Sensitivity and Identifiability in Dissolution-Precipitation Models

Why parameter estimates need a support check

A dissolution-precipitation model can reproduce a concentration-time curve while leaving several parameter combinations plausible. For example, a faster dissolution rate may be offset by faster precipitation, or a particle-size assumption may compensate for an adjusted dissolution coefficient. A fitted value is therefore not, by itself, evidence that the data have determined that parameter.

This tutorial uses two complementary diagnostics. Local sensitivity measures how predictions change near one chosen parameter vector. Profile-based diagnostics ask how well the model can fit when one parameter is set to different values and the other parameters are re-optimized. Neither diagnostic replaces scientific judgment: both depend on the model, measured outputs, sampling times, and assumed observation-error model.

Local sensitivity on a meaningful scale

Let \(y_i\) be the measured concentration at time \(t_i\), and let \(f_i(\boldsymbol{\theta})\) be the model-predicted concentration at that time. The parameter vector \(\boldsymbol{\theta}\) may include particle radius, a dissolution-rate parameter, nucleation and crystal-growth parameters, and micellar parameters. Assume for this diagnostic that the model is evaluated at a specified parameter vector and that each observation has a known additive-error standard deviation \(\sigma_i\). A useful sensitivity for a positive parameter \(\theta_j\) is

This derivative describes the change in prediction, measured in observation-error standard deviations, for a small proportional change in the parameter. It is dimensionless and does not change merely because a parameter is expressed in different units. The formula applies to positive parameters; parameters that can be zero or negative need a suitable alternative transformation or a derivative on their natural scale.

A large absolute value of \(S_{ij}\) means observation \(i\) is locally responsive to parameter \(j\). But a column with large entries does not guarantee that the parameter can be estimated separately. If two sensitivity columns have nearly the same pattern across observation times, changing either parameter has nearly the same effect on the data. The data may then constrain a combination of the parameters well while leaving their individual values uncertain.

From local sensitivities to local information

For independent Gaussian errors with known standard deviations, define the weighted residual objective

Near a candidate value \(\boldsymbol{\theta}_0\), use a first-order Taylor approximation to the prediction:

After dividing each prediction change by \(\sigma_i\), the local change in the fitted curve is approximately \(\mathbf{S}\Delta\boldsymbol{\eta}\), where \(\mathbf{S}\) is the sensitivity matrix. Consequently, the local curvature of the weighted residual objective is related to \(\mathbf{S}^{\mathsf T}\mathbf{S}\). If columns of \(\mathbf{S}\) are nearly dependent, this matrix has a small eigenvalue: there is a direction in parameter space along which the predictions change little. This explains why high individual sensitivities can coexist with poor separation of parameters.

The approximation is local. It assumes the model predictions are sufficiently smooth near \(\boldsymbol{\theta}_0\), and it can be misleading when the model is strongly nonlinear over the plausible parameter range, when parameters are at bounds, or when a parameter changes which kinetic regime dominates. Sensitivities can be calculated analytically, by differentiating the model equations, or numerically, by perturbing one parameter slightly and checking that the result is stable to the perturbation size.

Profiles test support beyond one local point

A profile diagnostic fixes one parameter at a sequence of values and minimizes the objective over the remaining parameters. For parameter \(\theta_j\), define

where \(\boldsymbol{\theta}_{-j}\) denotes all parameters other than \(\theta_j\). Re-optimizing those parameters is essential: the profile asks whether a proposed value can fit the data after allowing compensation by other parameters. A curve that rises sharply away from its minimum indicates that the data constrain the parameter over that range. A broad, flat profile indicates weak support; a profile that remains essentially flat over its allowed range indicates that the data do not meaningfully constrain it under the model and error assumptions.

With independent Gaussian errors of known \(\sigma_i\), and under regularity conditions including an interior optimum and a locally adequate model, an approximate 95% likelihood-based interval for one parameter uses profile values satisfying \(Q_{\mathrm{prof}}-Q_{\min}\leq3.84\). This threshold is the 95th percentile of a chi-square distribution with one degree of freedom. It is an approximation, not a universal rule: boundaries, severe nonlinearity, sparse data, or uncertain error assumptions may make it unreliable. A profile is often more revealing than a local standard error because it can show asymmetry, flat regions, multiple minima, and bounds.

What concentration-time data can distinguish

Sensitivity and profile patterns should be interpreted in terms of the model and the observation design. Early measurements may be especially responsive to dissolution and particle-size assumptions, while later measurements may contain more information about precipitation onset and subsequent growth. These are tendencies, not guarantees: the precise pattern depends on the coupled equations, dose, medium, and sampling schedule.

A particle-size parameter and a dissolution coefficient may be confounded if both mainly change the rate at which dissolved drug appears. Nucleation parameters can be weakly supported when the sampling schedule misses the induction period or when precipitation is absent in the observed time window. Nucleation and crystal-growth parameters may trade off when the data show only the overall loss of dissolved drug, without enough temporal resolution to distinguish onset from the speed of subsequent removal. Micellar-capacity or exchange parameters may be weakly supported when only total drug is measured and the model predicts little change in that total as drug partitions between free and micelle-bound forms.

Diagnostics, alternatives, and limitations

A useful diagnostic sequence is to scale sensitivities consistently, inspect both their magnitudes and their similarity, and then compute profiles for parameters that appear weak or confounded. Profiles should be evaluated over scientifically plausible ranges, with nuisance parameters re-optimized at every fixed value. Check that the optimizer converges from multiple starting values and that the profile is not an artifact of a parameter bound or a failed optimization. If finite differences are used, repeat them at more than one perturbation size; a derivative that changes substantially with step size is not a reliable local diagnostic.

One-at-a-time perturbations are simpler than the sensitivity matrix, but they can miss parameter compensation because all other parameters remain fixed. Local curvature summarizes nearby behavior efficiently, but relies on a local approximation. Profiling is more computationally demanding and can reveal nonlinear trade-offs, although it still depends on the selected model, objective, parameter bounds, and successful optimization. Collecting measurements at additional times or under conditions that produce different sensitivity patterns can provide more separation than merely increasing precision at already redundant time points.

These tools diagnose support within the chosen model; they do not prove that the model is correct. A parameter may appear precisely estimated because a mechanistic process was omitted, a fixed input was treated as known, or the observation-error model was too restrictive. Nor does weak support mean that the physical process is unimportant. It means the particular data and model combination do not determine that parameter reliably. When support is weak, report the limitation, examine scientifically justified parameter combinations, and avoid interpreting an optimizer’s point estimate as direct evidence.