BASIC ASSUMPTION & EXPLANATION :-
1.) Basic Assumptions
To combat the intention to human a body appropriate dose of medicine is essential Because the amount of the drug in the human body decrease with time. So, medicine must be given in multiple does.
ASSUMPTION 1.)
​
In this model, it is assumed that the ingested drug is absorbed immediately which is not usally the case. However the time
of absorption is small compared with the time between does.
ASSUMPTION 2.)
​
The rate of decreasing of quantity of drug is directly proportional to that drug present quantity.
So, the rate at which the level of the drug in a patient’s blood decays can be moduled by the decay equation
Where k is a constant to be experimentally determined for each drug.
2.) Explanation:-
Let initially, at t = 0 a patient is given an initial dose y0, then drug level y at any time t is the solution of the below differential equation.
Now we find the solution of above differential equation.
Now integrating equation at both sides. at t = 0 , y = y0 and at
t = t , y = y.
Let at t = T the second dose of y0 is taken, Which increase the drug level to
Now this value becomes our initial value at t = T and from eq-(1), We write at time t = t drug level is
Then the third dose of y0 is taken at t = 2T and the drug just before this dose is given by
The dosage taken at t = 2T raises the drug level to
Continuing in this way, We find after (n+1)th dose is taken that the drug level is
We notice that the drug level after (n+1)th dose is the sum of the first n terms of a geometric series, With first term as y0 and the common ratio is e^(-kT) . This sum can be written as.
As n becomes larger, the drug level approaches a stedy state value, say ys, given by
The stedy state value ys, is called the saturation of the drug.