# Characterization of Surfactant Spheroidal Micelle Structure for Pharmaceutical Applications: A Novel Analytical Framework

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- ATSAS package [5], which includes various programs (PRIMUS, GNOM,…) for SAXS data analysis and structure modeling, provides information such as distance distribution function, gyration radius, molecular weight, flexibility, etc.
- SASView [7], an open-source software designed for scattering analysis with a focus on user-friendliness and data visualization.
- SCATTER [8], a program for the analysis, modeling, and fitting of 1D and 2D SAXS data of non-ordered, partially ordered, or fully ordered nano- and mesoscale structures.

## 2. The New Graphical-Analytical Approach

_{M}, R

_{sh}, ${\epsilon ,{N}_{agg},\mathsf{\Delta}\rho}_{core}=\rho -{\rho}_{s}$, ${\mathsf{\Delta}\rho}_{shell}$ = ${\rho}_{1}-{\rho}_{s}$, where (in the schematic model described in Figure 1) ρ, ${\rho}_{1}$ and ${\rho}_{s}$ are the core, shell, and buffer electron density values, respectively. D

_{M}and D

_{E}are the maximum sizes of the whole prolate spheroidal micelle and of its core, respectively, and ε is the ratio of the polar R

_{pol}to the equatorial R

_{eq}core half-sizes (for prolate spheroids, ε > 1; for spheres, ε = 1; for oblate spheroids, ε < 1). For prolate spheroids, D

_{M}coincides with the maximum size of the micelle, namely D

_{max}. For oblate spheroids (ε < 1), D

_{M}= εD

_{max}.

^{2}γ(r) [14]. The autocorrelation function of any box function of width t is a triangular function of half width t, and its first derivative has a distance between maximum and minimum equal to t. Therefore, for ideal micelles of spheroidal shape, made of a core of equatorial size R

_{eq}and a shell of size R

_{sh}< R

_{eq}, as schematically shown in Figure 1, due to the properties of the autocorrelation function, the distance between the maximum and the neighborhood minimum of the PDF first derivative should be equal to the minimum size between R

_{sh}and R

_{eq}, i.e., usually R

_{sh}(half width of the autocorrelation function of the shell). However, in real micelles, the region surrounding the hydrophobic core can be viewed as a polymer solution of hydrophilic chains and water (or buffer solution), characterized by the fact that the hydrophilic chains are fixed, at one end, to the hydrophobic core but can move freely at the other end, implying a continuous random chain deformation and a consequent local fluctuation of the shell size and the micelle’s shape [1].

_{sh}but to 2R

_{sh}. Similarly, the distance between the two minima closer to the maximum is equal to 4R

_{sh}instead of 2R

_{sh}. The values of R

_{sh}obtained either by the distance between the two minima closer to the maximum or by the distance between the maximum and the neighborhood left minimum are quite equivalent if the main peak’s shape is not too asymmetrical with respect to its maximum value. For this reason, the value of R

_{sh}obtained by the distance between the two minima closer to the maximum can be selected as the default. However, for very asymmetric main peaks, it is preferable to select the R

_{sh}value obtained by the distance between the maximum and the left neighborhood minimum.

_{sh}, caused by random movements of the hydrophilic chains into the buffer liquid. These random variations of the shell thickness influence the actual PDF size values. Indeed, if x, y are discrete independent random shell size fluctuations along two independent axes of the micelle, with probability functions f(x) and g(y), then the probability that x + y = d is the sum of possible values of x of the product f(x) × g(d − x). This is the definition of a convolution, which could be mathematically taken into account in the PDF by considering it convolved with a suitable function of width $\sqrt{1+1}$ × R

_{sh}= $\sqrt{2}$R

_{sh}, the variations of the shell thickness along the two considered axes, being two independent random processes. The same happens for three independent random shell size fluctuations along the three principal axes of the spheroid-shaped micelle, implying a convolution of PDF with a suitable function of width $\sqrt{3}$R

_{sh}. Therefore, the difference between the maximum and minimum of the derivative of the PDF, which is the integral of all angular directions over the whole solid angle, will be characterized by a width of the order of $\sqrt{3+1}$ × R

_{sh}= 2R

_{sh}, i.e., twice the shell size, i.e., twice the value that we would have expected for an ideal shell without any random size variation. This relationship is verified in Figure 2b for all three considered experimental cases [4,12,13], on micelles made by three different monomers. This property can be used for a quick and reliable estimate of the shell size directly from the distances between peaks of the PDF derivative, as previously shown. Moreover, given the maximum distance, D

_{max}, determined from the PDF profile where it falls to 0, the quantity D

_{max}/2 − R

_{sh}can then be readily calculated. This is the polar core half-size R

_{pol}for prolate spheroids or the equatorial core half-size R

_{eq}for oblate spheroids.

