Local Sensitivity as an Output Derivative

Why Specify an Output Before Asking About Sensitivity?

A PBPK model can produce concentration-time profiles in multiple tissues, plasma concentrations at chosen times, and summary quantities such as AUC. A parameter change need not affect all of these outputs in the same way. Therefore, a statement such as “clearance is sensitive” is incomplete: it does not specify which output, at what time or under what conditions, or how the change is measured.

Local sensitivity makes the question precise. It describes how a selected model output changes for a small change in a parameter near a specified nominal parameter set, with the other parameters and model inputs held fixed. The derivative is a property of that output, that parameter, and that nominal point—not a universal property of the parameter.

Definition: Let $g(\boldsymbol{\theta})$ be a specified scalar output from a PBPK model, where $\boldsymbol{\theta}$ is the vector of model parameters. The local sensitivity of $g$ to parameter $\theta_j$ at nominal parameter set $\boldsymbol{\theta}_0$ is the partial derivative $$ S_j(\boldsymbol{\theta}_0)=\left.\frac{\partial g(\boldsymbol{\theta})}{\partial \theta_j}\right|_{\boldsymbol{\theta}=\boldsymbol{\theta}_0}. $$ It describes the instantaneous rate of change in the selected output per unit change in $\theta_j$, evaluated at $\boldsymbol{\theta}_0$, with the other parameters fixed.

The selected output may be a concentration at a specified time, such as $g(\boldsymbol{\theta})=C_{\mathrm{plasma}}(t^\ast;\boldsymbol{\theta})$, or a summary endpoint, such as AUC. A time-course output can be considered at each specified time; its local sensitivity can vary over time and can have different signs at different times. A derivative of a scalar endpoint and a derivative of a time-varying concentration are both local sensitivities, but they answer different questions.

Meaning, Units, and a Local Approximation

The sign gives the direction of the output’s local response to the parameter. A positive derivative means that, sufficiently near the nominal point, increasing the parameter tends to increase the output. A negative derivative means that a small increase tends to decrease it. A derivative near zero means that the output changes little per unit parameter change at that point; it does not establish that the output is insensitive at other parameter values or for other outputs.

The units follow directly from the definition: output units divided by parameter units. If concentration is in mg/L and clearance is in L/h, then the derivative of concentration with respect to clearance has units $(\mathrm{mg/L})/(\mathrm{L/h})$. This is not itself a concentration change. To approximate a change in concentration, multiply the derivative by a parameter change in L/h.

The local interpretation follows from the definition of a derivative. For a small change $\Delta\theta_j$, with the other parameters fixed, the output change is approximately the derivative times that change. The approximation becomes more accurate as the parameter change approaches zero, provided the output is differentiable there.

g(\boldsymbol{\theta}_0+\Delta\theta_j\mathbf{e}_j)-g(\boldsymbol{\theta}_0) = S_j(\boldsymbol{\theta}_0)\Delta\theta_j+o(\Delta\theta_j), \qquad \Delta\theta_j\to 0.

Here $\mathbf{e}_j$ denotes a change in the $j$th parameter only, and $o(\Delta\theta_j)$ is a remainder whose ratio to $\Delta\theta_j$ tends to zero as the change tends to zero. This statement does not claim that the approximation is exact for a large change. Curvature can make the actual change differ substantially from the derivative-based approximation.

Worked Example: Concentration Sensitivity to Clearance

Consider a hypothetical model output at a fixed time, defined by $C(CL)=5\exp(-0.1CL)$ mg/L, where $CL$ is clearance in L/h. The coefficient $0.1$ has units h/L, so the exponent is dimensionless. Find the local sensitivity at $CL=4.0$ L/h and compare a small-change approximation with the model output at $CL=4.2$ L/h.

Differentiate the stated output with respect to clearance:

\frac{dC}{dCL}=-0.5\exp(-0.1CL) \quad \frac{\mathrm{mg/L}}{\mathrm{L/h}}.

At the nominal value, $C(4.0)=5\exp(-0.4)\approx 3.3516$ mg/L, and \[ \left.\frac{dC}{dCL}\right|_{CL=4.0} =-0.5\exp(-0.4)\approx -0.3352 \quad \frac{\mathrm{mg/L}}{\mathrm{L/h}}. \] The negative sign means that, near 4.0 L/h, increasing clearance tends to lower the concentration at this specified time.

For an increase of $0.2$ L/h, the first-order approximation is \[ \Delta C\approx (-0.3352)(0.2)=-0.06704\ \mathrm{mg/L}. \] The exact model value at $4.2$ L/h is $C(4.2)=5\exp(-0.42)\approx3.2852$ mg/L. Thus the exact change is $3.2852-3.3516\approx-0.0664$ mg/L. The approximation is close, but it is not identical to the finite model change.

Assumptions and a Short Justification

A local derivative is meaningful only after the model output and the point of evaluation have been specified. The model must define $g$ for parameter values near the nominal set, and the parameter must be varied within its admissible domain. The derivative also presumes that the selected output changes smoothly enough with that parameter at the nominal point. Other parameters, dose, subject characteristics, and protocol conditions are held fixed for this partial derivative.

For a differentiable scalar function of one varying parameter, the derivative is defined by the limit of output change divided by parameter change. Multiplying and dividing the output difference by $\Delta\theta_j$ gives the local approximation directly:

\frac{g(\boldsymbol{\theta}_0+\Delta\theta_j\mathbf{e}_j)-g(\boldsymbol{\theta}_0)} {\Delta\theta_j} \longrightarrow S_j(\boldsymbol{\theta}_0) \quad\text{as }\Delta\theta_j\to0.

Consequently, for sufficiently small changes, the output difference is approximately $S_j(\boldsymbol{\theta}_0)\Delta\theta_j$. This justification also shows why the derivative is not the same quantity as an observed or simulated change across a fixed, nonzero parameter interval: that ratio is a finite-interval average rate, whereas the derivative is the limiting rate at one point.

Worked Example: Sensitivity of AUC to Clearance

Suppose a hypothetical PBPK model under linear intravenous conditions predicts $AUC_{\infty}=D/CL$, where $D=100$ mg is the dose and $CL=5.0$ L/h is systemic clearance. The output is in mg·h/L. Find the local derivative with respect to clearance and use it to approximate the AUC change for an increase of 0.5 L/h.

At the nominal clearance, $AUC_{\infty}=100/5.0=20.0$ mg·h/L. Differentiating with dose fixed gives \[ \frac{dAUC_{\infty}}{dCL}=-\frac{D}{CL^2} =-\frac{100}{(5.0)^2} =-4.00 \quad \frac{\mathrm{mg\cdot h/L}}{\mathrm{L/h}}. \] Thus a small increase in clearance is associated with a decrease in this model-predicted AUC near 5.0 L/h.

For $\Delta CL=0.5$ L/h, the first-order approximation is $\Delta AUC_{\infty}\approx(-4.00)(0.5)=-2.00$ mg·h/L, giving an approximated AUC of 18.0 mg·h/L. The exact value at 5.5 L/h is $100/5.5\approx18.18$ mg·h/L, an exact change of about $-1.82$ mg·h/L. The approximation has the correct direction and is reasonably close, but the 0.5 L/h change is not infinitesimal.

The result is conditional on the stated model relationship and fixed dose. It should not be read as a universal AUC-clearance relationship for every PBPK setting or as evidence about an observed clinical response.

Interpreting a Different Parameter and Output

The derivative’s sign is determined by the specific model relationship, rather than by a general label such as “physiological parameter.” Even an intuitively plausible direction should be checked against the output definition and model equations. The following simple hypothetical relationship illustrates a positive local derivative.

Worked Example: Tissue Concentration Sensitivity to a Partition Coefficient

Let a specified tissue concentration output be $C_T(K)=12K/(2+K)$ mg/L, where $K$ is a dimensionless partition coefficient. At the nominal value $K=2.0$, find the derivative and estimate the change for an increase to $K=2.1$.

The nominal output is $C_T(2.0)=24/4=6.00$ mg/L. Differentiation gives \[ \frac{dC_T}{dK}=\frac{24}{(2+K)^2}\ \frac{\mathrm{mg/L}}{\text{unit }K}, \qquad \left.\frac{dC_T}{dK}\right|_{K=2.0}=\frac{24}{16}=1.50\ \frac{\mathrm{mg/L}}{\text{unit }K}. \] The derivative is positive, so a small increase in $K$, with other conditions fixed, increases this model output near $K=2.0$.

For $\Delta K=0.1$, the local approximation is $\Delta C_T\approx1.50(0.1)=0.150$ mg/L. The exact value at $K=2.1$ is $12(2.1)/(2+2.1)=25.2/4.1\approx6.146$ mg/L, a change of about $0.146$ mg/L from 6.00 mg/L. The actual finite change is close to, but not equal to, the first-order approximation.

