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Estimate parameters in a PBPK model

Metrum Research Group

Packages and setup

library(tidyverse)
library(PKPDmisc)
library(mrgsolve)
library(nloptr)
library(DEoptim)
library(GenSA)
library(hydroPSO)
source("script/functions.R")
source("script/global.R")
set.seed(10101)
theme_set(theme_bw() + theme(legend.position = "top"))
scale_colour_discrete <- function(...) scale_color_brewer(palette="Set2")

Models are located here:

model_dir <- "model"

Reference

Quantitative Analyses of Hepatic OATP-Mediated Interactions Between Statins and Inhibitors Using PBPK Modeling With a Parameter Optimization Method

  • T Yoshikado, K Yoshida, N Kotani, T Nakada, R Asaumi, K Toshimoto, K Maeda, H Kusuhara and Y Sugiyama

  • CLINICAL PHARMACOLOGY & THERAPEUTICS | VOLUME 100 NUMBER 5 | NOVEMBER 2016

  • https://www.ncbi.nlm.nih.gov/pubmed/27170342

Data

  • Example taken from figure 4a from the publication
  • Using this as example data to fit
data.file <- "data/fig4a.csv"

data <-
  data.file %>% 
  read_csv() %>% 
  mutate(
    profile = NULL, 
    type=ID, 
    typef=factor(ID, labels = c("Statin", "Statin+CsA")), 
    DV = ifelse(DV==-1, NA_real_, DV)
  )
  • The goal is to fit the pitavastatin data either alone or in combination with cyclosporin administered 1 hour before the pitavastatin
ggplot(data=data,aes(time,DV)) + 
  geom_point(aes(col = typef), size = 3) + 
  geom_line(col = "darkgrey", aes(group = typef)) + 
  scale_y_continuous(trans="log", limits=c(0.1,300), breaks=logbr())

PBPK model: pitavastatin / CsA DDI

  • Check out the model / data with a quick simulation
mod <- mread_cache("yoshikado", model_dir)

Make some persistent updates to the model

  • Simulate out to 14 hours
  • Only interested in CP, the pitavastatin concentration
mod <- mod %>% update(end=14, delta=0.1) %>% Req(CP)

A practice simulation

dose <- filter(data, evid==1) %>% mutate(typef=NULL)

sims <- 
  mod %>% 
  mrgsim_d(dose, obsaug=TRUE) %>% 
  mutate(type = typef(ID))

ggplot(sims, aes(time,CP,col=type)) + 
  geom_line(lwd = 1) + 
  scale_x_continuous(breaks = seq(0,12,2)) + 
  scale_y_log10(name = "Pitavastatin concentration")

sims %>% 
  group_by(type) %>% 
  summarise(auc = auc_partial(time,CP)) %>% 
  mutate(fold_increase = auc /first(auc))
. # A tibble: 2 x 3
.   type                 auc fold_increase
.   <fct>              <dbl>         <dbl>
. 1 Pitavastatin alone  44.1          1   
. 2 Pitavastatin + CsA 161.           3.65

Objective function

  • Least squares objective function
  • Weighted by the observations

Arguments:

  • dv the observed data
  • pred the predicted data
wss <- function(dv, pred, weight = 1/dv) {
  sum(((dv-pred)*weight)^2,na.rm=TRUE) 
}

Prediction function

  • Let’s go through step by step what each line is doing for us

Arguments:

  • p the parameters proposed by the optimizer
  • data the simulation template (doses and observation records)
  • pred logical; if TRUE, just return predicted data
sim_ofv <- function(p, data, pred = FALSE) {
  
  names(p) <- names(theta)
  
  p <- lapply(p,exp)
  
  out <- mod %>% param(p) %>% mrgsim_q(data, output="df")
  
  if(pred) return(out)
  
  ofv <- wss(data[["DV"]], out[["CP"]])
  
  return(ofv)
  
}

What this function does:

  1. Take in arguments; focus is on a new set of parameters p proposed by the optimizer; other arguments are just fixed data that we need
  2. Get the parameters out of log scale
  3. Also, put names on the list of parameters; this is crutial
  4. Update the model object with the new parameters
  5. (optionally simulate and return)
  6. Simulate from the data set, taking only observed values
  7. Calculate and return the objective function value

Data grooming

  • Drop the non-numeric columns
data <-  dplyr::select(data, -typef)

Optimize

First, set up the initial estimates

theta <- c(
  fbCLintall = 1, 
  ikiu = 1, 
  fbile = 0.5, 
  ka = 1, 
  ktr = 1
) %>% log()

nloptr::newuoa: minimization without derivatives

fit <- nloptr::newuoa(x0 = theta, fn = sim_ofv, data = data)
fit
. $par
. [1] -0.20428246 -4.51446804 -1.06769436 -0.01128541 -0.37159534
. 
. $value
. [1] 0.6860764
. 
. $iter
. [1] 386
. 
. $convergence
. [1] 4
. 
. $message
. [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."

fit_minqa <- minqa::newuoa(theta, fn = sim_ofv, data = data)

The final objective function value and estimates

sim_ofv(fit$par,data=data)
. [1] 0.6860764

exp(fit$par) %>% set_names(names(theta))
. fbCLintall       ikiu      fbile         ka        ktr 
. 0.81523207 0.01094943 0.34380028 0.98877803 0.68963325

optim: Nelder-Mead

fit1b <- optim(theta, sim_ofv, data=data, control = list(maxit = 1000))

neldermead: Alternate Nelder-Mead

fit1c <- nloptr::neldermead(x0=theta, fn=sim_ofv, data = data )

DEoptim: differential evolution algorithm

https://en.wikipedia.org/wiki/Differential_evolution

“Performs evolutionary global optimization via the Differential Evolution algorithm.”

lower <- rep(-6,length(theta)) %>% setNames(names(theta))
upper <- rep(5, length(theta)) %>% setNames(names(theta))

set.seed(330303)

decontrol <- DEoptim.control(
  trace = 10,
  NP=10*length(theta), 
  CR=0.925, 
  F=0.85,
  itermax=90, 
  storepopfrom=0
)

fit2 <- DEoptim(
  fn=sim_ofv, 
  lower=lower,
  upper=upper, 
  control=decontrol,
  data=data
)
. Iteration: 10 bestvalit: 2.359438 bestmemit:    0.090828   -4.810051   -1.032300   -0.596372   -0.748306
. Iteration: 20 bestvalit: 2.359438 bestmemit:    0.090828   -4.810051   -1.032300   -0.596372   -0.748306
. Iteration: 30 bestvalit: 0.745879 bestmemit:   -0.220189   -4.444982   -1.045812   -0.135432   -0.454597
. Iteration: 40 bestvalit: 0.733471 bestmemit:   -0.201757   -4.500855   -1.075498   -0.170296   -0.414804
. Iteration: 50 bestvalit: 0.691808 bestmemit:   -0.220169   -4.512342   -1.101670   -0.043517   -0.409125
. Iteration: 60 bestvalit: 0.688542 bestmemit:   -0.210497   -4.516279   -1.078079    0.011272   -0.380982
. Iteration: 70 bestvalit: 0.686352 bestmemit:   -0.203172   -4.509802   -1.066623   -0.004658   -0.362483
. Iteration: 80 bestvalit: 0.686121 bestmemit:   -0.203278   -4.514187   -1.064788   -0.010720   -0.370142
. Iteration: 90 bestvalit: 0.686081 bestmemit:   -0.204019   -4.514555   -1.067274   -0.011397   -0.372682

