COMSOL 6.4 - Drug Delivery System

This example describes the operation of a drug delivery system that supplies a variable concentration of a water soluble drug. A droplet with a fixed volume of water travels down a capillary tube at a constant velocity. Part of the capillary wall consists of a permeable membrane separating the interior of the capillary from a concentrated solution of the drug. As the drop passes by the membrane, the drug dissolves into the water. To model this process, a constant flux of the drug is assumed on the capillary wall for the duration of its contact with the membrane. By altering the droplet velocity, the final concentration of the drug in the drop can be adjusted. The principle of the device is illustrated in Figure 1.

Figure 1: Diagram showing the operating principle of the drug delivery device. The color in the droplet represents drug concentration, with red indicating a higher concentration and blue indicating zero concentration. Note that the membrane length is not shown to scale.

Model Definition

The axisymmetric model geometry is shown in Figure 2. The droplet is visible near the top of the geometry. The horizontal lines across the capillary are included to assist with meshing. The drop is initially stationary at the top of the domain, but accelerates rapidly to a constant velocity before it reaches the permeable membrane. The permeable part of the capillary is not visible as part of the geometry as it is represented by a function applied to the boundary condition. It is located between z = 0.6 mm and z = 0.8 mm.

The droplet consists of liquid water of density 1000 kg/m3 and viscosity 10-3 Pa⋅s. The remainder of the capillary is filled with air, with a density of 1.25 kg/m3 and a viscosity of 2×10−5 Pa⋅s. The water air surface tension coefficient is 70 mN/m. The contact angle of the droplet with the capillary wall is 135°, whilst that with the membrane is 157.5°. As the droplet passes the membrane the flux of the drug entering it is 1×10−3 mol/(m2⋅s). The diffusion coefficient of the drug in the water is 5×10−9 m2/s.

The droplet velocity past the membrane is varied between 0.1 and 1 mm/s to adjust the final concentration of the drug in the droplet.

Figure 2: Axisymmetric model geometry.

Results and Discussion

The flow velocity is shown for the drop moving at 0.25 mm/s in Figure 3. The flow pattern around the interface is complex as the flow must redistribute itself from a Poiseuille flow profile away from the droplet surface to a constant velocity flow at the surface of the droplet. Notice that the change in contact angle as the droplet passes the edge of the membrane at z = 8 mm is apparent.

Figure 3: Flow velocity around the droplet as it travels past the edge of the permeable membrane. The droplet velocity is 0.25 mm/s.

Figure 4 shows the concentration profile for the 0.25 mm/s at the same point in time. The drug is diffusing into the droplet and is also convected by the fluid flow. A marked change in concentration is apparent between the top and the bottom of the droplet.

The total amount of drug in the droplet as a function of time is shown in Figure 5, for the drop traveling at 0.1 mm/s. The dissolved drug quantity increases with an ‘S’ shaped profile as the drug travels down the capillary.

Figure 4: Drug concentration in the droplet as it travels past the edge of the permeable membrane. The droplet velocity is 0.25 mm/s.

Figure 6 shows the total amount of drug delivered against the droplet velocity. The number of moles delivered is approximately inversely proportional to the droplet velocity, which is expected as the amount of drug that diffuses into the drop depends on the time the drop takes to traverse the permeable part of the capillary.