- Research
- Open Access
Modeling fibrous cap formation in atherosclerotic plaque development: stability and oscillatory behavior
- Wanwarat Anlamlert^{1},
- Yongwimon Lenbury^{2, 3}Email author and
- Jonathan Bell^{4}
https://doi.org/10.1186/s13662-017-1252-9
© The Author(s) 2017
- Received: 24 January 2017
- Accepted: 20 June 2017
- Published: 10 July 2017
Abstract
Atherosclerosis usually occurs within the large arteries. It is characterized by the inflammation of the intima, which involves dynamic interactions between the plasma molecules; namely, LDL (low density lipoproteins), monocytes or macrophages, cellular components and the extracellular matrix of the arterial wall. This process is referred to as plaque formation. If the accumulation of LDL cholesterol progresses unchecked, atherosclerotic plaques will form as a result of increased number of proliferating smooth muscle cells (SMCs) and extracellular lipid. This can thicken the artery wall and interfere further with blood flow. The growth of the plaques can become thrombotic and unstable, ending in rupture which gives rise to many life threatening illnesses, such as coronary heart disease, cardiovascular diseases, myocardial infarction, and stroke. A mathematical model of the essential chemical processes associated with atherosclerotic plaque development is analyzed, considering the concentrations of LDLs, oxidized LDLs, foam cells, oxidized LDL-derived chemoattractant and macrophage-derived chemoattractant, the density of macrophages, smooth muscle cells (SMCs), and extracellular matrix (ECM). The positive invariant set is found and local stability is established. Oscillatory behavior of the model solutions is also investigated. Numerical solutions show various dynamic behaviors that can occur under suitable conditions on the system parameters.
Keywords
- atherosclerosis
- atherosclerotic plaque growth
- system stability
- Hopf bifurcation
1 Introduction
Atherosclerosis is a serious disease occurring in the major arteries, or blood vessels, caused by a formation of fatty lesions which contain cholesterol and cell debris in the arterial wall. Lesions, which is a region that has been damaged due to injury, can form quite early in life and develop throughout one’s lifetime.
Atherosclerotic plaque formation and growth in arteries are complex processes hemo-dynamically and mechanically. Some plaques, which are build-ups inside your arteries composed of fat, cholesterol, calcium, and other matters, remain stable throughout an individual’s life, or they become unstable and can grow to such a size that they pose a health risk from stenosis, which is a partial blockage of the artery, leading to disruption. The rupture of these vulnerable plaques is thought to be responsible for most fatalities. According to Davies [1], almost 73% of deaths from myocardial infarction (heart attack) are caused by plaque rupture.
According to recent statistics [2], atherosclerosis overwhelmingly causes more morbidity and mortality in the western world than other diseases. The prevention and treatment of atherosclerosis is one of the most important problems in medicine. For this reason, better understanding of the atherosclerotic plaque development has been a subject of intense investigation.
The disease process begins when LDL (‘bad’ cholesterol) in the bloodstream internalizes through the endothelial cells and enters the intima in the artery wall. The LDL particles then are modified by free receptors and can get oxidized into a modified form such as oxidized LDL (oxLDL). The immune response following the diffusion and oxidation of LDL cholesterol urges the endothelial cells near the inflammatory area to recruit monocytes from the bloodstream, which subsequently enter the intima [3]. Once in the intima, the monocytes differentiate (or become specialized) to turn into macrophages in the artery wall and ingest the oxLDLs through the scavenger receptors on their surfaces [4].
These macrophages eventually transform into foam cells, the hallmark of the artery lesion of fatty streak. This leads to trapping of cholesterol within the artery wall. The maturation of fatty streaks into more advanced plaques produces lesions that are usually covered with a fibrous cap composed of smooth muscle cells (SMCs) and extracellular matrix (ECM) components, such as elastin and collagen [5].
