On the computational modeling of the innate immune system
 Alexandre Bittencourt Pigozzo^{1}Email author,
 Gilson Costa Macedo^{2},
 Rodrigo Weber dos Santos^{1} and
 Marcelo Lobosco^{1}
https://doi.org/10.1186/1471210514S6S7
© Pigozzo et al.; licensee BioMed Central Ltd. 2013
Published: 17 April 2013
Abstract
In recent years, there has been an increasing interest in the mathematical and computational modeling of the human immune system (HIS). Computational models of HIS dynamics may contribute to a better understanding of the relationship between complex phenomena and immune response; in addition, computational models will support the development of new drugs and therapies for different diseases. However, modeling the HIS is an extremely difficult task that demands a huge amount of work to be performed by multidisciplinary teams. In this study, our objective is to model the spatiotemporal dynamics of representative cells and molecules of the HIS during an immune response after the injection of lipopolysaccharide (LPS) into a section of tissue. LPS constitutes the cellular wall of Gramnegative bacteria, and it is a highly immunogenic molecule, which means that it has a remarkable capacity to elicit strong immune responses. We present a descriptive, mechanistic and deterministic model that is based on partial differential equations (PDE). Therefore, this model enables the understanding of how the different complex phenomena interact with structures and elements during an immune response. In addition, the model's parameters reflect physiological features of the system, which makes the model appropriate for general use.
Introduction
The human immune system (HIS) consists of a wide and complex network of cells, tissues and organs. The HIS plays a crucial role in defending the body against disease. To achieve this goal, the HIS identifies and kills a wide range of external pathogens such as viruses and bacteria as well as the body's own abnormally behaving cells. The HIS is also responsible for removing dead cells and regenerating some of the body's structures [1].
A complete understanding of the HIS is therefore essential. However, its complexity and the intense interactions among several components on various different levels make this task extremely complex [2, 3]. However, we may better understand some properties of the HIS by applying a computational model, which allows researchers to test a large number of hypotheses in a short period of time [2, 3]. In the future, we can envision a computer program that will simulate the entire HIS, allowing scientists to develop and test new drugs against various diseases virtually, thus reducing the number of animals used in experiments.
In this study, our work aims to implement and simulate a mathematical model of the HIS. Due to the complexity of this task, our focus is to reproduce the spatiotemporal dynamics of an immune response to the injection of lipopolysaccharides (LPS) into a small section of tissue. To reproduce these dynamics, we introduce a mathematical model composed of a system of partial differential equations (PDEs) that extends our previous model [2] and defines the dynamics of representative cells and molecules of the HIS during the immune response to LPS. The model presented is descriptive, mechanistic and deterministic; therefore, it enables the understanding of how different complex phenomena, structures and elements interact during an immune response. In addition, the model's parameters reflect the physiological features of the system, making the model appropriate for general use.
The remainder of the paper is organized as follows. First, the necessary biological background is presented. Next, related works are briefly discussed. This exposition is followed by a description of both the mathematical model proposed in this work and its computational implementation. Then simulation results obtained from the proposed model are discussed, and finally, our conclusions and plans for future work are presented.
Biological background
"Human body surfaces are protected by epithelia, which provide a physical barrier between internal and external environments. Epithelia make up the skin and lining of the tubular structures of the body (i.e., the gastrointestinal, respiratory and genitourinary tracts), and they form an effective barrier against the external environment. At the same time, epithelia can be crossed or settled by pathogens, causing infections. After crossing the epithelium, the pathogens encounter cells and molecules of the innate immune system, which immediately develop a response" [4].
The body's initial response to an acute biological stress, such as a bacterial infection, is an acute inflammatory response [4]. The strategy of the HIS is to keep some resident macrophages on guard in tissues to look for any signal of infection. When they find such a signal, the macrophages alert neutrophils (also known as polymorphonuclear neutrophils (PMNs)) that their help is required. Because of this communication, the cooperation between macrophages and neutrophils is essential to mount an effective defense against disease. Without macrophages to herd neutrophils toward the location of infection, the latter would circulate indefinitely in the blood vessels, impairing the control of systemic infections [1].
