Reducing M2 macrophage in lung fibrosis by controlling anti-M1 agent

Abstract

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.

Introduction

Idiopathic pulmonary fibrosis (IPF) is a chronic and progressive lung disease characterized by the formation and accumulation of scar tissue (fibrosis) in the lungs. The term “idiopathic” denotes that the cause of the disease is unknown. In IPF, the lung tissue becomes thickened and stiff, resulting in a reduction in the lungs’ ability to expand and contract properly¹. The Extracellular Matrix (ECM) constitutes a complex network of proteins and other molecules providing structural support to tissues. In IPF, an abnormal accumulation of ECM components, particularly collagen, occurs, leading to fibrosis and scarring. Injury or inflammation can trigger the activation of alveolar epithelial cells (AECs) in the lungs. When stimulated by various factors, including injury or proinflammatory signals, AECs undergo a process called activation. During this activation, AECs can secrete several proinflammatory cytokines, such as tumor necrosis factor-alpha (TNF-a)^2,3 and Monocyte chemotactic protein-1 (MCP-1). MCP-1, a chemokine, plays a crucial role in recruiting circulating monocytes from the blood into damaged or inflamed lung tissue. Once monocytes enter the damaged lung tissue, they can differentiate into macrophages. In the context of lung injury or inflammation, they tend to transform into classically activated macrophages, commonly referred to as M1 macrophages. The presence of M1 macrophages in the damaged lung tissue contributes to the inflammatory response, potentially exacerbating the progression of fibrosis in IPF. Understanding these cellular and molecular processes is essential for developing targeted therapies that can modulate the immune response, potentially slowing down or reversing the fibrotic changes associated with IPF. One of the key factors in the progression of IPF is the activity of M2 macrophages. M2 macrophages, while normally involved in tissue repair and immune regulation, can contribute to fibrosis in IPF. Targeting M2 macrophages is crucial for the treatment of IPF due to their significant role in promoting fibrosis. Normally, M2 macrophages are involved in tissue repair, but in the context of IPF, they become profibrotic by secreting cytokines and growth factors such as Transforming Growth Factor-beta (TGF-$\beta$) and Platelet-Derived Growth Factor (PDGF). These factors stimulate collagen production and activate fibroblasts, leading to excessive scar tissue formation. By regulating M2 macrophage activity, it may be possible to slow or even halt the progression of fibrosis in IPF patients.

Importance of M2 Macrophages in IPF: M2 macrophages contribute to fibrosis by promoting the accumulation of extracellular matrix (ECM) and facilitating the activation of myofibroblasts, which are directly responsible for producing the fibrotic tissue characteristic of IPF. Controlling these cells is therefore crucial for preventing fibrosis and supporting normal tissue repair mechanisms.

Other therapeutic approaches for IPF, including anti-inflammatory therapies, anti-fibrotic drugs (e.g., pirfenidone and nintedanib), and TGF-$\beta$ inhibitors, primarily target inflammation or fibroblast activation. However, they do not specifically address the imbalance in macrophage populations. As none of these treatments modulate M2 macrophage activity directly, therapies targeting M2 macrophages fill a crucial gap in current IPF treatments, potentially improving patient outcomes. Specifically, pro-fibrotic subsets of M2 macrophages release cytokines and growth factors such as TGF-$\beta$ and PDGF, which promote collagen production and fibroblast activation, exacerbating fibrosis. Effectively regulating these macrophages, especially in inflamed and damaged tissues, is crucial for preventing the progression of IPF and improving patient outcomes.

M1 macrophages are commonly linked to proinflammatory responses, while M2 macrophages are generally associated with functions related to tissue repair, remodeling, and immunoregulation. Profibrotic M2 macrophages have been identified as contributors to the promotion of fibrosis through the secretion of different cytokines, growth factors, and matrix remodeling enzymes. These macrophages play a role in stimulating the activation and proliferation of fibroblasts, the cells responsible for generating excessive collagen and other components of fibrotic scar tissue. M2 macrophages actively contribute to the regulation of the inflammatory process and play a pivotal role in wound healing and tissue regeneration by releasing specific molecules. Some of the crucial molecules released by M2 macrophages include: Interleukin-13 (IL-13): IL-13 serves as an anti-inflammatory cytokine, playing a crucial role in immune response regulation. It possesses the ability to suppress the production of proinflammatory cytokines, such as TNF-a, while fostering tissue repair and remodeling processes. TGF-$\beta$: TGF-$\beta$ stands out as a critical fibrogenic cytokine that holds a central role in encouraging myofibroblast activation and the production of ECM. In the context of tissue repair and fibrosis, TGF-$\beta$ plays a stimulatory role in collagen and other ECM component production, actively contributing to tissue remodeling and the formation of scars. Matrix Metalloproteinases (MMPs): MMPs represent a family of enzymes crucially involved in the breakdown and remodeling of the ECM. M2 macrophages have the ability to release specific MMPs, such as MMP-9 and MMP-12, which play a significant role in processes like tissue remodeling and wound healing. Tissue Inhibitor of Metalloproteinases (TIMPs): TIMPs act as endogenous inhibitors of MMPs. M2 macrophages can release TIMPs, contributing to the control of MMP activity and maintaining a balance between ECM remodeling and tissue repair. PDGF: PDGF is a growth factor that stimulates fibroblast proliferation and recruitment. It plays a crucial role in promoting the accumulation of fibroblasts and myofibroblasts, thereby contributing to the development of fibrosis. PDGF, along with TGF-$\beta$, transforms fibroblasts into myofibroblasts^4,5,6,7, leading to the production of ECM. The dysregulation between MMP and its inhibitor TIMP contributes to ECM accumulation and the onset of fibrosis⁸. The term “anti-M1” typically refers to strategies intended to diminish the pro-inflammatory and fibrotic functions of M1 macrophages. In the context of IPF, considering M1 macrophages or their associated inflammatory pathways as potential therapeutic targets could be essential in mitigating disease progression. Various approaches have been explored to modulate M1 macrophage activity in IPF with the aim of reducing inflammation and fibrosis. These approaches encompass anti-inflammatory agents, targeted therapies, immunomodulatory strategies, and cell-based therapies. However, there is a gap in the development of mathematical models and optimal control strategies specifically for regulating M2 macrophages in the context of IPF. Existing models often focus on different aspects of inflammation or fibrosis but do not address the combined effect of PDE constraints and control strategies on M2 macrophage dynamics.

Motivation for study

Regulating M2 macrophage activity in IPF presents a complex challenge. These cells typically contribute to tissue repair and immune regulation, but in IPF, their profibrotic functions can worsen the disease. Existing treatments focus on controlling inflammation or preventing fibrosis but do not specifically target M2 macrophages. This study seeks to address this gap by developing a novel dynamic mathematical model and applying optimal control strategies to manage M2 macrophage activity in IPF effectively. The urgency for innovative therapeutic strategies in IPF, a progressive and debilitating lung disease with limited treatment options, motivates this study. IPF involves the buildup of fibrotic tissue, partially driven by M2 macrophages, which, while usually involved in repair, can exacerbate fibrosis through their profibrotic activities. Although significant progress has been made in understanding M2 macrophages, existing approaches lack a comprehensive strategy to regulate these cells effectively in IPF. This research introduces a novel mathematical model integrating a dynamical system with PDE constraints to simulate M2 macrophage diffusion and optimize anti-M1 control strategies. The innovative aspect of this approach lies in the application of advanced mathematical methods, such as Pontryagin’s minimum principle, canonical Hamiltonian equations, and extended Riccati differential equations, to address fibrosis prevention through precise modulation of macrophage activity. This study not only provides a theoretical framework for controlling fibrosis but also offers a potential pathway for developing more effective therapies. By designing an optimal control strategy to regulate M2 macrophages, this research aims to enhance tissue repair processes and potentially mitigate fibrosis, providing a promising approach to improving patient outcomes in IPF.

The application of PDE constraints in modeling M2 macrophage diffusion is innovative because it allows for a more accurate representation of the complex temporal and spatial behavior of these cells compared to simpler models. This enhanced accuracy can reveal optimal control strategies that previous studies may have overlooked. This study presents a novel approach by developing a dynamic system model with PDE constraints to describe M2 macrophage diffusion in inflamed tissues. The key innovations are:

• (i) From a medical perspective, a new dynamical system is proposed to model M2 macrophage diffusion in inflamed tissue, where anti-M1 medication is hypothetically involved. Identifying the optimal strategy to regulate M2 macrophages by controlling anti-M1 medication could potentially prevent fibrosis formation.

• (ii) To regulate M2 macrophages, a new hybrid optimal regulator problem with a PDE constraint and a single control variable is mathematically formulated.

• (iii) The canonical Hamiltonian equations and boundary conditions at the final time $t=t_f$ are established.

• (iv) The extended optimal feedback control law is assumed to solve the problem outlined in (ii).

• (v) The extended matrix Riccati differential equations are solved to compute the optimal solution.

Paper organization

This paper is divided into 6 sections. The first section focuses on lung simulation, presenting schematic representations of IPF and lung geometry in Figs. 1 and 2, respectively moreover, in Fig. 3, shows the process of M2 macrophage diffusion and the effect of the anti-M1 agent control function. In the second section, we present the diffusion equation for the M2 macrophage. We express and discretize this diffusion equation, proposing the M2 diffusion and homogenized equations, which we solved using the Galerkin spectral method. The functions, variables, and parameters are provided in Table 1. The third section introduces the optimal regulator problem, solved through a two-step process involving discretization and optimization techniques. Firstly, the M2 diffusion PDE is transformed into an algebraic system of ODEs using the Galerkin spectral method. Secondly, the optimal control problem is solved using Pontryagin’s minimum principle. The block diagram of the optimal regulator problem, employing the extended Riccati equation, is depicted in Fig. 4. In the fourth section, we present numerical results illustrated in Figs. 5, 6, 7, 8, 9. In Figs. 10, 11, and 12, we performed a sensitivity analysis on the parameters $\lambda _{M1}$, M1 macrophages, and the diffusion coefficient of macrophages M2. Additionally, we provide a comparative analysis of dynamical system and optimal regulator problem solvers, as detailed in Table 2. Area D In the fifth section, we conclude and discuss. Finally, in the last section, we outlined suggestions, discussed limitations, and proposed future directions for continued research.

Fig. 1

A schematic representation of the major players shows in Pulmonary Fibrosis (IPF). The blue rectangle is part of tissue that is under investigation including the system under control and the anti-M1 as contoroller.

Fig. 2

Alveolar and wound representation in lung tissue geometry. In (a), there is a square labeled AA, and the circles within it depict the alveolar air spaces in the lung. In (b), a smaller square with sides measuring 0.3 is displayed, representing the wound area within the lung geometry. In (c), the alveolar air space is not illustrated because the circles representing them are exceedingly small³⁹. Instead, the emphasis is on showcasing a wound area denoted as D, situated at the center of the square denoted as R.

