Evaluation of a rapid, generic human gestational dose model

Abstract

Chemical risk assessment considers potentially susceptible populations including pregnant women and developing fetuses. Humans encounter thousands of chemicals in their environments, few of which have been fully characterized. Toxicokinetic (TK) information is needed to relate chemical exposure to potentially bioactive tissue concentrations. Observational data describing human gestational exposures are unavailable for most chemicals, but physiologically based TK (PBTK) models estimate such exposures. Development of chemical-specific PBTK models requires considerable time and resources. As an alternative, generic PBTK approaches describe a standardized physiology and characterize chemicals with a set of standard physical and TK descriptors – primarily plasma protein binding and hepatic clearance. Here we report and evaluate a generic PBTK model of a human mother and developing fetus. We used a published set of formulas describing the major anatomical and physiological changes that occur during pregnancy to augment the High-Throughput Toxicokinetics (httk) software package. We simulated the ratio of concentrations in maternal and fetal plasma and compared to literature in vivo measurements. We evaluated the model with literature in vivo time-course measurements of maternal plasma concentrations in pregnant and non-pregnant women. Finally, we prioritized chemicals measured in maternal serum based on predicted fetal brain concentrations. This new model can be used for TK simulations of 859 chemicals with existing human-specific in vitro TK data as well as any new chemicals for which such data become available. This gestational model may allow for in vitro to in vivo extrapolation of point of departure doses relevant to reproductive and developmental toxicity.

1. Introduction

Human health chemical risk assessment involves consideration of potentially exposed or susceptible subpopulations, including pregnant women ([24], 2016). However, pregnant women are underrepresented in pharmaceutical clinical studies [29], and non-therapeutic chemicals are rarely studied in humans at any life-stage [13,32,43]. In any case, the developing fetus is likely to be exposed to the same environmental chemicals as the mother [3]. To assess the risk posed by a chemical, toxicokinetic (TK) information is needed to relate chemical exposure to doses delivered internally (to tissues and organs) as a prerequisite to potential toxicity [61]. Toxicokinetics describes chemical absorption, distribution, metabolism, and excretion by the body (O’Flaherty, [62, 102]) and allows for predictions of the tissue concentrations resulting from external doses whether they are controlled (for example, doses administered in a clinical trial or animal toxicity study [88]) or uncontrolled (for example, exposures to chemicals in the environment CDC, [15]).

Mathematical TK models can facilitate prediction of both maternal and fetal tissue chemical exposure [5,52]. Physiologically-based TK (PBTK) models, in particular, offer an attractive option for extrapolating information in applications such as human health risk assessment [16, 17,22]. Barton et al. [6] identified many ways in which PBTK models may be used for extrapolation: from in vitro to in vivo conditions [22,76], across exposure routes [19], across species [19,86], across chemicals [42,65], across populations [18,7,77,87], and across life-stages [20,52]. Before applying such a model for extrapolation and risk assessment, however, one should evaluate the model’s ability to predict quantities of interest, such as blood concentrations [16].

During the developmental stages experienced during gestation, humans may be susceptible to chemical perturbation, making the ability of PBTK to predict transient chemical exposures critical for understanding reproductive toxicology [4,5,82,9]. However, many parameters treated as constants in traditional PBTK models must be formulated as time-varying quantities when modeling pregnancy. That is, when simulating adult animals, relatively small changes in the values of parameters such as body mass, average cardiac output, and hematocrit, may be treated as negligible and the parameters treated as constants, but when simulating pregnant animals and developing fetuses for a period longer than a few days, the same parameters may undergo relatively large changes and should not be treated as constants. A variety of PBTK models have been developed to address animal and human gestational exposure [103,30,33,51,52]. Kapraun et al. [44] recently used a consistent set of statistical methods to build new empirical models for human maternal and fetal tissue volumes, blood flow rates, and other changing factors, and compared these with previously published and comparable models.

Here we describe a new time-varying maternal-fetal PBTK model that has been incorporated into the high-throughput toxicokinetics (httk) R software package. We define the methodology “HTTK” as the combination of generic TK models with standardized in vitro-measured and in silico-predicted chemical-specific parameters [12]. We distinguish between the specific R package “httk” and the methodology of high throughput toxicokinetics (HTTK) using capitalization. The R package is free, open-source software that provides TK models and data for a large collection of chemicals [67]. We used the models of Kapraun et al. [44] to estimate values of time-varying parameters, including blood flow rates through various fetal blood routes that have no counterparts in the typical adult such as the foramen ovale and the umbilical cord. While the current version of the httk pregnancy model is only available for humans, it can be parameterized for nearly 1000 chemicals for which in vitro measures of toxicokinetics are available [100,78,90,95,99]. The PBTK model simulates gestational week 13 until parturition using empirical models for time-dependent anatomical and physiological quantities. We evaluated the PBTK model by comparing its predictions to data from available observational studies and used these comparisons to quantify uncertainty in model predictions.

2. Materials and methods

2.1. Software implementation

All data processing and analyses described herein were performed using R version 4.2.0 (R [71]) on a Dell Latitude computer with Intel Core i7–7600 U 2.80 GHz CPI and 16 GB of RAM running Microsoft Windows 10. Dynamic models were implemented using the MCSim model specification language [11] and were subsequently translated to C and compiled for use in R. Supplemental source code and data files are freely available from the Comprehensive R Archive Network (https://CRAN.R-project.org/package=httk). R package “httk” [67] version 2.1.0 includes the model (“fetal_pbtk”) and evaluation data described here as well as the vignette “Kapraun2022” which provides the scripts for generating all figures and analysis.

2.2. Model overview

To investigate the kinetics of a given xenobiotic substance in a human mother and fetus, we have developed a general PBTK model that accounts for changing tissue volumes and blood flow rates, as a well as fundamental differences between the maternal (adult) and fetal circulatory systems. Fig. 1, which provides a schematic representation of this maternal-fetal PBTK model, adheres to the following conventions:

Fig. 1.

Schematic diagram of a PBTK model representing a human mother and fetus. Blood flow rates are represented as arrows labeled with symbols of the form $Q_y^x$ and tissue compartments and associated blood-tissue exchange areas are represented as rectangles. A chemical can enter the system through the maternal gut lumen after ingestion, and substance can be cleared from the system through maternal liver metabolism and glomerular filtration.

• Rectangles represent physiologically based compartments (which correspond to distinct tissues and organs in the mother and fetus);

• Rectangles with darkest (red) fill at the bottom of the diagram indicate heart compartments in the fetus;

• Solid line (black) arrows connecting boxes depict permanent blood flow routes;

• Dotted line (red) arrows connecting boxes depict temporary blood flow routes in the fetus;

• Wide, lighter colored (yellow) arrows indicate routes of elimination (for example, metabolism and renal clearance); and

• Wide, darker colored (purple) arrows indicate routes of dosing (for example, oral ingestion).

There is no direct mixing of maternal and fetal blood, so the blood circuits in the “maternal blood” (upper) and “fetal blood” (lower) parts of the schematic do not connect to one another. In the case of the placenta, through which both fetal and maternal blood pass, total blood flow is conserved separately on the maternal and fetal “sides” of the compartment.

We use mathematical symbols of the forms $A_y^x$, $Q_y^x$, and $V_y^x$ in Table 1 and elsewhere to represent amounts of substance in compartments, blood flow rates, and compartment volumes, respectively. For any given symbol, the superscript (x) is a one-letter code indicating whether the quantity represented describes an attribute of the mother (“m”) or the fetus (“f”). In most cases, the subscript (y) is a four-letter code indicating the compartment into which the flow occurs or for which we consider the volume of the compartment or amount of substance within the compartment. We provide a complete list of the four-letter subscript codes in Table 1. For symbols representing blood flow rates through temporary blood routes in the fetus, the subscript y is an upper-case two-letter code representing the specific blood route. All the subscript codes utilized for this case are shown in Table 2.

Table 1.

Four-letter codes used to represent compartments in the PBTK model. These codes appear as subscripts in mathematical symbols.

Code (y)	Compartment
artb	Arterial blood
venb	Venous blood
plas	Plasma
rbcs	Red blood cells
lung	Lungs
gutl	Gut lumen
gutx	Gut
livr	Liver
adip	Adipose
thyr	Thyroid
kidn	Kidneys
rest	“Rest” of body
plac	Placenta
amnf	Amniotic fluid
bran	Brain
ratm	Right atrium
rvtl	Right ventricle
latm	Left atrium
lvtl	Left ventricle

Table 2.

Two-letter codes used to represent (transient) fetal blood routes in the PBTK model. These codes appear as subscripts in mathematical symbols.

Code (y)	Blood Route
DA	Ductus arteriosus
DV	Ductus venosus
FO	Foramen ovale

2.3. Time-dependent parameters

In this section, we specify the time-dependent parameters in our PBTK model, including body masses W^m and W^f, blood flow rates $\left\{Q_y^x\right\}$, compartment volumes $\left\{V_y^x\right\}$, and hematocrits H^m and H^f. Some parameter values can change substantially during pregnancy and gestation, and so we do not treat them as having constant values. For example, according to the models proposed by Kapraun et al. [44], maternal and fetal body masses increase by 16 % and 12,547 %, respectively, between gestational ages 13 weeks and 40 weeks.

We used the empirical models of Kapraun et al. [44] for the body masses of the mother and fetus as shown in Table 3. Note that the body mass of the mother includes the mass of the fetus, placenta, and all other products of conception. We also used the empirical models of Kapraun et al. [44] for the maternal and fetal hematocrits as shown in Table 4.

Table 3.

Time-dependent body masses (kg). The symbol t denotes gestational age in weeks. These formulas are attributed to [44].

Parameter	Symbol	Formula
Maternal mass	BW	$W^{\text{m}}(t)=61.103-0.010614t+0.029161t^2-(5.0203\times {10}^{-4})t^3$
Fetal mass	fBW	$W^{\text{f}}(t)=\left(1.8282\times {10}^{-6}\right)\times \exp \left[\frac{1.1735}{0.077577}\times (1-\exp [-0.077577t])\right]$

Table 4.

Time-dependent hematocrits expressed as percentages (with values between 0 and 100). The symbol t denotes gestational age in weeks. These formulas are attributed to [44].

Parameter	Symbol	Formula
Maternal hematocrit	hematocrit	$H^{\text{m}}(t)=39.192-0.10562t-(7.1045\times {10}^{-4})t^2$
Fetal hematocrit	fhematocrit	$H^{\text{f}}(t)=4.5061t-0.18487t^2+0.0026766t^3$

As shown in Table 5, the volumes for most of the maternal compartments were calculated using the models of Kapraun et al. [44]. For the maternal arterial and venous blood compartments, however, we used fixed proportions of the total blood volume of the maternal blood at conception. This reflects two fundamental assumptions. First, while the total blood volume of a pregnant woman increases as pregnancy progresses, we assume that most of the new blood resides in the placenta, adipose tissue, and other tissues that grow in volume during pregnancy and that the core arterial and venous blood vessels, themselves, do not increase in size. That is, we assume that the volumes of the arterial and venous blood compartments do not increase and thus the quantities we denote $V_{\text{artb}}^{\text{m}}$ and $V_{\text{venb}}^{\text{m}}$ remain constant throughout pregnancy. Second, we assume that 16 % and 59.5 % of the total blood volume resides in arterial blood and venous blood, respectively. This is because 16 % of the total blood volume of an adult human resides in the aorta and the arteries (excluding vessels comprising the pulmonary circulatory system) and 59.5 % of this total volume resides in the veins (again excluding the pulmonary circulatory system) ICRP, [39]. Alternative assumptions have been applied for calculating volumes of arterial and venous blood for a human PBTK model [8,48], but there is general agreement that volume of the venous blood is several times larger than that of the arterial blood.

