Empirical models for anatomical and physiological changes in a human mother and fetus during pregnancy and gestation

Dustin F. Kapraun; John F. Wambaugh; R. Woodrow Setzer; Richard S. Judson

doi:10.1371/journal.pone.0215906

Abstract

Many parameters treated as constants in traditional physiologically based pharmacokinetic models must be formulated as time-varying quantities when modeling pregnancy and gestation due to the dramatic physiological and anatomical changes that occur during this period. While several collections of empirical models for such parameters have been published, each has shortcomings. We sought to create a repository of empirical models for tissue volumes, blood flow rates, and other quantities that undergo substantial changes in a human mother and her fetus during the time between conception and birth, and to address deficiencies with similar, previously published repositories. We used maximum likelihood estimation to calibrate various models for the time-varying quantities of interest, and then used the Akaike information criterion to select an optimal model for each quantity. For quantities of interest for which time-course data were not available, we constructed composite models using percentages and/or models describing related quantities. In this way, we developed a comprehensive collection of formulae describing parameters essential for constructing a PBPK model of a human mother and her fetus throughout the approximately 40 weeks of pregnancy and gestation. We included models describing blood flow rates through various fetal blood routes that have no counterparts in adults. Our repository of mathematical models for anatomical and physiological quantities of interest provides a basis for PBPK models of human pregnancy and gestation, and as such, it can ultimately be used to support decision-making with respect to optimal pharmacological dosing and risk assessment for pregnant women and their developing fetuses. The views expressed in this article are those of the authors and do not necessarily represent the views or policies of the U.S. Environmental Protection Agency.

Citation: Kapraun DF, Wambaugh JF, Setzer RW, Judson RS (2019) Empirical models for anatomical and physiological changes in a human mother and fetus during pregnancy and gestation. PLoS ONE 14(5): e0215906. https://doi.org/10.1371/journal.pone.0215906

Editor: Frank T. Spradley, University of Mississippi Medical Center, UNITED STATES

Received: October 18, 2018; Accepted: April 10, 2019; Published: May 2, 2019

This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Data Availability: All relevant data are within the manuscript and its Supporting Information files.

Funding: This work was supported in part by an appointment (DFK) to the Research Participation Program at the National Center for Computational Toxicology administered by the Oak Ridge Institute for Science and Education (ORISE). through an interagency agreement between the US Department of Energy and the US Environmental Protection Agency. ORISE did not play any role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. More information about ORISE is available at https://orise.orau.gov/.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Human health chemical risk assessments frequently consider pregnant women as a subpopulation of interest based on the relatively high exposure rates and/or susceptibility of this group to various compounds [1, 2]. Unfortunately, pregnant women are generally underrepresented in pharmaceutical clinical studies [3] and non-therapeutic chemicals are rarely studied in humans at any life-stage [4–6]. In order to assess the risk posed by a chemical, pharmacokinetic (PK) modeling can be used to relate chemical exposure to potential toxicity in tissues [7]. PK models describe chemical absorption, distribution, metabolism, and elimination by the body [8, 9]. Furthermore, such models allow one to quantify the tissue concentrations resulting from external doses, whether they are controlled (e.g., doses administered in a clinical trial or animal toxicity study [10]) or uncontrolled (e.g., through complex environmental exposures [11]).

During gestation there are windows of toxic susceptibility during which chemical insults may induce life-long adverse effects [2, 9, 12–14]. Mathematical PK models provide a means for predicting fetal tissue exposures to chemicals [12, 15]. Given that PK data in pregnant women and in utero infants are unavailable for most chemicals, models are needed to estimate doses of concern based on data collected for non-pregnant adults or from animals [12]. Physiologically based PK (PBPK) models offer an attractive option for extrapolating information in applications such as human health risk assessments [16–18]. Furthermore, PBPK models can be used to understand and potentially replace some of the default uncertainty factors that are typically applied when using toxicity data to establish a reference dose [12, 19].

Taking into account an external dose (such as an amount that is consumed orally, applied dermally, or inhaled from ambient air), a PBPK model utilizes information about absorption into the body, distribution and storage throughout the body’s tissues, metabolism within particular tissues, and excretion from the body to estimate internal doses (amounts or concentrations) at various sites within the body [20]. In other words, a PBPK model predicts concentrations in various tissues (e.g., adipose and brain tissues) represented by “compartments” of known volume that are connected by “flows”, most typically of blood [21, 22]. Barton et al. [23] identified many ways in which PBPK models may be used for extrapolation, including across life-stages [15, 24]. For the current manuscript, we constructed mathematical models describing anatomical and physiological changes associated with human pregnancy and gestation, as we contend that such models must be an integral part of any PBPK model that describes the kinetics of a chemical in a mother and her fetus over a significant portion of the pregnancy.

