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Bulletin of Mathematical Biology

, Volume 64, Issue 4, pp 673–702 | Cite as

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

  • S. R. McDougall
  • A. R. A. Anderson
  • M. A. J. Chaplain
  • J. A. Sherratt
Article

Abstract

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial cell migration, and organize themselves into a branched, connected network structure. Subsequent cell proliferation near the sprout-tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we initially generate theoretical capillary networks (which are morphologically similar to those networks observed in vivo) using the discrete mathematical model of Anderson and Chaplain. This discrete model describes the formation of a capillary sprout network via endothelial cell migratory and proliferative responses to external chemical stimuli (tumour angiogenic factors, TAF) supplied by a nearby solid tumour, and also the endothelial cell interactions with the extracellular matrix.

The main aim of this paper is to extend this work to examine fluid flow through these theoretical network structures. In order to achieve this we make use of flow modelling tools and techniques (specifically, flow through interconnected networks) from the field of petroleum engineering. Having modelled the flow of a basic fluid through our network, we then examine the effects of fluid viscosity, blood vessel size (i.e., diameter of the capillaries), and network structure/geometry, upon: (i) the rate of flow through the network; (ii) the amount of fluid present in the complete network at any one time; and (iii) the amount of fluid reaching the tumour. The incorporation of fluid flow through the generated vascular networks has highlighted issues that may have major implications for the study of nutrient supply to the tumour (blood/oxygen supply) and, more importantly, for the delivery of chemotherapeutic drugs to the tumour. Indeed, there are also implications for the delivery of anti-angiogenesis drugs to the network itself. Results clearly highlight the important roles played by the structure and morphology of the network, which is, in turn, linked to the size and geometry of the nearby tumour. The connectedness of the network, as measured by the number of loops formed in the network (the anastomosis density), is also found to be of primary significance. Moreover, under certain conditions, the results of our flow simulations show that an injected chemotherapy drug may bypass the tumour altogether.

