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A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, plateletderived growth factorB, and pericytes
Using fractal geometry and universal growth curves as diagnostics for comparing tumor vasculature and metabolic rate with healthy tissue and for predicting responses to drug therapies
1.  David Geffen School of Medicine at UCLA, Department of Biomathematics, Los Angeles, CA 900951766, United States, United States 
2.  University of California, San Francisco, Medical Sciences Training Program, San Francisco, CA 94143, United States 
3.  Santa Fe Institute, Sante Fe, NM 87501, United States 
References:
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J. W. Baish, Y. Gazit, D. A. Berk. M. Nozue, L. T. Baxter and R. K. Jain, Role of tumor vascular architecture in nutrient and drug delivery: an invasion percolationbased network model,, Microvasc Res., 51 (1996), 327. 
[2] 
J. W. Baish, T. Stylianopoulos, R. M. Lanning, W. S. Kamoun, D. Fukumura, L. L. Munn, and R. K. Jain, Scaling rules for diffusive drug delivery in tumor and normal tissues,, Proc. Natl. Acad. Sci.., 108(5) (2011), 1799. 
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I. D. Bassukas, Evidence for a Narrow Range of Growth Patterns of Malignant Tumors and Embryos of Different Species,, Naturwissenschaften, 80 (1993), 366. 
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L. E. Benjamin and E. Keshet, Conditional switching of vascular endothelial growth factor (VEGF) expression in tumors: Induction of endothelial cell shedding and regression of hemangioblastomalike vessels by VEGF withdrawal,, Proc. Natl. Acad. Sci. USA, 94 (1997), 8761. 
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show all references
References:
[1] 
J. W. Baish, Y. Gazit, D. A. Berk. M. Nozue, L. T. Baxter and R. K. Jain, Role of tumor vascular architecture in nutrient and drug delivery: an invasion percolationbased network model,, Microvasc Res., 51 (1996), 327. 
[2] 
J. W. Baish, T. Stylianopoulos, R. M. Lanning, W. S. Kamoun, D. Fukumura, L. L. Munn, and R. K. Jain, Scaling rules for diffusive drug delivery in tumor and normal tissues,, Proc. Natl. Acad. Sci.., 108(5) (2011), 1799. 
[3] 
G. M. Baker, H. L. Goddard, M. B. Clarke and W. F. Whimster, Proportion of necrosis in transplanted murine adenocarcinoma and it's relationship to tumor growth,, Growth, 54 (1990), 85. 
[4] 
I. D. Bassukas, Evidence for a Narrow Range of Growth Patterns of Malignant Tumors and Embryos of Different Species,, Naturwissenschaften, 80 (1993), 366. 
[5] 
L. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors: I. Role of interstitial pressure and convection,, Microvascular Research, 37 (1989), 77. 
[6] 
L. E. Benjamin and E. Keshet, Conditional switching of vascular endothelial growth factor (VEGF) expression in tumors: Induction of endothelial cell shedding and regression of hemangioblastomalike vessels by VEGF withdrawal,, Proc. Natl. Acad. Sci. USA, 94 (1997), 8761. 
[7] 
R. R. Berges, J. Vukanovic, J. I. Epstein, M. CarMIchel, L. Cisek, D. E. Johnson, R. W. Veltri, P. C. Walsh, and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer,, Clinical Cancer Research, 1 (1995), 473. 
[8] 
B. Blonder, C. Violle, L. Patrick and B. Enquist, Leaf venation networks and the origin of the leaf economics spectrum,, Ecology Letters, 14 (2011), 91. 
[9] 
C. J. W. Breward, H. M. Byrne, and C. E. Lewis, A multiphase model describing vascular tumor growth,, Bulletin of Mathematical Biology, 65 (2003), 609. 
[10] 
H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model,, Nature Reviews Cancer, 10 (2010), 221. 
[11] 
M. A. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neurooncology, 50 (2000), 37. 
[12] 
E. H. Cooper, The biology of cell death in tumours,, Cell Tissue Kinet., 6 (1973), 87. 
[13] 
O. I. Craciunescu, S. K. Das and S. T. Clegg, Dynamic contrastenhancecd MRI and fractal characteristics of percolation clusters in twodimensional tumor blood perfusion,, Transactions of the ASME, 121 (1999), 480. 
[14] 
P. P. Delsanto, C. Guiot, P. G. Degiorgis, C. A. Condat, Y. Mansury and T. S. Deisboeck, Growth model for multicellular tumor spheroids,, Applied Physics Letter, 85 (2004), 4225. 
[15] 
L. A. Dethlefsen, J. M. S. Prewitt and M. L. Mendelsohn, Analysis of tumor growth curves,, Journal of the National Cancer Institute, 40 (1967), 389. 
[16] 
J. Folkman, What is the evidence that tumors are angiogenesis dependent?,, Journal of the National Cancer Institute, 83 (1989), 4. 
[17] 
J. Folkman and M. Hochberg, Selfregulation of growth in three dimensions,, The Journal of Experimental Medicine, 138 (1973), 745. 
[18] 
J. Folkman and M. Hochberg, Selfregulation of growth in three dimensions,, The Journal of Experimental Medicine, 138 (1973), 745. 
