Biology Reference
In-Depth Information
(b) Show that at t
¼
t
1
þ
D, dP(t)/dt
¼
0, and P(t
1
þ
D)
>
K.
(c) Assume now that t
K,1-P(t - D)/K
< 0. Use Eq. (1-28) to show that population size will be
decreasing for t
1
þ
>
t
1
þ
D. Because P(t - D)
>
D<t<t
2
þ
¼
D, where t
t
2
is the first time
¼
after t
1
at which P(t
2
)
K.
X. MODELING PHYSIOLOGICAL MECHANISMS
OF DRUG ELIMINATION
A primary purpose of this chapter is to study how single processes
evolve with time. One of our first assumptions was that quantities could
be expected to change at a rate proportional to the amount of quantity
present. We have demonstrated that this may work well for modeling
population growth for relatively short periods of time, but
environmental limitations will eventually cause the growth rate to abate.
In other situations, however, quantities diminish rather than increase, in
proportion to the amount present. While not extremely common for
population changes, there are many biological processes that do change
in this way. For example, in many organisms, foreign materials are
excreted at a rate proportional to their concentration.
To illustrate this phenomenon of exponential decay, we shall study how
the body eliminates drugs by modeling concentrations of physiologically
active substances in the bloodstream. Each drug dose received
increases its concentration in the bloodstream, but, simultaneously, the
kidneys are working to remove the drug. It has been experimentally
determined, in fact, that substances entering the bloodstream are
eliminated by the kidneys at a rate proportional to their concentration.
Clearly, the factors governing this physiological process are different
from the factors that determine population growth but, as we shall see,
the underlying mathematical models are very similar.
Ask yourself: Why do you need to take two acetaminophen tablets every
4 to 6 hours when you have a headache? Why is there a warning label
that cautions you not to take any more than four doses in a given
24-hour period? And why does your head start to ache again after four
hours when the warning suggests you really ought to wait six hours
before you take the next dose?
When a physician administers a drug to a patient, he or she has two
important aims—ensuring that the dosage is high enough to provide the
desired effect while ensuring that it is not so high that the drug becomes
toxic. These aims illustrate two critical drug concentrations—the
minimum effective concentration (MEC) and the minimum toxic
concentration (MTC). Between these limits lies the therapeutic window
