A new interpretation of the Keller-Segel model based on multiphase modelling

Journal of Mathematical Biology

Volumn 49, Issue 6, 2004, Pages 604-626

  Byrne, Helen M ^a   Owen, Markus R ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

Chemotaxis; Keller Segel; Multiphase; Pattern formation

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; CHEMOTAXIS; COMPARATIVE STUDY; COMPUTER SIMULATION; CYTOLOGY; MORPHOGENESIS; MYXOMYCETES; PHYSIOLOGY;

BODY PATTERNING; CHEMOTAXIS; COMPUTER SIMULATION; MODELS, BIOLOGICAL; MYXOMYCETES;

MYXOGASTRIA;

EID: 11144223026     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00285-004-0276-4     Document Type: Article

References (61)

1. Advani, S.G.: Flow and rheology in polymeric composites manufacturing. Elsevier, 1994
2. Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol 9, 147-177 (1980)
3. Serini, G., Ambrosi, D., Giraudo, E., Gamba, A., Preziosi, L., Bussolini, F.: Modeling the early stages of vascular network assembly. EMBO J. 22, 1771-1779 (2003)
4. Anderson, A.R.A., Chaplain, M.A.J.: Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857-899 (1998)
5. Barker, M.K., Seedhom, B.B.: Articular cartilage deformation under physiological cycling loading. J. Biomech 30, 377-381 (1997)
6. Bearon, R.N., Pedley, T.J.: Modelling run-and-tumble chemotaxis in a shear flow. Bull. Math. Biol. 62, 775-791 (2000)
7. Bennet, N.T., Schultz, G.S.: Growth factors and wound healing: Part II role in normal and chronic wound healing. Am. J. Surgery 166, 74-81 (1993)
8. Boyden, S.V.: The chemotactic effect of mixtures of antibody and antigen on polymorphonuclear leukocytes. J. Exp. Med. 115, 453-466 (1962)
9. Bray, D.: Cell movements: from molecules to motility. Garland Publishing, 2001
10. Breward, C.J.W., Byrne, H.M., Lewis, C.E.: The role of cell-cell interactions in a two-phase of solid tumor growth. J. Math. Biol. 45, 125-152 (2002)
11. Buettner, H.M., Lauffenburger, D.A., Zigmond, S.H.: Measurement of leukocyte motility and chemotaxis parameters with the Millipore filter assay. J. Immunol. Meth. 123, 25-37 (1989)
12. Byrne, H.M., Cave, G., McElwain, D.L.S.: The effects of chemokinesis on leukocyte locomotion: a new interpretation of experimental results. IMA J. Math. Appl. Med. Biol. 15, 235-256 (1998)
13. Byrne, H.M., King, J.R., McElwain, D.L.S., Preziosi, L.: A two-phase model of solid tumor growth. Appl. Math. Lett. 16, 567-573 (2003)
14. Byrne, H.M., Preziosi, L.: Modelling solid tumor growth using the theory of mixtures. IMA J. Math. Appl. Med. Biol. 20, 341-366 (2003)
15. Dallon, J.C., Othmer, H.G.: A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum. Phil. Trans Roy. Soc. B 352, 391-417 (1997)
16. Drew, D.A., Segel, L.A.: Averaged equations for two-phase flows. Stud. Appl. Math. 50, 205-231 (1971)
17. Early, A., Abe, T., William, J.: Evidence for positional differentiation of prestalk cells and for a morphogenetic gradient in dictyostelium. Cell 83, 91-99 (1995)
18. Farina, A., Preziosi, L.: Deformable porous media and composites manufacturing. In: Heterogeneous Media: Micromechanics, Modelling, Methods and Simulations, Markov, M., Preziosi, L. (eds.), Birkhäuser, 2000
19. Folkman, J., Brem, H.: Angiogenesis and inflammation. In: Inflammation: Basic Principles and Clinical Correlates, Second Edition. Gallin, J.I., Goldstein, I.M. and Dnyderman, R. (eds.), New York, Raven Press, 1992
20. Fowler, A.C.: Mathematical models in the applied sciences. Cambridge University Press, Cambridge, 1997
21. Franks, S.J.: Mathematical modelling of tumour growth and stability. PhD thesis, University of Nottingham, England, 2001
22. Franks, S.J., King, J.R.: Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties. IMA J. Math. Med. Biol., 2003 (submitted)
23. Fung, Y.C.: Biomechanics: motion, flow, stress and growth. Spinger-Verlag, New York, 1990
24. Gamba, A., Ambrosi, D., Coniglio, A., de Candia, A., di Talia, S., Giraudo, E., Serini, G., Preziosi, E., Bussolini, F.: Percolation, morphogenesis, and Burgers dynamics in blood vessel formation. Phys. Rev. Lett. 90 (11), 118101 (2003)
25. Haddox, J.L., Pfister, R.R., Sommers, C.I.: A visual assay for quantitating neutrophil chemotaxis in a collagen gel matrix. J. Immunol. Meth. 141, 41-52 (1991)
