Cefamandole: PK after IV with six subjects in JAGS

April 12, 2015
By

(This article was first published on Wiekvoet, and kindly contributed to R-bloggers)

After last week’s post on Theoph I noticed the MEMSS library contained more PK data sets. The Cefamandole data is an IV data set with six subjects. It is somewhat challenging, since there seem to be several elimination rates.

Data

Data are displayed below. Notice that the lines seem somewhat curved, there is clearly more going on that just simple elimination.

Models

The first model fits concentration as function of two eliminations. The problem here is that the eliminations need to be ordered. All faster eliminations should have one prior, all slower another. After much fiddling around I found that I cannot force the ordering at subject level with priors or limits. Hence I chose a different way out. The model gets an extra component, it will predict 0 if the parameters are incorrectly ordered. This indeed gave the desired result in terms of model behavior.

The model itself though, was less to my linking. It goes way too low at the later times. Upon consideration, since the model assumes homoscedastic error it does not matter to the likelihoood if the fit at the lower end is not so good. After all, an error of 1 is nothing compared to errors of 10 or 20 at the upper end of the scale. In addition it can be argued that the error is not homoscedastic. If it were homoscedastic, the underlying data would show way more variation at the lower end of the scale. It would need confirmation from the measuring team, but for now I will assume proportional error.

Model 2: Proportional error

The proportional error will be implemented by using a log normal distribution for concentration. This was pretty simple to code. The log of the expected concentration is used in combination with dlnorm(). The only issue remaining is that expectation 0 for incorrectly ordered parameters has to be shifted a bit, log(0) is a bit of an issue.

This model looks pretty decent. There are still some things to do, similar to last week, translate the parameters c1star and c2star into the parameters which can be interpreted from PK perspective.

Model outputs

Model 1

Inference for Bugs model at “C:/Users/Kees/AppData/Local/Temp/RtmpSe5nAn/modele207c187661.txt”, fit using jags,
 4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 10
 n.sims = 2000 iterations saved
          mu.vect sd.vect    2.5%     25%     50%     75%    97.5%  Rhat n.eff
C[1]       91.473  24.508  51.690  76.444  88.708 103.600  144.987 1.016   160
C[2]      324.947 132.978 147.584 253.433 304.913 366.175  626.711 1.007   460
Ke[1]       0.020   0.002   0.016   0.018   0.019   0.021    0.024 1.045    76
Ke[2]       0.141   0.192   0.081   0.116   0.132   0.147    0.202 1.072   180
c1star[1]  58.783   8.381  43.892  53.222  58.514  63.406   76.601 1.047    64
c1star[2]  59.841  18.962  32.887  47.487  56.324  67.606  108.087 1.048    72
c1star[3]  92.337  10.569  70.647  85.720  92.805  99.469  112.338 1.041    74
c1star[4]  89.554  12.223  67.975  81.086  88.831  96.932  116.699 1.027   110
c1star[5] 146.419  21.504 106.395 132.153 145.199 159.149  193.558 1.032    84
c1star[6] 121.228  14.872  87.712 112.706 122.379 130.819  148.106 1.092    42
c2star[1] 426.241 107.777 278.572 355.121 406.845 471.583  703.121 1.038    92
c2star[2] 201.118  35.861 149.478 179.179 197.081 217.084  278.248 1.033   830
c2star[3] 489.294 218.162 268.493 349.208 419.178 537.325 1125.062 1.088    42
c2star[4] 449.402  68.164 340.627 402.410 440.666 487.652  603.535 1.038    75
c2star[5] 402.518  43.391 335.760 373.724 397.703 423.902  498.253 1.017   380
c2star[6] 140.893  46.737  79.548 114.343 133.240 157.967  237.080 1.024   650
ke1[1]      0.020   0.002   0.016   0.018   0.020   0.021    0.025 1.042    74
ke1[2]      0.021   0.004   0.016   0.019   0.020   0.023    0.032 1.060    57
ke1[3]      0.018   0.002   0.014   0.017   0.018   0.019    0.022 1.031    99
ke1[4]      0.020   0.002   0.017   0.019   0.020   0.021    0.025 1.023   120
ke1[5]      0.020   0.002   0.016   0.019   0.020   0.021    0.024 1.036    74
ke1[6]      0.019   0.002   0.015   0.018   0.019   0.020    0.022 1.071    46
ke2[1]      0.167   0.025   0.128   0.150   0.164   0.181    0.225 1.050    64
ke2[2]      0.101   0.026   0.067   0.085   0.096   0.111    0.164 1.050    91
ke2[3]      0.181   0.042   0.122   0.151   0.173   0.200    0.283 1.076    43
ke2[4]      0.143   0.019   0.110   0.130   0.141   0.155    0.181 1.040    74
ke2[5]      0.112   0.017   0.086   0.100   0.110   0.121    0.151 1.027   130
ke2[6]      0.118   0.038   0.062   0.093   0.114   0.137    0.205 1.051    61
deviance  464.708   7.996 451.264 458.919 464.029 469.608  482.206 1.006   550

