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Mathematical model of cancer chemotherapy. Periodic schedules of phase- specific cytotoxic-agent administration increasing the selectivity of therapy. (English) Zbl 0565.92006
Considered is the problem of optimization of the protocols of cancer chemotherapeutic treatments. The model employed includes two cell populations, normal and malignant, the latter having longer mean interdivision times. The interdivision times are assumed to be distributed according to a delayed noncentral gamma law. The chemotherapeutic agent is assumed to be applied periodically and to have a cell cycle phase specific, purely killing, action.
The basic question is what should be the interval between two successive doses of the drug, to minimize the number of tumor cells, keeping the number of normal cells at a predetermined level (other, equivalent formulations of this control problem are also provided).
The problem is dealt with numerically. Over the range of parameters considered (approximating the usually observed cell cycle characteristics), the study indicates that the best interdosage period is approximately equal to the mean interdivision time of normal cells. A thorough discussion of applicability of this result is provided.
Reviewer: M.Kimmel

MSC:
92C50 Medical applications (general)
92D25 Population dynamics (general)
65K99 Numerical methods for mathematical programming, optimization and variational techniques
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[1] Aroesty, J.; Lincoln, T.; Shapiro, N.; Boccia, G., Tumor growth and chemotherapy: mathematical methods, computer simulations, and experimental foundations, Math. biosci., 17, 243-300, (1973) · Zbl 0257.92002
[2] Bertuzzi, A.; Gandolfi, A.; Giovenco, M.A., Mathematical models of the cell cycle with a view to tumor studies, Math. biosci., 53, 159-188, (1981) · Zbl 0525.92005
[3] Bruce, W.R.; Meeker, B.E.; Valeriote, F.A., Comparison of the sensitivity of normal hematopoietic and transplanted lymphoma colony-forming cells to chemotherapeutic agents administered in vivo, J. natl. cancer inst., 37, 233-245, (1966)
[4] Bruce, W.R.; Meeker, B.E., Comparison of the sensitivity of hematopoietic colony-forming cells in different proliferative states to 5-fluorouracil, J. natl. cancer inst., 38, 401-405, (1967)
[5] Bruce, W.R.; Meeker, B.E.; Powers, W.E.; Valeriote, F.A., Comparison of the dose- and time-survival curves for normal hematopoietic and lymphoma colony-forming cells exposed to vinblastine, vincristine, arabinosylcytosine and amethopterin, J. natl. cancer inst., 42, 1015-1025, (1969)
[6] Cairnie, A.B., Homeostasis in the small intestine, (), 67-77
[7] Chertkov, I.L.; Friedenstein, A.J., Cellular basis of hemopoiesis, (1976), Meditsina Moscow
[8] Cowan, R.; Culpin, D.; Morris, V.B., A method for the measurement of variability in cell lifetimes, Math. biosci., 54, 249-263, (1981) · Zbl 0459.92005
[9] Dibrov, B.F.; Zhabotinsky, A.M.; Krinskaya, A.V.; Neyfakh, Yu.A.; Churikova, L.I., The dependence of hemopoietic stem cell survival on the interval between multiple regular injections of hydroxyurea (in Russian), Bull. eksp. biol. med., 97, 345-347, (1984)
[10] Dibrov, B.F.; Zhabotinsky, A.M.; Neyfakh, Yu.A.; Orlova, M.P.; Churikova, L.I., Optimal scheduling for cell synchronization by cycle-phase specific blockers, Math. biosci., 66, 167-185, (1983) · Zbl 0526.92002
[11] Eisen, M., Mathematical models in cell biology and cancer chemotherapy, (1979), Springer Berlin · Zbl 0414.92005
[12] Frankfurt, O.S., Cellular mechanisms of cancer chemotherapy, (1976), Meditsina Moscow
[13] Goldin, A.; Schabel, F.M., Clinical concepts derived from animal chemotherapy studies, Cancer treatm. rep., 65, 11-19, (1981), (Suppl. 3)
[14] Graham, F.L.; Whitmore, G.F., The effects of α-β-D-arabinofuranosylcytosine on growth, viability and DNA synthesis of mouse L-cells, Cancer res., 30, 2627-2635, (1970)
[15] Gray, J.W., Flow cytometry and cell kinetics: relation to cancer therapy, (), 485-491
[16] Hersig, R.H.; Wolff, S.N.; Lazarus, H.M.; Phillips, G.L.; Karanes, C.; Herzig, G.P., High-dose cytosine-arabinoside therapy for refractory leukemia, Blood, 62, 361-369, (1983)
