Robert G Pearce, R Woodrow Setzer, Cory L Strope, Nisha S Sipes, and John F Wambaugh

Abstract

Thousands of chemicals have been profiled by high-throughput screening programs such as ToxCast and Tox21; these chemicals are tested in part because most of them have limited or no data on hazard, exposure, or toxicokinetics. Toxicokinetic models aid in predicting tissue concentrations resulting from chemical exposure, and a “reverse dosimetry” approach can be used to predict exposure doses sufficient to cause tissue concentrations that have been identified as bioactive by high-throughput screening. We have created four toxicokinetic models within the R software package httk. These models are designed to be parameterized using high-throughput in vitro data (plasma protein binding and hepatic clearance), as well as structure-derived physicochemical properties and species-specific physiological data. The package contains tools for Monte Carlo sampling and reverse dosimetry along with functions for the analysis of concentration vs. time simulations. The package can currently use human in vitro data to make predictions for 553 chemicals in humans, rats, mice, dogs, and rabbits, including 94 pharmaceuticals and 415 ToxCast chemicals. For 67 of these chemicals, the package includes rat-specific in vitro data. This package is structured to be augmented with additional chemical data as they become available. Package httk enables the inclusion of toxicokinetics in the statistical analysis of chemicals undergoing high-throughput screening.

This vignette was created from the R replication code v79i04.R for the 2017 paper that introduced httk.

Examples

To get a list of parameters for the pbtk model of Triclosan in a rat:

parameters <- try(parameterize_pbtk(chem.name = "triclosan", 
                                    species = "rat"))
#> Error in check_model(chem.cas = chem.cas, chem.name = chem.name, dtxsid = dtxsid,  : 
#>   Chemical CAS: 3380-34-5, DTXSID: DTXSID5032498, named: triclosan has insufficient parameters for model pbtk. Try setting default.to.human=TRUE. See also help(get_cheminfo)

We do not have a full set of in vitro parameters for rat for Triclosan, so we fill in the in vitro measured values with human-derived measurements and use rat physiological parameters by setting “default.to.human = TRUE”:

parameters <- parameterize_pbtk(chem.name = "triclosan", 
                                species = "rat", 
                                default.to.human = TRUE)
#> Warning in apply_clint_adjustment(Clint.point, Fu_hep = Fu_hep,
#> suppress.messages = suppress.messages): Clint adjusted for in vitro
#> partitioning (Kilford, 2008), see calc_hep_fu.
#> Warning in calc_ma(chem.cas = chem.cas, chem.name = chem.name, dtxsid = dtxsid,
#> : Membrane affintity (MA) predicted with method of Yun and Edginton (2013), see
#> calc_ma.
#> Warning in get_fup(dtxsid = dtxsid, chem.name = chem.name, chem.cas = chem.cas,
#> : Fraction unbound below limit of detection for rat replaced with human value.
#> Warning in get_fup(dtxsid = dtxsid, chem.name = chem.name, chem.cas = chem.cas,
#> : Fraction unbound is provided as a distribution.
#> Warning in apply_fup_adjustment(fup.point, fup.correction = fup.adjustment, :
#> Fup adjusted for in vivo lipid partitioning (Pearce, 2017), see
#> calc_fup_correction.
#> Warning in available_rblood2plasma(chem.cas = chem.cas, species = species, :
#> Human in vivo measured Rblood2plasma substituted.
#> Warning in get_caco2(chem.cas = chem.cas, chem.name = chem.name, dtxsid =
#> dtxsid, : Default value of 1.6 used for Caco2 permeability.

To change the \(f_{up}\) in the previous parameters list to 0.001 from the human value of 0.003044 and use it in a simulation of the PBTK model for a single dose of 1 mg/kg BW of Triclosan in a rat:

parameters["Funbound.plasma"] <- 0.001
out <- solve_pbtk(parameters = parameters, suppress.messages = TRUE)

Making a table

In the example below, setting model to “pbtk” in “get_cheminfo” removes the chemicals from the list with \(f_{up}\) below the limit of detection which have been set to zero because the model uses partition coefficients that are calculated with \(f_{up}\). However, we could include all chemicals that work with the models without partition coefficients by using the default option of “3compartmentss” or setting exclude.fup.zero to false, and \(f_{up}\) would then automatically be set to 0.005.

