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

Department of Mechanical Engineering, University of Texas at San Antonio, USA

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

Abstract

Background

Progressive remodeling of the left ventricle (LV) following myocardial infarction (MI) can lead to congestive heart failure, but the underlying initiation factors remain poorly defined. The objective of this study, accordingly, was to determine the key factors and elucidate the regulatory mechanisms of LV remodeling using integrated computational and experimental approaches.

Results

By examining the extracellular matrix (ECM) gene expression and plasma analyte levels in C57/BL6J mice LV post-MI and ECM gene responses to transforming growth factor (TGF-β_{1}) in cultured cardiac fibroblasts, we found that key factors in LV remodeling included macrophages, fibroblasts, transforming growth factor-β_{1}, matrix metalloproteinase-9 (MMP-9), and specific collagen subtypes. We established a mathematical model to study LV remodeling post-MI by quantifying the dynamic balance between ECM construction and destruction. The mathematical model incorporated the key factors and demonstrated that TGF-β_{1 }stimuli and MMP-9 interventions with different strengths and intervention times lead to different LV remodeling outcomes. The predictions of the mathematical model fell within the range of experimental measurements for these interventions, providing validation for the model.

Conclusions

In conclusion, our results demonstrated that the balance between ECM synthesis and degradation, controlled by interactions of specific key factors, determines the LV remodeling outcomes. Our mathematical model, based on the balance between ECM construction and destruction, provides a useful tool for studying the regulatory mechanisms and for predicting LV remodeling outcomes.

Background

Myocardial infarction (MI) is a leading cause of congestive heart failure (CHF)

Previous studies have shown that matrix metalloproteinases (MMPs) regulate ECM degradation and fibroblasts regulate ECM synthesis _{1 }(TGF-β_{1}), tissue inhibitor of metalloproteinase-1 (TIMP-1), and collagen I levels are significantly elevated from day 1 to day 7 post-MI

Results

Identifying Key Factors

The key factors were pre-targeted by examining the most significant changes in ECM gene expression in the infarct region at day 7 post-MI, compared to gene expression in the remote non-infarcted region of the same LV and in the LV from control group. In the ECM gene array analysis, total RNA yield was 1.0 ± 0.1, 1.9 ± 0.2, and 2.9 ± 0.3 μg/mg LV tissue for control, remote, and infarct samples, respectively (p < 0.05 for control vs remote and infarct, and for remote vs infarct). Of the 84 genes examined, 51 genes were differentially expressed among control, remote, and infarcted groups (all p < 0.05). The most prevalent pattern of gene expression changes was an increased expression level in the infarct tissue, compared to both control and remote groups. Of the 51 genes, 17 genes showed > 2.5-fold change in the infarct region, and these genes are listed in Table

ECM Gene Array (Data are Mean ± SD normalized levels.)

