Lateral Transport during Membrane Permeation in Diffusion Cell: In Silico Study on Edge Effect and Membrane Blocking

Abstract

Membrane transport in diffusion cell studies is not one-dimensional from the donor to the receptor. Lateral diffusion within the membrane into the surrounding clamped region can lead to edge effect. Lateral diffusion can also affect the impact of an object blocking the membrane in a diffusion cell. The effects of lateral transport on permeation across a two-layer membrane in diffusion cells were investigated in this study under edge effect and membrane blocking conditions that could be encountered in previous gingiva and hypothetical skin permeation studies. Model simulations of time-dependent and steady-state transport were performed using COMSOL Multiphysics. The simulations indicated edge effect could increase the steady-state flux across the membrane up to 35% with a relatively thick membrane and small diffusion cell opening (e.g., gingiva study). The edge effect decreased when the relative thickness and permeability of the major barrier (top layer in the two-layer membrane) decreased. When the membrane was partially blocked by an object, lateral diffusion within the membrane could mitigate its impact: e.g., when the object was in the receptor, the impact caused by membrane blocking was reduced more than half. Therefore, membrane lateral transport should be considered under certain circumstances in permeation studies using diffusion cells.

Introduction

Diffusion cells are widely used to evaluate drug (or chemical) release and permeation across synthetic and biological membranes ^1–5. There are two main types of diffusion cells commonly used in permeation studies: side-by-side diffusion cell and vertical diffusion cell (Franz diffusion cell). Whereas the side-by-side diffusion cell is normally used in steady-state transport studies to determine the permeability coefficient of a membrane, the vertical diffusion cell is commonly used to determine the performance of topical formulations in drug delivery. An example of the applications of vertical diffusion cell is the in vitro permeation testing (IVPT) for skin to evaluate drug formulations in topical and transdermal permeation studies and dermatopharmacokinetics assessments ^6–9. In this example, the skin is sandwiched between the donor and receptor chambers of the diffusion cell, the formulation is applied onto the skin surface, and samples are removed from the receptor to measure the rate and extent of drug permeation across the membrane. For side-by-side diffusion cells and other similar diffusion cell settings (e.g., Ussing chambers), permeation of drugs across eye tissues such as cornea, conjunctival, and sclera for ocular drug delivery ^10–12, mucosal membranes such as buccal and intestinal tissues ^13–16, skin in steady state transport studies ^17–19, and hard tissues such as enamel for dental applications and nail plate for transungual delivery ^20–22 can be evaluated.

In typical diffusion cell setups for permeation studies, the membrane is placed between two diffusion cell chambers in which a portion of a membrane is clamped between two opposite impermeable surfaces of the chambers (the donor and receptor chamber ground glass joints facing the membrane). The concentric part of the membrane connected to the aperture (diffusional opening) of the diffusion cell provides the pathway for drug permeation across the membrane. This pathway is assumed to be a one-dimensional pathway for drug diffusion from the donor chamber to the receptor chamber across the membrane, but in reality, diffusion within the membrane is three-dimensional such that the membrane portion clamped between the impermeable surfaces of the chambers can be involved in membrane transport (edge effect).

The edge effect describes the mass transport pathway from the donor to the receptor via membrane lateral diffusion into the surrounding clamped region of the membrane beyond the edge of the diffusional opening in the diffusion cell. In other words, the conventional assumption of a one-dimensional diffusion process from the donor to the receptor across the membrane may not be correct when the mass transport pathway spreads into the surrounding clamped region of the diffusion cell away from the central area. The “extra” mass transport via the additional pathway can increase the total flux and transport lag time, and this can introduce errors in the measurement of membrane permeability. For a relatively thick membrane and in permeation studies with a small diffusion cell opening, the edge effect can be significant. This is because the thicker membrane provides larger cross-sectional area for lateral diffusion and it has smaller differences between the transmembrane pathway length (membrane thickness) and the length of the additional pathway around the membrane edge. Previous studies have investigated the edge effect and provided the analytical solution of the mathematical equations describing the mass transport ^23,24. It has been suggested that, for a uniform circular membrane with thickness that is equal or less than 1/5 the radius of the diffusion cell opening (i.e., membrane thickness divided by cell opening radius ≤ 0.2), the edge effect may safely be neglected ²³. When membrane thickness divided by cell opening radius is > 0.2, the extent of edge effect on flux is a function of the thickness, opening radius, and the size of the circular membrane. Most biological membranes in diffusion cell experiments are not single-layer uniform membranes but are composed of multiple layers with different permeabilities and thicknesses (e.g., epidermis and dermis in skin and epithelium and underlying connective tissues in gingiva) ^9,25. The impact of edge effect on a two-layer membrane has not been investigated.

Recently, a modified diffusion cell was used to study drug permeation across the gingiva ²⁵. The modification is necessary due to the sizes of the gingiva compared to the diffusional opening of conventional diffusion cells such as those for skin studies; the sizes (surface area) of the tissue samples obtained from the gingiva are limited and smaller than those of other biological membranes such as the skin. Therefore, the new diffusion cell design in the recent gingiva study had a relatively small diffusional opening to accommodate the small gingiva samples. The modified diffusion cell had an opening of 0.2 cm in diameter (area of 0.03 cm²) for the gingiva membrane. This diffusion cell opening is significantly smaller than the smallest diffusional opening of a typical commercial diffusion cell (i.e., 0.5 cm diameter and area of 0.2 cm²). When a small diffusion cell opening (relative to the thickness of the membrane) is used, edge effect should be considered.

Another situation that can introduce errors in diffusion cell permeation studies is the formation of air bubbles at the interface between the membrane and the medium in the diffusion cell chamber, particularly in the receptor chamber. This is a potential problem in IVPT and can be important in the modified diffusion cell with the small diffusional opening in the previous gingiva study ²⁵. Membrane blocking in drug permeation studies is not limited to air bubbles in the receptor chamber and can occur in the donor chamber. For example, air can be trapped between the formulation and the membrane surface in the donor chamber when the formulation is applied onto the surface. When a membrane is partially blocked by an impermeable object (e.g., an air bubble), it can lead to transport phenomenon similar to an edge effect around the blocked area. Lateral diffusion within the membrane can contribute to additional pathways for mass transport across the membrane from the donor to the receptor when an object blocks a portion of the membrane surface, and hence reducing the impact of the object. The examination of the effects of different “membrane blocking” conditions can provide useful information for assessing the quality of permeation studies when the situations arise.

