Adaptive non-linear control for cancer therapy through a Fokker–Planck observer

• Published in: IET systems biology Vol. 12; no. 2; pp. 73 - 82
• Main Authors: Shakeri, Ehsan, Latif-Shabgahi, Gholamreza, Esmaeili Abharian, Amir
• Format: Journal Article
• Language: English
• Published: England The Institution of Engineering and Technology 01.04.2018
• Subjects: adaptive nonlinear control; Algorithms; cancer; cancer therapy; Cell Proliferation; cellular biophysics; empirical data; Fokker‐Planck equation; Fokker‐Planck observer; Humans; logarithmic function; mathematical modelling; Models, Theoretical; Neoplasms - therapy; nonlinear stochastic observer; optimal control theory; patient treatment; physiological models; probability; probability density function; Research Article; statistical analysis; statistical methods; stochastic Gompertz model; Stochastic Processes; tumour cell growth behavior; tumours; tumour‐cell population dynamics; logarithmic function; cellular biophysics; Fokker-Planck observer; probability; tumour cell growth behavior; tumour-cell population dynamics; Fokker-Planck equation; nonlinear stochastic observer; cancer therapy; mathematical modelling; physiological models; patient treatment; adaptive nonlinear control; empirical data; cancer; optimal control theory; stochastic processes; statistical analysis; tumours; stochastic Gompertz model; statistical methods; probability density function
• ISSN: 1751-8849; 1751-8857; 1751-8857
• DOI: 10.1049/iet-syb.2017.0032

In recent years, many efforts have been made to present optimal strategies for cancer therapy through the mathematical modelling of tumour-cell population dynamics and optimal control theory. In many cases, therapy effect is included in the drift term of the stochastic Gompertz model. By fitting the model with empirical data, the parameters of therapy function are estimated. The reported research works have not presented any algorithm to determine the optimal parameters of therapy function. In this study, a logarithmic therapy function is entered in the drift term of the Gompertz model. Using the proposed control algorithm, the therapy function parameters are predicted and adaptively adjusted. To control the growth of tumour-cell population, its moments must be manipulated. This study employs the probability density function (PDF) control approach because of its ability to control all the process moments. A Fokker–Planck-based non-linear stochastic observer will be used to determine the PDF of the process. A cost function based on the difference between a predefined desired PDF and PDF of tumour-cell population is defined. Using the proposed algorithm, the therapy function parameters are adjusted in such a manner that the cost function is minimised. The existence of an optimal therapy function is also proved. The numerical results are finally given to demonstrate the effectiveness of the proposed method.