A membrane serves as a selective barrier between different phases, such as inside and outside a cell or between two compartments within a cell. Material can cross this membrane through various mechanisms:

###### 1. Passive Transport:

This involves the movement of substances across the membrane without the input of energy. Diffusion, osmosis, and facilitated diffusion are examples of passive transport.

###### 2. Active Transport:

Active transport, on the other hand, requires energy to move substances against their concentration gradient, specifically from an area of lower concentration to an area of higher concentration. Moreover, this process is typically facilitated by protein pumps embedded within the membrane.

###### 3. Facilitated Transport:

Facilitated transport involves the movement of substances across the membrane with the help of specific carrier proteins, but it does not require energy input. This process helps substances move down their concentration gradient more quickly than simple diffusion.

# Passive Diffusion Through Membrane

Passive diffusion is a fundamental process wherein molecules move across a membrane from areas of high concentration to areas of low concentration, solely driven by the inherent kinetic energy of the molecules. Moreover, this process is vital for various physiological functions, including the exchange of gases in the lungs and the uptake of nutrients by cells. Unlike active transport, passive diffusion does not require energy expenditure by the cell. Instead, it relies on the natural tendency of molecules to move down their concentration gradient until equilibrium is reached. Various factors, such as membrane permeability, molecular size, and solubility, influence the rate of passive diffusion.

In Figure 1, depicting a membrane dividing two distinct solutions, under steady-state conditions, we can infer from Fick’s first law that

$$ J=-D\frac{dc}{dx}=-D\left(\frac{c_2-c_1}h\right)=\frac Dh(c_1-c_2) $$

The expression h/D is commonly referred to as the diffusional resistance, represented by R. Consequently, the flux equation can be expressed as:

$$ J=\frac{c_1-c_2}R $$

While resistance to diffusion is a core concept in science, permeability is a term frequently utilized in pharmaceutical sciences.Resistance and permeability exhibit an inverse relationship; consequently, as the resistance to diffusion increases, the permeability of the diffusing substance decreases.

## Permeability

When a membrane divides the two sections of a diffusion cell with a cross-sectional area S and thickness h, and if the concentrations within the membrane on the left (donor) and right (receptor) sides are C1 and C2 respectively (as shown in Figure 2.), Fick’s first law can be expressed as:

$$ J=\frac1S\frac{dM}{dt}=\frac Dh\left(c_1-c_2\right) $$

This equation assumes that the aqueous boundary layers (unstirred aqueous layers) on both sides of the membrane have negligible impact on the overall transport process.

The concentrations C1 and C2 within the membrane are typically unknown but can be substituted by the partition coefficient multiplied by the concentration Cd on the donor side or Cr on the receiver side. This can be expressed as follows:

The distribution or partition coefficient, K, is defined by

$$ K=\frac{c_1}{c_d}=\frac{c_2}{c_r} $$

So,

$$ \frac{dM}{dt}=\frac{DSK}h\left(c_d-c_r\right) $$

Additionally, if sink conditions prevail in the receptor compartment, approximately Cr ≈ 0,

$$ \frac{dM}{dt}=\frac{DSKc_d}h=PSc_d $$

where,

$$ P=\frac{DK}h $$

It’s worth noting that the permeability coefficient, also known as permeability, P, is measured in units of (cm/sec).

### Passive Diffusion Through a Membrane-The Multilayer Diffusion

Multilayer diffusion refers to the process of diffusion occurring across multiple layers of materials or barriers. In this phenomenon, particles or molecules move through different layers, each with its own unique properties and thicknesses, influencing the overall rate and mechanism of diffusion. Understanding multilayer diffusion is crucial in various fields such as material science, chemistry, and engineering, where it plays a vital role in processes like drug delivery and coating technologies.

Consider two barriers in series with thickness h1 and h2, respectively,(as shown in figure 3). The total resistance to diffusion (R) can be expressed as the sum of the individual resistances:

$$ R_{total}=R_1+R_2+…+R_n $$

Since the resistance (R) is equal to the reciprocal of the permeability coefficient (P) for each layer, we have:

$$ R_i=\frac1{P_i}=\frac{h_i}{K_iD_i} $$

Therefore, the total resistance (R) can be written as:

$$ R_{total}=\frac1{P_{total}}=\frac1{P_1}+\frac1{P_2}+…+\frac1{P_n} $$

and,

$$ \frac1{P_{total}}=\frac{h_1}{K_1D_1}+\frac{h_2}{K_2D_2}+…+\frac{h_n}{K_nD_n} $$

To find the total permeability (P) of the system, we take the reciprocal of the total resistance. And the total permeability for two layers is:

$$ P_{total}=\frac1{{\displaystyle\frac1{P_2}}+{\displaystyle\frac1{P_2}}}=\frac{D_1K_1D_2K_2}{h_1K_2D_2+h_2K_1D_1} $$

This equation represents the total permeability of the two barriers in series.

**Reference**:

- Sinko, P. (2011). Martin’s Physical Pharmacy and Pharmaceutical Sciences. Baltimore, : Lippincott Williams & Wilkins, a Wolters Kluwer business.
- Robinson, J. Lee, V. (1997). Controlled drug delivery fundamentals and applications. New york, NY: Taylor & Francis Group, LLC.