The study of combined action of agents using differential geometry of dose-effect surfaces

Although graphic surfaces have been used routinely in the study of combined action of agents, they are mainly used for display purposes. In this paper, it is shown that useful mechanistic information can be obtained from an analytical study of these surfaces using the tools of differential geometry. From the analysis of some simple dose-effect surfaces, it is proposed that the intrinsic curvature, referred to in differential geometry as the Gaussian curvature, of a dose-effect surface can be used as a general criterion for the classification of interaction between different agents. This is analogous to the interpretation of the line curvature of a dose-effect curve as an indication of self-interaction between doses for an agent. In this framework, the dose-effect surface would have basic uniform fabric with zero curvature in the absence of interaction, tentatively referred to as null-interaction. Pictorially speaking, this fabric is distorted locally or globally like the stretching and shrinking of a rubber sheet by the presence of interaction mechanisms between different agents. Since self-interaction with dilution dummies does not generate intrinsic curvature, this criterion of null-interaction would describe the interaction between two trulydifferent agents. It is shown that many of the published interaction mechanisms give rise to dose-effect surfaces with characteristic curvatures. This possible correlation between the intrinsic geometric curvature of dose-effect surfaces and the biophysical mechanism of interaction presents an interesting philosophical viewpoint for the study of combined action of agents.     

The differential aspects of the linear isobole in the study of combined action of agents

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.     

The combined effects of mixtures of ionizing radiations

Ionizing radiation is a special group of toxic agents whose general interaction can be calculated. This was demonstrated using a radiation interaction model previously published. In this paper, this model is refined and mathematically reformulated using a unified set of assumptions. It postulates the existence of a common intermediate lesion and the relative action of lesions before, at and after this common stage. General quantitative dose-effect relationships of mixed radiations can be derived from the dose-effect relationships of the components in the mixture.     

Application of the isobologram technique for the analysis of combined effects with respect to additivity as well as independence

Combined actions of two substances with similar effects are frequently expressed by pairs of doses that produce a fixed response, usually 50%, in so-called isobolograms (ED50 isobolograms). In addition to the dose scales in such graphs we propose the addition of effect scales, where possible, to indicate the effect at certain doses, e.g., the ED30. We further propose to construct isoboles for expected independent interaction, in addition to the additivity line, for which purpose a simple procedure is delineated. In practice, an independent isobole for 50% effect passes through the point formed by the ED30s of A and of B in ED50 isobolograms. Thus, the ED30s constitute the "zenith" of an independent isobole in ED50 isobolograms. It is shown that theoretical independent isoboles can either represent additive, overadditive, or underadditive interactions, depending on the steepness of the dose-response curves of the components. Hence, drugs with shallow dose-response curves exhibit overadditive independent effects, compounds with exponentially steep curves show additive independent interactions. Substances with very steep dose-response curves, producing lethal effects, exhibited marked underadditive effects which could be ascribed largely to an independent mechanism of action of the components. Hence, the inclusion of independent isoboles into conventional isobolograms provides new insights into the mechanisms of interactions and into the actions of the components. Interactions can thus be characterized better and more completely, and misinterpretations appear less likely than with conventional isoboles.     

A general formulation of the concept of independent action for the combined effects of agents

Analysis schemes for the classification of synergism and antagonism for mixed agents operate on the discrepancies between observed and calculated results. As such they cannot be confirmed by experiments and therefore have to be tested in terms of mathematical and logical self-consistency. The concept of independent action is close to the literal meaning of the term “non-interaction”. Since this concept does not depend on the mechanisms of actions nor on the type of effect scale used, it is suitable as one of the basic criterion for the definition of synergism and antagonism. A general mathematical framework of independent action is presented in this paper based on the concept of “relative effect” as used in the literature. The, different equations for independent action currently used in various areas are shown to be manifestations, of a general formula under different sets of boundary conditions, which are the natural limiting values of the effects of the corresponding system observed at low and at high doses of the agents. The framework can, be generalized to the combined action ofn-agents as well as to the interaction of an agent with itself. In addition, the differential form of the formula for independent action is derived. This framework of systematic definitions and derived equations enable a more in-depth study of the implications of the concept of independent action and its relation to other concepts of non-interaction.     

Non-monotonous dose-response relationship in the region of low doses of ionizing radiation

The statement about nonmonotony of dose-effect curves as a result of nonmonotony of the time-effect relationship including the field of low doses is discussed. The living cells possess a fundamental property to response to action of different stress agents by oscillatory--nonmonotonous--hanges of metabolism. The systems keeping up homeostasis by direct and feed-back regulation return metabolism to norm. In the fixed temporary point a dose-effect dependence may take the nonmonotonous character e.g. reverse dose-response relationship. The changes of the oscillation parameters suggested the inclusion of the different pathway for homeostasis keeping. Radiation hormesis does not focused on the metabolic and functional nonmonotonous response. Radiation stimulation is considered as consequence of the peculiarity of the homeostasis maintenance pathways in the certain interval of the low doses of ionizing radiation.     

EFFICIENT DESIGNS FOR STUDYING SYNERGISTIC DRUG COMBINATIONS

Distinguishing between pharmacologically additive and synergistic drug combinations requires experimental designs and statistical analyses that often require appreciable numbers of animals and much experimenter time. The current study employed a design in which individual dose-effect data from each drug were translated into theoretically additive total dose combinations, in a fixed drug proportion, in order to produce a composite additive dose-effect relation that could be compared with that of an actual mixture having the same proportion. Results from this approach, using a combination of intrathecal doses of morphine and clonidine, were virtually identical to those using isobolographic analysis of the same data set. Both analyses showed significant synergism for this combination and, in each method, it was not necessary to constrain the drug regression lines to parallelism. In contrast to the isobole approach, the use of the composite additive dose-effect relation also allows observation of the interaction over a range of effects while reducing the size of the data sets needed. © 1997 Elsevier Science Inc.     

Modeling physiological resistance in bacterial biofilms

A mathematical model of the action of antimicrobial agents on bacterial biofilms is presented. The model includes the fluid dynamics in and around the biofilm, advective and diffusive transport of two chemical constituents and the mechanism of physiological resistance. Although the mathematical model applies in three dimensions, we present two-dimensional simulations for arbitrary biofilm domains and various dosing strategies. The model allows the prediction of the spatial evolution of bacterial population and chemical constituents as well as different dosing strategies based on the fluid motion. We find that the interaction between the nutrient and the antimicrobial agent can reproduce survival curves which are comparable to other model predictions as well as experimental results. The model predicts that exposing the biofilm to low concentration doses of antimicrobial agent for longer time is more effective than short time dosing with high antimicrobial agent concentration. The effects of flow reversal and the roughness of the fluid/biofilm are also investigated. We find that reversing the flow increases the effectiveness of dosing. In addition, we show that overall survival decreases with increasing surface roughness.     

Chemical induction of streptomycin-resistant mutations in Escherichia coli. Dose and mutagenic effects of dichlorvos and methyl methanesulfonate

A procedure for the quantitative determination of induced streptomycin-resistant mutants in E. coli was applied to study and compare mutation induction by the organophosphate dichlorvos and by methyl methanesulfonate (MMS). Both compounds increased the frequency of mutants even under conditions where no inactivation of cell was observed. Mutation induction by these agents as a function of both concentration and exposure time was measured. The dose-response curves found with both mutagens were non-linear; atp higher doses more mutants were induced per unit dose than at lower doses. Possible relationships between dose-effect curves and the chemical nature of alkylating mutagenic agents are discussed.     

