---
title: "Examples"
author: "Giulia Lestini, Martin Fink and Andrew Hooker"
date: "`r format(Sys.time(), '%d %B, %Y')`"
output: 
  rmarkdown::html_vignette:
  # rmarkdown::html_document:
  #   toc: true
  #   toc_depth: 3
  #   number_sections: true
    fig_width: 7
    fig_height: 5
  #bookdown::html_document2:
    toc: true
    # toc_depth: 3
  #  toc_float:
   #   collapsed: false
  #    smooth_scroll: true  
   # number_sections: true
vignette: >
  %\VignetteIndexEntry{Examples}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
#pkgdown:
#  as_is: true
---

```{r setup, include = FALSE, cache=FALSE}
if(Sys.getenv("LOGNAME")=="ancho179") devtools::load_all("~/Documents/_PROJECTS/PopED/repo/PopED/")
library(deSolve)
library(Rcpp)

knitr::opts_chunk$set(
  collapse = TRUE
  , comment = "#>"
  #, fig.width=6
  , cache = TRUE
  , fig.align = "center"
  , out.width = "80%"
  , autodep=TRUE
)
```

# Introduction

In this vignette, we try to  highlight PopED features that may be useful.  Only code related to specific features we would like to highlight is described here in this vignette. These features (and more) are presented as r-scripts in the "examples" folder in the PopED installation directory. You can view a list of these example files using the commands:
```{r,eval=TRUE}
ex_dir <- system.file("examples", package="PopED")
list.files(ex_dir)
```

You can then open one of the examples (for example, `ex.1.a.PK.1.comp.oral.md.intro.R`) using the following code
```{r,eval=FALSE}
file_name <- "ex.1.a.PK.1.comp.oral.md.intro.R"

ex_file <- system.file("examples",file_name,package="PopED")
file.copy(ex_file,tempdir(),overwrite = T)
file.edit(file.path(tempdir(),file_name))
```

The table below provides a check list of features for each of the 15 available examples.

| Features  | Ex1   | Ex2 |Ex3|Ex4|Ex5|Ex6|Ex7|Ex8|Ex9|Ex10|Ex11|Ex12|Ex13|Ex14|Ex15|
|  -----   |  ----   | ----  |---- |---- |---- |---- |---- |---- |---- |---- |---- |---- |---- |---- |---- |
| Analytic model       |X|X|X|X|X|X|X|-|-|-|X|X|X|X|X|
| ODE model    |X|X|-|-|-|X|-|X|X|X|-|-|-|-|-|
| Irregular dosing       |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
| Full cov matrix W      |-|-|-|-|-|-|-|-|-|-|-|-|-|-|X|
| Inter-occ variability  |-|-|-|-|-|-|-|-|-|-|-|-|-|X|-|
| Discrete covariates    |-|-|-|-|-|-|X|-|-|-|X|-|-|-|-|
| Continuous covariates  |X|X|X|X|-|X|X|X|X|X|X|X|X|X|X|
| Multiple arms          |X|-|X|X|-|-|X|X|-|-|X|X|-|X|-|
| Multi response models  |-|-|X|X|-|-|-|X|-|X|-|-|-|-|-|
| Designs differ <br> across responses         |-|-|-|X|-|-|-|X|-|-|-|-|-|-|-|
| Calculate precision <br>of derived parameters|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
| Power calculation      |-|-|-|-|-|-|-|-|-|-|X|-|-|-|-|
| Include previous FIM   |-|-|-|-|-|-|-|-|-|-|X|-|-|-|-|
| Shrinkage/Bayesian FIM |X|X|X|X|-|-|X|-|-|X|-|-|X|-|-|
| Discrete optimization  |X|X|X|-|-|X|-|X|-|-|-|-|-|X|-|
| Optimization <br> of multi-group designs<br> (same response)   |X|-|X|X|-|-|X|X|-|-|-|-|-|X|-|
| Different optimal<br> sampling times<br> between groups       |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
| Optimization with constraining <br> sampling times            |X|-|X|-|-|-|-|-|-|-|-|-|-|X|-|
| Optimization of subjects <br>per group                        |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|

*Note:* All features are available in PopED but some are not demonstrated in the supplied examples.


#	Analytic solution of PKPD model, multiple study arms 

The full code for this example is available in `ex.4.PKPD.1.comp.emax.R`.

Here we define a PKPD mode using analytical equations. The PK is a one compartment model with intravenous bolus administration and linear elimination.  The PD is an ordinary Emax model driven by the PK concentrations. The expected output of each measurement (PK or PD) is given in the vector `model_switch` (see below for details).
```{r}
library(PopED)
f_pkpdmodel <- function(model_switch,xt,parameters,poped.db){
  with(as.list(parameters),{
    y=xt
    MS <- model_switch
    
    # PK model
    CONC = DOSE/V*exp(-CL/V*xt) 
    
    # PD model
    EFF = E0 + CONC*EMAX/(EC50 + CONC)
    
    y[MS==1] = CONC[MS==1]
    y[MS==2] = EFF[MS==2]
    
    return(list( y= y,poped.db=poped.db))
  })
}
```

The error model also has to accommodate both response models.
```{r}
## -- Residual Error function
## -- Proportional PK + additive PD
f_Err <- function(model_switch,xt,parameters,epsi,poped.db){
  returnArgs <- do.call(poped.db$model$ff_pointer,list(model_switch,xt,parameters,poped.db)) 
  y <- returnArgs[[1]]
  poped.db <- returnArgs[[2]]
  
  MS <- model_switch
  
  prop.err <- y*(1+epsi[,1])
  add.err <- y+epsi[,2]
  
  y[MS==1] = prop.err[MS==1]
  y[MS==2] = add.err[MS==2]
  
  return(list( y= y,poped.db =poped.db )) 
}
```

```{r, echo=FALSE}
f_etaToParam <- function(x,a,bpop,b,bocc){
  parameters=c( 
    CL=bpop[1]*exp(b[1]),
    V=bpop[2]*exp(b[2]),
    E0=bpop[3]*exp(b[3]),
    EMAX=bpop[4],
    EC50=bpop[5]*exp(b[4]),
    DOSE=a[1]
  )
  return( parameters ) 
}
```

