biexp: Two-phase half-life estimation by biexponential model

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Description

Estimation of initial and terminal half-life by fitting a biexponential model.

Usage

1
biexp(conc, time, log.scale=FALSE, tol=1E-9, maxit=500)

Arguments

conc

Levels of concentrations as a vector.

time

Time points of concentration assessment as a vector. One time point for each concentration measured needs to be specified.

log.scale

Logical value indicating whether fitting is performed on the observed or log-scale (default=FALSE).

tol

Relative error tolerance (default=1E-9).

maxit

Maximum number of iterations (default=500).

Details

Estimation of initial and terminal half-life using the biexponential y=a1*exp(-b1*x)+a2*exp(-b2*x) model with a parameterization to ensure b1 > b2 > 0 fitted by the least squares criteria with function optim of package base with method "Nelder-Mead". Curve peeling (Foss, 1969) is used get start values for nonlinear model fitting. When no adequate starting values are determined by curve peeling, a single exponential model is fitted with starting values obtained from an OLS regression on log transformed values with a parameterization to ensure a slope > 0.

Fitting on the log-scale is based on the transform-both-sides approach described for example in chapter 4 of Bonate (2006) which is useful for some error distributions. An additional discussion regarding weighting schemes can be found in Gabrielsson and Weiner (2000, pages 368-374).

Value

A list of S3 class "halflife" containing the following components:

parms

half-life and model estimates.

time

time points of concentration assessments.

conc

levels of concentrations.

method

"biexp".

Note

Records including missing values and values below or equal to zero are omitted.

Author(s)

Martin J. Wolfsegger and Thomas Jaki

References

Bonate P. L. (2006). Pharmacokinetic-Pharmacodynamic Modeling and Simulation. Springer, New York.

Gabrielsson J. and Weiner D. (2000). Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Applications. 4th Edition. Swedish Pharmaceutical Press, Stockholm.

Foss S. D. (1969). A Method for Obtaining Initial Estimates of the Parameters in Exponential Curve Fitting. Biometrics, 25:580-584.

Pinheiro J. C. and Bates D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York.

Wolfsegger M. J. and Jaki T. (2009). Non-compartmental Estimation of Pharmacokinetic Parameters in Serial Sampling Designs. Journal of Pharmacokinetics and Pharmacodynamics, 36(5):479-494.

See Also

lee

Examples

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
#### example from Pinheiro J.C. and Bates D.M. (2000, page 279)
#### dataset Indometh of package datasets
require(datasets)
data <- subset(Indometh, Subject==2) 
time <- data$time
conc <- data$conc 

## fitting on observed and log-scale
res.obs <- biexp(conc=conc, time=time, log.scale=FALSE)
res.log <- biexp(conc=conc, time=time, log.scale=TRUE)

print(res.obs$parms)
print(res.log$parms)

plot(res.obs, ylim=c(0,5), xlim=c(0, max(time)), las=1)
plot(res.log, ylim=c(0,5), xlim=c(0, max(time)), las=1, add=TRUE, lty=2)
legend(x=0, y=5, lty=c(1,2), legend=c("fitted on observed scale", "fitted on log-scale"))

## get residuals using function nls with tol=Inf
parms.obs <- list(a1=res.obs$parms[3,1], b1=res.obs$parms[2,1], a2=res.obs$parms[3,2], 
                  b2=res.obs$parms[2,2])
parms.log <- list(a1=res.log$parms[3,1], b1=res.log$parms[2,1], a2=res.log$parms[3,2], 
                  b2=res.log$parms[2,2])

mod.obs <- nls(conc ~ a1*exp(-b1*time) + a2*exp(-b2*time), start=parms.obs, 
               control=nls.control(tol=Inf))
mod.log <- nls(conc ~ a1*exp(-b1*time) + a2*exp(-b2*time), start=parms.log, 
               control=nls.control(tol=Inf))

## identical estimates to mod.log but different SEs  
summary(nls(log(conc)~log(a1*exp(-b1*time) + a2*exp(-b2*time)), start=parms.log, 
        control=nls.control(tol=Inf)))

## different approach using weighted least squares (WLS) in nls 
mod.ols <- nls(conc ~ a1*exp(-b1*time) + a2*exp(-b2*time), start=parms.obs)
mod.wls1 <- nls(conc ~ a1*exp(-b1*time) + a2*exp(-b2*time), start=parms.obs, 
                weight=1/predict(mod.ols)^1)
mod.wls2 <- nls(conc ~ a1*exp(-b1*time) + a2*exp(-b2*time), start=parms.obs, 
                weight=1/predict(mod.ols)^2)