The objectives of the present study were to (a) assess the influence of edge effect on mass transport across a two-layer membrane in a diffusion cell by varying the permeabilities and thicknesses of the two layers and diffusion cell parameters and (b) determine the effect of membrane blocking on permeation in a diffusion cell by varying the size of the blocking object, permeabilities and thicknesses of the two layers in the membrane composite, and diffusion cell parameters. The determining factors on the flux across a two-layer membrane and their relationships were investigated for the edge effect and membrane blocking. Model simulations of time-dependent and steady-state transport were performed using COMSOL Multiphysics, and the simulation results were analyzed for their significance under the IVPT situations similar to those encountered in gingiva and skin permeation studies.

Experimental

COMSOL model setup

COMSOL Multiphysics finite element analysis software (version 6.0) was used to create a series of 3D-diffusion models with a circular pathway (diffusional opening) and square-shape structure to mimic a membrane sandwiched between two chambers representing the donor and receptor chambers in a diffusion cell (Fig. 1a). The membrane was a composite of two homogeneous layers (a two-layer membrane) that each layer had its own properties. In two of the model testing studies, a single-layer homogeneous membrane was also used. Several anisotropic diffusion conditions where the lateral diffusion coefficient was larger than the transverse diffusion coefficient in the membranes were also examined. The Chemical Engineering module with diffusion of diluted species was used in COMSOL, which operated under the 2^nd Fick’s law of diffusion without convective transport:

$$\frac{\partial C_i}{\partial t}=D_i{\nabla }^2{C}_i$$ (1)

where ▽ is the spatial derivative operator, D_i is the diffusion coefficient, C_i is the concentration, and t is time. An assumption was that the partition coefficients between the membrane layers and between the membranes and donor and receptor solutions equal to unity. The diffusion coefficients in the donor and receptor chambers were set to a high value (1.0E-5 to 10 cm²/s) to mimic a well-mixed medium in the chambers. The well-mixed chambers effectively made the concentration at the donor/membrane interface the same as that at the top of the donor chamber and the membrane/receptor interface the same as the bottom of the receptor chamber. Therefore, the effect of the unstirred aqueous boundary layer near the surface of the membrane was assumed to be negligible. In addition to using a well-mixed solution in the donor and receptor chambers, diffusion coefficients (1.0E-8 to 2E-6 cm²/s) within the same order of magnitude as those of the membranes were used to examine the impact of the aqueous boundary layer on the edge effect. The concentration at the boundary on the top of the donor chamber was a constant value (91.7 mol/m³) to provide an infinite dose condition. The sink condition was used for the boundary at the bottom of the receptor chamber. Hence, the model mimicked the infinite dose condition. Except the surfaces at the top of the donor and the bottom of the receptor, all outside surfaces were impermeable boundaries. The concentrations in all objects were set as zero at the start of the simulation (time t = 0). The mesh size was extremely fine, and adaptive refinement function was used in the steady-state mode. Time-dependent and stationary (steady-state) transport simulations were generated using the software and a laptop computer (Dell Latitude E5540, Intel Core i7). It should be noted that the results of the simulations were likely affected by the assumptions in the models (e.g., homogeneous membrane layers, membrane partition coefficients, unstirred aqueous boundary layer, and infinite dose condition) and therefore, there were limitations in the model simulations (see Results and Discussion section).

Fig. 1.

(a) 3D-diffusion models of a membrane sandwiched between the donor and receptor chambers (circular objects at the top and bottom, respectively). The interfaces between the chambers and the membrane represent the diffusion cell opening and create a circular pathway from the donor to the receptor across the membrane. (b) From left to right: a single impermeable object in the donor chamber (on the top of the membrane) blocking the opening of the diffusion cell, and multiple impermeable objects forming arrays of 4, 9, and 16 objects (of smaller sizes, from left to right) of the same total area of membrane blocking, respectively.

Model parameters: edge effect

To evaluate the edge effect on membrane transport in the diffusion cell using COMSOL, different conditions were used: dimension of the diffusion cell opening, number of membrane layer, and membrane geometry, permeability, size, and thickness. The two layers in the membrane had different diffusion coefficients and thicknesses. These parameters are summarized in Table 1. The theoretical flux without edge effect was calculated by:

$$J=P_t{C}_d$$ (2)

$$\frac{1}{P_t}=\frac{1}{P_e}+\frac{1}{P_u}$$ (3)

where J is the flux, P_t is the total permeability of the membrane (or membrane composite), C_d is the donor concentration, P_e is the permeability of the top layer representing the epithelial layer in the membrane, and P_u is the permeability of the underlying tissue bottom layer. The permeability coefficients of these layers were calculated by:

$$P_e=\frac{K_e{D}_e}{h_e}$$ (4)

$$P_u=\frac{K_u{D}_u}{h_u}$$ (5)

where h_e and h_u are the thicknesses, K_e and K_u are the partition coefficients, and D_e and D_u are the diffusion coefficients of the top epithelial layer (subscript e) and the underlying tissue bottom layer (subscript u), respectively, and h is the total membrane thickness. K_e and K_u were assumed to be unity in the present study. The transport lag time (t_l) without edge effect under the studied conditions was calculated by ²⁶:

$$t_l={\left(\frac{h_e}{D_{\text{e}}}+\frac{h_u}{D_{\text{u}}}\right)}^{-1}\left[\frac{h_e{}^2}{D_{\text{e}}}\left(\frac{h_e}{6D_{\text{e}}}+\frac{h_u}{2D_{\text{u}}}\right)+\frac{h_u{}^2}{D_u}\left(\frac{h_u}{6D_{\text{u}}}+\frac{h_e}{2D_{\text{e}}}\right)\right]$$ (6)

Table 1:

Model parameters for evaluating the edge effect in the diffusion cell.

