Mathematical modelling of flow through vascular networks: Implications for tumour-induced angiogenesis and chemotherapy strategies

Bulletin of Mathematical Biology

Volumn 64, Issue 4, 2002, Pages 673-702

  McDougall, S R ^a   Anderson, A R A ^b   Chaplain, M A J ^b   Sherratt, J A ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF DUNDEE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036664608     PISSN: 00928240     EISSN: None     Source Type: Journal    
DOI: 10.1006/bulm.2002.0293     Document Type: Article

References (74)

1. Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson (1994). Molecular Biology of the Cell, 3rd edn, New York: Garland Publishing.
2. Anderson, A. R. A. and M. A. J. Chaplain (1998). Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857-899.
3. Anderson, A. R. A., B. D. S. Sleeman, I. M. Young and B. S. Griffiths (1997). Nematode movement along a chemical gradient in a structurally heterogeneous environment: II. Theory. Fundam. Appl. Nematol. 20, 165-172.
4. Anderson, A. R. A., M. A. J. Chaplain, A. L. Newman, R. J. C. Steele and A. M. Thompson (2000). Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129-154.
5. Andreasen, P. A., L. Kjøller, L. Christensen and M. J. Duffy (1997). The urokinase-type plasminogen activator system in cancer metastasis: a review. Int. J. Cancer 72, 1-22.
6. Armitstead, J. P., C. D. Bertram and O. E. Jensen (1996). A study of the bifurcation behaviour of a model of flow through a collapsible tube. Bull. Math. Biol. 58, 611-641.
7. Arnold, F. and D. C. West (1991). Angiogenesis in wound healing. Pharmacol. Ther. 52, 407-422.
8. Ausprunk, D. H. and J. Folkman (1977). Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14, 53-65.
9. Baish, J. W., Y. Gazit, D. A. Berk, M. Nozue, L. T. Baxter and R. K. Jain (1996). Role of tumour vascular architecture in nutrient and drug delivery: An invasion percolation-based network model. Microvasc. Res. 51, 327-346.
10. Baish, J. W., P. Netti and R. K. Jain (1997). Transmural coupling of fluid flow in microcirculatory network and interstitium in tumours. Microvasc. Res. 53, 128-141.
11. Balding, D. and D. L. S. McElwain (1985). A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114, 53-73.
12. Bikfalvi, A. (1995). Significance of angiogenesis in tumour progression and metastasis. Euro. J. Cancer 31A, 1101-1104.
13. Bowersox, J. C. and N. Sorgente (1982). Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 42, 2547-2551.
14. Chaplain, M. A. J. and B. D. Sleeman (1990). A mathematical model for the production and secretion of tumour angiogenesis factor in tumours. IMA J. Math. Appl. Med. Biol. 7, 93-108.
15. Chaplain, M. A. J. and A. M. Smart (1993). A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149-168.
16. Chaplain, M. A. J. (1996). Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development. Math. Comp. Modell. 23, 47-87.
17. Chaplain, M. A. J. and A. R. A. Anderson (1999). Modelling the growth and form of capillary networks, in On Growth and Form: Spatio-temporal Pattern Formation in Biology, M. A. J. Chaplain, G. D. Singh and J. C. McLachlan (Eds), Chichester: Wiley, pp. 225-250.
18. Cliff, W. J. (1963), Observations on healing tissue: A combined light and electron microscopic investigation. Trans. Roy. Soc. Lond. B246, 305-325.
19. Dodd, C. G. and O. G. Kiel (1959). Evaluation of Monte Carlo methods in studying fluid-fluid displacement and wettability in porous rock. J. Phys. Chem. 63, 1646.
20. Dullien, F. A. L. (1992). Porous Media, Fluid Transport and Pore Structure, 2nd edn, New York: Academic Press Inc.
21. Dutta, A. and J. M. Tarbell (1996). Influence of non-Newtonian behaviour of blood on flow in an elastic artery model. J. Biomech. Eng.-Trans. ASME 118, 111-119.
22. Ellis, L. E. and I. J. Fidler (1995). Angiogenesis and breast cancer metastasis. Lancet 346, 388-389.
23. Fatt, I. (1956). The network model of porous media. Trans. AIME 207, 144.
