From a PBPK Model to a Screening Analysis
A PBPK model may contain dozens or hundreds of uncertain inputs, but not all are equally consequential for a prediction of interest. A screening analysis is a structured first pass: it varies inputs across prespecified ranges, evaluates selected outputs, and identifies which inputs warrant closer examination. Its value depends less on producing a complicated ranking than on making the choices behind that ranking explicit and defensible.
The analysis must begin with a question about a particular output in a particular model context. For example, a team might ask which uncertain inputs most affect the predicted plasma AUC from 0 to 24 hours after a specified dose in a specified population. The ranking answers that question under the ranges and design used. It is not a universal ordering of the parameters, and it does not establish that a high-ranked input is identifiable from available data or that a low-ranked input can safely be fixed.
A useful record therefore states the model version, the scenario being evaluated, the output targets, each parameter's range and scale, the screening design, the calculation used to rank inputs, and any failed or excluded simulations. The steps below use Morris screening, a practical global screening method for many inputs. Earlier material on Morris screening establishes its elementary-effect framework; here the emphasis is on building a complete, auditable analysis.
Define Ranges and Targets Before Running the Model
For every candidate input, specify a lower and upper bound and explain why those bounds are scientifically plausible. Sources can include measured values, literature ranges, informed expert elicitation, or a documented uncertainty analysis. Bounds should also respect model constraints: for example, a clearance or fraction may need to remain positive, and physiological quantities may have to satisfy a mass-balance or consistency condition. A convenient interval is not automatically a defensible interval.
Choose the scale on which a change is meaningful. For a parameter \(x_i\) with bounds \(L_i\) and \(U_i\), linear rescaling gives \(u_i=(x_i-L_i)/(U_i-L_i)\), where \(0\leq u_i\leq1\). If proportional changes are more appropriate for a positive parameter, a log scale can instead be used: \(u_i=(\ln x_i-\ln L_i)/(\ln U_i-\ln L_i)\). The selected scale changes what counts as an equal step and can therefore change the ranking.
Output targets should likewise be defined before looking at the screening results. They may be summary endpoints, such as AUC or peak concentration, or prespecified values of a time course. State the units, time window, dose, population, and any normalization. If outputs have very different units, their raw effects are not directly comparable. A ratio to a reference prediction, for example, can put an endpoint on a dimensionless scale, but the denominator and reference run must be stated. Multiple targets are generally ranked separately rather than silently combined.
Worked Example: Setting a Defensible Screening Scope
Suppose a hypothetical PBPK model is being screened for its prediction of plasma AUC from 0 to 24 hours after one specified oral dose in a defined adult population. The team selects four uncertain inputs and expresses each as a multiplier of its nominal value.
| Input multiplier | Proposed range | Scale and reason |
|---|---|---|
| Hepatic intrinsic clearance | 0.5 to 1.5 | Linear multiplier range, based on a prespecified uncertainty assessment |
| Plasma unbound fraction | 0.7 to 1.3 | Linear multiplier range, subject to the model's valid fraction bounds |
| Tissue:plasma partition multiplier | 0.6 to 1.4 | Linear multiplier range for a bounded sensitivity exercise |
| Renal clearance | 0.75 to 1.25 | Linear multiplier range around the nominal value |
These bounds are hypothetical and would need supporting evidence in an actual analysis. The team defines one target: AUC from 0 to 24 hours, in concentration-times-time units, divided by the AUC from the nominal-input run. The target is therefore dimensionless and equals 1 at the reference run. The target, dose, population, and time window remain fixed during this screen. If any input is correlated with another, the team must not treat those inputs as freely and independently varying factors without justification; it should use a design that respects the dependence or clearly limit the interpretation to an appropriate conditional analysis.
Constructing a Morris Screen
Morris screening evaluates inputs on a grid in the unit hypercube. For a factor \(u_i\), an elementary effect is the change in the model output caused by changing that factor by a fixed grid step \(\Delta\), while holding the other factors at their current design values:
Here \(f\) is the selected model output, \(\mathbf{u}\) is a vector of scaled inputs, and \(\mathbf{e}_i\) is a vector that is 1 in position \(i\) and 0 elsewhere. The step must stay within the grid bounds. In a standard Morris design with \(p\) grid levels, a commonly used step is \(\Delta=p/[2(p-1)]\). A trajectory changes each of the \(k\) factors once, so it requires \(k+1\) model evaluations. With \(r\) trajectories, the design uses \(r(k+1)\) evaluations, before any extra runs for reference checks or retries.