In the `poped.db` object the vector we specify  `model_switch` in order to assign the sampling times defined in the vector `xt` to the PK (=1) or PD (=2) model. 
```{r}
poped.db <- create.poped.database(
  
  # Model
  ff_fun=f_pkpdmodel,
  fError_fun=f_Err,
  fg_fun=f_etaToParam,
  sigma=diag(c(0.15,0.015)),
  bpop=c(CL=0.5,V=0.2,E0=1,EMAX=1,EC50=1),  
  d=c(CL=0.09,V=0.09,E0=0.04,EC50=0.09), 
  
  # Design
  groupsize=20,
  m=3,
  xt = c(0.33,0.66,0.9,5,0.1,1,2,5),
  model_switch=c(1,1,1,1,2,2,2,2),
  a=list(c(DOSE=0),c(DOSE=1),c(DOSE=2)),

  # Design space
  minxt=0,
  maxxt=5,
  bUseGrouped_xt=1,
  maxa=c(DOSE=10),
  mina=c(DOSE=0))
```

The model predictions below show typical PK and PD profiles for three dose groups and the expected 95% prediction interval of the data. The initial design, as shown in the `poped.db` object, consists of 3 arms with doses of 0, 1, and 2 mg; PK sampling times are 0.33, 0.66, 0.9, and 5 hours/days; PD sampling times are 0.1, 1, 2, and 5 hours/days. With `model.names=c("PK","PD")` one can name the outputs in the graph.

```{r simulate_multi-response_model}
plot_model_prediction(
  poped.db,PI=TRUE,
  facet_scales="free",
  separate.groups=TRUE,
  model.names=c("PK","PD")) 
```


#	ODE solution of PK model, multiple dosing

The full code for this example is available in `ex.9.PK.2.comp.oral.md.ode.compiled.R`. 

In this example, the `deSolve` library needs to be installed for computing solutions to a system of differential equations. For faster solutions one can use pre-compiled code using the `Rcpp` library (see below). 
```{r,eval=TRUE}
library(deSolve)
```

Here we define the two compartment model in R using deSolve notation
```{r}
PK.2.comp.oral.ode <- function(Time, State, Pars){
  with(as.list(c(State, Pars)), {    
    dA1 <- -KA*A1 
    dA2 <- KA*A1 + A3* Q/V2 -A2*(CL/V1+Q/V1)
    dA3 <- A2* Q/V1-A3* Q/V2
    return(list(c(dA1, dA2, dA3)))
  })
}
```

Now we define the initial conditions of the ODE system `A_ini` with a named vector, in this case all compartments are initialized to zero `c(A1=0,A2=0,A3=0)`. The dosing input is defined as a data.frame `dose_dat` referring to the named compartment `var = c("A1")`, the specified `dose_times` and `value=c(DOSE*Favail)` dose amounts. Note that the covariates `DOSE` and the regimen `TAU` can differ by arm and be optimized (as shown in `ex.1.a.PK.1.comp.oral.md.intro.R`).  For more information see the help pages for `?deSolve::ode` and `?deSolve::events`. 
```{r}
ff.PK.2.comp.oral.md.ode <- function(model_switch, xt, parameters, poped.db){
  with(as.list(parameters),{
    
    # initial conditions of ODE system
    A_ini <- c(A1=0, A2=0, A3=0)

    #Set up time points to get ODE solutions
    times_xt <- drop(xt) # sample times
    times_start <- c(0) # add extra time for start of study
    times_dose = seq(from=0,to=max(times_xt),by=TAU) # dose times
    times <- unique(sort(c(times_start,times_xt,times_dose))) # combine it all
    
    # Dosing
    dose_dat <- data.frame(
      var = c("A1"), 
      time = times_dose,
      value = c(DOSE*Favail), 
      method = c("add")
    )
    
    out <- ode(A_ini, times, PK.2.comp.oral.ode, parameters,
               events = list(data = dose_dat))#atol=1e-13,rtol=1e-13)
    y = out[, "A2"]/V1
    y=y[match(times_xt,out[,"time"])]
    y=cbind(y)
    return(list(y=y,poped.db=poped.db))
  })
}

```

```{r, echo=FALSE}
fg <- function(x,a,bpop,b,bocc){
  parameters=c( CL=bpop[1]*exp(b[1]),
                V1=bpop[2],
                KA=bpop[3]*exp(b[2]),
                Q=bpop[4],
                V2=bpop[5],
                Favail=bpop[6],
                DOSE=a[1],
                TAU=a[2])
  return( parameters ) 
}
```

When creating a PopED database. `ff_fun` should  point to the function providing the solution to the ODE. Further, the names in the parameter definition (`fg`) function should match the parameters used in the above two functions.

```{r}
poped.db <- create.poped.database(
  
  # Model
  ff_fun="ff.PK.2.comp.oral.md.ode",
  fError_fun="feps.add.prop",
  fg_fun="fg",
  sigma=c(prop=0.1^2,add=0.05^2),
  bpop=c(CL=10,V1=100,KA=1,Q= 3.0, V2= 40.0, Favail=1),
  d=c(CL=0.15^2,KA=0.25^2),
  notfixed_bpop=c(1,1,1,1,1,0),
  
  # Design
  groupsize=20,
  m=1,      #number of groups
  xt=c( 48,50,55,65,70,85,90,120),
  
  # Design space 
  minxt=0,
  maxxt=144,
  discrete_xt = list(0:144),
  a=c(DOSE=100,TAU=24),
  discrete_a = list(DOSE=seq(0,1000,by=100),TAU=8:24))
```

We plot the population prediction of the model for the initial design
```{r simulate_ODE_model}
plot_model_prediction(poped.db,model_num_points = 500)
```

**Faster computations with Rcpp:**  We could also define the system using Rcpp, which will produce compiled code that should run faster (further examples in `ex.2.c.warfarin.ODE.compiled.R`).  First we redefine the ODE system using Rcpp. 

```{r}
library(Rcpp)
cppFunction(
  'List two_comp_oral_ode_Rcpp(double Time, NumericVector A, NumericVector Pars) {
     int n = A.size();
     NumericVector dA(n);
            
     double CL = Pars[0];
     double V1 = Pars[1];
     double KA = Pars[2];
     double Q  = Pars[3];
     double V2 = Pars[4];
            
     dA[0] = -KA*A[0];
     dA[1] = KA*A[0] - (CL/V1)*A[1] - Q/V1*A[1] + Q/V2*A[2];
     dA[2] = Q/V1*A[1] - Q/V2*A[2];
     return List::create(dA);
  }')
```


