GNU LESSER GENERAL PUBLIC LICENSE Version 3, 29 June 2007
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Copyright (C) 2007 Free Software Foundation, Inc. http://fsf.org/ Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
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Copyright (C) 2007 Free Software Foundation, Inc. http://fsf.org/ Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
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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:
+
+
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:
The table below provides a check list of features for each of the 15
+available examples.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Features
Ex1
@@ -524,328 +553,389 @@
-
Note: All features are available in PopED but some are not demonstrated in the supplied examples.
+
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).
+
+
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).
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.
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.
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.
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.
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).
+
+
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).
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.
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.
-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))
- })
-}
-
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.
+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))
+})
+}
+
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.
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.
+
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.
Next we add the compiled function
+(two_comp_oral_ode_Rcpp) in the ODE solver.
-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.
+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.
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.
-
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.
+
+
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.
+
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.
-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))
- })
-}
+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))
+})
+}
-
Two different sub-studies are defined, with different sampling times per arm - in terms of total number of samples and the actual times1. 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.
+
Two different sub-studies are defined, with different sampling times
+per arm - in terms of total number of samples and the actual times1. 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.
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.
+
+
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.
-evaluate_design(poped_db)
-#> Problems inverting the matrix. Results could be misleading.
-#> Warning: The following parameters are not estimable:
-#> WT_CL, WT_V
-#> Is the design adequate to estimate all parameters?
-#> $ofv
-#> [1] -Inf
-#>
-#> $fim
-#> CL V WT_CL WT_V d_CL d_V sig_prop
-#> CL 65.8889583 -0.7145374 0 0 0.00000 0.00000 0.000
-#> V -0.7145374 2.2798156 0 0 0.00000 0.00000 0.000
-#> WT_CL 0.0000000 0.0000000 0 0 0.00000 0.00000 0.000
-#> WT_V 0.0000000 0.0000000 0 0 0.00000 0.00000 0.000
-#> d_CL 0.0000000 0.0000000 0 0 9052.31524 29.49016 1424.255
-#> d_V 0.0000000 0.0000000 0 0 29.49016 8316.09464 2483.900
-#> sig_prop 0.0000000 0.0000000 0 0 1424.25450 2483.90024 440009.144
-#>
-#> $rse
-#> CL V WT_CL WT_V d_CL d_V sig_prop
-#> 3.247502 3.317107 NA NA 21.026264 21.950179 10.061292
-
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)).
+evaluate_design(poped_db)
+#> Problems inverting the matrix. Results could be misleading.
+#> Warning: The following parameters are not estimable:
+#> WT_CL, WT_V
+#> Is the design adequate to estimate all parameters?
+#> $ofv
+#> [1] -Inf
+#>
+#> $fim
+#> CL V WT_CL WT_V d_CL d_V sig_prop
+#> CL 65.8889583 -0.7145374 0 0 0.00000 0.00000 0.000
+#> V -0.7145374 2.2798156 0 0 0.00000 0.00000 0.000
+#> WT_CL 0.0000000 0.0000000 0 0 0.00000 0.00000 0.000
+#> WT_V 0.0000000 0.0000000 0 0 0.00000 0.00000 0.000
+#> d_CL 0.0000000 0.0000000 0 0 9052.31524 29.49016 1424.255
+#> d_V 0.0000000 0.0000000 0 0 29.49016 8316.09464 2483.900
+#> sig_prop 0.0000000 0.0000000 0 0 1424.25450 2483.90024 440009.144
+#>
+#> $rse
+#> CL V WT_CL WT_V d_CL d_V sig_prop
+#> 3.247502 3.317107 NA NA 21.026264 21.950179 10.061292
+
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)).
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.
+
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.
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).
+
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 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.
+
+
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.
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).
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).
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.
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.
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.
The full code for this example is available in ex.15.full.covariance.matrix.R.
-
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).
+
+
Full omega matrix
+
+
The full code for this example is available in
+ex.15.full.covariance.matrix.R.
+
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).
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.
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.
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)
+
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)
-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.
+
+
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.
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.
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.
To store the FIM from the adult design we evaluate this design
-(outAdult=evaluate_design(poped.db))
-#> Problems inverting the matrix. Results could be misleading.
-#> Warning: The following parameters are not estimable:
-#> pedCL
-#> Is the design adequate to estimate all parameters?
