Provenance for Derived and Model-Dependent Inputs

Why Derived Inputs Need a Traceable Lineage

A PBPK input may look like a single number in a parameter table, but it can represent a chain of evidence and decisions. A value reported by an assay may be transformed into a tissue-level quantity, scaled to an organ, or estimated conditional on a model equation. If that chain is hidden, a reviewer cannot tell what the number means, whether its assumptions apply, or which part should change when new evidence becomes available.

The key distinction is not simply “published” versus “unpublished.” A published value can still be a model-derived estimate, while a quantity calculated from an assay may be reproducible if its inputs, equation, and assumptions are recorded. The provenance of an input is its traceable account of where the evidence came from and how it became the value used in the model.

A useful provenance record therefore identifies the value actually used, its unit and status, the primary evidence for every source quantity, the calculation or model that connects those quantities to the input, and the assumptions that make that connection defensible. Earlier material on Documenting Transformations From Measurement to Input addresses the general transformation record; here the emphasis is on making the derivation’s evidentiary and model dependence explicit.

What to Record for a Derived or Model-Dependent Value

A record should let another analyst reconstruct the input without guessing. For each quantity in the calculation, preserve its reported value and qualifier, unit, experimental system, and source. Then give the equation and substitution, including unit conversions. State the assumptions that connect the source measurement to the target quantity, and say whether the input is measured, calculated, scaled, fitted, or some combination. Finally, record applicability limits: for example, the enzyme, species, tissue, pH, or concentration range to which the source evidence applies.

The source of an equation is not necessarily the source of its inputs. A publication might provide a scaling equation, while separate assay reports provide the activity and abundance values inserted into it. Both links belong in the lineage. If a value is chosen from among plausible alternatives, record that decision and its rationale rather than presenting the choice as though it were measured.

Scaling an Enzyme-Dependent Assay Result

A common derivation translates an intrinsic clearance measured per amount of microsomal protein to a target tissue abundance. If the assay clearance is attributable to a particular enzyme, and activity per unit of that enzyme is assumed unchanged between the assay preparation and target tissue, an enzyme-abundance ratio can scale the assay value. Further physiological factors can then scale the result to an organ.

Here \(A\) is enzyme abundance per milligram of microsomal protein in the stated preparation. The ratio is dimensionless only when both abundances use compatible units and refer to the same enzyme. Multiplication by microsomal protein per gram of tissue and tissue mass then yields an organ-level clearance. This scaling is not itself a new measurement of organ clearance; it is a calculation conditional on the abundance and transfer assumptions.

Documenting Equation-Based Physicochemical Inputs

A physicochemical input can also be derived from measured or estimated molecular properties using an equation. The resulting value depends on both the source quantities and the chemical assumptions built into that equation. A record should not label the output as a direct measurement merely because its inputs came from experiments.

When an Input Depends on the PBPK Model

Some quantities used as inputs are inferred through model equations rather than observed directly. In that case, a numerical estimate is conditional on the model structure and on the values of other parameters. This conditionality should be visible in the record, especially if the estimated quantity is later treated as fixed in a simulation.

For example, under the well-stirred liver model, hepatic availability \(F_h\) and intrinsic clearance \(CL_{\mathrm{int}}\) are related by:

Here \(Q_h\) is hepatic blood flow and \(f_{u,b}\) is the unbound fraction in blood. Rearranging gives an estimate of intrinsic clearance from an estimated \(F_h\):

Justifying the Scaling and Recognizing Failure

The abundance scaling equation follows from a proportionality assumption. If enzyme-specific intrinsic clearance per milligram of microsomal protein is proportional to enzyme abundance, write \(CL_{\mathrm{int}}=kA\), where \(k\) is activity per unit abundance under the relevant assay conditions. For assay and target preparations, this gives \(CL_{\mathrm{int,assay}}=kA_{\mathrm{assay}}\) and \(CL_{\mathrm{int,target}}=kA_{\mathrm{target}}\). Dividing the second relation by the first cancels \(k\), provided it is the same in both preparations, yielding the abundance-ratio formula. If substrate conditions, enzyme activity, or contributions from other enzymes differ, that cancellation is not justified and the ratio alone does not establish the target clearance.

Several checks can expose a weak or broken provenance chain:

• Check dimensions. Confirm that units cancel as claimed and that the final quantity has the unit required by the model. An abundance ratio should be dimensionless only after compatible units are verified.
• Check definitions and scope. A value for one enzyme, species, tissue, or assay condition cannot silently become a value for another. Preserve qualifiers such as “total,” “unbound,” or “per milligram.”
• Check arithmetic independently. Recalculate each step from the recorded values, including conversions between microliters and milliliters or between minutes and hours.
• Check sensitivity to assumptions. If modest changes in abundance, binding, or a model estimate cause large changes in the derived input, report that dependence rather than presenting excess precision.
• Check for circularity. If a quantity was estimated by fitting the same model that later uses it, identify the data and model connection. Do not imply that the estimate is independent experimental confirmation.

A direct measurement may avoid some scaling assumptions but may not exist for the target tissue or population. An empirical scaled value may be more feasible, but depends on transferability assumptions. A mechanistic or fitted estimate can use several evidence sources coherently, yet is conditional on model structure and other inputs. These are alternatives with different evidence and limitations, not interchangeable labels for the same certainty.

Common Provenance Pitfalls

Another frequent error is recording only the final value and a citation. That does not reveal which reported quantity was selected, how it was transformed, whether the units were compatible, or what assumptions were added by the analyst. A citation is a path to evidence, not a substitute for documenting the derivation. Similarly, a value labeled “scaled” is not fully interpretable unless the reference system, target system, scaling factor, and transfer assumptions are stated.

The practical standard is reproducibility with honest qualification: another modeler should be able to reconstruct the calculation and understand what would invalidate it. This does not require claiming that every assumption is certain. It requires distinguishing observed evidence from analyst choices and model conditions, and stating where uncertainty or applicability limits enter the lineage.