Stochastic geometry and related statistical problems in biomedicine

Complex Systems in Biomedicine

Volumn , Issue , 2006, Pages 35-69

  Capasso, V ^a   Micheletti, A ^a  

^a UNIVERSITY OF MILAN   (Italy)

Author keywords

angiogenesis; birth and growth processes; geometric densities; statistics of stochastic geometries; stochastic geometry; tumor growth

Indexed keywords

EID: 84900088439     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/88-470-0396-2_2     Document Type: Chapter

References (54)

1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford: Clarendon Press 2000
2. Anderson, A.R.A.: The effects of cell adhesion on solid tumour geometry. In: Sekimura, T. et al. (eds.): Morphogenesis and pattern formation in biological systems. Tokyo: Springer 2003, pp. 315-325
3. Anderson, A.R.A., Chaplain, M.A.J.: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857-899 (1998) (Pubitemid 28427465)
4. Barles, G., Soner, H.M., Souganidis, P.E.: Front propagation and phase field theory. SIAM J. Control Optim. 31, 439-469 (1993)
5. Barndorff-Nielsen, O.E., Kendall,W.S., van Lieshout, M.N.M. (eds.): Stochastic geometry. Likelihood and computation. Boca Raton, FL: Chapman & Hall/CRC 1999
6. Burger, M.: Growth fronts of first-order Hamilton-Jacobi equations. SFB Report 02-8. Linz: J. Kepler Universit? 2002
7. Burger, M., Capasso,V., Pizzocchero, L.: Mesoscale averaging of nucleation and growth models. CAM Report 05-19. Los Angeles: Dept. of Math., UCLA 2005
8. Burger, M., Capasso,V., Salani, C.: Modelling multi-dimensional crystallization of polymers in interaction with heat transfer. Nonlinear Anal. Real World Appl. 3, 139-160 (2002) (Pubitemid 40030739)
9. CapassoV. (ed.): Mathematical modelling for polymer processing. Berlin: Springer 2003
10. Capasso,V., Micheletti, A.: Local spherical contact distribution function and local mean densities for inhomogeneous random sets. Stochastics Stochastics Rep. 71, 51-67 (2000)
11. Capasso,V., Micheletti, A.: On the hazard function for inhomogeneous birth-And-growth processes. Quaderno 39/2001. Milano: Dip. di Mat., Uni. degli Studi di Milano 2001
12. Capasso,V., Micheletti, A.: Stochastic geometry of spatially structured birth and growth processes. Application to crystallization processes. In: Capasso, V. et al. (eds.): Topics in spatial stochastic processes. (Lecture Notes in Math. 1802) Berlin: Springer 2003, pp. 1-39
13. Capasso, V., Micheletti, A., Burger, M.: Densities of n-facets of incomplete Johnson-Mehl tessellations generated by inhomogeneous birth-And-growth processes. Quaderno 38/2001. Milano: Dip. di Mat., Uni. degli Studi di Milano 2001
14. Capasso,V., Morale, D., Salani, C.: Polymer crystallization processes via many particle systems. In: Capasso, V. (ed.): Mathematical modelling for polymer processing. Berlin: Springer 2003, pp. 243-259
15. Capasso,V.,Villa, E.: On the evolution equation of mean geometric densities for a class of space and time inhomogeneous stochastic birth-And-growth processes. In: Weil, W. (ed.): Stochastic geometry. (Lecture Notes in Math.) Berlin: Springer 2006
16. Capasso, V., Villa, E.: On the continuity and absolute continuity of random closed sets. Preprint. Milano: Dip. di Mat., Uni. degli Studi di Milano 2005
17. Carmeliet, P., Jain, R.K.:Angiogenesis in cancer and other diseases. Nature 407, 249-257 (2000)
18. Chaplain, M.A.J., Anderson, A.R.A.: Modelling the growth and form of capillary networks. In: Chaplain, M.A.J. et al. (eds.): On growth and form: Spatio-Temporal pattern formation in biology. Chichester:Wiley 1999, pp. 225-250
19. Chen, C.S., Mrksich, M., Huang, S., Whitesides, G.M., Ingber, D.E.: Geometric control of cell life and death. Science 276, 1425-1428 (1997) (Pubitemid 27235413)
20. Cressie, N.A.C.: Statistics for spatial data. (Wiley Series in Probability and Mathematical Statistics) NewYork:Wiley 1993
21. do Carmo, M.P.: Differential geometry of curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall 1976
22. Drasdo, D., Hoehme, S.: Individual-based approaches to birth and death in avascular tumors. Math. Comput. Modelling 37, 1163-1175 (2003) (Pubitemid 36761623)
23. Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Boca Raton, FL: CRC Press 1992
