UseCase3_1 : PKPD model for warfarin population pharmacokinetics and PCA

Variant on UseCase3 : PK parameters passed as model inputs via “use is variable”

This example script is intended to illustrate how to use the 'ddmore' R package to perform a M&S workflow using the DDMoRe Standalone Execution Environment (SEE).

The following steps are implemented in this workflow:

To run a task, select with the cursor any code lines you wish to execute and press CTRL+R+R in your keyboard. An HTML file containing the commands in this file and associated output will be provided to allow the user to compare the results

Initialisation

Clear workspace and set working directory under 'models' folder

Clear workspace and set working directory under 'UsesCasesDemo' project

rm(list=ls(all=F))
mydir <- file.path(Sys.getenv("MDLIDE_WORKSPACE_HOME"),"UseCasesDemo")
setwd(mydir)

Set name of .mdl file and dataset for future tasks

uc <- "UseCase3_1"
datafile <- "warfarin_conc_pca_PKparam.csv"
mdlfile <- paste0(uc,".mdl")

Create a new folder to be used as working directory and where results will be stored

wd <- file.path(mydir,uc)
dir.create(wd)

Copy the dataset and the .mdl file available under “models” into the working directory

file.copy(file.path(mydir,"models", datafile),wd)
## [1] TRUE
file.copy(file.path(mydir,"models",mdlfile),wd)
## [1] TRUE

Set the working directory.

setwd(file.path(mydir,uc))

The working directory needs to be set differently when knitr R package is used to spin the file

library(knitr)
opts_knit$set(root.dir = file.path(mydir,uc)) 
cache.path <- file.path(mydir,uc, 'cache')
dir.create(cache.path)
opts_chunk$set(cache.path=cache.path)
figure.path <- file.path(mydir,uc, 'figure')
dir.create(figure.path)
opts_chunk$set(figure.path=figure.path)

List files available in working directory

list.files()
## [1] "cache"                         "figure"                       
## [3] "UseCase3_1.mdl"                "warfarin_conc_pca_PKparam.csv"

Introduction to 'ddmore' R package

View objects within the .mdl file

Use 'ddmore' function getModelObjects() to retrieve model object(s) from an existing .mdl file. This function reads the MDL in an .mdl file and parses the MDL code for each MDL Object into an R list of objects of appropriate types with names corresponding to the MDL Object names given in the file.

myMDLObj <- getMDLObjects(mdlfile)
length(myMDLObj)
## [1] 4
names(myMDLObj)
## [1] "warfarin_PKPD_turnover_dat"  "warfarin_PKPD_turnover_par" 
## [3] "warfarin_PKPD_turnover_mdl"  "warfarin_PKPD_turnover_task"

Use 'ddmore' function getDataObjects() to retrieve only data object(s) from an existing .mdl file. This function returns a list of Parameter Object(s) from which we select the first element. Hover over the variable name to see its structure

myDataObj <- getDataObjects(mdlfile)[[1]]

Use 'ddmore' function getParameterObjects() to retrieve only parameter object(s) from an existing .mdl file

myParObj <- getParameterObjects(mdlfile)[[1]]

Use 'ddmore' function getModelObjects() to retrieve only model object(s) from an existing .mdl file.

myModObj <- getModelObjects(mdlfile)[[1]]

Use 'ddmore' function getTaskPropertiesObjects() to retrieve only task properties object(s) from an existing .mdl file

myTaskObj <- getTaskPropertiesObjects(mdlfile)[[1]]

Exploratory Data Analysis

Recall that getDataObjects only reads the MDL code from the .mdl file. Use 'ddmore' function readDataObj() to create an R object from the MDL data object.

myData <- readDataObj(myDataObj)

Let's look at the first 6 lines of the data set

head(myData)
##   ID TIME   WT AGE  SEX AMT DVID DV MDV    CL    V    KA  TLAG
## 1  1    0 66.7  50 male 100    0 NA   1 0.278 8.26 0.395 0.976
## 2  1    0 66.7  50 male  NA    2 NA   1 0.278 8.26 0.395 0.976
## 3  1   24 66.7  50 male  NA    2 44   0 0.278 8.26 0.395 0.976
## 4  1   36 66.7  50 male  NA    2 27   0 0.278 8.26 0.395 0.976
## 5  1   48 66.7  50 male  NA    2 28   0 0.278 8.26 0.395 0.976
## 6  1   72 66.7  50 male  NA    2 31   0 0.278 8.26 0.395 0.976

