wang1, Huihui Ye3, Shi Su1, Xin Liu1, and Dong Liang1,4

1Paul C. Lauterbur Research Center for Biomedical Imaging, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China, 2Shenzhen College of Advanced Technology, University of Chinese Academy of Sciences, Shenzhen, China, 3State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, Zhejiang, China, 4Research Center for Medical AI, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

### Synopsis

Magnetic resonance fingerprinting is a time-efficient acquisition and reconstruction framework to provide simultaneous measurements of multiple parameters including the T1 and T2 maps. The accuracy of the mapping dictionary of MRF is very important for its clinical applications. In this work, we validated the dictionary performance of the augmented fractional order Bloch equations on MRF in the experimental phantom study. Representative results of experimental phantom demonstrate that the utilization of the augmented fractional model is able to improve the accuracy of the T1 and T2 values.

### Introduction

Magnetic resonance fingerprinting (MRF)1 is a fast quantitative imaging framework for simultaneous quantification of T1 and T2 maps with pseudo-randomized acquisition patterns. A challenge in MRF is the accuracy of the resulting multi-parameters2-4. Many groups have proposed various methods including optimized sequence design2, B1 correction3, and extension the concept of Bloch equations4 to improve T1 and T2 accuracy. Since MRI is a more complex system, there are many anomalous cases being observed such as stretched-exponential or power-law behavior5-7. Fractional order generalization of Bloch equations, thus, is more flexible to describe the dynamics of complex phenomenon including the anomalous NMR relaxation phenomenon. An extensive number of researchers5-7 have proposed applying the fractional calculus in MRI, but rare in MRF. Our group4 have attempted to utilize the fractional order Bloch equations on MRF in a preliminary simulation study. The simulations show that the fractional order method (frac-MRF) is able to improve the evaluation accuracy of the T1 and T2 maps comparing with the conventional dictionary used in MRF (con-MRF). In this work, we try to explore the dictionary performance of the augmented fractional order Bloch equations in the experimental phantom study, which has more value ranges than the conventional fractional order Bloch equations.

### Theory

The fractional order Bloch equations, as developed in Ref.4, adopt the Magin’s fractionalizing approach with incorporating the Caputo derivative into the left side of the Bloch equations. The definition and properties of the fractional derivative can refer to Ref.5-7. The solutions to the fractional order Bloch equations can be solved as $$M_{z}(t)=M_{z}(0)E_{\alpha}(-(\frac{t}{T_{1}})^{\alpha})+M_{0}( \frac{t}{T_{1}})^{\alpha}E_{\alpha,\alpha+1}(- (\frac{t}{T_{1}})^{\alpha}), (1a)$M_{xy}(t)= M_{xy}(0)E_{\beta}(-(\frac{t}{T_{2}})^{\beta}), (1b)Where$E_{*}(t)$and$E_{\alpha,\alpha+1}(t)$are the single and two-parameter Mittag-Leffler function4-7, respectively.$*$$\$ represents α or β. When α and β equals to one, the Mittag-Leffler function corresponds to the classical mono-exponential function and the conventional Bloch equations emerge. Here, the conventional fractional order Bloch equations usually require that α and β are the range from 0 to 1, but the value range of α and β in the model has been augmented to be larger than 1 for MR fingerprinting.

### Methods

Numerical Simulation:

We firstly plotted the longitude and transverse relaxation curves using the augmented fractional order Bloch equations as Eq. (1), where α and β were set to range from 0.6 to 1 with step 0.1. T1 and T2 were set as 1000ms and 80ms, respectively. Signal evolutions were generated using both classical and augmented fractional order models with parameters as Ref.8.

Phantom study:

Dictionary entries used for MRF matching were generated using the two models mentioned above. The dictionary was generated for a wide range of possible T1 values (range from 100 to 4500 ms), T2 values (range from 10 to 1000 ms), α and β values (increase from 0.96 to 1.1 with step 0.01). The MRF image series (full sampled with 600 time points) of the phantom (12 tubes; mixtures of Agar and MnCl2) were acquired on a commercial 3 Tesla Prisma scanner (Siemens Healthcare, Erlangen, Germany) with a 16-channel head coils. The resolution of the images was 1×1 mm2 in a field of view (FOV) 220×220 mm2. The resulting T1 and T2 values were compared to the standard values (Figure 1), which were calculated by conventional spin echo sequence.

### Results and Discussion

As shown in Figure 2a, longitudinal magnetization recovers sharply at the beginning and then slows down to reach the thermal equilibrium when fractional order α goes smaller. Transverse relaxation (Figure 2b) conversely has a rapid decay at the beginning then attenuates slowly with small β. Signal evolutions with variation of fractional order are illustrated in Figure 3a. The process in the complex dynamic system ( when α=β=0.9 ) is interpreted to underestimate the T1 and T2 values when using the conventional dictionary (Figure 3b). α=0.98 and β=1.08 were selected as the best fractional order for the augmented fractional order model to approach T1 and T2 standards. Figure 4 shows frac-MRF with the best α and β improves the accuracy of T1 values perfectly. It also improves the T2 value accuracy compared to con-MRF, but a deviation is shown for the scattered values. As can be seen in Figure 5, T1 and T2 values deviation from the standard values become severe when tubes in phantom are off the magnetic field center, which can be corrected by simulating the B1 into the dictionary.

