Pharmacodynamics is the study of the relationship between the concentration of a drug and the response obtained in a patient. Originally, investigators examined the dose–response relationship of drugs in humans but found that the same dose of a drug usually resulted in different concentrations in individuals because of pharmacokinetic differences in clearance and volume of distribution. Examples of quantifiable pharmacodynamic measurements include changes in blood pressure during antihypertensive drug therapy, decreases in heart rate during -blocker treatment, and alterations in prothrombin time or international normalized ratio during warfarin therapy.
For drugs that exhibit a direct and reversible effect, the following diagram describes what occurs at the level of the drug receptor:
According to this scheme, there is a drug receptor located within the target organ or tissue. When a drug molecule "finds" the receptor, it forms a complex that causes the pharmacologic response to occur. The drug and receptor are in dynamic equilibrium with the drug–receptor complex.
The Emax and Sigmoid Emax Models
The mathematical model that comes from the classic drug receptor theory shown previously is known as the Emax model:
where E is the pharmacologic effect elicited by the drug, Emax is the maximum effect the drug can cause, EC50 is the concentration causing one-half the maximum drug effect (Emax/2), and C is the concentration of drug at the receptor site. EC50 can be used as a measure of drug potency (a lower EC50, indicating a more potent drug), whereas Emax reflects the intrinsic efficacy of the drug (a higher Emax, indicating greater efficacy). If pharmacologic effect is plotted against concentration in the Emax equation, a hyperbola results with an asymptote equal to Emax (Fig. 8–14). At a concentration of zero, no measurable effect is present.
The Emax model [E = (Emax x C)/(EC50 + C)] has the shape of a hyperbola with an asymptote equal to Emax. EC50 is the concentration where effect = Emax/2.
When dealing with human studies in which a drug is administered to a patient, and pharmacologic effect is measured, it is very difficult to determine the concentration of the drug at the receptor site. Because of this, serum concentrations (total or unbound) usually are used as the concentration parameter in the Emax equation. Therefore, the values of Emax and EC50 are much different than if the drug were added to an isolated tissue contained in a laboratory beaker.
The result is that a much more empirical approach is used to describe the relationship between concentration and effect in clinical pharmacology studies. After a pharmacodynamic experiment has been conducted, concentration–effect plots are generated. The shape of the concentration–effect curve is used to determine which pharmacodynamic model will be used to describe the data. Because of this, the pharmacodynamic models used in a clinical pharmacology study are deterministic in the same way that the shape of the serum-concentration-versus-time curve determines which pharmacokinetic model is used in clinical pharmacokinetic studies.
Sometimes a hyperbolic function does not describe the concentration–effect relationship at lower concentrations adequately.
When this is the case, the sigmoid Emax equation may be superior to the Emax model:
where n is an exponent that changes the shape of the concentration– effect curve. When n >1, the concentration–effect curve is S- or sigmoid-shaped at lower serum concentrations. When n <1, the concentration–effect curve has a steeper slope at lower concentrations (Fig. 8–15).
The sigmoid Emax model [E = (Emax x Cn)/(ECn50 + Cn)] has an S-shaped curve at lower concentrations. In this example, Emax and EC50 have the same values as in Fig. 8–14.
With both the Emax and sigmoid Emax models, the largest changes in drug effect occur at the lower end of the concentration scale. Small changes in low serum concentrations cause large changes in effect. As serum concentrations become larger, further increases in serum concentration result in smaller changes in effect. Using the Emax model as an example and setting Emax = 100 units and EC50 = 20 mg/L, doubling the serum concentration from 5 to 10 mg/L increases the effect from 20 to 33 units (a 67% increase), whereas doubling the serum concentration from 40 to 80 mg/L only increases the effect from 67 to 80 units (a 19% increase). This is an important concept for clinicians to remember when doses are being titrated in patients.
When serum concentrations obtained during a pharmacodynamic experiment are between 20% and 80% of Emax, the concentration–effect curve may appear to be linear (Fig. 8–16). This occurs often because lower drug concentrations may not be detectable with the analytic technique used to assay serum samples, and higher drug concentrations may be avoided to prevent toxic side effects. The equation used is that of a simple line: E = S x C + I, where E is the drug effect, C is the drug concentration, S is the slope of the line, and I is the y intercept. In this situation, the value of S can be used as a measure of drug potency (the larger the value of S, the more potent the drug). The linear model can be derived from the Emax model. When EC50 is much greater than C, E = (Emax/EC50)C = S x C, where S = Emax/EC50.
The linear model (E = S x C + I ) is often used as a pharmacodynamic model when the measured pharmacologic effect is 20% to 80% of Emax. In this situation, the determination of Emax and EC50 is not possible. To illustrate this, effect measurements from Fig. 8–14 between 20% and 80% of Emax are graphed using the linear pharmacodynamic model.
The linear model allows a nonzero value for effect when the concentration equals zero. This may be a baseline value for the effect that is present without the drug, the result of measurement error when determining effect, or model misspecification. Also, this model does not allow the prediction of a maximum response.