**Gene**

**Control**

**Remote**

**Infarct**

Cdh3

1.0E-05 ± 9.8E-06

7.4E-05 ± 5.5E-05

2.8E-04 ± 1.3E-04

Col1a1

1.5E-02 ± 2.6E-03

1.9E-01 ± 1.0E-01

8.2E-01 ± 3.4E-01

Col2a1

3.1E-06 ± 6.3E-07

1.1E-05 ± 1.1E-05

2.3E-04 ± 1.8E-04

Col3a1

5.0E-03 ± 1.5E-03

8.0E-02 ± 5.5E-02

2.3E-01 ± 9.7E-02

Col5a1

8.0E-3 ± 2.0E-3

2.3E-2 ± 1.1E-2

4.1E-2 ± 1.6E-2

Ctgf

2.5E-02 ± 4.3E-03

7.6E-02 ± 5.0E-02

1.7E-01 ± 9.1E-02

Fn1

9.0E-03 ± 2.5E-03

4.3E-02 ± 2.7E-02

1.6E-01 ± 6.9E-02

Mmp2

3.5E-03 ± 1.2E-03

1.2E-02 ± 7.9E-03

1.8E-02 ± 7.5E-03

Mmp14

1.9E-03 ± 4.6E-04

6.8E-03 ± 3.2E-03

1.5E-02 ± 6.8E-03

Ncam1

6.2E-04 ± 1.3E-04

1.8E-03 ± 1.3E-03

4.1E-03 ± 1.7E-03

Postn

5.3E-03 ± 1.6E-03

1.5E-01 ± 9.2E-02

2.9E-01 ± 1.2E-01

Sparc

1.3E-02 ± 3.7E-03

7.0E-02 ± 3.7E-02

1.1E-01 ± 4.5E-02

Spp1

1.3E-05 ± 5.7E-06

3.1E-04 ± 4.5E-04

2.6E-03 ± 2.2E-03

Tgfbi

2.0E-03 ± 3.3E-04

3.3E-03 ± 1.2E-03

7.1E-03 ± 3.6E-03

Thbs1

7.2E-03 ± 6.7E-03

1.2E-02 ± 7.6E-03

4.0E-02 ± 1.6E-02

Thbs2

9.0E-03 ± 2.3E-03

1.1E-02 ± 3.6E-03

3.9E-02 ± 1.4E-02

Timp1

2.0E-04 ± 9.4E-05

4.1E-03 ± 3.1E-03

6.1E-03 ± 3.3E-03

Control is unoperated mice; MI is 7d post-MI mice.

We further examined plasma profiles at day 7 post-MI to determine if gene level changes were mirrored in the post-MI plasma. In the plasma profiles, 11 proteins increased in the day 7 post-MI samples (Table

Multi-analyte Profiling of Control and 7 day Post-MI Plasma

**Control n = 6**

**7 d MI n = 7**

**P value**

Clusterin (μg/mL)

330 ± 40

510 ± 200

0.046

Cystatin-C (ng/mL)

360 ± 20

530 ± 100

0.008

Eotaxin (pg/mL)

1500 ± 200

1900 ± 400

0.049

Fibrinogen (mg/mL)

12 ± 2

19 ± 5

0.012

Haptoglobin (μg/mL)

83 ± 30

180 ± 50

0.002

Macrophage Inflammatory Protein-1 gamma (ng/mL)

24 ± 4

33 ± 9

0.042

Matrix Metalloproteinase-9 (ng/mL)

71 ± 9

96 ± 20

0.009

Myeloperoxidase (ng/(mL)

58 ± 10

100 ± 30

0.006

Osteopontin (ng/mL)

250 ± 60

390 ± 100

0.041

Serum Amyloid Protein (μg/mL)

21 ± 3

33 ± 9

0.013

TIMP-1 (ng/mL)

0.8 ± 0.1

3.7 ± 2.0

0.002

In addition, we examined the ECM productions in isolated cardiac fibroblasts stimulated with TGF-β_{1}. Fibroblast ECM array analysis showed that TGF-β_{1 }stimulation of cardiac fibroblasts up regulated 5 genes and down regulated 7 genes, which are shown in Table _{1 }stimulation at the concentration of 10 ng/mL. These experimental data indicated a primary regulatory effect of TGF-β_{1 }on fibroblast ECM production. Interestingly, the 5 up regulated genes are among the 17 ECM genes that were significantly expressed in the LV, indicating that the cardiac fibroblast is likely the major tissue source for these ECM genes (Table

Fibroblast ECM array in serum free control and 10 ng/ml TGF-β stimulated fibroblasts