Fig. 3

In this figure, the M2 macrophage diffusion process and the impact of the anti-M1 agent control function are illustrated. In (a), normal wound healing is depicted where M1 and M2 macrophages are in biological equilibrium⁵⁷. In (b), fibrotic wound healing is shown. In the early stages, M1 macrophages predominate and contribute to inflammation and tissue damage²⁷. In later stages, the number of M2 macrophages increases, leading to fibrosis formation and tissue repair. In fibrotic wounds, the balance between M1 and M2 macrophages is disrupted, with the number of M2 macrophages being higher than M1 macrophages. In (c), we used anti-M1 as a control agent. The use of anti-M1 reduces the pro-inflammatory activity of M1 macrophages and shifts the balance towards M2 macrophages, which can help improve tissue repair processes and reduce inflammation and fibrosis²⁵. In summary, the anti-M1 agent directly regulates M2 macrophages by reducing the pro-inflammatory effects of M1 macrophages and promoting a balanced immune response. This balance is crucial in preventing the excessive fibrosis associated with IPF, as it helps shift the immune environment towards tissue repair rather than scar tissue formation⁵³.

Table 1 Nomenclator.

Fig. 4

Block diagram of the optimal regulator problem, using the extended Riccati equation.

Fig. 5

Dynamical system for M2 density with different constant scalar values for $\eta _{M1}(t)$. The dynamical system (2) for M2 at point p, located at the center of square D, is solved using the Lagrangian spectral method for N=32, with different $\eta _{M1}(t)$ values, for fibrosis scars in lung tissue (see section “Legendre spectral method in two-dimensions”). For $\eta _{M1}(t)=0$ we have Case I, while for Case II different valuse of $\eta _{M1}(t)= 0.4 \times 10^{-3}, 0.4 \times 10^{-2}, 0.1 \times 10^{-1}$, and $0.4 \times 10^{-1}$ are considered. It is clear that the closer $\eta _{M1}(t)$ gets to zero, the higher M2 density becomes. This figure uses different colors to show the decreasing levels of M2 density for various increasing values of $\eta _{M1}(t)$ ( $\eta _{M1}(t) \in [0, 0.4\times 10^{-1} ]$). These graphs demonstrate that as $\eta _{M1}(t)$ decreases over time, M2 density increases, however, the graphs of M2 density over time continue to trend upward.

Fig. 6

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.

Fig. 7

Objective function. For the optimal regulator problem, a graph of the objective functional ${\hat{J}}(M2,\eta _{M1},t)$ in (33) versus time has been plotted, revealing a gradual decrease over time. It is evident that the objective function diminishes progressively. This implies that the objective function is being minimized or optimized as time advances, indicating that the optimal control problem is making progress toward achieving the desired goal.

Fig. 8

Analysis of the solutions in the two inflammation areas. In Fig. 8, we analyzed and plotted the dynamical system solutions (4) and the optimal regulator problem solutions (29) in different inflammation areas. In Fig. 8a, the wound area is $0.3 \times 0.3$, and in Fig. 8b, the wound area is $0.5 \times 0.5$.

Fig. 9

Comparing the solutions in the two inflammation areas. It is observed that when we consider the same dynamical system for both inflammation areas and do not change it for the larger inflammation area, the solutions remain unchanged.

Fig. 10

Sensitivity analysis of the dynamic system and ORP at different $\lambda _{M1}$. The polarization rate ($\lambda _{M1}$) shows a significant effect on the dynamic system graph but minimal impact on the ORP graph. For values of $\lambda _{M1}=0.0006$ and 0.06, the dynamic system graph (blue) decreases, while it increases for$\lambda _{M1}=0.6$. This indicates that a higher polarization rate from M1 to M2 macrophages makes the dynamic system more active. However, the ORP graph (red) shows only minimal change and consistently decreases across all tested values, suggesting that the polarization rate has a limited effect on the ORP parameter.

Fig. 11

Sensitivity analysis of the dynamic system and ORP at different M1. In this figure, we adjusted the value of M1 to 0.00037, 0.037, and 0.37. We observed significant changes in both the dynamic system graph (blue) and the ORP graph (red). These results suggest that the M1 parameter plays a critical role in the diffusion of M2 macrophages.

Fig. 12

Sensitivity analysis of the dynamic system and ORP at different $D_M$. The diffusion coefficient of M2 macrophages significantly affects system behavior, particularly at higher values. While minor changes are observed at lower diffusion coefficients (0.000864 and 0.00864) in both the dynamic system (blue) and ORP (red) graphs, a higher coefficient (0.864) causes pronounced changes, with the ORP graph shifting from a downward to an upward trend. This suggests that increased diffusion of M2 macrophages markedly alters system dynamics and ORP responses.

Table 2 Comparative analysis of dynamical system and optimal regulator problem solvers in midpoint wound area D.

Contributions of the study

This study offers three key contributions to the field of IPF treatment. First, it develops a novel mathematical model that captures the dynamic behavior of M2 macrophages and their critical role in fibrosis progression in IPF. Second, the study introduces an optimal control strategy aimed at regulating M2 macrophage activity using advanced mathematical techniques such as Pontryagin’s minimum principle and Riccati equations, specifically designed to prevent fibrosis. Finally, the detailed numerical analysis demonstrates the feasibility and clinical potential of the proposed control strategy, providing a promising new approach for more effective treatment of IPF.

Literature review

Artificial intelligence (AI) has had a profound impact across multiple sectors, particularly in healthcare, by enabling computers to perform tasks traditionally managed by humans. Notable applications of AI include finger vein recognition⁹, diabetic retinopathy detection^{10,11,12,13,14}, and RNA engineering^15,16. These innovations underscore the growing role of AI, made possible by advancements in computing power, large datasets, and sophisticated algorithms. In healthcare, AI has proven transformative, with innovations such as machine vision for tuberculosis diagnosis from CT scans¹⁷ and YOLO-based systems for colorectal cancer detection¹⁸. AI’s integration into medical diagnostics and treatment planning is reshaping the field, improving accuracy and efficiency across various applications, including expert systems for diagnosing heart disease¹⁹. In the context of IPF and fibrosis-related diseases, AI algorithms, such as machine learning (ML) techniques, can help in the automatic analysis of large datasets, such as patient-specific data, to predict disease progression and response to treatments. AI can also assist in refining mathematical models by learning from real-world data and identifying patterns that may not be easily captured by traditional methods. While our study primarily focuses on a mathematical control strategy, future models could benefit from integrating AI techniques for enhanced parameter estimation, model validation, and the development of personalized treatment protocols. For example, AI-driven algorithms could optimize the dosing of anti-M1 agents in real-time based on patient-specific factors, leading to more accurate and adaptive treatments. By combining AI with the proposed mathematical control methods, the overall predictive power and precision of fibrosis management could be improved without adding significant computational burden. Hao et al. initially introduced a mathematical model in 2014 focusing on sarcoidosis as a biomedical issue²⁰. This model was later extended for application in chronic pancreatitis²¹, laying the foundation for exploring macrophage dynamics in fibrosis through mathematical models. Hao and his team proposed a model for interstitial fibrosis of the immune system, assessing the efficacy of anti-fibrotic drugs and those under clinical trials for renal fibrosis²². In tackling pulmonary fibrosis, this model considered both M1-derived inflammatory macrophages and M2 anti-inflammatory alveolar macrophages²³. In 2017, Hao and colleagues evaluated potential drugs targeting liver fibrosis prevention²⁴. Further studies by Isshiki focused on developing therapeutic strategies targeting profibrotic macrophages in interstitial lung disease²⁵. Ge²⁶ and Deng et al.²⁷ emphasized the role of M2 macrophages as key regulators in pulmonary diseases, while Cheng²⁸ addressed lung injury, repair, and fibrosis. Recent research by Bahram Yazdroudi and Malek in 2022 and 2023 introduced optimal control models aiming to counteract pulmonary fibrosis through the regulation of TGF-$\beta$ and PDGF^29,30. Their work demonstrated that targeting multiple pathways can effectively reduce fibrosis progression, although these models primarily focused on growth factors rather than directly addressing the regulation of M2 macrophages. Building on these studies, recent works by Ge³¹ and Cheng³² have highlighted macrophage polarization’s role in lung disease, but did not incorporate dynamic control strategies for regulating macrophages in real-time clinical settings. To fill this gap, our study introduces a novel mathematical model combining dynamic systems with PDE constraints to regulate M2 macrophages, aiming to prevent fibrosis in IPF. In contrast to previous approaches, this study applies advanced control techniques such as Pontryagin’s minimum principle and Riccati equations to regulate M2 macrophage activity dynamically. This novel method integrates real-time optimization strategies to adjust treatment interventions dynamically, providing an approach that was lacking in earlier models. Additionally, optimal control theory has been crucial in addressing disease dynamics and solving complex optimization problems, especially those involving PDEs. In 2018, Mehrali-Varjani, Shamsi, and Malek explored a class of Hamilton-Jacobi-Bellman equations as an optimal control problem using pseudospectral methods³³. Following this, in 2019, Abbasi and Malek introduced hyperthermia cancer therapy using domain decomposition methods³⁴, and in 2020, they applied pointwise optimal control for hyperthermia involving thermal wave bioheat transfer³⁵. Finally, in 2023, Alimirzaei and Malek developed an optimal control problem centered around anti-angiogenesis and radiation treatments for cancerous tumors³⁶.

Keno examines the applications of optimal control strategies in the dynamic modeling of COVID-19 in³⁷. In the context of medical applications, many researchers have successfully applied optimal control problems to address issues related to cancer^38,39,40 and infectious diseases^{41,42,43,44,45,46}.

Mathematical configuration

Methodology flow

This methodology involves several key stages: First, the mathematical model of the dynamic system and the objective function are defined. Mathematical modeling involves formulating a dynamic system to describe the distribution of M2 macrophages in inflamed tissue, incorporating the impact of anti-M1 drugs. The objective function aims to reduce fibrosis by controlling M2 macrophage activity with anti-M1 drugs, balancing the reduction of M2 macrophages with the management of treatment side effects.

Next, the PDE is discretized into an ODE system using the Galerkin spectral method. The Galerkin spectral method was chosen for this study due to its superior accuracy and efficiency, particularly for problems with smooth solutions, such as the diffusion of M2 macrophages in lung tissue^47,48. This method transforms PDEs into a system of ODEs by projecting the PDE onto a set of global basis functions, such as Legendre or Chebyshev polynomials⁴⁸. These global basis functions allow the spectral method to capture the solution’s behavior across the entire domain with fewer grid points, leading to higher accuracy and reduced computational costs compared to other numerical methods⁴⁹. In contrast, the Finite Difference Method (FDM), while simple to implement, may suffer from numerical instability and lower accuracy, especially in high-dimensional problems. The Finite Element Method (FEM), which uses local basis functions and is effective for complex geometries, typically requires more computational resources to achieve the same level of accuracy as the spectral method⁵⁰. The global nature of the basis functions in the Galerkin spectral method ensures a more precise approximation of the PDEs governing M2 macrophage diffusion, capturing subtle spatial and temporal changes efficiently⁴⁷. Thus, the Galerkin spectral method offers an ideal balance between accuracy and computational efficiency, making it the optimal choice for modeling M2 macrophage dynamics in this study⁴⁸. The primary goal of the optimization process is to find the most effective strategy for controlling M2 macrophage activity to prevent or reduce fibrosis progression in IPF.

