# vLiverPBPK
Robert Pearce, Cory Strope, Woodrow Setzer, John Wambaugh



## 1 Introduction
Every day we are exposed to low levels of hundreds of environmental chemicals, most of which have not been tested for toxicity [14, 23, 3, 8]. However, in vitro data for thousands of chemicals has been generated in the Tox21 [1] and ToxCast [9] programs using a high-throughput robotic screening system where chemicals are tested in at least 50 assays at varying concentrations. Reverse dosimetry has been applied to this data in Wetmore et al. (2012)[25] in estimating oral equivalent doses that yield steady state concentrations corresponding to various levels of biological activity. We introduce vLiverPBPK, an R package for statistical analysis of pharmacokinetics for chemicals undergoing high throughput screening (HTS) and other toxicity studies. We provide tools for Monte Carlo simulation and reverse dosimetry methods [20] with physiological and physicochemical pharmacokinetic data [25] for 369 chemicals to simulate four pharmacokinetic models and solve for their steady state concentrations and other pharmacokinetic summary

1statistics.
Pharmacokinetics (PK) is a field of study for determining the absorption, distribution, metabolism, and excretion of substances in the body [13]. The necessary data for PK can be collected in rats and other animals, but if we want to collect it for the thousands of untested chemicals, there is a high cost in terms of time, money, and animals [17]. Creating a few harmonized models for most chemicals that can be parameterized with in vitro PK data that are more easily obtained than in vivo PK data can help address these problems. Once we have an estimate of the amount of a chemical someone is ordinarily exposed to, we can use this model to predict the concentrations in various parts of the body and then compare these concentrations to known concentrations that cause biological activity in the cells of a given tissue. We can also use the model in a reverse manner to see what dose is needed to produce a specific concentration such as the in vitro AC 50 or other levels of biological activity; as done in Wetmore et al. (2012) whose results are contained in the package. If we find that many people are exposed to an amount of a chemical that is likely to create a biological effect, we will place a high priority on more testing of the chemical [21]. Although the packagehas been developed primarily for the prioritization of testing for environmental chemicals, 75 of the chemicals contained are pharmaceuticals.

The reverse toxicokinetic model used in Wetmore et al. assumes a constantinfusion dose in reaching steady state where the dose rate is both continuous and constant and the concentration is the same for the entire body, but

2this is not necessarily an accurate representation of reality. In our physiologically based pharmacokinetic (PBPK) models, we can use discrete doses which create a fluctuating steady state that oscillates around the infusion dose prediction as well as multiple compartments with different partition coefficients and flows between them. These models are expressed as a set of mass balance differential equations describing the rate of change of the amount of a substance, and they use in vitro and physiological data incalculating the partition coefficients, clearance, and flows between compartments or out of the kidneys. This in vitro data includes the intrinsic hepatic clearance, the fraction of a chemical unbound to plasma ( f up ), and other physicochemical properties. With this R package we provide data, models, and examples to allow the inclusion of PK in statistical analysis of chemical exposure and toxicity for 369 chemicals. The package is structured to be modular and expandable to allow new modeling approaches chemical data to be added as they become available.