First we make up a list of chemicals with both literature \(C_{ss}\) values and HTTK data for making new \(C_{ss}\) predictions:

chem.list <- intersect(get_cheminfo(model = "pbtk", 
                                    suppress.messages = TRUE)[seq(
                                      1, NUM.CHEMS, SKIP.CHEMS)],
                       get_wetmore_cheminfo())
#> Warning in get_wetmore_cheminfo(): Function "get_wetmore_cheminfo" has been
#> renamed to "get_lit_cheminfo".

Then we initialize a NULL table and build it up one row at a time:

css.table <- NULL
for (this.cas in chem.list)
{
    ids <- get_chem_id(chem.cas=this.cas)
    this.row <- data.frame(Compound=ids$chem.name,
                              DTXSID=ids$dtxsid,
                              CAS=this.cas)
    this.row <- cbind(this.row,
                      as.data.frame(calc_analytic_css(chem.cas = this.cas,
                                                      model = "pbtk",
                                                      output.units = "uM",
                                                      suppress.messages=TRUE)))
    this.row <- cbind(this.row,
                      as.data.frame(get_wetmore_css(chem.cas = this.cas,
                                                    which.quantile = 0.50,
                                                    output.units = "uM",
                                                    suppress.messages=TRUE)))
    this.row <- cbind(this.row,
                      as.data.frame(calc_mc_css(chem.cas = this.cas,
                                                which.quantile = 0.50,
                                                output.units = "uM",
                                                samples = NSAMP,
                                                suppress.messages=TRUE)))
    css.table <- rbind(css.table, this.row)
}
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
colnames(css.table) <- c("Compound","DTXSID","CAS", "PBTK", "Wetmore", "MC")

Now display the table and plot two columns against each other:

knitr::kable(css.table, caption = "Literature and HTTK Plasma Css", 
             floating.environment="sidewaystable",
             row.names=FALSE,
             digits=3)

Literature and HTTK Plasma Css
Compound	DTXSID	CAS	PBTK	Wetmore	MC
2,4-d	DTXSID0020442	94-75-7	40.550	43.320	56.530
Diethyltoluamide	DTXSID2021995	134-62-3	0.958	0.847	2.445
Isoxaben	DTXSID8024159	82558-50-7	0.962	24.460	1.234
Rotenone	DTXSID6021248	83-79-4	2.232	1.833	3.170
Genistein	DTXSID5022308	446-72-0	1.109	0.310	1.844
Coumarin	DTXSID7020348	91-64-5	0.430	0.424	17.780

litvshttkcss <- ggplot(css.table, aes(Wetmore, MC)) +
    geom_point() + geom_abline() +
    scale_x_log10() + scale_y_log10() +
    xlab(bquote("Literature"~C[ss]~"("*mu*"M)")) +
    ylab(bquote("HTTK"~C[ss]~"("*mu*"M)")) +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16))
print(litvshttkcss)

Plotting example

To see how \(C_{ss}\) resulting from discrete dosing deviates from the average steady state concentration, we can make a plot with ggplot2 that includes a horizontal line through the \(y\)-axis at the predicted \(C_{ss}\) for oral infusion dosing (Figure 2 in Pearce et al., 2017).

out <- solve_pbtk(chem.name = "Bisphenol A", 
                  days = 50, 
                  doses.per.day = 3,
                  daily.dose=1,
                  output.units = "uM")
#> Warning in get_clint(dtxsid = dtxsid, chem.name = chem.name, chem.cas =
#> chem.cas, : Clint is provided as a distribution.
#> Warning in apply_clint_adjustment(Clint.point, Fu_hep = Fu_hep,
#> suppress.messages = suppress.messages): Clint adjusted for in vitro
#> partitioning (Kilford, 2008), see calc_hep_fu.
#> Warning in get_fup(dtxsid = dtxsid, chem.name = chem.name, chem.cas = chem.cas,
#> : Fraction unbound is provided as a distribution.
#> Warning in apply_fup_adjustment(fup.point, fup.correction = fup.adjustment, :
#> Fup adjusted for in vivo lipid partitioning (Pearce, 2017), see
#> calc_fup_correction.
#> Warning in available_rblood2plasma(chem.cas = chem.cas, species = species, :
#> Human in vivo measured Rblood2plasma used.
#> Warning in get_caco2(chem.cas = chem.cas, chem.name = chem.name, dtxsid =
#> dtxsid, : Default value of 1.6 used for Caco2 permeability.
#> None of the monitored components undergo unit conversions  (i.e. conversion factor of 1).
#> 
#> AUC is area under the plasma concentration curve in uM*days units with Rblood2plasma = 0.795.
#> The model outputs are provided in the following units:
#>  umol: Agutlumen, Atubules, Ametabolized
#>  uM: Cgut, Cliver, Cven, Clung, Cart, Crest, Ckidney, Cplasma
#>  uM*days: AUC
plot.data <- as.data.frame(out)