Parameter	Condition A	Condition B	Condition C
Membrane dimensions
- barrier top layer
width*length (cm*cm)	0.5*0.5	0.3*0.3	1.0*1.0
height (cm)	0.01, 0.03, and 0.05	0.01 and 0.03	0.01 and 0.03
- underlying tissue bottom layer
width*length (cm*cm)	0.5*0.5	0.3*0.3	1.0*1.0
height (cm)	0.05, 0.07, and 0.09	0.07 and 0.09	0.07 and 0.09
- total thickness (cm)	0.1	0.1	0.1
Thickness ratio (he/h) x100%	10%, 30%, and 50%	10% and 30%	10% and 30%
Permeability ratio (P t /P e ) x100%	50%, 70%, and 90%	50%, 70%, and 90%	50%, 70%, and 90%
Diameter of diffusion cell opening (cm)	0.2	0.2	0.5
Donor concentration (mol/m 3 )	91.7	91.7	91.7
Diffusion coefficient of barrier top layer (cm 2 /s)	1.00E-07, 7.20E-08, 5.58E-08, 6.00E-08, 4.32E-08, 3.35E-08, 2.00E-08, 1.43E-08, and 1.11E-08	6.00E-08, 4.32E-08, 3.35E-08, 2.00E-08, 1.43E-08, and 1.11E-08	6.00E-08, 4.32E-08, 3.35E-08, 2.00E-08, 1.43E-08, and 1.11E-08
Diffusion coefficient of underlying tissue bottom layer (cm 2 /s)	1.00E-07, 1.66E-07, 5.00E-07, 1.40E-07, 2.30E-07, 6.70E-07, 1.80E-07, 3.00E-07, and 9.10E-07	1.40E-07, 2.30E-07, 6.70E-07, 1.80E-07, 3.00E-07, and 9.10E-07	1.40E-07, 2.30E-07, 6.70E-07, 1.80E-07, 3.00E-07, and 9.10E-07

The theoretical flux and lag time without edge effect served as the control for comparison. The diffusion coefficients and thicknesses of the two layers were selected to provide the range of P_t/P_e ratios (expressed as percentage from 50% to 99%) and h_e/h ratios (expressed as percentage from 10% to 50%) as the key conditions to be examined in this study.

The dimensions of gingiva samples (length, width, and thickness) in the diffusion cell were different from those of skin ²⁵. The smaller size membrane and smaller diffusion cell opening compared to skin in the present model reflected the experimental conditions for the gingiva in the previous study, and although the epithelium of the gingiva was thicker than the stratum corneum and epidermis, the total thickness of the gingiva was thinner than that of full-thickness skin. Condition A was to mimic the dimensions of the diffusion cell encountered in the gingiva study ²⁵, Condition B was the same as Condition A except that the membrane was smaller (smaller area) than that in Condition A, and Condition C was to mimic the larger diffusion cells used in skin, nail, and eye studies ^9,27–29, which are commercially available ³⁰. A range of permeabilities and thicknesses of the epithelium and underlying tissue extended from those of the gingiva was also investigated. The edge effect on gingiva permeation in the previous modified diffusion cell setup was first evaluated (Condition A). The flux from the simulation of edge effect was compared with the theoretical flux without edge effect (control), and the ratio of the flux (J ratio) with edge effect to that without was calculated. After the simulations using the parameters for the gingiva, the diffusion cell setup representing the experimental conditions similar to those for the skin was evaluated (Condition C). In this setting, the diffusion cell had the smallest diffusional opening (0.2 cm²) available among the commercial diffusion cells (e.g., PermeGear, Inc.) ³⁰.

Model parameters: blocking on membrane

To evaluate the effect of an object, such as an air bubble, blocking the membrane (“membrane blocking” effect), an impermeable circular object of varying sizes blocking 20%, 50%, 70%, and 90% opening area of the membrane was created in the COMSOL model. The object was placed on the surface at the center of the membrane either in the donor (top of the membrane) or receptor chamber (bottom of the membrane). The diffusion of molecule at the interface between the object and the surrounding medium was assumed to be equal to that in the medium. In addition to using a single object, multiple circular objects (4, 9, and 16 objects) were also used. When multiple objects were used, the total area blocked by the objects was set at 50% opening area. The effect of a single object blocking 50% area was compared to that of multiple objects at the same total blocking area (i.e., dividing the single object into 4, 9, and 16 objects of smaller sizes, Fig. 1b). When the number of objects increased, the total circumference of these objects increased, and an increase in edge effect due to lateral diffusion in the membrane was expected. The parameters of these model simulations are summarized in Table 2. Similar to the dimensions in the edge effect study (“Model parameters: edge effect” section), Condition A was to mimic membrane blocking in the previous diffusion cell experiments with gingiva ²⁵, and Condition C was to mimic the situation with commercial diffusion cells such as those in skin studies. The flux from the simulation with membrane blocking was compared with the theoretical flux without blocking (control). The ratio of the flux with blocking to that without blocking was calculated to examine the impact of membrane blocking under these conditions.

Table 2:

Model parameters for evaluating the effect of membrane blocking (membrane blocked by an object or objects) in the diffusion cell.

Parameter	Condition A	Condition C
Membrane dimensions
- barrier top layer
width*length (cm*cm)	0.5*0.5	1.0*1.0
height (cm)	0.01, 0.03, and 0.05	0.01 and 0.03
- underlying tissue bottom layer
width*length (cm*cm)	0.5*0.5	1.0*1.0
height (cm)	0.05, 0.07, and 0.09	0.0233, 0.04, and 0.09
- total thickness (cm)	0.1	0.0333, 0.05, and 0.1
Thickness ratio (h e /h) x100%	10%, 30%, and 50%	10% and 30%
Permeability ratio (P t /P e ) x100%	90% and 95%	90%, 95%, and 99%
Diameter of diffusion cell opening (cm)	0.2	0.5 and 0.9
% area of opening of membrane blocked by the object	20%, 50%, 70%, and 90%	20%, 50%, 70%, and 90%
Thickness of object (cm)	0.025	0.025
Donor concentration (mol/m 3 )	91.7	91.7
Multiple object conditions (including single object condition)
- arrangement	-	array of 1*1, 2*2, 3*3, and 4*4
- number	-	1, 4, 9, and 16
- total area of all objects (cm2)	-	0.3181
- area of each object (cm2)	-	0.3181, 0.0795, 0.0353, and 0.0199
- radius of each object (cm)	-	0.3181, 0.1591, 0.1061, and 0.0796
- distance between each object (cm)	-	0.01
- circumference of each object (cm)	-	2.00, 1.00, 0.67, and 0.50
- total circumference of all objects (cm)	-	2.00, 4.00, 6.00, and 8.00

