Free Versus Micelle-Bound Drug in Dissolution Models

Why the Dissolved Pool Must Be Partitioned

A dissolution sample can contain drug molecules freely dispersed in the aqueous phase and drug associated with bile-salt micelles. Both contribute to a measurement of total dissolved drug, but they need not have the same availability at a solid surface or the same thermodynamic activity relevant to precipitation. Treating the measured total as though every molecule were free can therefore distort both dissolution and precipitation predictions.

The key distinction is between how much drug is dissolved in total and how much is present in the free pool. Micellar binding can increase apparent solubility by storing drug in the bound pool. If binding and release are sufficiently rapid, that pool can replenish free drug as it is removed. It does not follow that bound drug directly contributes to the interfacial concentration gradient or should be counted as free when evaluating supersaturation.

These concentrations refer to dissolved drug; undissolved solid is not included in \(C_T\). This tutorial first treats the bulk medium as well mixed and assumes that micelle-bound drug does not directly cross the aqueous boundary layer to the solid surface. That is a modeling assumption, not a universal property of all micellar systems. The introductory material on dynamic micellar solubilization and drug exchange motivates why exchange timescale matters: an equilibrium partition is appropriate only when exchange is fast relative to the process being represented.

Equilibrium Partitioning and Its Assumptions

For a dilute system in which the micellar binding capacity is not appreciably saturated, a useful approximation is a linear partition relation \(C_b=\beta C_f\), where \(\beta\) is a nonnegative, dimensionless binding ratio for the specified medium. It summarizes the effects of micelle abundance and drug–micelle association under those conditions; it is not necessarily constant when medium composition changes.

The linear relation assumes that the relevant micellar environment is effectively constant, that the bound pool responds rapidly enough to remain in equilibrium with the free pool, and that one effective bound pool is an adequate description. It should not be extrapolated beyond the range in which the relation holds. If binding sites or micellar capacity are finite, a saturable relation is more appropriate. One common equilibrium form is

Here \(B_{\max}\) is the maximum concentration of drug that can be represented in the bound pool, and \(K_d\) is the free concentration at which the bound concentration is half of \(B_{\max}\). Both have concentration units. The relation is monotone in \(C_f\), so a nonnegative total concentration corresponds to one nonnegative free concentration. The capacity and affinity parameters must apply to the medium and conditions being modeled.

Worked Example: Recovering the Free Pool From a Linear Partition

Which Concentration Drives Dissolution?

For a simple mass-transfer representation, let \(S_f\) be the free-drug concentration at equilibrium with the solid at its surface, and let \(G\) be an effective mass-transfer conductance. If the bulk free concentration is \(C_f\), the net dissolution rate can be written

With \(G\) in L/min and concentrations in mg/L, \(r_{\mathrm{diss}}\) is in mg/min. A positive value represents net dissolution; a negative value represents net transfer toward the solid under this simplified reversible-flux description. The equation assumes the same free-drug basis for the surface equilibrium concentration and bulk concentration, and it omits complications such as changing particle area or a spatially varying boundary layer.

The corresponding driving difference is \(S_f-C_f\), not \(S_f-C_T\), under the assumption that bound drug does not directly contribute to transport across the boundary layer. Bound drug can still matter indirectly: if exchange is fast, release from micelles can replenish free drug and sustain dissolution. That replenishment is captured through the relationship between \(C_T\) and \(C_f\), rather than by inserting \(C_T\) as the free concentration.

Free Concentration and Precipitation

Supersaturation relevant to precipitation should likewise be evaluated using the concentration or activity of the precipitating free species, not automatically the total dissolved concentration. For the simple concentration-based model used here, let \(S_{\mathrm{eq}}\) be the equilibrium free-drug solubility for the specified solid form and medium. Define the free supersaturation ratio as \(C_f/S_{\mathrm{eq}}\). A ratio greater than 1 indicates that the free concentration exceeds this equilibrium reference; it does not, by itself, determine when precipitation begins.

For illustration, a precipitation-rate expression might use

Here \(k_{\mathrm{prec}}\) has units of concentration per time, \(n\) is a specified exponent, and \(r_{\mathrm{prec}}\) is a concentration removal rate. This is an illustrative rate law, not a universal precipitation mechanism. Nucleation barriers, induction periods, and growth kinetics can alter the time course even when the free concentration is supersaturated. The important point for this tutorial is that replacing \(C_f\) with \(C_T\) can create a false supersaturation signal or exaggerate a real one.

Why Binding Can Increase Apparent Solubility

Micellar binding can raise the total concentration that coexists with a solid without raising the equilibrium free concentration. At equilibrium with the solid, set \(C_f=S_f\) in the partition relation. For linear binding, the apparent total solubility is \(S_{\mathrm{app}}=(1+\beta)S_f\). For saturable binding, it is \(S_{\mathrm{app}}=S_f+B_{\max}S_f/(K_d+S_f)\). These are total concentrations at the specified free equilibrium concentration; neither expression means the free solubility itself has increased by the same factor.

Deriving the Effect of Binding on Total-Concentration Changes

When partitioning is at equilibrium, changes in total concentration do not translate one-for-one into changes in free concentration. For the saturable model, differentiate the total concentration with respect to \(C_f\):

To see the result, write \(C_T=C_f+B_{\max}C_f/(K_d+C_f)\). The derivative of the bound term follows from the quotient rule:

The derivative is at least 1 for nonnegative parameters. Thus a change in free concentration can require a larger change in total concentration because the bound pool changes at the same time. This relationship is a local equilibrium result; it does not by itself specify the time required for binding and unbinding. When exchange is slow relative to dissolution or precipitation, separate dynamic equations for \(C_f\) and \(C_b\) are needed rather than an instantaneous partition formula.

Diagnostics, Alternatives, and Limitations

A practical diagnostic is to check whether the model’s reported free and bound concentrations sum to the reported total, and whether each pool is nonnegative. Also check that the partition parameters and free-solubility reference apply to the same medium, temperature, pH, and drug species. A mismatch can appear as an implausible free fraction or as a predicted total concentration that cannot be reconciled with the assumed capacity.

Compare predictions based on \(C_f\) with those from a total-only calculation. A large difference in dissolution direction, supersaturation ratio, or precipitation rate is a warning that the pool distinction materially affects the prediction. It is not a reason to select whichever prediction better matches expectations; it is a reason to verify the partition assumption and the concentration basis of the measured data.

An explicit free–bound equilibrium model is a useful middle ground when partitioning is defensible but the detailed exchange timescale is not the target. A total-only representation may be adequate for reporting measured apparent solubility, but it is insufficient for a process whose driving force depends on free drug unless a validated conversion is supplied. A kinetic two-pool model is preferable when exchange is not rapid, but it should not be added without data or a justified timescale.

Finally, concentration is a practical proxy for chemical activity. Non-ideal solutions, changes in ionization, multiple drug species, or changes in micellar composition may require a more careful thermodynamic treatment. The simple formulas here also do not establish that micelle-bound drug can never reach a surface or contribute to transport: they state the assumption used in the illustrated flux and precipitation calculations. If evidence supports direct micellar transport or a different interfacial mechanism, that mechanism needs to be represented explicitly rather than hidden by substituting total for free concentration.

Common Mistake

• Wrong reasoning: “The assay reports 10 mg/L dissolved drug, so the interfacial model should use 10 mg/L as the bulk concentration.”
• Correction: The assay’s total may include micelle-bound drug. First determine \(C_f\) from an appropriate partition model or measurement; use that free concentration wherever the assumed driving force or supersaturation definition is based on free drug.
• Another pitfall: “If apparent solubility rises, free solubility must rise by the same amount.” Apparent solubility can rise because the bound pool increases while the equilibrium free concentration remains fixed.
• Check before interpreting: Confirm that total equals free plus bound, that the model’s equilibrium assumption is plausible, and that the free solubility and partition parameters refer to compatible conditions.