To solve the optimal control problem, Pontryagin’s Minimum Principle and the principal Hamiltonian equations are employed. In the subsequent stage, the developed Riccati differential equations are solved to obtain the optimal control strategy. Finally, implementation and optimization are carried out using numerical methods.

Lung tissue simulation into IPF

In Fig. 1, a simplified schematic network is presented, highlighting key components such as epithelial cells, fibroblasts, extracellular matrix, growth factors, and matrix metalloproteinases involved in IPF. The lung tissue is considered as a square with an edge size of 1 cm, divided into small squares named $T_{\epsilon }$, where the edge of each square is $\epsilon$ (extremely small and close to zero). A basic representation of the lung geometry in two dimensions, denoted by x and y, is depicted. The alveolar air space is illustrated by concentric circles within each small square, denoted as $A_{\epsilon }$. The alveolar tissue is displayed between the squares and circles.

The homogenized alveolar tissue ($T_{\epsilon }/A_{\epsilon }$) is represented as an R square. In this scenario, the lung tissue is a square without alveolar space, as illustrated in Fig. 2c. A square labeled as D symbolizes tissue inflammation within the homogenized alveolar tissue R, where R is defined as the square region $[0,1] \times [0,1]$. For a mild case of IPF, it is assumed that the area of tissue inflammation, represented by square D, is equal to $0.3 \times 0.3$$\hbox { cm}^{2}$⁵¹. This dimension reflects the size of the wound area within the lung tissue model.

In the next section, a dynamical system for M2 macrophages is presented. This dynamical system involves PDE in two dimensions. Subsequently, the dynamical system has been discretized using the Legendre spectral method in two dimensions.

Diffusion equation for M2 macrophages

The diffusion of M2 macrophages in this model is described using a PDE that captures the spatial and temporal behavior of M2 macrophages across inflamed lung tissue. We utilize $\eta _{M1}(t)$ as the anti-M1 control function. The problem at hand is to determine M2(q, t) for $q= x$ or y, satisfying the following diffusion equation,

$$\begin{aligned} \begin{aligned} \dfrac{ \partial M2(q,t)}{ \partial t}- D_M {\nabla }^2 M2(q,t)&=\underbrace{\lambda _{M1}M1(1-\eta _{M1}(t))}_{M1\rightarrow M2}- \underbrace{d_{M2}M2(q,t)}_{degradation }-\underbrace{\lambda _{M2T}\dfrac{T_{\alpha }}{T_{\alpha }+K_{T_{\alpha }}}M2(q,t)}_{M2\xrightarrow {T_{\alpha }} M1} -\underbrace{\lambda _{QM2}M2(q,t)}_{production \ Q}-\underbrace{\lambda _{Q_rM2}M2(q,t)}_{production \ Q_r}-\\&\underbrace{\lambda _{GM2}M2(q,t)}_{production \ G }-\underbrace{(1+\dfrac{\lambda _{T_{\beta }I_{13}}I_{13}}{K_{I_{13}}+I_{13}})\lambda _{T_{\beta }M2}M2(q,t)}_{ production \ T_{\beta }} -\underbrace{\lambda _{I_{13}}M2(q,t)}_{production \ I_{13} },\quad t\in [t_0,t_f] \end{aligned} \end{aligned}$$

(1)

where functions, variables, and parameters are outlined in Table 1. The term $\lambda _{M1}M1$ represents the polarization from M1 to M2 through the processes mentioned and possibly other processes (e.g.⁵²), and $\eta _{M1}(t)$ serves as the anti-M1. The second term accounts for the death rate of macrophages, the third term represents the transformation from M2 to M1 induced by $T_{\alpha }$⁵³, the fourth term involves MMP (Q) activation by M2 macrophages, the fifth term involves TIMP ($Q_r$) activation by M2 macrophages, the sixth term involves PDGF (G) production and activation by M2 macrophages, the seventh term, $T_{\beta }$, is produced and activated by M2 macrophages while enhanced by IL-13^54,55,56, and the last term, IL-13 is produced by M2 macrophages^54,55.

The diffusion equation, along with the initial and boundary conditions for the dynamical system for M2 macrophages in two dimensions within domain D, is as follows:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \dfrac{ \partial M2(x,y,t)}{ \partial t}- D_M {\nabla }^2 M2(x,y,t)=c_{M1}(1-\eta _{M1}(t))+A_MM2(x,y,t)\\ M2(x,y,0) = 0. 5 \times 10^{-3}, \quad \dfrac{ \partial M2(x,y,t)}{ \partial x}=0, \quad \dfrac{ \partial M2(x,y,t)}{ \partial y}=0, \end{array}\right. } \end{aligned} \end{aligned}$$

(2)

where in

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} c_{M1}& =\lambda _{M1}M1\\ A_M& =-d_{M2}-\lambda _{M2T}\dfrac{T_{\alpha }}{T_{\alpha }+K_{T_{\alpha }}} -\lambda _{QM2}-\lambda _{Q_rM2}-\lambda _{GM2}-\lambda _{T_{\beta }M2} -(1+\dfrac{\lambda _{T_{\beta }I_{13}}I_{13}}{K_{I_{13}} +I_{13}})\lambda _{T_{\beta }M2}-\lambda _{I_{13}}. \end{array}\right. } \end{aligned} \end{aligned}$$

(3)

If $\eta _{M1}(t)=0$ (no control), then the dynamical system for M2 is as follows:

$$\begin{aligned} \dfrac{ \partial M2(x,y,t)}{ \partial t}- D_M {\nabla }^2 M2(x,y,t)=c_{M1}+A_MM2(x,y,t). \end{aligned}$$

(4)

The key assumptions made in this model are as follows:

Uniform Diffusion Coefficient: In this model, the diffusion coefficient $D_M$is assumed to be uniform across the tissue, meaning that M2 macrophages are treated as diffusing at the same rate throughout the lung. While real biological tissues have heterogeneous structures, such as varying cell densities, extracellular matrix compositions, and tissue stiffness, this uniform diffusion coefficient is used to capture the overall trend of macrophage movement. The assumption of homogeneity at the macroscopic scale simplifies the model, making it mathematically manageable. However, future work may incorporate spatial variations to better reflect the complexity of biological environments^23,24.

M1 to M2 Polarization: The equation models the polarization of M1 macrophages into M2 macrophages through a term $\lambda _{M1} M1(1-\eta _{M1}(t))$, where $\eta _{M1}(t)$ is the control variable (anti-M1 agent). This term assumes that M1 macrophages can convert into M2 macrophages in response to the anti-M1 agent, which reduces the pro-inflammatory activity of M1 macrophages^25,27.

Degradation of M2 macrophages: The degradation of M2 macrophages is represented by the term $d_{M2} M2(q,t)$, which assumes a natural rate of decay for these cells, independent of external factors. This simplifies the model by not accounting for additional biological processes that may affect M2 macrophage lifespan^57,58.

Boundary conditions: The model assumes zero-flux boundary conditions, meaning that M2 macrophages do not flow out of the tissue domain. This simplifies the mathematical representation, although in real biological systems, cells could potentially migrate across boundaries.

Initial conditions: The initial concentration of M2 macrophages is assumed to be very low, simulating the early stages of inflammation before the accumulation of fibrotic tissue^23,42.

The homogenized equations

For the homogenized equations, we first have the following equation in the region $T_\epsilon / A_\epsilon$:

$$\begin{aligned} \dfrac{ \partial M2(x,y,t)}{ \partial t}- D_M {\nabla }^2 M2(x,y,t)=c_{M1}(1-\eta _{M1}(t))+A_MM2(x,y,t). \end{aligned}$$

(5)

After homogenization the equation is as follows:

$$\begin{aligned} \gamma \dfrac{ \partial M2(x,y,t)}{ \partial t}- D_M { \tilde{\nabla }}^2M2 (x,y,t)=\gamma \big (c_{M1}(1-\eta _{M1}(t))+A_MM2(x,y,t)\big ) \quad \text {in} \quad {\textbf {R}} \end{aligned}$$

(6)

$\gamma ={\frac{127}{343}}$, and ${\tilde{\nabla }}$ is computed with coefficientst $a_{ij}$. For simplicity, the homogenized equation is written as (To read more, refer to^23,29,30):

$$\begin{aligned} \dfrac{ \partial M2(x,y,t)}{ \partial t}- r { \nabla }^2M2(x,y,t) = c_{M1}(1-\eta _{M1}(t))+A_MM2(x,y,t) \quad \text {in} \quad {\textbf {R}}, \end{aligned}$$

(7)

where $r=\dfrac{aD_M}{\gamma }$and $a_{ii}=0.11$.

Spectral method

Below are the fundamental formulas for Lagrangian polynomials, the Lagrangian spectral method in both one and two dimensions, and the discretization technique^47,49,50,59.

Legendre polynomials

Legendre polynomial $P_k(\xi )$ are orthogonal on the interval $[-1, 1]$, where $k = 0, 1, 2, \ldots$ and defined by:

Orthogonality: $\int _{-1}^{1} P_k(\xi ) P_j(\xi ) d\xi = \frac{2}{2k+1} \delta _{kj}$, where $\delta _{kj}$ represents the Kronecker delta symbol.

Normalization: $\int _{-1}^{1} P_k(\xi ) P_k(\xi ) d\xi = \frac{2}{2k+1}$. The Legendre polynomials emerge as solutions to Legendre’s differential equation, a second-order linear ordinary differential equation. This equation and its associated Sturm-Liouville problem are typically defined as follows:

$$\begin{aligned} ( (1-\xi ^2)P^{\prime }_k(\xi ) )^{\prime }+ k(k+1)P_k(\xi )=0 \quad -1<\xi <1. \end{aligned}$$

(8)

$P_k(\xi )$ is even when k is even and odd when k is odd. If $P_k(\xi )$ is normalized such that $P_k(1)= 1$. For each k, we get:

$$\begin{aligned} P_k(\xi )=\dfrac{1}{2^k}\sum ^{[k/2]}_{j=0} (-1)^j \genfrac(){0.0pt}0{k}{j}\genfrac(){0.0pt}0{2k-2j}{k}\xi ^{k-2j}, \end{aligned}$$

(9)

$L0_k$, derives from the $(k + 1)$-degree Legendre polynomial, $P_{k+1}$, as:

Legendre polynomials can be computed using the recursive relations⁴⁹ as follows:

$$\begin{aligned} P_{k+1}(\varvec{\xi })=\dfrac{2k+1}{k+1}\varvec{\xi } P_k(\varvec{\xi })-\dfrac{k}{k+1}P_{k-1}(\varvec{\xi }), \end{aligned}$$

(10)

Legendre spectral method for M2 macrophage in one-dimension

The approximate M2 macrophage function defined at Gauss-Lobatto integration points, denoted as $M2_N(x,t)$ and $N=32$ can be expressed as a linear combination of these basis functions:

$$\begin{aligned} M2_N(x,t)=\sum ^N_{i=0} {\hat{M}}2_i\phi _i(x), \end{aligned}$$

