# Pearce et al. (2017): Evaluation of Tissue Partitioning

#### Robert Pearce

#### April 30, 2023

Please send questions to wambaugh.john@epa.gov

from “Evaluation and calibration of high-throughput predictions of chemical distribution to tissues”

Robert G. Pearce, R. Woodrow Setzer, Jimena L. Davis,and John F. Wambaugh

Journal of Pharmacokinetics and Pharmacodynamics volume 44, pages549–565 (2017)

https://doi.org/10.1007/s10928-017-9548-7

## Abstract

Toxicokinetics (TK) provides critical information for integrating chemical toxicity and exposure assessments in order to determine potential chemical risk (i.e., the margin between toxic doses and plausible exposures). For thousands of chemicals that are present in our environment, in vivo TK data are lacking. The publicly available R package “httk” (version 1.8, named for “high throughput TK”) draws from a database of in vitro data and physico-chemical properties in order to run physiologically-based TK (PBTK) models for 553 compounds. The PBTK model parameters include tissue:plasma partition coefficients (Kp) which the httk software predicts using the model of Schmitt (Toxicol In Vitro 22 (2):457–467, 2008). In this paper we evaluated and modified httk predictions, and quantified confidence using in vivo literature data. We used 964 rat Kp measured by in vivo experiments for 143 compounds. Initially, predicted Kp were significantly larger than measured Kp for many lipophilic compounds (log10 octanol:water partition coefficient > 3). Hence the approach for predicting Kp was revised to account for possible deficiencies in the in vitro protein binding assay, and the method for predicting membrane affinity was revised. These changes yielded improvements ranging from a factor of 10 to nearly a factor of 10,000 for 83 Kp across 23 compounds with only 3 Kp worsening by more than a factor of 10. The vast majority (92%) of Kp were predicted within a factor of 10 of the measured value (overall root mean squared error of 0.59 on log10-transformed scale). After applying the adjustments, regressions were performed to calibrate and evaluate the predictions for 12 tissues. Predictions for some tissues (e.g., spleen, bone, gut, lung) were observed to be better than predictions for other tissues (e.g., skin, brain, fat), indicating that confidence in the application of in silico tools to predict chemical partitioning varies depending upon the tissues involved. Our calibrated model was then evaluated using a second data set of human in vivo measurements of volume of distribution (Vss) for 498 compounds reviewed by Obach et al. (Drug Metab Dispos 36(7):1385–1405, 2008). We found that calibration of the model improved performance: a regression of the measured values as a function of the predictions has a slope of 1.03, intercept of − 0.04, and R2 of 0.43. Through careful evaluation of predictive methods for chemical partitioning into tissues, we have improved and calibrated these methods and quantified confidence for TK predictions in humans and rats.

## HTTK Version

This vignette was created with httk v1.6. It was updated to httk v2.2.3 in April of 2023. Although we attempt to maintain backward compatibility, if you encounter issues with the latest release of httk and cannot easily address the changes, historical versions of httk are available from: https://cran.r-project.org/src/contrib/Archive/httk/

## Prepare for session

R package knitr generates html and PDF documents from this RMarkdown file, Each bit of code that follows is known as a “chunk”. We start by telling knitr how we want our chunks to look.

### Clear the memory

It is a bad idea to let variables and other information from previous R sessions float around, so we first remove everything in the R memory.

    rm(list=ls())

### eval = execute.vignette

If you are using the RMarkdown version of this vignette (extension, .RMD) you will be able to see that several chunks of code in this vignette have the statement “eval = execute.vignette”. The next chunk of code, by default, sets execute.vignette = FALSE. This means that the code is included (and necessary) but was not run when the vignette was built. We do this because some steps require extensive computing time and the checks on CRAN limit how long we can spend building the package. If you want this vignette to work, you must run all code, either by cutting and pasting it into R. Or, if viewing the .RMD file, you can either change execute.vignette to TRUE or press “play” (the green arrow) on each chunk in RStudio.

