Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis

Mathematical Biosciences

Volumn 214, Issue 1-2, 2008, Pages 20-31

  Capasso, Vincenzo ^a   Micheletti, Alessandra ^a   Morale, Daniela ^a  

^a UNIVERSITY OF MILAN   (Italy)

Author keywords

Multiple scales; Stochastic distributions; Stochastic geometry; Tumor angiogenesis

Indexed keywords

BLOOD VESSELS; ESTIMATION; FIBERS; GEOMETRY; MATHEMATICAL MODELS; MODAL ANALYSIS; REAL TIME SYSTEMS; STATISTICAL METHODS; STATISTICS; STOCHASTIC PROGRAMMING; TUMORS;

(060.2410) FIBERS; (ALGORITHMIC) COMPLEXITY; ANGIOGENESIS; BIOCHEMICAL SIGNALS; CLOSED SETS; COUPLED SYSTEMS; ELSEVIER (CO); EVOLUTION (CO); GEOMETRIC DENSITIES; GEOMETRIC MODELLING; GROWTH PROCESSES; HAUSDORFF DIMENSIONS; HYBRID MODEL (HM); MATHEMATICAL MODELLING; MESO SCALE; MESOSCALES; QUANTITATIVE DESCRIPTION; STATISTICAL ANALYSIS (IGC: E4/K7); STATISTICAL ESTIMATION; STOCHASTIC BRANCHING; VASCULAR SYSTEM;

STOCHASTIC MODELS;

CARDIOVASCULAR DISEASE; GEOMETRY; STATISTICAL ANALYSIS; STOCHASTICITY; TUMOR;

ANGIOGENESIS; ARTICLE; BLOOD VESSEL; CARCINOGENESIS; CARDIOVASCULAR SYSTEM; CHEMICAL ANALYSIS; GEOMETRY; MATHEMATICAL MODEL; STATISTICAL ANALYSIS; STOCHASTIC MODEL;

ALGORITHMS; ANIMALS; BLOOD VESSELS; CHEMOTAXIS; MODELS, BIOLOGICAL; NEOPLASMS; NEOVASCULARIZATION, PATHOLOGIC; STOCHASTIC PROCESSES;

EID: 47549094092     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2008.03.009     Document Type: Article

References (49)

