Pattern formation in a model of acute inflammation

SIAM Journal on Applied Dynamical Systems

Volumn 11, Issue 2, 2012, Pages 629-660

  Penner, Kevin ^a   Ermentrout, Bard ^b   Swigon, David ^b  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Chemotaxis; Immunology; Mathematical modeling; Spatial patterns

Indexed keywords

ANTI-INFLAMMATORIES; CHEMOTAXIS; CYTOKINES; DYNAMICAL INSTABILITIES; KELLER-SEGEL CHEMOTAXIS MODEL; NON-HOMOGENEOUS; PATHOGENIC STIMULUS; PATTERN FORMATION; ROTATING WAVES; SPATIAL PATTERNS; STEADY STATE; TRAVELING PULSE;

BIOCHEMISTRY; IMMUNOLOGY; MATHEMATICAL MODELS; NONLINEAR ANALYSIS;

PATHOLOGY;

EID: 84865683304     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/110834081     Document Type: Article

References (40)

1. A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumorinduced angiogenesis, Bull. Math. Biol., 60 (1998), pp. 857-899.
2. J. C. Arciero, Q. Mi, M. F. Branca, D. J. Hackam, and D. Swigon, Continuum model of collective cell migration in wound healing and colony expansion, Biophys. J., 100 (2011), pp. 535-543.
3. M. Bode, A.W. Liehr, C. P. Schenk, and H. G. Purwins, Interaction of dissipative solitons: Particlelike behaviour of localized structures in a three-component reaction-diffusion system, Phys. D, 161 (2002), pp. 45-66.
4. R. G. Casten, H. Cohen, and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quart. Appl. Math., 32 (1975), pp. 365-402.
5. M. A. J. Chaplain, S. R. McDougall, and A. R. A. Anderson, Mathematical modeling of tumorinduced angiogenesis, Annu. Rev. Biomed. Eng., 8 (2006), pp. 233-257.
6. A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), pp. 169-205.
7. J. Day, J. Rubin, Y. Vodovotz, C. C. Chow, A. Reynolds, and G. Clermont, A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration, J. Theoret. Biol., 242 (2006), pp. 237-256.
8. B. Ermentrout, Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square, Proc. R. Soc. Lond. A, 434 (1991), pp. 413-417.
9. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, New York, Berlin, 1979.
10. P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), pp. 335-361.
11. P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves, Clarendon Press, Oxford, UK, 1996.
12. T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), pp. 183-217.
13. J. M. Hyman, L. A. Segel, A. S. Perelson, and S. N. Klaus, Rash theory, in Theoretical and Experimental Insights into Immunology, A. S. Perelson and G. Weisbuch, eds., NATO ASI Series, Springer-Verlag, New York, Berlin, 1992, pp. 333-352.
14. E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1975), pp. 309-317.
15. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), pp. 399-415.
16. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), pp. 225-234.
17. K. Krischer and A. Mikhailov, Bifurcation to traveling spots in reaction-diffusion systems, Phys. Rev. Lett., 73 (1994), pp. 3165-3168.
18. I. R. Lapidus and R. Schiller, Model for the chemotactic response of a bacterial population, Biophys. J., 16 (1976), pp. 779-789.
19. S. R. Lubkin, R. Tyson, and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proc. Roy. Soc. London, 266 (1999), pp. 299-304.
20. N. V. Mantzaris, S. Webb, and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), pp. 111-187.
21. M. Meixner, A. De Wit, S. Bose, and E. Schöll, Generic spatiotemporal dynamics near codimensiontwo Turing-Hopf bifurcations, Phys. Rev. E, 55 (1997), pp. 6690-6697.
22. A. S. Moskalenko, A. W. Liehr, and H. G. Purwins, Rotational bifurcation of localized dissipative structures, Europhys. Lett., 63 (2003), pp. 361-367.
23. L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer-Verlag, New York, Berlin, 2006.
24. H. G. Purwins, H. U. Bödeker, and S. Amiranashvili, Dissipative solitons, Adv. Phys., 59 (2010), pp. 485-701.
25. A. Reynolds, J. Rubin, G. Clermont, J. Day, Y. Vodovotz, and G. B. Ermentrout, A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of antiinflammation, J. Theoret. Biol., 242 (2006), pp. 220-236.
26. A. Rovinsky and M. Menzinger, Interaction of Turing and Hopf bifurcations in chemical systems, Phys. Rev. A, 46 (1992), pp. 6315-6322.
27. R. C. Schugart, A. Friedman, R. Zhao, and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model, Proc. Natl. Acad. Sci. USA, 105 (2008), pp. 2628-2633.
28. H. Schwetlick, Traveling waves for chemotaxis-systems, PAMM, 3 (2003), pp. 476-478.
29. L. A. Segel, Incorporation of receptor kinetics into a model for bacterial chemotaxis, J. Theoret. Biol., 57 (1976), pp. 23-42.
30. L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J. Appl. Math., 32 (1977), pp. 653-665.
31. J. A. Sherratt, P. Martin, J. D. Murray, and J. Lewis, Mathematical models of wound healing in embryonic and adult epidermis, Math. Med. Biol., 9 (1992), pp. 177-196.
32. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren Math. Wiss. 258, Springer- Verlag, New York, 1994.
33. A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 237 (1952), pp. 37-72.
34. N. V. Valeyev, C. Hundhausen, Y. Umezawa, N. V. Kotov, G. Williams, A. Clop, C. Ainali, C. Ouzounis, S. Tsoka, and F. O. Nestle, A systems model for immune cell interactions unravels the mechanism of inflammation in human skin, PLoS Comput. Biol., 6 (2010), e1001024.
35. Y. Vodovotz, Deciphering the complexity of acute inflammation using mathematical models, Immunol. Res., 36 (2006), pp. 237-245.
36. Y. Vodovotz, G. Clermont, C. Chow, and G. An, Mathematical models of the acute inflammatory response, Curr. Opin. Crit. Care, 10 (2004), pp. 383-390.
37. Y. Vodovotz, M. Csete, J. Bartels, S. Chang, and G. An, Translational systems biology of inflammation, PLoS Comput. Biol., 4 (2008), e1000014.
38. K. Wendelsdorf, J. Bassaganya-Riera, R. Hontecillas, and S. Eubank, Model of colonic inflammation: Immune modulatory mechanisms in inflammatory bowel disease, J. Theoret. Biol., 264 (2010), pp. 1225-1239.
39. C. Xue, A. Friedman, and C. K. Sen, A mathematical model of ischemic cutaneous wounds, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 16782-16787.
40. M. Zhu and J. D. Murray, Parameter domains for generating spatial pattern: A comparison of reactiondiffusion and cell-chemotaxis models, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), pp. 1503- 1524.