3Function Description calc analytic css Calculates the analytic steady state plasma and blood concentrations for the four models used in the package. calc css Finds the day, max and average steady state concentrations along with the fraction of the analytic steady state reached. calc elimination rate Calculates the elimination rate for a one compartment model due to the liver and kidneys, dividing the total clearance by volume of distribution. calc hepatic clearance Calculates the hepatic clearance for a well-stirred model or other type if specified. calc mc css Monte Carlo simulation of steady state three compartment model. calc mc oral equiv Converts a μ M steady state plasma concentration to an oral equivalent dose using a concentration obtained from calc mc css
. calc ratioblood2plasma Calculates the ratio of chemical concentration in blood to plasma. calc stats Calculates the area under the curve, mean, and peak values for the blood or plasma concentration of a specified chemical or all chemicals. calc total clearance Calculates the total clearance rate for a one compartment model whereclearance is entirely due to metablism by the liver and glomerular filtration in the kidneys. calc volume of distribution Calculates the volume of distribution for a one compartment model. export pbpk jarnac Exports the PBPK model to jarnac. export pbpk sbml Exports the PBPK model to sbml. get cheminfo Provides the CAS numbers and other specified information for chemicals that work with a given model. get wetmore cheminfo Provides the names and CAS numbers of chemicals with information from Wetmore et al. (2012) and (2013) [26]. get wetmore css Retrieves a steady state plasma concentration as a result of infusion dosing from Wetmore et al. (2012) and (2013). get wetmore oral equiv Converts a μ M steady state plasma concentration to an oral equivalent dose using the values from Wetmore et al. (2012) and (2013). monte carlo Runs a monte carlo simulation of a given model. parameterize 1comp Parameterizes the one compartment model. parameterize 3comp Parameterizes the three compartment model. parameterize pbpk Parameterizes the PBPK model. parameterize steadystate Parameterizes the three compartment steady state model.
solve 1comp Solves the one compartment model. solve 3comp Solves the three compartment model. solve pbpk Solves the PBPK model.

Table 1: List of functions in package.

4Model Hepatic Clearance Partition Coefficients Fraction Unbound Hematocrit Molecular Weight Ratio of Blood to Plasma Elimination Rate Volume of Distribution* Dynamic Prediction Steady State Prediction Plasma without Blood

PBPK Yes Yes Yes Yes Yes Yes No No Yes Yes NoOne Compartment No No No No Yes No Yes Yes Yes Yes Yes

Three Compartment Yes Yes Yes Yes Yes Yes No No Yes Yes No

Three Compartment Steady State Yes No Yes No Yes No No No No Yes Yes

Table 2: Model parameter and prediction comparison. *Partition coefficients needed in calculating volume of distirbution.



## 2 Methods

2.1 Models Used Three general models are included in the package. The first, being the most extensive, is referred to in the package as “pbpk” and contains separatetissue compartments for the gut, liver, lungs (unused for dosing in our examples), arteries, veins, and kidneys. The rest of the tissues in the body are aggregated into a single compartment termed “rest”. Tissue descriptors, including volume, blood flow, cellular fraction of total volume, water, lipid and protein fraction of cellular volume, and distribution of types of lipid are taken from the corrected table in Schmitt (2008) [18]. Tissue flows and volumes are scaled to cardiac output and body weights, respectively, from Davies and Morris (1993) [2], which also provides values for the glomerularfiltration rate and hematocrit. Absorption from the gut lumen into gut tissue is modeled as a first order process with a rate of 1/h. Venous blood from

5the gut and arterial blood feed separately into the liver, where metabolismis modeled with the “well-stirred” approximation [27, 4] with two other options available from Ito and Houston (2004) [5].

Tissue to unbound fraction in plasma partition coefficients are predicted using Schmitt’s method [18] from the in vitro fraction of unbound chemical in plasma, f

up , the predicted hydrophobicity (EPI Suite), and any acid/basedissociation constants available in the literature. We assume perfusionlimited tissue (i.e., tissue, red blood cells, and plasma come to equilibrium rapidly with respect to the flow of blood). A constant ratio of blood to plasma concentration is used throughout the body and is predicted using the hematocrit and the predicted partitioning between red blood cells andplasma from Schmitt (2008). With these assumptions, the chemical concentration in any tissue divided by that tissue’s partition coefficient is equal to the unbound fraction of chemical in the plasma of that tissue. The rest of body compartment is created by weighting the partition coefficients for all tissues described in Schmitt (2008) by their volumes.

