Sensitivity Across Doses and Physiological Scenarios

Why Sensitivity Can Change Across Conditions

A parameter can have a modest influence on a PBPK prediction at one dose and a much larger—or smaller—influence at another. The same can happen when physiological inputs change. Saturable metabolism can alter the role of enzyme capacity as concentration changes; hepatic blood flow can determine whether clearance is flow-limited; and a change in unbound fraction can shift the balance between flow and intrinsic clearance. A sensitivity calculated at one setting therefore does not automatically describe another.

The comparison must be made for a clearly specified output, dose, and physiological scenario. Here, a scenario means a defined set of physiological model inputs, such as hepatic blood flow or unbound fraction. It is a conditional model setting, not necessarily a sampled population or a probability distribution. The goal is to see whether a parameter’s local influence changes between these conditions, not to rank endpoints at different times.

Definition: Let $Y(D,g,\theta)$ be a positive PBPK output at dose $D$, physiological scenario $g$, and parameter value $\theta$. Its dimensionless local sensitivity is $$ \mathcal{S}_{Y,\theta}(D,g) = \frac{\partial \ln Y(D,g,\theta)}{\partial \ln \theta} = \frac{\theta}{Y(D,g,\theta)} \frac{\partial Y(D,g,\theta)}{\partial \theta}. $$ It describes the approximate proportional change in $Y$ for a small proportional change in $\theta$, holding dose, scenario, other parameters, and the output definition fixed. A positive value means $Y$ increases locally as $\theta$ increases; a negative value means it decreases.

For example, a sensitivity of $-0.6$ means that a small 1% increase in the parameter corresponds approximately to a 0.6% decrease in the output, locally and with the other conditions fixed. It is not a universal property of the parameter: it is conditional on $D$, $g$, the output $Y$, and the specified parameter point.

Comparing Sensitivities Across Dose Levels

A useful comparison holds the physiological scenario and output constant while changing dose. In a nonlinear PBPK model, changing dose may alter concentrations in the range where an enzyme or transporter operates. Consequently, the local slope with respect to a parameter can change even though the parameter itself is unchanged. The earlier material on dimensionless sensitivity coefficients establishes why proportional, rather than raw-unit, changes are useful for comparing parameters with different units.

When derivatives are not available, a central finite difference on the log scale estimates the local sensitivity. For a positive parameter $\theta$, use a multiplicative perturbation $e^h$, where $h>0$:

\widehat{\mathcal{S}}_{Y,\theta}(D,g) = \frac{ \ln Y(D,g,\theta e^h)-\ln Y(D,g,\theta e^{-h}) }{2h}.

This estimates the derivative of $\ln Y$ with respect to $\ln\theta$, centered at the baseline parameter value. The perturbations must be applied to the same parameter and model in both runs; other inputs, the dose, scenario, solver settings, and output calculation should be held fixed. The finite-difference step should be small enough to represent a local change but large enough that numerical solver error does not dominate the difference.

Worked Example: A Metabolic-Capacity Sensitivity at Two Doses

Suppose a hypothetical PBPK model predicts parent-drug plasma concentration at 8 hours, $C_8$, under the same physiological scenario at two intravenous doses. A saturable metabolic pathway is represented in the model, and $V_{\max}$ is the parameter being perturbed. The table gives hypothetical model outputs; all runs use the same model, endpoint, solver tolerances, and scenario. $V_{\max}^{-}$ and $V_{\max}^{+}$ are the baseline value divided and multiplied, respectively, by 1.1.

Dose	Baseline $C_8$ (mg/L)	$C_8$ at $V_{\max}/1.1$ (mg/L)	$C_8$ at $1.1V_{\max}$ (mg/L)
10 mg	2.00	2.13	1.88
100 mg	8.00	8.30	7.75

For both doses, $h=\ln(1.1)=0.09531$, so $2h=0.19062$. At 10 mg, the estimate is

\widehat{\mathcal{S}}_{C_8,V_{\max}} = \frac{\ln(1.88)-\ln(2.13)}{0.19062} = \frac{\ln(1.88/2.13)}{0.19062} \approx \frac{-0.1248}{0.19062} = -0.655.

At 100 mg, the estimate is

\widehat{\mathcal{S}}_{C_8,V_{\max}} = \frac{\ln(7.75/8.30)}{0.19062} \approx \frac{-0.06856}{0.19062} = -0.360.

In these hypothetical simulations, increasing metabolic capacity decreases the 8-hour concentration at both doses, but the estimated local proportional influence is smaller in magnitude at 100 mg. This is consistent with a dose-dependent change in the model’s response, potentially associated with saturation. The comparison alone does not prove that saturation is the cause: that explanation should be checked against the model’s rate laws and concentration profiles. Nor do the estimates say that concentration changes by exactly these percentages for large changes in $V_{\max}$; they describe local proportional changes near the specified baseline.

