Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, USA

Department of Medicine, University of Texas Health Science Center at San Antonio, San Antonio, USA

San Antonio Cardiovascular Proteomics Center, University of Texas Health Science Center at San Antonio, San Antonio, USA

Barshop Institute for Longevity and Aging Studies, University of Texas Health Science Center at San Antonio, San Antonio, USA

Abstract

Background

About 6 million Americans suffer from heart failure and 70% of heart failure cases are caused by myocardial infarction (MI). Following myocardial infarction, increased cytokines induce two major types of macrophages: classically activated macrophages which contribute to extracellular matrix destruction and alternatively activated macrophages which contribute to extracellular matrix construction. Though experimental results have shown the transitions between these two types of macrophages, little is known about the dynamic progression of macrophages activation. Therefore, the objective of this study is to analyze macrophage activation patterns post-MI.

Results

We have collected experimental data from adult C57 mice and built a framework to represent the regulatory relationships among cytokines and macrophages. A set of differential equations were established to characterize the regulatory relationships for macrophage activation in the left ventricle post-MI based on the physical chemistry laws. We further validated the mathematical model by comparing our computational results with experimental results reported in the literature. By applying Lyaponuv stability analysis, the established mathematical model demonstrated global stability in homeostasis situation and bounded response to myocardial infarction.

Conclusions

We have established and validated a mathematical model for macrophage activation post-MI. The stability analysis provided a possible strategy to intervene the balance of classically and alternatively activated macrophages in this study. The results will lay a strong foundation to understand the mechanisms of left ventricular remodelling post-MI.

Background

Myocardial infarction is defined by pathology as myocytes necrosis and apoptosis due to prolonged ischemia. Since myocytes cannot divide and replace themselves, myocytes in the infarct area deprived of oxygen die and are replaced by a collagen scar. There is a series of cellular and molecular activities respond to MI in the myocardium. Myocytes apoptosis appears in the first 6 to 8 hours post-MI, and necrosis occurs in 12 hrs to 4 days post-MI

A large amount of experimental research has been conducted to elucidate the underlying mechanisms of macrophage activation, and an abundant accumulation of experimental results define on macrophage responses to different stimuli. There is a need, however, to systemically analyze the accumulated data and integrate the results into a framework that will allow a more complete understanding. To address this need, several mathematical models have been established to characterize the effects of macrophages on wound healing, inflammatory responses, and collagen synthesis post-MI

Results

We have collected experimental data from adult C57 mice and built a framework to represent the regulatory relationship among cytokines and macrophages. Based on this framework, we established a set of nonlinear differential equations to characterize the regulatory relationship for macrophage activation in the left ventricle post-myocardial infarction using physical chemistry laws. Our framework and the mathematical model were established based on the following three assumptions.

1) All monocytes that migrate to the infarct region are differentiated to unactivated macrophages

2) All activated macrophages are differentiated from unactivated macrophage since previous studies have shown that <5% of macrophages undergo mitotic division

3) All parameters and coefficients in this model are constant.

Framework of regulatory relationship for macrophage activation

In this framework, myocytes and monocytes were considered as inputs to the system. Cellular densities of M1 and M2 macrophages were considered as the outputs of the system. We chose IL-1, IL-10, and TNF-

A framework representing regulatory relationship macrophage activation post-MI

**A framework representing regulatory relationship macrophage activation post-MI**. Myocytes and monocytes are two inputs as shown in the blue dotted box. The red dotted box denotes outputs of the system, including M1 and M2 densities. Secretion function is denoted by thick arrow, activation and differentiation are denoted by thin arrow. The regulatory relationship shown in the framework was determined by published results

Input to the framework

Temporal profiles of monocytes and myocytes densities were used as inputs to our mathematical framework (Figure ^{9 }cells/ml as an initial value. Myocytes numbers monotonically decreases post-MI and is directly associated with LV wall thickness. We have measured the LV wall thickness at days 0, 1, 3, 5, and 7 post-MI. The temporal profile of myocytes was determined by combining the initial value and the monotone progression trend (the crosses) as shown in Figure

Temporal profiles of monocytes and myocytes

**Temporal profiles of monocytes and myocytes**. The wall thickness of left ventricle and observed a monotone decrease of wall thickness at day 1, 3, 5, 7 post-MI, suggesting the same trend of myocytes density based on the experimental results ^{3}cells/ml. Temporal profiles of monocytes are shown in Figure 2(B). The estimation of monocytes for each day are shown in Figure 2(C).

