Abstract
Although the isobologram is presently the most widely used method of analysis for combined effects of agents, there are several different interpretations of the linear isobole isobole in regard to its use as a criterion of interaction. An investigation of the differential aspects of the linear isobole relation may cast some light in this regard. By conceptual extension of the present single effect level (i.e. effect-point) relation of the linear isobole to an effect-neighbourhood relation in which the linear isobole holds over a small continuous range of effect levels, the mathematical differential of the linear isobole can be developed and investigated. This differential aspect provides some useful insights into the implication and interpretation of the linear isobole relation when used as a general criterion in agent interaction studies. it can also serve as the mathematical basis for the formulation of analytic schemes in which the linear isobole relation is applicable over a continuous range of effect levels.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature
Ashford, J. R. 1981. General models for the joint action of mixtures of drugs.Biometrics 37, 457–474.
Berenbaum, M. C. 1978. A method for testing for synergy with any number of agents.J. infect. Dis. 137, 122–130.
Berenbaum, M. C. 1981. Criteria for analyzing interactions between biologically active agents.Adv. Concer Res. 35, 269–335.
Berenbaum, M. C. 1985. The expected effect of a combination of agents: the general solution.J. theor. Biol. 114, 413–431.
Chou, T.-C and P. Talalay. 1981. Generalized equations for the analysis of inhibitions of Michaelis-Menten and higher-order kinetic systems with two or more mutually exclusive and nonexclusive inhibitors.Eur. J. Biochem. 115, 207–216.
Chou, T.-C. and P. Talalay. 1984. Quantitative analysis of dose-effect relationships: The combined effects of multiple drugs or enzyme inhibitors.Adv. Enzyme Reg. 22, 27–55.
Chou, T.-C. and P. Talalay. 1987. Applications of the median-effect principle for the assessment of low-dose risk or carcinogens and for the quantitation of synergism and antagonism of chemotherapeutic agents. In:New Avenues in Developmental Cancer Chemotherapy, K. R. Harrap and T. A. Connors (Eds), pp. 37–64. Orlando, FL: Academic Press.
Dewey, W. C. 1979.In vitro systems: standardization of endpoints.Int. J. Radiat. Oncol. Biol. Phys. 5, 1165–1174.
Fraser, T. R. 1869–1872. An experimental research on the antagonism between the actions of physiostigma and atropia.Proc. R. Soc. Edinburgh 7, 506–511.
Fraser, T. R. 1872. The antagonism between the actions of active substances.Br. Med. J. 2, 485–487.
Hewlett, P. S. 1969. Measurement of the potencies of drug mixtures.Biometrics 25, 477–487.
ICRU (International Commission on Radiation Units and Measurements). 1979.Quantitative Concepts and Dosimetry in Radiobiology. ICRU Report No. 30, Bethesda, MD.
King, T. C., D. Schlessinger and D. J. Krogstad. 1981. The assessment of antimicrobial combinations.Rev. Infect. Dis. 3, 627–633.
Krogstad, D. J. and R. C. Moellering, Jr. 1980. Combinations of antibiotics, mechanisms of interaction against bacteria. In:Antibiotics in Laboratory Medicine, V. Lorian (Ed.), pp. 298–341. Baltimore, MD: Williams and Wilkins.
Lam, G. K. Y. 1987. The interaction of radiations of different LET.Phys. Med. Biol. 32, 1291–1309.
Lam, G. K. Y. 1988. The combined effects of mixtures of ionizing radiations.J. theor. Biol. 134, 531–546.
Lam, G. K. Y. 1989a. Analysis of interaction for mixtures of agents using the linear isobole.Bull. math. Biol. 51, 293–309.
Lam, G. K. Y. 1989b. An isoeffect approach to the study of combined effects of mixed radiations-the nonparametric analysis ofin vivo data.Radiat. Res. 119, 424–431.
Lam, G. K. Y. 1992. The study of combined action of agents using differential geometry of dose-effect surfaces.Bull. math. Biol., in press.
Loewe, S. 1928. Die quantitativen Problems der Pharmakologie.Ergeb. Physiol. 27, 47–187.
Loewe, S. 1953. The problem of synergism and antagonism of combined drugs.Arzneim.-Forsch. 3, 285–290.
Loewe, S. and H. Muischnek. 1926. Uber Kombinationswirkungen. I. Mitteilung: Hilfsmittel der Fragestellung.Arch. Exp. Pathol. Pharmakol. 114, 313–326.
Steel, G. G. 1979. Terminology in the description of drug-radiation interactions.Int. J. Radiat. Oncol. Biol. Phys. 5, 1145–1150.
Tyrell, R. M. 1978. Radiation synergism and antagonism.Photochem. Photobiol. Rev. 3, 35–113.
Unkelbach, H.-D. and G. Pöch. 1988. Comparison of independence and additivity in drug combinations.Arzneim.-Forsch. Drug. Res. 38(I), 1–6.
UNSCEAR (United Nations Scientific Committee on the Effects of Atomic Radiation). 1982.Ionizing Radiation: Sources and Biological Effects. United Nations Publications: New York.
Wessinger, W. D. 1986. Approaches to the study of drug interactions in behavioral pharmacology.Neurosci. Biobehav. Rev. 10, 103–113.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lam, G.K.Y. The differential aspects of the linear isobole in the study of combined action of agents. Bltn Mathcal Biology 55, 295–313 (1993). https://doi.org/10.1007/BF02460885
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02460885