# Optimal control problem in pulmonary fibrosis and its solution by mathematical methods

Document Type : Original Paper

Authors

Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares, Tehran, Iran

Abstract

In this paper, we presented an optimal control problem for the least myofibroblast diffusion

in order to prevent the formation of fibrotic tissue in the repair process in lung tissue. Since transforming

growth factor-beta causes the proliferation and activation of myofibroblast, we considered this factor as

the control function of the optimal control problem. By presenting a theorem, the optimal control of the

above issue is guaranteed. In order to solve the problem of optimal control, there is a need to convert

an equation with partial derivatives into a system of linear ordinary differential equations. Therefore,

we solved this problem by using the central finite difference method. Then, using two methods of the

maximum principle (Hamiltonian equations ) and the calculus of variations (Euler–Lagrange equation) in

optimal linear regulator problem, we controlled the most important factor, i.e. transforming growth factor-beta, which is common to all fibrotic tissues. In a theorem, we prove that the solution of the optimal linear

regulator problem is the same for both methods. Finally, we compared the calculation results with the two

presented methods.

Keywords

Main Subjects

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### History

• Receive Date: 13 September 2022
• Revise Date: 26 December 2022
• Accept Date: 10 June 2023