Turbidity Spectroscopy for Characterization of Submicroscopic Drug Carriers, Such as Nanoparticles and Lipid Vesicles: Size Determination

ABSTRACT

Purpose

To apply UV/Vis spectrometry for characterization of submicroscopic drug carriers, such as nanoparticles and lipid vesicles.

Methods

We first investigated theoretically, within the framework of the Rayleigh-Gans-Debye approximation (RGDA), parameters affecting turbidity spectrum, τ(λ), of nanosized light scatterers. We then analyzed, within the framework of the RGDA, experimental turbidity spectra (λ = 400–600 nm) of extruded unilamellar vesicle (70 nm ≤ 2r ≤ 110 nm) suspensions to derive vesicle size, using dynamic light scattering results for comparison. We similarly studied the preparations polydispersity and lamellarity and monitored vesicle size changes.

Results

Turbidimetry suffices for accurate, fast, and viscosity-independent characterization of submicroscopic particles. Analysis of turbidity spectra, or more precisely wavelength exponent spectra (derivatives of logarithmic turbidity spectra), yielded similar average radii (r = 54.2 ± 0.2 nm; 46.0 ± 0.2 nm; 35.5 ± 0.1 nm) as dynamic light scattering (r = 55.9 ± 1.5 nm; 46.1 ± 0.4 nm; 36.1 ± 0.4 nm). Both methods also revealed similar suspension polydispersity and cholate-induced vesicle size changes in a few nanometer range.

Conclusion

Despite its experimental simplicity, the widely accessible turbidimetric method provides accurate size values and is suitable for (continuous) monitoring size stability, or sameness, of submicroscopic drug carriers.

Appendices

APPENDIX A: RANGE OF VALIDITY OF THE RAYLEIGH-GANS-DEBYE APPROXIMATION

Kerker and colleagues (67) tested the range of validity of the Rayleigh-Gans-Debye approximation for homogeneous spheres by comparing the outcome of the exact Mie calculations and of the approximate calculation. The comparison included the scattering function at different angles as well as the scattering coefficient, Q _sca, defined as the total radiation scattered by a particle relative to the incident radiation intensity intercepted by the particle, i.e. Q _sca = τ/N _P πr ².

As we are only concerned with turbidity in this work, we present in Fig. 10 just the results for the scattering coefficient. Region I in the figure defines the range of kr = 2πnr/λ and m values for which the RGDA agrees with the exact Mie theory to within 10%. In region II, the RGDA deviates from the Mie theory by no more than 100%, except in small islands where the actual deviation may be less than 10%. In region III, the error exceeds 100%.

Fig. 10

The error contour chart for the scattering coefficient, Q _sca (modified from (45,67)). In region I, the accuracy of the Rayleigh-Gans-Debye approximation (RGDA) is better than 10%. In region II, the RGDA accuracy is between 10% and 100%, whereas in region III, the error resulting from using the RGDA exceeds 100%, except in small islands. k = 2πn/λ is the propagation constant in the dispersion medium with refractive index n in which scatterers with average diameter 2r and refractive index n _S are dispersed. Relative refractive index is described as m = n _S/n.

It is noteworthy that the area just above the abscissa with m > 1.25 is part of region II in the originally published chart (67). The failure to obtain 10% agreement in this area is not due to limitation of the RGDA but rather due to the numerical approximation made by the authors about the refractive index, lim_m→1 (m ²−1)/(m ² + 2) = 2(m−1)/3. The error due to this approximation is 10% at m = 1.25. We avoided making such an approximation in our calculations as well as in the Theory section, and consequently included the area just above the abscissa with m > 1.25 into region I.

For a suspension of homogeneous spheres with m = 1.10 the 10% contour line is located at kr = 9.2. For a suspension of homogeneous spheres in water with m = 1.10, the RGDA analysis of turbidity spectra is consequently correct to within 10% when r ≤ λ.

APPENDIX B: RANGE OF VALIDITY OF THE ANALYTICAL APPROACH

Fig. 11 illustrates scatterer size effect on turbidity and wavelength exponent spectra over an extended range of r values. It reveals a monotonous increase of turbidity with r for homogeneous spheres and a nearly monotonous increase for vesicles. In contrast, wavelength exponent decreases with increasing scatterer size only up to certain r-limit that depends on geometry and potentially refractive index (data not shown) of the scatterer. Above such limit, wavelength exponent oscillates as a function of r, mainly owing to size-dependency of form factors. This precludes unambiguous size determination for scatterers larger than such limit based just on a non-linear regression analysis of wavelength exponent spectra. The restriction is more severe for hollow spheres, such as vesicles, than for homogeneous spheres, such as nanoparticles (Fig. 11). The r-limit generally increases with the scattered light wavelength.

Fig. 11

Effect of the light scatterer diameter (2r = 50.0–500.0 nm, 25 nm intervals, monodisperse) on the scatterer suspension turbidity (upper panels) and wavelength exponent (lower panels) spectra (λ = 400–600 nm, 20 nm intervals). The curves were calculated for suspensions comprising either homogeneous spheres (left two panels, number concentration N _P = 1.1 × 10¹² mL⁻¹) or hollow spheres/spherical shells/lipid vesicles (right two panels, number concentration N _v = 3.5 × 10¹³ mL⁻¹, shell thickness d _shell = 3.6 nm). The homogeneous spheres and the spherical shells/lipid bilayers were assumed to have the refractive index of dipalmitoylphosphatidylcholine (Eq. D1) and the dispersion medium to have the refractive index of water (Eqs. C1–C2).

Particle size derivation via turbidity spectrum analysis is feasible so long as the Rayleigh-Gans-Debye approximation may be applied. One can therefore employ such analysis for the particles larger than the r-limit of the otherwise more convenient wavelength exponent spectrum analysis. An even better solution is to combine turbidity and wavelength exponent spectra analyses.