css <- calc_analytic_css(chem.name = "Bisphenol A",
                  output.units = "uM")
#> Warning in get_clint(dtxsid = dtxsid, chem.name = chem.name, chem.cas =
#> chem.cas, : Clint is provided as a distribution.
#> Warning in apply_clint_adjustment(Clint.point, Fu_hep = Fu_hep,
#> suppress.messages = suppress.messages): Clint adjusted for in vitro
#> partitioning (Kilford, 2008), see calc_hep_fu.
#> Warning in get_fup(dtxsid = dtxsid, chem.name = chem.name, chem.cas = chem.cas,
#> : Fraction unbound is provided as a distribution.
#> Warning in apply_fup_adjustment(fup.point, fup.correction = fup.adjustment, :
#> Fup adjusted for in vivo lipid partitioning (Pearce, 2017), see
#> calc_fup_correction.
#> Warning in available_rblood2plasma(chem.cas = chem.cas, species = species, :
#> Human in vivo measured Rblood2plasma used.
#> Warning in get_caco2(chem.cas = chem.cas, chem.name = chem.name, dtxsid =
#> dtxsid, : Default value of 1.6 used for Caco2 permeability.
#> Plasma concentration returned in uM units.

c.vs.t <- ggplot(plot.data, aes(time, Cplasma)) +
    geom_line() + geom_hline(yintercept = css) +
    ylab(bquote("Plasma Concentration ("*mu*"M)")) +
    xlab("Day") +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16),
          plot.title = element_text(size = 17, hjust=0.5)) +
    ggtitle("Bisphenol A")
print(c.vs.t)

Suppressing Messages

Note that since in vitro-in vivo extrapolation (IVIVE) is built upon many, many assumptions, *httk** attempts to give many warning messages by default. These messages do not usually mean something is wrong, but are rather intended to make the user aware of the assumptions involved. However, they quickly grow annoying and can be turned off with the “suppress.messages=TRUE” argument. Proceed with caution…

out <- solve_pbtk(chem.name = "Bisphenol A", 
                  days = 50, 
                  doses.per.day = 3,
                  daily.dose=1,
                  output.units = "uM",
                  suppress.messages = TRUE)

css <- calc_analytic_css(chem.name = "Bisphenol A",
                  output.units = "uM",
                  suppress.messages = TRUE)

TK Study Summary Statistics

“calc_tk_stats” allows summary TK statistics to be calculated for a specific study design (including dose regiment, duration, and species). The following code calculates the peak plasma concentration for all chemicals in httk simulated for 10 days at 1 mg/kg BW/day with 3 doses per day. Note that the function “calc_stats” was renamed “calc_tk_stats” in a later release of httk.

all.peak.stats <- calc_stats(days = 10, doses.per.day = 3, stats = "peak")

The same function can be used for a single chemical. For example, the AUC, peak, and mean for a single 1 mg dose of Triclosan over 10 day:

triclosan.stats <- calc_stats(days = 10, chem.name = "triclosan")
#> Warning in calc_stats(days = 10, chem.name = "triclosan"): Function
#> "calc_stats" has been renamed to "calc_tkstats".
#> Human plasma concentrations returned in uM units.
#> AUC is area under plasma concentration curve in uM * days units with Rblood2plasma = 0.6436 .

Steady-state Plasma Concentration

Below are examples of two functions,comparing the medians of the Wetmore data in humans for 1 mg/kg BW/day of Bisphenol A with the “calc_mc_css” simulation with probability distributions containing a third of the standard deviation, half the limit of detection for \(f_{up}\), and the same number of samples of the parameters used in Wetmore et al. (2012).

We use make use of the default Monte Carlo sampler by setting argument “httkpop=FALSE” to turn off httk-pop (Ring et al., 2017) and “invitrouv=FALSE” to turn off uncertainty/variability for the in vitro parameters (Wambaugh et al., 2019).

