All Title Author
Keywords Abstract

PLOS ONE  2014 

The LDL-HDL Profile Determines the Risk of Atherosclerosis: A Mathematical Model

DOI: 10.1371/journal.pone.0090497

Full-Text   Cite this paper   Add to My Lib


Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart attack. It has long been recognized that the formation of a plaque relates to the cholesterol concentration in the blood. For example, individuals with LDL above 190 mg/dL and HDL below 40 mg/dL are at high risk, while individuals with LDL below 100 mg/dL and HDL above 50 mg/dL are at no risk. In this paper, we developed a mathematical model of the formation of a plaque, which includes the following key variables: LDL and HDL, free radicals and oxidized LDL, MMP and TIMP, cytockines: MCP-1, IFN-γ, IL-12 and PDGF, and cells: macrophages, foam cells, T cells and smooth muscle cells. The model is given by a system of partial differential equations with in evolving plaque. Simulations of the model show how the combination of the concentrations of LDL and HDL in the blood determine whether a plaque will grow or disappear. More precisely, we create a map, showing the risk of plaque development for any pair of values (LDL,HDL).


[1]  Hoyert DL, Xu J (2012) Deaths: Preliminary Data for 2011, National Vital Statistics Reports 61.
[2]  The top 10 causes of death. Available: Accessed 2013 Dec. 18.
[3]  Frink RJ (2002) Inflammatory Atherosclerosis, Heart Research Foundation.
[4]  Moreno PR (2010) Vulnerable Plaque: Definition, Diagnosis, and Treatment. Cardiology Clinics 28: 1–30. doi: 10.1016/j.ccl.2009.09.008
[5]  Libby P, Ridker PM, Maseri A (2002) Inflammation and Atherosclerosis. Circulation 105: 1135–1143. doi: 10.1161/hc0902.104353
[6]  Cohen A, Myerscough MR, Thompson RS (2012) Athero-protective effects of High Density Lipoproteins (HDL): An ODE model of the early stages of atherosclerosis. Preprint doi: 10.1007/s11538-014-9948-4
[7]  McKay C, McKee S, Mottram N, Mulholland T, Wilson S (2005) Towards a model of atherosclerosis. Strathclyde Mathematics Research Report
[8]  Ryu BH (2000) Low Density Lipoprotein (LDL), Atherosclerosis and Antioxidants. Biotechnol. Bioprocess Eng 5: 313–319. doi: 10.1007/bf02942205
[9]  Harrington JR (2000) The Role of MCP-1 in Atherosclerosis. Stem Cells 18: 65–66. doi: 10.1634/stemcells.18-1-65
[10]  Reape TJ, Groot P (1999) Chemokines and atherosclerosis. Atherosclerosis 147: 213C225. doi: 10.1016/s0021-9150(99)00346-9
[11]  Osterud B, Bjorklid E (2003) Role of Monocytes in Atherogenesis. Physiol Rev 83: 1069–1112.
[12]  Gui T, Shimokado A, Sun Y, Akasaka T, Muragaki Y (2012) Diverse Roles ofMacrophages in Atherosclerosis: From Inflammatory Biology to Biomarker Discovery. Mediators of Inflammation 2012: ID693083. doi: 10.1155/2012/693083
[13]  Johnson JL, Newby AC (2009) Macrophage heterogeneity in atherosclerotic plaques. Curr Opin Lipidol 20: 370–378. doi: 10.1097/mol.0b013e3283309848
[14]  Little MP, Gola A, Tzoulaki I (2009) A Model of Cardiovascular Disease Giving a Plausible Mechanism for the Effect of Fractionated Low-Dose Ionizing Radiation Exposure. PLoS Computational Biology 5: e1000539. doi: 10.1371/journal.pcbi.1000539
[15]  Calvez V, Ebde A, Meunier N, Raoult A (2009) Mathematical modelling of the atherosclerotic plaque formation. CEMRACS 2008 - Modelling and Numerical Simulation of Complex Fluids 28: 1–12. doi: 10.1051/proc/2009036
[16]  Ross R, Masuda J, Raines EW, Gown AM, Katsuda S, et al. (1990) Localization of PDGF-B protein in macrophages in all phases of atherogenesis. Science 248: 1009–1012. doi: 10.1126/science.2343305
[17]  Raines EW, Ross R (1993) Smooth muscle cells and the pathogenesis of the lesions of atherosclerosis. Br Heart J 69: 30–37. doi: 10.1136/hrt.69.1_suppl.s30
