Choosing Titration Step Size From Exposure-Response Sensitivity

Why Titration Step Size Depends on Sensitivity

A dose increase of 10 mg/day does not imply a particular response increase on its own. The expected response change depends on how dose changes exposure and how sensitive response is to exposure at the current operating point. A modest dose step may produce a large response change where the exposure–response curve is steep, but little change where it is nearly flat.

This tutorial develops a local calculation for translating a specified response increment into a dose step. It complements the earlier material on writing titration rules as state-update equations: here the focus is how to choose the size of the dose change from a model’s local sensitivity. A calculated step is a model-based expectation, not a guarantee that an individual’s response will change by exactly the requested amount.

Units are part of the calculation. If response is measured in response-score points and exposure in mg·h/L, then \(S_E\) has units of response-score points per mg·h/L. If dose is measured in mg/day, \(S_D\) has units of response-score points per mg/day. Consequently, dividing a response increment by \(S_D\) gives a dose increment in mg/day.

The exposure metric must be stated and used consistently. It might be an average concentration, an AUC over a defined interval, or another model-based exposure summary. The dose-to-exposure mapping must correspond to the same regimen and exposure definition. The calculation does not mean that a dose increment and a response increment have matching numerical values: they are different quantities with different units.

Deriving the Local Dose-Step Formula

Suppose the current dose is \(D_0\), with predicted response \(R(D_0)\). For a small proposed step \(\Delta D\), a Taylor expansion around \(D_0\) gives

for some \(\xi\) between \(D_0\) and \(D_0+\Delta D\), when the second derivative exists on that interval. Subtracting the current prediction shows that the response change is approximately \(S_D\Delta D\) when the second-order term is small. Setting this approximate change equal to the desired increment \(\Delta R\) and solving yields

This derivation also identifies the approximation’s main condition: curvature must be small enough over the proposed step that the second-order contribution does not materially alter the predicted response change. A derivative at one point describes local sensitivity, not necessarily sensitivity across a large dose range.

Signs matter. If response increases with dose, then \(S_D>0\); a desired response increase has \(\Delta R>0\) and gives a positive dose step. If response decreases with dose, a desired response decrease has \(\Delta R<0\), and the quotient again gives a positive step only when the sign of the desired change and slope are treated consistently. The sign of \(\Delta D\) says whether dose should increase or decrease.

Nonlinear Curves and the Limits of a Local Step

In many models, sensitivity changes with exposure. For example, an Emax response curve becomes progressively flatter as exposure approaches saturation. A derivative-based step can still be useful, but a proposed step should be checked against the model’s predicted response at the new dose. If the local calculation is intended to be approximate, this check reveals whether curvature makes the approximation poor.

Estimating Sensitivity From Model Predictions

A fitted model may be available as a prediction function or numerical program rather than a convenient symbolic derivative. Sensitivity can then be estimated with a finite difference: evaluate predicted response at nearby doses and divide the response difference by the dose difference. A centered difference uses predictions on both sides of the current dose and can reduce one-sided approximation error, provided both doses are in a scientifically meaningful region.

Assumptions, Diagnostics, and Alternatives

The calculation is meaningful only when its inputs represent the same modeling context. In particular, the exposure definition, regimen, response scale, and current operating point must be clear. The dose–exposure derivative should reflect the relevant dose-to-exposure relationship; assuming exposure is proportional to dose without justification can produce a misleading step. The response model and exposure mapping should also be differentiable locally, with a slope that is sufficiently far from zero for division to be stable.

Useful diagnostics include comparing slopes just below and above the current dose, evaluating the full model prediction at the proposed new dose, and checking the size of the second-order change when curvature is available. A very large proposed step relative to the dose range used to estimate the local slope is a warning that the calculation is extrapolating a local quantity. A slope near zero is also a warning: dividing by a small number can produce an impractically large step, while small errors in the estimated slope can cause large changes in the calculated step.

Sensitivity itself may be uncertain because model parameters, individual exposure, and response measurements are uncertain. If the estimated slope varies substantially across plausible parameter values or individuals, a single step computed from a point estimate should not be described as reliably achieving a fixed response increment. The calculated change is an expected model-based change under the stated inputs, not an individual-level promise.

A practical choice is to use the local derivative to identify a candidate step, then use the full model to check the predicted response at the candidate dose. If a specific response target must be met under the model and inversion is feasible, solving \(R(E(D))=R(D_0)+\Delta R\) is more direct than relying on a tangent approximation. Neither method removes uncertainty in the model or guarantees the outcome for a particular patient.