Predicting Apparent Solubility in Mixed-Micelle Media

Why Mixed-Micelle Composition Matters

Bile salts and phospholipids can assemble into mixed micelles that incorporate poorly water-soluble drug. At equilibrium, the amount of drug present in the solution may therefore exceed the amount that can remain in the aqueous phase alone. This total dissolved amount is the apparent solubility; it includes both aqueous drug and drug associated with micelles. It does not mean that the drug's intrinsic aqueous solubility has increased.

For a PBPK dissolution input, this distinction matters because the predicted solubilization capacity can depend on both the total amount of micellized material and its composition. A model that treats every micelle mixture as having the same capacity per unit concentration can miss a substantial composition effect. The calculation below makes that effect explicit while keeping the model simple enough to inspect and check.

Definition: Let $S_0$ be the drug's equilibrium aqueous solubility in the specified medium in the absence of micelles. Let $C_B^{\mathrm{mic}}$ and $C_P^{\mathrm{mic}}$ be the micellized concentrations of bile salt and phospholipid. The apparent solubility $S_{\mathrm{app}}$ is the total equilibrium dissolved-drug concentration at saturation, including aqueous and micelle-associated drug. In the linear mixed-micelle partition model, $$ S_{\mathrm{app}}=S_0\left(1+K_B C_B^{\mathrm{mic}}+K_P C_P^{\mathrm{mic}}\right), $$ where $K_B$ and $K_P$ are component-specific partition coefficients with units reciprocal to the concentration units used for $C_B^{\mathrm{mic}}$ and $C_P^{\mathrm{mic}}$.

Model, Assumptions, and Composition

The model treats micellized bile salt and phospholipid as two contributions to the drug's capacity to reside outside the aqueous phase. Its assumptions are that the system is at equilibrium; the aqueous phase at saturation has concentration $S_0$; drug loading is sufficiently low that the micelle composition and the partition coefficients do not change appreciably; and the contributions of the two components are additive over the concentration range used. The pH, temperature, and other medium conditions defining $S_0$ and the coefficients must be consistent.

The concentrations in the equation are micellized concentrations, not automatically the total amphiphile concentrations added to the vessel. If total concentrations and monomer concentrations are available, a simple mass balance gives $C_B^{\mathrm{mic}}=C_B^{\mathrm{tot}}-C_B^{\mathrm{mon}}$, with the analogous relation for phospholipid. If monomer levels are not negligible, or the mixture has a composition-dependent micellization threshold, they should not be silently counted as micellar material. The micellized concentrations should come from measurements or from an explicitly justified micellization approximation.

For a mixture containing only these two micellized components, define $C_M=C_B^{\mathrm{mic}}+C_P^{\mathrm{mic}}$, the total micellized amphiphile concentration, and $x_B=C_B^{\mathrm{mic}}/C_M$, $x_P=C_P^{\mathrm{mic}}/C_M$. When $C_M>0$, $x_B+x_P=1$. The weighted coefficient $K_{\mathrm{mix}}=x_BK_B+x_PK_P$ then gives the equivalent form $S_{\mathrm{app}}=S_0(1+K_{\mathrm{mix}}C_M)$. Thus composition changes the prediction even when $C_M$ is held constant, if $K_B$ and $K_P$ differ.

Why the Apparent-Solubility Equation Follows

The partition model states that, at equilibrium and within its linear range, the ratio of micelle-associated drug to aqueous drug is the sum of the two component contributions. If $C_{D,\mathrm{aq}}$ denotes aqueous drug concentration and $C_{D,\mathrm{mic}}$ denotes micelle-associated drug concentration, then

\frac{C_{D,\mathrm{mic}}}{C_{D,\mathrm{aq}}} =K_B C_B^{\mathrm{mic}}+K_P C_P^{\mathrm{mic}}.

At saturation, the aqueous concentration is $C_{D,\mathrm{aq}}=S_0$. Total dissolved drug is the sum of the two compartments, so $S_{\mathrm{app}}=C_{D,\mathrm{aq}}+C_{D,\mathrm{mic}}$. Substituting the partition relation yields

\begin{aligned} S_{\mathrm{app}} &=C_{D,\mathrm{aq}}+C_{D,\mathrm{mic}}\\ &=S_0+S_0\left(K_B C_B^{\mathrm{mic}}+K_P C_P^{\mathrm{mic}}\right)\\ &=S_0\left(1+K_B C_B^{\mathrm{mic}}+K_P C_P^{\mathrm{mic}}\right). \end{aligned}

The leading 1 is essential: it is the aqueous contribution. Each product $K_iC_i^{\mathrm{mic}}$ is dimensionless, so the expression in parentheses is a dimensionless solubilization factor and $S_{\mathrm{app}}$ retains the concentration units of $S_0$. This derivation also shows why the result is a total dissolved concentration, not a replacement value for the free aqueous saturation concentration.

Worked Example: Combining Bile-Salt and Phospholipid Contributions

Suppose, hypothetically, that a drug has $S_0=0.0200\ \mathrm{mg/mL}$. A medium contains $C_B^{\mathrm{mic}}=4.00\ \mathrm{mM}$ bile salt and $C_P^{\mathrm{mic}}=1.00\ \mathrm{mM}$ phospholipid. Let $K_B=0.800\ \mathrm{mM}^{-1}$ and $K_P=2.00\ \mathrm{mM}^{-1}$. Calculate its predicted apparent solubility.

The units are compatible: each coefficient is in $\mathrm{mM}^{-1}$, and each micellized concentration is in $\mathrm{mM}$. The partition-model factor is $$ 1+(0.800\ \mathrm{mM}^{-1})(4.00\ \mathrm{mM})+(2.00\ \mathrm{mM}^{-1})(1.00\ \mathrm{mM}) =1+3.20+2.00=6.20. $$

Therefore, $$ S_{\mathrm{app}}=(0.0200\ \mathrm{mg/mL})(6.20)=0.124\ \mathrm{mg/mL}. $$ The predicted total dissolved concentration at saturation is $0.124\ \mathrm{mg/mL}$. Under this model, the aqueous contribution is $0.0200\ \mathrm{mg/mL}$, and the additional micelle-associated contribution is $0.124-0.0200=0.104\ \mathrm{mg/mL}$.

Comparing Mixtures at Fixed Micellized Concentration

A useful way to isolate the composition effect is to hold $C_M$ fixed and change the component fractions. Since $K_{\mathrm{mix}}=x_BK_B+x_PK_P$, a mixture richer in the component with the larger partition coefficient has a larger predicted capacity in this linear model. This is a conditional model prediction, not a general claim that one component always solubilizes every drug better: the coefficients are drug- and medium-specific.

Worked Example: Changing the Mixture Ratio

For a hypothetical drug, let $S_0=0.0200\ \mathrm{mg/mL}$, $K_B=0.600\ \mathrm{mM}^{-1}$, and $K_P=1.80\ \mathrm{mM}^{-1}$. Compare two media, each with $C_M=5.00\ \mathrm{mM}$ of micellized amphiphile. In medium A, $x_B=0.800$ and $x_P=0.200$. In medium B, $x_B=0.400$ and $x_P=0.600$.

For medium A, $C_B^{\mathrm{mic}}=0.800(5.00)=4.00\ \mathrm{mM}$, and $C_P^{\mathrm{mic}}=0.200(5.00)=1.00\ \mathrm{mM}$. Thus $$ K_{\mathrm{mix,A}}=(0.800)(0.600)+(0.200)(1.80)=0.840\ \mathrm{mM}^{-1}, $$ and $$ S_{\mathrm{app,A}}=0.0200[1+(0.840)(5.00)]=0.0200(5.20)=0.104\ \mathrm{mg/mL}. $$

For medium B, $C_B^{\mathrm{mic}}=2.00\ \mathrm{mM}$ and $C_P^{\mathrm{mic}}=3.00\ \mathrm{mM}$. Its weighted coefficient is $$ K_{\mathrm{mix,B}}=(0.400)(0.600)+(0.600)(1.80)=1.32\ \mathrm{mM}^{-1}. $$ Therefore $S_{\mathrm{app,B}}=0.0200[1+(1.32)(5.00)]=0.0200(7.60)=0.152\ \mathrm{mg/mL}$. The prediction is higher in medium B because the phospholipid fraction is larger and $K_P>K_B$, while the total micellized concentration and $S_0$ are unchanged.

Worked Example: Converting Total Concentrations to Micellized Concentrations

Suppose a hypothetical medium has total bile-salt concentration $6.00\ \mathrm{mM}$ and total phospholipid concentration $2.00\ \mathrm{mM}$. Assume measured monomer concentrations are $1.00\ \mathrm{mM}$ and $0.200\ \mathrm{mM}$, respectively. For a drug with $S_0=0.0150\ \mathrm{mg/mL}$, $K_B=0.500\ \mathrm{mM}^{-1}$, and $K_P=1.00\ \mathrm{mM}^{-1}$, calculate $S_{\mathrm{app}}$.

First apply the component mass balances: $$ C_B^{\mathrm{mic}}=6.00-1.00=5.00\ \mathrm{mM},\qquad C_P^{\mathrm{mic}}=2.00-0.200=1.80\ \mathrm{mM}. $$ Then the dimensionless contributions are $(0.500)(5.00)=2.50$ and $(1.00)(1.80)=1.80$. The predicted factor is $1+2.50+1.80=5.30$, so $$ S_{\mathrm{app}}=(0.0150\ \mathrm{mg/mL})(5.30)=0.0795\ \mathrm{mg/mL}. $$ The calculation uses the micellized, not total, amphiphile concentrations. Substituting the total values would yield a different result and would incorrectly count the monomer pools as micellar capacity under this model.

Checking Predictions and Choosing an Approach

A prediction is most useful when its inputs and behavior can be checked. Confirm that each $C_i^{\mathrm{mic}}$ is nonnegative and no greater than its corresponding total concentration, that $x_B+x_P=1$ when using composition fractions, and that the coefficient and concentration units cancel in every product. With nonnegative partition coefficients and micellized concentrations, the model also requires $S_{\mathrm{app}}\geq S_0$. A value below $S_0$ signals an arithmetic, unit, or model-implementation problem.

Compare predictions with equilibrium solubility measurements across more than one concentration or mixture ratio when possible. Systematic underprediction or overprediction, curvature as amphiphile concentration rises, or a composition effect inconsistent with the fitted coefficients can indicate that the linear additive assumption is inadequate. Check whether the assumed monomer concentrations and micellized composition are plausible for the actual medium. Apparent discrepancies can also arise if measurements are not at equilibrium or if $S_0$ and the coefficients refer to different pH or temperature conditions.

The component-specific model is more informative than a bile-salt-only capacity expression when phospholipid levels vary and the two components have distinct effects. A single empirical slope against total micellized amphiphile is simpler, but it effectively assumes the same response across compositions, or is restricted to a fixed mixture ratio. Directly measured solubility values or interpolation between measurements can avoid imposing a partition form, but provide less mechanistic extrapolation beyond the tested conditions. The appropriate choice depends on the available data and the range of compositions the model must represent.

The linear model is not a universal description of mixed micelles. It can fail when micelle structure or composition changes substantially with concentration, when interactions between components make their effects nonadditive, or when the drug loading is high enough to alter the micelles. It also does not describe the time course of drug exchange or micelle formation. Those are distinct questions from the equilibrium capacity calculation here. In PBPK use, treat $S_{\mathrm{app}}$ as a model-based saturation input under specified medium conditions, not as a guaranteed in vivo solubility or a direct measure of absorbed drug.

Common Pitfall

Common mistake: “The mixture contains $6.00\ \mathrm{mM}$ bile salt and $2.00\ \mathrm{mM}$ phospholipid, so those are the concentrations to put into the partition equation.” This reasoning counts monomeric amphiphile as if it were micellized. Use $C_i^{\mathrm{mic}}$, or explicitly state an approximation that justifies equating total and micellized concentration. Also retain the leading 1 in the solubilization factor: omitting it drops the aqueous contribution and predicts only the micelle-associated amount, not apparent solubility.

Check Your Understanding

Use the linear partition model and state units in your calculations.

A hypothetical drug has $S_0=0.0100\ \mathrm{mg/mL}$, $K_B=0.400\ \mathrm{mM}^{-1}$, $K_P=1.00\ \mathrm{mM}^{-1}$, $C_B^{\mathrm{mic}}=2.00\ \mathrm{mM}$, and $C_P^{\mathrm{mic}}=1.00\ \mathrm{mM}$. What is $S_{\mathrm{app}}$?
At fixed $C_M$, why can two mixtures with the same total micellized amphiphile concentration predict different apparent solubilities?
Total phospholipid is $3.00\ \mathrm{mM}$ and its measured monomer concentration is $0.500\ \mathrm{mM}$. What concentration belongs in the phospholipid term?
Under the model, what does a prediction of $S_{\mathrm{app}}<S_0$ indicate if both partition coefficients and micellized concentrations are nonnegative?

Solutions. (1) The factor is $1+(0.400)(2.00)+(1.00)(1.00)=2.80$, so $S_{\mathrm{app}}=(0.0100)(2.80)=0.0280\ \mathrm{mg/mL}$. (2) The weighted coefficient $K_{\mathrm{mix}}=x_BK_B+x_PK_P$ changes with the fractions when $K_B\ne K_P$, so $S_{\mathrm{app}}=S_0(1+K_{\mathrm{mix}}C_M)$ changes. (3) Use $C_P^{\mathrm{mic}}=3.00-0.500=2.50\ \mathrm{mM}$. (4) It violates the model's $S_{\mathrm{app}}\geq S_0$ implication; check the arithmetic, units, inputs, or assumptions.