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

Van M. Savage; Alexander B. Herman; Geoffrey B. West; Kevin Leu

doi:10.3934/dcdsb.2013.18.1077

Article Contents

2013, Volume 18, Issue 4: 1077-1108. Doi: 10.3934/dcdsb.2013.18.1077 open access

This issue Previous Article B cell chronic lymphocytic leukemia - A model with immune response Next Article A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, platelet-derived growth factor-B, 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 90095-1766
2.
University of California, San Francisco, Medical Sciences Training Program, San Francisco, CA 94143
3.
Santa Fe Institute, Sante Fe, NM 87501

Received: January 2012

Revised: April 2012

Published: June 2013

Abstract Related Papers Cited by

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:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

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 percolation-based network model, Microvasc Res., 51 (1996), 327-346.
[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), 1799-1803.
[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, Development and Aging, 54 (1990), 85-93.
[4]	I. D. Bassukas, Evidence for a Narrow Range of Growth Patterns of Malignant Tumors and Embryos of Different Species, Naturwissenschaften, 80 (1993), 366-368.
[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), 77-104.
[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 hemangioblastoma-like vessels by VEGF withdrawal, Proc. Natl. Acad. Sci. USA, 94 (1997), 8761-8766.
[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), 473-480.
[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), 91-100.
[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), 609-640.
[10]	H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230.
[11]	M. A. Chaplain, Mathematical modelling of angiogenesis, Journal of Neurooncology, 50 (2000), 37-51.
[12]	E. H. Cooper, The biology of cell death in tumours, Cell Tissue Kinet., 6 (1973) 87-95.
[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), 480-486.
[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), 4225-4227.
[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), 389-405.
[16]	J. Folkman, What is the evidence that tumors are angiogenesis dependent?, Journal of the National Cancer Institute, 83 (1989), 4-6.
[17]	J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions, The Journal of Experimental Medicine, 138 (1973), 745-753.
[18]	J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions, The Journal of Experimental Medicine, 138 (1973), 745-753.
[19]	S. A. Frank, "Dynamics of Cancer: Incidence, Inheritance, and Evolution," Princeton University Press, Princeton, 2007.
[20]	Y. C. Fung, "Biomechanics: Circulation," Springer Verlag, New York, 1996.
[21]	M. P. Gallee, J. E. Visser-de Jong, K. F. J. ten, F. H., Schroeder and T. H. Van der Kwast, Monoclonal anitbody Ki-67 defined growth fraction in benign prostatic hyperplasia and prostatic cancer, J Urol, 142 (1989), 1342-1346.
[22]	R. A. Gatenby and P. K. Maini, Cancer Summed Up, Nature, 421 (2003), 321.
[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) 395-402.
[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), 147-151.doi: 10.1016/S0022-5193(03)00221-2.
[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), 459-463.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), 6858-6863.
[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), 679-693.
[28]	P. M. Gullino and F. H. Grantham, The vascular space of growing tumors, Cancer Research, 24 (1964), 1727-1732.
[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), 1465-1491.
[30]	M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev. Aging, 53 (1989), 25-33.
[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), e22973.
