## Introduction

This is a simple example on how to couple PopED with external R based PKPD simulation tools. Typically, these tools might be R packages that can simulate from ordinary differential equation (ODE) based models. In this document you will see how to couple PopED to models, defined with ODEs, implemented using:

• deSolve (with native R ODE models)
• deSolve (with compiled C ODE models)
• deSolve (with compiled C++ ODE models using Rcpp)
• PKPDsim
• mrgsolve
• rxode2
library(PopED)
library(PKPDsim)
library(mrgsolve)
#>
#> Attaching package: 'mrgsolve'
#> The following object is masked from 'package:stats':
#>
#>     filter
library(deSolve)
library(Rcpp)
library(rxode2)
#>   no cache: create with rxCreateCache()

## The model

We will use a one-compartment with linear absorption population pharmacokinetic (PK) model as an example (see below).

This model can be described with the following set of ODEs:

$\begin{split} \frac{dA_{0}}{dt} &= - k_{a} \cdot A_{0}\\ \frac{dA_{1}}{dt} &=-(CL/V_1)\cdot A_1 + k_{a} \cdot A_{0} \\ f(t) &= A_1/V_1 \end{split}$

All compartment amounts are assumed to be zero at time zero ($$\boldsymbol{A}[t=0]=0$$). Inputs to the system come in tablet form and are added to the amount in $$A_{0}$$ according to

$\text{Input}(t,D,t_D) = \begin{cases} D, &\text{if} \quad t = t_D \\ 0, &\text{otherwise} \end{cases}$

Parameter values are defined as:

$\begin{split} k_a &= \theta_{k_a} \cdot e^{\eta_{k_a}} \\ CL &= \theta_{CL} \cdot e^{\eta_{CL}} \\ V_1 &= \theta_{V_1} \cdot e^{\eta_{V_1}} \\ \end{split}$ where elements of the between subject variability (BSV), $$\eta_{j}$$, vary across individuals and come from normal distributions with means of zero and variances of $$\omega^2_{j}$$.

The residual unexplained variability (RUV) model has a proportional and additive component

$y = f(t) \cdot (1+\varepsilon_{prop}) + \varepsilon_{add}$

elements of $$\boldsymbol{\varepsilon}_{j}$$ vary accross observations and come from normal distributions with means of zero and variances of $$\sigma^2_{j}$$.

Parameter values are assumed to be the following:
Parameter Value
$$k_a$$ 0.2500
CL 3.7500
$$V_1$$ 72.8000
$$\omega^2_{k_a}$$ 0.0900
$$\omega^2_{CL}$$ 0.0625
$$\omega^2_{V_1}$$ 0.0900
$$\sigma^2_{prop}$$ 0.0400
$$\sigma^2_{add}$$ 0.0025

## Model implementation

Below we implement this model using a number of different methods. For the ODE solvers, if possible, we set the tuning parameters to be the same values (atol, rtol, etc.).

### Analytic solution

First we implement an analytic solution to the model in a function that could be used in PopED. Here we assume a single dose or multiple dosing with a dose interval of TAU time units. The named vector parameters defines the values of KA, CL, V, DOSE and TAU used to compute the value of f at each time point in the vector xt.

ff_analytic <- function(model_switch,xt,parameters,poped.db){
with(as.list(parameters),{
y=xt
N = floor(xt/TAU)+1
f=(DOSE/V)*(KA/(KA - CL/V)) *
(exp(-CL/V * (xt - (N - 1) * TAU)) * (1 - exp(-N * CL/V * TAU))/(1 - exp(-CL/V * TAU)) -
exp(-KA * (xt - (N - 1) * TAU)) * (1 - exp(-N * KA * TAU))/(1 - exp(-KA * TAU)))
return(list( f=f,poped.db=poped.db))
})
}

### ODE solution using deSolve

The same model can be implemented using ODEs. Here the ODEs are defined in deSolve:

PK_1_comp_oral_ode <- function(Time, State, Pars){
with(as.list(c(State, Pars)), {
dA1 <- -KA*A1
dA2 <- KA*A1 - (CL/V)*A2
return(list(c(dA1, dA2)))
})
}

Then, just as in the analytic solution, the named vector parameters defines the values of KA, CL, V, DOSE and TAU used to compute the value of f at each time point in the vector xt. The inputs to the system (dosing amounts and times) need to be added as events in the deSolve ODE solver called deSolve::ode().

ff_ode_desolve <- function(model_switch, xt, parameters, poped.db){
with(as.list(parameters),{
A_ini <- c(A1=0, A2=0)

#Set up time points for the ODE
times_xt <- drop(xt)
times <- c(0,times_xt) ## add extra time for start of the experiment
dose_times = seq(from=0,to=max(times_xt),by=TAU)
times <- c(times,dose_times)
times <- sort(times)
times <- unique(times) # remove duplicates

eventdat <- data.frame(var = c("A1"),
time = dose_times,
value = c(DOSE), method = c("add"))

out <- deSolve::ode(A_ini, times, PK_1_comp_oral_ode, parameters,
events = list(data = eventdat),
atol=1e-8, rtol=1e-8,maxsteps=5000)

# grab timepoint values
out = out[match(times_xt,out[,"time"]),]

f = out[,"A2"]/V

f=cbind(f) # must be a column matrix
return(list(f=f,poped.db=poped.db))
})
}

### ODE solution using deSolve and compiled C code

We can use compiled C code with deSolve to speed up computing solutions to the ODEs. The C code is written in a separate file is that needs to be compiled and looks like this:

/* file one_comp_oral_CL.c */
#include <R.h>
static double parms[3];
#define CL parms[0]
#define V parms[1]
#define KA parms[2]

/* initializer  */
void initmod(void (* odeparms)(int *, double *))
{
int N=3;
odeparms(&N, parms);
}

/* Derivatives and 1 output variable */
void derivs (int *neq, double *t, double *y, double *ydot,
double *yout, int *ip)
{

if (ip[0] <1) error("nout should be at least 1");

ydot[0] = -KA*y[0];
ydot[1] = KA*y[0] - CL/V*y[1];
yout[0] = y[0]+y[1];
}

/* END file one_comp_oral_CL.c */

This code is available as a file in the PopED distribution, and is compiled with the following commands:

file.copy(system.file("examples/one_comp_oral_CL.c", package="PopED"),"./one_comp_oral_CL.c")
#> [1] TRUE
system('R CMD SHLIB one_comp_oral_CL.c')
dyn.load(paste("one_comp_oral_CL", .Platform$dynlib.ext, sep = "")) The function used to compute the value of f at each time point in the vector xt, given the inputs to the system (dosing amounts and times), needs to be changed slightly, updating the arguments to deSolve::ode(). ff_ode_desolve_c <- function(model_switch, xt, parameters, poped.db){ with(as.list(parameters),{ A_ini <- c(A1=0, A2=0) #Set up time points for the ODE times_xt <- drop(xt) times <- c(0,times_xt) ## add extra time for the start of the experiment dose_times = seq(from=0,to=max(times_xt),by=TAU) times <- c(times,dose_times) times <- sort(times) times <- unique(times) # remove duplicates eventdat <- data.frame(var = c("A1"), time = dose_times, value = c(DOSE), method = c("add")) out <- deSolve::ode(A_ini, times, func = "derivs", parms = c(CL,V,KA), dllname = "one_comp_oral_CL", initfunc = "initmod", nout = 1, outnames = "Sum", events = list(data = eventdat), atol=1e-8, rtol=1e-8,maxsteps=5000) # grab timepoint values out = out[match(times_xt,out[,"time"]),] f = out[, "A2"]/V f=cbind(f) # must be a column matrix return(list(f=f,poped.db=poped.db)) }) } ### ODE solution using deSolve and compiled C++ code (via Rcpp) Here we define the ODE system using inline C++ code that is compiled via Rcpp cppFunction('List one_comp_oral_rcpp(double Time, NumericVector A, NumericVector Pars) { int n = A.size(); NumericVector dA(n); double CL = Pars[0]; double V = Pars[1]; double KA = Pars[2]; dA[0] = -KA*A[0]; dA[1] = KA*A[0] - (CL/V)*A[1]; return List::create(dA); }') Again, the arguments to deSolve::ode() need to be updated: ff_ode_desolve_rcpp <- function(model_switch, xt, p, poped.db){ A_ini <- c(A1=0, A2=0) #Set up time points for the ODE times_xt <- drop(xt) times <- c(0,times_xt) ## add extra time for start of integration dose_times = seq(from=0,to=max(times_xt),by=p[["TAU"]]) times <- c(times,dose_times) times <- sort(times) times <- unique(times) # remove duplicates eventdat <- data.frame(var = c("A1"), time = dose_times, value = c(p[["DOSE"]]), method = c("add")) out <- deSolve::ode(A_ini, times, one_comp_oral_rcpp, c(CL=p[["CL"]],V=p[["V"]], KA=p[["KA"]]), events = list(data = eventdat), atol=1e-8, rtol=1e-8,maxsteps=5000) # grab timepoint values for central comp f = out[match(times_xt,out[,"time"]),"A2",drop=F]/p[["V"]] return(list(f=f,poped.db=poped.db)) } ### ODE solution using PKPDsim We can use PKPDsim to describe this set of ODEs. We then adjust the function used to compute the value of f at each time point in the vector xt, given the inputs to the system (dosing amounts and times), using the ODE solver PKPDsim::sim_core(). pk1cmtoral <- PKPDsim::new_ode_model("pk_1cmt_oral") # take from library ff_ode_pkpdsim <- function(model_switch, xt, p, poped.db){ #Set up time points for the ODE times_xt <- drop(xt) dose_times <- seq(from=0,to=max(times_xt),by=p[["TAU"]]) times <- sort(unique(c(0,times_xt,dose_times))) N = length(dose_times) regimen = PKPDsim::new_regimen(amt=p[["DOSE"]],n=N,interval=p[["TAU"]]) design <- PKPDsim::sim( ode = pk1cmtoral, parameters = c(CL=p[["CL"]],V=p[["V"]],KA=p[["KA"]]), regimen = regimen, only_obs = TRUE, t_obs = times, checks = FALSE, return_design = TRUE) tmp <- PKPDsim::sim_core(sim_object = design, ode = pk1cmtoral) f <- tmp$y
m_tmp <- match(round(times_xt,digits = 6),tmp[,"t"])
if(any(is.na(m_tmp))){
stop("can't find time points in solution\n",
"try changing the digits argument in the match function")
}

f <- f[m_tmp]
return(list(f = f, poped.db = poped.db))
}

### ODE solution using mrgsolve

We can also use mrgsolve to describe this set of ODEs.

code <- '
$PARAM CL=3.75, V=72.8, KA=0.25$CMT DEPOT CENT
$ODE dxdt_DEPOT = -KA*DEPOT; dxdt_CENT = KA*DEPOT - (CL/V)*CENT;$TABLE double CP  = CENT/V;
$CAPTURE CP ' We then compile and load the model with mcode moda <- mrgsolve::mcode("optim", code, atol=1e-8, rtol=1e-8,maxsteps=5000) #> Building optim ... done. Finally, we adjust the function used to compute the value of f at each time point in the vector xt, given the inputs to the system (dosing amounts and times), using the ODE solver mrgsolve::mrgsim_q(). ff_ode_mrg <- function(model_switch, xt, p, poped.db){ times_xt <- drop(xt) dose_times <- seq(from=0,to=max(times_xt),by=p[["TAU"]]) time <- sort(unique(c(0,times_xt,dose_times))) is.dose <- time %in% dose_times data <- tibble::tibble(ID = 1, time = time, amt = ifelse(is.dose,p[["DOSE"]], 0), cmt = ifelse(is.dose, 1, 0), evid = cmt, CL = p[["CL"]], V = p[["V"]], KA = p[["KA"]]) out <- mrgsolve::mrgsim_q(moda, data=data) f <- out$CP

