'Boeckmann, Sheiner and Beal (1994) report data from a study by Dr. Robert Upton of the kinetics of the anti-asthmatic drug theophylline. Twelve subjects were given oral doses of theophylline then serum concentrations were measured at 11 time points over the next 25 hours.
These data are analyzed in Davidian and Giltinan (1995) and Pinheiro and Bates (2000) using a two-compartment open pharmacokinetic model, for which a self-starting model function,
SSfol, is available.'Model
The model which is used in the example for the data is SSfol. Its value is:Dose * exp(lKe+lKa-lCl) * (exp(-exp(lKe)*input) - exp(-exp(lKa)*input))
/ (exp(lKa) - exp(lKe))
With input the time variable. This looks very complex, but there seems to be a trick here. What is estimated is not Ke (elimination rate) but its natural logarithm. The same is true for Ka and CL. This is probably done to keep the parameters positive. So in terms of ordinary parameters:Dose * (Ke+Ka)/CL * (exp(-Ke*input) - exp(-Ka*input)) / (Ka - Ke)
This looks much more simple.
model1 <- function() {
tau <- 1/pow(sigma,2)
sigma ~ dunif(0,100)
for (i in 1:n) {
pred[i] <- c0star[subject[i]]*
(exp(-ke[subject[i]]*time[i])-exp(-ka[subject[i]]*time[i]))
conc[i] ~ dnorm(pred[i],tau)
}
for(i in 1:nsubject) {
ka[i] ~ dlnorm(kam,katau)
ke[i] ~ dlnorm(kem,ketau)
c0star[i] ~ dunif(0,50)
# c0star=dose*((ke*ka)/CL)/(ka-ke)
CL[i] <- Dose[i]*ka[i]*ke[i]/((ka[i]-ke[i])*c0star[i])
CLr[i] <- CL[i]-CLm
z[i] ~ dnorm(CLr[i],CLtau)
}
kam ~ dunif(-10,10)
katau <- 1/pow(kasd,2)
kasd ~ dunif(0,100)
Kam <- exp(kam)
# ta0_5 <- log(2)/Kam
kem ~ dunif(-10,10)
ketau <- 1/pow(kesd,2)
kesd ~ dunif(0,100)
Kem <- exp(kem)
# te0_5 <- log(2)/Kem
CLm ~ dunif(0,100)
CLtau <- 1/pow(CLsd,2)
CLsd ~dunif(0,100)
}
Inference for Bugs model at "C:/Users/.../modeld8059bf37f0.txt", fit using jags,
4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 2
n.sims = 10000 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
CL[1] 0.026 0.002 0.022 0.025 0.026 0.028 0.030 1.003 1100
CL[2] 0.035 0.002 0.031 0.034 0.035 0.037 0.039 1.001 4200
CL[3] 0.051 0.004 0.044 0.048 0.051 0.053 0.058 1.002 2300
CL[4] 0.038 0.002 0.034 0.037 0.038 0.040 0.044 1.006 460
CL[5] 0.041 0.003 0.036 0.039 0.041 0.043 0.047 1.005 620
CL[6] 0.041 0.003 0.035 0.039 0.041 0.042 0.046 1.002 2200
CL[7] 0.037 0.003 0.033 0.036 0.037 0.039 0.043 1.003 990
CL[8] 0.043 0.003 0.038 0.041 0.043 0.044 0.048 1.004 960
CL[9] 0.046 0.004 0.039 0.044 0.046 0.048 0.053 1.002 1600
CL[10] 0.046 0.003 0.040 0.044 0.046 0.049 0.053 1.002 1600
CL[11] 0.044 0.003 0.038 0.042 0.044 0.046 0.051 1.002 2000
CL[12] 0.033 0.002 0.029 0.031 0.033 0.034 0.038 1.007 430
CLm 0.040 0.003 0.035 0.038 0.040 0.042 0.046 1.004 1200
Kam 1.673 0.450 0.998 1.386 1.616 1.885 2.700 1.001 6400
Kem 0.086 0.005 0.076 0.082 0.085 0.089 0.096 1.010 420
ka[1] 1.482 0.205 1.127 1.338 1.466 1.607 1.934 1.001 9300
ka[2] 0.656 0.089 0.507 0.593 0.649 0.710 0.854 1.006 510
ka[3] 3.964 0.860 2.657 3.372 3.836 4.407 5.973 1.001 5700
ka[4] 0.954 0.125 0.730 0.869 0.948 1.032 1.219 1.009 310
ka[5] 2.110 0.325 1.555 1.882 2.080 2.302 2.827 1.003 1100
ka[6] 2.385 0.441 1.671 2.073 2.336 2.637 3.411 1.002 2200
ka[7] 1.208 0.171 0.910 1.092 1.195 1.309 1.584 1.002 2000
ka[8] 1.502 0.183 1.171 1.376 1.493 1.615 1.890 1.003 1100
ka[9] 1.287 0.245 0.878 1.112 1.261 1.429 1.834 1.002 1600
ka[10] 0.787 0.139 0.551 0.690 0.774 0.871 1.094 1.004 910
