New Aspects of Solution Feasibility in a Context of Personalized Therapy Optimization

Abstract

In this work, the feasibility of a personalized therapy design is considered. We attempt to determine whether all of the obtained results of computer simulations should be presented to medical personnel. For this purpose, a two-drug displacement problem was used, which is the starting point of this research work. The relationships that can be used to characterize the progress as well as the efficiency of treatment in advanced cases can be modeled by a system of nonlinear dynamical equations with additional algebraic dependencies (differential-algebraic equations, DAEs). Then, to improve the efficiency of the therapy, an optimization task needs to be formulated and solved. The solution should meet all the assumed requirements and expectations. Therefore, a control vector parametrization (CVP) procedure for a DAE model is often suggested as an appropriate tool for solving the optimization-based therapy design tasks. In this work, a general iterative optimization framework is discussed in detail together with the proposed three levels of solution feasibility which try to decide if the iteratively obtained solution is trustworthy. The CVP optimization procedure with the designed levels of solution feasibility are implemented and tested. The obtained results are discussed from the perspective of their practical use in the treatment process. It is worth noting that solutions that are valuable from the perspective of creating new optimization algorithms may be rejected by the final recipient as devoid of application possibilities. Some of the presented solutions can be considered as a reference in further clinical research.

1. Introduction

Patient treatment is a complex process made up of many elements. While the basis of treatment is the knowledge and experience of medical personnel, nowadays the importance of other aspects is also emphasized, including screening tests, quick and accurate diagnostics, as well as psychological elements: trust in the patient–doctor relationship, professional, modern hospital equipment, cleanliness of the rooms, etc. In this way, a patient who feels surrounded by specialist care, experiences psychological comfort that has a positive impact. The presented views constitute a wide field of possible actions aimed at appropriate communication between patients, medical staff and technical staff. This is especially important in clinical and experimental treatment, where new solutions and procedures are tested.

The problem undertaken in this work is an attempt to determine what features should have new methods of computer data processing so that the returned results do not create unnecessary information noise but on the contrary constitute additional professional support. The features that a solution should have before it is presented, proposed in this work, are not limited to a specific case study. They can be implemented wherever we deal with data acquisition, digital data analysis and their final presentation.

In general, an application of new computational methods gives a hope for a significant increase in the quality of a life in various areas. The ability to simulate and optimize even complex phenomena can result in better and more accessible solutions [1]. The development of new, computer-aided design methods can be observed in social life, where optimization-based approaches can be efficiently applied to solve general and specialized problems. In particular, the application of computer programs built of specialized numerical procedures can have a positive impact on the treatment process design. This is especially true in a context of individual healing therapies, where unique individual solutions can be applied.

Computer-assisted medicine has many specific applications. Some of the examples of applications include:

• Designing three-dimensional models of bones and joints [2],
• Precise additive tissue production [3],
• Scheduling activities during complex, multi-team surgical operations [4],
• Planning the course of operations in a dedicated simulation environment [5],
• Improvement of selected activities using physical simulators [6],
• An attempt at algorithmization of activities and a numerical evaluation of the quality of their execution [7].

Moreover, depending on the available mathematical models and data, the observed phenomena can be personalized. Then, the obtained results, conclusions and observations can be generalized and extrapolated to other therapeutic processes. Therefore, all research in computationally personalized therapy design, such as conceptual considerations, experimental research and computer-based simulations, have an extremely important application potential.

The positive effects of the work in this area, undertaken by numerous researchers, have been presented in recent literature. In particular, Simpson et al. [8] presented a new approach to optimize a thiazole series of novel small molecule Abelson kinase activators. They proposed and tested a new methodology to increase a number of circulating neutrophils in order to improve treatment for chemotherapy-induced neutropenia. Wrobleski et al. [9] considered a method for optimization of a structure-guided design and water displacement strategy. Moreover, they indicated JAK inhibitors as a major therapeutic advancement in treating autoimmune disease. As a result, a discovered JAK inhibitor BMS-986165 (11) has been advanced as a pseudokinase-directed therapeutic in clinical development. Recently, Gutti et al. [10] tested a cholinergic hypothesis of Alzheimer’s disease as a tool for drug development. They presented a de novo fragment growing strategy for hit optimization spiropyrazoline derivatives. Moreover, the obtained pharmacokinetic assessment of optimized hit molecules can be used for further drug development. In their research, Colter, Wirostko and Coats [11],used a three-dimensional computational finite element model of the eye to determine the effect of geometry and surface friction on film retention in the inferior fornix. The researchers indicate, that drug-loaded hydrogel devices can be used as an effective means of localized and sustained drug delivery for the treatment of corneal conditions and injuries. The authors considered and presented a new approach: that thiolated cross-linked carboxymethylated, hyaluronic acid-based hydrogel film retention can be achieved through modifications of geometry and manipulation of surface interaction with the eye. An optimization-based approach for treatments for intervertebral disc degeneration and herniation were presented in the work [12]. Although some approaches are palliative only and cannot restore disc structure and function, nucleus pulposus replacements by cellulose-based hydrogel systems have been considered as a promising strategy. Finally, the  further optimization of the considered solution hydrogel system can improve the clinical solution for disc degeneration and herniation.

In the presented research, it is worth paying attention to the close connection between the subject of optimization and the way the optimization is carried out. The object is often in the form of a spatial numerical model or is a substance with known properties. Optimization, on the other hand, consists of changing the numerical structure of an object or features of a substance in such a way as to achieve the assumed goal. Moreover, optimization progress is often determined experimentally. In this way, it is clear that mathematical models can only play a supporting role. All layout properties and constraints are visual or even informal, especially when the consecutive optimization steps mean successive experiments and observations. Thus, it is possible to note various approaches used, including those in which the description of the problem in the form of mathematical equations is not clearly presented. The comparison of optimization, which combines experimental research with numerical optimization based on a mathematical equations model, encourages us to think about how to understand the constraints present in the task and what actions to take to be able to state that the obtained solution is indeed feasible.

The use of already developed numerical optimization algorithms for therapy design and optimization is strictly related to new types of requirements and risks that are specific to this area of application. Therefore, some general approaches that are successfully used in other industries cannot be directly transferred.

The usual expected conditions that need to be met in the process of solving the optimization task are reaching the optimum with an assumed accuracy while meeting continuous and point-wise constraints. It is also a good practice to archive intermediate results that can be used in future research. In particular, an intermediate solution may be either acceptable (feasible) or unacceptable (unfeasible). Then, reaching a mathematically correct solution through the area of unfeasible solutions makes the algorithm computations more flexible, but also makes them more incomprehensible. In other words, the obtained intermediate results can have no medical interpretation. This often results in a justified sense of distrust in the obtained solution. As a direct consequence of this approach, the entire series of results can also be rejected when:

-

The unfeasible solutions are close to the area of feasible solutions,

-

The solution meets all the constraints but does not meet some additional conditions which, due to the assumed obviousness, are not explicitly included in the task’s description.

In order to emphasize the importance of this problem and indicate ways that can unify the view and improve the current state of affairs, the following aims have been set in this work:

• Present a modified mathematical model of the drug displacement design task to indicate a general optimization-based formulation of design problems (Section 2).
• Introduce new possibilities of understanding the term "feasible solution". In this work, in Section 3, three different views on the issue are presented. It can be assumed that, depending on the application area, there may be more proposed definitions.
• Design an extended iterative optimization procedure, which is based on the control vector parametrization approach and direct approach for optimization (Section 4).
• Discuss, in Section 5, the obtained results, taking into account the feasibility of the proposed solution.
• Indicate new elements, that should be taken into account when creating new methods of searching for a solution (Section 6).

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, Aarons and Rowland 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

$$\underset{u\left(t\right)}{min}\rho {\left(y_1\left(t_f\right)-0.02\right)}^2+{\left(y_2\left(t_f\right)-2.00\right)}^2$$

(1)

with $\rho =100$, as well as the dynamical relations

$$\begin{array}{ccc}d{y}_1/dt& =& \frac{D^2}{C_3}[C_2(0.02-y_1)+46.4y_1(u-2y_2)]\hfill \\ \\ d{y}_2/dt& =& \frac{D^2}{C_1}[C_2(u-2y_2)+46.4y_2(0.02-y_1)]\hfill \end{array}$$

(2)

the algebraic relations

$$\begin{array}{ccc}0& =& D-(1+0.2y_1+0.2y_2)\hfill \\ \\ 0& =& C_1-(D^2+232+46.4y_2)\hfill \\ \\ 0& =& C_2-(D^2+232+46.4y_1)\hfill \\ \\ 0& =& C_3-(C_1{C}_2-2152.96y_1{y}_2).\hfill \end{array}$$