The inflammation of an infectious tissue has many benefits for the control of the infection. In addition to recruiting cells and molecules of innate immunity from blood vessels to the location of the infected tissue, inflammation increases the lymph flux, which contains microorganisms and cells that carry antigens to neighboring lymphoid tissues; there, these cells will present the antigens to the lymphocytes and initiate the adaptive response. Once the adaptive response has been activated, the inflammation also shuttles the effector cells of the adaptive immune system to the location of infection [4].
A component of the cellular wall of Gramnegative bacteria, such as LPS, can trigger an inflammatory response through the interaction with receptors on the surface of some cells [1]. For example, the macrophages that reside in tissue recognize a bacterium through the binding of TLR4 (Tolllike receptor 4) with LPS. When receptors on the surface of macrophages bind to LPS, the macrophage starts to phagocytose, internally weakening the bacterium and secreting proteins known as cytokines and chemokines, as well as other molecules.
In many inflammatory conditions, neutrophils dominate the initial influx of leukocytes into the inflamed tissue. The first wave of extravasated neutrophils is soon replaced by a second wave of monocytes [1]. A study presented initial proofs of the existence of this sequence of events [5]. In that study, neutrophils dominated the leukocyte extravasation three hours after the beginning of the inflammation, and some time later, the extravasated cells were predominantly monocytes [5].
The resolution of the inflammatory response is a complex process that includes the production of antiinflammatory mediators and the apoptosis (or programmed death) of effector cells of the HIS, such as neutrophils [6]. Antiinflammatory cytokines form a set of immunoregulatory molecules that control the inflammatory response. These cytokines work together with specific inhibitors and cytokines' soluble receptors to regulate the immune response [6]. A previous work [6] demonstrated the participation of cytokines in inflammatory states. Primary antiinflammatory cytokines include the antagonist receptor of IL1 (Interleukin 1) in addition to IL4, IL6, IL10, IL11 and IL13 [6]. Specifically, IL10 is a strong inhibitor of many proinflammatory cytokines [7], including IL8 and TNFα (tumor necrosis factor α), which are produced both by monocytes [8] and by neutrophils [9, 10].
Apoptotic cells maintain membrane integrity for a small period of time and therefore need to be quickly removed to prevent a secondary necrosis and the consequent release of cytotoxic molecules, which cause inflammation and tissue damage [11]. As a consequence of the phagocytosis of apoptotic cells by macrophages or dendritic cells, these phagocytic cells produce antiinflammatory cytokines. For example, macrophages secrete TGFβ (transforming growth factor β), which prevents the release of proinflammatory cytokines induced by LPS [12]. Additionally, the binding of apoptotic cells to macrophage receptor CD36 (cluster of differentiation 36) inhibits the production of proinflammatory cytokines such as TNFα, IL1β and IL12; this binding also increases the secretion of TGFβ and IL10 [13].
Related work
This section presents and discusses other models found in the literature to model the innate HIS. Essentially, two distinct approaches are used: ordinary differential equations (ODEs) and partial differential equations (PDEs).
Models based on ODEs
The authors of [14] presented a model of inflammation that is based on ODEs and considers three types of cells/molecules: the pathogen and two inflammatory mediators. This model was able to reproduce some experimental results depending on the values used for initial conditions and parameters. The authors described the results of the sensitivity analysis, which suggests some therapeutic strategies. Their work was then extended [15] to investigate the influence of time on an antiinflammatory response. The mathematical model presented in [15] consists of a system of ODEs with four equations that model: a) the pathogen; b) the active phagocytes; c) tissue damage; and d) antiinflammatory mediators. The source term of the phagocytes, in other words, a term that models the entry of new phagocytes into the infected tissue, is a function that depends on a) the concentration of phagocytes; b) the concentration of pathogens; and c) tissue damage. This term models the different interactions that phagocytes can undergo during an immune response, whether the interactions are direct or mediated by cytokines. In the interaction mediated by cytokines, they consider only the implicit presence of cytokines. For example, in an immune response, the interaction of phagocytes with tissue is mediated by proinflammatory cytokines produced by infected epithelial tissue cells, and this relationship is modeled directly in the source term of the phagocytes. This representation contrasts with the model proposed in the current work, where cytokines and all their interactions are explicitly represented.
