# Angiogenesis and vascular remodelling in normal and cancerous tissues

- 851 Downloads
- 119 Citations

## Abstract

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.

## Keywords

Blood flow Multiscale modelling Tumour angiogenesis Vascular adaptation Vascularisation VEGF## Mathematics Subject Classification (2000)

9208 92C15 92C17 92C35 62P10## Supplementary material

Movie corresponding to Fig. 3: Evolution, from an initial pair of straight vessels (both with inflow at the left and outflow at the right), of an irregular vascular network via angiogenesis. Note that the normal cells most distant from the original two parent vessels initially die, until angiogenesis provides sufficient oxygen to sustain them. Parameter values are as in Tables 1-4 in the manuscript. ESM (mov 997 kb)

Movie corresponding to Fig. 5A: Pruning followed by angiogenesis. The chemotactic sensitivity gamma=8x10^{4}, and the new vessels are poorly directed and hence lead to poorer vascularisation. ESM (mov 1,204 kb)

Movie corresponding to Fig. 5B: Increasing the chemotaxis coefficient to gamma=8x10^{5} gives more rapid and better directed sprout growth, hence effectively remodelling the vasculature and oxygenating the whole tissue region. ESM (mov 990 kb)

Movie corresponding to Fig. 7B: Angiogenesis from a single initial vessel. Parameters as for Figure 5B, including an inflow pressure Pin=22 mmHg. From a single initial vessel the whole tissue region is fully oxygenated after 800 time units. At first, normal cells far from the parent vessel die, and then the whole region is repopulated in a wave-like manner as new vessels and normal cells grow together. ESM (mov 1,887 kb)

Movie corresponding to Fig. 7C: For a lower inflow pressure, Pin=19 mmHg, the balance between angiogenesis and vessel regression is altered, so that the vessel network cannot extend across the whole region, and hypoxia induced VEGF production is not eliminated. ESM (mov 2,994 kb)

Movie corresponding to Fig. 10: A simulation with tumour cells implanted in a tissue with normal cells and two linear initial vessels (as in Figure 3). Notice how the tumour remains confined close to the upper vessel until connections are made that allow it to spread fully into the lower half of the domain. Also worth noting is that tumour cells' increased oxygen consumption triggers VEGF expression by more normal cells as well, since they also experience resultant low oxygen levels. The final vascular density is significantly higher than with normal cells only (see Figure 12). Parameter values are as in Tables 1-4 in the manuscript. ESM (mov 882 kb)

## References

- 1.Addison-Smith B, McElwain S, Maini PK (2008) A simple mechanistic model of sprout spacing in tumour-associated angiogenesis. J Theor Biol 250(1): 1–15CrossRefGoogle Scholar
- 2.Alarcón T, Byrne HM, Maini PK (2003) A cellular automaton model for tumour growth in an inhomogeneous environment. J Theor Biol 225: 257–274CrossRefGoogle Scholar
- 3.Alarcón T, Byrne HM, Maini PK (2004) A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells. J Theor Biol 229(3): 395–411CrossRefGoogle Scholar
- 4.Alarcón T, Byrne HM, Maini PK (2005) A multiple scale model for tumor growth. Multiscale Model Sim 3: 440–475zbMATHCrossRefGoogle Scholar
- 5.Alarcón T, Owen MR, Byrne HM, Maini PK (2006) Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy. Comput Math Methods Med 7(2–3): 85–119zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Anderson ARA (2005) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math Med Biol 22: 163–186zbMATHCrossRefGoogle Scholar
- 7.Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60(5): 857–899zbMATHCrossRefGoogle Scholar
- 8.Arakelyan L, Merbl Y, Agur Z (2005) Vessel maturation effects on tumour growth: validation of a computer model in implanted human ovarian carcinoma spheroids. Eur J Cancer 41(1): 159–167CrossRefGoogle Scholar
- 9.Arakelyan L, Vainstein V, Agur Z (2002) A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth. Angiogenesis 5(3): 203–214CrossRefGoogle Scholar
- 10.Balding D, McElwain DLS (1985) A mathematical model of tumor induced capillary growth. J Theor Biol 114: 55–73CrossRefGoogle Scholar
- 11.Bartha K, Rieger H (2006) Vascular network remodeling via vessel cooption, regression and growth in tumors. J Theor Biol 241(4): 903–918MathSciNetGoogle Scholar
- 12.Bentley K, Gerhardt H, Bates PA (2008) Agent-based simulation of Notch-mediated tip cell selection in angiogenic sprout initialisation. J Theor Biol 250(1): 25–36CrossRefGoogle Scholar
- 13.Bertuzzi A, Gandolfi A (2000) Cell kinetics in a tumour cord. J Theor Biol 204(4): 587–599CrossRefGoogle Scholar
- 14.Betteridge R, Owen MR, Byrne HM, Alarcón T, Maini PK (2006) The impact of cell crowding and active cell movement in vascular tumour growth. Netw Heterog Media 1: 515–535zbMATHMathSciNetGoogle Scholar
- 15.Byrne HM, Alarcón T, Owen MR, Webb SD, Maini PK (2006) Modelling aspects of cancer dynamics: a review. Philos Trans R Soc A 364: 1563–1578CrossRefGoogle Scholar
- 16.Byrne HM, Chaplain MAJ (1995) Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull Math Biol 57(3): 461–486zbMATHGoogle Scholar
- 17.Carmeliet P (2003) Angiogenesis in health and disease. Nat Med 9: 653–660CrossRefGoogle Scholar
- 18.Chaplain MAJ, McDougall SR, Anderson ARA (2006) Mathematical modeling of tumor-induced angiogenesis. Ann Rev Biomed Eng 8: 233–257CrossRefGoogle Scholar
- 19.Clark ER (1918) Studies on the growth of blood-vessels in the tail of the frog larva—by observation and experiment on the living animal. Am J Anat 23(1): 37–88CrossRefGoogle Scholar
- 20.Dorman S, Deutsch A (2002) Modeling of self-organized avascular tumor growth with cellular automata. In Silico Biol 2: 35Google Scholar
- 21.Folkman J (1971) Tumour angiogenesis—therapeutic implications. New Engl J Med 285: 1182–1186Google Scholar
- 22.Folkman J, Klagsburn M (1987) Angiogenic factors. Science 235: 442–447CrossRefGoogle Scholar
- 23.Gevertz JL, Torquato S (2006) Modeling the effects of vasculature evolution on early brain tumor growth. J Theor Biol 243(4): 517–531CrossRefMathSciNetGoogle Scholar
- 24.Hellstrom M, Phng LK, Hofmann JJ, Wallgard E, Coultas L, Lindblom P, Alva J, Nilsson AK, Karlsson L, Gaiano N, Yoon K, Rossant J, Iruela-Arispe ML, Kalen M, Gerhardt H, Betsholtz C (2007) Dll4 signalling through Notch1 regulates formation of tip cells during angiogenesis. Nature 445(7129): 776–780CrossRefGoogle Scholar
- 25.Jain RK (1987) Determinants of tumour blood flow: a review. Cancer Res 47: 2461–2468Google Scholar
- 26.Jain RK (2003) Molecular regulation of vessel maturation. Nat Med 9: 685–693CrossRefGoogle Scholar
- 27.Jain RK (2005) Normalization of tumour vasculature: an emerging concept in antiangiogenic therapy. Science 307: 58–62CrossRefGoogle Scholar
- 28.Lee DS, Rieger H, Bartha K (2006) Flow correlated percolation during vascular remodeling in growing tumors. Phys Rev Lett 96(5): 058,104–Google Scholar
- 29.Levine HA, Sleeman BD, Nilsen-Hamilton M (2001) Mathematical modeling of the onset of capillary formation initiating angiogenesis. J Math Biol 42(3): 195–238zbMATHCrossRefMathSciNetGoogle Scholar
- 30.Levine HA, Tucker AL, Nilsen-Hamilton M (2002) A mathematical model for the role of cell signal transduction in the initiation and inhibition of angiogenesis. Growth Factors 20(4): 155–175CrossRefGoogle Scholar
- 31.Mantzaris N, Webb SD, Othmer HG (2004) Mathematical modelling of tumour angiogenesis: a review. J Math Biol 49: 111–187zbMATHCrossRefMathSciNetGoogle Scholar
- 32.McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J Theor Biol 241(3): 564–89CrossRefMathSciNetGoogle Scholar
- 33.McDougall SR, Anderson ARA, Chaplain MAJ, Sherratt JA (2002) Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull Math Biol 64(4): 673–702CrossRefGoogle Scholar
- 34.Meeson A, Palmer M, Calfon M, Lang R (1996) A relationship between apoptosis and flow during programmed capillary regression is revealed by vital analysis. Development 122(12): 3929–3938Google Scholar
- 35.Othmer HG, Stevens A (1997) Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J Appl Math 57(4): 1044–1081zbMATHCrossRefMathSciNetGoogle Scholar
- 36.Owen MR, Byrne HM, Lewis CE (2004) Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites. J Theor Biol 226: 377–391CrossRefMathSciNetGoogle Scholar
- 37.Patel A, Gawlinski E, Lemieux S, Gatenby R (2001) A cellular automaton model of early tumor growth and invasion: the effects of native tissue vascularity and increased anaerobic tumor metabolism. J Theor Biol 213: 315–331CrossRefMathSciNetGoogle Scholar
- 38.Plank M, Sleeman B (2004) Lattice and non-lattice models of tumour angiogenesis. Bull Math Biol 66(6): 1785–1819CrossRefMathSciNetGoogle Scholar
- 39.Plank MJ, Sleeman BD (2003) A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies. Math Med Biol 20(2): 135–181zbMATHCrossRefGoogle Scholar
- 40.Pries A, Reglin B, Secomb T (2001) Structural adaptation of microvascular networks: functional roles of adaptive responses. Am J Physiol 281: H1015–H1025Google Scholar
- 41.Pugh C, Rattcliffe P (2003) Regulation of angiogenesis by hypoxia: role of the HIF system. Nat Med 9: 677–684CrossRefGoogle Scholar
- 42.Resnick N, Yahav H, Shay-Salit A, Shushy M, Schubert S, Zilberman LCM, Wofovitz E (2003) Fluid shear stress and the vascular endothelium: for better and for worse. Progress Biophys Mol Biol 81(3): 177–199CrossRefGoogle Scholar
- 43.Risau W (1997) Mechanisms of angiogenesis. Nature 386: 871–875Google Scholar
- 44.Siekmann AF, Lawson ND (2007) Notch signalling limits angiogenic cell behaviour in developing zebrafish arteries. Nature 445(7129): 781–784CrossRefGoogle Scholar
- 45.Stephanou A, McDougall SR, Anderson ARA, Chaplain MAJ (2005) Mathematical modelling of flow in 2d and 3d vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies. Math Comput Modell 41(10): 1137–1156zbMATHCrossRefMathSciNetGoogle Scholar
- 46.Stephanou A, McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis. Math Comput Modell 44(1–2): 96–123zbMATHCrossRefMathSciNetGoogle Scholar
- 47.Stokes CL, Lauffenburger DA (1991) Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J Theor Biol 152(3): 377–403CrossRefGoogle Scholar
- 48.Suchting S, Freitas C, le Noble F, Benedito R, Breant C, Duarte A, Eichmann A (2007) The Notch ligand Delta-like 4 negatively regulates endothelial tip cell formation and vessel branching. Proc Natl Acad Sci 104(9): 3225–3230CrossRefGoogle Scholar
- 49.Teicher B (1996) A systems approach to cancer therapy. Cancer Metastasis Rev 15: 247–272CrossRefGoogle Scholar
- 50.Tong R, Boucher Y, Kozin S, Winkler F, Kicklin D, Jain R (2004) Vascular normalisation by VEGFR2 blockade induces a pressure gradient accros the vasculature and improves drug penetration in tumours. Cancer Res 64: 3731–3736CrossRefGoogle Scholar
- 51.Tyson JJ, Novák B (2001) Regulation of the eukaryotic cell-cycle: molecular anatagonism, hysteresis, and irreversible transitions. J Theor Biol 210: 249–263CrossRefGoogle Scholar
- 52.Welter M, Bartha K, Rieger H (2008) Emergent vascular network inhomogeneities and resulting blood flow patterns in a growing tumor. J Theor Biol 250(2): 257–280CrossRefGoogle Scholar
- 53.Winkler F, Kozin S, Tong R, Chae S-S, Booth M, Garkavtsev I, Xu L, Hicklin D, Fukumura D, di Tomaso E, Munn L, Jain R (2004) Kinetics of vascular normalisation by VEGFR2 blockade governs brain tumour response to radiation: role of oxyxgenation, angiopoietin-1 and matrix metalloprotienases. Cancer Cell 6: 553–563Google Scholar
- 54.Yen RT, Fung YC (1978) Effect of velocity distribution on red cell distribution in capillary blood vessels. AJP Heart Circ Physiol 235(2): H251–H257Google Scholar