Lead in the Body

Part 1. Background

No responses are required in Part 1.

Part 2. The Matrix-Vector Formulation

Answer questions 1 and 2 in the text cells below.

Answer to question 1:

Answer to question 2:

Enter the matrix A and the vector b:

A  =  {{-(a01 + a21 + a31), a12, 
            a13}, {a21, -(a02 + a12), 0}, {a31, 
            0, -a13}};
% // MatrixForm
b  =  {L, 0, 0}

Det[A]
Factor[%]

Do whatever calculations are necessary using the input cell below, and answer questions 3 and 4 in the text cell following.

Part 3. Ingestion of Lead

Enter the transfer coefficients and environmental lead burden from the Rabinowitz study.

a01  =  0.021;
a02  =  0.016;
a12  =  0.012;
a13  =  0.000035;
a21  =  0.011;
a31  =  0.0039;
L  =  49.3;

Mathematica automatically updates A and b with numerical values.

A  // MatrixForm
b

Here is (Subscript[x, e]) with numerical values.

xe

Answer to question 1:

Compute the eigenvalues and eigenvectors of A.

eigen  =  Eigensystem[A]

Assign eigenvalues to a vector lambda, and construct eigenvectors v[1], v[2], v[3].

lambda  =  ?;

v1   =  ?;
v2  =  ?;
v3  =  ?;

Describe the solution of the homogeneous system in terms of the eigenvalues and eigenvectors (question 2).

Answer:

Describe the solution of the nonhomogeneous system (dx/dt) = A x + b.

Answer:

Construct the matrix of eigenvectors.

P  =  Transpose[{v1, v2, v3}];
% // MatrixForm

Solve for the coefficients in the solution of the initial value problem, (dx/dt) = A x + b, x(0) = 0.

\(\(\( (*\ 
      Enter\ formula\ to\ define\ the\ coefficients\ as\ entries\ of\ a\ vector\ c\ *) \)\(
\)\(c\  = \ \(?;\)\)\)\)

The following command will define the blood lead function x1 if you have the coefficient vector set up.  Copy and edit to define x2 and x3.

x1[t_] :=  
        c[[1]]*P[[1, 1]]*Exp[lambda[[1]]*t] + 
          c[[2]]*P[[1, 2]]*Exp[lambda[[2]]*t] + 
          c[[3]]*P[[1, 3]]*Exp[lambda[[3]]*t] + 
          xe[[1]];

Plot the blood, tissue, and bone functions together.

xplot  =  
        Plot[{x1[t], x2[t], x3[t]}, {t, 0, 400}, 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1], 
              RGBColor[0, 1, 0]}];

Answer to the rest of question 5:

Part 4. A Lead-Free Environment

Answer to question 1:

Think about how to set up the linear system that determines the coefficients in the solution of the new initial value problem.  In the following commands, we repeat the matrix P and then multiply its first column by a certain constant.  Think about how you can use this type of operation to find the matrix for your linear system.  Note that in Mathematica, it's better to transpose the matrix first and think in terms of row operations.

P  //  MatrixForm

Transpose[P];
ReplacePart[%, Exp[lambda[[1]]*400]*%[[1]], 1];

Define the vector of constants for your linear system.

p  =  {x1[400], x2[400], x3[400]}

Enter a formula that will compute the vector of coefficients for the solution of your initial value problem:

q  =  ?;

As we did with the x's, we provide the first function definition.  Copy and edit to get the other two.

y1[t_] :=  
        q[[1]]*P[[1, 1]]*Exp[lambda[[1]]*t] + 
          q[[2]]*P[[1, 2]]*Exp[lambda[[2]]*t] + 
          q[[3]]*P[[1, 3]]*Exp[lambda[[3]]*t];

Plot the component functions for y, and describe what you see.

yplot  =  
        Plot[{y1[t], y2[t], y3[t]}, {t, 400, 800}, 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1], 
              RGBColor[0, 1, 0]}];

Combine the x and y plots:

Show[xplot, yplot];

Answers to the rest of the questions in Part 4:

Part 5. Summary

Answers to the questions in part 5: