Identifiability of Ke0 and Response Parameters

Why Identifiability Matters

An effect-compartment model links the changing plasma concentration to an unobserved effect-site concentration, then maps that latent concentration to a measured response. Estimating both links from the same response data creates a central difficulty: a change in the equilibration rate can alter the predicted effect-site concentration, while a change in the response curve can compensate for that alteration. The fitted responses may look similar even when the parameter estimates differ substantially.

This is not merely a numerical nuisance. If the data do not distinguish the parameters, the estimated Ke0 and response parameters cannot each be interpreted as well determined by those data. A small residual error or a visually convincing fit does not, by itself, establish that the individual parameters have been identified.

Definition: A parameter is structurally identifiable under a specified model and input if ideal, noise-free observations over the stated observation design determine its value uniquely. It is practically identifiable if the finite, noisy data provide enough information to estimate it with useful precision. A parameter can be structurally identifiable yet practically weakly identified.

Write the effect-compartment equation and a standard sigmoidal response model as

\frac{dC_e(t)}{dt}=K_{e0}\bigl(C_p(t)-C_e(t)\bigr), \qquad E(t)=E_0+E_{\max}\frac{C_e(t)^\gamma}{EC_{50}^\gamma+C_e(t)^\gamma}.

Here \(C_p(t)\) and \(C_e(t)\) are plasma and effect-site concentrations, respectively; \(K_{e0}\) has units of inverse time; \(E_0\) is the baseline response; \(E_{\max}\) is the maximum increase above baseline; \(EC_{50}\) is the effect-site concentration giving half of that increase; and \(\gamma\) is the Hill coefficient. This tutorial assumes the model form is appropriate, the measured response follows the stated concentration-effect relationship without an additional delay, and the effect-site initial condition is specified. These assumptions do not guarantee identifiability; they define the model whose parameters are being assessed.

At constant plasma concentration maintained to equilibrium, \(C_e=C_p\). The response then depends on the response-curve parameters and concentration, but not on \(K_{e0}\). Equilibrium observations can inform the curve, but cannot alone estimate how quickly the effect site equilibrates. Information about \(K_{e0}\) must come from transient behavior and its timing.

How the Parameters Become Confounded

For a constant plasma concentration \(C_p\), beginning at time zero from \(C_e(0)=0\), the effect-site solution is

C_e(t)=C_p\bigl(1-e^{-K_{e0}t}\bigr).

At a single sampling time, a different \(K_{e0}\) implies a different \(C_e(t)\). The response curve can offset that difference: for example, with \(E_0\), \(E_{\max}\), and \(\gamma=1\) fixed, changing \(EC_{50}\) can give the same response at that altered effect-site concentration. Thus a response observation does not directly reveal the latent concentration or the equilibration rate.

The trade-off can be stated exactly for \(\gamma=1\). Given a response \(E\) between \(E_0\) and \(E_0+E_{\max}\), the response equation implies

EC_{50}=C_e\frac{E_0+E_{\max}-E}{E-E_0}.

For each possible positive \(C_e\), this expression gives an \(EC_{50}\) that reproduces the same response, provided the other curve parameters are held fixed. If \(C_e\) itself depends on an unknown \(K_{e0}\), one isolated response therefore cannot generally separate the two.

The problem also arises when several effect-site concentrations occupy only a narrow range. Over a small interval, the response curve is observed only locally. Different combinations of \(E_{\max}\), \(EC_{50}\), and \(\gamma\) can have similar values and slopes there, even if their predictions differ outside that interval. If the sampling times are sparse, the transient trajectory is also poorly constrained, allowing changes in \(K_{e0}\) to be absorbed by changes in curve shape.

Worked Example: One Transient Observation, Two Parameter Combinations

Suppose plasma concentration is held at \(10\ \text{mg/L}\) for one hour, with \(C_e(0)=0\). A hypothetical response observed at one hour is \(55\) units. Assume \(E_0=5\) units, \(E_{\max}=100\) units, and \(\gamma=1\). Show that two different \(K_{e0}\) and \(EC_{50}\) combinations reproduce the observation.

First take \(K_{e0}=\ln(2)=0.6931\ \text{h}^{-1}\). The effect-site concentration after one hour is

C_e(1)=10\bigl(1-e^{-0.6931(1)}\bigr) \approx 10(1-0.5000)=5.000\ \text{mg/L}.