Some investigators have used a log-linear model in pharmacodynamic experiments: E = S x (log C) + I, where the symbols have the same meaning as in the linear model. The advantages of this model are that the concentration scale is compressed on concentration–effect plots for experiments where wide concentration ranges were used, and the concentration values are transformed so that linear regression can be used to compute model parameters. The disadvantages are that the model cannot predict a maximum effect or an effect when the concentration equals zero. With the increased availability of nonlinear regression programs that can compute the parameters of nonlinear functions such as the Emax model easily, use of the log-linear model has been discouraged.53
At times, the effect measured during a pharmacodynamic study has a value before the drug is administered to the patient. In these cases, the drug changes the patient's baseline value. Examples of these types of measurements are heart rate and blood pressure. In addition, a given drug may increase or decrease the baseline value. Two basic techniques are used to incorporate baseline values into pharmacodynamic data. One way incorporates the baseline value into the pharmacodynamic model; the other transforms the effect data to take baseline values into account.
Incorporation of the baseline value into the pharmacodynamic model involves the addition of a new term to the previous equations. E0 is the symbol used to denote the baseline value of the effect that will be measured. The form that these equations takes depends on whether the drug increases or decreases the pharmacodynamic effect. When the drug increases the baseline value, E0 is added to the equations:
When E0 is not known with any better certainty than any other effect measurement, it should be estimated as a model parameter similar to the way that one would estimate the values of Emax, EC50, S, or n.54,55 If the baseline effect is well known and has only a small amount of measurement error, it can be subtracted from the effect determined in the patient during the experiment and not estimated as a model parameter. This approach can lead to better estimates of the remaining model parameters.55 Using the linear model as an example, the equation used would be E – E0 = S x C.
If the drug decreases the baseline value, the drug effect is subtracted from E0 in the pharmacodynamic models:
where Emax represents the maximum reduction in effect caused by the drug, and IC50 is the concentration that produces a 50% inhibition of Emax. These forms of the equations have been called the inhibitory Emax and inhibitory sigmoidal Emax equations, respectively. In this arrangement of the pharmacodynamic model, E0 is a model parameter and can be estimated. If the baseline effect is well known and has little measurement error, the effect in the presence of the drug can be subtracted from the baseline effect and not estimated as a model parameter. Using the inhibitory Emax model as an example, the formula would be E0 – E = (Emax x C)/(IC50 + C).
When using the inhibitory Emax model, a special situation occurs if the baseline effect can be obliterated completely by the drug (e.g., decreased premature ventricular contractions during antiarrhythmic therapy). In this situation, Emax = E0, and the equation simplifies to a rearrangement known as the fractional Emax equation:
This form of the model relates drug concentration to the fraction of the maximum effect.
An alternative approach to the pharmacodynamic modeling of drugs that alter baseline effects is to transform the effect data so that they represent a percentage increase or decrease from the baseline value.55 For drugs that increase the effect, the following transformation equation would be used: percent effectt = [(treatmentt – baseline)/baseline] x 100. For drugs that decrease the effect, the following formula would be applied to the data: percent inhibitiont = [(baseline – treatmentt)/baseline] x 100. The subscript indicates the treatment, effect, or inhibition that occurred at time t during the experiment. If the study included a placebo control phase, baseline measurements made at the same time as treatment measurements (i.e., heart rate determined 2 hours after placebo and 2 hours after drug treatment) could be used in the appropriate transformation equation.55 The appropriate model (excluding E0) then would be used.
Concentration–effect curves do not always follow the same pattern when serum concentrations increase as they do when serum concentrations decrease. In this situation, the concentration–effect curves form a loop that is known as hysteresis. With some drugs, the effect is greater when serum concentrations are increasing, whereas with other drugs, the effect is greater while serum concentrations are decreasing (Fig. 8–17). When individual concentration–effect pairs are joined in time sequence, this results in clockwise and counterclockwise hysteresis loops.
Hysteresis occurs when effect measurements are different at the same concentration. This is commonly seen after short-term IV infusions or extravascular doses where concentrations increase and subsequently decrease. Counterclockwise hysteresis loops are found when concentration–effect points are joined as time increases (shown by arrows) and effect is larger at the same concentration but at a later time. Clockwise hysteresis loops are similar, but the concentration–effect points are joined in clockwise order, and the effect is smaller at a later time.
Clockwise hysteresis loops usually are caused by the development of tolerance to the drug. In this situation, the longer the patient is exposed to the drug, the smaller is the pharmacologic effect for a given concentration. Therefore, after an extravascular or short-term infusion dose of the drug, the effect is smaller when serum concentrations are decreasing compared with the time when serum concentrations are increasing during the infusion or absorption phase.
Accumulation of a drug metabolite that acts as an antagonist also can cause clockwise hysteresis.
Counterclockwise hysteresis loops can be caused by the accumulation of an active metabolite, sensitization to the drug, or delay in time in equilibration between serum concentration and concentration of drug at the site of action. Combined pharmacokinetic/pharmacodynamic models have been devised that allow equilibration lag times to be taken into account.