Table 5.

Volumes (L) of maternal compartments. In formulas representing volumes that change during pregnancy, the symbol t denotes gestational age in weeks. For compartment volumes that do not change during pregnancy, values are given as constant values (L) based on an assumed pre-pregnant body mass of W^m(0) = 61.103 kg.

Compartment	Symbol	Formula
Lungs	Vlung	$V_{\text{lung}}^{\text{m}}=0.91945$
Gut	Vgut	$V_{\text{gutx}}^{\text{m}}=1.1110$
Liver	Vliver	$V_{\text{livr}}^{\text{m}}=1.3559$
Adipose	Vadipose	$V_{\text{adip}}^{\text{m}}(t)=\frac{1}{0.95}\cdot (17.067-0.14937t)$
Thyroid	Vthyroid	$V_{\text{thyr}}^{\text{m}}=0.016469$
Kidneys	Vkidney	$V_{\text{kidn}}^{\text{m}}=0.26653$
Plasma	Vplasma	$V_{\text{plas}}^{\text{m}}(t)=\frac{1.2406}{1+\exp [-0.31338(t-17.813)]}+2.4958$
Red blood cells	Vrbcs	$V_{\text{rbcs}}^{\text{m}}(t)=\frac{H^{\text{m}}(t)}{100-H^{\text{m}}(t)}\cdot V_{\text{plas}}^{\text{m}}(t)$
Arterial blood*	Vart	$V_{\text{artb}}^{\text{m}}=0.16\cdot \left(V_{\text{plas}}^{\text{m}}(0)+V_{\text{rbcs}}^{\text{m}}(0)\right)$
Venous blood*	Vven	$V_{\text{venb}}^{\text{m}}=0.595\cdot \left(V_{\text{plas}}^{\text{m}}(0)+V_{\text{rbcs}}^{\text{m}}(0)\right)$
Placenta	Vplacenta	$V_{\text{plac}}^{\text{m}}(t)=0.001\cdot \left(-1.7646t+0.91775t^2-0.011543t^3\right)$
Amniotic fluid	Vamnf	$V_{\text{amnf}}^{\text{m}}(t)=\frac{0.001\cdot 822.34}{1+\exp [-0.26988(t-20.150)]}$
Fat-free mass	Vffmx	$V_{\text{ffmx}}^{\text{m}}(t)=\frac{1}{1.1}\left[W^{\text{m}}(t)-0.95\cdot V_{\text{adip}}^{\text{m}}(t)-(W^f(t)+1.02\cdot V_{\text{plac}}^{\text{m}}(t)+1.01\cdot V_{\text{amnf}}^{\text{m}}(t))\right]$
All except adipose*	Vallx	$V_{\text{allx}}^{\text{m}}=V_{\text{artb}}^{\text{m}}+V_{\text{venb}}^{\text{m}}+V_{\text{thyr}}^{\text{m}}+V_{\text{kidn}}^{\text{m}}+V_{\text{gutx}}^{\text{m}}+V_{\text{livr}}^{\text{m}}+V_{\text{lung}}^{\text{m}}$
Rest of body	Vrest	$V_{\text{rest}}^{\text{m}}(t)=V_{\text{ffmx}}^{\text{m}}(t)-V_{\text{allx}}^{\text{m}}$

^“*”All formulas and values given here, except those labeled with an, are attributed to [44].

Formulas describing blood flow rates to maternal compartments are shown in Table 6. All these formulas are attributed to Kapraun et al. [44]. Table 7 shows that volumes for most of the fetal compartments were calculated using the models of Kapraun et al. [44]. Note that the blood compartments of the fetus are assumed to grow in volume as pregnancy and gestation proceed while those of the mother are not. To obtain values for the volumes of the fetal arterial blood and venous blood compartments, we assumed (as for the corresponding maternal blood compartments) that these volumes can be computed as (constant) proportions of the total blood volume of the fetus (which increases as the fetus grows). According to Kiserud [45], blood volume accounts for approximately 10–12 % of the mass of a human fetus; in a human adult, on the other hand, blood makes up only 7–8 % of the total body mass ([45], ICRP, 2002). The reason for this considerable discrepancy is that a large proportion of the fetal blood occupies the placenta at any given moment; the placenta contains both maternal and fetal blood, and the two do not directly mix with one another. For the volume of blood contained within the fetal body, Kiserud [45] provided an estimate of 80 milliliters of blood per kilogram of fetal body mass. This ratio (80 mL/kg) is only marginally higher than that observed in adults. For the maternal blood compartments, 16 % the total blood volume was apportioned to arterial blood and 59.5 % to venous blood, yielding the formulas shown in Table 7. We also identified and corrected a typographical error in Equation 35 of Kapraun et al. [44] (by referring to their Table 20) to arrive at the formula for fetal kidney volume shown in Table 7.

Table 6.

Blood flow rates (L/d) to maternal compartments. The symbol t denotes gestational age in weeks. These formulas are attributed to [44].

Compartment	Symbol	Formula
Arterial Blood	Qcardiac	$Q_{\text{artb}}^{\text{m}}(t)=24\cdot (301.78+3.2512t+0.15947t^2-0.0047059t^3)$
Gut	Qgut	$Q_{\text{gutx}}^{\text{m}}(t)=0.01\cdot \left[17.0+\left(\frac{12.5-17.0}{40}\right)t\right]\cdot Q_{\text{artb}}^{\text{m}}(t)$
Liver	Qliver	$Q_{\text{livr}}^{\text{m}}(t)=0.01\cdot \left[27.0+\left(\frac{20.0-27.0}{40}\right)t\right]\cdot Q_{\text{artb}}^{\text{m}}(t)$
Adipose	Qadipose	$Q_{\text{adip}}^{\text{m}}(t)=0.01\cdot \left[8.5+\left(\frac{7.8-8.5}{40}\right)t\right]\cdot Q_{\text{artb}}^{\text{m}}(t)$
Thyroid	Qthyroid	$Q_{\text{thyr}}^{\text{m}}(t)=0.01\cdot \left[1.5+\left(\frac{1.1-1.5}{40}\right)t\right]\cdot Q_{\text{artb}}^{\text{m}}(t)$
Kidney	Qkidney	$Q_{\text{kidn}}^{\text{m}}(t)=24\cdot (53.248+3.6447t-0.15357t^2+0.0016968t^3)$
Placenta	Qplacenta	$Q_{\text{plac}}^{\text{m}}(t)=24\cdot 0.059176\cdot V_{\text{plac}}^{\text{m}}(t)$
Rest of body	Qrest	$Q_{\text{rest}}^{\text{m}}(t)=Q_{\text{artb}}^{\text{m}}(t)-(Q_{\text{gutx}}^{\text{m}}(t)+Q_{\text{livr}}^{\text{m}}(t)+Q_{\text{adip}}^{\text{m}}(t)+Q_{\text{thyr}}^{\text{m}}(t)+Q_{\text{kidn}}^{\text{m}}(t))$

Table 7.

Volumes (L) of fetal compartments. The symbol t denotes gestational age in weeks.

Compartment	Symbol	Formula
Liver	Vfliver	$V_{\text{livr}}^{\text{f}}(t)=\frac{0.001}{1.05}\cdot \left(0.0074774\cdot \exp \left[\frac{0.65856}{0.061662}\cdot (1-\exp [-0.061662t])\right]\right)$
Gut	Vfgut	$V_{\text{gutx}}^{\text{f}}(t)=\frac{0.001}{1.045}\cdot \left(\left(8.1828\times {10}^{-4}\right)\cdot \exp \left[\frac{0.65028}{0.047724}\cdot (1-\exp [-0.047724t])\right]\right)$
Kidney†	Vfkidney	$V_{\text{kidn}}^{\text{f}}(t)=\frac{0.001}{1.05}\cdot \left(\left(6.3327\times {10}^{-5}\right)\cdot \exp \left[\frac{1.0409}{0.076435}\cdot (1-\exp [-0.076435t])\right]\right)$
Thyroid	Vfthyroid	$V_{\text{thyr}}^{\text{f}}(t)=\frac{0.001}{1.05}\cdot \left(0.0038483\cdot \exp \left[\frac{0.30799}{0.039800}\cdot (1-\exp [-0.039800t])\right]\right)$
Brain	Vfbrain	$V_{\text{bran}}^{\text{f}}(t)=\frac{0.001}{1.04}\cdot \left(0.01574\cdot \exp \left[\frac{0.70707}{0.064827}\cdot (1-\exp [-0.064827t])\right]\right)$
Lung	Vflung	$V_{\text{lung}}^{\text{f}}(t)=\frac{0.001}{1.05}\cdot \left(\left(3.0454\times {10}^{-4}\right)\cdot \exp \left[\frac{1.0667}{0.084604}\cdot (1-\exp [-0.084604t])\right]\right)$
Arterial blood*	Vfart	$V_{\text{artb}}^{\text{f}}(t)=0.001\cdot 0.16\cdot 80\cdot W^{\text{f}}(t)$
Venous blood*	Vfven	$V_{\text{venb}}^{\text{f}}(t)=0.001\cdot 0.595\cdot 80\cdot W^{\text{f}}(t)$
Rest of body*	Vfrest	$V_{\text{rest}}^{\text{f}}(t)=W^{\text{f}}(t)-\left(V_{\text{artb}}^{\text{f}}(t)+V_{\text{venb}}^{\text{f}}(t)+V_{\text{livr}}^{\text{f}}(t)+V_{\text{gutx}}^{\text{f}}(t)+V_{\text{kidn}}^{\text{f}}(t)+V_{\text{thyr}}^{\text{f}}(t)+V_{\text{bran}}^{\text{f}}(t)+V_{\text{lung}}^{\text{f}}(t)\right)$

^“*”All formulas and values given here, except those labeled with an, are attributed to [44].

^“†”The formulas labeled with an include corrections to typographical errors identified in formulas of [44].

Formulas describing blood flow rates to fetal compartments are shown in Table 8. All these formulas are attributed to Kapraun et al. [44]. We assumed that the compartments representing the chambers of the fetal heart in Fig. 1 can be treated as “nodes” at which blood flows converge and/or diverge rather than storage compartments in which the substance may partition into tissue. Thus, the fetal heart and lung compartments illustrated in Fig. 1 can be effectively simplified as shown in Fig. 2.

Table 8.

Blood flow rates (L/d) to fetal compartments. The symbol t denotes gestational age in weeks. These formulas are attributed to [44].