[17] Hill, B.T., Cancer chemotherapy. the relevance of certain concepts of cell cycle kinetics, BBA (rev. canc.), 516, 389-417, (1978)
[18] Jansson, B., Simulation of cell-cycle kinetics based on a multicompartmental model, Simulation, 25, 99-108, (1975) · Zbl 0318.68068
[19] Kirk, J.; Gray, W.M.; Watson, E.R., Cumulative radiation effect I. fractionated treatment regimens, Clin. radiol., 22, 145-155, (1971)
[20] Madoc-Jones, H.; Mauro, F., Site of action of cytotoxic agents in the cell life cycle, (), 205-219
[21] Necas, E.; Neuwirt, J., Proliferation rate of haemopoietic stem cells after damage by several cytotoxic agents, Cell tissue kinet., 9, 479-487, (1976)
[22] Pardee, A.B.; Shilo, B.-Z.; Koch, A.L., Variability of the cell cycle, (), 373-392
[23] Potten, C.S.; Schofield, R.; Laitha, L.G., A comparison of cell replacement in bone marrow, testis and three regions of surface epithelium, BBA (rev. canc.), 560, 281-299, (1979)
[24] Prescott, D.M., Variations in the individual generation times of tetrahymena geleii HS, Exp. cell res., 16, 279-284, (1959)
[25] Prescott, D.M., The cell cycle and control of cellular reproduction, Adv. genet., 18, 99-177, (1976)
[26] Rubinow, S.I.; Lebowitz, J.L., A mathematical model of the chemotherapeutic treatment of acute myeloblastic leukemia, Biophys. J., 16, 1257-1271, (1976)
[27] Shin, K.G.; Pado, R., Design of optimal cancer chemotherapy using a continuous-time state model of cell kinetics, Math. biosci., 59, 225-248, (1982) · Zbl 0486.92007
[28] Sinclair, W.K., Hydroxyurea: differential lethal effects on cultured Mammalian cells during the cell cycle, Science, 150, 1729-1731, (1965)
[29] Skipper, H.E.; Schabel, F.M.; Mellett, L.B.; Montgomery, J.A.; Wilkoff, L.J.; Lloyd, H.H.; Bruckman, R.W., Implications of biochemical, cytokinetic, pharmacologic and toxicologic relationships in the design of optimal therapeutic schedules, Cancer chemother. rep., 54, 431-450, (1970)
[30] Smith, J.A.; Martin, L., Do cells cycle?, Proc. nat. acad. sci. U.S.A., 70, 1263-1267, (1973)
[31] Steel, G.G., The cell cycle in tumors: an examination of data gained by the technique of labelled mitoses, Cell tissue kinet., 5, 87-100, (1972)
[32] Steward, P.C.; Hahn, G.M., The application of age response functions to the optimization of treatment schedules, Cell tissue kinet., 4, 279-291, (1971)
[33] Stokes, A., A Floquet theory for functional differential equations, Proc. nat. acad. sci. U.S.A., 48, 1330-1334, (1962) · Zbl 0109.06301
[34] Stuart, R.N.; Merkle, T.C., The calculation of treatment schedules for cancer chemotherapy (part II), ()
[35] Takahashi, M., Theoretical basis for cell cycle analysis. I. labelled mitosis wave method, J. theoret. biol., 13, 202-211, (1966)
[36] Tannock, I., Cell kinetics and chemotherapy: A critical review, Cancer treatm. rep., 62, 1117-1133, (1978)
[37] Till, J.E.; McCulloch, E.A., A direct measurement of the radiation sensitivity of normal mouse bone marrow cells, Rad. res., 14, 213-222, (1961)
[38] Tubiana, M.; Malaise, E., Comparison of cell proliferation kinetics in human and experimental tumors: response to irradiation, Cancer treatm. rep., 60, 1887-1911, (1976)
[39] Valeriote, F.A.; Bruce, W.R., Comparison of the sensitivity of hematopoietic colony-forming cells in different proliferative states to vinblastine, J. natl. cancer inst., 38, 393-399, (1967)
[40] Vassort, F.; Frindel, E.; Tubiana, M., Effects of hydroxyurea on the kinetics of colony-forming units of bone marrow in the mouse, Cell tissue kinet., 4, 423-431, (1971)
[41] Vassort, F.; Winterholer, M.; Frindel, E.; Tubiana, M., Kinetic parameters of bone marrow stem cells using in vivo suicide by tritiated thymidine or by hydroxyurea, Blood, 41, 789-796, (1973)
[42] Wright, N.A., Cell population in the normal gastrointestinal tract. implications for proliferative responses, (), 3-21
[43] Zietz, S.; Nicolini, C., Mathematical approaches to optimization of cancer chemotherapy, Bull. math. biol., 41, 305-324, (1979) · Zbl 0404.92004
[44] Sofiina, Z.P.; Syrkin, A.B.; Goldin, A.; Kline, A., Experimental evaluation of anti-tumor drugs in the USSR and in the USA, (1980), Meditsina Moscow
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