[32]	D. E. Hilmas and E. L. Gillette, Morphometric analysis of the microvasculature of tumors during growth and after X-irradiation, Cancer, 33 (1973), 103-110.
[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), 109-111.
[34]	R. E. Horton, Erosional development of streams and their drainage basins: Hydro-physical approach to quantitative morphology, Geological Society of America Bulletin, 56 (1945), 275-370.
[35]	W. Huang, R. T. Yen, M. McLaurine and G. Bledsoe, Morphometry of the human pulmonary vasculature, Journal of Applied Physiology, 81(5) (1996), 2123.
[36]	E. Katifori, G. J. Szollosi and M. O. Magnasco, Damage and fluctuations induce loops in optimal transport networks, Physical Review Letters, 104 (2010), 048704.
[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), H350-365.
[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), H366-375.
[39]	A. J. Kerkhoff and B. J. Enquist, Multiplicative by nature: Why logarithmic transformation is necessary in allometry, Journal of Theoretical Biology, 257 (2009), 519-521.
[40]	N. L. Komarova, M. A. Sengupta and M. A. Nowak, Mutation-selection networks of cancer initiation: Tumor suppressor genes and chromosomal instability, Journal of Theoretical Biology, 223 (2003), 433-450.doi: 10.1016/S0022-5193(03)00120-6.
[41]	R. K. Jain, Determinants of tumor blood flow: A review, Cancer Research, 48 (1988), 2641-2658.
[42]	R. K. Jain, Delivery of molecular medicine to solid tumors, Science, 271 (1996), 1079-1080
[43]	R. K. Jain, Normalization of tumor vasculature: An emerging concept in antiangiogenic therapy, Science, 307 (2005), 58-62.
[44]	R. K. Jain and K. Ward-Hartley, Tumor blood flow-characterization, modifications, and role in hyperthermia, IEEE Transactions on Sonics and Ultrasonics, SU-31 (1984), 504-524.
[45]	Z. L. Jiang, G. S. Kassab and Y. C. Fung, Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree, J. Appl. Physiol., 76 (1994), 882-892.
[46]	A. Krogh, The rate of diffusion of gases through animal tissues, with some remarks on the coefficient of invasion, J. Physiol., 52 (1919), 391-408.
[47]	A. Krogh, The supply of oxygen to the tissues and the regulation of the capillary circulation, J. Physiol., 52 (1919), 457-474.
[48]	A. K. Laird, Dynamics of tumor growth, Growth, 13 (1964), 490-502.
[49]	U. Ledzewicz, H. Maurer and H.Schaettler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.doi: 10.3934/mbe.2011.8.307.
[50]	U. Ledzewicz, H. Maurer and H. Schaettler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring, W. Michiels, Eds., Springer Verlag, (2010), 267-276.
[51]	U. Ledzewicz, J. Marriot, H. Schaettler and H. Maurer, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 159-179.doi: 10.1093/imammb/dqp012.
[52]	U. Ledzewicz, J. Munden and H. Schaettler, Scheduling of anti-angiogenic inhibitors for Gompertzian and logistic tumor growth models, Discrete and Continuous Dynamical Systems, Series B, Special Issue Application of Dynamical Systems to Biology and Medicine, 12 (2009), 415-439doi: 10.3934/dcdsb.2009.12.415.
[53]	U. Ledzewicz and H. Schaettler, Minimization of the tumor volume and endothelial support for a system describing tumor anti-angiogenesis, WSEAS Transactions on Biology and Biomedicine, 5 (2008), 23-33
[54]	U. Ledzewicz and H. Schaettler, On a class of systems describing tumor anti-angiogenesis under Gompertzian growth, WSEAS Transactions on Systems, 4 (2007), 758-766.
[55]	J. R. Less, M. C. Posner, T. C. Skalak, N. Wolmark and R. K. Jain, Geometric resistance and microvascular network architecture of human colorectal carcinoma, Microcirculation, 4 (1997), 25-33.
[56]	J. R. Less , T. C. Skalak, E. M. Sevick and R. K. Jain, Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions, Cancer Research, 51 (1991), 265-273.