Next we add the compiled function (`two_comp_oral_ode_Rcpp`) in the ODE solver.
```{r}
ff.PK.2.comp.oral.md.ode.Rcpp <- function(model_switch, xt, parameters, poped.db){
  with(as.list(parameters),{
    
    # initial conditions of ODE system
    A_ini <- c(A1=0, A2=0, A3=0)

    #Set up time points to get ODE solutions
    times_xt <- drop(xt) # sample times
    times_start <- c(0) # add extra time for start of study
    times_dose = seq(from=0,to=max(times_xt),by=TAU) # dose times
    times <- unique(sort(c(times_start,times_xt,times_dose))) # combine it all
    
    # Dosing
    dose_dat <- data.frame(
      var = c("A1"), 
      time = times_dose,
      value = c(DOSE*Favail), 
      method = c("add")
    )
    
    # Here "two_comp_oral_ode_Rcpp" is equivalent 
    # to the non-compiled version "PK.2.comp.oral.ode".
    out <- ode(A_ini, times, two_comp_oral_ode_Rcpp, parameters,
               events = list(data = dose_dat))#atol=1e-13,rtol=1e-13)
    y = out[, "A2"]/V1
    y=y[match(times_xt,out[,"time"])]
    y=cbind(y)
    return(list(y=y,poped.db=poped.db))
  })
}

```

Finally we create a poped database to use these functions by updating the previously created database.
```{r, }
poped.db.Rcpp <- create.poped.database(
  poped.db,
  ff_fun="ff.PK.2.comp.oral.md.ode.Rcpp")
```

We can compare the time for design evaluation with these two methods of describing the same model.
```{r}
tic(); eval <- evaluate_design(poped.db); toc()
tic(); eval <- evaluate_design(poped.db.Rcpp); toc()
```

The difference is noticeable and gets larger for more complex ODE models.

#	ODE solution of TMDD model with 2 outputs, Multiple arms, different dose routes, different number of sample times per arm

The full code for this example is available in `ex.8.tmdd_qss_one_target_compiled.R`.  

```{r, echo=FALSE, results="hide"}
cppFunction('List tmdd_qss_one_target_model_ode
            (double Time, NumericVector A, NumericVector Pars) {
            int n = A.size();
            NumericVector dA(n);
            
            double CL = Pars[0];
            double V1 = Pars[1];
            double Q  = Pars[2];
            double V2 = Pars[3];
            double FAVAIL = Pars[4];
            double KA = Pars[5];
            // double VMAX = Pars[6];
            // double KMSS = Pars[7];
            double R0 = Pars[8];
            double KSSS = Pars[9];
            double KDEG = Pars[10];
            double KINT = Pars[11];
            // double DOSE = Pars[12];
            // double SC_FLAG = Pars[13];
            
            double RTOT, CTOT, CFREE;
            RTOT = A[3];
            CTOT= A[1]/V1;
            CFREE = 0.5*((CTOT-RTOT-KSSS)+sqrt((CTOT-RTOT-KSSS)*(CTOT-RTOT-KSSS)+4*KSSS*CTOT));
            
            dA[0] = -KA*A[0];
            dA[1] = FAVAIL*KA*A[0]+(Q/V2)*A[2]-
            (CL/V1+Q/V1)*CFREE*V1-RTOT*KINT*CFREE*V1/(KSSS+CFREE);
            dA[2] = (Q/V1)*CFREE*V1 - (Q/V2)*A[2];
            dA[3] = R0*KDEG - KDEG*RTOT - (KINT-KDEG)*(RTOT*CFREE/(KSSS+CFREE));
            return List::create(dA);
            }')
```

```{r,echo=FALSE, results="hide"}
sfg <- function(x,a,bpop,b,bocc){
  parameters=c( CL=bpop[1]*exp(b[1])  ,
                V1=bpop[2]*exp(b[2])	,
                Q=bpop[3]*exp(b[3])	,
                V2=bpop[4]*exp(b[4])	,
                FAVAIL=bpop[5]*exp(b[5])	,
                KA=bpop[6]*exp(b[6])	,                       
                VMAX=bpop[7]*exp(b[7])	,
                KMSS=bpop[8]*exp(b[8])	,
                R0=bpop[9]*exp(b[9])	,
                KSSS=bpop[10]*exp(b[10])	,
                KDEG=bpop[11]*exp(b[11])	,
                KINT=bpop[12]*exp(b[12])	,
                DOSE=a[1]	,
                SC_FLAG=a[2])   
  return(parameters) 
}
```

In the function that defines the dosing and derives the ODE solution, the discrete covariate `SC_FLAG` is used to give the dose either into `A1` or `A2`, the sub-cutaneous or the IV compartment.
```{r}
tmdd_qss_one_target_model_compiled <- function(model_switch,xt,parameters,poped.db){
  with(as.list(parameters),{
    y=xt
    
    #The initialization vector for the compartment
    A_ini <- c(A1=DOSE*SC_FLAG,
               A2=DOSE*(1-SC_FLAG),
               A3=0,
               A4=R0)
    
    #Set up time points for the ODE
    times_xt <- drop(xt)
    times <- sort(times_xt)
    times <- c(0,times) ## add extra time for start of integration
    
    # solve the ODE
    out <- ode(A_ini, times, tmdd_qss_one_target_model_ode, parameters)#,atol=1e-13,rtol=1e-13)
    
    
    # extract the time points of the observations
    out = out[match(times_xt,out[,"time"]),]
    
    # Match ODE output to measurements
    RTOT = out[,"A4"]
    CTOT = out[,"A2"]/V1
    CFREE = 0.5*((CTOT-RTOT-KSSS)+sqrt((CTOT-RTOT-KSSS)^2+4*KSSS*CTOT))
    COMPLEX=((RTOT*CFREE)/(KSSS+CFREE))
    RFREE= RTOT-COMPLEX
    
    y[model_switch==1]= RTOT[model_switch==1]
    y[model_switch==2] =CFREE[model_switch==2]
    #y[model_switch==3]=RFREE[model_switch==3]
    
    return(list( y=y,poped.db=poped.db))
  })
}

```

<!-- Finally we define the residual unexplained variability model. -->

```{r, echo=FALSE}
tmdd_qss_one_target_model_ruv <- function(model_switch,xt,parameters,epsi,poped.db){
  returnArgs <- do.call(poped.db$model$ff_pointer,list(model_switch,xt,parameters,poped.db))
  y <- returnArgs[[1]]
  poped.db <- returnArgs[[2]]
  
  y[model_switch==1] = log(y[model_switch==1])+epsi[,1]
  y[model_switch==2] = log(y[model_switch==2])+epsi[,2]
  
  return(list(y=y,poped.db=poped.db))
}