**Serum Free**

**TGF-β**

**p Value**

ECM/Growth Factors

Col1a1

4.842 ± 1.399

9.614 ± 3.324

0.028

Col5a1

0.301 ± 0.085

0.605 ± 0.230

0.038

Fbln1

0.012 ± 0.005

0.008 ± 0.004

0.032

Sparc

2.709 ± 0.204

4.932 ± 0.379

<0.001

Tgfbi

0.022 ± 0.016

0.013 ± 0.015

0.009

Cell Adhesion Molecules

Ncam1

0.043 ± 0.016

0.169 ± 0.054

0.011

Pecam1

0.000060 ± 0.000012

0.000022 ± 0.000011

0.015

Sgce

0.187 ± 0.035

0.146 ± 0.018

0.030

Vcam1

0.301 ± 0.136

0.161 ± 0.088

0.031

MMPs/TIMPs

Mmp7

0.0000075 ± 0.0000011

0.0000056 ± 0.0000004

0.026

Timp1

0.052 ± 0.022

0.272 ± 0.136

0.035

Timp2

0.326 ± 0.109

0.188 ± 0.050

0.027

Data are Avg ± SEM levels (2^{-ΔCT}) normalized to 5 housekeeping genes for n = 4 paired fibroblast sets.

We also examined the correlations between LV wall thickness with 6 genes that were over expressed post-MI. The R^{2 }values were 0.76 for collagen 1α1, 0.64 for collagen 2α1, 0.75 for collagen 5α1, 0.60 for periostin, 0.61 for osteopontin, and 0.63 for TGF-β_{1}.

In summary, the key factors identified were macrophages, fibroblasts, TGF-β_{1}, MMP-9, and collagen. Based on these experimental results, we developed a framework of the interaction loops among the identified key factors (Figure

A working model of post-MI scar formation and remodeling with constructive and destructive modules

**A working model of post-MI scar formation and remodeling with constructive and destructive modules**. Red arrows represent the destruction pathway and green arrows represent the construction pathway. The regulation scheme includes lines in black, which can be a driving or a triggering stimulus. LV remodeling outcomes are denoted as collagen concentrations in the scar tissue.

Linking Experimental Results to Mathematical Modeling Framework

In the ECM construction pathway, collagen is secreted by fibroblasts. Growth and secretion of fibroblasts are stimulated by TGF-β_{1 }(Table _{1 }
_{1 }is the macrophage. Meanwhile, there are interactions between the ECM construction and destruction pathways: a) MMP-9 regulates ECM construction by activating TGF-β_{1 }which stimulates collagen synthesis; b) TGF-β_{1 }induces TIMP-1, which inhibits collagen degradation by blocking MMP-9 activity. Linking these key factors with their sources and effects allows us to develop the mathematical model for quantitative analysis.

Mathematical Modeling

We established a set of nonlinear differential equations to model the temporal interactions among the key factors identified by our experimental results. The model incorporated the following variables: macrophage cell density (M_{Φ}, cells/mm^{3}), fibroblast cell density (F, cells/mm^{3}), collagen concentration (C, μg/μL), TGF-β_{1 }concentration (T_{β}, pg/μL), and activated MMP-9 concentration (M_{9A}, pg/μL). Rates of cell number change were determined by the summation of constructive effects (migration rate or proliferation rate) and destructive effects (death rate or removal rate). Rates of chemical factors (TGF-β_{1}, MMP-9, collagen, etc) change were determined by the net difference between the synthesis rate and degradation rate.

Four assumptions were used based on experimental results: 1) All monocytes that migrate to the infarct region are differentiated to macrophages _{1 }secreted at the injured site becomes activated

Accordingly, the scar formation post-MI was modeled by the following set of differential equations

The parameters used in these equations with their biological meanings, experimental values, units, and references were listed in Table _{β}), _{
g
}(_{
β
}), and Fc(T_{β}), were established based on

Pre-determined parameters in the mathematical model

**Symbol**

**Biological meaning**

**Value**

**Units**

**Ref**

d_{MΦ}

Macrophage removal rate**(eqn 1)

0.6

day^{-1}

ρ_{MΦ}

maximal macrophage density (eqn 2)

2500

cells/mm^{3}

ρ_{F}

maximal fibroblast density(eqn 2)

1250

cells/mm^{3}

ρ_{C}

maximal collagen density (eqn 2)

