An intravenous line provides a continuous flow of a drug directly into the blood

Question

An intravenous line provides a continuous flow of a drug directly into the blood. Assuming no initial drug in the blood, the amount of drug in the blood i hours after the dosing begins is m(t) = A/kl - e, for t 0. where k is the rate constant again related to the half-life and A is the rate at which drug flows into the blood in units of mg/hr. Suppose an antibiotic with a half-life of 12 hr is given to a patient intravenously at a rate of A = 50 mg/hr. Find the rate constant k and graph the drug function m for 0 / 48. What is the steady-state level of the antibiotic in Step 7? That is. evaluate lim, infinity, m(i). Verify that the limit is consistent with the graph in Step 7. In general, what is the steady-state level of a drug delivered by infusion in terms of A and k? In general, at what time does the drug level reach 90% of the steady-state level, in terms of A and k? Based on a patient's weight, a doctor targets a steady state level of tetracycline of 100 mg through infusion. What infusion rate A should be used? The half-life of tetracycline is 9 hr. In Step 10, at what time docs the drug level reach 90% of the steady-state level? At that time, how much drug has actually been delivered? . Suppose a patient has been on infusion of tetracycline for 72 hours with infusion rate as found in Step 10. when the delivery is terminated. How long does it take for the drug level in the blood to reach 2 mg?

Explanation / Answer

m(t) = (a/k) (1-e^-kt)
a=50
(50/k)(1-e^-12k) = 1/2

(100/k)(1-e^-12k)=1
100(1-e^-12k)=k
100 - 100e^-12k -k = 0

Solve this equation graphically (using a calculator). I used Newton's method:
k=100

8)
m(t) = 100-100e^(-100t) - 100 = 0
-100e^(-100t)=0
As t approaches infinity m(t) approaches 0.

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