Blood Flow and the Well-Stirred Liver Model

A compact derivation shows how physiological conditions can change sensitivity even in a simple clearance relationship. In the well-stirred liver model, let $Q_H$ be hepatic blood flow, $f_u$ the unbound fraction in blood, and $CL_{\mathrm{int}}$ intrinsic hepatic clearance. Assume these quantities are positive and constant over the concentration range considered. The hepatic clearance is

CL_H = \frac{Q_H f_u CL_{\mathrm{int}}} {Q_H+f_u CL_{\mathrm{int}}}.

Write $x=f_u CL_{\mathrm{int}}$. Taking logarithms gives $\ln(CL_H)=\ln Q_H+\ln x-\ln(Q_H+x)$. Differentiating with respect to $\ln Q_H$, while holding $x$ fixed, gives

\frac{\partial \ln CL_H}{\partial \ln Q_H} = 1-\frac{Q_H}{Q_H+x} = \frac{x}{Q_H+x}.

Differentiating with respect to $\ln CL_{\mathrm{int}}$, while holding $Q_H$ and $f_u$ fixed, gives $\partial\ln x/\partial\ln CL_{\mathrm{int}}=1$ and therefore

\frac{\partial \ln CL_H}{\partial \ln CL_{\mathrm{int}}} = 1-\frac{x}{Q_H+x} = \frac{Q_H}{Q_H+x}.

These elasticities sum to 1 for clearance in this model. When $x$ is much larger than $Q_H$, hepatic clearance approaches blood flow and is more sensitive to $Q_H$. When $x$ is much smaller than $Q_H$, clearance approaches $f_u CL_{\mathrm{int}}$ and is more sensitive to intrinsic clearance. This result is conditional on the well-stirred model and its assumptions; it is not a general identity for every PBPK model.

Assumption check for AUC: For an intravenous dose $D$, $AUC=D/CL_{\mathrm{total}}$ under linear systemic elimination with a defined systemic clearance. It becomes $AUC=D/CL_H$ only if hepatic clearance is the sole route of systemic elimination, so that $CL_{\mathrm{total}}=CL_H$. If extrahepatic elimination is present, use total systemic clearance rather than hepatic clearance alone.

Worked Example: Clearance Sensitivity Under Different Blood Flows

Consider a hypothetical drug with $f_u CL_{\mathrm{int}}=30$ L/h. Compare two physiological scenarios, with $Q_H=90$ L/h and $Q_H=45$ L/h. Assume the well-stirred model applies and hepatic clearance is the sole route of systemic elimination. For a 240 mg intravenous dose, the model-based AUC is therefore $D/CL_H$.

At $Q_H=90$ L/h, hepatic clearance is

CL_H = \frac{90\times30}{90+30} = \frac{2700}{120} = 22.5\ \mathrm{L/h}.

Thus $AUC=240/22.5=10.67$ mg·h/L. The dimensionless sensitivities of AUC are the negatives of the corresponding clearance sensitivities, because $\ln(AUC)=\ln(D)-\ln(CL_H)$ with dose fixed. Therefore,

\mathcal{S}_{AUC,Q_H} = -\frac{30}{90+30} = -0.250, \qquad \mathcal{S}_{AUC,CL_{\mathrm{int}}} = -\frac{90}{90+30} = -0.750.

At $Q_H=45$ L/h, the same calculations give

CL_H = \frac{45\times30}{45+30} = 18.0\ \mathrm{L/h}, \qquad AUC = \frac{240}{18.0} = 13.33\ \mathrm{mg\cdot h/L}.

The sensitivities are now $\mathcal{S}_{AUC,Q_H}=-30/75=-0.400$ and $\mathcal{S}_{AUC,CL_{\mathrm{int}}}=-45/75=-0.600$. Lowering hepatic blood flow changes both the predicted AUC and the relative local influence of the two parameters. These are conditional model results, not evidence that a particular patient group has these exact values.

Physiological Scenarios and Population Characteristics

A scenario comparison should specify which physiological inputs differ and which remain fixed. For instance, a scenario might represent a changed unbound fraction while keeping hepatic blood flow and intrinsic clearance fixed. That change alters $x=f_u CL_{\mathrm{int}}$, which in turn changes the sensitivities derived above. A scenario may instead change several correlated physiological inputs together to represent a defined model setting. In that case, the sensitivity is conditional on the entire setting, and the comparison cannot attribute the difference to one input unless that input is changed in isolation or otherwise examined.

Worked Example: A Change in Unbound Fraction

Use the same hypothetical 240 mg intravenous dose, hepatic-only elimination assumption, and well-stirred model. In a reference scenario, let $Q_H=90$ L/h and $f_u CL_{\mathrm{int}}=30$ L/h. In a second defined scenario, suppose a change in unbound fraction doubles $f_u CL_{\mathrm{int}}$ to 60 L/h, while $Q_H$ and $CL_{\mathrm{int}}$ are otherwise held fixed for this comparison.