The second, an empirical, “three-compartment” model assumes no tissuepartitioning, and features elimination due to metabolism of the parent compound in the liver and renal elimination by passive glomerular filtration. Hepatic metabolism was modeled using the “well-stirred” approximationfrom the flow into the liver and the in vitro measured values for fraction unbound, intrinsic hepatic clearance, and the density of hepatocytes in the liver (1.1 x 108 hepatocytes per gram liver and a liver density of 1.05 g/mL). This

6is the only model used exclusively in calculating the analytic steady state, and is referred to in the package as “3compartmentss”, where “ss” stands for steady state. The analytic solution to the 3compartmentss model is the same equation used for determining steady state plasma concentrations, C ss , in previous work [16, 25, 26, 27]. The model referred to in the package as “3compartment” and simulated with the function solve 3comp pbpk is avariation of the 3compartmentss model that includes the partition coefficients from Schmitt’s method.

Finally, the one compartment model uses an elimination rate derived from dividing the total clearance by the volume of distribution. The volume of distribution is calculated by summing the products of the volumes of the compartments in the pbpk model and their partition coefficients and is also used as the volume the compartment. The total clearance is calculated by summing the hepatic clearance from the well-stirred approximation and the clearance from the kidneys, assumed to be the product of the glomerular filtration rate and f up . The clearances for the one compartment model and the 3compartmentss model are identical.

All models contain a compartment for absorption with an absorption rate of 1/h.

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### 2.2 Model Equations

Figure 1: (A) Three compartment model and (B) PBPK model.
2.2.1 PBPK Model Equations

d dt Agutlumen = − kgutabs Agutlumen

d dt C gut =  kgutabs Agutlumen + Q gut ( C art − C

gut ) . V

gut

d dt C liver =  Q liver C

art + Q gut C

gut − ( Q liver + Q gut ) Rb2p f up K liver 2plasma C

liver − CL metabolism K liver 2plasma C

liver  . V

liver

d dt C ven =  Q liver + Q gut

K liver 2plasma C

liver + Q kidney Kkidney 2plasma C

kidney
+ Q rest K rest 2plasma C

rest  Rb 2p f up − Q cardiac C

ven  . V

ven

d dt C lung = Q cardiac ( C ven − C

lung )/ V

lung

d dt C art = Q cardiac ( C

lung − C

art )/ V art

8d dt C rest = Q rest  C

art − Rb 2p K rest 2plasma f

up  . V

rest

d dt C kidney =  Q kidney C art − Q Kidney Rb 2p Kkidney 2plasma f

up C kidney − Q gfr Kkidney 2plasma C

kidney  . V

kidney
2.2.2 Three Compartment Model Equations

d dt Agutlumen = − kgutabs Agutlumen d dt C gut =  kgutabs Agutlumen + Q gut Rb2p K rest 2plasma f

up C syscomp − Q gut C

gut  V

gut

d dt C liver =  Q gut C

portven + Q liver Rb2p

K rest 2plasma f

up C rest

−  ( Q liver + Q gut ) Rb 2p f up K liver 2plasma + CL metabolism

K liver 2plasma  C

liver  . V

liver

d dt C rest =  ( Q gut + Q liver ) Rb 2p

f up K liver 2plasma C

liver −  Rb2p

f up  Q gut + Q liver  + Q gfr  C rest K rest 2plasma  . V

rest
2.2.3 One Compartment Model Equations

d dt Agutlumen = − kgutabs Agutlumen

d dt C rest = kgutabs V dist Agutlumen − kelimination C rest

92.2.4 Steady State Three Compartment Model
C ss = kdose  f

up Q gfr + ( Q liver + Q gut ) f up CL metabolism ( Q liver + Q gut ) + f up CL metabolism 



### 2.3 In Vitro Chemical Data Two types of in vitro data are used to parameterize the PK models: CL int ,the intrinsic metabolic clearance of the parent compound by primary hepatocytes (loss of parent over time) in a well on a multi-compound plate [19]; and f up , the fraction of chemical unbound in the presence of plasma protein as assessed using rapid equilibrium dialysis (RED) in which two wells are separated by a membrane that is permeable by smaller molecules but prevents the plasma protein added to one well from migrating to the other well (the relative chemical concentration in the two linked wells gives the free fraction of chemical) [24]. For environmental chemicals, data on these two parameters are primarily from Wetmore et al. (2012) and Tonnelieret al. (2012) [22]. For pharmaceutical compounds these values are compiled from Obach et al. (1999) [11], Jones et al. (2002) [7], Naritomi et al. (2003) [10], Ito and Houston (2004), Riley et al. (2005) [15], Schmitt (2008),and Obach et al. (2008) [12]. Where available, acid/base dissociation constants are curated from this literature; otherwise chemicals are treated as neutral/unionized regardless of tissue pH.


