UseCase4_2 : Warfarin population pharmacokinetic model for oral and infusion administration routes

Variant on UseCase4 : Using depot and direct COMPARTMENTS to define input process alongside ODEs

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 'UsesCasesDemo' project

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

Create a working directory under 'models' folder where results are stored

uc<-"UseCase4_2"
datafile <- "warfarin_infusion_oral.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)) 

List files available in working directory

list.files()
## [1] "UseCase4_2.mdl"             "warfarin_infusion_oral.csv"

Model Development

ESTIMATE model parameters using Monolix

mlx <- estimate(mdlfile, target="MONOLIX", subfolder="Monolix")
## -- Wed Aug 17 18:45:50 2016
## New
## Submitted
## Job 8da60e4c-ed3e-4853-b010-37e4a56a494c progress:
## Running [ ........ ]
## Importing Results
## Copying the result data back to the local machine for job ID 8da60e4c-ed3e-4853-b010-37e4a56a494c...
## From C:\Users\zparra\AppData\Local\Temp\RtmpSKgKTU\DDMORE.job1dec502542b to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase4_2/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
## -- Wed Aug 17 18:48:38 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/UseCase4_2.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")
print(parameters_mlx)
## $MLE
##      POP_V  BETA_V_WT     POP_KA     POP_CL BETA_CL_WT   POP_TLAG 
##    7.98883    1.00000    0.35758    0.10082    0.75000    1.00247 
##  POP_FORAL      PPV_V     PPV_KA     PPV_CL   PPV_TLAG  PPV_FORAL 
##    0.99160    0.12470    0.07619    0.11930    0.10000    0.32991 
##  CORR_CL_V    RUV_ADD   RUV_PROP 
##    0.03330    0.00000    0.12281
print(getPopulationParameters(mlx, what="precisions"))
## $MLE
##     Parameter     MLE       SE        RSE
## 1  BETA_CL_WT 0.75000  0.00000       0.00
## 2   BETA_V_WT 1.00000  0.00000       0.00
## 3   CORR_CL_V 0.03330  0.18907     567.86
## 4      POP_CL 0.10082  0.00222       2.21
## 5   POP_FORAL 0.99160  0.01030       1.04
## 6      POP_KA 0.35758  0.01258       3.52
## 7    POP_TLAG 1.00247  0.02850       2.84
## 8       POP_V 7.98883  0.18518       2.32
## 9      PPV_CL 0.11930  0.01583      13.27
## 10  PPV_FORAL 0.32991 16.10903    4882.78
## 11     PPV_KA 0.07619  0.04729      62.07
## 12   PPV_TLAG 0.10000  0.00000       0.00
## 13      PPV_V 0.12470  0.01684      13.51
## 14    RUV_ADD 0.00000  0.02167 5739954.00
## 15   RUV_PROP 0.12281  0.00458       3.73
print(getEstimationInfo(mlx))
## $OFMeasures
## $OFMeasures$LogLikelihood
## $OFMeasures$LogLikelihood[[1]]
## [1] -1336.86
## 
## 
## $OFMeasures$IndividualContribToLL
##    Subject ICtoLL
## 1        1 -34.19
## 2        2 -36.09
## 3        3 -27.25
## 4        4 -46.05
## 5        5 -43.52
## 6        6 -46.11
## 7        7 -37.45
## 8        8 -42.92
## 9        9 -34.60
## 10      10 -59.68
## 11      11 -42.45
## 12      12 -41.58
## 13      13 -49.62
## 14      14 -44.32
## 15      15 -48.82
## 16      16 -42.70
## 17      17 -39.41
## 18      18 -45.19
## 19      19 -41.56
## 20      20 -46.66
## 21      21 -40.62
## 22      22 -36.42
## 23      23 -51.37
## 24      24 -45.40
## 25      25 -43.13
## 26      26 -45.22
## 27      27 -39.98
## 28      28 -37.67
## 29      29 -32.96
## 30      30 -33.57
## 31      31 -45.91
## 32      32 -34.42
## 
## $OFMeasures$InformationCriteria
## $OFMeasures$InformationCriteria$AIC
## [1] 2697.72
## 
## $OFMeasures$InformationCriteria$BIC
## [1] 2715.31
## 
## 
## 
## $Messages
## list()

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

Use 'ddmore' function as.xpdb() to create an Xpose database object from the Standardised Output object, regardless of target software used for estimation.

