Running PK models with nlmixr

2024-09-18

nlmixr

Running PK models with nlmixr

nlmixr uses a unified interface for specifying and running models. Let’s start with a very simple PK example, using the single-dose theophylline dataset generously provided by Dr. Robert A. Upton of the University of California, San Francisco:

## Load libraries
library(nlmixr2)
str(theo_sd)
#> 'data.frame':    144 obs. of  7 variables:
#>  $ ID  : int  1 1 1 1 1 1 1 1 1 1 ...
#>  $ TIME: num  0 0 0.25 0.57 1.12 2.02 3.82 5.1 7.03 9.05 ...
#>  $ DV  : num  0 0.74 2.84 6.57 10.5 9.66 8.58 8.36 7.47 6.89 ...
#>  $ AMT : num  320 0 0 0 0 ...
#>  $ EVID: int  101 0 0 0 0 0 0 0 0 0 ...
#>  $ CMT : int  1 2 2 2 2 2 2 2 2 2 ...
#>  $ WT  : num  79.6 79.6 79.6 79.6 79.6 79.6 79.6 79.6 79.6 79.6 ...

We can try fitting a simple one-compartment PK model to this small dataset. We write the model as follows:

one.cmt <- function() {
  ini({
    ## You may label each parameter with a comment
    tka <- 0.45 # Log Ka
    tcl <- log(c(0, 2.7, 100)) # Log Cl
    ## This works with interactive models
    ## You may also label the preceding line with label("label text")
    tv <- 3.45; label("log V")
    ## the label("Label name") works with all models
    eta.ka ~ 0.6
    eta.cl ~ 0.3
    eta.v ~ 0.1
    add.sd <- 0.7
  })
  model({
    ka <- exp(tka + eta.ka)
    cl <- exp(tcl + eta.cl)
    v <- exp(tv + eta.v)
    linCmt() ~ add(add.sd)
  })
}

f <- nlmixr(one.cmt)

We can now run the model…

fit <- nlmixr(one.cmt, theo_sd, est="focei",
              control=list(print=0))
#> calculating covariance matrix
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00 
#> done

print(fit)
#> ── nlmixr² FOCEi (outer: nlminb) ──
#> 
#>           OBJF      AIC     BIC Log-likelihood Condition#(Cov) Condition#(Cor)
#> FOCEi 116.8096 373.4094 393.589      -179.7047        67.96527        9.332554
#> 
#> ── Time (sec $time): ──
#> 
#>            setup optimize covariance table compress    other
#> elapsed 0.019001 0.204398   0.204399 0.044    0.009 1.650202
#> 
#> ── Population Parameters ($parFixed or $parFixedDf): ──
#> 
#>        Parameter  Est.     SE %RSE Back-transformed(95%CI) BSV(CV%) Shrink(SD)%
#> tka              0.469  0.195 41.6        1.6 (1.09, 2.34)     70.0      1.40% 
#> tcl               1.01 0.0751 7.43       2.75 (2.37, 3.18)     26.7      4.16% 
#> tv         log V  3.46 0.0436 1.26       31.9 (29.2, 34.7)     14.2      11.1% 
#> add.sd           0.695                               0.695                     
#>  
#>   Covariance Type ($covMethod): r,s
#>   No correlations in between subject variability (BSV) matrix
#>   Full BSV covariance ($omega) or correlation ($omegaR; diagonals=SDs) 
#>   Distribution stats (mean/skewness/kurtosis/p-value) available in $shrink 
#>   Information about run found ($runInfo):
#>    • gradient problems with initial estimate and covariance; see $scaleInfo 
#>    • ETAs were reset to zero during optimization; (Can control by foceiControl(resetEtaP=.)) 
#>    • initial ETAs were nudged; (can control by foceiControl(etaNudge=., etaNudge2=)) 
#>   Censoring ($censInformation): No censoring
#>   Minimization message ($message):  
#>     relative convergence (4) 
#> 
#> ── Fit Data (object is a modified tibble): ──
#> # A tibble: 132 × 20
#>   ID     TIME    DV      PRED    RES   WRES IPRED   IRES  IWRES    CPRED   CRES
#>   <fct> <dbl> <dbl>     <dbl>  <dbl>  <dbl> <dbl>  <dbl>  <dbl>    <dbl>  <dbl>
#> 1 1      0     0.74 -1.78e-15  0.740  1.06   0     0.74   1.06  5.64e-16  0.740
#> 2 1      0.25  2.84  3.27e+ 0 -0.431 -0.230  3.85 -1.01  -1.45  3.23e+ 0 -0.388
#> 3 1      0.57  6.57  5.84e+ 0  0.731  0.294  6.79 -0.215 -0.310 5.78e+ 0  0.786
#> # ℹ 129 more rows
#> # ℹ 9 more variables: CWRES <dbl>, eta.ka <dbl>, eta.cl <dbl>, eta.v <dbl>,
#> #   ka <dbl>, cl <dbl>, v <dbl>, tad <dbl>, dosenum <dbl>