ka[11] 1.446 0.248 1.021 1.274 1.422 1.593 2.006 1.002 2400
ka[12] 8.409 5.846 4.277 5.818 7.078 8.998 20.996 1.001 7100
ke[1] 0.080 0.008 0.061 0.075 0.081 0.085 0.093 1.004 990
ke[2] 0.084 0.007 0.069 0.080 0.084 0.089 0.099 1.003 1100
ke[3] 0.085 0.007 0.072 0.081 0.085 0.089 0.101 1.002 2500
ke[4] 0.089 0.009 0.076 0.083 0.088 0.094 0.110 1.017 210
ke[5] 0.089 0.008 0.076 0.083 0.087 0.093 0.109 1.007 440
ke[6] 0.085 0.007 0.071 0.081 0.085 0.089 0.100 1.003 1300
ke[7] 0.086 0.007 0.073 0.082 0.086 0.090 0.104 1.006 530
ke[8] 0.086 0.007 0.073 0.081 0.085 0.090 0.101 1.008 420
ke[9] 0.086 0.008 0.071 0.081 0.085 0.090 0.103 1.005 970
ke[10] 0.086 0.008 0.071 0.081 0.085 0.090 0.104 1.009 480
ke[11] 0.086 0.008 0.072 0.081 0.085 0.090 0.103 1.005 630
ke[12] 0.088 0.008 0.075 0.083 0.087 0.092 0.106 1.008 340
deviance 200.433 9.015 184.713 193.957 199.825 206.049 220.328 1.002 2200
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 = 40.6 and DIC = 241.0
DIC is an estimate of expected predictive error (lower deviance is better).
Estimates using nls
The example code uses nls() to fit the model. Below the model is wrapped in an apply. In addition, the exponent is taken.
library(MEMSS)
la <- lapply(levels(Theoph$Subject), function(iSubject)
{Theoph.i <- subset(Theoph, Subject == iSubject)
fm1 <- nls(conc ~ SSfol(Dose, Time, lKe, lKa, lCl),
data = Theoph.i)
ee <- exp(coef(fm1))
dd <- data.frame(Ke=ee[1],Ka=ee[2],CL=ee[3])
row.names(dd) <- iSubject
dd
}
)
nlsres <- do.call(rbind,la)
nlsres
Ke Ka CL
A 0.05395450 1.7774170 0.01992348
B 0.07396616 0.6955019 0.03244300
C 0.09812331 3.8490404 0.05724601
D 0.10557584 0.8328979 0.04199695
E 0.10166133 1.9426573 0.04476553
F 0.08142497 2.4535654 0.03955890
G 0.08746697 1.1714752 0.03739991
H 0.08843514 1.4715045 0.04360427
I 0.09952647 1.1637219 0.05113727
J 0.10224639 0.6797358 0.05159475
K 0.09195675 1.3755229 0.04646244
Model in JAGS
In JAGS there are more direct mechanisms to keeping parameters positive. However, there is one problem, an alternative set of parameters for subject A, Ka=0.053, Ke=1.77 does equally well. Combining MCMC samples from these two solutions just results in nonsense. To avoid this, I have placed the whole scaling in a parameter C0star, which is positive. In subsequent code individual CL and its hyper parameter are estimated from C0star. To keep this as a directed acyclic graph (DAG) additional variable z and parameters CLr are introduced. In addition, dose has been made a subject level variable, rather than associated with time points.model1 <- function() {
tau <- 1/pow(sigma,2)
sigma ~ dunif(0,100)
for (i in 1:n) {
pred[i] <- c0star[subject[i]]*
(exp(-ke[subject[i]]*time[i])-exp(-ka[subject[i]]*time[i]))
conc[i] ~ dnorm(pred[i],tau)
}
for(i in 1:nsubject) {
ka[i] ~ dlnorm(kam,katau)
ke[i] ~ dlnorm(kem,ketau)
c0star[i] ~ dunif(0,50)
# c0star=dose*((ke*ka)/CL)/(ka-ke)
CL[i] <- Dose[i]*ka[i]*ke[i]/((ka[i]-ke[i])*c0star[i])
CLr[i] <- CL[i]-CLm
z[i] ~ dnorm(CLr[i],CLtau)
}
kam ~ dunif(-10,10)
katau <- 1/pow(kasd,2)
kasd ~ dunif(0,100)
Kam <- exp(kam)
# ta0_5 <- log(2)/Kam
kem ~ dunif(-10,10)
ketau <- 1/pow(kesd,2)
kesd ~ dunif(0,100)
Kem <- exp(kem)
# te0_5 <- log(2)/Kem
CLm ~ dunif(0,100)
CLtau <- 1/pow(CLsd,2)
CLsd ~dunif(0,100)
}
The usual code for running the model. I have noticed quite some sensitivity to inits while writing this post.
