## 2 Materials and Methods
### Evaluation
#### 2.2.3 Comparison with State-of-the-Art Methods  
Maps estimated with the proposed approach were compared to the CAR approach without clustering, already published in [15], and to a state-of-the-art Bayesian approach without spatial regularization [20]. As for the CAR without clustering, it follows the description reported in Section 2.1.2 but neglecting the clustering solution. Indeed, the weights \(w_{i,j}^{\alpha,\beta}\) of the CAR prior specification are set equal to 1 for all voxels \((\alpha,\beta)\) bordering on \((i,j)\), and 0 for the others:  
\[w_{i,j}^{\alpha,\beta}=\begin{cases}1&(\alpha,\beta)=\left\{(i-1,j-1),(i-1,j),( i-1,j+1),\right.\\ &\left(i,j-1\right),(i,j+1),(i+1,j-1),\\ &\left(i+1,j\right),(i+1,j+1)\right\}\\ 0&\text{otherwise}\end{cases} \tag{5}\]  
As for the standard Bayesian approach, we adopted the implementation provided by [PERSON] et al. [20], considering the default priors and the hyperparameters proposed by the authors. The IVIM signal was fitted using the bi-exponential decay in (1) and considering the whole set of \(b\)-values. The three IVIM coefficients and each signal \(SI(i,j,0)\) at \(b=0\) were estimated under the following priors [20]: a uniform prior for \(f\) and each \(SI(i,j,0)\); a lognormal prior for \(D\) and \(D^{*}\). Moreover, all prior distributions were truncated to the following intervals: \(\left[0,\,3\cdot 10^{-3}\ \frac{\text{mm}^{2}}{\text{s}}\right]\) for \(D\); \(\left[0,\,0.15\ \frac{\text{mm}^{2}}{\text{s}}\right]\) for \(D^{*}\); \(\left[0,1\right]\) for \(f\); \(\left[0,1\right]\) for \(SI(i,j,0)\) being the simulated images in this range. The parameter estimates were given in terms of the mean value from the respective marginalized posterior distribution.  
Following the procedures already adopted in previous works [10; 14; 15], the accuracy of each method on simulated images was evaluated in a voxel in terms of the error with respect to the true value:  
\[\varepsilon=\frac{\lambda_{est}-\lambda_{true}}{\lambda_{true}} \tag{6}\]  
where \(\lambda_{est}\) is the estimated value of the coefficient and \(\lambda_{true}\) the corresponding true value. The median of the errors over the voxels of a ROI was finally calculated for each coefficient.  
Moreover, the noisiness of the map was evaluated in terms of the coefficient of variation (\(CV\)), calculated as the ratio between the standard deviation \(StD(\lambda_{est})\) and the mean value \(Mean(\lambda_{est})\) of the estimated coefficients in a ROI with constant \(\lambda_{true}\):  
\[CV=\frac{StD(\lambda_{est})}{Mean(\lambda_{est})} \tag{7}\]  
The average value of the \(CV\) over the ROIs was then considered.  
A statistical analysis was performed to evaluate the differences among the three approaches. A Kruskal-Wallis test was applied to \(\varepsilon\) and \(CV\) estimated with the three methods for each \(SNR\) value, in order to identify significant differences among groups. For those groups presenting significant differences, a Tuckey post hoc test was next performed to determine which couples were significant. Significance was set to \(p<0.05\), and Bonferroni correction for multiple comparisons was applied.  
IVIM maps estimated on real cases were qualitatively evaluated and mean values of IVIM coefficients were calculated within each structure.