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Computational and Mathematical Methods in Medicine
Volume 2012 (2012), Article ID 809864, 11 pages
http://dx.doi.org/10.1155/2012/809864
Research Article

Stability Analysis of a Model for Foreign Body Fibrotic Reactions

1Department of Mathematics and Statistics, Texas Tech University, P.O. Box 41042, Lubbock, TX 79409-1042, USA
2Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA
3Department of Bioengineering, The University of Texas at Arlington, Arlington, TX 76019, USA

Received 30 April 2012; Revised 1 August 2012; Accepted 6 August 2012

Academic Editor: Gary C. An

Copyright © 2012 A. Ibraguimov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Implanted medical devices often trigger immunological and inflammatory reactions from surrounding tissues. The foreign body-mediated tissue responses may result in varying degrees of fibrotic tissue formation. There is an intensive research interest in the area of wound healing modeling, and quantitative methods are proposed to systematically study the behavior of this complex system of multiple cells, proteins, and enzymes. This paper introduces a kinetics-based model for analyzing reactions of various cells/proteins and biochemical processes as well as their transient behavior during the implant healing in 2-dimensional space. In particular, we provide a detailed modeling study of different roles of macrophages (𝑀Φ) and their effects on fibrotic reactions. The main mathematical result indicates that the stability of the inflamed steady state depends primarily on the reaction dynamics of the system. However, if the said equilibrium is unstable by its reaction-only system, the spatial diffusion and chemotactic effects can help to stabilize when the model is dominated by classical and regulatory macrophages over the inflammatory macrophages. The mathematical proof and counter examples are given for these conclusions.

1. Introduction

Recently, intensive research efforts have been focusing on developing mechanistic computational models for wound healing related processes. Wound healing is a very complicated biochemical and biophysical phenomenon, with many facets and subprocesses, including the inflammatory response process, angiogenesis as well associated fibrotic reactions. Many cells, enzyme, growth factors, and proteins participate at different stages of the wound healing reactions, and they form a network of signaling pathways that in turn leads to inflammatory, angiogenesis, and fibrotic reactions. We refer to the review by Diegelmann and Diegelmann and Evans 2004 [1] for a brief review of the recent scientific work.

As a subarea of general wound healing research, healing processes involved in medical implantations are of significant application for modern medicine [24]. It is commonly accepted that implants may cause foreign body reactions that are initiated with implant-mediated fibrin clot formation, followed by acute inflammatory responses [4, 5]. The inflammatory chemokines released by adherent immune cells serve as strong signals for triggering the migration of macrophages and fibroblasts from the surrounding tissues and circulation toward the implant surface [5]. The implant-recruited fibroblasts consequently synthesize chains of amino acids called procollagen, a process that is activated by growth factors, including in particular type-𝛽 transforming growth factor (TGF-𝛽) [6, 7] to become collagen, the dominant ingredient of the extracellular matrix (ECM) [8]. These processes may, however, differ slightly between dermal wound healing and implantation when it comes to specific activation and inhibition loops of reactions.

Among inflammatory cells, macrophages (𝑀Φ) are found to reside in the wound [9]. The roles of macrophages are multiple and stand prominent in the activations and inflammations during implantation. 𝑀Φ are known to remove damaged tissue and foreign debris via phagocytosis. In addition, 𝑀Φ often release a variety of chemokines to recruit other cell types, such as fibroblasts, which participate in the remodeling of ECM. The specific roles of 𝑀Φ vary significantly at different stages of healing process. Work by Mosser and Edwards 2008 [10] has shown there to be at least 3 phenotypes of 𝑀Φ, each of which displays a different functionality. Classically activated 𝑀Φ represent the effector 𝑀Φ that are produced during cell-mediated immune responses. Two signals, interferon-𝛾 and tumor-necrosis factor-𝛼, give rise to these effector 𝑀Φ which have enhanced microbicidal or tumoricidal capacity and secrete high levels of proinflammatory cytokines and mediators. Assisted in part by the production of transforming growth factor type 𝛽 (TGF-𝛽), the clearance of apoptotic inflammatory, as well as noninflammatory cells by classical 𝑀Φ, can lead to an inhibition of inflammation [11, 12]. Wound healing 𝑀Φ (or inflammatory 𝑀Φ) can develop in response to innate or adaptive signals through interleukin-4. In turn, interleukin-4 stimulates arginase activity in 𝑀Φ, allowing them to convert arginine to ornithine, a precursor of polyamine and collagen, thereby contributing to the production of extracellular matrix (ECM) [13]. Regulatory 𝑀Φ can also arise during the later stages of adaptive immune responses, the primary role of which dampen, the immune response and limits inflammation through production of interleukin-10 [14]. Although all three phenotypes were observed experimentally within the dermal wound healing context, the phagocyte biomaterial interactions are known to be similar here for foreign body reactions.

While experiments are still the main stay in the studying of wound healing related process, significant progress has also been made in detail predictive modeling based on biochemical and biophysics principles. For dermal wound healing, basic reactions were first considered in studies by Dale et al. 1996 [15], 1997 [16]; Dallon et al. 2001 [17] and many others. Their models incorporated the key features of kinetics which are essential to dermal wound healing. Their results have successfully described the dynamics and compare favorably with experiments, in terms of healed ECM fiber ratio, spatial orientation, and other features. Recently, the work of Schugart et al. 2008 [18] and Xue et al. 2009 [19] further included angiogenesis equations to the healing process and examined the positive effects of increased oxygen level in accelerating the healing and closure of open wound, suggesting new insights for the healing. Furthermore, a wound healing model based more on cell migration was considered in Arciero et al. 2011 [20].

Inflammatory reactions are important to wound healing as they activate many key agents for the healing process, however, prolonged inflammation may cause excessive scars and chronic wounds. Through interactions between immune mediators, phagocytes in the blood and tissue, the acute inflammatory response was modeled and analyzed by reduced compartmental models in Reynolds et al. 2006 [21] and Day et al. 2006 [22]. Closely related to dermal wound healing and implantation, atherogenesis in blood vessels was modeled by continuum equations in Ibragimov et al. [23]. The concept of debris and phagocytosis in [21, 22, 24] is analogous to our current model, which assumes that the digestion of dead cells (or tissues) initiates the entire healing process. Further, addition of stem cells can create a new dimension to the healing and implantation process; we mention Lemon et al. 2009 [24] for their new mathematical tool in providing quantitative analysis for this growing field.

Our primary goal in this paper is to use computational modeling to study the fibrotic reaction process following implantation with specific attention given to the effects caused by varying the mix of different phenotypes of 𝑀Φ. Our modeling results indicate trends for these variations, serving as a plausible clue for developing new experiments.