Elementary effects can be positive or negative. For each input, the mean absolute elementary effect, \(\mu_i^*\), summarizes the typical magnitude of the change across the design. The sample standard deviation of its signed elementary effects, \(s_{EE,i}\), summarizes how much those effects vary across the sampled locations. A high \(\mu_i^*\) indicates a substantial effect over the specified range and design; a high \(s_{EE,i}\) can indicate nonlinearity or interactions, but can also arise from other features of the response or from numerical error.
The signed mean \(\overline{EE}_i\) can be informative about an overall direction, but it can be near zero when effects of opposite signs cancel. This is why \(\mu_i^*\), rather than the signed mean alone, is commonly used for a screening ranking. These summaries are design-based screening measures. They are not variance-based fractions of output variance and do not, by themselves, estimate a probability distribution for the model prediction.
Worked Example: Calculating One Elementary Effect
In the hypothetical analysis, hepatic intrinsic clearance is represented by a multiplier \(u_1\) on the interval from 0 to 1. Use a four-level Morris grid, so \(p=4\) and \(\Delta=4/[2(4-1)]=2/3\). On one trajectory, the model gives a normalized AUC target of 1.60 at \(u_1=1/3\), then 0.40 at \(u_1=1\), with all other inputs held at their trajectory values.
The elementary effect is
Thus, on this trajectory, increasing the clearance multiplier by one unit on the scaled input axis corresponds to a change of \(-1.80\) in the normalized AUC target. The negative sign describes the direction for this particular change. It does not mean every trajectory must have the same direction, nor does it mean the output changes by \(-1.80\) for every physically possible unit change in clearance.
From Model Runs to a Ranked Table
For a transparent analysis, retain the input coordinates and output values for every run, not just the final summary. Compute each elementary effect from the paired evaluations that differ in exactly one factor, then calculate \(\mu_i^*\) and \(s_{EE,i}\) for each input and target. State the number of trajectories and the grid specification. Ranking by \(\mu_i^*\) gives a simple ordering by typical absolute effect on the chosen output scale; reporting \(s_{EE,i}\) alongside it helps reveal effects that vary substantially across the design.
Worked Example: Producing and Interpreting a Ranked Result Table
For illustration, suppose four hypothetical trajectories produce the following signed elementary effects for the dimensionless AUC target. Each value has units of target change per unit change in the scaled input. The values are illustrative results, not observations from a real study.
| Input | Elementary effects from four trajectories | \(\mu_i^*\) | \(s_{EE,i}\) | Rank by \(\mu_i^*\) |
|---|---|---|---|---|
| Hepatic intrinsic clearance | 2.0, 1.8, 2.2, 2.0 | 2.000 | 0.1633 | 1 |
| Renal clearance | 1.2, 1.1, 1.3, 1.2 | 1.200 | 0.08165 | 2 |
| Tissue:plasma partition multiplier | 0.4, -0.3, 0.5, -0.4 | 0.400 | 0.4655 | 3 |
| Plasma unbound fraction | 0.1, 0.1, 0.2, 0.1 | 0.125 | 0.05000 | 4 |
For hepatic intrinsic clearance, the mean absolute effect is \((2.0+1.8+2.2+2.0)/4=2.000\). Its signed mean is also 2.000, and the sample standard deviation is \(\sqrt{[(0)^2+(-0.2)^2+(0.2)^2+0^2]/3}=0.1633\). For the partition multiplier, the mean absolute effect is \((0.4+0.3+0.5+0.4)/4=0.400\). Its signed mean is \((0.4-0.3+0.5-0.4)/4=0.0500\), despite the larger average absolute magnitude. The sample standard deviation of its signed effects is \(\sqrt{0.65/3}=0.4655\).