```

Two different sub-studies are defined, with different sampling times per arm - in terms of total number of samples and the actual times^[Study 1 and 2 from table 2 in: Gibiansky, L., Gibiansky, E., & Bauer, R. (2012). Comparison of Nonmem 7.2 estimation methods and parallel processing efficiency on a target-mediated drug disposition model. Journal of Pharmacokinetics and Pharmacodynamics, 39(1), 17–35. https://doi.org/10.1007/s10928-011-9228-y].
Due to this difference in numbers and the relatively complicated study design we define the sample times (`xt`), what each sample time will measure (`model_switch`) and which samples should be taken at the same study time (`G_xt`) as matrices.  Here three variables `xt`, `model_switch`, and `G_xt` are matrices with each row representing one arm, and the number of columns is the maximum number of samples (for all endpoints) in any of the arms (i.e., `max(ni)`). To be clear about which elements in the matrices should be considered we specify the number of samples per arm by defining the vector `ni` in the `create.poped.database` function.

```{r}
xt <- zeros(6,30)
study_1_xt <- matrix(rep(c(0.0417,0.25,0.5,1,3,7,14,21,28,35,42,49,56),8),nrow=4,byrow=TRUE)
study_2_xt <- matrix(rep(c(0.0417,1,1,7,14,21,28,56,63,70,77,84,91,98,105),4),nrow=2,byrow=TRUE)
xt[1:4,1:26] <- study_1_xt
xt[5:6,] <- study_2_xt

model_switch <- zeros(6,30)
model_switch[1:4,1:13] <- 1
model_switch[1:4,14:26] <- 2
model_switch[5:6,1:15] <- 1
model_switch[5:6,16:30] <- 2

G_xt <- zeros(6,30)
study_1_G_xt <- matrix(rep(c(1:13),8),nrow=4,byrow=TRUE)
study_2_G_xt <- matrix(rep(c(14:28),4),nrow=2,byrow=TRUE)
G_xt[1:4,1:26] <- study_1_G_xt
G_xt[5:6,] <- study_2_G_xt
```

These can then be plugged into the normal `poped.db` setup.
```{r}
poped.db.2 <- create.poped.database(
  
  # Model
  ff_fun=tmdd_qss_one_target_model_compiled,
  fError_fun=tmdd_qss_one_target_model_ruv,
  fg_fun=sfg,
  sigma=c(rtot_add=0.04,cfree_add=0.0225),
  bpop=c(CL=0.3,V1=3,Q=0.2,V2=3,FAVAIL=0.7,KA=0.5,VMAX=0,
         KMSS=0,R0=0.1,KSSS=0.015,KDEG=10,KINT=0.05),
  d=c(CL=0.09,V1=0.09,Q=0.04,V2=0.04,FAVAIL=0.04,
      KA=0.16,VMAX=0,KMSS=0,R0=0.09,KSSS=0.09,KDEG=0.04,
      KINT=0.04),
  notfixed_bpop=c( 1,1,1,1,1,1,0,0,1,1,1,1),
  notfixed_d=c( 1,1,1,1,1,1,0,0,1,1,1,1),
  
  # Design
  groupsize=rbind(6,6,6,6,100,100),
  m=6,      #number of groups
  xt=xt,
  model_switch=model_switch,
  ni=rbind(26,26,26,26,30,30),
  a=list(c(DOSE=100, SC_FLAG=0),
         c(DOSE=300, SC_FLAG=0),
         c(DOSE=600, SC_FLAG=0),
         c(DOSE=1000, SC_FLAG=1),
         c(DOSE=600, SC_FLAG=0),
         c(DOSE=1000, SC_FLAG=1)),
  
  # Design space
  bUseGrouped_xt=1,
  G_xt=G_xt,
  discrete_a = list(DOSE=seq(100,1000,by=100),
                    SC_FLAG=c(0,1)))

```

Now we can plot population predictions for each group and evaluate the design.
```{r simulate_different_dose_regimen}
plot_model_prediction(poped.db.2,facet_scales="free")
```

```{r,results='hide'}
eval_2 <- evaluate_design(poped.db.2)
round(eval_2$rse) # in percent
```

```{r,echo=FALSE}
knitr::kable(round(eval_2$rse),col.names = c("RSE in %")) #%>%  
  #kableExtra::kable_styling("striped",full_width = F) 
```


#	Model with continuous covariates

The R code for this example is available in `ex.12.covariate_distributions.R`.  

Let's assume that we have a model with a covariate included in the model description.  Here we define a one-compartment PK model that uses allometric scaling with a weight effect on both clearance and volume of distribution. 

```{r model}
mod_1 <- function(model_switch,xt,parameters,poped.db){
  with(as.list(parameters),{
    y=xt
    
    CL=CL*(WT/70)^(WT_CL)
    V=V*(WT/70)^(WT_V)
    DOSE=1000*(WT/70)
    y = DOSE/V*exp(-CL/V*xt) 
    
    return(list( y= y,poped.db=poped.db))
  })
}

par_1 <- function(x,a,bpop,b,bocc){
  parameters=c( CL=bpop[1]*exp(b[1]),
                V=bpop[2]*exp(b[2]),
                WT_CL=bpop[3],
                WT_V=bpop[4],
                WT=a[1])
  return( parameters ) 
}
```

Now we define a design.  In this case one group of individuals, where we define the individuals' typical weight as 70 kg (`a=c(WT=70)`). 
```{r design}
poped_db <- 
  create.poped.database(
    ff_fun=mod_1,
    fg_fun=par_1,
    fError_fun=feps.add.prop,
    groupsize=50,
    m=1,
    sigma=c(prop=0.015,add=0.0015),
    notfixed_sigma = c(1,0),
    bpop=c(CL=3.8,V=20,WT_CL=0.75,WT_V=1), 
    d=c(CL=0.05,V=0.05), 
    xt=c( 1,2,4,6,8,24),
    minxt=0,
    maxxt=24,
    bUseGrouped_xt=1,
    a=c(WT=70)
  )
```

We can create a plot of the model prediction for the typical individual
```{r}
plot_model_prediction(poped_db)
```

And evaluate the initial design
```{r}
evaluate_design(poped_db)
```

From the output produced we see that the covariate parameters can not be estimated according to this design calculation (RSE of WT_CL and WT_V are `NA`).  Why is that? Well, the calculation being done is assuming that every individual in the group has the same covariate (to speed up the calculation).  This is clearly a poor assumption in this case!

