Skip to main content
SpringerLink
Log in

Analysis of a model of a virus that replicates selectively in tumor cells

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract.

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. This process is modelled as a free boundary problem for a nonlinear system of hyperbolic-parabolic differential equations, where the free boundary is the surface of the tumor. The unknowns are the densities of uninfected cells, infected cells, necrotic cells and free virus particles, and the velocity of cells within the tumor as well as the free boundary r=R(t). The purpose of this paper is to establish a rigorous mathematical analysis of the model, and to explore the reduction of the tumor size that can be achieved by this therapy.

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. Adam, J.: A simplified mathematical model of tumor growth. Math. Biosci. 81, 224–229 (1986)

    Article  Google Scholar 

  2. Bazaliy, B.V., Friedman, A.: A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth. Commun. PDE, 2003. To appear

  3. Bazaliy, B.V., Friedman, A.: Global existence and stability for an elliptic-parabolic free boundary problem; An application to a model of tumor growth. Indian Univ. Math. J. To appear

  4. Bischoff, J.R.: et al, An adenovirus mutant that replicates selectively in p53-deficient human tumor cells. Science 274, 373–376 (1996)

    Article  Google Scholar 

  5. Britton, N., Chaplain, M.: A qualitative analysis of some models of tissue growth. Math. Biosci. 113, 77–89 (1993)

    Article  MATH  Google Scholar 

  6. Byrne, H.M.: A weakly nonlinear analysis of a model of vascular solid tumour growth. J. Math. Biol. (39), 59–89

  7. Byrne, H.M., Chaplain, M.A.J.: Growth of nonnecrotic tumours in the presence and absence of inhibitors. Math. Biosciences 181, 130–151 (1995)

    MATH  Google Scholar 

  8. Byrne, H., Chaplain, M.: Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187–216 (1996)

    Article  MATH  Google Scholar 

  9. Byrne, H.M., Chaplain, M.A.J.: Free boundary value problems associated with growth and development of multicellular spheroids. European J. Appl. Math. 8, 639–358 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, X., Cui, S., Friedman, A.: A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior. To appear

  11. Chen, X., Friedman, A.: A free boundary problem for elliptic-hyperbolic system: An application to tumor growth. To appear

  12. Coffey, M.C., Strong, J.E., Forsyth, P.A., Lee, P.W.K.: Reovirus therapy of tumors with activated Ras pathways. Science 282, 1332–1334 (1998)

    Article  Google Scholar 

  13. Cui, S., Friedman, A.: Analysis of a mathematical model of the effect of inhibitors on the growth of tumors. Math. Biosci. 164, 103–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cui, S., Friedman, A.: Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. Appl. 255, 636–677 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cui, S., Friedman, A.: A free boundary problem for a singular system of differential equations: An application to a model of tumor growth. Trans. Amer. Math. Soc. To appear

  16. Cui, S., Friedman, A.: A hyperbolic free boundary problem modeling tumor growth. To appear in Interfaces and Free Boundaries

  17. Fontelos, M., Friemdan, A.: Symmetry-breaking bifurcations of free boundary problems in three dimensions. To appear in Asymptotic Analysis

  18. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, 1964

  19. Friedman, A., Reitich, F.: Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262–284 (1999)

    Article  MATH  Google Scholar 

  20. Friedman, A., Reitich, F.: Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth. Trans. Amer. Math. Soc. 353, 1587–1634 (2000)

    Article  MATH  Google Scholar 

  21. Friedman, A., Reitich, F.: On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumor. Math. Models and Methods in Appl. Sciences 77, 1–25 (2001)

    MATH  Google Scholar 

  22. Greenspan, H.: Models for the growth of solid tumor by diffusion. Stud. Appl. Math 51, 317–340 (1972)

    MATH  Google Scholar 

  23. Greenspan, H.: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–242 (1976)

    Google Scholar 

  24. Heise, C., Sampson-Johannes, A., Williams, A., McCormick, F., Von Hoff, D.D., Kirn, D.H.: ONYX-015, an E1B gene-attenuated adenovirus, causes tumor-specific cytolysis and antitumoral efficacy that can be augmented by standard chemotherapeutic agents. Nature Med. 3, 639–645 (1997)

    Google Scholar 

  25. Jain, R.: Barriers to drug delivery in solid tumors. Sci. Am. 271, 58–65 (1994)

    Google Scholar 

  26. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Amer. Math. Soc. Transl., Vol. 23, American Mathematics Society, Providence, RI, 1968

  27. Ta-tsien, L.: Global Classical Solutions for Quasilinear Hyperbolic System. John Wiley and Sons, New York, 1994

  28. Oelschläger, K.: The spread of a parasitic infection in a spatially distributed host. J. Math. Biol. 30, 321–354 (1992)

    Google Scholar 

  29. Pettet, G., Please, C.P., Tindall, M.J., McElwain, D.: The migration of cells in multicell tumor spheroids. Bull. Math. Biol. 63, 231–257 (2001)

    Article  Google Scholar 

  30. Rodriguez, R., Schuur, E.R., Lim, H.Y., Henderson, G.A., Simons, J.W., Henderson, D.R.: Prostate attenuated replication competent adenovirus (ARCA) CN706: a selective cytotoxic for prostate-specific antigen-positive prostate cancer cells. Cancer Res. 57, 2559–2563 (1997)

    Google Scholar 

  31. Sherrat, J., Chaplain, M.: A new mathematical model for avascular tumor growth. J. Math. Biol. 43, 291–312 (2001)

    Article  Google Scholar 

  32. Swabb, E.A., Wei, J., Gullino, P.M.: Diffusion and convection in normal and neoplastic tissues. Cancer Res. 34, 2814–2822 (1974)

    Google Scholar 

  33. Ward, J.P., King, J.R.: Mathematical modelling of avascular-tumor growth II: Modelling growth saturation. IMA J. Math. Appl. Med. Biol. 15, 1–42 (1998)

    MATH  Google Scholar 

  34. Wu, J.T., Byrne, H.M., Kirn, D.H., Wein, L.M.: Modeling and analysis of a virus that replicates selectively in tumor cells. Bull. Math. Biol. 63, 731–768 (2001)

    Article  Google Scholar 

  35. Yoon, S.S., Carroll, N.M., Chiocca, E.A., Tanabe, K.K.: Cancer gene therapy using a replication-competent herpes simplex virus type I vector. Ann. Surg. 228, 366–374 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, U.S.A

    Avner Friedman

  2. Department of Applied Mathematics, Dong Hua University, Shanghai, 200051, P. R. China

    Youshan Tao

Authors
  1. Avner Friedman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youshan Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Avner Friedman.

Additional information

Mathematics Subject Classification (2000): 35R35; 92A15

Rights and permissions

Reprints and permissions

About this article

Cite this article

Friedman, A., Tao, Y. Analysis of a model of a virus that replicates selectively in tumor cells. J. Math. Biol. 47, 391–423 (2003). https://doi.org/10.1007/s00285-003-0199-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-003-0199-5

Key words or phrases:

Search

Navigation