The application of drug dose equivalence in the quantitative analysis of receptor occupation and drug combinations

Abstract

In this review we show that the concept of dose equivalence for two drugs, the theoretical basis of the isobologram, has a wider use in the analysis of pharmacological data derived from single and combination drug use. In both its application to drug combination analysis with isoboles and certain other actions, listed below, the determination of doses, or receptor occupancies, that yield equal effects provide useful metrics that can be used to obtain quantitative information on drug actions without postulating any intimate mechanism of action. These other drug actions discussed here include (1) combinations of agonists that produce opposite effects, (2) analysis of inverted U-shaped dose-effect curves of single agents, (3) analysis on the effect scale as an alternative to isoboles and (4) the use of occupation isoboles to examine competitive antagonism in the dual receptor case. New formulas derived to assess the statistical variance for additive combinations are included, and the more detailed mathematical topics are included in the Appendix.

Figure 1

Dose a alone produces a negative effect of magnitude E* which nullifies a dose of drug B that is shown here as b_eq(a).

Figure 2

The broken curve is the isobole for a specified effect. Also shown are paths (a) and (b), not necessarily linear, that represent the relation between the agonist’s fractional occupations x at receptor #1 and y at receptor #2. Path (a) represents the bound agonist when it is the sole agent, while path (b) shows the agonist binding when there is an antagonist with fixed concentration [B] with higher affinity for receptor #2. The high antagonist affinity for receptor #2 means that the agonist binding for receptor #1 denoted by x is increased; hence it shows as path (b) on the right. Each intersection is on the isobole curve and each gives the agonist fractional occupation values of the two receptors that produce the specified effect magnitude for the two cases. These occupation values, in turn, allow a calculation of the agonist concentration for this effect in the blocked and un-blocked cases. These agonist concentrations are denoted by [A′] and [A] respectively in expression (8), (9) and (10). Path (b) is to the right of path (a), as in this illustration, because K_B1 > K_B2 but would be to the left of curve (a) if K_B2 < K_B1. If the experiment yields termination points that are off the contour, there is an interaction between the agonist-occupied receptors as described in the text.

Fig.3

(upper) Inverted U-shaped dose-effect curves (black squares) refer to situations in which the monotone increasing dose-effect curve (shown broken) shows decreasing effects (dark squares) as the dose b is increased above a certain value. (lower) The trend in the monotone increasing curve allows approximate values of the decrease ΔE which, in turn, corresponds to a decrease Δb in the dose of drug B at some value b*. When the decrease Δb is referred to drug B’s dose-effect curve it is seen to occur at an effect level E*, thereby locating the point (b*, E*). This point represents the magnitude of the negative effect component of drug B’s action. This point is shown (for visual convenience) as a function with a positive slope, but it should be noted that its height indicates the magnitude of the negative effect. Points corresponding to other effect drops are similarly analyzed and together these (shown as diamond shape) give the solid curve for this negative component.

Fig. A-1

A competitive antagonist in concentration [B] diminishes agonist binding at both receptors with consequent reduction in effect. The restoration of the effect requires a greater agonist concentration and consequent increase in binding (shown as one of the radial lines) that must terminate on the isobole (smooth curve) to restore the specified effect. Each radial line represents the fractional agonist occupation at receptor #1 (denoted x) and at receptor #2 (denoted y) that accompany the increase in agonist concentration. (See Gaddum equation, text, section 7.) For purposes of illustration we show this x–y relation (as a solid radial line) for the special (and unlikely) case in which the antagonist affinity is the same at each receptor. Two other cases are illustrated. In case (1) the antagonist affinity is greater for receptor #2, whereas in case (2) the antagonist affinity is greater for receptor #1. The arrows indicate for each case how these radial lines would be displaced if [B] were increased. (Although shown as radial lines, the actual x–y binding curve, determined from mass action, could be curvilinear.)