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Nesting in rxode2

More than one level of nesting is possible in rxode2; In this example we will be using the following uncertainties and sources of variability:

Level Variable Matrix specified Integrated Matrix
Model uncertainty NA thetaMat thetaMat
Investigator inv.Cl, inv.Ka omega theta
Subject eta.Cl, eta.Ka omega omega
Eye eye.Cl, eye.Ka omega omega
Occasion iov.Cl, occ.Ka omega omega
Unexplained Concentration prop.sd sigma sigma
Unexplained Effect add.sd sigma sigma

Event table

This event table contains nesting variables:

  • inv: investigator id
  • id: subject id
  • eye: eye id (left or right)
  • occ: occasion
#> rxode2 2.1.3 using 2 threads (see ?getRxThreads)
#>   no cache: create with `rxCreateCache()`
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union
et(amountUnits="mg", timeUnits="hours") %>%
  et(amt=10000, addl=9,ii=12,cmt="depot") %>%
  et(time=120, amt=2000, addl=4, ii=14, cmt="depot") %>%
  et(seq(0, 240, by=4)) %>% # Assumes sampling when there is no dosing information
  et(seq(0, 240, by=4) + 0.1) %>% ## adds 0.1 for separate eye
  et(id=1:20) %>%
  ## Add an occasion per dose
  mutate(occ=cumsum(!is.na(amt))) %>%
  mutate(occ=ifelse(occ == 0, 1, occ)) %>%
  mutate(occ=2- occ %% 2) %>%
  mutate(eye=ifelse(round(time) == time, 1, 2)) %>%
  mutate(inv=ifelse(id < 10, 1, 2)) %>% as_tibble ->
  ev

rxode2 model

This creates the rxode2 model with multi-level nesting. Note the variables inv.Cl, inv.Ka, eta.Cl etc; You only need one variable for each level of nesting.

mod <- rxode2({
  ## Clearance with individuals
  eff(0) = 1
  C2 = centr/V2*(1+prop.sd)
  C3 = peri/V3
  CL =  TCl*exp(eta.Cl + eye.Cl + iov.Cl + inv.Cl)
  KA = TKA * exp(eta.Ka + eye.Ka + iov.Cl + inv.Ka)
  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
  ef0 = eff + add.sd
})
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

Uncertainty in Model parameters

theta <- c("TKA"=0.294, "TCl"=18.6, "V2"=40.2,
           "Q"=10.5, "V3"=297, "Kin"=1, "Kout"=1, "EC50"=200)

## Creating covariance matrix
tmp <- matrix(rnorm(8^2), 8, 8)
tMat <- tcrossprod(tmp, tmp) / (8 ^ 2)
dimnames(tMat) <- list(names(theta), names(theta))

tMat
#>               TKA           TCl           V2            Q           V3
#> TKA   0.173571236 -0.1003204607  0.038185010 -0.004108928 -0.095032973
#> TCl  -0.100320461  0.2195710868 -0.043849095  0.013295549 -0.007477895
#> V2    0.038185010 -0.0438490948  0.129784612 -0.017270432 -0.038004762
#> Q    -0.004108928  0.0132955493 -0.017270432  0.022145634  0.020376451
#> V3   -0.095032973 -0.0074778948 -0.038004762  0.020376451  0.165568340
#> Kin  -0.040867119 -0.0492597458 -0.003056722 -0.033468634 -0.003021883
#> Kout  0.035469225  0.0275087955  0.033725901  0.027668205  0.005497301
#> EC50  0.026158042  0.0009434711  0.039426946 -0.036283167 -0.093134292
#>               Kin         Kout          EC50
#> TKA  -0.040867119  0.035469225  0.0261580416
#> TCl  -0.049259746  0.027508796  0.0009434711
#> V2   -0.003056722  0.033725901  0.0394269465
#> Q    -0.033468634  0.027668205 -0.0362831667
#> V3   -0.003021883  0.005497301 -0.0931342922
#> Kin   0.226735493 -0.083447793  0.0659884544
#> Kout -0.083447793  0.117195570 -0.0291684598
#> EC50  0.065988454 -0.029168460  0.0928611407

Nesting Variability

To specify multiple levels of nesting, you can specify it as a nested lotri matrix; When using this approach you use the condition operator | to specify what variable nesting occurs on; For the Bayesian simulation we need to specify how much information we have for each parameter; For rxode2 this is the nu parameter.