-#> $ofv
-#> [1] -Inf
-#>
-#> $fim
-#> V KA CL pedCL d_V d_KA
-#> V 0.05854391 -6.815269 -0.01531146 0 0.0000000 0.00000000
-#> KA -6.81526942 2963.426688 -1.32113719 0 0.0000000 0.00000000
-#> CL -0.01531146 -1.321137 37.50597895 0 0.0000000 0.00000000
-#> pedCL 0.00000000 0.000000 0.00000000 0 0.0000000 0.00000000
-#> d_V 0.00000000 0.000000 0.00000000 0 1203.3695137 192.31775149
-#> d_KA 0.00000000 0.000000 0.00000000 0 192.3177515 428.81459138
-#> d_CL 0.00000000 0.000000 0.00000000 0 0.2184104 0.01919009
-#> d_CL
-#> V 0.000000e+00
-#> KA 0.000000e+00
-#> CL 0.000000e+00
-#> pedCL 0.000000e+00
-#> d_V 2.184104e-01
-#> d_KA 1.919009e-02
-#> d_CL 3.477252e+03
-#>
-#> $rse
-#> V KA CL pedCL d_V d_KA d_CL
-#> 6.634931 8.587203 4.354792 NA 33.243601 55.689432 27.133255
-
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):
+(outAdult=evaluate_design(poped.db))
+#> Problems inverting the matrix. Results could be misleading.
+#> Warning: The following parameters are not estimable:
+#> pedCL
+#> Is the design adequate to estimate all parameters?
+#> $ofv
+#> [1] -Inf
+#>
+#> $fim
+#> V KA CL pedCL d_V d_KA
+#> V 0.05854391 -6.815269 -0.01531146 0 0.0000000 0.00000000
+#> KA -6.81526942 2963.426688 -1.32113719 0 0.0000000 0.00000000
+#> CL -0.01531146 -1.321137 37.50597895 0 0.0000000 0.00000000
+#> pedCL 0.00000000 0.000000 0.00000000 0 0.0000000 0.00000000
+#> d_V 0.00000000 0.000000 0.00000000 0 1203.3695137 192.31775149
+#> d_KA 0.00000000 0.000000 0.00000000 0 192.3177515 428.81459138
+#> d_CL 0.00000000 0.000000 0.00000000 0 0.2184104 0.01919009
+#> d_CL
+#> V 0.000000e+00
+#> KA 0.000000e+00
+#> CL 0.000000e+00
+#> pedCL 0.000000e+00
+#> d_V 2.184104e-01
+#> d_KA 1.919009e-02
+#> d_CL 3.477252e+03
+#>
+#> $rse
+#> V KA CL pedCL d_V d_KA d_CL
+#> 6.634931 8.587203 4.354792 NA 33.243601 55.689432 27.133255
+
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):
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.2.
-
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:
+
+
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.2.
+
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 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).
+
column 3 defines the variance of the distribution (or length of
+uniform distribution).
Here we define a log-normal distribution
-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
-#> bpop_vals
-#> CL 4 0.15 0.000225
-#> V 4 8.00 0.640000
-#> KA 4 1.00 0.010000
-#> Favail 0 1.00 0.000000
-
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.
+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
+#> bpop_vals
+#> CL 4 0.15 0.000225
+#> V 4 8.00 0.640000
+#> KA 4 1.00 0.010000
+#> Favail 0 1.00 0.000000
+
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.
-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:
-
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.
+
+
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:
+
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.
The full code for this example is available in “ex.13.shrinkage.R”.
+
+
Shrinkage
+
+
The full code for this example is available in
+“ex.13.shrinkage.R”.
-
To evaluate the estimation quality of the individual random effects in the model (the b’s) we use the function shrinkage().
+
To evaluate the estimation quality of the individual random effects
+in the model (the b’s) we use the function shrinkage().
-shrinkage(poped.db)
-#> # A tibble: 3 x 5
-#> d_KA d_V `D[3,3]` type group
-#> <dbl> <dbl> <dbl> <chr> <chr>
-#> 1 0.504 0.367 0.424 shrink_var grp_1
-#> 2 0.295 0.205 0.241 shrink_sd grp_1
-#> 3 0.710 0.303 0.206 se grp_1
-
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.
