Ch 2.7 Drug Assimilation into Blood

• We readily take pills without necessarily having a good understanding of how these drugs are absorbed into the bloodstream or for how long they have an effect on us.

• The warnings on the packaging list some of the effects and are intended to ensure safety for all users.

• The GI-tract compartment has a single input and output.
• The bloodstream compartment has a single input and output.

• Balance law yields two word equations, one for each compartment.

For the our variables, let

• x(t) be the amount of a drug in the GI-tract at time t
  
• y(t) the amount in the bloodstream at time t.

Consider two scenarios to model:

• A single cold pill taken once (Model I).
• A course of cold pills where drug intake occurs continuously (Model II).

• \( x_d(t) = \) amount of decongestant (mg) in GI-tract at \( t \) (hours).
• \( x_a(t) = \) amount of antihistamine (mg) in GI-tract at \( t \) (hours).

• \( y_d(t) = \) amount of decongestant (mg) in blood at \( t \) (hours).
• \( y_a(t) = \) amount of antihistamine (mg) in blood at \( t \) (hours).

• In GI-tract, pill has been swallowed and enters GI-tract.
• Pill dissolves quickly and enters bloodstream from GI-tract.
• Thus for GI-tract there is only an output term.

• \( x_d(t) = \) amount of decongestant (mg) in GI-tract at \( t \) (hours).
• \( x_a(t) = \) amount of antihistamine (mg) in GI-tract at \( t \) (hours).
• \( y_d(t) = \) amount of decongestant (mg) in blood at \( t \) (hours).
• \( y_a(t) = \) amount of antihistamine (mg) in blood at \( t \) (hours).

\[ \begin{align*} \frac{dx}{dt} &= -k_1x, \,\,\, x(0)= x_0 \\ \frac{dy}{dt} &= k_1x - k_2 y, \,\,\, y(0)= y_0 \end{align*} \]

• System of equations for both drugs and both compartments:

\[ \begin{align*} \frac{dx_d}{dt} &= -k_{1d}x_d, \,\,\, x_d(0)= x_{d0} \\ \frac{dx_a}{dt} &= -k_{1a}x_a, \,\,\, x_a(0)= x_{a0} \\ \frac{dy_d}{dt} &= k_{1d}x_d - k_{2d} y_d, \,\,\, y_d(0)= y_{d0} \\ \frac{dy_a}{dt} &= k_{1a}x_a - k_{2a} y_a, \,\,\, y_a(0)= y_{a0} \end{align*} \]

• Both \( x \) and \( y \) for the two drugs approach zero as \( t \) increases.
• See analytical solutions in reading:

\[ \begin{align*} \frac{dx}{dt} &= -k_1x, \,\,\, x(0)= x_0 \\ & \implies x(t) = x_0 e^{-k_1 t} \\ \frac{dy}{dt} &= k_1x - k_2 y, \, \,\, y(0)= y_0 \\ & \implies y(t) = \frac{k_1 x_0}{k_1-k_2}\left( e^{-k_2 t} - e^{-k_1 t} \right) \end{align*} \]

Ch27model1 <- function(T,n) {

#Chapter2.7 Model I
#Perform Rk4 for decongestant & antihistamine

#T is the time length for [0, T]
#n is the number of time steps
h = T/n  #This is the time step size

#System Parameters
t0 <- 0
xd0 <- 1   #initial GI decongestant
xa0 <- 1   #initial GI antihistamine
yd0 <- 0   #initial blood decongestant
ya0 <- 0   #initial blood antihistamine
k1d <- 1.3860  #k1 for GI decongestant
k1a <- 0.6931  #k1 for GI antihistamine
k2d <- 0.1386  #k2 for blood decongestant  
k2a <- 0.0231  #k2 for blood antihistamine

#System of ODEs
f1d <- function(x) {-k1d*x}   
f1a <- function(x) {-k1a*x}   
f2d <- function(x,y) {k1d*x - k2d*y}   
f2a <- function(x,y) {k1a*x - k2a*y}   

#Initialize time, GI-tract x, and bloodstream y
t <- rep(0, n)
xd <- rep(0, n)
xa <- rep(0, n)
yd <- rep(0, n)
ya <- rep(0, n)
t[1] <- t0
xd[1] <- xd0
xa[1] <- xa0
yd[1] <- yd0
ya[1] <- ya0