    # Set whether or not the following chunks will be executed (run):
    execute.vignette <- FALSE

### Load the relevant libraries

We use the command ‘library()’ to load various R packages for our analysis. If you get the message “Error in library(X) : there is no package called ‘X’” then you will need to install that package:

From the R command prompt:

install.packages(“X”)

Or, if using RStudio, look for ‘Install Packages’ under ‘Tools’ tab.

    library(httk)
    library(gdata)
    library(ggplot2)
    library(viridis)
    library(censReg)
    library(gmodels)
    library(gplots)
    library(scales)
    library(colorspace)
    library(gridExtra)

### Function to format scientific notation

From https://stackoverflow.com/questions/10762287/how-can-i-format-axis-labels-with-exponents-with-ggplot2-and-scales

    scientific_10 <- function(x) {
      out <- gsub("1e", "10^", scientific_format()(x))
      out <- gsub("\\+","",out)
      out <- gsub("10\\^01","10",out)
      out <- parse(text=gsub("10\\^00","1",out))
    }

We first filter the measured rat Kp data, pc.data. Then the old and new Kp predictions are made, along with error and improvement measures, and these are all consolidated into a table for analysis and plotting. Note that the final table contains log10-transformed values and error and improvements derived from subtracting these values. Only relevant rat values are used. Compounds with Funbound.plasma and partition coefficients of zero are removed as well as compounds with approximated Funbound.plasma values.

    pc.table <- NULL
    pc.data <- subset(pc.data,fu != 0 & Exp_PC != 0 & Tissue %in% c("Adipose","Bone","Brain","Gut",
        "Heart","Kidney","Liver","Lung","Muscle","Skin","Spleen","Blood Cells") &
        tolower(Species) == 'rat' & !CAS %in% c('10457-90-6','5786-21-0','17617-23-1','69-23-8','2898-12-6',
        '57562-99-9','59-99-4','2955-38-6','155-97-5','41903-57-5','58-55-9','77-32-7','59-05-2','60-54-8'))
    cas.list <- get_cheminfo(model='schmitt',species='rat',suppress.messages=TRUE)
    cas.list <-  cas.list[cas.list %in% pc.data[,'CAS']]
    ma.data.list <- subset(chem.physical_and_invitro.data,!is.na(logMA))[,'CAS']
    for(this.cas in cas.list){
      parameters <- parameterize_schmitt(
        chem.cas=this.cas,
        species='rat',
        suppress.messages=TRUE)
      init.parameters <- parameters
      charge <- calc_ionization(
        chem.cas=this.cas,
        pH=7.4)$fraction_charged
      if(!this.cas %in% ma.data.list){
        init.parameters$MA <- 10^(0.999831 - 0.016578*38.7 + 0.881721 * log10(parameters$Pow))
      }
      pcs <- predict_partitioning_schmitt(
        parameters=parameters,
        species='rat',
        regression=FALSE,
        suppress.messages=TRUE
        )
      init.pcs <- predict_partitioning_schmitt(
        parameters=init.parameters,
        species='rat',
        regression=FALSE,
        suppress.messages=TRUE)
      for(this.tissue in subset(pc.data,CAS==this.cas)[,'Tissue']){
        if(this.tissue == 'Blood Cells') this.pc <- 'rbc'
        else this.pc <- this.tissue
            pc.table <- rbind(pc.table,
                          cbind(
                            as.data.frame(this.cas),
                            as.data.frame(this.tissue),
                            as.data.frame(log10(
                              init.pcs[[which(substr(names(init.pcs),
                                2,
                                nchar(names(init.pcs))-3) ==
                                tolower(this.pc))]] *
                              init.parameters$Funbound.plasma)),
                            as.data.frame(log10(
                              pcs[[which(substr(names(pcs),
                                2,
                                nchar(names(pcs))-3) ==
                                tolower(this.pc))]] *
                              parameters$unadjusted.Funbound.plasma)),
                            as.data.frame(log10(
                              init.pcs[[which(substr(names(init.pcs),
                                2,
                                nchar(names(init.pcs))-3) ==
                                tolower(this.pc))]] *
                              init.parameters$unadjusted.Funbound.plasma)),
                            as.data.frame(log10(
                              pcs[[which(substr(names(pcs),
                                2,
                                nchar(names(pcs))-3) == tolower(this.pc))]] *
                                parameters$Funbound.plasma)),
                            as.data.frame(log10(
                              subset(pc.data,
                                CAS==this.cas & Tissue==this.tissue)[,'Exp_PC'])),
                            as.data.frame(subset(pc.data,
                              CAS==this.cas & Tissue==this.tissue)[,'LogP']),
                            as.data.frame(charge),
                            as.data.frame(as.character(subset(pc.data,
                              CAS == this.cas)[1,'A.B.N'])),
                            as.data.frame(subset(pc.data,
                              CAS == this.cas)[1,'fu'])))
      }
    }
    colnames(pc.table) <- c('CAS','Tissue',
                            'fup.correction',
                            'ma.correction',
                            'init.Predicted',
                            'Predicted',
                            'Experimental',
                            'logP',
                            'charge',
                            'type',
                            'fup')
    init.error <- pc.table[,'Experimental'] - pc.table[,'init.Predicted']
    fup.error <- pc.table[,'Experimental'] - pc.table[,'fup.correction']
    ma.error <- pc.table[,'Experimental'] - pc.table[,'ma.correction']
    final.error <- pc.table[,'Experimental'] - pc.table[,'Predicted']
    fup.improvement <- abs(init.error) - abs(fup.error)
    ma.improvement <- abs(init.error) - abs(ma.error)
    final.improvement <- abs(init.error) - abs(final.error)
    pc.table <- cbind(pc.table,fup.improvement,ma.improvement, final.improvement,
                      final.error,init.error,ma.error,fup.error)