[18]  Rodriguez JA, Orbe J, Paramo JA (2007) Metalloproteases, Vascular Remodeling, and Atherothrombotic Syndromes. Rev Esp Cardiol 60: 959–967. doi: 10.1157/13109649
[19]  Fabunmi RP, Sukhova GK, Sugiyama S, Libby P (1998) Expression of Tissue Inhibitor of Metalloproteinases-3 in Human Atheroma and Regulation in Lesion-Associated Cells A Potential Protective Mechanism in Plaque Stability. Circ Res 83: 270–278. doi: 10.1161/01.res.83.3.270
[20]  King IL, Segal BM (2005) Cutting edge: IL-12 induces CD4+CD25- T cell activation in the presence of T regulatory cells. J Immunol 175: 641–645. doi: 10.4049/jimmunol.175.2.641
[21]  Hansson GK, Holm J, Jonasson L (1989) Detection of activated T lymphocytes in the human atherosclerotic plaque, Am. J Pathol 135: 169–175.
[22]  Kosaka C, Masuda J, Shimokado K, Zen K, Yokota T, et al. (1992) Interferon-gamma suppresses PDGF production from THP-1 cells and blood monocyte-derived macrophages. Atherosclerosis 97: 75–87. doi: 10.1016/0021-9150(92)90053-j
[23]  Barter P (2005) The role of HDL-cholesterol in preventing atherosclerotic disease, European Heart Journal Supplements. 7: : 4–8.
[24]  Barter P, Brandrup-Wognsen G, Palmer M, Nicholls S (2010) Effect of statins on HDL-C: a complex process unrelated to changes in LDL-C: analysis of the VOYAGER Database. J Lipid Res 51: 1546–1553. doi: 10.1194/jlr.p002816
[25]  Mayo Clinic Staff, Cholesterol levels: What numbers should you aim for? Available: Accessed 2013 Dec. 12.
[26]  Rahdert DA, Sweet WL, Tio FO, Janicki C, Duggan DM (1999) Measurement of density and calcium in human atherosclerotic plaque and implications for arterial brachytherapy. Cardiovasc Radiat Med 1: 358–367. doi: 10.1016/s1522-1865(00)00030-5
[27]  Orbe J, Rodriguez JA, Arias R (2003) Antioxidant vitamins increase the collagen content and reduce MMP-1 in a porcine model of atherosclerosis: Implications for plaque stabilization. Atherosclerosis 167: 45–53. doi: 10.1016/s0021-9150(02)00392-1
[28]  Kim Y, Roh S, Lawler S, Friedman A (2011) miR and AMPK mutual antagonism in glioma cell migration and proliferation: a mathematical model. PLoS One 6: e28293. doi: 10.1371/journal.pone.0028293
[29]  Byrne HM (1997) The importance of intercellular adhesion in the development of carcinomas. IMA Journal of Mathematics Applied in Medicine & Biology 14: 305–323. doi: 10.1093/imammb/14.4.305
[30]  Byrne HM (1999) A weakly nonlinear analysis of a model of avascular solid tumour growth. J Math Biol 39: 59–89. doi: 10.1007/s002850050163
[31]  Byrne HM, Chaplain MA (1996) Modelling the role of cell-cell adhesion in the growth and development of carcinomas. Mathematical and Computer Modelling 24: 1–7. doi: 10.1016/s0895-7177(96)00174-4
[32]  Chothia C (1976) The nature of the accessible and buried surfaces in proteins. J Mol Biol 105: 1–12. doi: 10.1016/0022-2836(76)90191-1
[33]  Janin J, Chothia C (1978) Role of Hydrophobicity in the Binding of Coenzymes. Biochemistry 17: 2943–2948. doi: 10.1021/bi00608a001
[34]  Cobbold C, Sherratt J, Maxwell S (2002) Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bulletin of Mathematical Biology 64: 65–95. doi: 10.1006/bulm.2001.0267
[35]  Ingold KU, Bowry VW, Stocker R, Walling C (1993) Autoxidation of lipids and antioxidation by a-tocopherol and ubiquinol in homogeneous solution and in aqueous dispersions of lipids: unrecognised consequences of lipid particle size as exemplified by oxidation of human low density lipoprotein. Proc Natl Acad Sci 90: 45–49. doi: 10.1073/pnas.90.1.45
[36]  Kim Y, Lawler S, Nowicki M, Chiocca E, Friedman A (2009) A mathematical model for pattern formation of glioma cells outside the tumor spheroid core. J Theor Biol 260: 359–371. doi: 10.1016/j.jtbi.2009.06.025
