all users chapter 8 project: lead in the body name___________________________ name ____________________________ introduction: lead

All users

Chapter 8 Project: Lead in the Body

Name___________________________ Name ____________________________

Introduction:

Lead is a heavy metal that can damage the neurological system and

kidneys in humans. Although lead is not used anymore in most paints

and gasoline products, it is still present in many drinking water

systems. The source of lead in drinking water is typically the solder

holding the pipes together, not the pipes themselves. Water quality

tests conducted in 2004 in the Seattle Public School system revealed

that several elementary schools had high lead levels in water from

drinking fountains.1 Many fountains that were tested had lead

concentrations over 100 parts per billion (ppb), with lows of around 5

ppb to a high value of over 1,600 ppb. The U.S. Environmental

Protection Agency (EPA) standard for lead in drinking water is 15 ppb.2

In this project you will mathematically model the blood lead levels in

Katrina, a 10-year-old who enrolls in a school that has lead in the

drinking water. How does the amount of lead in Katrina’s bloodstream

increase? How long does it take for Katrina to become “lead poisoned”?

1. Daily intake of lead

Suppose Katrina enrolls at a school and is placed into a classroom

near a drinking fountain where the lead concentration in the water is

at the EPA standard of 15 parts per billion (ppb) by mass. This means

that there are 15 grams of lead for every billion grams of water:

a) Assume that each day Katrina drinks about ¾ of a liter of water

from the drinking fountain near her classroom. Determine the number of

grams of lead that Katrina consumes each day from the school’s water.

Recall that 1 liter of water has a mass of 1,000 grams.

b) Micrograms are a more common unit of measure when working with

small masses such as the amount of lead in the body. Convert your last

answer into micrograms ( ). Recall that there are a million

micrograms in a gram. Show unit conversion.

c) Not all of the lead that people consume is absorbed into the

bloodstream. Adults absorb less than 5% of consumed lead, whereas

children can absorb up to 50%.3 This is one reason that children are

more susceptible to lead poisoning. Suppose that Katrina absorbs 50%

of her ingested lead. How many micrograms of lead will Katrina’s blood

system absorb each day from drinking ¾ of a liter of the school’s

water?

2. Blood lead half-life and elimination rate

a) Lead in the bloodstream is gradually removed by the kidneys, and is

excreted in the urine. If no additional lead is ingested, then the

lead in the blood will decrease in an exponential manner. Estimates

for the amount of time for 50% of the lead to be passed out of an

adult’s bloodstream range from 28 to 36 days.4 In children the amount

of time can be even longer. Assume that for a 10-year-old girl the

amount of time is about 50 days. With a half-life of 50 days, show

that the daily decay multiplier is about . Hint: 10 micrograms

of lead will exponentially decrease to 5 micrograms of lead in 50

days—what is the value of M?

b) With a half-life of 50 days, what percent of the lead in a child’s

bloodstream is removed each day? What percent is retained each day?

3. Creating the model

You have almost enough information to create an affine model of how

lead in Katrina’s bloodstream will change from day to day. You have

the exponential decay multiplier for lead in the blood, and you know

how much lead Katrina’s blood is absorbing each school day. There are

a few problems though. You do not know how much lead Katrina consumes

on weekends and holidays when she is away from school. To overcome

this difficulty, assume that the ¾ liter of school water she drinks

daily is an average for all 7 days of the week.

You also need a standard time to measure Katrina’s blood lead level

(BLL) each day, and you must decide the order in which the exponential

decay and the linear absorption of lead take place (for writing your

difference equation). Assume that Katrina’s school day starts at 8am,

and that is when her BLL is measured. Also assume that the exponential

decay happens throughout the day, and Katrina drinks her water at the

end of the day, right before 8am.

a) Let represent the amount of lead in Katrina’s bloodstream

in micrograms on day n. Using the above assumptions, write the affine

difference equation in terms of .

b) Recall that Katrina is just starting to attend this particular

school, so assume that she has no lead in her bloodstream at the start

of day . Write the initial condition. Include units!

c) Enter the difference equation and initial condition into your

technology device. Use a table to explore what happens to Katrina’s

blood lead level (BLL) over the 10-month (300-day) school year. In the

space below describe what you find. Hint: set your table to jump by

increments of 50 days.

d) Using technology, plot the difference equation over the 300-day

period, to illustrate Katrina’s projected BLL over a full school year.

Set the graphing window dimensions equal to those of the graph below.

Make a sketch below; include labels.

e) Determine the equilibrium value for the difference equation using

algebra. Add a horizontal dashed line to your graph to indicate the

equilibrium level. Include units!

