# Robust Control of Repeated Drug Administration with Variable Doses Based on Uncertain Mathematical Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Continuous Time Model

**Remark 1.**

_{i}and b

_{i}include the rate constants ${k}_{a}$, ${k}_{e1}$, ${k}_{23}$, and ${k}_{e3}$.

_{i}, b

_{i}are often easily identifiable, the situation for the constants ${k}_{a}$, ${k}_{e1}$, ${k}_{23}$, and ${k}_{e3}$ is more complicated. If we cannot determine them from the known values of a

_{i}, b

_{i}unambiguously, the system is considered “unidentifiable” and the values of the ${k}_{a}$, ${k}_{e1}$, ${k}_{23}$, ${k}_{e3}$ must be identified directly from the in vivo samples while considering the system (2). The details can be found in [17,18] and the references therein.

#### 3.2. Discrete Time Counterpart of the Continuous Time Model

#### 3.3. Uncertain Discrete Time Model

#### 3.4. Drug Dosing in the Open Loop

#### 3.5. Drug Dosing in the Closed Loop

#### 3.6. Drug Dosing in the Case of an Unstable Subject

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- The size of the doses administered at the particular time instants;
- The order in which the variable doses are administered.

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**Figure 1.**The considered three-compartment pharmacokinetic model showing the drug transport mechanisms, state variables, and model parameters. Variables ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ are the drug concentrations in the gastrointestinal tract (abbr. GIT), central compartment, and tissue compartment, respectively, while $u$ and $y$ are the input and output of the model, respectively.

**Figure 2.**Cybernetic abstraction of a living organism (treated subject) in the case of open-loop dosing approach.

**Figure 3.**Trajectories of the drug concentrations $y\left(k\right)$ in the case of repeated constant dose showing poor robustness of the open-loop approach resulting in either ineffective or toxic treatment (AUC = 6615.7 h × mg/mL).

**Figure 4.**Block diagram of drug dosing in the closed loop that involves the integral controller with feedback on the drug concentration to determine the new dose size.

**Figure 5.**Trajectories of the drug doses $u\left(k\right)$ for the integral controller with ${k}_{i}=0.3$ showing a relatively fast response.

**Figure 6.**Trajectories of the drug concentration $y\left(k\right)$ for the integral controller with ${k}_{i}=0.3$ showing a relatively fast but robust response with slightly periodic behavior and overshoots (AUC = 6419.7 h × mg/mL).

**Figure 7.**Trajectories of the drug doses $u\left(k\right)$ for the integral controller with ${k}_{i}=0.1$ showing a relatively fast response.

**Figure 8.**Trajectories of the drug concentrations $y\left(k\right)$ for the integral controller with ${k}_{i}=0.1$ showing a relatively slow but robust response with aperiodic behavior and no overshoots (AUC = 4647.5 h × mg/mL).

**Figure 9.**Trajectories of the drug concentrations $y\left(k\right)$ for the integral controller with ${k}_{i}=0.1$ and application of a high loading dose $u\left(0\right)=50$ mg (AUC = 5956.0 h × mg/mL).

**Figure 10.**Trajectories of the drug concentrations $y\left(k\right)$ for the integral controller with ${k}_{i}=0.1$ and underestimated initial doses (AUC = 4605.0 h × mg/mL).

**Figure 11.**Trajectories of the drug concentrations $y\left(k\right)$ for the integral controller with ${k}_{i}=0.1$ and overestimated initial doses (AUC = 4751.2 h × mg/mL).

**Figure 12.**Block diagram of the inner stabilizing closed loop for the drug dosing that involves the state-feedback controller to determine the new stabilizing dose size.

**Figure 13.**Trajectories of the drug doses $u\left(k\right)$ for an unstable subject showing unfeasible negative drug doses to be administered.

**Figure 14.**Block diagram of the cascade control loop for the robustly stabilizing drug dosing that involves the state-feedback controller and the integral controller to determine the new dose size.

**Figure 15.**Robustly controlled drug concentrations $y\left(k\right)$ of an unstable subject stabilized by the state feedback showing stable behavior despite the unstable nature of the system (AUC = 3737.0 h × mg/mL).

t [h] | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

c [mg/mL] | 0 | 0.0715 | 0.0855 | 0.0780 | 0.0735 | 0.0490 | 0.0535 |

**Table 2.**Drug doses $u\left(k\right)$ and trajectory of drug concentrations $y\left(k\right)$ for the nominal case.

k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

u(k) | 15.00 | 28.78 | 39.71 | 47.14 | 51.26 | 52.77 | 52.52 | 51.31 | 49.77 | 48.30 |

y(k) | 0 | 4.05 | 13.55 | 25.24 | 36.25 | 44.96 | 50.82 | 54.03 | 55.15 | 54.88 |

k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

fs(k) | 0 | −3.02 | −5.25 | −7.53 | −9.64 | −11.65 | −13.54 | −15.32 | −17.00 | −18.58 |

k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

u(k) | 15.00 | 19.70 | 25.17 | 29.21 | 32.55 | 35.23 | 37.40 | 39.15 | 40.56 | 41.70 |

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**MDPI and ACS Style**

Vitková, Z.; Dodek, M.; Miklovičová, E.; Pavlovičová, J.; Babinec, A.; Vitko, A.
Robust Control of Repeated Drug Administration with Variable Doses Based on Uncertain Mathematical Model. *Bioengineering* **2023**, *10*, 921.
https://doi.org/10.3390/bioengineering10080921

**AMA Style**

Vitková Z, Dodek M, Miklovičová E, Pavlovičová J, Babinec A, Vitko A.
Robust Control of Repeated Drug Administration with Variable Doses Based on Uncertain Mathematical Model. *Bioengineering*. 2023; 10(8):921.
https://doi.org/10.3390/bioengineering10080921

**Chicago/Turabian Style**

Vitková, Zuzana, Martin Dodek, Eva Miklovičová, Jarmila Pavlovičová, Andrej Babinec, and Anton Vitko.
2023. "Robust Control of Repeated Drug Administration with Variable Doses Based on Uncertain Mathematical Model" *Bioengineering* 10, no. 8: 921.
https://doi.org/10.3390/bioengineering10080921