My wife is a pharmacist at a hospital. She and her co-workers often provide me with some really interesting math problems. Clinical pharmacists often need mathematics to effectively work with complex patient cases. Modeling drug interactions becomes tricky as the number of prescriptions for a particular patient increases. As the number of prescriptions increases, the amount of time a pharmacist must spend doing drug interaction research also increases. Time is money, so herein lies the problem.
One of my wife’s co-workers asked her to ask me, “What is the number of possible permutations of seven days?” After looking at a handwritten note and talking to the pharmacist directly, here’s a more thorough description of the problem.
Coumadin is a drug used to treat issues associated with blood clots.
Risks come with any anti-clotting drug. If the patient has a car accident, for example, bleeding risk increases dramatically and can have dire consequences. The benefits associated with Coumadin usage must be weighed with the bleeding risk. Computing the correct treatment scheme – a schedule and selection of doses – is important to the patient’s safety. Doctors and pharmacists determine dosage based on a target value: INR target value of 2.5, target range of 2.0 – 3.0 (a confidence interval!!), and if the INR goes above 4.0, there is no greater therapeutic benefit to the dose and patient bleeding risk increases beyond any benefit.
Dosages for this particular drug vary dependent on many factors. However, for the sake of this problem, the pharmacist in question wishes to investigate how much time it will take to write an Excel spreadsheet to determine the different possible treatment schemes. We will assume the simple case: either a patient takes a dose (pill) on a particular day, or a patient does not.
Back to the original question: why does the pharmacist ask for possible “permutations” of the days of the week? Because these would correspond to patient dosage schedules. For example, if the patient takes a dose three days a week, they might take that dose on Monday – Tuesday – Wednesday and not take a pill the rest of the days of the week. Or they might take a dose Tuesday – Thursday – Saturday. All the possibilities would not correspond to permutations, despite the wording of the original question. What we really need to consider are combinations. Permutation means order matters, so we would treat Tuesday – Thursday – Saturday and Thursday – Tuesday – Saturday as different events, when in reality they would be the same treatment schedule in a given week.
If the patient takes a dosage of Coumadin three days a week, then the possible number of treatment schedules would be 7 choose 3.
There are 35 different possible ways to choose 3 days out of the 7 days in a week.
How would we count the number of possible treatment schedules assuming a patient either takes a dose or does not take a dose each day? We need to consider all the ways to select different groupings of 7 days. Enter Pascal’s Triangle & binomial coefficients.
The bottom row above corresponds to each case. 7 choose 0 would correspond to a patient not taking any Coumadin. There’s only one way for that to happen: the patient takes no doses. This would be the trivial case. We aren’t concerned with the patients not taking any doses. The blue number, 128, is the sum of the values in the row with 1, 7, 21, 35, 35, 21, 7, and 1.
Written a little more formally, we have
This value, 127, is 128 – 1, which is , where the 2 corresponds to the number of daily outcomes, either dose or no dose. The 7 corresponds to the number of days in week, and the subtracted 1 corresponds to the trivial case where of the 7 days, the patient takes doses on 0 days.
(Tangential side note: This value, 127, is the difference of a power of 2 and 1. I am immediately reminded of the original Legend of Zelda game on the Nintendo Entertainment System. The maximum number of rupees – currency – a player could obtain is 255, which is 256 – 1, and seems to be related to binary storage limitations of the game.)
This was the answer the pharmacist needed to communicate the number of different outcomes his Excel spreadsheet would need to consider. Some of the treatment schemes might be impractical, so instead of considering 127 different possibilities, he will argue they should boil the cases down to 10 or 12 common dosage schemes.
But for me, the math doesn’t stop there.
What if a person is taking a drug with a more elaborate dosing scheme? I wanted to put a structure on the next case up in complexity, the case where a patient might take one of two different doses on any given day. Here is the structure I used when reasoning through this case initially.
Case 2: Two different dosages on any given day
A = 5 mg dose of a drug (an arbitrary concentration)
B = 2.5 mg dose of a drug
1 = Monday
2 = Tuesday
3 = Wednesday
4 = Thursday
5 = Friday
6 = Saturday
7 = Sunday
A patient might miss a dose. Let C = no dose. Or, they might be instructed to take nothing on a particular day. Either way, the patient takes no dose. The table below describes all the cases for a week.
To count all the possible outcomes efficiently, we have 3 independent choices each day. At least, we will treat each treatment choice as independent although this may or may not be practically true. The total number of possible treatment schedules, then, would be
But we would also throw out the trivial case (no treatments on any day) by subtracting 1. So our total number of possible treatment schedules would be 2,186.
I then wondered if I could write a function to count the number of treatment schedules for any possible number of different dosages.
d = the number of different dosages a patient may take
t = number of days in the timeframe of reference
7 above for days in a week; we could change this to 30 for days in a given month
The number of cases grows quickly as the number of different dosages increases.
This problem is a great practical example of mathematics used in an authentic example. I can extend this problem to exponential decay and dosing with antibiotics. Kids need to know there's a reason a person needs to take an entire course of antibiotics, even if they are feeling better midway through the treatment course.
I will pose this question to my students when we start our unit on counting theory this coming school year. I need to spend more time this summer finding authentic applications locally.