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 31.8 and DIC = 496.6
DIC is an estimate of expected predictive error (lower deviance is better).

Model 2

Inference for Bugs model at “C:/Users/Kees/AppData/Local/Temp/RtmpSe5nAn/modele2045997fac.txt”, fit using jags,
 4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 10
 n.sims = 2000 iterations saved
          mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
C[1]       23.273  10.942   8.344  16.933  21.737  27.259  48.399 1.004   710
C[2]      159.545  32.361 106.916 141.343 157.165 172.618 230.957 1.004  1200
Ke[1]       0.010   0.001   0.009   0.010   0.010   0.011   0.012 1.010   270
Ke[2]       0.040   0.005   0.032   0.036   0.039   0.042   0.050 1.006   660
c1star[1]  14.141   3.164   8.481  11.970  13.923  16.049  20.826 1.012   230
c1star[2]  13.177   4.777   6.178   9.622  12.276  16.064  24.394 1.020   210
c1star[3]  34.030   6.876  20.963  29.102  33.778  38.458  47.854 1.006   450
c1star[4]  12.399   4.038   6.525   9.482  11.712  14.524  22.656 1.019   130
c1star[5]  40.359   8.756  24.398  34.248  40.002  45.836  58.609 1.007   360
c1star[6]  38.415   9.194  21.683  32.043  37.885  44.595  57.469 1.006   440
c2star[1] 123.292  17.872  93.250 110.418 121.625 134.114 164.930 1.000  2000
c2star[2] 134.174  18.465 103.312 121.677 132.580 145.038 176.803 1.004   680
c2star[3] 138.541  24.539  97.522 122.254 135.654 151.958 195.143 1.000  2000
c2star[4] 184.399  20.868 148.465 170.310 183.027 197.048 229.782 1.002  1500
c2star[5] 247.359  38.113 178.403 222.054 244.317 269.412 330.096 1.003  2000
c2star[6] 150.684  22.359 111.345 135.290 149.545 164.626 198.568 1.002  2000
ke1[1]      0.010   0.001   0.008   0.010   0.010   0.011   0.012 1.013   210
ke1[2]      0.012   0.001   0.009   0.011   0.011   0.012   0.014 1.022   180
ke1[3]      0.010   0.001   0.008   0.009   0.010   0.010   0.011 1.005   540
ke1[4]      0.011   0.001   0.009   0.010   0.010   0.011   0.013 1.020   130
ke1[5]      0.010   0.001   0.008   0.010   0.010   0.011   0.012 1.007   350
ke1[6]      0.010   0.001   0.009   0.010   0.010   0.011   0.012 1.005   540
ke2[1]      0.042   0.005   0.034   0.038   0.041   0.044   0.055 1.004   690
ke2[2]      0.041   0.006   0.032   0.037   0.040   0.044   0.056 1.013   230
ke2[3]      0.042   0.008   0.032   0.037   0.040   0.045   0.062 1.006   440
ke2[4]      0.037   0.003   0.031   0.034   0.036   0.038   0.044 1.008   320
ke2[5]      0.040   0.005   0.032   0.037   0.040   0.043   0.053 1.003  1100
ke2[6]      0.036   0.006   0.027   0.033   0.036   0.040   0.048 1.003  1200
deviance  372.564   7.382 359.811 367.354 372.140 376.777 388.833 1.001  2000

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 27.3 and DIC = 399.8
DIC is an estimate of expected predictive error (lower deviance is better).