We can alternatively express the same model by ordinary differential equations (ODEs):

one.compartment <- function() {
  ini({
    tka <- 0.45 # Log Ka
    tcl <- 1 # Log Cl
    tv <- 3.45    # Log V
    eta.ka ~ 0.6
    eta.cl ~ 0.3
    eta.v ~ 0.1
    add.sd <- 0.7
  })
  model({
    ka <- exp(tka + eta.ka)
    cl <- exp(tcl + eta.cl)
    v <- exp(tv + eta.v)
    d/dt(depot) = -ka * depot
    d/dt(center) = ka * depot - cl / v * center
    cp = center / v
    cp ~ add(add.sd)
  })
}

We can try the Stochastic Approximation EM (SAEM) method to this model:

fit2 <- nlmixr(one.compartment, theo_sd,  est="saem",
               control=list(print=0))
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
print(fit2)
#> ── nlmixr² SAEM OBJF by FOCEi approximation ──
#> 
#>  Gaussian/Laplacian Likelihoods: AIC() or $objf etc. 
#>  FOCEi CWRES & Likelihoods: addCwres() 
#> 
#> ── Time (sec $time): ──
#> 
#>            setup covariance  saem table compress   other
#> elapsed 0.001745   0.010005 4.402 0.064    0.019 1.15325
#> 
#> ── Population Parameters ($parFixed or $parFixedDf): ──
#> 
#>         Est.     SE %RSE Back-transformed(95%CI) BSV(CV%) Shrink(SD)%
#> tka    0.464  0.195   42       1.59 (1.09, 2.33)     71.1   -0.0900% 
#> tcl     1.01  0.085 8.43       2.74 (2.32, 3.24)     27.4      4.80% 
#> tv      3.46 0.0447 1.29         31.7 (29, 34.6)     13.1      8.77% 
#> add.sd 0.696                               0.696                     
#>  
#>   Covariance Type ($covMethod): linFim
#>   No correlations in between subject variability (BSV) matrix
#>   Full BSV covariance ($omega) or correlation ($omegaR; diagonals=SDs) 
#>   Distribution stats (mean/skewness/kurtosis/p-value) available in $shrink 
#>   Censoring ($censInformation): No censoring
#> 
#> ── Fit Data (object is a modified tibble): ──
#> # A tibble: 132 × 19
#>   ID     TIME    DV  PRED    RES IPRED   IRES  IWRES eta.ka eta.cl   eta.v    cp
#>   <fct> <dbl> <dbl> <dbl>  <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl> <dbl>
#> 1 1      0     0.74  0     0.74   0     0.74   1.06  0.0839 -0.477 -0.0849  0   
#> 2 1      0.25  2.84  3.28 -0.437  3.83 -0.991 -1.42  0.0839 -0.477 -0.0849  3.83
#> 3 1      0.57  6.57  5.86  0.715  6.76 -0.194 -0.278 0.0839 -0.477 -0.0849  6.76
#> # ℹ 129 more rows
#> # ℹ 7 more variables: depot <dbl>, center <dbl>, ka <dbl>, cl <dbl>, v <dbl>,
#> #   tad <dbl>, dosenum <dbl>

And if we wanted to, we could even apply the traditional R method nlme method to this model:

fitN <- nlmixr(one.compartment, theo_sd, list(pnlsTol=0.5), est="nlme")
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> 
#> **Iteration 1
#> LME step: Loglik: -183.2083, nlminb iterations: 1
#> reStruct  parameters:
#>       ID1       ID2       ID3 
#> 0.2195819 0.9924330 1.6502972 
#>  Beginning PNLS step: ..  completed fit_nlme() step.
#> PNLS step: RSS =  64.59841 
#>  fixed effects: 0.4443024  1.038584  3.449959  
#>  iterations: 3 
#> Convergence crit. (must all become <= tolerance = 1e-05):
#>      fixed   reStruct 
#> 0.03715048 0.91006943 
#> 
#> **Iteration 2
#> LME step: Loglik: -182.0743, nlminb iterations: 1
#> reStruct  parameters:
#>       ID1       ID2       ID3 
#> 0.1149602 0.9686071 1.6508466 
#>  Beginning PNLS step: ..  completed fit_nlme() step.
#> PNLS step: RSS =  64.59843 
#>  fixed effects: 0.4443024  1.038584  3.449959  
#>  iterations: 1 
#> Convergence crit. (must all become <= tolerance = 1e-05):
#>        fixed     reStruct 
#> 0.000000e+00 5.047516e-06
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
#> [====|====|====|====|====|====|====|====|====|====] 0:00:00
print(fitN)
#> ── nlmixr² nlme by maximum likelihood ──
#> 
#>          OBJF      AIC      BIC Log-likelihood Condition#(Cov) Condition#(Cor)
#> nlme 121.5489 378.1487 398.3283      -182.0743        18.17386               1
#> 
#> ── Time (sec $time): ──
#> 
#>            setup table compress    other
#> elapsed 0.002096 0.069    0.005 2.860904
#> 
#> ── Population Parameters ($parFixed or $parFixedDf): ──
#> 
#>          Est.      SE  %RSE Back-transformed(95%CI) BSV(CV%) Shrink(SD)%
#> tka    0.4443  0.1916 43.12     1.559 (1.071, 2.27)     68.7     -3.49% 
#> tcl     1.039 0.08335 8.026    2.825 (2.399, 3.327)     27.0      7.85% 
#> tv       3.45 0.04494 1.303      31.5 (28.84, 34.4)     13.5      7.29% 
#> add.sd 0.6979                                0.6979                     
#>  
#>   Covariance Type ($covMethod): nlme
#>   No correlations in between subject variability (BSV) matrix
#>   Full BSV covariance ($omega) or correlation ($omegaR; diagonals=SDs) 
#>   Distribution stats (mean/skewness/kurtosis/p-value) available in $shrink 
#>   Censoring ($censInformation): No censoring
#> 
#> ── Fit Data (object is a modified tibble): ──
#> # A tibble: 132 × 19
#>   ID     TIME    DV  PRED    RES IPRED   IRES  IWRES eta.ka eta.cl   eta.v    cp
#>   <fct> <dbl> <dbl> <dbl>  <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl> <dbl>
#> 1 1      0     0.74  0     0.74   0     0.74   1.06  0.0756 -0.483 -0.0896  0   
#> 2 1      0.25  2.84  3.24 -0.401  3.78 -0.943 -1.35  0.0756 -0.483 -0.0896  3.78
#> 3 1      0.57  6.57  5.81  0.760  6.72 -0.146 -0.210 0.0756 -0.483 -0.0896  6.72
#> # ℹ 129 more rows
#> # ℹ 7 more variables: depot <dbl>, center <dbl>, ka <dbl>, cl <dbl>, v <dbl>,
#> #   tad <dbl>, dosenum <dbl>

This example delivers a complete model fit as the fit object, including parameter history, a set of fixed effect estimates, and random effects for all included subjects.

The UI

The nlmixr modeling dialect, inspired by R and NONMEM, can be used to fit models using all current and future estimation algorithms within nlmixr. Using these widely-used tools as inspiration has the advantage of delivering a model specification syntax that is instantly familiar to the majority of analysts working in pharmacometrics and related fields.

Overall model structure

Model specifications for nlmixr are written using functions containing ini and model blocks. These functions can be called anything, but often contain these two components. Let’s look at a very simple one-compartment model with no covariates.

f <- function() {
  ini({   # Initial conditions/variables
    # are specified here
  })
  model({ # The model is specified
    # here
  })
}

The ini block

The ini block specifies initial conditions, including initial estimates and boundaries for those algorithms which support them (currently, the built-in nlme and saem methods do not). Nomenclature is similar to that used in NONMEM, Monolix and other similar packages. In the NONMEM world, the ini block is analogous to $THETA, $OMEGA and $SIGMA blocks.

f <- function() { # Note that arguments to the function are currently
  # ignored by nlmixr
  ini({
    # Initial conditions for population parameters (sometimes
    # called THETA parameters) are defined by either '<-' or '='
    lCl <- 1.6      # log Cl (L/hr)
    
    # Note that simple expressions that evaluate to a number are
    # OK for defining initial conditions (like in R)
    lVc = log(90)  # log V (L)
    
    ## Also, note that a comment on a parameter is captured as a parameter label
    lKa <- 1       # log Ka (1/hr)
    
    # Bounds may be specified by c(lower, est, upper), like NONMEM:
    # Residuals errors are assumed to be population parameters
    prop.err <- c(0, 0.2, 1)
    
    # IIV terms will be discussed in the next example
  })
  
  # The model block will be discussed later
  model({})
}

As shown in the above example:

• Simple parameter values are specified using an R-compatible assignment
• Boundaries may be specified by c(lower, est, upper).
• Like NONMEM, c(lower,est) is equivalent to c(lower,est,Inf)
• Also like NONMEM, c(est) does not specify a lower bound, and is equivalent to specifying the parameter without using R’s c() function.