^{2}from its center [4]. Other exponents in Equation (1) would give complementary information, with respect to the gyration radius, about the electron-density distribution inside the micelle, like moments do for a probability distribution.

_{M}and V

_{E}are the micelle and core volume, respectively, and $K$ is a quantity related to the concentration of monomers in solution (see Appendix A). Adding this further equation to the previous four, we obtain an overdetermined system of equations in the four unknowns. In turn, this gives us the possibility of defying a Figure of Merit (FOM), described in Appendix A, useful to find the more probable solution of the system of analytical equations. This is fundamental, considering that the analytical equations are polynomial ratios depending on powers of the unknowns, thereby allowing for more than one possible physical solution. The non-linearity of the analytical equations derived by the PDF visualizes in mathematical terms the reason why different software gives different solutions, even starting from the same experimental SAXS dataset.

_{min}should not exceed the first Shannon channel (π/D

_{M}), with a total number of channels proportional to D

_{M}× s

_{max}/π, where s

_{max}is the maximum measured scattering vector not affected by excessive levels of noise [14].

## 3. Application of the Graphical-Analytical Approach to Micelles

#### 3.1. PS20 Micelles

^{3}(data reported for 25 °C). The PS20 properties summarized in Table 1 have been derived by using the above mass density’s value.

_{agg}= 34 was estimated as I(0)/I

_{PS20}, where I

_{PS20}= 0.227 × 10

^{−3}cm

^{−1}is the computed forward SAXS scattering by PS20 molecules dispersed in water and I(0) is the measured forward SAXS scattering of the entire PS20 micelle, placed on an absolute scale. From Equation (A27), reported in Appendix A, for a unit concentration, we obtain I

_{PS20}= 0.235 × 10

^{−3}cm

^{−1}/unit concentration, very close to the value reported in [11], but a quite different value of N

_{agg}. Indeed, from [12], we have: I(0) = 0.054 cm

^{−1}; ${c}_{mon}$ = 5 mg/mL = 0.005 g/cm

^{3}; $cmc$ = 0.06 mg/mL (0.049 mM), $K=\frac{\left({c}_{mon}-cmc\right){N}_{A}{{r}_{e}}^{2}}{{MW}_{mon}}=1.93\times {10}^{-7}\text{}{\mathrm{c}\mathrm{m}}^{-1}{{n}_{e}}^{-2},$ ${r}_{g}=34.0\pm 1.0\text{}\mathbf{\AA}$. The I(0) value, normalized per unit concentration, is 0.0108 cm

^{−1}/unit concentration. This value divided by I

_{PS20}= 0.227 × 10

^{−3}cm

^{−1}/unit concentration gives N

_{agg}~ 48, and not 34 as indicated in [12]. In any case, by using I

_{PS20}= 0.235 × 10

^{−3}cm

^{−1}/unit concentration, here derived, we obtain N

_{agg,ini}= 46. Alternatively, as a second estimate of N

_{agg}, inserting the length of the alkyl chain (lauric acid), made of 12 carbon atoms, constituting the PS20 monomers, into Equations (A24) and (A25) of Appendix A, we obtain N

_{agg,ini}= 55. Averaging these two values (N

_{agg,ini}= 46 and N

_{agg,ini}= 55), we have N

_{agg,ini}= 50 ± 5. This is the first estimate of the number of monomers constituting the micelles, which is inserted in Equation (A22) of Appendix A for calculating the N

_{agg}derived by solving the analytical equations.

_{M}/2 − R

_{sh}, published in [12], seems to be overestimated and not compatible with the maximum size of the micelles (D

_{M}= 86.0 Å), as evidenced in the note of Table 2.

#### 3.2. DPC Micelles

^{3}.