In this case: - id, nu=100 or the model came from 100 subjects - eye, nu=200 or the model came from 200 eyes - occ, nu=200 or the model came from 200 occasions - inv, nu=10 or the model came from 10 investigators

To specify this in lotri you can use | var(nu=X), or:

omega <- lotri(lotri(eta.Cl ~ 0.1,
                     eta.Ka ~ 0.1) | id(nu=100),
               lotri(eye.Cl ~ 0.05,
                     eye.Ka ~ 0.05) | eye(nu=200),
               lotri(iov.Cl ~ 0.01,
                     iov.Ka ~ 0.01) | occ(nu=200),
               lotri(inv.Cl ~ 0.02,
                     inv.Ka ~ 0.02) | inv(nu=10))
omega
#> $id
#>        eta.Cl eta.Ka
#> eta.Cl    0.1    0.0
#> eta.Ka    0.0    0.1
#> 
#> $eye
#>        eye.Cl eye.Ka
#> eye.Cl   0.05   0.00
#> eye.Ka   0.00   0.05
#> 
#> $occ
#>        iov.Cl iov.Ka
#> iov.Cl   0.01   0.00
#> iov.Ka   0.00   0.01
#> 
#> $inv
#>        inv.Cl inv.Ka
#> inv.Cl   0.02   0.00
#> inv.Ka   0.00   0.02
#> 
#> Properties: nu

Unexplained variability

The last piece of variability to specify is the unexplained variability

sigma <- lotri(prop.sd ~ .25,
               add.sd~ 0.125)

Solving the problem

s <- rxSolve(mod, theta, ev,
             thetaMat=tMat, omega=omega,
             sigma=sigma, sigmaDf=400,
             nStud=400)
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
#> unhandled error message: EE:[lsoda] 70000 steps taken before reaching tout
#>  @(lsoda.c:751
#> Warning: some ID(s) could not solve the ODEs correctly; These values are
#> replaced with 'NA'
#> -- Solved rxode2 object --
#> -- Parameters ($params): --
#> # A tibble: 8,000 x 24
#>    sim.id id    `inv.Cl(inv==1)` `inv.Cl(inv==2)` `inv.Ka(inv==1)`
#>     <int> <fct>            <dbl>            <dbl>            <dbl>
#>  1      1 1              0.00900           -0.117            0.312
#>  2      1 2              0.00900           -0.117            0.312
#>  3      1 3              0.00900           -0.117            0.312
#>  4      1 4              0.00900           -0.117            0.312
#>  5      1 5              0.00900           -0.117            0.312
#>  6      1 6              0.00900           -0.117            0.312
#>  7      1 7              0.00900           -0.117            0.312
#>  8      1 8              0.00900           -0.117            0.312
#>  9      1 9              0.00900           -0.117            0.312
#> 10      1 10             0.00900           -0.117            0.312
#> # i 7,990 more rows
#> # i 19 more variables: `inv.Ka(inv==2)` <dbl>, `eye.Cl(eye==1)` <dbl>,
#> #   `eye.Cl(eye==2)` <dbl>, `eye.Ka(eye==1)` <dbl>, `eye.Ka(eye==2)` <dbl>,
#> #   `iov.Cl(occ==1)` <dbl>, `iov.Cl(occ==2)` <dbl>, `iov.Ka(occ==1)` <dbl>,
#> #   `iov.Ka(occ==2)` <dbl>, V2 <dbl>, V3 <dbl>, TCl <dbl>, eta.Cl <dbl>,
#> #   TKA <dbl>, eta.Ka <dbl>, Q <dbl>, Kin <dbl>, Kout <dbl>, EC50 <dbl>
#> -- Initial Conditions ($inits): --
#> depot centr  peri   eff 
#>     0     0     0     1 
#> 
#> Simulation with uncertainty in:
#> * parameters ($thetaMat for changes)
#> * omega matrix ($omegaList)
#> 
#> -- First part of data (object): --
#> # A tibble: 976,000 x 21
#>   sim.id    id time  inv.Cl inv.Ka  eye.Cl  eye.Ka iov.Cl iov.Ka    C2     C3
#>    <int> <int>  [h]   <dbl>  <dbl>   <dbl>   <dbl>  <dbl>  <dbl> <dbl>  <dbl>
#> 1      1     1  0   0.00900  0.312 -0.330  -0.0547 0.0370  0.139  0    0     
#> 2      1     1  0.1 0.00900  0.312  0.0792  0.185  0.0370  0.139  8.04 0.0108
#> 3      1     1  4   0.00900  0.312 -0.330  -0.0547 0.0370  0.139 55.4  5.58  
#> 4      1     1  4.1 0.00900  0.312  0.0792  0.185  0.0370  0.139 23.2  5.69  
#> 5      1     1  8   0.00900  0.312 -0.330  -0.0547 0.0370  0.139 19.6  7.95  
#> 6      1     1  8.1 0.00900  0.312  0.0792  0.185  0.0370  0.139 20.0  7.97  
#> # i 975,994 more rows
#> # i 10 more variables: CL <dbl>, KA <dbl>, ef0 <dbl>, depot <dbl>, centr <dbl>,
#> #   peri <dbl>, eff <dbl>, occ <fct>, eye <fct>, inv <fct>

There are multiple investigators in a study; Each investigator has a number of individuals enrolled at their site. rxode2 automatically determines the number of investigators and then will simulate an effect for each investigator. With the output, inv.Cl(inv==1) will be the inv.Cl for investigator 1, inv.Cl(inv==2) will be the inv.Cl for investigator 2, etc.

inv.Cl(inv==1), inv.Cl(inv==2), etc will be simulated for each study and then combined to form the between investigator variability. In equation form these represent the following:

inv.Cl = (inv == 1) * `inv.Cl(inv==1)` + (inv == 2) * `inv.Cl(inv==2)`

If you look at the simulated parameters you can see inv.Cl(inv==1) and inv.Cl(inv==2) are in the s$params; They are the same for each study:

print(head(s$params))
#>   sim.id id inv.Cl(inv==1) inv.Cl(inv==2) inv.Ka(inv==1) inv.Ka(inv==2)
#> 1      1  1    0.008995575     -0.1165083      0.3118205   -0.001926179
#> 2      1  2    0.008995575     -0.1165083      0.3118205   -0.001926179
#> 3      1  3    0.008995575     -0.1165083      0.3118205   -0.001926179
#> 4      1  4    0.008995575     -0.1165083      0.3118205   -0.001926179
#> 5      1  5    0.008995575     -0.1165083      0.3118205   -0.001926179
#> 6      1  6    0.008995575     -0.1165083      0.3118205   -0.001926179
#>   eye.Cl(eye==1) eye.Cl(eye==2) eye.Ka(eye==1) eye.Ka(eye==2) iov.Cl(occ==1)
#> 1    -0.33023666    0.079190835    -0.05471051     0.18487157    0.037021761
#> 2    -0.42211135   -0.006773841    -0.15399939     0.22463048   -0.007762485
#> 3    -0.05128596    0.123504458     0.18243012     0.13361259   -0.168921159
#> 4    -0.06360621    0.340443016     0.65606292     0.04807945    0.157027939
#> 5     0.08009126   -0.027627640     0.06662396    -0.05838949   -0.226341946
#> 6    -0.11937190    0.090540884    -0.17320340     0.02039840   -0.211648032
#>   iov.Cl(occ==2) iov.Ka(occ==1) iov.Ka(occ==2)       V2       V3      TCl
#> 1    -0.19726803     0.13932112    0.088763868 40.26476 296.4748 19.25043
#> 2     0.04555116     0.07716277    0.049148937 40.26476 296.4748 19.25043
#> 3     0.17916513     0.08347085   -0.002990168 40.26476 296.4748 19.25043
#> 4    -0.13444417     0.24736740   -0.077647794 40.26476 296.4748 19.25043
#> 5     0.02049190    -0.09568733    0.054658457 40.26476 296.4748 19.25043
#> 6     0.12619350    -0.01547841    0.040206608 40.26476 296.4748 19.25043
#>        eta.Cl       TKA      eta.Ka        Q       Kin     Kout     EC50
#> 1  0.09788109 0.2354877 -0.21526790 10.43157 0.9256189 1.044937 200.3679
#> 2  0.33302598 0.2354877 -0.01028112 10.43157 0.9256189 1.044937 200.3679
#> 3  0.08504325 0.2354877  0.37253705 10.43157 0.9256189 1.044937 200.3679
#> 4  0.16521678 0.2354877 -0.01859974 10.43157 0.9256189 1.044937 200.3679
#> 5  0.53131380 0.2354877 -0.37109644 10.43157 0.9256189 1.044937 200.3679