Extract only observation records

myEDAData<-myData[myData$MDV==0,]

Open an R window to record and access all your plots

windows(record=TRUE) 

Now plot the data using xyplot from the lattice library

plot3 <- xyplot(DV~TIME,groups=ID,data=myEDAData,subset=DVID==2,type="b",ylab="PCA",xlab="Time (h)")
print(plot3)

plot of chunk unnamed-chunk-15

plot4 <- xyplot(DV~TIME|ID,data=myEDAData,subset=DVID==2,type="b",layout=c(3,4),ylab="PCA",xlab="Time (h)",scales=list(relation="free"))
print(plot4)

plot of chunk unnamed-chunk-15 plot of chunk unnamed-chunk-15 plot of chunk unnamed-chunk-15

Export the results in a PDF file

pdf(paste0(uc,"_EGA.pdf"))
 print(plot3)
 print(plot4)
dev.off()
## png 
##   2

Model Development

ESTIMATE model parameters using Monolix

The ddmore “estimate” function translates the contents of the .mdl file to a target language and then estimates parameters using the target software. After estimation, the output is converted to a Standardised Output object which is saved in a .SO.xml file.

Translated files and Monolix output will be returned in the ./Monolix subfolder. The Standardised Output object (.SO.xml) is read and parsed into an R object called “mlx” of (S4) class “StandardOutputObject”.

mlx <- estimate(mdlfile, target="MONOLIX", subfolder="Monolix")
## -- Tue Aug 16 17:48:54 2016
## New
## Submitted
## Job 19773e24-a8b9-4d6c-ae14-619e0c7d65c0 progress:
## Running [ ...... ]
## Importing Results
## Copying the result data back to the local machine for job ID 19773e24-a8b9-4d6c-ae14-619e0c7d65c0...
## From C:\Users\zparra\AppData\Local\Temp\Rtmpk5tG0t\DDMORE.job1c843c7d1eea to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase3_1/Monolix
## Done.
## 
## 
## The following main elements were parsed successfully:
##   ToolSettings
##   RawResults
##   Estimation::PopulationEstimates::MLE
##   Estimation::PrecisionPopulationEstimates::MLE
##   Estimation::IndividualEstimates::Estimates
##   Estimation::IndividualEstimates::RandomEffects
##   Estimation::Residuals::ResidualTable
##   Estimation::Predictions
##   Estimation::OFMeasures::IndividualContribToLL
##   Estimation::OFMeasures::InformationCriteria
##   Estimation::OFMeasures::LogLikelihood
## 
## Completed
## -- Tue Aug 16 17:50:59 2016
slotNames(mlx)
## [1] "ToolSettings"     "RawResults"       "TaskInformation" 
## [4] "Estimation"       "ModelDiagnostic"  "Simulation"      
## [7] "OptimalDesign"    ".pathToSourceXML"

The ddmore “LoadSOObj” reads and parsed existing Standardise Output Objects

mlx <- LoadSOObject(“Monolix/UseCase3_1.SO.xml”)

The ddmore function “getPopulationParameters” extracts the Population Parameter values from an R object of (S4) class “StandardOutputObject” and returns the estimates. See documentation for getPopulationParameters to see other arguments and settings for this function.