[37]  Kim Y, Friedman A (2010) Interaction of tumor with its micro-environment: A mathematical model. Bull Math Biol 75: 1029–1068. doi: 10.1007/s11538-009-9481-z
[38]  Mercapide J, Cicco R, Castresana J, Klein-Szanto A (2003) 450 Stromelysin-1/matrix metalloproteinase-3 (MMP-3) expression accounts for invasive properties of human astrocytoma cell lines. Int J Cancer 106: 676–682. doi: 10.1002/ijc.11286
[39]  Friedman A, Turner J, Szomolay B (2008) A model on the influence of age on immunity to infection with Mycobacterium tuberculosis. Exp Gerontol 43: 275–285. doi: 10.1016/j.exger.2007.12.004
[40]  Orme IM (1987) Aging and immunity to tuberculosis: increased susceptibility of old mice reflects a decreased capacity to generate mediator T lymphocytes. J Immunol 138: 4414–4418.
[41]  Barrett T, Benditt E (1988) Platelet-derived growth factor gene expression in human atherosclerotic plaques and normal artery wall. Proc Natl Acad Sci 85: 2810–2814. doi: 10.1073/pnas.85.8.2810
[42]  Xue C, Friedman A, Sen CK (2009) A mathematical model of ischemic cutaneous wounds. Proc Natl Acad Sci 106: 16782–16787. doi: 10.1073/pnas.0909115106
[43]  Dollery CM, Libby P (2006) Atherosclerosis and proteinase activation. Cardiovascular Research 69: 625–635. doi: 10.1016/j.cardiores.2005.11.003
[44]  Chen D, Roda JM, Mash CB, Eubank TD, Friedman A (2012) Hypoxia inducible factors mediated-inhibition of cancer by GM-CSF: A mathematical model. Bulletin of Mathematical Biology 74: 2752–2777. doi: 10.1007/s11538-012-9776-3
[45]  Day J, Friedman A, Schlesinger LS (2009) Modeling the immune rheostat of macrophages in the lung in response to infection. Proc Natl Acad Sci 106: 11246–11251. doi: 10.1073/pnas.0904846106
[46]  Yamamoto S, Nguyen JH (2006) TIMP-1/MMP-9 imbalance in brain edema in rats with fulminant hepatic failure. J Surg Res 134: 307–314. doi: 10.1016/j.jss.2005.11.588
[47]  Olson MW, Gervasi DC, Mobashery S, Fridman R (1997) Kinetic analysis of the binding of human matrix metalloproteinase-2 and -9 to tissue inhibitor of metalloproteinase (TIMP)-1 and TIMP-2. J Biol Chem 21: 29975–29983. doi: 10.1074/jbc.272.47.29975
[48]  Palosaari H, Yliopisto O (2003) Matrix Metalloproteinases (MMPs) and Their Specific Tissue Inhibitors (TIMPs) in Mature Human Odontoblasts and Pulp Tissue. Oulun yliopisto doi: 10.1034/j.1600-0722.2003.00026.x
[49]  Bofill M, Janossy G, Lee CA, MacDonald-Burns D, Phillips AN, et al. (1992) Laboratory 475 control values for CD4 and CD8 T lymphocytes Implications for HIV-1 diagnosis. Clin Exp Immunol 88: 243–252. doi: 10.1111/j.1365-2249.1992.tb03068.x
[50]  Persson PO, Strang G (2004) A Simple Mesh Generator in MATLAB. SIAM Review 46: 329–345. doi: 10.1137/s0036144503429121
[51]  Alberty J, Carstensen C, Funken SA (1999) Remarks around 50 lines of Matlab: short finite element implementation. Numerical Algorithms 20: 117–137. doi: 10.1023/a:1019155918070
[52]  Z Li. The finite element method for two dimensional problems. Chapter 9 Avialable Accessed 2013 Oct. 10.
[53]  Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysisin systems. J Theor Biol 254: 178–196. doi: 10.1016/j.jtbi.2008.04.011
[54]  Arsenault BJ, Rana JS, Stroes ES, Despres JP, Shah PK, et al. (2009) Beyond low-density lipoprotein cholesterol: respective contributions of non-high-density lipoprotein cholesterol levels, triglycerides, and the total cholesterol/high-density lipoprotein cholesterol ratio to coronary heart disease risk in apparently healthy men and women. J Am Coll Cardiol 55: 35–41.
[55]  Mayo Clinic Staff, How important is cholesterol ratio? Available: Accessed 2013 Dec. 20.
[56]  Crestor Managing Choleserol. Available:
[57]  Otani H (2013) Site-Specific Antioxidative Therapy for Prevention of Atherosclerosis and Cardiovascular Disease. Oxid Med Cell Longev
