Population models

Introduction

This describes the structure of prior population models compatible with posologyr and illustrates how to define new models from published population models.

General structure

Most models can be written as a model function which can be parsed by rxode2. The models are written in two block statements:

• ini({}) defining parameter values
• model({}) for the ODE-based model specification.

For example, the gentamicin model of Xuan et al. 2003 (doi:10.1016/j.ijantimicag.2003.07.010) can be written as:

mod_gentamicin_Xuan2003 <- function() {
  ini({
    #Fixed effects: population estimates 
    THETA_Cl  = 0.047
    THETA_V   = 0.28
    THETA_k12 = 0.092
    THETA_k21 = 0.071
    
    #Random effects: inter-individual variability
    ETA_Cl  ~ 0.084
    ETA_V   ~ 0.003
    ETA_k12 ~ 0.398
    ETA_k21 ~ 0.342
    
    #Unexplained residual variability
    add_sd  <- 0.230
    prop_sd <- 0.237
  })
  model({
    #Individual model and covariates
    TVl   = THETA_Cl*ClCr
    TVV   = THETA_V*WT
    TVk12 = THETA_k12
    TVk21 = THETA_k21
    Cl    = TVl*exp(ETA_Cl)
    V     = TVV*exp(ETA_V)
    k12   = TVk12*exp(ETA_k12)
    k21   = TVk21*exp(ETA_k21)
    
    #Structural model defined using ordinary differential equations (ODE)
    ke    = Cl/V
    Cp    = centr/V

    d/dt(centr)  = - ke*centr - k12*centr + k21*periph
    d/dt(periph) =            + k12*centr - k21*periph

    #Model for unexplained residual variability
    Cp ~ add(add_sd) + prop(prop_sd) + combined1()
  })
}

The rxode2 mini-language syntax is detailed in the rxode2 documentation.

Individual model, random effects

Inter-individual variability can also be defined as a symmetric matrix to integrate the covariance between random effects.

For example, the amikacin model of Burdet et al. 2015 (doi:10.1007/s00228-014-1766-y) can be written as:

mod_amikacin_Burdet2015 <- function() {
    ini({
      #Fixed effects: population estimates 
      THETA_Cl=4.3
      THETA_Vc=15.9
      THETA_Vp=21.4
      THETA_Q=12.1
      
      #Random effects: inter-individual variability
      ETA_Cl + ETA_Vc + ETA_Vp + ETA_Q ~
        c(0.1,
          0.01     ,    0.05 ,
          0.01     ,    0.02 ,   0.2  ,
         -0.06     ,    0.004,   0.003,    0.08)
          
      #Unexplained residual variability
      add_sd <- 0.2
      prop_sd <- 0.1
    })
    model({
      #Individual model and covariates
      TVCl  = THETA_Cl*(CLCREAT4H/82)^0.7
      TVVc  = THETA_Vc*(TBW/78)^0.9*(PoverF/169)^0.4
      TVVp  = THETA_Vp
      TVQ   = THETA_Q
      Cl    = TVCl*exp(ETA_Cl)
      Vc    = TVVc*exp(ETA_Vc)
      Vp    = TVVp*exp(ETA_Vp)
      Q     = TVQ *exp(ETA_Q)
      
      #Structural model defined using ordinary differential equations (ODE)
      ke    = Cl/Vc
      k12   = Q/Vc
      k21   = Q/Vp
      Cp    = centr/Vc      
      
      d/dt(centr)  = - ke*centr - k12*centr + k21*periph
      d/dt(periph) =            + k12*centr - k21*periph

      #Model for unexplained residual variability
      Cp ~ add(add_sd) + prop(prop_sd) + combined1()
    })
  }

The estimates of the variances of the random effects can be given under different parameterizations depending on the authors.

• Standard deviation (SD): the square root of the variance, as returned by Monolix
• Coefficient of variation (CV): calculated as sqrt(exp(SD^2)-1), the variance can be computed back with log((CV^2)+1)
• Full covariance matrix: the easiest to reuse, but less common in the literature

The estimates of covariance (off diagonal) are sometimes given as coefficients of correlation between ETAs. The covariance between ETA_a and ETA_b can be computed with the following product: standard_deviation(ETA_a) * standard_deviation(ETA_b) * correlation(ETA_a and ETA_b).

Bioavailability, lag-time

Special model event changes including bioavailability (f(depot)=1), and lag time (alag(depot)=0) can be used in the model({}) block. For example, the ganciclovir model of Caldès et al. 2009 (doi:10.1128/aac.00085-09) can be written as:

mod_ganciclovir_Caldes2009 <- function() {
  ini({
    #Fixed effects: population estimates 
    THETA_cl  <- 7.49
    THETA_v1  <- 31.90
    THETA_cld <- 10.20
    THETA_v2  <- 32.0
    THETA_ka  <- 0.895
    THETA_baf <- 0.825
      
    #Random effects: inter-individual variability
    ETA_cl ~ 0.107
    ETA_v1 ~ 0.227
    ETA_ka ~ 0.464
    ETA_baf ~ 0.049
    
    #Unexplained residual variability
    add.sd <- 0.465
    prop.sd <- 0.143
  })
  model({
    #Individual model and covariates
    TVcl  = THETA_cl*(ClCr/57);
    TVv1  = THETA_v1;
    TVcld = THETA_cld;
    TVv2  = THETA_v2;
    TVka  = THETA_ka;
    TVbaf = THETA_baf;
    cl  = TVcl*exp(ETA_cl);
    v1  = TVv1*exp(ETA_v1);
    cld = TVcld;
    v2  = TVv2;
    ka  = TVka*exp(ETA_ka);
    baf = TVbaf*exp(ETA_baf);

    #Structural model defined using ordinary differential equations (ODE)
    k10 = cl/v1;
    k12 = cld / v1;
    k21 = cld / v2;
    Cc = centr/v1;

    d/dt(depot)  = -ka*depot
    d/dt(centr)  =  ka*depot - k10*centr - k12*centr + k21*periph;
    d/dt(periph) =                         k12*centr - k21*periph;
    d/dt(AUC)    = Cc;

    #Special model event changes
    f(depot)=baf;
    alag(depot)=0.382;
    
    #Model for unexplained residual variability
    Cc ~ add(add.sd) + prop(prop.sd) + combined1()
  })
}

Classic posologyr models

Some models cannot be described by an rxode2 model function alone. It is then possible to define a classic posologyr model, see vignette ("classic_posologyr_models"). Models falling in this category are :

• models with inter-occasion variability (IOV), sometimes called intra-individual variability
• models with an unexplained residual error model other than additive, proportional, or combined models