### 2.4 Physicochemical dataPhysico-chemical properties (e.g., logP) using EPAs Estimation Program Interface (EPI) Suite (4.11, November, 2012, http://www.epa.gov/opptintr/ex10posure/pubs/episuite.htm). EPI Suite quantitative structure activity relationships (QSARs) were used to estimate physico-chemical properties if Simplified Molecular Input Line Entry System (SMILES) descriptions of chemical structure were available and the QSARs did not fail for that structure.In addition to QSAR model estimates, EPI Suite contains a database of experimentally obtained physico-chemical properties that were used in place ofestimated values when available. EPI Suite does not predict acid/base dissociation, so chemicals were assumed to be neutral unless data was available from the literature. Molecular weight and structure are determined from the DSStox database (http://www.epa.gov/dsstox) and hydrophobicity is predicted for all compounds using EPI Suite.


### 2.5 Determination of Steady State In our model, we can simulate discrete doses to approximate real world exposure scenarios. However, we do assume an equivalent constant infusion dosing in the calculation of the analytic steady state concentrations from the differential equations. The day a chemical reaches steady state is found by determining when the average concentration for a given day comes within 1% of the analytic solution. The concentration resulting from discrete dosing oscillates around the analytic steady state, and the lower the concentration is, the greater the oscillation is relative to the analytic concentration, as can be seen in the examples.

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### 2.6 Monte Carlo SamplerThe package contains the results in Wetmore et al. (2012) and (2013) obtained with the SimCYP population simulator [6], which uses Monte Carlo to simulate human variability in steady state plasma concentrations usingthe 3compartmentss model. Normal distributions, truncated to ensure positive values, were used with mean values drawn from the references above and a coefficient of variation of 0.2. Body weight, liver volume, blood flow to the liver, and CL int were all varied in this manner. The f up was drawn from a censored distribution with identical properties to the other distributions if the mean was above the limit of detection (assumed to be 1% unbound) and a uniform distribution between 0% and 1% unbound otherwise. Foreach chemical, 1000 different combinations of parameters were used to determine C

ss , allowing estimation of the 5%, median, and 95% quantiles forhumans but only the median for rats. These concentrations were determined with doses of 1 mg/kg bw/day but, assuming a linear concentration response, can be extrapolated to other doses with get wetmore css . Using get wetmore oral equiv , we can, in a reverse manner, back calculate the dose for a given concentration. The package also includes a function, calc mc css , that runs a Monte Carlo simulation of the 3compartmentss model, emulating the results in Wetmore et al. (2012) and (2013). This function implements the simulation with another function, monte carlo , that allows the user touse his own PK model or a model contained in the package besides the 3compartmentss model. Analogously to get wetmore oral equiv , get mc oral equivcalculates the oral equivalent dose through division of the given concentra12tion by the steady state concentration generated by calc mc css with a dose of 1 mg/kg bw/day.


## 3 Examples
3.1 Accessing and Changing Model Parameters VLiverPBPK allows the user to access and change the parameters used in each of the models. The models each contain their own parameterize function that generates a list of the parameters required by the model. For example, to get a list of parameters for the pbpk model of triclosan in a rat: R> parameters <- parameterize_pbpk(chem.name = "triclosan", + species = "rat") To see the effect a change in parameters has on the model, we can modify the desired entries in the list and use the new parameter list as an input for the parameters argument of a function that uses that model. For example, to change the f

up in the previous parameters list to 0.1 from the default of 0.005 and use it in a simulation of the pbpk model for a single dose of 1 mg/kg bw of triclosan in a rat: R> parameters["Fraction_unbound_plasma"] <- 0.1 R> out <- solve_pbpk(parameters=parameters) Individual parameters, such as the ratio of blood to plasma, total clearance, volume of distribution, hepatic clearance, and elimination rate, can also be calculated using the functions with the prefix calc followed by the parameter name and the same arguments as the above parameterize function.

