Ronald J. Tallarida

Manual of pharmacologic calculations with computer programs

Manual of Pharmacologic Calculations with Computer Programs. By R. J. Tallarida and R. B. Murray. New York, Heidelberg and Berlin, Springer‐Verlag, 1981. ix, 150 p. 24·5 cm. Unpriced.

pA2 and receptor differentiation: A statistical analysis of competitive antagonism

Abstract The pA 2 of a competitive antagonist was defined by Schild as the negative logarithm of the molar concentration of an antagonist which reduces the effect of a dose of agonist to that of half the dose. The relationship of the pA 2 to the dissociation constant, K, of the antagonist-receptor complex is discussed within the framework of competitive theory. Various methods for the determination of pA 2 , and the accuracy and precision of these methods, are presented. Special problems associated with the determination of pA 2 in vivo are also discussed, with particular attention being given to narcotic analgesics and their antagonists.

Statistical analysis of drug-drug and site-site interactions with isobolograms.

The use of more than one drug to achieve a desired effect has been a common practice in pharmacologic testing and in clinical practice. For example, combinations of analgesics are frequently prescribed with a view to enhancing pain relief and reducing adverse effects. It is also well established that administration of more than one drug may give effects that are greater than, or less than, the additive effect of each drug given individually. A non-mechanistic method of characterizing the effect resulting from the administration of two compounds is the isobologram. It is relatively simple to draw and interpret isobolograms. However, this graphical technique, which employs equieffective concentrations of individual drugs and combinations of these, obtains the concentrations as random variables from concentration-effect data, usually transformed to a parallel line assay. Thus, statistical confidence limits from such assays, as well as from non-parallel designs, must be expressed on the isobologram if this diagram is to establish superadditive, subadditive, or merely additive effects. We now present a detailed statistical analysis of the isobolographic method illustrated with examples of the statistical procedures, a rational basis for selecting proportions of each drug in the combination, and a relatively novel application of the isobolographic concept, i.e., interactions involving different anatomical sites.

An Overview of Drug Combination Analysis with Isobolograms

Drugs given in combination may produce effects that are greater than or less than the effect predicted from their individual potencies. The historical basis for predicting the effect of a combination is based on the concept of dose equivalence; i.e., an equally effective dose (a) of one will add to the dose (b) of the other in the combination situation. For drugs with a constant relative potency, this leads to linear additive isoboles (a-b curves of constant effect), whereas a varying potency ratio produces nonlinear additive isoboles. Determination of the additive isobole is a necessary procedure for assessing both synergistic and antagonistic interactions of the combination. This review discusses both variable and constant relative potency situations and provides the mathematical formulas needed to distinguish these cases.

Statistical analysis of drug combinations for synergism.

Testing a pair of drugs for synergism requires determination of the potency of the combination and comparison of this potency with that of a theoretically additive combination. The potency of a drug or of a mixture is measured as a dose (or concentration) that produces a specified level of effect such as one-half the maximum effect (E,,,). Most often this dose, denoted either D,, (for graded data) or ED50 (for quanta1 data), is obtained from an appropriate analysis of dose-effect data that yields a smooth curve. Because doses have frequently been found to be log-normally distributed, the smooth curve analyzed is often plotted with log(dose) or log(concentration) on the abscissa for both graded and quanta1 curves. When data are quantal, the proportion or percent of subjects responding is often converted to probits. Probit analysis, a weighted regression procedure, is deeply rooted in the pharmacologic literature. It permits the use of 0 and 100% responses in the regression analysis and thus uses all the data. Discussions of probit analysis are contained in Hewlett and Plackett (1979) and Tallarida and Murray (19871, the latter provided a computational algorithm and computer program for getting the mean ED50 and confidence limits. The most complete discussion of probit analysis is in the standard work by Finney 0971). In contrast to the weighted regression procedure of probit analysis, analysis of graded data is usually conducted in the mid-range of effects where the error variance is approximately constant. Thus, simple linear regression of effect on log dose is employed and yields the mean log(D,,) and variance, V[log(D,,)], from which the mean D,, and VCD,,) may be calculated. It is convenient to have a simple notation for potency, whether ED50 or D,,, or doses based on other levels of effect (e.g., D,, and ED70 when the level of effect is 70% of E,,,). We here denote this dose by z”, with appropriate subscripts to identify each individual drug or the mixture. Of course, in all comparisons and analyses that utilize these quantities, it is understood that the same common level of effect applies. Most often this will be one-half of E,,, so that z * will be a D,, or ED50 value, but other levels of effect are sometimes used (Ossipov et al. 1990). It should also be mentioned that non-linear curve fitting algorithms that compute z *, its variance and confidence limits from dose-response data may be employed instead of simple linear regression or probit analysis. However, the latter is especially suitable to the analysis of quanta1 data because it allows the use of 0 and 100% responses in a method of estimation based on maximum likelihood (Finney 1971). In the following it will be assumed that potency estimates (z *) and related statistical quantities have been obtained by a valid analysis of the data.