^{−1}; ${c}_{mon}$ = 5 mg/mL; $cmc$ = 0.31 mg/mL, $K=\frac{\left({c}_{mon}-cmc\right){N}_{A}{{r}_{e}}^{2}}{{MW}_{mon}}=6.4\times {10}^{-7}\text{}{\mathrm{c}\mathrm{m}}^{-1}{{n}_{e}}^{-2}$. Moreover, in [13], N

_{agg}= 56 is obtained by the fitting model (Table 2 of [13]). Inserting the length of the alkyl chain made of 12 carbon atoms, constituting the DPC monomers, into Equations (A24) and (A25) of Appendix A, we obtain N

_{agg,ini}= 55. The gyration radius can be computed by the PDF by means of Equation (A11), obtaining ${r}_{g}=32.1\pm 0.1$ Å, a value very different from the published value of $37.5\pm 2.0$ Å, derived by the low-angle SAXS intensity’s analysis (Guinier approximation), also called “reciprocal space” gyration radius [17]. The PDF-derived gyration radius, also called “real-space” gyration radius, given by Equation (A11), has the advantage of being derived from the entire scattering curve and not just the lowest-resolution data [17]. Therefore, it is more representative of the atoms’ distribution within the micelle. The electron density values are strongly affected by the value of the gyration ratio if the low-resolution (reciprocal-space) estimation is too different from the real-space value.

_{e}/Å

^{3}is 80% larger than that of lipid tails (−0.036 n

_{e}/Å

^{3}) [12], leading to a too low electron density of the core: 0.268 n

_{e}/Å

^{3}.

_{pol}= 20.2 ± 0.5 Å, R

_{sh}= 6.9 ± 2.0 Å, in good agreement with all the values in Table 5. In addition, in [11], the authors determined an ellipticity value of ε = 1.22 ± 0.07, which is in good agreement with the ellipticity value here derived (1.15 ± 0.07), confirming that ε = 1.52 ± 0.014, reported in [13], is too large, probably caused by the overestimation of the gyration radius previously described.

#### 3.3. VitE-TPGS Micelles

_{sh}= 16.1 ± 0.5 Å, in agreement with the value reported for sample S5 in [4], and D

_{M}= 121.0 ± 0.5 Å. In [4], we have assumed a spherical shape for the VitE-TPGS micelles and convolved the PDF theoretical predictions, given the model, with a gaussian function of width σ = 28 ± 0.5 Å to describe non-ideal core-shell and shell-buffer interfaces. In the comparison of the results published in [4] with those here obtained, we also need to consider the widening of the PDF profiles due to this convolution. Table 8 summarizes the core and shell sizes obtained for VitE-TPGS micelles [4] versus the values here derived from the graphical approach (derivative of the PDF).

#### 3.4. VitE-TPGS Micelles with Eltrombopag (PSC)

_{sh}= 17.7 ± 0.5 Å, in agreement with the value reported for sample S7 in [4], and D

_{M}= 117.0 ± 0.5 Å. In [4], we have assumed a spherical shape for the VitE-TPGS micelles and convolved the PDF theoretical predictions, given the model, with a gaussian function with a width σ = 29 ± 0.5 Å to describe non-ideal core-shell and shell-buffer interfaces. By comparing the results published in [4] with those here obtained, we also need to consider the widening of the PDF profiles due to this convolution. Table 10 summarizes the core and shell sizes obtained for VitE-TPGS micelles with PSC [4] versus the values derived here from the graphical approach (derivative of the PDF).

#### 3.5. SDS Micelles

^{3}.

^{−1}; ${c}_{mon}$ = 6.25 mg/mL; $cmc$ = 0.21 mg/mL, $K=\frac{\left({c}_{mon}-cmc\right){N}_{A}{{r}_{e}}^{2}}{{MW}_{mon}}=4.2\times {10}^{-7}\text{}{\mathrm{c}\mathrm{m}}^{-1}{{n}_{e}}^{-2}$. Moreover, in [13], N

_{agg}= 90 has been obtained from the I(0) value and N

_{agg}= 118 from the model (Table 2 of [13]). The gyration radius can be computed by the PDF by means of Equation (A11), obtaining ${r}_{g}=33.0\pm 0.1$ Å, which coincides with the value reported in [13].

## 4. Conclusions and Perspectives

^{−1}, so the total number of Shannon channels is typically of the order of 5–10. In turn, this finding limits the number of free parameters of the micelles’ structure under study that can be readily determined by SAXS data. The spheroidal core-shell model presented here, with six free parameters, is a good compromise to handle the above-described limits, providing already several structural details that are very useful for studying new surfactants for pharmaceutical applications.

