In this third PK posting I move to chapter 10, study problem 4 of Rowland and Tozer (Clinical pharmacokinetics and pharmacodynamics, 4th edition). In this problem one subject gets a 24 hours continuous dose. In many respects the Jags calculation is not very different from before, but the relation between parameters in the model and PK parameters is a bit different.

Data

Data is from infusion at constant rate, (1.5 µm/hr, but the calculations come out on the expected values with 1.5 µg/hr, and no molecular weight is given in the question) for 24 hours. Drug concentration (µg/L) in plasma is measured for 28 hours.
ch10sp4 <- data.frame(
    time=c(0,1,2,4,6,7.5,10.5,14,18,24,25,26,27,28),
    conc=c(0,4.2,14.5,21,23,19.8,22,20,18,21,18,11.6,7.1,4.1))

Model

The model is straightforward, but does suffer a bit from sensitivity of init values. The construction (time<24 and time>=24 is just a quick way to ensure that the up and down doing parts of the model sit at the correct spots in time.
library(R2jags)
datain <- as.list(ch10sp4)
datain$n <- nrow(ch10sp4)
datain$Rinf <- 1.5*1000
model1 <- function() {
  for (i in 1:n) {
    pred[i] <- Css*((1-exp(-k*time[i]))*(time[i]<24)
                  +exp(-k*(time[i]-24))*(time[i]>=24))
    conc[i] ~ dnorm(pred[i],tau)
  }
  tau <- 1/pow(sigma,2)
  sigma ~ dunif(0,100)
  Css <- Rinf/CL
  k <- CL/V
  V ~ dlnorm(2,.01)
  CL ~ dlnorm(1,.01)
  t0_5 <- log(2)/k
  Cssalt <- 3000/CL
  h1 <- Cssalt*(1-exp(-k))
  h2 <- Cssalt*(1-exp(-2*k))
  CLpermin <- CL/60
}

parameters <- c('k','CL','V','t0_5','Cssalt','h1','h2','Css','CLpermin')
inits <- function() 
  list(V=rlnorm(1,log(100),.1),CL=rlnorm(1,log(70),.1),sigma=rlnorm(1,1,.1))
jagsfit <- jags(datain, model=model1, 
    inits=inits, 
    parameters=parameters,progress.bar=”gui”,
    n.chains=4,n.iter=14000,n.burnin=5000,n.thin=2)
jagsfit
Expected values: CL=1.2 L/min, t_1/2=1.6 hr, V=164 L. The values are a bit off, it was given in the text that Css is 21 mg/L while the model finds 22.
Three additional parameters; Cssalt, h1, h2 correspond to part (c) of the question; what concentrations if the dosage is doubled. Expected answers: Cssalt=42, h1=14.8, h2=24.

Plot of the model

part (b) Is the curve in the up and down going part equally steep?

This requires a different model, since right now by definition they are equally shaped. The parameter k12, difference between k1 and k2, is the parameter of interest.
datain2 <- as.list(ch10sp4)
datain2$n <- nrow(ch10sp4)
model2 <- function() {
  for (i in 1:n) {
    pred[i] <- Css*((1-exp(-k1*time[i]))*(time[i]<24)
          +exp(-k2*(time[i]-24))*(time[i]>=24))
    conc[i] ~ dnorm(pred[i],tau)
  }
  tau <- 1/pow(sigma,2)
  sigma ~ dunif(0,100)
  k1 ~ dlnorm(0,0.001)
  k2 ~ dlnorm(0,0.001)
  Css ~ dnorm(21,.01)
  k12 <- k1-k2
}

parameters2 <- c('k1','k2','k12','Css')
jagsfit2 <- jags(datain2, model=model2,  
    parameters=parameters2,progress.bar=”gui”,
    n.chains=4,n.iter=14000,n.burnin=5000,n.thin=2)

Previous posts in this series:

Simple Pharmacokinetics with Jags