Quantifying Tolerance as Potency or Efficacy Change

Why Quantify Tolerance From Concentration–Response Curves?

A smaller response after repeated or sustained treatment is not, by itself, evidence that the drug has become less potent. The response may have fallen because the concentration at the effect site changed, because the response system adapted, or because the drug’s apparent potency or maximal effect changed. Once exposure has been appropriately accounted for, dose-ranging observations can help describe the pharmacodynamic change: an increase in the concentration needed for a given effect, a reduction in the maximum effect, or both.

The distinction is useful because two patients—or two times in the same patient—can show the same response at one concentration for different parameter reasons. A single concentration–response observation generally cannot tell those explanations apart. A range of concentrations, especially one that includes both low concentrations and concentrations near saturation, provides more information.

A Time-Dependent Concentration–Response Model

Let \(C(t)\) be the relevant drug concentration at time \(t\), and let \(\Delta R(t)\) be the drug-related response after subtracting the appropriate baseline. For a stimulatory effect with a fixed Hill coefficient \(n\), a standard model is:

Here \(E_{\max}(t)>0\) is the maximum effect magnitude, \(EC_{50}(t)>0\) is the half-maximum concentration, and \(n>0\) controls the steepness of the curve. If \(n=1\), the model is the simple hyperbolic Emax model. For an inhibitory outcome, the same curve can describe the magnitude of inhibition; the direction and units of the measured response must be stated explicitly.

A convenient reference-time parameterization is \(EC_{50}(t)=EC_{50,0}\,q(t)\) and \(E_{\max}(t)=E_{\max,0}\,r(t)\). Here \(q(t)\) is the potency-shift factor and \(r(t)\) is the efficacy fraction. Under tolerance represented as a rightward shift or a reduction in maximum effect, \(q(t)\geq1\) and \(0<r(t)\leq1\). Potency-only tolerance sets \(r(t)=1\); efficacy-only tolerance sets \(q(t)=1\); a combined model allows both to change. These constraints describe the specific tolerance hypotheses being compared, not all possible forms of adaptation.

The model assumes that the concentration used is an adequate measure of the exposure driving the response; that the response definition and baseline are comparable over time; and that, within each time-specific curve, the chosen concentration–response function is appropriate. To interpret curve differences as pharmacodynamic changes, concentration differences must not be mistaken for response-system changes. In particular, a declining concentration at a fixed dose can lower the response without any change in \(EC_{50}\) or \(E_{\max}\).

Why Dose Ranging Separates the Parameters

The two parameters have different signatures across the concentration range. At very low concentrations, where \(C\ll EC_{50}\), the model is approximately:

Thus, low-concentration observations mainly inform a combination of \(E_{\max}\) and \(EC_{50}\), rather than cleanly identifying each one. At concentrations far above \(EC_{50}\), the response approaches \(E_{\max}(t)\). Near the half-maximum point, the response is \(E_{\max}(t)/2\), and the corresponding concentration is \(EC_{50}(t)\).

This gives a practical separation: a well-observed plateau helps estimate \(E_{\max}\), while the concentration at half that plateau helps estimate \(EC_{50}\). If the observed concentrations never approach a plateau, a change in the curve’s slope or level over the sampled range can often be explained by multiple combinations of the two parameters. That is a limitation of the information in the data, not a reason to label one parameter as changed based on a single response.

Fitting and Comparing Tolerance Descriptions

For repeated concentration–response measurements, fit the same response model to each time point or fit one joint model in which \(EC_{50}(t)\) and \(E_{\max}(t)\) are allowed to vary. A joint fit can compare three useful descriptions: potency-only, with changing \(EC_{50}\) and shared \(E_{\max}\); efficacy-only, with changing \(E_{\max}\) and shared \(EC_{50}\); and combined, with both parameters changing. The Hill coefficient may be held fixed when data are limited, or estimated if the concentration range and sample size support it. Changing several curve parameters at once can make estimates unstable.

Compare the candidate models using their fit to the data and uncertainty in the parameter estimates, not only the visual impression of a curve. Inspect residuals—the differences between observed and predicted responses—across both concentration and time. A consistent pattern, such as underprediction at low concentrations and overprediction at high concentrations, suggests that the chosen curve or its time structure may be inadequate. Wide uncertainty in both \(EC_{50}\) and \(E_{\max}\), especially when their estimates trade off against one another, indicates that the dose range may not distinguish them well.

A time-varying shift can also be described directly. For example, \(EC_{50}(t)=EC_{50,0}q(t)\) summarizes the shift relative to a reference time, while \(E_{\max}(t)=E_{\max,0}r(t)\) summarizes the relative plateau. These expressions quantify a curve change; they do not specify the adaptation dynamics that generated it. When exposure and response vary over time, use the appropriate exposure and response measurements at each time rather than interpreting a dose label as if it were a concentration.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to plot observed and predicted responses against concentration separately by time, while also displaying the observations themselves. Check whether early and late curves cover overlapping concentration ranges; if they do not, an apparent shift may be driven by extrapolation. Check that high concentrations adequately inform the plateau and that observations bracket the half-maximum region. Examine residuals for systematic concentration-dependent or time-dependent patterns, and check whether the estimated parameters are implausibly extreme or poorly determined.

Alternative explanations should be considered before assigning a pharmacodynamic parameter change. A concentration–response curve based on dose rather than measured exposure may confound pharmacokinetic tolerance with pharmacodynamic tolerance. A time-varying baseline can make uncorrected responses look smaller even when the drug-related effect is unchanged. A delayed or indirect response can also mean that the current response is related to prior exposure rather than the current concentration. In those settings, a time-specific Emax curve is a compact empirical description, not necessarily a complete dynamic model.

The main advantage of potency and efficacy parameterization is interpretability: \(EC_{50}\) describes the concentration scale and \(E_{\max}\) describes the modeled ceiling. Its limitations are equally important. A finite dose range may not reveal a true plateau; measurement noise can make the apparent half-maximum uncertain; and a fitted shift does not prove receptor desensitization, internalization, or any other mechanism. Curve comparisons quantify how response changed under stated assumptions. Mechanistic interpretation requires additional evidence.