3300

μg/mm^{3}

k_{F}

Fibroblast growth rate* (eqn 3)

0.924

day^{-1}

d_{F}

Fibroblast apoptosis rate (eqn 3)

0.12

day^{-1}

k_{MΦT}

Macrophage TGF-β production rate (eqn 4)

0.07

pg/cell/day

k_{FT}

Fibroblast TGF-β production rate (eqn 4)

0.004

pg/cell/day

d_{Tβ}

TGF-β degradation rate^{¥}(eqn 4)

15

day^{-1}

K_{MφM9}

Macrophage secretion MMP9 rate (eqn 5)

3

pg/cell/day

estimated

d_{M9}

MMP-9 degradation rate(eqn 5)

0.875

day^{-1}

k_{on}

Kinetic reaction speed (eqn 5)

3 × 10^{-4}

1/(μg/mm^{3})s^{-1}

k_{off}

Kinetic reaction speed (eqn 5)

4 × 10^{-4}

s^{-1}

k_{onc}

Kinetic reaction speed (eqn 6)

0.004

s^{-1}

k_{FC}

Fibroblast collagen production rate (eqn 6)

20

μg/cell/day

*The growth rate of cells was determined by the population doubling time (T_{2}) via equation k = ln 2/T_{2 }

** Since macrophages emigrate from the scar tissue to lymph node system instead of dying locally in the scar tissue, the removal rate of macrophage, d_{MA}, was used in our model.

¥ The decaying rate of chemical factors was calculated from their half-life (T_{1/2}) via the equation d = ln 2/T_{1/2 }

Plot of these constructed functions and the available experimental data were shown in Figure

Macrophage migration rate M(T_{β}) _{β}) _{β}) _{1 }concentration

**Macrophage migration rate M(T _{β}) **

Equation 1 determines the rate of macrophage (M_{Φ}) infiltration. The function M(T_{β}) describes the migration rate of macrophages to the scar tissue _{β}) also represents the migration of monocytes stimulated by TGF-β_{1}. Parameter d_{MΦ }denotes the emigration rate of macrophages

Equation 2 determines the crowding effect of myocytes, endothelial cells, vascular smooth muscles cells, fibroblasts, macrophages, and collagen in the myocardium, which are affected by total environment density _{mc }= 0.05 represents the rate of myocyte cell death since ischemic myocytes undergo necrosis in the infarct region post-MI. The crowding effect of macrophages, fibroblasts, and collagen was considered by calculating the normalized density with respect to their maximum density in scar tissue _{
mem
}(

Plots of crowding effects (equation 2) and TGF-β_{1 }activation function _{T }(equation 4)

**Plots of crowding effects (equation 2) and TGF-β**_{1 }**activation function **_{T }**(equation 4)**. A: Crowding effects (_{mem }(_{1 }activation function, _{T }according to small scar size. Activation peak time is at day 2 post-MI with amplitude at 15 pg/μL/day.

Equation 3 determines the rate of fibroblast (F) density changes based on the assumption that majority of fibroblasts come from the proliferation of resident cells (assumption 2). The function _{g}(_{
β
}) denotes the stimulating effects of TGF-β_{1 }on the growth rate of fibroblasts _{F }represents the apoptosis rate of the fibroblast

Equation 4 determines the rate of TGF-β_{1 }concentration change, wherein k_{FT }denotes the TGF-β_{1 }secretion rate of fibroblasts _{MΦT }denotes the TGF-β_{1 }secretion rate of macrophages _{1 }in the scar tissue is the activated macrophage. Parameter d_{Tβ }represents the degradation rate of TGF-β_{1}, which can be calculated from the half life data

TGF-β_{1 }gene levels demonstrated temporal progression at the early stage post-MI. Gene expression profile of TGF-β_{1 }increased post-MI, peaked at day 2, and returned to normal levels after day 7 in mice post-MI _{1 }secreted in the infarct is activated (Assumption 3), gene expression profile can be used as an activation pattern of TGF-β_{1}. The function, _{1 }activation post-MI

