Comments on Using the Time of Maximum Effect Site Concentration to Combine Pharmacokinetics and Pharmacodynamics: In Reply

On behalf of the our coauthors of the article “Using the time of maximum effect site concentration to combine pharmacokinetics and pharmacodynamics,”1we welcome the comments and observations in the letter by Drs. Nigrovic and Amann and the letter by Dr. Van Meurs, Mr. Nikkelen, and Dr. Good. The letter by Drs. Nigrovic and Amann raises several points, which are addressed in the order presented:

1. Polyexponential equations are, of course, completely interchangeable with compartmental models mathematically. Although Sheiner et al.  2postulated that the effect site was “negligibly small,” both the Sheiner model and our model readily permit drug to flow into the effect site simply by assigning an appreciable volume to the effect site. Of course, the amount is limited by the amount of drug in the plasma.

2. Our mathematical development of the effect site has some different nuances than that of Sheiner et al. , 2but the definition of the effect site is identical:

   

   Sheiner et al.  used the term k^eo, with o  meaning “outside.” We have always believed that was a poor choice because it is anglocentric. Compartment 0 is uniformly recognized in pharmacokinetic models as the external compartment. Hence is our preference for “k^eo.”

3. t^peakis the time of maximum effect site concentration after a submaximal intravenous bolus dose when there is no drug initially in the system. By definition, the time of maximum effect is the same as the time of maximum effect site concentration for direct-acting monotonic pharmacodynamic models (as opposed to indirect effect models, or biphasic effect models, where none of this applies). Therefore, t^peakcan be directly observed by observing the time of peak effect. Because no model is required to simply look at the raw data and identify the time of peak effect, this suggests that t^peakis model independent. We do not share the concern expressed by Drs. Nigrovic and Amann as to whether it is the dose or effect that is “submaximal” or “maximal.” We presume that Drs. Nigrovic and Amann would consider k^eoa pharmacodynamic parameter. Because t^peakis directly calculable from k^eoand vice versa , we do not understand their issue with labeling t^peaka pharmacodynamic parameter. Of course, the observed time of peak effect is limited by the resolution of the observations. Perhaps that is their point.

4. Drs. Nigrovic and Amann have correctly identified the reason that we focus on submaximal effect.

5. We interpret the question as saying that the “peak” effect should not be the last point observed in the study because it is not possible to know whether the effect has truly peaked if observations stop at or before the time of peak effect. We agree, which is why the period of observation extended well beyond the time of peak effect in all of the studies cited in our article.

6. We agree that the curves overlap because t^peakvalues predicted by the integrated pharmacokinetic–pharmacodynamic models were identical. In fact, that is exactly the point. As Drs. Nigrovic and Amann observe, the pharmacokinetic models are very different, and so the fact that the calculated t^peakvalues are nearly identical provides validation for the concept that t^peakis model independent. (As noted by Dr. Van Meurs, Mr. Nikkelen, and Dr. Good, this result could be serendipity, and so even though it provides validation, it is certainly not a proof). We agree that it is mathematically and conceptually incorrect to substitute k^eofrom one study to another, but to the best of our knowledge, this is the current standard in every target-controlled anesthetic drug delivery system except for STANPUMP*and RUGLOOP.†Drs. Nigrovic and Amann observe that the magnitude of the time course of effect site concentrations predicted by Shanks et al.  3is modestly different from that predicted by Stanski and Maitre. 4k^eo(and t^peak) only relates to the time delay between plasma and effect site and would not be expected to account for differences in the magnitude of the six parameter pharmacokinetic models. The purpose of the t^peakapproach is to provide a means of building an integrated pharmacokinetic–pharmacodynamic model from a pharmacokinetic analysis that did not concurrently include estimation of k^eo(which is the majority of pharmacokinetic analyses).

7. The pharmacodynamics of remifentanil are fully described by Minto et al . 5(reference 9 of the article).

8. Simulated time courses of drug effect can be readily calculated from the simulated effect site concentrations using standard pharmacodynamic models.

In response to the letter from Dr. Van Meurs, Mr. Nikkelen, and Dr. Good, we acknowledge their excellent manuscript on k^eoand t^peakand appreciate their consistency with the definition of t^peakintroduced by Shafer and Gregg 6(reference 5 of their manuscript). Although we share many of the same conclusions, in particular, their observation on page 586 that “this result justifies the use of t^peakas a parameter to base a k^eoestimate on.” What particularly distinguishes their article is the thoughtful and novel sensitivity analysis. Their article describes development of their own software for this purpose. STANPUMP, which has been available over the Internet since 1989, provides a mean for nonprogrammers to perform the same simulations.

We do not know how the authors responded to their reviewer, but the reviewer is not quite correct. Assuming that effect has been continuously sampled (getting around the issue of discrete sample times that concerned Drs. Nigrovic and Amann), there is a one-to-one relation between k^eoand t^peak. When analyzing a population of patients, variability in k^eocan be determined directly from variability in t^peak. What the reviewer may have intended was that when analyzing an individual patient, the uncertainty (e.g. , SE) about the estimate of t^peakmay be greater than the uncertainty about an estimate of k^eo, because t^peakis a single observation that may have considerable variability, whereas an estimate of k^eois based on analysis of all points in the hysteresis curve and thus may be known with greater certainty. If one has access to multiple concentrations and measures of drug effect, the standard approach should be used to estimate k^eo. We would never advocate using t^peakto estimate k^eowhen one has access to a full set of concentration and effect measures. Nevertheless, comparing the calculated t^peakfrom the resulting model with the observed time of peak effect provides a quick verification that the modeling was done correctly.

We certainly agree that the fact that different k^eos, combined with different pharmacokinetic parameter sets, demonstrated nearly identical values for t^peak(figs. 1 and 2) in our article could be serendipitous. The purpose of our article was to present a theoretical concept and show some examples of the concept. It was not known before our analysis that the t^peakwould be the same for figures 1 and 2. It is not clear how a sensitivity analysis could address the question of serendipity.

Again, on behalf of our coauthors, we appreciate the thoughtful letter of Drs. Nigrovic and Amann and that of Dr. Van Meurs, Mr. Nikkelen, and Dr. Good and hope that this discussion on a moderately arcane topic is of interest to the readers of Anesthesiology.

Palo Alto Veterans Affairs Health Care System, Palo Alto, California; Stanford University School of Medicine, Stanford, California; and University of California at San Francisco, San Francisco, California. steven.shafer@stanford.edu