Keywords

Bolus Injection Blood Viscosity Vascular Network Capillary Network Parent Vessel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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.Google Scholar
  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.CrossRefzbMATHGoogle Scholar
  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.Google Scholar
  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.zbMATHGoogle Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefzbMATHGoogle Scholar
  7. Arnold, F. and D. C. West (1991). Angiogenesis in wound healing. Pharmacol. Ther. 52, 407–422.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  11. Balding, D. and D. L. S. McElwain (1985). A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114, 53–73.Google Scholar
  12. Bikfalvi, A. (1995). Significance of angiogenesis in tumour progression and metastasis. Euro. J. Cancer 31A, 1101–1104.CrossRefGoogle Scholar
  13. Bowersox, J. C. and N. Sorgente (1982). Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 42, 2547–2551.Google Scholar
  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.MathSciNetzbMATHGoogle Scholar
  15. Chaplain, M. A. J. and A. M. Stuart (1993). A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168.zbMATHGoogle Scholar
  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.zbMATHCrossRefGoogle Scholar
  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.Google Scholar
  18. Cliff, W. J. (1963). Observations on healing tissue: A combined light and electron microscopic investigation. Trans. Roy. Soc. Lond. B246, 305–325.Google Scholar
  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.CrossRefGoogle Scholar
  20. Dullien, F. A. L. (1992). Porous Media, Fluid Transport and Pore Structure, 2nd edn, New York: Academic Press Inc.Google Scholar
  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.Google Scholar
  22. Ellis, L. E. and I. J. Fidler (1995). Angiogenesis and breast cancer metastasis. Lancet 346, 388–389.CrossRefGoogle Scholar
  23. Fatt, I. (1956). The network model of porous media. Trans. AIME 207, 144.Google Scholar
  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.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Folkman, J. (1985). Tumor angiogenesis. Adv. Cancer Res. 43, 175–203.CrossRefGoogle Scholar
  26. Folkman, J. and M. Klagsbrun (1987). Angiogenic factors. Science 235, 442–447.Google Scholar
  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.Google Scholar
  28. Folkman, J. (1995). Angiogenesis in cancer, vascular, rheumatoid and other disease. Nat. Med. 1, 21–31.CrossRefGoogle Scholar
  29. Gasparini, G. (1995). Tumour angiogenesis as a prognostic assay for invasive ductal breastcarcinoma. J. Natl. Cancer Inst. 87, 1799–1801.Google Scholar
  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.Google Scholar
  31. Gödde, R. and H. Kurz (2001). Structural and biophysical simulation of angiogenesis and vascular remodelling. Dev. Dyn. 220, 387–401.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.Google Scholar
  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.CrossRefGoogle Scholar
  35. Harris, A. L. (1997). Antiangiogenesis for cancer therapy. Lancet 349(suppl. II), 13–15.CrossRefGoogle Scholar
  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.Google Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.Google Scholar
  40. Levick, J. R. (1998). An Introduction to Cardiovascular Physiology, Oxford: Butterworth-Heinemann.Google Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  43. Liotta, L. A., C. N. Rao and S. H. Barsky (1983). Tumour invasion and the extracellular matrix. Lab. Invest. 49, 636–649.Google Scholar
  44. Madri, J. A. and B. M. Pratt (1986). Endothelial cell-matrix interactions: in vitro models of angiogenesis. J. Histochem. Cytochem. 34, 85–91.Google Scholar
  45. Matrisian, L. M. (1992). The matrix-degrading metalloproteinases. Bioessays 14, 455–463.CrossRefGoogle Scholar
  46. Mitchell, A. R. and D. F. Griffiths (1980). The Finite Difference Method in Partial Differential Equations, Chichester: Wiley.zbMATHGoogle Scholar
  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.Google Scholar
  48. Muthukkaruppan, V. R., L. Kubai and R. Auerbach (1982). Tumor-induced neovascularization in the mouse eye. J. Natl. Cancer Inst. 69, 699–705.Google Scholar
  49. Nekka, F., S. Kyriacos, C. Kerrigan and L. Cartilier (1996). A model of growing vascular structures. Bull. Math. Biol. 58, 409–424.CrossRefzbMATHGoogle Scholar
  50. Norton, J. A. (1995). Tumor angiogenesis: the future is now. Ann. Surg. 222, 693–694.CrossRefGoogle Scholar
  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.zbMATHGoogle Scholar
  52. Paweletz, N. and M. Knierim (1989). Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197–242.Google Scholar
  53. Pedley, T. J. and X. Y. Luo (1998). Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277–294.CrossRefzbMATHGoogle Scholar
  54. Pepper, M. S. (2001). Role of the matrix metalloproteinase and plasminogen activator-plasmin systems in angiogenesis. Arterioscler. Thromb. Vasc. Biol. 21, 1104–1117.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.CrossRefGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  60. Rose, W. (1957). Studies of waterflood performance, in Use of Network Models Illinois State Geology Survey, Vol. 3, Illinois: Circ. No. 237, Urbana.Google Scholar
  61. Schoefl, G. I. (1963). Studies on inflammation III. Growing capillaries: Their structure and permeability. Virchows Arch. Pathol. Anat. 337, 97–141.CrossRefGoogle Scholar
  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.Google Scholar
  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)Google Scholar
  64. Schmid-Schonbein, G. W., R. Skalak, S. Usami and S. Chien (1980). Cell distribution in capillary networks. Microvasc. Res. 19, 18–44.CrossRefGoogle Scholar
  65. Schmid-Schonbein, G. W. (1999). Biomechanics of microcirculatory blood perfusion. Ann. Rev. Biomed. Eng. 1, 73–102.CrossRefGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.CrossRefGoogle Scholar
  72. Warren, B. A. (1966). The growth of the blood supply to melanoma transplants in the hamster cheek pouch. Lab. Invest. 15, 464–473.Google Scholar
  73. Wilkinson, D. and J. F. Willemsen (1983). Invasion percolation—a new form of percolation. J. Phys. A. 16, 3365.MathSciNetCrossRefGoogle Scholar
  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.Google Scholar

Copyright information

© Society for Mathematical Biology 2002

Authors and Affiliations

  • S. R. McDougall
    • 1
  • A. R. A. Anderson
    • 2
  • M. A. J. Chaplain
    • 2
  • J. A. Sherratt
    • 3
  1. 1.Department of Petroleum EngineeringHeriot-Watt UniversityEdinburghUK
  2. 2.Department of Mathematics, The SIMBIOS CentreUniversity of DundeeDundeeUK
  3. 3.Department of MathematicsHeriot-Watt UniversityEdinburghUK

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