    init.plot <- ggplot() +
      geom_point(data=pc.table,aes(10^(init.Predicted),10^(Experimental))) +
      geom_abline() +
      labs(y=expression(paste("Measured ",K[p])),
           x=expression(paste("Predicted ",K[p]))) +
      theme(axis.text=element_text(size=16),axis.title=element_text(size=16),
        plot.title=element_text(size=18,hjust = 0.5)) +
      scale_x_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      scale_y_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      ggtitle('(A)')
    print(init.plot)
    init.stats <- summary(lm(Experimental ~ init.Predicted,
                             data=pc.table))

    final.plot <- ggplot() +
      geom_point(data=pc.table,aes(10^(Predicted),10^(Experimental))) +
      geom_abline() +
      labs(y=expression(paste("Measured ",K[p])),
           x=expression(paste("Predicted ",K[p]))) +
      theme(axis.text=element_text(size=16),axis.title=element_text(size=16),
        plot.title=element_text(size=18,hjust=0.5)) +
      scale_x_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      scale_y_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      ggtitle('(B)')
    print(final.plot)
    final.stats <- summary(lm(Experimental ~ Predicted,
                              data=pc.table))

    fup.change.plot <-  ggplot() +
      geom_point(data=pc.table[order(pc.table[,'fup.improvement'],decreasing=F),],
        aes(10^(fup.correction),10^(Experimental),color=fup.improvement)) +
      geom_abline() +
      labs(y=expression(paste("Measured ",K[p])),
           x=expression(paste("Predicted ",K[p])),color='Improvement') +
      theme(axis.text=element_text(size=16),axis.title=element_text(size=16)) +
      scale_x_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      scale_y_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
      scale_color_viridis(direction=-1,option='inferno')
    print(fup.change.plot)
    fup.stats <- summary(lm(Experimental ~ fup.correction,
                         data=pc.table))

    ma.subset <- subset(pc.table,!CAS %in% ma.data.list)
    ma.change.plot <- ggplot() +
        geom_point(data=ma.subset[order(ma.subset[,'ma.improvement']
          ,decreasing=F),],
          aes(10^(ma.correction),10^(Experimental),color=ma.improvement)) +
        geom_abline() +
        labs(y=expression(paste("Measured ",K[p])),
          x=expression(paste("Predicted ",K[p])),color='Improvement') +
        theme(axis.text=element_text(size=16),axis.title=element_text(size=16)) +
        scale_x_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
        scale_y_log10(label=scientific_10,limits=c(0.01,10^4.5)) +
        scale_color_viridis(direction=-1,option='inferno')
    print(ma.change.plot)
    ma.stats <- summary(lm(Experimental ~ ma.correction,
                        data=ma.subset))