+shrinkage(poped.db)
+#> # A tibble: 3 × 5
+#> d_KA d_V `D[3,3]` type group
+#> <dbl><dbl><dbl><chr><chr>
+#> 1 0.504 0.367 0.424 shrink_var grp_1
+#> 2 0.295 0.205 0.241 shrink_sd grp_1
+#> 3 0.710 0.303 0.206 se grp_1
+
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
+
+
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
+
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)
+
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
@@ -1897,11 +2097,19 @@
Symbolic differentiation
-
+
-
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↩︎
-
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↩︎
+
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↩︎
+
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↩︎
@@ -1917,11 +2125,13 @@
@@ -1930,5 +2140,7 @@
+
+
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+++ b/docs/articles/handling_LOQ.html
@@ -26,6 +26,8 @@
+
+
We will use an additive and proportional residual unexplained
+variability (RUV) model, predefined in PopED as the function
+feps.add.prop.
-
-
-Define an initial design and design space
-
Now we define the model parameter values, the initial design and design space for optimization.
-
We define model parameters similar to the Warfarin example from the software comparison in Nyberg et al. (2015) and an arbitrary design of two groups of 20 individuals.
+
+
Define an initial design and design space
+
+
Now we define the model parameter values, the initial design and
+design space for optimization.
+
We define model parameters similar to the Warfarin example from the
+software comparison in Nyberg et al. (2015) and an arbitrary design of two
+groups of 20 individuals.
First it may make sense to check your model and design to make sure you get what you expect when simulating data. Here we plot the model typical values for the two groups:
+
+
Simulation
+
+
First it may make sense to check your model and design to make sure
+you get what you expect when simulating data. Here we plot the model
+typical values for the two groups:
Next, we evaluate the initial design. We see that the relative standard error of the parameters (in percent) are relatively well estimated with this design except for the proportional RUV.
+
+
Design evaluation
+
+
Next, we evaluate the initial design. We see that the relative
+standard error of the parameters (in percent) are relatively well
+estimated with this design except for the proportional RUV.
We use optimization criteria based on the D6 (loq_method=1 which is the default) and D2 (loq_method=2) methods from Vong et al. (2012). For the D6 method, which has been shown to be a better method in comparison to SSE studies, we have updated the method to only investigate points where the 95% PI overlaps LOQ, otherwise we set the design point to either no information or full information. Further we filter out situations with very low probabilities (loq_prob_limit = 0.001). Both the CI% and the low probability limit can be specified in the calculation (default values are loq_PI_conf_level = 0.95 and loq_prob_limit = 0.001). In this way we can get the D6 method to compute in reasonable time-frames even for larger number of design samples.
-
Here we can evaluate the design with both methods and test the speed of computation. We see that D6 is significantly slower than D2 (but D6 should be a more accurate representation of the RSE expected using M3 estimation methods).
+
We use optimization criteria based on the D6
+(loq_method=1 which is the default) and D2
+(loq_method=2) methods from Vong et
+al. (2012). For the D6 method, which
+has been shown to be a better method in comparison to SSE studies, we
+have updated the method to only investigate points where the 95% PI
+overlaps LOQ, otherwise we set the design point to either no information
+or full information. Further we filter out situations with very low
+probabilities (loq_prob_limit = 0.001). Both the CI% and
+the low probability limit can be specified in the calculation (default
+values are loq_PI_conf_level = 0.95 and
+loq_prob_limit = 0.001). In this way we can get the D6
+method to compute in reasonable time-frames even for larger number of
+design samples.
+
Here we can evaluate the design with both methods and test the speed
+of computation. We see that D6 is significantly slower than D2 (but D6
+should be a more accurate representation of the RSE expected using M3
+estimation methods).
Predicted parameter uncertainty for the three methods is shown below. We see that the uncertainty is generally higher with the LOQ evaluations (as expected). We also see that because the D2 method ignores data that is below LOQ (the last observation in the design), then the predictions of uncertainty are significantly larger.
Predicted parameter uncertainty for the three methods is shown below.
+We see that the uncertainty is generally higher with the LOQ evaluations
+(as expected). We also see that because the D2 method ignores data that
+is below LOQ (the last observation in the design), then the predictions
+of uncertainty are significantly larger.
+
-RSE (in %) for the initial design using different methods of handling LOQ.
+RSE (in %) for the initial design using different methods of handling
+LOQ.
@@ -434,31 +472,34 @@
-
-
-ULOQ handling
-
If needed we can also handle upper limits of quantification. Lets assume we have an ULOQ at 7 units in addition to the LLOQ of 2 units:
+
+
ULOQ handling
+
+
If needed we can also handle upper limits of quantification. Lets
+assume we have an ULOQ at 7 units in addition to the LLOQ of 2
+units:
-eval_ul_D6<-evaluate_design(poped_db,
- loq=2,
- uloq=7)
-
-eval_ul_D2<-evaluate_design(poped_db,
- loq=2,
- loq_method=2,
- uloq=7,
- uloq_method=2)
-#> Problems inverting the matrix. Results could be misleading.
