Auditing Permeability and Tissue-Exchange Flux Units

Why Permeability-Flux Units Deserve a Separate Audit

A permeability-limited tissue model describes transfer driven by a concentration difference across a barrier. Its exchange term can look simple, but it combines quantities that are easy to mix: a permeability coefficient, a membrane area, concentrations that may use different volume units, and compartment amounts or concentrations. A formula can have the right general shape and still produce a flux that is wrong by a factor of 1,000—or more—if one conversion is omitted.

The earlier material on scaling membrane permeability by area and volume establishes the role of surface area in the exchange term. Here the task is to audit the units of that term from end to end: first as amount transferred per time, then as a contribution to each compartment’s concentration rate. The focus is dimensional consistency, not estimating permeability from data or choosing a tissue structure.

Definition: Let \(P\) be a permeability coefficient with units of length per time, \(A\) a membrane area, and \(C_1\) and \(C_2\) concentrations expressed in the same amount-per-volume units. The permeability-surface-area product is \(PS=PA\), with units of volume per time. For a simple concentration-driven exchange, define the signed amount flux from compartment 1 to compartment 2 as \(J=PS(C_1-C_2)\). Thus \(J\) has units of amount per time; \(J>0\) means transfer from compartment 1 to compartment 2 under this sign convention.

From Permeability to Amount Flux

The units of permeability depend on the length and time units chosen. If \(P\) is in cm/s and \(A\) is in cm\(^2\), then \(PA\) is in cm\(^3\)/s. Because one cm\(^3\) is one mL, this is also mL/s. Multiplying that volume-per-time quantity by a concentration in amount per volume gives amount per time. The volume units must match: mL/s multiplied by amount/mL works directly, whereas mL/s multiplied by amount/L requires a conversion.

[P]=\frac{\mathrm{cm}}{\mathrm{s}},\qquad [A]=\mathrm{cm}^2,\qquad [PS]=\frac{\mathrm{cm}^3}{\mathrm{s}}=\frac{\mathrm{mL}}{\mathrm{s}}

[J]=\frac{\mathrm{volume}}{\mathrm{time}} \times\frac{\mathrm{amount}}{\mathrm{volume}} =\frac{\mathrm{amount}}{\mathrm{time}}

In some formulations, the driving force is a partition-adjusted concentration difference rather than \(C_1-C_2\). For example, if a dimensionless partition coefficient \(K\) is defined so that equilibrium requires \(C_2=KC_1\), a consistent driving force is \(C_1-C_2/K\). This expression still has concentration units, so multiplying it by \(PS\) still gives amount per time. The definition and direction of \(K\) must be stated; using a partition coefficient with the opposite convention changes the required expression.

From Amount Flux to Concentration Rates

An exchange flux is an amount rate, but a PBPK ordinary differential equation may be written for a concentration state. Let \(V_1\) and \(V_2\) be the corresponding compartment volumes, expressed in the same volume units used to define concentrations. For exchange alone, conservation requires that the amount leaving compartment 1 enters compartment 2. Consequently, the concentration-rate contributions are the signed amount rates divided by their respective volumes.

\frac{dC_1}{dt}=-\frac{J}{V_1},\qquad \frac{dC_2}{dt}=+\frac{J}{V_2}

The units on each right-hand side are \((\text{amount}/\text{time})/(\text{volume})\), or concentration per time. These equations assume well-mixed compartments with fixed volumes for this exchange calculation, the same amount basis on both sides of the membrane, and a flux defined using compatible concentration units. Other physiological processes may add terms to the full ODE, but they do not change the units required of this exchange contribution.

A short conservation check follows directly. The compartment amounts are \(A_1=V_1C_1\) and \(A_2=V_2C_2\). For fixed volumes, the exchange-only amount rates are \(V_1(dC_1/dt)=-J\) and \(V_2(dC_2/dt)=+J\). Their sum is zero, so exchange redistributes amount without creating or removing it.

\frac{d(A_1+A_2)}{dt} =V_1\frac{dC_1}{dt}+V_2\frac{dC_2}{dt} =-J+J=0

Worked Example: Converting a Permeability and Area into Flux

Suppose a hypothetical membrane has \(P=2.5\times10^{-5}\) cm/s and \(A=80\) cm\(^2\). The concentrations are \(C_1=4.0\) μmol/mL and \(C_2=1.5\) μmol/mL. Calculate the signed exchange flux.

First calculate the permeability-surface-area product:

PS=(2.5\times10^{-5}\ \mathrm{cm/s})(80\ \mathrm{cm}^2) =2.0\times10^{-3}\ \mathrm{cm}^3/\mathrm{s} =2.0\times10^{-3}\ \mathrm{mL/s}

The concentration difference is \(4.0-1.5=2.5\) μmol/mL. Therefore,

J=(2.0\times10^{-3}\ \mathrm{mL/s})(2.5\ \mathrm{\mu mol/mL}) =5.0\times10^{-3}\ \mathrm{\mu mol/s}

Converting seconds to minutes gives \(J=(5.0\times10^{-3})(60)=0.300\) μmol/min. The positive sign indicates net transfer from compartment 1 to compartment 2. The area is essential: \(P\) alone has units of length per time and cannot be multiplied by a concentration to obtain an amount rate.

Worked Checks for Volume and Concentration Units

Worked Example: Turning Flux into Concentration Rates

In a hypothetical exchange, \(P=0.006\) cm/min, \(A=25\) cm\(^2\), \(C_1=12\) μmol/L, and \(C_2=5\) μmol/L. The receiving compartment has volume \(V_2=0.35\) mL. Find the flux and the exchange contribution to \(dC_2/dt\).

Calculate \(PS\) first:

PS=(0.006\ \mathrm{cm/min})(25\ \mathrm{cm}^2) =0.15\ \mathrm{cm}^3/\mathrm{min} =0.15\ \mathrm{mL/min}

The concentrations are given per litre, while \(PS\) is in mL/min. Convert the difference to μmol/mL. Since \(1\) L is \(1000\) mL, \(12\) μmol/L is \(0.012\) μmol/mL and \(5\) μmol/L is \(0.005\) μmol/mL. Hence the difference is \(0.007\) μmol/mL, and

J=(0.15\ \mathrm{mL/min})(0.007\ \mathrm{\mu mol/mL}) =0.00105\ \mathrm{\mu mol/min}

The receiving compartment gains this amount, so its concentration rate is

\frac{dC_2}{dt} =\frac{0.00105\ \mathrm{\mu mol/min}}{0.35\ \mathrm{mL}} =0.00300\ \mathrm{\mu mol/(mL\,min)} =3.00\ \mathrm{\mu mol/(L\,min)}

The positive rate is consistent with \(C_1>C_2\) and the chosen flux direction. Dividing by volume is required because the ODE is for concentration, not amount. A smaller receiving volume would give a larger concentration rate for the same amount flux.

Worked Example: Finding a Hidden Concentration Conversion Error

A hypothetical calculation uses \(P=1.2\times10^{-4}\) cm/s, \(A=50\) cm\(^2\), \(C_1=2.4\) mg/L, and \(C_2=0.9\) mg/L. A draft calculation multiplies \(PS\), in mL/s, directly by the difference \(1.5\) mg/L and reports the result as mg/s. Audit and correct it.

The area-scaled permeability is

PS=(1.2\times10^{-4}\ \mathrm{cm/s})(50\ \mathrm{cm}^2) =0.0060\ \mathrm{cm}^3/\mathrm{s} =0.0060\ \mathrm{mL/s}

The reported multiplication would have units \((\mathrm{mL/s})(\mathrm{mg/L})\), not mg/s. Convert concentrations to mg/mL: \(2.4\) mg/L is \(0.0024\) mg/mL, and \(0.9\) mg/L is \(0.0009\) mg/mL. The difference is \(0.0015\) mg/mL. Thus,

J=(0.0060\ \mathrm{mL/s})(0.0015\ \mathrm{mg/mL}) =9.0\times10^{-6}\ \mathrm{mg/s}

Equivalently, the concentration difference is \(1.5\times10^{-3}\) mg/mL, not \(1.5\) mg/mL. Failing to convert litres to millilitres overstates the flux by a factor of \(1000\). The corrected positive flux is \(9.0\times10^{-6}\) mg/s from compartment 1 to compartment 2.

A Practical Audit and Its Failure Modes

A reliable check proceeds through the physical dimensions rather than relying on the appearance of the formula. The following sequence catches most errors in tissue-exchange terms.

Identify what the state represents.
Determine whether each ODE state is an amount or a concentration. An amount derivative requires amount/time; a concentration derivative requires concentration/time.

Build the area-scaled permeability.
Multiply permeability by membrane area. Confirm that length times area gives volume, so \(PS\) is volume/time.

Make the driving concentrations compatible.
Use the same amount unit and volume unit for both concentrations, and verify any partition-adjusted term has concentration units.

Check the flux and state conversion.
Confirm that \(PS\) times the concentration difference is amount/time. Divide by the appropriate compartment volume only when converting that amount rate into a concentration rate.

Several errors have recognizable signatures. If a purported flux has units of concentration/time, a required volume-per-time factor may have been omitted. If multiplying \(P\) directly by a concentration produces the claimed amount rate, the membrane area has likely been omitted. If units are written as mL/s times mg/L and declared mg/s, the litre-to-millilitre conversion is missing. If the amount flux is added directly to an equation for concentration, the compartment volume has not been used. Finally, if the receiving compartment loses concentration while \(J\) is defined as positive into it, the sign convention and the equations disagree.

Expression or approach	Required units	What it represents
\(P\)	length/time	Permeability coefficient alone; not an amount flux.
\(PS=PA\)	volume/time	Area-scaled permeability.
\(PS(C_1-C_2)\)	amount/time	Signed amount transfer rate when concentrations use compatible units.
\(J/V_i\)	concentration/time	Contribution to a concentration ODE for compartment \(i\).
\(Q(C_{\mathrm{in}}-C_{\mathrm{out}})\)	amount/time	A flow-driven exchange term; its flow factor is volume/time, rather than permeability times area.

The flow-driven expression is a useful comparison, not a substitute for auditing a permeability-limited term. Both expressions produce amount/time when the concentration units match the volume units in the flow or \(PS\) factor. But their physical rate factors differ: a flow term uses a volumetric flow rate, whereas a membrane term uses permeability times area. Using one in place of the other changes the model mechanism as well as the parameter interpretation.

Dimensional consistency is necessary, but it cannot show that a numerical permeability is physiologically appropriate, that the chosen area represents the effective exchange surface, or that the tissue is well mixed. Nor does a correct unit check establish that the model’s partition convention or flux direction is biologically appropriate. It verifies the arithmetic structure and units of the stated model; physiological assumptions require separate justification.

Common Mistake

Common pitfall: “The concentration difference is 1.5 mg/L, and \(PS\) is in mL/s, so their product is a mass rate.” This is incorrect because mL and L do not cancel. Convert the concentration to mg/mL (or convert \(PS\) to L/s) before multiplying. The same principle applies when converting amount flux to a concentration rate: divide by a compartment volume expressed compatibly with the concentration basis.

A second useful check is to set \(C_1=C_2\). For the simple unpartitioned exchange expression, this makes \(J=0\), as expected when the driving concentration difference vanishes. If the two compartment equations do not then have equal and opposite amount-rate contributions, the sign or volume conversion has been applied incorrectly.

Check Your Understanding

Try each question, then compare your reasoning with the brief solutions below.

A hypothetical membrane has \(P=0.004\) cm/min and \(A=30\) cm\(^2\). What are the units and value of \(PS\)?
If \(PS=0.20\) mL/min and the compatible concentration difference is \(3.0\) μmol/mL, what is the amount flux?
An amount flux is \(0.006\) mg/min into a compartment of volume \(2.0\) mL. What is the concentration-rate contribution in mg/L/min?
Why is it dimensionally wrong to multiply a value in mL/s directly by a concentration in μmol/L and report μmol/s?
For exchange alone, \(dC_1/dt=-J/V_1\) and \(dC_2/dt=+J/V_2\). What condition makes total exchanged amount conserved?

Solutions. (1) \(PS=(0.004)(30)=0.12\) cm\(^3\)/min \(=0.12\) mL/min. (2) \(J=(0.20\ \mathrm{mL/min})(3.0\ \mathrm{\mu mol/mL})=0.60\) μmol/min. (3) Divide by volume: \(0.006/2.0=0.003\) mg/(mL·min), which is \(3.0\) mg/L/min. (4) The volume units do not cancel: mL/L contributes a factor of \(1/1000\). Convert one volume unit before multiplying. (5) The compartments must receive equal and opposite amount-rate contributions, \(-J\) and \(+J\); then their sum is zero. For the concentration equations, this requires using each compartment’s own volume when translating the shared flux into a concentration rate.