(3)

The control is the rate of infusion of phenylbutazone and is bounded by

$$0\le u\left(t\right)\le 8.$$

(4)

The initial concentrations of warfarin and phenylbutazone are given by

$$y\left(0\right)=\left[\begin{array}{c}{y}_1\left(0\right)\\ \\ y_2\left(0\right)\end{array}\right]=\left[\begin{array}{c}0.02\\ \\ 0.00\end{array}\right].$$

(5)

The system is to be taken to the desired final state

$$y\left(t_f\right)=\left[\begin{array}{c}{y}_1\left(t_f\right)\\ \\ y_2\left(t_f\right)\end{array}\right]=\left[\begin{array}{c}0.02\\ \\ 2.00\end{array}\right].$$

(6)

The toxic concentration level is represented by a inequality constraint

$$y_1\left(t\right)\le 0.026.$$

(7)

The design task is considered on the following range of an independent variable

$$t\in \left[0.0250.0\right].$$

(8)

The presented optimization task can be rewritten in a general form according to the given substitutions. Then, a real-valued performance index can be considered

$$\underset{u\left(t\right)}{min}F(y\left(t\right),z\left(t\right),u\left(t\right),p,t)$$

(9)

with a system of dynamical constraints

$$dy/dt=f\left(y\right(t),z(t),u(t),p,t)$$

(10)

and continuous algebraic relations

$$0=g\left(y\right(t),z(t),u(t),p,t),$$

(11)

with

$$F:{\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to \mathcal{R}$$

(12)

$$f:{\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to {\mathcal{R}}^{n_y}$$

(13)

$$g:{\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to {\mathcal{R}}^{n_z}$$

(14)

where $\mathcal{R}$ denotes a set of real numbers. Moreover,

$$y\left(t\right)=\left[\begin{array}{c}{y}_1\left(t\right)\\ ⋮\\ y_{n_y}\left(t\right)\end{array}\right]\in {\mathcal{R}}^{n_y}$$

(15)

is a vector of differential state trajectories;

$$z\left(t\right)=\left[\begin{array}{c}{z}_1\left(t\right)\\ ⋮\\ z_{n_z}\left(t\right)\end{array}\right]\in {\mathcal{R}}^{n_z}$$

(16)

is a vector of algebraic state trajectories;

$$u\left(t\right)=\left[\begin{array}{c}{u}_1\left(t\right)\\ ⋮\\ u_{n_u}\left(t\right)\end{array}\right]\in {\mathcal{R}}^{n_u}$$

(17)

is a vector of control functions;

$$p=\left[\begin{array}{c}{p}_1\left(t\right)\\ ⋮\\ p_{n_p}\left(t\right)\end{array}\right]\in {\mathcal{R}}^{n_p}$$

(18)

is a vector of global constant parameters; $t\in \mathcal{R}$ is the independent variable; and

$$t\in \left[t_0{t}_f\right].$$

(19)

A vector of initial conditions for state trajectories $y\left(t\right)$ at the initial point $t=t_0$ is given

$$y\left(t_0\right)=y_0.$$

(20)

The constraints on the control function take a form of lower and upper bounds

$$u_L\le u\left(t\right)\le u_U.$$

(21)

The pointwise constraints related to the final state trajectories $y\left(t\right)$, in particular

$$y_i\left(t_f\right)=y_{i,t_f}\mathrm{for}i=1,\cdots ,n_y$$

(22)

can be reflected in the performance index in the compact form

$$\underset{u\left(t\right)}{min}\sum _{i=1}^{n_y}{\rho }_i{\left(y_i\left(t_f\right)-y_{i,t_f}\right)}^2$$

(23)

where ${\rho }_i\in \mathcal{R},i=1,\cdots ,n_y$ are problem-related adjustable penalty parameters.

