Fractional order vs. exponential fitting in UTE MR imaging of the patellar tendon

Quantification of the T₂^∗ relaxation time constant is relevant in various magnetic resonance imaging applications. Mono- or bi-exponential models are typically used to determine these parameters. However, in case of complex, heterogeneous tissues these models could lead to inaccurate results. We compared a model, provided by the fractional-order extension of the Bloch equation with the conventional models.

Axial 3D ultra-short echo time (UTE) scans were acquired using a 3.0 T MRI and a 16-channel surface coil. After image registration, voxel-wise T₂^∗ was quantified with mono-exponential, bi-exponential and fractional-order fitting. We evaluated all three models repeatability and the bias of their derived parameters by fitting at various noise levels. To investigate the effect of the SNR for the different models, a Monte-Carlo experiment with 1000 repeats was performed for different noise levels for one subject. For a cross-sectional investigation, we used the mean fitted values of the ROIs in five volunteers.

Comparing the mono-exponential and the fractional order T₂^∗ maps, the fractional order fitting method yielded enhanced contrast and an improved delineation of the different tissues. In the case of the bi-exponential method, the long T₂^∗ component map demonstrated the anatomy clearly with high contrast. Simulations showed a nonzero bias of the parameters for all three mathematical models. ROI based fitting showed that the T₂^∗ values were different depending on the applied method, and they differed most for the patellar tendon in all subjects.

In high SNR cases, the fractional order and bi-exponential models are both performing well with low bias. However, in all observed cases, one of the bi-exponential components has high standard deviation in T₂^∗. The bi-exponential model is suitable for T₂^∗ mapping, but we recommend using the fractional order model for cases of low SNR.