## 4 Discussion and Conclusions  
In this work, we have proposed an improvement to the Bayesian CAR fitting of the IVIM model [15] by embedding a \(k\)-means clustering procedure to specify the weights in the CAR prior specification. The aim was to improve the estimation of the parametric maps for the IVIM coefficients. In fact, in the previous work [15], it was highlighted that the main limitation of the CAR approach was in the way the neighbor voxels were considered in the fitting procedure, since all the eight surrounding voxels were included with the same weight, thus neglecting the heterogeneity of the surrounding tissues. The simplest solution to this issue from a theoretical point of view is to also estimate all weights of all voxels by means of the Bayesian approach; however, this approach proved to be ineffective due to the huge number of parameters to be estimated and the \begin{table}
\begin{tabular}{c l c c c c c} \hline \hline \multicolumn{1}{c}{**HN Patients**} & \multicolumn{2}{c}{\(D\) **[mm\({}^{2}\)/s]**} & \multicolumn{2}{c}{\(f\)} & \multicolumn{2}{c}{\(D^{*}\) **[mm\({}^{2}\)/s]**} \\  & & **Mean** & **StD** & **Mean** & **StD** & **Mean** & **StD** \\ \hline \multirow{4}{*}{**Parotid**} & BAYES & 0.0010 & 0.0003 & 0.1975 & 0.1275 & 0.0318 & 0.0168 \\  & CAR-cluster & 0.0011 & 0.0004 & 0.1974 & 0.1127 & 0.0378 & 0.0298 \\  & CAR-orig & 0.0011 & 0.0004 & 0.1957 & 0.1084 & 0.0379 & 0.0286 \\ \multirow{4}{*}{**Tumor**} & BAYES & 0.0009 & 0.0002 & 0.1600 & 0.0856 & 0.0292 & 0.0126 \\  & CAR-cluster & 0.0009 & 0.0003 & 0.1679 & 0.0843 & 0.0424 & 0.0305 \\  & CAR-orig & 0.0009 & 0.0003 & 0.1666 & 0.0824 & 0.0421 & 0.0309 \\ \multirow{4}{*}{**Muscle**} & BAYES & 0.0013 & 0.0003 & 0.2094 & 0.1054 & 0.0259 & 0.0113 \\  & CAR-cluster & 0.0014 & 0.0004 & 0.1771 & 0.0930 & 0.0332 & 0.0184 \\  & CAR-orig & 0.0015 & 0.0004 & 0.1747 & 0.0871 & 0.0347 & 0.0189 \\ \hline \multicolumn{2}{l}{**PELVIC patients**} & BAYES & 0.0012 & 0.0002 & 0.1614 & 0.0740 & 0.0192 & 0.0090 \\ \multirow{4}{*}{**Prostate**} & BAYES & 0.0013 & 0.0002 & 0.1456 & 0.0481 & 0.0136 & 0.0081 \\  & CAR-orig & 0.0013 & 0.0002 & 0.1498 & 0.0445 & 0.0116 & 0.0046 \\ \multirow{4}{*}{**Tumor**} & BAYES & 0.0010 & 0.0003 & 0.1711 & 0.0969 & 0.0242 & 0.0118 \\  & CAR-cluster & 0.0010 & 0.0004 & 0.1856 & 0.1114 & 0.0189 & 0.0161 \\ \multirow{4}{*}{**Muscle**} & BAYES & 0.0010 & 0.0004 & 0.1838 & 0.1092 & 0.0189 & 0.0159 \\  & BAYES & 0.0011 & 0.0002 & 0.1606 & 0.0754 & 0.0231 & 0.0091 \\ \multirow{4}{*}{**Muscle**} & CAR-cluster & 0.0012 & 0.0002 & 0.1488 & 0.0474 & 0.0155 & 0.0046 \\  & CAR-orig & 0.0012 & 0.0002 & 0.1473 & 0.0407 & 0.0160 & 0.0043 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of \(D\), \(f\) and \(D^{*}\) estimated from the real images in each structure of interest by the three methods, reported in terms of mean and standard deviation over the subjects. Figure 6: Parametric maps of \(D\) (**first row**), \(f\) (**second row**) and \(D^{*}\) (**third row**), estimated from a selected slice of a pelvic patient using the three methods (columns). The tumor, the prostate and the internal obturator muscle are delineated in red, green and blue, respectively. Red arrows indicate improvement in boundary definition, yellow arrows indicate reduced blurred effects. too high computational effort. In fact, this is not pursued in the literature, while only examples are available in other applications in which the weight of the CAR component is tuned as a whole without distinguishing the contributions of the specific elements in the neighborhood [21]. To overcome this issue, the weights have been provided by embedding a \(k\)-means clustering approach in the procedure, based on a distance function that considers the whole set of images at different \(b\)-values. This allows us to group the voxels with similar exponential decay and to give them higher weights.