Could john doe have been drugged prior to his brutal #2 pencil murder? this is a question the investigators want to answer. unfortunately, at his time of discovery, no trace of a drug was found. at minimum, understanding how the body processes chemicals will allow them to rule out drugs that would otherwise be detectable in the bloodstream after the duration of time between when he was murdered and when he was found. a system of differential equations can be used to us understand this scenario. as a bit of background, when we consume something, that “something” travels through our gastrointestinal (gi) tract. the gi tract is made up of, among other things, the small intestine, the large intestine, and the stomach.
chemicals are then absorbed from the gi tract into the bloodstream. the following (real) model shows the interaction between the two:
= ( ? ? / ? ? ? / )
= ? ? 1 ( ? ? / ? ? ? / ) ? ? ? 2 ? ? / , where,
? ? = amount of the chemical that enters the body orally into gi tract, mg
? ? = amount of the chemical that is in the bloodstream, mg
? ? = time, in hours, since the chemical was orally taken
= volume of gi tract, liters
= volume of blood in the body, liters
? ? 1 = diffusion rate parameter 1, ? ? 1 > 0