These parameters can be named using almost any R-compatible name. Please note that:

• Residual error estimates should be coded as population estimates (i.e. using = or <-, not ~).
• Variable names that start with _ are not supported. Note that R does not allow variable starting with _ to be assigned without quoting them.
• Naming variables that start with rx or nlmixr is not suggested, since rxode2() and nlmixr use these prefixes internally for certain estimation routines and for calculating residuals.
• Variable names are case-sensitive, just like they are in R. CL is not the same as Cl.

In mixture models, multivariate normal individual deviations from the normal population and parameters are estimated (in NONMEM these are called “ETA” parameters). Additionally, the variance/covariance matrix of these deviations are is also estimated (in NONMEM this is the “OMEGA” matrix). These also take initial estimates. In nlmixr, these are specified by the ~ operator. This that is typically used in statistics R for “modeled by”, and was chosen to distinguish these estimates from the population and residual error parameters.

Continuing from the prior example, we can annotate the estimates for the between-subject error distribution…

f <- function() {
  ini({
    lCl <- 1.6      ; label("log Cl (L/hr)")
    lVc = log(90)   ; label("log V (L)")
    lKa <- 1        ; label("log Ka (1/hr)")
    prop.err <- c(0, 0.2, 1)
    
    # Initial estimate for ka IIV variance
    # Labels work for single parameters
    eta.ka ~ 0.1    ## BSV Ka

    # For correlated parameters, you specify the names of each
    # correlated parameter separated by a addition operator `+`
    # and the left handed side specifies the lower triangular
    # matrix initial of the covariance matrix.
    eta.cl + eta.vc ~ c(0.1,
                        0.005, 0.1)
    
    # Note that labels do not currently work for correlated
    # parameters.  Also, do not put comments inside the lower
    # triangular matrix as this will currently break the model.
  })
  
  # The model block will be discussed later
  model({})
}

As shown in the above example:

• Simple variances are specified by the variable name and the estimate separated by ~
• Correlated parameters are specified by the sum of the variable labels and then the lower triangular matrix of the covariance is specified on the left handed side of the equation. This is also separated by ~.
• The initial estimates are specified on the variance scale, and in analogy with NONMEM, the square roots of the diagonal elements correspond to coefficients of variation when used in the exponential IIV implementation.

The model block

The model block specifies the model, and is analogous to the $PK, $PRED and $ERROR blocks in NONMEM.

Once the initialization block has been defined, you can define a model in terms of the variables defined in the ini block. You can also mix rxode2() blocks into the model if needed.

The current method of defining a nlmixr model is to specify the parameters, and then any required rxode2() lines. Continuing the annotated example:

f <- function() {
  ini({
    lCl <- 1.6       # log Cl (L/hr)
    lVc <- log(90)   # log Vc (L)
    lKA <- 0.1       # log Ka (1/hr)
    prop.err <- c(0, 0.2, 1)
    
    eta.Cl ~ 0.1     # BSV Cl
    eta.Vc ~ 0.1     # BSV Vc
    eta.KA ~ 0.1     # BSV Ka
  })
  model({
    # Parameters are defined in terms of the previously-defined
    # parameter names:
    Cl <- exp(lCl + eta.Cl)
    Vc =  exp(lVc + eta.Vc)
    KA <- exp(lKA + eta.KA)
    
    # Next, the differential equations are defined:
    kel <- Cl / Vc;
    
    d/dt(depot)  = -KA*depot;
    d/dt(centr)  =  KA*depot-kel*centr;
    
    # And the concentration is then calculated
    cp = centr / Vc;
    # Finally, we specify that the plasma concentration follows
    # a proportional error distribution (estimated by the parameter 
    # prop.err)
    cp ~ prop(prop.err)
  })
}

A few points to note:

• Parameters are defined before the differential equations. Currently directly defining the differential equations in terms of the population parameters is not supported.
• The differential equations, parameters and error terms are in a single block, instead of multiple sections.
• Additionally state names, calculated variables, also cannot start with either rx_ or nlmixr_ since these are used internally in some estimation routines.
• Errors are specified using the tilde, ~. Currently you can use either add(parameter) for additive error, prop(parameter) for proportional error or add(parameter1) + prop(parameter2) for combined additive and proportional error. You can also specify norm(parameter) for additive error, since it follows a normal distribution.
• Some routines, like saem, require parameters expressed in terms of Pop.Parameter + Individual.Deviation.Parameter + Covariate*Covariate.Parameter. The order of these parameters does not matter. This is similar to NONMEM’s mu-referencing, though not as restrictive. This means that for saem, a parameterization of the form Cl <- Cl*exp(eta.Cl) is not allowed.
• The type of parameter in the model is determined by the ini block; covariates used in the model are not included in the ini block. These variables need to be present in the modeling dataset for the model to run.