Equation 5 determines the rate of activated MMP-9 concentration change. Proteolytic collagen degradation with activated MMP-9 is described in equation 5a, where M_{9A}, C, CM_{9}, and CID denote activated MMP-9, collagen, binding of MMP-9 and collagen, and degraded collagen peptide concentration, respectively. MMP-9 is inhibited primarily by TIMP-1, and TIMP-1 is induced by TGF-β_{1}. Thus, we established an inhibition function h(T) = 1/(1+T_{β}/T_{βN}) with T_{βN }= 6.0 pg/μL to represent the inhibition effect.

Equation 6 determines the rate of collagen concentration changes. Collagen secretion rate by fibroblasts was denoted by parameter_{
FC
}. Meanwhile, the function, _{
c
}(_{
β
}), characterizes effects of TGF-β_{1 }on collagen secretion rate by fibroblasts _{
c
}(

Equation 7 determines the concentration change of CM_{9}, based on the theoretical model for collagen degradation by MMPs proposed by Popel's group

Computational simulations

Computational simulations of scar formation (collagen deposition) were carried out by solving the nonlinear differential equations with MATLAB. Initial conditions of the fibroblast and macrophage densities were chosen as F(0) = 20 cells/mm^{3 }and M_{Φ}(0) = 5 cells/mm^{3}. Accordingly, T_{β}(0) = 0.21 pg/μL, M_{9A}(0) = 7.1 pg/μL, C(0) = 839.5 μg/μL, CM_{9}(0) = 447.6 μg/μL were calculated by the equilibriums of equations 1-7 for normal LV. All the simulations shown in this study used the same initial conditions. The initial conditions were chosen based on measurements in the normal LV for both the control and MI groups (MI induced at day 0). The simulations covered the LV remodeling process from day 0 to day 30 post-MI.

Model validation

To validate our mathematical model, we compared our simulation results to experimental data from our lab or reported in the literature

The relative ratio changes of fibroblast density, macrophage density, and MMP-9 concentrations

**The relative ratio changes of fibroblast density, macrophage density, and MMP-9 concentrations**. Computational results were normalized to initial conditions and are shown in solid lines. Previously published experimental results were normalized to the corresponding measurements in control group and are shown as x (Mean ± SD). All experiments were carried out in mice with MI induced by coronary artery ligation. The fibroblast and macrophage densities were collected from C57BL/6J mice

In addition, our simulations correctly predicted MMP-9 responses to three TGF-β_{1 }stimuli corresponding to reduced, normal, and elevated post-MI activation strength. Others have reported an early increase of MMP-9 levels of 78 ± 19 pg/μL for small infarcts and 195 ± 63 pg/μL for large infarcts

Temporal profiles of TGF-β_{1}, macrophages, MMP-9, fibroblasts, collagen, and collagen peptides from days 0 to 30 post-MI in response to low (····), median (blue line), and elevated (red dash) TGF-β_{1 }stimuli

**Temporal profiles of TGF-β**_{1}**, macrophages, MMP-9, fibroblasts, collagen, and collagen peptides from days 0 to 30 post-MI in response to low (**····**), median (blue line), and elevated (red dash) TGF-β**_{1 }**stimuli**. The initial conditions were set to F(0) = 20 cells/mm^{3 }and M_{Φ}(0) = 5 cells/mm^{3}, T_{β}(0) = 0.21 pg/μL, M_{9A}(0) = 7.1 pg/μL, C(0) = 839.5 μg/μL, CM_{9}(0) = 447.6 μg/μL, according to measurements in normal myocardium. The symbol ↑ indicates the peak time of TGF-β_{1 }stimulus.