(11)

Where ${\hat{M}}2_j$ represents the discrete polynomial coefficient, and $\phi _i(x,t)$ denotes the Lagrangian interpolation polynomials of order N. $\{{\xi _j }\}^N_{j=0}$ are the Gauss-Legendre-Lobatto (GLL) points, and

$$\begin{aligned} \phi _j(\varvec{\xi })=\dfrac{1}{N(N+1)P_N(\varvec{\xi }_j)} \dfrac{(\varvec{\xi }^2-1)P^{\prime }_N(\varvec{\xi })}{\varvec{\xi }-\varvec{\xi }_j}\quad \quad 0\le j \le N, \quad -1\le \varvec{\xi } \le 1,\quad \varvec{\xi } \ne \varvec{\xi }_j, \end{aligned}$$

(12)

in this expression, the Legendre polynomial of order N is represented by $P_N(\xi )$, and its derivative is denoted as $P^{\prime }_N({\xi)}$. To map $[-1, 1]$ to the interval [a, b], the mapping function and its inverse can be expressed as:

$$\begin{aligned} x(\varvec{\xi })&=\dfrac{(x_b-x_a)\varvec{\xi }}{2}+\dfrac{x_b+x_a}{2}\quad -1\le \varvec{\xi }\le 1, \\ \varvec{\xi }(x)&=\dfrac{2x-(x_b+x_a)}{x_b-x_a}\quad \quad x_a\le x\le x_b. \end{aligned}$$

For $h=x_b-x_a$, the stiffness⁵⁰ ($S_q$), mass ($M_q$), and constant coefficients ($C_q$) matrices for $q=x, y$ are as follows:

$$\begin{aligned}&{S_q}_{i,j}=\int ^{x_b}_{x_a}\phi ^{\prime }_i(x) \phi ^{\prime }_j(x)dx=\dfrac{2}{h}\int ^ 1_{-1}\phi ^{\prime }_i(\varvec{\xi }) \phi ^{\prime }_j(\varvec{\xi })d\varvec{\xi }, \end{aligned}$$

(13)

$$\begin{aligned}&{M_q}_{i,j}=\int ^{x_b}_{x_a}\phi _i(x) \phi _j(x)dx=\dfrac{h}{2}\int ^ 1_{-1}\phi _i(\varvec{\xi }) \phi _j(\varvec{\xi })d\varvec{\xi }, \end{aligned}$$

(14)

$$\begin{aligned}&{C_q}_{i,j}=\int ^{x_b}_{x_a}\phi _i(x)d\varvec{\xi }. \end{aligned}$$

(15)

Utilizing Gauss quadrature, we have⁴⁹

$$\begin{aligned} {S_q}_{ij}&=\dfrac{2}{h}\sum ^ N_{k=0} d_{ik}d_{jk}w_k, \end{aligned}$$

(16)

$$\begin{aligned} {M_q}_{ij}&=\dfrac{h}{2}\delta _{ij}w_i, \end{aligned}$$

(17)

where the GLL quadrature weights $\{{w_k}\}^N_{k=1}$ are provided in the following:

$$\begin{aligned} w_k=\dfrac{2}{N(N+1)[L_N(\varvec{\xi })]^2} \quad 0\le k \le N, \end{aligned}$$

(18)

$$\begin{aligned} d_{ij}= {\left\{ \begin{array}{ll} \dfrac{-N(N+1)}{4} & \quad i=j=0\\ 0 & \quad i=j\in [1,N-1] \\ \\ \dfrac{N(N+1)}{4}& \quad i=j=N\\ \\ \dfrac{L_N(\varvec{\xi }_i)}{L_N(\varvec{\xi }_j)} \dfrac{1}{\varvec{\xi }_ i -\varvec{\xi }_ j}& \quad i\ne j \end{array}\right. } \end{aligned}$$

(19)

The mass matrix is diagonal when the nodal points coincide with the quadrature points due to the cardinality properties of Lagrange polynomials^47,59.

For $q=x$ or y, we define the spaces $H^1 (D_{q})$ and $H^1 0(D{q})$ as follows:

$$\begin{aligned} H^1 (D_{q})= \left\{ v \in D_{q}, \quad \dfrac{\partial v}{\partial q}\in D_{q} \right\} , \quad H^1 _0(D_{q})=\left\{ v \in H^1(D_{q}), \quad v|_{ \partial D_{q} }=0 \right\} \end{aligned}$$

(20)

For Eq. (7), proper approximation for $M2_N(x,t)$ applies as a weighted Galerkin method. Find ${\hat{M}}2_N(q,t) \in H^1 _0 (D_{q})$ such that for all $\phi \in H^1 _0(D_{q})$.

$$\begin{aligned} \int _{D_{x}} \phi _i\bigg (\dfrac{ \partial M2_N(x,t)}{ \partial t}- \dfrac{ \partial }{\partial x}(r \dfrac{ \partial M2_N(x,t) }{\partial x})-A_MM2_N(x,t)-c_{M1}(1-\eta _{M1}(t))\bigg )dD_{x}=0. \end{aligned}$$

(21)

The weak form as follows:

$$\begin{aligned} \begin{aligned}&\int \phi _i\dfrac{ \partial M2_N(x,t)}{\partial t}dx-\int _{\partial D_{x}}\phi _i \dfrac{ \partial M2_N(x,t)}{\partial x}dx-\int r\dfrac{\partial \phi _i}{\partial x}\dfrac{\partial M2_N(x,t)}{\partial x}dx-A_M\int \phi _i M2_N(x,t)dx\\&-c_{M1}(1-\eta _{M1}(t))\int \phi _i dx=0. \end{aligned} \end{aligned}$$

(22)

From the boundary condition $\left( \dfrac{\partial M2_N(x,t)}{\partial x}=0 \right)$,

$$\begin{aligned} \int \phi _i\dfrac{ \partial M2_N(x,t)}{\partial t}dx-\int r\dfrac{ \partial \phi _i}{\partial x}\dfrac{\partial M2_N(x,t)}{\partial x}dx-A_M\int \phi _i M2_N(x,t)dx-c_{M1}(1-\eta _{M1}(t))\int \phi _i dx=0. \end{aligned}$$

(23)

We substitute Eq. (11) in Eq. (23), ${\hat{M}}2_N(x,t)$ can be determined by solving the following ODE systems where the entries of the $c_{M1}$, $S_x$, $M_x$, and $C_x$ are defined in Eqs. (3), (13), (14), and (15).

$$\begin{aligned}&M_x\dot{{\hat{M}}}2_N(x,t)(t)-rS_x{\hat{M}}2_N(x,t)-A_MM_x\hat{M}2_N(x,t)-c_{M1}(1-\eta _{M1}(t))C_x=0, \end{aligned}$$

(24)

$$\begin{aligned}&\dot{{\hat{M}}}2_N(x,t)=M_x^{-1}((A_MM_x+rS_x)\hat{M}2_N(x,t)+c_{M1}(1-\eta _{M1}(t))C_x). \end{aligned}$$

(25)

Legendre spectral method in two-dimensions

The domain considered is divisioned into the quadrilateral where $[-1,1] \times [-1,1]$ is the reference square. The local approximating functions are the tensor product of the one-dimensional Legendre polynomials. The approximation $M2_N(x,y,t)$ from order N is as follows:

$$\begin{aligned} M2_N(x,y,t)=\sum ^N_{j=0} {\hat{M}}2_j(x,y,t)\phi _ j(x) \phi _ j(y). \end{aligned}$$

(26)

The stiffness matrix S, the mass matrix M, and constant coefficients C matrices⁵⁰, respectively, are defined as:

$$\begin{aligned} \begin{aligned} S&=S_x\otimes S_y,\\ M&=M_x\otimes M_y,\\ C&=C_x\otimes C_y, \end{aligned} \end{aligned}$$

(27)

where the entries of the $S_x$ and $S_y$ are defined in Eq. (13), the entries of the $M_x$ and $M_y$ are defined in Eq. (14), and the entries of the $C_x$ and $C_y$ are defined in Eq. (15). For Eq. (2), a appropriate approximation performs for $M2_N(x,y,t)$ as a weighted Galerkin method therefore

$$\begin{aligned} \dot{{\hat{M}}}2_N(x,y,t)=M^{-1}((A_MM+rS){\hat{M}}2_N(x,y,t)+c_{M1}(1-\eta _{M1}(t))C). \end{aligned}$$

(28)

For simplicity, we use the notion ${\hat{M}}2_N(x,y,t) = M2(t)$ for $N=32$.

In the next section, an optimal control problem for M2 macrophages is presented. The optimal regulator formulation tuning method is described, and the block diagram of the optimal regulator problem, using the extended Riccati equation is provided. Subsequently, we address the optimal control problem solution by first discretizing it and then determining the optimal solution for this discrete problem.

Optimal regulator method

Significance of choosing the anti-M1 agent

Studies indicate that in the advanced stages of IPF, M2 macrophages known for their pro-fibrotic properties can exacerbate fibrosis by driving excessive collagen production and activating myofibroblasts. Consequently, reducing the population of pro-fibrotic M2 macrophages and shifting the macrophage balance toward an anti-fibrotic phenotype may offer a promising therapeutic approach. In such scenarios, implementing a control strategy to restore M2 macrophage concentrations to their equilibrium levels is desirable. Therefore, the anti-M1 agent is selected as the control function. This choice is based on the agent’s role as an anti-inflammatory and immune system regulator. The proposed model suggests that the anti-M1 agent can suppress abnormal repair processes and help return M2 macrophages to their natural equilibrium. This choice is based on biological studies showing that anti-inflammatory agents can aid in reducing inflammation and promoting tissue repair.

Optimal regulator formulation

The optimal regulator problem (ORP) described in the context of the dynamical system for M2 can be summarized as follows:

Objective: The goal is to find an optimal treatment strategy that minimizes the objective function while satisfying certain constraints.

Constraints: The treatment strategy must adhere to the fixed final time and ensure that the M2 density remains nonnegative.

Objective Function: The objective function includes both linear and quadratic terms to control doses. By incorporating the quadratic term, the aim is to achieve continuous and low doses of therapeutic drugs, as previous studies have shown improved outcomes with such an approach.

Mathematical Challenges: The inclusion of both linear and quadratic terms in the objective function allows for more control over the doses. Ledzewicz et al.⁶⁰, observed that optimal control in this context often involves a singular arc, where bang-bang control fails to be optimal or desirable due to the need for dose continuity. This mathematical challenge necessitates the consideration of the quadratic term to achieve continuous and low doses.

Clinical Challenges: In the clinical aspect, treatment strategies for reducing M2 face the challenge of ensuring dose continuity while aiming for low and continuous doses. Administering continuous and low doses of therapeutic drugs has been shown to lead to improved outcomes in studies⁶¹. Therefore, the inclusion of the quadratic term in the objective function helps address this clinical challenge by controlling doses to achieve continuous and low levels. By formulating the ORP with the appropriate objective function and constraints, it becomes possible to explore treatment strategies that optimize the control of doses, considering both mathematical and clinical challenges faced in the context of M2 reduction.