The Dose—Response Relation in Pharmacology

1. The Dose-Response Relation.- The Dose-Response Relation.- Methods of Plotting Dose-Response Curves.- Drug Antagonism.- Use of Dose-Response Curves.- Enhancement of Drug Action.- References.- 2. Functions and Relations.- Mathematical Symbols and Conventions.- Relations and Functions.- The Linear Function.- Equations in Linear Form: Scatchard and Lineweaver-Burk Plots.- Power Functions.- Exponential Functions: Half-Life.- Logarithms and Logarithmic Functions: The Henderson-Hasselbach Equation.- Rate of Change and Drug Action.- Integration.- References.- Additional Readings.- 3. Kinetics of Drug-Receptor Interaction: Interpreting Dose-Response Data.- Pharmacological Receptor.- Formation of the Drug-Receptor Complex.- Classical Theory.- Modification of Classical Theory.- Dissociation Constants of Competitive Antagonists.- Dissociation Constants of Agonists: Method of Partial Irreversible Blockade.- Dissociation Constants of Agonists: Method of Partial Agonists.- Perturbation Methods.- Allosteric Theory.- Rate Theory.- References.- 4. Construction of Dose-Response Curves: Statistical Considerations.- Mean Dose and Mean Response.- Mean and Standard Deviation.- Samples and Populations.- Distributions.- Normal Distribution.- Estimation.- Tests of Significance.- Linear Regression.- Parallel Lines-Assays and Antagonism.- Quantal Dose-Response Relation.- Probit Diagram.- References.- Additional Readings.- 5. Drug Binding and Drug Effect.- Receptor Interaction and Effect.- Binding Constants and Dissociation Constants.- Desensitization.- Molecularity and Order.- Pharmacokinetic Considerations.- In Vivo Considerations.- Protein Binding.- Receptor Status and Disease States.- References.- 6. Isolated Preparations: Dose-Response Data.- Rabbit Thoracic Aorta.- Guinea Pig Ileum.- Isolated Taenia Ceca.- Ductus Deferens Preparation of the Guinea Pig and Rat.- Rat Fundus Strip.- Phrenic Nerve Diaphragm Preparation of the Rat.- Rat Uterus Preparation.- Frog Rectus Abdominus.- Isolated Rabbit Heart.- References.- Appendix A. Mathematical Tables.- Appendix B. Molecular Weights of Selected Drugs and Composition of Solutions.- Appendix C. Calculus.

The interaction index: a measure of drug synergism.

&NA; Two drugs used in combination may produce enhanced or reduced effects. The degree of enhancement or reduction is measured from the interaction index (&ggr;), a quantity that indicates the changed potency of the combination. The index is therefore a quantitative marker for the drug combination and effect metric used. Methodology for measuring the interaction index utilizes the combination and individual drug dose–effect data suitably modeled by regression techniques that most often produce linear plots of effect on log dose from which isobolar analysis is employed. The isobologram provides a simple and convenient graphical assessment of the interaction index but an independent statistical analysis is needed to assess its precision. In some cases, the relative potency of the constituent drugs is the same at all effect levels. When this is so, it is shown that the interaction index can be measured by either an isobolar or an alternate method that is illustrated here. These calculations demonstrate that these different methods of analysis yield the same value of &ggr;, and do so with comparable precision.