#> 6 -0.30786713 0.2354877  0.14897588 10.43157 0.9256189 1.044937 200.3679
print(head(s$params %>% filter(sim.id == 2)))
#>   sim.id id inv.Cl(inv==1) inv.Cl(inv==2) inv.Ka(inv==1) inv.Ka(inv==2)
#> 1      2  1    -0.07227613    -0.01884171     0.07620864    -0.01962499
#> 2      2  2    -0.07227613    -0.01884171     0.07620864    -0.01962499
#> 3      2  3    -0.07227613    -0.01884171     0.07620864    -0.01962499
#> 4      2  4    -0.07227613    -0.01884171     0.07620864    -0.01962499
#> 5      2  5    -0.07227613    -0.01884171     0.07620864    -0.01962499
#> 6      2  6    -0.07227613    -0.01884171     0.07620864    -0.01962499
#>   eye.Cl(eye==1) eye.Cl(eye==2) eye.Ka(eye==1) eye.Ka(eye==2) iov.Cl(occ==1)
#> 1    -0.25102634      0.1354374   -0.135230741    -0.01605664    0.081212198
#> 2    -0.24832622      0.2233348   -0.380770273    -0.37101991    0.207985292
#> 3     0.24081384     -0.3826770    0.133361609    -0.32496203   -0.099983807
#> 4    -0.11021791      0.3655308    0.009423897     0.08736599   -0.002580804
#> 5     0.21155058      0.3626918    0.194977611    -0.19452347    0.058392202
#> 6     0.03663028     -0.1003754    0.068047203     0.32368023    0.167658084
#>   iov.Cl(occ==2) iov.Ka(occ==1) iov.Ka(occ==2)       V2       V3      TCl
#> 1    -0.06556217   -0.032678168   -0.061277262 40.21091 296.6424 18.86433
#> 2    -0.05016896    0.130258486   -0.103721118 40.21091 296.6424 18.86433
#> 3     0.06144127   -0.262726137   -0.104186144 40.21091 296.6424 18.86433
#> 4     0.03954265   -0.037289852   -0.021344468 40.21091 296.6424 18.86433
#> 5     0.12713682   -0.007530921    0.081238197 40.21091 296.6424 18.86433
#> 6    -0.02811353    0.026801132   -0.002702762 40.21091 296.6424 18.86433
#>        eta.Cl       TKA      eta.Ka        Q       Kin     Kout   EC50
#> 1  0.19368101 0.5686675  0.26184170 10.54673 0.5434136 1.323825 200.09
#> 2 -0.16262369 0.5686675  0.12969487 10.54673 0.5434136 1.323825 200.09
#> 3  0.22164455 0.5686675  0.24501105 10.54673 0.5434136 1.323825 200.09
#> 4  0.05997662 0.5686675  0.03059399 10.54673 0.5434136 1.323825 200.09
#> 5  0.45691421 0.5686675 -0.20098716 10.54673 0.5434136 1.323825 200.09
#> 6 -0.19943300 0.5686675 -0.36846082 10.54673 0.5434136 1.323825 200.09

For between eye variability and between occasion variability each individual simulates a number of variables that become the between eye and between occasion variability; In the case of the eye:

eye.Cl = (eye == 1) * `eye.Cl(eye==1)` + (eye == 2) * `eye.Cl(eye==2)`

So when you look the simulation each of these variables (ie eye.Cl(eye==1), eye.Cl(eye==2), etc) they change for each individual and when combined make the between eye variability or the between occasion variability that can be seen in some pharamcometric models.