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Design evaluation and optimization indiscrete space

Overview

Data for this example result from a PKPD population study on anon-steroidal molecule(Flores-Murrieta et al. 1998).Single doses of 1, 3.2, 10, 31.6, 56.2, or 100mg/kg per os were givenrespectively to 6 groups, each of which contained at least 6 rats(weighing 200g on average), following a parallel design. Blood samplingand drug response (DI score) evaluation were conducted at 0, 15, 30, and45 min and at 1, 1.25, 1.5, 2, 3 and 4 hours after administration. Basedon the evaluation results, we choose to optimize a design for 30 ratsreceiving a single dose of either 100mg/kg or 320 mg/kg, selecting only3 from previously defined time points for blood sampling and responseevaluation by using Fedorov-Wynn method and Multiplicative Algorithm.Reports of the design evaluation and optimization are available athttps://github.com/packagePFIM/PFIM

# Save the evaluation and optimization results in the default folderoutputPath=getwd()

Design evaluation

Define the model equations of the PKPD model

modelEquations=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

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.

errorModelRespPK=Combined1(output ="RespPK",sigmaInter =0,sigmaSlope =0.21 )errorModelRespPD=Constant(output ="RespPD",sigmaInter =9.6 )modelError=list( errorModelRespPK, errorModelRespPD )

Define the arms and for each arm

# sampling timessamplingTimesRespPK=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 administrationsadministrationRespPK1=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 designdesign1

design1=Design(name ="design1",arms =list( arm1, arm2, arm3, arm4, arm5, arm6 ) )

Evaluate the population FIM

evaluationPop=Evaluation(name =" ",modelEquations = modelEquations,modelParameters = modelParameters,modelError = modelError,outputs =list("RespPK"="Cc","RespPD"="E" ),designs =list( design1 ),fimType ="population",odeSolverParameters =list(atol =1e-8,rtol =1e-8 ) )evaluationPop=run( evaluationPop )

Display the results of the design evaluation

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

plotOptions=list(unitTime =c("hour"),unitOutcomes =c("mcg/mL","DI%")  )plotEvaluation=plotEvaluation( evaluationPop, plotOptions )plotSensitivityIndices=plotSensitivityIndices( evaluationPop, plotOptions )SE=plotSE( evaluationPop )RSE=plotRSE( evaluationPop )# examplesplotOutcomesEvaluationRespPK= plotEvaluation[["design1"]][["20mg Arm"]][["RespPK"]]plotOutcomesEvaluationRespPD= plotEvaluation[["design1"]][["20mg Arm"]][["RespPD"]]plotSensitivityIndice_RespPK_Cl= plotSensitivityIndices[["design1"]][["20mg Arm"]][["RespPK"]][["Cl"]]

Report for the design evaluation

outputFile="Example01_EvaluationPopFIM.html"plotOptions=list(unitTime=c("hour"),unitOutcomes =c("mcg/mL","DI%") )Report( evaluationPop, outputPath, outputFile, plotOptions )

Evaluate the individual and Bayesian FIMs

evaluationInd=Evaluation(name =" ",modelEquations = modelEquations,modelParameters = modelParameters,modelError = modelError,outputs =list("RespPK"="Cc","RespPD"="E" ),designs =list( design1 ),fimType ="individual",odeSolverParameters =list(atol =1e-8,rtol =1e-8 ) )evaluationInd=run( evaluationInd )evaluationBay=Evaluation(name =" ",modelEquations = modelEquations,modelParameters = modelParameters,modelError = modelError,outputs =list("RespPK"="Cc","RespPD"="E" ),designs =list( design1 ),fimType ="Bayesian",odeSolverParameters =list(atol =1e-8,rtol =1e-8 ) )evaluationBay=run( evaluationBay )

Display the results of the design evaluation

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

plotOptions=list(unitTime=c("hour"),unitOutcomes=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

administrationRespPK=Administration(outcome ="RespPK",timeDose =c(0),dose =c(6.24 ) )

We create sampling times that will be used in the initial design forcomparison during the optimization process.

