From tested conditions to untested ones
A QSP model can be used to ask what might happen at a dose or dosing schedule that has not been tested. That ability is useful when experiments cannot cover every possible regimen, but it does not make a model’s prediction equally reliable everywhere. A prediction between tested dose levels is generally closer to the available evidence than one far outside them; a prediction for a new schedule also depends on assumptions about how the system responds over time.
The distinction is about the conditions being predicted, not about whether the model has equations. A model may generate precise numerical outputs for an untested condition, while those outputs remain uncertain because its assumptions have not been checked there. As in the earlier material on counterfactual questions for mechanistic models, a prediction is conditional: it describes what follows if the modeled relationships and assumptions are adequate for the question.
For dose, the tested range is the span between the lowest and highest dose levels with relevant evidence. For schedule, it is not enough to compare total amount of drug: the interval, timing, and pattern of administration can change exposure over time. A schedule prediction may therefore be extrapolation even when its total daily dose matches a tested regimen.
Why interpolation is usually less demanding
Suppose responses \(y_1\) and \(y_2\) were observed at doses \(D_1\) and \(D_2\), with \(D_1 < D_2\). Under a straight-line interpolation, the predicted response at a dose \(D\) between them is a weighted average of the two observed responses:
Because both weights are between zero and one and sum to one, this estimate lies between \(y_1\) and \(y_2\). The estimate still relies on an assumption about the shape of the response between the measured points; it is not automatically correct simply because the dose lies within the range.
The same calculation shows why extrapolation needs more caution. Outside the tested interval, \(w\) is less than zero or greater than one. One weight is then negative, so the calculation is no longer a weighted average of the observed responses. Its prediction can move well beyond them and can violate biological limits, such as predicting more than 100% inhibition.
Worked Example: Interpolating Between Two Tested Doses
In a hypothetical experiment, pathway inhibition is 20% at 10 mg and 40% at 30 mg. Estimate inhibition at 20 mg using straight-line interpolation.
The dose 20 mg lies between 10 mg and 30 mg, so this is interpolation. The weight on the 30 mg response is \(w=(20-10)/(30-10)=10/20=0.5\).
Substitute the observed responses: \(\hat y(20)=(1-0.5)(20\%)+(0.5)(40\%)=10\%+20\%=30\%\).
The interpolated estimate is 30% inhibition at 20 mg. It is a calculation under a straight-line assumption, not proof that the biological response is linear. If the response changes sharply between the tested doses, an additional measurement or a better-supported mechanistic model may give a different estimate.
Extrapolation depends on the mechanism assumed
A mechanistic model can make an extrapolation more credible when its structure represents relevant biological processes and those processes are supported by evidence. For example, a saturable relationship may be more plausible than an indefinitely increasing straight line when drug action approaches a maximum. But a biologically plausible equation is not, by itself, evidence that its parameters or structure are right for an untested dose.
A useful way to examine this dependence is to compare predictions from different models that fit the tested observations. If those predictions diverge outside the observed range, the data have not yet established which model gives the better extrapolation. Additional evidence can help: measurements near the proposed dose, evidence about target engagement, or observations of downstream responses can test parts of the assumed chain. Each measurement supports only the links it actually informs.
Worked Example: Two Dose Models, Two Extrapolations
Suppose a hypothetical study observes 20% pathway inhibition at 10 mg and 40% at 30 mg. Compare a straight line through these points with a saturable model \(E(D)=80D/(30+D)\), where \(E(D)\) is inhibition in percentage points and \(D\) is dose in mg. Both models reproduce the two observations.
For the straight line, the slope is \((40-20)\text{ percentage points}/(30-10)\text{ mg}=1\) percentage point per mg. Its equation is \(E(D)=10+D\), with \(D\) in mg and \(E\) in percentage points. At 60 mg, it predicts \(E(60)=10+60=70\%\).
For the saturable model, at 10 mg it predicts \(80(10)/(30+10)=800/40=20\%\); at 30 mg it predicts \(80(30)/(30+30)=2400/60=40\%\). At 60 mg, it predicts \(E(60)=80(60)/(30+60)=4800/90=53.33\%\).
The two models match the tested points but predict 70% versus 53.33% inhibition at 60 mg, which is outside the tested range. The difference arises from the saturable model’s assumption of a maximum inhibition of 80%, whereas the straight line has no such bound. Neither prediction is established by the two observations alone. Evidence that the response approaches a finite maximum would support considering the saturable form, but the 60 mg prediction remains conditional until the relevant behavior is checked.
Schedules change the time course
A dosing schedule specifies not only how much drug is given but also when it is given. A simple pharmacokinetic calculation illustrates why equal total daily amounts need not imply equal concentration profiles. Consider a hypothetical one-compartment model with instantaneous doses, linear elimination, and a 12-hour half-life. Assume each mg administered increases concentration by 0.1 mg/L. These assumptions are deliberately simple; they are what make the following calculation possible.