Compartment	Symbol	Formula
Right Ventricle	Qfrvtl	$Q_{\text{rvtl}}^{\text{f}}(t)=60\cdot 24\cdot 0.001\cdot \frac{2466.5}{1+\exp [-0.14837\cdot (t-43.108)]}$
Left Ventricle	Qflvtl	$Q_{\text{lvtl}}^{\text{f}}(t)=60\cdot 24\cdot 0.001\cdot \frac{506.30}{1+\exp [-0.21916\cdot (t-30.231)]}$
Ductus Arteriosus	Qfda	$Q_{\text{DA}}^{\text{f}}(t)=60\cdot 24\cdot 0.001\cdot \frac{1125.3}{1+\exp [-0.18031\cdot (t-35.939)]}$
Arterial Blood	Qfcardiac	$Q_{\text{artb}}^{\text{f}}(t)=Q_{\text{lvtl}}^{\text{f}}(t)+Q_{\text{DA}}^{\text{f}}(t)$
Lung	Qflung	$Q_{\text{lung}}^{\text{f}}(t)=Q_{\text{rvtl}}^{\text{f}}(t)-Q_{\text{DA}}^{\text{f}}(t)$
Placenta	Qfplacenta	$Q_{\text{plac}}^{\text{f}}(t)=60\cdot 24\cdot 0.001\cdot \frac{262.20}{1+\exp [-0.22183\cdot (t-28.784)]}$
Ductus Venosus	Qfdv	$Q_{\text{DV}}^{\text{f}}(t)=60\cdot 24\cdot 0.001\cdot \left(1.892\cdot \exp \left[\frac{0.098249}{0.0064374}\cdot (1-\exp [-0.0064374t])\right]\right)$
Gut	Qfgut	$Q_{\text{gutx}}^{\text{f}}(t)=\frac{6.8}{75}\cdot \left(1-\frac{Q_{\text{plac}}^{\text{f}}(t)}{Q_{\text{artb}}^{\text{f}}(t)}\right)\cdot Q_{\text{artb}}^{\text{f}}(t)$
Kidney	Qfkidney	$Q_{\text{kidn}}^{\text{f}}(t)=\frac{5.4}{75}\cdot \left(1-\frac{Q_{\text{plac}}^{\text{f}}(t)}{Q_{\text{artb}}^{\text{f}}(t)}\right)\cdot Q_{\text{artb}}^{\text{f}}(t)$
Brain	Qfbrain	$Q_{\text{bran}}^{\text{f}}(t)=\frac{14.3}{75}\cdot \left(1-\frac{Q_{\text{plac}}^{\text{f}}(t)}{Q_{\text{artb}}^{\text{f}}(t)}\right)\cdot Q_{\text{artb}}^{\text{f}}(t)$
Liver	Qfliver	$Q_{\text{livr}}^{\text{f}}(t)=\frac{6.5}{54}\cdot \left(1-\frac{26.5}{75}\right)\cdot \left(1-\frac{Q_{\text{plac}}^{\text{f}}(t)}{Q_{\text{artb}}^{\text{f}}(t)}\right)\cdot Q_{\text{artb}}^{\text{f}}(t)$
Thyroid	Qfthyroid	$Q_{\text{thyr}}^{\text{f}}(t)=\frac{1.5}{54}\cdot \left(1-\frac{26.5}{75}\right)\cdot \left(1-\frac{Q_{\text{plac}}^{\text{f}}(t)}{Q_{\text{artb}}^{\text{f}}(t)}\right)\cdot Q_{\text{artb}}^{\text{f}}(t)$
Rest of Body	Qfrest	$Q_{\text{rest}}^{\text{f}}(t)=Q_{\text{artb}}^{\text{f}}(t)-\left(Q_{\text{plac}}^{\text{f}}(t)+Q_{\text{livr}}^{\text{f}}(t)+Q_{\text{gutx}}^{\text{f}}(t)+Q_{\text{kidn}}^{\text{f}}(t)+Q_{\text{thyr}}^{\text{f}}(t)+Q_{\text{bran}}^{\text{f}}(t)\right)$
Lung Bypass	Qfbypass	$Q_{\text{lbyp}}^{\text{f}}(t)=Q_{\text{artb}}^{\text{f}}(t)-Q_{\text{lung}}^{\text{f}}(t)$

Fig. 2.

Simplified partial schematic diagram illustrating effective blood flows in the vicinity of the fetal heart. Compare with the lower portion of Fig. 1.

The ratio of the concentrations of substance in whole blood and plasma, is calculated as

$$R_{\text{b}:\text{p}}^x(t)=1-H^x(t)+H^x(t)\cdot K_{\text{rbcs}}^x\cdot f_{\text{up}}^x,$$

where H^x(t) is the hematocrit, $K_{\text{rbcs}}^x$ is the constant partition coefficient describing the ratio of concentrations in red blood cells and plasma, and $f_{\text{up}}^x$ is the fraction of substance available (or “unbound”) in the plasma [67]. As with other symbols in this manuscript, the superscript x in the symbol $R_{\text{b:p}}^x$ indicates whether the ratio is for the mother (“m”) or fetus (“f”). The parameters $f_{\text{up}}^{\text{m}}$ and $f_{\text{up}}^{\text{f}}$ and partition coefficients (including $K_{\text{rbcs}}^x$) are described in more detail in the next section. The symbols used for the blood-to-plasma concentration ratio parameters are shown in Table 9.

Table 9.

Other time-varying parameters in the PBTK model. Conventions for the superscript x are defined in the text.

Mathematical Symbol	Code Symbol	Units	Description
$R_{\text{b}:\text{p}}^x(t)$	Rblood2plasma	—	Ratio of concentrations of substance in whole blood and plasma at time t.
$k_{\text{kidn}}^{\text{m}}(t)$	Qgfr	L/d	Rate of glomerular filtration (in the maternal kidney) at time t.

The time-dependent glomerular filtration rate, denoted $k_{\text{kidn}}^{\text{m}}(t)$, is given by Equation 26 of Kapraun et al. [44]. The glomerular filtration rate (L/d) is multiplied by the concentration (mg/L) of unbound chemical in the blood leaving the kidney compartment to determine an excretion rate (mg/d) from the kidney. The symbols used for this glomerular filtration rate parameter are shown in Table 9. We remark that glomerular filtration in the fetus is limited, with the fetal glomerular filtration rate only about 5 % of the adult rate at gestation week 36 [35]. Therefore, in developing our model, we assumed that there is no glomerular filtration in the fetus.

2.4. Constant parameters

All model parameters not described in the preceding section are treated as constants for any given chemical. The symbols used to represent these constant parameters are provided in Table 10.

Table 10.

Constant parameters in the PBTK model. Conventions for the superscript x and subscript y are defined in the text.

Mathematical Symbol	Code Symbol	Units	Description
$k_{\text{gutl}}^x$	kgutabs	d−1	First-order rate constant describing absorption from the gut lumen into the gut tissue.
$f_{\text{up}}^x$	Fraction_unbound_plasma	—	Fraction of substance that is “unbound” in the plasma.
$K_y^x$	K[tissue]2pu	—	Partition coefficient representing the equilibrium ratio of concentration of substance in compartment (or “tissue”) y to concentration of unbound substance in the plasma.
$k_{\text{livr}}^{\text{m}}$	Clmetabolism	L/d	Rate of metabolic clearance (in the maternal liver).

Absorption from gut lumen to gut tissue is assumed to be a first-order process with the rate of absorption equal to the amount in the gut lumen multiplied by a constant of proportionality denoted $k_{\text{gutl}}^x$. In the absence of model for predicting this rate, the median rate of absorption (2.2 1/h) observed by Wambaugh et al. [93] for 37 structurally diverse chemicals was used (this is the httk default).

The fraction of substance available (or “unbound”) in the maternal plasma, denoted $f_{\text{up}}^{\text{m}}$, is assumed to be constant and is calculated based on in vitro measured plasma protein binding data for humans [100]. Our PBTK model assumes that $f_{\text{up}}^{\text{m}}$ influences many key processes, including rate of metabolism (restrictive clearance), glomerular filtration (passive kidney excretion), and tissue partitioning [66]. Fraction unbound in plasma is a function of multiple proteins in plasma and kinetics including both on- and off-binding rates. We approximate the kinetics as steady-state – meaning that the on- and off-binding must be relatively rapid such that a constant fraction $f_{\text{up}}^{\text{m}}$ is maintained. For HTTK, the adult human value of $f_{\text{up}}^{\text{m}}$ is measured in vitro using plasma proteins from a pool of human donors [100]. We make different estimates for fetal and maternal values of $f_{\text{up}}^{\text{m}}$ depending on which plasma protein is assumed to dominate in binding the chemical.

Unfortunately, the data available from the in vitro plasma protein binding experiments do not determine which protein is responsible for the observed binding, so we follow McNamara and Meiman [58] and assume that neutral and negatively charged compounds (that is, acids) bind to albumin and that positively charged compounds (that is, bases) bind to alpha-1-acid glycoprotein (AAG). Because the concentrations of both albumin and AAG vary between adults and fetuses, f_up is expected to vary between the mother and fetus. Further, we were not aware of any time-varying models for the protein components of blood comparable to those used elsewhere in our PBTK model and so we assumed the ratio of fetal to maternal binding is constant in time.

The fetal fraction unbound ($f_{\text{up}}^{\text{f}}$) is calculated from the maternal fraction unbound and the serum protein concentration ratio in infants vs. mothers based on Eq. 6 of McNamara and Meiman [58]; that is,

$$f_{\text{up}}^{\text{f}}=\frac{f_{\text{up}}^{\text{m}}}{f_{\text{up}}^{\text{m}}+\left(P^{\text{f}}/P^{\text{m}}\right)\times \left(1-f_{\text{up}}^{\text{m}}\right)},$$ (1)

where the maternal fraction unbound, $f_{\text{up}}^{\text{m}}$, is assumed to be equal to the in vitro measured value for fraction unbound in plasma and the ratio of protein concentrations P^f_/P^m depends on the identity of the dominant binding protein for the chemical (assumed to be either albumin or AAG). Lacking data to model the gestational kinetics of albumin and AAG concentrations, we used the concentrations at birth McNamara and Alcorn, [57] to calculate a constant $f_{\text{up}}^{\text{f}}$, using P^f_/P^m = 0.777 for albumin and P^f_/P^m = 0.456 for AAG. We determine the charge state of a compound separately for maternal and fetal plasma as a function of plasma pH (7.38 for maternal and 7.28 for fetal [47]) and chemical-specific predictions for ionization affinity (that is, pKa [85]) using the “httk” function “calc_ionization” [66]. If the fraction of a chemical that is predicted to be in positive ionic form is greater than 50 %, we treat the chemical as a base (which is in its conjugate acid form) and use only the maternal-to-infant ratio of AAG concentrations. Otherwise, we use the ratio of albumin concentrations, presuming that both neutral and acidic compounds preferentially bind to albumin. Fig. 3 shows results obtained when using Eq. 1 to compare fetal and maternal values of fraction unbound in plasma for all 856 chemicals for which in vitro HTTK data were available to parameterize our PBTK model in “httk” version 2.1.0.

Fig. 3.