Note that the function “get_wetmore_css” was renamed “get_lit_css” in a later release of httk.

get_wetmore_css(chem.cas = "80-05-7", daily.dose = 1,
                which.quantile = 0.5, output.units = "uM")
#> Warning in get_wetmore_css(chem.cas = "80-05-7", daily.dose = 1, which.quantile
#> = 0.5, : Function "get_wetmore_css" has been renamed to "get_lit_cheminfo".
#> Human plasma concentrations returned in uM units.
#> Retrieving Css from literature based on 1 uM intrinsic clearance data for the 0.5 quantile in Human.
#> [1] 0.8569

set.seed(654321)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            censored.params = list(
              Funbound.plasma = list(cv = 0.1, lod = 0.005)),
            vary.params = list(
              BW = 0.15, 
              Vliverc = 0.15, 
              Qgfrc = 0.15,
              Qtotal.liverc = 0.15, 
              million.cells.per.gliver = 0.15,
              Clint = 0.15),
            output.units = "uM", 
            samples = NSAMP,
            httkpop=FALSE,
            invitrouv=FALSE,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>    50% 
#> 0.8689

Using more sophisticated Monte Carlo

httk uses Monte Carlo to simulate population variability and propagate parameter uncertainty. The algorithm for population variability, “httk-pop” (Ring et al., 2017) simulates population variability in TK by predicting distributions of physiological parameters based on distributions of demographic and biometric quantities from the National Health and Nutrition Examination Survey (NHANES), conducted by the U.S. Centers for Disease Control and Prevention (CDC). MEthods for propagating uncertainty in in vitro measurements were described by Wambaugh et al. (2019).

Both httk-pop and in vitro uncertainty/variability simulation, are on by default:

set.seed(654321)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.342

Controlling the random number generator

Any of the Monte Carlo functions in httk, often indicated by the inclusion of “mc” in the function name, make use of random draws from distributions to simulate uncertainty and variability. The random number generator in any computer is actually a pseudo-random number generator that creates a sequence of nearly uncorrelated numbers, but that sequence can be recreated at any time if we set a random number generator “seed”. In R we control the seed with the function “set.seed”, which allows us to get the same Monte Carlo output again and again (and in many cases on different computers). The seed can take any value, and if you use the same seed followed by the same functions called in the same order with the same arguments, you will get the same answer again and again:

# Random number generator seed 1:
set.seed(654321)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.342

# Continuing to draw random numbers without resetting the seed:
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.202

# Random number generator seed 2:
set.seed(123456)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.108

# Continuing to draw random numbers without resetting the seed:
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 1.034

# Random number generator seed 2 gives the same answer as it did above:
set.seed(123456)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.108

# Random number generator seed 1 gives the same answer as it did above:
set.seed(654321)
calc_mc_css(chem.cas = "80-05-7", 
            daily.dose = 1, 
            which.quantile = 0.5,
            output.units = "uM", 
            samples = NSAMP,
            suppress.messages = TRUE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>   50% 
#> 2.342

IVIVE with HTTK

Simple “reverse dosimetry” in vitro-in vivo extrapolation (IVIVE) determines an administered equivalent dose (AED) by taking an in vitro concentration \(C_{\text{in vitro}}\) (50 uM in the example below) and dividing it by the steady-state plasma concentration \(C_{ss}\) produced by a dose rate of 1 mg/kg bw/day:

\[ AED = \frac{C_{\text{in vitro}}}{C_{ss}}\]

We can also calculate the AED using literature values or sampling with httk:

get_wetmore_oral_equiv(50, chem.cas = "80-05-7")
#> Warning in get_wetmore_oral_equiv(50, chem.cas = "80-05-7"): Function
#> "get_wetmore_oral_equiv" has been renamed to "get_lit_oral_equiv".
#> Retrieving Css from literature based on 10 uM intrinsic clearance data for the 0.95 quantile in Human.
#> Human uM concentration converted to mg/kg bw/day dose.
#> [1] 37.06
set.seed(654321)
calc_mc_oral_equiv(50, chem.cas = "80-05-7", samples = NSAMP)
#> Warning in get_clint(dtxsid = dtxsid, chem.name = chem.name, chem.cas =
#> chem.cas, : Clint is provided as a distribution.
#> Warning in get_fup(dtxsid = dtxsid, chem.name = chem.name, chem.cas = chem.cas,
#> : Fraction unbound is provided as a distribution.
#> Warning in get_caco2(chem.cas = chem.cas, chem.name = chem.name, dtxsid =
#> dtxsid, : Default value of 1.6 used for Caco2 permeability.
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#> Human plasma concentration returned in uM units for 0.95 quantile.
#> uM concentration converted to mgpkgpday dose for 0.95 quantile.
#>   95% 
#> 9.713

