Absorption of MedicationsWhen you take a pill to obtain medication, the pill first goes into your stomach and themedication passes into your GI tract. From there the medication is absorbed into yourbloodstream and circulated through your body before being eliminated from the blood by thekidneys and other organs. If we let xtdenote the amount of medication in your GI tract attime t, then we can model the movement of the medication out of the GI tract with theequationx′t= −k1xt, x0= A.1This is the assertion that after taking the pill, an amount A of medication is in the GI tractand, as the drug is absorbed by the bloodstream, the amount decreases at a rateproportional to the amount currently present in the GI tract. If the amount of medication inthe bloodstream at time t is denoted by yt, theny′t= k1xt− k2yt, y0= 0,2expresses the fact that medication is coming into the bloodstream at exactly the rate it isleaving the GI tract and it is leaving the bloodstream at a rate proportional to the amountcurrently present in the bloodstream. The constant here is denoted by k2. Also, we areassuming that there is no medication in the bloodstream initially. Now this is two equationsfor the two unknown functions xtand yt, but the first equation can be solvedindependently and the solution substituted into the equation (2).The solution of (1) is easily found to bext= Ae−k1t,3and then y′t+ k2yt= k1xt= k1Ae−k1t.We will find a particular solution for the y-equation by guessing (i.e., the method ofundetermined coefficients). We guess that ypt= ae−k1tand thenyp′t+ k2ypt= −k1ae−k1t+ k2ae−k1t= k1Ae−k1t.This leads to a =k1Ak2− k1and ypt=k1Ak2− k1e−k1t.Thenyt= Ce−k2t+k1Ak2− k1e−k1tand, using the initial condition to evaluate C, we getyt=k1Ak2− k1 e−k1t− e−k2t4Plotting xt and yt versus t for some representative values of the constants gives thefollowing figure1x(t) and y(t) vs t for A=1Note that the solution3, for ytis only valid if k1≠ k2. However, the problem remainssolvable in the case that k1= k2so how do we find the solution? One approach is to takethe limit in4as k1approaches k2. Since this leads to an indeterminate form of the type0/0, we can apply L’Hopital’s rule. That is,limk1→k2k1Ak2− k1 e−k1t− e−k2t = k1A limk1→k2e−k1t− e−k2tk2− k1= k1A limk1→k2−te−k1t−1= k1Ate−k2tThen when k1= k2the solution4must be replaced byyt= k1A te−k2t4′Now consider the situation in which, after taking a pill at time t = 0, a second pill is taken ata later time t = T, followed by a third pill taken at t = 2T and so on. Then the model for thedrug level in the GI system becomes:x1′t= −k1x1t, x10= A, 0 < t < Tx2′t= −k1x2t, x2T= A + x1T, T < t < 2Tx3′t= −k1x3t, x32T= A + x22T, 2T < t < 3TetcA similar set of problems will describe the drug level in the bloodstream.Project:(a) Solve these two initial value problems for xtand ytand plot them on the same graphin the following three cases: k1< k2, k1= k2, and k1> k2. In each case, describe what theplot tells you about the drug levels in the two systems.(b) Consider the situation in which a pill is taken every T hours (i.e. a pill is taken att= 0,T,2T,... etc). Write out the sequence of initial value problems that must be solved tofind the drug levels in the GI system and the bloodstream for t between 0 and 4T. Assume2k1≠k2.(c) Solve the initial value problems in (b) and graph xtand yton the same axes for thefollowing parameter values:k1=.1 k2=.2 T = 1k1=.3 k2=.2 T = 1k1=.2 k2=.1 T = 2A = 1 in all casesDiscuss your results with respect to maintaining the drug level in the bloodstream betweenmax and min limiting values.If we wish to consider a model that represents taking a continuously acting pill, (a pill thatreleases medication continuously so as to maintain a constant level of medication in the GItract for a sustained period of time), we might modify the previous model1,2to readx′t= X0− k1xt, x0= 0.y′t= k1xt− k2yt, y0= 0.In this case, we findxt=X0k11 − e−k1tyt=1k21 +1k2− k1k1e−k2t− k2e−k1tPlotting these solutions gives,uuper curve = xtlower curve =