26. Hader, D.P.: Polarotaxis, gravitaxis and vertical phototaxis in the green flagellate, Euglena-gracilis. Arch. Microbiol. 147, 179-183 (1987)
27. Hill, N.A., Plumpton, L.A.: Control strategies for the polarotactic orientation of the microorganism Euglena gracilis. J. Theor. Biol. 203, 357-365 (2000)
28. Hillen, T., Painter, K.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 280-301 (2001)
29. p-estimates for a class of reaction-diffusion systems. Coll. Math. 87, 113-127 (2001)
30. Hou, J.S., Holmes, M.H., Lai, W.M., Mow, V.C.: Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Eng. 111, 78-87 (1989)
31. Jackson, T.J., Byrne, H.M.: A mathematical model of tumour encapsulation. Math. Biosci. 180, 307-328 (2002)
32. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399-415 (1970)
33. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225-234 (1971a)
34. Keller, E.F., Segel, L.A.: Travelling bands of chemotactic bacteria: a theoretical analysis J. Theor. Biol. 30, 235-248 (1971b)
35. Keller, E.F.: Assessing the Keller-Segel model: how has it fared? In: Biological growth and spread, Jager, W., Rost, H. and Tautu, P. (eds.), Lecture Notes in Biomathematics, 38, Berlin, Springer-Verlag, 1980, pp. 379-387
36. King, J.R., Franks, S.J.: Mathematical analysis of some multi-dimensional tissue growth models. Eur. J. Appl. Maths., 2003 (in press)
37. Lai, W.M., Hou, J.S., Mow, V.C.: A triphasic theory for the swelling and deformation behaviours of articular cartilage. J. Biomech. Eng. 113, 245-258 (1991)
38. Landman, K., Please, C.P.: Tumor dynamics and necrosis: Surface tension and stability. IMAJ. Maths. Appl. Med. Biol. 18, 131-158 (2001)
39. Lega, J., Passot, T.: Hydrodynamics of bacterial colonies: a model. Phys. Rev. E 67(3), 31906 (2003)
40. Lubkin, S.R., Jackson, T.: Multiphase mechanics of capsule formation in tumours. J. Biomech. Eng. 124, 237-243 (2002)
41. Mow, V.C., Lai, W.M.: Mechanics of animal joints. Annu. Rev. Fluid Mech. 11, 247-288 (1979)
42. Murray, J.D.: Mathematical Biology. Springer-Verlag, New York, 1989
43. Nicholson, C.: Diffusion from an injected volume of a substance in brain tissues with arbitrary volume fraction and tortuosity. Brain Res. 333, 325-329 (1985)
44. Othmer, H.G., Stevens, A.: Aggregation, blowup and collapse: the ABCs of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1041-1081 (1997)
45. Owen, M.R., Sherratt, J.A.: Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions. J. Theor. Biol. 189, 63-80 (1997)
46. Painter, K.J., Hillen, T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. App. Math. Quart., 2003 to appear
47. Patlak, C.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311-338 (1953)
48. Please, C.P., Pettet, G., McElwain, D.L.S.: A new approach to modelling the formation of necrotic regions in tumors, Appl. Math. Lett. 11, 89-94 (1998)
49. Primicerio, M., Zaltzman, B.: A free boundary problem arising in chemotaxis. Adv. Math. Sci. Appl. 12, 685-708 (2002)
50. Primicerio, M., Zaltzman, B.: Free boundary in radial-symmetric chemotaxis Proceedings WASCOM 2001, World Scientific, 2002
51. Segel, L.A.: A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32, 653-665 (1977)
52. Sherratt, J.A., Sage, E.H., Murray, J.D.: Chemical Control of Eukaryotic Cell Movement: A New Model. J. Theor. Biol. 162, 23-40 (1993)
53. Sleeman, B.D., Levine, H.A.: A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683-730 (1997)
54. Sorek, S., Sideman, S.: A porous medium approach for modelling heart mechanics: Part 1 theory. Math. Biosci. 81, 1-14 (1986)
55. Spencer, A.J.M.: Deformations of fibre-reinforced materials. Clarendon Press, 1972
56. Spencer, A.J.M.: Continuum theory of the mechanics of fibre-reinforced composites. Springer Verlag, 1984
57. Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183-212 (2000)
58. Stokes, C.I., Lauffenburger, D.A.: Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377-403 (1991)
59. Tyson, R., Lubkin, S.R., Murray, J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol. 38, 359-375 (1999)
60. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B237, 37-72 (1952)
61. Ward, J.P., King, J.R.: Mathematical modelling of avascular-tumour growth. IMA. J. Math. Appl. Med. 14, 39-69 (1997)