[58]  Ozkanlar S, Akcay F (2012) Antioxidant vitamins in atherosclerosis–animal experiments and clinical studies. Adv Clin Exp Med 21: 115–123.
[59]  Tardif J (2005) Antioxidants and atherosclerosis: emerging drug therapies. Curr Atheroscler Rep 7: 71–77. doi: 10.1007/s11883-005-0078-1
[60]  Levonen A, Vahakangas E, Koponen JK, Herttuala S (2008) Antioxidant Gene Therapy for Cardiovascular Disease Current Status and Future Perspectives. Circulation 117: 2142–2150. doi: 10.1161/circulationaha.107.718585
[61]  Ishigaki Y, Oka Y, Katagiri H (2009) Circulating oxidized LDL: a biomarker and 501 a pathogenic factor. Curr Opin Lipidol 20: 363–369. doi: 10.1097/mol.0b013e32832fa58d
[62]  Tsimikas S, Witztum JL (2001) Measuring circulating oxidized low-density lipoprotein to evaluate coronary risk. Circulation 103: 1930–1932. doi: 10.1161/01.cir.103.15.1930
[63]  Rajman I, Eacho P, Chowienczyk P, Ritter J (1999) LDL particle size: an important drug target? Br J Clin Pharmacol 48: 125–133. doi: 10.1046/j.1365-2125.1999.00991.x
[64]  Hokland B, Mendez A, Oram J (1992) Cellular localization and characterization of proteins that bind high density lipoprotein. J Lipid Res 33: 1335–1342.
[65]  Mitsuhashi T, Ono K, Fukuda M, Hasegawa Y (2013) Free radical scavenging ability and structure of a 90-kDa protein from the scallop shell. Fisheries Science 79: 495–502. doi: 10.1007/s12562-013-0616-7
[66]  Garcia-Tunon I, Ricote M, Ruiz A, Fraile B, Paniagua R, et al. (2007) Influence of IFN-gamma and its receptors in human breast cancer. BMC Cancer 7: 158. doi: 10.1186/1471-2407-7-158
[67]  Antoniades H (1981) Human platelet-derived growth factor (PDGF): purification of PDGF-I and PDGF-II and separation of their reduced subunits. Proc Natl Acad Sci 78: 7314–7317. doi: 10.1073/pnas.78.12.7314
[68]  Yokochi S, Hashimoto H, Ishiwata Y, Shimokawa H, Haino M, et al. (2001) An Anti-Inflammatory Drug, Propagermanium, May Target GPI-Anchored Proteins Associated with an MCP-1 Receptor, CCR2. Journal of Interferon and Cytokine Research 21: 389–398. doi: 10.1089/107999001750277862
[69]  Hamza T, Barnett J, Li B (2010) Interleukin 12 a Key Immunoregulatory Cytokine in Infection Applications. Int J Mol Sci 11: 789–806. doi: 10.3390/ijms11030789
[70]  O'Keefe JH, Cordain L, Harris WH, Moe RM, Vogel R (2004) Optimal low-density lipoprotein is 50 to 70 mg/dl: lower is better and physiologically normal. Journal of the American College of Cardiology 43: 2142–2146. doi: 10.1016/j.jacc.2004.03.046
[71]  Miller DC, Thapa A, Haberstroh KM, Webster TJ (2004) Endothelial and vascular smooth muscle cell function on poly(lactic-co-glycolic acid) with nano-structured surface features. Biomaterials 25: 53–61. doi: 10.1016/s0142-9612(03)00471-x
[72]  Bowen-Pope DF, Malpass TW, Foster DM, Ross R (1984) Platelet-derived growth factor in 526 vivo: levels, activity, and rate of clearance. Blood 64: 458–469.
[73]  Rhodes J, Sharkey J, Andrews P (2009) Serum IL-8 and MCP-1 concentration do not identify patients with enlarging contusions after traumatic brain injury. J Trauma 66: 1591–1597. doi: 10.1097/ta.0b013e31819a0344
[74]  Gattorno M, Picco P, Vignola S, Stalla F, Buoncompagni A, et al. (1998) Serum interleukin 12 concentration in juvenile chronic arthritis. Ann Rheum Dis 57: 425–428. doi: 10.1136/ard.57.7.425
[75]  Tietz NW (1999) Clinical Guide to Laboratory Tests. Philadelphia 3rd Ed.
[76]  Liao KL, Bai XF, Friedman A (2013) The role of CD200-CD200R in tumor immune evasion. J Theor Biol 328: 65–76. doi: 10.1016/j.jtbi.2013.03.017
[77]  Tsukaguchi K, Balaji KN, Boom WH (1995) CD4+ alpha beta T cell and gamma delta T cell responses to Mycobacterium tuberculosis. J Immunol 154: 1786–1796.


comments powered by Disqus