A new adaptation of the first model [14] was proposed to simulate many scenarios involving repeated doses of endotoxin [16]. This work applied results obtained through experiments using mice to guide in silico experiments seeking to reproduce these results qualitatively. The mathematical model represents the key aspects of an acute inflammatory response, specifically when repeated doses of endotoxin are administered. This model replaces the pathogen equation proposed in the authors' previous work [15] with an equation incorporating the endotoxin. In their simulations, they observed that the timing and magnitude of endotoxin doses, as well as the dynamics between pro and antiinflammatory mediators, are key to distinguishing between potentiation and tolerance phenomena [16]. The authors also argued that their model, although simplified, nevertheless incorporates sufficiently complex dynamics to qualitatively reproduce a set of experimental results associated with different endotoxin administrations in mice.
One final work [17] developed a more complete system of ODEs that models acute inflammation. This model includes macrophages, neutrophils, dendritic cells, TH1 cells, blood pressure, tissue trauma, effector elements such as iNOS, ${\mathsf{\text{NO}}}_{\mathsf{\text{2}}}^{}$ and ${\mathsf{\text{NO}}}_{3}^{}$, proinflammatory and antiinflammatory cytokines, and coagulation factors. In this model, as well as our own (described in detail in the next section), neutrophils and macrophages are directly activated by LPS. Moreover, activation also occurs indirectly by way of various stimuli consistently elicited after a trauma or hemorrhage. However, the model proposed by [17] does not explicitly include initial events of inflammation such as mast cell degranulation and complement activation, although these factors were incorporated implicitly into cytokine and endotoxin dynamics. The model also includes antiinflammatory cytokines such as IL10 and TGF β, in addition to soluble receptors for proinflammatory cytokines. The authors argued that their model proved useful in simulating the inflammatory response induced in mice by endotoxin, trauma and surgery or surgical bleeding, as it can predict levels of TNF, IL10, IL6 and reactive products of NO (${\mathsf{\text{NO}}}_{\mathsf{\text{2}}}^{}$ and ${\mathsf{\text{NO}}}_{3}^{}$) to some extent.
Models based on PDEs
The model proposed by Su et al [18] uses a system of PDEs to represent the spatial dynamics of the innate and adaptive immune systems. It considers the simplest form of antigens, the molecular constituents of pathogen patterns, taking into account all the basic factors of an immune response: antigens, cells of the immune system, cytokines and chemokines. This model captures the following stages of immune response: recognition, initiation, effector response and resolution of infection or change to a new steady state. Accordingly, it can reproduce important phenomena such as a) the temporal order of cell arrival at the site of infection; b) antigen presentation by dendritic cells, macrophages and the involvement of regulatory T cells (Treg) in the resolution of the immune response; c) the production of proinflammatory and antiinflammatory cytokines; and d) chemotaxis. This model has formed the basis for the development of our work.
Mathematical model
The complete modeling of the HIS demands that a huge amount of work be performed by a large multidisciplinary team. In this work, we focus on a specific task: the development of a mathematical model of the innate immune response to the injection of LPS in a section of tissue, as well as such a model's computational implementation. One motivation for developing a model of the innate immune system is the fact that few such models are available in the literature; the majority of available models solely focus on the adaptive immune system. Another reason in favor of modeling the innate immune system is that many diseases result from the malfunction of the innate immune system; for these diseases, our proposed model could contribute to the definition of therapeutic strategies. In addition, a better comprehension of the inner workings of the separate parts composing the innate immune system is fundamental to a better understanding of immune response as a whole, as the innate immune system is responsible for both initiating the immune response and triggering the adaptive immune system.
Our objective is to develop a parameterized mathematical model of the human innate immune system that simulates the immune response occurring in a generic tissue. To achieve this goal, we first build a model of the immune response to LPS. We have chosen to use LPS because it is the major component of the outer membrane of Gramnegative bacteria, acting as an endotoxin substance that elicits strong immune responses; thus, it represents a vast number of inflammatory diseases. However, our proposed model is generic in the sense that it can be easily adapted to specific pathogens and distinct types of tissue through the adjustment of its parameters.
The mathematical model simulates the temporal and spatial behavior of lipopolysaccharide (LPS), macrophages, neutrophils (N), apoptotic neutrophils (ND), proinflammatory cytokines (CH), antiinflammatory cytokine (AC) and protein granules (G). Macrophages are present in two states of readiness: resting (RM) and hyperactivated (AM). The different subsets of protein granules [19] released by neutrophils during their extravasation from blood vessels to the tissues are represented by a unique variable. Additionally, we must stress that the equations modeling pro and antiinflammatory cytokines are generic in the sense that they model the role of distinct cytokines taking part in the inflammatory process. Equation parameters can be adjusted to model the role of a specific pro or antiinflammatory cytokine.
The main characteristics of the proposed model are:

Macrophages and neutrophils cooperate to mount a more effective and intense response against the LPS;

The endothelium's permeability may vary with time and space and also depends on the local concentration of proinflammatory cytokine and protein granules, as depicted by Figure 1;

Active macrophages regulate immune responses through the production of antiinflammatory cytokines and the phagocytosis of apoptotic neutrophils;

Antiinflammatory cytokines perform a key role in the control of the inflammatory response, avoiding a state of persistent inflammation after the complete elimination of LPS.
The calculation of RM_{ P } involves the following parameters: a) ${P}_{RM}^{max}$, the maximum endothelium permeability induced by the proinflammatory cytokine; b) ${P}_{RM}^{min}$, the minimum endothelium permeability induced by the proinflammatory cytokine; and c) keqch, the number of proinflammatory cytokines that exert 50% of the maximum effect on permeability.
RM_{ Q } denotes the increase in endothelium permeability induced by protein granules, and its calculation is similar to that of RM_{ P }, except for the parameters involved: ${Q}_{RM}^{max}$, ${Q}_{RM}^{min}$ and keq_g. source_{ RM } denotes the source term of macrophages, which is related to the number of monocytes that will enter into the tissue from the blood vessels. This number depends on the endothelium permeability RM_{ P } + RM_{ Q } and on the number of monocytes appearing in the blood (M^{ max }).
μ_{ RM } RM denotes resting macrophage apoptosis, where μ_{ RM } is the apoptosis rate. RM_{ activation }, as explained above, models the activation of resting macrophages and denotes the number of resting macrophages that are becoming active. The term D_{ RM }ΔRM denotes the resting macrophage diffusion, where D_{ RM } is the diffusion coefficient. ∇.(χ_{ RM }RM∇CH) denotes the resting macrophage chemotaxis, where χ_{ RM } is the chemotaxis rate.
Above, μ_{ AM }AM, D_{ AM }ΔAM, and ∇.(χ_{ AM }AM∇CH) denote the active macrophage apoptosis, diffusion, and chemotaxis, respectively, whereas μ_{ AM }, D_{ AM }, and χ_{ AM } are the apoptosis rate, diffusion coefficient, and chemotaxis rate, respectively.
In this equation, μ_{ CH }CH denotes the proinflammatory cytokine decay, where μ_{ CH } is the decay rate. β_{CHN}.N denotes the proinflammatory cytokine production by the neutrophils, where β_{CHN}is the production rate. β_{CHAM}.AM denotes the proinflammatory cytokine production by active macrophages, where β_{ CHAM } is the production rate. The saturation of cytokine production by active macrophages is calculated by the equation $\left(1\frac{CH}{chInf}\right)$, where chInf is an estimate of the maximum quantity of proinflammatory cytokine supported by the tissue. The production of proinflammatory cytokine decreases when antiinflammatory cytokine acts on the producing cells. This influence of antiinflammatory cytokine is denoted by the expression 1/(1 + θ_{ AC }.AC). D_{ CH }ΔCH models proinflammatory cytokine diffusion, where D_{ CH } is the diffusion coefficient.
In this equation, P_{ N } denotes the increase in endothelium permeability and its effects on neutrophil extravasation. In the top equation, ${P}_{N}^{max}$ is the maximum endothelium permeability induced by proinflammatory cytokines, ${P}_{N}^{min}$ is the minimum endothelium permeability induced by proinflammatory cytokines and keqch is the number of proinflammatory cytokines that exert 50% of the maximum effect on endothelium permeability.