The presented infinitely dimensional task can be transformed into a finite-dimensional one with, e.g., a control parametrization method. This approach is based on dividing the range of the independent variable into an assumed number N sub-intervals such that

$$t_0=t_0^1<t_f^1=t_0^2<\cdots =t_0^N<t_f^N=t_f.$$

(24)

The parametrization of the control function $u\left(t\right)$ is carried out separately in each sub-interval. In particular, a step function is often used, which causes the control function to take a constant value in each subinterval

$$u\left(t^i\right)=u^i=const\mathrm{for}{t}^i\in \left[t_0^i{t}_f^i\right]\mathrm{and}i=1,\cdots ,N.$$

(25)

The values of the state variables at the end of a subinterval are treated as starting values for the next subinterval. Therefore, Equations (9) and (11) can take the form of a finite-dimensional optimization task with an objective function

$$\underset{\mathbf{U}\in {\mathcal{R}}^{n_u\times N}}{min}\tilde{F}\left(\mathbf{U}\right)$$

(26)

with a matrix of decision variables, which combine relation (25) with the vector representation (17)

$$\mathbf{U}=\left[\begin{array}{ccc}{u}_1^1& \cdots & u_1^N\hfill \\ ⋮& \ddots & ⋮\hfill \\ u_{n_u}^1& \cdots & u_{n_u}^N\hfill \end{array}\right]\in {\mathcal{R}}^{n_u\times N}$$

(27)

as well as the constraints vector

$$c\left(\mathbf{U}\right)=0$$

(28)

with

$$c:{\mathcal{R}}^{n_u\times N}\to {\mathcal{R}}^{n_c}$$

(29)

The vector of decision variables ${\mathbf{U}}^{\star }$, which is considered to be a solution to the given optimization problem, should meet all of the imposed constraints (28). The key problem at this point is how to understand the fulfillment of the constraints and what kind of constraints should be taken into account.

3. Three Levels of Feasibility

Research experience can suggest that a feasible solution may be understood from at least several perspectives. Therefore, it is worth considering in detail what restrictions are actually met, what conditions have been set and, finally, what the main expectations are. For these reasons, the feasibility of a given solution can be presented on various levels.

• Level 1. A mathematical feasibility
  The first perspective taken into account is to view the considered model or set of constraints as a pure mathematical construct, such as, e.g., (28). Such systems of equations can only be considered in clearly defined sets or their indicated subsets. However, the very origin of the model under consideration and the way it was constructed can be completely arbitrary and may not reflect the full physics of the phenomenon.

Definition 1. 

The mathematical feasibility denotes whether the problem constraints (28) have been met.

Proposition 1. 

If the constraints (28) are not met, then the constraints’ violation can be measured as

$$\parallel c\left({\mathbf{U}}^{\star }\right)\parallel \to [0,+inf).$$

(30)

Remark 1. 

Fulfillment of the constraints (28), understood as the pure mathematical relations, does not guarantee that the obtained solution is of an application nature.

• Level 2. A physical rationality
  In practice, such limitations are difficult to grasp because they are taken for granted, and therefore, often not even symbolic attention is paid to them. For example, surface area and volume are evidently considered as positive values from the set of real numbers. Likewise, the concentration of a substance cannot be expressed as a negative number. In addition, the temperature, expressed in degrees Celsius, is greater than or equal to −273.15${}^{\circ }$.

Definition 2. 

The physical rationality indicates whether the constraints that were used to determine the physical feasibility of the proposed solution are met.

Remark 2. 

A computational experience can suggest that taking the above physical properties as granted and not expressed explicitly in the model may lead to results which, despite the mathematical correctness, will not meet the most obvious expectations.

• Such a phenomenon can be observed at the beginning of the iterative calculations when the starting point may differ significantly from the final solution. Then, the intermediate results do not have any reasonable interpretation even though the dynamical equations of the model were resolved correctly.
• However, deficiencies of this type result in a significant distrust of the applied computing procedure, which allows for an intermediate solution that does not comply with the fundamental laws of physics and expectation of common sense. For the same reason, the received final solution, although factually correct and valuable, may be disqualified.   

• Level 3. A technological applicability
  Specifying this area causes the main goal of the performed computer simulations to increase the numerical accuracy of the proposed solution, which is approximately known a priori.

Definition 3. 

The technological applicability denotes the highest level of details in the constraints that reflect real technological expectations and possibilities.

Remark 3. 

The technological applicability’s constraints reflect the experiences of practitioners, technologists and experimenters that must be met and may not result explicitly from the model equations.

This can be the narrowest area of the solution feasibility.