parameters_mlx <- getPopulationParameters(mlx, what="estimates")$MLE
print(parameters_mlx)
## POP_PCA0  POP_TEQ  POP_C50 POP_EMAX PPV_PCA0  PPV_TEQ  PPV_C50 PPV_EMAX 
## 96.22244 13.09847  1.17781  1.00000  0.05236  0.10754  0.43963  0.00000 
##   RUV_FX 
##  3.75425
print(getPopulationParameters(mlx, what="precisions"))
## $MLE
##   Parameter      MLE      SE   RSE
## 1   POP_C50  1.17781 0.09410  7.99
## 2  POP_EMAX  1.00000 0.00000  0.00
## 3  POP_PCA0 96.22244 1.08568  1.13
## 4   POP_TEQ 13.09847 0.39148  2.99
## 5   PPV_C50  0.43963 0.05787 13.16
## 6  PPV_EMAX  0.00000 0.00000  0.00
## 7  PPV_PCA0  0.05236 0.00972 18.57
## 8   PPV_TEQ  0.10754 0.03035 28.22
## 9    RUV_FX  3.75425 0.22135  5.90
print(getEstimationInfo(mlx))
## $OFMeasures
## $OFMeasures$LogLikelihood
## $OFMeasures$LogLikelihood[[1]]
## [1] -722.45
## 
## 
## $OFMeasures$IndividualContribToLL
##    Subject ICtoLL
## 1        1 -24.98
## 2        2 -20.82
## 3        3 -31.60
## 4        4 -23.18
## 5        5 -25.86
## 6        6 -22.88
## 7        7 -24.16
## 8        8 -23.44
## 9        9 -24.56
## 10      10 -23.06
## 11      12 -22.97
## 12      13 -16.06
## 13      14 -23.37
## 14      15 -24.65
## 15      16 -22.71
## 16      17 -24.19
## 17      18 -19.90
## 18      19 -19.54
## 19      20 -21.70
## 20      21 -20.15
## 21      22 -24.54
## 22      23 -22.63
## 23      24 -18.99
## 24      25 -30.98
## 25      26 -19.10
## 26      27 -19.06
## 27      28 -23.09
## 28      29 -19.21
## 29      30 -22.19
## 30      31 -20.78
## 31      32 -20.85
## 32      33 -21.23
## 
## $OFMeasures$InformationCriteria
## $OFMeasures$InformationCriteria$AIC
## [1] 1458.9
## 
## $OFMeasures$InformationCriteria$BIC
## [1] 1469.16
## 
## 
## 
## $Messages
## list()

Perform model diagnostics for the base model using Xpose functions (graphs are exported to PDF)

There is currently a bug with as.xpdb and Monolix in UseCase3_1. Therefore, goodness-of-fit plots are created manually from the standardised output. Later in the script, Xpose functionality on UseCase3 is shown for NONMEM.

Use 'ddmore' function as.xpdb() to create an Xpose database object from the standardised output object, regardless of target software used for estimation. Users can then call xpose functions directly.

 mlx.xpdb<-as.xpdb(mlx,datafile)
## 
## Removed dose rows in rawData slot of SO to enable merge with Predictions data.

We can then call Xpose functions referencing this mlx.xpdb object as the input. Perform some basic goodness of fit (graphs are exported to PDF file)

pdf("GOF_MLX.pdf")
basic.gof(mlx.xpdb,by="DVID",subset="DVID==2")
ind.plots(mlx.xpdb,subset="DVID==2")
dev.off()
## png 
##   2
myPredictions <- mlx@Estimation@Predictions@data
myPredictions <- apply(myPredictions,2,as.numeric)

myXPDB <- merge(myEDAData, myPredictions) 
plot(ggplot() + ggtitle("PCA level") +
                geom_abline() +
                geom_point(data=myXPDB[myXPDB$DVID==2,], aes(x=PRED, y=DV)) +
                xlab("Population predictions") + ylab("Observations") )

plot of chunk GOFforMonolixEstimation

plot(ggplot() + ggtitle("PCA level") +
                geom_abline() +
                geom_point(data=myXPDB[myXPDB$DVID==2,], aes(x=IPRED, y=DV)) +
                xlab("Individual predictions") + ylab("Observations") )

plot of chunk GOFforMonolixEstimation

Change estimation method to FOCEI (for speed)

MDL Objects can be manipulated from R to change for example the estimation algorithm

myTaskProperties <- getTaskPropertiesObjects(mdlfile)[[1]]
myNewTaskProperties <- myTaskProperties
myNewTaskProperties@ESTIMATE$algo <- "focei"

Assembling the new MOG. Note that we reuse the data and model from the previous run.

myNewerMOG <- createMogObj(dataObj = getDataObjects(mdlfile)[[1]], 
        parObj = getParameterObjects(mdlfile)[[1]], 
        mdlObj = getModelObjects(mdlfile)[[1]], 
        taskObj = myNewTaskProperties)

We can then write the MOG back out to an .mdl file.

mdlfile.FOCEI <- paste0(uc,"_FOCEI.mdl")
writeMogObj(myNewerMOG,mdlfile.FOCEI)

Test estimation using this new MOG.

