A certain drug is being administered intravenously to a hospital patient. Let M

Question

A certain drug is being administered intravenously to a hospital patient. Let M be the total amount of the drug (in milligrams) in the patient's body at any given time t in hours. Assuming that the drug is always uniformly distributed throughout the bloodstream, a differential equation for the amount of drug that is present in the bloodstream at any time is given by the following expression. dM dt = 2500 0.2 · M mg h How many milligrams of the drug will be in the body after a very long time? Hint: After a very long time the rate of change of M approaches zero.

Explanation / Answer

Let M(t) be the amount of the drug in the blood stream at the time t. Let's take M in mg and t in h. Consider a small time interval of duration t. Let's characterize the change in the amount of the drug M during this interval.

M = (amount in per time unit) t - (amount out per time unit) t.

the amount in is the product of the amount of fluid coming in by IV times the concentration of the drug in that fluid

(amount in per time unit) t = (100 cm^3/h)(5 mg/cm^3) t = (500 mg/h) t.

The amount out is proportional to the current drug level M with a given constant of proportionality.

(amount out per time unit) t = (M(t) mg)(0.2 /h) t = (0.2M mg/h) t.

Plug this into the conceptual equation, divide by t, take the limit as t -> 0 to get a differential equation.

M/t = (2500 - 0.2M) mg/h ==> dM/dt = 2500 - 0.2M.

Note that the units indicate we have the right equation.

(b) You can answer this question using a couple different approaches. One is to solve the DE. It will have a transient part that depends on the initial condition (which isn't known but could be labeled as M_0.)

You can also suppose that M --> Constant as t --> . This implies that dM/dt --> 0. From the DE, letting M_ denote the long time amount

0 = 2500 (mg/h) - 0.2 (1/h) M_ ==> M_ = 2500/0.2(mg h/h) = 12500 mg

If you solve the equation and then take t -> , you'll get the same answer.

M(t) = 12500 + (M_0 - 12500)e^(-0.2t).

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