×

zbMATH — the first resource for mathematics

The competitive dynamics between tumor cells, a replication-competent virus and an immune response. (English) Zbl 1066.92035
Summary: Replication-competent viruses have been used as an alternative therapeutic approach for cancer treatment. However, new clinical data revealed an innate immune response to virus that may mitigate the effects of treatment. Recently, L. M. Wein, J. T. Wu and D. H. Kirn [Cancer Research 63, 1317–1324 (2003)] have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response . The purpose of this paper is to extend their model from the view points of mathematics and biology and then prove global existence and uniqueness of solutions to this new model, to study the dynamics of this novel therapy for cancers, and to explore an explicit threshold of the intensity of the immune response for controlling the tumor.
We also study a time-delayed version of the model. We analytically prove that there exists a critical value \(\tau_{0}\) of the time-delay \(\tau\) such that the system has a periodic solution if \(\tau > \tau_{0}\). Numerical simulations are given to verify the analytical results. Furthermore, we numerically study the spatio-temporal dynamics of the model. The effects of the diffusivity of the immune response on the tumor growth are also discussed.

MSC:
92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C05 Biophysics
35R35 Free boundary problems for PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adam, J.: A simplified mathematical model of tumor growth. Math. Biosci. 81, 224–229 (1986) · Zbl 0601.92007 · doi:10.1016/0025-5564(86)90119-7
[2] Adam, J., Bellomo, N.: A survey of models for tumor-immune system dynamics. Boston, MA: Birkhäuser, 1997 · Zbl 0874.92020
[3] Ambrosi, D., Bellomo, N., Preziosi, L.: Modelling the immune response to tumor etherogenity and progression. J. Thero. Medicine 1, 51–61 (2002) · Zbl 1050.92026 · doi:10.1080/10273660290015206
[4] Becciolini, A., Balzi, M., Barbarisi, M., Faraoni, P., Biggeri, A., Potten, C.S.: 3H-thymidine labelling index (TLI) as a marker of tumour growth heterogeneity: evaluation in human solid carcinomas. Cell Prolif. 30, 117–126 (1997) · doi:10.1111/j.1365-2184.1997.tb00928.x
[5] Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York, 1963 · Zbl 0105.06402
[6] Bellomo, N., Bellouquid, A., De Angelis, E.: The modelling of the immune competition by generalized kinetic (Boltzmann) models: review and research perspectives. Math. Comut. Modelling 37, 65–86 (2003) · Zbl 1022.92012 · doi:10.1016/S0895-7177(03)80007-9
[7] Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comput. Modelling 20, 107–122 (1994) · Zbl 0811.92014 · doi:10.1016/0895-7177(94)90223-2
[8] Bischoff, J.R. et al.: An adenovirus mutant that replicates selectively in p53-deficient human tumor cells. Science 274, 373–376 (1996) · doi:10.1126/science.274.5286.373
[9] Britton, N., Chaplain, M.: A qualitative analysis of some models of tissue growth. Math. Biosci. 113, 77–89 (1993) · Zbl 0786.92011 · doi:10.1016/0025-5564(93)90009-Y
[10] Byrne, H.M.: A weakly nonlinear analysis of a model of vascular solid tumour growth. J. Math. Biol. 39, 59–89 (1999) · Zbl 0981.92011 · doi:10.1007/s002850050163
[11] Byrne, H.M., Chaplain, M.A.J.: Growth of nonnecrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences 181, 130–151 (1995) · Zbl 0836.92011
[12] Byrne, H., Chaplain, M.: Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187–216 (1996) · Zbl 0856.92010 · doi:10.1016/0025-5564(96)00023-5
[13] 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) · Zbl 0906.92016 · doi:10.1017/S0956792597003264
[14] Chaplain, M.A.J.: Reaction-diffusion prepatterning and its potential role in tumor invasion. J. Bio. Sys. 3, 929–936 (1995) · doi:10.1142/S0218339095000824
[15] Chaplain, M.A.J., Ganesh, M., Graham, I.G.: Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biol. 42, 387–423 (2001) · Zbl 0988.92003 · doi:10.1007/s002850000067
