So Much Coffee, So Little Time: More on Caffeine in the Body
Filtration of the blood by the kidneys is a major process in removing chemicals
from our bodies. Usually, the kidneys filter a fixed fraction of the chemical from theblood each time period. A second method for removal of chemicals from our body ismetabolism by enzymes from the liver; often, this method results in a nearly constantamount of the chemical in the blood being broken down each time period. There areother mechanisms that eliminate chemicals from the body, such as through the hair andnails, and through the respiratory system.
In this lesson, we study caffeine and cadmium, two chemicals that are eliminated
primarily by filtration by the kidneys; that is, in which an approximately fixed percent
eliminated each time period. In your earlier study of the elimination of chemicals such as
caffeine and cadmium (and lead) from the body, you have discovered that a function that
reasonably models the amount of the chemical left in the body is an exponential of the
Here, represents the amount of time, represents the amount of the chemical that was
in our body at the start, represents the
fraction of the chemical that was not
during the time period, and represents the amount of the chemical left in
our plasma after time units. For caffeine, we measured in hours while for cadmiumand lead, was measured in days. For cadmium, we could easily have used years for ,since the elimination was so slow.
In each of our examples, we assumed that no more of the chemical was being added
to the body. However, many of us will still continue to consume caffeine. Furthermore,no matter how careful we are, more cadmium and lead will get into our bodies. Thus, wewant to model situations in which there is a continuing entry of the chemical into ourbodies.
To model effectively, one usually tries to model a simplified situation; thus, we
started by modeling a situation where a person drank only one cup of coffee. Once thesimple situation has been effectively modeled, one can try to model a slightly morerealistic, but still contrived, situation. That is what we are going to do for our study ofcaffeine. Suppose someone drinks a small cup of coffee every hour all day long, startingat 6 am. Let us say that there is 100 mg of caffeine per cup. We want to know how muchcaffeine is in this person's body immediately after drinking the first cup, second cup, thirdcup, and so on. Recall that 13% of the caffeine in our bodies is eliminated each hour. (It
Copyright by Rosalie A. Dance & James T. Sandefur, 1997
This project was supported, in part, by the National Science Foundation. Opinions
expressed are those of the authors and not necessarily those of the Foundation
is unrealistic to assume that a person will drink one cup of coffee every hour. But theapproach developed for this model applies to medicines that are taken periodically, sayonce every 6 hours or once every day.)
Find the amount of caffeine in this person's body, immediately after drinking the first,
second, third, and fourth cups of coffee.
Let's think about how you might have answered problem 1. You first ingest some
caffeine, so you began with 100. Then, because 13% of the caffeine is eliminated eachhour, you multiplied by 0.87 to get the amount of caffeine present in the body an hourlater, 87 mg. You then added another cup of coffee (100 mg of caffeine) to the newamount giving a total of 187 mg in the body in one hour (after the second cup of coffee).
Next, you multiplied the result by 0.87 and then add another 100 mg to get the amount ofcaffeine in the body two hours later (just after the third cup of coffee). You continuedmultiplying by 0.87 and adding 100 mg.
We want a function for , the amount of caffeine in the body after hours. To
find such a function, let's think about how we are computing the amount of caffeine in thebody after hours. As we did for our elimination model which resulted in an exponential,we repeatedly multiply by the same number. This suggests that there may be anexponential function for this model, too. But our model must also take into account theaddition of a constant amount of caffeine to the system each hour. The exponentialfunction that describes the amount of caffeine in your body after hours
where in this case is the time in hours since 6am, when we had our first cup of coffee.
Since we multiplied by 0.87 each hour to find the amount of the previous hour's caffeineleft in the body, we know an appropriate base for the exponential is 0.87, so our functionmight be
In fact, the function that models the amount of caffeine in the body is of this form.
Unlike before, though, is
the amount of caffeine we started with and
amount of caffeine that is consumed each hour. We now need to determine reasonablechoices for and
. We will do this by fitting this function to the known data.
knowledge of the situation and your answers to problem 1,
This information can be used to find and
Solve the equations
problem 1. Then graph the function
and describe what eventuallyhappens to the amount of caffeine in this person's body. In particular, discuss whethera person who actually drank a cup of coffee every hour for a long period would die ofan overdose of caffeine (5000 mg).
As was discussed before, cadmium is an extremely dangerous chemical. It tends to
be in the soil and in plants we eat, at very low levels. One of the properties of cadmiumis that it is eliminated from our bodies at a very slow rate. In fact, only about 4% of thecadmium is eliminated from our bodies each year.
Cadmium is most toxic when inhaled. Initial symptoms include chest pains and
nausea. Symptoms can progress to emphysema and fatal pulmonary edema. One sourceof inhalation of cadmium is through cigarettes. Each cigarette contains about 0.001 to0.002 mg of cadmium. Smokers absorb between 10% and 40% of this cadmium.
Suppose a person smokes one pack of 20 cigarettes every day, with each cigarettecontaining 0.0015 mg of cadmium. Assume this person absorbs 25% of the cadmium inthe cigarettes. This means this person would absorb an additional
Suppose this person starts smoking when 20 years old. Let represent the amount
of cadmium in this person's body after years, from smoking alone. Since the body iseliminating cadmium at the rate of 4% per year but some constant amount of cadmiumis being continually added, we expect to be an ex
At the time of starting smoking (when ) we
into the formula for the amount of cadmium to g
. to get another equation. Solve these
to get the function that approximates the amount ofcadmium in this person's body which results from smoking.
Suppose this person smokes one pack a day for 30 years, that is, until reaching the
age of 50. How much additional cadmium is in this person's body due to smoking?
What is the horizontal asymptote of the function as gets large without bound?
Suppose a person smokes one pack of cigarettes each day for 30 years. Each year,
between 1.4% and 7% of the cadmium is removed from the body.
Assume the best case scenario, that 7% of the cadmium is removed from the body
each year. Also assume the best case that only 0.001 mg of cadmium is in eachcigarette and that this person only absorbs 10% of this. Find the function for ,the amount of cadmium in this person's body as a result of smoking, after years.
Assume the worst case scenario, that only 1.4% of the cadmium is removed from
the body each year. Also assume the worst case that 0.002 mg of cadmium is ineach cigarette and that this person absorbs 40% of this. Find the function for ,the amount of cadmium in this person's body as a result of smoking, after years.
It is estimated that a smoker has twice the exposure to cadmium as a nonsmoker.
Does this estimate seem reasonable when the answers to problems 3 and 4 arecompared to the result that the average 50-year-old adult has about 30 mg of cadmiumin her or his body? Discuss what other information you would need to know if youhad to make an accurate assessment of the relationship between smokers' andnonsmokers' exposure to cadmium.
Information in this activity was obtained from Goodman and Gilman's The Pharmacological Basis of
: A.G. Gilman, L. S. Goodman, T.W. Rall, and F. Murad, Macmillan Pub. Co., NY, 1985.
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