Why Quantities Need Harmonizing
A model handoff can be mathematically correct and still be scientifically wrong if the two components use different meanings for the same label. One component may report total plasma concentration in milligrams per litre after the first dose; another may require unbound concentration in micromoles per litre at steady state. Copying the number across does not make those quantities equivalent. The same problem occurs when one model uses milligrams of salt per administration and another expects milligrams of active moiety per day, or when two reported averages cover different time intervals.
Harmonizing quantities means establishing whether a value has the same definition at both ends of a model interface, or applying a justified transformation that makes the definitions agree. It is not a request to force unlike outputs into a common label. If the information needed for a valid conversion is unavailable, the mismatch should be retained and made explicit rather than hidden.
The relevant details depend on the quantity. For concentration, distinguish such features as parent drug versus metabolite, total versus unbound drug, and plasma versus blood. For dose, distinguish active moiety versus salt or formulated product, amount per administration versus amount per day, and mass versus molar units. For a time-dependent quantity, name the clock origin and the interval. For a population quantity, name the summary operation, such as an arithmetic mean or a percentile.
Define the Quantity Before Converting It
A conversion is justified only if the source and target quantities are related by known information under stated assumptions. A unit conversion changes the numerical scale while preserving the underlying quantity. A conversion between total and unbound concentration, or between salt mass and active-moiety mass, additionally requires a scientifically supported factor. Those factors may depend on the compound, formulation, matrix, or conditions; they are not universal unit conversions.
For a fixed mass-to-molar conversion, the molecular weight must refer to the same chemical entity represented by the mass concentration. If \(C_m\) is a mass concentration in mg/L and \(MW\) is in g/mol, then the numerical concentration in micromoles per litre is:
This follows because mg/L is converted to g/L by multiplying by \(10^{-3}\), and mol/L is converted to micromol/L by multiplying by \(10^6\). The combined numerical factor is \(10^3/MW\). The formula does not resolve a mismatch in analyte, matrix, or binding basis; those must be addressed separately.
Worked Example: Converting Total Plasma Concentration to an Unbound Molar Quantity
Suppose a hypothetical PBPK output is 2.50 mg/L total plasma concentration of a parent drug. A receiving model requires unbound parent-drug concentration in micromoles per litre. Assume, for this illustration, that the molecular weight is 500 g/mol and the unbound fraction in plasma is 0.080 at the relevant conditions.
First convert total mass concentration to total molar concentration:
Then apply the stated unbound fraction:
The value passed forward is therefore 0.400 micromoles per litre of unbound parent drug, under the assumptions that the stated molecular weight matches the measured chemical entity and that the unbound fraction applies to these conditions. Passing 2.50 directly, or converting it to 5.00 micromoles per litre without applying the unbound fraction, would not supply the requested quantity.
Dose conversion also requires care. A mass of salt is not necessarily the same mass of active moiety. Likewise, a dose per administration is not the same quantity as a daily dose. Converting between them requires, at minimum, the product-to-active-moiety factor and the dosing schedule. A mass-to-mole conversion requires the molecular weight of the relevant dose basis.
Worked Example: Reconciling Salt Dose and Daily Active-Moiety Dose
A hypothetical regimen administers 120 mg of a salt formulation every 12 hours. The product information for this hypothetical example states that each milligram of salt corresponds to 0.75 mg of active moiety. A receiving model requires daily active-moiety dose in mg/day.
The active-moiety amount per administration is:
A 12-hour interval gives two administrations per 24 hours, so:
Thus the model input is 180 mg/day of active moiety, given the stated conversion factor and schedule. If the receiving model instead required amount per administration, its corresponding input would be 90 mg/dose. These values describe different dose quantities; neither should be substituted for the other without checking the model’s input definition.
Time Origins and Averaging Windows
A time value is meaningful only relative to a defined origin. Possible origins include first administration, the most recent administration, the start of an observation period, or a model-specific reference event. If two models use different origins, a known clock offset can translate the time coordinate. That translation does not change the underlying biological or treatment condition: for example, relabelling steady-state time as time since first dose does not make a steady-state profile an initial-dose profile.
An interval average must also be tied to its interval and state. For a concentration profile \(C(t)\), the time-average concentration over an interval from \(a\) to \(b\) is the area under the concentration-time curve on that interval divided by its duration:
This definition assumes the profile and time units are defined consistently, so the average has the same concentration units as \(C(t)\). An average over 0–24 hours is not interchangeable with one over 0–12 hours merely because both are called “average concentration.” Nor does changing the time origin alone make averages comparable if their windows or dosing states differ.
The following example shows why both the window and the dosing state must be checked before interpreting a difference.
Worked Example: Distinguishing a First-Day Average from a Steady-State Average
Suppose a hypothetical model reports a first-day AUC of 18 mg·h/L over hours 0–12 after the first dose and 30 mg·h/L over hours 12–24. Its average concentration over the full first day is:
A second output reports an AUC of 30 mg·h/L over one 12-hour dosing interval at steady state. Its interval-average concentration is:
The averages differ by 0.500 mg/L, but the difference cannot be attributed to a clock-origin mismatch alone: one output describes the first 24 hours after the first dose and the other describes a steady-state 12-hour interval. Even if a steady-state regimen repeats every 12 hours, averaging its interval AUC across 24 hours would give \(60/24=2.50\) mg/L, not the first-day value. The two outputs therefore remain distinct until the intended treatment state and window are made consistent.
Population Summaries Are Operations, Not Labels
A population summary is defined not just by the values being summarized but by the order of operations. For example, the arithmetic mean of individual peak concentrations is generally different from the peak of the mean concentration-time profile. The reason is that taking a maximum and taking an average do not generally commute. Similarly, arithmetic and geometric means answer different summary questions, even when computed from the same positive values.
Worked Example: Mean Individual Peaks Versus Peak of the Mean Profile
Consider two hypothetical individuals with concentrations, in mg/L, at three shared times:
| Individual | Time 0 h | Time 4 h | Time 8 h | Individual maximum |
|---|---|---|---|---|
| A | 0 | 8 | 0 | 8 |
| B | 0 | 0 | 6 | 6 |
The mean of the individual maxima is:
The mean profile at the shared times is 0, 4, and 3 mg/L, respectively, so its maximum is 4 mg/L. Thus the mean individual maximum is 7 mg/L, whereas the maximum of the mean profile is 4 mg/L. They differ because the individuals peak at different times and because averaging is performed before the maximum in one calculation and after it in the other. The appropriate summary depends on the quantity the receiving model is intended to use.
A Practical Harmonization Procedure
A disciplined handoff treats harmonization as a sequence of explicit checks. Resolve mismatches before passing a number onward; do not rely on variable names or units alone to establish equivalence.
Record the analyte or dose basis, matrix, units, time origin, interval or evaluation time, and summary operation for each side of the interface.
Separate straightforward unit changes from changes in chemical form, binding basis, treatment state, or statistical summary. List the information needed to justify each transformation.
Show conversion factors, time mappings, or summary calculations, along with the assumptions under which they hold. Keep the original quantity available for traceability.
Confirm that its meaning, units, time reference, window, and summary operation match what the receiving model requires. If they do not, revise the handoff or retain the quantities as non-equivalent.
This process offers a useful cross-check against alternative approaches. A direct handoff is appropriate when source and target definitions already match. Explicit conversion is appropriate when the relationship is known and its assumptions are defensible. Re-expressing the source model output may be preferable when it can produce the target quantity directly. If none of these options is supported, preserving separate quantities and documenting the incompatibility is more defensible than applying an assumed conversion.
Diagnostics, Failure Modes, and Limits
Several practical checks can reveal that a quantity has not been harmonized successfully:
- Definition check: Confirm that analyte, matrix, and binding basis match. A correct numerical unit does not make plasma concentration equivalent to blood concentration or total drug equivalent to unbound drug.
- Dimensional check: Track units through every calculation. For an AUC divided by hours, the result should have concentration units; for a daily dose, the result should carry amount-per-day units.
- Time check: Confirm the event that defines time zero, the dosing state, and the start and end of every interval. A time shift should not be used to disguise a change in the condition being summarized.
- Summary check: State whether the operation is performed for each individual before summarizing, or on a population profile. When the operations differ, do not assume the resulting values are interchangeable.
- Factor check: Verify the source and applicability of molecular weights, unbound fractions, formulation factors, and other conversion constants. A factor valid for one matrix, product, or condition may not apply elsewhere.
A common mistake is to reason, “Both models report concentration in mg/L, so the values can be compared directly.” Matching units establishes only that the values share a numerical scale; it does not establish matching analyte, matrix, binding basis, time window, dosing state, or population summary. The correction is to compare the complete quantity specifications first, then convert only the differences for which a valid relationship is available.
Harmonization also has limits. A conversion cannot recover information that was never recorded, such as the relevant unbound fraction, the original time origin, or individual-level profiles needed to compute a different population summary. Some transformations rely on assumptions that may be reasonable for a particular handoff but not generally true. These assumptions should be stated and assessed in context. If their validity is uncertain, report that uncertainty rather than presenting a converted number as exact.
Check Your Understanding
For each question, identify the relevant mismatch and determine what can be concluded from the information given.
- A receiving model requires unbound concentration in micromoles per litre. A source reports total concentration in mg/L, but no molecular weight or unbound fraction is available. Can the requested quantity be calculated?
- A regimen uses 40 mg of active moiety every 8 hours. What is the daily active-moiety dose?
- A model reports AUC of 36 mg·h/L over 0–12 hours. What is the average concentration over that interval?
- Two individuals have maximum concentrations of 10 and 4 mg/L. Is their mean individual maximum necessarily equal to the maximum of their mean profiles?
- One output is indexed to time since first dose and another to time since the most recent dose. What must be established before aligning their time coordinates?
Solutions. (1) No. Both a mass-to-molar factor and a justified unbound fraction are needed, and the analyte and matrix basis must also be confirmed. (2) There are \(24/8=3\) administrations per day, so \(40\ \mathrm{mg/dose}\times 3\ \mathrm{doses/day}=120\ \mathrm{mg/day}\). (3) \(36\ \mathrm{mg\cdot h/L}/12\ \mathrm{h}=3.00\ \mathrm{mg/L}\). (4) Not necessarily. The mean individual maximum is 7 mg/L, but the maximum of the mean profile depends on the individuals’ concentration values at shared times; the two operations need not agree. (5) Establish the event times and dosing schedule that relate the two clocks, and confirm that both outputs describe comparable dosing states and intervals. A coordinate shift alone does not establish that their underlying conditions match.