Skip to main content
SpringerLink
Log in

A Hybrid Discrete-Continuum Model of Tumour Induced Angiogenesis

  • Chapter
  • First Online:
Modeling Tumor Vasculature

  • 1085 Accesses

Abstract

Angiogenesis, the formation of new blood vessels, is a crucial step in solid tumour development, as it links the relatively harmless avascular and the potentially fatal vascular phases of tumour growth. Tumour induced angiogenesis follows a specific sequence of events that are initiated by the tumour secreting angiogenic factors into the surrounding tissue; upon contact with nearby blood vessels these angiogenic factors induce the endothelial cells lining the vessels to degrade their basement lamina. Endothelial cells then degrade the surrounding tissue, migrate towards the tumour and proliferate, in so doing they generate new vessels which can branch and link with one another to form loops (anastomosis). As soon as the first anastomosis forms, blood begins to flow and then all subsequent growth evolves in response to the combined effects of migratory cues within the tumour microenvironment and haemodynamic forces produced by the vasculature. In this chapter, we first present a brief review of previous models of angiogenesis and then focus our attention on the original hybrid discrete-continuum model of tumour induced angiogenesis. This model has been extensively cited and developed in a multitude of ways, specifically via the incorporation of blood flow and as a component of larger multiscale models of cancer development. Some of that work is discussed in subsequent chapters of this book, however, our primary focus here is on the details of the original no-flow model and its implementation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alarcon, T., Byrne, H. M., Maini, P. K. (2005). A multiple scale model for tumor growth. Multiscale Model Simul.3, 440–475.

    CAS  Google Scholar 

  • Alarcon T, Owen MR, Byrne HM, Maini PK (2006) Multiscale modeling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy. Comput Math Method M 7: 85119.

    Google Scholar 

  • Albini A, Allavena G, Melchiori A, Giancotti F, Richter H, Comoglio PM, Parodi S, Martin GR, Tarone G. Chemotaxis of 3T3 and SV3T3 cells to fibronectin is mediated through the cell-attachment site in fibronectin and fibronectin cell surface receptor. J. Cell Biol. 1987; 105: 1867–1872.

    CAS  PubMed  Google Scholar 

  • Anderson, A. R. A., Sleeman, B.D., Young, I.M. & Griffiths, B.S. (1997) Nematode movement along a chemical gradient in a structurally heterogeneous environment: II. Theory. Fundam. appl. Nematol., 20, 165–172.

    Google Scholar 

  • Anderson ARA, Chaplain MAJ. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 1998; 60: 857–899.

    CAS  PubMed  Google Scholar 

  • Anderson ARA, Chaplain MAJ. A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Letters 1997; 11: 109–114.

    Google Scholar 

  • Anderson ARA, Chaplain MAJ, Garcia-Reimbert C, Vargas CA. A Gradient-Driven Mathematical Model of Anti-Angiogenesis. Math. Comp. Mod. 2000; 32: 1141–1152.

    Google Scholar 

  • Anderson, A. R. A., Chaplain, M. A. J., Newman, E. L., Steele, R. J. C. & Thompson, A. M. (2000) Mathematical Modeling of Tumour Invasion and Metastasis. J. Theoret. Med., 2, 129–154.

    Google Scholar 

  • Anderson, A. R. A. A Hybrid Discrete-Continuum Technique for Individual Based Migration Models in Polymer and Cell Dynamics, eds. W. Alt, M. Chaplain, M. Griebel, J. Lenz, 2003, Birkhauser.

    Google Scholar 

  • Anderson, A. R. A. & Pitcairn, A. Application of the Hybrid Discrete-Continuum Technique in Polymer and Cell Dynamics, eds. W. Alt, M. Chaplain, M. Griebel, J. Lenz, 2003, Birkhauser.

    Google Scholar 

  • Anderson, A. R. A. (2005) A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion. IMA J. Math. Med. and Biol., 22, 163–186.