_{M}). If the size of the scattering nano-objects increases, the first Shannon channel’s constraints become more stringent, limiting the maximum nano-object size to a few tens of nanometers. However, collecting SAXS data at dedicated beamlines where sample-to-detector distances can reach several tens of meters may permit us to overcome this limitation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CMC | Critical Micelle Concentration |

DPC | Dodecyl phosphocholine |

Pair Distribution Function | |

PEG | Polyethylene Glycol |

PSC | Poorly Soluble Compound |

PS20 | Polysorbate 20 |

SAXS | Small-Angle X-ray Scattering |

SDS | Sodium Dodecyl Sulfate |

VitE-TPGS | D-α-tocopherol polyethylene glycol 1000 succinate |

## Appendix A. Derivation of the Analytic Formulae

_{M}and D

_{E}, respectively, and are schematically shown in Figure 1. Here, ε = R

_{pol}/R

_{eq}is the ratio between the polar R

_{pol}and equatorial R

_{eq}radius of the micelle’s core, and D

_{M}= 2R

_{pol}+ 2R

_{sh}, D

_{E}= 2R

_{pol}, where R

_{sh}is the shell size. For prolate spheroids (ε > 1), D

_{M}coincides with the maximum size of the micelle, namely D

_{max}. For oblate spheroids (ε < 1), D

_{M}= εD

_{max}.

^{−1}/distance], its integral for a two-component spherical micelle can be expressed as follows [2]:

_{H}is the gyration radius for a homogenous elliptical spheroid if the micelle had a constant electron density without any difference between core and shell values. For R = R

_{H}, one would have the average electron density within the micelle.

_{M}, R

_{sh}, R, $\mathsf{\epsilon}$ and $I\left(0\right)$. For prolate spheroids ${D}_{M}$ in Equation (A7) should coincide with the maximum distance ${D}_{max}$ measured in the PDF, i.e., ${D}_{M}={D}_{max}$. For oblate ($\epsilon <1$) spheroids ${D}_{M}=\epsilon {D}_{max}$. ${D}_{max}$ and ${R}_{sh}$ can be determined graphically from the PDF derivative, as discussed in the main section.

^{2}for a single tail and 0.042 nm

^{2}for a double tail. This quantity is fundamental in determining the packing of the monomer part constituting the micelles in their core. For more complex surfactant molecules with a head characterized by a lateral size larger than the tail (a single hydrocarbon chain), the above ratio can assume values ranging from 0.021 to 0.042 nm

^{2}. A lateral surfactant head size larger than the corresponding value of its tail will influence both its packing in the core and the core-shell interface order, causing the ellipticity of the aggregate when the equilibrium aggregation number would require a spherical shape with a radius larger than the maximum extended hydrophobic chain length.

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**Figure 2.**(

**a**) PDF(r) derived from the SAXS data measured/published on micelles made of three different types of monomers: the blue curve for PS20 [12], the red curve for DPC [13], and the green curve for VitE-TPGS [4]. (

**b**) First derivative of the PDF(r) reported in (

**a**): the blue curve for PS20 [12], the red curve for DPC [13], and the green curve for VitE-TPGS [4].

**Figure 3.**Flow chart of the proposed new method for determining the structural parameters of core-shell spheroidal micelles.

**Figure 4.**Figure of Merit (FOM) (Equation (A33) of Appendix A) for PS20 micelles. The minimum of the FOM indicates the solution.

**Figure 5.**Figure of Merit (FOM) (Equation (A33) of Appendix A) for DPC micelles. The minima of the FOM indicate possible solutions.

**Figure 6.**Figure of Merit (FOM) (Equation (A33) of Appendix A) for VitE-TPGS micelles. The minimum of the FOM indicates the solution.

**Figure 7.**(

**a**) PDI derived from the SAXS data measured on micelles made by VitE-TPGS with Eltrombopag [4]. (

**b**) First derivative of the PDF reported in (

**a**) (VitE-TPGS with Eltrombopag).

**Figure 8.**(

**a**) PDF derived from the SAXS data for SDS micelles published in [13]. (

**b**) First derivative of the PDF reported in (

**a**) (SDS micelles).

**Table 1.**Physical/chemical properties of the PS20 surfactant. N

_{A}is the Avogadro number. Mass density = 1.1507 g/cm

^{3}[16].

