__ Shahana Moideen XB __
Pharmacokinetics is the process whereby substances (like food and
drugs) are ingested into the body (via mouth or needles) and processed. We’ll
concentrate on drugs.
The process of pharmacokinetics has 5 steps:
Liberation – the drug is released from the formulation.
Absorption – the drug enters the body.
Distribution – the drug disperses throughout the body
Metabolism – the drug is broken down by the body.
Excretion – the drug is eliminated from the body.
Of course, each drug needs to act on the body in a different way.
Some drugs need to be absorbed quickly (like nitroglycerin if we are having a
heart attack) and preferably eliminated quickly (otherwise toxins build up in
the blood). For other drugs, we want slow absorption so we get maximum benefit
and don’t lose a lot of it from excretion.
So when your doctor prescribes (say) "take 2 tablets every
meal time", this is based on the desirable levels of drug concentration
and known levels of distribution, metabolism and excretion in the body.
What’s
the math?
When the nurse first administers the drug, the concentration of
the drug in the blood stream is zero. As the drug moves around the body and is
metabolized, the concentration of the drug increases.
There comes a point when the concentration no longer increases and
begins to decline. This is the period when the drug is fully distributed and
metabolism is taking place. As time goes on, the drug concentration gets less
and less and falls below a certain effective amount. Time to take some more
pills.
We can model such a situation mathematically with a differential equation It has 2 parts – an absorption part
and an elimination part. At first, absorption (increasing drug concentration)
takes precedence and over time, elimination (decreasing concentration) is the
most important element.
We have the following variables:
D = drug dose given
V = volume distributed in the
body
C = concentration of the drug at time t
F = fraction of dose which has been absorbed (also called
bioavailability)
A = absorption rate constant
E = elimination rate constant
t = time
Absorption part: This depends on the amount of
the drug given, the fraction that has been absorbed and the absorption rate
constant. It decreases as time goes on. The expression for absorption is given
by:
A × F × D × e-At
Elimination part: The elimination dynamic is affected by the elimination constant,
the volume distributed in the body and the concentration left of the drug. The
expression for this part is:
E × V × C
For our model, we need to subtract the elimination part from the
absorption part (since the absorption part increases the concentration of drug
and the elimination part decreases it). Our differential equation is as
follows:
We now substitute some typical values for our variables (without
units to keep things simple. Note C is a variable, the one for which we
seek an expression in t.)
Solving this differential equation (using a computer algebra
system), gives the concentration at time t as:
C(t)
= 533.3(e−0.4t − e−0.5t)
We can see in the graph the portion where the concentration
increases (up to around t = 3) and levels off. The concentration
then decreases to almost zero at t = 24.
Pharmacokinetics is yet another interesting “real life”
application of math.
