Abstract
We present a simple model that describes ovarian tumor growth and tumor induced angiogenesis, subject to on and off anti-angiogenesis treatment. The tumor growth is governed by Droop’s cell quota model, a mathematical expression developed in ecology. Here, the cell quota represents the intracellular concentration of necessary nutrients provided through blood supply. We present mathematical analysis of the model, including proving positivity of the solutions so that they are biologically meaningful, as well as discussing local and global stability. The mathematical model can be employed to fit both on-treatment and off-treatment preclinical data using the same biologically relevant parameters. We also state an open mathematical question.
Similar content being viewed by others
References
Agur, Z., Arakelyan, L., Daugulis, P., Ginosar, Y.: Hopf point analysis for angiogenesis models. Discret. Contin. Dyn. Syst. 4(1), 29–38 (2004)
Aravantinos, G., Pectasides, D.: Bevacizumab in combination with chemotherapy for the treatment of advanced ovarian cancer: a systematic review. J. Ovarian Res. 7(57), 1–13 (2014)
Basanta, D., Anderson, A.R.A.: Exploiting ecological principles to better understand cancer progression and treatment. Interface Focus 3, 20130020 (2013)
Bast Jr, R.C., Hennessy, B., Mills, G.B.: The biology of ovarian cancer. Nat. Rev. Cancer 9, 415–528 (2009)
Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential dystems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)
Bickel, S.T., Juliano, J.D., Nagy, J.D.: Evolution of proliferation and the angiogenic switch in tumors with high clonal diversity. PLoS One 9(4), e91992 (2014)
Droop, M.R.: Vitamin B12 and marine ecology, iv: the kinetics of uptake, growth and inhibition in monochrysis lutheri. J. Mar. Biol. Assoc. UK 48(3), 689–733 (1968)
Elser, J.J., Kyle, M.M., Smith, M.S., Nagy, J.D.: Biological stoichiometry in human cancer. PLoS One 10, e1028 (2007)
Folkman, J.: Tumor angiogenesis: therapeutic implications. N. Engl. J. Med. 118, 1182–1186 (1971)
Folkman, J.: What is the evidence that tumors are angiogenesis dependent? J. Natl. Cancer Inst. 82(1), 4–6 (1990)
Folkman, J.: Role of angiogenesis in tumor growth and metastasis. Semin. Oncol. 29(6), 15–18 (2002)
Gerber, H.-P., Ferrara, N.: Pharmacology and pharmacodynamics of bevacizumab as monotherapy or in combination with cytotoxic therapy in preclinical studies. Cancer Res. 65(3), 671–680 (2005)
Geva, E., Jaffe, R.B.: Role of vascular endothelial growth factor in ovarian physiology and pathology. Fertil. Steril. 74(3), 429–438 (2000)
Goff, B.A., Mandel, L., Muntz, H.G., Melancon, C.H.: Ovarian carinoma diagnosis. Cancer 89(10), 2068–2075 (2000)
Gómez-Raposo, C., Mendiola, M., Barriuso, J., Casado, E., Hardisson, D., Redondo, A.: Angiogenesis and ovarian cancer. Clin. Transl. Oncol. 11, 564–571 (2009)
Limin, H., Zaloudek, C., Mills, G.B., Gray, J., Jaffe, Robert B.: In Vivo and in Vitro ovarian carcinoma growth inhibition by a phosphatidylinositol 3-kinase inhibitor (LY294002). Clin. Cancer Res. 6, 880–886 (2000)
Jain, H.V., Nör, J.E., Jackson, T.L.: Modeling the VEGF-Bcl-2-CXCL8 pathway in intratumoral angiogenesis. Bull. Math. Biol. 70, 89–117 (2008)
Korolev, K.S., Xavier, J.B., Gore, J.: Turning ecology and evolution against cancer. Nat. Rev. Cancer 14, 371–380 (2014)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc, San Diego (1993)
Kuang, Y., Nagy, J.D., Elser, J.J.: Biological stoichiometry of tumor dynamics: mathematical models and analysis. Discret. Cont. Dyn. Syst. 4(1), 221–240 (2004)
Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J. Optim. 9(1), 112–147 (1998)
Leunig, M., Yuan, F., Menger, M.D., Boucher, Y., Goetz, A.E., Messmer, K., Jain, R.K.: Angiogenesis, microvascular architecture, microhemodynamics, and interstitial fluid pressure during early growth of human adenocarcinoma LS174T in SCID mice. Cancer Res. 52, 6553–6560 (1992)
Lauren, L.M.F., Pepper, J.W., Reid, B.J., Maley, C.C.: Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6, 924–935 (2006)
Mesiano, S., Ferrara, N., Jaffe, R.B.: Role of vascular endothelial growth factor in ovarian cancer: inhibition of ascites formation by immunoneutralization. Am. J. Pathol. 153(4), 1249–1256 (1998)
Nagy, J.D.: Competition and natural selection in a mathematical model of cancer. Bull. Math. Biol. 66, 663–687 (2004)
Nagy, J.D.: The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity. Math. Biosci. Eng. 2(2), 381–418 (2005)
Nagy, J.D.: Hypertumors in cancer can be caused by tumor phosphorus demand. Proc. Appl. Math. Mech. 7, 1121703–1121704 (2007)
Nagy, J.D., Armbruster, D.: Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Math. Biosci. Eng. 9(4), 843–876 (2012)
Panetta, J.C.: A mathematical model of breast and ovarian cancer treated with paclitaxel. Math. Biosci. 146, 89–113 (1997)
Pienta, K.J., McGregor, N., Axelrod, R., Axelrod, D.E.: Ecological therapy for cancer: defining tumors using an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl. Oncol. 1(4), 158–164 (2008)
Siegel, R., Ma, J., Zou, Z., Jemal, A.: Cancer statistics, 2014. Cancer J. Clin. 64(1), 9–29 (2014)
Siegel, R., Miller, K.D., Jemal, A.: Cancer statistics, 2015. Cancer J. Clin. 65(1), 5–29 (2015)
Sitochy, B., Nagy, J.A., Dvorak, H.F.: Anti-VEGF/VEGFR therapy for cancer: reassessing the target. Cancer Res. 72(8), 1909–1914 (2012)
Sterner, R.W., Elser, J.J.: Ecological Stoichiometry The Biology of Elements from Molecules to the Biosphere. Princeton University Press, Princeton (2002)
Waldner, M.J., Neurath, M.F.: Targeting the VEGF signaling pathway in cancer therapy. Expert Opin. Ther. Targets 16(1), 5–13 (2012)
Ye, Q., Chen, H.-L.: Bevacizumab in the treatment of ovarian cancer: a meta-analysis from four phase iii randomized controlled trials. Arch. Gynecol. Obstet. 288(3), 655–666 (2013)
Acknowledgments
We would like to thank the referee for many helpful suggestions. This work is partially supported by an ARCS scholarship to Rebecca Everett and by NSF Grants DMS 1148771 and DMS-1518529.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Everett, R.A., Nagy, J.D. & Kuang, Y. Dynamics of a Data Based Ovarian Cancer Growth and Treatment Model with Time Delay. J Dyn Diff Equat 28, 1393–1414 (2016). https://doi.org/10.1007/s10884-015-9498-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-015-9498-y