The main mathematical contribution of this paper is as follows. The nonzero equilibrium of our model represents an inflamed state. If it is linearly stable in terms of the corresponding ODE system (the reactions network of the model), then it is also stable for the full system (which includes spatial diffusion and chemotaxis). In other words, spatial effects cannot destabilize the equilibrium if it is stable in its pure reactions. However, even if the equilibrium is unstable by its reaction system, the spatial diffusion and chemotactic effects can help to stabilize the equilibrium under several conditions. These conditions suggest the need for the model to be dominated by classical and regulatory macrophages over the inflammatory macrophages. The mathematical proof and counter examples are given for these results.

We organize the paper as follows. In Section 2, we introduce the model and the modeling considerations. In Section 3, we discuss the spatially uniform equilibria and their stability in relation to the ODE system representing the reaction system without spatial variations. In Section 4, we prove that if the equilibrium is stable in ODE sense, then it is stable for the full system with respect to any small spatial perturbation measured in 𝐿2. In Section 5, we provide a set of sufficient conditions for the equilibrium to be stable for the full system, which allows us to explicitly give a counter example where PDE solutions can be conditionally stable without requiring stability in the ODE system. A brief summary and discussion are presented in Section 6.

2. Modeling Based on Chemical Kinetics Equations

Our foreign body reaction model is partially from the mass-action kinetics framework developed by Schugart et al. 2008 [18], which modeled wound healing under oxygen pressure. In medical implantation processes, new kinetics of 𝑀Φ reactions were added in the framework. The main biological question that we hope to address is the variance between tissue responses at different percentages of classical, inflammatory, and regulatory 𝑀Φ cells during foreign body fibrotic reaction processes. We model the following: 𝜕𝐷𝜕𝑡=𝐷𝑑2𝐷𝑓0𝜆1𝑓𝑀𝐷+0𝜆3𝑀,(1)𝜕𝐶𝜕𝑡=𝐷𝑐2𝐶+𝑓1𝐷+𝑓2𝜆3𝑀𝑓3𝜆2𝑀𝐶𝑓4𝐶,(2)𝜕𝐹𝜕𝑡=𝐷𝑓2𝐹𝜒0(𝐹𝐶)+𝑎1𝜆1𝑀+𝑎2𝐹𝑎2𝑎3𝑎2𝐹𝐹0+𝑎12𝐹𝐶𝐹𝐻0,𝐹(3)𝜕𝑀𝜕𝑡=𝐷𝑚2𝑀𝜒1𝑀𝑀𝐻0𝑀𝐶𝑎0𝑀+𝑎11𝑀𝐶𝑀𝐻0,𝑀(4)𝜕𝐸𝜕𝑡=𝐷𝑒2𝐸Φ+𝑎16𝐹𝐸1𝐸0,(5) where 2=, and the vector field Φ=𝐵𝐷𝑓𝐹0𝐸𝐹+𝐵𝜒𝑗𝐹0𝐹𝐸𝐹𝐻0,𝐹𝐶(6) and all coefficients are positive. The form of the logistic terms in (3) is for representing biological meanings of the coefficients.

In the system (1)–(5), the debris cell population 𝐷 represents dead tissue cells following implantation. Abnormal white blood cells and molecules caused by the surgery are also included in this debris term, which is assumed to be the initiation point of reactions. We assume that they are digested by 𝑀1-classical 𝑀Φ and that 𝑀3-inflammatory 𝑀Φ contribute to the accumulation of debris during the healing process (as modeled in (1)).

The chemoattractant consists mainly of various forms of growth factors including tissue growth factors type 𝛽 (TGF𝛽) released during the tissue injury. The chemoattractant field 𝐶 is enhanced by the presence of debris and 𝑀3-inflammatory 𝑀Φ cell, but is inhibited by 𝑀2-regulatory 𝑀Φ cells. In (2), we assume that cell spatial migration occurs through diffusion and chemotactic migration based on the gradient field of 𝐶.

Fibroblast density 𝐹 represents a main cell type in secreting collagen (a major component of ECM). Fibroblast proliferation and collagen synthesis are upregulated by the chemoattractant gradient field 𝐶. Thus fibroblast population 𝐹 (shown in (3)) can be approximated by a chemically enhanced logistic growth 𝐹(1(𝐹/𝐹0)) with a threshold 𝐹0, along with its diffusion in space modeled by 𝐷𝑓2𝐹, chemotactic migration by 𝜒0(𝐹𝐶) and its natural decay according to time as shown in (3). New experimental data also shows autocrine upregulation of fibroblast by TGF𝛽 without chemotaxis [25]; this effect is also included in the modeling. The term 𝑎3𝐹 is the decaying factor.

Macrophage density, 𝑀, is the summation of 𝑀1-classical 𝑀Φ, 𝑀2-regulatory 𝑀Φ, and 𝑀3-inflammatory 𝑀Φ. We assume that they each take on a proportion 𝜆1, 𝜆2, and 𝜆3 of 𝑀Φ, respectively. Each phenotype 𝑀𝑗, 𝑗=1,2,3, may take a different share of 𝑀Φ at different stages of foreign body fibrotic reactions. However, our model simplifies the situation in that (a) the proportions 𝜆1, 𝜆2, and 𝜆3 for different phenotypes of 𝑀Φ are fixed, and (b) the total 𝑀Φ population is set to share one common biochemical reaction equation (4), since its basic biochemical properties are similar. The proliferation of 𝑀Φ at the field is through diffusion and migration upregulated by the chemotactic gradient field 𝐶, but the production does reach a limiting value once the 𝑀Φ population reaches its saturation of 𝑀0. 𝑀Φ cell apoptosis and proliferation caused by the direct interaction with chemoattractants are also assumed.

Finally in (5), fibroblasts secrete procollagen which is then activated by the chemoattractant TGF𝛽s into collagen (or ECM) represented by the quantity 𝐸. We also incorporate the effects of ECM diffusion, fibroblast movement, chemotactic migration, and ECM saturation in mass-action law. In all discussions, 𝐻 is the Heaviside function, and 𝑀0 is the 𝑀Φ saturation level.

We assume in our implant model that the computational domain is large enough and also the cell changes are slow enough (measured in days) that there is no significant boundary flux, allowing us to take homogeneous Neumann boundary conditions as a reasonable approximation.

