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title: "Design evaluation and optimization in discrete space"
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```{r global_options, echo = FALSE, include = FALSE}
backup_options <- options()
options(width = 3000)
knitr::opts_chunk$set(collapse = TRUE,
comment = "#>",echo = FALSE, warning = FALSE, message = FALSE,
cache = FALSE, tidy = FALSE, size = "small")
library(PFIM)
```
# Overview
Data for this example result from a PKPD population study on a non-steroidal molecule [@FloresMurrieta1998]. Single doses of 1, 3.2, 10, 31.6, 56.2, or 100mg/kg per os were given respectively to 6 groups, each of which contained at least 6 rats (weighing 200g on average), following a parallel design. Blood sampling and drug response (DI score) evaluation were conducted at 0, 15, 30, and 45 min and at 1, 1.25, 1.5, 2, 3 and 4 hours after administration. Based on the evaluation results, we choose to optimize a design for 30 rats receiving a single dose of either 100mg/kg or 320 mg/kg, selecting only 3 from previously defined time points for blood sampling and response evaluation by using Fedorov-Wynn method and Multiplicative Algorithm. Reports of the design evaluation and optimization are available at https://github.com/iame-researchCenter/PFIM
# Design evaluation
### Define the model equations of the PKPD model
```{r, echo = TRUE, comment=''}
modelEquations = list(
outcomes = list( "RespPK" = "Cc",
"RespPD" = "E" ),
equations = list( "Deriv_Cc" = "dose_RespPK/V*ka*exp(-ka*t) - Cl/V*Cc",
"Deriv_E" = "Rin*(1-Imax*(Cc**gamma)/(Cc**gamma + IC50**gamma))-kout*E" ) )
```
### Set mu and omega for each parameter
```{r, echo = TRUE, comment=''}
modelParameters = list(
ModelParameter( name = "V",
distribution = LogNormal( mu = 0.74, omega = 0.316 ) ),
ModelParameter( name = "Cl",
distribution = LogNormal( mu = 0.28, omega = 0.456 ) ),
ModelParameter( name = "ka",
distribution = LogNormal( mu = 10, omega = sqrt( 0 ) ), fixedMu = TRUE ),
ModelParameter( name = "kout",
distribution = LogNormal( mu = 6.14, omega = 0.947 ) ),
ModelParameter( name = "Rin",
distribution = LogNormal( mu = 614, omega = sqrt( 0 ) ), fixedMu = TRUE ),
ModelParameter( name = "Imax",
distribution = LogNormal( mu = 0.76, omega = 0.439 ) ),
ModelParameter( name = "IC50",
distribution = LogNormal( mu = 9.22, omega = 0.452 ) ),
ModelParameter( name = "gamma",
distribution = LogNormal( mu = 2.77, omega = 1.761 ) ) )