Compound | Mole Mass (g) | N_{e} = n_{e}/Molecule | Monomer or Molecule Volume (Å ^{3}) | Mole/L | n_{e}/L | Electron Density ρ (n _{e}/Å^{3}) |
---|---|---|---|---|---|---|

PS20 C _{58}H_{114}O_{26} | 1227.54 | 670 | 1771 | 0.938 | 616.4 × N_{A} | 0.378 |

Water H _{2}O | 18.016 | 10 | 29.9 | 55.51 | 555.1 × N_{A} | 0.334 |

**Table 2.**Core (D

_{M}/2 − R

_{sh}) half-size, shell (R

_{sh}) half-size, and maximum micelle size (D

_{M}) obtained by the graphical analysis of the PDF of PS20 micelles versus the values published in [12]. Concentration of the monomer: 5 mg/mL. The last column indicates the maximum core ellipsoid semiaxis, D

_{M}/2 − R

_{sh}, which is the polar core half-size R

_{pol}for prolate spheroids and the equatorial core half-size R

_{eq}for oblate spheroids.

Method | R_{sh} (Å) | D_{M} (Å) | D_{M}/2 − R_{sh} (Å) |
---|---|---|---|

Published [12] | 8.8 ± 0.7 | 86.0 ± 0.5 | 36.8 ± 0.7 (*) |

Graphical | 9.0 ± 0.5 | 86.0 ± 0.5 | 34.1 ± 0.5 |

_{M}= 86 Å. From the maximum micelle size and shell size published in [12], reported in the first two columns, we should expect D

_{M}/2 − R

_{sh}= 34.2 Å as the value of the third column, and not 36.8.

**Table 3.**Summary of the ellipticity, aggregation number, and the core and shell electron density contrasts’ values obtained in [12] versus the values here derived by the analytical formulae.

Method | ε | N_{agg} | $\mathsf{\Delta}{\mathit{\rho}}_{\mathit{E}}$ (n_{e}/Å^{3}) | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{P}}$ (n_{e}/Å^{3}) |
---|---|---|---|---|

Published [12] | 1.50 ± 0.06 | 34 | −0.035 ± 0.002 | 0.060 ± 0.004 |

Published [12] | 1.50 ± 0.06 | 34 | −0.031 ± 0.002 | 0.064 ± 0.003 |

Analytical | 1.47 ± 0.01 | 35 ± 1 | −0.030 ± 0.002 | 0.050 ± 0.005 |

**Table 4.**Summary of the chemical/physical properties of the dodecyl phosphocholine (DPC) experiment. N

_{A}is the Avogadro number. Assumed mass density of 1 g/cm

^{3}.

Compound | Mole Mass (g) | N_{e} = n_{e}/Molecule | Monomer or Molecule Volume (Å^{3}) | Mole/L | n_{e}/L | Electron Density ρ (n _{e}/Å^{3}) |
---|---|---|---|---|---|---|

DPC C _{17}H_{38}NO_{4}P | 351.5 | 194 | 548 | 3.03 | 587.9 × N_{A} | 0.354 |

H_{2}O | 18.016 | 10 | 29.9 | 55.51 | 555.1 × N_{A} | 0.334 |

**Table 5.**Core (D

_{M}/2 − R

_{sh}) half-size, shell (R

_{sh}) half-size, and maximum micelle size (D

_{M}) obtained by the graphical analysis of the PDF of DPC micelles compared with the values published in [13]. Concentration of the monomer: 5 mg/mL. The last column indicates D

_{M}/2 − R

_{sh}, which is the polar core half-size R

_{pol}for prolate spheroids the equatorial core half-size R

_{eq}for oblate spheroids.

Method | R_{sh} (Å) | D_{M} (Å) | D_{M}/2 − R_{sh} (Å) |
---|---|---|---|

Published [13] | 6.83 ± 0.22 | 58.1 ± 1.2 | 22.21 ± 0.38 |

Graphical | 6.8 ± 0.5 | 58.0 ± 0.5 | 22.2 ± 0.5 |

**Table 6.**Summary of the ellipticity, aggregation number, and the core and shell electron density contrasts’ values obtained in [13] versus the values here derived by the analytical formulae for prolate and oblate shapes.