  #Runge-Kutta Loop: GI-Tract Compartment
  for(i in 1:n) {
    a1 <- h*f1d(xd[i])    #f1d = slope of xd
    a2 <- h*f1a(xa[i])    #f2a = slope of xa
    b1 <- h*f1d(xd[i]+0.5*a1) #Half-step predictor 
    b2 <- h*f1a(xa[i]+0.5*a2) #Half-step predictor 
    c1 <- h*f1d(xd[i]+0.5*b1) #Half-step predictor 
    c2 <- h*f1a(xa[i]+0.5*b2) #Half-step predictor 
    d1 <- h*f1d(xd[i]+c1) #Full-step predictor 
    d2 <- h*f1a(xa[i]+c2) #Full-step predictor 
    xd[i+1] <- xd[i]+(1/6)*(a1+2*b1+2*c1+d1) 
    xa[i+1] <- xa[i]+(1/6)*(a2+2*b2+2*c2+d2) 
    t[i+1] <- t[i] + h
  }

  #Runge-Kutta Loop: Bloodstream Compartment
  for(i in 1:n) {
    a1 <- h*f2d(xd[i],yd[i])          
    a2 <- h*f2a(xa[i],ya[i])          
    b1 <- h*f2d(xd[i]+0.5*a1,yd[i]+0.5*a1) 
    b2 <- h*f2a(xa[i]+0.5*a2,ya[i]+0.5*a2) 
    c1 <- h*f2d(xd[i]+0.5*b1,yd[i]+0.5*b1)  
    c2 <- h*f2a(xa[i]+0.5*b2,ya[i]+0.5*b2)  
    d1 <- h*f2d(xd[i]+c1,yd[i]+c1)          
    d2 <- h*f2a(xa[i]+c2,ya[i]+c2)          
    yd[i+1] <- yd[i]+(1/6)*(a1+2*b1+2*c1+d1) 
    ya[i+1] <- ya[i]+(1/6)*(a2+2*b2+2*c2+d2) 
  }

#Plot results
plot(t,xd,               
        main = "Model I: GI-Tract",
        xlab = "t (hours)",           
        ylab = "Amount (mg)",       
        type="l",col="blue", 
        xlim=c(0,T),                  
        ylim=c(0,1)                   
       )
lines(t,xa, type="l",col="red")  
legend("topright",
         legend = c("Decongestant", "Antihistamine"),
         col=c("blue","red"),  
         lty=c(1,1)  
         )

plot(t,yd,               
       main = "Model I: Bloodstream",
       xlab = "t (hours)",           
       ylab = "Amount (mg)",       
       type="l",col="blue", 
       xlim=c(0,T),                  
       ylim=c(0,1.5)                   
       )
lines(t,ya, type="l",col="red")   
legend("topright",
         legend = c("Decongestant", "Antihistamine"),
         col=c("blue","red"),  
         lty=c(1,1.5)  
         )
}

#System Parameters
k1d <- 1.3860    #k1 for GI decongestant
k1a <- 0.6931    #k1 for GI antihistamine
k2d <- 0.1386    #k2 for blood decongestant  
k2a <- 0.0231    #k2 for blood antihistamine
Id  <- 1
Ia  <- 1

#System of ODEs
f1d <- function(x) {Id - k1d*x}   
f1a <- function(x) {Ia - k1a*x}   
f2d <- function(x,y) {k1d*x - k2d*y}   
f2a <- function(x,y) {k1a*x - k2a*y}   

• Both \( x(t) \) and \( y(t) \) approach \( I/k_1 \) and \( I/k_2 \), respectively, as \( t \) increases (see analytical solutions in reading):
  

\[ \begin{align*} \frac{dx}{dt} &= I -k_1x, \,\,\, x(0)= 0 \\ & \implies x(t) = \frac{I}{k_1} \left( 1- e^{-k_1 t}\right) \\ \frac{dy}{dt} &= k_1x - k_2 y, \,\,\, y(0)= 0 \\ & \implies y(t) = \frac{I}{k_2} \left[ 1 - \frac{1}{k_2-k_1}\left( k_2 e^{-k_1 t} - k_1 e^{-k_2 t} \right) \right] \end{align*} \]