Model testing

COMSOL model simulations were performed to check the quality of the simulations in the computer program by changing the mesh size from extremely fine mesh to extra fine mesh and changing the software function from adaptive refinement mode to non-adaptive refinement mode. The results from these simulations were compared. Simulations were also performed using different membrane configurations and parameters to evaluate the model settings in COMSOL. One of these tests was the simulation without edge effect, in which the size of the membrane was equal to the size of the diffusion cell opening, i.e., with no membrane area beyond the diffusion cell opening to represent the membrane section sandwiched between the diffusion cell chambers. The flux and lag time results of this simulation were compared to the values calculated from transport theory. In another test, because it was expected that edge effect was independent of the total permeability coefficient of the membrane, simulations were performed with membranes of different total permeability coefficients while fixing the other parameters. Particularly, the diffusion coefficients of the two layers in the membrane were modified by the same multiplers while maintaining the other parameters (h_e/h, P_t/P_e, and membrane/diffusion cell dimensions) constant to examine the dependency of edge effect on the total permeability coefficient. In addition, to evaluate whether the donor and receptor chambers could behave as a well-mixed medium in the model simulations, the diffusion coefficient of the donor and receptor solution was increased while fixing the other parameters, and the results were compared before and after the modification. The diffusion coefficient of the donor and receptor solution was also decreased to within the same order of magnitude as those of the membranes to examine the effect of the unstirred aqueous boundary layer. Furthermore, the two-layer membrane geometry setup was tested by comparing the simulation results of a single-layer membrane (single-layer membrane geometry) with those of a two-layer membrane when the two layers had the same diffusion coefficient, making the two-layer composite essentially a single uniform membrane. These two geometry settings should provide the same results. The present study also compared the results between the time-dependent and steady-state modes in COMSOL. Significant differences between the flux obtained in the steady-state mode simulation and that in the time-dependent mode simulation when it approached steady state would indicate discrepancy and possible errors in the computer simulations.

The analytical solution to the mathematical model that describes the edge effect of a uniform single-layer circular membrane and the resulting equations have been discussed previously ²³. In the present study, simulations were performed to compare the edge effect results from the COMSOL model and those from the equations derived from the previous mathematical model. The following conditions were evaluated in this comparison: membrane thickness divided by the radius of diffusion cell opening = 0.2, 0.4, and 0.8 and membrane radius divided by the radius of diffusion cell opening = 1.05, 1.1, 1.15, 1.2, and 1.25. Although assumptions were made in the derivation of the equations ^23,24 and only the single-layer circular membrane conditions could be examined, similar results between the simulations and derived equations would support the validity of the edge effect analyses using COMSOL.

Results and Discussion

Flux increase due to edge effect

For a homogeneous membrane of 0.1 cm thick in a diffusion cell of 0.2 cm diameter opening similar to the diffusion cell setup in the previous gingiva study (Condition A), the edge effect would result in approximately 35% increase in flux compared to that without the edge effect (theoretical flux of membrane permeation without lateral diffusion into the membrane beyond the diffusion cell opening). In the analyses of this study, flux ratio (J ratio) was defined as the flux with edge effect divided by that without the effect to evaluate its influence. For a membrane consisting of two layers, the top layer representing the epithelium and bottom layer the underlying connective tissue, the edge effect decreased when the permeability and thickness of the top layer decreased. Fig. 2 shows the relationship between the flux ratio (i.e., the increase in flux due to the edge effect from the control) and the ratio of the permeability coefficients and thicknesses of the two layers in the membrane composite. By maintaining the total permeability of the membrane and its total thickness constant, increasing the ratio of P_t/P_e (i.e., a decrease in the permeability of the top layer) would result in the decrease of edge effect. Decreasing the ratio of h_e/h (i.e., a decrease in the thickness of the top layer) at the same P_t/P_e would also result in the decrease of edge effect. The ratio of P_t/P_e indicates the contribution of the top layer to the total barrier of the two-layer membrane. The h_e/h indicates the thickness of the top layer relative to the total membrane thickness. When the top layer became the rate-limiting barrier (e.g., P_t/P_e = 90%), an effective and thinner barrier was created by the top layer and the edge effect was reduced. In general, the edge effect is related to the cross-sectional area of the rate-limiting barrier available for lateral diffusion in the membrane (i.e., the thickness of this barrier). When the relative thickness of the rate-limiting barrier (the top layer in this case) decreases, the flux ratio and edge effect decrease.

Fig. 2.

(a) Relationships between flux ratio (J ratio), P_t/P_e ratio, and h_e/h ratio under Condition A (closed symbols) and Condition C (open symbols). The J ratio is a function of both P_t/P_e and h_e/h under the conditions studied. The dash-dot line indicates the condition of no edge effect (control, flux ratio = 1.0). (b) Representative images of steady-state concentration profiles in the membrane sandwiched between the donor and receptor chambers in cross section (left) and horizontal plane (right) views. (c) Representative image of flux lines across the membrane. The conditions of Figs. 2b and 2c were h_e/h = 50% and P_t/P_e = 70% in the COMSOL simulations.

Edge effect under conventional configuration

In addition to the conditions encountered in gingiva permeation studies (Condition A), the conditions that could be encountered in commercial diffusion cell studies (Condition C) such as those with skin were evaluated by modifying the size of the membrane and diameter of the diffusion cell opening. The results of these simulations are also presented in Fig. 2. When the diameter of the diffusion cell opening and the size of the membrane increased without changing the membrane thickness, the flux ratios were smaller than those under Condition A. The edge effect was significantly smaller due to the larger opening of the diffusion cell and the decrease in the contribution of lateral transport to transmembrane transport (i.e., smaller edge effect compared to Condition A). Despite the smaller edge effect, similar trends of decreasing flux ratio with increasing P_t/P_e ratio and decreasing h_e/h ratio were observed.

Edge effect of square vs. circular membranes

Possible effects of membrane geometry were investigated by changing the geometry of the membrane from square to circular. When maintaining all other parameters constant and changing only the geometry of the membrane to circular, the edge effects from the square and circular membranes were not significantly different under the conditions studied (less than 0.2% difference in the flux ratio). For example, the flux ratios were 1.186 and 1.184 for the square and circular membranes at the same thickness (h_e/h = 30%) and permeability ratios (P_t/P_e = 90%), respectively. Decreasing the P_t/P_e ratio to 50%, the flux ratios were 1.279 and 1.280 for the square and circular membranes, respectively. The lack of impact of membrane geometry between square and circular membranes was likely due to the long lateral distance to the location of the geometry difference (square vs. circle) from the diffusion cell opening. The contribution of lateral diffusion of the “corners” of the square membrane to membrane transport was minimal as it was limited by the distance even though the area of the square membrane was larger than that of the circular membrane.

