Skip to main content
SpringerLink
Log in

Global boundedness in an oncolytic virotherapy model with generalized logistic source

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is devoted to investigating a haptotactic cross-diffusion system with generalized logistic source in two-dimensional domain, which describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic virus particles. It is mainly concerned with the global boundedness of classical solutions. Indeed, we rigorously proved that an associated spatially two-dimensional initial-boundary value problem has a unique global bounded classical solution under some suitably conditions. Our work generalizes and improves the partial of the results in the literature.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alzahrani, T., Eftimie, R.R., Trucu, D.D.: Multiscale modelling of cancer response to oncolytic viral therapy. Math. Biosci. 310, 76–95 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Alemany, R.: Viruses in cancer treatment. Clin. Transl. Oncol. 15, 182–188 (2013)

    Google Scholar 

  3. Anderson, A.R., Chaplain, M.A.J., Newman, E.L., Steele, R.J.C., Thompson, A.M.: Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129–154 (2000)

    MATH  Google Scholar 

  4. Bellomo, N., Painter, K.J., Tao, Y., Winkler, M.: Occurrence vs. absence of taxis-driven instabilities in a May–Nowak model for virus infection. SIAM J. Appl. Math. 79, 1990–2010 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Bellomo, N., Outada, N., Soler, J., Tao, Y., Winkler, M.: Chemotaxis and cross-diffusion models in complex environments: models and analytic problems toward a multiscale vision. Math. Models Methods Appl. Sci. 32, 713–792 (2022)

    MathSciNet  MATH  Google Scholar 

  6. Biler, P., Hebisch, W., Nadzieja, T.: The Debye system: existence and large time behavior of solutions. Nonlinear Anal. 23, 1189–1209 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Breitbach, C.J., Parato, K., et al.: Pexa-Vec double agent engineered vaccinia: oncolytic and active immunotherapeutic. Curr. Opin. Virol. 13, 49–54 (2015)

    Google Scholar 

  8. Cao, X.: Boundedness in a three-dimensional chemotaxis–haptotaxis model. Z. Angew. Math. Phys. 67, 11 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 18, 1685–1734 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Chen, Z.: Dampening effect of logistic source in a two-dimensional haptotaxis system with nonlinear zero-order interaction. J. Math. Anal. Appl. 492, 124435 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Fontelos, M.A., Friedman, A., Hu, B.: Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33, 1330–1355 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Goldsmith, K., Chen, W., Johnson, D.C., Hendricks, R.L.: Infected cell protein (ICP) 47 enhances herpes simplex virus neurovirulence by blocking the \(\text{ CD8}^{+}\) T cell response. J. Exp. Med. 187, 341–348 (1998)

    Google Scholar 

  13. Ganly, I., Kirn, D.: A phase I study of Onyx-015, an E1B-attenuated adenovirus, administered intratumorally to patients with recurrent head and neck cancer. Clin. Cancer Res. 6, 798–806 (2000)

    Google Scholar 

  14. Hu, B., Lankeit, J.: Boundedness of solutions to a virus infection model with chemotaxis. J. Math. Anal. Appl. 468, 344–358 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Jackson, T.L., Byrne, H.M.: A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math. Biosci. 164, 17–38 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Jin, C.: Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread. J. Differ. Equ. 269, 3987–4021 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Jin, H.Y., Xiang, T.: Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model. Math. Models Methods Appl. Sci. 31, 1373–1417 (2021)

    MathSciNet  MATH  Google Scholar 

  18. Lawler, S., Speranza, M., Cho, C., Chiocca, E.: Oncolytic viruses in cancer treatment: a review. JAMA Oncol. 3, 841–849 (2017)

    Google Scholar 

  19. Liţcanu, G., Morales-Rodrigo, C.: Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20, 1721–1758 (2010)

  20. Li, J., Wang, Y.: Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Differ. Equ. 270, 94–113 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Msaouel, P., Opyrchal, M., Musibay, E.D., Galanis, E.: Oncolytic measles virus strains as novel anticancer agents. Expert Opin. Biol. Ther. 13, 483–502 (2013)