**Distribution of covariates:** We can improve the computation by assuming a distribution of the covariate (WT) in the individuals in the study. We set `groupsize=1`, the number of groups to be 50 (`m=50`) and assume that WT is sampled from a normal distribution with mean=70 and sd=10 (`a=as.list(rnorm(50, mean = 70, sd = 10)`).
```{r}
poped_db_2 <- 
  create.poped.database(
    ff_fun=mod_1,
    fg_fun=par_1,
    fError_fun=feps.add.prop,
    groupsize=1,
    m=50,
    sigma=c(prop=0.015,add=0.0015),
    notfixed_sigma = c(prop=1,add=0),
    bpop=c(CL=3.8,V=20,WT_CL=0.75,WT_V=1), 
    d=c(CL=0.05,V=0.05), 
    xt=c(1,2,4,6,8,24),
    minxt=0,
    maxxt=24,
    bUseGrouped_xt=1,
    a=as.list(rnorm(50, mean = 70, sd = 10))
  )

```

```{r}
ev <- evaluate_design(poped_db_2) 
round(ev$ofv,1)
```


```{r,results='hide'}
round(ev$rse)
```

```{r,echo=FALSE}
knitr::kable(round(ev$rse),col.names = c("RSE in %")) #%>%  
  #kableExtra::kable_styling("striped",full_width = FALSE) 
```


Here we see that, given this distribution of weights, the covariate effect parameters (`WT_CL` and `WT_V`) would be well estimated.

However, we are only looking at one sample of 50 individuals.  Maybe a better approach is to look at the distribution of RSEs over a number of experiments given the expected weight distribution. 

```{r, cache=TRUE,results='hide'}
nsim <- 30
rse_list <- c()
for(i in 1:nsim){
  poped_db_tmp <- 
    create.poped.database(
      ff_fun=mod_1,
      fg_fun=par_1,
      fError_fun=feps.add.prop,
      groupsize=1,
      m=50,
      sigma=c(prop=0.015,add=0.0015),
      notfixed_sigma = c(1,0),
      bpop=c(CL=3.8,V=20,WT_CL=0.75,WT_V=1), 
      d=c(CL=0.05,V=0.05), 
      xt=c( 1,2,4,6,8,24),
      minxt=0,
      maxxt=24,
      bUseGrouped_xt=1,
      a=as.list(rnorm(50,mean = 70,sd=10)))
  rse_tmp <- evaluate_design(poped_db_tmp)$rse
  rse_list <- rbind(rse_list,rse_tmp)
}
(rse_quant <- apply(rse_list,2,quantile))
```

```{r,echo=FALSE}
knitr::kable(as.data.frame(rse_quant),digits = 2)#,col.names = c("RSE in %")) #%>%  
  #kableExtra::kable_styling("striped",full_width = FALSE) 
```

Note, that the variance of the RSE of the covariate effect is in this case strongly correlated with the variance of the weight distribution (not shown).

#	Model with discrete covariates	

See `ex.11.PK.prior.R`. This has the covariate `isPediatric` to distinguish between adults and pediatrics.  Alternatively,  `DOSE` and `TAU` in the first example can be considered as discrete covariates.

#	Model with Inter-Occasion Variability (IOV)

The full code for this example is available in `ex.14.PK.IOV.R`.

The IOV is introduced with `bocc[x,y]` in the parameter definition function as a matrix with the first argument `x` indicating the index for the IOV variances, and the second argument `y` denoting the occasion. This is used in the example to derive to different clearance values, i.e., `CL_OCC_1` and `CL_OCC_2`.
```{r}
sfg <- function(x,a,bpop,b,bocc){
  parameters=c( CL_OCC_1=bpop[1]*exp(b[1]+bocc[1,1]),
                CL_OCC_2=bpop[1]*exp(b[1]+bocc[1,2]),
                V=bpop[2]*exp(b[2]),
                KA=bpop[3]*exp(b[3]),
                DOSE=a[1],
                TAU=a[2])
  return( parameters ) 
}
```

These parameters can now be used in the model function to define the change in parameters between the occasions (here the change occurs with the 7th dose in a one-compartment model with first order absorption).
```{r}
cppFunction(
  'List one_comp_oral_ode(double Time, NumericVector A, NumericVector Pars) {
   int n = A.size();
   NumericVector dA(n);
            
   double CL_OCC_1 = Pars[0];
   double CL_OCC_2 = Pars[1];
   double V = Pars[2];
   double KA = Pars[3];
   double TAU = Pars[4];
   double N,CL;
            
   N = floor(Time/TAU)+1;
   CL = CL_OCC_1;
   if(N>6) CL = CL_OCC_2;
   
   dA[0] = -KA*A[0];
   dA[1] = KA*A[0] - (CL/V)*A[1];
   return List::create(dA);
   }'
)

ff.ode.rcpp <- function(model_switch, xt, parameters, poped.db){
  with(as.list(parameters),{
    A_ini <- c(A1=0, A2=0)
    times_xt <- drop(xt) #xt[,,drop=T] 
    dose_times = seq(from=0,to=max(times_xt),by=TAU)
    eventdat <- data.frame(var = c("A1"), 
                           time = dose_times,
                           value = c(DOSE), method = c("add"))
    times <- sort(c(times_xt,dose_times))
    out <- ode(A_ini, times, one_comp_oral_ode, c(CL_OCC_1,CL_OCC_2,V,KA,TAU), 
               events = list(data = eventdat))#atol=1e-13,rtol=1e-13)
    y = out[, "A2"]/(V)
    y=y[match(times_xt,out[,"time"])]
    y=cbind(y)
    return(list(y=y,poped.db=poped.db))
  })
}

```

The within-subject variability variances (`docc`) are defined in the poped database as a 3-column matrix with one row per IOV-parameter, and the middle column giving the variance values.
```{r} 
poped.db <- 
  create.poped.database(
    ff_fun=ff.ode.rcpp,
    fError_fun=feps.add.prop,
    fg_fun=sfg,
    bpop=c(CL=3.75,V=72.8,KA=0.25), 
    d=c(CL=0.25^2,V=0.09,KA=0.09), 
    sigma=c(prop=0.04,add=5e-6),
    notfixed_sigma=c(0,0),
    docc = matrix(c(0,0.09,0),nrow = 1),
    m=2,
    groupsize=20,
    xt=c( 1,2,8,240,245),
    minxt=c(0,0,0,240,240),
    maxxt=c(10,10,10,248,248),
    bUseGrouped_xt=1,
    a=list(c(DOSE=20,TAU=24),c(DOSE=40, TAU=24)),
    maxa=c(DOSE=200,TAU=24),
    mina=c(DOSE=0,TAU=24)
  )
```

We can visualize the IOV by looking at an example individual.  We see the PK profile changes at the 7th dose (red line) due to the change in clearance.
```{r simulate_IOV_with_IIV}
library(ggplot2)
set.seed(123)
plot_model_prediction(
  poped.db, 
  PRED=F,IPRED=F, 
  separate.groups=T, 
  model_num_points = 300, 
  groupsize_sim = 1,
  IPRED.lines = T, 
  alpha.IPRED.lines=0.6,
  sample.times = F
) + geom_vline(xintercept = 24*6,color="red")
```

We can also see that the design is relatively poor for estimating the IOV parameter:

```{r,results='hide'}
ev <- evaluate_design(poped.db)
round(ev$rse)
```

```{r,echo=FALSE}
knitr::kable(round(ev$rse),col.names = c("RSE in %")) #%>%  
  #kableExtra::kable_styling("striped",full_width = FALSE) 
```

#	Full omega matrix	

The full code for this example is available in `ex.15.full.covariance.matrix.R`.

```{r, echo=FALSE, results="hide"}
ff <- function(model_switch,xt,parameters,poped.db){
  ##-- Model: One comp first order absorption
  with(as.list(parameters),{
    y=xt
    y=(DOSE*Favail*KA/(V*(KA-CL/V)))*(exp(-CL/V*xt)-exp(-KA*xt))
    return(list(y=y,poped.db=poped.db))
  })
}

sfg <- function(x,a,bpop,b,bocc){
  ## -- parameter definition function 
  parameters=c(CL=bpop[1]*exp(b[1]),
               V=bpop[2]*exp(b[2]),
               KA=bpop[3]*exp(b[3]),
               Favail=bpop[4],
               DOSE=a[1])
  return(parameters) 
}

feps <- function(model_switch,xt,parameters,epsi,poped.db){
  ## -- Residual Error function
  ## -- Proportional 
  returnArgs <- ff(model_switch,xt,parameters,poped.db) 
  y <- returnArgs[[1]]
  poped.db <- returnArgs[[2]]
  y = y*(1+epsi[,1])
  
  return(list(y=y,poped.db=poped.db)) 
}

## -- Define initial design  and design space
poped.db <- 
  create.poped.database(
    ff_file="ff",
    fg_file="sfg",
    fError_file="feps",
    bpop=c(CL=0.15, V=8, KA=1.0, Favail=1), 
    notfixed_bpop=c(1,1,1,0),
    d=c(CL=0.07, V=0.02, KA=0.6), 
    sigma=c(prop=0.01),
    groupsize=32,
    xt=c( 0.5,1,2,6,24,36,72,120),
    minxt=0,
    maxxt=120,
    a=70
  )
```

The `covd` object is used for defining the covariances of the between subject variances (off-diagonal elements of the full variance-covariance matrix for the between subject variability).

```{r}
poped.db_with <- 
  create.poped.database(
    ff_file="ff",
    fg_file="sfg",
    fError_file="feps",
    bpop=c(CL=0.15, V=8, KA=1.0, Favail=1), 
    notfixed_bpop=c(1,1,1,0),
    d=c(CL=0.07, V=0.02, KA=0.6), 
    covd = c(.03,.1,.09),
    sigma=c(prop=0.01),
    groupsize=32,
    xt=c( 0.5,1,2,6,24,36,72,120),
    minxt=0,
    maxxt=120,
    a=70
  )
```

What do the covariances mean?

```{r}
(IIV <- poped.db_with$parameters$param.pt.val$d)
cov2cor(IIV)
```
They indicate a correlation of the inter-individual variabilities, here of ca. 0.8 between clearance and volume, as well as between volume and absorption rate.

We can clearly see a difference in the variance of the model predictions. 
```{r simulate_with_cov_matrix}
library(ggplot2)
p1 <- plot_model_prediction(poped.db, PI=TRUE)+ylim(-0.5,12) 
p2 <- plot_model_prediction(poped.db_with,PI=TRUE) +ylim(-0.5,12)
gridExtra::grid.arrange(p1+ ggtitle("No covariance in BSV"), p2+ ggtitle("Covariance in BSV"), nrow = 1)
```

Evaluating the designs with and without the covariances:

```{r}
ev1 <- evaluate_design(poped.db)
ev2 <- evaluate_design(poped.db_with)
```

```{r,results="hide"}
round(ev1$rse)
round(ev2$rse)
```

```{r,echo=FALSE,message=FALSE}
tb1 <- tibble::tibble(" "=names(ev1$rse),
         "Diagonal BSV"=ev1$rse)

tb2 <- tibble::tibble(" "=names(ev2$rse),
         "Covariance in BSV"=ev2$rse)

tb_final <- dplyr::right_join(tb1,tb2,by=" ")

knitr::kable(tb_final,digits = 0) #%>%  
  #kableExtra::kable_styling("striped",full_width = FALSE) 
```

Note, that the precision of all other parameters is barely affected by including the full covariance matrix. This is likely to be different in practice with more ill-conditioned numerical problems.

**Evaluate the same designs with full FIM (instead of reduced)**

```{r,results='hide'}
ev1 <- evaluate_design(poped.db, fim.calc.type=0)
ev2 <-evaluate_design(poped.db_with, fim.calc.type=0)

round(ev1$rse,1)
round(ev2$rse,1)
```

```{r,echo=FALSE,message=FALSE}
tb1 <- tibble::tibble(" "=names(ev1$rse),
         "Diagonal BSV"=ev1$rse)

tb2 <- tibble::tibble(" "=names(ev2$rse),
         "Covariance in BSV"=ev2$rse)

tb_final <- dplyr::right_join(tb1,tb2,by=" ")

knitr::kable(tb_final,digits = 0) #%>%  
  #kableExtra::kable_styling("striped",full_width = FALSE) 
```

#	Include a prior FIM, compute power to identify a parameter

In this example we incorporate prior knowledge into a current study design calculation.
First the expected FIM obtained from an experiment in adults is computed. Then this FIM is added to the current experiment in children. One could also use the observed FIM when using estimation software to fit one realization of a design (from the $COVARIANCE step in NONMEM for example). The full code for this example is available in `ex.11.PK.prior.R`.

