# Predictive drug dosage control through a Fokker–Planck observer

## Abstract

During the years, various techniques have been presented for optimal control of tumor growth for which its stochastic behavior has rarely been considered. This article uses the well-known Gompertz stochastic model, for describing tumor growth, and by receding horizon model predictive control (RH-MPC) scheme computes the drug dosage for minimizing the tumor-cell population. To do this, a predictive control problem is defined based on the difference between the probability density function of tumor-cell population and the desired probability density function. In the model, both the drug dosage limitation and the Fokker–Planck equation have been considered as constraints. By solving this problem, the drug dosage which is considered as an external input to tumor model, is computed. In this article, the Fokker–Planck equation is used (1) as a nonlinear observer of probability density function of tumor-cell population and (2) a mapping vehicle from stochastic to deterministic domain. The Fokker–Planck equation is written for the Gompertz stochastic model of tumor growth. The equation is then solved through the path integral method. In this way, the probability density function of tumor-cell population, which contains the entire stochastic characteristics of the tumor growth, is obtained for any instance of time. The simulation results havemore »

- Authors:

- Shahid Beheshti University, A.C, Faculty of Electrical Engineering (Iran, Islamic Republic of)
- Islamic Azad University, Department of Electrical Enginnering, Garmsar Branch (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 22769261

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; NEOPLASMS; OPTIMAL CONTROL; PATH INTEGRALS; PROBABILITY DENSITY FUNCTIONS; SIMULATION; STOCHASTIC PROCESSES; TUMOR CELLS

### Citation Formats

```
Shakeri, Ehsan, Latif-Shabgahi, Gholamreza, and Abharian, Amir Esmaeili, E-mail: aeabharian@srbiau.ac.ir.
```*Predictive drug dosage control through a Fokker–Planck observer*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0542-X.

```
Shakeri, Ehsan, Latif-Shabgahi, Gholamreza, & Abharian, Amir Esmaeili, E-mail: aeabharian@srbiau.ac.ir.
```*Predictive drug dosage control through a Fokker–Planck observer*. United States. doi:10.1007/S40314-017-0542-X.

```
Shakeri, Ehsan, Latif-Shabgahi, Gholamreza, and Abharian, Amir Esmaeili, E-mail: aeabharian@srbiau.ac.ir. Sun .
"Predictive drug dosage control through a Fokker–Planck observer". United States. doi:10.1007/S40314-017-0542-X.
```

```
@article{osti_22769261,
```

title = {Predictive drug dosage control through a Fokker–Planck observer},

author = {Shakeri, Ehsan and Latif-Shabgahi, Gholamreza and Abharian, Amir Esmaeili, E-mail: aeabharian@srbiau.ac.ir},

abstractNote = {During the years, various techniques have been presented for optimal control of tumor growth for which its stochastic behavior has rarely been considered. This article uses the well-known Gompertz stochastic model, for describing tumor growth, and by receding horizon model predictive control (RH-MPC) scheme computes the drug dosage for minimizing the tumor-cell population. To do this, a predictive control problem is defined based on the difference between the probability density function of tumor-cell population and the desired probability density function. In the model, both the drug dosage limitation and the Fokker–Planck equation have been considered as constraints. By solving this problem, the drug dosage which is considered as an external input to tumor model, is computed. In this article, the Fokker–Planck equation is used (1) as a nonlinear observer of probability density function of tumor-cell population and (2) a mapping vehicle from stochastic to deterministic domain. The Fokker–Planck equation is written for the Gompertz stochastic model of tumor growth. The equation is then solved through the path integral method. In this way, the probability density function of tumor-cell population, which contains the entire stochastic characteristics of the tumor growth, is obtained for any instance of time. The simulation results have also been presented for the evaluation of the suggested approach. The results show that the tumor-cell population can be controlled within a number of time windows (15 time windows in our case study) if an appropriate desired PDF is selected.},

doi = {10.1007/S40314-017-0542-X},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 3,

volume = 37,

place = {United States},

year = {2018},

month = {7}

}