From repeated-dose observations to competing models
Repeated-dose data can show a falling response even when concentration is similar across dosing occasions. That pattern is compatible with tolerance, but it does not by itself identify how tolerance develops. A potency shift, a changing maximum effect, delayed response dynamics, or an incorrectly specified concentration profile can produce different patterns. A useful analysis therefore compares explicit candidate models against the same concentration and response observations.
This tutorial works through three alternatives: a no-adaptation model, a potency-shift model indexed by accumulated exposure, and a dynamic exposure-driven adaptation model. The comparison is conditional on the specified response equation, the observed concentration history, and the assumed error model. A smaller residual sum of squares does not establish a causal mechanism.
Specify the candidate models before fitting
For a concrete comparison, let \(R(t)\) be a response measured in response units, with larger values representing more of the measured outcome. The drug reduces the response from a baseline \(R_0\). Use the inhibitory Emax form
where \(C(t)\) is concentration, \(E_{\max}>0\) is the maximum drug-related reduction, \(EC_{50}>0\) is the concentration producing half of that maximum reduction when \(A(t)=0\), and \(\gamma\geq 0\) scales the potency shift. The residual \(\varepsilon(t)\) represents unexplained observation error. The equation assumes that, conditional on concentration and adaptation, the response follows the stated concentration–response curve; this example treats observations as independent with constant-variance Gaussian errors. Those assumptions are for the worked comparison, not universal properties of repeated-dose data.
For the no-adaptation model, set \(A(t)=0\), so that \(EC_{50}\) stays constant. For a simple cumulative-exposure potency-shift model, set \(A(t)=q(t)\), where \(q(t)\) is a prespecified, dimensionless exposure index calculated from dosing history. The model estimates \(\gamma\), but does not estimate the time course of adaptation. In the exposure-driven model, let adaptation evolve according to
where \(C_{\mathrm{ref}}\) is a fixed reference concentration and \(k_a>0\) is an adaptation-rate constant. This formulation makes \(A(t)\) a smoothed history of normalized exposure: the state approaches the current normalized concentration, rather than changing instantly with it. The three models differ in their adaptation assumptions; all use the same response equation and concentration observations. Parameters designated as fixed must be chosen or supported independently of the response data being compared.
Worked Analysis of Repeated-Dose Tolerance
Worked Example: Comparing the Three Candidate Models
Suppose a hypothetical repeated-dose experiment records six concentration–response pairs. Response is in arbitrary response units, and concentration is in \(\mu\text{g/L}\). The fixed values are \(R_0=100\) response units and \(E_{\max}=80\) response units. The observations are:
| Observation | Concentration (\(\mu\text{g/L}\)) | Response (units) |
|---|---|---|
| 1 | 0.5 | 92 |
| 2 | 1.0 | 84 |
| 3 | 2.0 | 70 |
| 4 | 4.0 | 58 |
| 5 | 2.0 | 62 |
| 6 | 1.0 | 74 |
The same concentrations recur later with lower responses: at \(2.0\ \mu\text{g/L}\), response changes from 70 to 62 units; at \(1.0\ \mu\text{g/L}\), it changes from 84 to 74 units. Under this inhibitory model, those lower responses indicate larger drug-related reductions at the same concentrations, the opposite of the response increase expected from a potency shift representing tolerance. These differences motivate testing adaptation, but do not prove it. The model fits use the full concentration history to calculate the exposure-driven state, not just the concentration in the row.
Assume nonlinear least squares was used under the stated error assumptions. For this hypothetical data set, suppose the fitted residual sums of squares are 176, 15.13, and 37.23 response-units squared for the no-adaptation, cumulative-exposure potency-shift, and exposure-driven models, respectively. The candidate models estimate \(K=1,2,\) and \(3\) response-model parameters, respectively; the common residual-variance parameter is omitted from \(K\) because its penalty is identical for all candidates. These are stipulated illustrative fit outputs, not values reconstructed from the rounded table alone.
A compact comparative score, with constants common to all models omitted, is
Here \(SSE\) is the residual sum of squares, \(n=6\), and \(K\) is the number of estimated response-model parameters. Substitution gives:
The cumulative-exposure potency-shift model has the lowest score among these three candidates. The differences from its score are approximately 12.72 for no adaptation and 7.40 for the exposure-driven model. Under this criterion and these assumptions, the data favor the cumulative-index model among the specified candidates. The result does not establish that cumulative exposure is the true biological cause of the response pattern; omitted structures and error assumptions can change the comparison.
Why the exposure state changes gradually
The dynamic state equation has a useful exact solution over an interval where concentration is constant. Write \(u=C/C_{\mathrm{ref}}\). Then \(dA/dt=k_a(u-A)\), so \(d(A-u)/dt=-k_a(A-u)\). Solving this first-order equation over an interval of length \(\Delta t\) gives
Thus \(A\) moves a fraction \(1-e^{-k_a\Delta t}\) of the way from its previous value toward the interval’s normalized concentration. A large \(k_a\Delta t\) implies faster adjustment; a small value implies slower adjustment. This derivation also shows why the order and spacing of concentrations matter: two regimens with the same total exposure can produce different state values at a particular observation time.