Definition 1. Let us define inflammatory equilibrium as a strictly nonzero constant vector 𝑈𝑒 in 5-dimensional space 𝑈𝑒=(𝑑𝑒,𝑐𝑒,𝑓𝑒,𝑚𝑒,𝑒𝑒) with 𝑑𝑒>0,𝑐𝑒>0,𝐹𝑜𝑓𝑒>0,𝑀𝑜𝑚𝑒>0,𝑒𝑒=𝐸0>0, which solves system of the equations (1)–(5).

Remark 2. In the case of a no-flux boundary condition, the spatially uniform steady state is often used when modeling inflammatory response in tissue (see e.g., [23, 26] and reference therein). A physically realistic, nonnegative set of equilibriums can easily be obtained by letting the RHS of the original system (1)–(5) equal to zero. It is natural to define the trivial (zero) equilibrium as ground or healthy state and study its stability. Instability of the ground state is usually interpreted as unfavorable development of the disease. In this paper we take a different approach and are interested in analyzing the stability of the abnormal/inflammatory equilibrium which is nonzero for all five components of the unknown. This equilibrium can be stable or unstable depending on the parameters of the model. In this case, instability of the equilibrium does not necessarily mean an unhealthy response of the immune system. An instability of a nonzero equilibrium can lead to a ground healthy state (best case scenario), to another steady state (uncertain developments), or to infinity (acute development). If in contrary, the perturbation of 𝑈𝑒 is linearly stable and vanishes at time infinity, then 𝑈𝑒 can be interpreted as sustainable. All these make linear stability analysis very appealing from both a theoretical and applied point of view. It is worth mentioning that from a biological point of view, a strictly positive steady state 𝑈𝑒 can be transitioned from some other nonstrictly positive state. We believe that this type of interpretation of the inflammatory equilibrium stability conditions is logical and presents an example of a sustainable wound which does not heal over the course of a long time period (see [1921]). An indirect analogy of such an inflammatory (chronically) stable equilibrium has been introduced and applied for studying biological dynamic system in virology for some years (see e.g., [27]). At this stage of the research, we are studying stability of the strictly positive state 𝑈𝑒 mostly as a model of inflammatory equilibrium, without analysis of its genesis. As commonly occurs in biomedical research, the mathematical model can often provide nonintuitive insights into dynamics of inflammatory responses in the wound healing processes and can suggest new avenues for experimentation. In the forthcoming sections, sufficient conditions on the parameters of the system of the equation guarantee stability of nonzero equilibrium.

2.1. Linearized System

Let perturbation near this equilibrium be as following: 𝑑=𝐷𝑑𝑒,𝑐=𝐶𝑐𝑒,𝑓=𝐹𝑓𝑒,𝑚=𝑀𝑚𝑒,𝑒=𝐸𝑒𝑒.(7) Denote vector field of the perturbation by 𝑣(𝑥,𝑡)=(𝑑,𝑐,𝑓,𝑚,𝑒). Then the linearized system for 𝑣(𝑥,𝑡) will take the following form: 𝜕𝑑𝜕𝑡=𝐷𝑑2𝑑𝑏11𝑑𝑏14𝑚𝜕𝑐𝜕𝑡=𝐷𝑐2𝑐𝑏21𝑑𝑏22𝑐𝑏24𝑚,𝜕𝑓𝜕𝑡=𝐷𝑓2𝑓𝜒𝑓2𝑐𝑏32𝑐𝑏33𝑓𝑏34𝑚,𝜕𝑚𝜕𝑡=𝐷𝑚2𝑚𝜒𝑚2𝑐𝑏42𝑐𝑏44𝑚,𝜕𝑒𝜕𝑡=𝐷𝑒2𝑒𝜒𝑒12𝑓𝜒𝑒22𝑐𝑏53𝑓𝑏55𝑒.(8) Here, 𝜒𝑓=𝑓𝑒𝜒0,𝜒𝑚=𝜒1𝑚𝑒,𝜒𝑒1=𝐵𝐷𝑓𝑒0𝐹0,𝜒𝑒2=𝐵𝜒𝑗𝑒0𝑓𝑒𝐹0,𝑏11=𝑓0𝜆1𝑚𝑒,𝑏12=0,𝑏13𝑏=0,14𝑓=0𝜆3𝑓0𝜆1𝑑𝑒,𝑏15𝑏=0,21=𝑓1,𝑏22=𝑓3𝜆2𝑚𝑒+𝑓4,𝑏24𝑓=2𝜆3𝑓3𝜆2𝑐𝑒,𝑏23=𝑏25𝑏=0,31=0,𝑏32=𝑎12𝑓𝑒,𝑏33𝑎=2𝑓12𝑒𝐹0+𝑎12𝑐𝑒𝑎3,𝑏34=𝑎1𝜆1,𝑏35𝑏=0,41=0,𝑏42=𝑎11𝑚𝑒,𝑏43𝑏=0,44=𝑎0𝑎11𝑚𝑒,𝑏45𝑏=0,51=0,𝑏52=0,𝑏53=𝑎16𝑒1𝑒𝐸0,𝑏54=0,𝑏55=𝑎16𝑓𝑒𝐸0.(9)

3. Spatially Uniform Equilibrium States and Linear Stability in ODE System

We now focus on equilibrium states that are uniform in space for this Neumann problem. By removing the spatial variations, (1)–(5) reduce to the following ODE system: 𝑑𝐷𝑑𝑡=𝑓0𝜆1𝑓𝑀𝐷+0𝜆3𝑀,𝑑𝐶𝑑𝑡=𝑓1𝐷+𝑓2𝜆3𝑀𝑓3𝜆2𝑀𝐶𝑓4𝐶,𝑑𝐹𝑑𝑡=𝑎1𝜆1𝑀+𝑎2𝐹𝐹1𝐹0𝑎3𝐹+𝑎12𝐹𝐶𝐹𝐻0,𝐹𝑑𝑀𝑑𝑡=𝑎0𝑀+𝑎11𝑀𝐶𝑀𝐻0,𝑀𝑑𝐸𝑑𝑡=𝑎16𝐹𝐸1𝐸0.(10)

In looking for the equilibrium of the simplified system, (10), we assume that our values are taken to be below threshold and therefore we ignore the Heaviside functions. There are several possible equilibrium states, but as it was pointed out earlier, we focus on what one can call the interior equilibrium, one in which none of the components of the equilibrium are zero. We let the right-hand side of (10) to be zero. After some algebraic work, one can obtain the following explicit formula for a unique, nonzero solution 𝑈𝑒=(𝑑𝑒,𝑐𝑒,𝑒𝑒,𝑚𝑒,𝑓𝑒): 𝑑𝑒=𝑓0𝜆3𝑓0𝜆1,𝑐𝑒=𝑎0𝑎11,𝑒𝑒=𝐸0,𝑚𝑒=𝑓4𝑓0𝜆1𝑎0𝑎11𝑓1𝑓0𝜆3𝑓0𝜆1𝑓2𝜆3𝑎11𝑓3𝑎0𝜆2,𝑓𝑒=𝐹02𝑎2𝑎2𝑎3+𝑎12𝑎0𝑎11+𝐿1.(11) Here, 𝐿1=(𝑎2𝑎3+𝑎12(𝑎0/𝑎11))2+4(𝑎2/𝐹0)𝑎1𝜆1𝑚𝑒.

