The Emax model is a nonlinear model frequently used in dose-response analyses. The model is shown in Eq. (9.1)
where i = The patient indicator
Ri = The value of the response for patient i
Di = The level of the drug for patient i, the concentration may also be used in many settings E0 = The basal effect, corresponding to the response when the dose of the drug is zero Emax = The maximum effect attributable to the drug ED50 = The dose, which produces half of Emax
N = The slope factor (Hill factor), measures sensitivity of the response to the dose range of the drug, determining the steepness of the dose-response curve (N > 0) ei = The random error term for patient i. A standard assumption, adopted here, is that the ei terms are independent identically distributed with a mean of 0 and variance a2. An additional standard assumption is that the error terms are normally distributed.
The Emax model dose-response curve shown in Eq. (9.2) is the expected value of the Emax model.
The Emax model dose-response curve can be either increasing or decreasing relative to an increase in dose. If the response is decreasing, the value of the Emax parameter will be negative. Figure 9.1 illustrates the Emax model dose-response curve where the response increases with increasing dose. Note the difference between Emax, the maximal effect attributable to the drug, and (E0 + Emax), the asymptotic response with increasing dose. This difference is particularly relevant when the drug of interest is being administered along with concurrent therapy, as the response at maximum dose includes both the maximal effect of the drug (Emax) plus the effect of the concurrent therapy (E0).
Dose
Figure 9.1. Emax Model dose-response curve. Salient features of the model are:
Dose
Figure 9.1. Emax Model dose-response curve. Salient features of the model are:
• The Emax model has four parameters: E0, Emax, ED50, and N.
• The Emax model predicts the maximum effect a drug can have (Emax).
• The Emax model predicts a zero-dose effect (E0) when no drug is present.
• The Emax model follows the "law of diminishing returns" at higher doses.
• The Emax model parameters are readily interpretable.
A particular case of the Emax model is given by Eq. (9.3):
Here the slope factor, N, is not included in the model and as such has an implicit value of one. The Emax model without the slope factor parameter is sometimes referred to as the hyperbolic Emax model, while the model including the slope factor is referred to as the sigmoidal Emax model. Although both the hyperbolic and sigmoidal Emax models can be justified on the relationship of drug-receptor interactions (Boroujerdi, 2002), they are primarily used for empirical reasons.
Another common modification of the Emax model is to not include the E0 parameter if the response at the no drug level is known to be zero.
The Emax model is a common descriptor of dose-response relationships. However, as in all modeling, there are situations where it may not be appropriate to apply. These scenarios include:
• Where the dose-response relationship is not monotonic. The Emax model assumes a monotonic dose response.
• Where the number of different doses for the model is small. As the Emax is a four-parameter model, it desirable to have at least five different doses across the effective dose-response range.
• When the response should not be modeled as a continuous outcome.
Dose estimates other than ED50 are often of interest in dose-response analysis. For example ED90, the dose which produces 90% of Emax, is sometimes used as an estimate of the maximal effective dose (MaxED). From the Emax model, the ED90 is estimated by the formula in Eq. (9.4) where p (0 < p < 1) is the percentile of interest. For example if the slope factor N is 0.5, the estimate of ED90 is (EDsox 81).
9.2 Sensitivity of the Emax Model Parameters
This section reviews the sensitivity of the Emax model dose-response curve to changes in its four parameters: E0, Emax, ED50, and N. In addition, study design criteria based on these parameters will be reviewed.
As described in Section 9.1, the E0 parameter is the response when there is no drug present, and the Emax parameter is the maximum attributable drug effect. These two parameters define the upper and lower asymptotic values for the dose-response curve. Changes in the E0 parameter affect the "starting value" of the dose-response curve (i.e., when there is no drug present). Changes in the Emax parameter affect the range of the dose-response curve.
Figure 9.2 illustrates the change in response as the E0 and Emax parameters vary. In Figure 9.2, the dotted line illustrates the change in the Emax model curve relative to the solid line when the E0 and Emax parameters are increased from 0 to 10 and 100 to 110, respectively.
1201
100-
Dotted line:
Solid line:
0.01
10 100 1000
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Figure 9.2. Emax Model dose-response curves with differing E0 and emax values
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