The migration of SMCs responds to a chemical signal produced during the accumulation of oxLDL, foam cells, and debris. This results in the formation of a fibrous cap, a layer of connective tissue that forms an atherosclerotic plaque, which shields the lesion from the lumen. The fibrous cap encloses a lipid-rich necrotic core composed of oxLDLs, cholesterol and apoptotic or necrotic cells that are unable to obtain sufficient nutrients for survival [6]. As the inflammatory process progresses, apoptosis (cell death that occurs normally) and matrix degradation by matrix metalloproteinases (MMPs) become apparent. As inflammation escalates, accompanied by persistent foam cell recruitment and the ever more necrotic environment within the atherosclerotic plaque, further development and maturation of the atherosclerotic lesion ensue. This assists in the enlargement of the lipid rich core and thickening of the fibrous cap [6].
Atherosclerotic plaques can be classified into two types: stable and unstable plaques. Stable atherosclerotic plaques are characterized by a thicker layer of fibrous cap, which protects the plaque from rupture. Unstable atherosclerotic plaques are characterized by a lipid core covered by a relatively thin fibrous cap containing less extracellular matrix and vascular smooth muscle cells, often with inflammatory cells and secretion of proteinases. This may lead to rupture or fissure of the surface of the plaque, thus exposing the lipid core to the bloodstream causing thrombosis (local clotting of the blood) and many cardiovascular diseases [7].
Most modeling of plaque development involves spatial-temporal constructs because there are also fluid stresses that come into play [3, 8, 9]. Because of the effort spent in dealing with the partial differential equations, the chemistry gets reduced drastically. Our purpose is to develop the basic dynamic model that describes the interaction among vital components in the temporal dynamics of the plaque formation process to which the spatial aspect could be readily incorporated later. First, the positive invariant set is found and local stability is established. Then, Hopf bifurcation analysis is carried out to illustrate the existence of sustained oscillatory behavior of the model solutions. Next, numerical simulations are carried out to verify our theoretical predictions concerning various dynamic behaviors that can occur under suitable conditions on the system parameters. Finally, clinical interpretation and conclusion are given.
2 System model
Atherosclerotic plaques start with some ‘insult’ to the intimal layer of the cardiac artery that initiates an inflammatory response. LDLs and immune cells (mainly monocytes and T-cells) migrate from the lumen into the intima. The monocytes quickly mature into macrophages, and through a rather complicated process the LDLs are oxidized, mainly due to the pressure of the free radicals. The macrophages are now able to ingest the oxidized LDLs and become fat-ladened foam cells.
We assume there are chemotactic mechanisms, like a macrophage colony stimulating factor, that facilitate the migration of smooth muscle cells into the intima to augment the native smooth muscle cell population. A late process is the formation of the cap separating the plaques from the lumen.
The ‘health’ of the cap is our main concern because it is what determines stable from unstable plaque.
Variable | Meaning | Unit |
---|---|---|
\(L_{l}\) | Low density lipoprotein concentration | g cm^{−3} |
\(L_{O}\) | Oxidized low density lipoprotein concentration | g cm^{−3} |
M | Macrophage concentration | g cm^{−3} |
F | Foam cells density | g cm^{−3} |
\(C_{O}\) | Concentration of oxidized-LDL-derived chemoattractant | g cm^{−3} |
\(C_{M}\) | Concentration of macrophage-derived chemoattractant | g cm^{−3} |
S | Smooth muscle cell density | g cm^{−3} |
E | Extracellular matrix concentration | g cm^{−3} |
In the next section, we show that the model consisting of equations (1)-(8) admits positive and bounded solutions under suitable conditions on the system parameters.
3 Stability analysis
First, we need to show the following lemma which ensures that, under suitable conditions, all the solutions of system (1)-(8) are nonnegative. We also identify a set \(\mathcal{B}\) in \(\mathbb {R}_{+}^{8}\) such that all solutions starting from \(\mathcal{B}\) remain bounded.
Lemma 1
Proof
By using the Routh-Hurwitz criteria, the following theorem can be shown.
