Sensitivities of Time Courses and Summary Endpoints

From a Time Course to an Endpoint

A PBPK model can predict concentration in each compartment over time, but decisions often depend on a smaller set of outputs: concentration at a specified time, area under the concentration-time curve (AUC), maximum concentration (Cmax), a predose trough, or exposure in a tissue. Each is a different mathematical function of the model output. Its sensitivity to a parameter therefore depends not only on the simulated profile but also on how the endpoint is defined.

Let the model state \(y(t,\theta)\) satisfy an ordinary differential equation \(\dot{y}=f(t,y,\theta)\), where \(\theta\) is a model parameter. A concentration output is \(C(t,\theta)=g(t,y(t,\theta),\theta)\). We consider local sensitivities: derivatives evaluated at a specified parameter value, with the other parameters and model conditions held fixed. This is a local description, not a prediction of the effect of large parameter changes.

For a time-course output, the sensitivity is itself a function of time: \(\partial C(t,\theta)/\partial\theta\). At a fixed time, the time coordinate is held constant during differentiation. This differs from differentiating an endpoint whose time of evaluation depends on \(\theta\), as occurs for a continuously defined peak. Earlier material on forward and adjoint sensitivities provides ways to calculate parameter derivatives for model outputs; here the emphasis is how those derivatives map to time courses and summary endpoints.

Time-Course Sensitivity at a Fixed Time

Consider a hypothetical concentration curve after an oral dose, \(C(t)=10(e^{-k_e t}-e^{-k_a t})\) mg/L, with \(k_e=0.20\) h\(^{-1}\) and \(k_a=1.00\) h\(^{-1}\). The coefficient 10 mg/L is fixed for this calculation. This is a simple illustrative profile, not a claim that every PBPK concentration follows this form. The assumptions are that the stated curve applies over the time considered, the parameters are positive, and the derivative is evaluated locally at the stated parameter values.

The sign and size depend on time. For this profile, \(\partial C/\partial k_a=10t e^{-k_at}\) is positive for \(t>0\), but its magnitude eventually becomes small as \(e^{-k_at}\) decays. The derivative at one time cannot stand in for the sensitivity of the whole profile: the profile may be more responsive at some times than others, and a parameter can affect different time regions in different directions in more complex models.

Sensitivity of AUC and Integration Limits

For a concentration profile, finite-window AUC from \(t_0\) to \(T\) is \(Q(\theta)=\int_{t_0}^{T}C(t,\theta)\,dt\). If \(C\) and its parameter derivative are continuous over the interval, and the limits are fixed, differentiation under the integral gives \(\partial Q/\partial\theta=\int_{t_0}^{T}\partial C(t,\theta)/\partial\theta\,dt\). If a limit also depends on \(\theta\), endpoint terms must be added; changing the window changes the endpoint being studied.

For a simple illustration of finite-window AUC and its derivative, use a hypothetical mono-exponential concentration \(C(t)=10e^{-k_e t}\) mg/L, with fixed initial concentration 10 mg/L. This curve is intentionally distinct from the oral profile above. It isolates the effect of \(k_e\) on exposure over a stated interval.

Peak Concentration and a Moving Peak Time

A continuous Cmax is the maximum of \(C(t,\theta)\) over a specified time domain. Its maximizing time can change with the parameter. For a smooth profile with a unique interior maximum at \(t_{\max}(\theta)\), the peak satisfies \(\partial C/\partial t=0\) there. Differentiating \(C(t_{\max}(\theta),\theta)\) by the chain rule gives a useful result.

A sampled maximum is not necessarily the continuous Cmax. If the endpoint is defined as the largest model prediction at a prespecified set of sampling times, the maximizing time can jump from one sample time to another as parameters change. Within a region where the same sample remains the largest, differentiate the concentration at that fixed time. At a switch between tied maxima, the sampled maximum may not have a single ordinary derivative.

Troughs and Tissue Exposure

A trough must be tied to a dosing regimen and an observation rule. For example, a predose steady-state trough after repeated instantaneous doses is not the same endpoint as concentration at a fixed time after one dose. Similarly, tissue exposure may mean tissue concentration at one time, tissue AUC over a specified window, or another defined quantity. The compartment, interval, units, and any tissue-to-plasma relationship must be stated before interpreting its sensitivity.

Calculation Choices, Diagnostics, and Limitations

The appropriate calculation depends on the endpoint. A fixed-time concentration uses the time-course derivative at that time. A finite-window AUC integrates the time-course derivative over its stated limits when those limits are fixed. A smooth continuous maximum uses the parameter derivative at the peak time, while a sampled maximum follows the specified sampling grid. A predose trough requires a dosing schedule and a definition of steady state. Tissue exposure requires a tissue output and a defined window or other summary rule.

Directly differentiating a closed-form endpoint, propagating forward sensitivities through the model, and using an adjoint calculation are alternative routes to derivatives. Their computational costs differ, particularly with many parameters or outputs. Regardless of route, the endpoint definition must be identical. A numerical finite-difference check can be helpful, but its result depends on the perturbation size: a step that is too large reflects nonlocal curvature, while a step that is too small can amplify numerical solver error. A derivative should be checked for units, sign, and stability to reasonable numerical settings.

Potential warning signs include a reported AUC whose units do not include time, a peak value evaluated at a time that is not actually maximal, a trough that changes meaning across scenarios, or a sensitivity that changes substantially when the ODE solver tolerances are tightened. For an AUC calculation, compare numerical integration with the model’s integrated output or a finer time grid. For a peak, inspect the profile around the proposed maximum and verify whether the endpoint is continuous Cmax or a maximum over recorded times. For tissue exposure, confirm that the reported output is tissue concentration rather than plasma concentration and that its units and integration window are correct.

Local derivatives have limitations. They describe behavior near one parameter vector and one scenario; they do not establish that a finite parameter change will produce the same proportional response. A derivative can also be poorly defined at a nonsmooth endpoint, such as a sampled maximum at a switch between tied times. Finally, endpoint sensitivities answer how a selected model prediction changes locally; they do not by themselves establish predictive accuracy or clinical importance.

Common Mistake

A common error is to differentiate \(C(t_{\max}(\theta),\theta)\) as though the peak time were fixed, then add a nonzero time-shift contribution without checking the peak condition. For a smooth interior maximum, the chain rule gives \(\frac{dC_{\max}}{d\theta}=\frac{\partial C}{\partial\theta}+\frac{\partial C}{\partial t}\frac{dt_{\max}}{d\theta}\). Since \(\partial C/\partial t=0\) at that maximum, the second term is zero. The correction is not to ignore the moving peak when locating it: calculate the parameter-dependent peak time, then evaluate the partial derivative there. For a sampled maximum or a boundary maximum, apply the endpoint’s actual definition instead.