Developing a new approach in computer simulations, the results of which could be used to design dedicated therapies, require consideration of the various constraints and their impact on the computation, as well as the confidence of those who will benefit from these results. The experience so far shows that the final results of the simulation are characterized by trust and approval, especially when the subsequent calculation steps are accessible to understand and an aura of secrecy and inaccessibility is not created. Moreover, intermediate results, which are often the result of numerous fundamental operations, should indicate in a clear and universally understandable way the path to the result.

4. The Diagram of a General Solution Procedure

One of the most frequently used approaches for solving dynamic optimization problems is a sequential approach [18]. The sequential approach to optimization is based on iteratively performing the following computational steps:

(1)

Parameterize the control function $u\left(t\right)$, such that

$$\mathcal{P}\left\{u\right(t\left)\right\}=\mathbf{U}$$

(31)

where $\mathcal{P}$ is a chosen parametrization operator defined as in [19] and try to indicate reasonable values of the parameters that constitute the matrix $\mathbf{U}$.

(2)

For a given matrix of parameters $\mathbf{U}$, the trajectories of the state variables $y\left(t\right)$ and $z\left(t\right)$, as well as the quality index (26) and the constraints functions (28), are calculated.

(3)

An influence of the selected parameters on the value of the quality indicator can be examined.

(4)

Choosing one of the available numerical gradient methods [20], the  values of the matrix $\mathbf{U}$ can be improved according to the rule

$$\mathbf{U}=\mathbf{U}+\Delta \mathbf{U}.$$

(32)

The described steps of the procedure can be performed iteratively until the solution is found with the required accuracy or until the computational resources are used [21].

An important issue in any constrained optimization task is the problem of determining, whether a proposed solution is feasible/acceptable or not. The term "feasible" should clearly state whether a given solution meets the limitations and whether it is a potential application. The scheme of the proposed general solution procedure is presented in Figure 1. It takes into account the required input data in the form of the objective function (Equation (9)) and model Equations (10) and (11) with the given initial conditions (20), as well as additional inequality constraints (such as, e.g., (21) and (22)) that can be used to determine the type of feasibility of the obtained solution.

It should be emphasized, that the main elements of the presented scheme include:

• Choice of the parametrization method,
• A method for solving systems of differential-algebraic equations,
• A gradient numerical optimization algorithm,
• Defining the rules, the fulfillment of which means that the solution is feasible, depending on the particular issue.

The presented results were obtained after the implementation of the presented procedure in the Matlab environment. The constant function was used to parameterize the control function. The system of the DAE equations was solved using the ode15s solver. The fmincon function was chosen as the numerical optimization algorithm. The trajectories obtained as a result of the performed calculations were presented and discussed from the perspective of different levels of the solution feasibility.

5. Results and Discussion

The presented procedure was used to solve the two-drug displacement problem according to the description presented in Section 2. At the beginning, all the most important elements of the task were introduced into the program, i.e., the objective function (1) and constraints in the form of the differential and algebraic Equations (2) and (3). Then, the given initial (5) and final (6) conditions for the state variables were attached to the task. The last elements are the inequality constraints for the input function (4), as well the inequality constraint (7), representing the toxic concentration level. In the second step, which corresponds to the second rectangle in the first line of the flowchart in Figure 1, the control function was parameterized. The parametrization was conducted in such a way that the given range of the independent variable t was divided into 25 sub-intervals, where each interval had the same length. It was assumed that within each of the sub-intervals the control function can have only a constant value. In this way, the inequality constraints in the form (4) were directly included in the problem.

As already mentioned, as a result of the control vector parametrization, the range of the independent variable was divided into 25 sub-intervals. Thus, the nonlinear optimization problem took into account 25 decision variables that corresponded to the value of the control function $u\left(t\right)$. The state variables $y_1\left(t\right)$ and $y_2\left(t\right)$ at the beginning of the next sub-interval took the final values from the previous calculations.

To calculate the gradient of the objective function with respect to the decision variables, it is necessary to execute 26 objective function calls, which is within the capabilities of a personal computer. The required matrix of second derivatives was calculated by the Broyden–Fletcher–Goldfarb–Shanno method, which guarantees well-conditioning of the optimization task. Then, the proposed starting solution was iteratively improved and the changes in the value of the decision variables were determined by the fmincon algorithm. The dynamic optimization task, approximated by a finite dimensional form, was attacked by the numerical optimization algorithm—Sequential Quadratic Programming. Numerical calculations were performed using a personal computer with an 11th Gen Intel(R) Core(TM) i7-1165G7 2.80 GHz processor with 8.00 GB RAM and a 64-bit operating system.

