Why Fix a Parameter?
A PBPK model may contain parameters whose values are uncertain or difficult to estimate from the available data. Fixing a parameter means holding it at a specified value rather than allowing the fitting procedure to estimate it. This can make a model easier to fit, but it also removes the opportunity for the data to adjust that parameter. A poor fixed value can therefore shift predictions or force other estimated parameters to compensate.
Sensitivity evidence can help identify candidates for fixing, but “low influence” is not an intrinsic property of a parameter. It is relative to a model output, the parameter range considered, and the physiological scenarios being studied. A parameter with little effect on plasma area under the curve may still materially affect a tissue concentration or a peak concentration. The aim is not simply to select the smallest entries in a sensitivity ranking. It is to decide whether fixing a parameter at a defensible value has acceptably small consequences for the outputs that matter.
Scale Sensitivity to the Question
For a positive parameter \(p\) and a positive model output \(Y\), the scaled sensitivity, also called the elasticity of \(Y\) with respect to \(p\), is
It is dimensionless: locally, a 1% change in \(p\) corresponds to approximately an \(E_{Y,p}\)% change in \(Y\). The sign indicates direction. For example, \(E_{Y,p}=-0.2\) means that a small increase in \(p\) is associated with a local decrease in \(Y\). This is a model sensitivity, not a causal claim about a biological intervention.
The output \(Y\) must be specified. It might be concentration at a particular time, an area under the concentration-time curve, a maximum concentration, or a tissue exposure measure. For time-course questions, sensitivities at multiple times may be needed. If an output can be zero or negative, a logarithmic scaling is not suitable; use an appropriate absolute or otherwise scaled sensitivity instead.
From Local Sensitivity to a Range-Based Check
A local sensitivity describes behaviour near one parameter value. To assess fixing at \(p_0\) when another plausible value is \(p\), express the output change on the logarithmic scale. Since \(d\log Y=E_{Y,p}\,d\log p\), integrating between the two parameter values gives the following justification.
If the magnitude of the elasticity is no greater than \(M\) throughout the interval, then
This bound makes the decision depend on both sensitivity and the distance from the selected value to the plausible range. A small local sensitivity does not guarantee a small change if the range is wide, or if sensitivity grows away from the nominal value. In practice, the local calculation is a screening approximation; rerunning the model at plausible values checks whether it remains adequate.
Worked Example: Screening a Clearance Parameter for AUC
Suppose a hypothetical PBPK model predicts a plasma AUC of \(120\ \text{mg}\cdot\text{h}/\text{L}\) at a clearance value of \(10\ \text{L}/\text{h}\). The local scaled sensitivity of this AUC to clearance is \(-0.08\). Physiological review gives a plausible clearance range of \(8\) to \(12\ \text{L}/\text{h}\), and the proposed fixed value is \(10\ \text{L}/\text{h}\). Use a constant-elasticity approximation to screen the effect across that range.
Under this approximation, \(Y(p)=120(p/10)^{-0.08}\). At \(8\ \text{L}/\text{h}\),
At \(12\ \text{L}/\text{h}\),
Relative to the value at the proposed fixed setting, these are changes of about \(+1.80\%\) and \(-1.45\%\), respectively. This suggests that fixing clearance at \(10\ \text{L}/\text{h}\) may have little effect on this AUC over the stated range, if that size of change is acceptable for the purpose. The calculation does not establish that clearance is unimportant for other outputs, or that the constant-elasticity approximation holds throughout the range.
Choose Fixed Values Using Physiology and Bounds
Sensitivity evidence indicates the potential impact of changing a parameter; it does not determine the value at which to fix it. A fixed value should be justified for the population, dose, and physiological scenario represented by the model. Depending on the parameter, relevant sources of justification may include a well-supported physiological measurement, a model-specific reference value, or a value selected in advance for a defined scenario. A convenient numerical value alone is not a physiological justification.
Use plausible bounds as part of the decision, not merely as a fitting constraint. They should reflect the intended application and any relevant relationships among physiological quantities. A value can be plausible in isolation but inconsistent with the model’s scenario when combined with other fixed inputs. The sensitivity assessment should cover the plausible range around the proposed fixed value, not just an arbitrarily narrow interval chosen to make the parameter appear unimportant.
A practical decision is output-specific: identify the outputs that the model must support; examine each candidate parameter’s sensitivity for those outputs; choose a defensible fixed value; and compare predictions at that value with predictions across plausible alternatives. If the choice affects a fitted analysis, also refit the remaining parameters and examine the resulting fit. These checks answer different questions: a direct model rerun shows the effect of changing the fixed value while holding other inputs constant, whereas refitting shows how much the estimated parameters and fit can adjust in response.
Worked Example: A Partition Coefficient Has Different Endpoint Effects
Consider a hypothetical dimensionless tissue-to-plasma partition coefficient \(K_p\), with proposed fixed value \(1.5\) and plausible range \(1.2\) to \(1.8\). At \(K_p=1.5\), the model gives a plasma AUC of \(80\ \text{mg}\cdot\text{h}/\text{L}\) and a kidney maximum concentration of \(4.0\ \text{mg}/\text{L}\). The local elasticities are \(0.03\) for plasma AUC and \(0.65\) for kidney maximum concentration.
