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Lead in the Body

Part 3: Ingesting Lead

In a study published in 1973 (see title page for reference), Rabinowitz and colleagues measured over an extended period of time the lead levels in bones, blood, and tissue of a healthy male volunteer living in Los Angeles. Their measurements produced the following transfer coefficients for movement of lead between various parts of the body and for excretion from the body. Note that, relatively speaking, lead is somewhat slow to enter the bones and very slow to leave them.

Lead Transfer Coefficients (Rabinowitz, et al.)

Units: days-1

a21 = 0.011 a12 = 0.012 from blood to tissue and back
a31 = 0.0039 a13 = 0.000035 from blood to bone and back
a01 = 0.021 a02 = 0.016 excretion from blood and tissue

The Rabinowitz study also showed that the average rate of ingestion of lead in Los Angeles over the period studied was 49.3 micrograms per day.

In your worksheet, set the values of the transfer coefficients aij and the ambient lead level L to those given above. In Part 2 you computed an equilibrium solution xe of the model system

x' = Ax + b.

The equilibrium solution is a particular solution of the nonhomogeneous system. Thus, to find the general solution. we need to find the general solution xh of the corresponding homogeneous system x' = Ax, and then add xh and xe.

  1. Evaluate xe with the given numbers. Interpret the components of xe in terms of what would be expected to happen to an individual who lived in Los Angeles for a long time (before clean-up of lead in the atmosphere).
  2. Use eigenvalues and eigenvectors of A to construct the general solution xh of the homogeneous system x' = Ax.
  3. Construct the general solution of x' = Ax + b.
  4. Solve the initial value problem
  5. x' = Ax + b, x(0) = 0.

    That is, find the vector function that describes the state of lead in the volunteer's body if there were no lead in his body when he moved to Los Angeles.

  6. Define the individual component functions x1, x2, x3 that give lead levels in the blood, tissue, and bones of the volunteer. Plot these functions over a span of 400 days. How do the levels at the end of that time compare with the levels in the equilibrium solution? About how long would it take to reach equilibrium levels? ("They are never reached" is not an adequate answer -- that's true only in theory.)
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