ppk_model

A model defined in the rxode2::rxode() mini-language. posologyr needs a structural model, defined with either differential or algebraic equations, and an individual model.

The concentration in the central compartment must be named Cc.

The differential function d/dt(AUC) = Cc; is needed for the optimisation function poso_dose_auc().

ppk_model   = rxode2::rxode({
    centr(0) = 0;
    TVCl  = THETA_Cl*(CLCREAT/120)^0.8*(0.7^DIAL);
    TVVc  = THETA_Vc*(WT/70)          *(0.5^DIAL);
    TVVp  = THETA_Vp;
    TVQ   = THETA_Q;
    Cl    = TVCl*exp(ETA_Cl);
    Vc    = TVVc*exp(ETA_Vc);
    Vp    = TVVp*exp(ETA_Vp);
    Q     = TVQ;
    ke    = Cl/Vc;
    k12   = Q/Vc;
    k21   = Q/Vp;
    Cc    = centr/Vc;
    d/dt(centr)  = - ke*centr - k12*centr + k21*periph;
    d/dt(periph) =            + k12*centr - k21*periph;
    d/dt(AUC)    =   Cc;
  })

error_model

A function of the residual error model, taking two arguments: the simulated concentrations, and a vector sigma of the estimates of the parameters for the residual error model.

error_model <- function(f,sigma){     #additive model if sigma[2] == 0
  g <- sigma[1] + sigma[2]*f          #proportional model if sigma[1] == 0
  return(g)
}

Alternatively, the function can take the simulated concentrations, and a matrix sigma of the estimates of the parameters for the residual error model, as in the following example:

error_model <- function(f,sigma){
  dv <- cbind(f,1)
  g  <- diag(dv%*%sigma%*%t(dv))     #sigma is the square matrix of the residual
  return(sqrt(g))                    #errors
}

For multiple endpoint models, error_model must be a named list with a function for each endpoint vignette("multiple_endpoints").

theta

The estimations of the parameters for the fixed effects of the model (THETA), in a named vector. The names must match the names used in ppk_model.

theta = c(THETA_Cl=4.5, THETA_Vc=58.4, THETA_Vp=38.4, THETA_Q=6.5)

omega

The variance-covariance matrix of the random effects (ETA) for the individual model. A symmetric matrix. The names must match the names used in ppk_model. An easy way to define it is using lotri::lotri().

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 standard deviation can be computed back with sqrt(log((CV^2)+1))
• Full covariance matrix: the easiest to reuse, but rarely seen in articles

In the case of the vancomycin model, the estimates of between subject variability (BSV) are given as CV%. They must be converted to variances prior to their inclusion in omega.

omega = lotri::lotri({ETA_Cl + ETA_Vc + ETA_Vp + ETA_Q ~
                          c(0.147,
                            0    ,  0.510 ,
                            0    ,  0     ,   0.282,
                            0    ,  0     ,   0    ,    0)})

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).

In this example, all covariances are equal to zero.

sigma

The estimates of the parameters for the residual error model, either in a vector:

sigma       = c(additive_a = 3.4, proportional_b = 0.227)

in a matrix:

sigma       = lotri::lotri({prop + add ~ c(0.227,0.0,3.4)})

or in a named list, see vignette("multiple_endpoints"):

sigma       = list(
    cp=c(additive_a = 0.144, proportional_b = 0.15),
    pca=c(additive_a = 3.91, proportional_b = 0.0)
    )

depending on the residual error model.

pi_matrix

Optional: only needed for models with inter-occasion variability (IOV). The variance-covariance matrix of the random effects (KAPPA) for the IOV. As for the omega matrix, the names must match the names used in ppk_model. An easy way to define it is using lotri::lotri().

pi_matrix = lotri::lotri({KAPPA_Cl + KAPPA_Vc ~
      c(0.1934626,
        0.00     ,  0.05783106)})

covariates

The names of every covariate defined in ppk_model, in a character vector.

covariates  = c("CLCREAT","WT","DIAL")

Full model

The posologyr model is the list of all these objects. Note: This model does not include inter-occasion variability, so the pi_matrix is omitted.

mod_vancomyin_Goti2018 <- list(
  ppk_model   = rxode2::rxode({
     centr(0) = 0;
    TVCl  = THETA_Cl*(CLCREAT/120)^0.8*(0.7^DIAL);
    TVVc  = THETA_Vc*(WT/70)          *(0.5^DIAL);
    TVVp  = THETA_Vp;
    TVQ   = THETA_Q;
    Cl    = TVCl*exp(ETA_Cl);
    Vc    = TVVc*exp(ETA_Vc);
    Vp    = TVVp*exp(ETA_Vp);
    Q     = TVQ;
    ke    = Cl/Vc;
    k12   = Q/Vc;
    k21   = Q/Vp;
    Cc    = centr/Vc;
    d/dt(centr)  = - ke*centr - k12*centr + k21*periph;
    d/dt(periph) =            + k12*centr - k21*periph;
    d/dt(AUC)    =   Cc;
  }),
  error_model = function(f,sigma){
    g <- sigma[1] + sigma[2]*f
    return(g)
  },
  theta = c(THETA_Cl=4.5, THETA_Vc=58.4, THETA_Vp=38.4,THETA_Q=6.5),
  omega = lotri::lotri({ETA_Cl + ETA_Vc + ETA_Vp + ETA_Q ~
                          c(0.147,
                            0    ,  0.510 ,
                            0    ,  0     ,   0.282,
                            0    ,  0     ,   0    ,    0)}),
  sigma       = c(additive_a = 3.4, proportional_b = 0.227),
  covariates  = c("CLCREAT","WT","DIAL"))

mod_vancomyin_Goti2018
#> $ppk_model
#> rxode2 2.1.2 model named rx_274fc254fbfa90fbebccd323b4a60983 model (✔ ready). 
#> $state: centr, periph, AUC
#> $params: THETA_Cl, CLCREAT, DIAL, THETA_Vc, WT, THETA_Vp, THETA_Q, ETA_Cl, ETA_Vc, ETA_Vp
#> $lhs: TVCl, TVVc, TVVp, TVQ, Cl, Vc, Vp, Q, ke, k12, k21, Cc
#> 
#> $error_model
#> function(f,sigma){
#>     g <- sigma[1] + sigma[2]*f
#>     return(g)
#>   }
#> 
#> $theta
#> THETA_Cl THETA_Vc THETA_Vp  THETA_Q 
#>      4.5     58.4     38.4      6.5 
#> 
#> $omega
#>        ETA_Cl ETA_Vc ETA_Vp ETA_Q
#> ETA_Cl  0.147   0.00  0.000     0
#> ETA_Vc  0.000   0.51  0.000     0
#> ETA_Vp  0.000   0.00  0.282     0
#> ETA_Q   0.000   0.00  0.000     0
#> 
#> $sigma
#>     additive_a proportional_b 
#>          3.400          0.227 
#> 
#> $covariates
#> [1] "CLCREAT" "WT"      "DIAL"