The reference scenario has $CL_H=90\times30/(90+30)=22.5$ L/h and $AUC=240/22.5=10.67$ mg·h/L. Its sensitivities are $\mathcal{S}_{AUC,Q_H}=-30/120=-0.250$ and $\mathcal{S}_{AUC,CL_{\mathrm{int}}}=-90/120=-0.750$.

In the altered scenario,

CL_H = \frac{90\times60}{90+60} = 36.0\ \mathrm{L/h}, \qquad AUC = \frac{240}{36.0} = 6.667\ \mathrm{mg\cdot h/L}.

The new sensitivities are $\mathcal{S}_{AUC,Q_H}=-60/150=-0.400$ and $\mathcal{S}_{AUC,CL_{\mathrm{int}}}=-90/150=-0.600$. Thus the scenario changes the baseline AUC and shifts the local balance of influence: blood flow has a greater relative sensitivity, while intrinsic clearance has a smaller one. The assumed doubling of $f_u CL_{\mathrm{int}}$ is a hypothetical input change, not a general claim about a specific physiological group. The calculation also assumes $f_u$ is constant over the relevant concentration range.

Practical Comparison, Diagnostics, and Limitations

The comparison strategy depends on the question. Local derivatives or finite differences are useful when the aim is to compare nearby parameter influence at specified doses and scenarios. A broad global sensitivity analysis, such as the earlier material on Morris screening or Sobol indices, addresses variation over defined input ranges rather than just the local slope at one parameter point. Neither replaces the other: a local sensitivity can identify a change in nearby influence, while a global measure summarizes behavior over the chosen ranges and distributions. A simple one-at-a-time comparison can be easy to interpret, but it does not represent joint parameter variation or dependence among changing inputs.

For a fair cross-condition comparison, use the same output definition and units, the same baseline parameterization where appropriate, and documented scenario values. If different output scales are compared, dimensionless sensitivities may help with scale, but they do not make different biological endpoints interchangeable. Check that model outputs are positive before taking logarithms and that perturbations remain within valid parameter ranges.

A practical diagnostic is to repeat finite differences with more than one reasonable perturbation size. Estimates that change sharply as the step shrinks may indicate strong curvature, numerical noise, solver error, or a parameter boundary. Check that the unperturbed model reproduces its baseline output, and inspect whether all paired runs truly differ only in the intended parameter. For dose comparisons, confirm that the same physiological scenario is used; for scenario comparisons, list every changed input. A ranking reversal should be investigated rather than treated as an automatic model error: it can reflect genuine nonlinearity, but also inconsistent runs or unstable numerical derivatives.

Local sensitivities have important limits. They describe behavior near one parameter point, do not guarantee the response to a large parameter change, and do not by themselves quantify uncertainty in predictions. A change in a population characteristic may affect multiple parameters together; separately calculated single-parameter sensitivities do not capture that joint change. Saturable pathways, changing binding, or other nonlinear mechanisms may require evaluating sensitivities at several conditions rather than reporting one value as representative.

Common pitfall: “The parameter ranked highest at the reference dose, so it must be the most influential at every dose and in every population.” This does not follow. Sensitivities are conditional on dose, physiology, model assumptions, and output. Recalculate them under the conditions of interest and verify that the comparisons use consistent definitions and numerical settings.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

At a specified dose and scenario, an output has dimensionless sensitivity $0.7$ to a parameter. Interpret this value for a small 1% parameter increase.
In the well-stirred model, let $Q_H=80$ L/h and $f_u CL_{\mathrm{int}}=20$ L/h. Calculate the hepatic clearance and the sensitivities of clearance to $Q_H$ and $CL_{\mathrm{int}}$.
Why is an AUC comparison using $D/CL_H$ invalid if extrahepatic elimination contributes to systemic clearance?
Name two checks that can help determine whether a change in finite-difference sensitivity across doses is reliable.

Solutions. (1) Locally, a 1% increase in the parameter corresponds approximately to a 0.7% increase in the output, holding dose, scenario, and other parameters fixed. (2) $CL_H=80\times20/(80+20)=16.0$ L/h. Its sensitivities are $20/100=0.200$ to $Q_H$ and $80/100=0.800$ to $CL_{\mathrm{int}}$. (3) With extrahepatic elimination, $CL_{\mathrm{total}}\ne CL_H$ in general, so the intravenous relationship is $AUC=D/CL_{\mathrm{total}}$, not $D/CL_H$. (4) Repeat the calculation using another reasonable perturbation size and check solver tolerances; also verify that only the intended parameter changed and that the same endpoint and scenario were used.