Equilibrium and Apparent Solubility

Why Solubility Needs a Precise Definition

A solubility value is not a property of a drug alone. It depends on the solid form, temperature, pH, and medium composition, among other conditions. For an ionizable drug, the concentration of neutral molecules in equilibrium with a crystal can remain fixed while the total concentration in the surrounding water changes with pH. Surfactant micelles can raise the total amount measured in solution further, without raising the concentration of freely dissolved drug by the same factor.

These distinctions matter when a PBPK model uses a solubility value to describe luminal drug availability. A saturation concentration is an equilibrium limit for specified conditions; it does not state how quickly that concentration is reached. The calculations here concern crystalline drug in aqueous media and assume that the stated solid form remains the phase in equilibrium.

The word “free” here means unassociated with micelles. For an ionizable drug, free aqueous drug includes both neutral and ionized dissolved species. In contrast, intrinsic solubility refers only to the neutral species. These concentrations are not interchangeable.

Deriving the pH-Dependent Saturation Concentration

Consider a monoprotic weak acid, \(HA\), in equilibrium with its neutral crystal. Assume the neutral dissolved concentration at saturation is \(S_0\), the solution is sufficiently dilute that concentration ratios approximate activity ratios, and the ionization equilibrium is described by a pKa applicable to the stated conditions. The Henderson–Hasselbalch relation gives the ratio of ionized to neutral dissolved acid. Adding the two dissolved species gives the total free aqueous concentration.

For a monoprotic weak base, \(B\), in equilibrium with its neutral crystal, the protonated-to-neutral ratio is \(10^{\mathrm{p}K_a-\mathrm{pH}}\). The same addition of neutral and ionized species therefore gives a different pH dependence.

For a weak acid, the predicted free saturation concentration rises as pH rises above the pKa; for a weak base, it rises as pH falls below the pKa. When pH equals pKa, the neutral and ionized concentrations are equal, so the total free concentration is \(2S_0\). Far from the pKa in the direction favoring ionization, the total can be much larger than \(S_0\).

Adding Micelle-Associated Drug

In a medium containing surfactant above its critical micelle concentration (CMC), micelles can associate with dissolved drug. A simple linear association model treats the free aqueous saturation concentration as fixed by the solid and pH, then adds a micelle-associated pool. Let \([M]\) be the micelle concentration and \(K_m\) an effective association coefficient. In the model below, \(K_m\) has units reciprocal to those of \([M]\). It assumes that association is linear over the concentration range considered and that the micelle concentration is not appreciably changed by drug loading.

If the total surfactant concentration \(C_{\mathrm{surf}}\) exceeds the CMC and each micelle contains, on average, \(N_{\mathrm{agg}}\) surfactant molecules, a simplified estimate is \([M]\approx(C_{\mathrm{surf}}-\mathrm{CMC})/N_{\mathrm{agg}}\). This estimate requires consistent concentration units and treats aggregation number as fixed. The association equation is an effective model, not a universal law: different drug species may associate differently, and micelle loading may become nonlinear or saturable.

Recovering pH or Intrinsic Solubility From Measurements

The same equations can be rearranged to infer a pH or an intrinsic solubility, but only when the measured concentration represents free aqueous saturation and the solid-phase assumptions are appropriate. For example, for a weak acid, if the measured free saturation concentration is \(S_{\mathrm{free}}\), then

The condition \(S_{\mathrm{free}}>S_0\) is required for this inversion to return a finite pH under the model. A measurement that includes micelle-associated drug cannot be substituted directly for \(S_{\mathrm{free}}\).

Assumptions, Diagnostics, and Alternatives

These equations describe equilibrium with a specified crystalline phase. Their use requires that excess solid remains, that the pH is known at equilibrium, and that the drug behaves as the assumed monoprotic acid or base over the relevant range. The equations also use an idealized concentration treatment. In nonideal media, activities and conditional pKa values may be needed; temperature, ionic strength, buffer composition, and surfactant composition can all affect the result.

A calculated saturation concentration is not necessarily the concentration present in a formulation or lumen. If the available drug mass divided by liquid volume is below the calculated saturation concentration, the solution may be unsaturated and the actual concentration is limited by the dose. Conversely, if the solid is a salt, a different polymorph, a hydrate, or another phase, its equilibrium relationship may not be represented by the neutral-solid equations. A change in the stable solid phase can invalidate the assumed \(S_0\).

Useful checks include verifying the final, not merely initial, pH; confirming that excess solid remained during an equilibrium measurement; and checking that the measured concentration is not being mistaken for free drug when it includes micelle-associated material. In micellar experiments, separation and assay procedures can disturb the association equilibrium. A measured total concentration should therefore be interpreted according to the actual sampling and analytical method.

There are several levels of approach. A mechanistic ionization calculation predicts a pH profile from \(S_0\) and pKa. An empirical solubility-pH profile instead measures concentrations across pH and may be preferable when activity effects or additional equilibria matter, though it may not identify the underlying mechanism by itself. A micelle-association model adds a medium-specific term, while a direct measurement in the intended medium incorporates its combined effects but may not distinguish free from associated drug. The appropriate choice depends on the question and on which concentrations the PBPK model requires.

A diagnostic pattern is disagreement between predicted and measured concentrations that changes systematically with pH or surfactant concentration. Such disagreement can indicate an incorrect pKa or \(S_0\), nonideal solution behavior, a change in solid phase, additional ionization or complexation, or failure of the simple linear micelle model. It should not automatically be resolved by adjusting \(S_0\), because doing so can conceal a mistaken assumption about what the assay measured.

Common Pitfall

For example, inserting a micelle-enhanced \(S_{\mathrm{app}}\) into the weak-acid equation to infer pH is invalid unless the micelle contribution has been removed or explicitly modeled. Likewise, a large calculated saturation concentration does not imply that a finite dose will attain it; the dose must supply enough mass to reach saturation in the available volume.