APPENDIX C: THE REFRACTIVE INDEX OF WATER

The refractive index of water at 25°C under atmospheric pressure is described as a function of wavelength in the visible wavelength range with the formula

$$ {n_{\text{w}}}\left( \lambda \right) = {\left( {\frac{{2{u_{\text{w}}}\left( \lambda \right) + 1}}{{1 - {u_{\text{w}}}\left( \lambda \right)}}} \right)^{0.5}}, $$

(C1)

where

$$ {u_{\text{w}}}\left( \lambda \right) = {b_0} + {b_1}{\lambda^{*2}} + \frac{{{b_2}}}{{{\lambda^{*2}}}} + \frac{{{b_3}}}{{{\lambda^{*2}} - \lambda_{\text{UV}}^{*2}}} + \frac{{{b_4}}}{{{\lambda^{*2}} - \lambda_{\text{IR}}^{*2}}}. $$

(C2)

b ₀ = 0.232602194, b ₁ = +0.294685133 × 10⁻³, b ₂ = +0.163176785 × 10⁻², b ₃ = +0.241520886 × 10⁻², b ₄ = +0.897025499, $\lambda_{\text{UV}}^* = 0.{2292}0{2}0,\lambda_{\text{IR}}^* = {5}.{432937}$, and λ ^* = λ/589 nm. Eqs. C1–C2 provide absolute accuracy of ±1 × 10⁻⁵.

We derived Eqs. C1–C2 by simplifying the more general expression published by Schiebener and colleagues (68), which covers wide ranges of wavelengths, temperatures, densities, and pressures. We reached the goal by taking water density at 25°C and atmospheric pressure to be ρ = 997.0480 kg m⁻³ (69).

APPENDIX D: THE REFRACTIVE INDEX OF LIPID

Khlebtsov and colleagues (37) proposed the following parametric description of dipalmitoylphosphatidylcholine (DPPC) refractive index as a function of light wavelength at 20°C, based on the data measured by Chong and Colbow with visible light above 400 nm (36):

$$ {n_{\text{L}}}\left( \lambda \right) = 1.4713 + 1.31\,{\text{nm}}\,{\lambda^{ - 1}} + 4309\,{\text{n}}{{\text{m}}^2}\,{\lambda^{ - 2}}. $$

(D1)

The result of Eq. D1 at λ = 632.8 nm, n _L = 1.484, compares favorably with the experimental values reported for DPPC at T = 25°C by Erbe and Sigl (70), n _L = 1.478. The result of Eq. D1 at λ = 589 nm, n _L = 1.486, likewise resembles acceptably the value extrapolated for DPPC to T = 25°C from the data published by Yi and McDonald (71), n _L = 1.475.

At the specified temperature, DPPC forms one particular type of the ordered-gel, L_ß-phase. Eq. D1 thus does not strictly apply to any other temperature or lipid. The former restriction is especially important, as temperature not only gradually decreases n _L but moreover can trigger even more influential lipid bilayer phase transitions. Chain fluidization, for example, lowers lipid refractive index abruptly, the reported difference for DPPC being approximately −0.008 units (71,72).

Polar lipid headgroups contribute relatively little to the refractive index difference between lipid bilayers and water. The influence of lipid chain-length and unsaturation is bigger. Both these parameters increase lipid refractive index and thus can “compete” with the temperature- and fluidization-induced n _L changes.

In all experiments reported herein we were using soybean phosphatidylcholine (SPC). This lipid has roughly two more methylene groups per chain than DPPC and contains mainly di-unsaturated chains. SPC melts below the water freezing point, and the lipid is consequently in the fluid lamellar phase, ${{\text{L}}_\alpha }$, at T = 25°C. To the best of our knowledge, results of the kind reported for DPPC by Chong and Colbow (36) are unavailable for soybean phosphatidylcholine to date. We only found some information on the refractive index of soybean oil wavelength dependency (73). Fortunately, soybean oil has arguably similar chain composition as soybean phosphatidylcholine. We therefore used the parametrization published by van Staveren and colleagues for such oil to check Eq. D1 applicability to SPC, and thus to our illustrative experimental system. Between 500 nm and 800 nm, the calculated difference between the two parametric equations amounts to −0.01203 ± 0.00084. The two underlying expressions have therefore quite similar slope dn _L/dλ in the compared wavelength region. Van Staveren expression may not be applied below 500 nm (where it predicts dn _L/dλ to change sign) but is essentially equivalent to Eq. D1 at longer wavelengths. We therefore applied Eq. D1 herein to cover the entire analyzed wavelength range: 400 nm ≤ λ ≤ 600 nm. Shifting results of Eq. D1 downward (e.g. by subtracting the above-mentioned difference of 0.01203) from the constant in Eq. D1 ⁶ would merely affect turbidity spectrum analysis and leave the results of wavelength exponent spectrum analysis practically unchanged. As this work has a focus on the latter option, we refrained from making such a correction herein.

Any cautious users of the analytical method advocated in this work should always check applicability of Eq. D1 to their particular experimental system. More likely than not, the expression will need to be adjusted and/or generalized. This will require knowledge of at least some reliable n _L vs. λ data pairs. If such information is missing, the appropriate refractive index dependency should be measured (e.g. with an Abbè refractometer). Alternatively, the n _L vs. λ dependency could be determined by, first, reversing the experimental sequence used in this work with the aim of generating a calibration data set for further applications. For this purpose, at least three suspensions of differently large vesicles should be prepared from the same batch of lipid and then assessed with wavelength exponent analysis and with dynamic light scattering. The results should be compared and the parameters needed for the former kind of analysis iteratively adjusted until the two size characterization methods give the same result.