Question

(Pills) An antibiotic drug is contained in a solid inner core and is surrounded by an outer coating that makes it palatable. The outer coating and the drug are dissolved at different rates in the stomach, owing to their differences in equilibrium solubilities. (a) If \(D_{2}=4 \mathrm{mm}\) and \(D_{1}=3 \mathrm{mm}\), calculate the time necessary for the pill to dissolve completely. (b) Assuming first-order kinetics \(\left(k_{\mathrm{A}}=10 \mathrm{h}^{-1}\right)\) for the absorption of the di solved drug (i.e... in solution in the stomach) into the bloodstream, pl the concentration in grams of the drug in the blood per gram of boc weight as a function of time when the following three pills are take simultaneously: $$\begin{aligned} &\text { Pill } 1: \quad D_{2}=5 \mathrm{mm} . \quad D_{1}=3 \mathrm{mm}\\\ &\text { Pill 2: } \quad D_{2}=4 \mathrm{mm}, \quad D_{1}=3 \mathrm{mm}\\\ &\text { Pill 3: } \quad D_{2}=3.5 \mathrm{mm} . \quad D_{1}=3 \mathrm{mm} \end{aligned}$$ (c) Discuss how you would maintain the drug level in the blood at a constat level using different-size pills? (d) How could you arrange a distribution of pill sizes so that the concentri tion in the blood was constant over a period (e.g.. 3 hr) of time? Additional information: Amount of drug in inner core \(=500 \mathrm{mg}\) Solubility of outer layer at stomach conditions \(=1.0 \mathrm{mg} / \mathrm{cm}^{3}\) Solubility of inner layer at stomach conditions \(=0.4 \mathrm{mg} / \mathrm{cm}^{3}\) Volume of fluid in stomach \(=1.2 \mathrm{L}\) Typical body weight \(=75 \mathrm{kg}\) \(\mathrm{Sh}=2 . . D_{\mathrm{AB}}=6 \times 10^{-4} \mathrm{cm}^{2} / \mathrm{min}\)

Short answer

In summary, to have a constant concentration of the antibiotic drug in the blood over time using different-sized pills, we first find the time required for each pill to dissolve completely. Then, we determine a suitable combination of fast-dissolving and slow-dissolving pills to maintain a constant drug level in the blood. Finally, we arrange the pill sizes in such a way that the combination of their dissolution rates ensures a continuous release of the drug into the bloodstream.

Step-by-step solution

Calculate the time necessary for the pill to dissolve completely.

First, we need to find the volume of the outer layer of the pill to determine how much of the outer layer needs to be dissolved. Given dimensions are \(D_{2}=4 mm\) (outer diameter) and \(D_{1}=3 mm\) (inner diameter). The solid pill is cylindrical in shape, so using the formula for the volume of a cylinder: \(V=\pi r^{2}h\) and considering it as two concentric cylinders (outer coating and inner core), we get: Outer layer volume: \(V_{2}=\pi \left(\frac{D_{2}}{2}\right)^{2}h - \pi \left(\frac{D_{1}}{2}\right)^{2}h = \pi h\left(\frac{D_{2}^2}{4} - \frac{D_{1}^2}{4}\right)\) Substituting the given dimensions, we have: \(V_{2} = \pi h \left(\frac{16}{4} - \frac{9}{4}\right) = \pi h(7/4) cm^{3}\) Now let's find the time required to dissolve the outer layer completely: The Sherwood number (Sh) is given, \(Sh = 2\). From the equation \(Sh=\frac{K_c r}{D} \Rightarrow K_c = \frac{Sh \times D}{r}\), where \(K_c\) is the mass transfer coefficient. Given values, \(D_{\mathrm{AB}}=6 \times 10^{-4} cm^{2} / min\), and the radius of the outer layer is \(r=\frac{D_{2}}{2} = 2\). We found that \(K_c = \frac{2 \times 6 \times 10^{-4}}{2} = 6 \times 10^{-4}\) cm/min. To find the time required to dissolve the outer layer, we can use the equation \(t= \frac{V_{2}}{K_c S}\), where \(S\) is the solubility of the outer layer. Given: \(S=1.0 mg/cm^{3}\), So, the time required to dissolve the outer layer is, \(t=\frac{\pi h(7/4)}{6\times10^{-4}\times1}= \frac{7\pi h}{24\times10^{-4}}\) min This is the time necessary for the pill to dissolve completely.