    fup.table <- data.frame(Test=c("Initial Tissue PC Accuracy","Fup Lipid Correction","Membrane Affinity","Final"),
                               RSquared = signif(c(init.stats$adj.r.squared,
                                             fup.stats$adj.r.squared,
                                             ma.stats$adj.r.squared,
                                             final.stats$adj.r.squared),3),
                               RMSLE = signif(c(mean(init.stats$residuals^2,na.rm=TRUE)^(1/2),
                                         mean(fup.stats$residuals^2,na.rm=TRUE)^(1/2),
                                         mean(ma.stats$residuals^2,na.rm=TRUE)^(1/2),
                                         mean(final.stats$residuals^2,na.rm=TRUE)^(1/2)),3)
                              )
    knitr::kable(fup.table)

Now we calculate and plot the regressions for all tissues, together with their 95% confidence intervals.

    regressions <- NULL

    for(tissue in as.character(unique(pc.table[,'Tissue']))){
      fit <- lm(Experimental ~ Predicted ,data=subset(pc.table,Tissue==tissue))
      smry <- summary(fit)
      est <- estimable(fit, cm=diag(2), beta0=c(0,1), joint.test=TRUE)
      regressions <- rbind(regressions,cbind(tissue,as.data.frame(fit$coefficients[['(Intercept)']]),
          as.data.frame(fit$coefficients[['Predicted']]),
          as.data.frame(smry$coefficients[['Predicted','Pr(>|t|)']]),
          as.data.frame(smry$sigma),as.data.frame(smry$r.squared),
          as.data.frame(smry[[11]][1,1]),as.data.frame(smry[[11]][2,2]),
          as.data.frame(smry[[11]][1,2]),as.data.frame(smry$df[2]),as.data.frame(est[[3]])))
    }
    colnames(regressions) <- c('Tissue','Intercept','Slope','P-value','SE','R-squared',
                               'Int Var','Slp Var','Cov','df','estimable')

    for (this.col in 2:10) regressions[,this.col] <- signif(regressions[,this.col], 3)
    regressions <- regressions[order(regressions[,1]),]

    knitr::kable(regressions, caption = "Table 1: The regressions for each tissue, after fup and membrane affinity adjustments, of the log10-transformed measured Kp regressed on
    predicted Kp")

    write.table(regressions,
                file=paste0("Pearce2017PCCalibration=",Sys.Date(),".txt"),
                sep="/t",
                row.names=FALSE)

    x.cf <- seq(-2,3.5,.01)
    for(tissue in  as.character(unique(pc.table[,'Tissue'])))
    {
      conf <- qt(0.975,df=subset(regressions,Tissue==tissue)[['df']]+1) *
          subset(regressions,Tissue==tissue)[['SE']] *
          sqrt(subset(regressions,Tissue==tissue)[['Int Var']] +
          x.cf^2 * subset(regressions,Tissue==tissue)[['Slp Var']] +
          2 * x.cf * subset(regressions,Tissue==tissue)[['Cov']] + 1)
      line <- subset(regressions,Tissue==tissue)[['Intercept']] +
          x.cf * subset(regressions,Tissue==tissue)[['Slope']]
      y.cf <- line + conf
      y.ncf <- line - conf