The table places hepatic intrinsic clearance first for this target and design. The partition multiplier's relatively large \(s_{EE}\), together with effects of both signs, flags substantial variation in its elementary effects across the sampled locations. That pattern merits inspection but does not prove a specific interaction or mechanism. The four trajectories are also few; a real analysis should choose a trajectory count adequate for stable screening, inspect rank stability as the design is enlarged, and avoid presenting the illustrative precision as evidence of a precise population quantity.
Justification, Diagnostics, and Alternatives
The reason for using \(\mu_i^*\) rather than the signed mean follows directly from the summary. If an input has effects \(a\) and \(-a\) on two trajectories, its signed mean is \([a+(-a)]/2=0\), although the magnitude on both trajectories is \(|a|\). Its mean absolute effect is \([|a|+|-a|]/2=|a|\). Thus cancellation can hide a consequential input if ranking relies only on the signed mean. The accompanying spread measure preserves a different piece of information: how much the signed effects vary across design locations.
Confirm that model version, population, dose, output definition, time window, and reference run match the stated question.
Verify units, valid bounds, transformations, grid values, and any dependence between inputs. Confirm that every paired run used for an elementary effect differs in exactly one factor.
Inspect convergence, mass balance where applicable, valid output units, and failed runs. Repeat evaluations or tighten numerical tolerances when computational noise could be similar in size to an elementary effect.
Increase the number of trajectories or repeat the design with different valid trajectories. If the leading ranks change substantially, report the ordering as unstable rather than implying a definitive ranking.
A ranking may be misleading if the ranges are too narrow, omit plausible values, cross invalid model regions, or use a scale that makes the grid steps scientifically inappropriate. Outputs close to zero, discontinuities, solver failures, and noisy simulations can also produce unstable effects. Check scatterplots or trajectory-level results when a summary looks surprising. A large \(s_{EE,i}\) is a prompt for investigation, not a diagnostic that uniquely distinguishes nonlinearity from interaction, and a small value does not prove either is absent.
The main alternatives answer different questions. Local sensitivity evaluates changes near a nominal parameter set and is efficient when the question concerns small perturbations around that point. Morris screening explores a broader specified range at comparatively modest computational cost, but its summaries are not variance decompositions. Sobol indices quantify variance contributions under their sampling assumptions and generally require more model evaluations; ordinary independent-input Sobol designs are not valid descriptions of correlated inputs. One-at-a-time analysis is easy to communicate but can miss effects that vary across the input space and can be misleading when several inputs change together in realistic ways. For correlated PBPK inputs, a screen must preserve or explicitly account for the dependence; otherwise the sampled combinations and resulting interpretation may not represent plausible parameter sets.
A screening table is a prioritization aid, not a parameter-reduction decision. It does not establish identifiability, prove that low-ranked parameters are negligible for other outputs, or show that a high-ranked parameter can be estimated from the available data. A ranking applies only to the stated model, target, ranges, and scenario. Changes in dose or physiological scenario can change the ordering and require a separately defined analysis.
Check Your Understanding
Answer each question using the definitions and hypothetical calculations in this tutorial.
- Why must a screening report state both the output target and the parameter ranges?
- For two elementary effects \(0.8\) and \(-0.8\), calculate the signed mean and mean absolute effect. Which better describes typical magnitude?
- A screen uses five uncertain inputs and six Morris trajectories. How many model evaluations are needed for the trajectories, excluding reference or retry runs?
- What can a high \(s_{EE,i}\) suggest, and why does it not prove that an input interaction is present?
- Why might an independent-input screening design be inappropriate for two correlated PBPK parameters?
Solutions. (1) The ranking is conditional on the output being evaluated and the range and scale over which each input changes; changing either can change the result. (2) The signed mean is \((0.8-0.8)/2=0\), while the mean absolute effect is \((0.8+0.8)/2=0.8\). The mean absolute effect captures the nonzero typical magnitude. (3) Each trajectory requires \(k+1=5+1=6\) evaluations, so six trajectories require \(6\times6=36\) model evaluations. (4) A high spread can flag effects that vary across the sampled design, potentially because of nonlinearity or interaction. It is not unique to interaction; numerical error or other response features can also contribute, so further inspection is needed. (5) Independent sampling can generate combinations that violate the dependence structure or are implausible. The resulting ranking may then describe an inappropriate input space rather than the intended joint uncertainty.