For repeated equal doses with interval \(\tau\), let \(A\) be the concentration increase per dose and \(r=2^{-\tau/12}\) the fraction remaining after one interval. At steady state, the peak is the sum of the current dose increment and the residual contributions from earlier doses. This is a geometric series:
The infinite series converges because \(0 \leq r < 1\). The trough concentration is the peak multiplied by the fraction remaining over one interval. This result depends on the stated assumptions; it does not describe delayed absorption, nonlinear elimination, or time-varying clearance.
Worked Example: Predicting a Change in Dosing Interval
A hypothetical patient receives 24 mg once every 24 hours. Consider a prediction for 12 mg every 12 hours, keeping the total daily dose at 24 mg. Assume the one-compartment model and parameters just described, and assume drug occupancy responds rapidly according to \(C/(C+K_d)\), with \(K_d=1.2\) mg/L.
For 24 mg every 24 hours, the concentration increment is \(A=24(0.1)=2.4\) mg/L and \(r=2^{-24/12}=2^{-2}=0.25\). Thus \(C_{\max}=2.4/(1-0.25)=2.4/0.75=3.2\) mg/L, and \(C_{\min}=0.25(3.2)=0.8\) mg/L.
For 12 mg every 12 hours, \(A=12(0.1)=1.2\) mg/L and \(r=2^{-12/12}=0.5\). Thus \(C_{\max}=1.2/(1-0.5)=2.4\) mg/L, and \(C_{\min}=0.5(2.4)=1.2\) mg/L.
At the trough, predicted occupancy is \(0.8/(0.8+1.2)=0.40\), or 40%, for once-daily dosing and \(1.2/(1.2+1.2)=0.50\), or 50%, for twice-daily dosing. Under this simplified model, splitting the same daily amount lowers the peak and raises the trough. This is a schedule extrapolation, not a clinical outcome prediction: its credibility depends on the assumed concentration-time behavior and rapid occupancy relationship, among other links that would need support.
Assessing credibility and recognizing failure
Before using an extrapolation, state what is being predicted and what must remain true for the prediction to hold. For dose predictions, ask whether the modeled relationship is expected to remain valid beyond the tested doses. For schedule predictions, ask whether the model captures the time course that matters, including accumulation and any relevant delay between concentration, target engagement, and downstream response. The more the prediction depends on a link that has not been measured or supported, the more cautiously it should be interpreted.
Several practical checks help reveal when an extrapolation is going wrong:
- Check the boundary. Mark the tested dose range and schedule conditions. Confirm whether the requested prediction is inside or outside them rather than describing every model output as an interpolation.
- Check constraints. Ask whether the output respects meaningful bounds, such as a fraction between 0 and 1 or a maximum possible inhibition. A violation may indicate an unsuitable model form or an invalid extrapolation.
- Check competing assumptions. Compare reasonable alternative structures. Large differences in predictions beyond the evidence indicate that the extrapolation is sensitive to assumptions.
- Check the relevant time course. For a schedule change, matching total daily dose is not sufficient evidence that peaks, troughs, or biological responses will match.
- Check against new evidence when possible. A measured result near the proposed condition can directly challenge the prediction. A mismatch should prompt review of assumptions and model structure, not an automatic adjustment of the measurement to fit the model.
An alternative to mechanistic extrapolation is to use empirical interpolation only where data support it, or to collect data at the proposed dose or schedule. These approaches may be more defensible when the biological links needed for a model prediction are poorly supported. Conversely, a mechanistic model can organize evidence about processes that a simple curve does not represent. Neither approach removes uncertainty: interpolation depends on its local shape assumption, new experiments have scope limits, and a mechanistic model depends on the adequacy of its structure and inputs.
Common pitfall
Wrong reasoning: “The model gives a numerical answer at 60 mg, so the dose is covered by the model and the prediction is reliable.” A model can calculate outside the tested range, but calculation alone does not validate the assumptions there. Correction: distinguish the model’s ability to produce an output from evidence that the output is credible. State which assumptions support the prediction, identify where the data stop, and check whether plausible alternatives give materially different answers.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Responses of 15% and 35% are observed at 5 mg and 25 mg. Under straight-line interpolation, what response is estimated at 15 mg?
- Using the same observations, what does the straight line predict at 35 mg? Why is this extrapolation rather than interpolation?
- A model predicts the same total daily exposure for two schedules. Is that alone enough to conclude their biological responses will be the same? Explain.
- Two models fit all tested doses but predict substantially different responses at a higher untested dose. What does that tell you about the evidence for the higher-dose prediction?
Solutions. (1) The weight is \((15-5)/(25-5)=0.5\), so the estimate is \(0.5(15\%)+0.5(35\%)=25\%\). (2) The slope is \((35-15)/(25-5)=1\) percentage point per mg. At 35 mg, the prediction is \(15+1(35-5)=45\%\). This is extrapolation because 35 mg exceeds the highest tested dose, 25 mg. (3) No. The schedules may produce different concentration-time profiles, and responses may depend on peaks, troughs, timing, or delays. Equal total exposure alone does not establish equal biological response. (4) The tested data do not distinguish the models’ behavior at that dose. The prediction depends on model assumptions and should be treated as uncertain unless further evidence supports one structure or directly tests the higher dose.