Lixian Zou^{1,2}, Haifeng Wang^{1}, Huihui Ye^{3}, Shi Su^{1}, Xin Liu^{1}, and Dong Liang^{1,4}

Magnetic resonance
fingerprinting is a time-efficient acquisition and reconstruction framework to
provide simultaneous measurements of multiple parameters including the T_{1} and
T_{2} 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 T_{1} and T_{2} values.

Introduction

Magnetic resonance fingerprinting (MRF)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 function^{4-7},
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. T_{1} and T_{2} 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 T_{1} values
(range from 100 to 4500 ms), T_{2} 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 MnCl_{2})
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 mm^{2} in a field of view (FOV) 220×220 mm^{2}. The
resulting T_{1} and T_{2} values were compared to the standard values (Figure 1),
which were calculated by conventional spin echo sequence.

Results and Discussion

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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.

Figure 1. (a) T_{1} and T_{2} standard values of the phantom.
(b) Reference T_{1} map. (c) Reference T_{2} map.

Figure 2. (a) Fractional
order longitudinal relaxation curves with T_{1}=1000ms, α range from 0.6 to 1 with
an increment of 0.1. (b) Fractional order transverse relaxation curves with
T_{2}=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
T_{1}=1000ms and T_{2}=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 T_{1}=1000ms and T_{2}=80ms, and the complexity of environment performed as α=β=0.9.

Figure 4. Accuracy of the phantom scan. (a) T_{1} values
from 12 tubes from the phantom using conventional dictionary (con-MRF) and the
proposed dictionary (frac-MRF). (b) T_{2} 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
T_{1} (a) and T_{2} (b) in 12 tubes from the phantom using conventional and augmented fractional Bloch equations to
generate dictionary.