NM.FOCEI <- estimate(mdlfile.FOCEI, target="PsN", subfolder="NONMEM_FOCEI")
## -- Tue Aug 16 17:51:17 2016
## New
## Submitted
## Job a0b8c2a7-e948-4bf9-af6b-d7d1dd022b4e progress:
## Running [ ......... ]
## Importing Results
## Copying the result data back to the local machine for job ID a0b8c2a7-e948-4bf9-af6b-d7d1dd022b4e...
## From C:\Users\zparra\AppData\Local\Temp\Rtmpk5tG0t\DDMORE.job1c8477a27221 to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase3_1/NONMEM_FOCEI
## Done.
## 
## 
## The following main elements were parsed successfully:
##   RawResults
##   Estimation::PopulationEstimates::MLE
##   Estimation::IndividualEstimates::Estimates
##   Estimation::IndividualEstimates::RandomEffects
##   Estimation::Residuals::ResidualTable
##   Estimation::Predictions
##   Estimation::OFMeasures::Deviance
## 
## The following MESSAGEs were raised during the job execution:
##  estimation_successful: 1
##  covariance_step_run: 0
##  rounding_errors: 0
##  hessian_reset: 0
##  zero_gradients: 0
##  final_zero_gradients: 0
##  estimate_near_boundary: 0
##  s_matrix_singular: 0
##  significant_digits: 3.8
##  nmoutput2so_version: This SOBlock was created with nmoutput2so version 4.5.27
## 
## Completed
## -- Tue Aug 16 17:54:22 2016

Load previous results

NM.FOCEI <- LoadSOObject(“NONMEM_FOCEI/UseCase3_1_FOCEI.SO.xml”)

Results from NONMEM should be comparable to previous results

print(getPopulationParameters(NM.FOCEI,  what="estimates"))
## $MLE
##   POP_PCA0    POP_C50    POP_TEQ     RUV_FX   POP_EMAX   PPV_PCA0 
## 96.6348000  1.1755700 13.0066000  3.7638300  1.0000000  0.0532667 
##   PPV_EMAX    PPV_C50    PPV_TEQ 
##  0.0000000  0.4394390  0.1015090

VPC of model

Before running the VPC with PsN we must update the (initial) values in the MDL Parameter Object MLE estimates from previous step can be used

structuralPar <- getPopulationParameters(NM.FOCEI, what="estimates",block='structural')$MLE
variabilityPar <- getPopulationParameters(NM.FOCEI, what="estimates",block='variability')$MLE

Update the parameter object using the ddmore “updateParObj” function. This function updates an R object of (S4) class “parObj”. The user chooses which block to update, what items within that block, and what to replace those items with. NOTE: that updateParObj can only update attributes which ALREADY EXIST in the MDL Parameter Object for that item. This ensures that valid MDL is preserved.

myParObj <- getParameterObjects(mdlfile)[[1]]
myParObjUpdated <- updateParObj(myParObj,block="STRUCTURAL",
        item=names(structuralPar),
        with=list(value=structuralPar))
myParObjUpdated <- updateParObj(myParObjUpdated,block="VARIABILITY",
        item=names(variabilityPar),
        with=list(value=variabilityPar))

Check that the appropriate initial values have been updated to the MLE values from the previous fit.

print(myParObjUpdated@STRUCTURAL)
## $POP_PCA0
## $POP_PCA0$value
## [1] "96.6348"
## 
## $POP_PCA0$lo
## [1] "0.01"
## 
## $POP_PCA0$hi
## [1] "200"
## 
## 
## $POP_EMAX
## $POP_EMAX$value
## [1] "1"
## 
## $POP_EMAX$lo
## [1] "0"
## 
## $POP_EMAX$fix
## [1] "true"
## 
## 
## $POP_C50
## $POP_C50$value
## [1] "1.17557"
## 
## $POP_C50$lo
## [1] "0.01"
## 
## $POP_C50$hi
## [1] "10"
## 
## 
## $POP_TEQ
## $POP_TEQ$value
## [1] "13.0066"
## 
## $POP_TEQ$lo
## [1] "0.01"
## 
## $POP_TEQ$hi
## [1] "100"
## 
## 
## $RUV_FX
## $RUV_FX$value
## [1] "3.76383"
## 
## $RUV_FX$lo
## [1] "0"
print(myParObjUpdated@VARIABILITY)
## $PPV_PCA0
## $PPV_PCA0$value
## [1] "0.0532667"
## 
## $PPV_PCA0$type
## [1] "sd"
## 
## 
## $PPV_EMAX
## $PPV_EMAX$value
## [1] "0"
## 
## $PPV_EMAX$type
## [1] "sd"
## 
## $PPV_EMAX$fix
## [1] "true"
## 
## 
## $PPV_C50
## $PPV_C50$value
## [1] "0.439439"
## 
## $PPV_C50$type
## [1] "sd"
## 
## 
## $PPV_TEQ
## $PPV_TEQ$value
## [1] "0.101509"
## 
## $PPV_TEQ$type
## [1] "sd"