[57]	A. J. Leu, D. A. Berk, A. Lymboussaki, K. Alitalo and R. K. Jain, Absence of functional lymphatics within a murine sarcoma: A molecular and functional evaluation, Cancer Research, 60 (2000), 4324-4327.
[58]	M. Leunig, F. Yuan, M. D. Menger, Y. Boucher, A. E. Goetz, K. Messmer and R. K. Jain, Angiogenesis, microvascular architecture, microhemodynamics, and interstitial fluid pressure during early growth of human adenocarcinoma LS174T in SCID mice, Cancer Research, 52 (1992), 6553-6560.
[59]	D. Lyden, D. R. Welch and B. Psaila, "Cancer Metastasis: Biologic Basis and Therapeutics," Cambridge University Press, New York, 2011.
[60]	H. Lyng, A. Skretting and E. K. Rofstad, Blood flow in six human melanoma xenograft lines with different growth characteristics, Cancer Research, 52 (1992), 584-592.
[61]	B. B. Mandelbrot, "The Fractal Geometry of Nature," W. H. Freeman, 1982.
[62]	S. Masunaga, H. Nagasawa, Y. Liu, Y. Sakurai, H. Tanaka, G. Kashino, M. Suzuki. Y. Kinashi and K. Ono, Evaluation of radiosensitivity of the oxygenated tumor cell fractions in quiescent cell populations within solid tumors, Radiation Research, 174 (2010), 459-466.
[63]	H. R. Mellor, D. J. P. Ferguson and R. Callaghan, A model of quiescent tumor microregions for evaluating multicellular resistance to chemotherapeutic drugs, British Journal of Cancer, 93 (2005), 302-309.
[64]	L. M. Merlo, J. W. Pepper. B. J. Reid, C. C. Maley, Cancer as an evolutionary and ecological process, Nature Reviews. Cancer, 612 (2006), 924-935.
[65]	R. J. Metzger, O. D. Klein, G. R. Martin and M. A. Krasnow, The branching programme of mouse lung development, Nature, 453 (2008), 745-751.
[66]	F. Michor, S. A. Frank, R. M. May, Y. Iwasa and M. A. Nowak, Somatic selection for and against cancer, Journal of Theoretical Biology, 225 (2003), 377-382.doi: 10.1016/S0022-5193(03)00267-4.
[67]	C. D. Murray, The physiological principle of minimum work. I. The vascular system and the cost of blood volume, Proc. Natl. Acad. Sci. USA, 12 (1926), 207-214.
[68]	L. N. Owen and G. G. Steel, The growth and cell population kinetics of spontaneous tumours in domestic animals, Br. J. Cancer, 23 (1969), 493-509.
[69]	F. William Orr, Michael R. Buchanan and L. Weiss, "Microcirculation in Cancer Metastasis," CRC Press, Inc., Boca Raton, 1991.
[70]	C. A. Price, S. Wing and J. S. Weitz, "Scaling and Structure of Dicotyledonous Leaf Venation Networks," Ecology Letters, In Press.
[71]	C. A. Price, B. J. Enquist and V. M. Savage, A general model for allometric covariation in botanical form and function, Proceedings of the National Academy of Sciences, USA, 104 (2007), 13204-13209.
[72]	M. R. Owen, T. Alarcon, P. K. Maini and H. M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol., 58 (2009), 689-721.doi: 10.1007/s00285-008-0213-z.
[73]	J. Panovska, H. M. Byrne and P. K. Maini, A theoretical study of the response of vascular tumours to different types of chemotherapy, Math. Comp. Mod., 47 (2008), 560-579.doi: 10.1016/j.mcm.2007.02.028.
[74]	W. Rogers, "Tissue Blood Flow in Transplantable Tumors of the Mouse and Hamster," University of Minnesota, 1967.
[75]	E. Sabo, A. Boltenko, Y. Sova, A. Stein, S. Kleinhaus and M. B. Resnick, Microscopic analysis and significance of vascular architectural complexity in renal cell carcinoma, Clin Cancer Res, 7 (2001), 533-537.
[76]	V. M. Savage, E. J. Deeds and W. Fontana, Sizing up allometric scaling theory, Public Library of Science: Computational Biology, 4 (2008), e1000171.doi: 10.1371/journal.pcbi.1000171.
[77]	V. M. Savage, A. P. Allen, J. F. Gillooly, A. B. Herman, J. H. Brown and G. B. West, Scaling of number, size, and metabolic rate of cells with body size in mammals, Proceedings of the National Academy of Sciences, USA, 104 (2007), 4718-4723.
