3. (12 pts) A patient is given 30 mg of a drug every 8 hours. Immediately before

Question

3. (12 pts) A patient is given 30 mg of a drug every 8 hours. Immediately before each dose is given the concentration of the drug has been reduced by 90%. Let Cn be the amount of the drug present in the patient just after the nth dose is given (a) Provide a difference equation to model Cn+i in terms of Cn (b) How much of the drug is present right after the third dose is given? (c) How much of the drug remains in the patient in the long run? (d) Suppose the drug is harmful when the amount in the body exceeds 40 mg. In this case, what is the maximum safe dose for long term use?

Explanation / Answer

Part (a)

Let Dn be the nth dose. Then,

D1 = 30

D2 = 3 = 30 x 0.1

D3= 0.3 = 30 x 0.12

D4 = 0.03 = 30 x 0.13

Or, in general, Dn = 30 x 0.1n – 1 ……………………………………………. (1)

Now, Cn+ 1 = Cn + Dn+ 1

= Cn + (30 x 0.1n) [from (1) above]

Thus, Cn+ 1 = Cn + (30 x 0.1n) ANSWER

Part (b)

Drug in the patient after the 3rd dose = C3 = C2 + (30 x 0.12) [from answer of Part (a)]

= C2 + 0.3

= {C1 + (30 x 0.1)} + 0.3

= C1 + 3.3

= 30 + 3.3

= 33.3 ANSWER

Part (c)

The drug in the patient in the long run

= 30 + (30 x 0.1) + (30 x 0.12) + (30 x 0.13) + ………..

= (30)/(1 – 0.1) [applying the formula {a/(1 - r)} for the sum of an infinite geometric series – here, a= 30 and r = 0.1]

= 100/3

= 33.3 mg ANSWER

Part (d)

From the above answer, the drug in the patient in the long run = D1/0.9

We want this not to exceed 40.

i.e., D1 < (0.9 x 40) =36.

Thus, maximum dose must be less than 36 mg ANSWER

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