Here, μ_{ N }N denotes neutrophil apoptosis, where μ_{ N } is the rate of apoptosis. λ_{ LPSN } LPS.N denotes the neutrophil apoptosis induced by phagocytosis, where λ_{ LPSN } represents the rate of this induced apoptosis. The term D_{ N }ΔN denotes neutrophil diffusion, where D_{ N } is the diffusion coefficient. source_{ N } represents the source term of neutrophil, i.e., the number of neutrophils entering the tissue from the blood vessels. This number depends on the endothelium permeability (P_{ N }) and on the number of neutrophils in the blood (N^{ max }). The term ∇.(χ_{ N }N∇CH) denotes the chemotaxis process of the neutrophils, where χ_{ N } represents the chemotaxis rate.
Here, note that μ_{ N }N and λ_{ LPSN }LPS.N were defined previously, whereas λ_{ NDAM }ND.AM denotes the phagocytosis of the apoptotic neutrophil carried out by active macrophages, and λ_{ NDAM } is the rate of this phagocytosis. D_{ ND }ΔND models the apoptotic neutrophil diffusion, where D_{ ND } is the diffusion coefficient.
μ_{ G }G models the decay of the granules, where μ_{ G } is the decay rate. α_{ GN }.source_{ N } denotes the production of protein granules by neutrophils extravasating from the blood into inflamed tissue, where α_{ GN } is a dimensionless constant. The saturation of protein granule production is calculated by the expression $\left(1\frac{G}{gInf}\right)$, where gInf is the maximum number of protein granules. D_{ G }ΔG models protein granule diffusion, where D_{ G } is the diffusion coefficient.
In this equation, μ_{ AC }AC denotes the antiinflammatory cytokine decay, where μ_{ AC } represents the decay rate. β_{ RMND }.RM.ND denotes the antiinflammatory cytokine production by the resting macrophages in the presence of apoptotic neutrophils, where β_{ RMND } is the rate of this production. α_{ ACAM }.AM denotes the antiinflammatory cytokine production by active macrophages, where this production has rate α_{ ACAM } and saturation $\left(1\frac{AC}{acInf}\right)$, where acInf is the maximum number of antiinflammatory cytokines in the tissue. D_{ AC }ΔAC models the antiinflammatory cytokine diffusion, where D_{ AC } is the diffusion coefficient.
Implementation
The numerical method that we have applied to our mathematical model is presented in our previous work [2].
We executed some convergence tests to test the implementation of our numerical method. In short, we assumed that the correct solution derived from the results of a very refined mesh, where the refinement was in terms of time (dt = 10^{6}day) and space (deltaX = 0.1mm). To show convergence with respect to time, we selected two new values for dt, dt 1 = 4.0 × 10^{6}day and dt 2 = 8.0 × 10^{6}day. We applied the L2norm to compute the errors when using dt 1 and dt 2 for our refined mesh. We observed that the error when using dt 2 was 2.3 times greater than the error obtained with dt 1. Therefore, as theoretically predicted, our numerical scheme is firstorder accurate with respect to time. We then conducted the same analysis for convergence with respect to space, choosing two new values of deltaX, dx 1 = 0.4mm and dx 2 = 0.8mm. The L2norm error when using dx 2 was 2.03 times greater than the error obtained with dx 1. Once again, the values obtained were as expected, as we were using a firstorder discretization (upwind) in space. These results gave us confidence that our numerical solver had been correctly implemented.
Numerical experiments
Initial Conditions
Parameter  Value  Unit 