Monte Carlo Simulation of \(C_{ss}\)

We can perform a Monte Carlo simulation on Zoxamide with the model pbtk with the same limit of detection and coefficients of variation (cv). Note, the function “monte_carlo” was replaced with a similar function “calc_mc_tk” for simulation concentration timecourses, while arguments were added to “calc_mc_css” that allowed the functionality seen in the 2017 paper. Separately, a new function “monte_carlo” was created to complement “httkpop_mc” and “invitro_mc” as part of “create_mc_samples”. “monte_carlo” varies parameters according to a normal distribution with given mean and coefficient of variation or according to a censored distribution with an additional limit of detection parameter. Note that we again turn off httkpop and invitrouv to replicate the 2017 version of Monte Carlo.

First we create a list of parameters and then assign a coefficient of variation to each parameter we wish to vary:

vary.params <- NULL
params <- parameterize_pbtk(chem.name = "Zoxamide")
#> Warning in apply_clint_adjustment(Clint.point, Fu_hep = Fu_hep,
#> suppress.messages = suppress.messages): Clint adjusted for in vitro
#> partitioning (Kilford, 2008), see calc_hep_fu.
#> Warning in calc_ma(chem.cas = chem.cas, chem.name = chem.name, dtxsid = dtxsid,
#> : Membrane affintity (MA) predicted with method of Yun and Edginton (2013), see
#> calc_ma.
#> Warning in apply_fup_adjustment(fup.point, fup.correction = fup.adjustment, :
#> Fup adjusted for in vivo lipid partitioning (Pearce, 2017), see
#> calc_fup_correction.
#> Warning in available_rblood2plasma(chem.cas = chem.cas, species = species, :
#> Human in vivo measured Rblood2plasma used.
#> Warning in get_caco2(chem.cas = chem.cas, chem.name = chem.name, dtxsid =
#> dtxsid, : Default value of 1.6 used for Caco2 permeability.
for (this.param in names(subset(params, names(params) != "Funbound.plasma")))
  # Only want to vary numerical parameters:
  if (is.numeric(params[[this.param]])) 
      vary.params[this.param] <- 0.2

We can draw \(f_{up}\) from a censored distribution with a limit of dextion (LOD) of 0.01:

censored.params <- list(Funbound.plasma = list(cv = 0.2, lod = 0.01))

Solve the full TK time-course

set.seed(1)
out <- calc_mc_tk(parameters=params, 
                  vary.params = vary.params,
                  censored.params = censored.params,
                  return.samples = TRUE, 
                  model = "pbtk",
                  suppress.messages = TRUE,
                  httkpop = FALSE,
                  invitrouv = FALSE,
                  samples = NSAMP,
                  solvemodel.arg.list = list(times = seq(0,0.5,0.025))
                  )

Make a table of concentration vs. time with confidence intervals:

zoxtable <- cbind(out[["means"]][,"time"],
                  out[["means"]][,"Cplasma"],
                  out[["means"]][,"Cplasma"] - 1.96*out[["sds"]][,"Cplasma"],
                  out[["means"]][,"Cplasma"] + 1.96*out[["sds"]][,"Cplasma"])
colnames(zoxtable) <- c("time","Cplasma","lcl","ucl")
knitr::kable(zoxtable, caption = "Zoxamide plasma concentration vs. time",
             floating.environment="sidewaystable")

