Fraction Unbound In Plasma And Binding Equilibria

Why distinguish free and bound drug?

A plasma concentration reported as “total” combines drug that is free in plasma with drug bound to plasma constituents. The distinction matters in PBPK because the free concentration is the concentration available to move independently of those binding interactions. A fraction unbound is therefore not another concentration: it is a dimensionless ratio that relates free concentration to total concentration under specified conditions.

The introductory material on unbound fractions as inputs to Kp methods establishes why an unbound fraction can enter partition predictions. Here the focus is the plasma quantity itself: its definition, its relation to binding at equilibrium, and how to calculate free and bound concentrations. The binding equations below describe an idealized equilibrium model; they do not identify which plasma constituent is responsible for binding.

Both fractions are dimensionless and range from 0 to 1. A small \(f_{u,p}\) means a large proportion of the measured total concentration is bound; it does not, by itself, say that the free concentration is small in absolute units. That also depends on \(C_{p,\mathrm{tot}}\). When a fraction is known, the corresponding concentrations follow directly:

Equilibrium affinity and a finite set of binding sites

A simple mechanistic model treats free drug, \(D\), as reversibly binding to an available site, \(S\), to form a drug–site complex, \(DS\). At equilibrium, the association and dissociation processes balance. The dissociation constant \(K_d\) is the ratio of free drug and free site concentrations to complex concentration. A smaller \(K_d\) indicates greater binding affinity in this model. The association constant \(K_a\) is its reciprocal.

For one class of independent, equivalent sites, let \(B_{\max}\) be the total concentration of binding sites, including both occupied and unoccupied sites. The model assumes reversible binding has reached equilibrium, each site binds at most one drug molecule, and the sites have the same affinity. It also assumes concentrations can be used in the mass-action relationship, as is standard for this simplified treatment. These assumptions give a useful model, not a claim that plasma binding always follows one homogeneous site class.

The fraction of sites occupied is \(\theta=[DS]/B_{\max}\). Solving the equilibrium relationship shows how occupancy depends on the free, not the total, drug concentration:

Rearrange the last equation to collect the terms involving \([DS]\):

Because \([DS]=C_{b,p}\) in this model, the bound concentration is \(C_{b,p}=B_{\max}C_{u,p}/(K_d+C_{u,p})\), and \(\theta=C_{u,p}/(K_d+C_{u,p})\). Total concentration is then free plus bound. This derivation also explains why the same \(f_{u,p}\) need not apply at every concentration: the bound concentration does not increase proportionally without limit when sites can become occupied.

Using equilibrium parameters to calculate binding

When \(B_{\max}\) and \(K_d\) are given, the binding equation calculates bound concentration from free concentration. If instead total concentration is given, first use the mass balance \(C_{p,\mathrm{tot}}=C_{u,p}+C_{b,p}\). Substitution produces an equation for \(C_{u,p}\). In the one-site model, this equation can be solved algebraically. The physically meaningful solution must be nonnegative and cannot exceed the total concentration.

When is fraction unbound constant?

A constant \(f_{u,p}\) is a useful approximation when binding behaves linearly over the concentration range of interest. In the one-site equation, if free concentration is much smaller than \(K_d\), then \(K_d+C_{u,p}\) is approximately \(K_d\). Consequently,

This approximation follows from the model’s low-occupancy condition, \(C_{u,p}\ll K_d\); it is not a universal property of plasma binding. As sites become occupied, the bound concentration approaches \(B_{\max}\), while free concentration can continue increasing. The total-to-free relationship then becomes nonlinear and \(f_{u,p}=C_{u,p}/C_{p,\mathrm{tot}}\) generally changes with concentration. Binding affinity and capacity have distinct roles: \(K_d\) sets the concentration scale for occupancy, while \(B_{\max}\) sets the maximum bound concentration for the site class.

Checks, alternatives, and limitations

A sound calculation begins by identifying which concentration is supplied. If \(f_{u,p}\) is supplied, multiply it by total concentration to obtain free concentration. If free concentration and equilibrium binding parameters are supplied, use the binding equation to obtain bound concentration. If only total concentration and \(B_{\max}\) and \(K_d\) are supplied, combine the binding equation with the mass balance and solve for the physically meaningful free concentration.

• Check the mass balance. Under the stated two-form definition, \(C_{u,p}+C_{b,p}\) must equal \(C_{p,\mathrm{tot}}\), apart from rounding.
• Check bounds and units. Fractions must lie between 0 and 1; concentrations must be nonnegative and use compatible units. \(K_d\) must have concentration units, and \(B_{\max}\) must be a concentration.
• Check the equilibrium assumptions. A single-site prediction may be inadequate if binding sites are heterogeneous, more than one binding process matters, or equilibrium is not a reasonable description.
• Check concentration dependence. A fraction measured or assumed at one concentration should not automatically be treated as constant if binding may saturate over the range being modelled.

The simplest alternative to the explicit site model is to use a supplied or measured \(f_{u,p}\) directly. This is convenient when a constant fraction is an adequate representation in the relevant range, but it does not explain affinity or saturation. A mechanistic equilibrium model represents those features under stated assumptions, but its parameters and structure must be appropriate. Neither approach removes the need to define total and free concentrations consistently. In particular, a one-site equation cannot establish that real plasma binding consists of one uniform site class.

A second frequent error is to call a small fraction unbound “a small free concentration.” For example, a fraction of 0.01 applied to a total concentration of 1,000 concentration units gives a free concentration of 10 units. The fraction describes a proportion; the absolute concentration depends on the total. For PBPK input selection, this distinction helps prevent a ratio from being mistaken for a concentration and makes the assumptions behind a constant unbound fraction explicit.