Since we do not have fetal-specific models for variation in plasma protein binding, we use the McNamara and Meiman [58]₍ model for neonatal ₎ differences to adjust the adult fraction unbound in plasma $\left(f_{up}^{\text{m}}\right)$ to predict fetal $f_{\text{up}}^{\text{f}}$. Across the library of chemicals with HTTK data, the difference in fraction unbound in plasma for neonates from the adult value was small and depended upon ionization state, with a larger difference for positively charged chemicals, which we assumed bound to AAG. The identity line (solid) indicates a perfect (1:1) relationship between fetal and adult values.

For each tissue the partition coefficient is a constant ratio between the chemical concentration in tissue and the unbound chemical concentration in plasma. The symbols denoting partition coefficients have the form $K_y^x$ as shown in Table 10, with superscript x denoting maternal (“m”) or fetal (“f”) and subscript y denoting the tissue (indicated by the abbreviations in Table 1). We calculate each partition coefficient using the method of Schmitt [81] as described by Pearce et al. [66]. The partition coefficient for any given type of tissue (for example, liver tissue) depends on fraction unbound in plasma ($f_{\text{up}}^{\text{m}}$ or $f_{\text{up}}^{\text{f}}$), so in general these differ for mother and fetus.

In our model there are two routes of elimination, both from the mother: first order hepatic metabolism and excretion by passive glomerular filtration in the kidney. We characterize metabolism using a chemical-specific in vitro measured intrinsic hepatic clearance (μL/min/10⁶ hepatocytes). While we account for changes in GFR over time (as described previously) we hold metabolism to be constant because we do not in general have data or a method to ascribe metabolism to specific metabolizing enzymes, despite being aware that expression of these enzymes varies significantly during gestation in both the fetus and mother (Hines and McCarver, [37], McCarver and Hines, [10,55]). In part because xenobiotic metabolizing enzymes are substrates for diverse ligands, it is very difficult to predict which enzyme might be the primary driver of metabolism. Enzyme-specific data could be obtained, either through literature curation or empirical screening, for example by repeating the in vitro clearance assays for each enzyme. However, enzyme-specific measurement would decrease the throughput of in vitro toxicokinetic testing in proportion to the number of enzymes examined. Computationally ascribing in vitro whole hepatocyte clearance to the appropriate enzymes would not have this limitation but such tools are currently limited.

For the mother, hepatic (metabolic) clearance is assumed to be a first-order process with the rate of removal of the parent compound from the liver compartment equal to the concentration of unbound chemical in the blood leaving that compartment multiplied by a chemical-specific constant of proportionality. Because we cannot ascribe the in vitro clearance data to specific enzymes and we anticipate that the fetus will express lower amounts of metabolizing enzymes [4] the model currently assumes that there is no metabolic or renal clearance from the fetal compartments – this is a conservative error with respect to overestimating fetal concentrations. Similarly, we neglect metabolism by the placenta [4]. We do not currently model any metabolite formation – transformed chemical is considered as having been removed from the system.

2.5. Toxicokinetic equations

We describe the amounts of substance in the various maternal and fetal compartments using a system of coupled ordinary differential equations (ODEs). As for any PBTK model, these equations describe flows into and out of each compartment (via blood), other ingress terms (for example, absorption from one compartment to another), and other egress terms (for example, filtration from the kidney) [14]. The complete set of equations is provided as the supplemental file “fetalpbtk. model” (MCsim).

2.6. Chemical-specific data

No new chemical-specific data were measured for this analysis. The R package “httk” includes chemical-specific in vitro measurements of plasma protein binding ($f_{\text{up}}^x$, fraction_unbound_plasma) and intrinsic hepatic clearance ($k_{\text{livr}}^{\text{m}}$, Clmetabolism) [12,67]. Both $f_{\text{up}}^x$ and $k_{\text{livr}}^{\text{m}}$ must be successfully measured to allow parameterization of a PBTK model. As of “httk” version 2.1.0 there were 859 chemicals with both parameters measured in human tissues (in R, use: length (get_cheminfo(model=“fetal_pbtk”)) to see how many chemicals have data in the domain of the model). The chemicals include many from the ToxCast chemical testing library (that is, chemicals found in commerce and the environment) [72] as well as pharmaceuticals. In addition to the chemicals with in vitro measured values for $f_{\text{up}}^x$ and $k_{\text{livr}}^{\text{m}}$, simulations can be performed for an additional ~7000 chemicals for which in silico predictions of those quantities are available [83] via the function “load_sipes2017”. Physico-chemical properties for chemicals were predicted using two-dimensional, desalted SMILES [98] structures and the OPERA quantitative-structure activity relationships [53] as provided by the CompTox Chemicals Dashboard (https://comptox.epa.gov/dashboard) [101]. Chemicals with a log₁₀ Henry’s law constant greater than or equal to − 4.5 were considered volatile while PFAS identity was determined based on whether the chemical was included in the EPA “master list” of PFAS substances (https://comptox.epa.gov/dashboard/chemical-lists/PFASMASTER).

2.7. Evaluation data

Data sets were curated from the literature to allow evaluation of the gestational PBTK model. In all cases chemical identities from the original publications were mapped to unique identifiers (that is, DTXSIDs) from the CompTox Chemicals Dashboard (https://comptox.epa.gov/dashboard) [101]. Statistical testing for correlation between predictions and observations was performed using R function “lm” and p-values were calculated according to an F-distribution.

2.7.1. Maternal-to-fetal blood concentration ratios

Aylward et al. [2] compiled measurements on the ratio of maternal to cord blood chemical concentrations at birth (R_mat:fet) for a range of chemicals with environmental routes of exposure, including bromodiphenyl ethers, fluorinated compounds, organochlorine pesticides, polyaromatic hydrocarbons, tobacco smoke components, and vitamins. The PBTK model does not have an explicit cord blood compartment, so the ratio between maternal and fetal venous plasma concentrations was used as a surrogate comparisons between model predictions and these data. For each chemical three daily oral doses (every eight hours) totaling 1 mg/kg/day were simulated starting from the 13th week of gestation until full term (40 weeks). Three daily doses have been found to reasonably approximate exposure to chemicals in the environment from a variety of sources (not just food, but food-web accumulated compounds and drinking water) and also produces steady-state values similar to constant-infusion dosing [67]. Simulations were made both with and without the adjustment (Eq. 1) to $f_{\text{up}}^{\text{f}}$ for differences in neonate plasma proteins (note that, without the adjustment $f_{\text{up}}^{\text{f}}=f_{\text{up}}^{\text{m}}$. All data analyzed here are available in the httk version 2.1.0 table “httk:: aylward2014”.

2.7.2. Tissue-to-Blood Partition Coefficients

Partition coefficients were measured for tissues, including placenta, in vitro by Csanády et al. [25] for Bisphenol A and Daidzein.

Curley et al. [26] compiled data on the concentration of a variety of pesticides in the cord blood of newborns and in the tissues of infants that were stillborn. In order to make predictions to compare with these data, we assumed the ratio of chemical concentration in tissue $y\left(C_{\text{y}}^{\text{f}}\right)vs$. blood $\left(C_{\text{venb}}^{\text{f}}\right)$ was related to the tissue-to-unbound-plasma concentration partition coefficients [66,81] used in the PBTK model as

$$\frac{C_{\text{y}}^{\text{f}}}{C_{\text{venb}}^{\text{f}}}=\frac{C_{\text{y}}^{\text{f}}}{R_{\text{b}:\text{p}}^{\text{f}}\times C_{\text{plas}}^{\text{f}}}=\frac{C_{\text{y}}^{\text{f}}\times f_{\text{up}}^{\text{f}}}{R_{\text{b}:\text{p}}^{\text{f}}\times C_{\text{up}}^{\text{f}}}=\frac{f_{\text{up}}^{\text{f}}}{R_{\text{b}:\text{p}}^{\text{f}}}\times K_{\text{y}}^{\text{f}},$$ (2)

where $C_{\text{up}}^{\text{f}}$ denotes the concentration of substance unbound in the fetal plasma and all other symbols have been defined previously (cf. Table 9 and Table 10).

All data analyzed here are available in httk v2.1.0 table “httk:: fetalpcs”.

Three of the chemicals studied by Curley et al. [26] were modeled by Weijs et al. [97] using the same partition coefficients for mother and fetus. The values used by Weijs et al. [97] summarized the available literature as Bayesian prior distributions.

2.7.3. Concentration time course summaries

Dallmann et al., [28] compiled literature descriptions of toxicokinetic summary statistics, including time-integrated plasma concentrations (that is, area under the concentration vs. time curve or AUC) for drugs administered to a sample of subjects including both pregnant and non-pregnant women. These studies have been augmented with additional literature data [27,36,59,64,84]. The circumstances of the dosing varied slightly between drugs and pregnancy status required variation in simulated dose regimen as summarized in Table 12. The httk function “solve_fetal_pbtk” was used to determine the time-integrated plasma concentration (AUC) for the mothers both when pregnant and non-pregnant. All data analyzed here are available in httk version 2.1.0 table “httk::pregnonpregaucs”.

Table 12.

Observed time-integrated plasma concentration (AUC, area under the curve in uM•h) normalized to 1 mg/kg bodyweight, parameters, and HT-PBTK predictions for studies compiled from [27,28,59,36,84,64].

Drug	DTXSID	Observed Non-Pregnant	Observed Pregnant	Gestational Age (Weeks)	NonPreg Dosing Duration (Days)	Preg Dosing Duration (Days)	Predicted Non-Pregnant	Predicted Pregnant	Observed Preg:Non Ratio	Predicted Preg:Non Ratio
Caffeine	DTXSID0020232	24.50	71.00	36.00	0.50	1.00	42.48	42.60	2.90	1.00
Midazolam	DTXSID5023320	17.90	9.50	30.00	0.25	0.25	0.21	0.16	0.53	0.77
Nifedipine	DTXSID2025715	326.00	272.00	32.00	1.00	0.33	3.08	1.73	0.83	0.56
Metoprolol	DTXSID2023309	256.00	121.00	37.00	0.50	0.50	0.66	0.49	0.47	0.73
Ondansetron	DTXSID8023393	234.00	164.00	39.00	0.33	0.33	0.61	0.47	0.70	0.77
Granisetron	DTXSID0023111	125.00	113.00	15.00	1.00	1.00	0.12	0.10	0.90	0.84
Diazepam	DTXSID4020406	2.33	0.70	39.00	1.00	0.42	16.87	4.65	0.30	0.28
Metronidazole	DTXSID2020892	151.00	102.00	39.00	2.00	2.00	102.98	70.94	0.68	0.69
Acetaminophen	DTXSID2020006	101.71	74.30	30.90	0.25	0.25	824.88	551.28	0.73	0.67
Digoxin	DTXSID5022934	9.30	7.30	30.50	2.00	2.00	0.07	0.04	0.78	0.59
Lorazepam	DTXSID7023225	560.00	175.25	38.41	1.50	2.00	2.44	1.83	0.31	0.75
Amoxicillin	DTXSID3037044	103.25	87.00	35.10	0.25	0.17	201.84	119.69	0.84	0.59

2.7.4. Case study

To evaluate the degree to which the uncertainty estimated in the previous section agrees with other estimates [66], we attempted to prioritize chemicals observed in maternal blood based on predicted concentrations in an important developmental tissue: the fetal brain. Although it is known that the blood brain barrier (BBB) develops during gestation week 8 [79], our model does not currently account for differential or delayed partitioning into the brain due to BBB.

