Structural Identifiability With Sparse Response Sampling

Why Sparse Sampling Can Hide Parameters

An indirect response can change slowly even when drug exposure changes quickly. If only a few response measurements are collected, different combinations of baseline, turnover rate, and drug effect may predict exactly the same observed values. In that case, fitting an equation to the data does not by itself establish that each parameter has been identified: the model may have several parameter sets that agree perfectly at the sampled times.

This tutorial examines that issue for one deliberately simple setting: a response initially at baseline, followed by a known constant exposure that inhibits response production. The setting makes it possible to see directly how the number and timing of observations affect unique recovery. The conclusions depend on the stated model; they are not universal rules that can be applied to every indirect response model.

Model and Assumptions

Let \(R(t)\) be the response, in response units, and let \(B\) be its pre-dose baseline. Let \(k\) be the turnover rate, in inverse time, and let \(E\) be the fractional inhibition of production under the constant exposure. Thus \(0\leq E\leq 1\): \(E=0\) means no inhibition, and \(E=1\) means complete inhibition. Assume the pre-dose system is at steady state, the exposure begins at time zero and remains constant, and the drug acts only by multiplying baseline production by \(1-E\). The loss process remains first order with rate \(k\).

Because the pre-dose response is at steady state, baseline production equals \(kB\). After exposure begins, the response therefore follows

The assumptions are substantive. The exposure and its onset time must be known; the initial value must be the model baseline; and the model must have no unaccounted baseline drift, delay in the drug effect, or change in the loss rate. If these conditions do not hold, a failure to recover parameters could reflect model misspecification as well as sparse sampling.

To justify the solution, rewrite the differential equation as \(dR/dt+kR=kB(1-E)\). Multiplying by the integrating factor \(e^{kt}\) gives \(\frac{d}{dt}(e^{kt}R)=kB(1-E)e^{kt}\). Integrating from 0 to \(t\), using \(R(0)=B\), yields \(e^{kt}R(t)-B=B(1-E)(e^{kt}-1)\). Dividing by \(e^{kt}\) and collecting terms gives the stated response. The derivation also shows why the rate \(k\) controls the approach to the drug-affected steady response \(B(1-E)\), while \(E\) determines the size of that change.

One Post-Dose Observation: Two Parameters, One Constraint

Suppose \(B\) is known and one response is observed at a post-dose time \(t_1>0\). The observed deficit \(d_1\) gives the single equation \(d_1=E(1-e^{-kt_1})\), which involves both \(E\) and \(k\). Unless a parameter is fixed by outside information, this equation generally defines multiple compatible pairs rather than a unique pair.

Two Timed Observations With Known Baseline

A second observation can provide the extra information needed, but its timing matters. When responses are measured at \(t\) and \(2t\), define \(d_1=d(t)\), \(d_2=d(2t)\), and \(z=e^{-kt}\). For positive \(k\), \(0<z<1\), and the model gives \(d_1=E(1-z)\) and \(d_2=E(1-z^2)\). If \(d_1>0\), dividing the equations gives \(d_2/d_1=1+z\). This determines \(z\), then \(k\), and then \(E\).

This is a direct identifiability argument: it constructs the parameters from the observations. The observed values must also be compatible with the assumed model. In particular, for an inhibitory response with \(0<E\leq 1\), the deficits should be positive and should satisfy the parameter bounds after the calculations.

The special spacing makes the algebra especially transparent; it is not a claim that every pair of sampling times is equally informative. More generally, for two distinct positive times, the ratio \[ \frac{d(t_2)}{d(t_1)}=\frac{1-e^{-kt_2}}{1-e^{-kt_1}} \] depends on \(k\), and \(E\) can then be obtained from either deficit. Whether this equation has one valid solution must be checked for the actual times and observations. If the observations are nearly at the plateau, small changes in \(k\) may barely change the predicted values, so estimates can be imprecise even when the exact model has a unique solution.

When Baseline Is Not Directly Observed

If no pre-dose measurement is available, \(B\) is also unknown. A pair of post-dose observations then supplies only two equations for the three quantities \(B\), \(E\), and \(k\). Under ordinary, non-boundary conditions, that is not enough to uniquely determine all three. With three equally spaced post-dose observations, however, the model has a useful geometric structure. Write the spacing as \(\Delta\), set \(z=e^{-k\Delta}\), and let \(C=B(1-E)\) and \(D=BE\). At the first three sample times, \(R_j=C+Dz^j\), for \(j=1,2,3\). The successive differences have ratio \((R_3-R_2)/(R_2-R_1)=z\), provided the denominator is nonzero. This gives a way to recover the parameters when the observations satisfy the model and the resulting values are in range.

Diagnostics, Failure Modes, and Limits

A useful first diagnostic is to write the prediction for each actual observation time and count which unknown quantities enter. Counting equations and unknowns is not a proof of identifiability, but it quickly flags designs that cannot generally identify all parameters. Then check whether the equations can be inverted uniquely within the allowed parameter domain. A local sensitivity matrix, whose entries are derivatives of predicted observations with respect to parameters, can also reveal a local rank deficiency: if its columns are linearly dependent, small parameter changes can offset one another in the sampled predictions. Full rank is a local diagnostic, not a guarantee of global uniqueness.

Several boundary or near-boundary cases deserve attention. If \(E=0\), the response does not change and the observations contain no information about \(k\). If \(k=0\), there is no turnover-driven movement toward the drug-affected steady response. If samples are taken only at times when the response is effectively at its plateau, the data may mostly reveal \(B(1-E)\), not the separate values of \(B\), \(E\), and \(k\). In exact mathematics a finite-time exponential is not fully at its plateau when \(k\) is finite, but in noisy data the remaining difference may be too small to be useful.

These cases distinguish structural identifiability from practical identifiability. Structural identifiability asks whether exact model predictions at the specified times uniquely determine parameters. Practical identifiability asks whether the available, noisy measurements estimate them with useful precision. Parameters can be structurally identifiable but practically very uncertain when predicted values change only slightly across a wide range of parameter values.

A common mistake is to reason, “There are as many observations as parameters, so every parameter is identified.” The number of equations alone does not establish that the equations distinguish the parameters. For example, the one-observation case above has a continuum of valid pairs \((E,k)\), even though a fitting routine can return a single pair. That single numerical answer may be one point on a set of equally compatible solutions, selected by starting values, constraints, or optimization details—not evidence of unique recovery.

Analytical inversion, as used in the examples, is especially informative when it is available because it exposes ambiguity directly. For more complicated models, one can examine parameter-to-prediction mappings, sensitivities, or model fits across parameter values. These numerical checks can help diagnose potential problems but should not be mistaken for a proof of global uniqueness. Nor does an identifiable model guarantee that its biological assumptions are correct: a sparse schedule may simply be unable to reveal a mismatch between the assumed mechanism and the true response process.