Assembling the new MOG. Note that we reuse the data, model and tasks from the previous run.

myVPCMOG <- createMogObj(dataObj = getDataObjects(mdlfile)[[1]], 
        parObj = myParObjUpdated, 
        mdlObj = getModelObjects(mdlfile)[[1]], 
        taskObj = getTaskPropertiesObjects(mdlfile)[[1]])

We can then write the MOG back out to an .mdl file.

mdlfile.VPC <- paste0(uc,"_VPC.mdl")
writeMogObj(myVPCMOG,mdlfile.VPC)

Similarly as above, ddmore “VPC.PsN” function can be used to run a VPC using PsN as target tool

vpcFiles <- VPC.PsN(mdlfile.VPC,samples=20, seed=12345,
        vpcOptions ="-n_simulation=10 -auto_bin=5,8",
        subfolder="VPC", plot=FALSE) 
## -- Tue Aug 16 17:54:38 2016
## New
## Submitted
## Job 63de699a-a129-48d0-8874-b25a22546bcb progress:
## Running [ ..... ]
## Importing Results
## Copying the result data back to the local machine for job ID 63de699a-a129-48d0-8874-b25a22546bcb...
## From C:\Users\zparra\AppData\Local\Temp\Rtmpk5tG0t\DDMORE.job1c846d0050f3 to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase3_1/VPC
## Done.
## 
## 
## The following main elements were parsed successfully:
##   RawResults
##   SimulationSimulationBlock
##   SimulationSimulationBlock
## 
## The following MESSAGEs were raised during the job execution:
##  nmoutput2so_version: This SOBlock was created with nmoutput2so version 4.5.27
## 
## Completed
## -- Tue Aug 16 17:56:24 2016

To replay the visualisation using information from the VPC SO file

pdf(paste0(uc,"_VPC.pdf"))
 print(xpose.VPC(vpc.info= file.path("./VPC",vpcFiles@RawResults@DataFiles$PsN_VPC_results$path),
        vpctab= file.path("./VPC",vpcFiles@RawResults@DataFiles$PsN_VPC_vpctab$path),
        main="VPC warfarin PK/PD"))
dev.off()
## png 
##   2

Simulation using simulx

The mlxR package has been developed to visualize and explore models that are encoded in MLXTRAN or PharmML. The ddmore function as.PharmML translates an MDL file (extension .mdl) to its PharmML representation. The output file (extension .xml) is saved in the working directory.

myPharmML <- as.PharmML(mdlfile)

Use parameter values from the Monolix estimation

parValues <- getPopulationParameters(mlx, what="estimates")$MLE

Simulate for the typical weight of 70. Recall that logtWT = log(WT/70).

p <- c(parValues,WT=70,CL=0.139,V=8,KA=2.64,TLAG=0.976)

Parameter values used in simulation

print(p)
## POP_PCA0  POP_TEQ  POP_C50 POP_EMAX PPV_PCA0  PPV_TEQ  PPV_C50 PPV_EMAX 
## 96.22244 13.09847  1.17781  1.00000  0.05236  0.10754  0.43963  0.00000 
##   RUV_FX       WT       CL        V       KA     TLAG 
##  3.75425 70.00000  0.13900  8.00000  2.64000  0.97600

Simulate for a dose of 100mg given at time 0 into the GUT (oral administration)

adm <- list( time = 0, amount = 100)

Simulate PK parameters for individuals

ind <- list(name = c('TEQ','C50','PCA0','EMAX'))

Simulate predicted (CC) and observed concentration values (CP_obs), predicted (PCA) and observed PCA (PCA_obs)

f2   <- list( name = c('PCA'), time = seq(0,to=50,by=1))
y2   <- list( name = c('PCA_obs'), time = c(0, 0.5, 1, 2, 3, 4, 6, 8, 12, 24, 36, 48))

