Local Sensitivity of Human Predictions

Why Local Sensitivity Matters

A human prediction can depend on quantities translated from preclinical evidence, such as clearance, bioavailability, or potency. A prediction at one chosen set of values gives a result, but not how responsive that result is to a small change in one of those inputs. Local sensitivity quantifies that response near the specified parameter set. It can help identify which inputs deserve careful review when assessing a prediction.

The key word is local. A sensitivity describes what happens for small changes around one point, with the model structure and other inputs held fixed. It does not, by itself, describe uncertainty across a wide range of values or establish which input contributes most to total predictive uncertainty.

Definition: Let \(y=f(\boldsymbol{\theta})\) be a scalar human PK or PD prediction, where \(\boldsymbol{\theta}=(\theta_1,\ldots,\theta_p)\) is a vector of model inputs. The local derivative with respect to \(\theta_j\), evaluated at a specified parameter set, is \(\partial y/\partial\theta_j\). It is the first-order change in the prediction per unit change in that input, with the other inputs held fixed. If \(\theta_j\) and \(y\) are positive, the local elasticity is \(E_j=(\theta_j/y)(\partial y/\partial\theta_j)\). It approximates the proportional change in \(y\) for a proportional change in \(\theta_j\).

The derivative has units of prediction divided by units of parameter. The elasticity is dimensionless: near the evaluation point, a 1% increase in \(\theta_j\) corresponds to approximately an \(E_j\)% change in \(y\). A negative elasticity indicates an opposing direction of change; a positive one indicates the same direction. If the output or parameter is zero or can be negative, the elasticity may be undefined or unhelpful, and the derivative or another explicitly scaled measure should be used instead.

First-Order Justification

If \(f\) is differentiable near the parameter set \(\boldsymbol{\theta}_0\), a small change \(\Delta\boldsymbol{\theta}\) gives the first-order approximation

f(\boldsymbol{\theta}_0+\Delta\boldsymbol{\theta}) = f(\boldsymbol{\theta}_0) + \sum_{j=1}^{p} \left.\frac{\partial f}{\partial\theta_j}\right|_{\boldsymbol{\theta}_0} \Delta\theta_j + o(\|\Delta\boldsymbol{\theta}\|).

The final term is smaller than the size of the change as that change approaches zero. For a change in just one input, the approximation reduces to \(\Delta y\approx(\partial y/\partial\theta_j)\Delta\theta_j\). Dividing by \(y\) and writing the input change as a fraction of \(\theta_j\) gives \(\Delta y/y\approx E_j(\Delta\theta_j/\theta_j)\), which justifies the percentage interpretation of elasticity. These are local approximations; a larger change can produce appreciable curvature and a different result.

When the inputs have estimated uncertainty, a derivative alone is not a measure of how much uncertainty an input creates. For small parameter uncertainty with covariance matrix \(\boldsymbol{\Sigma}\), the first-order approximation to the prediction variance is \(\mathbf{g}^{T}\boldsymbol{\Sigma}\mathbf{g}\), where \(\mathbf{g}\) is the gradient of \(f\) at the evaluation point. Thus both sensitivity and the size and correlation of input uncertainty matter. This approximation is not a general attribution of predictive uncertainty, especially when uncertainty is large or the model is strongly nonlinear.

Worked Example: Clearance Sensitivity of a Concentration Prediction

Consider a hypothetical concentration prediction \(C=5.00\exp(-0.300CL)\) mg/L, where \(CL\) is in L/h. The factor \(0.300\) has units h/L, so the exponent is dimensionless. Evaluate sensitivity at \(CL=2.00\) L/h, holding the other model quantities fixed.

The predicted concentration is \(C=5.00\exp(-0.600)=2.744\) mg/L. Differentiating with respect to clearance gives

\frac{\partial C}{\partial CL}=-0.300C.

At the specified point, this is \(-0.300(2.744)=-0.8232\) mg/L per (L/h). The negative sign means that a small increase in clearance lowers the predicted concentration. The elasticity is \(E_{CL}=(CL/C)(-0.300C)=-0.300CL=-0.600\). Therefore, near this point, a 1% increase in clearance corresponds to approximately a 0.600% decrease in concentration.