The ratio of the fetal brain concentrations to the maternal blood $\left(R_{\text{bran:matblood}}^{\text{f}}\right)$ can be calculated from the partition coefficient $\left(K_{\text{bran}}^{\text{f}}\right)$ for fetal brain to free fraction in fetal plasma by using 1/R_mat:fet to give the concentration in fetal blood:

$$R_{\text{bran:matblood}}^{\text{f}}=K_{\text{bran}}^{\text{f}}\times f_{\text{up}}^{\text{f}}\times \frac{1}{R_{\text{mat:fet}}}$$ (3)

2.7.5. Data

Wang et al. [96] screened the blood of 75 pregnant women for the presence of environmental organic acids (EOAs) and identified 480 chemical suspect candidates of which 48 could be mapped to likely structures by the CompTox Chemicals Dashboard (https://comptox.epa.gov/dashboard) [101]. Among those 48 chemical candidates with structures, standards had been available to Wang et al. [96] for six, allowing the confirmation of their identities. These EOAs included substances categorized as weak acids in the sense that they were negatively charged after ionization by the mass spectrometry but were neutral biologically. Bisphenol A, for example, ionizes around pH 10, which not very relevant physiologically.

The chemicals analyzed here are available in httk version 2.1.0 table “httk::wang2018”.

2.7.6. Uncertainty propagation

While our predictive methods largely give us point estimates (that is, specific quantitative values), the empirically estimated uncertainties indicate that the true values are likely to lie somewhere within a range about these predictions. For the equilibrium partition coefficients $\left(K_y^x\right)$ between chemical concentration in tissues and the unbound (free) chemical fraction in plasma we use a physico-chemical property driven algorithm derived from Schmitt [81]. Pearce et al. [66] calibrated these predictions using a data set of measured tissue partition coefficients and characterized tissue-specific residual errors. We can convert the partition coefficient predictions to distributions by assuming each predicted value is the mean of a normal distribution and approximating the residual as a standard deviation, ${\sigma }_y^x$, for tissue y where x indicates a maternal (m“) or fetal (f“) parameter. As an upper limit on the range of possible true values we use the 97.5th percentile of the normal distribution as given by 1.96 standard deviations above the mean. The range from the 2.5th to the 97.5th percentiles is the 95 % confidence interval. The Pearce et al. [66] calibration was performed on the base 10 logarithmic scale, so we expect 97.5 % of the log-transformed values for tissue y to be below ${\log }_{10}\left(K_y^x\right)+1.96\times {\sigma }_y^x$. For example, the residual standard deviation for the brain partition coefficient $\left(K_{\text{bran}}^x\right)$ was 0.647 Pearce et al. [66], so we can estimate the upper limit of the 95 % confidence interval estimate (UCL₉₅) for that partition coefficient as

$${\text{UCL}}_{95}\left(K_{\text{bran}}^{\text{f}}\right)={10}^{{\log }_{10}\left(K_{\text{bran}}^{\text{f}}\right)+1.96\times 0.647}.$$ (4)

Similarly, we can use the residual standard deviation (${\text{Error}}_{R_{\text{mat:fet}}}$) for $R_{\text{mat:fet}}$ to estimate a lower limit on the true value (since we want the inverse, $R_{\text{fet:mat}}=1/R_{\text{mat:fet}}$). (Note that we estimate the residual errors through comparing the observed values to the predictions and summarize with the standard deviation of those residuals.) The upper limit of a 95 % confidence interval estimate of the ratio is given by

$${\text{UCL}}_{95}\left(R_{\text{fet:mat}}\right)=1/\left(R_{\text{mat:fet}}-1.96\times {\text{Error}}_{R_{\text{mat:fet}}}\right).$$ (5)

For $f_{\text{up}}^{\text{f}}$, [95] found that the typical in vitro measurement has a coefficient of variation of 0.4, and so ${\sigma }_{f_{\text{up}}^{\text{f}}}=0.4\times f_{\text{up}}^{\text{f}}$ is used unless a chemical-specific value is available:

$${\text{UCL}}_{95}\left(f_{\text{up}}^{\text{f}}\right)=f_{\text{up}}^{\text{f}}+1.96\times {\sigma }_{f_{\text{up}}^{\text{f}}}.$$ (6)

$R_{\text{bran:matblood}}^{\text{f}}$ is a function three parameters that we are assuming have normally distributed uncorrelated errors. We propagated error by adding them in quadrature: for a function of three parameters with normally distributed error, $F(X,Y,Z))$, ${\sigma }_F=({\left(\frac{\partial \text{F}}{\partial \text{X}}\right)}^2{\sigma }_X^2+{\left(\frac{\partial \text{F}}{\partial \text{Y}}\right)}^2{\sigma }_Y^2+{{\left(\frac{\partial \text{F}}{\partial \text{Z}}\right)}^2{\sigma }_Z^2)}^{1/2}$ [46]. In this case $F\left(lk,f_{\text{up}}^{\text{f}},R_{\text{mat:fet}}\right)={10}^{lk}\times f_{\text{up}}^{\text{f}}\times \frac{1}{R_{\text{mat:fet}}}$, where $lk={\log }_{10}{K}_{\text{bran}}^{\text{f}}\cdot \frac{\partial \text{F}}{\partial lk}=\ln 10\times {10}^{lk}\times f_{\text{up}}^{\text{f}}\times \frac{1}{R_{\text{mat:fet}}},\frac{\partial \text{F}}{\partial f_{\text{up}}^{\text{f}}}={10}^{lk}\times \frac{1}{R_{\text{mat:fet}}}$, and $\frac{\partial \text{F}}{\partial R_{\text{mat:fet}}}=-{10}^{l\text{k}}\times f_{\text{up}}^{\text{f}}\times \frac{1}{R_{\text{mat:fet}}{}^2}$. So, we have

$${\sigma }_{R_{\text{bran:matblood}}^{\text{f}}}=R_{\text{bran:matblood}}^{\text{f}}{\left({(\ln 10)}^2{\sigma }_{lk}^2+\frac{{\left({\sigma }_{f_{\text{up}}^{\text{f}}}\right)}^2}{{\left(f_{\text{up}}^{\text{f}}\right)}^2}+\frac{{\left({\text{Error}}_{R_{\text{mat:fet}}}\right)}^2}{R_{\text{mat:fet}}{}^2}\right)}^{1/2}$$ (7)

which means that the upper confidence level is:

$${\text{UCL}}_{95}\left(R_{\text{bran:matblood}}^{\text{f}}\right)=R_{\text{bran:matblood}}^{\text{f}}{(1+1.96\times \left({(\ln 10)}^2{\sigma }_{lk}^2+\frac{{\left({\sigma }_{f_{\text{up}}^{\text{f}}}\right)}^2}{{\left(f_{\text{up}}^{\text{f}}\right)}^2}+\frac{{\left({\text{Error}}_{R_{\text{mat:fet}}}\right)}^2}{R_{\text{mat:fet}}{}^2}\right)}^{1/2}$$ (8)

This approach is based upon a Taylor series expansion of the likelihood assuming that the errors are approximately normally distributed and uncorrelated [46]. We acknowledge that the errors are in fact likely correlated and that there may be departures from normality, but we presently lack the data needed for more sophisticated treatments.

With the HT-PBTK model we can predict the fetal-to-maternal ratio of chemical concentration in plasma. In the Results (Fig. 8) we use the values described above to indicate the relative contribution of various error terms to the predicted ratio between fetal brain and maternal plasma. In that figure squares indicate $R_{\text{fet:mat}}$, circles indicate ${\text{UCL}}_{95}\left(R_{\text{fet:mat}}\right)$, triangles indicate $K_{\text{bran}}^{\text{f}}\times f_{\text{up}}^{\text{f}}\times {\text{UCL}}_{95}\left(R_{\text{fet:mat}}\right)$ and diamonds indicate ${\text{UCL}}_{95}\left(R_{\text{bran:matblood}}^{\text{f}}\right)$.

Fig. 8.

Prioritizing chemicals detected in maternal plasma, ordering from the top those chemicals with the highest predicted fetal brain concentrations relative to maternal blood. Wang et al. [96] detected xenobiotic chemicals in the plasma of expectant mothers – here we prioritize those chemicals with respect to potential concentration in the fetal brain (Numeric values are provided in Table 13). With the HT-PBTK model we can predict the fetal-to-maternal ratio of chemical concentration in plasma (squares indicate $R_{\text{fet:mat}}$). However, as shown in Fig. 4 there is uncertainty (error) in these predictions which we indicate here with circles ${\text{UCL}}_{95}\left(R_{\text{fet:mat}}^{\text{f}}\right)$). We can then multiply the predicted fetal:maternal plasma ratio, including uncertainty, by the predicted brain-to-plasma partition coefficient to obtain an estimated ratio of fetal brain tissue to maternal blood, indicated as triangles (calculated as $K_{\text{bran}}^{\text{f}}\times f_{\text{up}}^{\text{f}}\times {\text{UCL}}_{95}\left(R_{\text{fet:mat}}\right)$. Finally, we can consider the Pearce et al. [66] estimate of uncertainty in the partition coefficient, as indicated by the diamonds $\left(UC{L}_{95}\left(R_{\text{bran:matblood}}^f\right)\right)$. Despite the uncertainties, we can see clear discrimination between chemicals based upon how likely they are to concentrate in the fetal brain.

3. Results

We evaluated a generic mathematical PBTK model describing absorption, distribution, metabolism, and excretion in a human mother and fetus. The model consists of a system of time-dependent ordinary differential equations in which time, representing gestational age, is a single independent parameter. Key parameters, including tissue volumes and blood flows, vary as a function of time, and our model includes descriptions of these functions starting at gestation week 13 through full term (week 40). In this study, we did not simulate biological variability of the mother or fetus, and the developmental sequence is fixed. Our model predicts different distribution, metabolism, and excretion for different chemicals based on physico-chemical properties and in vitro measures of plasma protein binding $\left(f_{\text{up}}^x\right)$ and intrinsic hepatic clearance $\left(k_{\text{livr}}^{\text{m}}\right)$, which are currently available for 859 chemicals. We evaluated our model for cases in which we were able to obtain both in vivo measures of maternal-fetal TK and chemical-specific in vitro $f_{\text{up}}^x$ and $k_{\text{livr}}^{\text{m}}$.

3.1. Model evaluation

3.1.1. Maternal-to-fetal blood concentration ratios

Determination of chemical concentration typically requires destructive analysis of samples [89]; therefore, for the maternal-fetal system, monitoring is often performed using samples of cord blood, amniotic fluid, or meconium [4]. Presence of a chemical in a sample provides evidence of exposure, but a PBTK model provides a framework for interpreting the relationship between exposure, samples, and sampling times [4]. Blood compartments in PBTK models are typically considered to be “flow-limited” or “well-mixed” meaning that the tissue and blood are assumed to be instantaneously at equilibrium as characterized by partition coefficients; this assumes that rate of tissue uptake of the chemical is rapid relative to the flow of the chemical to the tissue from blood [14]. In our model we do not have amniotic fluid or meconium but have assumed that cord blood has the same concentration as fetal venous blood in a direct extension of the “well-mixed” assumption.