Discretization and optimization process

The goal of the optimal control problem in this study is to reduce fibrosis by controlling the activity of M2 macrophages, which play a significant role in the fibrotic process in IPF^25,27. The model used in this research simulates the behavior of M2 macrophages in the lung tissue and introduces an anti-M1 agent as the control variable to modulate M2 activity^25,31.

Mathematical modeling and discretization: The distribution of M2 macrophages is modeled using a system of PDEs^23,24. These equations describe how M2 macrophages diffuse in inflamed lung tissue and contribute to fibrosis²³. The PDEs are then discretized using a numerical method called the Galerkin spectral method, which transforms the PDE system into ODEs that are easier to solve numerically^47,48.

Optimization process: After discretization, the optimal control problem is formulated to minimize M2 macrophage activity and, thus, reduce fibrosis^29,30. The objective function represents the total number of M2 macrophages over time, and the control variable is the dose of the anti-M1 agent²⁵. The Pontryagin’s Minimum Principle and Hamiltonian equations are used to derive the optimal control strategy, and extended Riccati differential equations are solved to compute the optimal solution^62,63. The optimal control aims to reduce the production of M2 macrophages over time, which in turn reduces fibrotic tissue formation.

Results: Numerical simulations show that, without control, M2 macrophage levels increase, leading to increased fibrosis. However, the optimal control strategy effectively reduces M2 macrophage levels, preventing fibrosis formation within a 120-day period. The control strategy involves gradually reducing the dosage of the anti-M1 agent to minimize side effects while still achieving therapeutic efficacy²⁵. The objective is to minimize fibrosis by controlling the anti-M1 agent. The objective function J is defined as: $J=\dfrac{1}{2}M2(x,y,t_f)^2+\dfrac{1}{2}\int _{t_0}^{t_f} \int _{y}\int _{x}{M2(x,y,t)^2}dxdydt+\dfrac{1}{2}\int _{t_0}^{t_f}{\eta _{M1}(t)^2}dt$

where $M2(x,y,t_f)$ represent M2 macrophages and $\eta _{M1}(t)$represent the anti-M1 agent. The Pontryagin’s Minimum Principle and Hamiltonian equations are used to derive the optimality conditions. The Hamiltonian function H incorporates state dynamics, the control function, and adjoint variables (Lagrange multipliers) as follows: $H=L(M2,\eta _{M1},t)+vf(M2,\eta _{M1},t)$ where L is the integrand of the objective function and f represents the system dynamics. The Hamiltonian equations, which include state equations, adjoint equations, and optimality conditions, are solved. Finally, the extended Riccati differential equations are used to compute the optimal feedback control law. These equations provide a recursive method for determining the optimal control strategy that minimizes the objective function and addresses the mathematical and clinical challenges in reducing M2 and preventing fibrosis. The formulation of the ORP is outlined as follows:

$$\begin{aligned} \text {ORP}: {\left\{ \begin{array}{ll} {\min _{M2(x,y,t),\eta _{M1}(t)}} \quad J(M2(x,y,t),\eta _{M1}(t),t) =\dfrac{1}{2}M2(x,y,t_f)^2+\dfrac{1}{2}\int _{t_0}^{t_f}\int _{y}\int _{x} {M2(x,y,t)^2}dxdydt+\dfrac{1}{2}\int _{t_0}^{t_f}\eta _{M1}(t)^2dt\\ \\ \text {subject to:} \\ \\ \dfrac{ \partial M2(x,y,t)}{ \partial t}- r {\nabla }^2 M2(x,y,t)=F(M2(x,y,t),\eta _{M1}(t)), \quad r=\dfrac{aD_M}{\gamma } \\ \\ M2(x,y,0) = 0. 5 \times 10^{-3}, \quad \dfrac{ \partial M2(x,y,t)}{ \partial x}=0, \quad \dfrac{ \partial M2(x,y,t)}{ \partial y}=0, \quad \text {initial and boundary conditions }\\ \end{array}\right. } \end{aligned}$$

(29)

where

$$\begin{aligned} \begin{aligned} F(M2(x,y,t),\eta _{M1}(t))=c_{M1}(1-\eta _{M1}(t))+A_MM2(x,y,t). \end{aligned} \end{aligned}$$

(30)

$J(M2(x,y,t),\eta _{M1}(t),t): {\mathbb {R}}^3\times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is the objective functional consists of two terms $M2(x,y,t): {\mathbb {R}}^3\rightarrow {\mathbb {R}}$ (is the state function), $\eta _{M1}(t): {\mathbb {R}}\rightarrow {\mathbb {R}}$ (is the control function).

Solving ORP

There are two primary numerical approaches to solving optimal control problems, and the choice depends on whether the problem is initially optimized and then discretized or vice versa. The direct method entails discretizing the optimal control problem first and subsequently determining the optimal solution for this discrete problem. On the other hand, the indirect method involves optimizing and then discretizing to find a solution to the ODE system resulting from Pontryagin’s minimum principle. Both approaches come with their own set of advantages and disadvantages⁶⁴. In this paper, we solve with the direct method to address the related problem.

Discretized ORP by spectral method

For the direct method, spectral method is used to discretize Eq (7) (see “Legendre spectral method in two-dimensions”). Thus, we deal with the following ODE.

$$\begin{aligned} \begin{aligned} {\dot{M}}2(t)=AM2(t)-C_{M1}\eta _{M1}(t)+C_{M1}, \end{aligned} \end{aligned}$$

(31)

where using (15), (27), and (28), we get A, and $C_{M1}$ as follows

$$\begin{aligned} \begin{aligned} A=M^{-1}(A_MM+rS),\quad C_{M1}=c_{M1}M^{-1}C, \end{aligned} \end{aligned}$$

(32)

The discrete form of the ORP is given by the following Discretized Optimal Regulator Problem (DORP):

$$\begin{aligned} \text {DORP}: {\left\{ \begin{array}{ll} {\min _{M2(t),\eta _{M1}(t)}} {\hat{J}}(M2(t),\eta _{M1}(t) ,t) =\dfrac{1}{2}M2(t_f)^2+\dfrac{1}{2}\int _{t_0}^{t_f}M2(t)^2dt+\dfrac{1}{2}\int _{t_0}^{t_f}\eta _{M1}(t)^2 dt,\\ \\ \text {subject to:} \\ \\ {\dot{M}}2(t)=AM2(t)-C_{M1}\eta _{M1}(t)+C_{M1}\\ \\ M2(t_0) = 0. 5 \times 10^{-3} \end{array}\right. } \end{aligned}$$

(33)

The objective function ${\hat{J}}$ will be minimized using Pontryagin’s minimum principle^65,66. Next, we will address the solution of the DORP problem using optimization methods.

Solution of the DORP

The fundamentals of optimal control theory include Pontryagin’s Minimum Principle, Canonical Hamiltonian Equations, and Riccati Equations. We will explain each of these concepts.

Pontryagin’s minimum principle: Pontryagin’s Minimum Principle is fundamental in optimal control theory. This principle provides the necessary conditions for optimality by introducing the Hamiltonian, which combines the system dynamics with the cost function. The goal of this principle is to derive control laws that minimize the cost while satisfying the system’s constraints. In other words, Pontryagin’s Minimum Principle helps us find control strategies that deliver the most efficient performance under given conditions.

Canonical Hamiltonian equations: Canonical Hamiltonian equations, derived from the Hamiltonian function, describe the changes in state and cost variables. These equations are essential for understanding how the system’s state evolves over time under optimal control. They assist in deriving optimal control policies and ensure that the Hamiltonian is minimized. Essentially, these equations tell us how to adjust the control to achieve optimal system performance.

Extended Riccati differential equations: Riccati equations are crucial for solving optimal control problems, especially for linear quadratic regulators (LQR). Extended Riccati differential equations help determine optimal control coefficients and assess system stability. These equations, derived from the Hamiltonian, assist in computing control strategies that minimize the total system cost. They also provide insights into the stability of the system, enabling us to find optimal control for linear systems with quadratic costs.

In addressing the optimal control problem for M2, both the Hamiltonian and extended Riccati equations have been employed. The following sections will provide a detailed exploration of these concepts to clearly explain their roles and applications.

To achieve this, define the Hamiltonian as follows:

$$\begin{aligned} H(M2(t),\eta _{M1}(t),\lambda (t),t)=\dfrac{1}{2}M2(t)^2+\dfrac{1}{2} \eta _{M1}(t)^2+ \lambda ^{T}(t) \left[ AM2(t)-C_{M1}\eta _{M1}(t)+C_{M1}\right] \end{aligned}$$

(34)

where $\lambda ^{T}(t)$ is the vector of the Lagrange multipliers. The new objective function ${\tilde{J}}$ is :

$$\begin{aligned} {\tilde{J}}&= \dfrac{1}{2}M2(t_f)^2+ \dfrac{1}{2}\int _{t_0}^{t_{f}}{\left[ M2(t)^2+ \eta _{M1}(t)^2 \right] +\lambda ^{T}(t) \left[ AM2(t)-C_{M1}\eta _{M1}(t)+C_{M1}-{\dot{M}}2(t)\right] }dt. \end{aligned}$$

(35)

By substituting Eq. (34) into Eq. (35), we have

$$\begin{aligned} {\tilde{J}}&=\dfrac{1}{2}M2(t_f)^2+ \int _{t_0}^{t_{f}}\left[ H(M2(t),\eta _{M1}(t)^2,\lambda (t),t)-\lambda ^{T}(t) {\dot{M}}2(t)\right] dt. \end{aligned}$$

(36)

Using the integration by parts method, the objective functional ${\tilde{J}}$ becomes

$$\begin{aligned} {\tilde{J}}&=\dfrac{1}{2} M2(t_f)^2-\lambda (t_f)M2(t)+\int _{t_0}^{t_{f}}\left[ H(M2(t),\eta _{M1}(t),\lambda (t), t)+{\dot{\lambda }}^{T}(t) M2(t)\right] dt \end{aligned}$$

(37)

The first differential, $\delta {J}$, with respect to the vectors M2(t) and $\eta _{M1}(t)$ is given by:

$$\begin{aligned} {\tilde{J}}&= M2(t_f)-\lambda (t_f)+\int _{t_0}^{t_{f}}\delta M2^T \left[ \dfrac{\partial H(M2(t),\eta _{M1}(t),\lambda (t), t)}{\partial M2}+{\dot{\lambda }}\right] +\delta \eta _{M1}\left[ \dfrac{\partial H(M2(t),\eta _{M1}(t),\lambda (t), t)}{\partial \eta _{M1}}\right] dt \end{aligned}$$

(38)

A necessary condition for the performance index ${\tilde{J}}$ to be at a minimum is that $\delta {{\tilde{J}}}=0$. Thus, the vectors $\varvec{M2^*}(t)$, $\eta _{M1}^*(t)$ must satisfy the following equations:

$$\begin{aligned}&{\textbf { State equations:}} \quad \dfrac{\partial H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)}{\partial {\varvec{M}}{\varvec{2}}}=-\varvec{{\dot{\lambda }}^*}(t), \end{aligned}$$

