Let \(a_t\) be the sequence representing the amount of Adderall XR in the patient’s gut \(t\) hours after the pill was taken. Write down a recursion relation for \(a_t\). Find an explicit formula for \(a_t\). Calculate the time at which the amount of drug remaining in the gut drops to \(1\%\) of its starting value; that is, find \(t\) for which \[a_t = 0.01a_0\] Find \(\lim_{t\to\infty}a_t\), what does it represent? Now let’s build a model for the amount of drug in the patient’s blood. Define another sequence \(b_t\) that represents the amount of Adderall XR in the patient’s blood at time \(t\). At time \(t = 0\) there is no Adderall XR in the blood. To model \(b_t\) we start with a word equation: \[b_{t + 1} = b_t + (\text{amount of drug that enters blood from the gut})\]\[\qquad - (\text{amount of drug that is eliminated from blood})\] We need mathematical expressions for the terms in this equation. Using this and (b), write the recursion relation for \(b_t\). Any drug that leaves the gut enters the blood. We know that \(42\%\) of the Adderall XR in the gut leaves the gut each hour. So the amount of drug that enters the blood is \(0.42 a_t\). Adderall XR has first-order elimination kinetics; \(6\%\) of the Adderall XR present in the blood leaves the blood each hour. So the amount that leaves the blood is \(0.06 b_t\). Calculate how the amount of drug present in the blood changes over a \(6\)-hour interval starting at \(t = 0\); that is, calculate \(b_0, b_1, b_2,\ldots,b_6\). What is the maximum amount of drug in the blood? At what time is the amount of drug in the blood highest? Using a spreadsheet calculate and plot the amount of drug present in the blood over a $24$-hour interval (i.e., calculate \(b_0, b_1, b_2, \ldots,b_{24})\). Suppose that, instead of being slowly released, the drug entered the bloodstream immediately after the pill was taken at time \(t = 0\). Explain why the amount of Adderall XR in the blood would then be modeled by a recursion relation: \(b_{t+1} = 0.94\,b_t,\ b_0 = 10\,mg\). Calculate \(b_{24}\), the amount of drug present after \(24\) hours under the new model.
First, we will build a model for the total amount of drugs in the gut. At time \(t = 0\), \(10mg\) of the drug is present in the gut. From the gut, it passes slowly into the blood. The passage of the drug to the blood has first-order kinetics; that is, every hour \(42\%\) of the drug remaining in the gut is passed from the gut into the blood. No more pills are taken, so no extra drug is added to the gut.
Let \(a_t\) be the sequence representing the amount of Adderall XR in the patient’s gut \(t\) hours after the pill was taken. Write down a recursion relation for \(a_t\).
Find an explicit formula for \(a_t\).
Calculate the time at which the amount of drug remaining in the gut drops to \(1\%\) of its starting value; that is, find \(t\) for which \[a_t = 0.01a_0\]
Find \(\lim_{t\to\infty}a_t\), what does it represent?
Now let’s build a model for the amount of drug in the patient’s blood. Define another sequence \(b_t\) that represents the amount of Adderall XR in the patient’s blood at time \(t\). At time \(t = 0\) there is no Adderall XR in the blood.
To model \(b_t\) we start with a word equation:
\[b_{t + 1} = b_t + (\text{amount of drug that enters blood from the gut})\]\[\qquad - (\text{amount of drug that is eliminated from blood})\]
We need mathematical expressions for the terms in this equation. Using this and (b), write the recursion relation for \(b_t\).
Any drug that leaves the gut enters the blood. We know that \(42\%\) of the Adderall XR in the gut leaves the gut each hour. So the amount of drug that enters the blood is \(0.42 a_t\).
Adderall XR has first-order elimination kinetics; \(6\%\) of the Adderall XR present in the blood leaves the blood each hour. So the amount that leaves the blood is \(0.06 b_t\).
Calculate how the amount of drug present in the blood changes over a \(6\)-hour interval starting at \(t = 0\); that is, calculate \(b_0, b_1, b_2,\ldots,b_6\). What is the maximum amount of drug in the blood? At what time is the amount of drug in the blood highest?
Using a spreadsheet calculate and plot the amount of drug present in the blood over a $24$-hour interval (i.e., calculate \(b_0, b_1, b_2, \ldots,b_{24})\).
Suppose that, instead of being slowly released, the drug entered the bloodstream immediately after the pill was taken at time \(t = 0\).
Explain why the amount of Adderall XR in the blood would then be modeled by a recursion relation: \(b_{t+1} = 0.94\,b_t,\ b_0 = 10\,mg\).
Calculate \(b_{24}\), the amount of drug present after \(24\) hours under the new model.
Compare your answer from (ii) with your answer to (g) above. Which strategy (slow-release or immediate absorption) gives the highest amount of drug present in the blood 24 hours after the pill was taken?
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