Note that we define  the parameters for a one-compartment first-order absorption model using a covariate called `isPediatric` to switch between adult and pediatric models, and `bpop[5]=pedCL` is the factor to multiply the adult clearance `bpop[3]`  to obtain the pediatric one.

```{r}
sfg <- function(x,a,bpop,b,bocc){
  parameters=c( 
    V=bpop[1]*exp(b[1]),
    KA=bpop[2]*exp(b[2]),
    CL=bpop[3]*exp(b[3])*bpop[5]^a[3], # add covariate for pediatrics
    Favail=bpop[4],
    isPediatric = a[3],
    DOSE=a[1],
    TAU=a[2])
  return( parameters ) 
}
```

The design and design space for adults is defined below (Two arms, 5 sample time points per arm, doses of 20 and 40 mg, `isPediatric = 0`). As we want to pool the results (i.e. add the FIMs together), we also have to provide  the `pedCL` parameter so that both the adult and children FIMs have the same dimensions.
```{r}
poped.db <- 
  create.poped.database(
    ff_fun=ff.PK.1.comp.oral.md.CL,
    fg_fun=sfg,
    fError_fun=feps.add.prop,
    bpop=c(V=72.8,KA=0.25,CL=3.75,Favail=0.9,pedCL=0.8), 
    notfixed_bpop=c(1,1,1,0,1),
    d=c(V=0.09,KA=0.09,CL=0.25^2), 
    sigma=c(0.04,5e-6),
    notfixed_sigma=c(0,0),
    m=2,
    groupsize=20,
    xt=c( 1,8,10,240,245),
    bUseGrouped_xt=1,
    a=list(c(DOSE=20,TAU=24,isPediatric = 0),
           c(DOSE=40, TAU=24,isPediatric = 0))
  )
```

Create plot of model without variability
```{r simulate_adult}
plot_model_prediction(poped.db, model_num_points = 300)

```

To store the FIM from the adult design we
evaluate this design
```{r}
(outAdult = evaluate_design(poped.db))
```
It is obvious that we cannot estimate the pediatric covariate from adult data only; hence the warning message.
You can also note the zeros in the 4th column and 4th row of the FIM indicating that `pedCL` cannot be estimated from the adult data.

We can evaluate the adult design without warning, by setting the `pedCL` parameter to be fixed (i.e., not estimated):
```{r}
evaluate_design(create.poped.database(poped.db, notfixed_bpop=c(1,1,1,0,0)))
```
One obtains good estimates for all parameters for adults (<60% RSE for all).

For pediatrics the covariate `isPediatric = 1`. We define one arm, 4 sample-time points. 
```{r}
poped.db.ped <- 
  create.poped.database(
    ff_fun=ff.PK.1.comp.oral.md.CL,
    fg_fun=sfg,
    fError_fun=feps.add.prop,
    bpop=c(V=72.8,KA=0.25,CL=3.75,Favail=0.9,pedCL=0.8), 
    notfixed_bpop=c(1,1,1,0,1),
    d=c(V=0.09,KA=0.09,CL=0.25^2), 
    sigma=c(0.04,5e-6),
    notfixed_sigma=c(0,0),
    m=1,
    groupsize=6,
    xt=c( 1,2,6,240),
    bUseGrouped_xt=1,
    a=list(c(DOSE=40,TAU=24,isPediatric = 1))
  )
```

We can create a plot of the pediatric model without variability 
```{r simulate_pediatrics}
plot_model_prediction(poped.db.ped, model_num_points = 300)
```

Evaluate the design of the pediatrics study alone. 
```{r}
evaluate_design(poped.db.ped)

```
Clearly the design has problems on its own.

We can add the prior information from the adult study and evaluate that design (i.e., pooling adult and pediatric data).
```{r}
poped.db.all <- create.poped.database(
  poped.db.ped,
  prior_fim = outAdult$fim
)

(out.all <- evaluate_design(poped.db.all))
```
The pooled data leads to much higher precision in parameter estimates compared to either study separately.

One can also obtain the power for estimating the pediatric difference in clearance (power in estimating bpop[5] as different from 1).
```{r}
evaluate_power(poped.db.all, bpop_idx=5, h0=1, out=out.all)
```
We see that to clearly distinguish this parameter  one would need 14 children in the pediatric study (for 80% power at $\alpha=0.05$).

# Design evaluation including uncertainty in the model parameters (robust design)	

In this example the aim is to evaluate a design incorporating uncertainty around parameter values in the model.
The full code for this example is available in `ex.2.d.warfarin.ED.R`.
This illustration is one of the Warfarin examples from software comparison in: Nyberg et al.^[Nyberg, J., Bazzoli, C., Ogungbenro, K., Aliev, A., Leonov, S., Duffull, S., Hooker, A.C. and Mentré, F. (2014). Methods and software tools for design evaluation for population pharmacokinetics-pharmacodynamics studies. British Journal of Clinical Pharmacology, 79(1), 1–32. https://doi.org/10.1111/bcp.12352]. 

```{r, echo=FALSE, results="hide"}
sfg <- function(x,a,bpop,b,bocc){
  ## -- parameter definition function 
  parameters=c(
    CL=bpop[1]*exp(b[1]),
    V=bpop[2]*exp(b[2]),
    KA=bpop[3]*exp(b[3]),
    Favail=bpop[4],
    DOSE=a[1])
  return(parameters) 
}

ff <- function(model_switch,xt,parameters,poped.db){
  ##-- Model: One comp first order absorption
  with(as.list(parameters),{
    y=xt
    y=(DOSE*Favail*KA/(V*(KA-CL/V)))*(exp(-CL/V*xt)-exp(-KA*xt))
    return(list(y=y,poped.db=poped.db))
  })
}
```

We define the fixed effects in the model and add a 10% uncertainty to all but Favail.  To do this we use a  
Matrix defining the fixed effects, per row (row number = parameter_number) we should have:

* column 1 the type of the distribution for E-family designs (0 = Fixed, 1 = Normal, 2 = Uniform, 3 = User Defined Distribution, 4 = lognormal and 5 = truncated normal)
* column 2  defines the mean.
* column 3 defines the variance of the distribution (or length of uniform distribution).

