# New Aspects of Solution Feasibility in a Context of Personalized Therapy Optimization
## 2 Problem Formulation  
The mathematical model of the two-drug displacement problem reflects the kinetics of warfarin \(y_{1}\) and phenylbutazone \(y_{2}\) interactions in the patient's bloodstream. Originally, [PERSON] and [PERSON] considered a computational model of the kinetics of drug displacement kinetics [13]. The studied mathematical formulation was confronted with the available experimental data. Researchers have noted that for highly bound lowly cleared drugs, displacement interactions are transitory. Consequently, the kinetics of the interaction as well as the in vitro interaction have to be considered. Moreover, they indicated that it is possible to moderate drug displacement by adjusting the rate and the timing of administration of the displacing agent. In this way, the first mathematical model was created, which was then developed in subsequent studies. Finally, a time-optimal control task was constructed in the presence of kinetic equations in the form of ordinary differential equations (ODEs) [14]. The ODEs model distinguishes specific parameters, \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(D\), which, in general, may change over time and are dedicated to the considered issue. Choosing the selected parameters and presenting them in the form of algebraic equations result in a differential-algebraic model (DAE). Depending on the solution method used, one can talk about an optimization problem with the DAE model or differential-algebraic constraints [15]. The main difference is that the model is solved during numerical calculations, and constraints can be fulfilled or violated. Then, the additional goal is to minimize the violation of the constraints [16]. The task considered in this work is a modified formulation presented in the article [17]. The duration of the observation is not a decision variable here but has been assumed in advance. During this time, the state variables, \(y_{1}\) and \(y_{2}\), must be influenced in such a way that the boundary conditions are met. In the considered task, the solution needs to meet the following requirements:  
* In an assumed time range, drugs need to reach a desired levels,
* A warfarin concentration cannot exceed an assumed level of toxicity. The presented expectations were taken into account in a performance index formulation  
\[\min_{u(t)}\quad\rho\Big{(}y_{1}(t_{f})-0.02\Big{)}^{2}+\Big{(}y_{2}(t_{f})-2. 00\Big{)}^{2} \tag{1}\]  
with \(\rho=100\), as well as the dynamical relations  
\[ \begin{array}{lll}dy_{1}/dt&=&\frac{D^{2}}{C_{3}}[C_{2}(0.02-y_{1})+46.4y_{1 }(u-2y_{2})]\\ dy_{2}/dt&=&\frac{D^{2}}{C_{1}}[C_{2}(u-2y_{2})+46.4y_{2}(0.02-y_{1})]\end{array}  \tag{2}\]  
the algebraic relations  
\[ \begin{array}{lll}0&=&D-(1+0.2y_{1}+0.2y_{2})\\ 0&=&C_{1}-(D^{2}+232+46.4y_{2})\\ 0&=&C_{2}-(D^{2}+232+46.4y_{1})\\ 0&=&C_{3}-(C_{1}C_{2}-2152.96y_{1}y_{2}).\end{array}  \tag{3}\]  
The control is the rate of infusion of phenylbutazone and is bounded by  
\[0\leq u(t)\leq 8. \tag{4}\]  
The initial concentrations of warfarin and phenylbutazone are given by  
\[y(0)=\left[ \begin{array}{c}y_{1}(0)\\ y_{2}(0)\end{array} \right]=\left[ \begin{array}{c}0.02\\ 0.00\end{array} \right]. \tag{5}\]  
The system is to be taken to the desired final state  
\[y(t_{f})=\left[ \begin{array}{c}y_{1}(t_{f})\\ y_{2}(t_{f})\end{array} \right]=\left[ \begin{array}{c}0.02\\ 2.00\end{array} \right]. \tag{6}\]  
The toxic concentration level is represented by a inequality constraint  
\[y_{1}(t)\leq 0.026. \tag{7}\]The design task is considered on the following range of an independent variable  
\[t\in[0.0\quad 250.0]. \tag{8}\]  
The presented optimization task can be rewritten in a general form according to the given substitutions.