A mathematical model for atherosclerotic plaque formation and arterial wall remodelling

Atherosclerosis is a condition whereby fatty material is deposited in the walls of arteries (plaque) resulting in a thickening of the wall. Here we develop a mathematical model describing the biochemical processes of the formation of atherosclerotic plaque, which involves the interaction between pro-inflammatory mediators, modified low density lipoprotein, monocytes, macrophages, foam cells and high density lipoprotein. In addition, based on the outcomes of the biochemical model, we develop a plaque growth model that takes into account both the inward and outward expansion of the arterial walls. We examine the stability and bifurcations of this model in order to explore the clinical and medical implications of plaque growth. References M. J. van Gils and D. Vukadinovic, A. C. van Dijk, D. W. J. Dippel, W. J. Niessen and A. van der Lugt. Carotid atherosclerotic plaque progression and change in plaque composition over time: A 5-year follow-up study using serial CT angiography. Am. J. Neuroradiol., 37(7):1267–1273, 2012. doi:10.3174/ajnr.A2970 M. Naghavi, P. Libby, E. Falk, et al. From vulnerable plaque to vulnerable patient; A call for new definitions and risk assessment strategies: Part I. Circulation, 108(14):1664–1672, 2003. doi:10.1161/01.CIR.0000087480.94275.97 T. F. Lucher and M. Barton. Biology of the Endothelium. Clin Cardiol., 20(Supp. 2):3–10, 1997. https://www.researchgate.net/profile/Matthias_Barton/publication/13807118_Biology_of_the_endothelium/links/0046352c54fe700895000000.pdf E. Mannarino and M. Pirro. Endothelial injury and repair: A novel theory for atherosclerosis. Angiology, 59:69S-72S, 2008. doi:10.1177/0003319708320761 C. Davis, J. Fischer, K. Ley and I. J. Sarembock. The role of inflammation in vascular injury and repair. J. Thromb. Haemost., 1(8):1699-1709, 2003. doi:10.1046/j.1538-7836.2003.00292.x G. K. Hansson and P. Libby. The immune response in atherosclerosis: a double-edged sword. Nat. Rev. Immunol., 6(7):508–519, 2006. doi:10.1038/nri1882 R. Ross. Atherosclerosis–-an inflammatory disease, New Engl. J. Med., 340(2):115-126, 1999. doi:10.1056/NEJM199901143400207 B. F. Asztalos. High-density lipoprotein metabolism and progression of atherosclerosis: new insights from the HDL atherosclerosis treatment study. Curr. Opin. Cardiol., 19(4):385–391, 2004. http://journals.lww.com/co-cardiology/Abstract/2004/07000/High_density_lipoprotein_metabolism_and.16.aspx N. El Khatib, S. Genieys and V. Volpert. Atherosclerosis initiation modeled as an inflammatory process. Math. Model. Nat. Phenom., 2(2):126–141, 2007. doi:10.1051/mmnp:2008022 P.-W. Fok. Mathematical model of intimal thickening in atherosclerosis: Vessel stenosis as a free boundary problem. J. Theor. Biol., 314:23–33, 2012. doi:10.1016/j.jtbi.2012.07.029 M. A. K. Bulelzai and J. L. A. Dubbeldam. Long time evolution of atherosclerotic plaques, J. Theor. Biol., 297:1–10, 2012. doi:10.1016/j.jtbi.2011.11.023 A. D. Chalmers, A. Cohen, C. A. Bursill and M. R. Myerscough. Bifurcation and dynamics in a mathematical model of early atherosclerosis. J. Math. Biol., 71:1451–1480, 2015. doi:10.1007/s00285-015-0864-5 A. Friedman and W. Hao. A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors. B. Math. Biol., 77(5):758–781, 2015. doi:10.1007/s11538-014-0010-3 G. M. Chisolm, S. L. Hazen, P. L. Fox and M. K. Cathcart. The oxidation of lipoproteins by monocytes-macrophages: Biochemical and biological mechanisms. J. Biol. Chem., 274(37):25959–25962, 1999. doi:10.1074/jbc.274.37.2595 A. Daugherty, N. R. Webb, D. L. Rateri and V. L. King. Thematic review series: The immune system and atherogenesis. Cytokine regulation of macrophage functions in atherogenesis, J. Lipid Res., 46(9):1812-1822, 2005. doi:10.1194/jlr.R500009-JLR200 L. J. H. van Tits, R. Stienstra, P. L. van Lent, M. G. Netea, L. A. B. Joosten and A. F. H. Stalenhoef. Oxidized LDL enhances pro-inflammatory responses of alternatively activated M2 macrophages: A crucial role for Kruppel-like factor 2. Atherosclerosis, 214(8):345–349, 2011. doi:10.1016/j.atherosclerosis.2010.11.018 Z. Mallat, S. Besnard, M. Duriez, et al. Protective role of interleukin-10 in atherosclerosis. Circ. Res., 85(8):e17-e24, 1999. doi:10.1161/01.RES.85.8.e17 P. J. Barter, S. Nicholls, K.-A. Rye, G. M. Anantharamaiah, M. Navab and A. M. Fogelman. Antiinflammatory properties of HDL, Circ. Res., 95(8):764–772, 2004. doi:10.1161/01.RES.0000146094.59640.13 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, 8(8):e74676, 2013. doi:10.1371/journal.pone.0074676 L. Tilling, J. Hunt, A. Donald, B. Clapp and P. Chowienczyk. Arterial injury and endothelial repair: Rapid recovery of function after mechanical injury in healthy volunteers. Card. Res. Prac., 2014:367537, 2014. doi:10.1155/2014/367537 F. Alexandrea, V. H. S. Zagoa, N. B. Panzoldo, et al. Reference values for high-density lipoprotein particle size and volume by dynamic light scattering in a Brazilian population sample and their relationships with metabolic parameters. Clin. Chim. Acta, 442:63–72, 2015. doi:10.1016/j.cca.2015.01.006 T. Khamdaeng, J. Luo, J. Vappou, P. Terdtoon, E. E. Konofagou. Arterial stiffness identification of the human carotid artery using the stress-strain relationship in vivo. Ultrasonics, 52(3):402–411, 2012. doi:10.1016/j.ultras.2011.09.006 T. G. Kuznetsova, M. N. Starodubtseva, N. I. Yegorenkov, S. A. Chizhik, R. I. Zhdanov. Atomic force microscopy probing of cell elasticity, Micron., 38(8):824–833, 2007, doi:10.1016/j.micron.2007.06.011 J. Bell, C. Breward, T. Chou, P.-W. Fok, J. M. Haugh, Q. Li, L. Rossi, A. Walter, X. Yang, A. Zemlyanova and N. Zhang. Mathematical models for vulnerable plaques. Technical Report, 2009. http://www.maths-in-industry.org/miis/272/ A. Dhooge, W. Govaerts and Y. A. Kuznetsov. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM T. Math. Softw. (TOMS), 29(2):141–164, 2003. doi:10.1145/779359.779362. R. Ross. The pathogenesis of atherosclerosis: A perspective for the 1990s. Nature, 362:801–809, 1993. doi:10.1038/362801a0