4. A new classroom

Half-way through the school year, at the start of day ,

Katrina must move to a different classroom. She’ll stay in that

classroom for 2 days, then return to her original classroom at the

start of day . The drinking fountain near this new classroom

has extensive lead-based solder, polluting the water at a lead

concentration of 200 ppb. After drinking ¾ of a liter of water from

this fountain for 2 days, what will happen to Katrina’s BLL? After she

moves back to her original classroom, what will happen to her BLL? To

answer these questions, forge on!

a) Begin by recording the amount of lead in Katrina’s blood when she

moves into the new classroom. Round to 2 decimal places.

b) Now find the amount of lead that Katrina’s blood will absorb daily

from the new classroom’s drinking water. Express your answer in

micrograms. You’ll need to repeat several of the computations from

question 1.

c) Modify your old difference equation to create a new model for the

lead dynamics in Katrina’s blood while she is in the new classroom.

What is the new difference equation?

d) Previously you determined Katrina’s BLL when she entered the new

classroom (day ). Use that amount, and the new difference

equation, to determine the amount of lead in Katrina’s blood one day

later (day ). Show the calculation.

e) Now use the difference equation once again to find Katrina’s BLL

one day later (day ). Again show the calculation.

f) At the start of day , Katrina moves back into her original

classroom. Her BLL has been elevated substantially over the past 2

days. Re-draw the graph that you sketched previously, but stop after

you reach day . On this new graph indicate the spike in

Katrina’s BLL on days and . Also, indicate the

previous equilibrium level with a horizontal dashed line. Include

labels.

g) Now determine what happens to Katrina’s BLL for days to

, after she has returned to her old classroom. Make use of the

BLL on day and the original difference equation. Add the

results to the above graph. Describe below how you determined what

would happen to her BLL.

h) What does the last graph that you sketched tell you about the

stability of the original equilibrium level? In other words, is the

equilibrium level stable or unstable? Explain.

5. Blood poisoning

You know that Katrina’s BLL became very high during the school year.

But was that level high enough to be dangerous? The Centers for

Disease Control and Prevention (CDC) considers it dangerous if a

child’s blood lead concentration (BLC) reaches or exceeds 10

micrograms of lead per deciliter of blood ( ). It is common to

express hazard levels in terms of concentrations.

To determine if Katrina’s BLC reached a dangerous level, you’ll need

an estimate of the amount of blood in a typical 10-year-old girl.

Generally speaking, the bigger the person, the more blood he or she

will have. One rule of thumb is that the blood volume for 10-year-old

children is 75 milliliters (mL) for every 1 kilogram (kg) of body

mass.5 This means that the blood volume can be computed by the

formula:

a) The CDC publishes growth charts that display weight-for-age graphs

(see attached sheet). Read the chart carefully to estimate Katrina’s

weight (mass) to the nearest whole kilogram. Assume that she is at the

50th percentile by weight, meaning that half of the girls her age

weigh less, and half weigh more.

b) Now use the formula above to estimate the volume of blood in

Katrina’s bloodstream. Express answer in milliliters.

c) BLCs are typically computed using blood volumes that are expressed

in deciliters. How many deciliters of blood does Katrina have? Recall

that there are 10 deciliters in 1 liter.

d) Now that you know how many deciliters of blood are in Katrina’s

body, you can determine her BLC on any day of the school year. Start

by computing Katrina’s BLC on day , immediately before she

moved to her temporary classroom. How much lead was in her blood on

that day? What was the BLC in micrograms per deciliter?

e) Did Katrina’s BLC reach a dangerous level before the 150th day of

the school year? Explain.

f) The CDC would like to know how many days a typical 10-year-old girl

(such as Katrina) can drink water with a lead concentration of 15 ppb

before becoming lead poisoned (BLC = ). You could work with

your graph, but a more precise computation can be made using a

solution equation. Compute the solution equation to Katrina’s

difference equation and initial condition. Then use the solution

equation to determine the exact day when Katrina’s BLC would reach

. Show all work below.

g) You should have found that Katrina becomes lead poisoned well

before the 150th day of the school year. But recall that she was

drinking water with a lead concentration of 15 ppb—the maximum

allowable concentration set by the EPA. It’s possible that the EPA

limit is too high, but it’s also possible that something in your model

is either wrong or missing. List one aspect of your model (an

assumption, a constant, etc) that you think might be erroneous. Then

modify your difference equation model and investigate how Katrina’s

BLL would change over time. Does your new model predict that Katrina

will become lead poisoned? Explain.

References

1 Deborah Bach, “Lead Woes Years in the Making,” Seattle Post

Intelligencer, August 10, 2004.

2 U.S. Environmental Protection Agency PA, “Lead in Drinking Water,”

http://www.epa.gov/safewater/lead/index.html.

3 Gabriel M. Filippelli et al., “Urban Lead Poisoning and Medical

Geology: An Unfinished Story,” GSA Today, January, 2005.

4 Agency for Toxic Substances and Disease Registry, “Case Studies in

Environmental Medicine: Lead Toxicity,”

http://www.atsdr.cdc.gov/HEC/CSEM/lead/

5 Geigy Scientific Tables, 7th ed.:1971. As cited at the website:

University of Michigan Medical School: Pediatric Research.

“Guidelines: Children as Research Subjects,”

http://www.med.umich.edu/irbmed/InformationalDocuments/childguide.html.

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