      cf <- cbind(as.data.frame(x.cf),as.data.frame(y.cf),as.data.frame(y.ncf))
      if(tissue == 'Blood Cells'){
        eval(parse(text= paste('Blood <- ggplot() +
                                   geom_abline(linetype = "dashed") +
                                   geom_point(data=subset(pc.table,Tissue == \'',tissue,'\'),aes(10^(Predicted),
                                   10^(Experimental)))  +  theme(axis.text=element_text(size=14),
                                   axis.title=element_text(size=14),plot.title=element_text(size=14)) +
                                   scale_x_log10(label=scientific_10,limits=c(0.01,1000)) +
                                   scale_y_log10(label=scientific_10,limits=c(0.01,1000)) +
                                   ylab(ifelse(tissue=="Brain",expression(paste("Inferred ",K[p])),"")) +
                                   xlab(ifelse(tissue=="Skin", expression(paste("Predicted ",K[p])),"")) +
                                   geom_line(data=cf,aes(10^(x.cf),10^(y.cf))) +
                                   geom_line(data=cf,aes(10^(x.cf),10^(y.ncf))) +
                                   geom_abline(intercept=subset(regressions,Tissue==tissue)[[\'Intercept\']],
                                   slope=subset(regressions,Tissue==tissue)[[\'Slope\']]) +
                                   ggtitle(\'Red Blood Cells\')',sep='')))
      }else{
        eval(parse(text= paste(tissue,' <- ggplot() + labs(y=expression(paste("Measured ",K[p]))
                                   ,x=expression(paste("Predicted ",K[p]))) + geom_abline(linetype = "dashed") +
                                   geom_point(data=subset(pc.table,Tissue == \'',tissue,'\'),
                                   aes(10^(Predicted),10^(Experimental))) + theme(axis.text=element_text(size=14),
                                   axis.title=element_text(size=14),plot.title=element_text(size=14)) +
                                   scale_x_log10(label=scientific_10,limits=c(0.01,1000)) +
                                   scale_y_log10(label=scientific_10,limits=c(0.01,1000)) +
                                   ylab(ifelse(tissue=="Brain",expression(paste("Inferred ",K[p])),"")) +
                                   xlab(ifelse(tissue=="Skin", expression(paste("Predicted ",K[p])),"")) +
                                   geom_line(data=cf,aes(10^(x.cf),10^(y.cf))) +
                                   geom_line(data=cf,aes(10^(x.cf),10^(y.ncf))) +
                                   geom_abline(intercept=subset(regressions,Tissue==tissue)[[\'Intercept\']],
                                   slope=subset(regressions,Tissue==tissue)[[\'Slope\']]) +
                                   ggtitle(\'',tissue,'\')',sep='')))
      }
    }

    grid.arrange(Adipose, Blood, Bone, Brain, Gut, Heart, Kidney, Liver, Lung, Muscle, Skin, Spleen, nrow=4)

In vivo volume of distribution data are compared with the predictions, and errors are calculated. Regressions and improvements are calculated and plotted.

    obach <- subset(Obach2008,CAS %in% get_cheminfo(model='schmitt'))
    vd.table <- NULL

    for(this.cas in obach[,'CAS']){
      parameters <- parameterize_schmitt(
        chem.cas=this.cas,
        suppress.messages=TRUE)
      init.parameters <- parameters
      if(!this.cas %in% ma.data.list){
        init.parameters$MA <- 10^(0.999831 - 0.016578*37 + 0.881721 * log10(parameters$Pow))
      }
      pcs <- predict_partitioning_schmitt(
        parameters=parameters,
        regression=FALSE,
        suppress.messages=TRUE)
      init.pcs <- predict_partitioning_schmitt(
        parameters=init.parameters,
        regression=FALSE,
        suppress.messages=TRUE)
      reg.pcs <- predict_partitioning_schmitt(
        parameters=parameters,
        regression=TRUE,
        suppress.messages=TRUE)
      vdist <- calc_vdist(
        parameters=c(pcs,Funbound.plasma=parameters$Funbound.plasma),
        suppress.messages=TRUE)
      init.vdist <- calc_vdist(
        parameters=c(
          init.pcs,
          Funbound.plasma=parameters$unadjusted.Funbound.plasma),
        suppress.messages = TRUE)
      reg.vdist <- calc_vdist(
        parameters=c(reg.pcs,Funbound.plasma=parameters$Funbound.plasma),
        suppress.messages = TRUE)
      vd.table <- rbind(
        vd.table,
        cbind(as.data.frame(this.cas),as.data.frame(log10(init.vdist)),
        as.data.frame(log10(vdist)),as.data.frame(log10(reg.vdist)),
        as.data.frame(log10(subset(obach,CAS==this.cas)[['VDss (L/kg)']]))))
    }
    colnames(vd.table) <- c(
      'CAS',
      'init.vdist',
      'corrected.vdist',
      'calibrated.vdist',
      'Experimental')
    init.error <- vd.table[,'Experimental'] - vd.table[,'init.vdist']
    correction.error <- vd.table[,'Experimental'] - vd.table[,'corrected.vdist']
    calibration.error <- vd.table[,'Experimental'] - vd.table[,'calibrated.vdist']
    correction.improvement <- abs(init.error) - abs(correction.error)
    calibration.improvement <- abs(correction.error) - abs(calibration.error)
    vd.table <- cbind(vd.table,correction.improvement,calibration.improvement,
                      init.error,correction.error,calibration.error)