Effects of TGF-β_{1 }levels

We employed activation of TGF-β_{1}, _{
T
}in equation 4, at reduced, normal, and elevated post-MI strength. The activation function peaked at 15, 30, and 60 pg/μL/day, according to active expression levels observed in small, median, and very large infarcts, respectively. Temporal profiles of cell densities of macrophages, fibroblasts, concentrations of MMP-9, collagen, and TGF-β_{1 }responses to the stimuli _{
T
}are shown in Figure _{1 }stimulus level setting at 30 pg/μL/day (a normal post-MI level in mice), TGF-β_{1 }peaked after day 2 post-MI, macrophage density peaked at day 3, MMP-9 concentration peaked at day 4, and all were returned to normal levels at 30 days post-MI. In contrast, fibroblast density and collagen density continued to increase beginning at day 4, reached a stable value after day 20 post-MI, and then remained at a higher equilibrium level at day 30 post-MI (blue solid line in Figure

In the case of reduced TGF-β_{1 }levels (with an activation peak at 15 pg/μL/day in Figure

When the TGF-β_{1 }stimulus strength was elevated to a level 2-fold higher than normally seen post-MI in mice, more macrophages infiltrated to the infracted region in the early days (Figure

Effects of MMP-9 interventions on ECM destruction

With the validated parameter settings, we also used the mathematical model to predict the effects of MMP-9 interventions at different strengths and intervention times. Specifically, we simulated the LV remodeling responses to three different MMP-9 interventions post-MI: 1) elevation of MMP-9 level (200 pg/μL) beginning at 8 hours post-MI to mimic the earlier increase of MMP-9 levels seen with reperfusion, 2) elevation of MMP-9 levels (200 pg/μL) beginning at 7 days post-MI to mimic a prolonged macrophage infiltration, and 3) reduced elevation of MMP-9 levels (100 pg/μL) beginning at 7 day to mimic therapeutic targeting of MMP-9. The LV remodeling responses were shown in Figure

Temporal profiles of MMP-9, macrophages, TGF-β_{1}, fibroblasts, collagen, and collagen peptides from days 0 to 30 post-MI at different MMP-9 interventions

**Temporal profiles of MMP-9, macrophages, TGF-β**_{1}**, fibroblasts, collagen, and collagen peptides from days 0 to 30 post-MI at different MMP-9 interventions**. The profiles include progressive changes in response to MMP-9 intervention at day 1 (····) and day 7 (red dash), as well as reduced MMP-9 intervention strength at day 7 post-MI (green dot-dash), and no intervention (blue line). The initial conditions were set to F(0) = 20 cells/mm^{3 }and M_{Φ}(0) = 5 cells/mm^{3}, T_{β}(0) = 0.21 pg/μL, M_{9A}(0) = 7.1 pg/μL, C(0) = 839.5 μg/μL, CM_{9}(0) = 447.6 μg/μL according to the measurements in normal myocardium.

Discussion

This study is the first investigation to integrate mathematical modeling with ECM and fibroblast gene array data and plasma analytes to predict ECM remodeling post-MI. We have integrated _{1}, and TIMP-1 are critical biomarker candidates of LV remodeling outcomes.

Our experimental results on microarray and plasma data provided the foundations to build our computational framework. We examined 84 ECM genes and chose the 17 genes that were most highly over-expressed in the infarct region compared to both control and non-infarcted groups (>2.5-fold over-expression). The expression levels of several of these factors were further verified by our plasma data at protein level. One interesting finding was that SPP1 (osteopontin) gene expression levels increased 206-fold in the infarct region compared to the control group at day 7 post-MI, suggesting strong macrophage activation. The plasma protein levels of osteopontin increased from 250 ± 60 ng/mL in controls to 390 ± 100 ng/mL in post-MI samples, which adds support for the critical role of macrophages in our mathematical model. Therefore, the primary selection of the most highly changed genes allowed us to focus on the most significant factors at gene level and predict the possible interactions at protein level and cellular level.

