Skip to main content
Log in

Dynamics of a Data Based Ovarian Cancer Growth and Treatment Model with Time Delay

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Agur, Z., Arakelyan, L., Daugulis, P., Ginosar, Y.: Hopf point analysis for angiogenesis models. Discret. Contin. Dyn. Syst. 4(1), 29–38 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Google Scholar 

  3. Basanta, D., Anderson, A.R.A.: Exploiting ecological principles to better understand cancer progression and treatment. Interface Focus 3, 20130020 (2013)

    Article  Google Scholar 

  4. Bast Jr, R.C., Hennessy, B., Mills, G.B.: The biology of ovarian cancer. Nat. Rev. Cancer 9, 415–528 (2009)

    Article  Google Scholar 

  5. Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential dystems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Elser, J.J., Kyle, M.M., Smith, M.S., Nagy, J.D.: Biological stoichiometry in human cancer. PLoS One 10, e1028 (2007)

    Article  Google Scholar 

  9. Folkman, J.: Tumor angiogenesis: therapeutic implications. N. Engl. J. Med. 118, 1182–1186 (1971)

    Google Scholar 

  10. Folkman, J.: What is the evidence that tumors are angiogenesis dependent? J. Natl. Cancer Inst. 82(1), 4–6 (1990)

    Article  Google Scholar 

  11. Folkman, J.: Role of angiogenesis in tumor growth and metastasis. Semin. Oncol. 29(6), 15–18 (2002)

    Article  Google Scholar 

  12. 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)

    Google Scholar 

  13. Geva, E., Jaffe, R.B.: Role of vascular endothelial growth factor in ovarian physiology and pathology. Fertil. Steril. 74(3), 429–438 (2000)

    Article  Google Scholar 

  14. Goff, B.A., Mandel, L., Muntz, H.G., Melancon, C.H.: Ovarian carinoma diagnosis. Cancer 89(10), 2068–2075 (2000)

    Article  Google Scholar 

  15. Gómez-Raposo, C., Mendiola, M., Barriuso, J., Casado, E., Hardisson, D., Redondo, A.: Angiogenesis and ovarian cancer. Clin. Transl. Oncol. 11, 564–571 (2009)

    Article  Google Scholar 

  16. 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)

    Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Korolev, K.S., Xavier, J.B., Gore, J.: Turning ecology and evolution against cancer. Nat. Rev. Cancer 14, 371–380 (2014)

    Article  Google Scholar 

  19. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc, San Diego (1993)

    MATH  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    Article  Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. Nagy, J.D.: Competition and natural selection in a mathematical model of cancer. Bull. Math. Biol. 66, 663–687 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nagy, J.D.: Hypertumors in cancer can be caused by tumor phosphorus demand. Proc. Appl. Math. Mech. 7, 1121703–1121704 (2007)

    Article  Google Scholar 

  28. Nagy, J.D., Armbruster, D.: Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Math. Biosci. Eng. 9(4), 843–876 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Panetta, J.C.: A mathematical model of breast and ovarian cancer treated with paclitaxel. Math. Biosci. 146, 89–113 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  Google Scholar 

  31. Siegel, R., Ma, J., Zou, Z., Jemal, A.: Cancer statistics, 2014. Cancer J. Clin. 64(1), 9–29 (2014)

    Article  Google Scholar 

  32. Siegel, R., Miller, K.D., Jemal, A.: Cancer statistics, 2015. Cancer J. Clin. 65(1), 5–29 (2015)

    Article  Google Scholar 

  33. Sitochy, B., Nagy, J.A., Dvorak, H.F.: Anti-VEGF/VEGFR therapy for cancer: reassessing the target. Cancer Res. 72(8), 1909–1914 (2012)

    Article  Google Scholar 

  34. Sterner, R.W., Elser, J.J.: Ecological Stoichiometry The Biology of Elements from Molecules to the Biosphere. Princeton University Press, Princeton (2002)

    Google Scholar 

  35. Waldner, M.J., Neurath, M.F.: Targeting the VEGF signaling pathway in cancer therapy. Expert Opin. Ther. Targets 16(1), 5–13 (2012)

    Article  Google Scholar 

  36. 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)

    Article  MathSciNet  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to Y. Kuang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-015-9498-y

Keywords

Navigation