f <- f[match(times_xt,out$time)] return(list(f=matrix(f,ncol=1),poped.db=poped.db)) } ### ODE solution using rxode2 We can use rxode2 to describe this set of ODEs. modrx <- rxode2::rxode2({ d/dt(DEPOT) = -KA*DEPOT; d/dt(CENT) = KA*DEPOT - (CL/V)*CENT; CP=CENT/V; }) #> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’ We adjust the function used to compute the value of f at each time point in the vector xt, given the inputs to the system (dosing amounts and times), using the ODE solver rxode2::rxSolve(). ff_ode_rx <- function(model_switch, xt, p, poped.db){ times_xt <- drop(xt) et(0,amt=p[["DOSE"]], ii=p[["TAU"]], until=max(times_xt)) %>% et(times_xt) -> data out <- rxode2::rxSolve(modrx, p, data, atol=1e-8, rtol=1e-8,maxsteps=5000, returnType="data.frame") f <- out$CP[match(times_xt,out$time)] return(list(f=matrix(f,ncol=1),poped.db=poped.db)) } ### Common model elements Other functions are used to define BSV and RUV.  sfg <- function(x,a,bpop,b,bocc){ parameters=c( KA=bpop[1]*exp(b[1]), CL=bpop[2]*exp(b[2]), V=bpop[3]*exp(b[3]), DOSE=a[1], TAU=a[2]) return( parameters ) } feps <- function(model_switch,xt,parameters,epsi,poped.db){ f <- do.call(poped.db$model$ff_pointer,list(model_switch,xt,parameters,poped.db))[[1]] y = f*(1+epsi[,1])+epsi[,2] return(list(y=y,poped.db=poped.db)) } ## Create PopED databases Next we define the model to use, the parameters of those models, the intial design design and design space for any design calculation. Here we create a number of databases that correspond to different model implementations. The initial design is a 2 group design, with doses of 20 mg or 40 mg every 24 hours. Each group has the same sampling schedule, with 3 samples in the first day of the study and 2 on the 10th day of the study. poped_db_analytic <- create.poped.database( ff_fun =ff_analytic, fg_fun =sfg, fError_fun=feps, bpop=c(KA=0.25,CL=3.75,V=72.8), d=c(KA=0.09,CL=0.25^2,V=0.09), sigma=c(prop=0.04,add=0.0025), m=2, groupsize=20, xt=c( 1,2,8,240,245), minxt=c(0,0,0,240,240), maxxt=c(10,10,10,248,248), bUseGrouped_xt=1, a=cbind(DOSE=c(20,40),TAU=c(24,24)), maxa=c(DOSE=200,TAU=24), mina=c(DOSE=0,TAU=24)) poped_db_ode_desolve <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_desolve) poped_db_ode_desolve_c <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_desolve_c) poped_db_ode_desolve_rcpp <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_desolve_rcpp) poped_db_ode_pkpdsim <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_pkpdsim) poped_db_ode_mrg <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_mrg) poped_db_ode_rx <- create.poped.database(poped_db_analytic,ff_fun = ff_ode_rx) ## Model predictions So are there difference in the model predictions between the different implementations? Here is a visual representation of the model predictions for this study design, based on the analytic solution: plot_model_prediction(poped_db_analytic,model_num_points = 500,PI=T,separate.groups = T)  We can compare the different predictions in this plot accross model implementations. Here we see that the accuracy of the different methods are within machine precision (or very small). pred_std <- model_prediction(poped_db_analytic,model_num_points = 500,include_sample_times = TRUE,PI = TRUE) pred_ode_desolve <- model_prediction(poped_db_ode_desolve, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_desolve) #> [1] TRUE pred_ode_desolve_c <- model_prediction(poped_db_ode_desolve_c, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_desolve_c) #> [1] TRUE pred_ode_desolve_rcpp <- model_prediction(poped_db_ode_desolve_rcpp, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_desolve_rcpp) #> [1] TRUE pred_ode_pkpdsim <- model_prediction(poped_db_ode_pkpdsim, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_pkpdsim) #> [1] "Component \"PI_l\": Mean relative difference: 1.998734e-08" pred_ode_mrg <- model_prediction(poped_db_ode_mrg, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_mrg) #> [1] TRUE pred_ode_rx <- model_prediction(poped_db_ode_rx, model_num_points = 500, include_sample_times = TRUE, PI = TRUE) all.equal(pred_std,pred_ode_rx) #> [1] TRUE ## Evaluate the design Here we compare the computation of the Fisher Information Matrix (FIM). By comparing the $$ln(det(FIM))$$ (the lnD-objective function value, or ofv). (eval_std <- evaluate_design(poped_db_analytic)) #>$ofv
#> [1] 48.98804
#>
#> $fim #> KA CL V d_KA d_CL d_V #> KA 1695.742314 -11.73537527 -6.75450789 0.00000 0.00000 0.00000 #> CL -11.735375 29.99735715 -0.03288331 0.00000 0.00000 0.00000 #> V -6.754508 -0.03288331 0.04213359 0.00000 0.00000 0.00000 #> d_KA 0.000000 0.00000000 0.00000000 147.24270 1.52226 192.23403 #> d_CL 0.000000 0.00000000 0.00000000 1.52226 2254.55188 1.21987 #> d_V 0.000000 0.00000000 0.00000000 192.23403 1.21987 634.42055 #> sig_prop 0.000000 0.00000000 0.00000000 148.86724 844.57325 387.53816 #> sig_add 0.000000 0.00000000 0.00000000 6555.68433 14391.88132 8669.58391 #> sig_prop sig_add #> KA 0.0000 0.000 #> CL 0.0000 0.000 #> V 0.0000 0.000 #> d_KA 148.8672 6555.684 #> d_CL 844.5733 14391.881 #> d_V 387.5382 8669.584 #> sig_prop 7759.5374 110702.705 #> sig_add 110702.7045 4436323.946 #> #>$rse
#>         KA         CL          V       d_KA       d_CL        d_V   sig_prop
#>  16.285678   4.909749  11.209270 120.825798  34.448477  57.300408  36.104027
#>  24.339781