Simulate 12 subjects

g <- list( size = 12, level = 'individual',  treatment = adm)

Call simulx

res  <- simulx(model = myPharmML,
               parameter = p,
               group = g,
               output = list(ind,f2,y2))

Simulated parameter values for each individual

print(res$parameter)
##    id      TEQ       C50      PCA0 EMAX
## 1   1 13.13711 0.5850570  88.89440    1
## 2   2 12.63256 1.3696278  91.58011    1
## 3   3 13.13277 0.9444983  95.79183    1
## 4   4 12.17334 1.1199062  90.52653    1
## 5   5 13.42784 2.0796638 102.52597    1
## 6   6 14.15351 1.4592066  97.26957    1
## 7   7 13.23981 1.1176807  98.10918    1
## 8   8 13.49183 1.6380987  96.40673    1
## 9   9 13.13777 2.2158749 100.02220    1
## 10 10 14.74174 1.6561927 107.19870    1
## 11 11 14.05558 1.5601296 100.13338    1
## 12 12 13.30317 0.7213143  93.52413    1

Plot simulated results

plot(ggplot() + 
                geom_line(data=res$PCA, aes(x=time, y=PCA, colour=id)) +
                geom_point(data=res$PCA_obs, aes(x=time, y=PCA_obs, colour=id)) +
                xlab("time (h)") + ylab("PCA") )

plot of chunk unnamed-chunk-37

Simulate 1000 subjects - with simulx this is a QUICK process!

g <- list( size = 1000, level = 'individual',  treatment = adm)

Call simulx

res.1000  <- simulx(model =myPharmML,
                    parameter = p,
                    group = g,
                    output = list(ind,f2,y2))

Plot of predicted concentrations band defines the percentile bands displayed level = range of values to examine (in %) 100 = full range of values number = number of bins within the level range. Plot of observed PCA levels (with residual error)

prctilemlx(res.1000$PCA_obs,band=list(number=9, level=90))

plot of chunk unnamed-chunk-39

Table of the same information

prctilemlx(res.1000$PCA_obs,band=list(number=10, level=100), plot=F)$y
##    time       0%      10%      20%      30%      40%      50%      50%
## 1   0.0 80.80782 87.71609 90.62009 93.20755 94.70615 96.82203 96.82203
## 2   0.5 81.79883 88.29684 90.62239 92.72910 94.04895 96.93432 96.93432
## 3   1.0 82.19491 88.44517 90.80083 93.84492 95.09661 96.44032 96.44032
## 4   2.0 79.92890 85.66292 87.62960 89.18685 90.64459 92.33912 92.33912
## 5   3.0 72.67165 81.46284 84.46590 85.80091 87.62213 88.75165 88.75165
## 6   4.0 72.90332 77.39289 80.18231 81.42480 82.54670 84.29300 84.29300
## 7   6.0 63.21221 69.56848 71.62519 73.77040 74.83148 76.25319 76.25319
## 8   8.0 53.12278 63.62336 65.43826 67.21999 68.72875 70.32720 70.32720
## 9  12.0 43.31475 50.51841 52.78452 54.48037 56.68720 58.93206 58.93206
## 10 24.0 24.55448 28.92819 30.10429 32.10619 34.53137 36.17932 36.17932
## 11 36.0 12.03267 17.27569 18.58139 21.60372 24.43431 25.86370 25.86370
## 12 48.0  5.01268 13.67069 15.85161 17.55724 19.21846 20.55666 20.55666
##         60%       70%       80%       90%      100%
## 1  98.01844  99.00440 101.75152 105.09070 110.92809
## 2  98.59772  99.79860 101.52452 104.13297 115.22040
## 3  97.93200 100.13965 102.57501 104.98236 111.90698
## 4  93.66400  95.40340  98.42196  99.85229 108.71710
## 5  89.73239  91.16822  93.04803  95.02154  99.61950
## 6  85.60058  86.71709  88.79145  90.63862  97.11764
## 7  77.52433  78.92731  81.28792  85.38052  89.25445
## 8  71.30379  72.59227  73.98983  76.71863  85.31144
## 9  60.45358  61.99947  64.69160  66.96287  75.06890
## 10 37.56840  38.63930  41.08998  44.06710  56.63046
## 11 27.71709  29.37971  32.69246  36.59667  44.33175
## 12 22.19428  24.72106  27.76607  31.25445  38.38310