-
+eval_ul_D6<-evaluate_design(poped_db,
+ loq=2,
+ uloq=7)
+
+eval_ul_D2<-evaluate_design(poped_db,
+ loq=2,
+ loq_method=2,
+ uloq=7,
+ uloq_method=2)
+#> Problems inverting the matrix. Results could be misleading.
+
-RSE (in %) for the initial design using different methods of handling LOQ and ULOQ.
+RSE (in %) for the initial design using different methods of handling
+LOQ and ULOQ.
@@ -645,42 +686,49 @@
-
-
-Design optimization
-
We can then optimize the design using the different methods of computing the FIM. Here we optimize only using lower LOQ.
+
+
Design optimization
+
+
We can then optimize the design using the different methods of
+computing the FIM. Here we optimize only using lower LOQ.
-All designs together in one plot show how the different handling of BLQ data results in different optimal designs.
+optim_D6<-poped_optim(poped_db, opt_xt =TRUE,
+ parallel=T,
+ loq=2)
+
+optim_D2<-poped_optim(poped_db, opt_xt =TRUE,
+ parallel=T,
+ loq=2,
+ loq_method=2)
+
+optim_full<-poped_optim(poped_db, opt_xt =TRUE,
+ parallel=T)
+All designs together in one plot show how the different handling of BLQ
+data results in different optimal designs.
-Design points for the apraoch ignoring LOQ, using the D2 method, and using the D6 method
+Design points for the apraoch ignoring LOQ, using the D2 method, and
+using the D6 method
-
Predictions using the D6 method from each of the optimizations shows the expected %RSE of the parameters if each design is used and the LOQ is at 2 concentration units. We see that D2 may be a reasonable strategy to optimize designs that are “good enough” if the D6 method is too slow for optimization.
+
Predictions using the D6 method from each of the optimizations shows
+the expected %RSE of the parameters if each design is used and the LOQ
+is at 2 concentration units. We see that D2 may be a reasonable strategy
+to optimize designs that are “good enough” if the D6 method is too slow
+for optimization.
-Nyberg, Joakim, Caroline Bazzoli, Kay Ogungbenro, Alexander Aliev, Sergei Leonov, Stephen Duffull, Andrew C Hooker, and France Mentré. 2015. “Methods and software tools for design evaluation in population pharmacokinetics-pharmacodynamics studies.”British Journal of Clinical Pharmacology 79 (1): 6–17. https://doi.org/10.1111/bcp.12352.
+Nyberg, Joakim, Caroline Bazzoli, Kay Ogungbenro, Alexander Aliev,
+Sergei Leonov, Stephen Duffull, Andrew C Hooker, and France Mentré.
+2015. “Methods and software tools for design
+evaluation in population pharmacokinetics-pharmacodynamics
+studies.”British Journal of Clinical
+Pharmacology 79 (1): 6–17. https://doi.org/10.1111/bcp.12352.
-Nyberg, Joakim, Sebastian Ueckert, Eric A. Strömberg, Stefanie Hennig, Mats O. Karlsson, and Andrew C. Hooker. 2012. “PopED: An extended, parallelized, nonlinear mixed effects models optimal design tool.”Computer Methods and Programs in Biomedicine 108 (2): 789–805. https://doi.org/10.1016/j.cmpb.2012.05.005.
+Nyberg, Joakim, Sebastian Ueckert, Eric A. Strömberg, Stefanie Hennig,
+Mats O. Karlsson, and Andrew C. Hooker. 2012. “PopED: An extended, parallelized, nonlinear mixed effects
+models optimal design tool.”Computer Methods and
+Programs in Biomedicine 108 (2): 789–805. https://doi.org/10.1016/j.cmpb.2012.05.005.
-Vong, Camille, Sebastian Ueckert, Joakim Nyberg, and Andrew C. Hooker. 2012. “Handling Below Limit of Quantification Data in Optimal Trial Design.”PAGE, Abstracts of the Annual Meeting of the Population Approach Group in Europe. https://www.page-meeting.org/?abstract=2578.
+Vong, Camille, Sebastian Ueckert, Joakim Nyberg, and Andrew C. Hooker.
+2012. “Handling Below Limit of Quantification Data in Optimal
+Trial Design.”PAGE, Abstracts of the Annual Meeting of the
+Population Approach Group in Europe. https://www.page-meeting.org/?abstract=2578.
