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

Question

A certain drug is being administered intravenously to a hospital patient. Fluid containing k = 4 mg/cm3 of the drug enters the patient's bloodstream at a rate of r = 60 cm3/hr. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of p = 0.12 (hr)-1.

Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of the drug that is present in the bloodstream at any time. Your answer should be an expression in terms of q (in milligrams) and t (in hours).

Explanation / Answer

at any time t the amount present is q

we can write

dq/dt = 4 x60 - 0.12q = 240 - 0.12q

dq/(240-0.12q) = dt

on integrating both side

-1/0.12 ln(240-0.12q) = t + K

=> 240 - 0.12q = K' e-0.12t

=> q = 2000 - ce-0.12t

since at t= 0 q =0 therefore c = 2000

hence q = 2000 (1- e-0.12t )

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