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

Document Type : Original Paper


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


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.


Main Subjects

[1] Abbasi G. and Malek A., Hyperthermia cancer therapy by domain decomposition methods using strongly continuous semigroups, Mathematics and Computers in Simulation 165 (2019): 1-12.
[2] Abbasi G. and Malek A., Pointwise optimal control for cancer treatment by hyperthermia with thermal wave bioheat transfer, Automatica 111 (2020): 108579.
[3] Choudhury, Roy D., Modern control engineering, PHI Learning Pvt. Ltd., 2005.
[4] Darby, Ian A., Laverdet B., Bonté F. and Desmoulière A., Fibroblasts and myofibroblasts in wound healing, Clinical, cosmetic and investigational dermatology (2014): 301-311.
[5] Friedman A. and Hao W., Mathematical modeling of liver fibrosis, Mathematical Biosciences and Engineering 14.1 (2017): 143.
[6] Kisseleva, Tatiana., The origin of fibrogenic myofibroblasts in fibrotic liver, Hepatology 65.3 (2017): 1039-1043.
[7] Klingberg F., Hinz B. and White E.S., The myofibroblast matrix: implications for tissue repair and fibrosis, The Journal of pathology 229.2 (2013): 298-309.
[8] Hao W., Crouser E.D. and Friedman A., Mathematical model of sarcoidosis, Proceedings of the National Academy of Sciences 111.45 (2014): 16065-16070.
[9] Hao W., Komar H. M., Hart P. A., Conwell D. L., Lesinski G. B. and Friedman A., Mathematical model of chronic pancreatitis, Proceedings of the National Academy of Sciences 114.19 (2017): 5011-5016.
[10] Hao W., Marsh C. and Friedman A., A mathematical model of idiopathic pulmonary fibrosis, PLoS One 10.9 (2015): e0135097.
[11] Hao W., Rovin B.H. and Friedman A., Mathematical model of renal interstitial fibrosis, Proceedings of the National Academy of Sciences 111.39 (2014): 14193-14198.
[12] Henderson N.C., Rieder F. and Wynn T.A., Fibrosis: from mechanisms to medicines, Nature 587.7835 (2020): 555-566.
[13] Mehrali-Varjani M., Shamsi M. and Malek A., Solving a class of Hamilton-Jacobi-Bellman equations using pseudospectral methods, Kybernetika 54.4 (2018): 629-647.
[14] Prockop D.J., Inflammation, fibrosis, and modulation of the process by mesenchymal stem/stromal cells, Matrix Biology 51 (2016): 7-13.