Why a Calibration Freeze Matters
A PBPK model can only be interpreted in relation to the observations and parameter values that produced it. If records are added or removed after fitting, or if parameters are adjusted while results are being reviewed, the model being evaluated may no longer be the model that was originally calibrated. A clear freeze makes that boundary explicit: it records which observations informed estimation and the exact parameter set produced under a specified model and analysis configuration.
This is a matter of scientific traceability, not a claim that the frozen model is correct. A frozen calibration can still be inadequate, and a later analysis can legitimately improve it. The requirement is to distinguish the original, documented calibration from any subsequent version. The earlier material on calibration and validation distinguishes fitting a model from assessing its predictive performance; this tutorial specifies how to preserve that distinction operationally.
Define the Data Boundary Before Fitting
The calibration dataset should be defined at the level of the records actually used by the fitting procedure. In PBPK applications, a record may include a measured concentration at a specified time, dose, route, formulation, and participant or experimental system, together with units, assay information, and study context. A file name alone is not a sufficient definition: files can be overwritten, columns can be transformed, and records can be silently filtered.
Write the inclusion and exclusion rules before examining how individual observations affect the fitted result. Rules might specify the relevant dose or route, required units, acceptable assay status, or whether a documented protocol deviation makes a record unusable for the stated calibration question. Each excluded record should have an identifiable reason. A record set aside for a different role is not automatically invalid; record that status separately from a quality-based exclusion.
For reproducibility, record both the source and the analysis-ready form of the data. The record should identify the source file or database extract, the data version or date, the included record identifiers, the exclusions and their reasons, unit conversions, transformations, and any derived quantities. A checksum or other immutable identifier for the archived files can help establish that a later file is not being mistaken for the frozen one.
Worked Example: Making Record-Level Inclusion Decisions
Suppose a hypothetical calibration is intended to fit a PBPK model to plasma concentrations after a specified oral dose. Before fitting, the team states that records must have the specified dose and route, concentration and time units must be recoverable, and the assay must meet the documented validity criteria. One protocol-defined dataset is designated for later evaluation and is not authorized for parameter fitting. The table shows six hypothetical records.
| Record | Information | Decision under the stated rules |
|---|---|---|
| A1 | Specified dose and route; valid assay; 2 h, 8 ng/mL | Include in calibration |
| A2 | Specified dose and route; valid assay; 4 h, 4 ng/mL | Include in calibration |
| A3 | Specified dose and route; documented sample thaw outside validated limit; 2 h, 14 ng/mL | Exclude; document sample-handling reason |
| A4 | Specified dose and route; valid assay; concentration reported in a convertible unit | Include after documented conversion |
| B1 | Valid assay, but a different route from the calibration question | Do not include; record the scope reason |
| C1 | Specified dose and route; valid assay; belongs to the dataset designated for later evaluation | Do not use for fitting; preserve its designated role |
The calibration set consists of A1, A2, and A4 after its documented unit conversion. A3 is excluded because a prespecified quality condition failed, not because its concentration is inconvenient. B1 falls outside the stated calibration question, while C1 is kept out of fitting because its role was assigned in advance. These are different reasons and should not be collapsed into a generic “data removed” category.
Why Outcome-Dependent Exclusions Change the Fit
Let \(D_{\mathrm{cal}}\) denote the authorized calibration observations, \(M\) the specified model version, and \(\theta\) the vector of estimated parameters. In general, estimation chooses parameter values according to a stated fitting criterion \(J\):
The criterion might be a likelihood, a weighted residual objective, or another prespecified method. The important point here is that \(D_{\mathrm{cal}}\) is an argument of the criterion. Changing the observations changes the function being optimized and can therefore change the estimate, even if the model code and optimizer remain identical.
This can be seen directly with a simple squared-residual objective. Suppose a candidate parameter value has residuals \(r_1,\ldots,r_m\), and the objective is \(J=\sum_{i=1}^{m}r_i^2\). Removing observation \(k\) changes it to \(J_{-k}=\sum_{i\ne k}r_i^2=J-r_k^2\). Because the removed term depends on the model prediction at that observation, refitting the reduced objective can favor a different parameter value. Thus, excluding a record because it has a large residual is not a neutral correction: it changes which discrepancies the fit is asked to address. An exclusion can be justified by a documented data-quality or scope rule, but not merely by the fact that the observation disagrees with the model.
Worked Example: Checking a Fitting Criterion and Recording the Result
Consider a hypothetical calibration with three plasma concentrations, using a prespecified weighted squared-residual objective. The observed concentrations are 10, 5, and 2 ng/mL; the corresponding model predictions at the fitted parameter vector are 9.5, 5.5, and 1.8 ng/mL. The prespecified scales are 1, 0.5, and 0.4 ng/mL, respectively. The parameter vector contains \(CL_{\mathrm{int}}=18\) L/h and \(V_c=12\) L.
