Optimal minimum variance-entropy control of tumour growth processes based on the Fokker-Planck equation

Area of Science:

• Mathematical Oncology
• Stochastic Control Theory
• Computational Biology

Background:

• Cancer treatment efficacy is often limited by the complex dynamics of tumor growth.
• Mathematical models, such as the Gompertz model, are crucial for understanding and predicting tumor behavior.
• Optimal control strategies are needed to effectively manage cancer progression.

Purpose of the Study:

• To develop an optimal stochastic control algorithm for cancer treatment based on the Gompertz model.
• To simultaneously control the entropy and variance of cancerous cells.
• To investigate the impact of external treatment agents on tumor growth and death rates.

Main Methods:

• Utilized the Gompertz model to represent tumor dynamics.
• Introduced two time-dependent external forces to manipulate growth and death rates.
• Formulated a constrained optimization problem with variance as the cost function and entropy as a constraint.
• Employed a real-coded genetic algorithm to solve the optimization problem.

Main Results:

• Demonstrated an optimal stochastic control algorithm for cancer treatment.
• Identified logarithmic control signals to reduce cancerous cell growth rates.
• Identified hyperbolic tangent-like control functions to increase cancerous cell death rates.
• Provided mathematical justifications for the existence and uniqueness of the optimal control solution.

Conclusions:

• The developed algorithm offers a novel approach to cancer treatment by optimizing control over tumor cell populations.
• Simultaneous control of cancerous cell entropy and variance can lead to more desirable treatment outcomes.
• The study provides a robust mathematical framework for designing effective cancer therapies.