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

print(basic.gof(mlx.xpdb))

plot of chunk unnamed-chunk-10

print(ind.plots(mlx.xpdb))

plot of chunk unnamed-chunk-10 plot of chunk unnamed-chunk-10

Export results to PDF file

pdf("GOF_MLX.pdf")
print(basic.gof(mlx.xpdb))
print(ind.plots(mlx.xpdb))
dev.off()
## rj.GD 
##     2

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")
## -- Wed Aug 17 18:49:02 2016
## New
## Submitted
## Job 65bb9437-f27f-4e93-8368-6e4c36ba08e4 progress:
## Running [ ................. ]
## Importing Results
## Copying the result data back to the local machine for job ID 65bb9437-f27f-4e93-8368-6e4c36ba08e4...
## From C:\Users\zparra\AppData\Local\Temp\RtmpSKgKTU\DDMORE.job1dec586019ac to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase4_2/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 WARNINGs were raised during the job execution:
##  estimation_successful: 0
##  zero_gradients: 1
##  final_zero_gradients: 1
##  estimate_near_boundary: 1
## 
## The following MESSAGEs were raised during the job execution:
##  covariance_step_run: 0
##  rounding_errors: 0
##  hessian_reset: 0
##  s_matrix_singular: 0
##  nmoutput2so_version: This SOBlock was created with nmoutput2so version 4.5.27
## 
## Completed
## -- Wed Aug 17 18:54:51 2016

Load previous results

# NM.FOCEI <- LoadSOObject("NONMEM_FOCEI/UseCase4_1_FOCEI.SO.xml")

Results from NONMEM should be comparable to previous results

print(getPopulationParameters(NM.FOCEI,  what="estimates"))
## $MLE
##     POP_CL      POP_V     POP_KA   POP_TLAG  POP_FORAL   RUV_PROP 
## 0.10009300 7.93231000 0.34186300 0.95012900 0.99999000 0.12169300 
##    RUV_ADD BETA_CL_WT  BETA_V_WT     PPV_CL  CORR_CL_V      PPV_V 
## 0.00109900 0.75000000 1.00000000 0.12223000 0.00838324 0.12313400 
##     PPV_KA   PPV_TLAG  PPV_FORAL 
## 0.09522580 0.10000000 0.12487700
print(parameters_mlx)
## $MLE
##      POP_V  BETA_V_WT     POP_KA     POP_CL BETA_CL_WT   POP_TLAG 
##    7.98883    1.00000    0.35758    0.10082    0.75000    1.00247 
##  POP_FORAL      PPV_V     PPV_KA     PPV_CL   PPV_TLAG  PPV_FORAL 
##    0.99160    0.12470    0.07619    0.11930    0.10000    0.32991 
##  CORR_CL_V    RUV_ADD   RUV_PROP 
##    0.03330    0.00000    0.12281

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

Use 'ddmore' function as.xpdb() to create an Xpose database object from the Standard Output object, regardless of target software used for estimation.

nm.xpdb<-as.xpdb(NM.FOCEI,datafile)
## 
## Removed dose rows in rawData+Predictions slot of SO to enable merge with Residuals data.
## 
## Residuals data does not currently contain dose rows in output from Nonmem executions.

We can then call Xpose functions referencing this mlx.xpdb object as the input. Perform some basic goodness of fit

print(basic.gof(nm.xpdb))

plot of chunk unnamed-chunk-17

print(ind.plots(nm.xpdb))

plot of chunk unnamed-chunk-17 plot of chunk unnamed-chunk-17

Export results to PDF file

pdf("GOF_NM.FOCEI.pdf")
print(basic.gof(nm.xpdb))
print(ind.plots(nm.xpdb))
dev.off()
## rj.GD 
##     2

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_CL
## $POP_CL$value
## [1] "0.100093"
## 
## $POP_CL$lo
## [1] "0.001"
## 
## 
## $POP_V
## $POP_V$value
## [1] "7.93231"
## 
## $POP_V$lo
## [1] "0.001"
## 
## 
## $POP_KA
## $POP_KA$value
## [1] "0.341863"
## 
## $POP_KA$lo
## [1] "0.001"
## 
## 
## $POP_TLAG
## $POP_TLAG$value
## [1] "0.950129"
## 
## $POP_TLAG$lo
## [1] "0.001"
## 
## 
## $BETA_CL_WT
## $BETA_CL_WT$value
## [1] "0.75"
## 
## $BETA_CL_WT$fix
## [1] "true"
## 
## 
## $BETA_V_WT
## $BETA_V_WT$value
## [1] "1"
## 
## $BETA_V_WT$fix
## [1] "true"
## 
## 
## $POP_FORAL
## $POP_FORAL$value
## [1] "0.99999"
## 
## $POP_FORAL$lo
## [1] "0.001"
print(myParObjUpdated@VARIABILITY)
## $PPV_CL
## $PPV_CL$value
## [1] "0.12223"
## 
## 
## $PPV_V
## $PPV_V$value
## [1] "0.123134"
## 
## 
## $PPV_KA
## $PPV_KA$value
## [1] "0.0952258"
## 
## 
## $PPV_TLAG
## $PPV_TLAG$value
## [1] "0.1"
## 
## $PPV_TLAG$fix
## [1] "true"
## 
## 
## $PPV_FORAL
## $PPV_FORAL$value
## [1] "0.124877"
## 
## 
## $CORR_CL_V
## $CORR_CL_V$value
## [1] "0.00838324"
## 
## 
## $RUV_PROP
## $RUV_PROP$value
## [1] "0.121693"
## 
## $RUV_PROP$lo
## [1] "0"
## 
## 
## $RUV_ADD
## $RUV_ADD$value
## [1] "0.001099"
## 
## $RUV_ADD$lo
## [1] "1.0E-4"