(39)

$$\begin{aligned}&{\textbf {Costate equations:}} \quad \dfrac{\partial H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)}{\partial \varvec{\lambda }}=\varvec{{\dot{M2}}^*}(t), \end{aligned}$$

(40)

$$\begin{aligned}&{\textbf {Optimality condition:}} \quad H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)\le H(\varvec{M2^*}(t),\varvec{\eta _{M1}}(t),\varvec{\lambda ^*}(t),t) \end{aligned}$$

(41)

$$\begin{aligned}&{\textbf {Optimality condition:}}\quad H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)\le H(\varvec{M2^*}(t),\varvec{\lambda ^*}(t),t) \end{aligned}$$

(42)

$$\begin{aligned}&{\textbf {Transversality Conditions}}\quad M2(t_f)=\lambda (t_f). \end{aligned}$$

(43)

The above equations are referred to as canonical Hamiltonian equations.To find the optimal control $\eta _{M1}^{*}(t)$, the Hamiltonian must be minimized with respect to $\eta _{M1}(t)$ (Maximum Principle or Pontryagin’s Maximum Principle):

$$\begin{aligned} \dfrac{\partial H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)}{\partial \varvec{\eta _{M1}}}=0, \end{aligned}$$

(44)

By establishing Eq. (44) and positive definite of the matrices $\dfrac{\partial ^2 H(\varvec{M2^*}(t),\varvec{\eta _{M1}^*}(t),\varvec{\lambda ^*}(t),t)}{\partial \varvec{\eta _{M1}}^2 }$, in this case, $\varvec{\eta _{M1}^*}(t)$ ensures that $H({\varvec{M}}{\varvec{2}}(t),\varvec{\eta _{M1}}(t),\varvec{\lambda }(t),t)$ becomes a local minimum.

$$\begin{aligned} \begin{aligned} \varvec{\eta _{M1}^*}(t)=C_{M1}^{T}\varvec{\lambda ^*}(t). \end{aligned} \end{aligned}$$

(45)

In this case, Eqs. (45) and (39) are used to determine the optimal state trajectory $M2^*(t)$ and control $\eta _{M1}^{*}(t)$.

We employ the linear feedback format to determine the optimal control, wherein we seek functions $K_{M1}(t)$ and $\rho _{M1}(t)$.

$$\begin{aligned} \begin{aligned} \varvec{\eta _{M1}^*}(t)=K_{M1}(t)\varvec{M2^*}(t)+\rho _{M1}(t). \end{aligned} \end{aligned}$$

(46)

For the unknowns, $\rho _{M1}(t)$, and $K_{M1}(t)$, which represent the feedback matrices, we make the assumption that the Lagrange multiplier vector, $\varvec{\lambda ^*}(t)$, is linear in $\varvec{M2^*}(t)$, i.e.:

$$\begin{aligned} \varvec{\lambda ^*}(t)=\varvec{p}(t)\varvec{M2^*}(t)+c_{M1}g(t) \end{aligned}$$

(47)

For the unknowns $\varvec{p}(t)$ and g(t), upon substituting Eq. (47) into Eq. (45), we obtain:

$$\begin{aligned} \begin{aligned} \varvec{\eta ^*_ {M1}}(t)=C_{M1}^{T}(\varvec{p}(t)\varvec{M2^*}(t)+c_{M1}g(t)). \end{aligned} \end{aligned}$$

(48)

Upon comparing Eq. (46) with Eq. (48), we find:

$$\begin{aligned} \begin{aligned} K_{M1}(t)=C_{M1}^{T}\varvec{p}(t), \quad \rho _{M1}(t)=C_{M1}^{T} c_{M1}g(t). \end{aligned} \end{aligned}$$

(49)

Upon substituting Eq. (48) into Eq. (33), we obtain:

$$\begin{aligned} \begin{aligned} \varvec{{\dot{M}}2}^*(t)=A{\varvec{M}}{\varvec{2}}^*(t)-C_{M1}(C_{M1}^{T} \varvec{p}(t)\varvec{M2^*(t)}+c_{M1}g(t))+C_{M1}. \end{aligned} \end{aligned}$$

(50)

By differentiating Eq. (47) and applying (39), we have:

$$\begin{aligned} \begin{aligned} \varvec{{\dot{\lambda }}}^*(t)={\dot{p}}(t)M2^*(t)+p(t){\dot{M2}}^*(t) +c_{M1}{\dot{g}}(t)=-\varvec{M2^*}(t) - A^{T}\varvec{\lambda ^*}(t). \end{aligned} \end{aligned}$$

(51)

Finally, if we substitute (47) and (50) into Eq. (51):

$$\begin{aligned} \begin{aligned}&{\dot{p}}(t)M2^*(t)+p(t)\big (A{\varvec{M}}{\varvec{2}}^*(t)-C_{M1}(C_{M1}^{T} \varvec{p}(t)\varvec{M2^*(t)}+c_{M1}g(t))+C_{M1}\big ) +c_{M1}{\dot{g}}(t)=-M2^*(t)-A\big (p(t)M2^*(t)+c_{M1}g(t)\big ) \end{aligned} \end{aligned}$$

(52)

$C_{M1}$ from (32) is substituted.

$$\begin{aligned} \begin{aligned}&M2^*(t)\big ({\dot{p}}(t)+p(t)A-p(t)c_{M1}M^{-1}Cp(t)c_{M1}M^{-1}C+I+Ap(t)\big ) +c_{M1}\big (-c_{M1}p(t)M^{-1}Cg(t)+M^{-1}C+{\dot{g}}(t)+Ag(t)\big )=0 \end{aligned} \end{aligned}$$

(53)

In Eq (53), both $M2^*(t)$ and $c_{M1}$ are positive and non-zero. Consequently, the coefficient of M2(t) and the second term must be equal to zero simultaneously. Therefore, Eq. (53) reduces to two differential equations (Riccati equations) as follows.

$$\begin{aligned} \begin{aligned}&{\dot{p}}(t)+p(t)A-p(t)C_{M1}p(t)C_{M1}+ I + Ap(t) = 0 \\&{\dot{g}}(t)-p(t)C_{M1}g(t)+M^{-1}C+Ag(t)=0 \end{aligned} \end{aligned}$$

(54)

According to (43), the final conditions will be:

$$\begin{aligned} \begin{aligned} M2(t_f)=\lambda (t_f) \end{aligned} \end{aligned}$$

(55)

definition (47), will be

$$\begin{aligned} M2(t_f)=\varvec{p}(t_f)M2(t_f)+c_{M1}g(t_f) \end{aligned}$$

(56)

Consequently

$$\begin{aligned} \begin{aligned} \varvec{p}(t_f)=I\quad \text {and} \quad g(t_f)=0 \end{aligned} \end{aligned}$$

(57)

We solve (54) using the Euler approximation method to get p(t), g(t), and $\varvec{{\lambda }^*}(t)$ from (51) and (54).

If p is positive definite, the system is guaranteed to be stable. In other words, solving the Riccati equation ensures that the system under optimal control behaves properly and reaches a stable state.

In the following section, the numerical results of the previous parts are presented.

Numerical results

In this study, we formulated and solved two types of problems: an uncontrolled problem representing the natural dynamics of M2 macrophages and a controlled problem, known as the ORP, involving external intervention. Figure 6 compares the uncontrolled scenario, where fibrotic wound healing occurs naturally, with the optimal control scenario, where an anti-M1 drug is used to manage fibrotic wound healing. Results indicate that without a control agent, M2 macrophages proliferate, escalating fibrotic tissue formation. Introducing the anti-M1 drug significantly reduces M2 macrophage activity, highlighting the potential therapeutic benefit of such control strategies. The numerical simulations show a distinct reduction in M2 density when optimal control is applied, as demonstrated in Fig. 6. Specifically, the red graph (a) depicts how M2 macrophages converge to zero over time, confirming the effectiveness of the proposed control strategy in mitigating fibrotic progression. This finding suggests that the optimal regulator control problem formulation is both appropriate and effective, especially compared to baseline scenarios modeled by dynamical systems alone, which fail to suppress M2 macrophage density completely.

This study surpasses existing control methods by addressing a gap in the current research landscape, as there is no existing optimal control problem to compare our findings on the diffusion of M2 macrophages in lung fibrosis. It introduces a novel framework that could inform and guide future therapeutic strategies. The model’s numerical complexities are managed using advanced spectral methods and optimization principles The model’s numerical complexities are managed using advanced spectral methods and optimization principles, providing a robust approach for potential clinical applications.

The Algorithm for dynamical system for M2 (2) is as follows:

Step 1::

Convert Eqs. (7) to (28) using the spectral method.

Step 2::

Initials the control values from Table 1.

Step 3::

Solve (28) by using the Euler approximation.

Computational complexity dynamical system

Discretizing the M2 macrophage diffusion equation using the spectral Galerkin method with Legendre polynomials of order 32 has a complexity of $O(N^2)$, where N is 32⁴⁸. Solving the resulting ODE system with Euler’s method, which involves $N\times N$ matrices, has a per-time-step complexity of $O(N^2)$⁶². For T time steps, the overall complexity is $O(N^2)+O(T \cdot N^2)$. This combines the complexities of discretization and time stepping solution.

The Algorithm for ORP is as follows:

Step 1::

Solve ORP (29) with first discretize then optimize method. Discretize ORP by spectral discretization to get DORP (33).

Step 2::

Optimize DORP by Pontryagins minimum principle. (Use the linear feedback form to find an optimal control).

Step 3::

Update the control and states in each iteration by using values from previous iterations.

Step 4::

Go to Step 2 and continue until the convergent is achieved.

Computational complexity ORP

Solving an optimal control problem with a quadratic objective function and PDE constraints involves several steps. Discretizing the PDE using the spectral Galerkin method with Legendre polynomials of order 32 has a complexity of $O(N^2)$, where N is 32⁶³. Computing the Hamiltonian during optimization requires $O(n^2 + nm + m^2)$, with n as the number of state variables and m as the number of control variables. Solving the resulting ODE system with Euler’s method has a complexity of $O(n^2)$ per time step, leading to $O(T \cdot n^2)$ for T time steps. The overall complexity is $O(N^2) + O(T \cdot n^2)$.

In the following, based on the theatrical aspect in the previous sections the numerical results are discussed:

Case I: Only the Dynamical System (4) without any Medication is solved. The dynamical system solution is obtained by converting the associated PDE (4) into a system of ODEs (28) using 32nd-order Lagrangian interpolation polynomials (10) defined at Gauss-Legendre-Lobatto integration points (12). The M2 density is depicted in Fig. 5 for $\eta _{M1}(t)=0$. This baseline scenario demonstrates the natural progression of M2 macrophages without external intervention $\square$

In the next Case, we give positive values to $\varvec{\eta _{M1}(t)}$ in order to solve the dynamical system (2) and discuss the result.

