Numerical and analytical study of an atherosclerosis inflammatory disease model

Journal of Mathematical Biology

Volumn 68, Issue 7, 2014, Pages 1785-1814

  Hidalgo, A ^a   Tello, L ^a   Toro, E F ^b  

^a UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

^b UNIVERSITY OF TRENTO   (Italy)

Author keywords

Asymptotic behaviour; Atherosclerosis; Finite volume formulation; Parabolic reaction diffusion models

Indexed keywords

ALGORITHM; ANIMAL; ATHEROSCLEROSIS; BIOLOGICAL MODEL; COMPUTER SIMULATION; DISEASE COURSE; HUMAN; IMMUNOLOGY; INFLAMMATION; MATHEMATICAL COMPUTING; MATHEMATICAL PHENOMENA; MOUSE; NONLINEAR SYSTEM;

ALGORITHMS; ANIMALS; ATHEROSCLEROSIS; COMPUTER SIMULATION; DISEASE PROGRESSION; HUMANS; INFLAMMATION; MATHEMATICAL CONCEPTS; MICE; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS; NUMERICAL ANALYSIS, COMPUTER-ASSISTED;

EID: 84900000016     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-013-0688-0     Document Type: Article

References (38)

1. Balsara DS, Shu Ch-W (2000) Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly order of accuracy. J Comp Phys 160: 405-452.
2. Calvez V, Houot JG, Meunier N, Raoult A, Rusnakova G (2010) Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM: Proceedings 30: 1-14.
3. Cantrell RS, Cosner C (2003) Spatial ecology via reaction-diffusion equations. Wiley series in Mathematical and computational Biology. John Wiley & Sons, Ltd., Chichester.
4. Cobbold CA, Sherrat JA, Maxwell SRJ (2002) Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bull Math Biol 64: 65-95.
5. Chow CC, Clermont G, Kumar R, Lagoa C, Tawadrous Z, Gallo D, Betten B, Bartels J, Constantine G, Fink MP, Billiar TR, Vodovotz Y (2005) The acute inflammatory response in diverse shock states. Shock 24(1): 74-84.
6. Dumbser M, Enaux C, Toro EF (2008) Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Com Phys 227: 3971-4001.
7. Dumbser M (2010) Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-stokes equations. Comput Fluids 39(1): 60-76.
8. Edelstein-Keshet L, Spiros A (2002) Exploring the formation of Alzheimer's disease senile plaques in silico. J Theor Biol 216(3): 301-326.
9. El Khatib N, Genieys S, Volpert V (2007) Atherosclerosis initiation modeled as an inflammatory disease. Math Model Nat Phenom 2(2): 126-141.
10. El Khatib N, Genieys S, Kazmierczak B, Volpert V (2009) Mathematical modelling of atherosclerosis as an inflammatory disease. Phil Trans R Soc 367: 4877-4886.
11. El Khatib N, Genieys S, Kazmierczak B, Volpert V (2012) Reaction diffusion model of atherosclerosis development. J Math Biol 65(2): 349-374. doi: 10. 1007/s00285-011-0461-1.
12. Gassner G, Lorcher F, Munz CD (2007) A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J Comput Phys 224(2): 1049-1063. doi: 10. 1016/j. jcp. 2006. 11. 004.
13. Godunov SK (1959) A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat Sbornik 47: 357-393.
14. Herald MC (2010) General model of inflammation. Bull Math Biol 72: 765-779. doi: 10. 1007/s11538-009-9468-9.
15. Hidalgo A, Dumbser M (2011) ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J Sci Comput 48(1-3): 173-189. doi: 10. 1007/s10915-010-9426-6.
16. Hidalgo A, Tello L (2011) A finite volume scheme for simulating the coupling between deep ocean and an atmospheric energy balance model. Modern Mathematical Tools and Techniques capturing complexity. Springer, Berlin, pp 239-255.
17. Ibragimov AI, McNeal CJ, Ritter LR, Walton JR (2005) A mathematical model of atherogenesis as an inflammatory response. Math Med Biol 22(4): 305-333.
18. Ikeda N, Torii R (2005) When does atherosclerosis become irreversible? Chronological change from an early to an advanced atherosclerotic lesion observed by angioscopy. Angiology 56(4): 361-370.
19. Jiang GS, Shu Ch-W (1996) Efficient implementation of weighted ENO schemes. J Comp Phys 126: 202-228.
20. Levick JR (2010) An introduction to Cardiovascular Physiology. Fifth edition 2010 published in Great Britain by Hodder Arnold, a member of the Hodder Education, an Hachette UK company, London.
21. Li ZY, Howarth SPS, Tang T, Gillard JH (2006) How critical is fibrous cap thickness to carotid plaque stability? A flow-plaque interaction model. Stroke 37: 1195-1199. doi: 10. 1161/01. STR. 0000217331. 61083. 3b.
22. Libby P (2002) Inflammation in atherosclerosis. Nature 420: 19-26.
23. Murray JD (2002) Mathematical biology I: an introduction. Springer, Berlin.
24. Nakata M, Itou T, Sakai T (2009) Quantitative analysis of inflammatory cytokines expression in peripheral blood mononuclear cells of the ferret (Mustela putorius furo) using real-time PCR. Vet Immunol Immunopathol 15: 130(1-2): 88-91 doi: 10. 1016/j. vetimm. 2008. 12. 009.
25. Ougrinovskaia A, Thompson RS, Myerscough MR (2010) An ODE model of early stages of atherosclerosis: mechanisms of the inflammatory response. Bull Math Biol 72: 1534-1561. doi: 10. 1007/s11538-010-9509-4.
26. Pao CV (1992) Nonlinear parabolic and elliptic equations. Plenum Press, New York.
27. Poston RN, Poston RM (2007) Typical atherosclerotic plaque morphology. Math Model Nat Phenom 2(02): 142-149.
28. Qi Y, Yafeng L, Ahmed BW et al (2012) Temporal and quantitative analysis of atherosclerotic lesions in diet-induced hypercholesterolemic rabbits. J Biomed Biotech 2012 (Article ID 506159). doi: 10. 1155/2012/506159.
29. Ross R, Glomset JA (1973) Atherosclerosis and the arterial smooth muscle cell. Science 180: 1332-1339.
30. Ross R (1993) The pathogenesis of atherosclerosis: a perspective for 1990s. Nature 362: 801-809.
31. Ross R (1999) Atherosclerosis: an inflammatory disease. N Engl J Med 340(2): 115-126.
32. Schachter M (1997) The pathogenesis of atherosclerosis. Int J Cardiol 62(Suppl. 2): S3-S7.
33. Titarev VA, Toro EF (2004) Finite-volume WENO schemes for three-dimensional conservation laws. J Comp Phys 201: 238-260.
34. Titarev VA, Toro EF (2005) ADER schemes for three-dimensional non-linear hyperbolic systems. J Comp Phys 204(2): 715-736.
35. Toro EF, Millington RC, Nejad LAM (2001) Towards very high-order Godunov schemes. In: Toro EF (ed) Godunov methods: theory and applications. Edited Review. Conference in Honour of Godunov SK, vol. 1. Kluwer Academic/Plenum Publishers, New York, pp 897-902.
36. Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics: a practical introduction, 3rd edn. Springer, Berlin.
37. Toro EF, Hidalgo A (2009) ADER finite volume schemes for nonlinear reaction-diffusion equations. Appl Num Math 59(1): 73-100.
38. Ye D, Zhao Y, Hildebrand RB, Singaraja R, Hayden MR, van Berkel TJC, van Eck M (2011) The dynamics of macrophage infiltration into the arterial wall during atherosclerotic lesion development in low-density lipoprotein receptor knockout mice. Am J Pathol 178(1): 413-422.