    Google Scholar 

  22. Nemunaitis, J., Ganly, I.: Selective replication and oncolysis in p53 mutant tumors with ONYX-015, an E1B-55kD gene-deleted adenovirus, in patients with advanced head and neck cancer: a phase II trial. Cancer Res. 60, 6359–6366 (2000)

    Google Scholar 

  23. Pang, P.Y.H., Wang, Y.: Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling. Math. Models Methods Appl. Sci. 28, 2211–2235 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  25. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations, 2nd edn. Springer, Berlin (1984)

    MATH  Google Scholar 

  26. Stinner, C., Surulescu, C., Winkler, M.: Global weak solutions in a PDE–ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Tao, Y., Winkler, M.: Large time behavior in a multidimensional chemotaxis–haptotaxis model with slow signal diffusion. SIAM J. Math. Anal. 47, 4229–4250 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Tao, Y., Winkler, M.: Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant. J. Differ. Equ. 257, 784–815 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Tao, Y., Winkler, M.: A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy. Nonlinear Anal. TMA 198, 111870 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Tao, Y., Winkler, M.: Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete Contin. Dyn. Syst. Ser. 41, 439 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Tao, Y., Winkler, M.: Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Differ. Equ. 268, 4973–4997 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Tao, Y., Winkler, M.: Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. Proc. R. Soc. Edinb. Sect. A Math. 52, 81–101 (2022)

    MathSciNet  MATH  Google Scholar 

  33. Tao, Y., Winkler, M.: A critical virus production rate for efficiency of oncolytic virotherapy. Eur. J. Appl. Math. 32, 301–316 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Tao, X., Zhou, S.: Dampening effects on global boundedness and asymptotic behavior in an oncolytic virotherapy model. J. Differ. Equ. 308, 57–76 (2022)

    MathSciNet  MATH  Google Scholar 

  35. Tao, X.: Global weak solutions to an oncolytic viral therapy model with doubly haptotactic terms. Nonlinear Anal. Real World Appl. 60, 103276 (2021)

    MathSciNet  MATH  Google Scholar 

  36. Tao, X.: Global classical solutions to an oncolytic viral therapy model with triply haptotactic terms. Acta Appl. Math. 171, 5 (2021)

    MathSciNet  MATH  Google Scholar 

  37. Tao, Y., Wang, M.: A combined chemotaxis–haptotaxis system: the role of logistic source. SIAM J. Math. Anal. 41, 1533–1558 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y.: Boundedness in the higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. J. Differ. Equ. 260, 1975–1989 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Ward, J.P., King, J.R.: Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures. Math. Biosci. 181, 177–207 (2003)

    MathSciNet  MATH  Google Scholar 

  40. Walker, C., Webb, G.F.: Global existence of classical solutions for a haptotaxis model. SIAM J. Math. Anal. 38, 1694–1713 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Wei, Y.N., Wang, Y., Li, J.: Asymptotic behavior for solutions to an oncolytic virotherapy model involving triply haptotactic terms. Z. Angew. Math. Phys. 73, 1–20 (2022)

    MathSciNet  MATH  Google Scholar 

  42. Winkler, M.: Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity. Nonlinearity 30, 735–764 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Winkler, M.: Singular structure formation in a degenerate haptotaxis model involving myopic diffusion. J. Math. Pures Appl. 112, 118–169 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Winkler, M.: Boundedness in a chemotaxis-May–Nowak model for virus dynamics with mildly saturated chemotactic sensitivity. Acta Appl. Math. 163, 1–17 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Wong, H., Lemoine, N., Wang, Y.: Oncolytic viruses for cancer therapy: overcoming the obstacles. Viruses 2, 78–106 (2010)

    Google Scholar 

  46. Zhigun, A., Surulescu, C., Uatay, A.: Global existence for a degenerate haptotaxis model of cancer invasion. Z. Angew. Math. Phys. 67, 146 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the editor and the anonymous referee for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Qiang Wen & Bin Liu

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Qiang Wen & Bin Liu

Authors
  1. Qiang Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by NNSF of China (No. 11971185)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wen, Q., Liu, B. Global boundedness in an oncolytic virotherapy model with generalized logistic source. Z. Angew. Math. Phys. 74, 38 (2023). https://doi.org/10.1007/s00033-022-01920-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01920-8

Keywords

Mathematics Subject Classification

Search

Navigation