An intravenous line provides a continuous flow of drug directly into the blood. Assuming no initial drug in the blood, the amount of drug in the blood t hours after the dosing begins

in m(t) - (a/k)(1-e^-kt), for t (=>)0, where k is the rate constant (again related to half life) and A is the rate at which drug flows into the blood (in units of mg/hr)

7.
Suppose an antibiotic with a half life of 12 hours is given to a patient intravenously at a rate of A=50mg/hr.
find the rate constant k
8.
what is the steady-state level of the antibiotic in step 7? that is, evaluate lim(t-> infinity)m(t).
9.
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?
10.
Based on patients weight, a doctor targets a steady state level of tetracycline of 100mg through infusion.
what infusion rate A should be used? The half life of tetracycline is 9 hr.
11.
in step 10, at what time does the drug level reach 90% of the steady state level? At that time,
how much drug has actually been delivered?
12.
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 2mg?

7. To find the rate constant (k), we can use the half-life of the antibiotic. The half-life (t1/2) is given as 12 hours. The relationship between half-life and the rate constant is given by the formula: t1/2 = ln(2)/k.

Rearranging the formula, we have k = ln(2)/t1/2. Plugging in the values, k = ln(2)/12.

8. To find the steady-state level of the antibiotic, we can use the formula m(t) = (a/k)(1 - e^(-kt)).
As t approaches infinity, e^(-kt) approaches 0, so the steady-state level can be found by evaluating the limit as t approaches infinity: lim(t->infinity) m(t).

9. In general, the steady-state level (Mss) of a drug delivered by infusion can be found by setting t equal to infinity in the formula m(t) = (a/k)(1 - e^(-kt)). The steady-state level is given by Mss = a/k, where a is the rate at which the drug flows into the blood (in units of mg/hr), and k is the rate constant (related to the drug's half-life).

To find the time at which the drug level reaches 90% of the steady-state level, we can substitute 90% of the steady-state level (0.9*Mss) into the formula m(t) = (a/k)(1 - e^(-kt)) and solve for t.

10. To find the infusion rate (A) that should be used to reach a steady-state level of 100mg for tetracycline, we can use the formula Mss = a/k, where Mss is the desired steady-state level (100mg), and k is the rate constant (related to the half-life, given as 9 hours). Rearranging the formula, we have A = Mss * k.

11. To find the time at which the drug level reaches 90% of the steady-state level for tetracycline, we can substitute 90% of the steady-state level (0.9 * Mss) into the formula m(t) = (a/k)(1 - e^(-kt)) and solve for t. At that time, we can evaluate m(t) to find how much drug has actually been delivered.

12. To find how long it takes for the drug level in the blood to reach 2mg after terminating the delivery of tetracycline, we can use the formula m(t) = (a/k)(1 - e^(-kt)). Substitute 2mg for m(t) and solve for t, taking into account that the infusion rate (A) has already been determined from step 10.