In pharmacometrics the nonlinear-mixed effect modeling software (like nlmixr) characterizes the between-subject variability. With this between subject variability you can simulate new subjects.
Assuming that you have a 2-compartment, indirect response model, you can set create an RxODE model describing this system below:
library(RxODE) set.seed(32) rxSetSeed(32) mod <- RxODE({ eff(0) = 1 C2 = centr/V2*(1+prop.err); C3 = peri/V3; CL = TCl*exp(eta.Cl) ## This is coded as a variable in the model 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; })
The next step is to get the parameters into R so that you can start the simulation:
theta <- c(KA=2.94E-01, TCl=1.86E+01, V2=4.02E+01, # central Q=1.05E+01, V3=2.97E+02, # peripheral Kin=1, Kout=1, EC50=200, prop.err=0) # effects
In this case, I use lotri
to specify the omega since it uses similar
lower-triangular matrix specification as nlmixr (also similar to
NONMEM):
## the column names of the omega matrix need to match the parameters specified by RxODE omega <- lotri(eta.Cl ~ 0.4^2) omega
The next step to simulate is to create the dosing regimen for overall simulation:
ev <- et(amount.units="mg", time.units="hours") %>% et(amt=10000, cmt="centr")
If you wish, you can also add sampling times (though now RxODE can fill these in for you):
ev <- ev %>% et(0,48, length.out=100)
Note the et
takes similar arguments as seq
when adding sampling
times. There are more methods to adding sampling times and events to
make complex dosing regimens (See the event
vignette). This includes ways to add variability
to the both the sampling and dosing
times).
Once this is complete you can simulate using the rxSolve
routine:
sim <- rxSolve(mod,theta,ev,omega=omega,nSub=100)
To quickly look and customize your simulation you use the default
plot
routine. Since this is an RxODE object, it will create a
ggplot2
object that you can modify as you wish. The extra parameter
to the plot
tells RxODE
/R
what piece of information you are
interested in plotting. In this case, we are interested in looking at
the derived parameter C2
:
plot
library(ggplot2) ## The plots from RxODE are ggplots so they can be modified with ## standard ggplot commands. plot(sim, C2, log="y") + ylab("Central Compartment")
Of course this additional parameter could also be a state value, like eff
:
## They also takes many of the standard plot arguments; See ?plot plot(sim, eff, ylab="Effect")
Or you could even look at the two side-by-side:
plot(sim, C2, eff)
Or stack them with patchwork
library(patchwork) plot(sim, C2, log="y") / plot(sim, eff)
Usually in pharmacometric simulations it is not enough to simply simulate the system. We have to do something easier to digest, like look at the central and extreme tendencies of the simulation.
Since the RxODE
solve object is a type of data
frame
It is now straightforward to perform calculations and generate plots with the simulated data. You can
Below, the 5th, 50th, and 95th percentiles of the simulated data are plotted.
confint(sim, "C2", level=0.95) %>% plot(ylab="Central Concentration", log="y")
confint(sim, "eff", level=0.95) %>% plot(ylab="Effect")
Note that you can see the parameters that were simulated for the example
head(sim$param)
In addition to conveniently simulating between subject variability, you can also easily simulate unexplained variability.
mod <- RxODE({ eff(0) = 1 C2 = centr/V2; C3 = peri/V3; CL = TCl*exp(eta.Cl) ## This is coded as a variable in the model d/dt(depot) =-KA*depot; d/dt(centr) = KA*depot - CL*C2 - Q*C2 + Q*C3; d/dt(peri) = Q*C2 - Q*C3; d/dt(eff) = Kin - Kout*(1-C2/(EC50+C2))*eff; e = eff+eff.err cp = centr*(1+cp.err) }) theta <- c(KA=2.94E-01, TCl=1.86E+01, V2=4.02E+01, # central Q=1.05E+01, V3=2.97E+02, # peripheral Kin=1, Kout=1, EC50=200) # effects sigma <- lotri(eff.err ~ 0.1, cp.err ~ 0.1) sim <- rxSolve(mod, theta, ev, omega=omega, nSub=100, sigma=sigma) s <- confint(sim, c("eff", "centr")); plot(s)
Sometimes you may want to match the dosing and observations of
individuals in a clinical trial. To do this you will have to create a
data.frame using the RxODE
event specification as well as an ID
column to indicate an individual. The RxODE event vignette talks more about
how these datasets should be created.