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

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

initialElementaryProtocols=list(c(0.25,0.75,1,4 ),c(1.5,2,6,12 ) )

Create an administration constraint with the vector of allowedamount of doses for the response PK

administrationConstraintsRespK=AdministrationConstraints(outcome ="RespPK",doses =list(0.2,0.64,2,6.24,11.24,20 ) )

Create the constraint design to the project

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 ),numberOfArms =30 )

Set the vector of initial proportions or numbers of subjects foreach elementary design

numberOfSubjects=c(30)proportionsOfSubjects=c(30)/30

Set the the parameters of the FedorovWynn algorithm and run thealgorithm for the design optimization with a population FIM

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 ),fimType ="population",outputs =list("RespPK"="Cc","RespPD"="E" ),odeSolverParameters =list(atol =1e-8,rtol =1e-8 ) )optimizationFWPopFIM=run( optimizationFWPopFIM )saveRDS( optimizationFWPopFIM,paste0( outputPath,"optimizationFWPopFIM.RDS" ) )

Display and plot the results of the design optimization

show( optimizationFWPopFIM )fisherMatrix=getFisherMatrix( optimizationFWPopFIM )getCorrelationMatrix( optimizationFWPopFIM )getSE( optimizationFWPopFIM )getRSE( optimizationFWPopFIM )getShrinkage( optimizationFWPopFIM )getDeterminant( optimizationFWPopFIM )getDcriterion( optimizationFWPopFIM )# plot the frequenciesplotFrequencies=plotFrequencies( optimizationFWPopFIM )plotFrequencies

Reports for the design optimization

outputFile="Example01_OptimizationFWPopFIM.html"Report( optimizationFWPopFIM, outputPath, outputFile, plotOptions )

Set the the parameters of the Multiplicative Algorithm algorithm andrun the algorithm for the design optimization with a population FIM

optimizationMultPopFIM=Optimization(name ="PKPD_ODE_multi_doses_populationFIM",modelEquations = modelEquations,modelParameters = modelParameters,modelError = modelError,optimizer ="MultiplicativeAlgorithm",optimizerParameters =list(lambda =0.99,numberOfIterations =1000,weightThreshold =0.01,delta =1e-04,showProcess = T ),designs =list( designConstraint ),fimType ="population",outputs =list("RespPK"="Cc","RespPD"="E" ),odeSolverParameters =list(atol =1e-8,rtol =1e-8 ) )

Run the Multiplicative Algorithm algorithm for the optimization witha population FIM

optimizationMultPopFIM=run( optimizationMultPopFIM )saveRDS( optimizationMultPopFIM,paste0( outputPath,"optimizationMultPopFIM.RDS" ) )

Display and plot the results of the design optimization

show( optimizationMultPopFIM )fisherMatrix=getFisherMatrix( optimizationMultPopFIM )getCorrelationMatrix( optimizationMultPopFIM )getSE( optimizationMultPopFIM )getRSE( optimizationMultPopFIM )getShrinkage( optimizationMultPopFIM )getDeterminant( optimizationMultPopFIM )getDcriterion( optimizationMultPopFIM )# plot the weightplotWeights=plotWeights( optimizationMultPopFIM )plotWeights

Create and save the report for the design optimization

plotOptions=list(unitTime =c("hour" ),unitOutcomes=c("mcg/mL" ,"DI%" ) )outputFile="Example01_OptimizationMultPopFIM.html"Report( optimizationMultPopFIM, outputPath, outputFile, plotOptions )

References

Flores-Murrieta, Francisco, Holly Kimko, Dora Flores-Acevedo, FranciscoLópez-Muñoz, William Jusko, Mark Sale, and Gilberto Castañeda-Hernández.1998.“Pharmacokinetic–Pharmacodynamic Modeling of TolmetinAntinociceptive Effect in the Rat Using an Indirect Response Model: APopulation Approach.”Journal of Pharmacokinetics andBiopharmaceutics 26 (November): 547–57.
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