Propagating Uncertainty to AUC, Cmax, and Tmax

From Concentration-Time Curves to Exposure Metrics

A PBPK simulation produces a concentration-time profile for a specified set of model inputs. When those inputs are uncertain, propagating them through the model produces many possible profiles rather than one curve. AUC, Cmax, and Tmax reduce each profile to a different summary: integrated exposure, peak concentration, and the time of that peak. Their distributions therefore describe uncertainty in distinct aspects of the predicted pharmacokinetic response.

The calculation must be repeated separately for every simulated profile. Calculating a metric from an averaged concentration-time curve is generally not equivalent to calculating it for each curve and then summarizing those results. This difference is particularly important for peak metrics, because both the height and timing of a peak can vary across simulations.

Definition: For a concentration profile \(C(t)\) over a specified time horizon from \(0\) to \(T\), \(AUC_{0,T}\) is the area under the profile on that horizon, \(C_{\max,T}\) is the maximum concentration on that horizon, and \(T_{\max,T}\) is the time at which that maximum occurs. If the horizon or extraction method could affect the result, report it with the metric.

For simulation \(b\), write the resulting quantities as \(AUC_{0,T}^{(b)}\), \(C_{\max,T}^{(b)}\), and \(T_{\max,T}^{(b)}\). If concentrations are measured in mg/L and time in hours, AUC is in mg·h/L, Cmax is in mg/L, and Tmax is in hours. These units are not interchangeable: Tmax is not a concentration, and AUC is not a peak value.

Definitions and Calculation Assumptions

Suppose a profile is available at ordered times \(0=t_0<t_1<\cdots<t_m=T\), with concentrations \(C_0,C_1,\ldots,C_m\). A common numerical approximation to the area is the composite trapezoidal rule:

AUC_{0,T}\approx \sum_{j=0}^{m-1} \frac{C_j+C_{j+1}}{2}(t_{j+1}-t_j).

This rule treats concentration as changing linearly between each adjacent pair of recorded times. It is exact for the area under that piecewise-linear interpolation, but it is only an approximation to the area under a curved model profile. AUC on a finite horizon also excludes any area after \(T\); it is not automatically equivalent to total exposure to infinity.

On the evaluated grid, the corresponding direct calculations are:

C_{\max,T}^{\text{grid}}=\max_{0\leq j\leq m} C_j, \qquad T_{\max,T}^{\text{grid}}=t_{j^*}, \quad j^*=\min\{j:C_j=C_{\max,T}^{\text{grid}}\}.

The minimum index in the definition specifies an earliest-time rule if the grid has a tied maximum. With only grid values, Cmax and Tmax are grid-based quantities. A smooth simulated curve may peak between output times, so a denser evaluation grid or a validated peak-finding method may be needed when a continuous-time peak is the intended target. The extraction rule should be consistent across simulations.

These calculations presume that each profile corresponds to the intended model, dose, and input draw; that times are correctly ordered and use consistent units; and that concentrations are finite and defined across the analysis horizon. The model output grid should resolve the concentration changes well enough for the desired accuracy. Any interpolation, extrapolation, or handling of tied peaks must be specified rather than chosen after inspecting results.

Worked Example: Calculating AUC, Cmax, and Tmax for One Profile

Consider a hypothetical simulated profile with concentrations in mg/L at times in hours: \(t=(0,1,2,4)\) and \(C=(0,4,8,2)\). Use the composite trapezoidal rule for \(AUC_{0,4}\), and use the observed grid maximum for Cmax and Tmax.

The three trapezoid areas are:

\begin{aligned} AUC_{0,4} &\approx \frac{0+4}{2}(1-0) +\frac{4+8}{2}(2-1) +\frac{8+2}{2}(4-2)\\ &=2+6+10=18\ \text{mg·h/L}. \end{aligned}

The largest recorded concentration is \(8\) mg/L, attained at \(t=2\) h. Thus \(C_{\max,4}^{\text{grid}}=8\) mg/L and \(T_{\max,4}^{\text{grid}}=2\) h. The interval from 2 to 4 h is two hours long, so its trapezoid area is \(10\) mg·h/L, not \(5\) mg·h/L.

These results describe the numerical profile over 0–4 h. They do not establish that the continuous model's true peak is exactly at 2 h or that exposure after 4 h is zero.

Propagating the Metrics Across Input Uncertainty

For each simulated input set, run the PBPK model, obtain its concentration-time profile, and calculate all three metrics using the same time horizon and calculation rules. The resulting collections of metric values are empirical distributions induced by the specified input distributions, dependence assumptions, model, and numerical calculation. They are not distributions of observed patients unless the simulation explicitly represents that target population.

