Why a PBPK Sensitivity Ranking Depends on the Prediction
A PBPK model can rank one parameter as most influential for an early plasma concentration and another as most influential for a late tissue concentration. This is not necessarily a contradiction or a computational error. A parameter can strongly affect distribution into a tissue but have little effect on an early plasma sample; another can govern the initial concentration while having a smaller effect after distribution and elimination have proceeded.
The practical question is therefore not simply “Which parameter is most sensitive?” It is “Which parameter most affects the prediction that matters?” A ranking for a single concentration at one time answers a different question from a ranking for a tissue-to-plasma ratio, an exposure summary, or concentrations across a dosing interval. The analysis must identify the output, time or time window, and scenario before ranking parameters.
The earlier material on local sensitivity and dimensionless sensitivity coefficients defines a derivative-based measure of parameter influence. Here, the emphasis is on how that measure changes across time and endpoints, and on how to summarize it without losing information relevant to the prediction task.
This derivative is local: it describes behavior near one parameter vector, dose, and physiological scenario. It assumes that the model output is differentiable with respect to the parameter and that the output is positive when the log form is used. If an output is zero or sufficiently close to zero, the relative derivative can be undefined or numerically unstable; use a clearly defined absolute-scale measure instead, such as \(\partial Y/\partial\theta_j\) or a scaled derivative based on a meaningful reference concentration.
Why Rankings Change Across Time and Endpoints
Sensitivity is a property of a particular output under particular conditions, not an immutable label attached to a parameter. In a PBPK model, parameters can affect processes that become important at different stages. For example, a parameter affecting delivery from plasma to tissue may have a larger effect during distribution than during the initial rise in plasma concentration. A parameter affecting elimination may have a modest early effect but a larger influence on a late concentration. The direction of an effect can also change: a parameter may increase one output at one time while decreasing it at another as the competing processes evolve.
The output definition matters as well. Plasma concentration and tissue concentration are different model outputs, even when evaluated at the same time. So are a tissue-to-plasma ratio, a maximum concentration, and an area under the concentration-time curve. Their sensitivities need not have the same magnitude, sign, or ordering. Dimensionless sensitivities help compare relative effects across outputs with different units, but do not make the endpoints interchangeable.
A ranking should state at least the endpoint, time or time window, dose and physiological scenario, parameter values, and whether it ranks signed sensitivities or their magnitudes. Without these details, a list such as “clearance is first” is incomplete and can be misleading.
Worked Example: A Ranking Reverses Across a Concentration-Time Profile
Suppose a hypothetical PBPK model gives the following absolute dimensionless sensitivities of plasma concentration to three parameters at early, middle, and late times. The values are illustrative, not results from a real study.
| Parameter | Early | Middle | Late |
|---|---|---|---|
| A | 0.80 | 0.10 | 0.20 |
| B | 0.25 | 0.45 | 0.35 |
| C | 0.10 | 0.20 | 0.60 |
At the early time, A ranks first because \(0.80>0.25>0.10\). At the middle time, B ranks first because \(0.45>0.20>0.10\). At the late time, C ranks first because \(0.60>0.35>0.20\). Thus, no single parameter is most influential at every sampling time.
If the task is prediction at the late time only, the late-time ranking is the relevant one: C, B, A. If the task concerns the full profile, a summary across the times may be more useful. For example, assign weights \(0.5\), \(0.3\), and \(0.2\) to the early, middle, and late times, respectively, and compute the weighted root-mean-square (RMS) sensitivity \(R_j\):
For A, the calculation is
For B, \(R_B=\sqrt{0.5(0.25^2)+0.3(0.45^2)+0.2(0.35^2)}=\sqrt{0.1165}\approx0.3413\). For C, \(R_C=\sqrt{0.5(0.10^2)+0.3(0.20^2)+0.2(0.60^2)}=\sqrt{0.089}\approx0.2983\). Under these particular weights and this RMS definition, the profile ranking is A, B, C. It differs from the late-time ranking because the early time receives the greatest weight and A is especially influential there.
The numbers do not show that A is universally more important than C. They show that A has the larger weighted RMS sensitivity for this chosen plasma profile and these chosen time weights.