Increase in lag time due to edge effect

In addition to the impact on flux, the edge effect also affects the transport lag time in diffusion cell studies. Fig. 3a shows the cumulative amount vs. time profiles from the time-dependent simulations. The lag times with edge effect in the figure were significantly longer than that without the edge effect. Due to the nature of edge effect, the cumulative amount vs. time profile did not follow the conventional shape for the comparison using the common definition of lag time (the extrapolation of the linear region in the cumulative amount vs. time profile to the x-axis). Particularly, longer time was needed to reach steady state after the initial delay. In conventional cumulative amount vs. time profiles, according to transport theory (from 2^nd Fick’s law), it normally takes approximately 2.2x lag time for the flux to reach 95% of its steady-state value. Under the influence of edge effect, longer time was needed for the flux to reach the 95% steady-state value when comparing the lag time calculated from the extrapolation to the x-axis and the time to reach 95% steady state in Figs. 3a and 3b. Fig. 3b shows the results of the time-dependent simulations and the flux ratios from the stationary (steady-state) transport simulations. As time increased, the flux ratio increased in the time-dependent model (symbols) and approached the steady-state value (dash lines). The time for the flux to reach 95% steady state value with edge effect in the figure was significantly longer than that without the edge effect. In the model simulation of membrane transport without edge effect (i.e., when the size of the membrane = diffusion cell opening, without any membrane area sandwiched between the diffusion cell chambers), the flux reached 95% steady-state value at ~10 h (data not shown). This value is consistent with the theoretical lag time (~4.6 hours, Eq. 6) as it is 2.2x lag time. For comparison, the flux in the time-dependent model of P_t/P_e = 50% and h_e/h = 30% reached its 95% steady-state value at ~140 h, which was ~6x lag time. For the conditions of P_t/P_e = 70% and 90%, the differences between the time to reach 95% steady state and the lag times were ~6x and 4x, respectively. In general, when the P_t/P_e ratio increased, corresponding to smaller edge effect, shorter lag time was observed. The increase in lag time due to edge effect can be attributed to the longer diffusion pathway (lateral diffusion) into the surrounding membrane beyond the opening of the diffusion cell to establish a concentration gradient there before reaching steady state. Interestingly, in permeation studies of normal duration designed without considering the increase in lag time due to edge effect, the lower apparent flux due to the long lag time (see Fig. 3b) can partly negate the flux increase from edge effect; the flux ratio is closer to unity (i.e., without edge effect, flux ratio = 1.0) after the flux progresses past the initial delay (initial lag time) and before it slowly reaches steady state. Therefore, in diffusion cell studies of significant edge effect and long lag time, short duration permeation experiments before attaining steady state could mitigate the errors caused by the edge effect.

Fig. 3.

(a) Cumulative amount vs. time profiles from the time-dependent model and (b) flux ratio at different time points from the time-dependent model (circles, squares, and triangles) and steady state model (dash lines) of three different conditions (purple, red, and blue; P_t/P_e = 50%, 70%, and 90% and h_e/h = 30% under Condition A, respectively). The condition of the simulation without edge effect (open circles) was P_t/P_e = 50 and h_e/h = 30%. The transport lag times calculated using the linear regression lines and equations in Fig. 3a were 22, 19, and 9.4 h for P_t/P_e = 50%, 70%, and 90%, respectively (purple, red, and blue) and that for the simulation without edge effect was 4.6 h (black). The linear regression lines in the figure were generated using simulation data between ~140 h to 200 h. The arrows in Fig. 3b indicate the approximate time to reach 95% steady-state values (purple, red, and blue for P_t/P_e = 50%, 70%, and 90%, respectively, and black arrow for the simulation without edge effect).

Decrease in flux from membrane blocking

Lateral diffusion in a membrane can reduce the effect of an object blocking the membrane. The effect of an object (or multiple objects) blocking the membrane in a diffusion cell (“membrane blocking” effect) was first evaluated using the parameters of Condition A (see “Flux increase due to edge effect” section). Fig. 4 presents the relationships between the % decrease in flux and the % area of the membrane opening blocked by the object in the donor and receptor chambers. The % decrease in flux was calculated by: (1- flux ratio) × 100%, and the flux ratio was calculated by: (flux with membrane blocking) / (flux without blocking). As expected, when the area of the membrane blocked by the object increased, a higher % decrease in flux was observed, and the extent of the flux decrease due to membrane blocking was related to the permeabilities and thicknesses of the membrane layers and the size of the diffusion cell opening. When the membrane was blocked by the object at the bottom (i.e., at the underlying tissue bottom layer), the % decrease in flux deviated significantly from the % area blocked by the object. For example, the flux decreased by 4%, 13%, and 28% when the % area blocked by the object was 20%, 50%, and 90%, respectively. When the membrane was blocked by the object on the top (i.e., on the major barrier top layer), the % decrease in flux was more significant, and this effect was a function of h_e/h. When the top layer was thinner (decreasing ratio of h_e/h), corresponding to a decrease in edge effect (see “Flux increase due to edge effect” section), the % flux decrease approached the % area blocked by the object on the top of the membrane. In contrast, for the object at the bottom, the impact of h_e/h on the % decrease in flux was small. In addition, there was no significant difference between the results (less than 5% difference) when P_t/P_e was 90% and 95%. For instance, with 70% membrane blocking, the flux ratios were 0.796 and 0.807 when the object was at the bottom and 0.637 and 0.627 when the object was on the top for P_t/P_e of 90% and 95%, respectively.

Fig. 4.

(a) Relationship between % decrease in flux and % membrane area of diffusion cell opening blocked by an impermeable object at the top and bottom of the membrane (open and closed symbols, respectively) in the diffusion cell under Condition A. Model parameters: h_e/h = 10%, 30%, and 50% (triangles, squares, and circles, respectively) and P_t/P_e = 90%. The dash-dot line indicates the proportional relationship between % flux decrease and % blocked membrane area. (b) Representative images of steady-state concentration profiles in the membrane in the diffusion cell when the object is at the top in the donor chamber (left) and at the bottom in the receptor chamber (right). The condition was h_e/h = 30%, P_t/P_e = 90%, and 50% area of the opening blocked by the object in the COMSOL simulations. In the right image, the concentration color scale was adjusted to highlight the concentration profiles near the object.