The metrics need not respond to uncertain inputs in the same way. AUC accumulates concentration over time, so changes across a broad portion of the profile can affect it. Cmax depends on the highest part of the curve and can be especially sensitive to inputs that change absorption rate, distribution, or clearance near the peak. Tmax depends on the location of the maximum: small profile changes may have no effect on a grid-based Tmax until a different time point becomes highest, at which point the reported value can jump. The direction and size of these effects depend on the model and the input combinations; they should be assessed from the propagated simulations rather than assumed.

A useful justification for the trapezoidal calculation follows from its interpolation assumption. On an interval \([t_j,t_{j+1}]\), linear interpolation gives a concentration that changes from \(C_j\) to \(C_{j+1}\). The area is the interval width times the average of its endpoint heights, the area formula for a trapezoid. Summing those interval areas gives the composite rule. This also explains why changing the time grid can change the approximation: a curved profile is represented by a different sequence of straight segments on a different grid.

Worked Example: Comparing Metrics Across Hypothetical Input Draws

Suppose four hypothetical PBPK input draws produce the following profiles at \(t=(0,1,2,3,4)\) h. The profiles are assumed to be finite and evaluated on the same grid; the earliest-time rule is used for any tied maximum. Each AUC below uses the trapezoidal rule.

Draw	Example uncertain inputs	Concentrations at 0, 1, 2, 3, 4 h (mg/L)	AUC (mg·h/L)	Cmax (mg/L)	Tmax (h)
1	F = 0.80, CL = 4.0 L/h, ka = 1.2 h⁻¹	0, 4, 8, 5, 2	18.0	8	2
2	F = 0.70, CL = 4.5 L/h, ka = 0.6 h⁻¹	0, 3, 6, 6, 3	16.5	6	2
3	F = 0.90, CL = 5.0 L/h, ka = 0.4 h⁻¹	0, 2, 5, 7, 4	16.0	7	3
4	F = 0.60, CL = 4.2 L/h, ka = 1.0 h⁻¹	0, 3, 5, 4, 2	13.0	5	2

For draw 3, for example, the trapezoid areas are \((0+2)/2=1\), \((2+5)/2=3.5\), \((5+7)/2=6\), and \((7+4)/2=5.5\) mg·h/L, giving \(AUC_{0,4}=16.0\) mg·h/L. Its largest grid concentration is \(7\) mg/L at 3 h. For draw 2, the maximum \(6\) mg/L occurs at both 2 and 3 h; the stated earliest-time rule gives \(T_{\max,4}^{\text{grid}}=2\) h.

Across the four draws, the arithmetic mean AUC is \((18+16.5+16+13)/4=15.875\) mg·h/L. The sample standard deviation is \(\sqrt{13.1875/3}=2.097\) mg·h/L. The mean Cmax is \((8+6+7+5)/4=6.5\) mg/L, with sample standard deviation \(\sqrt{5/3}=1.291\) mg/L. The mean Tmax is \((2+2+3+2)/4=2.25\) h, with sample standard deviation \(0.500\) h. These are summaries of four simulated outputs, not precise estimates of the full induced distributions.

The example illustrates why the outputs should be inspected separately. Draw 3 has a later peak than the other draws, whereas draw 1 has the largest AUC and Cmax. The four rows alone do not isolate the effect of any one input: F, CL, and ka differ together, and the PBPK model maps their joint values into each profile. To assess an input's influence, use the specified joint input sampling design and examine the resulting output behavior; do not infer a universal causal effect from this small table.

Time Grids, Peak Extraction, and Diagnostics

A time grid that is adequate for AUC may still be inadequate for Cmax or Tmax. Integration uses contributions over all intervals, while a narrow peak can be missed if no output time lies near it. Conversely, grid-based Tmax is restricted to the times that were evaluated. Refining the grid near rapid absorption or a suspected peak can improve resolution, but grid refinement should be checked for stability rather than assumed to fix every error.

Worked Example: What a Coarse Grid Can Miss

Consider a hypothetical profile sampled at 0, 1, 2, and 3 h with concentrations \(0, 6, 10, 4\) mg/L. The grid calculation gives \(C_{\max,3}^{\text{grid}}=10\) mg/L and \(T_{\max,3}^{\text{grid}}=2\) h. Its trapezoidal AUC is \(3+8+7=18\) mg·h/L.

