Figure 1. A two compartment pharmacokinetics model with an effect compartment
Ordinary Differential Equation (ODE) models, with applications for pharmacometrics
RxODE is an R package that facilitates simulation with ODE models in
R. It is designed with pharmacometrics models in mind, but can be
applied more generally to any ODE model. Here, a typical
pharmacokinetics-pharmacodynamics (PKPD) model is used to illustrate
the application of RxODE. This model is illustrated in Figure 1. It is
assumed that all model parameters have been estimated previously.
Figure 1. A two compartment pharmacokinetics model with an effect compartment
The model equations are specified through a text string in R. Both
differential and algebraic equations are permitted. Differential
equations are specified by d/dt(var_name) =. Each
equation is separated by a semicolon.
ode <- "
C2 = centr/V2;
C3 = peri/V3;
d/dt(depot) =-KA*depot;
d/dt(centr) = KA*depot - CL*C2 - Q*C2 + Q*C3;
d/dt(peri) = Q*C2 - Q*C3;
d/dt(eff) = Kin - Kout*(1-C2/(EC50+C2))*eff;
"
To load RxODE package and compile the model:
library(RxODE)
work <- tempfile("Rx_intro-")
mod1 <- RxODE(model = ode, modName = "mod1", wd = work)
Model parameters are defined in named vectors. Names of parameters in the vector must be a superset of parameters in the ODE model, and the order of parameters within the vector is not important.
theta <-
c(KA=2.94E-01, CL=1.86E+01, V2=4.02E+01, # central
Q=1.05E+01, V3=2.97E+02, # peripheral
Kin=1, Kout=1, EC50=200) # effects
Initial conditions (ICs) are defined through a vector as well. The number of ICs must equal exactly the number of ODEs in the model, and the order must be the same as the order in which the ODEs are listed in the model. Elements may be named if desired:
inits <- c(depot=0, centr=0, peri=0, eff=1)
RxODE provides a simple and very flexible way to specify dosing and
sampling through functions that generate an event table. First, an
empty event table is generated through the “eventTable()” function:
ev <- eventTable(amount.units='mg', time.units='hours')
Next, use the add.dosing() and add.sampling() functions of the
EventTable object to specify the dosing (amounts, frequency and/or
times, etc.) and observation times at which to sample the state of the
system. These functions can be called multiple times to specify more
complex dosing or sampling regiments. Here, these functions are used
to specify 10mg BID dosing for 5 days, followed by 20mg QD dosing for
5 days:
ev$add.dosing(dose=10000, nbr.doses=10, dosing.interval=12)
ev$add.dosing(dose=20000, nbr.doses=5, start.time=120, dosing.interval=24)
ev$add.sampling(0:240)
The functions get.dosing() and get.sampling() can be used to
retrieve information from the event table.
head(ev$get.dosing())
## time evid amt
## 1 0 101 10000
## 2 12 101 10000
## 3 24 101 10000
## 4 36 101 10000
## 5 48 101 10000
## 6 60 101 10000
head(ev$get.sampling())
## time evid amt
## 16 0 0 NA
## 17 1 0 NA
## 18 2 0 NA
## 19 3 0 NA
## 20 4 0 NA
## 21 5 0 NA
The simulation can now be run by calling the model object's run function. Simulation results for all variables in the model are stored in the output matrix x.
x <- mod1$solve(theta, ev, inits)
head(x)
## time depot centr peri eff C2 C3
## [1,] 0 10000 0 0.0 1.000 0.00 0.0000
## [2,] 1 7453 1784 273.2 1.085 44.38 0.9198
## [3,] 2 5554 2206 793.9 1.181 54.88 2.6730
## [4,] 3 4140 2087 1323.6 1.229 51.90 4.4565
## [5,] 4 3085 1789 1776.3 1.235 44.50 5.9807
## [6,] 5 2299 1467 2131.7 1.215 36.48 7.1775
par(mfrow=c(1,2))
matplot(x[,"C2"], type="l", ylab="Central Concentration")
matplot(x[,"eff"], type="l", ylab = "Effect")
Variability in model parameters can be simulated by creating a matrix of parameter values for use in the simulation. In the example below, 40% variability in clearance is simulated.
nsub <- 100 #number of subproblems
CL <- 1.86E+01*exp(rnorm(nsub,0,.4^2))
theta.all <-
cbind(KA=2.94E-01, CL=CL, V2=4.02E+01, # central
Q=1.05E+01, V3=2.97E+02, # peripheral
Kin=1, Kout=1, EC50=200) # effects
head(theta.all)
## KA CL V2 Q V3 Kin Kout EC50
## [1,] 0.294 20.74 40.2 10.5 297 1 1 200
## [2,] 0.294 19.83 40.2 10.5 297 1 1 200
## [3,] 0.294 16.43 40.2 10.5 297 1 1 200
## [4,] 0.294 17.79 40.2 10.5 297 1 1 200
## [5,] 0.294 17.77 40.2 10.5 297 1 1 200
## [6,] 0.294 15.80 40.2 10.5 297 1 1 200
Each subproblem can be simulated by using an explicit loop (or the apply()
function) to run the simulation for each set of parameters of in the parameter
matrix.
nobs <- ev$get.nobs()
cp.all <- matrix(NA, nobs, nsub)
for (i in 1:nsub)
{
theta <- theta.all[i,]
x <- mod1$solve(theta, ev, inits=inits)
cp.all[, i] <- x[, "C2"]
}
matplot(cp.all, type="l", ylab="Central Concentration")
It is now straightforward to perform calculations and generate plots with the simulated data. Below, the 5th, 50th, and 95th percentiles of the simulated data are plotted.
cp.q <- apply(cp.all, 1, quantile, prob = c(0.05, 0.50, 0.95))
matplot(t(cp.q), type="l", lty=c(2,1,2), col=c(2,1,2), ylab="Central Concentration")
R Shiny applications (http://shiny.rstudio.com)
may be programmatically created with the experimental function
genShinyApp.template().
The above application includes widgets for varying the dose, dosing regimen, dose cycle, and number of cycles. However, the user may then edit and tailor the shiny app server.R and ui.R files as needed.
genShinyApp.template(appDir = "shinyExample", verbose=TRUE)
library(shiny)
runApp("shinyExample")