split.screen(c(2,2))
screen(1)
plot(ylim=c(-0.35,0.35), y=resid(mod.obs), x=predict(mod.obs), las=1, 
    main='Fitted using biexp on observed scale', xlab='Predicted', ylab='Residual')
abline(h=0)
screen(2)
plot(ylim=c(-0.35,0.35), y=resid(mod.log), x=predict(mod.log), las=1, 
     main='Fitted using biexp on log-scale', xlab='Predicted', ylab='Residual')
abline(h=0)
screen(3)
plot(ylim=c(-0.35,0.35), y=resid(mod.wls1), x=predict(mod.wls1), las=1, 
     main='Fitted using nls with weights 1/predict(mod.ols)^1', xlab='Predicted', ylab='Residual')
abline(h=0)
screen(4)
plot(ylim=c(-0.35,0.35), y=resid(mod.wls2), x=predict(mod.wls2), las=1, 
     main='Fitted using nls with weights 1/predict(mod.ols)^2', xlab='Predicted', ylab='Residual')
abline(h=0) 
close.screen(all.screens=TRUE)


#### example for a serial sampling data design from Wolfsegger and Jaki (2009)
conc <- c(2.01, 2.85, 2.43, 0.85, 1.00, 0.91, 0.46, 0.35, 0.63, 0.39, 0.32, 
          0.45, 0.11, 0.18, 0.19, 0.08, 0.09, 0.06)
time <- c(rep(5/60,3), rep(3,3), rep(6,3), rep(9,3), rep(16,3), rep(24,3))

res.biexp1 <- biexp(conc=conc, time=time, log=TRUE)
res.biexp2 <- biexp(conc=conc, time=time, log=FALSE)

print(res.biexp1$parms)
print(res.biexp2$parms)

split.screen(c(1,2)) 
screen(1)
plot(x=c(0,25), y=c(0,3), type='n', las=1, 
ylab='Plasma concentration (IU/mL)', xlab='Time (hours)')
points(x=time, y=conc, pch=21)
plot(res.biexp1, pch=NA, add=TRUE, lty=1)
plot(res.biexp2, pch=NA, add=TRUE, lty=2)
legend(x=25, y=3, xjust=1, col=c('black', 'black'), lty=c(1,2), 
        title='Nonlinear fitting with function biexp:', 
        legend=c('option: log=TRUE', 'option: log=FALSE'))
close.screen(1)
screen(2)
plot(x=c(0,25), y=c(0.01, 10), type='n', log='y', yaxt='n', 
ylab='Plasma concentration (IU/mL)', xlab='Time (hours)')
axis(side=2, at=c(0.01, 0.1, 1, 10), labels=c('0.01', '0.1', '1', '10'), las=1)
axis(side=2, at=seq(2,9,1), tcl=-0.25, labels=FALSE) 
axis(side=2, at=seq(0.2,0.9,0.1), tcl=-0.25, labels=FALSE) 
axis(side=2, at=seq(0.02,0.09,0.01), tcl=-0.25, labels=FALSE) 
points(x=time, y=conc, pch=21)
plot(res.biexp1, pch=NA, add=TRUE, lty=1)
plot(res.biexp2, pch=NA, add=TRUE, lty=2)
legend(x=25, y=10, xjust=1, col=c('black', 'black'), lty=c(1,2), 
        title='Nonlinear fitting with function biexp:', 
        legend=c('option: log=TRUE', 'option: log=FALSE'))
close.screen(all.screens=TRUE)


#### example from Gabrielsson and Weiner (2000, page 743) 
#### endogenous concentration is assumed to be constant over time  
dose <- 36630  
time <- c(-1, 0.167E-01, 0.1167, 0.1670, 0.25, 0.583, 0.8330, 1.083, 1.583, 2.083, 4.083, 8.083,
          12, 23.5, 24.25, 26.75, 32)
conc <- c(20.34, 3683, 884.7, 481.1, 215.6, 114, 95.8, 87.89, 60.19, 60.17, 34.89, 20.99, 20.54, 
          19.28, 18.18, 19.39, 22.72)
data <- data.frame(conc,time)

## get starting values using function biexp using naive adjustment for endogenous concentration 
## by subtraction of pre-value
data$concadj <- data$conc - data$conc[1]
data$concadj[min(which(data$concadj<0)):nrow(data)] <- NA
res.biexp <- biexp(conc=data$concadj[-1], time=data$time[-1])$parms 
start <- list(a1=res.biexp[3,1], k1=res.biexp[2,1], a2=res.biexp[3,2], k2=res.biexp[2,2])  

## specify indicator variable enabling inclusion of pre-dose concentration for fitting  
data$i1 <- ifelse(data$time <0, 1, 0)
data$i2 <- ifelse(data$time <0, 0, 1)

## assuming constant absolute error: ordinary least squares 
mod.ols <- nls(conc ~ i1*base + i2*(base + a1*exp(-k1*time) + a2*exp(-k2*time)), 
               start=c(base=20.34, start), data=data, trace=TRUE)
 
## assuming constant relative error (i.e. proportional error - weight of 2): weighted least
## squares
mod.wls <- nls(conc ~ i1*base + i2*(base + a1*exp(-k1*time) + a2*exp(-k2*time)), 
               start=c(base=20.34, start), data=data, weight=1/predict(mod.ols)^2, trace=TRUE) 