Case II: Dynamical System Solutions with Different Constant Scalar Values for $\varvec{\eta _{M1}(t)\ne 0}$ (Involving Medication). M2 density solutions are obtained by treating the function $\eta _{M1}(t)$ as a constant scalar value in the dynamical system (2) and compared the results with the uncontrolled scenario. Using this approach, M2 densities with different medication dosages are calculated, and the results are plotted in Fig. 5 when $\varvec{\eta _{M1} (t)\ne 0}$ note that in the Fig. 5, graphs for M2 density without control presented as an uncontrolled state ($\star$). The other graphs are depicted for control $\eta _{M1}(t)\in (0, 0.4 \times 10^{-1}]$. These graphs demonstrate that although the control values $\eta _{M1}(t)\in [0, 0.4 \times 10^{-1}]$ reduce the M2 density level over time, the M2 densities are still increasing. As shown in the simulation, whether using only the dynamical system without medication (4) or with medication (2), it is not possible to entirely cure the patient (since the M2 density is increasing). $\square$

Case I and II show that the M2 density never completely disappears and, therefore, never gets eradicated. We conclude that investigating the dynamical system, whether by adding a constant value as control or without control, cannot decrease the density of M2 over time. To address this, the problem formulation (2) needs to be extended and transformed into an optimal control problem. In the following the optimal regulator control problem (29) is solved.

Case III: Solution of the Optimal Regulator Control Problem (29). Rather than relying solely on the results of the dynamical systems (2) and (4), we extend the M2 density problem as an optimal control problem (29). Graph is depicted in Fig. 6. In the red graph in Fig. 6 (a), we show that the M2 density decreases over time and eventually converges to zero when we formulate the M2 density problem by the optimal regulator problem (29). In Fig. 6 (b), the optimal control functions $\eta _{M1}(t)$ are illustrated. It is evident that the control function (anti-M1) initially decreases and then vanishes. Consequently, in the repair tissue, the M2 macrophages disappear through apoptosis, inhibiting the formation of fibrotic tissue. Medications are prescribed in specific doses that decrease over time. This indicates that the treatment strategy entails a gradual reduction in medication dosage throughout the course of treatment. $\square$

In the following we compare Cases I, II and III.

Case IV: comparison.

In Fig. 6, we compare the outcomes of the scenarios without medication (Case I) and with medication (Cases II and III).The modeling of the problem using only the dynamical system (Cases I and II) results in an increase in M2 density that never completely vanishes. In contrast, the optimal regulator control problem (Case III) leads to a substantial reduction in M2 density, which eventually disappears after approximately 120 days. The numerical results shown in Fig. 6a support the authors’ hypothesis that transforming the model from a simple dynamical system to an optimal regulator control problem is both appropriate and effective.

$\square$

Additionally, we computed the objective function for the optimal regulator control problem, demonstrating that it decreases consistently over time.

Case V: Objective function. Figure 7 illustrates the objective function ${\hat{J}}(M2^*, \eta _{M1}^*, t)$ for the problem (33) plotted against time, serving as an approximation of the objective function for the optimal control problem (29). The figure shows a clear and consistent decline in the objective function over time, eventually approaching zero. This decreasing trend indicates effective minimization of the objective function, suggesting that the optimal control strategy is progressing correctly toward achieving the desired outcome.

Next, we explore the effects of acute lung fibrosis by increasing the wound area from $[0.3 \times 0.3]$ cm to $[ 0.5 \times 0.5]$ cm.

Case VI: Comparing solutions in the two Inflammation Areas. Following Hao’s suggestion²³, we expanded the wound area without altering the dynamical system. We examined both scenarios: a smaller wound area and a larger wound area, using the same dynamical system (2). Figures 8 and 9 demonstrate that the results remain consistent whether solving the dynamical system (4) alone or the Optimal Regulator Problem (ORP) (29). However, in real-world conditions, as the wound or inflammation area increases, patients are likely to develop acute and severe fibrosis, which is distinct from normal fibrosis. Consequently, we recommend that future research simulate this scenario using a new dynamical system that accurately represents the behavior of acute fibrosis $\square$

It is observed that over time, M2 macrophage diffusion increases with the dynamical system solver, while it decreases with the ORP solver.

Sensitivity analysis

Sensitivity analysis in biological models, particularly regarding macrophages, is crucial for understanding the impact of changes in key parameters on system behavior. In this analysis, three primary parameters are considered: the polarization rate of M1 to M2 macrophages ($\lambda _{M1}$), the number of M1 macrophages, and the diffusion coefficient of M2 macrophages. The polarization rate of M1 to M2 macrophages ($\lambda _{M1}$) determines the speed at which M1 macrophages convert to M2 macrophages. This rate plays a critical role in regulating immune and inflammatory responses. Changes in this rate can significantly alter the balance between M1 and M2 macrophages, affecting the intensity and type of inflammatory responses and patient outcomes. For instance, an increase in M1 macrophages can intensify inflammation and cause more tissue damage, which may subsequently result in a higher rate of M2 macrophage diffusion. Conversely, reducing the number of M1 macrophages might alleviate inflammation, leading to a lower rate of M2 macrophage diffusion. The number of M1 macrophages is another key parameter in biological models. Variations in this number can impact the severity of inflammatory responses. For example, an increase in M1 macrophages may intensify inflammation and tissue damage, whereas a decrease could reduce inflammation. Sensitivity analysis of this parameter helps us understand how changes in the number of M1 macrophages affect system behavior and inflammatory responses. The diffusion coefficient of M2 macrophages represents the rate and pattern of their spread in the tissue. Changes in this coefficient can have significant effects on the spatial distribution of M2 macrophages and their interactions with other cells and inflammatory factors. Overall, sensitivity analysis of these three parameters ( polarization rate of M1 to M2 macrophages, the number of M1 macrophages, and the diffusion coefficient of M2 macrophages )provides valuable insights into how variations in these parameters influence model behavior, immune responses, and treatment outcomes. Sensitivity analysis of these three parameters helps us gain a clearer understanding of how changes in them affect model outcomes and biological responses. It also aids in optimizing therapeutic strategies and improving predictions of treatment results. The anti-M1 agent acts as an immune modulator to shift the balance between pro-inflammatory and pro-repair responses, ultimately preventing excessive fibrosis. Classically activated macrophages (M1) are known for their role in initiating inflammatory responses, which can lead to tissue damage and contribute to the progression of fibrosis.

Anti-M1

Therapies are designed to target and inhibit M1 macrophages, a subset of immune cells known for their pro-inflammatory properties. These macrophages produce a variety of cytokines and inflammatory molecules that play critical roles in the body’s innate immune response and contribute to inflammation. In diseases like pulmonary fibrosis, M1 macrophages can worsen the condition by promoting excessive inflammation and fibrotic tissue formation. The primary goal of anti-M1 therapies is to reduce inflammation and manage fibrotic diseases by specifically targeting M1 macrophages or the inflammatory signals they produce. Various approaches include anti-inflammatory drugs, cytokine inhibitors, biological agents, and small molecule drugs. In treating pulmonary fibrosis and similar conditions, anti-M1 strategies aim to balance the immune response by reducing harmful inflammation and preventing excessive fibrotic tissue formation. However, there are challenges associated with these therapies. One major challenge is ensuring that inhibiting M1 macrophages does not compromise the immune system’s ability to respond to infections and other diseases. Additionally, individual responses to anti-M1 treatments can vary, necessitating personalized treatment approaches. Overall, while anti-M1 therapies hold promise for managing inflammatory and fibrotic diseases, ongoing research is crucial to fully understand their effectiveness, optimize treatment protocols, and ensure patient safety. Long-term use of anti-M1 agents can offer several benefits. These treatments may help reduce chronic inflammation and prevent the progression of fibrosis, potentially improving lung function and the overall quality of life for patients. However, long-term use also presents challenges. Inhibiting M1 macrophages might increase the risk of infections, as these cells play a crucial role in fighting pathogens. Additionally, long-term treatment could disrupt the immune system balance and lead to side effects or toxicity. To manage these risks, it is important to regularly monitor patients, adjust treatment doses according to individual needs, and combine therapies that support immune function.

Role of anti-M1 agent in modulating M2 macrophages and preventing fibrosis

The anti-M1 agent acts as an immune modulator to shift the balance between pro-inflammatory and pro-repair responses, ultimately preventing excessive fibrosis. Classically activated macrophages (M1) are known for their role in initiating inflammatory responses, which can lead to tissue damage and contribute to the progression of fibrosis^25,27.

Effects of the anti-M1 agent: Reducing Inflammation: The anti-M1 agent decreases the activity of M1 macrophages, thereby preventing severe inflammation and limiting further tissue damage. By doing so, it also reduces immune stimulation that could lead to the overactivation of M2 macrophages^25,53.

Shifting macrophage polarization: By downregulating M1 activity, the anti-M1 agent helps restore the balance between M1 and M2 macrophages. With less inflammatory stimulus from M1 macrophages, M2 macrophages are more likely to remain in their reparative role, rather than becoming overactivated in fibrotic processes.

Preventing M2 profibrotic activation: A key impact of the anti-M1 agent is its ability to prevent M2 macrophages from shifting to a profibrotic state, where they stimulate excessive collagen production and fibroblast activation^25,53. In the absence of excessive inflammatory signals, M2 macrophages can continue their normal reparative functions, reducing scar tissue formation⁵³.

Decreasing collagen production: In fibrotic conditions, M2 macrophages secrete cytokines like TGF-$\beta$ and PDGF, which promote fibroblast activation and collagen deposition. By modulating M1 macrophage activity and reducing inflammation, the anti-M1 agent indirectly limits the profibrotic activity of M2 macrophages, thereby reducing the excessive accumulation of ECM and preventing fibrosis^25,53.

Thus, by regulating M1 macrophage activity, the anti-M1 agent directly reduces inflammation and tissue damage, and indirectly prevents the profibrotic activation of M2 macrophages²⁷. This helps maintain a balance between repair and regeneration, ultimately preventing the progression of fibrosis in IPF.

Discussion of results

In the early stages of lung fibrosis, M2 macrophages are thought to play a beneficial role in tissue repair and inflammation resolution. They release substances that support matrix remodeling, angiogenesis, and the suppression of pro-inflammatory responses. These M2 macrophages are commonly known as “repair” or “alternatively activated” macrophages. However, as fibrosis advances, the polarization of macrophages may become disrupted. In more progressed stages, there is evidence indicating that M2 macrophages might contribute to ongoing fibrosis by encouraging excessive collagen production, tissue scarring, and myofibroblast activation. This pro-fibrotic characteristic of M2 macrophages is often termed “profibrotic” macrophages. The dysregulation of M2 macrophages in lung fibrosis underscores the potential importance of restoring their equilibrium as a therapeutic strategy. Certain drugs or compounds, such as Pirfenidone and Nintedanib, both FDA-approved for treating IPF, have demonstrated immunomodulatory effects that can influence macrophage behavior, inflammation, and lung fibrosis.

Dynamical system, ORP, and treatment strategies

Based on the numerical results presented in section “Diffusion equation for M2 Macrophages” concerning the dynamical system, ORP solver, and treatment strategy, we conclude that:

Dynamical system (only)

Our findings indicate that the M2 density increases to a certain level over time. After approximately 200 days, it stabilizes but does not vanish, suggesting the resistance of M2 density to apoptosis in repair tissue. Our results align with the fact that persistent activation of M2 contributes to tissue remodeling and the formation of fibrosis^57,58.