    Google Scholar 

  • Anderson, A. R. A, Weaver A.M., Cummings P.T. & Quaranta V. (2006) Tumor Morphology and Phenotypic Evolution Driven by Selective Pressure from the Microenvironment. Cell, 127, 111.

    Google Scholar 

  • Arnold F, West DC. Angiogenesis in wound healing. Pharmac. Ther. 1991; 52: 407–422.

    CAS  Google Scholar 

  • Ausprunk DH, Folkman J. Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 1977; 14: 53–65.

    CAS  PubMed  Google Scholar 

  • Bowersox JC, Sorgente N. Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 1982; 42: 2547–2551.

    CAS  PubMed  Google Scholar 

  • Balding D, McElwain DLS. A mathematical model of tumour-induced capillary growth. J. theor. Biol. 1985 114: 53–73.

    CAS  PubMed  Google Scholar 

  • Basanta D, Strand DW, Lukner RB, Franco OE, Cliffel DE, Ayala GE, Hayward SW, Anderson AR. The role of transforming growth factor-beta-mediated tumor-stroma interactions in prostate cancer progression: an integrative approach. Cancer Res. 2009 Sep 1;69(17):7111–20.

    CAS  PubMed  PubMed Central  Google Scholar 

  • Bauer AL, Jackson TL, Jiang Y. A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys J. 2007 May 1;92(9):3105–21. Epub 2007 Feb 2.

    Google Scholar 

  • Bentley, K., Mariggi, G., Gerhardt, H. and Bates, P. A. (2009) Tipping the balance: Robustness of Tip Cell Selection, Migration and Fusion in Angiogenesis. PLoS Computational Biology 5(10): e1000549.

    PubMed  PubMed Central  Google Scholar 

  • Bowersox JC, Sorgente N. Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 1982; 42: 2547–2551.

    CAS  PubMed  Google Scholar 

  • Bray D: Cell Movements. Garland Publishing, New York, 1992

    Google Scholar 

  • Byrne HM, Chaplain MAJ. Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull. Math. Biol. 1995 57: 461–486.

    CAS  PubMed  Google Scholar 

  • Carmeliet P, Jain RK. Angiogenesis in cancer and other diseases. Nature. 2000 Sep 14;407(6801):249–57.

    CAS  PubMed  Google Scholar 

  • Carter SB. Principles of cell motility: The direction of cell movement and cancer invasion. Nature 1965; 208: 1183–1187.

    CAS  PubMed  Google Scholar 

  • Carter SB. Haptotaxis and the mechanism of cell motility. Nature 1967; 213: 256–260.

    CAS  PubMed  Google Scholar 

  • Chaplain MAJ, Stuart AM. A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 1993 10: 149–168.

    CAS  PubMed  Google Scholar 

  • Chaplain MAJ. The mathematical modeling of tumour angiogenesis and invasion. Acta Biotheor. 1995; 43: 387–402.

    CAS  PubMed  Google Scholar 

  • Chaplain MAJ. Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modeling of the stages of tumour development. Math. Comput. Modeling 1996; 23: 47–87.

    Google Scholar 

  • Chaplain MAJ, Anderson ARA. The mathematical modeling, simulation and prediction of tumour-induced angiogenesis. Invas. Metast. 1997; 16: 222–234.

    Google Scholar 

  • Chaplain MAJ, Orme ME: Mathematical modeling of tumor-induced angiogenesis. In: Little CD, Mironov V, Sage EH (eds) Vascular morphogenesis: In vivo, in vitro, in mente. Birkhäuser, Boston, 1998, pp 205–240.

    Google Scholar 

  • Chaplain MA, McDougall SR, Anderson AR. Mathematical modeling of tumor-induced angiogenesis. Annu Rev Biomed Eng. 2006;8:233–57.

    CAS  PubMed  Google Scholar 

  • Folkman J (1971) Tumour angiogenesis therapeutic implications. New Engl J Med 285: 11821186.