13

### 3.2 Making Data Frames and Tables VLiverPBPK is capable of generating large amounts of data, and in order to compare chemicals or models, we need to construct tables or data frames. Suppose we want to look at how the steady state concentrations compare at 1 mg/kg bw/day for the pbpk model, the Monte Carlo simulation, and the Wetmore data, formatted as a data frame (used with ggplot2 in following examples). We can do this using a for loop and the as.data.frame and cheminfo functions. The intersection of get wetmore cheminfo and get cheminfo for the pbpk and 3compartmentss (used in calc mc css ) models contains all the CAS numbers that will work for all three functions. The for loop then binds the CAS number and function outputs into a row with cbind , and each row is added to the bottom of the data frame with rbind

R> table <- NULL R> for(this.cas in intersect(get_cheminfo(model="3compartmentss"), + intersect(get_cheminfo(model = "pbpk"),get_wetmore_cheminfo()))){ + this.row <- as.data.frame(this.cas) + this.row <- cbind(this.row,as.data.frame(calc_analytic_css( + chem.cas = this.cas,model = model))) + this.row <- cbind(this.row,as.data.frame(get_wetmore_css( + chem.cas = this.cas,which.quantile = .50))) + this.row <- cbind(this.row,as.data.frame(calc_mc_css( + chem.cas = this.cas,which.quantile = .50))) + table <- rbind(table,this.row) + }

14R> colnames(table) <- c("CAS","PBPK","Wetmore","MC")


### 3.3 Plotting 3.3.1 Days to Steady State Histogram Creating histograms can allow us to visualize how a given value varies across all of the chemicals contained within the package. To create a histogram using ggplot2 for the number of days to steady state for all chemicals with sufficient data, we must first set up a for loop with get cheminfo and calc css to generate a vector containing the data. Then the vector must be converted to a data frame before entering it into the ggplot command and adding all the additional features. The ggplot command is used to initially format the data for any type of figure in ggplot2 . We first enter the data frame we want to use in the plot and then the specific column or columns in aes . Then additional commands are added to the basic ggplot object. We can just as easily create a histogram containing the average or maximum steady state concentrations by typing avg or max in place of the.day

. R> days <- NULL R> for(this.cas in get_cheminfo(model="pbpk")){ + css.info <- calc_css(chem.cas=this.cas,model="pbpk") + days[["this.cas"]] <- css.info[["the.day"]] + } R> days.data <- as.data.frame(days) R> ggplot <- ggplot(days.data,aes(days)) R> hist <- ggplot + geom_histogram(fill="blue",binwidth=1/6) +

15+ scale_x_log10() + ylab("Number of Chemicals") + xlab("Days") + + theme(axis.text=element_text(size=13,type="bold"), + axis.title=element_text(size=15))

Figure 2: Days to steady state histogram.

3.3.2 Concentration vs Time
The function solve pbpk plots and returns values for each of the compartment concentrations vs. time, but we can make a plot with ggplot2 that includes a horizontal line through the y axis at the predicted steady state plasma or venous blood concentration for constant infusion dosing. We first turn the output into a data frame and enter it into the ggplot command along with the desired x and y axes in the aes argument. Then we use calc analytic css to get the steady state plasma concentration and enter it into geom hline as the y intercept and add all the other options to our ggplot object.