Membrane blocking under conventional configuration

To investigate the effect of membrane blocking under more conventional configurations such as those of commercial diffusion cells with skin, model simulations were performed using the parameters of Condition C. Fig. 5 presents the results of membrane blocking when the membrane size and diffusion cell opening are larger than those in Fig. 4 (Condition C vs. Condition A). Under this condition, when the % area of the opening blocked by the object at the bottom of the membrane increased from 20% to 90%, the % flux decrease increased from ~6% to 44% at h_e/h = 10% and from ~9% to 62% at h_e/h = 20%. These % decrease values were larger than those under Condition A. In addition, the % decrease in flux increased when h_e/h increased (i.e., when the relative thickness of the underlying tissue of the membrane decreased). This indicates that the thickness of the underlying tissue is related to the ability of the permeant to diffuse laterally in the underlying tissue when the bottom is blocked by the object. For an object blocking the top of the membrane, more significant % decrease in flux compared to that of the object at the bottom of the membrane was observed. Under this condition, the % decrease in flux almost overlapped with the % area blocked by the object on the top of the membrane and was relatively independent of the ratio h_e/h. Together, the results suggest that when there is an object blocking the bottom of the membrane in the diffusion cell in practice, it might not significantly affect the experiments whereas the effect of membrane blocking at the top of the membrane would likely be significant.

Fig. 5.

Relationship between % decrease in flux and % membrane area of diffusion cell opening blocked by an impermeable object at the top and bottom of the membrane (open and closed symbols, respectively) in the diffusion cell under Condition C. Model parameters: P_t/P_e = 90%, h_e/h = 10% and diffusion cell opening diameter = 0.5 cm (diamonds) and 0.9 cm (triangles) and h_e/h = 20% and diffusion cell opening diameter = 0.9 cm (squares). Total membrane thickness = 0.05 cm and 0.1 cm for h_e/h = 20% and 10% conditions, respectively. The dash-dot line indicates the proportional relationship between % flux decrease and % blocked membrane area.

Fig. 5 also shows the effect of the diffusion cell opening on the % flux decrease (opening diameter of 0.5 cm vs. 0.9 cm under Condition C). The results can also be compared with those under Condition A (opening diameter of 0.2 cm). In general, decreasing the size of the diffusion cell opening relative to the thickness of the membrane would lead to a smaller % decrease in flux (% decrease in flux would diverge further from the % area blocked). Fig. 6 shows the relationships between the % decrease in flux and % area blocked at P_t/P_e = 90% to 99%, i.e., when the relatively permeability of the top layer in the membrane decreased from the P_t/P_e = 90% condition to P_t/P_e = 99% condition, with the thinner membrane and large diffusion cell opening under Condition C. For an object blocking the bottom of the membrane, the % decrease in flux was smaller when the top layer became a more dominant barrier (i.e., higher P_t/P_e ratio). A higher permeability underlying tissue and unhindered lateral diffusion would reduce the blocking effect of the object at the bottom. For an object blocking the top of the membrane, the effect of increasing P_t/P_e ratio on the flux was small. These results are consistent with lateral diffusion within the membrane that reduces the impact of an object blocking the surface of the membrane.

Fig. 6.

Relationship between % decrease in flux and % membrane area of diffusion cell opening blocked by an impermeable object at the top and bottom of the membrane (open and closed symbols, respectively) in the diffusion cell under Condition C. Model parameters: P_t/P_e = 90%, 95%, and 99% (triangles, squares, and circles, respectively), h_e/h = 30%, total membrane thickness = 0.0333 cm, and diffusion cell opening diameter = 0.9 cm. The dash-dot line indicates the proportional relationship between % flux decrease and % blocked membrane area.

Flux decrease due to multiple objects blocking the membrane

As the impact of membrane blocking was related to membrane lateral diffusion near the edge of the object, it was hypothesized that the % flux decrease could be affected by the length of the object edge relative to its area. Model simulations were performed with multiple objects having the same total membrane area blocked by the objects, effectively increasing the total length of the edges of these objects (total circumference). With multiple objects providing the same 50% blocked area, a smaller % flux decrease was observed compared to those of single object blocking, albeit that the effect was small, when the total length of the object edges increased. The changes in % flux decrease due to increasing the number of objects on the top of the membrane were larger than those at the bottom of the membrane. The % decrease in flux for the objects on the top was 47.0%, 44.1%, 41.5%, and 39.2% when the numbers of objects were 1, 4, 9, and 16, respectively, whereas the % decrease in flux for the objects at the bottom was from 17.4% to 15.5% under the same conditions. The effect reached a plateau when the number of the objects increased from 9 to 16, possibly because the membrane areas near the object edges available for lateral diffusion began to overlap.

Model testing

The first test was performed to evaluate the impact of the mesh in the model simulations. In the steady-state models, the parameters such as the sizes, thicknesses, and permeabilities of the membrane layers and other functions in the software program were the same except that the mesh sizes were set at extremely fine and extra fine for comparison. In the evaluation, the differences in the flux results from the two mesh sizes were less than 5%. For example, the flux ratios of extremely fine and extra fine mesh sizes were 1.069 and 1.093, respectively, when h_e/h = 30% and P_t/P_e = 90% and 1.104 and 1.127, respectively, when h_e/h = 30% and P_t/P_e = 50% under Condition C. The small differences in the simulation results between the two mesh sizes suggest satisfactory meshing in the finite element simulations under the conditions studied.

In the second test, simulation results were generated with two different software functions, adaptive and non-adaptive refinement functions, while keeping the other model parameters the same. The flux results from these functions were essentially the same with less than 3% difference. For instance, the flux ratios of adaptive refinement and non-adaptive refinement functions were 1.086 and 1.096, respectively, when h_e/h = 30% and P_t/P_e = 70% under Condition C. The small differences in the simulation results from the adaptive refinement method indicated that the simulation data were not significantly affected by this function in COMSOL, further suggesting satisfactory meshing in the finite element simulations in the present study.

In the simulations without edge effect, the flux and lag time results were essentially the same (<0.1%) as those from Eqs. 2–6. This supports the validity of the COMSOL model simulations. For the next test, according to transport theory, edge effect (i.e., flux ratio results) is a function of h_e/h and P_t/P_e ratios and the ratio of membrane thickness to the diameter of the diffusion cell opening. The edge effect is independent of the total permeability of the membrane composite. By decreasing the diffusion coefficients of the two layers in the membrane by the same multiplier (e.g., decreasing the total permeability coefficient of the membrane to the range that can be encountered in skin permeation studies) without changing the other parameters, essentially the same flux ratio was observed (< 0.5% difference). Changing the total permeability coefficient of the membrane without changing the other parameters did not affect the edge effect. In another test, increasing the diffusion coefficient of the donor and receptor solution in the model did not affect the flux ratio under the conditions studied. This suggests that the donor and receptor chambers essentially acted as a well-mixed medium in the model simulations, consistent with the study design. To examine the influence of the unstirred aqueous boundary layer on edge effect, diffusion coefficients within the same order of magnitude as those of the membranes were used (i.e., the donor and receptor chambers were no longer a well-mixed medium). When the diffusion coefficient of the donor and receptor solution decreased from 2E-6 to 1.0E-8 cm²/s, the edge effect decreased; the flux ratios comparing the simulation data with and without edge effect were 1.279, 1.253, 1.110, and 0.991 at diffusion coefficients of 2E-6, 5E-7, 1.0E-7, and 1.0E-8 cm²/s, respectively. The influence can be attributed to the increase in the transport resistance of the aqueous boundary layer relative to the membrane. When the aqueous boundary layer became the dominant barrier, the edge effect became insignificant because the aqueous boundary layer (in the donor and receptor chambers) had the same cross-sectional area as the diffusional area of the diffusion cell. In other words, the aqueous boundary layer did not have any “edge” extending beyond the opening of the diffusion cell and hence did not have any edge effect.