Constructing a Ranking for a Time Window
The weighted RMS is useful when large sensitivities at any selected time matter, regardless of direction. Squaring before averaging prevents positive and negative sensitivities from cancelling. The square root returns the score to the scale of the original dimensionless sensitivities. The weights define the prediction task: they may represent the relative importance of selected times or a numerical approximation to an average over a specified interval.
For a continuous prediction window \([t_a,t_b]\), a corresponding weighted summary can be written as
This is a definition of a task-specific summary, not a universal measure of importance. In numerical work, the time points and weights should represent the target window rather than merely reflect where the model solver happened to save output. With irregularly spaced time points, equal weights can overemphasize densely sampled regions. Quadrature weights, or weights based on the actual observation schedule when that schedule defines the task, make the intended emphasis explicit.
The RMS score is not the only reasonable summary. A maximum absolute sensitivity highlights the worst local influence and can be appropriate when any time in a window must be protected against large local effects. A weighted mean of signed sensitivities summarizes net directional change, but can be close to zero when positive and negative effects cancel. A sensitivity of a defined summary endpoint, such as an area under the curve, directly addresses that endpoint; it need not equal an average of pointwise sensitivities. The choice should follow the prediction question, not a desire to produce one ranking for all purposes.
Worked Example: Plasma and Tissue Endpoints Give Different Rankings
At a specified hypothetical time and dose, suppose the dimensionless sensitivities for plasma and liver concentration are as follows. All values are illustrative.
| Parameter | Plasma concentration | Liver concentration |
|---|---|---|
| A | 0.60 | 0.15 |
| B | 0.25 | 0.70 |
| C | 0.10 | 0.30 |
For plasma, the absolute-sensitivity ranking is A, B, C because \(0.60>0.25>0.10\). For liver, it is B, C, A because \(0.70>0.30>0.15\). The ranking depends on the endpoint, even though the time and scenario are held fixed.
To interpret the scale, consider a small 1% relative increase in a parameter. The local approximation for plasma under A is a relative concentration change of about \(0.60\times1\%=0.60\%\). Under B, it is about \(0.25\times1\%=0.25\%\). For liver concentration under B, it is about \(0.70\times1\%=0.70\%\). These are first-order local approximations, not exact predictions for a finite change.
If the decision concerns plasma monitoring, the plasma ranking is pertinent. If it concerns liver exposure, the tissue ranking is pertinent. Combining the two endpoints would require a stated rule, such as weights representing their relative importance. Taking an unweighted average without a prediction rationale would silently impose equal importance.
Signs, Weights, and Diagnostics
Before reporting a ranking, inspect the signed sensitivity trajectories as well as the ranked summary. A parameter with sensitivities that change sign may have a modest signed average but a substantial effect at individual times. Conversely, a large RMS score can reflect alternating increases and decreases that are important for profile shape, even if they partly cancel in a particular exposure summary. The signs answer a directional question; the absolute or squared summaries answer questions about magnitude.
A defensible calculation also checks the following:
- Endpoint validity: confirm that each output is the intended model quantity and that relative sensitivities are stable. Near-zero concentrations can produce extreme relative derivatives that reflect the denominator rather than a practically large absolute change.
- Time coverage: include the times at which the prediction will be used. A ranking based only on early samples cannot establish which parameter matters most for a late tissue prediction.
- Weight definition: state how the weights were chosen and verify that they are nonnegative and sum to one for the discrete RMS definition. If times are irregularly spaced, check that the weights do not unintentionally favor dense regions.
- Locality and numerical stability: the ranking applies at the evaluated parameter values and scenario. Check that derivative estimates are stable to reasonable numerical settings and, where appropriate, small changes in the finite-difference step.
- Signs and crossings: plot or tabulate signed sensitivities across time. Sign changes and rank crossings explain why a single-time ranking may not represent the entire profile.
Large local sensitivities indicate potential local influence, not necessarily large uncertainty in a prediction. A parameter may be sensitive but known precisely; another may be less sensitive but poorly constrained. Sensitivity rankings alone also do not establish causation, guarantee parameter identifiability, or show what happens under large changes. Those questions require information beyond the derivative ranking.