Now suppose a denser evaluation of the same hypothetical model shows that concentration reaches 10.4 mg/L at 1.8 h. The coarse-grid peak is then lower than the resolved peak by \(10.4-10=0.4\) mg/L, and the coarse-grid Tmax is 0.2 h later than the resolved peak time. The coarse AUC remains the area estimate from its own grid; its accuracy cannot be concluded from the peak comparison alone. Re-evaluating at intermediate times and comparing AUC and peak metrics across successively finer grids provides a diagnostic for discretization error.

Additional checks should include confirming that all draws use the same time units and horizon; that the initial concentration and dose timing are represented as intended; and that concentrations and metric outputs are finite. If a simulation fails or produces missing values, determine and report how it was handled. Silently dropping failed draws can alter the apparent output distribution, especially if failures occur for particular input regions. Similarly, an unexpectedly constant Tmax may reflect a coarse grid, not genuine certainty about peak timing.

The area calculation can also be compared with an integration routine that uses denser solver output or a suitable adaptive method. For a peak between output points, a validated continuous-time optimization or local interpolation may be considered, but its assumptions should be stated and applied consistently. Interpolation does not recover an unresolved peak without adequate information about the curve shape. If two adjacent grid values tie for the maximum, report the tie rule or the tied time range; do not let software-specific behavior silently determine the result.

Choosing and Reporting the Metric Calculation

The trapezoidal rule is transparent and convenient when a common grid of model predictions is available. Adaptive numerical integration may be preferable when the model can be evaluated reliably at arbitrary times and the curve changes rapidly. For Cmax and Tmax, the alternatives are grid-based extraction, denser output, or a validated method for locating a continuous peak. These choices target somewhat different numerical quantities, so they should be documented rather than mixed across simulations.

AUC depends on the selected horizon and the concentration-time integration method. Cmax and Tmax depend on the horizon, grid resolution, and peak rule. Consequently, reporting just “AUC,” “Cmax,” or “Tmax” may leave important calculation choices unclear. State the dose context, horizon, units, integration method, peak extraction rule, and number of simulated profiles. Summarize the distributions of the three metrics separately and retain their joint pairing by simulation when comparing them: the AUC, Cmax, and Tmax from a given draw belong together.

These metrics are useful because they turn a set of simulated profiles into quantities that can be compared with exposure or peak-related objectives. They also lose information: two profiles can share an AUC while having different peaks, and two can share Cmax while differing in overall exposure. Retain the concentration-time profiles when the shape, timing of exposure, or behavior outside the metric summary matters.

Common pitfall: Calculating AUC, Cmax, and Tmax from the average concentration curve and treating those as the average metrics is generally incorrect. In general, the maximum of an average curve is not the average of the individual maxima, and its peak time need not equal the average peak time. Calculate each metric for every simulated profile first; then summarize the resulting metric values.

Fix the target and horizon.
Specify the dose context, time window, units, and whether the target is grid-based or intended to represent a continuous-time metric.

Calculate within each draw.
Integrate each profile with the selected method, extract its maximum, and record the corresponding time using a stated tie rule.

Check numerical and simulation quality.
Inspect grid resolution, convergence under grid refinement, failed runs, non-finite outputs, and consistency of the horizon and units.

Summarize the paired outputs.
Describe AUC, Cmax, and Tmax distributions separately, while preserving the fact that all three values from a draw arise from the same simulated profile.

Check Your Understanding

Use the definitions and calculation rules above. Solutions follow the questions.

At times 0, 2, and 5 h, a hypothetical profile has concentrations 0, 6, and 3 mg/L. Calculate the trapezoidal AUC over 0–5 h, and give the grid-based Cmax and Tmax.
A simulated profile has the same maximum concentration at 2 and 3 h. What is Tmax under the earliest-time rule, and what should be reported if the scientific intent is to describe a tied peak?
Why can the maximum of the mean concentration curve differ from the mean of the individual Cmax values?
A coarse-grid Tmax is identical in every simulation. Name one diagnostic that could help determine whether this reflects the model predictions or inadequate temporal resolution.

Solutions. (1) The two trapezoids have areas \((0+6)(2)/2=6\) and \((6+3)(3)/2=13.5\) mg·h/L, so AUC is \(19.5\) mg·h/L. Cmax is \(6\) mg/L and Tmax is \(2\) h. (2) The earliest-time rule gives \(2\) h; report the tie or tied time range as well if describing the peak rather than only applying the specified extraction rule. (3) Taking a maximum is a nonlinear operation: each profile may peak at a different time, so maximizing after averaging does not generally give the average of the profile-specific maxima. (4) Re-evaluate the curves on a finer grid near the peak and compare Cmax and Tmax across grids; a change indicates that the coarse grid did not fully resolve the peak.