Code used

library(MEMSS)
library(ggplot2)
library(R2jags)
png(‘Cefamandole.png’)
qplot(y=conc,x=Time,col=Subject,data=Cefamandole) +
    scale_y_log10(‘cefamandole (mcg/ml)’)+
    geom_line()+
    theme(legend.position=’bottom’)
dev.off()

library(R2jags)
datain <-  list(
    time=Cefamandole$Time,
    conc=Cefamandole$conc,
    n=nrow(Cefamandole),
    subject =c(1:nlevels(Cefamandole$Subject))[Cefamandole$Subject],
    nsubject = nlevels(Cefamandole$Subject)
)

model1 <- function() {
  tau <- 1/pow(sigma,2)
  sigma ~ dunif(0,100)
  for (i in 1:n) {
    pred[i] <- (ke1[subject[i]]
        (c1star[subject[i]]*exp(-ke1[subject[i]]*time[i])+
          c2star[subject[i]]*exp(-ke2[subject[i]]*time[i]))
    conc[i] ~ dnorm(pred[i],tau)
  }
  
  for(i in 1:nsubject) {
    ke1[i] ~ dlnorm(ke[1],kemtau[1])
    ke2[i] ~ dlnorm(ke[2],kemtau[2]) 
    c1star[i] ~ dlnorm(cm[1],ctau[1])
    c2star[i] ~ dlnorm(cm[2],ctau[2])
  }
  for (i in 1:2) {
    kem[i] ~ dunif(-10,10) 
    kemtau[i] <- 1/pow(kesd[i],2) 
    kesd[i] ~ dunif(0,10)
    Ke[i] <- exp(ke[i])
    cm[i] ~ dunif(-10,20)
    ctau[i] <- 1/pow(csd[i],2)
    csd[i] ~ dunif(0,100)
    C[i] <- exp(cm[i])    
  }
  ke <- sort(kem)


parameters <- c(‘Ke’,’ke1′,’ke2′,’c1star’,’c2star’,’C’)
inits <- function() {
  list(ke1 = rnorm(6,0.13,.03),
      ke2=  rnorm(6,0.0180,.005), 
      kem=  log(c(rnorm(1,0.13,.03),rnorm(1,0.0180,.005))),
      kesd =runif(2,.1,.4),
      cm = log(rnorm(2,25,5)),
      c1star=rnorm(6,25,3),
      c2star=rnorm(6,25,3)
  )}
jagsfit1 <- jags(datain, model=model1, 
    inits=inits,
    parameters=parameters,progress.bar=”gui”,
    n.iter=10000,
    n.chains=4,n.thin=10)
jagsfit1
#plot(jagsfit1)
#traceplot(jagsfit1)