### Conclusion

In sum, representative results demonstrate that the utilization of the augmented fractional model is able to improve the accuracy of the T1 and T2 values. In the future, it would be further studied for more complex dynamic NMR system in considering time delay, multi-term and transient chaos in applications of fractional calculus9-11.

### Acknowledgements

This work was supported in part by the grant from the National Science Foundation of China (No. 61871373, No. 61471350, No. 81729003, No. 81830056), National Key R&D Program of China (2016YFC0100100), Guangdong Provincial Key Laboratory of Medical Image Processing (No. 2017A050501026), and the National Science Foundation of Guangdong Province (No. 2018A0303130132).

### References

1. D. Ma, V. Gulani, N. Seiberlich, K. Liu, J. L. Sunshine, J. L. Duerk, and M. A. Griswold, “Magnetic Resonance Fingerprinting,” Nature, vol. 495, pp.187-192, 2013.D. Ma, S. Coppo, Y. Chen, D. F. McGivney, Y. Jiang, S. Pahwa, and M. A. Griswold, “Slice profile and B1 corrections in 2D magnetic resonance fingerprinting,” Magn. Reson. Med., vol. 78, pp. 1781-1789, 2017.

2. B. Zhao et al., "Optimal Experiment Design for Magnetic Resonance Fingerprinting: Cramér-Rao Bound Meets Spin Dynamics," in IEEE Transactions on Medical Imaging. doi: 10.1109/TMI.2018.2873704

3. D. Ma, S. Coppo, Y. Chen, D. F. McGivney, Y. Jiang, S. Pahwa, and M. A. Griswold, “Slice profile and B1 corrections in 2D magnetic resonance fingerprinting,” Magn. Reson. Med., vol. 78, pp. 1781-1789, 2017.

4. H. Wang, L. Ying, X. Liu, H. Zheng, and D. Liang. MRF-FrM: A Preliminary Study on Improving Magnetic Resonance Fingerprinting Using Fractional-order Models. Proc. 26th Annual Meeting of ISMRM, Paris, France, 2018.

5. R. L. Magin, X.Feng, D. Baleanu, “Solving the fractional order Bloch equation,” Concepts Magn. Reson., vol. 34,pp. 16–23, 2009.

6. R. L. Magin, Weiguo Li, M. P. Velasco, J. Trujillo, D. A. Reiter, A. Morgenstern, R. G. Spencer, “Anomalous NMR relaxation in cartilage matrix components and native cartilage: Fractional-order models”, Journal of Magnetic Resonance, vol. 210, pp. 184-191, 2011.

7. S. Qin, F. Liu, I. W. Turner, Q. Yu, Q. Yang, and V. Vegh, “Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T2*-weighted magnetic resonance imaging at 7 T,” Magn. Reson. Med., vol. 77, pp. 1485-1494, 2017.

8. Y. Jiang, D. Ma, N. Seiberlich, V. Gulani, and M. A. Griswold, “MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout,” Magn. Reson. Med., vol. 74, pp. 1621–1631, 2015.

9. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. L. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, pp. 1355-1365, 2011.

10. S.L. Qin, F.W. Liu, I. Turner, V. Vegh, Q. Yu, Q.Q. Yang, “Multi-term time-fractional Bloch equations and application in magnetic resonance imaging,” Journal of Computational and Applied Mathematics, vol. 319, pp. 308-319, 2017.

11. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. L. Magin, “Transient chaos in fractional Bloch equations,” Computers & Mathematics with Applications, vol. 64, pp. 3367-3376, 2012.

### Figures

Figure 1. (a) T1 and T2 standard values of the phantom. (b) Reference T1 map. (c) Reference T2 map.

Figure 2. (a) Fractional order longitudinal relaxation curves with T1=1000ms, α range from 0.6 to 1 with an increment of 0.1. (b) Fractional order transverse relaxation curves with T2=80ms, β range from 0.6 to 1 with an increment of 0.1. Note that when α and β equal one, fractional order relaxations correspond to the classical mono-exponential relaxation.

Figure 3. (a) Illustration of signal evolutions with T1=1000ms and T2=80ms when α sets to increase 0.6 to 1 with step 0.1 (supposed α=βfor example). (b) Using conventional dictionary to match the supposed complex environment in a voxel with T1=1000ms and T2=80ms, and the complexity of environment performed as α=β=0.9.

Figure 4. Accuracy of the phantom scan. (a) T1 values from 12 tubes from the phantom using conventional dictionary (con-MRF) and the proposed dictionary (frac-MRF). (b) T2 values from 12 tubes from the phantom using conventional dictionary (con-MRF) and the proposed dictionary (frac-MRF).

Figure 5. Performance comparison: the plots of bias for T1 (a) and T2 (b) in 12 tubes from the phantom using conventional and augmented fractional Bloch equations to generate dictionary.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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