For this example, the objective is
where \(y_i\) is the observed concentration, \(\hat{y}_i(\theta)\) is the predicted concentration, and \(s_i\) is the stated scale, all in ng/mL. Substitution gives
The arithmetic verifies the reported objective for these predictions; it does not by itself prove that the optimizer found a global minimum or that the model is adequate. The frozen record should identify the estimated values \(CL_{\mathrm{int}}=18\) L/h and \(V_c=12\) L, their units, the three authorized observations, the objective and scales, and the model and solver versions. It should also state which quantities were fixed rather than estimated. Without those details, the parameter numbers cannot be unambiguously tied to the calculation.
What the Freeze Record Should Contain
A useful freeze record lets another analyst reconstruct what was allowed to inform estimation and identify precisely which model was produced. It need not repeat every item in the scientific report, but it must refer unambiguously to the relevant archived materials. At a minimum, document:
- Data identity: source, version, included record identifiers, exclusions and reasons, and the analysis-ready dataset identifier.
- Data handling: units, conversions, transformations, derived quantities, and any prespecified rules applied before fitting.
- Model identity: model equations or code version, structural options, and relevant configuration settings.
- Estimation setup: the objective or estimation method, estimated and fixed quantities, initial or fixed settings when needed for reproducibility, and solver configuration.
- Final parameter set: every value at the precision retained for use, its unit, and whether it was estimated, fixed, or otherwise specified.
- Provenance: date, responsible analyst or team, software versions, saved fitting output, and an identifier linking the parameter set to the data and model versions.
Retain sufficient precision in machine-readable parameter files even if a report displays rounded values. For example, a value printed as 18 L/h may have been stored as 17.84 L/h; silently replacing the stored value with the rounded display value can change predictions. The report should make clear which representation was used for subsequent calculations.
Worked Example: Handling a Later Model Change
A hypothetical team freezes model version M1 using calibration dataset D1. Its parameter file records \(CL_{\mathrm{int}}=18.0\) L/h and \(V_c=12.0\) L. Later, new evidence motivates adding a renal elimination pathway, producing model version M2. The team estimates a renal clearance parameter of 1.2 L/h and obtains \(CL_{\mathrm{int}}=16.7\) L/h and \(V_c=12.3\) L when fitting M2 to D1.
These later values are not a correction to the archived M1 parameter file. They belong to a new model-and-parameter version, M2 with its own parameter vector. The team preserves M1, D1, and the original fitting record unchanged, then records M2's code change, the new estimated parameter and its unit, the new values, and the fitting configuration. If any observations or data-processing rules also changed, the team creates and identifies a new dataset version rather than calling it D1.
The earlier frozen results can still be reported as the results for M1. Any later assessment must state which version and data were used; results from M2 must not be attributed to M1. A change made after inspecting evaluation outcomes also needs to be disclosed as a change, rather than represented as an unchanged, previously frozen calibration. This preserves the meaning of the original record even when scientific revision is appropriate.
Diagnostics, Alternatives, and Limitations
A freeze is probably incomplete if analysts cannot answer basic audit questions consistently: Which exact records were used? Why was each omitted record omitted? Which units entered the objective? Can the saved parameter file reproduce the reported predictions under the stated model version? Disagreement between scripts about record counts, unexplained differences between stored and displayed parameters, or a dataset file that has changed since fitting are practical warning signs. Resolve these discrepancies before treating the fit as a fixed reference.
A common alternative is informal iterative updating: append newly available data, remove points that seem anomalous, and adjust parameters until plots look satisfactory. This may be useful during exploratory development, but it does not define one stable calibration to evaluate. The correction is not to forbid iteration; it is to label exploratory work as such, set explicit rules for a final calibration, and create a new version whenever its authorized data, model, or parameter values change. A frozen snapshot is also not a substitute for retaining the data and code needed to reproduce it.
A strict freeze has limits. It cannot guarantee that the included measurements are unbiased, that exclusion rules were scientifically appropriate, that the PBPK structure is correct, or that estimates are identifiable and precise. Nor should it prevent correction of a verified transcription or programming error. Such a correction should be logged, the affected data or model version revised, and any refitting recorded as a new analysis. The purpose is transparent change control, not preserving an error or implying that a version is beyond question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A concentration has a large residual but passed all prespecified assay and protocol checks. May it be removed from the calibration dataset solely because of that residual? Why?
- A valid record uses a different unit that can be converted exactly. What should be recorded if it is included?
- A team changes one parameter value after fitting but keeps the original dataset and model code. Is the original frozen parameter set still the set used for the revised predictions?
- Name two identifiers or records that help another analyst establish which data and parameter file were used.
Solutions. (1) No. Removing it because of its residual makes inclusion depend on model disagreement and changes the fitting objective; the record should remain unless a documented, applicable rule justifies exclusion. (2) Record the original unit, conversion, resulting unit, and the transformed analysis-ready value. (3) No. The revised predictions use a changed parameter set; preserve the original and record the modified values as a new version. (4) Examples include included record identifiers and an immutable dataset-file identifier, plus a versioned parameter file linked to the model code and fitting record.