    Google Scholar 

  • Folkman J: Tumor angiogenesis. Adv. Cancer Res. 1985; 43: 175–203.

    CAS  PubMed  Google Scholar 

  • Folkman J, Klagsbrun M. Angiogenic factors. Science 1987; 235: 442–447.

    CAS  PubMed  Google Scholar 

  • Folkman J, Brem H: Angiogenesis and inflammation. In: Inflammation: Basic Principles and Clinical Correlates Second Edition. (eds. JI Gallin, IM Goldstein and R Snyderman). New York:Raven Press, 1992

    Google Scholar 

  • Folkman J: Angiogenesis in cancer, vascular, rheumatoid and other disease. Nature Medicine 1995; 1: 21–31.

    Google Scholar 

  • Frieboes HB, Lowengrub JS, Wise S, Zheng X, Macklin P, et al. (2007) Computer simulation of glioma growth and morphology. Neuroimage 37(S1): 5970.

    Google Scholar 

  • Gerlee P, Anderson AR. An evolutionary hybrid cellular automaton model of solid tumour growth. J Theor Biol. 2007 Jun 21;246(4):583–603. Epub 2007 Feb 12.

    Google Scholar 

  • Gerlee P, Anderson AR. A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype. J Theor Biol. 2008 Feb 21;250(4):705–22.

    CAS  PubMed  Google Scholar 

  • Gerlee P, Anderson AR. Evolution of cell motility in an individual-based model of tumour growth. J Theor Biol. 2009 Jul 7;259(1):67–83.

    CAS  PubMed  PubMed Central  Google Scholar 

  • Gimbrone MA, Cotran RS, Leapman SB, Folkman J. Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Natn. Cancer Inst. 1974; 52: 413–427.

    Google Scholar 

  • Graham CH, Lala PK. Mechanisms of placental invasion of the uterus and their control. Biochem. Cell Biol. 1992; 70: 867–874.

    CAS  PubMed  Google Scholar 

  • Hanahan D. Signaling vascular morphogenesis and maintenance. Science 1997; 227: 48–50.

    Google Scholar 

  • Hanahan D, Weinberg RA. The hallmarks of cancer. Cell. 2000 Jan 7;100(1):57–70.

    CAS  PubMed  Google Scholar 

  • Hynes RO: Fibronectins. New York. Springer-Verlag, 1990

    Google Scholar 

  • Holmes MJ, Sleeman BD. A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects J. Theor. Biol. 2000; 202: 95–112.

    CAS  PubMed  Google Scholar 

  • Jain RK, Schlenger K, Höckel M, Yuan F: Quantitative angiogenesis assays: Progress and problems. Nature Med. 1997; 3: 1203–1208.

    CAS  PubMed  Google Scholar 

  • Kerbel RS. Tumor angiogenesis: past, present and the near future. Carcinogenesis. 2000 Mar;21(3):505–15.

    CAS  PubMed  Google Scholar 

  • Kiani M, Hudetz A. Computer simulation of growth of anastomosing microvascular networks. J. theor. Biol. 1991; 150: 547–560.

    CAS  PubMed  Google Scholar 

  • Kirchner LM, Schmidt SP, Gruber BS. 1996. Quantitation of angiogenesis in the chick chorioallantoic membrane model using fractal analysis. Microvasc. Res. 51:2–14.

    CAS  PubMed  Google Scholar 

  • Klagsbrun M, Soker S. VEGF/VPF: the angiogenesis factor found? Curr Biol. 1993 Oct 1;3(10):699–702.

    CAS  PubMed  Google Scholar 

  • Lacovara J, Cramer EB, Quigley JP. Fibronectin enhancement of directed migration of B16 melanoma cells. Cancer Res. 1984; 44: 1657–1663.

    CAS  PubMed  Google Scholar 

  • Landini G, Misson G. Simulation of corneal neo-vascularization by inverted diffusion limited aggregation. Invest. Opthamol. Visual Sci. 1993; 34: 1872–1875.