Theorem 2
Proof
Therefore, the equilibrium point \(\tilde{X}^{*}\) is locally asymptotically stable as claimed. □
4 Sustained oscillation
We next show that the model system (1)-(8) admits periodic solutions through a Hopf bifurcation of the equilibrium point \(\tilde{X}^{*} = (L_{l}^{*}, L_{O}^{*}, M^{*}, F^{*}, C_{O}^{*}, C_{M}^{*}, S^{*}, E^{*})\).
Theorem 3
Proof
5 Model simulation
Parameter | Meaning | Value (Figure 5 ) | Unit | |
---|---|---|---|---|
\(d_{1}\) | Reaction rate constant in (1) | 1.520 × 10^{4} | 1.520 × 10^{3} | \(\mbox{g}^{-1}~\mbox{cm}^{3}~\mbox{wk}^{-1}\) |
\(d_{2}\) | Variation constant in (2) | 1.429 | 0.8024 | \(\mbox{g}^{-1}~\mbox{cm}^{3}~\mbox{wk}^{-1}\) |
\(d_{O}\) | Degradation rate of \(L_{O}\) | 6.048 × 10^{−1} | 0.01 | wk^{−1} |
\(d_{M}\) | Death rate of M | 6.048 × 10^{−1} | 0.01 | wk^{−1} |
\(d_{F}\) | Death rate of F | 4.285 × 10^{−3} | 0.25 | wk^{−1} |
\(d_{C}\) | Degradation rate of \(C_{O}\) | 0.01 | 0.01 | wk^{−1} |
\(d_{c}\) | Degradation rate of \(C_{M}\) | 0.01 | 0.1 | wk^{−1} |
\(d_{S}\) | Intrinsic death rate of S | 1.08 | 0.1 | wk^{−1} |
\(d_{E}\) | Removal rate of E | 0.5 | 0.05 | wk^{−1} |
\(k_{c}\) | Saturation constant of migration | 10^{−4} | 10^{−2} | wk^{−1} |
\(m_{s}\) | Removal rate of S to form E | 1.68 | 0.5 | wk^{−1} |
\(s_{1}\) | Production rate of \(C_{O}\) | 6.05 × 10^{−3} | 1.520 × 10^{3} | wk^{−1} |
\(s_{2}\) | Immigration rate of M fostered by \(C_{O}\) | 10.2 | 0.8024 | wk^{−1} |
\(s_{3}\) | Production rate of \(C_{M}\) | 0.1814 | 0.01 | wk^{−1} |
\(s_{4}\) | Proportionality constant in production rate of S due to migration | 1.814 | 0.01 | wk^{−1} |
\(s_{5}\) | Variation constant in production rate of E | 6.0 | 0.25 | wk^{−1} |
\(R_{f}\) | Number of free radicals in oxidation process | 0.02772 | 0.01 | mg/ml |
α | Fraction of \(M L_{O}\) that goes into E production | 0.5 | 0.6 | - |
β | Fraction of \(M L_{O}\) that goes into F production | 0.5 | 0.4 | - |
μ | Proportionality constant in (8) | 6.05 | 0.00238 | wk^{−1} |
ρ | Migration rate of S at vanishing \(C_{M}\) | 7.5 | 0.001 | \(\mbox{g}^{-1}~\mbox{cm}^{-3}~\mbox{wk}^{-1}\) |
σ | Growth rate of \(L_{l}\) | 2.016 × 10^{3} | 2.016 × 10^{3} | \(\mbox{g}^{-1}~\mbox{cm}^{-3}~\mbox{wk}^{-1}\) |
p | Proliferation rate of S | [1.700,1.765] | 0.4 | wk^{−1} |
Physically, if there is a time \(\tau> 0\) such that for some predetermined \(\delta> 0\) ECM level decreases to the point that \(E(T) \leq\delta\), the plaque would be expected to reach a point of ‘rupture’. The rupture of these vulnerable plaques is discovered to be responsible for most human fatalities suffering from heart diseases. According to Davies (1992) [1], almost 73% of deaths from myocardial infarction (heart attack) are caused by plaque rupture.