Method | ε | N_{agg} | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{E}}$ (n_{e}/Å^{3}) | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{P}}$ (n_{e}/Å^{3}) |
---|---|---|---|---|

Published [13] | 1.52 ± 0.014 | 56 | −0.066 ± 0.003 | 0.054 ± 0.004 |

Analytical | 1.15 ± 0.07 | 57 | −0.043 ± 0.003 | 0.041 ± 0.007 |

Analytical | 0.90 ± 0.07 | 57 | −0.041 ± 0.003 | 0.038 ± 0.007 |

**Table 7.**Summary of the VitE-TPGS physical properties. N

_{A}is the Avogadro number. Mass density of 1.08 g/cm

^{3}[4].

Compound | Mole Mass (g) | N_{e} = n_{e}/Molecule | Monomer or Molecule Volume (Å ^{3}) | Mole/L | n_{e}/L | Electron Density ρ (n _{e}/Å^{3}) |
---|---|---|---|---|---|---|

vitE-TPGS C _{33}O_{5}H_{54}(CH_{2}CH_{2}O)_{n}n = 0.7 × 22 + 0.3 × 23 | 1513.1 | 827 | 2327 | 0.716 | 592.1 × N_{A} | 0.357 |

H_{2}O | 18.016 | 10 | 29.9 | 55.51 | 555.1 × N_{A} | 0.334 |

**Table 8.**Core (D

_{M}/2 − R

_{sh}) half-size, shell (R

_{sh}) half-size, and maximum micelle size (D

_{M}) obtained by the graphical analysis of the PDF of VitE-TPGS micelles versus the values published in [4]. Concentration of the monomer: 4.1 mg/mL. The last column indicates D

_{E}= D

_{M}− 2R

_{sh}, which is the polar core size for prolate spheroids and the equatorial core size for oblate spheroids.

Method | R_{sh} (Å) | σ (Å) | D_{M} + σ (*)(Å) | D_{M} − 2R_{sh} + σ (*)(Å) |
---|---|---|---|---|

Published [4] | 16.8 ± 1.0 | 28.0 ± 0.5 | 119.5 ± 1.0 | 85.9 ± 2.5 |

Graphical | 16.1 ± 0.5 | 0 | 121.0 ± 1.0 | 88.8 ± 1.5 |

_{sh}derived by the two different approaches. However, for the convolution theorem, we must compare D

_{E}and D

_{M}here, derived with D

_{E}+ σ and D

_{M}+ σ determined in [4].

**Table 9.**Summary of the structural values obtained for VitE-TPGS micelles in sample S5 of ref. [4] versus the analytical formulae’s values here obtained.

Method | $\mathsf{\Delta}{\mathit{\rho}}_{\mathit{E}}$ (n _{e}/Å^{3}) | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{P}}$ (n _{e}/Å^{3}) | $\mathit{\epsilon}$ | ${\mathit{N}}_{\mathit{a}\mathit{g}\mathit{g}}$ |
---|---|---|---|---|

Published [4] | −0.037 ± 0.001 | 0.037 ± 0.001 | 1 (assumed) | 116 ± 1 |

Analytical | −0.029 ± 0.004 | 0.033 ± 0.002 | 1.41 ± 0.002 | 125 ± 1 |

**Table 10.**Core (D

_{M}/2 − R

_{sh}) half-size, shell (R

_{sh}) half-size, and maximum micelle size (D

_{M}) obtained by the graphical analysis of the PDF of (VitE-TPGS with PSC)-micelles versus the values published in [4]. Concentration of the monomer: 4.1 mg/mL. The last column indicates D

_{E}= D

_{M}− 2R

_{sh}, which is the polar core size for prolate spheroids and the equatorial core size for oblate spheroids.

Method | R_{sh} (Å) | σ (Å) | D_{M} + σ (*)(Å) | D_{M} − 2R_{sh} + σ (*)(Å) |
---|---|---|---|---|

Published [4] | 18.7 ± 1.0 | 29.0 ± 0.5 | 116.5 ± 1.0 | 79.1 ± 2.5 |

Graphical | 17.7 ± 0.5 | 0 | 116.9 ± 1.0 | 81.5 ± 1.5 |

_{sh}derived by the two different approaches. However, for the convolution theorem, we must compare D

_{E}and D

_{M}derived here with D

_{E}+ σ and D

_{M}+ σ determined in [4].

