K br K br I br where V is the

K
K
I
where V is the volume of the tumor and its metastases, K is the ABT263 of the vasculature, and I is the immunocompe-tent cells density. The original constant terms were also modified in order to obtain the system following an approximation of a mRCC cancer clinical behavior, according to the clinical results described in [20,21], resulting in: = 0.025 day−1 , = 0.1 day−1 ,
2.1.2. Pharmacokinetics
Pharmacokinetics (PK) is the block responsible for determining the fate of substances administered to the organism and thus the drug-concentration time profiles in the body, being drug specific. This process is represented by a two-compartmental model, that quantitatively describes the pharmacokinetic behavior of a drug in
the organism [22], being the profile of the central compartment drug concentration Cp [mg/kg/ml], given by
where D is the dose given to the patient in mg/kg, and ˛, ˇ, k12 , and k21 are rates of distribution or elimination processes.
2.1.3. Pharmacodynamics
Pharmacodynamics (PD) relates the drug concentration Cp with its effect, denoted here as u. It has been widely represented by the Hill equation [26], given by 
Fig. 2. Toxicity levels Tg and Ti . The blue and red marks are representing the drug concentration for which toxicity level is maximum (grade 5).
maximum dose allowed. These doses are 15 and 20 mg/kg, respec-tively, for bevacizumab [31] and atezolizumab [32]. The organism total toxicity is then measured by the mean of Tg and Ti , given by
(5) Although the toxicity levels do not influence the drug effect – Fig. 1
– they are important variables that are going to be used in Section
where C50 is the drug concentration for which 50% of maximum
effect is obtained, and ˛ is the Hill coefficient determining the 2.2. Controller design for a patient model
steepness of the resulting sigmoid, considered here to be unitary. Considering that the patient model with state x = [VKI] is such
The C50 parameter for bevacizumab and atezolizumab was calcu-
feedback controller with proportional gains can be implemented,
as illustrated in Fig. 3, being its blocks explained in the following
subsections.
A model for drug resistance (DR) was considered for both drugs, 2.2.1. Controller implementation
taking advantage of the capacity of malignant cells to proliferate
into more resistant cells when low drug concentration is present in In order to develop a feedback control system, the tumor growth
the plasma. This means that when Cp is smaller than a threshold, model was linearized around the only real equilibrium point in the
drug resistance is acquired, being this situation simulated by an absence of therapy. Aside from that, the controllability and observ-
increase in the C50 parameter from PD [29], since it will directly ability matrices were computed for confirming that the linearized
decrease the drug effect. Thus, C50 is given by
system is fully controllable and observable.
process error. The linear feedback controller used incorporates an
where Cbase is the previously defined initial value, and f (t), which is observer that estimates the state xˆ for each model. The product
a function that increases C50 if the drug concentration Cp is below
the threshold Lr , is given by
is made in order to obtain three different therapies. Those are
described by a desired input effect vector U = [U 1 U 2 U 3 ], where
t
T
anti-angiogenesis and immunotherapy, respectively.
The estimation xˆ is computed by adding an additional term
The capacity of the malignant cells to resist depends on Kr .
to the state estimation differential equation. This term is propor-
tional to the estimation error, and the multiplication by a matrix
L ensures the asymptotic convergence of xˆ to x [33]. In this work,
Following the Common Terminology Criteria for Adverse Events the observer matrix L is calculated by using the Kalman estima-
(CTCAE) [30], toxicity in clinical trials can be graded as mild (grade tor design for continuous-time systems, with covariance matrices
The Linear Quadratic Regulator (LQR) controller [33] consists of
By evaluating the concentration of drug induced in the body a state feedback control law whose gains are selected in order to
during therapy, toxicity levels Tg for anti-angiogenesis and Ti for minimize the infinite horizon quadratic cost
immunotherapy, can be estimated by using a function yielding ∞
achieved. After this threshold, the function grows exponentially where Q0 and Ri0 are matrices that can be tuned.
The curves were selected so that grade 5 of toxicity was reached