Using the constant-elasticity approximation, the plasma AUC at the lower bound is
At the upper bound it is
The corresponding approximate changes are \(-0.67\%\) and \(+0.55\%\). For the kidney maximum concentration, the same calculations give
Those are changes of about \(-13.5\%\) and \(+12.6\%\). Fixing \(K_p\) at \(1.5\) may be reasonable for an analysis concerned only with plasma AUC, but the same evidence does not support treating it as low influence for kidney peak concentration. The endpoint determines the conclusion.
Assess the Consequences After Refitting
When a parameter is fixed in a fitted model, the remaining parameters may shift. Therefore, a low direct influence on one output is not by itself enough to show that fixing the parameter preserves a multi-output fit. The following example separates the direct effect of fixing a parameter from the effect of allowing another parameter to compensate.
Worked Example: One Fixed Parameter, Two Fitted Endpoints
Suppose a hypothetical model has two positive parameters, \(q\) (dimensionless) and \(r\) (\(\text{L}/\text{h}\)), and predicts
Hypothetical observations are \(Y_1=99.09\) and \(Y_2=45.45\), with standard deviations \(1.0\) and \(2.0\) in their respective output units. These values are consistent, to rounding, with \(q=12\) and \(r=5\). The local elasticity of \(Y_1\) with respect to \(q\) is \(-0.05\), so \(q\) has low influence on that endpoint near the stated values. Its elasticity for \(Y_2\) is \(0.7\), so it is much more influential for the second endpoint.
Now fix \(q=10\), and choose \(r\) to match \(Y_1=99.09\). Substitution into the first equation gives
At these fixed and refitted values, the second prediction is
The residual for \(Y_2\), defined here as prediction minus observation, is \(39.95-45.45=-5.50\). In standard-deviation units it is \(-5.50/2.0=-2.75\). Thus the fit to \(Y_1\) can be retained by adjusting \(r\), while the fit to \(Y_2\) becomes poor. If both parameters are estimated at \(q=12\), \(r=5\), both predictions match the hypothetical observations to rounding. This example shows why the consequence of fixing must be assessed against the full set of relevant outputs, including after refitting remaining parameters.
Diagnostics, Alternatives, and Limitations
Warning signs that a proposed fixing decision is not supported include large output changes when the fixed value is moved within its plausible bounds; sensitivity rankings that change across times, doses, or physiological scenarios; and a substantial deterioration in residuals after refitting. A remaining parameter estimate that moves markedly when another parameter is fixed can also indicate that the fit is compensating for the decision. These findings do not automatically rule out fixing, but they require a more limited claim or a different modeling choice.
The main alternatives serve different purposes. Estimating a parameter allows the data to inform it, but may be difficult when the data provide little information. Constraining it to a plausible interval allows some data-driven adjustment while preventing implausible estimates. Fixing it at a value supported by external physiological knowledge reduces the number of estimated quantities, but does not represent uncertainty about that value in the fitted model. The appropriate choice depends on the available evidence and the analysis objective.
Sensitivity-based selection is limited by the model and range used to calculate sensitivities. Local values can miss nonlinear behaviour or interactions; a parameter may be low influence at the nominal point but influential elsewhere. Screening results should therefore be checked over relevant plausible values and scenarios. Sensitivity evidence also cannot repair a misspecified model or turn a weakly justified fixed value into a known physiological constant.
A Practical Decision Sequence
Name the endpoints, times, doses, and physiological scenarios the model must represent.
Use scaled sensitivities where appropriate, and inspect how they behave across the target outputs and plausible parameter ranges.
Use physiological knowledge and scenario consistency; do not infer the value from low sensitivity alone.
Rerun the model across plausible alternatives and, for a fitted model, refit the remaining parameters. Compare output changes and fit diagnostics with the analysis requirements.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- An output has elasticity \(0.10\) with respect to a parameter. Approximately what output change corresponds to a 5% parameter increase near the evaluation point?
- Why does low sensitivity of plasma AUC not establish that a parameter can be fixed when kidney maximum concentration is also a target?
- A parameter has low local sensitivity at its nominal value but high sensitivity near one plausible bound. Is the local result sufficient to justify fixing it? Explain.
- What is the difference between changing a fixed value while holding other parameters constant and refitting the remaining parameters?
Solutions. (1) The approximate output change is \(0.10\times5\%=0.5\%\), in the positive direction. (2) Sensitivity is output-specific; the kidney maximum concentration may respond substantially even when plasma AUC does not. (3) No. The high sensitivity near the bound means the parameter is not uniformly low influence over the plausible range, so the range-based consequences need further assessment. (4) Holding other parameters constant isolates the direct model response to the fixed value; refitting allows estimated parameters to adjust and reveals the resulting fit and parameter changes.