Pharmacokinetics, Half-Life, and Antidepressants in Everyday Life
- Page ID
- 418898
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This exemplar will describe the following concepts from the ACS Examinations Institute General Chemistry ACCM:
- VII.A.2.a Chemical reactions may occur at a wide range of rates, and a key aspect of rate is related to the concentration of species involved in the reaction.
- VII.B.1.b Once known, a rate law can be used to quantitatively predict concentrations of species involved in a reaction as a function of time.
Antidepressants, Pharmaceutical Chemistry, and Half-Life 
The Increasing Usage (and Potential Misusage) of Antidepressants
Antidepressants have become a widely accepted form of treatment for individuals exhibiting symptoms of depression.2 While antidepressants can pose a wide number of benefits for many patients, there has been some recent controversy over their effectiveness, their long-term impacts, and their usage.3
Antidepressants are most prescribed for patients with moderate to severe depression.4 As their usage has spread, however, some scientists recommend that they should be more regulated and restricted, considering their side effects, unknown long-term effects, and the severity of withdrawal symptoms for those wishing to stop taking antidepressants.5,6
One consideration that many pharmaceutical chemists and medical professionals are concerned with is the clinical effects patients may experience from not taking their antidepressants as instructed. The World Health Organization estimates that approximately 50% of patients do not take their medications as prescribed, which becomes problematic considering that medications may last in the body for much longer than a patient may realize.7 Therefore, it is important that pharmacists have a system to calculate the rate at which medication should metabolize in a patient's system, so that medical providers can enforce safe medication procedures and dosages for their patients.
Using Kinetics to Calculate the Metabolism of Medications
Kinetics is the measure of the rate of reactions, following the pathways of reactions and the energy associated with these pathways. Reactions have certain orders that come from the reaction rate's dependence (or lack thereof) on the concentration of reactants — reaction orders help determine their half-life, or the amount of time it takes for half of the substance to decompose.5 Most half-lives are dependent on the concentration of reactant, whereas with first-ordered reactions, the half-life is consistent. Reactions that are dependent on the rate of reactions are said to be kinetically favored.8
A rate law is a mathematical expression that relates the concentration of a reactant to time with some respect to the way the reaction progresses. In respect to a particular reactant, chemical reactions tend to follow one of three most common patterns:
- Zero Order: When the reactant doubles in concentration, the reaction rate does not change. This is because the reaction rate is not proportional to the concentration of this reactant.
- First Order: When the reactant doubles in concentration, the reaction rate doubles.
- Second Order: When the reactant doubles in concentration, the reaction rate quadruples.9
Like with nuclear processes, the absorption rate of most medications occurs with first-order respect to initial concentration
The rate law for a chemical reaction A + B → AB would be written as: k = [A]X [B]Y where:
- A and B are reactants, and AB is the product of the chemical reaction
- k represents the rate constant, a unitless value that relates temperature to reaction speed
- X is the order of the reaction with respect to Reactant A
- Y is the order of the reaction with respect to Reactant B
To examine exactly how a reaction’s factors affect one another based on the overall reaction order, we can integrate the rate law.
- Zero Order: [A] = [A]0 - kt
- First Order: ln[A] = ln[A]0 - kt
- Second Order: 1/[A] = 1/[A]0 + kt
Notes:
(1) A represents the concentration of a given reactant
(2) t represents the time elapsed since the start of the reaction.
One important standardization in chemical kinetics is the concept of half-life. A half-life is the amount of time it takes for 50% of an initial reactant to decay.5 The half-life for each substance is different, and in medicinal chemistry, the half-life is frequently used to refer to the rate at which a substance is metabolized or absorbed into the body. We can use known half-life values for brand-name antidepressants to track the concentration of a medication that remains in a person’s body after some time has elapsed.9
Pharmaceutical chemists and medical providers must take half-life into consideration when they are creating and prescribing medications. You may notice that after several half-life cycles, the concentration of a substance becomes relatively small in comparison to its initial concentration, but it does not deplete completely, and less and less of the substance depletes with later half-life cycles.10 To be safe, a medication should reach a clinically insignificant concentration in the body after four or five half-life cycles.4 In comparison to newer antidepressants, which deal with neurotransmitters in the brain and are known as selective serotonin reuptake inhibitors (SSRIs), older tricyclic antidepressants (TCAs) are more toxic and are more likely to contribute to nonfatal overdoses.8 However, it is of utmost importance that a medication’s half-life is considered when a patient receives a prescription, to maintain patient safety above all else.
A standard dose of Zoloft consists of a 50 mg dosage to be taken at the same time, daily. When ingested, Zoloft has a half-life of 26 hours.2 After an initial dosage, how many milligrams of Zoloft remain in a person’s body by the time they take their next dose?
Solution
Because we know that Zoloft is metabolized at a first-order rate, we can use the first order rate law and the given half-life to calculate how much Zoloft remains in the patient’s system after 24 hours:
ln[A]t = ln[A]0 - kt
At is half of our initial concentration, 50 mg, because the rate of this reaction is only dependent on the concentration of Zoloft present.
ln[A0/2] = ln[A]0 - kt1/2
Our first step is to solve for our rate constant, k. By substituting in 50 mg for Ao and 25 mg for Ao, we can deduce that k = 0.027 hrs-1.
ln[25 mg] = ln[50 mg] - k(26 hrs)
ln[25 mg] - ln[50 mg] = -26k
(3.22 mg - 3.91 mg )/26 hrs = k
k = 0.027 hrs-1.
