# State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment
## 4 Discussion  
The goal was to evaluate the state and parameter estimation capabilities of an MHE and an EKF which has been applied as a benchmark. In general, state and parameter estimation in the case of physiological processes have major difficulties which are presented in this study related to tumor growth estimation as well. Furthermore, the problem formulation makes it even more difficult. In the model, only the sum of the living and dead tumor volumes can be measured, often inaccurately. This limited amount of data can easily lead to observability issues mostly coming from the fact that several state and parameter combinations may result in the same output. In order to limit these issues, we applied two key components: penalizing the velocity of changing of state variables and estimated parameters, moreover, we have selected a branch of key parameters to be estimated based on sensitivity analysis. Namely, the cost function of the MHE made it possible to use additional constraints and penalize the change of parameters. The penalization was designed to be close to the physiological rate of changes to be as as realisic as possible. According to the results, we have successfully limited the set of possible state and parameter sets leading to the same output without a downturn in the estimation accuracy.  
The scenario for the comparison was designed to be biased arbitrary. The results show that even with optimizing the tuning of the EKF, and not taking into account the initial state of the MHE as an additional free variable, a moving horizon estimation based observer can still outperform the former EKF. The MHE-based observers achieved lower RMSE until the horizon length was shorter than 30 days. Using horizons around 20 days, there can be cases, where the EKF outperforms the MHE, e.g., the estimation of PLD9. At 7 days the difference in RMSE is around 30 [mm\({}^{3}\)], and the MHE has lower RMSE for each mouse. The inferior performance of the EKF can rise from the low gains for parameters \(a\) and \(n\). Since the system is most sensitive to those parameters, having larger gains could result in divergence in the case of certain patients due to the high interpatient variability. The estimation accuracy of the MHE deteriorated in a quasilinear manner with respect to the window length. The increased window lengths resulted in an increased RMSE of the tumor volumes because the effects of the structural mismatch became dominant. In general, a longer window would mean that the given parameter set is valid for more data. However, the predictive capabilities are crucial in therapy optimization, thus the determination of the optimal choice for closed-loop applications is future work. The runtime of the observers are greatly different. The EKF averages 2.6 ms, while the MHE ranges from 300 ms to 413 ms depending on the horizon length. However, runtime is not a cornerstone as the injections and measurements are done once a day at a maximum rate. An important aspect of the study is that the observers are evaluated on real laboratory data, where two major difficulties are present: structural mismatch and intrapatient variability.  
The results indicate the great importance of applying observers for therapy optimization algorithms. Without constant feedback about the behavior of the patient the model using only a single parameter set can introduce large discrepancies. As the tumor shifts phases, and as the heterogenity of the tumor changes its behavior changes as well. These shifts in physiology can be tracked with the observers-particularly well with the MHE-. These changes can be interpreted as trends in the parameters. Additionally, despite the optimized tuning of the EKF the MHE shows a smoother estimation of the parameters and a more accurate approximation of the measurements. These attributes facilitate the application of the MHE in closed-loop algorithms.