Zoxamide plasma concentration vs. time
time	Cplasma	lcl	ucl
0.0000	0.0000000	0.0000000	0.0000000
0.0001	0.0000137	0.0000074	0.0000200
0.0250	0.4607000	0.3590348	0.5623652
0.0500	0.5514000	0.4394252	0.6633748
0.0750	0.5804000	0.4646228	0.6961772
0.1000	0.5972000	0.4789532	0.7154468
0.1250	0.6098000	0.4897108	0.7298892
0.1500	0.6198000	0.4982800	0.7413200
0.1750	0.6281000	0.5054040	0.7507960
0.2000	0.6350000	0.5111868	0.7588132
0.2250	0.6406000	0.5160028	0.7651972
0.2500	0.6453000	0.5199188	0.7706812
0.2750	0.6492000	0.5231720	0.7752280
0.3000	0.6525000	0.5258840	0.7791160
0.3250	0.6552000	0.5280548	0.7823452
0.3500	0.6573000	0.5298412	0.7847588
0.3750	0.6592000	0.5313492	0.7870508
0.4000	0.6607000	0.5325160	0.7888840
0.4250	0.6620000	0.5335612	0.7904388
0.4500	0.6630000	0.5343652	0.7916348
0.4750	0.6638000	0.5349692	0.7926308
0.5000	0.6645000	0.5355320	0.7934680

Plot Monte Carlo analysis of concentration vs. time:

zoxamide1 <- ggplot(as.data.frame(zoxtable), aes(x=time,y=Cplasma)) +
    geom_line(color = "dark blue",linewidth=2) +
      geom_ribbon(aes(ymin=lcl, ymax=ucl), alpha = 0.3,
                  fill = "light blue", color="black", linetype="dotted")+
    ylab("Plasma concentration") +
    xlab("Time (days)") +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16))
print(zoxamide1)

Examine \(C_{ss}\)

out <- calc_mc_css(parameters=params, 
                  vary.params = vary.params,
                  censored.params = censored.params,
                  return.samples = TRUE, 
                  model = "pbtk",
                  suppress.messages = TRUE,
                  httkpop = FALSE,
                  samples = NSAMP,
                  invitrouv = FALSE)
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose

zoxamide2 <- ggplot(as.data.frame(out), aes(out)) +
    geom_histogram(fill = "blue", binwidth = 1/6) +
    scale_x_log10() + ylab("Number of Samples") +
    xlab(bquote(C[ss]~"("*mu*"M)")) +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16))
print(zoxamide2)

Adding a tissue

The httk package contains a data.frame called tissue.data that describes the composition of various tissues in terms of the Schmitt (2008) mechanistic model for deriving tissue:plasma partition coefficients. Once described, these tissues are available for use as compartments in new models or to be lumped into the rest of body in existing models (using function “lump_tissues”). We can add a new tissue (for example, mammary) to the tissue data by replicating the data from another tissue and adjusting as appropriate.

new.tissue <- subset(tissue.data,Tissue == "adipose" & Species =="Human")
new.tissue[, "Tissue"] <- "mammary"
new.tissue[new.tissue$variable=="Vol (L/kg)","value"] <- 
  320/1000/60 # Itsukage et al. (2017) PMID: 29308107
new.tissue[new.tissue$variable %in% c('Flow (mL/min/kg^(3/4))'),'value'] <-
# We'll arbitrarily set flow to a tenth of total adipose:
  new.tissue[new.tissue$variable %in% c('Flow (mL/min/kg^(3/4))'),'value']/10
tissue.data <- rbind(tissue.data, new.tissue)
knitr::kable(subset(tissue.data,Tissue=="mammary"), 
             caption = "Approximate Mamary Tissue Parameters", 
             floating.environment="sidewaystable",
             row.names=FALSE,
             digits=3)

Approximate Mamary Tissue Parameters
Tissue	Species	Reference	variable	value
mammary	Human	Schmitt 2008	Fcell	0.860
mammary	Human	Schmitt 2008	Fint	0.140
mammary	Human	Schmitt 2008	FWc	0.020
mammary	Human	Schmitt 2008	FLc	0.934
mammary	Human	Schmitt 2008	FPc	0.046
mammary	Human	Schmitt 2008	Fn_Lc	0.936
mammary	Human	Schmitt 2008	Fn_PLc	0.056
mammary	Human	Schmitt 2008	Fa_PLc	0.008
mammary	Human	Schmitt 2008	pH	7.100
mammary	Human	EPA, Physiological Parameters Database for PBPK Modeling	Density (g/cm^3)	0.916
mammary	Human	ILSI-RSI 1994	Vol (L/kg)	0.005
mammary	Human	Davies and Morris 1993	Flow (mL/min/kg^(3/4))	1.074

If we thought this tissue had been included in the rest of body total previously we could consider removing it from those volumes and flows, but we won’t do that here (Note eval=FALSE):