[78]	V. M. Savage, L. P. Bentley, B. J. Enquist, J. S. Sperry, D. D. Smith, P. B. Reich and E. I. von Allmen, Hydraulic tradeoffs and space filling enable better predictions of vascular structure and function in plants, Proceedings of the National Academy of Sciences, USA, 107 (2010), 22722-22727.
[79]	V. M. Savage, J. F. Gillooly, W. H. Woodruff, G. B. West, A. P. Allen, B. J. Enquist and J. H. Brown, The predominance of quarter-power scaling in biology, Functional Ecology, 18 (2004), 257-282.
[80]	R. Schrek, "Further Quantitative Methods for the Study of Transplantable Tumors. The Growth of R39 Sarcoma And Brown-Pearce Carcinoma," 1936.
[81]	E. M. Sevick and R. K. Jain, Geometric resistance to blood flow in solid tumors perfused ex vivo: Effects of tumor size and perfusion pressure, Cancer Research, 49 (1989), 3506-3512.
[82]	S. Singhal, R. Henderson, K. Horsfield, K. Harding and G. Cumming, Morphometry of the human pulmonary arterial tree, Circ. Res., 33 (1973), 190-197.
[83]	P. Skehan, On the normality of growth dynamics of neoplasms in vivo: A database analysis, Growth, 50 (1986), 496-515.
[84]	M. Soltani and P. Chen, Numerical modeling of fluid flow in solid tumors, PLoS One, 6 (2011), e20344.
[85]	R. P. Spencer, Blood flow in transplanted tumors: Quantitative approaches to radioisotopic studies, Yale Journal of Biology and Medicine, 43 (1969), 22-30.
[86]	R. P. Spencer and B. J. Lang, Quantification of studies of blood flow in growing and irradiated tumors in C3H mice, Growth, 40 (1976), 211-216.
[87]	A. N. Strahler, Quantitative analysis of watershed geomorphology, Transactions of the American Geophysical Union, 8 (1957), 913-920.
[88]	I. J. Stamper, M. R. Owen, P. K. Maini and H. M. Byrne, Oscillatory dynamics in a model of vascular tumour growth: Implications for chemotherapy, Biology Direct, 5 (2010), 27.
[89]	I. J. Stamper, H. M. Byrne, M. R. Owen and P. K. Maini, Modelling the role of angiogenesis and vasculogenesis in solid tumour growth, Bull. Math. Biol., 69 (2007), 2737-2772.doi: 10.1007/s11538-007-9253-6.
[90]	G. G. Steel, Species-dependent growth patterns for mammalian neoplasms, Cell Tissue Kinet., 13 (1980), 451-453.
[91]	G. G. Steel and L. F. Lamerton, The growth rate of human tumours, British Journal of Cancer, 20 (1966), 74-86.
[92]	G. G. Steel, "Growth Kinetics of Tumours," Clarendon Press, 1977.
[93]	W. C. Summers, Dynamics of tumor growth: A mathematical model, Growth, 30 (1966), 333-338.
[94]	I. F. Tannock, Population kinetics of carcinoma cells, capillary endothelial cells, and fibroblasts in a transplantable mouse mammary tumor, Cancer Research, 30 (1970), 2470-2476.
[95]	M. Tubiana, The kinetics of tumour cell proliferation and radiotherapy, British Journal of Radiology, 44 (1971), 325-247.
[96]	I. M. van Leeuwen, C. Zonneveld and S. A. Kooijiman, The embedded tumour: Host physiology is important for the evaluation of tumour growth, British Journal of Cancer, 89 (2003), 2254-2263.
[97]	G. B. West, J. H. Brown and B. J. Enquist, A general model for ontogenetic growth, Nature, 413 (2001), 628-631.
[98]	G. B. West, J. H. Brown and B. J. Enquist, A general model for the origin of allometric scaling laws in biology, Science, 276 (1997), 122-126.
[99]	M. R. T. Yen, Y. C. Fung and N. Bingham, Elasticity of small pulmonary arteries in the cat, J. Biomech. Eng., 102 (1980), 170-177.
[100]	M. R. T. Yen, F. Y. Zhuang, Y. C. Fung, H. H. Ho, H. Tremer and S. S. Sobin, Morphometry of the cat's pulmonary arteries, J. Biomech. Eng., 106 (1984), 131-136.
[101]	M. Zamir, On fractal properties of arterial trees, J. Theor. Biol., 197 (1999), 517-526.
[102]	M. Zamir, Fractal dimensions and multifractality in vascular branching, J. Theor. Biol., 212 (2001), 183-190.
[103]	M. Zamir, "The Physics of Coronary Blood Flow," Springer, New York, 2005.