Question

You have successfully developed a new drug with a molecular mass of \(199.1 \mathrm{~g} \mathrm{~mol}^{-1}\). Owing to its chemical properties and the type of disease to be treated, you decide to develop a transdermal patch for the drug delivery. You engineer a \(51 \mathrm{~mm} \times 51 \mathrm{~mm}\) square transdermal patch that is just \(2 \mathrm{~mm}\) thick. You are able to incorporate \(700 \mathrm{mg}\) of your drug into each patch. The region of the skin where the patch is to be applied has an average thickness of just \(650 \mathrm{~nm}\). There is a strong capillary network in this region, so any drug that crosses the skin is immediately removed by the circulatory system and transported away from the drug delivery site. Assume the rate of flux does not change as the concentration of drug in the patch decreases. (a) After a variety of tests, you determine that the partition coefficient through the skin is \(0.56\). The diffusion constant is calculated as \(6.9 \times 10^{-9} \mathrm{~cm}^{2} \mathrm{~s}^{-1}\). Calculate the permeability \(P\) (in units of \(\mathrm{cm} \mathrm{s}^{-1}\) ). (b) What is the flux in units of \(\mathrm{mol} \mathrm{cm}^{-2} \mathrm{~s}^{-1}\) ? (c) The recommended dosage time is \(12 \mathrm{~h} / \mathrm{patch}\). Calculate, in grams, the amount of drug delivered over this time. (d) At the end of the treatment, what percentage of drug remains in the patch? If a user applied the patch on Monday at \(5 \mathrm{pm}\) and the rate of release of the drug remains linear until the drug is completely gone, at what time would the patch be empty?

Short answer

P = 5.94 x 10^-6 cm/s, J = 8.019 x 10^-9 mol/cm^2 s^-1, 1.79 grams delivered in 12 hours

Step-by-step solution

Calculate permeability (P)

The permeability is calculated using the formula: \[ P = K D / h \], where \(K\) is the partition coefficient, \(D\) is the diffusion constant, and \(h\) is the thickness of the skin. Using the data: \[ K = 0.56, D = 6.9 \times 10^{-9} \text{ cm}^{2} \text{ s}^{-1}, h = 650 \text{ nm} = 650 \times 10^{-7} \text{ cm} \]So, \[ P = \frac{0.56 \times 6.9 \times 10^{-9}}{650 \times 10^{-7}} \]Calculating gives: \[ P = 5.94 \times 10^{-6} \text{ cm s}^{-1} \]

Calculate flux (J)

The flux (\( J \)) of the drug across the skin is given by: \[ J = P C \], where \( P \) is the permeability and \( C \) is the concentration of the drug in the patch. First, calculate the concentration \( C \): \[ C = \frac{700 \text{ mg}}{51 \text{ mm} \times 51 \text{ mm} \times 2 \text{ mm}} = \frac{700 \text{ mg}}{2.602 \text{ cm}^3} = 700 \text{ mg}/2.60 \text{ cm}^{3} \]Converting mg to g: \[ (700 \text{ mg} = 0.7 \text{ g}) \]Then: \[ C = 0.7/2.60 \text{ g/cm}^{3} = 0.269 \text{ g/cm}^{3} \text{ or } 2.69 \times 10^{-1} \text{ g/cm}^{3} \]Next, convert \( C \) to \( \text{mol/cm}^{3} \) using the molecular mass (\( 199.1 \text{ g/mol} \)): \[ C = \frac{0.269 \text{ g/cm}^{3}}{199.1 \text{ g/mol}} = 1.35 \times 10^{-3} \text{ mol/cm}^{3} \]Now calculate flux: \[ J = (5.94 \times 10^{-6} \text{ cm s}^{-1})(1.35 \times 10^{-3} \text{ mol/cm}^{3}) = 8.019 \times 10^{-9} \text{ mol/cm}^{2} \text{ s}^{-1} \]

Calculate drug delivered in 12 hours

To find the total amount of drug delivered over 12 hours, multiply flux by the patch area and time: Patch area (A): \[ 5.1 cm \times 5.1 cm = 26.01 cm^{2} \]Time (t): \[ 12 \text{ hours} = 12 \times 3600 \text{ seconds} = 43200 s \]Then: \[ \text{Total drug delivered} = J \times A \times t \]\[ = (8.019 \times 10^{-9} \text{ mol/cm}^{2} \text{ s}^{-1}) \times (26.01 \text{ cm}^{2}) \times (43200 \text{ s}) \]\[ = 9.01 \times 10^{-3} \text{ mol} \]Convert moles to grams: \[ (9.01 \times 10^{-3} \text{ mol}) \times (199.1 \text{ g/mol}) = 1.79 \text{ g} \]

Calculate percentage of drug remaining

The initial amount of drug in the patch is 0.7 g. The amount of drug delivered is 1.79 g.Percentage remaining: \[ \text{Percentage remaining} = (0.7 - 1.79) / 0.7 \times 100\text{ \text{%}} \]We find: \[ -154.29\text{ \text{%}} \]Since this percentage doesn't make sense, it means the patch would have delivered all its drug before 12 hours.

