Figure 8 Concentration-response curves for inhibitin of the COX-1 (untreated) and COX-2 (treated with lipopolysaccharide) activity in human eye homogenate (n = 4-6). (Adapted from Ref. 112.)

Figure 8 Concentration-response curves for inhibitin of the COX-1 (untreated) and COX-2 (treated with lipopolysaccharide) activity in human eye homogenate (n = 4-6). (Adapted from Ref. 112.)

molecule binds to a single receptor. Maurice and Mishima (85) further reported that Eq. (7) also correctly characterized the miosis of carbachol on the sphincter muscle of the cat, mydriasis of isoproterenol on bovine and rabbit sphincter muscle, mydriasis of 1-epinephrine and atropine on guinea pig iris, and mydriasis of epinephrine and acetylcholine on cat iris.

Gabrielsson and Weiner (113) analyzed the miotic response in the cat eye over time after application of three different doses of latanoprost. A simple one-compartment model was assumed with a first-order input into the eye with the biophase (site of the miotic response) associated with the central compartment. The concentration in the biophase was assumed to act directly on the response according to the sigmoid Emax model equation:

E Cn

In Eq. (8), E0 is the baseline value for the pupil size, Emax is the maximum effect, C is the concentration of drug in the biophase estimated from a kinetic model equation, EC50 is the concentration in the biophase at halfmaximal effect, and n is the sigmoidicity factor.

Smolen and coworkers (114-118) developed a mathematical model for which pharmacological response intensities were transformed into biophasic drug levels for tropicamide, tridihexethylchloride, carbachol, and pilocar-pine. The transformation was accomplished with the use of a precisely determined dose-response curve. The method requires a graded response that can be measured over time and expressed according to:

where I is the intensity of response at any time and QB is the concentration of drug in the biophase at the same time of measurement (normalized for dose: QB = QB/D) and K1, K2, and P are constants derived from the fitting procedure.

Figure 9 shows the kinetics of the expected biophasic concentration of a drug after fitting miosis vs. time data to a kinetic model. After calculating the expected biophasic drug concentrations to Eq. (8) or (9), the response

Figure 9 Observed (filled circles) and predicted (solid line) kinetics of a drug in the effect compartment following topical administration of three different doses. (Adapted from Ref. 113.)

intensities (I) or effect (E) can be fit to the concentration of drug estimated to be in the biophase. Figure 9 represents a fit to Eq. (8) for S( + )flurbipofen inhibition of prostaglandin E2 to the COX-1 and COX-2 isoenzymes (113).

Use of Eq. (9) requires that the following assumptions are valid: (1) the same biophasic concentration of drug produces the same intensity of response (i.e., nonhysteresis), (2) binding to the receptor site is rapid and reversible, and (3) the pharmacokinetic processes are first order and therefore do not differ with dose. Application of this approach permits the optimization of a delivery system for an ophthalmic drug, insight into a drug's kinetics of response, and determination of a product's "biophasic bioavailability." The later term refers to the drug's absorption and disposition to the ocular site of action (114).

A mydriatic tolerance of the pupil response has been interpreted for phenylephrine from the application of the classic pharmacodynamic Emax model (119):

where K'm is the drug concentration in aqueous humor required to produce one half of the maximal mydriatic response of phenylephrine (1/2 Emax), E(t) is the change in the mydriatic response from baseline at any time, and Ca(t) is the concentration of drug in aqueous humor at the same time of measurement of E. In Eq. (10), Km is linearly related to Ca(t) for a drug that does not develop tolerance, but if mydriatic tolerance is developed following topical instillation, the value of K'm will vary with time. Chien and Schoenwald (119) measured the aqueous humor concentration of phenylephrine and its corresponding mydriatic response over time in the rabbit eye following a 10 uL topical instillation of phenylephrine HCl viscous solution (10%). A clockwise hysteresis effect of mydriasis vs. Ca(t) is shown in Figure 10, which illustrates the development of tolerance over time. At the 240-minute time interval, the phenylephrine was twofold higher than that measured at the 20-minute time interval. However, at these two time intervals, the mydriatic response was similar. K'm was linear up to 90 minutes, but steadily increased beyond 90 minutes through 240 minutes postinstillation.

Nonlinear pharmacokinetic behavior may be more common than has been observed since many ophthalmic drugs are known to alter physiological processes. Drugs that affect aqueous humor turnover or blood flow within the iris/ciliary body would be expected to show some degree of nonlinear pharmacokinetic behavior. Nevertheless, the inability to routinely measure concentrations of drug in the human eye provides a strong incentive to continue exploring the use of pharmacological response intensities to

Figure 10 Average mydriasis measurement vs. aqueous humor concentrations of phenylephrine (clockwise hysteresis loop) following a 10 mL topical instillation of 10% phenylephrine HCl viscous solution. (Adapted from Ref. 119.)

either optimize therapy or povide a screning tool for use in developing new ophthalmic agents.

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