library(R2jags)
datain <- list(
time=Theoph$Time,
conc=Theoph$conc,
n=nrow(Theoph),
subject =c(1:nlevels(Theoph$Subject))[Theoph$Subject],
nsubject = nlevels(Theoph$Subject),
z=rep(0, nlevels(Theoph$Subject)),
Dose=unique(subset(Theoph,, c(Subject,Dose)))$Dose
)
parameters <- c('Kem','Kam','ke','ka','CL','CLm')
inits <- function() {
list(kam = rnorm(1,log(1),.3),
ka= rnorm(12,1,.2),
kem= rnorm(1,log(.08),.01),
ke= rnorm(12,.08,.01),
c0star = runif(12,9,11)
)}
jagsfit <- jags(datain, model=model1,
inits=inits,
parameters=parameters,progress.bar="gui",
n.iter=10000,
n.chains=4,n.thin=2)
Results
It seems the estimated parameters are close to the parameters from nls(). This is for this blog post sufficient. The data has quite some shrinkage, almost all subjects have Ke close 0.86. CL and Ka also have shrinkage, but much less than Ke.Inference for Bugs model at "C:/Users/.../modeld8059bf37f0.txt", fit using jags,
4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 2
n.sims = 10000 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
CL[1] 0.026 0.002 0.022 0.025 0.026 0.028 0.030 1.003 1100
CL[2] 0.035 0.002 0.031 0.034 0.035 0.037 0.039 1.001 4200
CL[3] 0.051 0.004 0.044 0.048 0.051 0.053 0.058 1.002 2300
CL[4] 0.038 0.002 0.034 0.037 0.038 0.040 0.044 1.006 460
CL[5] 0.041 0.003 0.036 0.039 0.041 0.043 0.047 1.005 620
CL[6] 0.041 0.003 0.035 0.039 0.041 0.042 0.046 1.002 2200
CL[7] 0.037 0.003 0.033 0.036 0.037 0.039 0.043 1.003 990
CL[8] 0.043 0.003 0.038 0.041 0.043 0.044 0.048 1.004 960
CL[9] 0.046 0.004 0.039 0.044 0.046 0.048 0.053 1.002 1600
CL[10] 0.046 0.003 0.040 0.044 0.046 0.049 0.053 1.002 1600
CL[11] 0.044 0.003 0.038 0.042 0.044 0.046 0.051 1.002 2000
CL[12] 0.033 0.002 0.029 0.031 0.033 0.034 0.038 1.007 430
CLm 0.040 0.003 0.035 0.038 0.040 0.042 0.046 1.004 1200
Kam 1.673 0.450 0.998 1.386 1.616 1.885 2.700 1.001 6400
Kem 0.086 0.005 0.076 0.082 0.085 0.089 0.096 1.010 420
ka[1] 1.482 0.205 1.127 1.338 1.466 1.607 1.934 1.001 9300
ka[2] 0.656 0.089 0.507 0.593 0.649 0.710 0.854 1.006 510
ka[3] 3.964 0.860 2.657 3.372 3.836 4.407 5.973 1.001 5700
ka[4] 0.954 0.125 0.730 0.869 0.948 1.032 1.219 1.009 310
ka[5] 2.110 0.325 1.555 1.882 2.080 2.302 2.827 1.003 1100
ka[6] 2.385 0.441 1.671 2.073 2.336 2.637 3.411 1.002 2200
ka[7] 1.208 0.171 0.910 1.092 1.195 1.309 1.584 1.002 2000
ka[8] 1.502 0.183 1.171 1.376 1.493 1.615 1.890 1.003 1100
ka[9] 1.287 0.245 0.878 1.112 1.261 1.429 1.834 1.002 1600
ka[10] 0.787 0.139 0.551 0.690 0.774 0.871 1.094 1.004 910
ka[11] 1.446 0.248 1.021 1.274 1.422 1.593 2.006 1.002 2400
ka[12] 8.409 5.846 4.277 5.818 7.078 8.998 20.996 1.001 7100
ke[1] 0.080 0.008 0.061 0.075 0.081 0.085 0.093 1.004 990
ke[2] 0.084 0.007 0.069 0.080 0.084 0.089 0.099 1.003 1100
ke[3] 0.085 0.007 0.072 0.081 0.085 0.089 0.101 1.002 2500
ke[4] 0.089 0.009 0.076 0.083 0.088 0.094 0.110 1.017 210
ke[5] 0.089 0.008 0.076 0.083 0.087 0.093 0.109 1.007 440
ke[6] 0.085 0.007 0.071 0.081 0.085 0.089 0.100 1.003 1300
ke[7] 0.086 0.007 0.073 0.082 0.086 0.090 0.104 1.006 530
ke[8] 0.086 0.007 0.073 0.081 0.085 0.090 0.101 1.008 420
ke[9] 0.086 0.008 0.071 0.081 0.085 0.090 0.103 1.005 970
ke[10] 0.086 0.008 0.071 0.081 0.085 0.090 0.104 1.009 480
ke[11] 0.086 0.008 0.072 0.081 0.085 0.090 0.103 1.005 630
ke[12] 0.088 0.008 0.075 0.083 0.087 0.092 0.106 1.008 340
deviance 200.433 9.015 184.713 193.957 199.825 206.049 220.328 1.002 2200
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 = 40.6 and DIC = 241.0
DIC is an estimate of expected predictive error (lower deviance is better).
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