How Local Derivatives Differ From Alternatives

Several analyses can describe model response, but they answer different questions. A local derivative is useful when the question concerns the immediate direction and rate of change at a chosen nominal set. A finite change compares two particular parameter settings; it can reflect curvature across the interval. A global analysis considers variation over a broader parameter range or joint uncertainty and cannot be inferred from one derivative alone. These approaches complement one another rather than being interchangeable.

Quantity or approach	What it describes	Key distinction
Local output derivative	Rate of change at one nominal parameter set	Has output-units per parameter-unit; depends on output and nominal point
Finite model change	Difference between outputs at two specified parameter values	Has output units; includes all response curvature across that interval
Global sensitivity analysis	Influence across a parameter range or distribution of inputs	Requires information beyond a derivative at one point

A derivative also does not by itself establish practical importance or identify which parameter is most influential. Its numerical magnitude depends on the units used for the parameter: expressing clearance in mL/min rather than L/h changes the derivative’s numerical value while leaving the underlying model response unchanged. Comparisons require care about scales and the particular output being considered; a raw derivative is not a universal ranking.

Diagnostics, Failure Modes, and Limitations

Before interpreting a computed derivative, check that the output and parameter are clearly named, the nominal parameter set is reported, and the units are consistent. Confirm that only the intended parameter is varied in the definition. The sign should agree with nearby model evaluations when smooth behavior is expected. A discrepancy can indicate a calculation error, an implementation problem, a parameter boundary, or strong curvature over the chosen comparison interval.

A derivative may fail to exist or be unstable where the model output is discontinuous, has a kink, or switches abruptly between computational branches. Outputs defined using thresholds or event rules can be especially problematic at a boundary. At a parameter bound, a two-sided local change may not be admissible; the relevant response may be one-sided, and it must be described as such. Numerical solver tolerances and convergence should also be checked if the derivative is computed from model runs, because numerical noise can obscure a small local response.

The derivative is conditional on the fixed values of all other parameters and inputs. If parameters are correlated or are expected to change together, a one-parameter partial derivative does not describe that joint movement. Nor does a local derivative establish the response far from the nominal set. The response can flatten, steepen, or reverse direction elsewhere. These limitations make the chosen target, nominal point, and intended use essential parts of any interpretation.

A Common Mistake

A common mistake is to say, “The derivative is $-0.3352$, so concentration falls by 0.3352 mg/L whenever clearance increases.” This is incorrect for two reasons. First, the derivative has units of concentration per clearance unit, not concentration alone. Second, it is a local rate at one nominal clearance, not a fixed change for every increase. For the hypothetical concentration example, an increase of 0.2 L/h gives an approximate decrease of $0.3352\times0.2=0.06704$ mg/L, not 0.3352 mg/L. For a larger change, the model should be evaluated across that change rather than assuming the local derivative remains constant.

Key takeaway: A local sensitivity is the derivative of a specified PBPK output with respect to a specified parameter at a specified nominal set. State its sign, units, output, and evaluation point; interpret it as a nearby rate of change, not as a universal or finite-change effect.

Check Your Understanding

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

If a concentration output has units mg/L and a parameter has units L/h, what are the units of its local derivative?
A local derivative is positive at a nominal parameter value. What does that say about a sufficiently small increase in the parameter, with other inputs fixed?
For $g(\theta)=10/\theta$, calculate the local derivative at $\theta=2$, and state its sign and units if $g$ is measured in mg·h/L and $\theta$ in L/h.
Why is a derivative at one nominal parameter set not enough to describe the model response over a wide parameter range?
What concern arises if the selected output jumps at a threshold as a parameter varies?

Solutions. (1) Its units are $(\mathrm{mg/L})/(\mathrm{L/h})$. (2) The output tends to increase for a sufficiently small increase near that nominal value; the derivative alone does not describe a large change. (3) $g'(\theta)=-10/\theta^2$, so $g'(2)=-10/4=-2.50\;(\mathrm{mg\cdot h/L})/(\mathrm{L/h})$, indicating a locally decreasing output. (4) Curvature can change the rate or even the direction of response away from the nominal point. (5) A discontinuity can prevent the derivative from existing there, so a local derivative interpretation may not be valid at the threshold.