An interesting observation was that MMP-9 mRNA levels did not increase in our gene array analysis, but MMP-9 protein levels increased in the plasma data. It is well known that MMP-9 protein levels and activation were increased at day 7 post-MI. This conflicting phenomena is caused by pre-formed MMP-9 proteins stored in leukocytes, which do not rely on increased gene expression to induce MMP-9 secretion and activation

There is limited data available to construct suitable functions of _{
β
}), _{
g
}(_{
β
}), _{
c
}(_{
β
}), and _{
c
}(_{
β
}) and _{
g
}(_{
β
}). Roberts et al have presented a set of data on _{
c
}(_{
β
}). Loftis and colleagues have studied effects of collagen density on cardiac fibroblast behavior and showed elevated fibroblast activities stimulated with higher collagen concentrations (750 μg/mL - 1250 μg/mL) _{
c
}(

Our computational results demonstrated that altering the strength of TGF-β_{1 }altered LV remodeling outcomes. Elevated TGF-β_{1 }levels at the early stage (day 3 post-MI) led to elevated macrophage density and MMP-9 levels, decreased fibroblast secretion of collagen and collagen deposition, and thereby, prolonged the progression of remodeling. Wetzler and colleagues have shown that the prolonged persistence of macrophages at the late phase (after day 7 post-MI) impairs the wound healing process _{1 }levels delays wound healing post-MI

Simulations of different TGF-β_{1 }strengths also shed light on the regulation scheme of ECM construction and destruction. For ECM construction regulation, active TGF-β_{1 }stimulated fibroblast proliferation and collagen secretion, which increased the crowding effect. The increased crowding coefficient ramped down fibroblast proliferation and TGF-β_{1 }secretion (negative feedback), which slowed down the stimulus for monocytes to migrate into the infarct region. The decrease of monocytes number led to less macrophage infiltration, which then reduced the crowding coefficient. Meanwhile, reduced macrophages lead to less TGF-β_{1 }secretion by macrophage, which further slowed down collagen synthesis. For collagen destruction regulation, TGF-β_{1 }induced macrophage infiltration, which lead to elevated MMP-9 secretion, elevated collagen degradation, and thereby reduced crowding coefficients. Smaller crowding coefficients lead to elevated fibroblast proliferation and collagen secretion. Notably, there are two types of negative feedback schemes in the mathematical model: degradation (apoptosis or emigration) rates associated with proteins (cells) and the crowding effects. Degradation rates are constants and determine how fast the proteins (cells) can respond to stimuli. Crowding effects are time varying impacts imposed by the environment. Through these regulation schemes, a dynamic balance of collagen construction and destruction can be maintained to generate a stable scar. Furthermore, profiles of crowding effects elucidated the transition from the normal LV to scar tissue with respect to cell types and collagen concentrations (Figure

It is worth mentioning that there exist biological negative feedbacks in our mathematical model. For macrophage density regulation, there was a positive feedback loop containing macrophage and TGF-β_{1}: TGF-β_{1 }stimulated monocytes migration, leading to macrophage infiltration; macrophages secreted TGF-β_{1 }which might attract more macrophages to the infarct site. We observed elevated macrophage density and MMP-9 concentrations corresponding to increased TGF-β_{1 }levels (Figure _{1}. In addition, Wahl et al pointed out that the monocyte chemotactic activity increased in response to low concentrations of TGF-β_{1 }stimuli, while the chemotactic activity decreased in response to higher concentrations of TGF-β_{1 }
_{1 }levels continuously increased, infiltration speed of macrophages decreased as shown in figure _{β})), suggesting a secondary biological inhibition scheme of the TGF-β_{1 }-- macrophage positive feedback loop.

There are a few limitations of the mathematical model that resulted to a large degree from the model assumptions. More research is needed to address these limitations and further enhance the models. First, our model calls for accurate determination of MMP-9 activation and inhibition functions. Complete time-course measurements of TIMP-1 and the other three TIMPs (for MMP-9 inhibition) and MMP-3 (for MMP-9 activation) would provide additional details on the regulators of MMP-9 function. Second, the large differences between simulation results and experimental measurement of MMP-9 concentrations before day 4 post-MI in Figure _{1 }concentrations post-MI.