LPS _{0}  100: 0<x < 1  10^{4}cells/mm^{3} 
LPS _{0}  0: 1 ≤ x < 5  10^{4}cells/mm^{3} 
RM _{0}  1: 0 <x < 5  10^{4}cells/mm^{3} 
AM _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
CH _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
N _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
ND _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
G _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
AC _{0}  0: 0 <x < 5  10^{4}cells/mm^{3} 
Parameters
Parameter  Value  Unit  Reference 

ϕ _{ RMLPS }  0.1  1/(cells/mm^{3}).day  [18]** 
θ _{ AC }  1  1/(cells/mm^{3})  estimated* 
μ _{ LPS }  0.005  1/day  [18] 
λ _{ NLPS }  0.55  1/(cells/mm^{3}).day  [18] 
λ _{ AMLPS }  0.8  1/(cells/mm^{3}).day  [18] 
D _{ LPS }  2000  μm^{2}/day  estimated* 
${P}_{RM}^{max}$  0.1  1/day  estimated* 
${P}_{RM}^{min}$  0.01  1/day  estimated* 
${Q}_{RM}^{max}$  0.5  1/day  estimated* 
${Q}_{RM}^{min}$  0  1/day  estimated* 
keqch  1  cells/mm ^{3}  estimated* 
keq_g  1  cells/mm ^{3}  estimated* 
M ^{ max }  6  cells/mm ^{3}  estimated* 
μ _{ RM }  0.033  1/day  [18] 
D _{RM}  4320  μm^{2}/day  
μ _{ RM }  3600  μm^{2}/day  
μ _{ AM }  0.07  1/day  [18] 
D _{AM}  3000  μm^{2}/day  
μ _{ AM }  4320  μm^{2}/day  
μ _{ CH }  7  1/day  [18]** 
β _{ CHN }  1  1/(cells/mm^{3}).day  [34] 
β _{ CHAM }  0.8  1/(cells/mm^{3}).day  [34] 
chInf  3.6  cells/mm ^{3}  [8]** 
D _{CH}  9216  μm^{2}/day  
${P}_{N}^{max}$  11.4  1/day  [35]** 
${P}_{N}^{min}$  0.0001  1/day  estimated* 
keqch  1  cells/mm ^{3}  estimated* 
N ^{ max }  8  cells/mm ^{3}  estimated* 
μ _{ N }  3.43  1/day  [36] 
λ _{ LPSN }  0.55  1/(cells/mm^{3}).day  [18] 
D _{ N }  12096  μm^{2}/day  [37] 
μ _{ N }  14400  μm^{2}/day  [38] 
λ _{ NDAM }  2.6  1/(cells/mm^{3}).day  [18] 
D _{ ND }  0.144  μm^{2}/day  [18]** 
μ _{ G }  5  1/day  estimated* 
α _{ GN }  0.6  dimensionless  estimated* 
gInf  3.1  cells/mm ^{3}  estimated* 
D _{ G }  9216  μm^{2}/day  estimated* 
μ _{ AC }  4  1/day  estimated* 
β _{ RMND }  1.5  1/(cells/mm^{3}).day  estimated* 
α _{ ACAM }  1.5  dimensionless  estimated* 
acInf  3.6  cells/mm ^{3}  [8]** 
D _{ AC }  9216  μm^{2}/day  [29] 
In this paper, we obtained parameter values for humans whenever they were available. We chose values for the initial concentrations of LPS according to the work of the authors in [22]. In their experiments, E. coli cells were inoculated intradermally (10^{8}) into normal and neutropenic rabbits. They reported that all bacteria and inflammatory cells were contained in this 1.5 cm diameter biopsy and restricted to its 0.2 cm thick layer of dermal collagen. Thus, the volume of dermis in which the E. coli cells were contained was approximately 0.35 cm^{3} [23]. This finding suggested us a value of LPS_{0} = 100.0 × 10^{4} cells/mm^{3} .
In Table 2, parameters marked with * were adjusted to qualitatively reproduce the results obtained in several studies of the immune response to LPS. In the case of LPS, we adjust the equation parameters to obtain an exponential decrease, as shown in [24]. The results of the concentration of proinflammatory cytokines over time are qualitatively similar to those obtained in some experimental works [25–27]. The time course for the antiinflammatory cytokine is qualitatively similar to the results in [25]. An important feature present in our model is the inhibition of the production of proinflammatory cytokines by neutrophils through the action of antiinflammatory cytokines [10]. The protein granule model behavior is based on existing work [28]. The parameters marked with ** were based on the values given in the references but were adjusted due to the use of distinct units (for example, from L to mm^{3}) or to fit in a 5 mm tissue.
Comparison of different scenarios
To show the importance of some cells, molecules and processes in the dynamics of the innate immune response, we performed a set of simulations under different scenarios. Each simulation begins with a simple scenario in which we assume that only macrophages participate in the immune response to LPS (Case 1). We then consider progressively more complex scenarios. In each subsequent scenario, a new set of equations and terms are added to the previous one until the complete scenario is obtained (Case 5).
A description of each case is given below:

Case 1: only macrophages participate in the immune response. Resting tissueresident macrophages are responsible for the initial response to LPS.

Case 2: considers a) the production of proinflammatory cytokines by active macrophages; and b) all effects of proinflammatory cytokines, such as the increase in permeability and chemotaxis.

Case 3: incorporates neutrophils into the model, which participate in the immune response as a major phagocytic leukocyte. They are also responsible for producing proinflammatory cytokines.

Case 4: incorporates protein granules into the model, which are produced by neutrophils and contribute to an increase in the endothelium's permeability, allowing more monocytes to enter into the tissues and differentiate in resting macrophages.

