Low- and High-Concentration Limits of Emax Curves

Why the Limits Matter

An Emax curve is nonlinear over its full concentration range, but its behavior at the extremes has simpler forms. At sufficiently low concentration, the drug effect is small and can be approximated by a power of concentration; for the ordinary Emax model, that power is one, so the curve is approximately linear. At sufficiently high concentration, the effect approaches its maximum and can be approximated by a plateau. These limits help interpret local behavior, but each approximation is reliable only when its concentration condition is met.

The key is to measure concentration relative to $EC_{50}$, and, for a sigmoid Emax model, to account for the Hill coefficient. A concentration that seems numerically small or large in isolation is not necessarily in the corresponding limit: its size relative to $EC_{50}$ matters.

Model and Assumptions

Definition: For a direct stimulatory sigmoid Emax model, let $C\geq 0$ be concentration, $E_0$ the response at zero concentration, $E_{\max}>0$ the maximum drug-related increase above $E_0$, $EC_{50}>0$ the concentration producing half that increase, and $\gamma>0$ the Hill coefficient. The model is $$ E(C)=E_0+E_{\max}\frac{C^\gamma}{EC_{50}^\gamma+C^\gamma}. $$ The ordinary Emax model is the special case $\gamma=1$. Define the dimensionless ratio $x=(C/EC_{50})^\gamma$. Then the drug-related response is $D(C)=E(C)-E_0=E_{\max}x/(1+x)$.

The limits below assume the model applies over the concentrations being considered: the effect is direct and described by this concentration-response relationship, $E_0$ and $E_{\max}$ are constant, and the response has no additional concentration-dependent component. For the high-concentration limit, the model also assumes that increasing concentration can approach the stated plateau. These are assumptions about the model, not conclusions established by observing a few concentrations.

If $\gamma\neq1$, the low-concentration approximation is a power law in $C$, not a straight line in $C$. Thus “approximately linear at low concentration” strictly describes the ordinary Emax model, or a plot against $C^\gamma$ for a fixed sigmoid model. The high-concentration plateau applies for any positive $\gamma$.

Low-Concentration Expansion

When $C$ is small relative to $EC_{50}$, $x=(C/EC_{50})^\gamma$ is small. For $x\geq0$, the identity $\frac{x}{1+x}=x-x^2+x^3-\cdots$ holds as a convergent geometric series when $x<1$. Keeping only the first term gives the low-concentration approximation.

D(C)=E_{\max}\frac{x}{1+x} =E_{\max}(x-x^2+x^3-\cdots) \approx E_{\max}x =E_{\max}\left(\frac{C}{EC_{50}}\right)^\gamma.

For $\gamma=1$, this becomes $D(C)\approx (E_{\max}/EC_{50})C$: the initial concentration-effect slope is $E_{\max}/EC_{50}$, in response units per concentration unit. For general $\gamma$, the leading term is proportional to $C^\gamma$. If $\gamma>1$, the effect rises more slowly than a straight line as concentration approaches zero; if $0<\gamma<1$, it rises more sharply near zero. Neither case changes the exact model into a linear-in-$C$ model.

The approximation error can be derived without relying on the series. Subtract the exact effect from the leading approximation:

E_{\max}x-D(C) =E_{\max}x-E_{\max}\frac{x}{1+x} =E_{\max}\frac{x^2}{1+x}.

Thus the leading approximation overpredicts the exact drug effect for every $x>0$. Relative to the exact drug effect, its error is especially simple:

\frac{E_{\max}x-D(C)}{D(C)} =\frac{E_{\max}x^2/(1+x)}{E_{\max}x/(1+x)} =x.

So if “relative error” means error divided by the true drug-related increment $D(C)$, the low approximation's relative error is exactly $x$, not merely approximately $x$. For example, $x=0.10$ means a 10% overprediction of the true increment. This is not the same as the percentage error relative to total response $E(C)$, which also depends on baseline $E_0$.

Worked Example: Ordinary Emax at Low Concentration

Suppose a hypothetical drug follows an ordinary Emax model with $E_0=10$ response units, $E_{\max}=60$ response units, and $EC_{50}=4$ mg/L. At $C=0.4$ mg/L, calculate the exact effect and the low-concentration approximation.