Time <- seq(0,max(Cefamandole$Time))
la1 <- sapply(1:nrow(jagsfit1$BUGSoutput$sims.matrix),
    function(i) {
      C1 <- jagsfit1$BUGSoutput$sims.matrix[i,’C[1]’]
      C2 <- jagsfit1$BUGSoutput$sims.matrix[i,’C[2]’]
      k1 <- jagsfit1$BUGSoutput$sims.matrix[i,’Ke[1]’]
      k2 <- jagsfit1$BUGSoutput$sims.matrix[i,’Ke[2]’]
      y=C1*exp(-k1*Time)+C2*exp(-k2*Time)
    })
res1 <- data.frame(
    Time=Time,
    Conc=apply(la1,1,mean),
    C05=apply(la1,1,FUN=function(x) quantile(x,0.05)),
    C95=apply(la1,1,FUN=function(x) quantile(x,0.95)))
png(‘cef1.png’)
ggplot(res1, aes(x=Time)) +
    geom_line(aes(y=Conc))+
    scale_y_log10(‘cefamandole (mcg/ml)’)+
    geom_line(aes(y=conc,x=Time,col=Subject),alpha=1,data=Cefamandole)+
    geom_ribbon(aes(ymin=C05, ymax=C95),alpha=.2)+
    theme(legend.position=’bottom’)
dev.off()
#########
#
# model 2: proportional error
#
#########
model2 <- function() {
  tau <- 1/pow(sigma,2)
  sigma ~ dunif(0,100)
  for (i in 1:n) {
    pred[i] <- log(
        (ke1[subject[i]]
            (c1star[subject[i]]*exp(-ke1[subject[i]]*time[i])+
              c2star[subject[i]]*exp(-ke2[subject[i]]*time[i]))
            +.001*(ke1[subject[i]]>ke2[subject[i]]))
    conc[i] ~ dlnorm(pred[i],tau)
  }
  
  
  for(i in 1:nsubject) {
    ke1[i] ~ dlnorm(ke[1],kemtau[1])
    ke2[i] ~ dlnorm(ke[2],kemtau[2]) 
    c1star[i] ~ dlnorm(cm[1],ctau[1])
    c2star[i] ~ dlnorm(cm[2],ctau[2])
  }
  for (i in 1:2) {
    kem[i] ~ dunif(-10,10) 
    kemtau[i] <- 1/pow(kesd[i],2) 
    kesd[i] ~ dunif(0,10)
    Ke[i] <- exp(ke[i])
    cm[i] ~ dunif(-10,20)
    ctau[i] <- 1/pow(csd[i],2)
    csd[i] ~ dunif(0,100)
    C[i] <- exp(cm[i])    
  }
  ke <- sort(kem)
  


parameters <- c(‘Ke’,’ke1′,’ke2′,’c1star’,’c2star’,’C’)
inits <- function() {
  list(ke1 = rnorm(6,0.13,.03),
      ke2=  rnorm(6,0.0180,.005), 
      kem=  log(c(rnorm(1,0.13,.03),rnorm(1,0.0180,.005))),
      kesd =runif(2,.1,.4),
      cm = log(rnorm(2,25,5)),
      c1star=rnorm(6,25,3),
      c2star=rnorm(6,25,3)
  )}
jagsfit2 <- jags(datain, model=model2, 
    inits=inits,
    parameters=parameters,progress.bar=”gui”,
    n.iter=10000,
    n.chains=4,n.thin=10)
jagsfit2
la2 <- sapply(1:nrow(jagsfit2$BUGSoutput$sims.matrix),
    function(i) {
      C1 <- jagsfit2$BUGSoutput$sims.matrix[i,’C[1]’]
      C2 <- jagsfit2$BUGSoutput$sims.matrix[i,’C[2]’]
      k1 <- jagsfit2$BUGSoutput$sims.matrix[i,’Ke[1]’]
      k2 <- jagsfit2$BUGSoutput$sims.matrix[i,’Ke[2]’]
      y=C1*exp(-k1*Time)+C2*exp(-k2*Time)
    })


res2 <- data.frame(
    Time=Time,
    Conc=apply(la2,1,mean),
    C05=apply(la2,1,FUN=function(x) quantile(x,0.05)),
    C95=apply(la2,1,FUN=function(x) quantile(x,0.95)))

png(‘cef2.png’)
ggplot(res2, aes(x=Time)) +
    geom_line(aes(y=Conc))+
    scale_y_log10(‘cefamandole (mcg/ml)’)+
    geom_line(aes(y=conc,x=Time,col=Subject),alpha=1,data=Cefamandole)+
    geom_ribbon(aes(ymin=C05, ymax=C95),alpha=.2)+
    theme(legend.position=’bottom’)
dev.off()

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