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For example, the optimal control study from Swan et al. [8] is critical for the understanding of the early modeling approaches for chemotherapy. Aside from that, Schättler et al. [9] applied geo-metrical optimal control to challenging problems. Subsequently, the same techniques were applied to a problem where a balance

Fig. 1. Summary of patient model, including the equations of each block.

between cell kill and toxicity level was considered [10]. In [11], solutions are found using optimal control with different cost func-tions and also receding horizon control. Several other techniques are used such as model predictive control [12], or model reference adaptive control [13].

To the best of the authors’ knowledge, there is no study approaching the therapy optimization problem with Multiple Model Adaptive Control (MMAC), despite being widely used for the control of neuromuscular blockade [14–16]. MMAC can also be used in combination with other techniques, such as model clus-tering, for finding a finite set of models that cover system dynamics [17]. Adaptation can be embedded in different ways, including the use of a control Lyapunov function, parameter estimation using a recursive estimator and the certainty equivalence principle [13], or the use of multiple models as proposed in this article. The Puromycin warranties provided by the Lyapunov approach are illusive due to the expected high level of non-structured un-modeled dynamics. Furthermore, the continuous adjustment of parameters induces a frequent adjustment of therapy that is not realistic and may lead to drift of controller gains to values that cause undesirable responses. The use of multiple models instead, allows a parsimonious descrip-tion of the system dynamics that reduces the possible number of controllers among which the best option given the observations is selected, as described below. This feature is possible by the use of model/controller clustering and by exploiting the robustness of each controller, i.e., its ability to control not just the nominal model, but a dense set of models around it according to a con-venient norm. MMAC is therefore selected as a natural choice for biomedical applications.

The objective of this work is to develop a control based frame-work to design therapies for reducing tumor volume, optimizing therapy while minimizing toxic effects.

A tumor growth model is developed, using metastatic renal cell carcinoma (mRCC) clinical results, through a combination of two different treatments: anti-angiogenesis and immunotherapy. To formulate a patient model, pharmacological aspects are also incor-porated.

The uncertainty around patient characteristics and his variabil-ity is taken into account by using MMAC, a control technique in which an optimal therapy is being constantly chosen by selecting from a variety of models the one that, simultaneously best fits the unknown patient model and has the lowest toxicity level. Once the model selected, the optimal control problem is solved using the Linear Quadratic Gaussian (LQG) approach.

Finally, to allow the selection of any model data set dimension, unsupervised learning in the form of clustering is used to agglomer-ate patient models that are similar. This objective implies a constant number of controllers regardless of the number of patient models in the data bank.

The outline of the paper is as follows. The material and methods are described in Section 2. In Section 3, the results are shown and discussed, and in Section 4, the conclusions are drawn.

2. Materials and methods

In order to develop the control algorithm, a patient model has to be developed, incorporating for a given drug the most important pharmacological and pharmacodynamic aspects. Briefly, species diversity can be summarized as in Fig. 1, and is described in the following subsec-tions. The letters inside the blocks stand for: T – Tumor growth model; PK – Pharmacokinetic model; PD – Pharmacodynamics model; DR – Drug resistance model; Tx – Toxicity model. The num-bers inside the blocks correspond to the equations in the text.

Anti-angiogenesis acts by blocking specific proteins, preventing the creation of new blood vessels in the tumor, limiting the num-ber of nutrients and oxygen, that results in tumor shrinkage. On the other hand, immunotherapy can act as a boost to trigger the protective defenses of the immune system, or by inhibiting specific immune checkpoint pathways.

In order to develop a dynamic model that describes the depen-dency of the tumor behavior on the previous therapies, a fusion between Hahnfeldt et al. [18] and Ledzewicz et al. [19] models was performed, resulting in the following equations

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