Here $\gamma=1$, so $x=C/EC_{50}=0.4/4=0.10$. The exact drug-related increment is $D=60(0.10)/(1+0.10)=6/1.10=5.455$ response units. The exact total response is $E=10+5.455=15.455$ response units.

The low approximation gives $D\approx E_{\max}x=60(0.10)=6.000$ response units, or total response $10+6=16.000$ response units. Its absolute error in the increment is $6.000-5.455=0.545$ response units. Relative to the exact increment, this is $0.545/5.455=0.100$, or 10%, matching the exact error formula $x=0.10$. The approximation is useful if a 10% error in the drug increment is acceptable, but it is not exact.

Worked Example: Low-Concentration Sigmoid Emax

Consider a hypothetical sigmoid Emax curve with $E_0=5$ response units, $E_{\max}=80$ response units, $EC_{50}=3$ mg/L, and $\gamma=2$. At $C=1.5$ mg/L, find the exact and approximate increments and explain why the approximation is not linear in concentration.

The dimensionless ratio is $x=(1.5/3)^2=(0.5)^2=0.25$. The exact increment is $D=80(0.25)/(1.25)=20/1.25=16.00$ response units, giving total response $E=5+16=21.00$ response units.

The low-concentration approximation is $D\approx80(0.25)=20.00$ response units. It overpredicts by $20-16=4.00$ response units, or $4/16=0.25$, a 25% relative error in the increment. Because $\gamma=2$, the leading approximation is $E_{\max}(C/EC_{50})^2=(80/9)C^2$, when $C$ is expressed in mg/L. It is quadratic, not linear, in concentration.

High-Concentration Expansion

At high concentration, $x$ is large. Rewrite the response fraction as $x/(1+x)=1-1/(1+x)$. For $x>1$, the geometric expansion of $1/(1+x)$ in powers of $1/x$ gives the approach to the plateau:

\frac{x}{1+x} =1-\frac{1}{x}+\frac{1}{x^2}-\frac{1}{x^3}+\cdots, \qquad E(C)\approx E_0+E_{\max}.

The exact shortfall from the plateau is

(E_0+E_{\max})-E(C) =E_{\max}\left(1-\frac{x}{1+x}\right) =\frac{E_{\max}}{1+x}.

This shortfall divided by the maximum drug-related increment $E_{\max}$ is $1/(1+x)$. If instead the plateau approximation error is divided by the exact drug increment $D(C)$, the relative error is $1/x$. These are two valid but distinct denominators: the first asks what fraction of the maximum increment remains unachieved; the second asks how large the plateau approximation's overstatement is relative to the effect actually achieved.

In terms of concentration, the leading high-concentration shortfall is approximately $E_{\max}(EC_{50}/C)^\gamma$, since $1/(1+x)\approx1/x$ for large $x$. The plateau approximation is always at or above the exact response for a positive stimulatory $E_{\max}$, and the gap decreases toward zero as concentration increases. A large concentration by itself is not enough to guarantee a small gap: $C/EC_{50}$, raised to $\gamma$, determines the relevant scale.

Worked Example: Quantifying the Plateau Shortfall

Suppose a hypothetical ordinary Emax model has $E_0=8$ response units, $E_{\max}=50$ response units, and $EC_{50}=2$ mg/L. At $C=18$ mg/L, compare the exact response with the plateau.

Here $x=18/2=9$. The exact increment is $D=50(9)/(1+9)=45.00$ response units, so the exact total response is $E=8+45=53.00$ response units. The plateau response is $E_0+E_{\max}=8+50=58.00$ response units.

The plateau overstates the exact response by $58-53=5.00$ response units, equal to $E_{\max}/(1+x)=50/10=5.00$. This gap is 10% of the maximum increment ($5/50=0.10$); it is $5/45=0.1111$, or 11.11%, of the exact drug increment. Therefore the response is near, but not identical to, the plateau. Which percentage is relevant depends on whether the question concerns the remaining fraction of the maximum effect or the error relative to the effect achieved.