    CAS  Google Scholar 

  • Levine HA, Sleeman BD, Nilsen-Hamilto M. Mathematical modeling of the onset of a capillary formation initiating angiogenesis. J. Math. Biol. 2001;42:195–238.

    CAS  PubMed  Google Scholar 

  • Macklin P, McDougall S, Anderson AR, Chaplain MAJ, Cristini V, et al. (2009) Multiscale modeling and nonlinear simulation of vascular tumour growth. J Math Biol 58: 765798.

    Google Scholar 

  • Madri JA, Pratt BM. Endothelial cell-matrix interactions: in vitro models of angiogenesis. J. Histochem. Cytochem. 1986; 34: 85–91.

    CAS  PubMed  Google Scholar 

  • Mannoussaki D, Lubkin SL, Vernon RB, Murray JD. A mechanical model for the formation of vascular networks in vitro. Acta Biotheor. 1996; 44: 271–282.

    Google Scholar 

  • Mantzaris N, Webb SD, Othmer HG (2004) Mathematical modeling of tumour angiogenesis: A review. J Math Biol 49: 111187.

    Google Scholar 

  • McCarthy JB, Furcht LT. Laminin and fibronectin promote the directed migration of B16 melanoma cells in vitro. J. Cell Biol. 1984; 98: 1474–1480.

    CAS  PubMed  Google Scholar 

  • McDougall SR, Anderson ARA, Chaplain MAJ, and Sherratt JA (2002) Mathematical modeling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 64(4), 673–702.

    CAS  PubMed  Google Scholar 

  • McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modeling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies. J Theor Biol 241: 56489.

    Google Scholar 

  • Merks RMH and Glazier JA. 2006. Dynamic mechanisms of blood vessel growth. Nonlinearity, 19, C1–C10.

    PubMed  PubMed Central  Google Scholar 

  • Merks RMH, Erica D. Perryn, Abbas Shirinifard and James A. Glazier. 2008. Contact-inhibited chemotactic motility in de novo and sprouting blood vessel growth. PLoS Computational Biology 4(9): e1000163, 2008.

    Google Scholar 

  • Mitchell, A. R. and D. F. Griffiths (1980). The Finite Difference Method in Partial Differential Equations, Chichester: Wiley.

    Google Scholar 

  • Moore A, Marecos E, Simonova M, et al. Novel gliosarcoma cell line expressing green fluorescent protein: A model for quantitative assessment of angiogenesis. Microvasc. Res. 1998; 56: 145–153.

    CAS  PubMed  Google Scholar 

  • Murray JD, Mannoussaki D, Lubkin SL, Vernon RB: A mechanical theory of in vitro vascular network formation. In: Little CD, Mironov V, Sage EH (eds) Vascular morphogenesis: In vivo, in vitro, in mente. Birkhäuser, Boston, 1998, pp 173–188

    Google Scholar 

  • Murray JD, Swanson KR: On the mechanochemical theory of biological pattern formation with applications to wound healing and angiogenesis. In: Chaplain MAJ, Singh GD, McLachlan JC (eds) On Growth and Form: Spatio-temporal Pattern Formation in Biology. Wiley, Chichester, 1999, pp 251–285

    Google Scholar 

  • Muthukkaruppan VR, Kubai L, Auerbach R: Tumor-induced neovascularization in the mouse eye. J. Natn. Cancer Inst. 69: 699–705, 1982

    CAS  Google Scholar 

  • Nekka F, Kyriacos S, Kerrigan C, Cartilier L. A model of growing vascular structures. Bull. Math. Biol. 1996; 58: 409–424.

    CAS  PubMed  Google Scholar 

  • Norrby K. Microvascular density in terms of number and length of microvessel segments per unit tissue volume in mammalian angiogenesis. Microvasc. Res. 1998; 55: 43–53.

