MRI signal loss has been shown to deviate from mono-exponential decay due to magnetic field inhomogeneities generated by magnetic susceptibility.¹ Therefore, classical models based on the traditional Bloch equation cannot fit the experimental data.² Models based on fractional order calculus show increasing promise in modelling of biophysical processes.³ Magin et al. have used time-fractional calculus to explore the MRI-based anomalous relaxation process.⁴ Their model was developed based on spin echo data and cannot account for frequency shift or susceptibility induced effects. We explore the utility of fractional calculus in the case when gradient recalled echo MRI data is acquired. This allows us to study susceptibility induced effects through a parameter accounting for frequency shift. We hypothesise that parameters of the anomalous relaxation equation may provide insight into tissue microstructure and susceptibility constituents within image voxels.

Three anomalous relaxation models were implemented: $$$S(t)=A_0 exp(-t/T_2^*)+C$$$ (MONOEXP), $$$S(t)=A_0 E_\alpha(-t^\alpha/T_2^*)+C$$$ (MAGIN) and $$$S(t)=A_0\sqrt{E_\alpha(-t^\alpha/T_2^*+i\Delta\omega t^\alpha)E_\alpha(-t^\alpha/T_2^*-i\Delta\omega t^\alpha)}+C$$$ (EXTENDED – the new model), where $$$A_0$$$ is the amplitude, $$$T_2^*$$$ is the relaxation time ($$$T_2^*=1/R_2^*$$$ and $$$R_2^*$$$ is the relaxation rate), $$$E_\alpha(t)$$$ is the Mittag-Leffler function,⁵ $$$\alpha$$$ is the order of the time-fractional derivative, $$$\Delta f$$$ is the frequency shift and $$$C$$$ is the constant offset. $$$C$$$ is used to account for the background noise in the acquired data.⁶ We evaluated the mean squared error (MSE) as a measure of quality of fit.

Ethics was granted by the University of Queensland human ethics committee. In vivo gradient recalled echo MRI brain imaging was performed in five healthy participants (age $$$33.6\pm4.4$$$) using a 7T human research scanner (Siemens Healthcare, Erlangen, Germany) equipped with a 32 channel head coil (Nova Medical, Wilmington, USA). 3D non-flow compensated scan was performed: $$$TE_1 = 2.04ms$$$, echo spacing = 1.53ms, 30 echoes, TR = 51ms, flip angle = 15, voxel size = $$$1mm\times 1mm\times 1mm$$$ and matrix size = $$$210\times 168\times 144$$$.

The caudate (CA), pallidum (PA), putamen (PU), thalamus (TH), internal capsule (IC), red nucleus (RN), insula (IN), substantia nigra (SN) and fornix (FO) were segmented manually with the aid of a human brain atlas,⁷ and an example is shown in Fig 1.

Fig 2 shows fitting results for four example voxels. The result shows that the EXTENDED model outperforms both the MONOEXP and MAGIN models (MSE obtained using the EXTENDED model is 1.83, 3.53, 3.93 and 1.53 times smaller at the four locations than obtained using the MAGIN model). Also, the EXTENDED and MAGIN models outperform the MONOEXP model. An average MSE across the entire image was computed as well. The MONOEXP model resulted in an MSE of $$$438.36\pm323.09$$$ and the MAGIN model in $$$318.49\pm228.03$$$, while the EXTENDED model in $$$245.26\pm190.45$$$.

Tab 1 provides calculated parameter values for the MONOEXP and EXTENDED model. In all cases, the MSE obtained using the EXTENDED model fits better than the MSE obtained using the MONOEXP model. In the pallidum, the MSE of the MONOEXP model is 76.71 times larger than calculated using the EXTENDED model. In the thalamus, on the other hand, the MONOEXP MSE is only 1.17 times larger than the MSE achieved using the EXTENDED model. Overall, the use of the MONOEXP model results in 3.65 times larger MSE than when the EXTENDED model is used. Hence, the EXTENDED model is able to fit the data more accurately.

The effect of magnetisation on high susceptibility tissue constituents, such as iron, generates a heterogeneity of magnetic fields, creating a frequency shift in the MRI signal.⁸ Previous studies have reported higher iron depositions in the pallidum, red nucleus, substantia nigra, putamen and caudate and relatively lower values in the thalamus, which correlates well with the frequency shift in our study.⁹ However, the general trend in frequency shift ($$$\Delta f$$$) obtained using our model is slightly different to previous studies. It seems that frequency shift is influenced not only by the amount of iron, but maybe also by other factors such as spatial distribution. A smaller $$$R_2^*$$$ is obtained using the EXTENDED model in comparison to the MONOEXP model. The fact that $$$R_2^*$$$ is consistent across all brain regions suggests validity of the EXTENDED model. The time-fractional derivative order ($$$\alpha$$$) varies with brain region, and the trend is different to the $$$R_2^*$$$ trend.The $$$R_2^*$$$ value calculated using the extended time-fractional model has the same trend as the one obtained using the classical monoexponential model, and therefore, may not provide additional information. However, $$$\alpha$$$ and $$$\Delta f$$$ vary differently in comparison to $$$R_2^*$$$. The variations in these parameters may be influenced by tissue composition and microstructure.