Remark 3. In order for the inflammatory equilibrium to exist, it is necessary and sufficient that macrophage percentages satisfy the following: 𝑓4𝑓0𝜆1𝑎0𝑎11𝑓1𝑓0𝜆3𝑓2𝜆3𝑎11𝑓3𝑎0𝜆2>0,(12) requiring either 𝑓2𝜆3𝑎11>𝑓3𝑎0𝜆2,𝑓4𝑓0𝜆1𝑎0>𝑎11𝑓1𝑓0𝜆3,(13) or 𝑓2𝜆3𝑎11<𝑓3𝑎0𝜆2,𝑓4𝑓0𝜆1𝑎0<𝑎11𝑓1𝑓0𝜆3.(14)

Condition on the parameters in (12) says that inflammatory macrophages dominate over either regulatory or classical macrophages and are guaranteeing existence of the inflamed steady state. This point will be expounded on further in the analysis of the conditions for stability of the nonzero equilibrium state. The illustration (Figure 1) provides a visualization of the necessary macrophage phenotype parameter ranges. “Hereafter we assume that the parameters of the original model satisfy condition (12).”

809864.fig.001
Figure 1: Illustration of the parameter range to ensure that 𝑚𝑒>0.

Turning now to satisfy the stability of the system at the equilibrium, we find the linearized system to be as follows: 𝑑𝑑𝑑𝑡=𝑏11𝑑𝑏14𝑚,𝑑𝑐𝑑𝑡=𝑏21𝑑𝑏22𝑐𝑏24𝑚,𝑑𝑓𝑑𝑡=𝑏32𝑐𝑏33𝑓𝑏34𝑚,𝑑𝑚𝑑𝑡=𝑏42𝑐𝑏44𝑚,𝑑𝑒𝑑𝑡=𝑏53𝑓𝑏55𝑒,(15) where 𝑏11=𝑓0𝜆1𝑚𝑒,𝑏14𝑓=0𝜆3𝑓0𝜆1𝑑𝑒,𝑏21=𝑓1,𝑏22=𝑓3𝜆2𝑚𝑒+𝑓4,𝑏24𝑓=2𝜆3𝑓3𝜆2𝑐𝑒,𝑏32=𝑎12𝑓𝑒,𝑏33𝑎=2𝑓12𝑒𝐹0+𝑎12𝑐𝑒𝑎3,𝑏34=𝑎1𝜆1,𝑏42=𝑎11𝑚𝑒,𝑏44=𝑎0𝑎11𝑐𝑒,𝑏53=𝑎16𝑒1𝑒𝐸0,𝑏55=𝑎16𝑓0𝐸0.(16)

Equations (32)–(39) in matrix form yields as follows: 𝑑𝑐𝑓𝑚𝑒𝑑𝑐𝑓𝑚𝑒,=𝐁(17) where 𝐁 is: 𝑏𝐁=1100𝑏140𝑏21𝑏220𝑏2400𝑏32𝑏33𝑏3400𝑏420𝑏44000𝑏530𝑏55.(18) For stability analysis, we look at the eigenvalues of matrix 𝐁; for convenience, we rearrange our equations in the following form: 𝑚𝑑𝑐𝑓𝑒=𝑏440𝑏4200𝑏14𝑏11000𝑏24𝑏21𝑏2200𝑏340𝑏32𝑏330000𝑏53𝑏55𝑚𝑑𝑐𝑓𝑒.(19) We break 𝐁 into a 3-block and a 2-block as follows: 𝐁𝟏=𝑏440𝑏42𝑏14𝑏110𝑏24𝑏21𝑏22,𝐁𝟐=𝑏330𝑏53𝑏55.(20) Since det(𝐁𝜎𝐈)= det(𝐁𝟏𝜎𝐈) det(𝐁𝟐𝜎𝐈), we find the eigenvalues by looking at the eigenvalues of the 3-block, 𝐁𝟏, and the two block, 𝐁𝟐, separately. We also simplify by noting that with the equilibrium values found above, 𝑏44=0 and 𝑏14=0 s.t. det𝐁𝟏=𝜎𝐈𝜎0𝑏420𝑏11𝜎0𝑏24𝑏21𝑏22𝑏𝜎=𝜎11𝑏+𝜎22+𝜎+𝑏24𝑏42𝑏11𝑏+𝜎=11𝜎+𝜎2+𝑏22𝜎𝑏24𝑏42,(21) solving for the roots we get the following eigenvalues: 𝜎1=𝑏11,𝜎(22)2=𝑏22𝑏222+4𝑏42𝑏242,𝜎(23)3=𝑏22+𝑏222+4𝑏42𝑏242.(24)

The lower triangular 𝐁𝟐 gives us our final two eigenvalues: 𝜎4=𝑏33,𝜎(25)5=𝑏55.(26)

ODE stability requires real parts of the 𝜎1,,𝜎5 to be negative. In the next remark, stability criteria are formulated in terms of the parameters of the model.

Remark 4. Under the model assumptions we have 𝑏11<0𝑏22<0,𝑏42<0,𝑏55<0.(27) Therefore, 𝜎1<0, 𝜎2<0, and 𝜎5<0. Next, if 𝑏33=𝑎2𝑓12𝑒𝐹0+𝑎12𝑐𝑒𝑎3>0,(28) then 𝜎4<0. Finally, because 𝑏42<0, real part of 𝜎3 is negative if and only if 𝑏24𝑓=2𝜆3𝑓3𝜆2𝑐𝑒>0.(29) Assumptions in (28) and (29) have clear biological interpretation.
Condition 𝑏33>0 requires 𝑎2𝑓12𝑒𝐹0+𝑎12𝑐𝑒<𝑎3,(30) suggesting the need for the logistic growth of fibroblasts combined with the direct proliferation resulting from the presence of chemoattractants to be overcome by the death rate of fibroblasts.
Condition 𝑏24>0 requires 𝑓3𝜆2𝑐𝑒>𝑓2𝜆3,(31) suggesting that stability is aided when the percentage of regulatory macrophages out-weighs the percentage of inflammatory macrophages.
Note that from a mathematical point of view, conditions in the form of a strict inequalities imply a stronger property of the solution, namely asymptotic stability of the equilibrium. Lyapunov stability follows from the less restrictive condition with nonstrict inequalities.

