## Population Simulations with rxode2

### Simulation of Variability with rxode2

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 rxode2 model describing this system below:

#### Setting up the rxode2 model

library(rxode2)
#>   no cache: create with rxCreateCache()

set.seed(32)
rxSetSeed(32)

mod <- function() {
ini({
KA <- 2.94E-01
TCl <- 1.86E+01
# between subject variability
eta.Cl ~ 0.4^2
V2 <- 4.02E+01
Q <- 1.05E+01
V3 <- 2.97E+02
Kin <- 1
Kout <- 1
EC50 <- 200
})
model({
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
eff(0) <- 1
})
}

#### Simulating

The next step to simulate is to create the dosing regimen for overall simulation:

ev <- et(amountUnits="mg", timeUnits="hours") %>%
et(amt=10000, cmt="centr")

If you wish, you can also add sampling times (though rxode2 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, ev, nSub=100)
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

To quickly look and customize your simulation you use the default plot routine. Since this is an rxode2 object, it will create a ggplot2 object that you can modify as you wish. The extra parameter to the plot tells rxode2/R what piece of information you are interested in plotting. In this case, we are interested in looking at the derived parameter C2:

#### Checking the simulation with plot

library(ggplot2)
## The plots from rxode2 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)

#### Processing the data to create summary plots

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 rxode2 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")
#> ! in order to put confidence bands around the intervals, you need at least 2500 simulations
#> summarizing data...done

confint(sim, "eff", level=0.95) %>%
plot(ylab="Effect")
#> ! in order to put confidence bands around the intervals, you need at least 2500 simulations
#> summarizing data...done

Note that you can see the parameters that were simulated for the example

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, dat) #> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’  sim %>% select(id, time, eff, C2) #> id time eff C2 #> 1 1 0.000000 [h] 1.000000 248.7562189 #> 2 1 5.333333 [h] 1.168348 19.1658271 #> 3 1 10.666667 [h] 1.040652 7.1764936 #> 4 1 16.000000 [h] 1.030849 6.0167596 #> 5 1 21.333333 [h] 1.027742 5.4446478 #> 6 1 26.666667 [h] 1.025204 4.9484199 #> 7 1 32.000000 [h] 1.022911 4.4984868 #> 8 1 37.333333 [h] 1.020828 4.0895169 #> 9 1 42.666667 [h] 1.018933 3.7177314 #> 10 1 48.000000 [h] 1.017211 3.3797461 #> 11 2 0.000000 [h] 1.000000 124.3781095 #> 12 2 6.857143 [h] 1.016301 1.9290018 #> 13 2 13.714286 [h] 1.006648 1.2918078 #> 14 2 20.571429 [h] 1.005617 1.0964981 #> 15 2 27.428571 [h] 1.004772 0.9317005 #> 16 2 34.285714 [h] 1.004055 0.7916739 #> 17 2 41.142857 [h] 1.003446 0.6726918 #> 18 2 48.000000 [h] 1.002928 0.5715923 ## Simulation of Clinical Trials 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. rxode2 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 rxode2s cvPost() function. ### Simulation from inverse Wishart correlations 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(mod2$theta)[1:8])

sim  <- rxSolve(mod2, ev, nSub=100, thetaMat=tMat, nStud=10,
dfSub=10, dfObs=100)

s <-sim %>% confint("sim")
#> ! in order to put confidence bands around the intervals, you need at least 2500 simulations
#> summarizing data...done

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$omegaList) #> [[1]] #> eta.Cl #> eta.Cl 0.1676778 #> #> [[2]] #> eta.Cl #> eta.Cl 0.2917085 #> #> [[3]] #> eta.Cl #> eta.Cl 0.1776813 #> #> [[4]] #> eta.Cl #> eta.Cl 0.1578682 #> #> [[5]] #> eta.Cl #> eta.Cl 0.1845614 #> #> [[6]] #> eta.Cl #> eta.Cl 0.3282268 head(sim$sigmaList)
#> [[1]]
#>            rxerr.eff   rxerr.C2
#> rxerr.eff 1.12416983 0.04197039
#> rxerr.C2  0.04197039 0.97293971
#>
#> [[2]]
#>             rxerr.eff    rxerr.C2
#> rxerr.eff  0.84311199 -0.06277998
#> rxerr.C2  -0.06277998  1.22140938
#>
#> [[3]]
#>           rxerr.eff  rxerr.C2
#> rxerr.eff 0.9834771 0.1060251
#> rxerr.C2  0.1060251 1.0024751
#>
#> [[4]]
#>            rxerr.eff   rxerr.C2
#> rxerr.eff 1.25556975 0.07690868
#> rxerr.C2  0.07690868 0.90991261
#>
#> [[5]]
#>           rxerr.eff  rxerr.C2
#> rxerr.eff  1.116261 -0.184748
#> rxerr.C2  -0.184748  1.320288
#>
#> [[6]]
#>            rxerr.eff   rxerr.C2
#> rxerr.eff 0.93539238 0.07270049
#> rxerr.C2  0.07270049 0.98648424

