Multiscale modelling and nonlinear simulation of vascular tumour growth

Alarcón T, Byrne HM, Maini PK (2005) A multiple scale model for tumor growth. Multiscale Model Simul 3: 440–475

Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Model Meth Appl Sci 12(5): 737–754

Anderson ARA (2005) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. IMA Math App Med Biol 22(2): 163–186

Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60(5): 857–900

Anderson ARA, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5): 905–915

Araujo RP, McElwain DLS (2004) A history of the study of solid tumor growth: the contribution of mathematical modeling. Bull Math Biol 66(5): 1039–1091

Araujo RP, McElwain DLS (2005) A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J Appl Math 65: 1261–1284

Araujo RP, McElwain DLS (2005) A mixture theory for the genesis of residual stresses in growing tissues II: solutions to the biphasic equations for a multicell spheroid. SIAM J Appl Math 66(2): 447–467

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

Bartha K, Rieger H (2007) Vascular network remodeling via vessel cooption, regression and growth in tumors. J Theor Biol 241(4): 903–918

Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20(4): 341–366

Byrne HM, Alarcón T, Owen MR, Webb SD, Maini PK (2006) Modeling aspects of cancer dynamics: a review. Phil Trans R Soc A 364(1843): 1563–1578

Byrne HM, Chaplain MAJ (1995) Growth of non-necrotic tumours in the presence and absence of inhibitors. Math Biosci 130: 151–181

Byrne HM, Chaplain MAJ (1996) Growth of necrotic tumours in the presence and absence of inhibitors. Math Biosci 135: 187–216

Byrne HM, Chaplain MAJ (1998) Free boundary problems arising in models of tumour growth and development. Eur J Appl Math 8: 639–658

Carmeliet P (2005) Angiogenesis in life, disease, and medicine. Nature 438: 932–936

Chaplain MAJ (1995) The mathematical modelling of tumour angiogenesis and invasion. Acta Biotheor 43: 387–402

Chaplain MAJ, McDougall SR, Anderson ARA (2006) Mathematical modelling of tumor-induced angiogenesis. Annu Rev Biomed Eng 8: 233–257

Chomyak OG, Sidorenko MV (2001) Multicellular spheroids model in oncology. Exp Oncol 23: 236–241

Cristini V, Frieboes HB, Gatenby R, Caserta S, Ferrari M, Sinek J (2005) Morphological instability and cancer invasion. Clin Cancer Res 11(19): 6772–6779

Cristini V, Li X, Lowengrub J, Wise S (2008) Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J Math Biol in press

Cristini V, Lowengrub JS, Nie Q (2003) Nonlinear simulation of tumor growth. J Math Biol 46: 191–224

Dickinson RB, Tranquillo RT (1993) A stochastic model for adhesion-mediated cell random motility and haptotaxis. J Math Biol 31: 563–600

DiMilla PA, Barbee K, Lauffenburger DA (1991) Mathematical model for the effects of adhesion and mechanics on cell migration speed. Biophys J 60: 15–37

Erler JT, Bennewith KL, Nicolau M, Dornhöfer N, Kong C, Le QT, Chi JTA, Jeffrey SS, Giaccia AJ (2006) Lysyl oxidase is essential for hypoxia-induced metastasis. Nature 440(27): 1222–1226

Frieboes HB, Lowengrub JS, Wise S, Zheng X, Macklin P, Cristini V (2007) Computer simulations of glioma growth and morphology. NeuroImage 37(S1): S59–S70

Frieboes HB, Zheng X, Sun CH, Tromberg B, Gatenby R, Cristini V (2006) An integrated computational/experimental model of tumor invasion. Can Res 66(3): 1597–1604

Friedl P, Wolf K (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat Rev Cancer 3: 362–374

Galaris D, Barbouti A, Korantzopoulos P (2006) Oxidative stress in hepatic ischemia–reperfusion injury: the role of antioxidants and iron chelating compounds. Curr Pharm Des 12(23): 2875–2890

Gerlee P, Anderson ARA (2007) Stability analysis of a hybrid cellular automaton model of cell colony growth. Phys Rev E 75: 0151,911

Graziano L, Preziosi L (2007) Mechanics in tumor growth. In: Mollica F, Rajagopal KR, Preziosi L (eds) Modelling of Biological Materials. Birkhäuser, Boston, pp 267–328

Greenspan HP (1976) On the growth and stability of cell cultures and solid tumors. J Theor Biol 56(1): 229–242

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

Hogea CS, Murray BT, Sethian JA (2006) Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method. J Math Biol 53(1): 86–134

Holash J, Weigand SJ, Yancopoulos GD (1999) New model of tumor-induced angiogenesis; dynamic balance between vessel regresion and growth mediated by angiopoietins and vegf. Oncogene 18: 5356–5362

Kaur B, Khwaja FW, Severson EA, Matheny SL, Brat DJ, Van Meir EG (2005) Hypoxia and the hypoxia-inducible-factor pathway in glioma growth and angiogenesis. Neuro-oncology 7(2): 134–153

Kim JB (2005) Three-dimensional tissue culture models in canceriology. J Biomol Screen 15: 365–377

Kloner RA, Jennings RB (2001) Consequences of brief ischemia: stunning, preconditioning, and their clinical implications: part 1. Circulation 104(24): 2981–2989

Kunz-Schughart LA, Freyer JP, Hofstaedter F, Ebner R (2004) The use of 3-d cultures for high-throughput screening: the multicellular spheroid model. J Biomol Screen 9: 273–285

Lauffenburger DA, Horwitz AF (1996) Cell migration: a physically integrated molecular process. Cell 84(3): 359–369

Lee D, Rieger H (2006) Flow correlated percolation during vascular remodeling in growing tumors. Phys Rev Lett 96: 058,104

Li X, Cristini V, Nie Q, Lowengrub J (2007) Nonlinear three-dimensional simulation of solid tumor growth. Disc Dyn Contin Dyn Syst B 7: 581–604

Liotta LA, Clair T (2000) Checkpoint for invasion. Ann Ital Med Int 15(3): 195–198

Liotta LA, Stetler-Stevenson WG (1991) Tumor cell motility. Sem Canc Biol 2(2): 111–114

Macklin P, Lowengrub JS (2005) Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth. J Comput Phys 203(1): 191–220

Macklin P, Lowengrub JS (2006) An improved geometry-aware curvature discretization for level set methods: application to tumor growth. J Comput Phys 215(2): 392–401

Macklin P, Lowengrub JS (2007) Nonlinear simulation of the effect of microenvironment on tumor growth. J Theor Biol 245(4): 677–704

Macklin P, Lowengrub JS (2008) A new ghost cell/level set method for moving boundary problems: Application to tumor growth. J Sci Comput (in press)

Mantzaris NV, Webb S, Othmer HG (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49: 111–187

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): 564–589

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): 673–702

Orme ME, Chaplain MAJ (1996) A mathematical model of vascular tumour growth and invasion. Math Comp Modell 23: 43–60

Palecek SP, Loftus JC, Ginsberg MH, Lauffenburger DA, Horwitz AF (1997) Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature 385: 537–540

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

Plank MJ, Sleeman BD (2004) Lattice and non-lattice models of tumour angiogenesis. Bull Math Biol 66: 1785–1819

Pouysségur J, Dayan F, Mazure NM (2006) Hypoxia signalling in cancer and approaches to enforce tumour regression. Nature 441(25): 437–443

Pries AR, Reglin B, Secomb TW (2001) Structural adaptation of microvascular networks: functional roles of adaptive responses. Am J Physiol Heart Circ Physiol 281: H1015–H1025

Pries AR, Reglin B, Secomb TW (2001) Structural adaptation of vascular networks: role of the pressure response. Hypertension 38: 1476–1479

Pries AR, Secomb TW, Gaehtgens P (1995) Design principles of vascular beds. Circ Res 77: 1017–1023

Pries AR, Secomb TW, Gaehtgens P (1996) Biophysical aspects of blood flow in the microvasculature. Cardivasc Res 32: 654–667

Pries AR, Secomb TW, Gaehtgens P (1998) Structural adaptation and stability of microvascular netwoks: theory and simulation. Am J Physiol Heart Circ Physiol 275(44): H349–H360

Quaranta V, Weaver AM, Cummings PT, Anderson ARA (2005) Mathematical modeling of cancer: the future of prognosis and treatment. Clin Chim Acta 357(2): 173–179

Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular cancer. SIAM Rev 49: 179–208

Sanga S, Sinek JP, Frieboes HB, Fruehauf JP, Cristini V (2006) Mathematical modeling of cancer progression and response to chemotherapy. Exp Rev Anticancer Ther 6(10): 1361–1376

Sinek J, Frieboes H, Zheng X, Cristini V (2004) Two-dimensional chemotherapy simulations demonstrate fundamental transport and tumor response limitations involving nanoparticles. Biomed Microdev 6(4): 197–309

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: 1137–1156

Stéphanou A, McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of the influence of blood rheological properties upon adaptive tumour-induced angiogenesis. Math Comp Model 44(1–): 96–123

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

Walles T, Weimer M, Linke K, Michaelis J, Mertsching H (2007) The potential of bioartificial tissues in oncology research and treatment. Onkologie 30: 388–394

Welter M, Bartha K, Rieger H (2008) Emergent vascular network inhomogenities and resulting blood flow patterns in a growing tumor. J Theor Biol 250: 257–280

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