Worked Example: Updating Adaptation During a Concentration Plateau
Suppose a hypothetical patient’s normalized concentration is \(u=2\), the state at the start of a constant-concentration interval is \(A(t)=0.5\), and \(k_a=0.2\ \text{h}^{-1}\). After \(\Delta t=3\) hours, the update is
The state has risen from 0.5 to about 1.177, but has not reached the normalized concentration of 2. The calculation checks the intended behavior of the model: with positive \(k_a\) and \(u>A(t)\), adaptation increases toward \(u\). If concentration subsequently falls, the same equation makes the state decline gradually toward the lower normalized concentration rather than vanish immediately.
Translate a potency shift into predicted response loss
Model selection is not the same as quantifying the predicted effect. A direct interpretation compares responses at a specified concentration and time under the fitted adaptation state against the corresponding no-adaptation prediction. This is a model-based contrast. It is most interpretable when the concentration is held fixed in the contrast, so the difference reflects the model’s adaptation term rather than a difference in exposure.
Worked Example: Calculating Response Loss at Matched Concentration
For illustration, suppose the no-adaptation fit predicts a response of 70.0 units at \(C=2.0\ \mu\text{g/L}\), while the fitted potency-shift model predicts 78.0 units at that same concentration and a later-dose adaptation state. The response loss relative to the no-adaptation prediction is
The fitted adapted model predicts an 8.0-unit increase in response at the matched concentration, equal to approximately 11.43% of the no-adaptation prediction of 70.0 units. With \(R_0=100\), the predicted drug-related reduction decreases from 30.0 to 22.0 units, a decrease of approximately 26.67%. These are contrasts between model predictions under specified assumptions, not an observed individual’s causal response loss.
Diagnostics, alternatives, and limits
A numerical score should be considered together with diagnostics. For each candidate, examine residuals against time, concentration, fitted response, and dose number. A systematic trend across dose number suggests that the model has not captured the time course. A concentration-dependent residual pattern can indicate a poor concentration–response shape. Large outliers or changing residual spread challenge the common-variance Gaussian error assumption. For the dynamic model, inspect whether the estimated state follows the known exposure history and whether parameter estimates are stable under reasonable changes in starting values.
The main alternatives answer different questions. The no-adaptation model is the simplest reference and is useful when repeated responses are adequately explained by concentration alone. The cumulative-index potency-shift model is parsimonious when a prespecified exposure summary captures the relevant ordering of adaptation, but it does not represent recovery or detailed timing unless those are built into the index. The exposure-driven state model represents gradual onset and decay, but requires a sufficiently informative concentration-time history and can be difficult to distinguish from other delayed processes. Other model structures may represent efficacy change or response delay; comparing these requires a common, clearly specified likelihood and careful accounting of their parameter counts.
This analysis has important limitations. Six observations provide little information for separating response-curve shape from adaptation dynamics, especially when \(R_0\) and \(E_{\max}\) are fixed rather than estimated. The concentration history between observations must be known or modeled; interpolation error can distort \(A(t)\). A low AIC-based score is relative to the candidates included and does not validate any of them. If participants contribute repeated measurements, treating all observations as independent can understate uncertainty. Finally, an apparent matched-concentration response difference can reflect time-varying disease, co-medication, measurement drift, or other changes as well as pharmacologic tolerance.
Use the same response observations, concentration history, fixed quantities, and error model for all candidate fits.
Estimate each model’s parameters, calculate residuals, and assess whether patterns remain over time, concentration, or dose number.
Calculate a complexity-penalized score, then report matched-concentration predicted contrasts with units and a clear reference prediction.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- In the state equation, what happens to \(A(t)\) when concentration is constant and \(k_a\Delta t\) becomes large?
- Why can two regimens with the same total exposure produce different adaptation states at a given time?
- Using the example scores, which candidate has the lowest \(AIC^*\), and what does that support?
- If the adapted prediction is 78 response units and the no-adaptation prediction is 70 units, what is the percentage response increase relative to the no-adaptation prediction?
- Name one diagnostic pattern that would challenge the assumed model or error structure.
Solutions. (1) The state approaches the normalized concentration \(C/C_{\mathrm{ref}}\), because \(e^{-k_a\Delta t}\) approaches zero. (2) The state depends on the time-ordered concentration history and interval spacing, not only total exposure. (3) The cumulative-exposure potency-shift candidate has the lowest score, 9.55; this favors it among the specified candidates under the stated assumptions, not as proof of mechanism. (4) \((78-70)/70\times100\%\approx11.43\%\). (5) Examples include residuals trending by dose number, residual spread increasing with fitted response, or unstable adaptation estimates across starting values.