# State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment
## 1 Introduction  
Cancer treatment and its related fields are intensively studied subjects, since cancer is a major cause of death globally. It is estimated that there will be 21.4 million cases annually by 2030 [1; 2]. The application of modern control algorithms for cancer treatment has much potential--especially in the adjustment of the dosages--however, there are difficulties as well [3]. The article investigates a facet of a larger project. The larger project aims to minimize the amount of administered chemotherapeutic drug--while retaining its effect--by applying personalized treatments. Lower amount of doses expected to reduce side effects and potentially postpone the emergence of tumor resistance. One approach is the utilization of feedback control, where the controller (which decides about the drug administration scheme) adapts to the patient. Based on mathematical models accurately describing the underlying physiological processes the professionals have the possibility to define appropriate therapies. System engineers thrive to provide a proper description of the tumor growth, however, the inter- and intrapatient variability and the effects of different drugs are cumbersome to model. Several models were introduced over the years [4; 5; 6; 7; 8; 9]. In particular, we are concerned with the use of a third-order model. The model describes the dynamics of the living and dead tumor cells, as well as drug concentration [7]. The output of the system is the total tumor volume (sum of the living and the dead tumor volumes). In tumor growth modeling there are variables that cannot be measured, or are not feasible to measure in practice on a day-to-day basis. Such variables in this particular case are the individual volume of the living and dead tumor cells and drug concentration or the model parameters characterizing the patient variability. A mathematical model gives the general behavior of the system. The general structure can be tailored to each individual by the model parameters. For instance, there isa parameter that corresponds to the clearence of the drug, which can vary from patient to patient. Although they cannot be measured, they play an important role in describing the dynamics of the system and are often required for specific therapy optimization or closed-loop algorithms. In order to extract those unmeasurable variables, usually, an observer is applied. The goal of an observer is to provide information about the inner state variables or parameters based on the measurements. The most common algorithm is the Kalman filter (KF) [10], however, other approaches like sliding mode observer, particle filter [11], linear observer [12] or moving horizon estimation (MHE) [13] are also employed. The latter one is an optimization-based approach, where the name suggests that the optimization is performed on a moving window. The window makes it possible to take into account a series of measurements, and the optimization makes it possible to define application-specific constraints and penalizations. In this paper, an MHE is developed and its performance is compared to an optimized extended Kalman filter (EKF). Often, the drawback of the studies, e.g., [10; 11; 12; 13] is that they are performed on simulated data which is favorable from several aspects. In contrast, this study is evaluated on laboratory measurements, where noise, irregular sampling times and most importantly, structural mismatch are present. The paper is structured as follows. Section 2.2 describes the utilized third-order tumor growth model and previously identified parameter sets. In Sections 2.3 and 2.4 first theoretical, general description for the EKF and MHE based observers are given. Thereafter the observers are applied to the specific tumor growth model. Measurement noise is characterized in Section 2.3.1. Sections describing the observers end with the tuning methods. In Section 3 the results are summarized. The observers are evaluated based on their capabilities following the trends of the measurements.