Modeling drug delivery from multiple emulsions

We present a mechanistic model of drug release from a multiple emulsion into an external surrounding fluid. We consider a single multilayer droplet where the drug kinetics are described by a pure diffusive process through different liquid shells. The multilayer problem is described by a system of diffusion equations coupled via interlayer conditions imposing continuity of drug concentration and flux. Mass resistance is imposed at the outer boundary through the application of a surfactant at the external surface of the droplet. The two-dimensional problem is solved numerically by finite volume discretization. Concentration profiles and drug release curves are presented for three typical round-shaped (circle, ellipse, and bullet) droplets and the dependency of the solution on the mass transfer coefficient at the surface analyzed. The main result shows a reduced release time for an increased elongation of the droplets.

Figure 1

Schematic representation of the cross section of the droplet comprising an internal circular core ${\mathrm{\Omega }}_0$ and an enveloping denser fluid shell ${\mathrm{\Omega }}_1$. A thin membrane (shown in blue) is present at the surface modeling the surfactant finite resistance.

Figure 2

A snapshot of a typical double emulsion (from Ref. [5]).

Figure 3

Notation used in the finite volume discretization. Depicted is the finite volume ($V_k$) corresponding to an arbitrary internal node $k$ with the boundary of the finite volume shown using a dashed line. The blue dot locates the midpoint of edge $\sigma$ (${\mathit{x}}_{\sigma }$) and the length of the red edge, labeled $\sigma$, is $L_{\sigma }$.

Figure 4

Schematic representation of the cross section of the droplet comprising an internal circular core ${\mathrm{\Omega }}_0$ and an enveloping shell ${\mathrm{\Omega }}_1$. Three 2D shapes are considered for ${\mathrm{\Omega }}_1$: circle, ellipse, and bullet, all with the same area. This two-layer emulsion-based vesicle is surrounded by a surfactant layer (blue boundary) having a mass transfer coefficient $P$ (m/s).

Figure 5

Triangular meshes for the three droplet shapes of equal area: circle (left column) consisting of 1769 nodes and 3408 triangular elements, an ellipse with $e=0.95$ (middle column) consisting of 1797 nodes and 3432 triangular elements, and a bullet (right column) consisting of 1785 nodes and 3438 triangular elements.

Figure 6

Concentration field at $t=1\text{min}$ (top row), $t=6\text{min}$ (middle row) and $t=1\text{h}$ (bottom row) for the three droplet shapes of equal area: circle (left column), an ellipse with $e=0.95$ $[\gamma =1.8]$ (middle column), and a bullet (right column) (cfr. Fig. 7). Results are produced using the meshes shown in Fig. 5 and the parameter values $R_0=40\mu \text{m}$, $R_1=80\mu \text{m}$, $D_1={10}^{-13}{\text{m}}^2/\text{s}$, $P=2\times {10}^{-4}\text{m}/\text{s}$.

Figure 7

Mass profiles for the three droplet shapes of equal area (cf. Fig. 6). $\text{Relative}\text{mass}=M_0\left(t\right)/M_T\left(0\right)$ (Core), $M_1\left(t\right)/M_T\left(0\right)$ (Shell), $M_T\left(t\right)/M_T\left(0\right)$ (Total), and $M_r\left(t\right)$ (Released) [cf. Eqs (3.4)–(3.7)]. Results are produced using the meshes shown in Fig. 5 and the parameter values used in Fig. 6. Legend applies across all three figures.

Figure 8

Reduction of the release time [RT, Eq. (4.1)] for the ellipsoidal-shaped droplet with increasing eccentricity ($e$) (cf. Table 3). All other parameters are held fixed: $R_0=40\mu \text{m}$, $R_1=80\mu \text{m}$, $D_0={10}^{-10}{\text{m}}^2/\text{s}$, $D_1={10}^{-13}{\text{m}}^2/\text{s}$, and $P=2\times {10}^{-4}\text{m}/\text{s}$.

Figure 9

Boundary flux $J=-D_1\mathbf{\nabla }{c}_1·{\mathit{n}}_{\mathrm{\Omega }}$ at $\mathit{x}\in \partial \mathrm{\Omega }$ and total boundary flux ${\int }_{\partial \mathrm{\Omega }}Jds$ at $t=1\text{h}$ for the three droplet shapes of equal area: circle (left column), an ellipse with $e=0.95$ $[\gamma =1.8]$ (middle column), and a bullet (right column) (cf. Fig. 6, bottom row). Results are produced using the meshes shown in Fig. 5 and the parameter values used in Fig. 6.