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.

Synergistic antiallodynic effects of spinal morphine with ketorolac and selective COX1- and COX2-inhibitors in nerve-injured rats

The possible role of spinal prostanoids in the tactile allodynia and thermal hyperalgesia associated with an experimental model of neuropathic pain was investigated. Neuropathic pain was induced by tight ligation of the L5 and L6 spinal nerves. Tactile allodynia was assessed 7 days after the surgery by measuring hindpaw withdrawal threshold to probing with von Frey filaments. Thermal hyperalgesia and nociception were determined by the 52 degrees C warm-water tail-flick test and by applying radiant heat to the plantar aspect of the hindpaw ipsilateral to the ligation. Minimal antiallodynic effect was produced by intrathecal (i.th.) administration of ketorolac or morphine up to the highest testable dose (100 microg) or by the (R)- or (S)-enantiomers of ketorolac (up to 6 microg) when administered alone. However, i.th. administration of a fixed ratio (1:1) of morphine plus racemic ketorolac or of morphine plus the (S)-enantiomer of ketorolac (S-ketorolac) produced a dose- and time-related antiallodynic effect: ED50 114 +/- 35.9 microg (total dose) for morphine plus ketorolac and 70.5 +/- 21.0 microg (total dose) for morphine plus S-ketorolac. The combination of i.th. morphine plus the (R)-enantiomer of ketorolac (R-ketorolac) (up to 200 microg total dose) was without effect. Similar antiallodynic activity was obtained for the co-administration of i.th. morphine and intravenous (i.v.) racemic ketorolac. In order to investigate the role of cyclooxygenase (COX) isozymes, relatively selective COX1 (piroxicam) and COX2 N-[2-cyclohexyloxy-4-nitrophenyl] metanesulfonamide (NS-398) inhibitors were administered i.th. (60 microg) alone or together with i.th. morphine. Piroxicam, NS-398, morphine and vehicle (90% DMSO) were without significant antiallodynic effect when administered alone, but moderate antiallodynic effects were produced by i.th. administration of fixed ratio (1:1) combinations of morphine with 60 microg each (highest soluble dose) of piroxicam (%MPE = 40.8 +/- 10.2) or NS-398 (%MPE = 32.4 +/- 9.5). Further, the combined i.th. administration of morphine, piroxicam and NS-398 in fixed 1:1:1 ratio (60 microg each) resulted in a supraadditive antiallodynic effect (%MPE = 70.4 +/- 10.8). Finally, morphine, but not ketorolac, given i.th. produced dose-dependent anti nociception in either the tail-flick or the paw-flick tests. However, there was no synergy between morphine and ketorolac against thermal nociception in either of the tests. These findings suggest that spinal prostanoids produced via both COX1 and COX2 pathways may play a role in neuropathic pain states and suggest the clinical utility of opioid plus COX-inhibitor combination therapy.

Modulation of morphine antinociception by peripheral [Leu5]enkephalin: a synergistic interaction.

This study determined whether the modulation of morphine antinociception by [Leu5]enkephalin, a compound which does not produce detectable antinociception when given i.p., was additive or synergistic. Co-administration of graded i.p. doses of morphine and [Leu5]enkephalin 20 min before testing resulted in a progressive leftward and parallel displacement of the i.p. morphine dose-response line. The increase in potency produced by i.p. [Leu5]enkephalin, but not the antinociception of i.p. morphine alone, was blocked by i.c.v. ICI 174,864. The data demonstrate strong and dose-related synergism between the two compounds, supporting the view that (a) peripheral administration of [Leu5]enkephalin can modulate morphine antinociception, apparently at the level of the brain and (b) that the interaction between mu and delta opioid agonists is not the result of simple additive action at the same (i.e., mu) receptor. Further evidence is thus provided for the existence of functional mu-delta interactions.