Obtain the concentration of the drug in the bloodstream as a function of time for each pill.

Given the following three pills data: Pill 1: \(D_{2}=5 mm\), \(D_{1}=3 mm\) Pill 2: \(D_{2}=4 mm\), \(D_{1}=3 mm\) Pill 3: \(D_{2}=3.5 mm\), \(D_{1}=3 mm\) Repeat Step 1 for all three pills to obtain the time required for each pill to dissolve. Concentration \(C(t)=Ae^{-kt}\), where \(A\) is the initial concentration (assuming it to be the same for all 3 pills), \(k\) is the first-order rate constant given as \(k_{A}=10\,h^{-1}\) and time (\(t\)) should be calculated for each pill as per Step 1.

Maintain constant drug level in the blood with different-pill sizes.

To maintain a constant drug level in the blood with different-pill sizes, we can adjust the dissolution rates of the pills. By maintaining a balance between the fast-dissolving and slow-dissolving pills, we can ensure a constant drug level is maintained in the blood.

Arrange pill sizes to have constant concentration in the blood over a certain period.

To have constant concentration in the blood over a period (e.g., 3 hours, as given), we can arrange the pill sizes in such a way that the combination of their dissolution rates leads to a constant release of the drug into the bloodstream. This means we need to find the optimum combination of pill sizes so that it continuously releases the drug at the target concentration for the given time period. One way to achieve this could be to organize different pills to be taken at different times, ensuring that, as the concentration from one pill begins to decrease, the next pill in the distribution begins to dissolve and release its drug into the bloodstream.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical processes occur and the factors that affect these rates. It's crucial in understanding how pharmaceutical pills dissolve in the body, impacting both the efficacy and the safety of the medication. For pills to work, their active ingredients must be released into the bloodstream, a process governed by chemical kinetics.

In the context of our exercise, the dissolution of the outer coating and the drug contained within the pill can be analyzed through the lens of chemical kinetics. The rate at which the outer layer dissolves is dependent on its chemical composition and how it interacts with the stomach environment. Kinetics also helps us understand the absorption of the dissolved drug into the bloodstream, which is modeled as a first-order reaction, where the rate of absorption is proportional to the drug concentration at a given time.

The kinetics of drug release can be manipulated by adjusting factors such as the pill's size and the solubility of its components. By doing so, pharmaceutical scientists can design pills to dissolve at desired rates, ensuring timely and consistent delivery of medication into the system.

Drug Solubility

Drug solubility refers to the ability of a substance to dissolve in a solvent, resulting in a homogeneous solution. It is a critical property in pharmacology because it dictates how readily a drug will become available to the body upon ingestion. In our exercise example, the solubility of the pill's outer layer and inner core in stomach fluids directly influences how quickly the pill dissolves.

Solubility depends on both the drug's chemical properties and the conditions in the stomach, like pH and temperature. A drug with high solubility in stomach conditions will dissolve quickly, leading to rapid absorption, whereas a drug with lower solubility will take longer. Thus, understanding solubility is essential for the control of drug release rates.

Improving drug solubility is often a goal in pharmaceutical development to increase the efficiency of drug delivery. Techniques such as micronization, which reduces the particle size, or the use of solubility enhancers are common strategies employed to enhance the rate at which a drug dissolves.

Pharmacokinetics

Pharmacokinetics involves the study of how drugs move through the body during absorption, distribution, metabolism, and excretion. It's a key concept in designing effective dosing regimens for medications and understanding the exercise's goal of maintaining drug levels in the blood at a constant level. The term encompasses multiple processes, one of which is the absorption of drugs into the bloodstream, as seen with the dissolution of pills.

The pharmacokinetics of a drug can be affected by its chemical properties, the form it's administered in, and the physiology of the patient. In the exercise, we see different-sized pills being used as a strategy to achieve a constant drug level in the bloodstream. By altering the sizes and, consequently, the dissolution rates of the pills, we can orchestrate a sequence of drug releases over time, ensuring that the concentration of medication remains steady.

Understanding pharmacokinetics is essential for the safe and effective use of pharmaceutical drugs. It allows health care providers to determine the correct dosage, frequency, and duration of treatment to ensure that the drug is therapeutic without being toxic.