4. ODE Linear Stability Implies PDE Linear Stability

Since the interior equilibrium solution represents the inflammatory state, one of the more biologically relevant questions is whether some modifications of conditions can cause the reactions to be away from the ill state and return to healthy state. Typically, the competition between diffusion and chemotaxis can aid the instability by creating spatial disturbance. One of the surprising findings for this system, however, is that if the equilibrium is stable by pure reactions, then it is stable for the whole reaction-diffusion-chemotactic system.

To start, we let 𝑣(𝑥,𝑡)=𝑒𝜎𝑡𝜙𝜇𝑛𝑢(𝑥)1,,𝑢5(32) to be a vector with unknown five components and function 𝜙𝑛(𝑥) to be the nth eigenfunction for Laplace equation with respect to Neumann boundary conditions: Δ𝜙𝑛(𝑥)=𝜇𝑛𝜙𝜇𝑛insidedomain,(33)𝜕𝜙𝜇𝑛𝜕𝑛=0ontheboundaryofthedomain.(34) Let us for simplicity assume that the domain is convex such that 𝜇𝑛0 for any 𝑛 is an eigenvalue for the eigenvalue problem, and 𝜙𝜇𝑛 is its corresponding eigenfunction. We will drop the subscripts 𝑛 in the text below. Substituting the function 𝑣(𝑥,𝑡) into equation one can get 𝜎𝑢1=𝐷𝑑𝜇𝑢1𝑏11𝑢1𝑏14𝑢4,𝜎𝑢2=𝐷𝑐𝜇𝑢2𝑏21𝑢1𝑏22𝑢2𝑏24𝑢4,𝜎𝑢3=𝐷𝑓𝜇𝑢3+𝜒𝑓𝜇𝑢2𝑏32𝑢2𝑏33𝑢3𝑏34𝑢4,𝜎𝑢4=𝐷𝑚𝜇𝑢4+𝜒𝑚𝜇𝑢2𝑏42𝑢2𝑏44𝑢4,𝜎𝑢5=𝐷𝑒𝜇𝑢5+𝜒𝑒1𝜇𝑢3+𝜒𝑒2𝜇𝑢2𝑏53𝑢3𝑏55𝑢5,(35) or 𝜎+𝐷𝑑𝜇+𝑏11𝑢1+𝑏14𝑢4𝑏=0,21𝑢1+𝜎+𝐷𝑐𝜇+𝑏22𝑢2+𝑏24𝑢4=0,𝜎+𝐷𝑓𝜇+𝑏33𝑢3𝜒𝑓𝜇𝑏32𝑢2+𝑏34𝑢4𝜒=0,𝑚𝜇𝑏42𝑢2+𝜎+𝐷𝑚𝜇+𝑏44𝑢4=0,𝜒𝑒2𝜇𝑢2𝜒𝑒1𝜇𝑏53𝑢3+𝜎+𝐷𝑒𝜇+𝑏55𝑢5=0.(36)

Then in matrix form it takes a form 𝐴(𝜎)𝑢=0,(37) with matrix 𝐴 defined as follows:𝜎+𝐷𝑑𝜇+𝑏1100𝑏140𝑏21𝜎+𝐷𝑐𝜇+𝑏220𝑏240𝜒0𝑓𝜇𝑏32𝜎+𝐷𝑓𝜇+𝑏33𝑏340𝜒0𝑚𝜇𝑏420𝜎+𝐷𝑚𝜇+𝑏4400𝜒𝑒2𝜒𝜇𝑒1𝜇𝑏530𝜎+𝐷𝑒𝜇+𝑏55.(38)

Below, we will show that if the real part of all eigenvalues of matrix 𝐁 is negative (corresponding ODE system is stable), then nontrivial solutions of (37) with parameter 𝜎 having negative real part exist. It is not difficult to see that the determinant of the matrix 𝐴 has aform as follows: 𝑃(𝜎)=𝜎+𝐷𝑒𝜇+𝑏55𝜎+𝐷𝑓𝜇+𝑏33𝐵det1.(39)

Here, 𝐵1 is a matrix associated to debris 𝑢1, chemotaxis 𝑢2, and macrophages 𝑢4 parameters only: 𝜎+𝐷𝑑𝜇+𝑏110𝑏14𝑏21𝜎+𝐷𝑐𝜇+𝑏22𝑏24𝜒0𝑚𝜇𝑏42𝜎+𝐷𝑚𝜇+𝑏44.(40)

Under the assumptions that the ODE part without diffusion is asymptotically stable, coefficients 𝑏55 and 𝑏33 should satisfy inequalities 𝑏44=𝑏14=0, 𝑏55>0 and 𝑏33<0.

We rearrange the matrix into a 𝑢4,𝑢1,𝑢2 order so that it is similar to the one addressed previously in the ODE stability analysis. Now, 𝐁det𝟏=+𝜎𝐈𝜎+𝐷𝑚𝜇0𝑏42𝜒𝑚𝜇0𝑏11+𝐷𝑑𝑏𝜇+𝜎024𝑏21𝑏22+𝐷𝑐=𝜇+𝜎𝜎+𝐷𝑚𝜇b11+𝐷𝑑𝑏𝜇+𝜎22+𝐷𝑐𝜇+𝜎𝑏24𝑏42𝜒𝑚𝜇𝑏11+𝐷𝑑=𝑏𝜇+𝜎11+𝐷𝑑𝜎𝜇+𝜎2+𝑏22+𝐷𝑐𝜇+𝐷𝑚𝜇𝜎+𝐷𝑚𝜇𝑏22+𝐷𝑐𝜇𝑏24𝑏42𝜒𝑚𝜇,(41) solving for the roots we get the following eigenvalues: 𝜎1=𝑏11𝐷𝑑𝜎𝜇,2=𝑏22+𝐷𝑐𝜇+𝐷𝑚𝜇𝑏22+𝐷𝑐𝜇+𝐷𝑚𝜇2+4𝜖2,𝜎(42)3=𝑏22+𝐷𝑐𝜇+𝐷𝑚𝜇+𝑏22𝐷𝑐𝜇+𝐷𝑚𝜇2+4𝜖2,(43) here 𝑏𝜖=42𝜒𝑚𝜇𝑏24𝐷𝑚𝜇𝑏22+𝐷𝑐𝜇.(44) The other two eigenvalues are 𝜎4=𝑏33𝜇𝐷𝑓,𝜎5=𝑏55𝜇𝐷𝑒.(45)

In the forthcoming remark, explicit representations for all possible 𝜎’s are explored for direct comparison between conditions of the stability of the linearized PDE (8) and ODE (15) systems.