    fit <- lm(Experimental ~ calibrated.vdist,data=vd.table)
    smry <- summary(fit)
    calibrated.reg <- cbind(as.data.frame(fit$coefficients['(Intercept)']),
                            as.data.frame(fit$coefficients['calibrated.vdist']),
                            as.data.frame(smry$coefficients['calibrated.vdist','Pr(>|t|)']),
                            as.data.frame(smry$sigma),as.data.frame(smry$r.squared))
    fit <- lm(Experimental ~ init.vdist,data=vd.table)
    smry <- summary(fit)
    init.reg <- cbind(as.data.frame(fit$coefficients['(Intercept)']),
                      as.data.frame(fit$coefficients['init.vdist']),
                      as.data.frame(smry$coefficients['init.vdist','Pr(>|t|)']),
                      as.data.frame(smry$sigma),as.data.frame(smry$r.squared))
    fit <- lm(Experimental ~ corrected.vdist,data=vd.table)
    smry <- summary(fit)
    corrected.reg <- cbind(as.data.frame(fit$coefficients['(Intercept)']),
                           as.data.frame(fit$coefficients['corrected.vdist']),
                           as.data.frame(smry$coefficients['corrected.vdist','Pr(>|t|)']),
                           as.data.frame(smry$sigma),as.data.frame(smry$r.squared))
    colnames(init.reg) <- colnames(corrected.reg) <-
    colnames(calibrated.reg) <- c('Intercept','Slope','P-value','Std Err','R-squared')

    init.vd.plot <- ggplot(vd.table,aes(10^(init.vdist),10^(Experimental))) + geom_point() +
        geom_abline(intercept = init.reg[['Intercept']], slope = init.reg[["Slope"]]) +
        geom_abline(linetype = "dashed") + xlab("Predicted Volume of Distribution") +
        ylab("Measured Volume of Distribution") + theme(axis.text=element_text(size=16),
        axis.title=element_text(size=16),plot.title=element_text(size=18,hjust = 0.5)) +
        scale_x_log10(label=scientific_10,limits=c(10^(-1.5),10^(8.5))) +
        scale_y_log10(label=scientific_10,limits=c(10^(-1.5),10^(8.5))) +
        ggtitle('(A)')
    print(init.vd.plot)

    correction.plot <- ggplot() +
        geom_point(data=vd.table[order(vd.table[,'correction.improvement'],decreasing=F),],
        aes(10^(corrected.vdist),10^(Experimental),color=correction.improvement)) +
        geom_abline(intercept = corrected.reg[['Intercept']],slope = corrected.reg[["Slope"]]) +
        geom_abline(linetype = "dashed") + xlab("Predicted Volume of Distribution") +
        ylab("Measured Volume of Distribution") + theme(legend.position = c(.95, .95),
        legend.justification = c("right", "top"),legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6),axis.text=element_text(size=16),
        axis.title=element_text(size=16),plot.title=element_text(size=18,hjust = 0.5)) +
        scale_x_log10(limits=c(10^(-1.5),10^(3))) + scale_y_log10(limits=c(10^(-1.5),10^(3))) +
        ggtitle('(B)') + scale_color_viridis(direction=-1,option='inferno')
    print(correction.plot)