    CAS  PubMed  Google Scholar 

  • Olsen L, Sherratt JA, Maini PK, Arnold, F. A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis. IMA J. Math. Appl. Med. Biol. 1997; 14: 261–281.

    CAS  PubMed  Google Scholar 

  • O’Reilly MS, Holmgren L, Shing Y, Chen C, Rosenthal RA, Moses M, Lane WS, Cao Y, Sage EH, Folkman J. Angiostatin: a novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma. Cell 1994; 79: 315–328.

    PubMed  Google Scholar 

  • O’Reilly MS, Holmgren L, Chen C, Folkman J. Angiostatin induces and sustains dormancy of human primary tumours in mice. Nature Med. 1996; 2: 689–692.

    PubMed  Google Scholar 

  • O’Reilly MS, Boehm T, Shing Y, Fukai N, Vasios G, Lane WS, Flynn E, Birkhead JR, Olsen BR, Folkman J. Endostatin: an endogenous inhibitor of angiogenesis and tumor growth. Cell. 1997 Jan 24;88(2):277–85.

    PubMed  Google Scholar 

  • Orme ME, Chaplain MAJ. A Mathematical model of the First Steps of Tumour-Related Angiogenesis: Capillary Sprout Formation and Secondary Branching. IMA J. Math. App. Med. Biol. 1996; 13: 73–98.

    CAS  Google Scholar 

  • Orme ME, Chaplain MAJ. Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies. IMA J. Math. App. Med. and Biol. 1997; 14: 189–205.

    CAS  Google Scholar 

  • Owen MR, Alarcn T, Maini PK, Byrne HM (2009) Angiogenesis and vascular remodelling in normal and cancerous tissues. J Math Biol 58: 689721.

    Google Scholar 

  • Paku S, Paweletz N. First steps of tumor-related angiogenesis. Lab. Invest. 1991; 65: 334–346.

    CAS  PubMed  Google Scholar 

  • Parsons-Wingerter P, Lwai B, Yang MC, Elliott KE, Milaninia A, Redlitz A, Clark JI, Sage EH. 1998. A novel assay of angiogenesis in the quail chorioallantoic membrane: stimulation by bFGF and inhibition by angiostatin according to fractal dimension and grid intersection. Microvasc. Res. 55:201–214.

    CAS  PubMed  Google Scholar 

  • Paweletz N, Knierim M. Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 1989; 9: 197–242.

    CAS  PubMed  Google Scholar 

  • Perfahl H, Byrne HM, Chen T, Estrella V, Alarcn T, et al. 2011 Multiscale Modeling of Vascular Tumour Growth in 3D: The Roles of Domain Size and Boundary Conditions. PLoS ONE 6(4): e14790.

    CAS  PubMed  PubMed Central  Google Scholar 

  • Plank, M. J. and Sleeman, B. D. (2003). Tumour-induced angiogenesis: a review. J. Theor. Med. 5, 137–153.

    Google Scholar 

  • Plank, M. J. and Sleeman, B. D. (2003). A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies. Math. Med. Biol. 20, 135–181.

    CAS  PubMed  Google Scholar 

  • Plank, M. J. and Sleeman, B. D. (2004). Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66, 1785–1819.

    CAS  PubMed  Google Scholar 

  • Quigley JP, Lacovara J, Cramer EB. The directed migration of B-16 melanoma-cells in response to a haptotactic chemotactic gradient of fibronectin. J. Cell Biol. 1983; 97: A450–451.

    Google Scholar 

  • Qutub AA, F. Mac Gabhann, E.D. Karagiannis; P. Vempati, A.S. Popel, Multiscale models of angiogenesis. Engineering in Medicine and Biology Magazine, IEEE, 28(2):14–31 2009

    Google Scholar 

  • Rupnick MA, Stokes CL, Williams SK, Lauffenburger, DA. Quantitative analysis of human microvessel endothelial cells using a linear under-agarose assay. Lab. Invest. 1988; 59: 363–372.