Case 5: incorporates antiinflammatory cytokines into the model. In this case, antiinflammatory cytokines block the production of proinflammatory cytokines by the neutrophils and active macrophages. In addition, antiinflammatory cytokines block the activation of resting macrophages.
In Case 3, the decrease in LPS has been accelerated due to the presence of neutrophils migrating into the tissue in huge quantities. The number of neutrophils in the tissue is enough to control the infection.
In Case 4, observe that the extravasation of a second wave of monocytes (a consequence of the presence of protein granules produced by the neutrophils) has no impact on the potentiation of the immune response because the LPS has been almost completely eliminated. Note that the LPS decrease is smaller in case 5 than in cases 3 and 4. This fact is a consequence of the presence of antiinflammatory cytokines in the model, which causes a decrease in the number of neutrophils and monocytes extravasating to the tissue.
Conclusions and future works
In this work, we have presented a computational model for the dynamics of representative types of cells and molecules of the HIS during an innate response to the injection of LPS into a small section of tissue. To achieve this objective, we have proposed a mathematical model that incorporates the main interactions occurring between LPS and some cells and molecules of the innate immune system. The model proposes a macroscopic or homogenized view of tissue composed of two different domains: one domain represents the concentration of immune cells in the vascular system (in our case, neutrophils, N_{ max }(x, t), and macrophages, M_{ max }(x, t)), whereas the other domain represents the different cells and molecules present in the tissue (our model considers lipopolysaccharide (LPS), neutrophils (N), apoptotic neutrophils (ND), proinflammatory cytokines (CH), antiinflammatory cytokines (AC), proteins granules (G), resting (RM) and hyperactivated (AM) macrophages). Communication between the two different domains is possible and is modeled by an endothelium permeability that varies in space and time and may depend also on the concentration of different cells and molecules (in our model, the endothelium's permeability to neutrophils depends on the concentration of CH, whereas its permeability to macrophages depends on CH and G).
The model proposed in this work has been able to reproduce several features present in immune responses, such as:

the order of arrival of cells at the site of infection, as shown in [39];

the coordination of macrophages and neutrophils to mount a more effective and intense response to LPS;

the endothelium's dynamic permeability, which may depend on local concentrations of proinflammatory cytokines and protein granules;

the important role of protein granules throughout the process of monocyte extravasation;

the regulation of immune response by macrophages through the production of antiinflammatory cytokines and the phagocytosis of apoptotic neutrophils;

the crucial role of antiinflammatory cytokines in the control of the inflammatory response, thus avoiding a state of persistent inflammation after the complete elimination of LPS.
In future work, we plan to implement a more complete mathematical model that includes new cells (such as natural killer and dendritic cells), molecules and other processes involved in the immune response. The model could be extended by any of the following methods: a) including the interaction between endothelial cells, LPS and some cytokines such as IL1β and TNFα [40]; b) incorporating the fact that high amounts of LPS also induce an increase in endothelium permeability [40]; c) considering the process of macrophage desensitization, in which high levels of LPS inhibit the production of TNFα by macrophages [41]; d) taking into account that the TNFα produced by macrophages induces the production of even more TNFα [1]; and e) considering that the TNFα has proapoptotic and antiapoptotic effects on macrophages and neutrophils. In low concentrations, TNFα delays the apoptosis of macrophages and neutrophils and induces the production of proinflammatory cytokines, whereas in high concentrations, it induces apoptosis [41].
An important final step is the validation of our proposed model using experimental data. Of particular interest is the spatiotemporal modeling of microabscess formation, a very important research topic. For instance, [42–45] presents animal studies detailing the formation of liver abscess and microabscess by different types of infections. Epidermal microabscess formation by neutrophils was also evaluated in [46–48] and [22]. Infection of the heart by bacteria (bacterial myocarditis [49]) or by viruses (viral myocarditis [50]) is also correlated with microabscess formation by neutrophils. The interaction between tumor cells and inflammatory cells plays an important role in cancer initiation and progression and was investigated in [51] for the case of tumorinfiltrating neutrophils in pancreatic neoplasia, where the pattern of microabscess formation by neutrophils was reported once again. We acknowledge that this distinct pattern of formation can only be numerically reproduced and studied by models that capture the spatiotemporal dynamics of the HIS. Therefore, in the near future, we plan to extend our PDE model and adjust its parameters in the hopes of reproducing some of these experimental findings.
Declarations
Acknowledgements
The authors would like to thank FAPEMIG, CNPq, CAPES and UFJF for supporting this study in addition to the anonymous reviewers who have helped to improve the quality of this work.
Declarations
This article has been published as part of BMC Bioinformatics Volume 14 Supplement 6, 2013: Selected articles from the 10th International Conference on Artificial Immune Systems (ICARIS). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S4.
Authors’ Affiliations
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