Question

Consider a problem in pharmacokinetics concerned with the dose-response relationship of a drug with its rate of clearance λC^2 , where C(t) denotes the concentration of the drug in a patient ’s body at time t. Let C_0 be the concentration at time t = 0, then dC / dt = −λC^2 .

(1) Now suppose constant doses C_0 are given at equal intervals of time T.

(a) Find the amount of drug in the body immedately after the n th dose.

(b) As n tends to infinity, what is the steady state concentration of the drug in the body?

(c) Consider the model of a continuous intravenous injection with rate I: dC / dt = −λC^2 + I.

(2) Solve the C(t) and compare C(∞) with the result in (b)

Answer 1

Question a) and b).

Question C)

dC/dt=-K*C^2+1 ---> dC/(K*C^2+1)=dt --> tan^-1(C*K^-1/2)=K*^-1/2*t+ Constant

C*K^-1/2=tan (K*^-1/2*t+ Constant)

to find the constat, t=0 C=Co

Co*K^-1/2=tan(Constant)---> Constant=tan^-1(Co*K^-1/2)

C=tan(K*^-1/2*t+tan^-1(Co*K^-1/2))/K*^-1/2

Cn=tan(K*^-1/2*t+tan^-1(Cn-1*K^-1/2))/K*^-1/2

As Cn is defined by a tangent funtion that is periodic the concentration the limit when the time tends to infinite isn't possible to calculate.