Abstract
A model of tumor growth, based on two-compartment cell population dynamics, and an overall Gompertzian growth has been previously developed. The main feature of the model is an inter-compartmental transfer function that describes the net exchange between proliferating (P) and quiescent (Q) cells and yields Gompertzian growth for tumor cell population N = P + Q. Model parameters provide for cell reproduction and cell death. This model is further developed here and modified to simulate antimitotic therapy. Therapy decreases the reproduction-rate constant and increases the death-rate constant of proliferating cells with no direct effect on quiescent cells. The model results in a system of two ODE equations (in N and P/N) that has an analytical solution. Net tumor growth depends on support from the microenvironment. Indirectly, this is manifested in the transfer function, which depends on the proliferation ratio, P/N. Antimitotic therapy will change P/N, and the tumor responds by slowing the transfer rate from P to Q. While the cellular effects of therapy are modeled as dependent only on antimitotic activity of the drug, the tumor response also depends on the tumor age and any previous therapies—after therapy, it is not the same tumor. The strength of therapy is simulated by the parameter λ, which is the ratio of therapy induced net proliferation rate constant versus the original. A pharmacodynamic factor inversely proportional to tumor size is implemented. Various chemotherapy regimens are simulated and the outcomes of therapy administered at different time points in the life history of the tumor are explored. Our analysis shows: (1) for a constant total dose administered, a decreasing dose schedule is marginally superior to an increasing or constant scheme, with more pronounced benefit for faster growing tumors, (2) the minimum dose to stop tumor growth is age dependent, and (3) a dose-dense schedule is favored. Faster growing tumors respond better to dose density.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bertuzzi, A., D'Onofrio, A., Fasano, A., Gandolfi, A., 2003. Regression and regrowth of tumour cords following single-dose anticancer treatment. Bull. Math. Biol. 65, 903–931.
Bonadonna, G., Zambetti, M., Moliterni, A., Gianni, L., Valagussa, P., 2004. Clinical relevance of different sequencing of doxorubicin and cyclophosphamide, methotrexate, and Fluorouracil in operable breast cancer. J. Clin. Oncol. 22, 1614–1620.
Bonadonna, G., Zambetti, M., Valagussa, P., 1995. Sequential or alternating doxorubicin and CMF regimens in breast cancer with more than three positive nodes. Ten-year results. JAMA 273, 542–547.
Byrne, H.T., 2003. Modeling avascular tumor growth. Cancer modeling and simulation. In: L. Preziosi (Ed.), Chapman & Hall/CRC, New York, pp. 80–86.
Citron, M.L., Berry, D.A., Cirrincione, C., Hudis, C., Winer, E.P., Gradishar, W.J., Davidson, N.E., Martino, S., Livingston, R., Ingle, J.N. et al., 2003. Randomized trial of dose-dense versus conventionally scheduled and sequential versus concurrent combination chemotherapy as postoperative adjuvant treatment of node-positive primary breast cancer: first report of Intergroup Trial C9741/Cancer and Leukemia Group B Trial 9741. J. Clin. Oncol. 21, 1431–1439.
Cojocaru, L., Agur, Z., 1992. A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs. Math. Biosci. 109, 85–97.
de Vladar, H.P., Gonzalez, J.A., 2004. Dynamic response of cancer under the influence of immunological activity and therapy. J. Theor. Biol. 227, 335–348.
Dyson, J., Villella-Bressan, R., Webb, G.F., 2002. Asynchronous exponential growth in an age structured population of proliferating and quiescent cells. Math. Biosci. 177/178, 73–83.
Jackson, T.L., Byrne, H.M., 2000. 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.
Kozusko, F., Bajzer, Z., 2003. Combining Gompertzian growth and cell population dynamics. Math. Biosci. 185, 153–167.
Kozusko, F., Chen, P., Grant, S.G., Day, B.W., Panetta, J.C., 2001. A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A. Math. Biosci. 170, 1–16.
Magni, P., Simeoni, M., Poggesi, I., Rocchetti, M., De Nicolao, G., 2006. A mathematical model to study the effects of drugs administration on tumor growth dynamics. Math. Biosci. 200, 127–151.
Norton, L., 2005. Conceptual and practical implications of breast tissue geometry: toward a more effective, less toxic therapy. Oncologist 10, 370–381.
Norton, L., Simon, R., 1977. Tumor size, sensitivity to therapy, and design of treatment schedules. Cancer Treat. Rep. 61, 1307–1317.
Norton, L., Simon, R., 1986. The Norton—Simon hypothesis revisited. Cancer Treat. Rep. 70, 163–169.
Norton, L., Simon, R., Brereton, H.D., Bogden, A.E., 1976. Predicting the course of Gompertzian growth. Nature 264, 542–545.
Panetta, J.C., 1996. A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425–447.
Panetta, J.C., 1997. A mathematical model of breast and ovarian cancer treated with paclitaxel. Math. Biosci. 146, 89–113.
Panetta, J.C., Kirstein, M.N., Gajjar, A.J., Nair, G., Fouladi, M., Stewart, C.F., 2003. A mechanistic mathematical model of temozolomide myelosuppression in children with high-grade gliomas. Math. Biosci. 186, 29–41.
Panetta, J.C., Webb, G.F., 1995. A mathematical model of cycle-specific chemotherapy. Math. Comput. Modell. 22, 67.
Parfitt, A.M., Fyhrie, D.P., 1997. Gompertzian growth curves in parathyroid tumours: further evidence for the set-point hypothesis. Cell. Prolif. 30, 341–349.
Pfreundschuh, M., Trumper, L., Kloess, M., Schmits, R., Feller, A.C., Rube, C., Rudolph, C., Reiser, M., Hossfeld, D.K., Eimermacher, H. et al., 2004a. Two-weekly or 3-weekly CHOP chemotherapy with or without etoposide for the treatment of elderly patients with aggressive lymphomas: results of the NHL-B2 trial of the DSHNHL. Blood 104, 634–641.
Pfreundschuh, M., Trumper, L., Kloess, M., Schmits, R., Feller, A.C., Rudolph, C., Reiser, M., Hossfeld, D.K., Metzner, B., Hasenclever, D. et al., 2004b. Two-weekly or 3-weekly CHOP chemotherapy with or without etoposide for the treatment of young patients with good-prognosis (normal LDH) aggressive lymphomas: results of the NHL-B1 trial of the DSHNHL. Blood 104, 626–633.
Ribba, B., Marron, K., Agur, Z., Alarcon, T., Maini, P.K., 2005. A mathematical model of Doxorubicin treatment efficacy for non-Hodgkin's lymphoma: investigation of the current protocol through theoretical modelling results. Bull. Math. Biol. 67, 79–99.
Sidorov, I.A., Hirsch, K.S., Harley, C.B., Dimitrov, D.S., 2003. Cancer treatment by telomerase inhibitors: predictions by a kinetic model. Math. Biosci. 181, 209–221.
Sullivan, P.W., Salmon, S.E., 1972. Kinetics of tumor growth and regression in IgG multiple myeloma. J. Clin. Invest. 51, 1697–1708.
Wheldon, T.E., 1988. Mathematical models in cancer research. Adam Hilger, Bristol, pp. 157–179.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Kozusko, F., Bourdeau, M., Bajzer, Z. et al. A Microenvironment Based Model of Antimitotic Therapy of Gompertzian Tumor Growth. Bull. Math. Biol. 69, 1691–1708 (2007). https://doi.org/10.1007/s11538-006-9186-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-006-9186-5