Macrophages density in the left ventricle of healthy adult mice is 2000 cells/ml, which will be used as initial values of unactivated macrophage density in this study

Based on our assumptions 1 and 2, all macrophages were differentiated from monocytes and emigrated from infarct area to the lymph node system. The estimation of unactivated macrophage based on the experimental results

where _{un }

Mathematical model for macrophage activation

The mathematical model of macrophage activations is a set of nonlinear differential equations represented by cellular densities (cell number/ml) of _{un}

where _{un}_{1}, _{2 }denote the cell densities of unactivated macrophages, M1 macrophages, and M2 macrophages, respectively. Variables _{10}, _{α}_{1 }denote the concentrations of IL-10, TNF-_{c }

**Table 1**. Pre-determined parameters from literature search.

Click here for file

Equation 2 determines the density of unactivated macrophages in the infarct area. For the construction part, the unactivated macrophages are differentiated from monocytes as shown in Figure _{1 }or _{2}. Additionally, inactivated macrophages do not die locally in the scar tissue but die out in the lymph node system

Equation 3 determines the activation rate of M1 macrophages. For the construction part, IL-1 and TNF-_{2 }and _{3 }denote the activation rates of M1 macrophages by IL-1 and TNF-_{IL1 }and _{1 }to denote the transition from M1 to M2 and parameter _{1})

Equation 4 determines the activation rate of M2 macrophages. The construction part is denoted by activation of M2 macrophages promoted by IL-10, and transition from M1 to M2. IL-10 promotes M2 activation and this activation rate has been approximated by parameters _{4 }based on the experimental results _{1}. The destruction part includes emigration of M2 macrophages (

Equation 5 determines the secretion rate of IL-10. For construction part, IL-10 is secreted by M2 macrophages, and parameter _{5 }denotes the secretion rate of IL-10 by M2 macrophages _{1 }denotes the self-inhibition effect of IL-10 post-MI

Equation 6 determines the deposition rate of TNF-

Equation 7 determines the deposition rate of IL-1. IL-1 is secreted by both M1 macrophage and myocytes. Parameter _{7 }denotes the secretion rate of IL-1 in cultured rat cardiac myocytes _{IL1 }represents the decay rate of IL-1 determined by its half-life time

Stability analysis

If there is no myocardial infarction, monocytes differentiation and myocytes apoptosis should be at a very low level, and the studied macrophage activation pathway should maintain homeostasis. We have calculated the equilibrium point of the system without any input and performed Lyapunov stability analysis. Our analysis showed that without any monocytes differentiation and myocytes secretion, the system would stay at the origin when

In the case of myocardial infarction, myocytes apoptosis and necrosis triggered inflammatory responses and significant monocytes differentiation, which will drive the system to a new equilibrium point. Correspondingly, the cell densities of M1 and M2 increase post-MI. We have obtained a steady state as

Computational results

Computational simulations of macrophage activation were carried out by solving the nonlinear differential equations with MATLAB. The initial conditions of unactivated, M1 and M2 macrophage densities were chosen as _{un }_{1}(0) = _{2}(0) = 0 cells/ml. The concentrations of IL-1, TNF-

Temporal profiles of cell densities of M1 (red line), M2 (blue line), and the total macrophage (black line) in LV 28 days post-MI

**Temporal profiles of cell densities of M1 (red line), M2 (blue line), and the total macrophage (black line) in LV 28 days post-MI**. Computational results were normalized to initial conditions and shown in solid lines. The initial cells number of unactivated macrophage is 2 × 10^{3 }cells/ml. The differentiated rate from monocytes to M1 macrophages