For a direct check, increase clearance from 2.00 to 2.20 L/h, a 10% change. The new prediction is \(5.00\exp(-0.660)=2.584\) mg/L. The decrease from 2.744 mg/L is \(100(2.744-2.584)/2.744\approx5.824\%\). The local elasticity predicts a 6.00% decrease for this 10% increase. The difference is consistent with using a first-order approximation for a finite change in a nonlinear model; it is not evidence that the derivative’s sign is wrong.

Alternative Scales and Worked PK and PD Calculations

A raw derivative is useful when changes in the parameter have a meaningful common unit. Across parameters with different units or numerical scales, raw derivatives cannot be compared directly. Elasticities provide a relative scale when both quantities are positive. Another local scale is the derivative multiplied by a chosen plausible parameter change, such as a standard deviation: \((\partial y/\partial\theta_j)s_j\), where \(s_j\) is the chosen change in \(\theta_j\). This gives a prediction change in the units of \(y\). Its interpretation depends on the chosen scale; it is not automatically a measure of uncertainty contribution.

Worked Example: Oral AUC Sensitivity to Bioavailability and Clearance

Assume a hypothetical linear PK model in which oral AUC from zero to infinity is \(AUC=FD/CL\). Here dose \(D=100\) mg, bioavailability \(F=0.60\), and clearance \(CL=0.75\) L/h. The model assumes the same dose and parameter values over the interval and no dose-dependent change in clearance.

The prediction is \(AUC=(0.60)(100)/0.75=80.0\) mg·h/L. Differentiating while holding the other inputs fixed gives

\frac{\partial AUC}{\partial F}=\frac{D}{CL}, \qquad \frac{\partial AUC}{\partial CL}=-\frac{FD}{CL^2}.

At the specified values, \(\partial AUC/\partial F=100/0.75=133.3\) mg·h/L per unit of \(F\), and \(\partial AUC/\partial CL=-60/(0.75)^2=-106.7\) (mg·h/L)/(L/h). The corresponding elasticities are \(E_F=1\) and \(E_{CL}=-1\). For this model, a small 1% increase in bioavailability produces a 1% increase in AUC, while a small 1% increase in clearance produces a 1% decrease.

For a finite 5% increase in clearance, the exact new value is \(60/0.7875=76.19\) mg·h/L, a decrease of \(100(76.19-80.0)/80.0=-4.762\%\). The elasticity approximation is \((-1)(0.05)=-0.050\), or a 5.00% decrease. The exact change differs slightly because the change is finite. This comparison also shows why the sign should be retained: the approximate relative change is \(-5.00\%\), not \(+5.00\%\).

Worked Example: Potency Sensitivity of a PD Prediction

Suppose a hypothetical inhibitory response is predicted by \(R=E_{\max}C/(EC_{50}+C)\), with concentration \(C=2.00\) mg/L, maximum response \(E_{\max}=80.0\) percentage points, and \(EC_{50}=1.00\) mg/L. Treat concentration and maximum response as fixed while evaluating sensitivity to potency parameter \(EC_{50}\).

The prediction is \(R=80.0(2.00)/(1.00+2.00)=53.33\) percentage points. Differentiation gives

\frac{\partial R}{\partial EC_{50}} = -\frac{E_{\max}C}{(EC_{50}+C)^2}.

At the specified point, this derivative is \(-160/9=-17.78\) percentage points per mg/L. The elasticity is

E_{EC_{50}} = \frac{EC_{50}}{R}\frac{\partial R}{\partial EC_{50}} = -\frac{EC_{50}}{EC_{50}+C} = -\frac{1}{3}.

A small 1% increase in \(EC_{50}\) therefore corresponds to approximately a 0.333% decrease in predicted response. For a 10% increase, from 1.00 to 1.10 mg/L, the local approximation gives \((-17.78)(0.10)=-1.778\) percentage points. Direct evaluation gives \(80.0(2.00)/(1.10+2.00)=51.61\) percentage points, a change of \(-1.720\) percentage points. The finite change is close to, but not identical to, the local approximation.

Calculating and Checking Local Sensitivities

For an analytically specified model, differentiate the prediction with respect to the input and evaluate that derivative at the stated parameter set. When analytic differentiation is inconvenient, a central finite difference can approximate the derivative:

\left.\frac{\partial f}{\partial\theta_j}\right|_{\boldsymbol{\theta}_0} \approx \frac{ f(\boldsymbol{\theta}_0+h\mathbf{e}_j) - f(\boldsymbol{\theta}_0-h\mathbf{e}_j) }{2h},

where \(\mathbf{e}_j\) changes only the \(j\)th input and \(h\) is a small step in that input’s units. The calculation holds all other inputs fixed. A step that is too large averages behavior over a wider region and may miss the local derivative; one that is too small can magnify numerical rounding or solver error. If the parameter is near a boundary, a central difference may require invalid values, so a valid one-sided difference or a suitable transformed parameter may be needed.