16R> out<-solve_pbpk(chem.name="Bisphenol A",days=25,doses.per.day=3) R> plot.data<-as.data.frame(out) R> ggplot <- ggplot(plot.data,aes(time,Cplasma)) R> css <- calc_analytic_css(chem.name="Bisphenol A") R> plot <- ggplot + geom_line() + geom_hline(yintercept=css) + + ylab("Plasma Concentration (uM)") + xlab("Day") + + theme(axis.text=element_text(size=16),axis.title=element_text(size=16), + plot.title=element_text(size=17)) + ggtitle("Bisphenol A") This example plots the concentration vs. time of 1 mg/kg bw/day of Bisphenol A broken into three doses per day. The same plots can be made for the other models by substituting one of the other two solve functions, solve 3comp pbpk or solve 1comp pk , for solve pbpk and setting the model argument of calc analytic css to the corresponding model. These three functions also have the option of simulating a single oral or i.v. dose and setting the initial values of each compartment with units matching the specified output units (defaults are μ M).

17Figure 3: Steady state plasma concentration at 3 doses per day, 1 mg/kg bw/day.

3.3.3 Average vs Maximum Concentration Using calc css and the same technique used to generate the histogram data,we can get a vector for the average concentration and the maximum concentration at steady state. Binding the columns together and converting it into a data frame will allow us to create another plot using ggplot2

. R> ggplot <- ggplot(data.frame,aes(avg,max)) R> Plot <- ggplot + geom_point() + geom_abline() + scale_x_log10() + + scale_y_log10() + xlab("Average Concentration at Steady State (uM)") + + ylab("Max Concentration at Steady State (uM)") + + theme(axis.text=element_text(size=16),axis.title=element_text(size=16))

18Figure 4: Average vs. maximum concentration at steady state for 3 doses per day, 1 mg/kg bw/day.


### 3.4 Calculating AUC, Peak, and Mean Values
The function calc stats gives the option to calculate the area under the curve (AUC), peak, or mean concentrations of any of the solve functions. If a chemical name or CAS number is specified, it will calculate the specified statistics for that chemical, and if not, it will calculate the values for all chemicals with sufficient data for the given model. To calculate the peak statistics for all chemicals that work with the pbpk model simulated for 10 days at 1 mg/kg bw/day with 3 doses per day and a list containing the AUC, peak, and mean for a single 1 mg dose of ibuprofen over 10 days, we have:
R> all.peak.stats <- calc_stats(days=10,doses.per.day=3,stats="peak")

19R> ibuprofen.stats <- calc_stats(days=10,chem.name="Ibuprofen")


### 3.5 Monte Carlo SamplerThe Monte Carlo data and simulations in the package use the 3compartmentss model. The published results from Wetmore et al. (2012) and (2013) are accessible with get wetmore css , and calc mc css runs an identical simulation. However, with this simulation, we can specify all of the parameters and any quantile we desire. Below are examples of these two functions, comparing the median of the Wetmore steady state μ M concentration in humans for 1 mg/kg bw/day of Bisphenol A with the calc mc css simulation but with probability distributions containing half the standard deviation, half the limit of detection for the fraction unbound to plasma, and double the number of samples of the default Wetmore parameters.
R> get_wetmore_css(chem.cas="80-05-7",daily.dose=1, + which.quantile=0.5,output.units="uM")

R> calc_mc_css(chem.cas="80-05-7",daily.dose=1,which.quantile=0.5, + censored.params=list(fub=list(cv=0.1,lod=0.005)),vary.params= + list(BW=0.1,liver.volume.per.kgBW=0.1,QGFRc=0.1,Qhc=0.1, + million.cells.per.gliver=0.1,CLint=0.1),output.units="uM",samples=2000)
The oral equivalent functions convert a plasma concentrations into a dose. Below is an example of a 50 μ M plasma concentration of Bisphenol A converted to an oral equivalent dose using the Wetmore data for the 95 th quantile of humans. We can call calc mc oral equiv in the same manner, passing

20additional arguments to calc mc css within the function and specifying any quantile we want. R> get_wetmore_oral_equiv(50,chem.cas="80-05-7")


### 3.6 Export Functions The package has the ability to export the multiple compartment pbpk model to sbml and jarnac files. Both functions, export pbpk sbml and export pbpk 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 typical call of an export function for a dose of 1 given to a rat. R> export_pbpk_sbml(chem.name="Bisphenol A",species="Rat", + initial.amounts=list(Agutlumen=1),filename="PBPKmodel.xml")



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