ORP

In this paper, we observe that the optimal regulator problem solutions (29) decrease over time, eventually approaching zero. This reduction leads to cell death and the cessation of the healing process. This outcome allows doctors to use drugs or control agents to restrict the activity of macrophage M1. We demonstrate that as M1 activity decreases, macrophage M2 activity gradually diminishes, ultimately resulting in the inactivity of M2 macrophages. Consequently, the inflammatory mediators causing fibroblast to myofibroblast transformation become inactive, leading to the cessation of the process of forming fibrotic tissue in the wound area.

Treatment strategy

In this study, we demonstrate that the optimal control problem exhibit a gradual decrease in medication dosage over time, suggesting a treatment strategy involving a progressive reduction in dosage throughout the cure duration. If pharmaceutical companies can develop a medication affecting macrophages (utilizing an anti-M1 agent), it could lead to a more effective treatment for the patient.

Discussion of results and real-time applications

The proposed optimal control strategy for regulating M2 macrophages in IPF has significant potential for real-time applications in clinical settings. Numerical simulations show that, without intervention, M2 macrophage levels increase and contribute to the stabilization of fibrosis. However, using the proposed control strategy, the density of M2 macrophages converges toward zero within 120 days, effectively preventing fibrosis formation. These results highlight the practical feasibility of this approach in controlling macrophage-driven fibrosis.

Key real-time applications of this strategy include personalized treatment, where doses of anti-M1 medication can be adjusted based on real-time monitoring to meet individual patient needs. Gradual dose adjustments using real-time data ensure an optimal balance between treatment effectiveness and minimizing side effects, improving patient safety. By intervening early, this approach can prevent excessive accumulation of M2 macrophages and subsequent fibrosis, leading to better treatment outcomes. Moreover, insights gained from this study can inform the development of new drug therapies that specifically target macrophages, further enhancing treatment efficacy and patient safety.

Comparison with other studies:

Hao et al. (2014, 2015)^22,23: Hao and colleagues developed a mathematical model focusing on M2 macrophage polarization and its contribution to fibrosis progression in pulmonary fibrosis. While their model emphasized the importance of macrophages, it lacked an optimal control component. Our study builds on their work by incorporating an optimal control strategy targeting M2 macrophages through anti-M1 interventions, resulting in a faster and more efficient reduction in fibrosis. Hao et al.’s model provided important insights, but our approach extends it by adding a real-time dynamic control mechanism that maximizes therapeutic effectiveness.

Bahram Yazdroudi et al. (2022, 2023)^29,30: Yazdroudi and colleagues applied optimal control theory to anti-TGF-$\beta$ and anti-PDGF strategies for fibrosis prevention, targeting multiple pathways. However, their focus was primarily on growth factors rather than macrophage-driven fibrosis. Our model shifts the focus to M2 macrophage regulation, which plays a critical role in the later stages of IPF. By reducing M2 macrophage density through anti-M1 regulation, our approach provides a more effective intervention for late-stage IPF management, where controlling macrophage activity is crucial.

Macrophage-Targeting Therapies (Various Studies): Studies such as those by Zhang et al.⁶⁷, Cheng et al.³¹ and Ge et al.³² highlighted the therapeutic potential of macrophage-targeting interventions in lung fibrosis. However, none of these studies incorporated control strategies. Our study enhances these findings by integrating advanced control methods, such as Pontryagin’s minimum principle and Riccati equations, to dynamically regulate macrophage activity and prevent fibrosis progression with greater precision. This integration represents a significant advancement over earlier approaches.

Previous studies primarily focused on fibroblast activation or inhibiting cytokines, such as TGF-$\beta$, without directly targeting macrophages. In contrast, our model specifically targets M2 macrophages, the critical drivers of fibrosis. By regulating M2 macrophages through an optimal control strategy, our model demonstrates a more pronounced reduction in fibrotic tissue within 120 days, as indicated by the lower M2 macrophage levels compared to baseline. This direct targeting of macrophages offers a more efficient approach to fibrosis prevention, distinguishing our model from prior work.

Novel contributions and clinical impact

Our study introduces several advancements over previous research by integrating optimal control strategies for the dynamic regulation of M2 macrophages. This directly addresses the root cause of fibrosis in the later stages of IPF. Compared to previous approaches, our method focuses on real-time applications and allows for dynamic dose adjustments, which are critical for personalized and precise treatment in clinical settings. Furthermore, by leveraging advanced computational techniques, such as the Galerkin spectral method, we achieve more accurate and efficient control of fibrosis progression, making this method highly suitable for clinical use with minimal computational burden.

Computational solutions and overhead reduction

Solving the optimal control problem for regulating M2 macrophages and preventing lung fibrosis using the anti-M1 agent involves significant computational challenges, including the complexities of PDEs, discretization, and optimization. To address these challenges, the Galerkin spectral method is employed to transform the PDE for M2 macrophage diffusion into an algebraic system of ODEs, which reduces computational overhead and improves accuracy. Optimization is further enhanced through Pontryagin’s minimum principle, canonical Hamiltonian equations, and extended Riccati differential equations, which collectively reduce computational time and increase result accuracy.

For real-time applications, it is essential to use fast and efficient optimization algorithms. Implementing effective numerical methods and optimizing algorithms can significantly reduce computational time and provide timely results.

To further reduce computational overhead, advanced algorithmic techniques and machine learning should be applied to improve efficiency and manage complexity. Utilizing advanced hardware, such as GPUs and multi-core processors, can accelerate computations and enhance performance. Additionally, dimensionality reduction techniques, including Singular Value Decomposition (SVD) and Principal Component Analysis (PCA), can lower data volume and computational complexity, thus improving overall efficiency.

Parameters for numerical simulations: The parameters used in the numerical simulations, such as the diffusion coefficient for M2 macrophages, polarization rates, and degradation rates, were selected based on data from the existing literature on lung fibrosis and macrophage biology. For example, the diffusion coefficient was derived from experimentally observed rates of M2 macrophage migration in inflamed tissues²². Additionally, the polarization rate from M1 to M2 macrophages was selected based on studies measuring immune responses during fibrosis progression²³. These rates were validated through sensitivity analysis, where key parameters were systematically varied to assess their impact on model behavior and accuracy. The simulations demonstrated that the model remained stable and robust across a range of parameter values, which strengthens the reliability of the proposed control strategy^29,30.

Sensitivity analysis further indicated that variations in the polarization rate and M2 macrophage diffusion coefficient significantly influenced fibrosis progression. This allowed us to fine-tune the parameters to optimize therapeutic outcomes. Each parameter was cross-referenced with experimental data to ensure that the numerical simulations aligned with real-world biological phenomena. The validation process also involved comparing the model’s predictions with established outcomes from clinical and preclinical studies on fibrosis, confirming the appropriateness of the selected parameters^25,27.

Potential limitations and future work

Assumptions and limitations in IPF modeling

In this study, a simplified lung tissue geometry is assumed to facilitate the mathematical modeling process, while the actual lung tissue structure is more complex and heterogeneous. The model also assumes a uniform diffusion coefficient for M2 macrophages across the inflamed tissue, whereas diffusion in biological tissues typically varies due to differences in ECM density and tissue stiffness. Additionally, the focus of the study is on M2 macrophages and the use of anti-M1 agents as a control strategy. However, fibrosis is a multifactorial process involving various cell types and molecular mediators. Excluding these additional factors may oversimplify the disease and limit the applicability of the results to broader therapeutic strategies. Furthermore, the model assumes a fixed timeline for the control problem, which may not reflect the real temporal dynamics of disease progression and treatment response in individual patients. In reality, the timing and duration of interventions could significantly affect their effectiveness.

Limitations of the model: The model has several limitations that may impact its ability to accurately represent real-world IPF conditions. First, the two-dimensional representation of lung tissue simplifies calculations but does not fully capture the complexity and heterogeneity of actual lung structures^23,24. The assumption of a uniform diffusion coefficient for M2 macrophages oversimplifies biological tissues, which often exhibit variable diffusion properties due to differences in extracellular matrix composition and tissue stiffness^25,27. Additionally, the model primarily focuses on controlling M2 macrophages, while fibrosis involves a complex interplay of multiple cell types and pathways. This narrow focus may limit the model’s applicability in more comprehensive cases of fibrosis. The assumption of a fixed treatment timeline does not account for variations in patient response to therapy, as disease progression and treatment efficacy can differ greatly across individuals. Future studies should aim to incorporate these biological complexities, including spatial heterogeneity and personalized immune responses, to better reflect real-world IPF conditions and improve the model’s clinical applicability.

Potential limitations

In a clinical setting, physicians would monitor a patient’s condition over time, adjusting dosages of anti-M1 drugs or other therapies as needed to suppress M1 activity. While this model provides a simplified representation of that process, it cannot capture the full complexity of real-world clinical situations. Simplifying M2 macrophage behavior by using fixed parameters can offer a basic solution, but further development in drug research and more sophisticated models would improve accuracy. Additionally, anti-M1 therapies, while effective in reducing inflammation, may have side effects such as increased susceptibility to infections or negative long-term impacts on the immune system. Therefore, balancing efficacy with safety remains a critical challenge. Finally, the model and control strategies require experimental and clinical validation to confirm their practical efficacy and safety in treating fibrosis.

Future research and clinical applications

Future research should focus on refining mathematical models to better capture the complexities of biological interactions between macrophages and other cellular components, such as fibroblasts, cytokines, and growth factors, which are crucial in IPF progression. Incorporating these detailed biological data will enhance model accuracy and predictive power. Additionally, strategies like Reduced Order Modeling (ROM) can simplify the system to a lower-dimensional model, reducing the computational complexity of optimal control problems with PDE constraints. The use of more efficient discretization methods, such as adaptive grids, and advanced optimization algorithms like quasi-Newton methods, can further reduce computation time, making the models more suitable for clinical applications.

Improvements in parameter estimation, sensitivity analyses, and implicit solvers for ODE systems can enhance the robustness of control strategies. The proposed control strategy, focusing on regulating M2 macrophages using anti-M1 agents, opens various avenues for personalized treatment protocols. By integrating patient-specific data, future models could optimize drug dosing to maximize therapeutic outcomes while minimizing side effects. This approach can be further developed into real-time decision-support systems in clinical settings, where physicians dynamically adjust treatments based on patient responses.

Moreover, exploring multi-targeted therapies that address other mediators of fibrosis, such as fibroblasts or extracellular matrix components, could enhance the efficacy of M2 macrophage regulation. Clinical trials will be essential to validate these strategies, particularly in comparison with existing treatments like pirfenidone and nintedanib, to assess the added benefits of macrophage-targeted interventions. Long-term studies on anti-M1 therapies, in combination with other treatments, may yield synergistic effects, improving overall treatment efficacy. Expanding these strategies to other fibrotic conditions, such as liver, kidney, and cardiac fibrosis, could demonstrate their generalizability. Additionally, incorporating stochastic elements, multiscale approaches, and artificial intelligence into future models will further enhance predictive capabilities and lead to more personalized therapeutic strategies for IPF and similar conditions.