All the computations give very similar results:

eval_ode_desolve <- evaluate_design(poped_db_ode_desolve)
all.equal(eval_std$ofv,eval_ode_desolve$ofv)
#> [1] "Mean relative difference: 2.493043e-08"

eval_ode_desolve_c <- evaluate_design(poped_db_ode_desolve_c)
all.equal(eval_std$ofv,eval_ode_desolve_c$ofv)
#> [1] "Mean relative difference: 2.493043e-08"

eval_ode_desolve_rccp <- evaluate_design(poped_db_ode_desolve_rcpp)
all.equal(eval_std$ofv,eval_ode_desolve_rccp$ofv)
#> [1] "Mean relative difference: 2.493043e-08"

eval_ode_pkpdsim <- evaluate_design(poped_db_ode_pkpdsim)
all.equal(eval_std$ofv,eval_ode_pkpdsim$ofv)
#> [1] TRUE

eval_ode_mrg <- evaluate_design(poped_db_ode_mrg)
all.equal(eval_std$ofv,eval_ode_mrg$ofv)
#> [1] "Mean relative difference: 2.361612e-08"

## Speed of FIM computation

We can compare the speed of the computations. Analytic solutions are fast, as expected, in this case more than 20 times faster than any of the ODE methods. mrgsolve is the fastest of the ODE solvers in this example. Note that much of the speed difference between mrgsolve, RxODE and PKPDsim has been found to be due to the overhead from pre- and post-processing of the simulation from ODE systems. Other ways of handling the pre- and post-processing may speed up these computations.

library(microbenchmark)
library(ggplot2)

compare <- microbenchmark(
evaluate_design(poped_db_analytic),
evaluate_design(poped_db_ode_desolve),
evaluate_design(poped_db_ode_desolve_c),
evaluate_design(poped_db_ode_desolve_rcpp),
evaluate_design(poped_db_ode_pkpdsim),
evaluate_design(poped_db_ode_mrg),
evaluate_design(poped_db_ode_rx),
times = 100L)

autoplot(compare)