1. L. Ambrosio, V. Capasso, E. Villa, On the approximation of geometric densities of random closed sets. RICAM Report N. 2006-14, Linz, Austria (2006).
2. Ambrosio L., Fusco N., and Pallara D. Functions of Bounded Variation and Free Discontinuity Problems (2000), Clarendon Press, Oxford
3. Anderson A.R.A., and Chaplain M. Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60 (1998) 857
4. Araujo A., and Giné E. The Central Limit Theorem for Real and Banach Valued Random Variables (1980), John Wiley & Sons, New York
5. Avrami M. Kinetics of phase change. Part I. J. Chem. Phys. 7 (1939) 1103
6. Baddeley A.J. A crash course in stochastic geometry. In: Barndorff-Nielsen O.E., et al. (Ed). Stochastic Geometry Likelihood and Computation (1999), Chapman& Hall/CRC, Boca Raton
7. Bellomo N., De Angelis E., and Preziosi L. Multiscale modelling and mathematical problems related to tumour evolution and medical therapy. J. Theor. Med. 5 (2004) 111
8. Beneš V., and Rataj J. Stochastic Geometry (2004), Kluwer, Dordrecht
9. Bosq D. Linear Processes in Function Spaces. Theory and Applications, Lecture Notes in Statistics 149 (2000), Springer-Verlag, New York
10. Bremaud P. Point Processes and Queues. Martingale Dynamics (1981), Springer-Verlag, New-York
11. Burger M., Capasso V., and Pizzocchero L. Mesoscale averaging of nucleation and growth models. Multiscale Model. Simulation: A SIAM Interdisciplinary J 5 (2006) 564
12. Burger M., Capasso V., and Salani C. Modelling multi-dimensional crystallization of polymers in interaction with heat transfer. Nonlinear Anal.: Real World Appl. 3 (2002) 139
13. Capasso V., and Bakstein D. An Introduction to Continuous-Time Stochastic Processes (2005), Birkhäuser, Boston
14. Capasso V., Dejana E., and Micheletti A. Methods of stochastic geometry and related statistical problems in the analysis and therapy of tumour growth and tumour-driven angiogenesis. In: Bellomo N., Chaplain M., and DeAngelis E. (Eds). Selected Topics on Cancer Modelling. Genesis-Evolution-Immune Competition-Therapy (2007), Birkhauser, Boston
15. Capasso V., and Micheletti A. Stochastic geometry and related statistical problems in Biomedicine. In: Quarteroni A., et al. (Ed). Complex Systems in Biomedicine (2006), Springer, Milano 36
16. V. Capasso, A. Micheletti, Kernel-like estimators of the intensity of inhomogeneous fibre processes, Preprint, 2008.
17. V. Capasso, A. Micheletti, On the mean geometric densities of random closed sets, and their estimation, Application to the estimation of the mean density of inhomogeneous fibre processes, in: Proceedings of ECMI06, L. Bonilla (Ed.), Springer, Heidelberg, 2008, pp. 3-34.
18. V. Capasso, A. Micheletti, On the estimation of mean geometric densities of non homogeneous fibre processes, manuscript in preparation.
19. V. Capasso, D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol. (2008), in press.
20. Capasso V., Morale D., and Salani C. Polymer crystallization processes via many particle systems. In: Capasso V. (Ed). Mathematical Modelling for Polymer Processing Polymerization Crystallization Manufacturing (2003), Springer-Verlag, Heidelberg
21. Capasso V., and Villa E. Continuous and absolutely continuous random sets. Stoch. Anal. Appl. 24 (2006) 381
22. Capasso V., and Villa E. On the geometric densities of random closed sets. Stoch. Anal. Appl. (2007)
23. Capasso V., and Villa E. On mean densities of inhomogeneous geometric processes arising in material science and medicine. Image Anal. Stereol. 26 (2007) 23
24. Chaplain M.A.J., and Anderson A.R.A. Modelling the growth and form of capillary networks. In: Chaplain M.A.J., et al. (Ed). On Growth and Form. Spatio-temporal Pattern Formation in Biology (1999), John Wiley & Sons, Chichester
25. Corada M., Zanetta L., Orsenigo F., Breviario F., Lampugnani M.G., Bernasconi S., Liao F., Hicklin D.J., Bohlen P., and Dejana E. A monoclonal antibody to vascular endothelial-cadherin inhibits tumor angiogenesis without side effects on endothelial permeability. Blood 100 (2002) 905
26. Crosby C.V., Fleming P., Zanetta L., Corada M., Giles B., Dejana E., and Drake C. VE-cadherin is essential in the de novo genesis of blood vessels (vasculogenesis) in the allantoids. Blood 105 (2005) 2771
27. Falconer K.J. The Geometry of Fractal Sets (1985), Cambridge University press, Cambridge
28. Federer H. Geometric Measure Theory (1996), Springer, Berlin
29. Folkman J. Tumour angiogenesis. Adv. Canc. Res. 19 (1974) 331
30. Folkman J., and Klagsbrun M. Angiogenic factors. Science 235 (1987) 442
31. Friedman A. Foundations of Modern Analysis (1982), Dover Pub., Mineola, New York
32. Gabrial A.S., and Krasnow M.A. Social interactions among epithelial cells during tracheal branching morphogenesis. Nature 441 (2006) 746
33. Hahn U., Micheletti A., Pohlink R., Stoyan D., and Wendrock H. Stereological analysis and modeling of gradient structures. J. Microscopy 195 (1999) 113
34. Hardle W. Smoothing Techniques with Implementation in S (1991), Springer-Verlag, New York
35. Harrington H.A., et al. A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors. Math. Comp. Modell. 46 (2007) 513
36. Jain R.K., and Carmeliet P.F. Vessels of death or life. Sci. Am. 285 (2001) 38
37. Kolmogorov A.N. Foundations of the Theory of Probability (1956), Chelsea Pub. Co., New York
38. Kolmogorov A.N. On the statistical theory of the crystallization of metals. Bull. Acad. Sci. USSR, Math. Ser. 1 (1937) 355
39. Kolmogorov A.N., and Fomin S.V. Introductory Real Analysis (1970), Prentice-Hall, Englewood Cliffs (N.J.)
40. Levine H.A., Pamuk S., Sleeman B.D., and Nilsen-Hamilton M. A mathematical model of capillary formation and development in tumour angiogenesis: penetration into the stroma. Bull. Math. Biol. 63 (2001) 801
41. Matheron G. Random Sets and Integral Geometry (1975), John Wiley & Sons, New York
42. McDougall S.R., Anderson A.R.A., and Chaplain M.A.J. Mathematical modelling of dynamic tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J. Theor. Biol. 241 (2006) 564
43. Møller J. Random Johnson-Mehl tessellations. Adv. Appl. Prob. 24 (1992) 814
44. Møller J. Lectures on Random Voronoi Tessellations Lecture Notes in Statistics (1994), Springer-Verlag, Neiw York
45. Pestman W.R. Mathematical Statistics. An Introduction (1998), Water de Gruyter, Berlin
46. Planck M.J., and Sleeman B.D. A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies. Math. Med. Biol. 20 (2003) 135
47. Preziosi L., and Astanin S. Modelling the formation of capillaries. In: Quarteroni A., et al. (Ed). Complex Systems in Biomedicine (2006), Springer, Milano 109
48. Stoyan D., Kendall W.S., and Mecke J. Stochastic Geometry and its Application (1995), John Wiley & Sons, New York
49. Sun S., et al. Nonlinear behaviors of capillary formation in a deterministic angiogenesis model. Nonlinear Anal. 63 (2005) e2237