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Bifurcation and stability analysis of atherosclerosis disease model characterizing the anti-oxidative activity of HDL during short- and long-time evolution

  • Asish Adak , Debasmita Mukherjee ORCID logo and Praveen Kumar Gupta ORCID logo EMAIL logo
Published/Copyright: January 9, 2024
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 79 Issue 5

Abstract

In this article, a partial differential equation (PDE) model for atherosclerosis disease is presented that analyzes the anti-oxidative activity of high-density lipoprotein (HDL) during the reverse cholesterol transport (RCT) process. The model thoroughly investigates the complex interplay between oxidized low-density lipoprotein (ox-LDL) and high-density lipoprotein in the context of atherosclerosis, emphasizing their combined impact on plaque formation, disease progression, and regression. In addition to this, we considered that monocytes are also attracted by the presence of ox-LDL within the intima. Detailed discussions on stability analyses of the reaction dynamical system at non-inflammatory and chronic equilibrium are provided, followed by a bifurcation analysis for the proposed system. Furthermore, stability analysis for the PDE model in the presence of diffusion is conducted. Our study reveals that the oxidation rate of LDL by monocytes (δ) and the influx rate of HDL (ϕ) due to drugs/diet are primarily responsible for the existence of bi-stability of equilibrium points. In the numerical results, we observe that non-inflammatory or chronic equilibrium points exist for either a short or a long time, and these findings are validated with existing results. The biological elucidation shows the novelty in terms of enhancing our ability to assess intervention efficacy to generate therapeutic strategies resulting in the reduction of the atherosclerotic burden and associated cardiovascular risks.

Keywords: atherosclerosis; reaction-diffusion equation; reverse cholesterol transport; stability analysis; bifurcation
Corresponding author: Praveen Kumar Gupta, Department of Mathematics, National Institute of Technology Silchar, Cachar, Assam, 788010, India, E-mail:

  1. Research ethics: Not applicable.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: The authors have no conflict of interest.

  4. Research funding: None declared.

  5. Data availability: Not applicable.

References

[1] R. Ross, “Atherosclerosis—an inflammatory disease,” N. Engl. J. Med., vol. 340, no. 2, pp. 115–126, 1999. https://doi.org/10.1056/nejm199901143400207.Search in Google Scholar PubMed

[2] P. Libby, P. M. Ridker, and A. Maseri, “Inflammation and atherosclerosis,” Circulation, vol. 105, no. 9, pp. 1135–1143, 2002. https://doi.org/10.1161/hc0902.104353.Search in Google Scholar PubMed

[3] D. Wolf and K. Ley, “Immunity and inflammation in atherosclerosis,” Circ. Res., vol. 124, no. 2, pp. 315–327, 2019. https://doi.org/10.1161/circresaha.118.313591.Search in Google Scholar

[4] T. Silva, W. Jäger, M. Neuss-Radu, and A. Sequeira, “Modeling of the early stage of atherosclerosis with emphasis on the regulation of the endothelial permeability,” J. Theor. Biol., vol. 496, p. 110229, 2020, https://doi.org/10.1016/j.jtbi.2020.110229.Search in Google Scholar PubMed

[5] N. A. Avgerinos and P. Neofytou, “Mathematical modelling and simulation of atherosclerosis formation and progress: a review,” Ann. Biomed. Eng., vol. 47, no. 8, pp. 1764–1785, 2019. https://doi.org/10.1007/s10439-019-02268-3.Search in Google Scholar PubMed

[6] E. T. Alexander, et al.., “Macrophage reverse cholesterol transport in mice expressing apoa-i milano,” Arterioscler., Thromb., Vasc. Biol., vol. 29, no. 10, pp. 1496–1501, 2009. https://doi.org/10.1161/atvbaha.109.191379.Search in Google Scholar PubMed PubMed Central

[7] M. G. Watson, H. M. Byrne, C. Macaskill, and M. R. Myerscough, “A two-phase model of early fibrous cap formation in atherosclerosis,” J. Theor. Biol., vol. 456, pp. 123–136, 2018. https://doi.org/10.1016/j.jtbi.2018.08.010.Search in Google Scholar PubMed

[8] M. Sanson, E. Distel, and E. A. Fisher, “Hdl induces the expression of the m2 macrophage markers arginase 1 and fizz-1 in a stat6-dependent process,” PloS One, vol. 8, no. 8, p. e74676, 2013. https://doi.org/10.1371/journal.pone.0074676.Search in Google Scholar PubMed PubMed Central

