## 3 Results and Discussion of Computational Experiments
### Contour-Propagation Algorithm  
Once a set of candidate parameters has been obtained to reach a DVF with an adequate registration, we propose to include Algorithm 1 in the matRad toolkit so that it reads the CT and SCT structures and estimates the contours of the organs in each scenario in such a way that it reduces the uncertainty in the dose calculation. For experimentation with the proposed algorithm, breast CT scans are taken in 4-D format and the contour propagation of the six structures is estimated. It is assumed that Scenario 1 is the fixed image while the other scenarios are objective. The scenarios correspond to five gray-scale cubes of the thoracic area and each scenario corresponds to an SCT with the six segmented structures. In the implementation, the algorithm assumes that there is only segmentation in Scenario 1 and the goal is to estimate the structures in the other scenarios from the registration. ```
Input: CT scans for the \(n\) scenarios and SCT for the fixed scenario. Output: CT scans for the \(n\) scenarios and SCT for the \(m\) structures. for scenario \(i=\{1,2,\ldots,n\}\)do  Calculate DVF \(\mathbf{u}_{i}\) endfor for structure \(j=\{1,2,\ldots,m\}\)do  Convert linear indices of SCT structure \(j\) for Scenario 1 into a cube (\(x\times y\times z\)) for scenario \(i=\{2,\ldots,n\}\)do  Apply \(\mathbf{u}_{i}\) to the structure \(j\) for scenario 1 to estimate structure \(j\) for scenario \(i\)  Convert the cube of structure \(j\) for Scenario \(i\) into linear indices  Store the structure \(j\) in the SCT for Scenario \(i\) endfor endfor
```  
**Algorithm 1**matRad-contourPropagation  
Note that the algorithm calculates the contours of the structures in the different scenarios, regardless of the registration method you choose. This allows us to find the DVF with both demon registration and B-spline, using the same parameters for both methods. Figure 8 shows the steps in which each one of the registrations intervenes and the dependence that transformation \(T^{*}\) has on the parameters found with the heuristic algorithm. The optimizer in the demons case is Equation (3) while for B-spline it is Equation (5) where the regularization term is Equation (8). The parameter \(\rho\) controls the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Scenario 1** & **Scenario 2** & **Scenario 3** & **Scenario 4** & **Scenario 5** \\ \hline \(\rho\) & \(1\) & \(1\) & \(1\) & \(1\) & \(1\) \\ \(N\) & \(100\) & \(100\) & \(100\) & \(100\) & \(100\) \\ \(\sigma\) & \(1.5\) & \(1.5\) & \(2.6\) & \(1.8\) & \(2.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Set of candidate parameters for contour estimation in the 5 scenarios using breast CT scans. amount of scaling to which the image is subjected to by its respective Gaussian filter, which allows for a noise reduction and thus obtains contours with minimal uncertainty. Therefore, Algorithm 1 is executed with both registrations and its performance is measured with the Dice coefficient, obtaining the results shown in Figure 9.  
With the demon-based Algorithm 1, the similarity value satisfies \(DICE(A,B)\geq 0.92\), for each of the six structures (skin, right lung, left lung, contralateral sinus, CTV, and heart) in the five scenarios with breast CT scans. On the other hand, as explained in the methodology section, the similarity obtained with the B-spline-based algorithm allows one to calculate the DVF in terms of control points, which are the ones that ultimately generate the deformation in the B-spline registration. Additionally, it allows one to control the parameters \(\rho\), \(\sigma\), and \(N\), which are part of the input values. If the experimentation is carried out with the default values for the distances between control points, and the parameters for level of smoothing, the number of iterations and number of pyramid levels are modified according to Table 3, then the values of Figure 9 are obtained.