You can also see the parameter realizations from the $params data frame. ### Simulate using variance/standard deviation standard errors Lets assume we wish to simulate from the nonmem run included in xpose First we setup the model; Since we are taking this from nonmem and would like to use the more free-form style from the classic rxode2 model we start from the classic model: rx1 <- rxode2({ 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 }) #> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’ 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); #> ℹ thetaMat has too many items, ignored: 'tlag'  print(sim) #> ── Solved rxode2 object ── #> ── Parameters ($params): ──
#> # A tibble: 10,000 × 9
#>    sim.id id      tcl crcl.cl  eta.cl    tv   eta.v   tka  eta.ka
#>     <int> <fct> <dbl>   <dbl>   <dbl> <dbl>   <dbl> <dbl>   <dbl>
#>  1      1 1      27.0   0.828 -0.884   1.73  0.531   5.38  0.114
#>  2      1 2      27.0   0.828  1.11    1.73 -0.906   5.38 -0.738
#>  3      1 3      27.0   0.828 -0.690   1.73  0.280   5.38 -0.365
#>  4      1 4      27.0   0.828  0.0751  1.73 -1.17    5.38 -0.575
#>  5      1 5      27.0   0.828 -0.299   1.73 -0.121   5.38 -0.601
#>  6      1 6      27.0   0.828 -0.638   1.73 -0.281   5.38 -0.309
#>  7      1 7      27.0   0.828  1.05    1.73 -0.0495  5.38  0.567
#>  8      1 8      27.0   0.828 -0.371   1.73 -0.372   5.38 -0.903
#>  9      1 9      27.0   0.828 -0.206   1.73  1.97    5.38  0.0203
#> 10      1 10     27.0   0.828  0.754   1.73  0.166   5.38 -1.73
#> # ℹ 9,990 more rows
#> ── Initial Conditions ($inits): ── #> named numeric(0) #> #> Simulation with uncertainty in: #> • parameters ($thetaMat for changes)
#> • omega matrix ($omegaList) #> • sigma matrix ($sigmaList)
#>
#> ── First part of data (object): ──
#> # A tibble: 50,000 × 10
#>   sim.id    id   time     cl     v    ka  ipred    obs    WT  CLCR
#>    <int> <int>    [h]  <dbl> <dbl> <dbl>  <dbl>  <dbl> <dbl> <dbl>
#> 1      1     1 0.0155   50.4 183.   6.03 0.0487  3.56   62.2  69.3
#> 2      1     1 0.749    50.4 183.   6.03 0.460  -1.59   62.2  69.3
#> 3      1     1 1.02     50.4 183.   6.03 0.431   1.15   62.2  69.3
#> 4      1     1 3.41     50.4 183.   6.03 0.224   1.92   62.2  69.3
#> 5      1     1 7.81     50.4 183.   6.03 0.0666  0.731  62.2  69.3
#> 6      1     2 0.0833 2816.   24.9  2.57 0.0754 -1.20   35.7 105.
#> # ℹ 49,994 more rows
## 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)
#> ℹ thetaMat has too many items, ignored: 'tlag'

s <-sim %>% confint(c("ipred"))
#> summarizing data...
#> done
plot(s)

### Simulate without uncertainty in omega or sigma parameters

If 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:


sim  <- rxSolve(mod2, ev, nSub=100, thetaMat=tMat, nStud=10,
simVariability=FALSE)

s <-sim %>% confint(c("centr", "eff"))
#> ! in order to put confidence bands around the intervals, you need at least 2500 simulations
#> summarizing data...done

plot(s)

Note since realizations of omega and sigma were not simulated, $omegaList and $sigmaList both return NULL.