# Fick’s Law of Diffusion Calculation Formula

All drugs must diffuse through various barriers when administered to the body. For example, some drugs must diffuse through the skin, gastric mucosa, or some other barrier to gain access to the interior of the body. Parenteral drugs must diffuse through muscle, connective tissue, and so on, to get to the site of action; even intravenous drugs must diffuse from the blood to the site of action. Drugs must also diffuse through various barriers for metabolism and excretion.

Considering all the diffusion processes that occur in the body (passive, active, and facilitated), it is not surprising that the laws governing diffusion are important to drug delivery systems. In fact, diffusion is important not only in the body but also in some quality control procedures used to determine batch-to-batch uniformity of products (dissolution test for tablets based on the Noyes-Whitney equation, which can be derived from Fick’s law).

When individual molecules move within a substance, diffusion is said to occur. This may occur as the result of a concentration gradient or by random molecular motion.

Probably the most widely used laws of diffusion are known as Fick’s first and second laws. Fick first law involving steady-state diffusion (where dc/dx does not change) is derived from the following expression for the quantity of material (M) flowing through a cross section of a barrier (S) in unit time (t) expressed as the flux (J):

J = dM/(Sdt)

Under a concentration gradient (dc/dx), Fick’s first law can be expressed thus:

J = D [(C1-C2)/h] or J = - D (dC/dx)

Where,

J is the flux of a component across a plane of unit area,

C1 and C2 are the concentrations in the donor and receptor compartments,

h is the membrane thickness, and

D is the diffusion coefficient (or diffusivity).

The sign is negative, denoting that the flux is in the direction of decreasing concentration. The units of J are grams per square centimeter; C, grams per cubic centimeter; M, grams or moles; S, square centimeters; x, centimeters; and D, square centimeters per second.

D is appropriately called a diffusion coefficient, not a diffusion constant, as it is subject to change. D may change in value with increased concentrations. Also, D can be affected by temperature, pressure, solvent properties, and the chemical nature of the drug itself. To study the rate of change of the drug in the system, one needs an expression that relates the change in concentration with time at a definite location in place of the mass of drug diffusing across a unit area of barrier in unit time; this expression is known as Fick’s second law. This law can be summarized as stating that the change in concentration in a particular place with time is proportional to the change in concentration gradient at that particular place in the system.

In summary, Fick’s first law relates to a steady-state flow, whereas Fick second law relates to a change in concentration of drug with time, at any distance, or an unsteady state of flow.

The diffusion coefficients (D × 10−6) of various compounds in water (25°C) and other media have been determined as follows: ethanol, 12.5 cm2/s; glycine, 10.6 cm2/s; sodium lauryl sulfate, 6.2 cm2/s; glucose, 6.8 cm2/s.

The concentration of drug in the membrane can be calculated using the partition coefficient (K) and the concentration in the donor and receptor compartments.

K = (C1/Cd) = (C2/Cr)

Where,

C1 and Cd are the concentrations in the donor compartment (g/cm3) and

C2 and Cr are the concentrations in the receptor compartment (g/cm3).

K is the partition coefficient of the drug between the solution and the membrane. It can be estimated using the oil solubility of the drug versus the water solubility of the drug. Usually, the higher the partition coefficient, the more the drug will be soluble in a lipophilic substance. We can now write the expression:

dM/dt = [DSK (Cd-Cr)]/h

or in sink conditions,

dM/dt = DSKCd/h = PSCd

The permeability coefficient (centimeters per second) can be obtained by rearranging to:

P = DK/h

Example

A drug passing through a 1-mm-thick membrane has a diffusion coefficient of 4.23 × 10^−7 cm2/s and an oil–water partition coefficient of 2.03. The radius of the area exposed to the solution is 2 cm, and the concentration of the drug in the donor compartment is 0.5 mg/mL. Calculate the permeability and the diffusion rate of the drug.

h = 1 mm = 0.1 cm

D = 4.23 × 10^−7 cm2/s

K = 2.03

r = 2 cm, S = π (2 cm)2 = 12.57 cm2

Cd = 0.5 mg/mL

P = [(4.23 × 10^−7 cm2/s) (2.03)]/0.1 cm = 8.59 × 10^−6 cm/s

dM/dt = (8.59 × 10^−6 cm/s) (12.57 cm2)(0.5 mg/mL) = 5.40 × 10^−5 mg/s (5.40 × 10^−5 mg/s)(3,600 s/h) = 0.19 mg/h