Lixian Zou^{1,2}, Haifeng Wang^{1}, Yuanyuan Liu^{1}, Weitian Chen^{3}, Yanjie Zhu^{1}, Dong Liang^{1,4}, and Xin Liu^{1}

T_{1ρ} imaging is a promising non-invasive
diagnostic tool for early detection of articular cartilage degeneration. A mono-exponential
model is normally used to describe the T_{1ρ} relaxation process. However, mono-exponentials
may not adequately to describe NMR relaxation in complex, heterogeneous, and
anisotropic materials, such as articular cartilage. Fractional-order models
have been successfully used to describe complex relaxation phenomena in the
laboratory frame in cartilage matrix components. In this work, we develop a
time-fractional order (T-FACT) model to analyze T_{1ρ} relaxation in human articular
cartilage. The results show the proposed method can better represent the T_{1ρ}
relaxation in human articular cartilage.

Theory and Methods

Fractional calculus aims to improve the power of
clinical diagnosis through improved modeling. It has been successfully used to
extend the classical Bloch equations and has been proposed to fit the experimental
bovine cartilage data with an accuracy that is not achievable with the
classical mono-exponential model in the laboratory frame^{2-4}. Similar
to T_{1} and T_{2} relaxation, the process of spin-lattice relaxation in the rotating
frame can be described as time-fractional order relaxation. Here, we adopt time-fractional order (T-FACT) model developed by Magin et al^{2, 4} to express the process of T_{1ρ}
relaxation. Thus, the T-FACT model used to fit voxel-wise image intensities
with different TSLs can be described as $$S_n=S_0 \cdot
E_{\alpha}(\frac{-TSL_n^{ \alpha}}{T_{1\rho}^{'}}),$$ where $$$E_{\alpha}(t)$$$ is the single-parameter
Mittag-Leffler function^{2-4}, $$$T_{1\rho}^{'}=\tau^{\alpha-1}T_{1\rho}$$$,
$$$\tau$$$ is fractional time
constant to maintain a consistent set of units on both sides of Bloch equation,
and α is the fractional-order in time. In the case of α=1, the Mittag-Leffler
function degrades to the conventional mono-exponential relaxation process.

Data sets were acquired from a Philips Achieva 3.0TX
scanner (Philips Healthcare, Best, the Netherlands) with an eight channel T/R
knee coil (Invivo Corp, Gainesville, USA). Sagittal knee scan of one subject
was conducted under the approval of the Institutional Review. A T_{1ρ}-prepared
fast spin echo with fat suppression was used for 39 slices data scanning. The
acquisition parameters include: resolution 1 x 1 x 3 mm^{3}, TR/TE
2200/23 ms, spin-lock frequency 500Hz, and 4 time-of-spinlock (TSL) [0 10 25
50] ms.

The data sets were fitted to the mono-exponential (MONO) model and the T-FACT model, respectively. $$$RMSE_{Model}=\sqrt{\sum_{n}(S_{Model}(TSL_{n})- S_{Acquired}(TSL_{n}))^{2}/N}$$$was performed to compare the two relaxation models voxel by voxel, where N the number of TSLs, $$$S_{Model}(TSL_{n})$$$ the intensity of $$$n$$$th TSL from $$$Model$$$ (means MONO or T-FACT model) and $$$S_{ Acquired }(TSL_{n})$$$ the acquired intensity of $$$n$$$th TSL.

Results

In the all slices of the subject, the T-FACT model shows improved fitting of the acquired data compared to the mono-exponential model. Figure 1 shows typical TDiscussion and Conclusion

Anomalous relaxation illustrated in the first plots on Figure 2(a) leads to the sharp deviation of acquired data from the mono-exponential decay, which results to abnormal T1. Y. J. Wang, Q. Zhang, X. Li, W. Chen, A. Ahuja, and J. Yuan. “T1ρ magnetic resonance: basic physics principles and applications in knee and intervertebral disc imaging”. Quantitative imaging in medicine and surgery, vol. 5, pp. 858-885, 2015.

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

3. 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.

4. 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.

Figure 1. Representative T_{1ρ} prepared images acquired
at TSL 0, 10, 25, 50 ms, respectively.

Figure 2. (a) The comparison of fitting results using
time-fractional order (T-FACT) and mono-exponential (MONO) models on three voxels
in the
articular cartilage shown in (b). (b) T_{1ρ}-weighted image at
TSL=0ms. (c) The measured parameters from two relaxation models and their mean
RMSE. The ROIs were selected around the asterisk on the articular cartilage
indicate the corresponding three regions with measurements shown in the table.
Mean RMSEs show that the T-FACT model fit the relaxation much better than the
MONO model.

Figure 3. A representative slice in the articular cartilage demonstrates:
T_{1ρ}–weighted image (a), T_{1ρ} maps (b, d) and RMSE maps (c, f) from two
relaxation models of MONO and T-FACT, and the α map from T-FACT model. White
arrows (articular cartilage of the patella) show a sharp deviation from the mono-exponential model, but follows
T-FACT better.

Figure 4. Another representative slice in the articular
cartilage demonstrates: T_{1ρ}–weighted image (a), T_{1ρ} maps
(b, d) and RMSE maps (c, f) from two relaxation models of MONO and T-FACT, and
the α map from T-FACT model. Orange arrows show variant results at
regions near the articular cartilage of the patella by using the
two fitting models. The T-FACT model is able to fit the experimental data with
smaller root mean squared error than the classical MONO model as shown in (c,
f).