The interaction between X-rays and 3 MeV neutrons in the skin of the mouse foot

Mouse feet were irradiated with mixtures of 3 MeV neutrons and 140 kVp X-rays given simultaneously or within 24 hours of each other. The effects of different treatments were contrasted by comparing the doses required to give equal skin reactions. Irradiation was given as 1, 2, 4 or 8 equal fractions, in order to assess r.b.e. and the shapes of the underlying dose-response curves for mixed beams over a wide range of dose per fraction. All dose-effect curves were well fitted by a linear-quadratic (alpha, beta) model. For X-rays and neutrons given simultaneously, the linear coefficient (alpha) decreased by a factor of 4.80 while the quadratic coefficient (beta) increased by a factor of only 1.44 when the proton contamination in the beam increased from 11 to 100 per cent, with alpha/beta changing from 95.0 to 13.8. The data from simultaneous X-ray and neutron irradiation were consistent with full interaction of those effects from the two radiations which give rise to the total quadratic component of effect. When the two radiations are separated by up to 24 h, this interaction decreases but does not entirely disappear.     

Interactions: Dose effect relationships and isoeffect curves

Summary Interaction from several agents, i.e., a greater or smaller effects than expected from the sum of the individual effects can be of essentially two types: a) The agents could act in paralell on the same target, and a term depending on the doses of the agents involved would have to be added to the individual dose effect relationships (parergic interaction). b) The agents could act at different points in the chain of events leading to the observed effect, and the action of the second agent, the promotor, would be, at least in part, contingent on the action of the first inducer agent so that the dose effect relationships are linked in a multiplicative manner (metergic interaction). The dose effect surfaces for interaction depend on the type of dose effect relationship of the individual agents. Formulas are given for the general cases and are exemplified in graphs. Typical isobolic diagrams are also illustrated. These formulas may be adapted to experimental data by means of non-linear regression or maximum-likelihood analysis.     

Analysis of translocations in stable cells and their implications in retrospective biological dosimetry

The aim of the present study was to evaluate the influence of the exclusion of cells with unstable aberrations in the elaboration of dose-effect curves for translocations and their implications in biological dosimetry of past exposures. To establish dose-effect curves, peripheral blood samples were irradiated with 60Co gamma rays at ten different doses and the yield of translocations analyzed by FISH was considered in all cells and in stable cells (those without dicentrics, acentrics or rings). To discriminate transmissible translocations, the dose- effect curve for total apparently simple translocations in stable cells was chosen as the reference. In stable cells, dose- effect curves for apparently simple translocations without pseudosimple and complex-derived one-way patterns, tAbtBa and total translocations were obtained. None of these curves differed from the reference curve. When all cells were considered, only the curve for total translocations was significantly different from the reference curve. From the results obtained it can be concluded that the use of dose-effect curves for apparently simple translocations in stable cells and in all cells will give similar dose estimates in retrospective biological dosimetry studies. However, the use of dose-effect curves for total translocations in all cells will lead to underestimations of the dose mainly at high doses.     

Combined effects of vasoactive intestinal peptide and dopamine on adenylate cyclase in prolactin-secreting cells

VIP stimulates adenylate cyclase activity of male and female rat anterior pituitaries and human prolactinomas, while dopamine inhibits the enzyme activity of female rat pituitaries and prolactinomas. A dopamine inhibited cyclase can be detected also in male rats provided the enzyme activity is increased by VIP. The analysis of the dose-response curves for one agent (VIP or dopamine) in the absence or in the presence of the other indicates that the two agents exhibit a different pattern of interaction in the different systems. In fact, in female rat pituitaries and in human prolactinomas, the curves for dopamine±VIP and for VIP±dopamine were parallel, indicating that the two agents exherted their effects independently from one another. On the contrary, in male rat pituitaries, the curves were definitively non parallel, that is, the inhibitory effect of dopamine was greatly amplified by VIP. In no case was the apparent affinity (EC₅₀) of one agent modified by the presence of the other. It is concluded that two different modes of interaction between stimulatory and inhibitory neurohormones might exist at the level of adenylate cyclase from anterior pituitary cells.     