A practical check is to repeat the finite difference with smaller and larger steps that remain valid. Estimates should be reasonably stable over a suitable range of step sizes. Where an analytic derivative is available, compare it with the numerical estimate. Large disagreement can indicate a coding error, an unsuitable step, numerical instability, or a discontinuity or kink in the prediction. Also verify units, the parameter set, and which quantities were held fixed.

Specify the prediction and evaluation point.
Name the human outcome, its units, the parameter values, and the fixed dose, time, or other conditions.

Choose an interpretable scale.
Use a derivative for unit changes, an elasticity for relative changes in positive quantities, or a scaled derivative for a stated plausible parameter change.

Calculate and verify.
Differentiate analytically or use a checked finite difference. Confirm the sign, units, and approximate response against a small direct perturbation.

Qualify the conclusion.
State that the result is local and conditional on the chosen model, parameter set, and fixed inputs. Consider parameter uncertainty and dependence separately.

Interpretation, Alternatives, and Limitations

Local sensitivity is useful for checking how a human prediction responds near a reference set, comparing inputs on an explicitly chosen scale, and identifying which translated quantities warrant scrutiny at that point. It complements, rather than replaces, uncertainty propagation. If an input is highly sensitive but known precisely, its influence on prediction uncertainty may be limited. A less sensitive input with substantial uncertainty may matter more. Correlated inputs also matter: changing one input alone may not represent plausible joint changes.

The main alternatives answer different questions. A raw derivative reports change per unit of input; elasticity reports approximate relative change per relative input change; a finite-difference calculation is a numerical way to estimate a derivative, not a distinct measure of uncertainty. Analytic differentiation can be precise but may be difficult to implement for complex models, while numerical differences are accessible but require step-size checks. Global sensitivity analysis addresses behavior over uncertain input ranges rather than only near one point; it is appropriate when local linear approximations may not represent the range of interest.

A local ranking can be misleading if the model is curved, if an input change is not small, or if important parameter combinations are constrained or correlated. Sensitivities can also change with time, dose, or the chosen prediction, so there is no universal ranking independent of the output and conditions. A large derivative does not establish a causal biological effect: it describes the mathematical response of the specified model to an input change, conditional on that model.

Ignoring units: derivatives for parameters measured in different units are not directly comparable. State units or use a justified relative scale.
Dropping the sign: reporting only a magnitude hides whether the prediction rises or falls as the input increases.
Overextending a local result: multiplying a derivative by a large parameter change can be inaccurate when curvature is substantial. Compare with direct model evaluation.
Confusing sensitivity with uncertainty: a derivative measures response, not how uncertain the input is. Consider the input’s uncertainty and its dependence on other inputs separately.

Key takeaway: A local derivative or elasticity is conditional on a particular model, output, and parameter set. Report the evaluation point, scale, sign, and units, and check the approximation with a small direct perturbation.

Check Your Understanding

Use the definitions and calculations above to answer these questions. Solutions follow each question.

For \(y=12\exp(-0.20\theta)\), with positive \(\theta\), derive the elasticity of \(y\) with respect to \(\theta\). Solution: The derivative is \(-0.20y\), so \(E_\theta=(\theta/y)(-0.20y)=-0.20\theta\).
For the AUC model \(AUC=FD/CL\), what is the first-order relative change in AUC for a 4% increase in clearance, holding dose and bioavailability fixed? Solution: \(E_{CL}=-1\), so the relative change is approximately \((-1)(0.04)=-0.04\), or a 4% decrease.
Why can a derivative not alone identify the parameter contributing most to predictive uncertainty? Solution: Its impact also depends on the amount of uncertainty in the parameter and on covariances with other inputs; the first-order variance depends on the gradient and the parameter covariance matrix.
A central finite-difference estimate changes substantially when the step size is reduced. Name two possible causes. Solution: The original step may be too large for a local approximation, or the smaller step may amplify numerical rounding or solver error. A discontinuity or kink is another possibility; step-size and model behavior should be checked.