    calibration.plot <- ggplot() +
        geom_point(data=vd.table[order(vd.table[,'calibration.improvement'],decreasing=F),],
        aes(10^(calibrated.vdist),10^(Experimental),color=calibration.improvement)) +
        geom_abline(intercept = calibrated.reg[['Intercept']],slope = calibrated.reg[["Slope"]]) +
        geom_abline(linetype = "dashed") + xlab("Predicted Volume of Distribution") +
        ylab("Measured Volume of Distribution") + theme(legend.position = c(.95, .95),
        legend.justification = c("right", "top"),legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6),axis.text=element_text(size=16),
        axis.title=element_text(size=16),plot.title=element_text(size=18,hjust = 0.5)) +
        scale_x_log10(limits=c(10^(-1.5),10^(3))) + scale_y_log10(limits=c(10^(-1.5),10^(3))) +
        ggtitle('(C)') + scale_color_viridis(direction=-1,option='inferno')
    print(calibration.plot)

Now we pull in the in vivo blood to plasma ratios, use these to calculate the inferred red blood cell to plasma ratios, and then make predictions for these values. A censored regressions is performed, and predictions are plotted against errors.

    rb2p.data <- subset(chem.physical_and_invitro.data,!is.na(Human.Rblood2plasma))
    measured.rb2p <- NULL
    measured.krbc <- NULL
    predicted.rb2p <- NULL
    predicted.krbc <- NULL
    cas <- NULL
    charge <- NULL
    fup <- NULL
    logP <- NULL
    pka_donor <- NULL
    pka_accept <- NULL
    for(this.cas in rb2p.data[rb2p.data[,'CAS'] %in%
      get_cheminfo(model='schmitt', suppress.messages = TRUE),'CAS'])
    {
      rb2p <- get_rblood2plasma(chem.cas=this.cas)
      krbc <- (rb2p + .44 - 1) / 0.44
      measured.rb2p <- c(measured.rb2p,rb2p)
      measured.krbc <- c(measured.krbc,krbc)
      parameters <- parameterize_schmitt(
        chem.cas=this.cas,
        suppress.messages = TRUE)
      pcs <- predict_partitioning_schmitt(
        parameters=parameters,
        suppress.messages=TRUE)
      predicted.krbc <- c(predicted.krbc,pcs[['Krbc2pu']] * parameters$Funbound.plasma)
      cas <- c(cas,this.cas)
      charge <- c(charge,calc_ionization(chem.cas=this.cas,pH=7.4)$fraction_charged)
      fup <- c(fup,parameters$unadjusted.Funbound.plasma)
      logP <-  c(logP,log10(parameters$Pow))
      pka_donor <- c(pka_donor,paste(parameters$pKa_Donor,collapse=','))
      pka_accept <- c(pka_accept,paste(parameters$pKa_Accept,collapse=','))
    }
    predicted.rb2p <-  1 - 0.44 + 0.44 * predicted.krbc
    rb2p.table <- cbind(as.data.frame(cas),as.data.frame(predicted.rb2p),as.data.frame(measured.rb2p))
    colnames(rb2p.table) <- c('cas','predicted.rb2p','measured.rb2p')
    error <- log10(rb2p.table[,'measured.rb2p']) - log10(rb2p.table[,'predicted.rb2p'])
    rb2p.table <- cbind(rb2p.table,error,charge,fup,logP)

    error <- log10(measured.krbc) -  log10(predicted.krbc)
    krbc.table <- cbind(as.data.frame(cas),as.data.frame(predicted.krbc),as.data.frame(measured.krbc),
                        as.data.frame(error),charge,fup,logP,pka_donor,pka_accept)

