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June  2013, 18(4): 1077-1108. doi: 10.3934/dcdsb.2013.18.1077

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 90095-1766, 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

Received  January 2012 Revised  April 2012 Published  February 2013

Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.
Citation: Van M. Savage, Alexander B. Herman, Geoffrey B. West, Kevin Leu. 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. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1077-1108. doi: 10.3934/dcdsb.2013.18.1077
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show all references

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 percolation-based network model,, Microvasc Res., 51 (1996), 327. Google Scholar

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[4]

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[5]

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[6]

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[7]

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[8]

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[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. Google Scholar

[10]

H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model,, Nature Reviews Cancer, 10 (2010), 221. Google Scholar

[11]

M. A. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neurooncology, 50 (2000), 37. Google Scholar

[12]

E. H. Cooper, The biology of cell death in tumours,, Cell Tissue Kinet., 6 (1973), 87. Google Scholar

[13]

O. I. Craciunescu, S. K. Das and S. T. Clegg, Dynamic contrast-enhancecd MRI and fractal characteristics of percolation clusters in two-dimensional tumor blood perfusion,, Transactions of the ASME, 121 (1999), 480. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

J. Folkman, What is the evidence that tumors are angiogenesis dependent?,, Journal of the National Cancer Institute, 83 (1989), 4. Google Scholar

[17]

J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions,, The Journal of Experimental Medicine, 138 (1973), 745. Google Scholar

[18]

J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions,, The Journal of Experimental Medicine, 138 (1973), 745. Google Scholar

[19]

S. A. Frank, "Dynamics of Cancer: Incidence, Inheritance, and Evolution,", Princeton University Press, (2007). Google Scholar

[20]

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[21]

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[22]

R. A. Gatenby and P. K. Maini, Cancer Summed Up,, Nature, 421 (2003). Google Scholar

[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. Google Scholar

[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/S0022-5193(03)00221-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

P. M. Gullino and F. H. Grantham, Studies on the exchcange of fluids between host and tumor. I. A method of growing "Tissue-Isolated" tumors in laboratory animals,, Journal of the National Cancer Institute, 27 (1961), 679. Google Scholar

[28]

P. M. Gullino and F. H. Grantham, The vascular space of growing tumors,, Cancer Research, 24 (1964), 1727. Google Scholar

[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. Google Scholar

[30]

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