A two-compartment pharmacokinetic model
restart:with(DEtools):
The non-homogeneous linear system for the model. d(t) is the drug dosage versus time
system1:=diff(x(t),t)=-a*x(t)+d(t),diff(y(t),t)=a*x(t)-b*y(t);
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The homogeneous version of the same system
system2:=diff(x(t),t)=-a*x(t),diff(y(t),t)=a*x(t)-b*y(t);
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The solution of the homogeneous system. Note that the eigenvalues are -a, -b , so as time increases, the solution of the homogeneous system will converge to zero
dsolve([system2],{x(t),y(t)});
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Let us first consider the example of a single dosage of the drug administered from t=0 to t=2:
a:=2;b:=1.5;d:=t->1-Heaviside(t-2);
IiIj
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plot(d(t),t=0..5,0..2,scaling=constrained);
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ics:=x(0)=0,y(0)=0;dsolve([system1,ics]);
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plot(-(8/3)*exp(-(3/2)*t)+2*exp(-2*t)+2/3-(2/3)*Heaviside(t-2)-2*Heaviside(t-2)*exp(-2*t+4)+(8/3)*Heaviside(t-2)*exp(-(3/2)*t+3),t=0..10);
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In the next example, the drug is administered periodically:
d:=t->0.3*(cos(t)+1);
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dsolve([system1,ics]);
PCQvLUkieEc2IjYjSSJ0R0YmLCotSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiZGJyMiIiQiI0QtSSRzaW5HRixGJyNGMCIjXSNGMCIjPyIiIi1JJGV4cEdGLDYjLCRGKCEiIyMhI0YiJCsiLy1JInlHRiZGJywsRiojIiNDIiREJEYyIyIjVUZHI0Y4IiImRjhGOSMiI0ZGMS1GOjYjLCRGKCMhIiQiIiMjISMpKSIjbA==
plot((16/65)*cos(t)+(28/65)*sin(t)+2/3+(18/5)*exp(-2*t)-(176/39)*exp(-(3/2)*t),t=0..40);
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