Remark 5. Similarly to criteria for ODE the stability for PDE, requires that real parts of the all 𝜎's to be negative. Under the natural constraints on the parameters of our original model 𝑏11, 𝑏55, 𝑏22, and 𝑏42 (see Remark 4) we already have 𝜎1<0,𝜎4<0 and 𝜎5<0. Therefore, our criteria for PDE stability reduce to conditions as follows: 𝑏22+𝐷𝑐𝜇+𝐷𝑚𝜇>0,𝜖<0.(46) It is obvious to see that if both inequalities hold, then 𝜎2and𝜎3 are negative. Since stability of the ODE system forces 𝑏24>0 and 𝑏42<0, these two inequalities for PDE stability hold for any 𝜒𝑚>0, 𝐷𝑚>0,𝐷𝑐>0, 𝜇>0.
From the above arguments it follows that if the ODE system is stable, then 𝑣(𝑥,𝑡) are converging to zero as time goes to infinity for any eigenfunction 𝜙𝑛. Therefore, since the 𝜙𝑛(𝑥) is complete in 𝐿2 space, one can conclude that the stability of the linearized PDE system (8) in 𝐿2 space follows from the stability of the ODE system (15).
As expected, the ODE stability and PDE stability are different. Let 𝐷𝑚=𝐷𝑐𝜒𝑚=0, then the first 5 eigenvalues of the PDE and ODE have the same sign. By definition of our original model 𝜎1, 𝜎2, and 𝜎5 are all negative. Assume 𝑏33>0 (in some sense reactive terms has stabilizing effect, with respect to 𝑈𝑒), then 𝜎4<0. However, now if one lets 𝑓2𝜆3>𝑓3𝜆2𝑐𝑒, which means that inflammatory macrophages dominate the regulatory macrophages, then 𝑏24<0 causing 𝜎3>0, and consequently the ODE system (15) is unstable. For the same set of the coefficients 𝑏's and given 𝜇>0, it is not difficult to find sufficient condition on 𝐷𝑚, 𝐷𝑐, and 𝜒𝑚 such that 𝜎3<0, which guarantee stability of the equilibrium state 𝑈𝑒. For example, any set with the same coefficients 𝑏’s with 𝐷𝑚𝐷𝑐>𝑏42𝜒𝜇𝑏24/𝜇(47) will have a real part of the 𝜎3<0 and consequently the solution of the corresponding IBVP with initial function to be 𝜙𝜇(𝑥)(𝑢1,,𝑢5) will be vanishing at time infinity. Condition (47) contains the following pattern in the biological interpretation. Assume that inflammatory macrophages dominate the regulatory macrophages and are characterized by the coefficient 𝑏24=(𝑓2𝜆3𝑓3𝜆2𝑐𝑒)<0. Then for any given value 𝑏24 if mobility of the macrophages and diffusion of the chemoattractant is high enough in comparison to the coefficient 𝑏24, then 𝑈𝑒 is stable for the class of perturbation which corresponds to eigenfunction 𝜙𝜇. In less strict wording, the system can be cleaned from dead cells by high “mobility/diffusivity” of the macrophages with respect to chemoattractant. This indicates vital impact of the key parameters 𝐷𝑚, 𝐷𝑐, and 𝜒𝑚 on “inflammatory” behavior both in space and in time of the system perturbed from equilibrium.
Obtained conclusion depends on 𝜇 and can be applied only if initial data is proportional to 𝜙𝜇. If in the Fourier extension of the initial data all coefficients are nonzero, then the sufficient condition for stability is the same as for ODE system.
In the next section, we will analyze conditional stability of the IBVP for (8) under assumption that 𝑣(𝑥,𝑡0) has zero average:𝑣(𝑥,𝑡0)𝑑𝑥=0. We will derive conditions on the coefficient of the system (8) such that the 𝐿2 norm of the solution is bounded by the 𝐿2 norm of the initial data, or it converges to zero at time infinity depending on the conditions on coefficients. Those conditions will depend only on coefficients of the model and Poincare constant (𝐶𝑝), which in turn depends only on the geometry of the domain. We will also show that there exists a specific initial distribution such that the corresponding IBVP solution is vanishing at time infinity while the corresponding solution of the ODE is unbounded at time infinity.

5. Stability of Equilibrium in the Linearized PDE System without ODE Stability

Let us rewrite the linearized system (8) as follows: 𝜕𝑑𝜕𝑡=𝐷𝑑2𝑑𝑓0𝜆1𝑚𝑒𝑑𝑏14𝜆𝑚,(48)1𝜕𝑐𝜕𝑡=𝜆1𝐷𝑐2𝑐𝑏21𝜆1𝑑𝑏22𝜆1𝑐𝑏24𝜆1𝜆𝑚,(49)1𝜕𝑓𝜕𝑡=𝐷𝑓𝜆12𝑓𝜒𝑓𝜆12𝑐𝑏32𝜆1𝑐𝑏33𝜆1𝑓𝑏34𝜆1𝜆𝑚,(50)1𝜕𝑚𝜕𝑡=𝐷𝑚𝜆12𝑚𝜒𝑚𝜆12𝑐𝑏42𝜆1𝑐𝑏44𝜆1𝜆𝑚,(51)1𝜕𝑒𝜕𝑡=𝐷𝑒𝜆12𝑒𝜒𝑒1𝜆12𝑓𝜒𝑒2𝜆12𝑐𝑏53𝜆1𝑓𝑏55𝜆1𝑒.(52)

Next multiplying equations (48) by 𝑑, (49) by 𝑐, (50) by 𝑓, (51) by 𝑚, and (52) by 𝑒 correspondingly and integrating by parts, one can easily get 12𝜕𝑑𝜕𝑡2𝐷=𝑑(𝑑)2𝑓0𝜆1𝑚𝑒𝑑2𝑏14𝜆𝑚𝑑,(53)12𝜕𝑐𝜕𝑡2𝜆=1𝐷𝑐(𝑐)2𝑏21𝜆1𝑑𝑐𝑏22𝜆1𝑐2𝑏24𝜆1𝜆𝑚𝑐,(54)12𝜕𝑓𝜕𝑡2𝐷=𝑓𝜆1(𝑓)2+Φ(𝑐,𝑓)𝑏32𝜆1𝑐𝑓𝑏33𝜆1𝑓2𝑏34𝜆1𝜆𝑚𝑓,(55)12𝜕𝑚𝜕𝑡2𝐷=𝑚𝜆1(𝑚)2+Φ(𝑐,𝑚)𝑏42𝜆1𝑐𝑚𝑏44𝜆1𝑚2,𝜆(56)12𝜕𝑒𝜕𝑡2𝐷=𝑒𝜆1(𝑒)2+Φ(𝑓,𝑒)+Φ(𝑐,𝑒)𝑏53𝜆1𝑓𝑒𝑏55𝜆1𝑒2.(57) Here, Φ(𝑓,𝑒)=𝜒𝑒1𝜆1𝑓𝑒, Φ(𝑐,𝑓)=𝜒𝑓𝜆1𝑐𝑓, Φ(𝑐,𝑚)=𝜒𝑚𝜆1𝑐𝑚, Φ(𝑐,𝑒)=𝜒𝑒2𝜆1𝑐𝑒.