library(dplyr) ev1 <- eventTable(amount.units="mg", time.units="hours") %>% add.dosing(dose=10000, nbr.doses=1, dosing.to=2) %>% add.sampling(seq(0,48,length.out=10)); ev2 <- eventTable(amount.units="mg", time.units="hours") %>% add.dosing(dose=5000, nbr.doses=1, dosing.to=2) %>% add.sampling(seq(0,48,length.out=8)); dat <- rbind(data.frame(ID=1, ev1$get.EventTable()), data.frame(ID=2, ev2$get.EventTable())) ## Note the number of subject is not needed since it is determined by the data sim <- rxSolve(mod, theta, dat, omega=omega, sigma=sigma) sim %>% select(id, time, e, cp)
By either using a simple single event table, or data from a clinical trial as described above, a complete clinical trial simulation can be performed.
Typically in clinical trial simulations you want to account for the uncertainty in the fixed parameter estimates, and even the uncertainty in both your between subject variability as well as the unexplained variability.
RxODE
allows you to account for these uncertainties by simulating
multiple virtual "studies," specified by the parameter nStud
. Each
of these studies samples a realization of fixed effect parameters and
covariance matrices for the between subject variability(omega
) and
unexplained variabilities (sigma
). Depending on the information you
have from the models, there are a few strategies for simulating a
realization of the omega
and sigma
matrices.
The first strategy occurs when either there is not any standard errors for standard deviations (or related parameters), or there is a modeled correlation in the model you are simulating from. In that case the suggested strategy is to use the inverse Wishart (parameterized to scale to the conjugate prior)/scaled inverse chi distribution. this approach uses a single parameter to inform the variability of the covariance matrix sampled (the degrees of freedom).
The second strategy occurs if you have standard errors on the
variance/standard deviation with no modeled correlations in the
covariance matrix. In this approach you perform separate simulations
for the standard deviations and the correlation matrix. First you
simulate the variance/standard deviation components in the thetaMat
multivariate normal simulation. After simulation and transformation
to standard deviations, a correlation matrix is simulated using the
degrees of freedom of your covariance matrix. Combining the simulated
standard deviation with the simulated correlation matrix will give a
simulated covariance matrix. For smaller dimension covariance matrices
(dimension < 10x10) it is recommended you use the lkj
distribution
to simulate the correlation matrix. For higher dimension covariance
matrices it is suggested you use the inverse wishart distribution
(transformed to a correlation matrix) for the simulations.
The covariance/variance prior is simulated from RxODE
s cvPost()
function.
An example of this simulation is below:
## Creating covariance matrix tmp <- matrix(rnorm(8^2), 8, 8) tMat <- tcrossprod(tmp, tmp) / (8 ^ 2) dimnames(tMat) <- list(NULL, names(theta)) sim <- rxSolve(mod, theta, ev, omega=omega, nSub=100, sigma=sigma, thetaMat=tMat, nStud=10, dfSub=10, dfObs=100) s <-sim %>% confint(c("centr", "eff")) plot(s)
If you wish you can see what omega
and sigma
was used for each
virtual study by accessing them in the solved data object with
$omega.list
and $sigma.list
:
head(sim$omega.list)
head(sim$sigma.list)
You can also see the parameter realizations from the $params
data frame.
Lets assume we wish to simulate from the nonmem run included in xpose
First we setup the model:
rx1 <- RxODE({ cl <- tcl*(1+crcl.cl*(CLCR-65)) * exp(eta.cl) v <- tv * WT * exp(eta.v) ka <- tka * exp(eta.ka) ipred <- linCmt() obs <- ipred * (1 + prop.sd) + add.sd })
Next we input the estimated parameters:
theta <- c(tcl=2.63E+01, tv=1.35E+00, tka=4.20E+00, tlag=2.08E-01, prop.sd=2.05E-01, add.sd=1.06E-02, crcl.cl=7.17E-03, ## Note that since we are using the separation strategy the ETA variances are here too eta.cl=7.30E-02, eta.v=3.80E-02, eta.ka=1.91E+00)
And also their covariances; To me, the easiest way to create a named
covariance matrix is to use lotri()
:
thetaMat <- lotri( tcl + tv + tka + tlag + prop.sd + add.sd + crcl.cl + eta.cl + eta.v + eta.ka ~ c(7.95E-01, 2.05E-02, 1.92E-03, 7.22E-02, -8.30E-03, 6.55E-01, -3.45E-03, -6.42E-05, 3.22E-03, 2.47E-04, 8.71E-04, 2.53E-04, -4.71E-03, -5.79E-05, 5.04E-04, 6.30E-04, -3.17E-06, -6.52E-04, -1.53E-05, -3.14E-05, 1.34E-05, -3.30E-04, 5.46E-06, -3.15E-04, 2.46E-06, 3.15E-06, -1.58E-06, 2.88E-06, -1.29E-03, -7.97E-05, 1.68E-03, -2.75E-05, -8.26E-05, 1.13E-05, -1.66E-06, 1.58E-04, -1.23E-03, -1.27E-05, -1.33E-03, -1.47E-05, -1.03E-04, 1.02E-05, 1.67E-06, 6.68E-05, 1.56E-04, 7.69E-02, -7.23E-03, 3.74E-01, 1.79E-03, -2.85E-03, 1.18E-05, -2.54E-04, 1.61E-03, -9.03E-04, 3.12E-01)) evw <- et(amount.units="mg", time.units="hours") %>% et(amt=100) %>% ## For this problem we will simulate with sampling windows et(list(c(0, 0.5), c(0.5, 1), c(1, 3), c(3, 6), c(6, 12))) %>% et(id=1:1000) ## From the run we know that: ## total number of observations is: 476 ## Total number of individuals: 74 sim <- rxSolve(rx1, theta, evw, nSub=100, nStud=10, thetaMat=thetaMat, ## Match boundaries of problem thetaLower=0, sigma=c("prop.sd", "add.sd"), ## Sigmas are standard deviations sigmaXform="identity", # default sigma xform="identity" omega=c("eta.cl", "eta.v", "eta.ka"), ## etas are variances omegaXform="variance", # default omega xform="variance" iCov=data.frame(WT=rnorm(1000, 70, 15), CLCR=rnorm(1000, 65, 25)), dfSub=74, dfObs=476); print(sim) ## Notice that the simulation time-points change for the individual ## If you want the same sampling time-points you can do that as well: evw <- et(amount.units="mg", time.units="hours") %>% et(amt=100) %>% et(0, 24, length.out=50) %>% et(id=1:100) sim <- rxSolve(rx1, theta, evw, nSub=100, nStud=10, thetaMat=thetaMat, ## Match boundaries of problem thetaLower=0, sigma=c("prop.sd", "add.sd"), ## Sigmas are standard deviations sigmaXform="identity", # default sigma xform="identity" omega=c("eta.cl", "eta.v", "eta.ka"), ## etas are variances omegaXform="variance", # default omega xform="variance" iCov=data.frame(WT=rnorm(100, 70, 15), CLCR=rnorm(100, 65, 25)), dfSub=74, dfObs=476, resample=TRUE) s <-sim %>% confint(c("ipred")) plot(s)
omega
or sigma
parametersIf you do not wish to sample from the prior distributions of either
the omega
or sigma
matrices, you can turn off this feature by
specifying the simVariability = FALSE
option when solving:
mod <- RxODE({ eff(0) = 1 C2 = centr/V2; C3 = peri/V3; CL = TCl*exp(eta.Cl) ## This is coded as a variable in the model 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; e = eff+eff.err cp = centr*(1+cp.err) }) theta <- c(KA=2.94E-01, TCl=1.86E+01, V2=4.02E+01, # central Q=1.05E+01, V3=2.97E+02, # peripheral Kin=1, Kout=1, EC50=200) # effects sigma <- lotri(eff.err ~ 0.1, cp.err ~ 0.1) sim <- rxSolve(mod, theta, ev, omega=omega, nSub=100, sigma=sigma, thetaMat=tMat, nStud=10, simVariability=FALSE) s <-sim %>% confint(c("centr", "eff")) plot(s)
Note since realizations of omega
and sigma
were not simulated,
$omega.list
and $sigma.list
both return NULL
.
RxODE now supports multi-threaded solving on OpenMP supported
compilers, including linux and windows. Mac OSX can also be supported
By default it uses all your available cores for solving as determined
by rxCores()
. This may be overkill depending on your system, at a
certain point the speed of solving is limited by things other than
computing power.
You can also speed up simulation by using the multi-cores to generate
random deviates with the threefry simulation engine. This is
controlled by the nCoresRV
parameter. For example:
sim <- rxSolve(mod, theta, ev, omega=omega, nSub=100, sigma=sigma, thetaMat=tMat, nStud=10, nCoresRV=2) s <-sim %>% confint(c("eff", "centr"))
The default for this is 1
core since the result depends on the
number of cores and the random seed you use in your simulation as well
as the work-load each thread is sharing/architecture. However, you can always
speed up this process with more cores if you are sure your
collaborators have the same number of cores available to them and have
OpenMP thread-capable compile.
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