24. Fister, K. R. and J. C. Panetta (2000). Optimal control applied to cell-cycle-specific cancer chemotherapy. Siam J. Appl. Math. 60, 1059-1072.
25. Folkman, J. (1985). Tumor angiogenesis. Adv. Cancer Res. 43, 175-203.
26. Folkman, J. and M. Klagsbrun (1987). Angiogenic factors. Science 235, 442-447.
27. Folkman, J. and H. Brem (1992). Angiogenesis and inflammation, in Inflammation: Basic Principles and Clinical Correlates, 2nd edn, J. I. Gallin, I. M. Goldstein and R. Snyderman (Eds), New York: Raven Press.
28. Folkman, J. (1995). Angiogenesis in cancer, vascular, rheumatoid and other disease. Nat. Med. 1, 21-31.
29. Gasparini, G. (1995). Tumour angiogenesis as a prognostic assay for invasive ductal breast-carcinoma. J. Natl. Cancer Inst. 87, 1799-1801.
30. Gasparini, G. and A. L. Harris (1995). Clinical importance of the determination of tumour angiogenesis in breast-cancer - Much more than a new prognostic tool. J. Clin. Oncol. 13, 765-782.
31. Gödde, R. and H. Kurz (2001). Structural and biophysical simulation of angiogenesis and vascular remodelling. Dev. Dyn. 220, 387-401.
32. Graham, C. H. and P. K. Lala (1992). Mechanisms of placental invasion of the uterus and their control. Biochem. Cell Biol. 70, 867-874.
33. Harris, A. L., S. Fox, R. Bicknell, R. Leek and K. Gatter (1994). Tumour angiogenesis in breast-cancer - prognostic factor and therapeutic target. J. Cellular Biochem. S18D SID, 225.
34. Harris, A. L., H. T. Zhang, A. Moghaddam, S. Fox, P. Scott, A. Pattison, K. Gatter, I. Stratford and R. Bicknell (1996). Breast cancer angiogenesis - new approaches to therapy via anti-angiogenesis, hypoxic activated drugs, and vascular targeting. Breast Cancer Res. Treat. 38, 97-108.
35. Harris, A. L. (1997). Antiangiogenesis for cancer therapy. Lancet 349 (suppl. II), 13-15.
36. Herblin, W. F. and J. L. Gross (1994). Inhibition of angiogenesis as a strategy for tumour-growth control. Mol. Chem. Neuropathol. 21, 329-336.
37. Holmes, M. J. and B. D. Sleeman (2000). A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects. J. Theor. Biol. 202, 95-112.
38. Honda, H. and K. Yoshizato (1997). Formation of the branching pattern of blood vessels in the wall of the avian yolk sac studied by a computer simulation. Dev. Growth Differ. 39, 581-589.
39. Itoh, J., K. Yasumura, T. Takeshita, H. Ishikawa, H. Kobayashi, K. Ogawa, K. Kawai, A. Serizana and R. Y. Osamura (2000). Three-dimensional imaging of tumor angiogenesis. Anal. Quant. Cytol. Histol. 22, 85-90.
40. Levick, J. R. (1998). An Introduction to Cardiovascular Physiology, Oxford: Butterworth-Heinemann.
41. Levin, M., B. Dawant and A. S. Popel (1986). Effect of dispersion on vessel diameters and lengths in stochastic networks. I. Modelling of microvascular haematocrit distribution. Microvasc. Res. 31, 223-234.
42. Levine, H. A., S. Pamuk, B. D. Sleeman and M. Nilsen-Hamilton (2001). Mathematical modelling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull. Math. Biol. 63, 801-863.
43. Liotta, L. A., C. N. Rao and S. H. Barsky (1983). Tumour invasion and the extracellular matrix. Lab. Invest. 49, 636-649.
44. Madri, J. A. and B. M. Pratt (1986). Endothelial cell-matrix interactions: in vitro models of angiogenesis. J. Histochem. Cytochem. 34, 85-91.
45. Matrisian, L. M. (1992). The matrix-degrading metalloproteinases. Bioessays 14, 455-463.
46. Mitchell, A. R. and D. F. Griffiths (1980). The Finite Difference Method in Partial Differential Equations, Chichester: Wiley.
47. McDougall, S. R. and K. S. Sorbie (1997). The application of network modelling techniques to multiphase flow in porous media. Petroleum Geosci. 3, 161-169.
48. Muthukkaruppan, V. R., L. Kubai and R. Auerbach (1982). Tumor-induced neovascularization in the mouse eye. J. Natl. Cancer Inst. 69, 699-705.
49. Nekka, F., S. Kyriacos, C. Kerrigan and L. Cartilier (1996). A model of growing vascular structures. Bull. Math. Biol. 58, 409-424.
50. Norton, J. A. (1995). Tumor angiogenesis: the future is now. Ann. Surg. 222, 693-694.
