{"id":89,"date":"2007-06-28T15:02:00","date_gmt":"2007-06-28T23:02:00","guid":{"rendered":"http:\/\/www.wilyness.com\/blog\/?p=89"},"modified":"2007-06-28T15:43:20","modified_gmt":"2007-06-28T23:43:20","slug":"math-for-pill-poppers","status":"publish","type":"post","link":"https:\/\/www.wilyness.com\/blog\/?p=89","title":{"rendered":"Math for pill poppers"},"content":{"rendered":"

I’ve always felt that mathematics is a useful tool in everyone’s daily life. Here is an example.<\/p>\n

Recently, I was prescribed some medicine, and it was to be taken twice daily: once in the morning, and once before bed. However, I made a couple mistakes. The first morning, I took twice the dosage in the morning. And, in the evening I dutifully took another dosage, but made the same error and took a double dosage again. I realized my hapless error just as I downed the requisite glass of water. So what was I to do. Clearly, the drug was now above the prescribed concentration in my bloodstream, so should I continue the normal schedule the next morning?<\/p>\n

We can use math to figure this out. Most drugs taken in orally can be considered to have a “half-life” in your bloodstream. Yes, this is the same concept of half-life in radioactivity. Basically, your body gets rid of stuff from your bloodstream at a constant rate, given the current concentration of the ‘stuff’ in your blood. This is modeled in math as an exponential decay function. To illustrate, have a look at these equations.<\/p>\n

\"equations\"
\nThe simplest form is (1), which illustrates the basic shape of the drug concentration. We add a step function (2), which allows us to make just the part after the dose is administered, along with a t0 time offset parameter to give a more accurate shape in (3). (4) adds a half-life parameter, which is scaled away from the irrational e to become (5). This is the concentration resulting from a single dose, over all time, parameterized by dose administration (6). While you are taking the medicine (t=0 to t=t_finish), we can represent your idealized concentration as (7). The summation variable i gets incremented by 0.5 each time, which flies in the face of normal convention. Sorry about that. The code farther down is correct.
\nAll simple, right? Let’s plot things. I used Mathematica to generate these plots. Equation (6) becomes:<\/p>\n

\"dose
\nNote that in this function, x is time (in days), and t is the time offset. I threw the equation together before thinking about how best to represent things for others… Sorry…. The 1.3333 factor is the the half-life constant. The half-life for my medicine was roughly 18 hours, which is 0.75 days, which is roughly 1.3333 (close enough for my (government’s) purposes).
\nAnd then we can plot our ideal concentration.<\/p>\n

\"dose,<\/p>\n

This is how the concentration would look like if I took the medicine exactly as directed. We can see that the concentration is roughly periodic after about two days of accumulation. But! I messed up, remember? How can I adjust my next doses to return to the ideal case?<\/p>\n

\"dose,<\/p>\n

You can see that at t=0, my medicine concentration is twice the ideal(conc = 2), since I took twice the dosage. Accordingly, the second dose, which was also twice too large, boosted my concentration to about 3.25. In this scenario, I skipped the third dosage (second day, morning), to bring my concentration roughly the same as the ideal case at t=1.5 . So there you go.<\/p>\n

The bottom line is that I should skip the third dose if I take two double-doses on the first day. Yay! Isn’t math fun? This amount of math is no more than what a high-school graduate should have, and probably less than what an overachieving middle schooler could do.
\nI’ll be sure to post other examples math in real life when I get the gumption. Have a nice day!<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve always felt that mathematics is a useful tool in everyone’s daily life. Here is an example. Recently, I was prescribed some medicine, and it was to be taken twice daily: once in the morning, and once before bed. However, …<\/p>\n

Read more »<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[],"_links":{"self":[{"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/89"}],"collection":[{"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=89"}],"version-history":[{"count":0,"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/89\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=89"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=89"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wilyness.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=89"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}