In class we derived the following general linear difference equation with consta

Question

In class we derived the following general linear difference equation with constant coefficients for the concentration of a drug c_t and its change over one dosing period: c_t+1 = ac_t + b b is the dosage amount, and a is the total decay in drug concentration in the patient's blood­stream over one dosing period. Suppose that the dosing period is fixed to be a particular amount of time, let's call it T, and also that we can write a = e^-rT where r is a decay rate per unit time the depends on drug and patient. I.e. a itself declines exponentially with the interval between doses T. This just tells us that if we wait longer between doses, there will be a lower concentration at the next time interval. (a) Derive an equilibrium solution, c_eq, in terms of b, r and T using the information above. (b) Suppose the decay rate r = 0.2 per hour, and a convenient dose size is b = 200 mg. What is c_eq if the dosing interval T is every 4 hours? Give your answer to the nearest mg. (Note that we are computing the total amount in the bloodstream rather than the concen­tration per ml.)

Explanation / Answer

(a)From the above question,

ct+1= act+b

at equilibrium point ceq=ct=ct+1

ceq=aceq +b

ceq(1-a)=b

ceq=b/(1-a)

ceq=b/(1- e-rT)

(b)b=200mg, r=0.2 per hour, T= 4hr

replacing the values,

ceq= 200/(1-e(-0.2 x 4))

=363.19mg

ceq=363mg

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