Concentrations of IL-10, IL-1 and TNF-

**Concentrations of IL-10, IL-1 and TNF- α from day 0 to day 30 post-MI**. The comparisons between the experimental measurements and temporal files were presented. In Figure 4 (A), experimental measurements of IL-10 from adult C57 mice at day 0.25, 3, 5 post-MI

Our computational results demonstrated that from days 0 to day 3 post-MI, cellular densities of the M1 phenotype increased at a faster rate than the M2 phenotype. At day 10, the M2 phenotype dominates over the M1 phenotype. This prediction agrees with the results reported by Troidl

Discussion

This study established a mathematical model for macrophage activation in the left ventricle post-MI by combining experimental and computational approaches. This is the first mathematical model focusing on the dynamic interactions among cytokines, myocytes, monocytes, and macrophages. Computational predictions based on this mathematical model match with experimental measurements, suggesting effectiveness of the model. In addition, our stability analysis provided insight for the activation pattern of macrophages post-myocardial infarction. In our mathematical model, there are two inputs, myocytes apoptosis and monocytes differentiation. In this study, we have predicted a stable equilibrium for homeostasis, which means without myocardial infarction or following small injury stimuli, macrophage densities and concentrations of IL-1, IL-10, and TNF-α should stay at the equilibrium. After MI, the monocytes differentiated into macrophage and apoptotic myocytes secreted significant amounts of cytokines to activate the macrophages. The strength of the monocytes differentiation and myocytes apoptosis (inputs of the system) drive the system to different states while all states will be bounded due to the bounded strength of the inputs.

However, there exist some differences between computational predictions and real experimental results. To address this issue and the variation in different experiments, a stochastic parameter distribution will need to be introduced to replace the constant parameters. In addition, more detailed measurements on monocytes and myocytes and concentration temporal profiles of IL-1, IL-10 and TNF-

Conclusions

Our study has established framework for macrophage activation and used ordinary differential equations to model the cellular interactions between macrophage activation types post-MI. The results on stability analysis can be used as a useful tool to predict the behaviour of biological systems.

Methods

To incorporate the experimental data, curve fitting algorithm was applied to obtain temporal continuous density profiles of myocytes and monocytes based on discrete experimental data.

Stability analysis of the established mathematical model

To analyse the stability of the proposed mathematical model, we have calculated the equilibrium point of the system and performed the Lyapunov stability analysis of the system.

In our mathematical model, equations (2-7) are six first-order equations with input M and _{c}_{1}, _{2}, _{3}, _{4}, _{5}, _{6 }to denote _{un}_{1}, _{2}, _{10}, _{α}_{1}. We first examined the stability of the system without any input and obtained an equilibrium of

and obtained its derivative as the following equation

By applying the boundary of the Hill equations, we got

Applying the parameters in Table 1 to equation (10), we got

Since the derivative of Lyapunov function is negative semi-definite and the semi-definite is satisfied with all states equal to zero, the system is globally asymptotically stable without any input. Given any bounded differentiate rate of monocytes and myocytes density as input, the system will have bounded states.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Y.F.J and M.L.L designed the research; Y.F.J and Y.W performed the computational analysis and simulation. Y.F.J, Y.W, T.Y, Y.M, G.V.H, M.Z, and M.L.L analyzed the results and wrote the manuscript.

Acknowledgements

Based on “Mathematical modeling of macrophage activation in left ventricular remodeling post-myocardial infarction”, by Yunji Wang, Yu-Fang Jin, Yonggang Ma, Ganesh V Halade and Merry L Linsey which appeared in

The authors acknowledge grant and contract support from NHLBI HHSN268201000036C (N01-HV-00244), NIH R01 HL75360, Veteran's Administration Merit Award, and the Max and Minnie Tomerlin Voelcker Fund (to M.L.L.), and NIH 1R03EB009496, and NIH SC2HL101430 (to Y.F.J.).

This article has been published as part of