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Assume all the cows are perfectly spherical

A while ago I posted the following math problem to some of my friends:

"A student strained her knee in an intramural volleyball game, and her doctor prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220-milligram tablets every 8 hours for 10 days. If her kidneys filtered 60% of this drug from her body every 8 hours, how much of the drug was in her system after 10 days? How much of the drug would have been in her system if she had continued to take the drug for a year?" [answer below]

Some people think most high school graduates should be able to answer question. Granted, if you see the trick it could be done pretty easily, with little more than a bit of algebra. However, it's not worded particularly well and with a little bit of elaboration, and some reasonable assumptions like: "Assume the human body is a perfectly mixed continuous flow reactor" It becomes a pretty standard introduction to chemical engineering question.

Some people do crossword puzzles or soduku to stimulate their mind. I do mass/energy balances. After some time pondering, I finally worked up the brainpower to set up the mass balance correctly, remember the integral of 1/x and solve for the resonance time of the system.

The full solution is hardly what one would consider basic high school math. Knowing the rate of elimation of drugs in her system is proportional to the amount of drugs in her system. This actually isn't that hard to phantom. It's based on the idea that the rate of elimination is directly proportional to the concentration of the substance within the system. And why there's detectable traces of drugs in your system even when you can't feel the effects anymore.

This got me thinking again about the horrors of my sophmore year in college. In particular the Intro to Chemical Engineering course. This was not an easy course and I distinctly remember the professor, eventually failed over half the class. Looking at the posted grade distribution was not a pretty site, 15% of the class got an A or B, there was a big hump for C's and D's, and quite a few people had "FAIL" marked next to the student ID's. In addition to that, he posted a disclaimer next to all the grades. Which read something to the order of:

"To all the students of Chem Eng 140: I was shockingly disappointed by the performance of this year's class of potential Chemical Engineers.



The intro to Chem E class was a pre-requisite for all other upper divison Chemical Engineering courses. I think this was his way of saying, that he didn't want to see all those students he flunked, in his class next year. I wonder if he got some sort of pleasure in doing that to the students. Now, after completing the aformentioned mass balance. I understand both the importance of course, and just how difficult it could be to grasp the concepts. There were a lot of students who griped, stuck it out, returned the following year and stayed in the major to graduate.

This was the professor who both stood for the intelligence and authority of the faculty, and provided ways to assail the exhaulted towers. Two years later during one of my presentations for a chemical processing course, while stuffing his mouth with a powdered jelly donut which I'd specifically brought, this same professor discredited my hard work saying "You cannot use Occam's Razor as justification, for your results." Sadly, I'd misunderstood his meaning and thought he was complimenting my rhetorical skills. My partner didn't apreciate the poor grade we got on that project. I'd simply done the analysis wrong and missed the point of the exercise, and had spent an hour trying to justify flawed logic, arguing just for the sake of arguing. I wonder if getting the correct analysis was what mattered in the end.

Being a Chemical Engineering major at Berkeley, was by no means easy, and the semseter I took 140 was probably the most difficult I would have. At the time I was doing so much, and changing so fast. I wonder how I ever managed to get through it, and I wonder what I really learned.

Febuary 22, 2006

Short Answer

With a small table, or a spreadsheet, one can approach the problem as follows

Initially After 8 hours 60% would have been removed.
So if you start with 440mg and after 8 hours there should be 40% left.
     40% of 440mg is 176.0

But you take 2 more pills, and add another 440 mg to the blood.
     176mg + 440mg = 616mg

Thus after another 8 hours there's 40% of 616 mg..
     40% of 616 = 246.4

Hours B A
0 440.0 176.0
8 616.0 246.4
16 686.4 274.6
24 714.6 285.8
32 725.8 290.3
40 730.3 292.1
48 732.1 292.9
56 732.9 293.1
64 733.1 293.3
72 733.3 293.3
80 733.3 293.3
88 733.3 293.3
96 733.3 293.3

it approaches limits of 733mg, or 293mg

The long answer is:

If you assume that the woman's body can be considered a perfectly mixed continuous flow reactor, and the flow in her kidneys is the rate of elimination of the drug in her system. Upon which at known intervals another instance of the drug is introduced.

And then you do a mass balance for the rate of elimination for the drugs in her system. You can calculate the amount of drugs in her system between dosages as well as immediately before she takes a dosage, and immediately afterwards.

It's a standard exponential decay problem.

After 10 days, or a year the amout of drug in her system will fluctuate according to the formula: (excuse me if I miss a minus sign) engineers seem to do things like that.

         D = Do exp ( -t / theta)

   Do - is the initial amount of drugs in her system when she takes the dosage (733mg)
   t - is the number of hours since the last dosage
   theta - is the resonance time of the drugs in her system (in this case it's 8.73 hours)

So plotting it out... we get as follows

Reason why we should ensure that we teach at least algebra to all high school students. It would prevent people from overdosing. And I was arguing with my father that they should require chemical engineering training to all the would be meth cooks out there. It's improve public safety.