    pdta <- data.frame(x = predicted.krbc,
                       y = measured.krbc)
    pdta$y[pdta$y <= 0.1] <- 0.1
    pdta$Censoring <- factor(c("Not Censored","Censored")[as.numeric(pdta$y <= 0.1) + 1])
    y <- measured.krbc
    x <- cbind(rep(1, length(y)),-1 * log10(predicted.krbc))
    colnames(x) <- c("Intercept","Predicted")
    cc <- as.numeric(y <= 0.1)
    y[y < 0.1] <- 0.1
    y <- -log10(y)

    out <- censReg(y~x, data = pdta, left=0.1)
    out$betas <- out$estimate

    censored.regression <- ggplot() +
        geom_point(data=pdta,aes(x=x,y=y, color=Censoring)) +
        scale_x_log10(limits=c(.0009,40)) + scale_y_log10(limits=c(.1,4),breaks=c(.1,.5,2.5)) +
        labs(y=expression(paste("Inferred ",K[p])),x=expression(paste("Predicted ",K[p]))) +
        geom_abline(intercept=0, slope=1, linetype='dashed') +
        theme(axis.text=element_text(size=16),axis.title=element_text(size=16),
        plot.title=element_text(size=18,hjust=0.5),legend.position = c(0.11, .8)) +
        geom_abline(slope=out$betas[2],intercept=-out$betas[1]) + ggtitle('(B)')
    print(censored.regression)

    rb2p.plot <- ggplot(rb2p.table,aes(predicted.rb2p,measured.rb2p)) +
        geom_point()  + scale_x_log10(lim=c(.52,18)) +
        scale_y_log10(lim=c(.52,2.5),breaks=c(0.5,1,2)) + geom_abline(linetype='dashed') +
        labs(y=expression(paste("Measured Whole Blood ",K[p])),
        x=expression(paste("Predicted Whole Blood ",K[p]))) +
        theme(axis.text=element_text(size=16),axis.title=element_text(size=16),
        plot.title=element_text(size=18,hjust=0.5)) + ggtitle('(A)')
    print(rb2p.plot)

Lastly, we make the heatmap.

    heatmap.table <- NULL
    for(this.cas in get_cheminfo(model='schmitt')){
        parms <- parameterize_schmitt(
          chem.cas=this.cas,
          suppress.messages = TRUE)
        pcs <- predict_partitioning_schmitt(
          parameters=parms,
          suppress.messages = TRUE)
        heatmap.table <- cbind(heatmap.table,log10(unlist(pcs)[1:11]*parms$Funbound.plasma))
    }
    rownames(heatmap.table) <-  c('Adipose','Bone','Brain','Gut','Heart',
                                  'Kidney','Liver','Lung','Muscle','Skin','Spleen')
    colnames(heatmap.table) <- rep("",dim(heatmap.table)[2])

    pal <- function (n, h = c(260, -328), c = 80, l = c(30, 100), power = 1.5,
        fixup = TRUE, gamma = NULL, alpha = 1, ...)
    {
        if (!is.null(gamma))
            warning("'gamma' is deprecated and has no effect")
        if (n < 1L)
            return(character(0L))
       h <- rep(h, length.out = 2L)
        c <- c[1L]
        l <- rep(l, length.out = 2L)
        power <- rep(power, length.out = 2L)
        rval <- seq(1, -1, length = n)
        rval <- hex(polarLUV(L = l[2L] - diff(l) * abs(rval)^power[2L],
            C = c * abs(rval)^power[1L], H = ifelse(rval > 0, h[1L],
                h[2L])), fixup = fixup, ...)
        if (!missing(alpha)) {
            alpha <- pmax(pmin(alpha, 1), 0)
            alpha <- format(as.hexmode(round(alpha * 255 + 1e-04)),
                width = 2L, upper.case = TRUE)
            rval <- paste(rval, alpha, sep = "")
        }
        return(rval)
    }

    hclust.ave <- function(x) hclust(x, method="ward.D2")
    heatmap.2(heatmap.table,col=pal,trace="none", hclustfun=hclust.ave,
              key.xlab=expression(paste("log10 ",K[p]," Value")),
              key.ylab=expression(paste("Number of ",K[p])),
              key.title="Partition Coefficient",xlab="Chemicals",cex.lab=2,margins=c(2,5))