Aylward et al. [2] summarized data from over 100 studies covering 88 unique chemical structures on the ratios of cord-to-maternal blood concentrations for a range of chemicals. The studies reviewed suggest that chemicals frequently detected in maternal blood will also be detectable in cord blood. For most chemical classes, cord blood concentrations were found to be similar to or lower than those in maternal blood, with reported cord-to-maternal blood concentration ratios generally between 0.1 and 1. Exceptions were observed for selected brominated flame-retardants, polyaromatic hydrocarbons (PAHs), and some metals, for which reported ratios consistently exceeded 1.

With the Aylward et al. [2] dataset we cannot make an absolute scale comparison between observations and predictions (that is, fetal-to-fetal or maternal-to-maternal) because we do not know the exposure histories (that is, doses) leading to the observed concentrations. However, if we assume that the maternal-to-fetal chemical concentration in plasma ratio is independent of dose, we can use the maternal-to-fetal ratio at full term (40 weeks) resulting from a 1 mg/kg/day exposure rate (oral) starting in week 13. In Fig. 4 we compare predictions made with our high throughput human gestational PBTK model for maternal-to-fetal plasma ratio with the Aylward et al. [2] maternal-to-cord blood experimental observations on a per chemical basis. There were 26 chemicals where we had both in vitro HTTK data (for parameterizing the model) and in vivo observations (for evaluating the model predictions). Multiple observations were available for 19 of the chemicals, so for these chemicals the median observation is plotted with a larger symbol in Fig. 4, while the 75 % interval is depicted with a vertical line and outliers beyond that range are plotted with smaller symbols. For a given chemical the model makes a single prediction of maternal-to-fetal ratio (as a function of physico-chemical properties and chemical-specific $f_{\text{up}}^x$ and $k_{\text{livr}}^{\text{m}}$ such that for any given chemical all observations (y-axis) are spread in a vertical line with the same predicted (x-axis) value.

Fig. 4.

Comparison between observed [2] and predicted maternal-to-fetal plasma concentration ratios at birth. The identity line (solid) indicates a perfect (1:1) correspondence between predictions and observations. For any one chemical there is a single prediction (x-value) but there are potentially multiple observations (y-values). The median observation is plotted with a larger symbol, while the 75 % interval is depicted with a vertical line and outliers beyond that range are plotted with smaller symbols. Chemical class is indicated by shape for bromodiphenylethers (solid square), organochlorine pesticides (open triangle), tobacco smoke components (open square), fluorinated compounds (solid circle), and polyaromatic hydrocarbons (open diamond).

The observed values for maternal-to-cord blood chemical concentration ratio span a greater range than the maternal-to-fetal predictions from the PBTK model. The minimum observed ratio was 0.11 for Dichlorodiphenyldichloroethane (that is, p,p’-DDD) and the maximum was 2.9 for Fluorene. However, among the chemicals with repeated observations, the median observations only ranged from 0.36 for Perfluorooctanesulfonic acid to 1.7 for Pyrene. The predictions for all chemicals ranged from 0.63 for Pentachlorophenol to 1.8 for 3,3’,5,5’-Tetrabromobisphenol A.

Because the blood concentration ratios span a relatively narrow range, caution is necessary in using these data to evaluate model performance. Since the plasma binding of chemicals is expected to vary between the mother and fetus, we attempted to evaluate the statistical performance of the predictions both without (not shown) and with (Fig. 4) the adjustment of fraction unbound in plasma $\left(f_{\text{up}}^{\text{f}}\right)$ to account for differences in neonate plasma proteins to in vitro (Eq. 1). For these data we did not observe a meaningful difference – there was relatively poor correlation both with (R² = 0.074, RMSE = 0.53) and without (R² = 0.0084, RMSE = 0.55) correcting for differences in plasma binding between neonates (used as a surrogate for fetal binding) and adults.

Summary statistics for our evaluation with various subsets of the Aylward et al. [2] are provided in Table 11. The Aylward et al. [2] in vivo data include some chemicals that are more volatile (that is, PAHs) and per- and poly-fluorinated alkyl substances (PFAS). We do not expect the model to work well for volatile or some semi-volatile chemicals since we only evaluated predictions based on oral doses and they are more likely to be associated with inhalation exposure. Additionally, the generic physiology we use does not include exhalation as a route of elimination, meaning that concentrations will tend to be overestimated for such chemicals. We restricted our model to chemicals for which the Henry’s law constant is less than 10^−4.5. In our model, the octanol-to-water partition coefficient is the most significant predictor for tissue partitioning [66]; therefore, we might expect our model to perform poorly for some PFAS, as these substances can be both hydrophobic and lipophobic [73]. When PAHs and PFAS are omitted, only 9 evaluation chemicals remain, but the R² is 0.21 and the RMSE is 0.2 without the correction. Although we know that maternal and fetal binding vary, for this subset of chemicals, when our prediction for the fetal plasma binding correction is used the predictivity decreases slightly: R² is 0.17 and the RMSE remains 0.2 for the non-volatile, non-PFAS chemicals.

Table 11.

Comparison with maternal-fetal blood concentration ratios as reported by.

	AllFig. 4	No PFAS/PAH	Replicates Only
Number of Chemicals	19	9	7
Number of Observations	64	28	7
Observed Mean (Min - Max)	0.75 (0.11–2.7)	0.71 (0.11–1.4)	0.76 (0.54–0.9)
Observed Standard Deviation	0.51	0.35	0.13
Predicted Mean (Min - Max)	1.3 (0.63–1.8)	1.3 (0.63–1.8)	1.2 (0.63–1.5)
RMSE	0.55	0.22	0.087

We can compare the RMSE for our predictions of maternal to fetal plasma concentration ratio to the standard deviation of the observations: 0.53 (0.25 for substances other than PAHs and PFAS). The RMSE of the predictive model is less than the standard deviation of the observations, indicating that there is less error in using the predictions than in simply using the mean observed ratio across all chemicals. However, the average standard deviation of the median observation for chemicals with repeated observations, which we assume to be more accurate, was 0.45, dropping to 0.14 for non-PAH and non-PFAS. The RMSE of the predictions for the 7 non-PAH and non-PFAS compounds with repeated observations is 0.12 with or without the fraction unbound in plasma correction. These values are close to the standard deviation of the mean for the chemicals with repeated observations, albeit evaluated across only 7 chemicals.

In Fig. 5, we examine the maternal to fetal plasma concentration ratios predicted for the 856 chemicals with HTTK data (both $f_{\text{up}}^x$ and $k_{\text{livr}}^{\text{m}}$) and properties amenable to this modeling exercise (non-volatile and no PFAS). Chemicals with a Henry’s law constant greater than 10^−4.5 were considered volatile while PFAS identity was determined based on whether the chemical was included in the EPA “master list” of PFAS substances (https://comptox.epa.gov/dashboard/chemical-lists/PFASMASTER). We observed a median ratio of 1.31, with values ranging from 0.429 for Bis(2-ethylhexyl) nonanedioate (DTXSID3026697) to 1.84 for 2’,4’,5’,7’-Tetrabromofluorescein (DTXSID3044590). There are 21 chemicals with ratios greater than 1.6, indicating a tendency to have higher concentrations in maternal plasma compared to fetal plasma. These chemicals tend to be hydrophobic (mean logP of 4.4) and highly bound (mean fetal plasma fraction unbound of 0.0066). Without the fetal correction to plasma binding (that is, using the same value for mother and fetus), the median ratio drops to 1.17, with the lowest ratio unchanged and the highest ratio (again for 2’,4’,5’,7’-Tetrabromo-fluorescein) dropping to 1.65.

Fig. 5.

Histogram of predicted maternal-to-fetal concentration ratios across the chemicals for which the HT-PBTK model can be parameterized (omitting volatile and PFAS compounds).

3.1.2. Tissue-to-blood partition coefficients

The Pearce et al. [66] calibrated version of the Schmitt [81] algorithm allows prediction of tissue-specific partition coefficients, in a manner like that of Dallmann et al. [28] (comparison across eight chemicals and nine tissues found significant correlation between their methods and httk with a p-value 1.5 × 10−10, see data in “httk::pksim. pcs”). Curley et al. [26] compiled data on chemical concentrations of six pesticides in cord blood of newborns and seven tissues of stillborn infants. Additionally, Csanády et al. [25] measured partition coefficients in vitro for various tissues including placenta for Bisphenol A and Daidzein. Fig. 6 provides a comparison of the predicted tissue partition coefficients and the measured values. The predictions greatly underestimate the observations derived from the Curley et al. [26] data: most measured partition coefficients, regardless of chemical or tissue, were greater than 100 while all of the predicted partition coefficients were less than 100. In a recent publication describing a PBTK model, Weijs et al. [97] summarized the literature on partition coefficients for three of these chemicals, finding that only the adipose partition coefficient should be greater than 100 – these estimated values are plotted with smaller points in Fig. 6. For both the non-adipose Weijs et al. [97] estimates and the two chemicals measured in vitro by Csanády et al. [25] the “httk” PBTK partition coefficient predictions are within ten-fold of the observations. For the two placental observations (1.4 for Bisphenol A and 1.1 for Daidzein) the predictions were 0.63 and 0.44, respectively.

Fig. 6.

Evaluation of predicted partition coefficients with measured data. Fetal tissue-to-blood partition coefficients were determined by Curley et al. [26] for six pesticides and seven tissues for which we can make predictions with the HT-PBTK model. The same data values are shown in both plots, but the data symbols reflect chemicals (pesticides) on the left and tissues on the right. Partition coefficients were measured for tissues, including placenta, in vitro by Csanády et al. [25] for Bisphenol A and Daidzein. The identity line (solid) indicates a perfect (1:1) prediction while the dotted lines indicate a ten-fold error. Small plot points indicate model-predicted, rather than measured, partition coefficients from Weijs et al. [97] for three of the Curley et al. [26] chemicals.

3.1.3. Concentration time course summaries

Having evaluated chemical distribution (that is, relative concentrations in different tissues at steady-state), we attempted to evaluate kinetics using the time integrated plasma concentration (AUC) for cohorts of women exposed to twelve pharmaceuticals including both those who were pregnant (typically the third trimester) and non-pregnant, as described by Table 12. Predictions of the AUC made with our PBTK model are also reported in Table 12 and are plotted in Fig. 7.

Fig. 7.

Comparison of observed and predicted time-integrated plasma concentration (AUC) for the data in Table 12 for non-pregnant (upper-left) and pregnant (upper-right) mothers across ten pharmaceuticals. At bottom-right we compare the observed and predicted ratios between the AUCs for non-pregnant and pregnant women. The identity line (solid) indicates a perfect (1:1) prediction while the dotted lines indicate a ten-fold error.