Evaluation of interaction in olfactory and taste mixtures

Models for the evaluation of interaction in olfactory and tastemixtures are compared to the isobole approach, which is widelyused in other fields of biomedical research. Analogies and differencesare described in detail. The isobole approach has a profoundtheoretical basis and can be applied to all possible types ofconcentration–response relationships, in particular toall values of exponents in the widely used power function, andeven if analytical expression of concentration–responserelationships are not known. Due to this generality it leadsto a substantial simplification of the evaluation proceduresas compared to other methods used in taste and olfaction research.It can be applied to any number of components of a mixture.Response surface modeling and computer graphics is recommededin appropriate cases because it can provide information on theconcentration dependence of interaction. Even though it hasto be pointed out that there is no general consensus on themost appropriate approach for the evaluation of interactionbetween biologically active agents so far, we are led to theconclusion that good arguments can be made in supporting theapplication of the isobole method for taste and olfactory mixtures.     

On the equivalence of meta-analysis using literature and using individual patient data

When data come from several independent studies for the purpose of estimating treatment control differences, meta-analysis can be carried out either on the best linear unbiased estimators computed from each study or on the pooled individual patient data modelled as a two-way model without interaction, where the two factors represent the different studies and the different treatments. Assuming that observations within and between studies are independent having a common variance, Olkin and Sampson (1998) have obtained the surprising result that the two meta-analytic procedures are equivalent, i.e., they both produce the same estimator. In this article, the same equivalence is established for the two-way fixed-effects model without interaction with the only assumption that the observations across studies be independent. A consequence of the equivalence result is that, regardless of the covariance structure, it is possible to get an explicit representation for the best linear unbiased estimator of any vector of treatment contrasts in a two-way fixed-effects model without interaction as long as the studies are independent. Another interesting consequence is that, for the purpose of best linear unbiased estimation, an unbalanced two-way fixed-effects model without interaction can be treated as several independent unbalanced one-way models, regardless of the covariance structure, when the studies are independent.     

Evaluation of experimental combined toxicity by use of dose-frequency curves: comparison with theoretical additivity as well as independence

Dose-frequency curves of toxic effects of a substance A were evaluated in the absence and in the presence of a fixed dose of a second substance B. Data were fitted by the curve-fitting program ALLFIT. Observed combined frequencies of A + B were compared statistically with the expected frequencies of additivity and (or) independence by the phi 2-square goodness-of-fit test. The theoretical dose-frequency curves expected for an additive response were obtained by a solely graphical procedure and the theoretical curves for independent effects were calculated from the effects of B and A at certain doses. In rotarod tests with trained mice, the combined deteriorating effect of ethanol and benzodiazepines were significantly over-additive. However, their lethal interaction appeared underadditive in mice. The lethal underadditive interaction of ethanol and phencyclidine (PCP) can be ascribed largely to independent actions of these compounds. Loss of righting reflex was additively enhanced by PCP, whereas PCP overadditively enhanced the effect of ethanol. The insecticidal action of the cholinesterase inhibitors malathion and parathion appeared additive and significantly different from independent interaction. A comparison of results from dose-response curves with isoboles showed good agreement. The method appears as an attractive alternative or as a complementary procedure to the isobolographic analysis. Combination experiments as described can be carried out and evaluated rather simply, with a minimum of expenditure and a maximum of information.     