Choosing a Tolerance and Checking the Approximation

A practical decision should begin by stating the error definition and acceptable tolerance. If the low approximation is judged by its relative error against the exact drug increment, requiring error no greater than $\epsilon$ means $x\leq\epsilon$. For example, a 10% tolerance requires $x\leq0.10$, or $C/EC_{50}\leq0.10^{1/\gamma}$.

For the plateau approximation, if the criterion is that no more than fraction $\epsilon$ of the maximum drug increment remains unachieved, require $1/(1+x)\leq\epsilon$. Solving gives $x\geq1/\epsilon-1$. If instead tolerance is measured relative to the exact increment, require $1/x\leq\epsilon$, or $x\geq1/\epsilon$. These conditions are similar when $\epsilon$ is small, but they are not identical. Stating the denominator avoids ambiguity.

Compute the dimensionless concentration.
Calculate $x=(C/EC_{50})^\gamma$, using concentration and $EC_{50}$ in the same units.

Select the candidate limit.
A small $x$ supports the low-concentration approximation; a large $x$ supports the plateau approximation. Values around one are not an extreme and generally require the full Emax expression.

Calculate the error on the relevant scale.
For the low limit, compare $E_{\max}x$ with $D(C)$. For the high limit, calculate $E_{\max}/(1+x)$, and state whether the denominator is the maximum or exact drug increment.

Check whether the model assumptions are credible.
Look for systematic departures from the assumed direct, monotone curve or evidence that baseline or plateau changes over the conditions being compared.

Diagnostics, Alternatives, and Limitations

The main alternative to either limit is the full Emax expression. It is preferable when $x$ is not very small or very large, when approximation error matters, or when concentrations span both transition and extreme regions. A local linear approximation may be convenient for describing a low-concentration trend, while a plateau approximation may succinctly describe a saturated response; neither should be substituted for the full curve outside its regime.

Useful checks are direct numerical comparisons at the concentrations of interest and inspection of residuals against concentration when an approximation is used in a fit. Curvature at low concentrations suggests the linear approximation is being used too far from zero; continuing response increases at concentrations treated as a plateau suggest saturation has not been reached. A systematic pattern can also indicate that the assumed Emax relationship is not suitable, rather than simply that a limit needs a slightly tighter cutoff.

These limits do not establish that the response is truly linear near zero or that a finite observed concentration range has reached the asymptote. A fitted curve can appear almost flat because data are sparse or noisy. Conversely, deviations from a proposed plateau can arise from model misspecification or changing conditions. Asymptotic calculations quantify the implications of the stated model; they do not verify the model or replace the full response curve when that curve is needed.

Common Pitfall

Common mistake: “The concentration is ten times $EC_{50}$, so the response is exactly at $E_{\max}$.” For the ordinary Emax model, $C=10EC_{50}$ gives $x=10$, so the drug-related increment is $10/11=90.91\%$ of $E_{\max}$, not 100%. The correction is to calculate the remaining fraction $1/(1+x)$ and report a plateau only as an approximation with an explicit error tolerance. Similarly, low-concentration linearity should not be assumed for a sigmoid curve with $\gamma\neq1$.

Check Your Understanding

Answer each question, then compare with the solutions below.

For an ordinary Emax model, $C=0.2EC_{50}$. What is $x$, and what is the relative error of the low approximation measured against the exact drug increment?
For a sigmoid model with $\gamma=2$, $C=0.5EC_{50}$, and $E_{\max}=40$ response units, calculate the exact increment and low approximation.
For any positive $\gamma$, what fraction of the maximum drug increment remains unachieved when $x=19$?
Why is the low-concentration approximation not generally linear in $C$ when $\gamma\neq1$?

Solutions. (1) $x=0.2$. The relative error is exactly $x=0.2$, or 20%. (2) $x=(0.5)^2=0.25$. The exact increment is $40(0.25)/(1.25)=8.00$ response units; the low approximation is $40(0.25)=10.00$ response units. (3) The unachieved fraction is $1/(1+x)=1/20=0.0500$, or 5.00%. (4) The leading term is $E_{\max}(C/EC_{50})^\gamma$, which is proportional to $C^\gamma$; it is linear in $C$ only when $\gamma=1$.