Question

A mathematical model for concentration of administered cortisone in humans over a 24-hour period uses the function
C= [(D x a) / V(a-b)] (e^(-bt)-e^(-at)
where C is the concentration, D is the dose given at time t=0, V is the volume of distribution (volume divided by bioavailability), a is the absorption rate, b is the elimination rate, and t is the time in hours.

a.)What is the value of C at t=0? Explain why this makes sense.
b.)What happens to the concentration as a large amount of time passes? Explain why this makes sense.
c.)The researchers used the values D=500 micrograms, a=8.5, b=0.09, and V=3,700 liters. Use these values and a graphing calculator to estimate when the concentration is greatest.

Answer 1

C(0) = 0
Naturally, at t=0, there has been no drug added

As t->∞, e^-bt and e^-at ->0, so we again have
C(∞) = 0
All the drug has been absorbed or eliminated.

If
C(t) = (500)(8.5)/(3700)(8.5-0.09) (e^-0.09t - e^-8.5t)
then max C is at t=0.54

Answer 2

I assume D x a means D a

I know nothing about medicine but

a) at t = 0, e^0 = 1
so
C(0) = something(1-1) = 0
It has not had time to be absorbed

after a long time, both e^-bt and e^-at approach zero so the concentration approaches zero. At first assuming a is much bigger than b the absorption will dominate, but after a long time the elimination rate balances the absorption.

go ahead, do C