Parallel dose-response curves in combination experiments

A possible experimental design for combination experiments is to compare the doseresponse curve of a single agent with the corresponding curve of the same agent using either a fixed amount of a second one or a fixed dose ratio. No interaction is then often defined by a parallel shift of these curves. We have performed a systematic study for various types of doseresponse relations both for the dose-additivity (Loewe additivity) and for the independence (Bliss independence) criteria for defining zero interaction. Parallelism between doseresponse curves of a single agent and those of the same agent in the presence of a fixed amount of another one is found for the Loewe-additivity criterion for linear doseresponse relations. For nonlinear relations, one has to differentiate between effect parallelism (parallel shift on the effect scale) and dose parallelism (parallel shift on the dose scale). In the case of Loewe additivity, zero-interaction dose parallelism is found for power, Weibull, median-effect and logistic doseresponse relations, given that special parameter relationships are fulfilled. The mechanistic model of competitive interaction exhibits dose parallelism but not effect parallelism for Loewe additivity. Bliss independence and Loewe additivity lead to identical results for exponential doseresponse curves. This is the only case for which dose parallelism was found for Bliss independence. Parallelism between single-agent doseresponse relations and Loewe additivity mixture relations is found for examples with a fixed doseratio design. However, this is again not a general property of the design adopted but holds only if special conditions are fulfilled. The comparison of combination doseresponse curves with single-agent relations has to be performed taking into account both potency and shape parameters. The results of this analysis lead to the conclusion that parallelism between zero interaction combination and single-agent doseresponse relations is found only for special cases and cannot be used as a general criterion for defining zero-interaction in combined-action assessment even if the correct potency shift is taken into account.     

Induction of chromosome aberrations and micronuclei in human peripheral blood lymphocytes at low dose of radiation

The chromosome damage induced by the doses of y-irradiation 6)Co in peripheral blood lymphocytes was studied using different cytogenetic assays. Isolated lymphocytes were exposed to 0.01-1.0 Gy, stimulated by PHA, and analysed for chromosome aberrations at 48 h postirradiation by metaphase method, at 49 h--by the anaphase method, at 58 h by micronucleus assay with cytochalasin B and, additionally, micronuclei were counted at 48 h on the slides prepared for the metaphase analysis without cytochalasin B. Despite of the quantitative differences in the amount of chromosome damage revealed by different methods all of them demonstrated complex nonlinear dose dependence of the frequency of aberrant cells and aberrations. At the dose range from 0.01 Gy to 0.05-0.07 Gy the cells had the highest radiosensitivity mainly due to chromatid-type aberration induction. With dose increasing the frequency of the aberrant cells and aberrations decreased significantly (in some cases to the control level). At the doses up to 0.5-0.7 Gy the dose-effect curves have become linear with the decreased slope compare to initial one (by factor of 5 to 10 for different criteria) reflecting the higher radioresistance of cells. These data confirm the idea that the direct linear extrapolation of high dose effect to low dose range--the procedure routinelly used to estimate genetic risk of low dose irradiation--cannot be effective and may lead to underestimation of chromosome damage produced by low radiation doses. Preferences and disadvantages of used cytogenetic assays and possible mechanisms of low ionising radiation doses action were discussed.     

Substitution of calorigenic effects of noradrenaline and adrenaline and differences in their inhibition by propranolol.

The calorigenic effect of infused adrenaline and noradrenaline was measured in cold-acclimated rats. The slopes of the dose-response curves for the two catecholamines and the maxima of the curves were the same. The adrenaline dose-response curve showed a shift to the right, towards higher infusion doses, compared with the noradrenaline curve. Thermogenesis due to the two catecholamines was not additive throughout the whole range of doses used. In interaction with noradrenaline, propranolol caused a parallel shift of the dose-response curve to the right, whereas in interaction with adrenaline it depressed the maximum. The concept that the two catecholamines act via different regulation sites on a common thermogenetic effector is discussed.     

Pathology of the reproductive apparatus of cells under the action of remantadine and deitiforin

The antiviral effect of chemical agents remantadine and deutephorine on the reproductive apparatus of MDCK-cells was studied. It was established that the action of the chemical agents during 3-4 days induced enhancement in pathologically transformed mitoses. This enhancement was in direct proportion to the doses used and to the cultivation time. Dose dependent enhancement of the total pathological mitoses number was simultaneous with the redistribution of mitoses into groups: the relative number of pathologies connected with chromosome damage became lower and the number of degenerating mitoses grew. Computer regression analysis showed the linear dependence between the chemical agent dose and the number of mitoses in the metaphase, that gives an indirect evidence of statmokinetic action of the agents under study.