Here we define a log-normal distribution  
```{r}
bpop_vals <- c(CL=0.15, V=8, KA=1.0, Favail=1)
bpop_vals_ed <- 
  cbind(ones(length(bpop_vals),1)*4, # log-normal distribution
        bpop_vals,
        ones(length(bpop_vals),1)*(bpop_vals*0.1)^2) # 10% of bpop value
bpop_vals_ed["Favail",] <- c(0,1,0)
bpop_vals_ed
```

With this model and parameter set we define the initial design and design space. Specifically note the `bpop=bpop_vals_ed` and the `ED_samp_size=20` arguments.  `ED_samp_size=20` indicates the number of samples used in evaluating the E-family objective functions.
```{r}
poped.db <- 
  create.poped.database(
    ff_fun=ff,
    fg_fun=sfg,
    fError_fun=feps.add.prop,
    bpop=bpop_vals_ed, 
    notfixed_bpop=c(1,1,1,0),
    d=c(CL=0.07, V=0.02, KA=0.6), 
    sigma=c(0.01,0.25),
    groupsize=32,
    xt=c( 0.5,1,2,6,24,36,72,120),
    minxt=0,
    maxxt=120,
    a=70,
    mina=0,
    maxa=100,
    ED_samp_size=20)
```

You can also provide `ED_samp_size` argument to the design evaluation or optimization arguments:
```{r}
tic();evaluate_design(poped.db,d_switch=FALSE,ED_samp_size=20); toc()
```

We can see that the result, based on MC sampling, is somewhat variable with so few samples.
```{r}
tic();evaluate_design(poped.db,d_switch=FALSE,ED_samp_size=20); toc()
```


# Design evaluation for a subset of model parameters of interest (Ds optimality)

Ds-optimality is a criterion that can be used if one is interested in estimating a subset "s" of the model parameters as precisely as possible.
The full code for this example is available in `ex.2.e.warfarin.Ds.R`. First we define initial design and design space:

```{r, echo=FALSE}
sfg <- function(x,a,bpop,b,bocc){
  ## -- parameter definition function 
  parameters=c(CL=bpop[1]*exp(b[1]),
               V=bpop[2]*exp(b[2]),
               KA=bpop[3]*exp(b[3]),
               Favail=bpop[4],
               DOSE=a[1])
  return(parameters) 
}

ff <- function(model_switch,xt,parameters,poped.db){
  ##-- Model: One comp first order absorption
  with(as.list(parameters),{
    y=xt
    y=(DOSE*Favail*KA/(V*(KA-CL/V)))*(exp(-CL/V*xt)-exp(-KA*xt))
    return(list(y=y,poped.db=poped.db))
  })
}
```

For Ds optimality we add the `ds_index` option to the `create.poped.database` function to indicate whether a parameter is interesting (=0) or not (=1). Moreover, we set the `ofv_calc_type=6` for computing the Ds optimality criterion (it is set to 4 by default, for computing the log of the determinant of the FIM). More details are available by running the command `?create.poped.database`.

```{r}
poped.db <- 
  create.poped.database(
    ff_fun=ff,
    fg_fun=sfg,
    fError_fun=feps.add.prop,
    bpop=c(CL=0.15, V=8, KA=1.0, Favail=1), 
    notfixed_bpop=c(1,1,1,0),
    d=c(CL=0.07, V=0.02, KA=0.6), 
    sigma=c(0.01,0.25),
    groupsize=32,
    xt=c( 0.5,1,2,6,24,36,72,120),
    minxt=0,
    maxxt=120,
    a=70,
    mina=0,
    maxa=100,
    ds_index=c(0,0,0,1,1,1,1,1), # size is number_of_non_fixed_parameters
    ofv_calc_type=6) # Ds OFV calculation
```

Design evaluation:

```{r}
evaluate_design(poped.db)
```

#	Shrinkage	

The full code for this example is available in "ex.13.shrinkage.R".  

```{r, echo=FALSE, results="hide"}

sfg <- function(x,a,bpop,b,bocc){
  ## -- parameter definition function 
  parameters=c(KA=bpop[1]*exp(b[1]),
               K=bpop[2]*exp(b[2]),
               V=bpop[3]*exp(b[3]),
               DOSE=a[1])
  return(parameters) 
}

ff <- function(model_switch,xt,parameters,poped.db){
  ##-- Model: One comp first order absorption
  with(as.list(parameters),{
    y<-(DOSE/V*KA/(KA-K)*(exp(-K*xt)-exp(-KA*xt)))
    return(list(y=y,poped.db=poped.db))
  })
}

## -- Residual unexplained variablity (RUV) function
## -- Additive + Proportional  
feps <- function(model_switch,xt,parameters,epsi,poped.db){
  returnArgs <- do.call(poped.db$model$ff_pointer,
                        list(model_switch,xt,parameters,poped.db)) 
  f <- returnArgs[[1]]
  y = f + (0.5 + 0.15*f)*epsi[,1]
  
  return(list( y= y,poped.db =poped.db )) 
}
## -- Define initial design  and design space
poped.db <- create.poped.database(
  ff_fun=ff,
  fg_fun=sfg,
  fError_fun =feps,
  bpop=c(KA=2, K=0.25, V=15), 
  d=c(KA=1, V=0.25,0.1), 
  sigma=c(1),
  notfixed_sigma = c(0),
  groupsize=1,
  xt=c( 1,3,8),
  minxt=0,
  maxxt=10,
  a=100)

plot_model_prediction(poped.db)

```

To evaluate the estimation quality of the individual random effects in the model (the b's) we use the function `shrinkage()`.

```{r} 
shrinkage(poped.db)
```
The output shows us the expected shrinkage on the variance scale ($shrink_{var}=1-var(b_j)/D(j,j)$) and on the standard deviation scale ($shrink_{sd}=1-sd(b_j)/sqrt(D(j,j))$), as well as the standard errors of the $b_j$ estimates.

# Further examples 

Available in PopED, but not shown in examples:

*	Espresso design
*	Handling BLQ data	
*	Irregular dosing more complex: e.g. switching between s.c. and i.v. within one arm.	
*	Constraining the optimization to different allowed sampling times for each group	
*	Constraining the optimization to different allowed sampling times for each response	
* Keep some sampling time fixed (they will be automatically part of the optimal design protocol)	
* Handling derived outputs	

To be implemented:

* Symbolic differentiation	