To generate the results shown in Fig. 7A, we examined data for non-pregnant women, finding the predicted AUCs to be within a factor of ten for ten of the twelve compounds. AUC was underestimated by our model for the two outliers, Digoxin and Lorazepam. For the Dallmann et al. [28] data we observed an average-fold error (AFE) of 1.1 and a RMSE of 12.6 for non-pregnant women. In Fig. 7B we again find the predictions for AUC to be within ten-fold of the observations for ten of the twelve compounds, with Digoxin and Lorazepam again underestimated. We again observe an AFE of 1.1 and a RMSE 12.6 for pregnant women. Finally, in Fig. 7C we compare the observed ratio between the pregnant and non-pregnant AUCs with predictions, finding the predicted ratio to differ by 60 % on average and at most 2.9 times (Caffeine). Among the twelve chemicals four are overestimated and eight are underestimated.

3.2. Case study: chemical prioritization with maternal biomonitoring data

Finally, we present a case study to evaluate the usefulness of the model in its current form. We examine how a generic PBTK model might inform public health chemical risk assessment. The underlying growth kinetics of the model are well established [44]. In the previous section we showed that we could predict plasma concentration metrics (for example, AUC) for pregnant (and non-pregnant) mothers (Fig. 7). We determined that for non-PFAS and non-volatile chemicals we can make predictions for the ratio of chemical concentrations between the maternal and fetal blood (and therefore plasma) and evaluated those predictions (Fig. 4). Although the tissue partitioning model was found to perform poorly for persistent chemicals (Fig. 6), the same model has previously been evaluated using adult data for many more, mostly non-persistent, chemicals (153) and was found to perform well for those chemicals [66]. In the previous analysis the relatively large number of chemicals further allowed tissue-specific prediction errors to be estimated [66]. Through propagation of the observed uncertainties for all relevant parameters, we can examine the implications of those uncertainties in the case of interpreting biomonitoring data to prioritize chemicals for further evaluation.

As a case study we examined a variety of chemicals observed in the blood of pregnant women and investigated the implications for the brain as a key toxicological target tissue. Wang et al. [96] screened the blood of 75 pregnant women for the presence of a broad list of chemicals and tentatively identified mass spectral features corresponding to 480 chemical candidates. Across the 480 chemical candidates, we used the CompTox Chemicals Dashboard (https://comptox.epa.gov/dashboard) [101] to map 48 to likely structures. Of the 48 chemicals with tentative or confirmed structures, 19 chemicals had in vitro measured HTTK data and were non-volatile and non-PFAS.

In Table 13 and Fig. 8 we propagate two sources of uncertainty: error in the fetal-to-maternal plasma ratio and error in the plasma-to-brain partition coefficient. Our model assumes that these two ratios are constant throughout gestation. We prioritize 19 chemicals from the Wang et al. [96] study based on predicted partitioning into the fetal brain relative to maternal plasma concentration. Given the qualitative nature of the chemical detections from Wang et al. [96], this prioritization assumes that each of the chemicals occurs in the maternal plasma at equal concentrations.

Table 13.

Uncertainty propagation for chemicals identified in maternal blood [96] on the basis of predicted partitioning into the fetal brain.

Compound	CAS	DTXSID	logP	Charge in Plasma	R fet:mat	f up	$UC{L}_{95}\left(K_{\text{bran}}^{\text{f}}\right)$	$UC{L}_{95}\left(R_{\text{bran:matblood}}\right)$
2,4-di-tert-butylphenol	96–76–4	DTXSID2026602	5.19	Neutral	0.70	1.44E-01	32,700	727.0
4-(1,1,3,3-tetramethylbutyl)phenol	140–66–9	DTXSID9022360	4.87	Neutral	1.44	1.90E-02	58,300	353.0
2,6-di-tert-butylphenol	128–39–2	DTXSID6027052	4.92	Neutral	0.68	1.95E-02	18,400	53.2
Bisphenol-a	80–05–7	DTXSID7020182	3.32	Neutral	1.44	3.85E-02	3990	47.7
4-tert-butylphenol	98–54–4	DTXSID1020221	3.31	Neutral	1.13	1.05E-01	1130	29.5
2-methylphenol	95–48–7	DTXSID8021808	1.95	Neutral	1.20	3.80E-01	115	11.5
2-tert-butylphenol	88–18–6	DTXSID2026525	3.31	Neutral	0.67	1.15E-04	9250	11.3
4-butylphenol	1638–22–8	DTXSID3047425	3.65	Neutral	0.87	3.58E-02	1320	10.1
4-methylphenol	106–44–5	DTXSID7021869	1.94	Neutral	1.17	3.26E-01	117	9.8
4-(butan-2-yl)phenol	99–71–8	DTXSID7022332	3.08	Neutral	0.92	6.08E-02	765	9.3
2,3-dihydro-2,2-dimethyl-7-benzofuranol	1563–38–8	DTXSID2027414	2.08	Neutral	0.82	3.64E-01	81	5.1
Eugenol	97–53–0	DTXSID9020617	2.27	Neutral	0.93	1.92E-01	120	4.7
Imazapyr	81,334–34–1	DTXSID8034665	0.22	100 % Zwitterion	1.01	9.23E-01	22	4.4
4-(hexyloxy)phenol	18,979–55–0	DTXSID4048195	3.36	Neutral	0.67	7.70E-04	3880	2.6
Methylparaben	99–76–3	DTXSID4022529	1.96	Neutral	0.83	7.06E-02	134	1.7
4-nitrophenol	100–02–7	DTXSID0021834	1.91	61 % Negative	0.74	1.41E-01	73	1.6
2,4-dinitrophenol	51–28–5	DTXSID0020523	1.67	100 % Negative	0.66	2.72E-02	313	1.2
Catechol	120–80–9	DTXSID3020257	0.88	Neutral	0.73	2.68E-02	122	0.6
4,4’-sulfonyldiphenol	80–09–1	DTXSID3022409	−0.10	45 % Negative	0.63	5.76E-02	56	0.4

Fig. 8 demonstrates how even uncertain HT-PBTK tools may be used for chemical prioritization if errors are quantified and propagated as in Table 13. That is, if we know (or can estimate) the magnitude of the errors, we can attempt to account for them in the prioritization. Sources of error may include inaccuracies of model framing, as with a lack of a description of placental transport dynamics beyond non-preferential linear partitioning; in vitro measurement; and physiological and chemical parameters.

The range of ratios for maternal-to-fetal partitioning, both as observed in the literature and predicted with a PBTK model, is dwarfed by the uncertainty in tissue-specific partition coefficients. However, despite the uncertainty we can, for example, prioritize tentative substance identifications in chemical screening analysis of maternal blood for confirmation based on potential exposure to the fetal brain. For instance, the median predicted ratio between fetal brain and maternal blood is 0.07 for 2-tert-butylphenol, with an upper 95 % confidence interval limit (Eq. 5) of 0.294. Meanwhile, the median prediction for 4- (1,1,3,3-Tetramethylbutyl) Phenol is 12.9 with an upper 95 % confidence interval limit of 57.1. Since we do not know the absolute concentrations in maternal blood or the relative potencies of any neurological effects, the compounds observed in maternal blood that are predicted to occur at higher concentrations in fetal brain (than in maternal blood) might be considered to be of higher priority for potential adverse health effects in fetal brains, given the assumption that the observed substances occur in maternal blood at similar concentrations and that they are associated with neurological effects for the similar fetal brain (point-of-departure) concentrations. Given these assumptions there are potentially meaningful differences predicted between the maternal plasma chemicals despite the uncertainties observed. As expected, the predicted fetal brain to maternal blood concentration ratio was strongly correlated with the predicted brain partition coefficient (R² 0.56, p-value 0.00013). However, the PBTK simulation impacted the plasma and tissue concentrations such that 9 chemicals were ranked higher than they would have been using partitioning alone. The predictions for fetal brain to maternal blood ratio with or without the fetal plasma binding correction (Eq. 1) were strongly correlated (R² = 0.97). There were 7 chemicals that were ranked higher with the correction than without, with an average increase of 1.2 times when the plasma binding change was included.

4. Discussion

Information on fetal exposure to chemicals is key to interpreting reproductive toxicology and epidemiologic birth cohort studies [4]. In 2018, the U.S. Food and Drug Administration issued draft guidance on potentially including pregnant women in pharmaceutical clinical trials in part because “Extensive physiological changes associated with pregnancy may alter drug pharmacokinetics and pharmacodynamics, which directly affects the safety and efficacy of a drug administered to a pregnant woman through alterations in drug absorption, distribution, metabolism, and excretion” U.S. FDA, [91]. The 2016 update of the U.S. Toxic Substances Control Act mandated that the U.S. Environmental Protection Agency evaluate chemicals to determine if they posed an “unreasonable risk to a potentially exposed or susceptible subpopulation” [1] including infants, children, and pregnant women [9].

For most chemicals in vivo TK data are unavailable [7]; this situation is even worse for gestational TK. However, a public library of in vitro TK data is available and growing – currently this library encompasses roughly one thousand chemicals [12] and can be augmented with quantitative structure-property model predictions (QSPRs) for chemicals without measured data [31,70,83]. Here we have implemented a chemical-independent PBTK model for the human maternal-fetal system that makes use of in vitro TK data and the time varying equations of Kapraun et al. [44] for human anatomical and physiological parameters during pregnancy and gestation. We selected the Kapraun et al. [44] equations because a standardized statistical approach was used to identify the parameters for each equation, and because most of the time-varying quantities needed for our PBTK model were included. This collection of equations contrasts with other comparable contemporary work, such as that of Dallmann et al. [29] in which the authors considered changes to fetal mass (as well as mass of total body water, intra- and extra-cellular water, proteins, and lipids), cord blood flow, and hematocrit, but they did not provide formulae for other fetal changes (that is, changes in volumes of and blood flow rates to specific fetal organs and tissues). In addition, the Dallmann et al. [29] model yields zero maternal blood flow to the placenta until 10.14 weeks (well after the placenta has formed).

While in some cases physiological processes have been completely omitted from this model, in other cases, we used constant values for physiological parameters describing time-varying processes – including the in vitro-measured TK parameters of plasma protein binding $\left(f_{\text{up}}^x\right)$ and metabolism (intrinsic hepatic clearance or $k_{\text{livr}}^{\text{m}}$). Generally, we do this when either the time-varying models themselves or the data needed to customize these models to thousands of chemicals are unavailable. For example, even if models for the time-variation of abundance of certain proteins (such as albumin) or enzymes (such as CYP3A4) were constructed, we currently lack chemical-specific data for confidently attributing both in vitro-measured chemical-specific plasma protein binding and metabolism to specific biological targets. That is, we do not know which enzymes are principally responsible for clearance of an arbitrarily chosen chemical in vivo, nor to which substances the chemical binds in in vitro assays for protein binding and metabolism.

