Checking Units in Compartmental Parameters

Why Units Are a Structural Check

A compartment model describes amounts moving between spaces and being eliminated from the body. Its parameters do not all have the same units: an amount, a concentration, a volume, a clearance, and a rate constant represent different kinds of quantities. Checking their units is a quick way to catch mistakes in equations and calculations before interpreting a concentration-time profile.

Dimensional consistency is necessary, but it is not sufficient to establish that a model or parameter value is correct. Two expressions can have matching units and still represent different biological processes. Here the focus is narrower: verify that the quantities in one- and two-compartment calculations can be combined as written.

In a well-mixed compartment, concentration is amount divided by volume. For the central compartment, \(C_c=A_c/V_c\). Multiplying a clearance by concentration gives an amount-per-time rate: \((\mathrm{L/h})(\mathrm{mg/L})=\mathrm{mg/h}\). This is why elimination clearance is often used in an amount-balance equation as \(CL\,C_c\).

From Clearances and Volumes to Rate Constants

A rate constant describes the fraction of an amount transferred or eliminated per unit time. In the standard linear two-compartment model, \(k_{10}\) is elimination from the central compartment, \(k_{12}\) is transfer from central to peripheral, and \(k_{21}\) is transfer from peripheral to central. The subscript identifies the direction and destination convention; the defining relationships below specify the denominators.

The denominator rule follows from the amount flux. Elimination from the central compartment is \(CL\,C_c=CL\,A_c/V_c\), so its coefficient multiplying \(A_c\) is \(CL/V_c\). Central-to-peripheral transfer is \(Q\,C_c=Q\,A_c/V_c\), giving \(Q/V_c\). Peripheral-to-central transfer is \(Q\,C_p=Q\,A_p/V_p\), giving \(Q/V_p\). Since \((\mathrm{L/h})/\mathrm{L}=\mathrm{h}^{-1}\), all three coefficients are rates per unit time.

Here \(R_{\mathrm{in}}\) is an input rate, with units of amount per time. Every term on the right side of either equation must have units of amount per time, matching the derivative on the left. For example, \((k_{10}+k_{12})A_c\) has units \((1/\mathrm{h})(\mathrm{mg})=\mathrm{mg/h}\). If an amount rather than a rate is entered as \(R_{\mathrm{in}}\), the equation is dimensionally inconsistent.

Worked Unit Checks

Concentrations, Doses, and Time Units

For an instantaneous intravenous bolus into the central compartment, the initial central amount is the dose \(D\), and the initial central concentration is \(C_c(0)=D/V_c\). The dose must be an amount and the volume must be in matching volume units. For example, mg divided by L gives mg/L. If a model output is requested in \(\mathrm{\mu g/mL}\), convert the concentration after calculating it or convert all inputs consistently; do not silently treat mg/L as \(\mathrm{\mu g/mL}\) without noting that their numerical values are equal.

Time units also carry through exponential expressions. If an amount or concentration profile contains \(\exp(-kt)\), then \(kt\) must be dimensionless. A rate in \(\mathrm{h}^{-1}\) therefore requires time in hours. Using \(k=0.4\ \mathrm{h}^{-1}\) with \(t=30\) minutes as though \(t=30\) hours changes the exponent by a factor of 60. Convert the time to \(0.5\) hours, or convert the rate to \(\mathrm{min}^{-1}\), before evaluating the expression.

Practical Checks and Common Failure Modes

A reliable check starts by writing the units next to every supplied value, rather than only checking the final answer. Reduce each expression to its basic dimensions and confirm that additions combine like units. Then check whether the numerical value is sensible for the stated time scale. A rate of \(0.4\ \mathrm{h}^{-1}\) is not numerically the same as \(0.4\ \mathrm{min}^{-1}\), even though both are rates.

• Mixing time units. Dividing a clearance in L/min by a volume in L gives a rate in min\(^{-1}\), not h\(^{-1}\). Convert one of the time units before reporting a result in the other.
• Using the wrong compartment volume. \(k_{12}\) uses \(V_c\), whereas \(k_{21}\) uses \(V_p\). Follow the source of the transferred amount.
• Confusing amount and concentration. A dose in mg is not a concentration in mg/L. Divide by a volume to obtain concentration.
• Adding unlike quantities. Clearance and volume cannot be added, nor can an amount be added directly to an amount-per-time rate.
• Assuming correct units prove a correct model. A rate constant can have valid units and still be assigned to the wrong pathway or calculated with the wrong compartment volume.

Unit checking complements, rather than replaces, the usual approaches to compartment calculations. Directly substituting parameters into the amount-balance equations makes flux directions explicit; dimensional analysis is a quick audit of those substitutions. Converting units before calculation is usually safer than trying to repair a result afterward, particularly when concentrations, doses, and times use different scales. Neither a correct unit check nor a plausible numerical magnitude establishes that parameters can be uniquely estimated or that the structural model is appropriate.