Problem Description
Four basic models for characterizing indirect pharmacodynamic responses after drug administration are based on drug effects (inhibition or stimulation) on the factors controlling either the input or the dissipation of drug response.
The assumptions, equations, and data patterns for the four basic indirect response models provide a starting point for evaluation of pharmacologic effects where the site of action precedes or follows the measured response variable.
Differential Equation Model
When the response to a drug is not directly related to the drug’s concentration (or to the concentration in an effect compartment), indirect response models have been proposed.
The response R is assumed to be produced by a zero-order synthesis rate and degraded via a first-order process:
\[{d R \over d t} = k_{in} \: - \: k_{out} R\]
The baseline response is \(R_0 \: = \: {k_{in} \over k_{out}}\)
The drug can act on \(k_{in}\) or \(k_{out}\) by inhibition or stimulation, leading to 4 cases\(^1\). \(C^{\gamma}\) represents the plasma concentration of the drug as a function of time (determined by a pharmacokinetic model).
A sample model is constructed for the case of a bolus administration of a drug and one compartment. The PK part of the model can easily be modified to represent more complex pharmacokinetics.
The resulting dynamics of the response for 3 different doses for each model is depicted below:
Comparison of Response Function Model to Indirect PD Response Models
Response functions are built to fit data resulting from indirect PD response models with inhibition and stimulation of \(k_{in}\).
Curves from the nonlinear response model are compared to the data points from the inhbition model in the following plot.
Curves from the nonlinear response model are compared to the data points from the stimulation model in the following plot.
The plots show that the nonlinear response function provides accurate fits to data derived from both the inhibition and stimulation indirect PD response models.
The nonlinear response function model requires no additional fitting of parameters beyond those which are contained in the response function.
Illustration of Ambiguity in the Indirect PD Response Models
Response functions are built to illustrate the ambiguities that can arise when using the indirect PD inhibition and stimulation models. This is done using two separate test cases defined below:
Case 1: An indirect PD model with nonlinear Michaelis-Menten PK and inhibition of \(k_{in}\) is compared to an indirect PD model with the same PK and stimulation of \(k_{out}\). Curves showing results from the two models are shown here.
Case 2: An indirect PD model with linear PK and stimulation of \(k_{in}\) is compared to an indirect PD model with the same PK and inhibition of \(k_{out}\). Curves showing results from the two models are shown here.
Model ambiguity can lead to difficulty in mapping system and input properties to model parameters, thus hindering the predictive capability of the model.