```
### Create the error model to the responses PK and PD.
```{r, echo = TRUE, comment=''}
errorModelRespPK = Combined1( outcome = "RespPK", sigmaInter = 0, sigmaSlope = 0.21 )
errorModelRespPD = Constant( outcome = "RespPD", sigmaInter = 9.6 )
modelError = list( errorModelRespPK, errorModelRespPD )
```
### Define the arms and for each arm
+ create the administration parameters for the response PK
+ create the sampling times for the responses PK and PD
```{r, echo = TRUE, comment=''}
# sampling times
samplingTimesRespPK = SamplingTimes( outcome = "RespPK",
samplings = c( 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4 ) )
samplingTimesRespPD = SamplingTimes( outcome = "RespPD",
samplings = c( 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4 ) )
# Define the arms and the administrations
administrationRespPK1 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 0.2 ) )
arm1 = Arm( name = "0.2mg Arm",
size = 6,
administrations = list( administrationRespPK1 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
administrationRespPK2 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 0.64 ) )
arm2 = Arm( name = "0.64mg Arm",
size = 6,
administrations = list( administrationRespPK2 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
administrationRespPK3 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 2 ) )
arm3 = Arm( name = "2mg Arm",
size = 6,
administrations = list( administrationRespPK3 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
administrationRespPK4 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 6.24 ) )
arm4 = Arm( name = "6.24mg Arm",
size = 6,
administrations = list( administrationRespPK4 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
administrationRespPK5 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 11.24 ) )
arm5 = Arm( name = "11.24mg Arm",
size = 6,
administrations = list( administrationRespPK5 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
administrationRespPK6 = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 20 ) )
arm6 = Arm( name = "20mg Arm",
size = 6,
administrations = list( administrationRespPK6 ) ,
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
initialCondition = list( "Cc" = 0,
"E" = 100 ) )
```
### Add the arms to the design `design1`
```{r, echo = TRUE, comment=''}
design1 = Design( name = "design1", arms = list( arm1, arm2, arm3, arm4, arm5, arm6 ) )
```
### Evaluate the population FIM
```{r, echo = TRUE, eval = FALSE, comment=''}
evaluationPop = Evaluation( name = "",
modelEquations = modelEquations,
modelParameters = modelParameters,
modelError = modelError,
outcomes = list( "RespPK" = "Cc", "RespPD" = "E" ),
designs = list( design1 ),
fim = "population",
odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 ) )
evaluationPop = run( evaluationPop )
```
### Display the results of the design evaluation
```{r, echo = TRUE, eval = FALSE, comment=''}
show( evaluationPop )
fisherMatrix = getFisherMatrix( evaluationPop )
getCorrelationMatrix( evaluationPop )
getSE( evaluationPop )
getRSE( evaluationPop )
getShrinkage( evaluationPop )
getDeterminant( evaluationPop )
getDcriterion( evaluationPop )
```
### Graphs of the results of the design evaluation
```{r, echo = TRUE, eval = FALSE, comment=''}
plotOptions = list( unitTime = c("hour"), unitOutcomes = c("mcg/mL","DI%") )
plotEvaluation = plotEvaluation( evaluationPop, plotOptions )
plotSensitivityIndice = plotSensitivityIndice( evaluationPop, plotOptions )
SE = plotSE( evaluationPop )
RSE = plotRSE( evaluationPop )
# examples
plotOutcomesEvaluationRespPK = plotEvaluation[["design1"]][["20mg Arm"]][["RespPK"]]
plotOutcomesEvaluationRespPD = plotEvaluation[["design1"]][["20mg Arm"]][["RespPD"]]
plotSensitivityIndice_RespPK_Cl = plotSensitivityIndice[["design1"]][["20mg Arm"]][["RespPK"]][["Cl"]]
SE = SE[["design1"]]
RSE = RSE[["design1"]]
```
### Report for the design evaluation
```{r, echo = TRUE, eval = FALSE, comment=''}
outputPath = "C:/...."
outputFile = "Example01_EvaluationPopFIM.html"
plotOptions = list( unitTime=c("hour"), unitResponses= c("mcg/mL","DI%") )
Report( evaluationPop, outputPath, outputFile, plotOptions )
```
### Evaluate the individual and Bayesian FIMs
```{r, echo = TRUE, eval = FALSE, comment=''}
evaluationInd = Evaluation( name = "",
modelEquations = modelEquations,
modelParameters = modelParameters,
modelError = modelError,
outcomes = list( "RespPK" = "Cc", "RespPD" = "E" ),
designs = list( design1 ),
fim = "individual",
odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 ) )
evaluationInd = run( evaluationInd )
evaluationBay = Evaluation( name = "",
modelEquations = modelEquations,
modelParameters = modelParameters,
modelError = modelError,
outcomes = list( "RespPK" = "Cc", "RespPD" = "E" ),
designs = list( design1 ),
fim = "Bayesian",
odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 ) )
evaluationBay = run( evaluationBay )
```
### Display the results of the design evaluation
```{r, echo = TRUE, eval = FALSE, comment=''}
show( evaluationInd )
show( evaluationBay )
fisherMatrix = getFisherMatrix( evaluationInd )
getCorrelationMatrix( evaluationInd )
getSE( evaluationInd )
getRSE( evaluationInd )
getShrinkage( evaluationInd )
getDeterminant( evaluationInd )
getDcriterion( evaluationInd )
fisherMatrix = getFisherMatrix( evaluationBay )
getCorrelationMatrix( evaluationBay )
getSE( evaluationBay )
getRSE( evaluationBay )
getShrinkage( evaluationBay )
getDeterminant( evaluationBay )
getDcriterion( evaluationBay )
```
### Reports of the results of the design evaluation
```{r, echo = TRUE, eval = FALSE, comment=''}
outputPath = "C:/...."
