Dorottya Papp^{1}, Gyula Kotek^{1}, Stephan Breda^{1}, Dirk H.J. Poot^{2}, Edwin H.G. Oei^{1}, and Juan Antonio Hernandez-Tamames^{1}

Determination of the T2* relaxation times with biexponential and monoexponential models has limitations, especially in case of complex, heterogeneous materials^{1}. Using fractional order fitting model we can overcome these limitations^{8,9}. In this model the introduced α parameter is the measure of the deviation from the monoexponential decay, and it accounts for micro-structural complexity. To evaluate the fractional order fitting, it was performed in patella tendon from UTE measurements, and we could demonstrate that compared to biexponential and monoexponential models it is less sensitive to variations to SNR.

**Introduction**

The $$$T_2^*$$$ values of highly organized tissues such as
cartilage or trabecular bone are extremely short (0.5ms-5ms). Hence, there is
an increasing interest for sequences with ultrashort echo time (UTE) or with
zero echo times (ZTE)^{2,3,4,5.} These short TE techniques in complex,
heterogeneous materials reveal stretched exponential $$$T_2^*$$$ decay^{6,7} (model 1) as opposed to a
simple exponential (model 2) or biexponential decay (model 3). Fractional order
generalization of the Bloch equation provides and alternative way for the
description of the relaxation processes. We compared the performance of all three
models on *in-vivo* data from 3D-UTE-MRI measurements.

**Methods**

As proposed by Magin et
al.^{8,9} in the fractional
order representation the transversal relaxation is described by the following
equation:

$$ M_{xy}(TE)=M_{xy}(0)E_{\alpha} \Big[ -\Big( \frac{TE}{T_2} \Big) ^{\alpha} \Big] +M_{xy}(\infty)$$

where $$$M_{xy}(0)$$$ is the transversal magnetization at TE=0, $$$M_{xy}(\infty)$$$ is the transversal magnetization at the steady state and $$$E_{\alpha}$$$ is the stretched Mittag-Leffler function: $$$E_{\alpha}(z)=\sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k+1)}$$$. In case of α=1 , the function is equivalent to the simple exponential function.

The basis of comparison between the different mathematical models were resulting $$$T_2^*$$$ maps of in-vivo volunteer scans.

Data was retrospectively collected from the Jumper Study (a randomized controlled trial in athletes with patellar tendinopathy who play tendon loading sports at least three times a week, aged 18-35 years). MRI of the symptomatic knee was performed using a 3T MR system (Discovery 750, General Electric, Boston, Massachusetts, USA) using a flexible 3.0 T 16-Channel surface coil (NeoCoil, Pewaukee, Wisconsin, USA). The center of the surface coil was aligned with the patellar apex. Regarding 3D-UTE-MRI, a total of 16 echoes were acquired at TEs of 0.032, 0.49, 0.97, 2.92, 4.87, 6.82, 8.77, 10.72, 13.6, 12.67, 16.57, 18.52, 18.7, 20.47, 22.42, 24.37, 26.32 msec. The 16 echoes were scanned in 4 separate multi-echo sequences with 4 echoes in interleaved order. For each multi-echo acquisition the same TR (83.4 ms) was used.

The fitting with all three methods was performed with an in-house developed Matlab script (R2011a; TheMathWorks). In order to test the effect of SNR on the derived $$$T_2^*$$$ values we retrospectively added extra Rician noise to the original magnitude data. The relative $$$T_2^*$$$ difference (difference=$$$\frac{|S_0-S_n|}{S_0}$$$) was evaluated as the function of the SNR ($$$SNR=\frac{mean signal}{\sigma (noise)}$$$).

**Results**

Figure 1. shows the result of the SNR dependence for all three fitting methods. It is visible that the monoexponential and the fractional order models are less influenced by the noise than the biexponential model.

The $$$T_2^*$$$ maps derived by the different mathematical models are shown in figure 2. The background of the images is masked.

**Discussion**

The fractional order model is less sensitive to low SNR compared to the biexponential model. The SNR is usually less controlled in longitudinal MR Imaging, the fractional order model outperforms the biexponential model from the perspective of repeatability.

The $$$\alpha$$$ is a novel parameter, potentially a new biomarker characterizing microscopic structure. The fractional order model could be the preferred fitting method in case of low SNR, for tissues which are only visible with UTE or ZTE techniques (e.g.: patella tendon).

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Figure 1. For each method showing the deviation from
ground truth as function of added noise
level. The level of the noise was added
from 1% to 30% of the initial signal
value , and the original SNR in the muscle was 31. The difference is defined in the methods. The
biexponential method gives three orders
of magnitudes bigger difference than the monoexponential and the fractional
order method.

Figure 2. T2^{*} maps of the biexponential method [a] and [b], the monoexponential method [c]
and the fractional order α and T2* maps are on [d] and [e] inside the tissue mask.