Set \(EC_{50}=5.000\ \text{mg/L}\). Then

E(1)=5+100\frac{5.000}{5.000+5.000} =5+50=55.00\ \text{units}.

Now take \(K_{e0}=-\ln(0.75)=0.2877\ \text{h}^{-1}\). This gives

C_e(1)=10\bigl(1-e^{-0.2877}\bigr) \approx 10(1-0.7500)=2.500\ \text{mg/L}.

With \(EC_{50}=2.500\ \text{mg/L}\), the predicted response is again

E(1)=5+100\frac{2.500}{2.500+2.500} =55.00\ \text{units}.

Both calculations use the same constant-input model and the same fixed baseline, maximum effect, and Hill coefficient. They demonstrate exact non-uniqueness for this single observation: the data point alone cannot choose between these parameter combinations. Additional observations at informative times or concentrations are needed.

Patterns That Help Separate Dynamics and Curve Shape

Separation improves when the data contain distinct information about the time course and the concentration-effect relationship. A useful data set typically includes observations during both changing concentrations and periods that cover a broad effect-site concentration range. Rising and falling observations at similar plasma concentrations can be particularly informative: because the effect site lags behind plasma, the corresponding \(C_e\) values can differ, and the response may exhibit hysteresis. The timing and direction of that difference inform the equilibration process, while responses across a broad range inform the curve shape.

The mechanism can be seen by differentiating the constant-input solution:

\frac{\partial C_e(t)}{\partial K_{e0}} =C_p t e^{-K_{e0}t}.

For \(C_p>0\) and \(t>0\), this derivative is positive: increasing \(K_{e0}\) raises \(C_e(t)\) toward \(C_p\) more quickly. Consequently, a change in \(K_{e0}\) changes the predicted response through the response curve. If observations cover only a short segment of that trajectory, or if the curve is nearly flat there, the response data may have little ability to distinguish the dynamic change from a change in curve parameters.

The following example shows why a narrow concentration range may leave response parameters weakly separated, even when \(K_{e0}\) is known and the effect-site concentrations are treated as given.

Worked Example: Similar Curves Over a Narrow Range

Consider hypothetical effect-site concentrations of \(4.8\), \(5.0\), and \(5.2\ \text{mg/L}\). Compare a curve with \(E_0=5\) units, \(E_{\max}=100\) units, \(EC_{50}=5\ \text{mg/L}\), and \(\gamma=1\), with an alternative curve that has \(E_{\max}=120\) units and \(EC_{50}=7\ \text{mg/L}\), keeping \(E_0=5\) and \(\gamma=1\).

For the first curve, at \(C_e=4.8\ \text{mg/L}\),

E=5+100\frac{4.8}{5+4.8} =5+\frac{480}{9.8} =53.98\ \text{units}.

At \(C_e=5.0\ \text{mg/L}\), it predicts \(E=5+100(5/10)=55.00\) units. At \(C_e=5.2\ \text{mg/L}\), it predicts \(E=5+100(5.2/10.2)=55.98\) units.

For the alternative curve, the predictions at those same concentrations are \(5+120(4.8/11.8)=53.81\) units, \(5+120(5/12)=55.00\) units, and \(5+120(5.2/12.2)=56.15\) units. Thus the two curves agree at \(5.0\ \text{mg/L}\), and differ by only about \(0.17\), \(0.00\), and \(0.17\) response units at the three points. If measurement error is of comparable or greater magnitude, these observations may not distinguish the parameter combinations reliably. This comparison does not claim the curves are identical; it shows why a narrow range can give weak practical identification.

A broader range, especially observations spanning the lower, middle, and upper portions of the response curve, can expose differences in plateau and curvature. In contrast, data confined near baseline may constrain only a combined low-concentration slope, while data confined near the upper plateau may provide little information about \(EC_{50}\) or \(\gamma\). Information about \(K_{e0}\) still requires transient observations; broad equilibrium coverage alone does not supply it.

Worked Effect-Site and Response Calculations

Worked Example: Reading a Rise-and-Washout Trajectory

For a hypothetical illustration, let plasma concentration rise abruptly to \(10\ \text{mg/L}\) for one hour and then fall to zero for one hour. Suppose \(C_e(0)=0\) and \(K_{e0}=1.000\ \text{h}^{-1}\). Use \(E_0=5\) units, \(E_{\max}=100\) units, \(EC_{50}=5\ \text{mg/L}\), and \(\gamma=1\). Calculate the effect-site concentrations and responses at the end of the rise and after the washout hour.