plotOptions = list( unitTime=c("hour"), unitResponses= c("mcg/mL","DI%") )
outputFile = "Example01_EvaluationIndFIM.html"
Report( evaluationInd, outputPath, outputFile, plotOptions )
outputFile = "Example01_EvaluationBayFIM.html"
Report( evaluationBay, outputPath, outputFile, plotOptions )
```
# Design optimization
### Create an administration for response PK and samplings times
```{r, echo = TRUE, comment=''}
administrationRespPK = Administration( outcome = "RespPK", timeDose = c(0), dose = c( 6.24 ) )
```
We create sampling times that will be used in the initial design for comparison during the optimization process.
```{r, echo = TRUE, comment=''}
samplingTimesRespPK = SamplingTimes( outcome = "RespPK", samplings = c( 0.25, 0.75, 1, 1.5, 2, 4, 6 ) )
samplingTimesRespPD = SamplingTimes( outcome = "RespPD", samplings = c( 0.25, 0.75, 1.5, 2, 3, 6, 8, 12 ) )
```
### Create a sampling constraint for the responses PK and PD
+ Set the vector of allowed sampling times for the responses PK and PD
+ Fix certain time values
+ Set the number of optimizable sampling times
```{r, echo = TRUE, comment=''}
samplingConstraintsRespPK = SamplingTimeConstraints( outcome = "RespPK",
initialSamplings = c( 0.25, 0.75, 1, 1.5, 2, 4, 6 ),
fixedTimes = c( 0.25, 4 ),
numberOfsamplingsOptimisable = 4 )
samplingConstraintsRespPD = SamplingTimeConstraints( outcome = "RespPD",
initialSamplings = c( 0.25, 0.75, 1.5, 2, 3, 6, 8, 12 ),
fixedTimes = c( 2, 6 ),
numberOfsamplingsOptimisable = 4 )
```
### Set the initial vectors for the FedorovWynn algorithm
```{r, echo = TRUE, comment=''}
initialElementaryProtocols = list( c( 0.25, 0.75, 1, 4 ), c( 1.5, 2, 6, 12 ) )
```
### Create an administration constraint with the vector of allowed amount of doses for the response PK
```{r, echo = TRUE, comment=''}
administrationConstraintsRespK = AdministrationConstraints( outcome = "RespPK",
doses = c( 0.2, 0.64, 2, 6.24, 11.24, 20 ) )
```
### Create the constraint design to the project
+ total number of individuals to be considered
+ administration constraint
+ sampling constraint
```{r, echo = TRUE, comment=''}
armConstraint = Arm( name = "armConstraint",
size = 30,
administrations = list( administrationRespPK ),
samplingTimes = list( samplingTimesRespPK, samplingTimesRespPD ),
administrationsConstraints = list( administrationConstraintsRespK ),
samplingTimesConstraints = list( samplingConstraintsRespPK, samplingConstraintsRespPD ),
initialCondition = list( "Cc" = 0,
"E" = "Rin/kout" ) )
designConstraint = Design( name = "designConstraint",
arms = list( armConstraint ) )
```
### Set the vector of initial proportions or numbers of subjects for each elementary design
```{r, echo = TRUE, comment=''}
numberOfSubjects = c(30)
proportionsOfSubjects = c(30)/30
```
### Set the the parameters of the FedorovWynn algorithm and run the algorithm for the design optimization with a population FIM
```{r, echo = TRUE, eval = FALSE, comment=''}
optimizationFWPopFIM = Optimization( name = "PKPD_ODE_multi_doses_populationFIM",
modelEquations = modelEquations,
