Bahram Yazdroudi Fatemeh, Malek Alaeddin
Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran.
Sci Rep. 2025 Feb 3;15(1):4120. doi: 10.1038/s41598-024-76561-0.
Idiopathic pulmonary fibrosis (IPF) is a chronic lung disease characterized by excessive scarring and fibrosis due to the abnormal accumulation of extracellular matrix components, primarily collagen. This study aims to design and solve an optimal control problem to regulate M2 macrophage activity in IPF, thereby preventing fibrosis formation by controlling the anti-M1 agent. The research models the diffusion of M2 macrophages in inflamed tissue using a novel dynamical system with partial differential equation (PDE) constraints. The control problem is formulated to minimize fibrosis by regulating an anti-M1 agent. The study employs a two-step process of discretization followed by optimization, utilizing the Galerkin spectral method to transform the M2 diffusion PDE into an algebraic system of ordinary differential equations (ODEs). The optimal control problem is then solved using Pontryagin/s minimum principle, canonical Hamiltonian equations, and extended Riccati differential equations. The numerical simulations indicate that without control, M2 macrophage levels increase and stabilize, contributing to fibrosis. In contrast, the optimal control strategy effectively reduces M2 macrophages, preventing fibrosis formation within 120 days. The results highlight the potential of the proposed optimal control approach in modulating tissue repair processes and mitigating the progression of IPF. This study underscores the significance of targeting M2 macrophages and employing mathematical methods to develop innovative therapies for lung fibrosis.
特发性肺纤维化（IPF）是一种慢性肺部疾病，其特征是由于细胞外基质成分（主要是胶原蛋白）异常积累导致过度瘢痕形成和纤维化。本研究旨在设计并解决一个最优控制问题，以调节IPF中的M2巨噬细胞活性，从而通过控制抗M1因子来预防纤维化形成。该研究使用具有偏微分方程（PDE）约束的新型动力学系统对M2巨噬细胞在炎症组织中的扩散进行建模。控制问题的制定是通过调节抗M1因子来使纤维化最小化。该研究采用了两步过程，即离散化然后优化，利用伽辽金谱方法将M2扩散PDE转化为常微分方程（ODE）的代数系统。然后使用庞特里亚金最小原理、正则哈密顿方程和扩展的里卡蒂微分方程来解决最优控制问题。数值模拟表明，在没有控制的情况下，M2巨噬细胞水平会升高并稳定下来，导致纤维化。相比之下，最优控制策略有效地减少了M2巨噬细胞，在120天内预防了纤维化形成。结果突出了所提出的最优控制方法在调节组织修复过程和减轻IPF进展方面的潜力。本研究强调了靶向M2巨噬细胞以及采用数学方法开发肺纤维化创新疗法的重要性。
