Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis
Abstract In the repair of injury, some transforming growth factor- $$\beta$$ β s (TGF- $$\beta$$ β s) and platelet-derived growth factors (PDGFs) bind to fibroblast receptors as ligands and cause the differentiation of fibroblasts into myofibroblasts. When the injury repair is repeated, the myofibro...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Nature Portfolio
2023-09-01
|
| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-023-41294-z |
| _version_ | 1797452737069187072 |
|---|---|
| author | Fatemeh Bahram Yazdroudi Alaeddin Malek |
| author_facet | Fatemeh Bahram Yazdroudi Alaeddin Malek |
| author_sort | Fatemeh Bahram Yazdroudi |
| collection | DOAJ |
| description | Abstract In the repair of injury, some transforming growth factor- $$\beta$$ β s (TGF- $$\beta$$ β s) and platelet-derived growth factors (PDGFs) bind to fibroblast receptors as ligands and cause the differentiation of fibroblasts into myofibroblasts. When the injury repair is repeated, the myofibroblasts proliferate excessively, forming fibrotic tissue. We goal to control myofibroblasts proliferation and apoptosis with anti-transforming growth factor- $$\beta$$ β (anti-TGF- $$\beta$$ β ) and anti-platelet-derived growth factor (anti-PDGF) medicines. The novel optimal regulator control problem with two controls (medicines) is proposed to simulate how to the preventing pulmonary fibrosis. Idiopathic pulmonary fibrosis (IPF) consists of restoring a system of cells, protein, and tissue networks with injury and scar. Myofibroblasts proliferation back to its equilibrium position after it has been disturbed by abnormal repair. Thus, the optimal regulator control problem with a parabolic partial differential equation as a constraint, zero flux boundary, and given specific initial conditions, is considered. The myofibroblast diffusion equation stands as a governing dynamic system while the objective function is the summation of myofibroblast, anti-TGF- $$\beta$$ β and anti-PDGF medicines for the fixed final time. Here, myofibroblast is a nonlinear state of time while anti-TGF- $$\beta$$ β and anti-PDGF are two unknown control functions. In order to solve the corresponding problem a weighted Galerkin method is used. Firstly, we convert the myofibroblast diffusion equation to a system of ordinary differential equations using the Lagrangian interpolation polynomials defined at Gauss-Lobatto integration points. Secondly, by the calculus of variations, the optimal control problem is solved successfully using canonical Hamiltonian and extended Riccati equations. Numerical results are given, and the plots are depicted. Moreover, solutions to the problem in which there is no control are compared. Numerical results show that, over time, the myofibroblast increases and then remains constant when there is no control. In contrast, the current solution decreases and vanishes after 300 days by prescribing controller medicines for anti-TGF- $$\beta$$ β and anti-PDGF. The optimal strategy proposed in this paper helps practitioners to reduce myofibroblasts by controlling both anti-TGF- $$\beta$$ β and anti-PDGF medicines. |
| first_indexed | 2024-03-09T15:12:58Z |
| format | Article |
| id | doaj.art-a6d575f197e64b8cb417a7699220e790 |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-03-09T15:12:58Z |
| publishDate | 2023-09-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-a6d575f197e64b8cb417a7699220e7902023-11-26T13:17:43ZengNature PortfolioScientific Reports2045-23222023-09-0113111510.1038/s41598-023-41294-zOptimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosisFatemeh Bahram Yazdroudi0Alaeddin Malek1Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares UniversityDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares UniversityAbstract In the repair of injury, some transforming growth factor- $$\beta$$ β s (TGF- $$\beta$$ β s) and platelet-derived growth factors (PDGFs) bind to fibroblast receptors as ligands and cause the differentiation of fibroblasts into myofibroblasts. When the injury repair is repeated, the myofibroblasts proliferate excessively, forming fibrotic tissue. We goal to control myofibroblasts proliferation and apoptosis with anti-transforming growth factor- $$\beta$$ β (anti-TGF- $$\beta$$ β ) and anti-platelet-derived growth factor (anti-PDGF) medicines. The novel optimal regulator control problem with two controls (medicines) is proposed to simulate how to the preventing pulmonary fibrosis. Idiopathic pulmonary fibrosis (IPF) consists of restoring a system of cells, protein, and tissue networks with injury and scar. Myofibroblasts proliferation back to its equilibrium position after it has been disturbed by abnormal repair. Thus, the optimal regulator control problem with a parabolic partial differential equation as a constraint, zero flux boundary, and given specific initial conditions, is considered. The myofibroblast diffusion equation stands as a governing dynamic system while the objective function is the summation of myofibroblast, anti-TGF- $$\beta$$ β and anti-PDGF medicines for the fixed final time. Here, myofibroblast is a nonlinear state of time while anti-TGF- $$\beta$$ β and anti-PDGF are two unknown control functions. In order to solve the corresponding problem a weighted Galerkin method is used. Firstly, we convert the myofibroblast diffusion equation to a system of ordinary differential equations using the Lagrangian interpolation polynomials defined at Gauss-Lobatto integration points. Secondly, by the calculus of variations, the optimal control problem is solved successfully using canonical Hamiltonian and extended Riccati equations. Numerical results are given, and the plots are depicted. Moreover, solutions to the problem in which there is no control are compared. Numerical results show that, over time, the myofibroblast increases and then remains constant when there is no control. In contrast, the current solution decreases and vanishes after 300 days by prescribing controller medicines for anti-TGF- $$\beta$$ β and anti-PDGF. The optimal strategy proposed in this paper helps practitioners to reduce myofibroblasts by controlling both anti-TGF- $$\beta$$ β and anti-PDGF medicines.https://doi.org/10.1038/s41598-023-41294-z |
| spellingShingle | Fatemeh Bahram Yazdroudi Alaeddin Malek Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis Scientific Reports |
| title | Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis |
| title_full | Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis |
| title_fullStr | Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis |
| title_full_unstemmed | Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis |
| title_short | Optimal controlling of anti-TGF- $$\beta$$ β and anti-PDGF medicines for preventing pulmonary fibrosis |
| title_sort | optimal controlling of anti tgf beta β and anti pdgf medicines for preventing pulmonary fibrosis |
| url | https://doi.org/10.1038/s41598-023-41294-z |
| work_keys_str_mv | AT fatemehbahramyazdroudi optimalcontrollingofantitgfbetabandantipdgfmedicinesforpreventingpulmonaryfibrosis AT alaeddinmalek optimalcontrollingofantitgfbetabandantipdgfmedicinesforpreventingpulmonaryfibrosis |