24. Federer, H.: Geometric measure theory. Berlin: Springer 1969
25. Galle, J., Loeffler, M., Drasdo, D.: Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys. J. 88, 62-75 (2005) (Pubitemid 40070658)
26. Ghyka, M.: The geometry of art and life. NewYork: Dover 1977
27. Huang, S., Ingber, D.E.: Shape-dependent control of cell growth, differentiation, and apoptosis: Switching between attractors in cell regulatory networks. Exp. Cell. Res. 261, 91-103 (2000)
28. Jain, R.K., Carmeliet, P.F.: Vessels of death or life. Sci. Amer. 285(6), 38-45 (2001)
29. Johnson, W.A., Mehl, R.F.: Reaction kinetics in processes of nucleation and growth. Trans. AIME 135, 416-458 (1939)
30. Kolmogorov, A.N.: On the statistical theory of the crystallization of metals. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 4, 355-359 (1937)
31. Kolmogorov, A.N.: Foundations of the theory of probability. NewYork: Chelsea 1956
32. Landini, G., Misson, G.: Simulation of corneal neovascularization by inverted diffusion limited aggregation. Invest. Ophtamol. Vis. Sci. 34, 1872-1875 (1993)
33. Li, X., Wang, W., Martin, R.R., Bowyer, A.: Using low-discrepancy sequences and the Crofton formula to compute surface areas of geometric models. Comp. Aided Design 35, 771-782 (2003)
34. Lorenz, T.: Set-valued maps for image segmentation. Comput.Vis. Sci. 4, 41-57 (2001)
35. Matheron, G.: Random sets and integral geometry. NewYork:Wiley 1975
36. Micheletti, A.: The surface density of a random Johnson-Mehl tessellation. Quaderno 17/2001. Milano: Dip. di Mat., Uni. degli Studi di Milano 2001
37. Micheletti,A., Capasso,V.:The stochastic geometry of polymer crystallization processes. Stochastic Anal. Appl. 15, 355-373 (1997) (Pubitemid 127370230)
38. Miller, R.T., Anderson, S.P., Corton, J.C., Cattley, R.C.: Apoptosis, mitosis and cyclophilin-40 expression in regressing peroxisome proliferator-induced adenomas. Carcinogenesis 21, 647-652 (2000) (Pubitemid 30214020)
39. Miller, J.: Random Johnson-Mehl tessellations. Adv. Appl. Prob. 24, 814-844 (1992)
40. Miller, J.: Lectures on randomVoronoi Tessellations. (Lecture Notes in Statist. 87) New York: Springer 1994
41. Morale, D.: A stochastic particle model for vasculogenesis: A multiple scale approach. In: Capasso, V. (ed.): Mathematical modelling & computing in biology and medicine. Bologna: Esculapio Pub. Co. 2003, pp. 616-621
42. Murray, J.D.: Mathematical biology. Berlin: Springer 1989
43. Muthukkaruppan, V.R., Kubai, L., Auerbach, R.: Tumor-induced neovascularization in the mouse eye. J. Natl. Cancer Inst. 69, 699-708 (1982)
44. Nanjundiah,V.:Alan Turing and "The Chemical Basis of Morphogenesis". In: Sekimura, T. et al. (eds.): Morphogenesis and pattern formation in biological systems. Tokyo: Springer 2003, pp. 33-44
45. Pettet, G., Chaplain, M.A.J, McElwain, D.L.S., Byrne, H.M.: On the role of angiogenesis in wound healing. Proc. Roy. Soc. London Ser. B 263, 1487-1493 (1996) (Pubitemid 26400215)
46. Santini, M.T., Rainaldi, G., Indovina, P.L.: Apoptosis, cell adhesion and the extracellular matrix in the three-dimensional growth of multicellular tumor spheroids. Crit. Rev. Oncol. Hematol. 36, 75-87 (2000)
47. Schiaparelli, G.V.: Studio comparativo tra le forme organiche naturali e le forme geometriche pure. Milano: Hoepli 1898
48. Sekimura,T., Noji, S., Ueno, N., Maini, P.K. (eds.): Morphogenesis and pattern formation in biological systems. Tokyo: Springer 2003
49. Serini, G., Ambrosi, D., Giraudo, E., Gamba, A., Preziosi, L., Bussolino, F.: Modeling the early stages of vascular network assembly. EMBO J. 22, 1771-1779 (2003) (Pubitemid 36460134)
50. Sokołowski, J., Zolio, J.-P.: Introduction to shape optimization. Berlin: Springer 1992
51. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its application. 2nd ed. NewYork:Wiley 1995
52. Thompson, D.W.: On growth and form. Cambridge: Cambridge University Press 1917; 2nd ed. 1968
53. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. London Ser. A 237, 37-72 (1952)
54. Ubukata, T.: Computer modelling of microscopic features of molluscan shells. In: Sekimura, T. et al. (eds.): Morphogenesis and pattern formation in biological systems. Tokyo: Springer 2003, pp. 355-367