[16] Chaplain, M.A.J., Kuznetsov, V.A., James, Z.H., Stepanova, L.A.: Spatio-temporal dynamics of the immune system response to cancer. Proceedings of the mathematical Models in Medical and Health Sciences Conference (eds. M. A. Horn, G. Simonett, G. Webb), Vanderbilt University Press, 1998, ISBN 0-8265-1310-7 · Zbl 0926.92022
[17] 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) · doi:10.1126/science.282.5392.1332
[18] Crampin, E.J., Mani, P.K.: Modelling biological pattern formation: the role of domain growth. Comments on Theoretical Biology 3, 229–249 (2001)
[19] 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) · Zbl 0998.92022 · doi:10.1016/S0025-5564(99)00063-2
[20] Cui, S., Friedman, A.: Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. Appl. 255, 636–677 (2001) · Zbl 0984.35169 · doi:10.1006/jmaa.2000.7306
[21] De Angelis, E., Jabin, P.E.: Analysis of a mean field modelling of tumor and immune system competition. Math. Models Meth. Appl. Sci. 13, 197–220 (2003) · Zbl 1043.92012
[22] Freyer, J.P., Sutherland, R.M.: Proliferative and clonogenic heterogeneity of cells from EMT6/R0 multicellular spheroids induced by the glucose and oxygen supply. Cancer Res. 46, 3513–3520 (1986)
[23] Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, 1964 · Zbl 0144.34903
[24] Friedman, A., Reitich, F.: Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262–284 (1999) · Zbl 0944.92018 · doi:10.1007/s002850050149
[25] Friedman, A., Reitich, F.: Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth. Trans. Am. Math. Soc. 353, 1587–1634 (2000) · Zbl 0983.35019 · doi:10.1090/S0002-9947-00-02715-X
[26] 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) · Zbl 1013.92024
[27] Friedman, A., Tao, Y.: Analysis of a model of a virus that selectively in tumor cells. J. Math. Biol. 47, 391–423 (2003) · Zbl 1052.92027 · doi:10.1007/s00285-003-0199-5
[28] Ganly, I., Kirn, D., Eckhardt, G., Rodriguez, G.I., Soutar, D.S., Otto, R., Robertson, A.G., Park, O., Gulley, M.L., Heise, C., Von Hoff, D.D., Kaye, S.B., Eckhardt, S.G.: A phase I study of Onyx-015, an E1B-attenuated adenovirus, administered intratumorally to patients with recurrent head and neck cancer. Clinical Cancer Res. 6, 798–806 (2000)
[29] Gerlowski, L.E., Jain, R.K.: Microvascular permeability of normal and neoplastic tissues. Microvas. Res. 31, 288–306 (1986) · doi:10.1016/0026-2862(86)90018-X
[30] Greenspan, H.: Models for the growth of solid tumor by diffusion. Stud. Appl. Math 51, 317–340 (1972) · Zbl 0257.92001
[31] Greenspan, H.: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–242 (1976) · doi:10.1016/S0022-5193(76)80054-9
[32] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theorey and Applications of Hopf Bifurcation. Cambridge University, Cambridge, 1981
[33] 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) · doi:10.1038/nm0697-639
[34] Hicks, K.O., Ohms, S.J., vanZijl, P.L., Hunter, P.J., Wilson, E.R.: An experimental and mathematical model for the extravascular transport of a DNA intercalator in tumours. Br. J. Canc. 76, 894–903 (1997) · doi:10.1038/bjc.1997.481
[35] 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) · Zbl 0947.92013 · doi:10.1016/S0025-5564(99)00062-0
[36] Jain, R.: Barriers to drug delivery in solid tumors. Sci. Am. 271, 58–65 (1994) · doi:10.1038/scientificamerican0794-58
[37] Jannink, I., Risberg, B., Vandiest, P.J., Baak, J.P.A.: Heterogeneity of mitotic-activity in breast-cancer. Histopathol. 29, 421–428 (1996) · doi:10.1046/j.1365-2559.1996.d01-509.x
[38] Kuang, Y.: Delay Differential Equations: with Application to Population Dynamics. Academic Press, Boston, 1993 · Zbl 0777.34002
[39] Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Am. Math. Soc. Transl., Vol. 23, American Mathematics Society, Providence, RI, 1968
[40] Levin, V.A., Patlak, C.s., Landahl, H.D.: Heuristic modelling of drug delivery to malignant brain tumours. J. Pharm. Biopharm. 8, 257–296 (1980) · doi:10.1007/BF01059646
[41] Li, T.: Global Classical Solutions for Quasilinear Hyperbolic System. John Wiley and Sons, New York, 1994 · Zbl 0841.35064
[42] Matzavinos, A., Chaplain, M.A.J., Kuznetsov, V.: Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol. IMA J. 21, 1–34 (2004) · Zbl 1061.92038 · doi:10.1093/imammb/21.1.1
