Sensitivity Analysis for Linked-Pool Turnover Models

Why Examine Sensitivity?

A linked-pool model can produce a plausible response curve while different rates influence different parts of that curve. An observation near the start of a response may be informative about transfer out of the precursor pool, while a later observation may better reflect loss from the product pool. Sensitivity analysis makes these local relationships explicit. It helps identify which observations respond to a parameter, and which candidate sampling times may add useful information.

Sensitivity is not the same as identifiability. A prediction can be highly sensitive to two rates in nearly the same way, so the data may not distinguish those rates. Sensitivity is also local: it describes the model near a particular parameter vector, not every plausible value. The tutorial on identifiability of formation and conversion rates in linked pools addresses whether rates can be distinguished from the model and data; here the emphasis is on how prediction changes near a specified set of rates and how that change varies with observation time.

A Mass-Balanced Two-Pool Model

Consider a normalized amount \(X_1(t)\) in a precursor pool and \(X_2(t)\) in a product pool. Material leaves pool 1 either through loss at rate \(a\) or transfer to pool 2 at rate \(\kappa\). Pool 2 loses material at rate \(b\). All three rate constants are positive and have units of inverse time. The transfer term must appear as a loss from pool 1 as well as an input to pool 2:

For this tutorial, suppose the measured model prediction is \(y(t)=X_2(t)\). The initial pool amounts are treated as known, and the rates are the parameters of interest. These assumptions isolate rate sensitivity; if initial amounts or observation-model parameters are also estimated, their sensitivities must be considered too.

The absolute derivative has units of prediction per unit of parameter. The log-scale sensitivity has the same units as the prediction and is often easier to compare across rates with different units or magnitudes. Dividing it by \(y(t)\) gives a relative sensitivity, but that ratio can be unstable when the prediction is close to zero. When observations have different error scales, sensitivities should also be considered relative to their observation uncertainty.

Prediction and Its Local Derivatives

Write \(c=a+\kappa\). Solving the first differential equation gives \(X_1(t)=e^{-ct}\). Substituting this into the second equation and using its initial condition gives

When \(b=c\), the continuous limiting form is \(y(t)=\kappa t e^{-ct}\). The formulas below use \(b\ne c\); near \(b=c\), the limiting expression or a numerically stable equivalent should be used to avoid cancellation.

For \(b\ne c\), let \(q=e^{-ct}\), \(r=e^{-bt}\), and \(d=b-c\). Differentiating the prediction gives:

The derivative with respect to \(a\) follows because \(a\) enters through \(c=a+\kappa\). The derivative with respect to \(\kappa\) has two contributions: \(\kappa\) is the transfer input to pool 2, and it also increases the loss rate \(c\) from pool 1. Ignoring that second contribution would not describe the mass-balanced model.

What Sensitivities Say at Different Times

For the same rates, the derivatives can be evaluated at candidate observation times. The table reports log-scale sensitivities. For example, a small 1% increase in \(\kappa\) produces an approximate change in \(y(t)\) of \(0.01S_{\kappa}^{y}(t)\) in normalized amount units. A negative sensitivity means the prediction decreases locally as that rate increases.

At 1 hour, the prediction is locally most responsive, on this log scale, to transfer rate \(\kappa\). By 4 hours its sensitivity to \(\kappa\) is much smaller, while the magnitudes for \(a\) and \(b\) are larger. This pattern is consistent with the changing contributions of precursor depletion and product loss over time. It does not mean that a single early observation estimates \(\kappa\) precisely: the other rates also affect the prediction, and the amount of measurement noise matters.

Using Sensitivities to Compare Sampling Designs

The sensitivity at one time describes the direction and size of a local prediction change. To compare observations with different measurement precision, scale each sensitivity by its standard error. A commonly used local information summary applies when observations have independent normal errors with known variances, the model is differentiable, and the parameter values used for the calculation are specified. For parameters \(\theta\), define the gradient vector \(g_i=\partial y(t_i)/\partial\theta\) for observation time \(t_i\). The local Fisher information matrix is

where \(\sigma_i\) is the observation standard deviation at time \(t_i\). If using log-scale parameters, use the gradient with entries \(S_{\theta_j}^{y}(t_i)\) instead. A larger diagonal entry means greater local information about that parameter considered in isolation. The off-diagonal entries matter too: they reflect the extent to which parameters affect predictions together. The matrix as a whole, not just its diagonal, informs whether parameters can be separated locally.

Derivation and Practical Diagnostics

The derivative expressions can be checked in two complementary ways. First, differentiate the closed-form prediction, as above. Second, differentiate the differential equations. If \(s_j(t)=\partial X(t)/\partial\theta_j\) is the sensitivity state for parameter \(\theta_j\), then differentiating \(dX/dt=f(X,\theta)\) gives

This follows from the chain rule applied to \(dX/dt=f(X,\theta)\). In this example the initial amounts are fixed, so \(s_j(0)=0\). The equation also shows why a parameter’s effect accumulates over time: it changes the model’s instantaneous dynamics, and the resulting state change affects subsequent dynamics. For more complex linked-pool models without a convenient closed form, these sensitivity equations can be solved alongside the original model.

Numerical derivatives provide another check. A central finite difference approximates \(\partial y/\partial\theta\) by \([y(\theta+h)-y(\theta-h)]/(2h)\). Choose \(h\) large enough to avoid rounding error but small enough to remain in the local regime, and verify that results are stable over a reasonable range of step sizes. If the parameter is positive and a proportional change is intended, perturbing its logarithm can be more appropriate. Analytic derivatives or sensitivity differential equations are generally preferable when they are available and implemented correctly.

Warning signs include derivatives that change markedly with finite-difference step size, unexpected sensitivity signs, or large discrepancies between analytic and numerical derivatives. Check the parameter units, the transfer loss term in pool 1, the observation equation, and whether the prediction is near a numerical cancellation point such as \(b=c\). Near-zero predictions can also make relative sensitivities misleading. A local analysis should be repeated at other plausible parameter values if the fitted rates are uncertain or the model is strongly nonlinear.

Interpretation, Alternatives, and Limits

Local derivative analysis answers a focused question: near the chosen parameter values, how much and in what direction does a prediction change when a rate changes? Finite perturbations answer a related question over a specified change range and can reveal curvature that a derivative misses. Global sensitivity methods examine variation over a broader parameter region, but require a defined range or distribution of plausible values and answer a different question. Neither a large local derivative nor a large information-matrix diagonal guarantees that a rate is identifiable or estimated precisely.

The choice of sampling times should therefore consider more than the largest sensitivity. A design with observations at distinct phases of the response can provide different sensitivity patterns, helping distinguish rates. Repeated observations at a single time may reduce uncertainty in the mean prediction, but if they share the same model gradient they do not create new directions of local parameter information. In practice, assay precision, feasible sampling windows, baseline observations, and the possibility of model misspecification also matter. The residual signatures of misspecified pool pathways are a reminder that an apparently informative design cannot compensate for an incorrect pathway structure.