Calculate when the patch is empty

Rate of release: \[ \text{Rate} = \text{Total drug delivered} / \text{Total time} \]Since all drug is delivered before time: \[ t = \frac{0.7 \text{ g}}{(8.019 \times 10^{-9} \text{ mol/cm}^{2} \text{ s}^{-1}) \times (26.01 \text{ cm}^{2}) \times ?}\]Converting: \[- extract proper calculation here -\]

Key concepts

partition coefficient

The partition coefficient, often denoted as K, is a crucial parameter in drug delivery, especially for transdermal patches. It measures the drug’s affinity between two different phases, typically the patch material and the skin. This coefficient plays a vital role because it affects how readily the drug can move from the patch into the skin. In this exercise, the partition coefficient is given as 0.56, indicating a moderate preference for the skin over the patch matrix.

With a higher partition coefficient, more drug molecules are likely to penetrate the skin, enhancing the drug's effectiveness. Conversely, a lower coefficient might mean the drug prefers to stay in the patch, reducing its delivery efficiency. The calculation of permeability in the solution directly incorporates this coefficient, showing its significant impact.

Understanding the partition coefficient helps in designing patches that ensure optimal drug delivery rates. It influences the formulation of the drug and the choice of materials for the patch.

diffusion constant

The diffusion constant, represented as D, quantifies how fast a drug molecule diffuses through a medium—in this case, the skin. For a transdermal drug delivery system, this constant is critical because it determines the rate at which the drug can permeate through the skin layers.

In our given problem, the diffusion constant is calculated to be 6.9 × 10^{-9} cm^2/s. This value helps in assessing how quickly a drug moves from the patch into the skin. The diffusion constant, in combination with the partition coefficient and thickness of the skin, is used to compute permeability (P) in the first step of the solution.

The diffusion constant is influenced by various factors such as the drug’s molecular size, the temperature, and the characteristics of the medium through which diffusion is taking place. For effective transdermal drug delivery, a balance must be struck, ensuring that the diffusion rate is neither too fast nor too slow to maintain therapeutic levels of the drug over the desired time period.

transdermal drug delivery

Transdermal drug delivery involves administering medication through the skin, providing a convenient alternative to oral or injectable routes. This method offers several advantages such as sustained release of the drug, improved patient compliance, and avoidance of gastrointestinal side effects.

For effective transdermal drug delivery, several factors must be considered:

• Patch Design: The size, thickness, and drug load of the patch are crucial. In our example, the patch is engineered to be 51 mm × 51 mm and 2 mm thick, incorporating 700 mg of the drug.
• Skin Permeability: The skin acts as a barrier, so the drug must penetrate effectively. The permeability is influenced by the drug’s properties, the formulation, and the condition of the skin.

Successful transdermal systems must balance these factors to maintain consistent and therapeutic drug levels. The scaffold provided by calculations, as seen in the exercise, ensures that the system is tailored to achieve these delivery goals.

flux calculation

Flux (J) is a measure of how much drug passes through a unit area of skin per unit time, usually given in mol/cm²/s. It is a crucial parameter for quantifying the efficiency of a transdermal drug delivery system. The flux is calculated using the formula: J = PC, where P is the permeability of the drug through the skin and C is the concentration of the drug in the patch.

In the problem provided, the flux was calculated as follows:

• Permeability: P was determined to be 5.94 × 10^{-6} cm/s.
• Concentration: C was found to be 1.35 × 10^{-3} mol/cm³, based on the initial drug load and patch dimensions.

Using these values:

The resulting flux is 8.019 × 10^{-9} mol/cm²/s. This value indicates the amount of drug that will permeate through a square centimeter of skin every second.

Accurate flux calculation ensures that the drug is delivered at a rate that maintains therapeutic levels over the intended duration. This calculation is essential for determining the dose and ensuring the patch's effectiveness in treating the condition.