# If we thought this tissue was included in the rest of the bod
tissue.data[tissue.data$Tissue == "rest", 'value'] <-
  tissue.data[tissue.data$Tissue == "rest", 'value'] -
  new.tissue[new.tissue$variable %in% c(
    'Vol (L/kg)',
    'Flow (mL/min/kg^(3/4))'),
    'value']

To generate the parameters for a model with kidneys, thyroid, brain, breast, liver compartment combining the liver and gut, and a rest of body compartment:

compartments <- list(liver = c("liver", "gut"), kidney = "kidney",
                     breast = "mammary", brain = "brain",
                     thyroid = "thyroid")
tissues.to.include <- unique(tissue.data$Tissue)
tissues.to.include <- tissues.to.include[!(tissues.to.include=="placenta")]
Kp <- predict_partitioning_schmitt(
  chem.name = "Nicotine", 
  tissues = tissues.to.include,
  suppress.messages = TRUE)
lumped.params <- lump_tissues(Kp, tissuelist=compartments)
knitr::kable(as.data.frame(lumped.params), 
             caption = "PCs, Volumes, and Flows for Lumped Tissues", 
             floating.environment="sidewaystable",
             row.names=FALSE,
             digits=3)

PCs, Volumes, and Flows for Lumped Tissues
Krbc2pu	Kliver2pu	Kkidney2pu	Kbreast2pu	Kbrain2pu	Kthyroid2pu	Krest2pu	Vliverc	Vkidneyc	Vbreastc	Vbrainc	Vthyroidc	Vrestc	Qtotal.liverf	Qkidneyf	Qbreastf	Qbrainf	Qthyroidf	Qrestf	Qgutf
0.802	4.749	5.149	11.61	1.451	1.448	1.916	0.04	0.004	0.005	0.019	0	0.764	0.464	0.221	0.005	0.125	0.016	0.376	0.205

Adding a species

The httk package contains a data.frame called physiology.data that describes the species-specific aspects of physiology necessary to parameterize a PBTK model. We can add a new species (for example, wolverines) to the physiology.data by replicating the data from another species and adjusting as appropriate.

new.species <- physiology.data[,"Rabbit"]
names(new.species) <- physiology.data[,"Parameter"]
rabbit.BW <- new.species["Average BW"] 
new.species["Average BW"] <- 31.2 # Rausch and Pearson (1972) https://doi.org/10.2307/3799057
new.species["Average Body Temperature"] <- 38.5 # Thiel et al. (2019) https://doi.org/10.1186/s12983-019-0319-8

physiology.data <- cbind(physiology.data, new.species)
colnames(physiology.data)[length(colnames(physiology.data))] <- "Wolverine"

knitr::kable(physiology.data[,c("Parameter","Units","Wolverine")], 
             caption = "Approximate Wolverine Physiological Parameters", 
             floating.environment="sidewaystable",
             row.names=FALSE,
             digits=3)

Approximate Wolverine Physiological Parameters
Parameter	Units	Wolverine
Total Body Water	ml/kg	716.000
Plasma Volume	ml/kg	44.000
Cardiac Output	ml/min/kg^(3/4)	212.000
Average BW	kg	31.200
Total Plasma Protein	g/ml	0.057
Plasma albumin	g/ml	0.039
Plasma a-1-AGP	g/ml	0.001
Hematocrit	fraction	0.360
Urine	ml/min/kg^(3/4)	0.042
Bile	ml/min/kg^(3/4)	0.083
GFR	ml/min/kg^(3/4)	3.120
Average Body Temperature	C	38.500
Plasma Effective Neutral Lipid Volume Fraction	unitless	0.002
Plasma Protein Volume Fraction	unitless	0.057
Pulmonary Ventilation Rate	l/h/kg^(3/4)	24.750
Alveolar Dead Space Fraction	unitless	0.330
Small Intestine Mean Residence Time	min	NA
Small Intestine Radius	cm	NA

The tissue volumes and flows are stored separately in tissue.data. Physiological parameters are in the table physiology.data.

new.tissue.data <- subset(tissue.data,Species=="Rabbit")
new.tissue.data$Species <- "Wolverine"

tissue.data <- rbind(tissue.data,new.tissue.data)

new.physiology.data <- physiology.data[,"Rabbit"]
physiology.data <- rbind(physiology.data, new.physiology.data)
#> Warning in rbind(deparse.level, ...): number of columns of result, 9, is not a
#> multiple of vector length 18 of arg 2
colnames(physiology.data)[length(colnames(physiology.data))] <- "Wolverine"

