Why PBPK Units Need to Be Audited
A PBPK model combines physiological quantities, drug quantities, and rates. A number such as 2.5 has no interpretable meaning until its quantity and units are clear: it could be 2.5 mg of drug, 2.5 mg/L, or 2.5 L/h. Confusing these can make a model internally inconsistent even when its equations look plausible. A reliable audit starts by identifying what each quantity represents and then recording its dimension and unit.
The central distinction is between a state variable, which describes the modelled system at a particular time, and a parameter, which specifies a property or rate used by the model. In an amount-based model, drug amounts in compartments are common states. Concentrations may be states in a concentration-based formulation, or they may be calculated from an amount and a volume. Physiological volumes, blood flows, clearances, and rate constants are usually parameters.
Quantity Types and Their Units
Use \(A_i\) for the drug amount in compartment \(i\), \(C_i\) for drug concentration, and \(V_i\) for compartment volume. The subscript identifies the compartment; it does not change the quantity’s type. Amount is commonly recorded in mg, concentration in mg/L, and volume in L. A model using molar quantities can instead use consistent units such as \(\mu\text{mol}\) and \(\mu\text{mol}/\text{L}\). Do not mix mass-based and molar quantities without an explicit molecular-weight conversion.
A blood or tissue flow, denoted here by \(Q_i\), is volume transported per unit time, such as L/h. A clearance, denoted by \(CL_i\), is also a volume per unit time, such as L/h. A first-order rate constant, denoted by \(k_i\), is an inverse time, such as h\(^{-1}\). These categories are not interchangeable: flow and clearance have the same dimensional form but different meanings, while a rate constant has a different dimension.
A unit expresses a quantity on a measurement scale, while a dimension describes its structural type. For this audit, amount, volume, and time are the basic dimensions. Thus concentration has dimension amount/volume, while flow and clearance each have dimension volume/time. Different units can express the same dimension: L/h and mL/min are both volume per time, but their numerical values are not interchangeable without conversion.
The concentration associated with an amount depends on the volume used to define it. Under the model’s stated concentration basis, the amount–concentration relationship is:
This relation is a useful unit check: \((\text{mg}/\text{L})\times\text{L}=\text{mg}\). It does not by itself determine which volume is physiologically appropriate. For example, plasma concentration and whole-blood concentration refer to different bases; the model must use the basis intended for its equations and data.
Worked Example: Classifying a Tissue Amount and Its Parameters
Hypothetical data: A tissue compartment contains \(A_i=12.0\) mg of drug and has volume \(V_i=0.600\) L. Its first-order elimination clearance is \(CL_i=0.120\) L/min. Assume the compartment is well mixed, its volume is fixed, and elimination is first order with respect to the stated tissue concentration.
First calculate the concentration on the specified tissue-volume basis:
The amount \(A_i\) is a state if the model tracks tissue drug amount over time. The volume \(V_i\) is a parameter in this example. The calculated \(C_i\) has units mg/L.
Clearance has units of volume per time, not amount per time. Multiplying it by concentration gives an amount-per-time elimination rate:
For this first-order elimination representation, the corresponding rate constant is \(k_i=CL_i/V_i\). The units and value are:
The amount-per-time rate calculated using this rate constant agrees: \(k_iA_i=(0.200\ \text{min}^{-1})(12.0\ \text{mg})=2.40\ \text{mg}/\text{min}\). This agreement follows from \(k_i=CL_i/V_i\) and \(C_i=A_i/V_i\). It relies on the stated first-order and volume assumptions; it is not a general identity for every elimination model.
Relating Clearances, Flows, and Rate Constants
Although clearance and flow both have units of volume per time, their meanings differ. A flow describes the movement of blood or another fluid through a physiological pathway. A clearance describes a volume-equivalent rate of drug removal relative to a specified drug concentration. The equality of their unit patterns does not make them the same model quantity.
A rate constant instead describes a fractional rate per unit time in a first-order process. For example, \(k_i=0.10\ \text{h}^{-1}\) indicates a first-order rate scale of 0.10 per hour under the model’s assumptions; it is not a volume of blood cleared per hour. If a clearance-based first-order representation is appropriate for a compartment of fixed volume, dividing clearance by that volume gives a reciprocal-time quantity. The dimensional justification is:
This check establishes that \(CL_i/V_i\) can have the units of a first-order rate constant. It does not establish that the biological process is actually first order, nor that a particular volume is appropriate. Those are modelling assumptions that must be justified separately.
Worked Example: Converting a Model Inventory to One Time Unit
Hypothetical model inventory: A model uses an infusion input of 2.00 mg/h for 3.00 h, a central amount state in mg, a physiological flow of 15.0 L/h, a compartment volume of 3.00 L, a clearance of 4.00 L/h, and a first-order rate constant of 0.0800 h\(^{-1}\). The implementation will use minutes as its time unit.
The amount supplied by the infusion over 3.00 h is \(2.00\ \text{mg}/\text{h}\times3.00\ \text{h}=6.00\) mg. This is an amount, not an infusion-rate parameter; the infusion rate itself remains 2.00 mg/h until converted.
Convert every quantity involving time to minutes. Since 1 h is 60 min:
The compartment volume remains 3.00 L, and the amount state remains measured in mg: neither has a time dimension. The converted infusion rate, flow, clearance, and rate constant now all use minutes. A numerical value cannot be copied unchanged merely because its symbol has not changed; the unit conversion changes the value when the time unit changes.
A Practical Audit for States and Parameters
A useful audit records, for every model quantity, its meaning, role, dimension, displayed unit, and basis. “Basis” matters for quantities such as concentration: a concentration should identify whether it is plasma, blood, or a particular tissue compartment. The audit should also state the model’s amount convention (for example, mg throughout) and its single time unit. These checks make it easier to find mismatches before interpreting simulation output.
Write whether it is an amount, concentration, volume, flow, clearance, rate constant, or another quantity. Do not infer type from the variable name alone.
Mark it as a state, parameter, or derived quantity. A concentration calculated from an amount and volume is derived; a concentration tracked directly can be a state.
Use the definitions of the quantity to assign its units, and record the amount, volume, and time conventions used by the model.
Confirm which compartment or physiological basis applies, and convert all quantities to compatible units before comparing values or using them together.
For a simple inventory check, an amount multiplied by a concentration is not a valid conversion; amount divided by volume produces concentration. A clearance multiplied by concentration has units of amount per time, whereas a clearance alone does not. A rate constant multiplied by an amount also has units of amount per time. These elementary checks often reveal a mislabeled parameter or an omitted volume.
Common Mistakes, Diagnostics, and Limitations
- Calling an amount-per-time rate a clearance. A value in mg/min is an elimination rate. Clearance has units such as L/min; dividing an amount-per-time rate by a compatible concentration gives a clearance.
- Treating flow and clearance as synonyms. Both have units of volume per time, but flow represents fluid movement and clearance represents drug removal relative to concentration.
- Using a rate constant as a clearance. Units of h\(^{-1}\) identify a reciprocal-time quantity, not a volume-per-time quantity.
- Mixing time units. Combining a flow in L/h with a rate constant in min\(^{-1}\) without conversion can create a numerical error by a factor of 60.
- Leaving the concentration basis unstated. A concentration value is not fully specified by its number and mass-per-volume unit if it is unclear which fluid or compartment it describes.
A dimensional audit can detect incompatible units, but it cannot prove that the physiological assumptions are correct. For example, a dimensionally valid clearance may still be inappropriate for the process it is meant to represent. Likewise, amount-based and concentration-based formulations can both be valid, but they require consistent definitions of states, volumes, and outputs. Detailed checks of complete concentration-based equations are a separate step; here the goal is to make the quantities entering those equations unambiguous.
Worked Example: Finding a Mislabeled Rate in a Parameter Table
Hypothetical audit: A tissue amount state is 4.00 mg, the tissue volume is 0.200 L, the associated concentration is 20.0 mg/L, and the model lists an elimination value of 0.120 mg/min under the column “clearance.” Determine whether the label is correct and, if the value is an elimination rate at the stated concentration, calculate the corresponding clearance.
The listed unit is mg/min, which is amount per time. Therefore, 0.120 mg/min is an elimination rate, not a clearance. If it is the rate at \(C_i=20.0\) mg/L, clearance is rate divided by concentration:
The units reduce to L/min, as required for clearance. The parameter table should either retain 0.120 mg/min and label it “elimination rate at the stated concentration,” or list 0.00600 L/min as clearance and state the concentration basis used to infer it. These are not numerically interchangeable entries: they describe different quantities.
Check Your Understanding
Answer each question by identifying the quantity type and showing the relevant unit reasoning. Then compare your work with the solutions below.
- A model tracks 8.0 mg of drug in a compartment with volume 0.40 L. What concentration follows on this volume basis, and is that concentration necessarily a state variable?
- A parameter is reported as 0.25 L/h. Give two quantity types it could represent, and explain what additional information distinguishes them.
- A first-order rate constant is 0.12 h\(^{-1}\). Express it in min\(^{-1}\).
- A reported drug-removal value is 3.0 mg/h. Is this a clearance or an amount-per-time rate? What additional quantity would be needed to calculate clearance?
- A clearance is 6.0 L/h and a fixed compartment volume is 2.0 L. What is the corresponding \(CL/V\) value and its unit, under a first-order representation?
Solutions. (1) \(C=A/V=8.0\ \text{mg}/0.40\ \text{L}=20\ \text{mg}/\text{L}\). It is derived if computed from the amount and volume, but it could instead be a state if the model directly tracks concentration. (2) It could be a physiological flow or a clearance; both have units of volume per time. Its meaning in the model—fluid transport versus drug removal relative to concentration—distinguishes them. (3) \(0.12\ \text{h}^{-1}/60=0.0020\ \text{min}^{-1}\), because one hour contains 60 minutes. (4) It is an amount-per-time rate because its units are mg/h. A compatible concentration, in amount per volume, is needed to calculate clearance by dividing the rate by concentration. (5) \(CL/V=(6.0\ \text{L}/\text{h})/(2.0\ \text{L})=3.0\ \text{h}^{-1}\). The result has reciprocal-time units and is a rate constant only if the stated first-order representation is appropriate.