We limit our attention here to several models that attempt to describe adverse effects arising from drug-drug interactions, an increasingly important area in pharmaceutical toxicology. The models involved in these studies cover a wide spectrum of endpoints, including effects on the central nervous system, kidney, cardiovascular, as well as antimicrobial activities. We look briefly at the following distinctive types of pharmacodynamic models: sigmoid Emax, isobolo-graphic, and response surface. In each case we briefly mention the purpose of the study and introduce the modeling approach without going into details of the outcome of the studies. The readers are encouraged to consult the original references for additional information.
Sigmoid Emax Model Jonkers and colleagues  studied the pharmacody-namics of racemic metoprolol, a cardioselective beta-blocker, and the active S-isomer in extensive metabolizers (EMs) and poor metabolizers (PMs). The drug effect studied was the antagonism by metoprolol of terbutaline-induced hypokalemia (abnormally low potassium concentration in the blood). The pharmacodynamic interaction was described by a sigmoidal function for competitive antagonism based on the earlier work of Holford and Sheiner :
where E0 is the potassium concentration in the absence of terbutaline, Emax is the maximum effect of hypokalemia or potassium concentration, Ce is the effect compartment concentration of terbutaline, n is the factor expressing sigmoidicity of the concentration effect relationship, EC50 is the Ce that corresponds to an effect equal to the mean of the sum of E0 and maximum effect, Cmeto is the metoprolol concentration, and IC50 is the metoprolol concentration that corresponds with 50% maximum receptor occupancy.
The sigmoid Emax model formed the basis for a number of subsequent pharmacodynamic analyses of drug-drug interactions. For instance, Mandema et al. , used quantitative electroencephalographic effect measurements to study pharmacodynamic interactions among benzodiazepines in male Wistar rats. In a separate study, but utilizing the same pharmacodynamic endpoint in the same animal, these investigators explored the interactions of antispastic agents, racemic baclofen and its enantiomers, which selectively bind GABAb receptor sites . Other drug interactions analyzed with similar pharmacodynamic modeling include alprazolam and caffeine  for central nervous system (CNS) effects, piperacillin/ciprofloxacin and piperacillin/tazobactam
 for antimicrobial combination activities, tiagabine and midazolam  for their antiepileptic effects, and irbesartan and hydrochlorothiazide  for their renal hypertensive effects.
Isobolographic Model Isobolographic analysis is a method to analyze dose-response curves (isobols) in a binary interaction study where the deviation from the line of additivity demonstrates antagonism or synergism. Levasseur et al. , in their studies on convulsant interactions between pefloxacin and theophylline in rats, developed a new approach for the isobolographic analysis of pharmacodynamic interactions. Their model for these interactions took the form of a quadratic equation for the combination index, CI, a measure indicative of whether the process is governed by additivity or synergism or antagonism:
and a is the interaction parameter. When a is positive, Loewe synergy is indicated; whereas a negative value reflects Loewe antagonism. When a is not significantly different from 0, the drug combination is Loewe additive. R represents the proportion of chemical 1, and C is the dose of drug in combination required to induce maximal seizures in rats. IC is the geometric mean dose of drug, which, when given alone, was required to induce maximal seizures. The subscripts 1 and 2 identify drug 1 (pefloxacin) and drug 2 (theophylline).
More recently Brochot et al.  reported an extension of the isobolo-graphic approach to interaction studies for convulsant interaction among pefloxacin, norfloxacin, and theophylline in rats. Their contribution is unique in that they started out by explaining pharmacodynamic interactions for two drugs, but then extended the approach to derive an isobol for three drug interaction. In addition they included Bayesian analysis and developed a population model with Markov chain Monte Carlo methods.
Response Surface Model A dose-response surface is an extension of dose-response lines (isobols) to three dimensions. In this representation there can be a dose-response surface representing additivity and surfaces above and below suggesting deviation from additivity. Tam et al.  studied the combined pharmacodynamic interactions of two antimicrobial agents, meropenem and tobramycin. Total bacterial density data, expressed as CFU (colony forming units), were modeled using a three-dimensional surface. Effect summation was used as the definition of additivity (null interaction hypothesis) and the pharmacodynamic model was assumed to take the functional form log10CFU/ml = Z
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