Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 1077 - 1108, Volume 18, Issue 4, June 2013      doi:10.3934/dcdsb.2013.18.1077

       Abstract        References        Full Text (1445.4K)              Related Articles       

Van M. Savage - David Geffen School of Medicine at UCLA, Department of Biomathematics, Los Angeles, CA 90095-1766, United States (email)
Alexander B. Herman - University of California, San Francisco, Medical Sciences Training Program, San Francisco, CA 94143, United States (email)
Geoffrey B. West - Santa Fe Institute, Sante Fe, NM 87501, United States (email)
Kevin Leu - David Geffen School of Medicine at UCLA, Department of Biomathematics, Los Angeles, CA 90095-1766, United States (email)

Abstract: 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.

Keywords:  Scaling theory, growth equations, fractal geometry.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: January 2012;      Revised: April 2012;      Available Online: February 2013.