Different methods to estimate ke0

J. Bruhn

Directly after application of an intravenous anaesthetic drug the plasma concentration increases, but a delay until the begin of the clinical response can be observed. This delay is caused by the time, which is needed for the drug to reach the effect site, e.g. for hypnotic drugs to reach the brain. Therefore, in the non-steady state the plasma concentration is not directly correlated with the effect. The same plasma concentration is associated with different responses, depending upon whether the plasma concentrations are increasing or decreasing. Targeting the effect site concentration has been shown to be superior to targeting the plasma concentration in TCI. For calculating the effect site concentration a reliable estimate of the effect site equilibration constant ke0 is needed.

There are 2 possible modelling approaches:

(1) sample the effect site (if possible)

(2) model the effect site as a hypothetical kinetic compartment (the “effect compartment” model), which involves incorporating concentrations at the effect site into a PD model.

ad (1). In contrast to the plasma concentration, the effect site concentration cannot be easily measured. In a laboratory setting, it would be possible to measure simultaneously plasma concentrations and brain concentrations of hypnotic drugs in an animal model. In humans certain neurosurgical procedures like epileptic surgery with excision of brain tissue or new technologies like microdialysis might offer the possibility to sample the effect site.

ad (2). Using pharmacokinetic/pharmacodynamic modelling the effect site equilibration constant ke0 [1/min] can be calculated when the time course of the plasma concentration and the time course of the effect are known. Ke0 describes a first order process caused by different drug concentrations in plasma and in the effect site.

dCe/dt= k_e0*(Cp(t)-Ce(t))                                                                                         (1)

Ce : effect site concentration                                   Cp : plasma concentration

k_e0 : constant of transfer between plasma and effect site.

In the full parametric approach all 3 models (PK, link and PD models) are fully parametrized. For example for propofol, a setting with frequent arterial blood samples and simultaneous use of the EEG to measure drug effect allows to determine the time course of plasma concentrations and the time course of effect. The link model links the time course of the plasma concentration with the model for the concentration effect relationship and ke0 is one of 7 parameters obtained as part of the PK/PD modelling to describe the time course of the effect site concentration for a drug with 3 compartment pharmacokinetic characteristic. [1,2]

In a combined pharmacokinetic-pharmacodynamic study the sequential approach is an alternative to the link approach. After calculating the pharmacokinetic parameters for a compartment model without an effect compartment, ke0 can be directly estimated as the value that collapses the hysteresis loop [3]. This hysteresis loop can be visualized by simply plotting effect (y axis) versus plasma concentration (x axis). Ke0 can mathematically be determined by the least squares method assuming a Emax model, by maximizing the prediction probability P_K for the dose response relationship or by minimizing the AUC of the loop. This approach is also adequate when the time course of concentration is easily obtainable without a PK model, e.g. the endtidal concentrations of volatile anaesthetics, or for a pharmacodynamic study without a pharmacokinetic study part, when the time course of plasma concentration is estimated by using the drug history and a given PK parameter set.

When only the PK model is known (like for the Marsh parameter) without time course of effect, it is not appropriate to simply take the ke0 from a pharmacodynamic study (with another PK model) and apply it naively to the pharmacokinetic study of interest. But, using the t_peak approach from Minto et al. [4], ke0 can be recalculated using the pharmacokinetics of interest to yield the correct time time of peak effect (as obtained in a different pharmacodynamic study).

Often, the drug response of interest (e.g. “hypnosis”) is difficult to quantitatively measure. Therefore, the effect of ultimate interest in a PK/PD trial may be replaced by a surrogate endpoint (e.g. the EEG). The limits of PK/PD approaches include (beneath the development of appropriate models) the validity of surrogate endpoints. A critical look at the processed EEG as a surrogate endpoint to estimate ke0 is needed. The length of epochs to calculate an EEG parameter induces an additional time delay. Data smoothing, which is averaging of data, and (possibly) a slow adapting background parameter as part of the artefact algorithm add to this. This additional time delay should be kept in mind while estimating ke0 using processed EEG.

1. Hull CJ, Van Beem HB, McLeod K, Sibbald A, Watson MJ: A pharmacodynamic model for pancuronium. Br.J.Anaesth. 1978; 50: 1113-1123

2. Sheiner LB, Stanski DR, Vozeh S, Miller RD, Ham J: Simultaneous modeling of pharmacokinetics and pharmacodynamics: application to d-tubocurarine. Clin.Pharmacol.Ther. 1979; 25: 358-371

3. Schnider TW, Minto CF, Shafer SL, Gambus PL, Andresen C, Goodale DB, Youngs EJ: The influence of age on propofol pharmacodynamics. Anesthesiology 1999; 90: 1502-16

4. Minto CF, Schnider TW, Gregg KM, Henthorn TK, Shafer SL: Using the time of maximum effect site concentration to combine pharmacokinetics and pharmacodynamics. Anesthesiology 2003; 99: 324-33.