Worked Example: Changing the Prediction Window Changes the Summary
A hypothetical plasma profile has the following signed sensitivities at three times. The first time is early and the third is late.
| Parameter | Early | Middle | Late |
|---|---|---|---|
| A | 0.80 | -0.20 | -0.30 |
| B | 0.20 | 0.35 | 0.40 |
| C | 0.10 | 0.25 | 0.55 |
With equal weights \(w_i=1/3\), the RMS score for A is \(R_A=\sqrt{(0.80^2+(-0.20)^2+(-0.30)^2)/3} =\sqrt{0.770/3}\approx0.5066\). For B it is \(R_B=\sqrt{(0.20^2+0.35^2+0.40^2)/3} =\sqrt{0.3225/3}\approx0.3279\). For C it is \(R_C=\sqrt{(0.10^2+0.25^2+0.55^2)/3} =\sqrt{0.375/3}\approx0.3536\). The equal-weight profile ranking is A, C, B.
For a late-time-only task, the scores reduce to the late-time absolute sensitivities: C is first at \(0.55\), B second at \(0.40\), and A third at \(0.30\). Thus the late-time ranking is C, B, A. The difference is expected because the profile summary gives the early value of A substantial influence, while the late-only task does not.
A signed average for A would be \((0.80-0.20-0.30)/3=0.10\), which is much smaller than its RMS score of about \(0.5066\). That signed average describes net direction across these equally weighted points; it does not mean A has little effect throughout the profile. The early positive and later negative sensitivities partly cancel in the signed average.
Choosing and Reporting the Ranking
The main alternatives answer distinct questions. A pointwise ranking is transparent and appropriate for a prediction at one specified time. A sensitivity of a summary endpoint directly targets that endpoint. A time-weighted RMS ranking summarizes local magnitude over a window and avoids sign cancellation. A maximum-magnitude ranking emphasizes the largest local effect. Global sensitivity measures, covered in earlier material, address variation over specified parameter distributions rather than derivatives at one nominal parameter vector. They can therefore produce different rankings without contradicting a local analysis.
No option is universally superior. A pointwise ranking can miss influence elsewhere in the profile. An RMS score depends on its time weights and can conceal where the influence occurs unless the time course is also shown. A maximum score can be dominated by a narrow time region. A summary-endpoint sensitivity is focused but does not describe all components of the profile. Global measures answer a broader uncertainty-based question but require specified input distributions and are not interchangeable with local sensitivities.
A concise report should name the endpoint, time or window, scenario, sensitivity definition, and ranking summary. For a window summary, report the weights or their construction and show enough time-resolved results to reveal sign changes or rank crossings. If rankings differ between plasma and tissue, present them separately unless there is a scientifically justified combined prediction target. The goal is not to eliminate task dependence; it is to make it visible and align the ranking with the use of the model.
Check Your Understanding
Answer each question, then compare with the short solutions below.
- If \(S_B^Y(t)=-0.7\) and \(S_A^Y(t)=0.4\) at one time, which parameter ranks higher by absolute local sensitivity, and what does the negative sign indicate?
- For two equally weighted time points, parameter D has sensitivities \(0.6\) and \(-0.6\). What are its signed mean and RMS sensitivity? Why do they answer different questions?
- Why might equal weights on output times give a misleading window summary when times are irregularly spaced?
- A parameter ranks first for plasma concentration but third for tissue concentration. Must one ranking be wrong? Explain.
- Name one limitation of a weighted RMS ranking and one diagnostic that can help reveal it.
Solutions. (1) B ranks higher because \(|-0.7|=0.7>0.4\). The negative sign indicates that a small increase in B locally decreases the output. (2) The signed mean is \((0.6-0.6)/2=0\); the RMS is \(\sqrt{(0.6^2+(-0.6)^2)/2}=0.6\). The mean describes net signed change and cancels opposing effects, whereas RMS summarizes their magnitude. (3) Equal weights can overrepresent densely sampled portions of the interval, so use weights that reflect the intended time integration or prediction task. (4) No. They refer to different outputs, and local sensitivities can legitimately differ by endpoint. (5) The RMS depends on the chosen weights and can hide where sensitivities are large or change sign; inspect signed sensitivity trajectories and report the weights.