Using chemical-specific data ascribing metabolism to specific enzymes for twelve chemicals, Dallmann et al. [28] included time-variation in metabolizing enzyme expression to obtain more accurate results than we report in Fig. 7. However, for not having described any of CYP ontogeny or other time-changes in proteins or enzymes we observed that the generic PBTK model performs decently: a RMSE of 10 for non-pregnant and 8.9 for pregnant woman. Simply by capturing the major physiological changes we obtain potentially useful predictions for 8 chemicals which may reflect the model performance for some of the 859 chemicals for which the model can be parameterized. Certainly, we would expect more accurate predictions (that is, lesser uncertainty) if chemical-specific data on key enzymes were available across the “httk” chemical library. However, given the available data, the performance of the human gestational PBTK model against the Dallmann et al. [28] data for the twelve pharmaceuticals (RMSE 0.94) is not too much worse than the performance observed for other, similar models for humans and animals using larger sets of evaluation chemicals (for example log₁₀ RMSE of 0.46 for AUC’s predicted with a generic inhalation PBTK model [49]).

Concentration variability in non-pregnant adult [38] and pediatric [41] human populations can span half to a full order of magnitude. It is well known that there is even greater variability in pregnant woman and gestating fetuses U.S. FDA, [91]; for example, variation in metabolism is significant [28]. The Kapraun et al. [44] equations and the PBTK model presented here do not address population variability. This is primarily due to a lack of data available for determining how the parameters of our model co-vary. For example, it might be expected that, within the mother or fetus there would be correlations in organ weight differences from the mean, and further that there would likely be correlations between the mother and fetus. We are unaware of any set of models for maternal-fetal parameters that account the interdependencies inherent in the corresponding anatomical and physiological parameters. To establish such models would require data sets of correlated observations. Lacking such data sets for most key parameters, we have instead focused on describing an “average” or typical human pregnancy and gestation process, while recognizing that biological variability drives many important phenomena including susceptibility [5].

Across the 856 non-volatile, non-PFAS chemicals with HTTK data, the PBTK model predicts a median ratio of 1.31 between maternal and fetal plasma concentrations; that is, higher maternal concentrations than fetal. Careful consideration of physico-chemical properties may eventually allow correlation of physico-chemical properties with the maternal-to-fetal ratio. For most chemical classes, Aylward et al. [2] found cord blood concentrations to be similar to or lower than those in maternal blood, though concentrations of most PAHs and some brominated flame-retardants and metals were observed to be higher in the fetus than the mother. Processes omitted by our model, such as active transport in the placenta [4], could impact the ratio for some chemicals and would be important to include if data to describe such processes become available.

While many chemicals were available for evaluating maternal-to-fetal plasma ratios, the evaluations of both kinetics and tissue partitioning were limited by the number of chemicals with available data. Unfortunately, there are chemicals where we have in vivo data but do not yet have in vitro data necessary to parameterize the model. Of the 88 chemical structures in the Aylward et al. [2] data set, we currently have in vitro HTTK data for only 26, limiting the extent of model evaluation using cord blood to maternal blood chemical concentration ratios. Similarly, peer reviewed scientific literature provides pharmaceutical exposure plasma concentrations for at least 28 chemicals, but of those only twelve had in vitro HTTK data to allow comparison. The collection of additional, chemical-specific HTTK data could improve estimation of the reliability of these predictions and potentially allow the evaluation of more sophisticated models that elaborate on the framework currently presented. Our analysis assumes that the cord blood is an appropriate surrogate for fetal blood with respect to chemical concentrations. If this is not correct, then other methods would be needed to determine fetal blood and tissue dosimetry.

For fetal tissues, we expect experimental data on chemical concentrations in those tissues to be especially rare. The evaluation of tissue-specific partitioning based on the six chemicals available from Curley et al. [26] was not promising – all tissues were observed by Curley et al. [26] to have concentrations much higher than in blood. Weijs et al. [97] developed a PBTK model for three of the same compounds and only assumed that the adipose tissue would have a much (>100 times) higher concentration than blood. The model used here reproduced the non-adipose Weijs et al. [97] partition coefficients well, but those values were not measurements (but rather predictions based on another model).

Predictive modeling that disregards data in favor of comparison to other models exists on perilously thin ice [63,69]. However, the chemicals covered by the Curley et al. [26] data set were all highly bioaccumulative with extremely high lipophilicity. For two non-bioaccumulative chemicals where in vitro partition coefficient measurements were available (Bisphenol A and Daidzein) the predictions for our model were much more similar to data. It may be that we need to handle chemicals differently depending on whether they are persistent or non-persistent (that is, based on their biological half-lives) [4]. The present model appears to be more appropriate for non-persistent chemicals. Under-predicting the tissue partition coefficients for persistent chemicals, as indicated by Fig. 6, would lead to under-estimating tissue fetal tissue exposure for those chemicals.

The algorithm we used to estimate tissue-to-blood partition coefficients is based on the method of Schmitt [81] for organic chemicals as harmonized by Peyret et al. [68]. Pearce et al. [66] showed that this method performed well for in vivo measured adult rat and human partition coefficients across 143 largely non-persistent chemicals and 12 tissues (964 total observations). Utsey et al. [92] recently reviewed five different models for tissue partitioning, including PK-Sim [50] and the three models (including the Schmitt [81] model) that were harmonized into the Peyret et al. [68] model. Using the method evaluated by Pearce et al. [66] allows for propagation of uncertainty into risk analyses. However, we did observe significant correlation between the partition coefficients we predicted and those predicted by Dallmann et al. [28] using PK-Sim.

We are unaware of general differences in the tissue-specific partitioning between adult and fetal tissue, and none of the chemicals from Curley et al. [26] were present in the Pearce et al. [66] evaluation, so it may be that the Curley et al. [26] chemical set is particularly challenging for Schmitt’s method. Further, it is difficult without paired data to know with any certainty how accurately we can estimate fetal tissue concentrations from cord blood – the Curley et al. [26] data are the only such data of which we are aware. We do know that cord blood represents a sample of the blood directly circulating into the fetus; that is, the cord blood is representative of blood traveling between the placenta and the fetus (Fig. 1). We have not determined a reason that the accuracy of tissue-to-blood partitioning should vary radically between the fetus and the mother. As discussed above, the differences in fetal plasma binding that we predicted (Fig. 3) were small. The predictions of tissue partitioning depend on the extent of chemical ionization; since the pH varies slightly between maternal and fetal plasma, for chemicals with ionization equilibria near 7.4 the tissue distribution might differ but would still be expected to be consistent with our model predictions. It is possible that distribution of the chemicals as observed by Curley et al. [26] occurred in the first trimester when the tissues were not differentiated or that the reported data are somehow inaccurate. Presently we expect the generic model here to perform well for chemicals and tissues like those examined by Pearce et al. [66] (that is, non-volatile, non-persistent chemicals).

Our prioritization case study demonstrates how the differences in model predictions (signal) compared to the estimated model uncertainty (noise). The observations of chemical occurrence from Wang et al. [96] must be thought of somewhat qualitatively because the concentrations of one chemical relative to another cannot be determined. Not knowing the absolute concentrations in maternal blood or the relative potencies of any neurological effects, the chemicals with highest predicted concentrations appear to have higher priority for potential adverse health effects in fetal brains despite the uncertainties. The Wang et al. [96] data are non-quantitative in part because the extraction efficiency (that is, the fraction of a chemical present in a sample that can be extracted) of chemicals from the mothers’ blood varies by chemical and is not yet known for all 19 chemicals. Extraction efficiency depends on the chemical, the media, the solvent used, and other sample preparation considerations [60]. These predictions could be easily weighted by relative exposure doses and toxic potency but for simplicity here we focused on TK. True risk prioritization data on exposures might be inferred from biomonitoring [75] or exposures estimated from chemical structure [74] could be used to augment non-quantitative chemical detections. In vitro or in silico models for identifying chemical-specific neurotoxic concentrations adversity would still be needed [34,54].

Fetal brain partitioning is included only as an example. The greatest value of the high throughput PBTK model presented here might be in the use of the dynamic model to estimate fetal exposure during windows of susceptibility [5]. Chemical exposures within key tissues during key developmental steps may have significant impact that could be at least tentatively estimated across large chemical libraries using this PBTK model.

A limitation of this PBTK model is that both the placenta and brain are described using equilibrium partition coefficients – that is, the chemical concentration in those tissues instantly reaches equilibrium with concentration in blood, presumedly through diffusion. While this may not be a bad first approximation for placenta since many chemicals do pass through the placenta by diffusion [4], it is known to be inappropriate for brain [80]. While Pearce et al. [66] did not examine placenta, in that analysis the largest residual standard deviation (that is, the greatest error) was observed for partitioning into the brain, based on evaluation with 98 measured brain tissue partition coefficients. This discrepancy is expected because the brain is protected by a semi-permeable barrier that, as with the placenta, includes multiple efflux and influx transporters [4,80]. For therapeutic compounds, PBTK models and even in vitro data have been observed to often fail to predict brain concentrations [80]. Potentially, the incorporation of more complex models for placental [21] and blood brain barrier transport [54] might improve predictions if a sufficient evaluation set could be established to demonstrate utility. However, as indicated by Fig. 8 and Table 13, among chemicals found to occur in maternal plasma Wang et al. [96], we predict differences in fetal brain concentrations that exceed the errors we have estimated. Chemicals with higher predicted fetal brain concentrations become higher priority for more detailed examination.

An additional limitation of this PBTK model is that it is human focused – we cannot currently use it to make predictions for animal species. The Kapraun et al. [44] physiological models are calibrated to human data and there are “fundamental differences” in the period of gestation and ordering of development between humans and various animal species [24]. However, much of the framework developed here could be reused if new species-specific models analogous to those of Kapraun et al. [44] were developed for anatomical and physiological changes during gestation.

We have not performed a sensitivity analysis here but have made all the parameters, including those for the Kapraun et al. [44] time-varying models accessible to user; that is, the values may be changed and simulations conducted with new values. Future work could examine the sensitivity of model predictions to input parameters, but here we have focused on ability to reproduce the (limited) in vivo data.

5. Conclusions

In an age of relatively fast computing, the challenge of mathematical modeling of biology is not whether one can make a model, but whether that model is free of errors and whether it appropriately represents key aspects of biology [17], McLanahan et al., [56]. Clark et al. [17] identified six aspects of PBTK models to be assessed for any model, particularly with respect to potential errors: 1) purpose, 2) structure and biology, 3) mathematical description, 4) computer implementation, 5) parameter values and model fitness, and 6) any specialized applications. Chemical-independent toxicokinetic models describing a fixed set of physiological processes potentially address these issues, particularly issues of implementation errors by using a consistent approach [12,40]. While such a model may not include all aspects of physiology that are relevant for a given chemical, there can be greater confidence that the processes included are appropriately described because the equations and their computational implementation is reused and evaluated across many chemicals and scenarios [12,40,94]. The amount of evaluation data available is typically a limiting factor that can be addressed with a chemical-independent model – data for all chemicals amenable to the model may be used for evaluation [23]. Given a larger and more diverse set of evaluation data, a chemical-independent model may describe the data for any one chemical less precisely but may describe the toxicokinetics for a class of chemicals overall more accurately. With sufficient evaluation the gestational model presented here may allow for better interpretation of biomonitoring data, improved analysis of toxicokinetics studies, and in vitro to in vivo extrapolation of bioactive concentrations relevant to reproductive and developmental toxicity.

Data Availability

All data and code are available through R package “httk” v2.1.0: https://cran.r-project.org/package=httk.