Running models

Models can be fitted several ways, including via the [magrittr] forward-pipe operator.

fit <- nlmixr(one.compartment) %>% saem.fit(data=theo_sd)

fit2 <- nlmixr(one.compartment, data=theo_sd, est="saem")

fit3 <- one.compartment %>% saem.fit(data=theo_sd)

Options to the estimation routines can be specified using nlmeControl for nlme estimation:

fit4 <- nlmixr(one.compartment, theo_sd,est="nlme",control = nlmeControl(pnlsTol = .5))

where options are specified in the nlme documentation. Options for saem can be specified using saemControl:

fit5 <- nlmixr(one.compartment,theo_sd,est="saem",control=saemControl(n.burn=250,n.em=350,print=50))

this example specifies 250 burn-in iterations, 350 em iterations and a print progress every 50 runs.

Model Syntax for solved PK systems

Solved PK systems are also currently supported by nlmixr with the ‘linCmt()’ pseudo-function. An annotated example of a solved system is below:

f <- function(){
  ini({
    lCl <- 1.6      ; label("log Cl (L/hr)")
    lVc <- log(90)  ; label("log Vc (L)")
    lKA <- 0.1      ; label("log Ka (1/hr)")
    prop.err <- c(0, 0.2, 1)
    eta.Cl ~ 0.1   # BSV Cl
    eta.Vc ~ 0.1   # BSV Vc
    eta.KA ~ 0.1   # BSV Ka
  })
  model({
    Cl <- exp(lCl + eta.Cl)
    Vc = exp(lVc + eta.Vc)
    KA <- exp(lKA + eta.KA)
    ## Instead of specifying the ODEs, you can use
    ## the linCmt() function to use the solved system.
    ##
    ## This function determines the type of PK solved system
    ## to use by the parameters that are defined.  In this case
    ## it knows that this is a one-compartment model with first-order
    ## absorption.
    linCmt() ~ prop(prop.err)
  })
}

A few things to keep in mind:

• The solved systems implemented are the one, two and three compartment models with or without first-order absorption. Each of the models support a lag time with a tlag parameter.
• In general the linear compartment model figures out the model by the parameter names. nlmixr2 currently knows about numbered volumes, Vc/Vp, Clearances in terms of both Cl and Q/CLD. Additionally nlmixr knows about elimination micro-constants (ie K12). Mixing of these parameters for these models is currently not supported.

For the most up-to-date information about linCmt() models see the rxode2 documentation.

Checking model syntax

After specifying the model syntax you can check that nlmixr is interpreting it correctly by using the nlmixr function on it. Using the above function we can get:

nlmixr(f)
#>  ── rxode2-based solved PK 1-compartment model with first-order absorption ────── 
#>  ── Initalization: ──  
#> Fixed Effects ($theta): 
#>      lCl      lVc      lKA prop.err 
#>  1.60000  4.49981  0.10000  0.20000 
#> 
#> Omega ($omega): 
#>        eta.Cl eta.Vc eta.KA
#> eta.Cl    0.1    0.0    0.0
#> eta.Vc    0.0    0.1    0.0
#> eta.KA    0.0    0.0    0.1
#>  ── μ-referencing ($muRefTable): ──  
#>   theta    eta level
#> 1   lCl eta.Cl    id
#> 2   lVc eta.Vc    id
#> 3   lKA eta.KA    id
#> 
#>  ── Model (Normalized Syntax): ── 
#> function() {
#>     ini({
#>         lCl <- 1.6
#>         label("log Cl (L/hr)")
#>         lVc <- 4.49980967033027
#>         label("log Vc (L)")
#>         lKA <- 0.1
#>         label("log Ka (1/hr)")
#>         prop.err <- c(0, 0.2, 1)
#>         eta.Cl ~ 0.1
#>         eta.Vc ~ 0.1
#>         eta.KA ~ 0.1
#>     })
#>     model({
#>         Cl <- exp(lCl + eta.Cl)
#>         Vc = exp(lVc + eta.Vc)
#>         KA <- exp(lKA + eta.KA)
#>         linCmt() ~ prop(prop.err)
#>     })
#> }

In general this gives you information about the model (what type of solved system/rxode2()), initial estimates as well as the code for the model block.