At the end of the one-hour constant-concentration period,

C_e(1)=10(1-e^{-1})=6.3212\ \text{mg/L}.

During washout, \(C_p=0\), so \(C_e\) decays exponentially from that value. After one more hour,

C_e(2)=6.3212e^{-1}=2.3254\ \text{mg/L}.

At the end of the rise, the response is

E(1)=5+100\frac{6.3212}{5+6.3212} =5+\frac{632.12}{11.3212} \approx 60.84\ \text{units}.

After the washout hour, it is

E(2)=5+100\frac{2.3254}{5+2.3254} =5+\frac{232.54}{7.3254} \approx 36.74\ \text{units}.

The paired concentration and response changes are consistent with a lagged effect-site trajectory under the assumed parameters. In real estimation, however, these response values alone would not prove that this particular \(K_{e0}\) and response curve are unique. Multiple time points across the rise and fall, and a response range that informs curve shape, are needed to assess separation.

Assumptions, Diagnostics, and Alternatives

Identifiability conclusions are conditional on the assumed model and the observation design. The effect-compartment equation presumes first-order equilibration driven by the stated plasma concentration. The response model presumes a stable concentration-effect relationship with the specified baseline and curve form. The initial effect-site concentration must also be known or appropriately accounted for. Unmodeled response delays, incorrect input histories, baseline drift, or an inappropriate response curve can mimic parameter trade-offs; confounding is not the only reason estimates may be unstable.

Warning signs include estimates that change substantially when a few time points are removed, very large standard errors or correlations among parameter estimates, estimates near imposed bounds, and distinct parameter sets with nearly indistinguishable fitted responses. A useful diagnostic is to inspect whether the observed concentrations cover a transient and a meaningful part of the response curve. If all observations are at equilibrium, they cannot identify \(K_{e0}\). If observations are all at nearly the same effect-site concentration, they provide little leverage on the curve's shape. A good visual fit should therefore be considered alongside parameter stability and the information content of the data.

Joint estimation of \(K_{e0}\) and response parameters is appropriate when the data contain both dynamic and concentration-range information. Fixing response parameters from an independent, informative concentration-effect data set can reduce the trade-off when the fixed values are credible. Conversely, fixing \(K_{e0}\) from external information can help estimate the response curve when transient data are weak. These approaches rely on the external values being applicable to the modeled population and conditions. Simply fitting the response curve to equilibrium data cannot estimate \(K_{e0}\), and fixing poorly supported values can conceal rather than resolve uncertainty.

The key distinction is between model fit and parameter identification. When the same response trajectory can be explained by different combinations of equilibration rate and curve parameters, report the limitation and avoid treating each fitted estimate as independently established. Identifiability analysis clarifies what the data support; it does not certify the physiological interpretation of a parameter or repair a misspecified model.

Key takeaway: Transient data carry information about \(K_{e0}\), while a sufficiently broad effect-site concentration range informs the response curve. Reliable separation generally requires both kinds of information; neither a good fit nor a large number of redundant observations guarantees it.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Why can observations collected only at equilibrium not identify \(K_{e0}\)?
With \(E_0=5\), \(E_{\max}=100\), \(\gamma=1\), \(C_e=4\ \text{mg/L}\), and \(E=45\) units, calculate the \(EC_{50}\) that reproduces the response.
What is one reason that a narrow effect-site concentration range can make \(E_{\max}\), \(EC_{50}\), and \(\gamma\) difficult to distinguish?
Name one data pattern that can help inform \(K_{e0}\), and explain why it is useful.

Solutions. (1) At equilibrium, \(C_e=C_p\), and the response equation contains no \(K_{e0}\); equilibrium responses therefore provide no direct information about the equilibration rate. (2) Rearranging gives \(EC_{50}=C_e(E_0+E_{\max}-E)/(E-E_0)=4(5+100-45)/(45-5)=4(60)/40=6.000\ \text{mg/L}\). Substitution confirms \(E=5+100(4)/(6+4)=45\) units. (3) Over a narrow range, different curves can have similar local values and slopes, so observed responses may not reveal their different plateaus or curvature. (4) Repeated observations during rising and falling concentrations can help: the effect-site lag produces time-dependent differences between plasma concentration and response, which contain information about equilibration speed.