Mapping Dosing Regimens to Predicted Dose Response

From a Proposed Regimen to a Predicted Response

A dose amount alone does not specify the concentration-time profile experienced by a patient. The dosing interval determines how much drug remains when the next dose is given, and the route determines how drug enters the systemic circulation. These differences can matter even when two regimens deliver the same nominal amount per day. To predict response from a proposed regimen, a PK/PD model first maps that regimen to exposure and then maps exposure to a response.

This mapping also clarifies the distinction between two kinds of relationship. A dose-response relationship describes the predicted response associated with a dose or regimen, after accounting for the PK processes connecting dose to exposure. An exposure-response relationship describes the predicted response conditional on a specified exposure measure, such as average concentration or peak concentration. The latter does not, by itself, say which dose or route will produce that exposure.

A useful conceptual representation is \(D \rightarrow C(t) \rightarrow E\), where \(D\) represents the regimen, \(C(t)\) is the resulting concentration-time profile, and \(E\) is the predicted response. The arrows encode model assumptions: the PK model must represent the relevant route and dosing schedule, and the PD model must use an exposure measure and response relationship appropriate to the drug and endpoint.

Building the Regimen-Specific Exposure Prediction

For linear PK at steady state, the average concentration over a dosing interval is determined by the systemic amount delivered per interval and clearance. For an intravenous dose \(D\) every \(\tau\) hours, the systemic amount is \(D\). For an oral dose with bioavailability \(F\), it is \(FD\). The average concentration is therefore:

Here \(CL\) is clearance, and \(F=1\) for an intravenous dose. Units must be consistent: if \(D\) is in mg, \(\tau\) in hours, and \(CL\) in L/h, then \(C_{\mathrm{avg},ss}\) is in mg/L. This average does not determine the full profile. For a given average, a short interval may produce smaller peak-to-trough fluctuations than a longer interval.

For a one-compartment model with instantaneous intravenous boluses, elimination rate constant \(k=CL/V\), volume \(V\), and repeated dose interval \(\tau\), the steady-state concentration immediately after a dose and immediately before the next dose are:

These expressions assume linear elimination, constant PK parameters, complete intravenous delivery, and doses repeated long enough to reach steady state. They do not apply unchanged to an oral regimen, because oral absorption takes time and may be incomplete.

For oral dosing, the model must represent absorption as well as elimination. In a one-compartment model with first-order absorption rate constant \(k_a\), linear elimination rate constant \(k\), bioavailability \(F\), and repeated doses every \(\tau\), the steady-state concentration \(t\) hours after a dose, for \(0\leq t<\tau\), is:

This expression assumes first-order absorption, no lag time, one-compartment disposition, constant bioavailability, linear elimination, and repeated dosing until steady state. A different formulation or route can require a different input model; it should not be treated as a simple change in dose unless its effect on absorption and bioavailability is adequately represented.

Mapping Exposure to Response, Including Variability

A PD model specifies how response changes with exposure. For example, a simple Emax relationship for a response that increases with concentration is:

Here \(E_0\) is the baseline response, \(E_{\max}\) is the maximum increase above baseline, and \(EC_{50}\) is the concentration producing half of that maximum increase. The formula presumes that the chosen concentration measure is the relevant predictor and that this direct relationship is suitable for the endpoint. If response is delayed, depends on concentration history, or is driven by peak or trough rather than average concentration, applying the formula to \(C_{\mathrm{avg},ss}\) may be inappropriate.

When individuals differ in PK parameters, the regimen generates an exposure distribution. Passing a population-average exposure through a nonlinear PD model generally does not give the average predicted response. Instead, the PK and PD calculations should be performed for representative individuals or parameter draws and then summarized across the target population. This distinction matters even if the model itself is deterministic for a given individual's parameters.

Justification, Diagnostics, and Alternatives

The average-concentration equation follows from mass balance at steady state. Under linear PK, the amount eliminated over one interval equals the systemically available dose \(FD\). If concentration is \(C(t)\), the amount eliminated over the interval is \(CL\int_0^\tau C(t)\,dt\). Dividing by the interval and equating average elimination rate to average systemic input rate gives:

The justification depends on steady state and linear clearance: the amount eliminated is proportional to concentration, with the same \(CL\) throughout the interval. Nonlinear elimination, time-varying clearance, nonadherence, or a changing disease state can invalidate this simple mapping.

Before interpreting a regimen-specific prediction, check whether the model reproduces observed concentration-time patterns for the route, formulation, and interval being considered. Systematic residual patterns, for example underprediction near peaks and overprediction during absorption, may indicate a misspecified input or absorption model. Check that dose units, time units, bioavailability, and route are implemented consistently. Predictions far outside the observed dose or exposure range are extrapolations and may be especially sensitive to model assumptions.

A mechanistic PK/PD model is not the only way to estimate dose-response. An empirical dose-response model can relate assigned dose directly to response and may be useful when the objective is prediction within the regimens studied. However, it may not explain how a new interval or route changes exposure, and it can confound differences in exposure with differences in regimen. An exposure-response model can describe response conditional on exposure, but does not by itself identify the exposure produced by a proposed dose. Integrating PK and PD makes the mapping between regimen, exposure, and response explicit, at the cost of assumptions in both parts of the model.

A common mistake is to assume that equal total daily dose means equal response. In the intravenous example, the two regimens had equal average concentration only under linear steady-state PK; their peaks and troughs differed. If PD depends on those profile features, the predicted responses may differ. Another mistake is to calculate response at the population-average exposure and call it the average response. With nonlinear PD, the worked example shows why predictions should instead be made across the relevant population distribution before summarizing.