We know that the amount of Zoloft remaining un-metabolized after 24 hours is dependent on this rate constant, the initial concentration of Zoloft, and Zoloft’s half-life. If we substitute these values into our equation and solve for A24 , we will identify the concentration of Zoloft in mg after 24 hours have passed.
ln[A]t = ln[A]0 - kt
ln[A]24 = ln[50 mg] - (0.027 hrs-1)(24 hrs)
ln[A]24 = 3.27
eln[A]24 = e3.27
[A]24 = 26.4 mg
After 24 hours, 26.4 mg of Zoloft remains in the body.
Say a patient has a 100 mg daily Zoloft prescription. At 8 a.m. on Day 1, he takes his prescription as normal. At 8 p.m. on Day 2, he realizes he forgot to take his medication that morning and takes another pill immediately. On the morning of Day 3, he takes the medication again at 8 a.m. Considering the maximum Zoloft dosage is 200 mg, does the patient overdose when the takes the medication on the third day?
Solution
To solve this, we need to calculate the metabolism of each of the patient’s three doses, and how much they have each metabolized by 8 a.m. on the third day.
Let’s consider the first dosage. There is a 72-hour time gap between the administration of Dose 1 and the final dosage. We can substitute our known values into the integrated rate law, relying on our previous calculations to determine the rate constant k:
Dose 1: ln[A]t = ln[A]0 - kt
ln[A]72 = ln[100 mg] - (0.027 mg/hr)(72 hrs)
ln[A]72 = 4.61 - 1.92
ln[A]72 = 2.69
eln[A]72 = e2.69
[A]72 = 14.7 mg Zoloft
By the final day, 14.7 mg of the initial 100 mg dosage remain in the patient’s system.
Next, we will repeat the process for the evening of the second day, 12 hours before the final dose:
Dose 2: ln[A]12 = ln[100 mg] - (0.027 mg/hr)(12 hrs)
ln[A]12 = 4.61 - 0.32
ln[A]12 = 4.29
eln[A]12 = e4.29
[A]12 = 72.6 mg Zoloft
So, 72.3 mg of the second day’s delayed dose remains in the patient’s body by the final dosage.
We do not need to perform an integration calculation for the final dosage, as we are observing whether the patient will overdose with that final administration.
Dose 3: 14.7 mg Zoloft
+ 72.6 mg Zoloft
+ 100 mg Zoloft
------------------------------------
187.3 mg Zoloft
Adding up our values, we have calculated that, at maximum, the patient will have 187.3 mg of Zoloft present in his system with his third dosage.
Therefore, since the overdose limit is 200 mg, the patient does not overdose with this 100 mg dose of Zoloft.
The decay of Zoloft, an antidepressant medication, is first ordered regarding kinetics. Using the half-life of Zoloft, 26 hours, calculate the following information with a concentration of 45 mg.
- The rate constant, k.
- Using the rate constant, k, and a concentration of 45 mg, calculate the rate of decay of Zoloft.
- How does this rate of decay change if you double the concentration of Zoloft to 90 mg?
Solution
First, we must use the integrated rate law for first-ordered reactions: ln[A]t = -kt + ln[A]0, where t represents time and k is the rate constant. To solve for k, we must use the half-life we are given. Half of 45 mg is 22.5 mg, and so that is our final value for concentration. Our initial value for concentration is the original 45 mg, and t is 26 hours (it takes 26 hours to reach 22.5 mg).
Substituting in those values and solving for k:
ln[A]t = ln[A]0 - kt
ln[22.5 mg]26 = ln[45 mg]0 - k(26 hrs)
ln[22.5 mg]26 - ln[45 mg]0 = -k(26 hrs)
-.693 = -k(26)
k = 0.0267 hrs-1
We find that the rate constant, k, is equal to 0.0267 hrs-1.
The next part of the question asks us to calculate the rate of decay, using the rate constant we calculated.
The rate law for any first-ordered substance can be written as rate = k[A]. Solving for x, which is the rate of decay for Zoloft, we substitute in our solved k value, 0.0267, and our concentration, 45 mg.
x = d(Zoloft)/dt
x = (0.0267 hrs-1)(45 mg)
x = 1.2015 mg/hrs
At 45 mg, the rate of decay is 1.2015 mg/s.
Finally, to find the effect of doubling concentration on rate, we will repeat the same process as before but with a doubled concentration, using 90 mg instead of 45 mg.
x = (0.0267 hrs-1)(90 mg)
x = 2.403 mg/hrs
When we do this, we find that dividing our new rate by our original rate produces a 2—this means, that since we doubled our concentration, the relationship is first ordered, and therefore concentration is directly proportional to the rate.
2.403/1.2015 = 2
If a 4 had been produced, this would have indicated a second-order reaction; a 1 would have indicated no change and a zero-ordered reaction, with no dependence on concentration.
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