Why Molecular Weight Matters in PBPK
PBPK models may represent drug in mass units, such as milligrams and milligrams per litre, or in molar units, such as micromoles and micromoles per litre. The choice is often practical: a dose label or assay result may be reported by mass, while a model state or physicochemical calculation may use amount of substance. Molecular weight is the bridge between these representations. If the molecular weight or its units are mishandled, a number can look plausible while representing the wrong dose or concentration.
A conversion changes the way a quantity is expressed; it does not change the amount of drug. That distinction matters when comparing model inputs with a dose record, translating a measured concentration for a simulation, or interpreting a predicted exposure. The earlier material on dimensional consistency supplies the general unit-checking principle: every conversion factor must carry units that cancel to the intended output units. Here we apply that principle specifically to molecular weight.
The last condition is essential. A molecular weight is not just a numerical setting: it identifies what entity the conversion counts. Before calculating, confirm what the dose or concentration mass represents and which molecular weight applies to it. A conversion is not a correction for a mismatched dose basis, an assay definition, or a different chemical entity.
Core Conversion Equations
Let \(MW\) be molecular weight in \(g/mol\). For a mass \(m\) in grams, the amount \(n\) in moles is \(m/MW\). For concentration, divide mass concentration by molecular weight after expressing the mass concentration in \(g/L\). The unit cancellation shows why the units must be handled, rather than treating conversion as a numerical shortcut.
For common PBPK and assay units, the following forms avoid repeated conversion between grams and smaller mass units:
The factor of 1000 follows from the metric prefixes. For the dose equation, \(D\) milligrams is \(D/1000\) grams; dividing by \(MW\) gives moles, and multiplying by \(10^6\) converts moles to micromoles. For the concentration equation, \(C_{\text{mass}}\) milligrams per litre is \(C_{\text{mass}}/1000\) grams per litre; dividing by \(MW\) gives moles per litre, then multiplying by \(10^6\) gives micromoles per litre. In both cases the combined factor is \(1000\).
A useful equivalent for a concentration reported in nanograms per millilitre is:
This works because \(1\ ng/mL=1\ \mu g/L\), and \(1\ \mu g/L\) divided by \(MW\) in \(g/mol\) gives the numerical concentration in \(\mu mol/L\). Equivalent mass-concentration units can also be converted first: \(1\ \mu g/mL=1\ mg/L\), and \(1\ ng/mL=0.001\ mg/L\).
Worked Example: Converting a Mass Dose to Micromoles
A hypothetical PBPK simulation represents dose amount in micromoles. The administered dose is 70.0 mg, and the applicable molecular weight is 350 \(g/mol\). Convert the dose to micromoles.
Use the dose conversion, checking that the mass and molecular weight refer to the same entity:
Thus the model input should be 200 \(\mu mol\), not the numerical value 70.0 with a different unit label. As a check, 200 \(\mu mol\) is 0.200 mmol, or \(0.000200\ mol\); multiplying by 350 \(g/mol\) gives 0.0700 g, which is 70.0 mg.
Worked Example: Converting a Plasma Concentration
A hypothetical plasma result is 240 \(ng/mL\). The molecular weight for the entity represented by that mass is 400 \(g/mol\). Express the concentration in both \(mg/L\) and \(\mu mol/L\).
First convert the mass concentration: \(1\ ng/mL=0.001\ mg/L\), so \(240\ ng/mL=0.240\ mg/L\). Next use the direct conversion for \(ng/mL\):
As an independent check using \(mg/L\), apply the corresponding formula:
The two routes agree. The reported concentration is 0.240 \(mg/L\), or 0.600 \(\mu mol/L\). These are two expressions of the same concentration, not two different exposure levels.
Following Unit Errors Into Model Predictions
Suppose a PBPK model receives a dose as an amount in micromoles. The dose must be converted before it enters the model if the source record gives milligrams. If the model instead expects a mass input, converting the dose to micromoles and then passing that number in as milligrams is also incorrect. The model’s parameter and state units determine which representation it expects; a number without its unit is not a complete input.
The same issue arises for concentration outputs. If the simulated concentration is in \(\mu mol/L\), comparing its numerical value directly with data in \(mg/L\) is not a valid comparison. Convert one side to the units of the other, using the correct molecular weight. Under a fixed molecular weight, a mass concentration and a molar concentration are proportional, but their numerical values generally differ.
Worked Example: Quantifying a Unit-Driven Exposure Error
A hypothetical model expects a dose in micromoles. The intended dose is 50.0 mg and the applicable molecular weight is 250 \(g/mol\). A modeller enters the numerical value 50.0 as though it were micromoles. Compare the entered amount with the correct amount.
The correct conversion is:
The entered value is 50.0 \(\mu mol\). Its ratio to the intended amount is \(50.0/200=0.250\), so the input is one quarter of the intended amount, a fourfold under-input. If all other model inputs and processes are unchanged and the model is linear in dose over this range, dose-proportional concentrations would also be one quarter as large. That proportionality is conditional; nonlinear processes or other dose-dependent model features can make the exposure response non-proportional. The certain error here is the fourfold discrepancy in the dose amount.
A separate common error is to treat an \(mg/L\) value as \(g/L\). For example, 2.00 \(mg/L\) at \(MW=500\ g/mol\) equals \(4.00\ \mu mol/L\). If the numerical value 2.00 is incorrectly read as \(g/L\), the result becomes \(4000\ \mu mol/L\), which is 1000 times too large. The error is not in the molecular-weight equation; it is in the mass unit supplied to it.
These examples show why checking only whether a simulation runs is insufficient. A unit mismatch can produce a smooth and numerically stable prediction at the wrong scale. A useful check is to convert a model dose or concentration back to the original reporting unit and verify that the result matches the documented quantity.
Practical Checks and Common Failure Modes
Before converting a PBPK input or output, write down the source quantity, its unit, the target quantity, and the molecular weight with units. Then check that the result has the target units. For dose, the result should be an amount such as \(\mu mol\); for concentration, it should be an amount per volume such as \(\mu mol/L\). If the units do not cancel to the intended dimension, stop before using the number.
- Check the molecular-weight basis. Confirm that the specified molecular weight corresponds to the entity whose mass is reported. A valid arithmetic calculation can still be conceptually wrong if those refer to different entities.
- Check prefixes explicitly. Distinguish \(mg/L\) from \(g/L\), and \(ng/mL\) from \(mg/L\). Convert the source units before applying a formula unless using a direct formula whose units are stated.
- Check the model’s expected units. Label dose, amount, concentration, volume, and model outputs with units at data import and export. Do not rely on a column name or numerical scale alone.
- Use a reverse conversion. Convert the result back to the original unit. A round trip should recover the starting value apart from rounding.
- Compare like with like. Before judging agreement between observed and simulated concentrations, express both in the same mass or molar units and confirm that they use the same concentration basis.
The principal alternative is to retain mass units throughout, or to retain molar units throughout. Neither is inherently more correct. Mass units can align naturally with dose labels and some assays; molar units are convenient when amounts of molecules or molar concentrations are needed. The safer choice is the representation that is explicit and consistent across the relevant model components and data. If units differ between components, document and test the conversion at each boundary rather than silently changing the numerical value.
A conversion alone does not establish that the chosen molecular weight, dose definition, or concentration measurement is appropriate. Nor does converting a total mass concentration determine a different concentration basis. Those are separate questions about what quantity was measured or specified. Keep this tutorial’s arithmetic task distinct: given a defined mass quantity, a matching molecular weight, and clear units, calculate its molar equivalent correctly.
Common Mistake
Correction: A model receives a number, not the intended unit meaning. Convert first using the matching molecular weight. For \(MW=250\ g/mol\), 50 mg is 200 \(\mu mol\), not 50 \(\mu mol\). Record the input unit and verify the conversion before interpreting simulated exposure.
Check Your Understanding
Try each conversion and unit check before reading the solutions.
- A hypothetical 30.0 mg dose has \(MW=300\ g/mol\). What amount is this in micromoles?
- A hypothetical concentration is 1.20 \(mg/L\), with \(MW=400\ g/mol\). Convert it to \(\mu mol/L\).
- A concentration is 80.0 \(ng/mL\), with \(MW=200\ g/mol\). Convert it to \(\mu mol/L\), showing why the direct formula applies.
- A model expects a dose in micromoles, but an input of 40.0 mg is entered as 40.0 \(\mu mol\). If \(MW=200\ g/mol\), is the input too high or too low, and by what factor?
Solutions. (1) \(n=1000(30.0)/300=100\ \mu mol\). (2) \(C=1000(1.20)/400=3.00\ \mu mol/L\). (3) Since \(1\ ng/mL=1\ \mu g/L\), \(C=80.0/200=0.400\ \mu mol/L\). (4) The intended amount is \(1000(40.0)/200=200\ \mu mol\), but 40.0 \(\mu mol\) was entered. The input is one fifth of the intended amount, so it is fivefold too low.