Characterization of VitE-TPGS Micelles Linked to Poorly Soluble Pharmaceutical Compounds Exploiting Pair Distribution Function’s Moments

Abstract

Background: Micelles have attracted significant interest in nanomedicine as drug delivery systems. This study investigates the morphology of micelles formed by the D-α-tocopherol polyethylene glycol 1000 succinate (VitE-TPGS) surfactant in the presence and absence of, respectively, a poorly soluble pharmaceutical compound (PSC), i.e., Eltrombopag (0.08 wt%) and CaCl₂ (0.03 wt%). The aim was to assess the micelles’ ability to solubilize the PSC and potentially shield it from Ca²⁺ ions, simulating in vivo conditions. Methods: For this purpose, we have developed a novel theoretical approach for analyzing Pair Distribution Function (PDF) data derived from Small-Angle X-ray Scattering (SAXS) measurements, based on the use of PDF’s moments. Results: Our spheroid-based model was able to characterize successfully the micellar morphology and their interactions with PSC and CaCl₂, providing detailed insights into their size, shape, and electron density contrasts. The presence of PSC significantly affected the shape and integral of the PDF curves, indicating incorporation into the micelles. This also resulted in a decrease in the micelle size, regardless of the presence of CaCl₂. When this salt was added, it reduced the amount of PSC within the micelles. This is likely due to a decrease in the overall PSC availability in solution, induced by Ca²⁺ ions. Conclusions: This advanced yet straightforward analytical model represents a powerful tool for characterizing and optimizing micelle-based drug delivery systems.

1. Introduction

Micelles have attracted significant attention in nanomedicine, particularly for their potential as drug delivery systems [1,2,3,4]. These nanoscale structures, typically formed by the self-assembly of amphiphilic molecules in an aqueous environment, usually surfactants, exhibit a core–shell architecture that is highly conducive to encapsulating hydrophobic drugs. The core of the micelle provides a hydrophobic environment suitable for the entrapment of water-insoluble drugs, while the hydrophilic shell stabilizes its structure in biological fluids, thereby enhancing the solubility, bioavailability, and targeted delivery of therapeutic agents. Among the various factors that influence the efficacy of micelles as drug carriers, their morphology—encompassing both size and shape—plays a pivotal role [5].

The morphology of micelles directly impacts their drug-loading capacity and stability, in particular upon dilution, circulation time, and biodistribution, which are crucial parameters for effective drug delivery [6,7,8,9,10,11]. The size of the micelles is particularly important because it determines the ability of the micelles to pass through biological barriers, including cellular membranes, and to evade the reticuloendothelial system (RES), which is responsible for clearing foreign particles from the bloodstream. Micelles ranging in size from approximately 10 to 100 nm are often ideal for prolonged circulation, as well as for enhancing permeability and retention (EPR) effect, a phenomenon that enables nanoparticles to accumulate preferentially in tumor tissues due to their leaky vasculature and inefficient lymphatic drainage [7].

Although micelles have been traditionally approximated as spherical systems, recent research has revealed the existence of other shapes, such as ellipsoidal, rod-like, worm-like or even disk-like structures. These different shapes are mainly due to the structure of the micelle-forming surfactants used and to the characteristics of the surrounding environment (temperature, pH, and chemical composition). Non-spherical micelles have been shown to exhibit different cellular uptake mechanisms, circulation times, and biodistribution patterns compared to their spherical counterparts. For instance, they may exhibit prolonged circulation times due to reduced recognition and clearance by the RES, or they may demonstrate enhanced tissue penetration and cellular uptake via mechanisms that are less dependent on size, influencing the drug release profile, which is a critical factor affecting therapeutic efficacy. The internal structure of micelles, which is influenced by its size and shape, dictates the distribution of the drug within the micellar core and the interactions between the drug and the micelle-forming molecules. This, in turn, affects the rate at which the drug is released from the micelles. For example, smaller micelles may release their encapsulated drug more rapidly due to a higher surface-area-to-volume ratio, while larger or elongated micelles may provide a more sustained release profile. The ability to fine-tune the release kinetics through careful control of micelle morphology is particularly important for applications requiring a controlled and sustained release of the drug over time, such as in chronic disease management and chemotherapy. In these cases, minimizing side effects while maximizing therapeutic efficacy is paramount. Additionally, sustained-release drug delivery systems can improve the patients’ quality of life by reducing dosing frequency, easing the burden of multiple daily administrations, and enhancing treatment adherence. This benefit is particularly important for patients with chronic conditions, as it ensures consistent medication delivery without the stress of managing multiple daily doses.