## assuming constant relative error (i.e. proportional error - weight of 2): iteratively 
## re-weighted least squares 
mod.irwls <- mod.wls 
for(i in 1:10){ 
   print(as.vector(coef(mod.irwls))) 
   mod.irwls <- nls(conc ~ i1*base + i2*(base + a1*exp(-k1*time) + a2*exp(-k2*time)), 
                    start=c(base=20.34, start), data=data, weight=1/predict(mod.irwls)^2)
}
 
summary(mod.ols)
summary(mod.wls)
summary(mod.irwls)

newdata <- data.frame(time=seq(0,32,0.01))
newdata$i1 <- ifelse(newdata$time <0, 1, 0)
newdata$i2 <- ifelse(newdata$time <0, 0, 1)
plot(conc ~ time, data=data, ylim=c(10,1E4), log='y', yaxt='n', 
     xlab='Time (hours)', ylab='Log of concentration (pmol/L)')
axis(side=2, at=c(10, 100, 1000, 10000), las=1)
axis(side=2, at=seq(1E1,1E2,1E1), tcl=-0.25, labels=FALSE) 
axis(side=2, at=seq(1E2,1E3,1E2), tcl=-0.25, labels=FALSE) 
axis(side=2, at=seq(1E3,1E4,1E3), tcl=-0.25, labels=FALSE) 
points(x=newdata$time, y=predict(mod.irwls, newdata), type='l')

## get total clearance (cls), inter-compartmental distribution (cld), and volume of distributions 
## from macro constant parametrization according to Gabrielsson and Weiner (2000) 
parm <- coef(mod.wls)[-1]
## get micro constants 
k21 <- as.double((parm[1]*parm[4]+parm[3]*parm[2])/(parm[1]+parm[3]))
k10 <- as.double(parm[2]*parm[4] / k21)
k12 <- as.double(parm[2]+parm[4] - k21 - k10)
## get cls, cld, vc, and vt 
cls <- as.double(dose / (parm[1]/parm[2] + parm[3]/parm[4])) 
vc <- as.double(dose / (parm[1] + parm[2]))
cld <- k12*vc 
vt <- cld / k21 
print(c(cls, cld, vc, vt))

## turnover model to account for endogenous baseline according to Gabrielsson and Weiner 
## using a biexponential (i.e. two-compartment) model parametrized in terms of clearance 

## Not run: require(rgenoud)
require(deSolve)

k <- 2 # assuming proportional error - weighting in function objfun 
tinf <- 1/60 # duration of bolus in hours 
data <- subset(data, time>0) 

defun <- function(time, y, parms) { 
	rte1 <- ifelse(time <= tinf, dose/tinf, 0)
	dCptdt1 <- (rte1 + parms["synt"] - parms["cls"]*y[1] - parms["cld"]*y[1] + 
                    parms["cld"]*y[2]) / parms["vc"]
	dCptdt2 <- (parms["cld"]*y[1] - parms["cld"]*y[2])/parms["vt"] 
	list(c(dCptdt1, dCptdt2))
}

modfun <- function(time, synt, cls, cld, vc, vt) {
	out <- lsoda(y=c(synt/cls, synt/cls), times=c(0, data$time), defun, 
                parms=c(synt=synt, cls=cls, cld=cld, vc=vc, vt=vt), rtol=1e-5, atol=1e-5)[-1,2]
} 

objfun <- function(par) {
        out <- modfun(data$time, par[1], par[2], par[3], par[4], par[5])
        gift <- which(data$conc != 0 )
        sum((data$conc[gift]-out[gift])^2 / data$conc[gift]^k)		
}        

## grid search to get starting values for Nelder-Mead 
## increase values of pop.size and max.generation to get better starting values  
## values of 10 are used for illustration purpose only 
options(warn = -1) # omit warning when hard maximum limit is hit
gen <- genoud(objfun, nvars=5, max=FALSE, pop.size=10, max.generation=10,
             starting.value=c(1500, cls, cld, vc, vt), BFGS=FALSE,
             print.level=1, boundary.enforcement=2,
             Domains=matrix(c(0,0,0,0,0,1E4,1E3,1E3,1E3,1E3),5,2),
             MemoryMatrix=TRUE)
options(warn = 0) # set back to default  

opt <- optim(gen$par, objfun, method="Nelder-Mead") 

trn.wls <- nls(conc ~ modfun(time, synt, cls, cld, vc, vt), data=data, 
               start=list(synt=opt$par[1], cls=opt$par[2], cld=opt$par[3], vc=opt$par[4], 
                          vt=opt$par[5]), 
               trace=TRUE, nls.control(tol=Inf))

summary(trn.wls) 

## End(Not run)

PK documentation built on April 3, 2020, 1:07 a.m.

Related to biexp in PK...