[9] A. Friedman, W. Hao, and B. Hu, “A free boundary problem for steady small plaques in the artery and their stability,” J. Diff. Eq., vol. 259, no. 4, pp. 1227–1255, 2015. https://doi.org/10.1016/j.jde.2015.02.002.Search in Google Scholar

[10] A. D. Chalmers, C. A. Bursill, and M. R. Myerscough, “Nonlinear dynamics of early atherosclerotic plaque formation may determine the efficacy of high density lipoproteins (hdl) in plaque regression,” PloS One, vol. 12, no. 11, p. e0187674, 2017. https://doi.org/10.1371/journal.pone.0187674.Search in Google Scholar PubMed PubMed Central

[11] K. M. Azzam and M. B. Fessler, “Crosstalk between reverse cholesterol transport and innate immunity,” Trends Endocrinol. Metab., vol. 23, no. 4, pp. 169–178, 2012. https://doi.org/10.1016/j.tem.2012.02.001.Search in Google Scholar PubMed PubMed Central

[12] X. Wang, et al.., “Macrophage abca1 and abcg1, but not sr-bi, promote macrophage reverse cholesterol transport in vivo,” J. Clin. Invest., vol. 117, no. 8, pp. 2216–2224, 2007. https://doi.org/10.1172/jci32057.Search in Google Scholar PubMed PubMed Central

[13] W. Anlamlert, Y. Lenbury, and J. Bell, “Modeling fibrous cap formation in atherosclerotic plaque development: stability and oscillatory behavior,” Adv. Diff. Eq., vol. 2017, no. 1, pp. 1–15, 2017. https://doi.org/10.1186/s13662-017-1252-9.Search in Google Scholar

[14] A. Cohen, M. R. Myerscough, and R. S. Thompson, “Athero-protective effects of high density lipoproteins (hdl): an ode model of the early stages of atherosclerosis,” Bull. Math. Biol., vol. 76, no. 5, pp. 1117–1142, 2014. https://doi.org/10.1007/s11538-014-9948-4.Search in Google Scholar PubMed

[15] M. A. Bulelzai and J. L. Dubbeldam, “Long time evolution of atherosclerotic plaques,” J. Theor. Biol., vol. 297, pp. 1–10, 2012. https://doi.org/10.1016/j.jtbi.2011.11.023.Search in Google Scholar PubMed

[16] A. Adak, A. Devi, and P. Gupta, “Dynamical behaviour of pro- and anti-inflammatory cytokines during pathogenesis of atherosclerosis,” Commun. Biomath. Sci., vol. 6, no. 2, pp. 108–125, 2023. https://doi.org/10.5614/cbms.2023.6.2.3.Search in Google Scholar

[17] P. Libby and P. M. Ridker, “Inflammation and atherothrombosis: from population biology and bench research to clinical practice,” J. Am. Coll. Cardiol., vol. 48, no. 9S, pp. A33–A46, 2006. https://doi.org/10.1016/j.jacc.2006.08.011.Search in Google Scholar

[18] P. Willeit, et al.., “Carotid intima-media thickness progression as surrogate marker for cardiovascular risk: meta-analysis of 119 clinical trials involving 100 667 patients,” Circulation, vol. 142, no. 7, pp. 621–642, 2020. https://doi.org/10.1161/circulationaha.120.046361.Search in Google Scholar PubMed PubMed Central

[19] K. Marušić, E. Marušić-Paloka, and M. Vrdoljak, “Effects of plaque shape on arterial blood flow,” Z. Naturforsch. A, vol. 78, no. 7, pp. 643–649, 2023. https://doi.org/10.1515/zna-2023-0059.Search in Google Scholar

[20] A. Ougrinovskaia, R. S. Thompson, and M. R. Myerscough, “An ode model of early stages of atherosclerosis: mechanisms of the inflammatory response,” Bull. Math. Biol., vol. 72, no. 6, pp. 1534–1561, 2010. https://doi.org/10.1007/s11538-010-9509-4.Search in Google Scholar PubMed

[21] N. El Khatib, S. Génieys, and V. Volpert, “Atherosclerosis initiation modeled as an inflammatory process,” Math. Modell. Nat. Phenom., vol. 2, no. 2, pp. 126–141, 2007. https://doi.org/10.1051/mmnp:2008022.Search in Google Scholar

