Predictive drug dosage control through a Fokker–Planck observer

• Published in: Computational & applied mathematics 2018-07, Vol.37 (3), p.3813-3831
• Main Authors: Shakeri, Ehsan, Latif-Shabgahi, Gholamreza, Abharian, Amir Esmaeili
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Applied physics; Computational mathematics; Computational Mathematics and Numerical Analysis; Computer simulation; Dosage; Drug dosages; Economic models; Fokker-Planck equation; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematical models; Mathematics; Mathematics and Statistics; Optimal control; Population; Predictive control; Probability density functions; Probability theory; Stochastic models; Tumors; Windows (intervals)
• ISSN: 0101-8205; 2238-3603; 1807-0302
• DOI: 10.1007/s40314-017-0542-x

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.