[19] 
S. A. Frank, "Dynamics of Cancer: Incidence, Inheritance, and Evolution,", Princeton University Press, (2007). 
[20] 
Y. C. Fung, "Biomechanics: Circulation,", Springer Verlag, (1996). 
[21] 
M. P. Gallee, J. E. Visserde Jong, K. F. J. ten, F. H., Schroeder and T. H. Van der Kwast, Monoclonal anitbody Ki67 defined growth fraction in benign prostatic hyperplasia and prostatic cancer,, J Urol, 142 (1989), 1342. 
[22] 
R. A. Gatenby and P. K. Maini, Cancer Summed Up,, Nature, 421 (2003). 
[23] 
Y. Gazit, J. W. Baish, N. Safabakhsh, M. Leunig, L. T. Baxter and R. K. Jain, Fractal characteristics of tumor vascular architecture during tumor growth and regression,, Microcirculation, 4 (1997), 395. 
[24] 
C. Guiot, P. G . Degiorgis, P. P. Delsanto, P. Gabriele and T. S. Deisboeck, Does tumor growth follow a "universal law"?,, Journal of Theoretical Biology, 225 (2003), 147. doi: 10.1016/S00225193(03)002212. 
[25] 
C. Guiot, P. P. Delsanto, A. Carpinteri, N. Pugno, Y. Mansury and T. S. Deisboeck, The dynamic evolution of the power exponent in a universal growth model of tumors,, Journal of Theoretical Biology, 240 (2006), 459. doi: 10.1016/j.jtbi.2005.10.006. 
[26] 
R. Glenny, S. Bernard, B. Neradilek and N. Polissar, Quantifying the genetic influence on mammalian vascular tree structure,, Proc. Natl. Acad. Sci. USA, 104 (2007), 6858. 
[27] 
P. M. Gullino and F. H. Grantham, Studies on the exchcange of fluids between host and tumor. I. A method of growing "TissueIsolated" tumors in laboratory animals,, Journal of the National Cancer Institute, 27 (1961), 679. 
[28] 
P. M. Gullino and F. H. Grantham, The vascular space of growing tumors,, Cancer Research, 24 (1964), 1727. 
[29] 
P. M. Gullino and F. H. Grantham, Studies on the exchange of fluids between host and tumor. II. The blood flow of hepatomas and other tumors in rats and mice,, Journal of the National Cancer Institute, 27 (1961), 1465. 
[30] 
M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth,, Growth Dev. Aging, 53 (1989), 25. 
[31] 
A. B. Herman, V. M. Savage and G. B. West, A quantitative theory of solid tumor growth, metabolic rate, and vascularization,, Public Library of Science One, 6 (2011). 
[32] 
D. E. Hilmas and E. L. Gillette, Morphometric analysis of the microvasculature of tumors during growth and after Xirradiation,, Cancer, 33 (1973), 103. 
[33] 
K. Hori, M. Suzuki, S. Tanda and S. Saito, Characterization of heterogeneous distribution of tumor blood flow in the rat,, Jpn. J. Cancer Res., 82 (1991), 109. 
[34] 
R. E. Horton, Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative morphology,, Geological Society of America Bulletin, 56 (1945), 275. 
[35] 
W. Huang, R. T. Yen, M. McLaurine and G. Bledsoe, Morphometry of the human pulmonary vasculature,, Journal of Applied Physiology, 81(5) (1996). 
[36] 
E. Katifori, G. J. Szollosi and M. O. Magnasco, Damage and fluctuations induce loops in optimal transport networks,, Physical Review Letters, 104 (2010). 
[37] 
G. S. Kassab, C. A. Rider, N. J. Tang and Y. C. Fung, Morphometry of pig coronary arterial trees,, Am. J. Physiol., 265 (1993), 350. 
[38] 
G. S. Kassab, K. Imoto, F. C. White, C. A. Rider, Y. C. Fung and C. M. Bloor, Coronary arterial tree remodeling in right ventricular hypertrophy,, Am. J. Physiol., 265 (1993), 366. 
[39] 
A. J. Kerkhoff and B. J. Enquist, Multiplicative by nature: Why logarithmic transformation is necessary in allometry,, Journal of Theoretical Biology, 257 (2009), 519. 
[40] 
N. L. Komarova, M. A. Sengupta and M. A. Nowak, Mutationselection networks of cancer initiation: Tumor suppressor genes and chromosomal instability,, Journal of Theoretical Biology, 223 (2003), 433. doi: 10.1016/S00225193(03)001206. 
[41] 
R. K. Jain, Determinants of tumor blood flow: A review,, Cancer Research, 48 (1988), 2641. 
[42] 
R. K. Jain, Delivery of molecular medicine to solid tumors,, Science, 271 (1996), 1079. 
[43] 
R. K. Jain, Normalization of tumor vasculature: An emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58. 
[44] 
R. K. Jain and K. WardHartley, Tumor blood flowcharacterization, modifications, and role in hyperthermia,, IEEE Transactions on Sonics and Ultrasonics, SU31 (1984), 504. 
[45] 
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