Fig. 6
From: Reducing M2 macrophage in lung fibrosis by controlling anti-M1 agent
Comparison between the optimal regulator problem and the dynamical system solutions. In Fig. 6a, we can see that the density of M2 macrophages increases over time in the absence of a control factor. This trend also holds when using different constant scalar values for \(\eta _{M1}(t)\) in the dynamical system (2). However, after the initial increase, the density remains constant. This indicates that wound repair is ongoing and not halted. Consequently, inflamed tissue transforms into fibrotic tissue over time, suggesting the presence of fibrosis. The optimal regulator problem solutions (29) are decreasing, and after some time, they reach zero, initiating the process of cell death. This signifies the cessation of the wound healing process, allowing doctors to restrict the activity of macrophage M1 using drugs or other control agents. As the activity of macrophage M1 decreases, the activity of macrophage M2 gradually diminishes over time. Eventually, with the process of cell death, the M2 macrophage becomes inactive. Consequently, the inflammatory mediators causing fibroblast to myofibroblast transformation are no longer active. Finally, the process of forming fibrotic tissue in the wound area ceases. In Fig. 6b, the optimal control function \(\eta _{M1}(t)\) are depicted. It is observed that the control function (anti-M1) decreases and then remains zero. Hence, in repair tissue, the M2 macrophages vanish through apoptosis, preventing the formation of fibrotic tissue. Medications are prescribed in specific doses that decrease over time. This implies that the treatment strategy involves a gradual reduction in medication dosage through the cure duration.