## Version information

devtools::session_info()
#> ─ Session info ───────────────────────────────────────────────────────────────
#>  setting  value
#>  version  R version 4.3.2 (2023-10-31)
#>  os       Ubuntu 22.04.3 LTS
#>  system   x86_64, linux-gnu
#>  ui       X11
#>  language en
#>  collate  C.UTF-8
#>  ctype    C.UTF-8
#>  tz       UTC
#>  date     2023-11-21
#>  pandoc   2.19.2 @ /usr/bin/ (via rmarkdown)
#>
#> ─ Packages ───────────────────────────────────────────────────────────────────
#>  package       * version    date (UTC) lib source
#>  backports       1.4.1      2021-12-13 [1] RSPM
#>  BH              1.81.0-1   2023-01-22 [1] RSPM
#>  bslib           0.5.1      2023-08-11 [1] RSPM
#>  cachem          1.0.8      2023-05-01 [1] RSPM
#>  callr           3.7.3      2022-11-02 [1] RSPM
#>  checkmate       2.3.0      2023-10-25 [1] RSPM
#>  cli             3.6.1      2023-03-23 [1] RSPM
#>  codetools       0.2-19     2023-02-01 [3] CRAN (R 4.3.2)
#>  colorspace      2.1-0      2023-01-23 [1] RSPM
#>  crayon          1.5.2      2022-09-29 [1] RSPM
#>  data.table      1.14.8     2023-02-17 [1] RSPM
#>  desc            1.4.2      2022-09-08 [1] RSPM
#>  deSolve       * 1.38       2023-09-05 [1] RSPM
#>  devtools        2.4.5      2022-10-11 [1] RSPM
#>  digest          0.6.33     2023-07-07 [1] RSPM
#>  dparser         1.3.1-10   2023-03-16 [1] RSPM
#>  dplyr           1.1.4      2023-11-17 [1] RSPM
#>  ellipsis        0.3.2      2021-04-29 [1] RSPM
#>  evaluate        0.23       2023-11-01 [1] RSPM
#>  fansi           1.0.5      2023-10-08 [1] RSPM
#>  farver          2.1.1      2022-07-06 [1] RSPM
#>  fastmap         1.1.1      2023-02-24 [1] RSPM
#>  fs              1.6.3      2023-07-20 [1] RSPM
#>  generics        0.1.3      2022-07-05 [1] RSPM
#>  ggplot2         3.4.4      2023-10-12 [1] RSPM
#>  glue            1.6.2      2022-02-24 [1] RSPM
#>  gtable          0.3.4      2023-08-21 [1] RSPM
#>  highr           0.10       2022-12-22 [1] RSPM
#>  htmltools       0.5.7      2023-11-03 [1] RSPM
#>  htmlwidgets     1.6.2      2023-03-17 [1] RSPM
#>  httpuv          1.6.12     2023-10-23 [1] RSPM
#>  httr            1.4.7      2023-08-15 [1] RSPM
#>  jquerylib       0.1.4      2021-04-26 [1] RSPM
#>  jsonlite        1.8.7      2023-06-29 [1] RSPM
#>  kableExtra    * 1.3.4      2021-02-20 [1] RSPM
#>  knitr         * 1.45       2023-10-30 [1] RSPM
#>  labeling        0.4.3      2023-08-29 [1] RSPM
#>  later           1.3.1      2023-05-02 [1] RSPM
#>  lattice         0.21-9     2023-10-01 [3] CRAN (R 4.3.2)
#>  lifecycle       1.0.4      2023-11-07 [1] RSPM
#>  lotri           0.4.3      2023-03-20 [1] RSPM
#>  magrittr        2.0.3      2022-03-30 [1] RSPM
#>  memoise         2.0.1      2021-11-26 [1] RSPM
#>  mime            0.12       2021-09-28 [1] RSPM
#>  miniUI          0.1.1.1    2018-05-18 [1] RSPM
#>  mrgsolve      * 1.1.1      2023-08-16 [1] RSPM
#>  munsell         0.5.0      2018-06-12 [1] RSPM
#>  nlme            3.1-163    2023-08-09 [3] CRAN (R 4.3.2)