In the comparison of the results from the simulations using the single-layer membrane geometry and two-layer membrane geometry of same diffusion coefficient in the two layers, the flux ratios were essentially the same (< 0.5% difference). For example, the flux ratio was 1.353 for the single-layer geometry and 1.350 for the two-layer geometry, respectively, when h_e/h = 50% and P_t/P_e = 50% under Condition A. The results did not indicate any errors with the geometry setup in the model simulations.

To compare the results between the model simulations using COMSOL and the previously derived equations for edge effect ²³, simulations of a uniform single-layer circular membrane were performed. The edge effect from these equations is a function of the parameters: membrane thickness (h), radius of diffusion cell opening (r), and radius of the circular membrane (r_m). The results from the COMSOL simulations overlapped with those from the equations when h/r = 0.2. When the membrane was thicker at h/r = 0.4, the edge effect from the simulations began to deviate from those of the equations (~2%). At h/r = 0.8, the results showed ~4–6% difference (e.g., flux ratios of 1.28 and 1.226 from the equations and simulation, respectively, under the condition of h/r = 0.8 and (r_m − r)/r = 0.25). This comparison supports the validity of the results obtained from the COMSOL model simulations.

Practical consideration of edge effect and membrane blocking in diffusion cell

Diffusion cells are commonly used in permeation studies to determine the permeability of biological and synthetic membranes for drug delivery. There are different types of commercial diffusion cells and custom-made apparatus in the industrial and academic settings. For example, the size of the diffusion cell opening (aperture) for drug transport can vary, ranging from the commercially available vertical diffusion cells of 0.5 cm diameter (0.2 cm²) to the often used 0.9 cm diameter (0.64 cm²) and 1.5 cm diameter (1.77 cm²) ³⁰ and Ussing chambers of 0.3 cm diameter (0.07 cm²) and 0.4 cm diameter (0.12 cm²) to 0.9 cm diameter (0.64 cm²) ^31,32. Custom-made diffusion cells and Ussing chambers of different diffusion cell openings (e.g., 0.3 cm and 0.55 cm diameter) and other systems have also been used ^33,34. Tissues that are commonly examined in permeation studies include skin, nail, cornea, and buccal mucosa ^14,20,33. The total thicknesses of these tissues vary: skin (~0.05 to 0.2 cm) ³⁵, nail (~0.05 to 0.16 cm) ³⁶, cornea (~0.05 cm) ^37,38, gingiva (~0.11 cm) ²⁵, and buccal mucosa (~0.05 to 0.08 cm) ³⁹. Most of these tissues consist of multiple cell layers with the thinner major barrier at the top (apical side) of the membrane. For example, the epithelium and underlying tissues of gingiva are ~0.4 and ~0.6 mm thick, respectively ⁴⁰. For skin, the epidermis and dermis are 0.03–0.07 mm and 0.5–2 mm thick, respectively ³⁵. The dimensions and parameters in the present model simulations were selected to mimic the diffusion cell opening and membrane thickness in the previous gingiva study ²⁵ and the larger diffusion cell openings with thinner tissues (or smaller h/r parameters) encountered in practice.

In conventional diffusion cell settings with a thin membrane and large diffusion cell opening (e.g., 0.1 cm membrane thickness and 0.9 cm diameter opening), edge effect is small. This condition is commonly encountered in skin permeation experiments. However, with relatively small openings of some commercial and custom-made diffusion cells (e.g., 0.2–0.3 cm diameter opening), edge effect might not be negligible and should be considered. An example is the previous gingiva permeation study ²⁵. Comparing the magnitudes of edge effect on flux and transport lag time, the edge effect on lag time is larger than that on transmembrane flux. For example, at P_t/P_e = 50% and h_e/h = 30% under Condition A, edge effect increased the flux by 28% and the lag time by more than two-fold (e.g., Figs. 2 and 3). However, the errors introduced by the longer than normal lag time (due to edge effect) in transport experiments are likely not critical in membrane permeability measurements; the longer lag time is attributed to the diffusion of the permeant into the “extra” membrane section clamped between the diffusion cell chambers beyond the diffusion opening. It should be noted that, for diffusion cell studies in practice, the clamped tissues beyond the diffusion cell opening are likely deformed due to the pressure created by clamping the tissues between the diffusion cell chambers, which can limit lateral diffusion into the clamped tissues. Tissue drying at the edge sandwiched between the chambers can also reduce lateral diffusion into the tissues beyond the diffusion cell opening. The present model simulations did not take into account these two factors. In addition, there are limitations in the COMSOL model due to the assumptions made in its derivation, which are discussed at the end of this section.

In diffusion cell studies, an object can be accidentally introduced into the donor or receptor chamber and “stuck” on the membrane blocking permeant transport. An example is an air bubble blocking the membrane at the membrane/receptor interface or air trapped between the test formulation and the membrane in the donor chamber. In the receptor chamber, air bubbles can form when air is released from the receptor medium or the bubbles can be introduced into the chamber during sampling. In the donor chamber, air can be trapped between the formulation and the membrane surface such as in the situations when (a) an air bubble is formed in a semisolid formulation and stuck on the membrane surface during dosing, (b) an air bubble is trapped between the adhesive of a transdermal delivery system and the membrane surface during application, and (c) the adhesive of a transdermal delivery system comes off from the membrane surface after application resulting in an air gap between the adhesive and membrane surface. Based on the analyses for the conditions in the present study, the % flux error from membrane blocking is smaller than the % area blocked by the object. For example, when a membrane with the top layer as the major transport barrier (P_t/P_e ≥ 90%) and relatively thick underlying tissue layer (h_e/h ≤ 30%) was blocked by an air bubble in the receptor chamber, the % flux decrease was ≤ 5% when 20% of the area was blocked by the object under Condition A. Increasing the blocked area to 50% would only decrease the flux by 17%. Membrane blocking in the receptor chamber generally has a small effect on transmembrane flux in diffusion cell studies when there is a relatively thick underlying tissue. In contrast, membrane blocking in the donor chamber generally has a significant impact on transmembrane flux unless the top layer is not a dominant barrier or is relatively thick (e.g., P_t/P_e ≤ 90% and h_e/h ≥ 50%).