This systems biology study for LV remodeling can be expanded to include proteomics and cardiac functions in future studies. We employed plasma data in this study since plasma proteins reflect the process of LV remodeling and plasma data are more directly translatable to the clinic. However, we are well aware that measuring tissue protein levels will provide a more direct evaluation of LV remodeling. Further investigation on the ECM proteomics in cardiac samples has been planned in our future research to establish a more complete mathematical model for LV remodeling. Though it is beyond the scope of the current paper, we have previously reported some data on cardiac function

The computational model for post-MI LV remodeling developed here illustrated the dynamic interactions among critical factors in LV remodeling. This is the first mathematical model focusing on the protein and cellular interactions post-MI. Thus, this model provides a strong foundation for future studies to build a more comprehensive model that takes into account a more complete set of parameters. The model also provides a tool to guide experimental designs by identifying candidate factors to intervene, the proper intervention time, and doses for effective interventions to achieve the most beneficial outcomes. As an example, we have shown the effects of MMP-9 intervention time and doses on LV remodeling outcomes in this study. Though this model was established based on

Conclusions

In conclusion, we developed a set of differential equations to quantitatively model the dynamic interactions and temporal changes of the key components identified from our experimental results. Predictions of the mathematical model fell well within experimental measurements, particularly with regard to macrophage infiltration and matrix remodeling. This mathematical model provides a powerful tool to better understand how the dynamic balance between ECM construction and ECM destruction influences LV remodeling outcomes.

Methods

Mice

All animal procedures were conducted in accordance with the Guide for the Care and Use of Laboratory Animals (National Research Council, 1996) and were approved by the Institutional Animal Care and Use Committee at the University of Texas Health Science Center at San Antonio. Male C57BL/6J wild type adult mice (n = 13) at age 8.0 ± 0.5 months were used. One group (n = 6) served as unoperated controls, while the other group (n = 7) underwent coronary artery ligation for 7 days as described previously

In Vivo Procedures

Blood was collected from the jugular vein and placed in a heparinized tube for plasma collection. Tissue was collected for the gene array analysis as described previously

Microarray and Plasma Analysis

Total RNA was isolated using the TRIzol plus Total RNA purification kit (Invitrogen). The RT^{2 }qPCR Primer Array for Extracellular Matrix and Adhesion Molecules (SuperArray catalog APMM-013A) was used for the gene array. Results were analyzed based on the ΔΔCt method with normalization of raw data to the GAPDH housekeeper gene. Data are presented as average 2^{-ΔCT }levels.

For the fibroblast ECM microarray analysis, cardiac fibroblasts were isolated from adult C57BL/6J mice and stimulated with or without 10 ng/ml TGF-β_{1 }for 24 hours

Statistical Analysis

Control, remote, and infarct LV groups were analyzed by ANOVA, with the Bonferroni post hoc test. Control and MI plasma was analyzed by Students t-test. Unstimulated and TGF-β_{1 }stimulated fibroblast groups were analyzed by paired t-test. A p < 0.05 was considered statistically significant.

Authors' contributions

YFJ, HCH, and MLL designed the research; JB, and QD performed animal experiments. YFJ performed the computational experiments. YFJ, HCH, and MLL analyzed the results and wrote the manuscript. All authors have read and approved the final manuscript.

Acknowledgements

The authors acknowledge grants support from NIH 1R03EB009496, NIH SC2HL101430, NSF 0649172, and AT&T foundation (to YFJ), from NSF 0644646, and NSF 0602834 (to HCH), and NHLBI HHSN268201000036C (N01-HV-00244), NIH R01 HL75360, AHA Grant-in-Aid 0855119F, and the Morrison Fund (to MLL).