6 Conclusion
Atherosclerosis is an extremely dangerous disease due to the fact that the process of narrowing and hardening of the arteries occurs slowly and can take several decades before it shows any symptoms. Apart from leading to heart attacks and strokes, which cause thousands of deaths annually, atherosclerosis also leads to kidney failure, blindness, and even impotence [9]. Mathematical modeling can shine some light on this stealthy disease in order to learn how to prevent its often hidden complications.
In this paper, we have constructed and analyzed a model of the process of atherosclerotic plaque formation. The system model is capable of simulating various dynamic behaviors in different cases which have been observed clinically. Ranges of physical parameters that delineate unstable from stable atherosclerotic plaques have been identified.
Including more chemistry considerations into our model, we are able to discover critical conditions that delineate different dynamic behaviors exhibited by the system. Specifically, there are two main types of atherosclerotic lesions: the stable plaque which simply impedes blood flows and an unstable one which is vulnerable to rupture. Our model identifies the proliferation rate p of the smooth muscle cells as the critical physical parameter that plays the most important role in maintaining the system’s stability. If \(m_{S}+d_{S}>p\), the system is stable, but if p increases beyond \(m_{S}+d_{S}\), the system may become unstable, posing higher risk of coronary heart diseases.
Insights gained from our investigation are expected to form a basis for further research to better understand the build-up of plaque in the arteries. Deeper comprehension of the process may lead us to a new way to treat this life threatening disease. Rather than trying to reduce the build-up by reducing LDL cholesterol levels in blood serum, regression and stabilization of plaque, not LDL cholesterol, may become the new approach in the treatment and prevention of cardiovascular disease, the leading cause of death in the U.S. and around the world [11].
Declarations
Acknowledgements
Both authors are supported by a research grant from the Centre of Excellence in Mathematics, Commission on Higher Education, Thailand.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Davies, MJ: Anatomic features in victims of sudden coronary death. Coronary artery pathology. Circulation 85(Suppl 1), I19-I24 (1992) Google Scholar
- Massera, D, Graf, L, Barba, S, Ostfeld, R: Angina rapidly improved with a plant-based diet and returned after resuming a western diet. J. Geriatr. Cardiol. 13(4), 364-366 (2016) Google Scholar
- Bulelzai, MAK, Dubbeldam, JL: Long time evolution of atherosclerotic plaques. J. Theor. Biol. 297, 1-10 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Winckers, K, ten Cate, H, Hackeng, TM: The role of tissue factor pathway inhibitor in atherosclerosis and arterial thrombosis. Blood Rev. 27(3), 119-132 (2013) View ArticleGoogle Scholar
- Hundal, RS, Salh, BS, Schrader, JW, Gómez-Muñoz, A, Duronio, V, Steinbrecher, UP: Oxidized low density lipoprotein inhibits macrophage apoptosis through activation of the PI 3-kinase/PKB pathway. J. Lipid Res. 42(9), 1483-1491 (2001) Google Scholar
- Fok, PW: Growth of necrotic cores in atherosclerotic plaque. Math. Med. Biol. 29(4), 301-327 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Finn, AV, Nakano, M, Narula, J, Kolodgie, FD, Virmani, R: Concept of vulnerable/unstable plaque. Arterioscler. Thromb. Vasc. Biol. 30(7), 1282-1292 (2010) View ArticleGoogle Scholar
- Ibragimov, A, Ritter, L, Walton, JR: Stability analysis of a reaction-diffusion system modeling atherogenesis. SIAM J. Appl. Math. 70(7), 2150-2155 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Calvey, V, Houot, JG, Meunier, N, Raoult, A, Rusnakova, G: Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM Proc. 30, 1-14 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Hao, W, Friedman, A: The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE 9(3), e90497 (2014) View ArticleGoogle Scholar
- Hassard, BD, Kazarinoff, ND, Wan, YH: Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar