MATH564 Midterm Examination

Model solution

Problem 1: Exercise 1, p312

The time-varying amount of lead (Pb) in a mammal is studied through a 3-compartment model composed of: (1) blood (Subscript[x, 1])(t), (2) soft tissue (Subscript[x, 2])(t), and (3) bone (Subscript[x, 3])(t). The model is specified by the first-order differential system x'=A x+f.

R={{21.1,16.2,0.},{0.,12.4,0.035},{11.1,0.,0.},{3.9,0.,0.}};
MatrixForm[R]

RT=Transpose[R];{m,n}=Dimensions[RT]

z=Table[0,m];o=Table[1,n];Id=IdentityMatrix[m];
A=Transpose[Insert[Id,z,1]].R-DiagonalMatrix[RT.o];
MatrixForm[A]

IL=49.3;f={IL,0.,0.};x0={0.,0.,0.}

(a) Long-range lead forecast

xinf=-Inverse[A].f

sol=NDSolve[{x'[t]==A.x[t]+f,x[0]==z},x[t],{t,0,1}]

(b) Lead forecast at 1 year

(c) Percentage of lead in bones

(d) Long range, 1 year forecast at double, half ingestion rate

Problem 2: Exercise 2, p313

(a) Time to accumulate 2500 μg Pb

(b) Decay of 2500 μg Pb after 1 year in clean environment

(c) Estimate of drug removal rate

(d) Pb accumulation with drug, contaminated environment

(e) Model half-life

(f) Lead elimnation after 100 days

Problem 3: Repeat Exercise 2 for a more detailed compartment model: environment, skeletal system, blood, lymph, muscle, liver, kidney, remaining soft tissue

(a) Time to accumulate 2500 μg Pb

(b) Decay of 2500 μg Pb after 1 year in clean environment

(c) Estimate of drug removal rate

(d) Pb accumulation with drug, contaminated environment

(e) Model half-life

(f) Lead elimnation after 100 days