[22] N. E. Khatib, S. Génieys, B. Kazmierczak, and V. Volpert, “Reaction–diffusion model of atherosclerosis development,” J. Math. Biol., vol. 65, no. 2, pp. 349–374, 2012. https://doi.org/10.1007/s00285-011-0461-1.Search in Google Scholar PubMed

[23] M. Bulelzai, J. Dubbeldam, and H. G. E. Meijer, “Bifurcation analysis of a model for atherosclerotic plaque evolution,” Phys. D: Nonlin. Phenom., vol. 278, pp. 31–43, 2014. https://doi.org/10.1016/j.physd.2014.04.005.Search in Google Scholar

[24] G. Lui and M. R. Myerscough, “Modelling preferential phagocytosis in atherosclerosis: delineating timescales in plaque development,” Bull. Math. Biol., vol. 83, no. 9, p. 96, 2021. https://doi.org/10.1007/s11538-021-00926-z.Search in Google Scholar PubMed

[25] A. Friedman and W. Hao, “A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors,” Bull. Math. Biol., vol. 77, no. 5, pp. 758–781, 2015. https://doi.org/10.1007/s11538-014-0010-3.Search in Google Scholar PubMed

[26] W. Hao and A. Friedman, “The ldl-hdl profile determines the risk of atherosclerosis: a mathematical model,” PloS One, vol. 9, no. 3, p. e90497, 2014. https://doi.org/10.1371/journal.pone.0090497.Search in Google Scholar PubMed PubMed Central

[27] M. Liu, Y. Cai, J. Pan, K. Peter, and Z. Li, “Macrophage polarization as a potential therapeutic target for atherosclerosis: a dynamic stochastic modelling study,” Roy. Soc. Open Sci., vol. 9, no. 8, p. 220239, 2022. https://doi.org/10.1098/rsos.220239.Search in Google Scholar PubMed PubMed Central

[28] G. A. Younes and N. E. Khatib, “Mathematical modeling of atherogenesis: atheroprotective role of hdl,” J. Theor. Biol., vol. 529, p. 110855, 2021. https://doi.org/10.1016/j.jtbi.2021.110855.Search in Google Scholar PubMed

[29] M. Navab, et al.., “Hdl and the inflammatory response induced by ldl-derived oxidized phospholipids,” Arterioscler., Thromb., Vasc. Biol., vol. 21, no. 4, pp. 481–488, 2001. https://doi.org/10.1161/01.atv.21.4.481.Search in Google Scholar PubMed

[30] F. Brites, M. Martin, I. Guillas, and A. Kontush, “Antioxidative activity of high-density lipoprotein (hdl): mechanistic insights into potential clinical benefit,” BBA Clin., vol. 8, pp. 66–77, 2017. https://doi.org/10.1016/j.bbacli.2017.07.002.Search in Google Scholar PubMed PubMed Central

[31] F. Ito and T. Ito, “High-density lipoprotein (hdl) triglyceride and oxidized hdl: new lipid biomarkers of lipoprotein-related atherosclerotic cardiovascular disease,” Antioxidants, vol. 9, no. 5, p. 362, 2020. https://doi.org/10.3390/antiox9050362.Search in Google Scholar PubMed PubMed Central

[32] S. Kumar, R. Kumar, C. Cattani, and B. Samet, “Chaotic behaviour of fractional predator-prey dynamical system,” Chaos, Solit. Fractals, vol. 135, p. 109811, 2020. https://doi.org/10.1016/j.chaos.2020.109811.Search in Google Scholar

[33] S. Kumar, A. Kumar, B. Samet, and H. Dutta, “A study on fractional host–parasitoid population dynamical model to describe insect species,” Numeric. Methods Part. Diff. Eqs., vol. 37, no. 2, pp. 1673–1692, 2021. https://doi.org/10.1002/num.22603.Search in Google Scholar

[34] S. Kumar, R. Kumar, M. S. Osman, and B. Samet, “A wavelet based numerical scheme for fractional order seir epidemic of measles by using genocchi polynomials,” Numeric. Methods Part. Diff. Eqs., vol. 37, no. 2, pp. 1250–1268, 2021. https://doi.org/10.1002/num.22577.Search in Google Scholar

[35] M. A. Khan, S. Ullah, and S. Kumar, “A robust study on 2019-ncov outbreaks through non-singular derivative,” Eur. Phys. J. Plus, vol. 136, no. 2, pp. 1–20, 2021. https://doi.org/10.1140/epjp/s13360-021-01159-8.Search in Google Scholar PubMed PubMed Central

[36] H. Mohammadi, S. Kumar, S. Rezapour, and S. Etemad, “A theoretical study of the caputo–fabrizio fractional modeling for hearing loss due to mumps virus with optimal control,” Chaos, Solit. Fractals, vol. 144, p. 110668, 2021. https://doi.org/10.1016/j.chaos.2021.110668.Search in Google Scholar

[37] S. Kumar, R. P. Chauhan, S. Momani, and S. Hadid, “Numerical investigations on covid-19 model through singular and non-singular fractional operators,” Numeric. Methods Part. Diff. Eqs., vol. 40, no. 1, p. e22707, 2024. https://doi.org/10.1002/num.22707.Search in Google Scholar

[38] A. Dutta and P. K. Gupta, “Stability analysis of hiv/aids dynamics: modelling the tested and untested populations,” Pramana, vol. 96, no. 1, p. 42, 2022. https://doi.org/10.1007/s12043-021-02288-6.Search in Google Scholar

[39] M. B. Devi, A. Devi, P. K. Gupta, and D. Tripathi, “Response of vaccination on community transmission of covid-19: a dynamical approach,” Eur. Phys. J. Spec. Top., vol. 231, no. 18, pp. 3749–3765, 2022. https://doi.org/10.1140/epjs/s11734-022-00652-0.Search in Google Scholar PubMed PubMed Central

[40] Y. M. Hong, “Atherosclerotic cardiovascular disease beginning in childhood,” Kor. Circ. J., vol. 40, no. 1, pp. 1–9, 2010. https://doi.org/10.4070/kcj.2010.40.1.1.Search in Google Scholar PubMed PubMed Central

[41] A. Dutta, A. Adak, and P. K. Gupta, “Analysis of fractional-order deterministic hiv/aids model during drug therapy treatment,” in Soft Computing for Problem Solving: SocProS 2018, vol. 1, Singapore, Springer, 2020, pp. 1–8.10.1007/978-981-15-0035-0_1Search in Google Scholar

[42] L. Perko, Differential Equations and Dynamical Systems, Switzerland, Springer Science & Business Media, 2013.Search in Google Scholar

[43] A. Hidalgo and L. Tello, “Numerical simulation of a porous medium type atherosclerosis initiation model,” Comput. Fluids, vol. 169, pp. 380–387, 2018. https://doi.org/10.1016/j.compfluid.2017.07.019.Search in Google Scholar

[44] D. Mukherjee, L. N. Guin, and S. Chakravarty, “Dynamical behavior of a mathematical model of early atherosclerosis,” Int. J. Model. Simul. Sci. Comput., vol. 11, no. 01, p. 2050006, 2020. https://doi.org/10.1142/s1793962320500063.Search in Google Scholar

[45] K. H. Cho, “The current status of research on high-density lipoproteins (hdl): a paradigm shift from hdl quantity to hdl quality and hdl functionality,” Int. J. Mol. Sci., vol. 23, no. 7, p. 3967, 2022. https://doi.org/10.3390/ijms23073967.Search in Google Scholar PubMed PubMed Central

Received: 2023-11-28
Accepted: 2023-12-14
Published Online: 2024-01-09
Published in Print: 2024-05-27

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

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  2. General
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https://doi.org/10.1515/zna-2023-0331
Keywords for this article
atherosclerosis; reaction-diffusion equation; reverse cholesterol transport; stability analysis; bifurcation

Articles in the same Issue

  1. Frontmatter
  2. General
  3. Magnetoacoustics and magnetic quantization of Fermi states in relativistic plasmas
  4. Atomic, Molecular & Chemical Physics
  5. Investigations on the EPR parameters and local structures for the substitutional Ti3+ and W5+ centers in stishovite
  6. Dynamical Systems & Nonlinear Phenomena
  7. The effects of viscosity on the structure of shock waves in a van der Waals gas
  8. Traveling wavefronts in an anomalous diffusion predator–prey model
  9. Bifurcation and stability analysis of atherosclerosis disease model characterizing the anti-oxidative activity of HDL during short- and long-time evolution
  10. Nuclear Physics
  11. Investigation of 90,92Zr(n,γ)91,93Zr in the s-process nucleosynthesis
  12. Quantum Theory
  13. Quantum-mechanical treatment of two particles in a potential box
  14. Solid State Physics & Materials Science
  15. Unveiling the luminescence property of Pr-incorporated barium cerate perovskites for white LED applications
  16. Electrical and magnetic properties of MF/CuAl nanocomposites
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