Adding LHS and RHS of the equations above: (53)+(54)+(55)+(56)+(57) and applying the Poincare inequality to the terms (𝑢)2𝑑𝑥 such that for 𝐶𝑝=𝐶𝑝(Ω)>0, 𝐶𝑝Ω𝑢2𝑑𝑥Ω(𝑢)2𝑑𝑥+Ω𝑢𝑑𝑥2,(58) one can easily get 12𝑑2+𝜆1𝑐2+𝑓2+𝑚2+𝑒2𝑡[][]𝑑𝐵(𝑑,𝑚)+𝐵(𝑐,𝑑)+𝐵(𝑐,𝑚)+𝐵(𝑐,𝑓)+𝐵(𝑓,𝑚)+𝐵(𝑓,𝑒)𝐵(𝑐,𝑚)+𝐵(𝑐,𝑓)+𝐵(𝑒,𝑐)+𝐵(𝑒,𝑓)+𝐶2+𝑐2+𝑓2+𝑚2+𝑒2,(59) where the bilinear forms are 𝐵(𝑑,𝑚)=0,(60)𝐵(𝑐,𝑑)=𝜆116𝐷𝑐𝐶𝑝+𝑏22𝑐2+𝑏21𝑓𝑑𝑐+0+𝐷𝑑𝐶𝑝𝑑2,(61)𝐵(𝑐,𝑚)=𝜆116𝐷𝑐𝐶𝑝+𝑏22𝑐2+𝑏2,4+1𝑐𝑚3𝐷𝑚𝐶𝑝+𝑏44𝑚2(62)𝐵(𝑓,𝑚)=𝜆114𝐷𝑓𝐶𝑝+𝑏33𝑓2+𝑏341𝑐𝑓+3𝐷𝑚𝐶𝑝𝑚2,(63)𝐵(𝑐,𝑓)=𝜆116𝐷𝑐𝐶𝑝𝑐2+𝑏321𝑐𝑓+3𝐷𝑓𝐶𝑝𝑓2𝐵(64)(𝑓,𝑒)=0,(65)𝐵(𝑐,𝑚)=𝜆116𝐷𝑐(𝑐)2𝜒𝑚1𝑐𝑚+3𝐷𝑚(𝑚)2,(66)𝐵(𝑐,𝑓)=𝜆116𝐷𝑐(𝑐)2𝜒𝑓1𝑐𝑓+4𝐷𝑓(𝑓)2,(67)𝐵(𝑓,𝑒)=𝜆114𝐷𝑓(𝑓)2𝜒𝑒21𝑓𝑒+2𝐷𝑒(𝑒)2,(68)𝐵(𝑐,𝑒)=𝜆116𝐷𝑐(𝑐)2𝜒𝑒11𝑐𝑒+2𝐷𝑒(𝑒)2.(69)

Imposing assumptions that all bilinear forms above are positively defined, one can then conclude that the system is stable. Below, we formulate a sufficient condition for the solution to be stable in 𝐿2 space. The formulation of the assumptions is presented in terms of the parameters of the original system where biological meanings are more evident.

Condition 1. If 16𝐷𝑐𝐶𝑝+𝑓3𝜆2𝑚𝑒+𝑓4𝑓0+𝐷𝑑𝐶𝑝1/212𝑓1,(70) then 𝐵(𝑐,𝑑)0.

Condition 2. If 14𝐷𝑓𝐶𝑝𝑎2𝑓12𝑒𝐹0+𝑎12𝑐𝑒𝑎313𝐷𝑚𝐶𝑝1/212𝑎1𝜆1,(71) then 𝐵(𝑓,𝑚)0.

Taking into account actual values for equilibriums 𝑐𝑒 and 𝑓𝑒 of the inflammatory equilibrium, one can reduce (71) to an inequality, which is easy to interpret.

Namely, assume that 14𝐷𝑓𝐷𝑚𝐶𝑝+𝐴𝐷𝑚1/212𝑎1𝜆1,(72) then 𝐵(𝑓,𝑚)0. From the previously mentioned, 𝑎𝐴=2𝑎3+𝑎12𝑎0𝑎112𝑎+42𝐹0𝑎1𝜆1𝑓4𝑓0𝜆1𝑎0𝑎11𝑓1𝑓0𝜆3𝑓0𝜆1𝑓2𝜆3𝑎11𝑓3𝑎0𝜆2.(73) Due to the assumption (12), parameter 𝐴 is well defined for all values of the coefficients of the original model. Biological meaning of constraint (12) was explained in Remark 3, and it is necessary for the existence of the inflammatory equilibrium. What we want to point out here is that for any set of the parameters there exist large enough diffusive constants 𝐷𝑚 and 𝐷𝑓 that inequality (72) holds, and consequently bilinear form 𝐵(𝑓,𝑚)0.

Condition 3. If 16𝐷𝑐𝐶𝑝+𝑓3𝜆2𝑚𝑒+𝑓413𝐷𝑚𝐶𝑝+𝑎11𝑚𝑒𝑎0212𝑓2𝜆3𝑓3𝜆2𝑐𝑒,(74) then 𝐵(𝑐,𝑚)0. For well posedness of the RHS in inequality (74), assume that 13𝐷𝑚𝐶𝑝+𝑎11𝑚𝑒𝑎0=13𝐷𝑚𝐶𝑝+𝑎11𝑓4𝑓0𝜆1𝑎0𝑎11𝑓1𝑓0𝜆3𝑓0𝜆1𝑓2𝜆3𝑎11𝑓3𝑎0𝜆2𝑎00.(75) We rewrite the above inequality in terms of the parameters of the original model to point out that for any given set of the parameters, there exists big enough coefficient 𝐷𝑚, characterizing macrophages mobility, such that inequality (74) holds.

Condition 4. If 16𝐷𝑐𝐶𝑝14𝐷𝑓𝐶𝑝1/212𝑎12𝑓𝑒,(76) then 𝐵(𝑐,𝑓)0.

Condition 5. If 16𝐷𝑐13𝐷𝑚1/212𝜒𝑚,(77) then 𝐵(𝑐,𝑚)0.

Condition 6. If 16𝐷𝑐14𝐷𝑓1/212𝜒𝑓,(78) then 𝐵(𝑐,𝑓)0.

Condition 7. If 14𝐷𝑓12𝐷𝑒1/212𝜒𝑒2,(79) then 𝐷(𝑓,𝑒)0.

Condition 8. If 16𝐷𝑐12𝐷𝑒1/212𝜒𝑒1,(80) then 𝐷(𝑐,𝑒)0.

We now assume that for all five components 𝑑(𝑥,0),𝑐(𝑥,0),𝑓(𝑥,0),𝑚(𝑥,0), and 𝑓(𝑥,0) is equal to 0 (initial data are orthogonal to 1. Then due to no-flux Neumann condition on the boundary for all times, 𝑈𝑑=𝑈𝑐=𝑈𝑓=𝑈𝑚=𝑈𝑒=0.(81) Therefore, the above Conditions (18) guarantee Lyapunov stability of the linearized system. If further for the same class of initial data we in addition assume strict inequalities in (70)–(76), then system will be asymptotically stable, and 𝐿2 norm of the solution will exponentially converge to zero as time goes to infinity.

Here, we do not assume the ODE stability conditions of the equilibrium in this section. It will be easy to construct a specially inhomogeneous solution of the initial-boundary value problem (IBVP) so that the solution of corresponding ODE for 𝑉=𝑣(𝑥,𝑡)𝑑𝑥 is identically zero, where the PDE solutions can be either stable or unstable by adjusting certain parameters. Indeed, let the domain be a segment [0,𝜋] and as in (32), with 𝜙=cos𝑥. Then, in as Section 4, in order for 𝑣(𝑥,𝑡) to be a solution of corresponding IBVP it is necessary and sufficient that 𝜎 to be a root of the characteristic polynomial equation 𝑃(𝜎)=0 in (39). To see Conditions (18) are essential, we show an example of the system with: (1) Conditions (18) are all met, and (2)𝑃(𝜎) has a positive root in (39). For selected domain, assume Poincare constant 𝐶𝑝=1. Assume that all coefficients are such that inequalities in all constraints except inequities in constraints Conditions 3 and 5 are satisfied. Let 𝑏224/5𝐷𝑐, 𝑎11𝑚𝑒𝑎0, and 0>𝑏24(𝐷𝑐𝐷𝑚/20)1/2. Obviously for these set of the parameter Condition 3 satisfies. Then if 𝐷𝑐𝐷𝑚/60𝜒𝑚 then Condition 5 holds and consequently 𝑣(𝑥,𝑡)0 as 𝑡. Furthermore, it is not difficult to see that if 𝑏22=4/5𝐷𝑐, and 𝑏24=(𝐷𝑐𝐷𝑚/20)1/2 then in (42) is positive provided 𝜒𝑚𝑏42𝐷𝑐𝐷𝑚201/295𝐷𝑚𝐷𝑐>0.(82) Inequality in (82) holds if 𝜒𝑚>102𝐷𝑐𝐷𝑚(83) Consequently, Condition (83) holds then 𝑣(𝑥,𝑡)𝐿2 as 𝑡. Comparing stability in (77) and instability in (83), the conditions are optimal unto discrepancy in coefficients. In the next remark, we want to highlight the impact of the diffusive parameter and chemotactic coefficients on the stability of the inflammatory equilibrium 𝑈𝑒.

Remark 6. In all above eight conditions inequalities hold for big enough values of diffusive coefficients 𝐷’s. This highlights the importance of the spatial distribution of the perturbation for the equilibrium. The major meaning of these condition is that for any set of the parameters if diffusivity coefficients are big enough then 𝑈𝑒 is stable. Another key parameter, which characterizes the behavior of the spatial distribution of the system is the chemotactic coefficient. From the example above, one can see that if the chemotactic sensitivity coefficient 𝜒 is relatively bigger than the diffusivity characteristic of the process, then 𝑈𝑒 is unstable. At the same time if it is relatively smaller, as in inequalities (77)–(80), then the inflammatory equilibrium is stable.

6. Conclusion and Discussion

To quantitatively study the processes governing inflammatory and fibrotic reactions against foreign bodies, we have built a mathematical model with the capability to predict the trends of macrophage migration, ECM production, and chemoattractant regulation by macrophages in these fibrotic reactions. The initiations of reactions are digestions of debris which are the natural responses of the immune system to damaged cells and tissues due to the implantation process. Our model is built based principally on biochemical mechanisms, and it has served its purpose in providing trends of reactions. The model is expressed by a system of partial differential equations with no flux boundary conditions.

We have considered an equilibrium state of the system and its stability conditions. We have provided a mathematical proof that when this equilibrium is stable in the corresponding ODEs, then it is also stable for the full system in 𝐿2(Ω). However, a system with a parameter set can be conditionally stable in the PDE sense when its ODE system is not necessarily stable. We provided some exclusive conditions for this to happen. These conditions correspond with feasible biological conditions, where the percentage of regulatory macrophages dominates that of the inflammatory macrophages.

We mention here that the system has infinitely many equilibria, all except for one containing at least one free parameter in it. The one under discussion here is called the interior equilibrium as it has 5 nonzero components. This particular equilibrium corresponds to an inflammatory state of the healing process, whose instability is an indicator of three possible dynamics: (1) best case scenario, returning to the healthy state; (2) uncertain development, transition to another “abnormal” equilibrium; (3) acute inflammatory response (worst case scenario), perturbations tend to infinity.

Our main mathematical result indicates that the inflammatory state’s stability mainly depends on the reaction dynamics and even that small spatial diffusion and big chemotaxis cannot destabilize the equilibrium which is stable in the reaction-only system. However, if the equilibrium is unstable by its reaction-only system, then spatial diffusion over chemotactic effects can help to stabilize the equilibrium if the initial perturbation is subjected to specific constraints. We did not discuss other equilibrium states due to the length of the paper, but there is no mathematical difficultly in accomplishing these tasks.

Acknowledgments

This work is supported by the National Institutes of Health Grant no. 1R01EB007271-01A2. The research of this paper was supported by the NSF grant DMS-0908177.

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