The size and shape of micelles influence their interaction with cell membranes, their uptake by cells, and their intracellular trafficking pathways [12]. These factors are crucial for ensuring that the drug reaches its target within the body, whether that target is a specific tissue, organ, or subcellular compartment. For example, smaller micelles are more readily taken up by cells via endocytosis, while larger or non-spherical micelles may interact differently with cell membranes, potentially leading to alternative uptake pathways or enhanced retention within specific tissues.

For the above reasons, an accurate and precise measurement of the micelle size and shape is essential. Various analytical techniques, such as dynamic light scattering (DLS), transmission electron microscopy (TEM), and cryogenic electron microscopy (cryo-EM), are commonly used for this purpose. They each have distinct advantages and provide complementary information about the size, shape, and internal structure of micelles. For instance, DLS is widely used for determining the hydrodynamic diameter of micelles in solution, while TEM and cryo-EM enable direct visualization of micelle morphology, providing detailed insights about their shape and internal organization.

In addition to these conventional techniques, emerging methods such as Small-/Wide-Angle X-ray Scattering (SAXS/WAXS) and atomic force microscopy (AFM) are being increasingly utilized to obtain more comprehensive and high-resolution data on micelle morphology [12]. These advanced techniques are particularly valuable for gaining a deeper understanding of the structural features that influence drug loading and release. SAXS is particularly suitable for the characterization of micelles in solution, under real conditions, requiring limited sample preparation.

In 2023, we developed a novel theoretical approach for analyzing Pair Distribution Function (PDF) data derived from SAXS measurements, based on a spherical core–shell model for micelles [13]. This method enables the determination of the core–shell and the shell–buffer electron density contrasts, as well as the core and shell sizes of micelles, by modeling the PDF data. The approach has been validated by applying it to VitE-TPGS micelles [13]. As a further development, we have extended our approach to model spheroidal core–shell micelles [14]. This new approach represents a significant improvement, as it uses the first derivative of the PDF and analytical equations to resolve the structure of spheroidal core–shell micelles. By applying these equations, we can determine key structural parameters, including the micelle’s aggregation number, ellipticity, and electron density contrast between core and shell regions. The method was applied successfully to micelles formed by different surfactants, such as Polysorbate 20, Dodecyl phosphocholine, Sodium Dodecyl Sulfate, and VitE-TPGS.

In this work, assuming spheroidal-shaped micelles, we exploit nth-order moments of the PDF, with n ranging from 1 to 6, to perform the structural characterization of VitE-TPGS micelles both in the presence and absence of a poorly soluble pharmaceutical compound (PSC). In our study, the PSC was the active pharmaceutical ingredient Eltrombopag, a thrombopoietin receptor agonist, which is used to treat low blood platelet counts in adults with chronic immune (idiopathic) thrombocytopenia. It was approved by the FDA in 2008 [15]. Additionally, samples with and without CaCl₂ in the buffer solution have been measured, to assess the potential role of micelles to prevent the interaction reported when the formulated drug was administered with high-calcium-content food [16]. Unlike the previous method [14], which solved the micelle structure with a hybrid approach, i.e., partially graphical (first derivative of the PDF) and partially analytical, we present here a fully analytical method. It is described in Section 2 and Appendix A, while the experimental data are summarized in Section 3 and discussed in Section 4.

2. Analytical Method

For a two-component spheroidal micelle, we can derive the analytical expression of the following integrals, for any n = 1, …, N; they are reported explicitly in Appendix A:

$$R^n=\left(\int r^n\left(\rho \left(r\right)-{\rho }_s\right)d^{3}r\right)/\left(\int \left(\rho \left(r\right)-{\rho }_s\right)d^{3}r\right).$$

(1)

Here, ${\rho }_s$ = solvent electron density, $\rho \left(r\right)={\rho }_E$, the electron density inside the core, and $\rho \left(r\right)={\rho }_P$, the electron density inside the shell. These integrals are a measure of the atomic electron density within the micelle as a function of the nth power distance rⁿ, like raw moments for a probability distribution calculated from zero and not with respect to the mean (central moments).

We can compare the analytical expression obtained by Equation (1) with the corresponding nth r-power integral of the PDF, derived from the experimental SAXS data:

$$R_{\mathrm{PDF}}^n ={\displaystyle \frac{{\int }_0^{D_{max}}PDF\left(r\right)r^{n}dr}{2^{n-1}{\int }_0^{D_{max}}PDF\left(r\right)dr}}.$$

(2)

Here $D_{max}$ is the maximum distance in the PDF. In this way, allowing n to range from 1 to N, we can derive a set of N independent equations with the following unknowns: the ellipticity $\epsilon$; the core-solvent electron contrast $\mathsf{\Delta }{\rho }_{ES}={\rho }_E-{\rho }_s$; the shell-solvent electron contrast $\mathsf{\Delta }{\rho }_{PS}={\rho }_P-{\rho }_s$; the micelle size $D_{max}$; and the shell radius $R_{\mathrm{SH}}$. Since we need at least 5 independent equations to determine the 5 unknown parameters defining the micelles’ structure, we can also overdetermine the solution by solving a set of N equations with N ≥ 5, by means of a least-square-minimum approach, as described in Appendix A. If N > 5, we would have an overdetermined system of equations.

3. Results

The micelles studied in this work are formed by VitE-TPGS surfactant monomers, with and without the presence in solution of a PSC, in order to study the level of incorporation of this compound into the micelles. SAXS data were collected at the SAXS Lab Sapienza with a Xeuss 2.0 Q-Xoom system (Xenocs SA, Grenoble, France). See details in [13]. We have considered four cases, summarized in

Table 1. Composition of samples 1–4, containing the VitE-TPGS surfactant in the buffer solution, without and with a poorly soluble compound (PSC), as well as without and with CaCl₂.

Sample		VitE-TPGS	PSC	CaCl2	pH Measured
1	19.9 g of buffer 50 mM	82.6 mg (0.415 wt%)	No	No	6.87
2	2.4 g of sample 1	Yes (0.415 wt%)	No	0.55 mg (0.024 wt%)	6.82
3	10 g of sample 1	Yes (0.415 wt%)	8 mg (0.08 wt%)	No	6.91
4	2.8 g of sample 3	Yes (0.415 wt%)	Yes (0.08 wt%)	0.8 mg (0.029 wt%)	6.84

. The nominal pH is 6.8. Samples 1 and 2 have been prepared with PSC, with and without CaCl₂ added to the buffer solution. Samples 3 and 4 have been prepared with PSC, with and without CaCl₂ added to the buffer. The nominal concentrations of VitE-TPGS and of the potassium phosphate buffer were kept constant across all experiments at 0.415% w/w and 50 mM, respectively. In subsequent solutions, concentration in PSC was maintained at 0.08% (w/w) and calcium chloride at about 0.03% (w/w). It corresponds to quite a large excess of calcium cations vs. PSC. The aim was to simulate digestive tract conditions, where high concentrations of Ca²⁺ from calcium-rich foods may be present and negatively affect the bioavailability and solubility of active pharmaceutical ingredients, and to evaluate the potential protective role of micelles. Samples 1–4 have been prepared according to the following protocol to maximize the sameness of the solution, minimizing experimental variability.

A buffer solution was prepared by mixing 682.7 mg of potassium hydrogen phosphate (K₂HPO₄) and 2.26 mmol of NaOH (added as 22.6 g of a 0.4 wt% solution) in Milli-Q water, to reach a final volume of 100 mL. The solution was stirred until a stable pH value was reached. The targeted value was 6.80 while the actual measured value was 6.81. The buffer solution was used to prepare all 4 samples, to ensure the same pH value. To prepare sample 1, 82.6 mg of VitE-TPGS was dissolved in 19.9 g of the pH 6.8 (50 mM) buffer solution to reach a concentration of 0.415 wt%. The sample was stirred at 500 rpm for 43 min to achieve complete dissolution. No significant variation was observed for the pH, in comparison to the buffer solution (as expected, since VitE-TPGS has no acidity/basicity). The final pH value was 6.87, before filtration. An aliquot of this sample was then filtered four times through Axiva sterile cellulose acetate syringe filters with pore diameters of 200 nm to remove any undissolved particulate matter and ensure a homogeneous solution. SAXS data collected on the unfiltered and filtered samples were superimposable, indicating that no undissolved surfactant in the form of particulate greater than 200 nm was present in the unfiltered sample. To prepare sample 2, 0.55 mg of CaCl₂ was dissolved in 2.4 g of the sample 1 solution, corresponding to 0.024 wt% of CaCl₂, and the solution was stirred at 500 rpm for 20 min. No significant variation was observed for the pH, as expected, since CaCl₂ has no particular acidity or basicity. To prepare sample 3, 8.0 mg of PSC was added to 10 g of the sample 1 solution to reach a concentration of 0.08 wt%. The solution was stirred at 500 rpm for 60 min to ensure complete dilution. After stirring, no significant deviation towards higher pH values was observed thanks to the buffer capacity. The final pH value was 6.91. To prepare sample 4, 0.8 mg of CaCl₂ was added to 2.8 g of the sample 3 solution, corresponding to a concentration of 0.415 wt% of VitE-TPGS, 0.08 wt% of PSC, and 0.029 wt% of CaCl₂. The solution was stirred at 500 rpm for 20 min. No significant variation was observed for the pH. The final pH value after filtration was 6.84. All preparations were performed at 25 °C. Table 1 summarizes the composition of the 4 samples.

The PDF curves, obtained for these samples, are shown in Figure 1; they describe the SAXS macroscopic cross-section as a function of the atom–atom distances r.

The PDFs, calculated by the software GNOM 47.5 [17], have been rescaled so that the integral of the PDFs plotted in Figure 1 is equal to the corresponding I(0) [cm⁻¹] value, i.e., the SAXS macroscopic cross-section for the whole sample extrapolated to the scattering vector value q = 0. Figure 1 shows that the presence of the PSC significantly affects the shape and the integral of the PDF curve, which indicates that some PSC molecules have been incorporated into the micelles. The presence of CaCl₂ in solution reduces the integral of the macroscopic cross-section (cf green curve versus black curve in Figure 1), indicating that a smaller quantity of PSC could be incorporated into the VitE-TPGS micelles. A possible explanation for the observed reduction of the amount of PSC in the micelles (and possibly also outside, in the same buffer solution) in the presence of CaCl₂ is the occurrence of some specific interactions, eventually exemplified by the drug. Cation interactions are indeed known to reduce the bioavailability of the drug [18].

After having determined $\epsilon , \mathsf{\Delta }{\rho }_{ES},$ $\mathsf{\Delta }{\rho }_{PS},$ $D_M$, $R_{SH}$, from Equations (1) and (2), via a least-square-minimum approach discussed in Appendix A, we can evaluate the aggregation number of monomers inside the micelles ($N_{agg}$), the number of water molecules inside the shell ($N_{H2O}),$and the number of PSC molecules per monomer inside the micelles ($x_{PSC}=N_{PSC}$/$N_{agg}$). All formulas are provided in Appendix A.

Table 2. Shell size ($R_{SH}$), maximum micelle size ($D_{max}$), ellipticity ($\epsilon$), $\mathsf{\Delta }{\rho }_{ES}=\mathsf{\Delta }{\rho }_{EP}+\mathsf{\Delta }{\rho }_{PS}$, $\mathsf{\Delta }{\rho }_{PS}$, $N_{agg},$ $N_{H2O}$/$N_{agg}$ obtained from the analysis of the PDF’s moments of VitE-TPGS micelles. $c_{mon}$ = 4.1 mg/mL. cmc = 0.02 mg/mL. $R_g$is the gyration radius. I(0) is the PDF integral on an absolute scale. For prolate micelles, the minimum core size is in the equatorial plane (equatorial core radius).

Sample	${\mathit{R}}_{\mathit{S}\mathit{H}}$ (Å)	${\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}}$ (Å)	Equatorial Core Radius $({\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}}$ − $2{\mathit{R}}_{\mathit{S}\mathit{H}}$$)/2\mathit{\epsilon }$ (Å)	$\Delta {\mathit{\rho }}_{\mathit{E}\mathit{S}}$ (ne/Å3)	$\mathit{\Delta }{\mathit{\rho }}_{\mathit{P}\mathit{S}}$ (ne/Å3)	$\mathit{\epsilon }$	${\mathit{N}}_{\mathit{a}\mathit{g}\mathit{g}}$	${\mathit{N}}_{\mathit{H}2\mathit{O}}$$/{\mathit{N}}_{\mathit{a}\mathit{g}\mathit{g}}$	${\mathit{R}}_{\mathit{g}}$(Å)	I(0) (cm−1)
1 (without PSC and CaCl2)	16.7 ± 0.1	122.0 ± 0.1	29.3 ± 0.2	−0.026 ± 0.001	0.032 ± 0.001	1.51 ± 0.01	127 ± 2	53 ± 4	47.3 ± 0.5	0.056 ± 0.001
2 (without PSC, with CaCl2)	14.2 ± 0.1	113.8 ± 0.1	30.5 ± 0.2	−0.031 ± 0.001	0.039 ± 0.001	1.40 ± 0.01	122 ± 3	36 ± 4	47.3 ± 0.4	0.055 ± 0.001
3 (with PSC, without CaCl2)	16.4 ± 0.1	116.7 ± 0.1	27.1 ± 0.2	−0.026 ± 0.001	0.036 ± 0.001	1.55 ± 0.01	118 ± 4	44 ± 4	43.5 ± 0.5	0.085 ± 0.001
4 (with PSC and CaCl2)	12.3 ± 0.1	107.7 ± 0.1	27.7 ± 0.2	−0.030 ± 0.001	0.055 ± 0.001	1.51 ± 0.01	125 ± 4	12 ± 5	42.8 ± 0.3	0.079 ± 0.001

summarizes the micelles’ parameters obtained for the 4 samples.

Figure 2 shows, for the four samples, the probability density function calculated by the SmoothDensityHistogram function implemented in Mathematica (Wolfram), associated with the histograms of least-square-root values (LSR) (cf Equation (A15)) obtained by the least-square approach discussed in Appendix A. We have set ten contour levels in the probability density function (P) at 10%, 20%, …, 80%, 90% of the histogram maximum. The red color is associated with 90% of the maximum. The left column on Figure 2 shows P as a function of the variables $D_M$ and $R_{sh}.$The enter column shows P as a function of the variables $\mathsf{\Delta }{\rho }_{PS}$ and $-\mathsf{\Delta }{\rho }_{EP}=-\mathsf{\Delta }{\rho }_{ES}+\mathsf{\Delta }{\rho }_{PS}.$The right column shows P as a function of the variables $\epsilon$ and $N_{agg}$.

It is interesting to note that in Figure 2, secondary maxima are occasionally observed in the two-dimensional P histograms. This is particularly evident for sample 4, suggesting that the presence of both PSC and CaCl₂ could cause a certain level of variability in the micelles’ structure, which, in turn, could imply the occurrence of a certain level of interaction between the PSC and Ca²⁺ cations in solution during the formation of micelles. Further investigations would be needed to validate this hypothesis.

4. Discussion

The comparison of the $D_{max}$ and $R_g$ values from Table 2 shows that the addition of PSC leads to the formation of smaller micelles, whether CaCl₂ is present or not. A similar effect can be seen on the shell size (R_SH) and the equatorial core radius. These observations are consistent with those made in our previous work [13], while our new spheroidal model provides here a more detailed characterization of the micelle morphological changes.

In our previous work [13], we assumed spherical-shaped micelles, following the suggestions in [19]. Under this hypothesis, we estimated, for sample 3, $x_{PSC,3}=0.30 \pm 0.01$. By introducing, with y, the fraction of the PSC molecules per monomer inside the core (see Appendix A) and by relaxing the spherical-shape constraint by solving Equation (A22), we obtain the results shown in Figure 3, where $x_{PSC}$ is represented as a function of y for samples 3 and 4. For sample 3 we obtain $x_{PSC,3}$ ranging between 0.47 and 0.53, as a function y. This difference with respect to the result obtained in [13] is mainly due to both the different number of monomers aggregated into the micelles and the different micelle shape given by the 2 models (spherical vs. spheroidal). Indeed in [13], from the difference of the PDF curves obtained with and without PSC in solution, we had estimated, under the hypothesis of spherical-shaped micelles, the localization of these molecules inside the micelles, close to the linker region, and a concentration $x_{PSC,3}=0.30 \pm 0.01$. For spheroidal-shaped micelles, this approach is not possible, because differences in intensity of the PDF at the same distance from the center could also be due to both different number of aggregated monomers and to different ellipticity values, which, in turn, influence the distribution of distances in the PDF profile and the estimated value for $x_{PSC}$.

The colored bands in Figure 3 are the constraints for samples 3 (in green) and 4 (in grey), derived from Equation (A16) and developed in Equations (3) and (4) below. Indeed, from Equation (A16), it follows that the square difference of electrons per monomer, between the samples with and without PSC, must be equal to the difference of $I\left(0\right)/K$:

$$\begin{gathered} {\left[827+232x_{PSC,3}-10{\displaystyle \frac{2327+554x_{PSC,3}}{29.9}}\right]}^2{N}_{agg,3}-{\left[827-10{\displaystyle \frac{2327}{29.9}}\right]}^2{N}_{agg,1} \\ ={\left[I\left(0\right)/K\right]}_3-{\left[I\left(0\right)/K\right]}_1; \end{gathered}$$

(3)

$$\begin{gathered} {\left[827+232x_{PSC,4}-10{\displaystyle \frac{2327+554x_{PSC,4}}{29.9}}\right]}^2{N}_{agg,4}-{\left[827-10{\displaystyle \frac{2327}{29.9}}\right]}^2{N}_{agg,2} \\ ={\left[I\left(0\right)/K\right]}_4-{\left[I\left(0\right)/K\right]}_2. \end{gathered}$$

(4)

The numerical values $827$, 232, and 10 correspond to the number of electrons of a VitE-TPGS, PSC, and water molecule, respectively; $2326$, $554$, and $29.9$ are the molecular volumes of these 3 species in ų [13]. From Equation (3), we have, for samples 3 and 4, $x_{PSC,3}=0.40 \pm 0.03$ and $x_{PSC,4}=0.29 \pm 0.03$, respectively, which are represented as colored bands in Figure 3.

For sample 3, the middle value of the green band is $0.40;$the upper value is $0.43$ and the lower value is $0.37$. We have obtained the grey band, for sample 4, similarly. The grey and green bands have to be compared with the black and green squares of Figure 3, respectively. The squares that are within the colored bands are possible solutions of all equations. By comparing these values with those obtained from Equation (A22), reported as squares in Figure 3, we obtain that for sample 3, the fraction of $x_{PSC}$ inside the core should be $y_3=0.65 \pm 0.15$. For sample 4, we have $y_4=0.35 \pm 0.35$, a range that includes also 0. Therefore, in the presence of CaCl₂ in buffer solution, the PSC molecules are much less segregated into the core and not isolated from the external environment with respect to the hydrophobic core. They are almost all in the shell, still in interaction with the buffer, due to its hydration. This result is also supported by the very low shell hydration of the micelles of sample 4 (see Table 2), because the available space in the shell is effectively already occupied by the PEG-1000 tails and the PSC molecules. Conversely, in the absence of CaCl₂ in solution, the PSC molecules are almost all localized in the core and, partially, in the region of the linker, being hence much better isolated from the buffer.

As calculated in Appendix A, the ratio of N_PSC and N_mon in sample 3 is about 2/3. The addition of PSC in the solution leads to micelles of a smaller diameter, as can be seen from the results of Table 2 for samples 3 and 4. The addition of CaCl₂ in the solution reduces the hydration of the PEG-1000 shell too. Moreover, in the presence of PSC, the CaCl₂ in solution leads to the formation of micelles with a smaller fraction of PSC linked to them, as can be clearly seen in Figure 3. This observation could be explained by the fact that the presence of Ca²⁺ in solution reduces the incorporation of PSC molecules into the hydrophobic core of the micelles. Additionally, from the $x_{PSC}$and the N_agg values (Table 2), we can estimate the total number of PSC molecules per micelle to 47 in sample 3 and 36 in sample 4. These combined results suggest that the presence of CaCl₂ in the buffer solution reduces the incorporation of PSC molecules both specifically into the hydrophobic core and in total into the micelles. One possible mechanism could be the overall reduction of available PSC in solution, induced by the presence of Ca²⁺ ions.

5. Conclusions

Under the assumption of a spheroidal model for micelles, we have developed a new fully analytical method, which exploits nth-order moments of the PDF derived from SAXS measurements to determine the micelles’ structural parameters. This model has been successfully applied to characterize the morphology of micelles with and without the presence of a poorly soluble API (PSC) and CaCl₂, respectively. We were able to confirm previous observations on the impact of PSC based on a simple spherical model, although now with many more details, thanks to our new spheroidal model. With this model, we were also able to estimate the relative fractions of the PSC incorporated in the core and in the shell of the micelles. The model also reveals that the presence of CaCl₂ reduces PSC incorporation, both within the hydrophobic core and the micelle as a whole. This advanced yet straightforward analytical model offers a powerful tool for characterizing micellar systems and guiding the development of optimized drug delivery strategies.

Future experiments, where the micelles’ morphology is characterized with our model as a function of the CaCl₂ concentration, could help us further understand the role of Ca²⁺ in influencing the micelles’ shape and size, and to which extent micelles can play a protective role for the PSC molecules. These studies could obviously be extended to other types of surfactants and PSC. The results of such in vitro studies could provide useful insights to understand in-vivo mechanisms. They could then help design drug delivery systems that can enhance the bioavailability of poorly soluble APIs and protect them against the detrimental effect of Ca²⁺ cations in vivo, which are known to impact significantly the absorption and effectiveness of some pharmaceuticals administered orally.