Abstract
Based on the logistic growth law for a tumour derived from enzymatic dynamics, we address from a physical point of view the phenomena of synergism, additivity and antagonism in an avascular anti-tumour system regulated externally by dual coupling periodic interventions, and propose a theoretical model to simulate the combinational administration of chemotherapy and immunotherapy. The in silico results of our modelling approach reveal that the tumour population density of an anti-tumour system, which is subject to the combinational attack of chemotherapeutical as well as immune intervention, depends on four parameters as below: the therapy intensities D, the coupling intensity I, the coupling coherence R and the phase-shifts Φ between two combinational interventions. In relation to the intensity and nature (synergism, additivity and antagonism) of coupling as well as the phase-shift between two therapeutic interventions, the administration sequence of two periodic interventions makes a difference to the curative efficacy of an anti-tumour system. The isobologram established from our model maintains a considerable consistency with that of the well-established Loewe Additivity model (Tallarida, Pharmacology 319(1):1–7, 2006). Our study discloses the general dynamic feature of an anti-tumour system regulated by two periodic coupling interventions, and the results may serve as a supplement to previous models of drug administration in combination and provide a type of heuristic approach for preclinical pharmacokinetic investigation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Appay, V., Voelter, V., Rufer, N., Reynard, S., Jandus, C., Gasparini, D., Lienard, D., Speiser, D. E., et al. (1997). Combination of transient lymphodepletion with busulfan and fludarabine and peptide vaccination in a phase I clinical trial for patients with advanced melanoma. J. Immunother. (Hagerstown, Md.), 30(2), 240–250.
Bunow, B., & Weinstein, J. N. (1990). COMBO: A new approach to the analysis of drug combinations in vitro. Ann. N.Y. Acad. Sci., 616(1), 490–494.
Butcher, J. C. (2003). Numerical methods for ordinary differential equations. New York: Wiley.
Byrne, H. M. (2003). Treatment of Homogeneous Solid Tumours. In L. Preziosi (Ed.), Cancer modelling and simulation. New York: Chapman & Hall/CRC. Section 4.2.3.
Chareyron, S., & Alamir, M. (2009). Mixed immunotherapy and chemotherapy of tumours: feedback design and model updating schemes. J. Theor. Biol., 258(3), 444–454.
Chou, T. (2006). Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies. Pharmacol. Rev., 58(3), 621–681.
Chou, T., & Talalay, P. (1984). Quantitative analysis of dose—effect relationships: the combined effects of multiple drugs or enzyme inhibitors. Adv. Enzyme Regul., 22, 27–55.
de Pillis, L. G., Gu, W., & Radunskaya, A. E. (2006). Mixed immunotherapy and chemotherapy of tumours: modeling, applications and biological interpretations. J. Theor. Biol., 238(4), 841–862.
de Pillis, L. G., Gu, W., Fister, K. R., Head, T., Maples, K., Murugan, A., Neal, T., & Yoshida, K. (2007). Chemotherapy for tumours: an analysis of the dynamics and a study of quadratic and linear optimal controls. Math. Biosci., 209(1), 292–315.
de Pillis, L. G., Fister, K. R., Gu, W., Head, T., Maples, K., Neal, T., Murugan, A., & Kozai, K. (2008). Optimal control of mixed immunotherapy and chemotherapy of tumours. J. Biol. Syst., 16(1), 51.
Engelhart, M., Lebiedz, D., & Sager, S. (2011). Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function. Math. Biosci., 229(1), 123–134.
Grem, J. (1999). Sequence-dependent antagonism between fluorouracil and paclitaxel in human breast cancer cells. Biochem. Pharmacol., 58(3), 477–486.
Isaeva, O. G., & Osipov, V. A. (2009). Different strategies for cancer treatment: mathematical modelling. Comput. Math. Methods Med., 10(4), 253–272.
Jemal, A., Bray, F., Center, M. M., Ferlay, J., Ward, E., & Forman, D. (2011). Global cancer statistics. CA: Cancer J. Clin., 61(2), 69–90.
Lefever, R., & Erneaux, T. (1984). On the growth of cellular tissue under constant and fluctuating environmental conditions. In W. Ross & A. Lawrence (Eds.), Nonlinear electrodynamics in biological systems (pp. 287–305). New York: Plenum Press.
Lefever, R., & Garay, R. P. (1978). Local description of immune tumour rejection. In A. J. Valleron & P. D. M. Macdonald (Eds.), Biomathematics and cell kinetics (pp. 333–344). North Holland, Elsevier: Amsterdam.
Ludwig, D., Jones, D. D., & Holling, C. S. (1978). Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Ecol., 47(1), 315.
Murray, J. D. (Ed.) (2002). Mathematical biology: I. An introduction (3rd ed.). Berlin: Springer.
Nowak, A. K., Robinson, B. W. S., & Lake, R. A., (2003). Synergy between chemotherapy and immunotherapy in the treatment of established murine solid tumours. Cancer Res., 63(15), 4490–4496.
Nowak, A. K., Lake, R. A., & Robinson, B. W. S. (2006). Combined chemoimmunotherapy of solid tumours: improving vaccines? Adv. Drug Deliv. Rev., 58(8), 975–990.
Shepard, D. (1968). A two-dimensional interpolation function for irregularly-spaced data. In Proceedings of the 1968 ACM national conference (pp. 517–524). doi:10.1145/800186.810616.
Tallarida, R. J. (2006). An overview of drug combination analysis with isobolograms. Pharmacology, 319(1), 1–7.
Tallarida, R. J. (2007). Interactions between drugs and occupied receptors. Pharmacol. Ther., 113(1), 197–209.
van Moorsel, C. J., Pinedo, H. M., Veerman, G., Bergman, A. M., Kuiper, C. M., Vermorken, J. B., van der Vijgh, W. J., & Peters, G. J. (1999). Mechanisms of synergism between cisplatin and gemcitabine in ovarian and non-small-cell lung cancer cell lines. Br. J. Cancer, 80(7), 981–990.
Voigt, W., Bulankin, A., Muller, T., Schoeber, C., Grothey, A., Hoang-Vu, C., & Schmoll, H.-J. (2000). Schedule-dependent antagonism of gemcitabine and cisplatin in human anaplastic thyroid cancer cell lines. Clin. Cancer Res., 6(5), 2087–2093.
Zhang, L., Dermawan, K.-t., Jin, M.-l., Xiong, S.-d., & Chu, Y.-w. (2008). Does chemotherapy augment anti-tumour immunotherapy by preferential impairment of regulatory T cells? Med. Hypotheses, 71(5), 802–804.
Zhong, W.-R., Shao, Y.-Z., & He, Z.-H. (2006a). Pure multiplicative stochastic resonance of a theoretical anti-tumour model with seasonal modulability. Phys. Rev. E, 73(6), 060902.
Zhong, W.-R., Shao, Y.-Z., & He, Z.-H. (2006b). Spatiotemporal fluctuation-induced transition in a tumour model with immune surveillance. Phys. Rev. E, 74(1), 011916.
Zhong, W.-R., Shao, Y.-Z., Li, L., Wang, F.-H., & He, Z.-H. (2008). Spatiotemporal noise triggering infiltrative tumour growth with immunosurveillance. Europhys. Lett., 82(2), 20003.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, WY., Zhong, WR., Wang, FH. et al. In Silico Synergism and Antagonism of an Anti-tumour System Intervened by Coupling Immunotherapy and Chemotherapy: A Mathematical Modelling Approach. Bull Math Biol 74, 434–452 (2012). https://doi.org/10.1007/s11538-011-9693-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-011-9693-x