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**Figure 23.2.1 Direct Reversible Response**

In general the receptor might be 'mathematically' within the central compartment, a peripheral compartment or a separate effect compartment. In each case the relationship between concentration at the receptor and the response might be described with a **Sigmoid E _{max} model** (Hill equation)

**Equation 23.2.1 Sigmoid E _{max} Response versus Concentration, C_{R} (Hill Equation)**

**Equation 23.2.2 Sigmoid E _{max} Response versus Concentration, C_{R} (Hill Equation) with Baseline Response**

In Equations 23.2.1 and 23.2.2, E_{max} is the maximum response, C_{R, 50%} is the concentration which produces a 50% maximum response (often called EC_{50%}) and γ is a slope factor. For responses which increase from zero, Equation 23.2.1 is more appropriate. In other cases where the response is an increase or decrease from a baseline value Equation 23.2.2 may be a better choice. Another version of this equation, Equation 23.2.1, is the **E _{max} model** but this is just the same except that γ is set to 1.

Equation 23.2.1 can be illustrated in Figure 23.2.2 as a sigmoid curve.

**Figure 23.2.2 Plot of Response versus Concentration (log scale)**

Here E_{max} was 80, C_{R, 50%} was 0.5 and γ was 1.

With data such as response versus concentration we could readily fit the data with a Hill equation model using Boomer, SAAM II or some other non-linear regression program. We could also add appropriate weight to the data and include a second response to a fitted model.

Another approach is to rearrange the Hill equation to produce a **Logarithmic model**, Equation 23.2.3.

**Equation 23.2.3 Linear version of the Hill equation**

Plotting log[Response/(E_{max} - Response)] *versus* log(C_{R}) should produce a straight line graph with a slope of γ and intercept of -log C^{γ}_{R, 50%}.

Another more commonly used Logarithmic model can be derived empirically from the Hill equation if we look at the response *versus* log concentration plot between 20 to 80% maximum response. Notice the straight line portion.

**Figure 23.2.3 Plot of Response versus Concentration with log-linear portion**

This region of the full response concentration curve can be described by Equation 23.2.4.

**Equation 23.2.4 Logarithmic model for the middle part of the curve**

This might be useful if the concentration isn't high enough to estimate E_{max}. It is also interesting to consider concentrations after an IV bolus, one compartment model in this log form, equating plasma concentration and receptor concentration assuming the receptor is close to the plasma or central compartment.

**Equation 23.2.5 log(C) versus time**

and

**Equation 23.2.6 Response versus time**

Note that with this scenario the response decreases linearly with time. One example of this is the results after the last dose of a one week regimen of labetalol (Derendorf and Hochhaus 1995).

With each of these variations we see a direct, immediate relationship between response and concentration. The higher the concentration the higher the response. Note, the receptor may be within the central compartment or a peripheral compartment. It might be in a region where there is little drug amount and thus doesn't show up as a distinct 'pharmacokinetic' compartment. For this situation we might need to include a 'small' effect compartment. We can explore this by looking at the relationship between response and concentration in various compartments.

**Figure 23.2.4 Plot of Cp, Ct and Response versus Time**

One way to explore this aspect of the model, that is, where is the receptor, is to plot response *versus* concentration. In Figure 23.2.5 we plot response *versus* the concentration in the central compartment.

**Figure 23.2.5 Response versus Concentration in the central compartment**

Since there is a significant difference in response between the increasing concentration and decreasing concentration (a counterclockwise hysteresis curve) we might assume that the receptor is not in the central compartment. We could construct the same plot for the concentration in the tissue or peripheral concentration and explore the possibility that the receptor might be linked to this compartment. Wagner et al. (1968) found the response to LSD could be correlated with peripheral drug concentration. A response to digoxin was also correlated with peripheral drug concentrations (Reuning, 1973).

In other case there may not be a correlation between central or peripheral concentrations. Here we might include a hypothetical effect compartment as shown in Figure 23.2.6 (Sheiner et al., 1979).

**Figure 23.2.6 Pharmacokinetic Model with Effect Compartment**

Want to explore this type of model!

Some PK/PD models included in the Adapt model library

- Lalonde, R.L. "Pharmacodynamics" Chapter 5 in
**Applied Pharmacokinetics and Pharmacodynamics: Principle of Therapeutic Drug Monitoring**, 4th ed., ed. Burton, M.E., Shaw, L.M., Schentag, J.J. and Evan,s W.E., LWW, Baltimore, MD, 2006 - Hill, A. V. 1910 The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves.
*J. Physiol. (Lond.)*, 40, iv-vii. - Hill Equation, Wikipedia entry
- Derendorf, H. and Hochhaus, G.
**Handbook of Pharmacokinetic/Pharmacodynamic Correlation**, CRC Press, 1995 page 207 - Sheiner, L.B., et al. Simultaneous modeling of pharmacokinetics and Pharmacodynamics: application to d-tubocurarine,
*Clin. Pharmacol. Therap.*, 25, 358-371 - Reuning, R.H. et al. 1973 Role of pharmacokinetics in drug dosage adjustment I Pharmacologic effect kinetics and apparent volume of distribution of digixon,
*J. Clin. Pharmacol.*, 13, 127-41 - Wagner, J.G. et al. 1968 Correlation of performance test scores with "tissue concentrations" of lysergic acid diethylamide in human subjects,
*Clin Pharmacol. Therap.*, 9, 635-8 - Nagashima, R., O'Reilly, R.A., Levy, G. 1969 Kinetics of pharmacologic response in man: the anti-coagulant action of warfarin,
*Clin. Pharmacol. Therap.*, 10, 22-35 - Jusko, W.J. 1971 Pharmacodynamics of chemotherapeutic effect: dose-time-response relationships for phase-specific agents,
*J. Pharm. Sci.*, 60, 892-5

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