#>  pillar          1.9.0      2023-03-22 [1] RSPM
#>  pkgbuild        1.4.2      2023-06-26 [1] RSPM
#>  pkgconfig       2.0.3      2019-09-22 [1] RSPM
#>  pkgdown         2.0.7      2022-12-14 [1] RSPM
#>  pkgload         1.3.3      2023-09-22 [1] RSPM
#>  PKPDsim       * 1.3.0      2023-03-02 [1] RSPM
#>  PopED         * 0.6.0.9002 2023-11-21 [1] local
#>  PreciseSums     0.6        2023-04-22 [1] RSPM
#>  prettyunits     1.2.0      2023-09-24 [1] RSPM
#>  processx        3.8.2      2023-06-30 [1] RSPM
#>  profvis         0.3.8      2023-05-02 [1] RSPM
#>  promises        1.2.1      2023-08-10 [1] RSPM
#>  ps              1.7.5      2023-04-18 [1] RSPM
#>  purrr           1.0.2      2023-08-10 [1] RSPM
#>  qs              0.25.5     2023-02-22 [1] RSPM
#>  R6              2.5.1      2021-08-19 [1] RSPM
#>  ragg            1.2.6      2023-10-10 [1] RSPM
#>  RApiSerialize   0.1.2      2022-08-25 [1] RSPM
#>  Rcpp          * 1.0.11     2023-07-06 [1] RSPM
#>  RcppParallel    5.1.7      2023-02-27 [1] RSPM
#>  remotes         2.4.2.1    2023-07-18 [1] RSPM
#>  rlang           1.1.2      2023-11-04 [1] RSPM
#>  rmarkdown       2.25       2023-09-18 [1] RSPM
#>  rprojroot       2.0.4      2023-11-05 [1] RSPM
#>  rstudioapi      0.15.0     2023-07-07 [1] RSPM
#>  rvest           1.0.3      2022-08-19 [1] RSPM
#>  rxode2        * 2.0.14     2023-10-07 [1] RSPM
#>  rxode2et        2.0.10     2023-03-17 [1] RSPM
#>  rxode2ll        2.0.11     2023-03-17 [1] RSPM
#>  rxode2parse     2.0.16     2023-03-28 [1] RSPM
#>  rxode2random    2.0.11     2023-03-28 [1] RSPM
#>  sass            0.4.7      2023-07-15 [1] RSPM
#>  scales          1.2.1      2022-08-20 [1] RSPM
#>  sessioninfo     1.2.2      2021-12-06 [1] RSPM
#>  shiny           1.8.0      2023-11-17 [1] RSPM
#>  stringfish      0.15.8     2023-05-30 [1] RSPM
#>  stringi         1.8.1      2023-11-13 [1] RSPM
#>  stringr         1.5.1      2023-11-14 [1] RSPM
#>  svglite         2.1.2      2023-10-11 [1] RSPM
#>  sys             3.4.2      2023-05-23 [1] RSPM
#>  systemfonts     1.0.5      2023-10-09 [1] RSPM
#>  textshaping     0.3.7      2023-10-09 [1] RSPM
#>  tibble          3.2.1      2023-03-20 [1] RSPM
#>  tidyselect      1.2.0      2022-10-10 [1] RSPM
#>  urlchecker      1.0.1      2021-11-30 [1] RSPM
#>  usethis         2.2.2      2023-07-06 [1] RSPM
#>  utf8            1.2.4      2023-10-22 [1] RSPM
#>  vctrs           0.6.4      2023-10-12 [1] RSPM
#>  viridisLite     0.4.2      2023-05-02 [1] RSPM
#>  webshot         0.5.5      2023-06-26 [1] RSPM
#>  withr           2.5.2      2023-10-30 [1] RSPM
#>  xfun            0.41       2023-11-01 [1] RSPM
#>  xml2            1.3.5      2023-07-06 [1] RSPM
#>  xtable          1.8-4      2019-04-21 [1] RSPM
#>  yaml            2.3.7      2023-01-23 [1] RSPM
#>
#>  [1] /home/runner/work/_temp/Library
#>  [2] /opt/R/4.3.2/lib/R/site-library
#>  [3] /opt/R/4.3.2/lib/R/library
#>
#> ──────────────────────────────────────────────────────────────────────────────
#sessionInfo()