## 5 Discussion  
The analysis of the example with real parathyroid glands shows rather good convergence of the PSO algorithm (Figure 4). It is not possible to achieve a zero value of MSE because of noise and because there are differences between the model and measurements of the real biological object. The minimal value is about 0.4 and 57% cases achieved \(\leq\)0.55 MSE values. Sometimes high values such as \(>\)2 are achieved and the solution for the improvement of results is the repetition of the estimation a few times for random starting parameters and the selection of the best result using minimal MSE value criterion. Multivalue histogram is an indicator of dependence between parameters and limits possibilities of reliable visualization of results for particular parameter, due to dependence between them. This is the reason why MSE values are presented in this paper. Slices (Figure 3) show the achieved separation of components. Maxima are located in different positions. The last component is spherical and two components are volumetric ellipsoids. The Monte Carlo test showed the most important result: the random topology was better in this application, because the particular distribution was shifted to the left (Figures 8-11 left) and a higher peak was observed for lower error values (Figure 8 left). The MSE histogram (Figure 12) demonstrated the advantages of random topology because lower MSE values were achieved more frequently (approximately two times more often). The mean shift was \(\sim\)1 (voxel size) (Figure 8 left), so relative improvement was \(\sim\)3% for the assumed size of volume. Global best and ring topologies gave similar results. This result shows  
Figure 8: Distributions of mean position errors (**left**) and box-and-whiskers plot of absolute position errors dependent on true position ranges (**right**). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Error Type** & **Without Noise** & **With Noise** \\ \hline Mean angle: global best—ring & 1 & 1 \\ \hline Mean angle: global best—random & 1 & 1 \\ Mean angle: ring—random & 1 & 1 \\ \hline Mean scale: global best—ring & 1 & 1 \\ Mean scale: global best—random & 1 & 1 \\ Mean scale: ring—random & 1 & 1 \\ \hline Mean \(\beta\): global best—ring & 1 & 1 \\ Mean \(\beta\): global best—random & 1 & 1 \\ Mean \(\beta\): ring—random & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._  
Achieved distributions are quite smooth for the selected resolution of error value (Figures 8-11 left), so the number of iterations assumed in Monte Carlo test is correct. The comparisons of absolute errors between amplitude and position, scale, and \(\beta\) are shown in Figures 5-7. There are significant differences between different topologies in such relations. Absolute mean amplitude error and scale were not correlated (Figure 6). Small positive correlation (but nonlinear) was found for absolute mean amplitude error and absolute mean position error (Figure 5). The analysis of particular parameters due to estimation errors was also important. The selection of random topology based on MSE was correct, but when it comes to estimation of particular parameters, quality needs to be checked as well. The hypotheses about similarity of distributions for all topology pairs were rejected (Table 5). The hypothesis about similarity of distributions cannot be rejected for most results (Table 4). The obtained means of position error values (3-6 voxels distance) are satisfactory results (Figure 8), but the reduction of this error is still an interesting avenue of research. Mean amplitude errors and mean scale errors are quite similar visually, but the Kolmogorov-Smirnov test rejects such hypothesis. Figure 11: Distributions of mean \(\beta\) errors (**left**) and box-and-whiskers plot of absolute \(\beta\) errors dependent on true \(\beta\) ranges (**right**). Figure 12: Histogram of MSE. Processing time is acceptable, further improvements are desired. Monte Carlo test is possible using parallel computations, so each core is assigned its own optimization test.