modelParameters = modelParameters,
modelError = modelError,
optimizer = "FedorovWynnAlgorithm",
optimizerParameters = list( elementaryProtocols = initialElementaryProtocols,
numberOfSubjects = numberOfSubjects,
proportionsOfSubjects = proportionsOfSubjects,
showProcess = T ),
designs = list( designConstraint ),
fim = "population",
outcomes = list( "RespPK" = "Cc","RespPD" = "E" ),
odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 ) )
optimizationFWPopFIM = run( optimizationFWPopFIM )
```
### Display and plot the results of the design optimization
```{r, echo = TRUE, eval = FALSE, comment=''}
show( optimizationFWPopFIM )
fisherMatrix = getFisherMatrix( optimizationFWPopFIM )
getCorrelationMatrix( optimizationFWPopFIM )
getSE( optimizationFWPopFIM )
getRSE( optimizationFWPopFIM )
getShrinkage( optimizationFWPopFIM )
getDeterminant( optimizationFWPopFIM )
getDcriterion( optimizationFWPopFIM )
# plot the frequencies
plotFrequencies = plotFrequencies( optimizationFWPopFIM )
plotFrequencies
```
### Reports for the design optimization
```{r, echo = TRUE, eval = FALSE, comment=''}
outputFile = "Example01_OptimizationFWPopFIM.html"
Report( optimizationFWPopFIM, outputPath, outputFile, plotOptions )
```
### Set the the parameters of the Multiplicative Algorithm algorithm and run the algorithm for the design optimization with a population FIM
```{r, echo = TRUE, eval = FALSE, comment=''}
optimizationMultPopFIM = Optimization( name = "PKPD_ODE_multi_doses_populationFIM",
modelEquations = modelEquations,
modelParameters = modelParameters,
modelError = modelError,
optimizer = "MultiplicativeAlgorithm",
optimizerParameters = list( lambda = 0.99,
numberOfIterations = 1000,
delta = 1e-04, showProcess = T ),
designs = list( designConstraint ),
fim = "population",
outcomes = list( "RespPK" = "Cc","RespPD" = "E" ),
odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 ) )
```
### Run the Multiplicative Algorithm algorithm for the optimization with a population FIM
```{r, echo = TRUE, eval = FALSE, comment=''}
optimizationMultPopFIM = run( optimizationMultPopFIM )
```
### Display and plot the results of the design optimization
```{r, echo = TRUE, eval = FALSE, comment=''}
show( optimizationMultPopFIM )
fisherMatrix = getFisherMatrix( optimizationMultPopFIM )
getCorrelationMatrix( optimizationMultPopFIM )
getSE( optimizationMultPopFIM )
getRSE( optimizationMultPopFIM )
getShrinkage( optimizationMultPopFIM )
getDeterminant( optimizationMultPopFIM )
getDcriterion( optimizationMultPopFIM )
# plot the weight
plotWeights = plotWeights( optimizationMultPopFIM, threshold = 0.001 )
plotWeights
```
### Create and save the report for the design optimization
```{r, echo = TRUE, eval = FALSE, comment=''}
outputPath = "C:/...."
plotOptions = list( unitTime = c( "hour" ), unitResponses= c( "mcg/mL" , "DI%" ), threshold = 0.01 )
outputFile = "Example01_OptimizationMultPopFIM.html"
Report( optimizationMultPopFIM, outputPath, outputFile, plotOptions )
```
# References
```{r global_options_end, echo = FALSE, include = FALSE}
options(backup_options)
```