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.
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
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
a) To find the value of C at t=0, we substitute t=0 into the given function:
C = [(D x a) / V(a-b)] (e^(-bt)-e^(-at))
When we substitute t=0, we get:
C = [(D x a) / V(a-b)] (e^(-b(0))-e^(-a(0)))
C = [(D x a) / V(a-b)] (e^(0)-e^(0))
Since any value raised to the power of 0 is equal to 1, the equation simplifies to:
C = [(D x a) / V(a-b)] (1-1)
C = 0
Therefore, the concentration at t=0 is 0. This makes sense because at time t=0, no time has passed since the dose of cortisone was administered. Thus, there has been no time for absorption or elimination to occur, resulting in a concentration of 0.
b) As a large amount of time passes, the concentration gradually decreases towards zero. This is because the equation for concentration includes the term (e^(-bt)-e^(-at)) which represents the rate of elimination and absorption.
The term e^(-bt) represents the rate of elimination, and as time passes, the value of e^(-bt) approaches 0. This means that over time, the elimination rate becomes more significant, reducing the concentration.
Similarly, the term e^(-at) represents the rate of absorption. As time passes and t increases, the rate of absorption decreases. Eventually, the value of e^(-at) also approaches 0, further reducing the concentration.
Therefore, as time passes, the concentration decreases asymptotically towards zero.
c) To estimate when the concentration is greatest, we need to find the maximum point on the concentration curve using the given values D=500 micrograms, a=8.5, b=0.09, and V=3,700 liters.
Using a graphing calculator, we can plot the concentration function:
C = [(D x a) / V(a-b)] (e^(-bt)-e^(-at))
By substituting the given values into the equation, we get:
C = [(500 micrograms x 8.5) / (3,700 liters x (8.5 - 0.09))] (e^(-0.09t)-e^(-8.5t))
From the graphing calculator, plot the function and find the highest point on the curve. The x-coordinate of this point will represent the time at which the concentration is greatest.