**Table 11.**Summary of the structural values obtained for (VitE-TPGS with PSC) micelles of sample S7 of ref. [4] versus the analytical formulae’s values obtained here.

Method | $\mathsf{\Delta}{\mathit{\rho}}_{\mathit{E}}$ (n _{e}/Å^{3}) | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{P}}$ (n _{e}/Å^{3}) | $\mathit{\epsilon}$ | ${\mathit{N}}_{\mathit{a}\mathit{g}\mathit{g}}$ |
---|---|---|---|---|

Published [4] | −0.055 ± 0.001 | 0.045 ± 0.001 | 1 (assumed) | 117 ± 1 |

Analytical | −0.046 ± 0.002 | 0.043 ± 0.002 | 1.45 ± 0.001 | 123 ± 1 |

**Table 12.**Summary of the chemical/physical properties of the sodium dodecyl sulfate (SDS) experiment. N

_{A}is the Avogadro number. Mass density of 1.1 g/cm

^{3}(https://pubchem.ncbi.nlm.nih.gov/compound/Sodium-dodecyl-sulfate, accessed on 20 April 2024).

Compound | Mole Mass (g) | N_{e} = n_{e}/Molecule | Monomer or Molecule Volume (Å ^{3}) | Mole/L | n_{e}/L | Electron Density ρ (n _{e}/Å^{3}) |
---|---|---|---|---|---|---|

SDS C _{12}H_{25}SO_{4}Na | 288.4 | 156 | 435.4 | 3.81 | 595.0 × N_{A} | 0.358 |

H_{2}O | 18.016 | 10 | 29.9 | 55.51 | 555.1 × N_{A} | 0.334 |

**Table 13.**Core (D

_{M}/2 − R

_{sh}) half-size, shell (R

_{sh}) half-size, and maximum micelle size (D

_{M}) obtained by the graphical analysis of the PDF of SDS micelles compared with the values published in [13]. Concentration of the monomer: 6.25 mg/mL. The last column indicates D

_{M}/2 − R

_{sh}, which is the polar core half-size R

_{pol}for prolate spheroids or the equatorial core half-size R

_{eq}for oblate spheroids.

Method | R_{sh} (Å) | D_{M} (Å) | D_{M}/2 − R_{sh} (Å) |
---|---|---|---|

Published [13] | 4.85 ± 0.17 | 72.07 ± 4.32 | 31.185 ± 1.99 |

Graphical | 6.6 ± 0.5 | 73.0 ± 0.5 | 29.9 ± 0.5 |

**Table 14.**Summary of the ellipticity, aggregation number, and core and shell electron density contrasts’ values obtained in [13] for SDS micelles versus the values derived here by the analytical formulae.

Method | ε | N_{agg} | $\mathsf{\Delta}{\mathit{\rho}}_{\mathit{E}}$ (n_{e}/Å^{3}) | ${\mathsf{\Delta}\mathit{\rho}}_{\mathit{P}}$ (n_{e}/Å^{3}) |
---|---|---|---|---|

Published [13] | 1.75 ± 0.11 | 90, 118 | −0.073 ± 0.006 | 0.138 ± 0.004 |

Analytical | 1.60 ± 0.07 | 96 ± 1 | −0.082 ± 0.010 | 0.105 ± 0.003 |

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## Share and Cite

**MDPI and ACS Style**

De Caro, L.; Stoll, T.; Grandeury, A.; Gozzo, F.; Giannini, C.
Characterization of Surfactant Spheroidal Micelle Structure for Pharmaceutical Applications: A Novel Analytical Framework. *Pharmaceutics* **2024**, *16*, 604.
https://doi.org/10.3390/pharmaceutics16050604

**AMA Style**

De Caro L, Stoll T, Grandeury A, Gozzo F, Giannini C.
Characterization of Surfactant Spheroidal Micelle Structure for Pharmaceutical Applications: A Novel Analytical Framework. *Pharmaceutics*. 2024; 16(5):604.
https://doi.org/10.3390/pharmaceutics16050604

**Chicago/Turabian Style**

De Caro, Liberato, Thibaud Stoll, Arnaud Grandeury, Fabia Gozzo, and Cinzia Giannini.
2024. "Characterization of Surfactant Spheroidal Micelle Structure for Pharmaceutical Applications: A Novel Analytical Framework" *Pharmaceutics* 16, no. 5: 604.
https://doi.org/10.3390/pharmaceutics16050604