Assembling the new MOG. Note that we reuse the data and model 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",
        subfolder="VPC", plot=TRUE) 
## -- Wed Aug 17 18:55:23 2016
## New
## Submitted
## Job ff0dd2e0-2d7d-47c2-85a3-7b81067e1282 progress:
## Running [ .... ]
## Importing Results
## Copying the result data back to the local machine for job ID ff0dd2e0-2d7d-47c2-85a3-7b81067e1282...
## From C:\Users\zparra\AppData\Local\Temp\RtmpSKgKTU\DDMORE.job1dec64a12e9c to D:/SEE-Prod5_RC4/MDL_IDE/workspace/UseCasesDemo/UseCase4_2/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
## -- Wed Aug 17 18:56:49 2016

plot of chunk VPC

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"))
dev.off()
## rj.GD 
##     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 FOCEI estimation

parValues <- getPopulationParameters(NM.FOCEI,what="estimates")$MLE

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

p <- c(parValues,logtWT=0)

Parameter values used in simulation

print(p) 
##     POP_CL      POP_V     POP_KA   POP_TLAG  POP_FORAL   RUV_PROP 
## 0.10009300 7.93231000 0.34186300 0.95012900 0.99999000 0.12169300 
##    RUV_ADD BETA_CL_WT  BETA_V_WT     PPV_CL  CORR_CL_V      PPV_V 
## 0.00109900 0.75000000 1.00000000 0.12223000 0.00838324 0.12313400 
##     PPV_KA   PPV_TLAG  PPV_FORAL     logtWT 
## 0.09522580 0.10000000 0.12487700 0.00000000

Simulate for a dose of 100mg given at time 0 into the CENTRAL (iv administration) and a dose of 150 mg given to the CENTRAL (iv dose). Note that we are using COMPARMTMENTS for dosing processes, which translate to PK macros, and therefore the type option needs to be used

adm1 <- list(type=2, time = 0, amount=100, rate=100) #iv dose
adm2 <- list(type=1, time = 168, amount=150) #oral dose

Simulate PK parameters for individuals

ind <- list(name = c('CL','V'))

Simulate predicted (CC) and observed concentration values (Y)

f   <- list( name = c('CC'), time = seq(0,to=300,by=1))
y   <- list( name = c('Y'), time = c(0, 0.5, 1, 4, 8, 12, 24, 36, 48,120,
                168,168.5,170,171,174,180,192,216,240,288))

Simulate 12 subjects

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

Call simulx

res  <- simulx(model = myPharmML,
        parameter = p,
        group = g,
        treatment = list(adm1,adm2),
        output = list(ind,f,y))

Simulated parameter values for each individual

print(res$parameter)
##    id         CL        V
## 1   1 0.11853300 7.693147
## 2   2 0.10916421 9.813362
## 3   3 0.11118152 7.440567
## 4   4 0.09540063 9.133591
## 5   5 0.09409975 6.327528
## 6   6 0.08808842 8.739982
## 7   7 0.12273784 9.646295
## 8   8 0.10070634 8.396366
## 9   9 0.09241264 9.403188
## 10 10 0.10417163 8.261747
## 11 11 0.11043235 8.509011
## 12 12 0.08736465 9.016890

Plot simulated results

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

plot of chunk unnamed-chunk-33

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

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

Call simulx

res.1000  <- simulx(model = myPharmML,
        parameter = p,
        group = g,
        treatment = list(adm1,adm2),
        output = list(ind,f,y))

Plot prediction intervals with prctilemlx. band defines the percentile bands displayed:

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

plot of chunk unnamed-chunk-35

Table of the same information

print(prctilemlx(res.1000$Y,band=list(number=10, level=100), plot=FALSE)$y)
##     time           0%          10%           20%           30%
## 1    0.0 -0.002434415 -0.001230859 -0.0006907394 -0.0004200868
## 2    0.5  3.699567360  4.784231583  5.1355278486  5.4656484887
## 3    1.0  8.462059863 10.406529370 10.9088929504 11.4284374639
## 4    4.0  8.026207746  9.575067110 10.4277315079 10.8513311650
## 5    8.0  7.824279559  9.457514508 10.0007730487 10.5010741736
## 6   12.0  6.707232305  8.636643943  9.3234437852  9.8019436809
## 7   24.0  6.760092899  7.730160768  8.1700421459  8.6205323158
## 8   36.0  5.331910727  6.472699351  7.0618593624  7.4557446406
## 9   48.0  4.310573071  5.258459931  5.9069358460  6.2129201131
## 10 120.0  1.547898477  2.073685943  2.2643115922  2.3826372314
## 11 168.0  0.796432423  0.953279896  1.2323513297  1.3593985668
## 12 168.5  0.767094852  0.993310769  1.2002944416  1.3249348729
## 13 170.0  5.042010982  5.562566754  5.9319510059  6.2414841111
## 14 171.0  6.799678398  8.670007058  9.3548631146  9.8382048598
## 15 174.0  8.980946344 13.097397490 14.0512572372 14.8467978707
## 16 180.0 11.553979374 14.690414285 15.2554575662 16.2787899935
## 17 192.0  9.587160280 12.627161548 13.9863829515 14.6720984542
## 18 216.0  7.641433237  9.635061483 10.1512436882 10.6384929135
## 19 240.0  4.859896150  6.940022392  7.2077916567  7.5285414347
## 20 288.0  2.622786107  3.523439683  3.9545377254  4.2374482747
##              40%           50%           50%          60%          70%
## 1  -0.0002055521 -7.926177e-05 -7.926177e-05 1.875618e-04 3.486082e-04
## 2   5.7719594447  5.986546e+00  5.986546e+00 6.247543e+00 6.699697e+00
## 3  11.8659035194  1.243564e+01  1.243564e+01 1.289813e+01 1.336204e+01
## 4  11.3723207559  1.160449e+01  1.160449e+01 1.195840e+01 1.256296e+01
## 5  11.0711281912  1.135717e+01  1.135717e+01 1.177366e+01 1.231765e+01
## 6  10.2622624180  1.079313e+01  1.079313e+01 1.116577e+01 1.154274e+01
## 7   8.9213350902  9.180900e+00  9.180900e+00 9.446969e+00 9.649599e+00
## 8   7.6614303399  7.908182e+00  7.908182e+00 8.205122e+00 8.686644e+00
## 9   6.3658616160  6.738689e+00  6.738689e+00 7.022017e+00 7.322612e+00
## 10  2.6607556809  2.863498e+00  2.863498e+00 2.959787e+00 3.104504e+00
## 11  1.4332180595  1.585081e+00  1.585081e+00 1.666304e+00 1.779183e+00
## 12  1.4340302475  1.493569e+00  1.493569e+00 1.580194e+00 1.687368e+00
## 13  6.6033727749  6.967273e+00  6.967273e+00 7.267045e+00 7.570171e+00
## 14 10.1927343290  1.046221e+01  1.046221e+01 1.098386e+01 1.148128e+01
## 15 15.4502120401  1.590675e+01  1.590675e+01 1.677786e+01 1.758867e+01
## 16 17.0823721805  1.779480e+01  1.779480e+01 1.829466e+01 1.894428e+01
## 17 15.1469564504  1.543846e+01  1.543846e+01 1.586982e+01 1.655826e+01
## 18 11.1503343917  1.148210e+01  1.148210e+01 1.189499e+01 1.251754e+01
## 19  8.0249442868  8.525774e+00  8.525774e+00 8.793255e+00 8.993150e+00
## 20  4.4043229874  4.616958e+00  4.616958e+00 4.890336e+00 5.103552e+00
##             80%          90%         100%
## 1  6.956913e-04  0.001552201  0.002077781
## 2  7.071120e+00  7.626427072  9.808802544
## 3  1.432685e+01 15.214651331 17.533738973
## 4  1.329814e+01 14.231605285 17.246800233
## 5  1.274449e+01 13.257028039 15.272249042
## 6  1.199748e+01 12.739487338 14.849527911
## 7  1.026644e+01 10.924402698 12.362065266
## 8  9.065670e+00  9.444225492 11.789462295
## 9  7.602882e+00  8.066878872  9.142123534
## 10 3.349665e+00  3.613078731  4.442882123
## 11 1.910945e+00  2.290155220  3.012997266
## 12 1.885443e+00  2.120144032  2.680734749
## 13 8.084254e+00  8.475212247 10.111259774
## 14 1.198187e+01 12.884703823 14.789141421
## 15 1.803176e+01 18.739138697 21.665636854
## 16 2.027715e+01 21.416393149 24.859522474
## 17 1.741120e+01 18.135591836 23.290245796
## 18 1.307620e+01 14.077267223 16.458148198
## 19 9.777029e+00 10.450793237 13.218800196
## 20 5.402829e+00  5.997519512  7.805624951