    CAS  PubMed  Google Scholar 

  • Schor SL, Schor AM, Brazill GW. The effects of fibronectin on the migration of human foreskin fibroblasts and syrian hamster melanoma cells into three-dimensional gels of lattice collagen fibres. J. Cell Sci. 1981; 48: 301–314.

    CAS  PubMed  Google Scholar 

  • Shing Y, Folkman J, Sullivan R, Butterfield C, Murray J, Klagsbrun M. Heparin affinity: purification of a tumor-derived capillary endothelial cell growth factor. Science. 1984 Mar 23;223(4642):1296–9.

    CAS  PubMed  Google Scholar 

  • Sholley MM, Ferguson GP, Seibel HR, Montour JL, Wilson JD. Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 1984; 51: 624–634.

    CAS  PubMed  Google Scholar 

  • Stephanou A, McDougall SR, Anderson ARA, Chaplain MAJ (2005a) Mathematical modeling of flow in 2d and 3d vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies. Math Comp Modeling 41: 1137–1156.

    Google Scholar 

  • Stephanou A, McDougall SR, Anderson ARA, Chaplain (2005b) Mathematical modeling of the influence of blood rheological properties upon adaptive tumour- induced angiogenesis. Math. Comp. Model. 44, 96–123.

    Google Scholar 

  • Stokes CL, Lauffenburger DA, Williams SK. Migration of individual microvessel endothelial cells: stochastic model and parameter measurement. J. Cell Sci. 1991; 99: 419–430.

    PubMed  Google Scholar 

  • Stokes CL, Lauffenburger DA. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol. 1991; 152: 377–403.

    CAS  PubMed  Google Scholar 

  • Terranova VP, Diflorio R, Lyall RM, Hic S, Friesel R, Maciag T. Human endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell Biol. 1985; 101: 2330–2334.

    CAS  PubMed  Google Scholar 

  • Thompson DW. On Growth and Form, Cambridge University Press, Cambridge, 1917

    Google Scholar 

  • Williams SK: Isolation and culture of microvessel and large-vessel endothelial cells; their use in transport and clinical studies. In: McDonagh P (ed) Microvascular Perfusion and Transport in Health and Disease. Karger, Basel, pp 204–245, 1987

    Google Scholar 

  • Zawicki DF, Jain RK, Schmid-Schoenbein GW, Chien S. Dynamics of neovascularization in normal tissue. Microvasc. Res. 1981; 21: 27–47.

    CAS  PubMed  Google Scholar 

  • Zhai Y, Ni J, Jiang GW, Lu J, Xing L, Lincoln C, Carter KC, Janat F, Kozak D, Xu S, Rojas L, Aggarwal BB, Ruben S, Li LY, Gentz R, Yu GL. VEGI, a novel cytokine of the tumor necrosis factor family, is an angiogenesis inhibitor that suppresses the growth of colon carcinomas in vivo. FASEB J. 1999 Jan;13(1):181–9.

    CAS  PubMed  Google Scholar 

  • Zheng X, Wise SM, Cristini V (2005) Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level set method. Bull Math Biol 67(2): 211–259.

    CAS  PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, 33612, USA

    Alexander R. A. Anderson

  2. University of Dundee, Dundee, DD1 4HN, Scotland

    Mark A. J. Chaplain

  3. Heriot-Watt University, Edinburgh, EH14 4AS, Scotland

    Stephen McDougall

Authors
  1. Alexander R. A. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark A. J. Chaplain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephen McDougall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander R. A. Anderson .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan, 48109, USA

    Trachette L. L. Jackson

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Anderson, A.R.A., Chaplain, M.A.J., McDougall, S. (2012). A Hybrid Discrete-Continuum Model of Tumour Induced Angiogenesis. In: Jackson, T.L. (eds) Modeling Tumor Vasculature. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0052-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation