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Mathematical Biosciences and Engineering (MBE)
 

On the mathematical modelling of tumor-induced angiogenesis
Pages: 45 - 66, Issue 1, February 2017

doi:10.3934/mbe.2017004      Abstract        References        Full text (1323.2K)           Related Articles

Luis L. Bonilla - G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)
Vincenzo Capasso - ADAMSS, Universitá degli Studi di Milano, 20133 MILANO, Italy (email)
Mariano Alvaro - G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)
Manuel Carretero - G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)
Filippo Terragni - G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)

1 L. Ambrosio, V. Capasso and E. Villa, On the approximation of mean densities of random closed sets, Bernoulli, 15 (2009), 1222-1242.       
2 A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-900.
3 C. Birdwell, A. Brasier and L. Taylor, Two-dimensional peptide mapping of fibronectin from bovine aortic endothelial cells and bovine plasma, Biochem. Biophys. Res. Commun., 97 (1980), 574-581.
4 L. L. Bonilla, V. Capasso, M. Alvaro and M. Carretero, Hybrid modelling of tumor-induced angiogenesis, Physical Review E, 90 (2014), 062716.
5 P. Bremaud, Point Processes and Queues. Martingale Dynamics, Springer-Verlag, New-York, 1981.       
6 M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958.       
7 M. Burger, V. Capasso and L. Pizzocchero, Mesoscale averaging of nucleation and growth models, Multiscale Modeling and Simulation, 5 (2006), 564-592.       
8 V. Capasso, Randomness and Geometric Structures in Biology, in Pattern Formation in Morphogenesis. Problems and Mathematical Issues (eds. V. Capasso, M. Gromov, A. Harel-Bellan, N. Morozova and L.L. Pritchard), Springer, Heidelberg, 2013.
9 V. Capasso and D. Bakstein, An Introduction to Continuous-time Stochastic Processes, $3^{rd}$ edition, Birkhäuser, Boston, 2015.       
10 V. Capasso and D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol., 58 (2009), 219-233.       
11 V. Capasso, D. Morale and G. Facchetti, The Role of Stochasticity for a Model of Retinal Angiogenesis, IMA J. Appl. Math., 77 (2012), 729-747.       
12 V. Capasso and E. Villa, On the geometric densities of random closed sets, Stoch. Anal. Appl., 26 (2008), 784-808.       
13 P. F. Carmeliet, Angiogenesis in life, disease and medicine, Nature, 438 (2005), 932-936.
14 P. Carmeliet and R. K. Jain, Molecular mechanisms and clinical applications of angiogenesis, Nature, 473 (2011), 298-307.
15 P. Carmeliet and M. Tessier-Lavigne, Common mechanisms of nerve and blood vessel wiring, Nature, 436 (2005), 193-200.
16 A. Carpio and G. Duro, Well posedness of a kinetic model for angiogenesis, Nonlinear Analysis; Real World Applications, 30 (2016), 184-212.       
17 N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., textbf{55} (2007), 147-188.       
18 M. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168.
19 M. Corada, L. Zanetta, F. Orsenigo, F. Breviario, M. G. Lampugnani, S. Bernasconi, F. Liao, D. J. Hicklin, P. Bohlen and E. Dejana, A monoclonal antibody to vascular endothelial-cadherin inhibits tumor angiogenesis without side effects on endothelial permeability. Blood, 100 (2002), 905-911.
20 S. L. Cotter, V. Klika, L. Kimpton, S. Collins and A. E. P. Heazell, A stochastic model for early placental development, J. R. Soc. Interface, 11 (2014), 20140149, Available from: http://rsif.royalsocietypublishing.org/content/11/97/20140149.
21 R. F. Gariano and T. W. Gardner, Retinal angiogenesis in development and disease, Nature, 438 (2004), 960-966.
22 J. Folkman, Tumour angiogenesis, Adv. Cancer Res., 19 (1974), 331-358.
23 J. W. Gibbs, Elementary Principles of Statistical Mechanics, Yale Bicentennial Publications, Scribner and Sons, New York, 1902.
24 H. A. Harrington, M. Maier, L. Naidoo, N. Whitaker and P. G. Kevrekidis, A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors, Mathematical and Computer Modelling, 46 (2007), 513-524.
25 R. K. Jain and P. F. Carmeliet, Vessels of death or life, Sci. Am., 285 (2001), 38-45.
26 S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.       
27 N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187.       
28 D. Morale, V. Capasso and K. Ölschlaeger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.       
29 K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theor. Relat. Fields, 82 (1989), 565-586.       
30 M. J. Plank and B. D. Sleeman, Lattice and non-lattice models of tumour angiogenesis, Bull. Math. Biol., 66 (2004), 1785-1819.       
31 M. Hubbard, P. F. Jones and B. D. Sleeman, The foundations of a unified approach to mathematical modelling of angiogenesis, Int. J. Adv. Eng. Sci. and Appl. Math., 1 (2009), 43-52.
32 P. E. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer-Verlag, Heidelberg, 2004.       
33 G. G. Roussas, A Course in Mathematical Statistics, $2^{nd}$ edition, Academic Press, San Diego, CA, 1997.
34 M. Scianna, L. Munaron and L. Preziosi, A multiscale hybrid approach for vasculogenesis and related potential blocking therapies, Prog. Biophys. Mol. Biol., 106 (2011), 450-462.
35 A. Stéphanou, S. R. McDougall, A. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis, Math. Comput. Modelling, 44 (2006), 96-123.       
36 C. L. Stokes and D. A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol., 152 (1991), 377-403.
37 S. Sun, M. F. Wheeler, M. Obeyesekere and C. W. Patrick Jr., A deterministic model of growth factor-induced angiogenesis, Bull. Math. Biol., 67 (2005), 313-337.       
38 S. Sun, M. F. Wheeler, M. Obeyesekere and C. W. Patrick Jr., A multiscale angiogenesis modeling using mixed finite element methods, Multiscale Model. Simul., 4 (2005), 1137-1167.       
39 K. R. Swanson, R. C. Rockne, J. Claridge, M. A. Chaplain, E. C. Alvord Jr and A. R. A. Anderson, Quantifying the role of angiogenesis in malignant progression of gliomas: In silico modeling integrates imaging and histology, Cancer Res., 71 (2011), 7366-7375.
40 A. S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math., Springer, Berlin, 1464 (1991), 165-251.       
41 F. Terragni, M. Carretero, V. Capasso and L. L. Bonilla, Stochastic model of tumor-induced angiogenesis: Ensemble averages and deterministic equations, Physical Review E, 93 (2016), 022413.
42 S. Tong and F. Yuan, Numerical simulations of angiogenesis in the cornea, Microvascular Research, 61 (2001), 14-27.

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