51. Orme, M. E. and M. A. J. Chaplain (1996). A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. IMA J. Math. App. Med. Biol. 13, 73-98.
52. Paweletz, N. and M. Knierim (1989). Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197-242.
53. Pedley, T. J. and X. Y. Luo (1998). Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277-294.
54. Pepper, M. S. (2001). Role of the matrix metalloproteinase and plasminogen activator-plasmin systems in angiogenesis. Arterioscler. Thromb. Vasc. Biol. 21, 1104-1117.
55. Pettet, G., M. A. J. Chaplain, D. L. S. McElwain and H. M. Byrne (1996). On the role of angiogenesis in wound healing. Proc. Roy. Soc. Lond. B 263, 1487-1493.
56. Perumpanani, A. J., J. A. Sherratt, J. Norbury and H. M. Byrne (1996). Biological inferences from a mathematical model of malignant invasion. Invasion Metastasis 16, 209-221.
57. Price, R. J. and T. C. Skalak (1995). A circumferential stress-growth rule predicts arcade arteriole formation in a network model. Microcirculation 2, 41-51.
58. Pries, A. R., T. W. Secomb and P. Gaehtgens (1998). Structural adaptation and stability of microvascular networks: theory and simulations. Am. J. Physiol. 275, H349-H360.
59. Pries, A. R., T. W. Secomb, P. Gaehtgens and J. F. Gross (1990). Blood flow in microvascular networks: experiments and simulations. Circ. Res. 67, 826-834.
60. Rose, W. (1957). Studies of waterflood performance, in Use of Network Models Illinois State Geology Survey, Vol. 3, Illinois: Circ. No. 237, Urbana.
61. Schoefl, G. I. (1963). Studies on inflammation III. Growing capillaries: Their structure and permeability. Virchows Arch. Pathol. Anat. 337, 97-141.
62. Schor, A. M., S. L. Schor and R. Baillie (1999). Angiogenesis experimental data relevant to theoretical analysis, in On Growth and Form: Spatio-temporal Pattern Formation in Biology, M. A. J. Chaplain, G. D. Singh and J. C. McLachlan (Eds), Chichester: Wiley, pp. 202-224.
63. Schor, A. M., S. L. Schor, A. R. A. Anderson and M. A. J. Chaplain (2002). Chemokinesis and chemotaxis within a three-dimensional collagen matrix: context modulation of fibroblast motogenic response to wound healing cytokines. J. Cell. Biol. (submitted)
64. Schmid-Schonbein, G. W., R. Skalak, S. Usami and S. Chien (1980). Cell distribution in capillary networks. Microvasc. Res. 19, 18-44.
65. Schmid-Schonbein, G. W. (1999). Biomechanics of microcirculatory blood perfusion. Ann. Rev. Biomed. Eng. 1, 73-102.
66. Secomb, T. W. and R. Hsu (1996). Motion of red blood cells in capillaries with variable cross-sections. J. Biomech. Eng. 118, 538-544.
67. Sholley, M. M., G. P. Ferguson, H. R. Seibel, J. L. Montour and J. D. Wilson (1984). Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51, 624-634.
68. Stokes, C. L., M. A. Rupnick, S. K. Williams and D. A. Lauffenburger (1990). Chemotaxis of human microvessel endothelial cells in response to acidic fibroblast growth factor. Lab. Invest. 63, 657-668.
69. Stokes, C. L. and D. A. Lauffenburger (1991). Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377-403.
70. Sutton, D. W. and G. W. Schmid-Schonbein (1994). The influence of pure erythrocyte suspensions on the pressure-flow relation in rat skeletal muscle. Biorheology 32, 107-120.
71. Terranova, V. P., R. Diflorio, R. M. Lyall, S. Hic, R. Friesel and T. Maciag (1985). Human endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell Biol. 101, 2330-2334.
72. Warren, B. A. (1966). The growth of the blood supply to melanoma transplants in the hamster cheek pouch. Lab. Invest. 15, 464-473.
73. Wilkinson, D. and J. F. Willemsen (1983). Invasion percolation - a new form of percolation. J. Phys. A. 16, 3365.
74. Williams, S. K. (1987). Isolation and culture of microvessel and large-vessel endothelial cells; their use in transport and clinical studies, in Microvascular Perfusion and Transport in Health and Disease, P. McDonagh (Ed.), Basel: Karger, pp. 204-245.