Concentration-Dependent Plasma Binding

Why Plasma Binding Can Depend on Concentration

A fraction unbound measured at one drug concentration is not necessarily a fixed property of a compound. If binding sites have finite capacity, increasing the drug concentration fills a larger share of those sites. The bound amount then approaches a maximum, and a progressively larger share of additional drug remains unbound. For PBPK parameterization, this means that a single fraction-unbound value may not describe the whole concentration range of interest.

The model below describes reversible binding to one class of identical sites at equilibrium. It separates the free drug concentration from the drug bound to protein and uses mass balance to connect them to the measured total plasma concentration. As in the earlier material on recovery and artifacts in plasma-binding assays, the concentration values used for interpretation need to represent the intended equilibrium system; apparent concentration dependence caused by assay problems is not evidence of saturation.

Deriving the Saturable-Binding Relationship

The model assumes that free drug binds reversibly to a fixed population of sites, that binding has reached equilibrium, and that all sites in the population have the same affinity. The plasma conditions and binding-protein abundance are treated as constant over the concentration range considered. There is no competition from other ligands, and there are no additional binding processes in the model. The bound concentration is given by the one-site equilibrium expression:

This expression is approximately proportional to \(C_u\) when \(C_u\) is small relative to \(K_d\), but it approaches \(B_{max}\) as \(C_u\) becomes large. Substituting it into the mass balance gives the total concentration as a function of the unbound concentration:

To calculate \(C_u\) from an observed or specified \(C_{tot}\), multiply both sides by \(K_d+C_u\) and collect terms. This produces a quadratic equation:

For positive \(C_{tot}\), the constant term is negative, so the two roots have opposite signs. Only the positive root is physically meaningful. Thus the unbound concentration and fraction unbound can be calculated as:

This solution gives \(C_u\) in the same concentration units as \(C_{tot}\), \(K_d\), and \(B_{max}\). If concentrations are expressed in mg/L, for example, all three model parameters must also use mg/L. \(B_{max}\) represents the binding capacity in drug-equivalent concentration units, not necessarily the molar concentration of the protein itself.

How Fraction Unbound Changes as Sites Fill

The model has two useful limits. At low unbound concentrations, the denominator \(K_d+C_u\) is approximately \(K_d\). The bound concentration is then nearly proportional to the free concentration, and the fraction unbound approaches a constant:

At high unbound concentrations, the binding capacity is nearly filled: \(C_b\) approaches \(B_{max}\). In that limit, \(C_{tot}\) is approximately \(C_u+B_{max}\), and \(f_u\) approaches 1 as total concentration increases. The fraction unbound therefore generally rises with concentration in this model. The change need not be large over every observed range; it depends on the relation between the concentrations and the binding parameters.

A short derivative makes the changing response explicit. Differentiating the mass-balance expression with respect to \(C_u\) gives \(dC_{tot}/dC_u=1+B_{max}K_d/(K_d+C_u)^2\). Reversing this derivative gives the local change in free concentration per unit change in total concentration:

For positive parameter values this slope is greater than 0 and less than 1. At low concentrations, binding absorbs a relatively large part of an increase in total concentration, so the free concentration rises more slowly. As sites approach saturation, the slope moves toward 1: less of an additional increase in total concentration can be accommodated by additional binding. The fraction unbound can rise even though the local absolute increase in \(C_u\) per unit increase in \(C_{tot}\) remains at most 1.

Assumptions, Alternatives, and Failure Modes

The model is useful when equilibrium binding is reversible and a single effective site class adequately describes the data. Its assumptions should be considered before using the quadratic solution. Changes in protein abundance, pH, competing ligands, or other binding mechanisms can alter the observed relationship. A measurement made before equilibrium, or one affected by the assay artifacts addressed in the earlier material, can also produce apparent concentration dependence that this equilibrium model does not explain.

A constant \(f_u\) is a simpler alternative. It may be a reasonable approximation over a concentration range where binding is effectively linear, such as when \(C_u\) is small relative to \(K_d\). It is not a general replacement for the saturable model across a range that approaches site capacity. Conversely, a one-site model may be too simple if different proteins or site classes contribute substantially. In that case, a model with multiple binding terms can represent more than one affinity or capacity, but it requires enough informative data to estimate the additional parameters.

Useful checks include verifying that predicted concentrations are nonnegative, that \(C_u\leq C_{tot}\), and that \(C_b=C_{tot}-C_u\) agrees with the binding equation. The bound concentration should approach, but not exceed, \(B_{max}\). A systematic change in measured \(f_u\) with concentration may be consistent with saturation, but by itself does not establish that the one-site mechanism is correct. Large inconsistencies with the model, or parameter estimates that change markedly when the sampled concentration range changes, call for examining the data range, measurement quality, and model adequacy.

• Pitfall: “The fraction unbound was 0.17 at one concentration, so multiplying every total concentration by 0.17 gives the free concentration.” This assumes that \(f_u\) stays constant. Under saturable binding, that fraction generally increases as total concentration increases.
• Correction: Use the one-site mass-balance equation and solve for \(C_u\) at each relevant \(C_{tot}\), or justify a constant-\(f_u\) approximation over the concentration range of interest.
• Pitfall: Treating \(B_{max}\) as an unlimited proportional binding coefficient. It is a capacity; the model predicts \(C_b\) cannot exceed it.