From Binding Equilibrium to a Measured Ratio
In PBPK models, plasma fraction unbound, \(f_{u,p}\), describes the fraction of drug in plasma that is not bound to plasma components. The earlier material on binding equilibria defines this fraction using free and total plasma concentrations. Measuring it requires an assay that separates, or otherwise estimates, the free concentration while preserving the relevant binding equilibrium. Equilibrium dialysis and ultrafiltration are two commonly used approaches, but they do not separate the free drug in the same way.
Both methods depend on measuring concentrations in compatible units and at conditions where the relationship between free and total drug is meaningful. In particular, binding can depend on the drug concentration and the sample environment. The reported result should therefore be associated with its tested concentration, plasma source, temperature, pH, and assay conditions, rather than treated as an unconditional property of a compound.
Equilibrium Dialysis
In equilibrium dialysis, a semipermeable membrane separates a plasma-containing chamber from a buffer chamber. The membrane permits free drug to pass but retains plasma proteins. Drug initially present in plasma can diffuse into the buffer until the free drug reaches equilibrium across the membrane. The buffer side contains no plasma proteins, so its measured drug concentration estimates the free concentration at equilibrium, provided the method’s assumptions hold.
This calculation assumes that free drug has equilibrated across the membrane; that proteins do not cross into the buffer; and that the measured concentrations represent the drug in the respective chambers. It also assumes that the assay environment preserves the binding state of interest, including suitable temperature and pH. Because equilibrium is essential, a fixed incubation duration should not be treated as proof that equilibrium was reached. The equilibration time should be selected and checked for the assay conditions.
A typical preparation starts by adding a known amount of drug to plasma, mixing it, and allowing the drug to interact with plasma components before loading the sample. Plasma is placed in one chamber and an appropriate buffer in the other; the apparatus is sealed and incubated under controlled conditions. After equilibration, samples are taken from both chambers for drug quantification. A matched analytical procedure and careful volume handling matter because the reported fraction is a ratio of concentrations, not a ratio of the amounts initially added.
Worked Example: Calculating Fraction Unbound by Equilibrium Dialysis
In a hypothetical assay, the equilibrium plasma-chamber concentration is \(8.00\ \mathrm{ng/mL}\), and the equilibrium buffer-chamber concentration is \(0.240\ \mathrm{ng/mL}\). Assume the membrane retained plasma proteins, the drug reached equilibrium across the membrane, and both concentrations were measured in the same units.
The dialysis estimate is: $$ f_{u,p}=\frac{C_{\mathrm{buffer,eq}}}{C_{\mathrm{plasma,eq}}} =\frac{0.240\ \mathrm{ng/mL}}{8.00\ \mathrm{ng/mL}} =0.0300. $$
The units cancel. The estimated plasma fraction unbound is \(0.0300\), or 3.00% under these assay conditions. This does not mean that 3.00% of the drug dose is free; it means that the measured free concentration is 3.00% of the measured total concentration in the plasma chamber.
Ultrafiltration
Ultrafiltration estimates the free concentration by physically separating a portion of the liquid phase from plasma. The drug–plasma mixture is first allowed to approach binding equilibrium. It is then placed in a filtration device and centrifuged under controlled conditions. A membrane retains plasma proteins while allowing water and sufficiently small solutes, including free drug, to pass into the filtrate. The concentration in the filtrate is used as an estimate of the free concentration in the original plasma sample.
This interpretation assumes that the filtrate represents the free drug in the plasma at the time of separation, that proteins are retained, and that the filtration procedure does not materially change the binding state represented by the measured concentrations. In practice, the operator needs to control sample temperature, centrifugation conditions, and the amount of filtrate collected. The procedure should not be interpreted as measuring binding equilibrium merely because a filtrate was obtained: the plasma must first be prepared and allowed to equilibrate, and the filtration conditions must be appropriate for the assay.
Preparation commonly involves spiking drug into plasma, mixing, and allowing time for binding before loading a defined volume into the filtration device. After centrifugation, the filtrate is collected and quantified. A matched aliquot is used to determine total plasma concentration. Using only the nominal spike concentration as the denominator can be misleading if the actual measured concentration differs from the nominal one; the concentration ratio should use measured values from the relevant sample.
Worked Example: Calculating Fraction Unbound by Ultrafiltration
In a hypothetical ultrafiltration assay, a matched aliquot of equilibrated plasma has a measured total drug concentration of \(12.0\ \mathrm{\mu g/mL}\). The filtrate concentration is \(0.900\ \mathrm{\mu g/mL}\). Assume the plasma proteins were retained and the collected filtrate is suitable for estimating the free concentration.
The calculation is: $$ f_{u,p}=\frac{C_{\mathrm{filtrate}}}{C_{\mathrm{plasma,tot}}} =\frac{0.900\ \mathrm{\mu g/mL}}{12.0\ \mathrm{\mu g/mL}} =0.0750. $$
Thus, the estimated plasma fraction unbound is \(0.0750\), or 7.50%, under the tested conditions. The calculation uses the matched measured total concentration, not the amount placed into the device or the volume of filtrate collected.
Why the Ratios Work—and When They Do Not
The central justification is the definition of \(f_{u,p}\), together with what each method uses as an estimate of the free concentration. If the total plasma concentration is the sum of free and bound drug, then the free concentration is \(f_{u,p}\) times the total concentration. In dialysis, a freely permeating drug at equilibrium has the same free concentration on the two sides of the membrane, under the assay’s conditions. Because the buffer side has no plasma binding proteins, its drug concentration estimates the free concentration in plasma. In ultrafiltration, the filtrate concentration estimates the free concentration present in the plasma before filtration, if the separation preserves the relevant state.
The practical distinction is how \(C_{u,p}\) is estimated: dialysis infers it from the equilibrium buffer concentration, whereas ultrafiltration infers it from the filtrate concentration. Neither ratio is self-validating. If equilibrium has not been reached in dialysis, the buffer concentration may not represent equilibrium free drug. If filtration conditions do not preserve the relevant state or the filtrate is not a suitable measure of free drug, its concentration may not represent the original plasma free concentration. In either case, the arithmetic may be correct while the estimate is not.
Worked Example: Accounting for the Measured Denominator
A hypothetical laboratory prepares \(1.00\ \mathrm{mL}\) of plasma by adding \(10.0\ \mathrm{\mu L}\) of a \(1.00\ \mathrm{mg/mL}\) drug stock to \(990\ \mathrm{\mu L}\) of plasma. The nominal concentration after mixing is: $$ \frac{(1.00\ \mathrm{mg/mL})(0.0100\ \mathrm{mL})}{1.00\ \mathrm{mL}} =0.0100\ \mathrm{mg/mL} =10.0\ \mathrm{\mu g/mL}. $$
After dialysis, suppose the measured plasma-chamber concentration is \(9.60\ \mathrm{\mu g/mL}\), while the buffer concentration is \(0.384\ \mathrm{\mu g/mL}\). For the dialysis calculation, use the measured equilibrium plasma concentration: $$ f_{u,p}=\frac{0.384\ \mathrm{\mu g/mL}}{9.60\ \mathrm{\mu g/mL}} =0.0400. $$
The estimated fraction unbound is \(0.0400\), or 4.00%. Using the nominal \(10.0\ \mathrm{\mu g/mL}\) in the denominator would instead give \(0.0384\), a different ratio. The relevant calculation uses the measured concentration in the plasma chamber at equilibrium, because that is the total concentration paired with the measured buffer concentration.
Choosing and Checking a Method
Equilibrium dialysis directly establishes a concentration gradient between plasma and protein-free buffer and is often useful when a well-defined equilibrium estimate is needed. Its practical cost is the time required to reach equilibrium and the need to sample both chambers. Ultrafiltration can produce a separated free-drug sample more quickly once the plasma mixture is prepared, but the separation step itself must be controlled so the filtrate remains interpretable as the free concentration in the starting plasma. Neither method is universally preferable: the choice depends on the compound, sample, analytical sensitivity, available equipment, and whether the method’s assumptions can be supported.
| Feature | Equilibrium dialysis | Ultrafiltration |
|---|---|---|
| How free drug is estimated | Drug concentration in the buffer chamber at equilibrium | Drug concentration in the filtrate after filtration |
| Total concentration used | Measured plasma-chamber concentration at equilibrium | Measured total concentration in a matched equilibrated plasma aliquot |
| Central condition | Free drug must reach equilibrium across the membrane | Filtration must provide a suitable estimate of the original plasma free concentration |
| Practical consideration | Requires incubation and sampling from both chambers | Requires controlled centrifugation and collection of filtrate |
A useful first diagnostic is to verify the identity of each concentration in the ratio. A buffer concentration should not be divided by an initial nominal concentration when the calculation calls for an equilibrium plasma concentration. A filtrate concentration should not be paired with a concentration measured in a different preparation or under incompatible conditions. Check that numerator and denominator have the same units and that the reported ratio is dimensionless.
Next, check whether the defining conditions are credible: the sample was prepared consistently, the assay temperature and pH were controlled, and the method had an adequate opportunity to reach or preserve the intended state. A result greater than 1 is inconsistent with the usual interpretation of a free fraction as a part of total concentration and should prompt investigation of the measurements or assumptions rather than a biological interpretation. Results that are difficult to quantify, especially when the free concentration is very low, also require attention to the analytical method’s ability to measure the numerator.
Common Mistakes and Limitations
- Using the spike concentration automatically. The nominal amount added is not necessarily the measured total concentration used in the ratio. Use the appropriate measured plasma concentration for the method.
- Swapping the numerator and denominator. Fraction unbound is free concentration divided by total plasma concentration, not the reverse.
- Assuming that separation proves equilibrium. Dialysis requires equilibrium across the membrane; ultrafiltration requires an equilibrated plasma sample before separation.
- Comparing results without comparing conditions. A fraction measured at one concentration, temperature, or plasma condition should not be assumed to match a value measured under different conditions.
- Expecting identical results from different methods. The methods estimate the same quantity through different procedures and have different assumptions. A discrepancy calls for a method- and condition-specific review; it is not resolved by automatically averaging the estimates.
Both methods provide assay-based estimates rather than direct access to an unobservable free concentration. A reported value is most useful when accompanied by the method, sample conditions, tested concentration, and measured concentrations used in the calculation. It should be carried into a PBPK application only with awareness of those conditions and the limits of the underlying measurement.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In equilibrium dialysis, which concentration estimates free drug: the buffer concentration or the plasma-chamber concentration?
- A dialysis assay gives \(C_{\mathrm{buffer,eq}}=0.150\ \mathrm{mg/L}\) and \(C_{\mathrm{plasma,eq}}=5.00\ \mathrm{mg/L}\). Calculate \(f_{u,p}\).
- An ultrafiltration assay reports a filtrate concentration of \(0.060\ \mathrm{\mu g/mL}\) and a matched plasma total concentration of \(3.00\ \mathrm{\mu g/mL}\). Calculate the estimated fraction unbound.
- Why is a nominal spike concentration not automatically the right denominator in the dialysis calculation?
- State one central assumption of ultrafiltration and one central assumption of equilibrium dialysis.
Solutions. (1) The equilibrium buffer concentration estimates free drug, because the buffer contains no plasma proteins and free drug has equilibrated across the membrane. (2) \(f_{u,p}=0.150/5.00=0.0300\), or 3.00%. (3) \(f_{u,p}=0.060/3.00=0.0200\), or 2.00%. (4) The calculation pairs the buffer measurement with the measured total concentration in the plasma chamber at equilibrium; the nominal spike value may differ from that measured concentration. (5) Ultrafiltration assumes the filtrate is a suitable estimate of free drug in the equilibrated starting plasma. Equilibrium dialysis assumes free drug reaches equilibrium across the membrane while plasma proteins are retained.