The model assumed that the two layers of the membrane were homogeneous, which was a limitation in the present study. This assumption implied isotropic diffusion. Differences between lateral and transverse (transmembrane) diffusion can affect the edge effect. While the isotropic diffusion assumption may be correct for some biological membranes such as the sclera, which is mainly composed of collagen ⁴¹, biological membranes such as the stratum corneum have different transmembrane and lateral diffusion coefficients ^42–44. When the lateral diffusion coefficient is larger than the transmembrane diffusion coefficient, a larger edge effect (larger flux increase) is expected. Table 3 presents the results of model simulations with lateral diffusion coefficients (D_x and D_y) that were larger than transmembrane diffusion coefficient (D_z) as examples to illustrate the effects of anisotropic diffusion. When the lateral diffusion coefficient was 5x larger than the transmembrane diffusion coefficient, the flux ratios increased from 1.025 to 1.091, 1.184 to 1.458, and 1.342 to 1.829 under the conditions of h_e/h = 10%, 30%, and 50% and P_t/ P_e = 90%, respectively. The higher lateral diffusion rate (relative to transmembrane diffusion) resulted in significant larger edge effects. When the lateral diffusion coefficients increased to 10x, 30x, and 50x larger than transmembrane diffusion, the flux ratios increased to 1.146, 1.292, and 1.416, respectively, under the condition of P_t/ P_e = 90% and h_e/h = 10%.

Table 3:

Comparison of edge effects (J ratio with edge effect to without) between isotropic and anisotropic diffusion in the membranes under steady state. Under isotropic diffusion, lateral diffusion in the x- and y-directions was the same as transverse (transmembrane) diffusion in the z-direction (D_x = D_y = D_z). Under anisotropic diffusion, lateral diffusion in the x- and y-directions was different from that in the z-direction (D_x = D_y ≠ D_z).

Isotropic/anisotropic condition	Parameters a	J ratio	% increase b	Ratio of % increase (anisotropic to isotropic) c
Dx = Dy = Dz	he/h = 10%	1.025	2.5	- d
Dx = Dy = 5*Dz	he/h = 10%	1.091	9.1	3.6
Dx = Dy = 10*Dz	he/h = 10%	1.146	15	6
Dx = Dy = 30*Dz	he/h = 10%	1.292	29	12
Dx = Dy = 50*Dz	he/h = 10%	1.416	42	17
Dx = Dy = Dz	he/h = 30%	1.184	18	- d
Dx = Dy = 5*Dz	he/h = 30%	1.458	46	2.6
Dx = Dy = Dz	he/h = 50%	1.342	34	- d
Dx = Dy = 5*Dz	he/h = 50%	1.829	83	2.4

^aAll simulations were performed under Condition A and P_t/P_e = 90%.

^bDefined as the % increase in flux relative to that without edge effect: % increase = (J ratio – 1) × 100%.

^cRatio of % increase = (% increase in flux due to edge effect with anisotropic diffusion) / (% increase in flux due to edge effect with isotropic diffusion).

^dNot applicable.

The model also assumed that membrane partition coefficients were equal to unity. Although this assumption was different from the conditions encountered in most biological membranes, the effect of membrane partitioning on the edge effect was likely not significant. For example, permeant partitioning would increase the concentration profile in the membrane as a whole by a constant factor. The relative magnitude of lateral transport compared to that of transmembrane transport would remain relatively the same with this increase in the concentration profile. As membrane partitioning was not the main focus in this study, it was not evaluated as a model parameter.

Conclusion

The present study evaluated the effects of lateral diffusion within a two-layer membrane that could impact the flux and lag time results of the membrane in diffusion cell studies. Two conditions of lateral transport influences that could be encountered in membrane transport studies of diffusion cells were investigated: (a) edge effect--lateral diffusion from the diffusion cell opening into the surrounding clamped region of the membrane contributing to membrane transport from the donor to the receptor in the diffusion cell and (b) membrane blocking--lateral diffusion within the membrane around the blocked area when the membrane is partially blocked by an impermeable object in either the donor or receptor chamber. COMSOL model simulations were performed in these investigations under the assumptions of homogeneous membrane layers, negligible unstirred aqueous boundary layer, transport under infinite dose, and membrane partition coefficient equal to unity. For the membrane and diffusion cell dimensions in the previous gingiva permeation study, edge effect could increase the flux across the membrane by up to 35%, and for the diffusion cell dimensions in conventional permeation studies with skin, edge effect was insignificant (< 1%). In general, edge effect decreased when the relative thickness and permeability of the major barrier (top layer in the two-layer membrane) decreased. In addition to increasing the flux, edge effect could significantly increase the transport lag time. In diffusion cell studies of significant edge effect and long lag time, the lower flux before reaching steady state due to the lag time in typical permeation experiments could reduce the transmembrane flux error caused by the edge effect. For membrane blocking, when a membrane was partially blocked by an impermeable object in the diffusion cell, the decrease in flux might not be proportional to the decrease in the area of diffusion cell opening caused by the object. Lateral diffusion within the membrane could reduce the impact of membrane blocking. The effect of lateral diffusion on membrane blocking was related to the relative thicknesses of the two layers in the membrane and their permeabilities. When the relative thickness and permeability of the barrier in contact with the object decreased, the influence of lateral transport on transmembrane flux decreased, and the % flux decrease from blocking would approach the % area blocked by the object. Although edge effect is small in conventional diffusion cell settings with thin membranes and large diffusion cell openings, caution must be exercised in the studies using a thick membrane barrier and custom-made diffusion cells with small diffusion cell openings when edge effect may not be negligible. For membrane blocking, the errors due to the accidental introduction of an impermeable object (e.g., air bubble) on the surface of the membrane during permeation experiments can be less significant than anticipated, particularly when the object is in the receptor, because lateral transport within the membrane around the blocked area can mitigate its impact.