DA for the plot

pops <- lapply(fit2$member$storepop, as.data.frame)
hx <- bind_rows(pops)
hx <- mutate(hx, iteration=rep(1:decontrol$itermax,each=decontrol$NP))
hx <- mutate(hx, pop = rep(1:decontrol$NP, time=decontrol$itermax))
hxm <- gather(hx, variable, value, 1:5) %>% mutate(value = exp(value))
best <- as_tibble(fit2$member$bestmemit) %>% 
  mutate(iteration = 1:decontrol$itermax)
bestm <- gather(best,variable,value,1:5) %>% mutate(value = exp(value))
ggplot(data=hxm) + 
  geom_line(aes(iteration,value,group=pop),col="darkslateblue") + 
  geom_line(data=bestm,aes(iteration,value),col="orange",lwd=1) + 
  scale_y_continuous(trans="log", breaks=10^seq(-4,4), name="Parameter value") + 
  facet_wrap(~variable, ncol=2, scales="free_y") + theme_bw()

GenSA: simulated annealing

set.seed(11001)

sacontrol <- list(maxit = 100, nb.stop.improvement = 20, verbose = TRUE)

fit3 <- GenSA(
  NULL, sim_ofv, lower=lower+1, upper=upper-1, data = data, control = sacontrol
)
. Initializing par with random data inside bounds
. It: 1, obj value: 3.470357641
. It: 25, obj value: 0.6860778737

hydroPSO: particle swarm optimization

https://en.wikipedia.org/wiki/Particle_swarm_optimization

set.seed(22022013)

fit4 <- hydroPSO(
  theta, fn = sim_ofv, lower = lower, upper = upper, 
  control = list(maxit = 100, REPORT = 5),
  data = data
)

nloptr::CRS: controlled random search

set.seed(11000222)

crs <- crs2lm(
  x0 = theta, 
  fn=sim_ofv, 
  lower = lower, 
  upper = upper, 
  data=data,
  maxeval=5500
)

Compare optimization methods

results <- list(theta, fit$par, fit1b$par, fit1c$par, fit2$optim$bestmem, fit3$par, fit4$par, crs$par)

results <- map(results, set_names, nm = names(theta))

results <- map(results, exp)

tibble(
  method = c("initial", "newuoa", "nelder", "nelder2", "DEoptim", "SA", "PSO","CRS"),
  fbCLintall = map_dbl(results, "fbCLintall"), 
  ikiu = map_dbl(results, "ikiu"), 
  fbile = map_dbl(results, "fbile"), 
  ka = map_dbl(results, "ka"), 
  ktr = map_dbl(results, "ktr")
) %>% mutate_if(is.double, list(signif), digits = 4)
. # A tibble: 8 x 6
.   method  fbCLintall   ikiu fbile    ka   ktr
.   <chr>        <dbl>  <dbl> <dbl> <dbl> <dbl>
. 1 initial      1     1      0.5   1     1    
. 2 newuoa       0.815 0.0110 0.344 0.989 0.690
. 3 nelder       0.815 0.0110 0.344 0.990 0.690
. 4 nelder2      0.815 0.0110 0.344 0.989 0.690
. 5 DEoptim      0.815 0.0110 0.344 0.989 0.689
. 6 SA           0.815 0.0110 0.344 0.989 0.689
. 7 PSO          0.815 0.0110 0.344 0.988 0.690
. 8 CRS          0.815 0.0110 0.344 0.989 0.690

value0 <- sim_ofv(theta,data)
results <- c(value0, fit$value, fit1b$value, fit1c$value, fit2$optim$bestval, fit3$value, fit4$value, crs$value)

tibble(
  method = c("initial", "newuoa", "nelder", "nelder2", "DEoptim", "SA", "PSO","CRS"),
  value = results
) 
. # A tibble: 8 x 2
.   method  value
.   <chr>   <dbl>
. 1 initial 5.00 
. 2 newuoa  0.686
. 3 nelder  0.686
. 4 nelder2 0.686
. 5 DEoptim 0.686
. 6 SA      0.686
. 7 PSO     0.686
. 8 CRS     0.686