# State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment
## 3 Results  
The observers were evaluated on the data described in Section 2.1. As mentioned each mouse is labeled with PLD followed by an ID number. Estimation horizons of the MHE were tested in the 7-35 days range, however for better visibility only the 14 day version is shown beside the EKF. Table 3 compares the RMSE of the measured and estimated volumes between the EKF and the investigated MHEs, given for each investigated mouse. When comparing the results two points must be emphasized. First, the assumptions made during the design of the observers. Secondly, the worst case scenario is investigated, in the sense that the EKF was tuned based on an optimization, where all the mice were taken into account. On the other hand, the weights of the MHE was not optimized only a penalizing function was introduced and applied instead of other tuning process. The RMSE indicates that for horizons shorter than 20 days the MHE can achieve better approximations on average. At horizon length of 28 days the results are varying from patient to patient. Only at 35 days the EKF has a clear edge over the MHE. Increasing the horizon of the MHE deteriorated the approximations as the actual parameter set had to describe longer sections, where structural mismatch and/or parameter variation occurs due to the change in the physiology of the tumor. At this point, it has to be emphasized again that while the MHE takes into account a series of measurements, the EKF works from sample to sample, thus various horizon lengths cannot be adapted. Figure 3 depicts the comparison of the time horizons using different measures. Three measures, namely mean absolute error (MAE), \"noise\" and standard deviation (SD) of each estimated parameter were selected in order to assess the performance of the observers and horizons. The MAE describes the general accuracy with respect to the measurements, the \"noise\" term quantifies the smoothness of the estimation by calculating the number of direction changes along the trajectory. The SD of each estimated parameter gives information about the range in which the parameter had to vary in order to provide an optimized fit. One can inspect that the MAE increases in a quasi-linear way with respect to the horizon. The smoothest trajectory is achieved by using 14-day-long horizon and the SD of parameter \(a\) is the lowest around 21 days. Concerning the penalized parameters \(b\), \(w\), \(n\) the lowest SD is achieved by using the 7-day-long horizon. The main difference between the EKF and MHE is that the optimization of the EKF resulted in very small gains for parameters \(a\) and \(n\). This is anticipated as those parameters directly affect the tumor growth, with larger gains it may be possible to achieve better approximation for one patient, but it could lead to bigger divergence for a population. The divergence is avoided by the MHE as the optimization is constrained, parameter sets producing trajectories that reach 2000 [mm\({}^{3}\)] are neglected. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **EKF** & **MHE 7 Days** & **MHE 14 Days** & **MHE 21 Days** & **MHE 28 Days** & **MHE 35 Days** \\ \hline PLD3 & 121.6 & 114.6 & 115.5 & 121.5 & 121.4 & 169.5 \\ PLD4 & 72.12 & 32.82 & 48.96 & 59.33 & 60.33 & 103.9 \\ PLD5 & 74.53 & 28.68 & 40.48 & 52.16 & 61.89 & 88.76 \\ PLD6 & 97.93 & 43.46 & 50.15 & 65.97 & 88.49 & 146.1 \\ PLD9 & 59.55 & 49.05 & 55.67 & 61.77 & 72.52 & 86.39 \\ MEAN & 85.14 & 53.73 & 62.16 & 72.16 & 80.94 & 118.9 \\ \hline \end{tabular}
\end{table} 
Table 3: RMSE between the measured and estimated volumes. In Figure 4, an example (PLD ID:5) is given about the effect of the rational penalizing function. The function was introduced in (21). The two estimated trajectories, denoted by _with_\(\omega(\cdot)\) and _without_\(\omega(\cdot)\) are the output of the MHE with 14 days of horizon length. For visualization only the 14 day scenario is given, however, the effect is similar in the other cases, and the delay is more pronounced towards the longer horizons.