One of the key issues considered in the article, is the method of presenting the obtained trajectories, which are also subject to change during subsequent iterations. In order for the results presented on the operator’s screen to be user-friendly, it was possible to display only those calculation results that fullfill the feasibility conditions. As already mentioned, this issue is important because often results obtained in an incomprehensible or questionable manner are not accepted and questioned. This applies primarily to those solutions obtained by algorithms that also take into account indirect unacceptable solutions. For this reason, a logic block was introduced into the program that can decide what kind of intermediate calculation results can be publicly presented.

The logic block checks the fulfillment of the conditions resulting from conditions which can be based on presented levels of the solution feasibility. The selection of the displayed results does not affect the numerical calculations. However, it introduces the final recipient to the course of calculations and allows understanding of the principle of iterative calculations, which try to improve the current solution. In this sense, information due to the failure to meet the constraints can be referred to as irrelevant.

In this section, some examples of intermediate results are presented. Figure 2 shows the situation when the control function does not influence the system. External input is within acceptable limits, i.e., $u\left(t\right)=0$ in the whole range of the performed simulations. The initial conditions of the state variables correspond to the given values. While the trajectory $y_1\left(t\right)$ satisfies the given boundary conditions, the trajectory $y_2\left(t\right)$ satisfies only the initial condition. The condition for the final state is unfulfilled, which is, however, in line with expectations when there is no active influence of the function $u\left(t\right)$. Therefore, it is legitimate to ask whether the obtained solution can be treated as a feasible solution. The presented trajectories meet the constraints resulting from the model. Moreover, they are in line with the expectations when the control function does not act on the system. Nevertheless, the assumed optimization goal was not met, which seems to be obvious, when the calculations did not even start. Moreover, the key question of whether the user will consider such a solution as worth presenting should be asked, that is whether the presented trajectories meet the assumed conditions sufficiently to be considered an acceptable solution.

The other situation, when $u\left(t\right)=8$, is shown in Figure 3. The control function is within the acceptable range. The obtained trajectories of state variables are starting at a given initial point. However, as can be seen, the values of the $y_1\left(t\right)$ function in the middle part exceed the acceptable level of toxicity. Additionally, none of the trajectories meets the given final condition. The conditions that were met, however, were the mathematical equations of the simulated model. In this situation, it can be concluded that the obtained solution is not acceptable because it has no application potential. It can rather be used as an example reference point for a future numerical research.

The obtained final solution is depicted in Figure 4. The presented trajectories meet the differential-algebraic equations of the model; the assumed boundary conditions and the toxicity level is not exceeded. It should be emphasized, that the considered issue did not take into account additional conditions that may be necessary when applying the obtained results in therapy. Issues, such as the minimization of possible additional side effects, the financial cost of the application and the actual accuracy with which the received control can be applied, are not considered. The conducted research shows new issues and limitations that can be added to the optimization task formulation and which can only result from good communication between the medical staff and the designer of the computing environment.

6. Conclusions

This work deals with the problem of computer-aided personalized therapy design. Two main problems were posed:

• What kind of results should be presented,
• What does it means, from the perspective of medical staff, that the solution is acceptable/feasible.

The paper presents a diagram of the general approach to iteratively obtaining a solution. Its essential element is the logical block, which should clearly define whether the received solution should be publicly presented. This is important because some of the results of measurements and calculations are presented to doctors and patients and thus create an atmosphere of trust and a professional and modern approach to treatment.

During the research, it was noticed that the concept of feasibility of a solution can have different meanings. The specific meaning is determined by the context in which it is used. For this reason, this work proposes three different meanings of the term under consideration. It can be assumed, that there may be more explanations.

As a conclusion from the conducted considerations and research, we would like to note that the mathematical models of the considered issues should be extended with additional dependencies and limitations that will clearly result from the context, in which the task was undertaken.

The obtained results and observations have a high application potential. Adding the appropriate application context to the calculation procedures in an explicit manner will not only result in personalized results, but also the calculation procedure will be adapted to the recipients’ requirements.