[43] Murray, J.D.: Mathematical Biology (Second Edition). Springer-Verlag, London, 1993 · Zbl 0779.92001
[44] Nemunaitis, J., Ganly, I., Khuri, F., Arseneau, J., Kuhn, J., McCarty, T., Landers, S., Maples, P., Romel, L., Randley, B., Reid, T., Kaye, S., Kirn, D.: 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)
[45] Nemunaitis, J., Cunningham, C., Buchanan, A., Blackburn, A., Edelman, G., Maples, P., Netto, G., Tong, A., Randley, B., Olson, S., Kirn, D.: Intravenous infusion of a replication-selective adenovirus (ONYX-015) in cancer patients: safety, feasibility and biological activity. Gene Therapy 8, 746–759 (2001) · doi:10.1038/sj.gt.3301424
[46] Oelschläger, K.: The spread of a parasitic infection in a spatially distributed host. J. Math. Biol. 30, 321–354 (1992) · Zbl 0756.92023
[47] Owen, M., Sherratt, J.A.: Pattern formation and spatio-temporal irregularity in a model for macrophage-tumour interactions. J. theor. Biol. 189, 63–80 (1997) · doi:10.1006/jtbi.1997.0494
[48] Owen, M., Sherratt, J.A.: Mathematical modelling of macrophage dynamics in tumours. Math. Models Methods Appl. Sci. 4, 513–539 (199) · Zbl 0932.92019
[49] Owen, M., Sherratt, J.A.: Modelling the macrophage invasion of tumours: Effects on growth and composition. IMA J. Math. Appl. Med. Biol. 15, 165–185 (1998) · Zbl 0909.92022 · doi:10.1093/imammb/15.2.165
[50] Palmqvist, R., Oberg, A., Bergstrom, C., Rutegard, J.N., Zackrisson, B., Stenling, R.: Systematic heterogeneity and prognostic significance of cell proliferation in colorectal cancer. Br. J. Canver 77, 917–925 (1998) · doi:10.1038/bjc.1998.152
[51] 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) · Zbl 1323.92037 · doi:10.1006/bulm.2000.0217
[52] 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)
[53] Routes, J.M., Ryan, S., Clase, A., Miura, T., Kuhl, A., Potter, T.A., Cook, J.L.: Adenovirus E1A oncogene expression in tumor cells enhances killing by TNF-related apoptosis-inducing ligand (TRAIL). J. Immunol. 165, 4522–4527 (2000)
[54] Sessa, F., Bonato, M., Bisoni, D., Bosi, F., Capella, C.: Evidence of a wide heterogeneity in cancer cell population in gallbladder adenocarcinomas. Lab. Invest. 76, 860 (1997)
[55] Sherratt, J.A.: Oscillatory and chaotic wakes behind moving boundaries in reaction-diffusion systems. Dynamics and Stability of Systems 4, 303–324 (1996) · Zbl 0877.35058
[56] Sherrat, J., Chaplain, M.: A new mathematical model for avascular tumor growth. J. Math. Biol. 43, 291–312 (2001) · Zbl 0990.92021 · doi:10.1007/s002850100088
[57] Swabb, E.A., Wei, J., Gullino, P.M.: Diffusion and convection in normal and neoplastic tissues. Cancer Res. 34, 2814–2822 (1974)
[58] Tao, Y., Yoshida, N., Guo, Q.: Nonlinear analysis of a model of vascular tumour growth and treatment. Nonlinearity 17, 867–895 (2004) · Zbl 1073.35215 · doi:10.1088/0951-7715/17/3/008
[59] 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) · doi:10.1093/imammb/15.1.1
[60] Ward, J.P., King, J.R.: Mathematical modelling of drug transport in tumour multicelll spheroids and monolayer cultures. Math. Biosci. 181, 177–207 (2003) · Zbl 1014.92021 · doi:10.1016/S0025-5564(02)00148-7
[61] Wein, L.M., Wu, J.T., Kirn, D.H.: Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: implications for virus design and delivery. Cancer Res. 63, 1317–1324 (2003)
[62] Wodarz, D.: Viruses as antitumor weapons: defining conditions for tumor remission. Cancer Res. 61, 3501–3507 (2001)
[63] 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) · Zbl 1323.92112 · doi:10.1006/bulm.2001.0245
[64] Wu, J.T., Kirn, D.H., Wein, L.M.: Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response. Bull. Math. Biol. 66, 605–625 (2004) · Zbl 1334.92243 · doi:10.1016/j.bulm.2003.08.016
[65] 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) · doi:10.1097/00000658-199809000-00009
[66] Yoshida, K., Kyo, E., Tsujino, T., Sano, T., Niimoto, M., Tahara, E.: Expression of epidermal growth factor, transforming growth factor-\(\alpha\) and their receptor genes in human carcinomas: implication for autocrine growth. Cancer Res. 81, 43–51 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.