Now we can make predictions for wolverines!

calc_mc_css(chem.cas="80-05-7",species="wolverine",
            parameterize.arg.list=list(default.to.human=TRUE),
            suppress.messages=TRUE,
            samples = NSAMP)
#> Warning in (function (chem.cas = NULL, chem.name = NULL, dtxsid = NULL, : httkpop model only available for human and thus not used.
#> 
#> Set species="Human" to run httkpop model.
#> Warning in (function (chem.name = NULL, chem.cas = NULL, dtxsid = NULL, :
#> calc_analytic_css deprecated argument daily.dose replaced with new argument
#> dose, value given assigned to dose
#>    95% 
#> 0.5492

Export functions

Jarnac (Sauro and Fell 2000) and SBML (Hucka et al. 2003) are commonly used languages for systems biology models of cellular and physiological processes. In the event that a modeler wishes to couple such a model to a toxicokinetic model, we provide functions to export model equations and chemical-specific parameters to these languages. The two functions, “export_pbtk_sbml” and “export_pbtk_jarnac”, have the same arguments and only differ in the file extension names (.xml and .jan) entered into the filename argument. Both use liters as the units for volume, but the amounts are unitless and to be determined by the user. If we suppose that we enter an initial amount of 1 mg in the gut lumen, then all the other compartments will contain amounts in mg.

Below is a call of an export function for a dose of 1 given to a rat:

export_pbtk_sbml(chem.name = "Bisphenol A", species = "Rat",
                 initial.amounts = list(Agutlumen = 1),
                 filename = "PBTKmodel.xml")

Days to steady state histogram

Creating histograms can allow us to visualize how a given value varies across all the chemicals contained within the package. To create a histogram using ggplot2 of the number of days to steady state, we must first set up a for loop with “get_cheminfo” and “calc_css” to generate a vector containing the data. Vectors containing the average and maximum concentrations at steady state are also generated in this example, avg and max. The data contained in the days vector are then plotted as a histogram (Pearce et al., (2017) Figure 3). We can just as easily create a histogram containing the average or maximum steady state concentrations by substituting avg or max for days.

css.data <- data.frame()
chem.list <- get_cheminfo(model='pbtk', suppress.messages = TRUE)[seq(
  1, NUM.CHEMS, SKIP.CHEMS)]
for (this.cas in chem.list) {
    css.info <- calc_css(chem.cas = this.cas,
                         doses.per.day = 1,
                         model="pbtk",
                         suppress.messages = TRUE)
    css.data[this.cas,"days"] <- css.info[["the.day"]]
    css.data[this.cas,"avg"] <- css.info[["avg"]]
    css.data[this.cas,"max"] <- css.info[["max"]]
}
#> Warning in min(out[which(out[, target]/max(out[, target]) == 1), "time"]): no
#> non-missing arguments to min; returning Inf

hist <- ggplot(css.data, aes(days+0.1)) +
    geom_histogram(fill = "blue", bins=5) +
    scale_x_log10() + ylab("Number of Chemicals") + xlab("Days") +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16))
print(hist)
#> Warning: Removed 1 row containing non-finite outside the scale range
#> (`stat_bin()`).

Average vs. maximum concentration

We can compare the average and maximum concentrations at steady state using the average and maximum concentration at steady state vectors, avg and max, from the previous example. The vectors are bound into a data frame and plotted with a line through the origin with a slope of 1 (Pearce et al. (2017) Figure 4).

avg.vs.max <- ggplot(css.data, aes(avg, max)) +
    geom_point() + geom_abline() +
    scale_x_log10() + scale_y_log10() +
    xlab(bquote("Avg. Conc. at Steady State ("*mu*"M)")) +
    ylab(bquote("Max. Conc. at Steady State ("*mu*"M)")) +
    theme(axis.text = element_text(size = 16),
          axis.title = element_text(size = 16))
print(avg.vs.max)
#> Warning in scale_x_log10(): log-10 transformation introduced infinite values.
#> Warning in scale_y_log10(): log-10 transformation introduced infinite values.

Pearce et al. (2017): Updated v79i04.R Examples