Several different collections of time-varying formulae describing quantities related to human pregnancy and gestation have already been published, and most recently Dallmann et al. [3] performed a meta-analysis of the literature in order to construct such a set of time-dependent formulae. Prior to this, Luecke and coauthors [9, 15, 25–27] developed a collection of formulae describing anatomical and physiological changes related to pregnancy and gestation. This latter collection of models has been particularly influential, with more than one hundred citations in peer-reviewed publications related to PBPK models. The following list describes some of the publications and research groups that have provided one or more formulae related to human gestation and pregnancy:

Wosilait et al. [27] constructed an empirical model for human embryonic and fetal growth during in utero development that was based on four data sets covering various periods of gestation that were published between 1909 and 1975.
Luecke et al. [15] published a general human pregnancy PBPK model that uses the fetal mass model of Wosilait et al. [27]. In the context of the PBPK model, Luecke et al. [15] provided a collection of models for changes in maternal and fetal tissue volumes and blood flow rates that are based upon allometric scaling of the fetal mass, but they did not describe the methods and data that were used to calibrate these models. Those authors cited a “submitted” article (“Luecke et al., ‘93b”) as the source of the models, but the manuscript of the same title that was ultimately published [25] only described the models and data sources for time-varying masses of fetal tissues (not for fetal blood flow rates, maternal tissue masses, or maternal blood flow rates).
Abduljalil et al. [28] and Gaohua et al. [29] described empirical models for many of the anatomical and physiological changes that occur in a human mother during pregnancy. For their maternal models, these authors compiled an extensive list of published studies to build a large combined data set for various anatomical and physiological changes that occur in human, singleton, low-risk, normal pregnancies. Abduljalil et al. [28] also developed a model for fetal volume and Gaohua et al. [29] constructed models for the volume of and blood flow rates to a “fetoplacental unit”, which they consider as a single lumped compartment in their human pregnancy PBPK model. For each quantity of interest, Abduljalil et al. [28] used the coefficient of determination (R²) to select an optimal model from among several polynomial models, but using this measure of goodness of fit tends to bias the model selection process by favoring models with more parameters [30].
El-Masri et al. [31] published a “life-stage” PBPK model for humans that includes a PBPK model for pregnancy and gestation. Therein, the authors provided some models for time-dependent parameters, but did not describe their methods for doing so. They also cited and used some models of Luecke et al. [25], which have already been described in this list.
Dallmann et al. [3] curated a data set and used it to fit models for anatomical and physiological changes that occur in women and their fetuses during pregnancy. In this analysis, 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. In contrast to models constructed by others, the models of Dallmann et al. [3] use “fetal age” (i.e., time since fertilization of the ovum) instead of “gestational age” (time since the first day of the final menstrual period prior to conception) as the independent variable.
Zhang et al. [32] published a PBPK model for a human mother and fetus and introduced novel models for volume of and blood flow to several fetal compartments not included in the other collections of formulae named here in this list. Like Abduljalil et al. [28], Zhang et al. [32] used the coefficient of determination (R²) to select optimal models; as asserted previously, using this measure of goodness of fit tends to bias the model selection process [30]. Furthermore, Zhang et al. [32] do not completely describe all fetal blood flow rates relevant to their PBPK model; for example, they do not provide formulae for blood flow through the “ductus venosus/foramen ovale” or to the fetal “rest of body”, both of which are depicted in their PBPK model schematic Fig 1 [32]. Finally, several of the models provided by Zhang et al. [32] are only valid during relatively brief periods of pregnancy (e.g., “10–20 weeks” for “fetal brain blood flow”) and they make no recommendation about extrapolating the models for use in other periods of pregnancy, though they, themselves, seem to have performed PBPK model simulations for times well outside these periods (e.g., up to “40 weeks”).
Abduljalil et al. [33] described empirical models for organ and tissue volumes of a human fetus during gestation. To construct the models, these authors compiled a data set from published studies on embryonic and fetal tissue growth and composition during in utero development. Like Dallmann et al. [3], but in contrast to their own previous work [28], these authors used fetal age rather than gestational age as the independent variable in their models.

Download:

Fig 1. Mass of a human mother vs. gestational age.

The cubic model (solid line) given by Eq 1 was selected as the most parsimonious model in our analysis. The models of Abduljalil et al. [28] and Gaohua et al. [29], both quadratic, were calibrated using the same curated data set [28] used by us. The maternal mass gain data (or “growth data”) depicted here were modified from the source [48] to account for an assumed initial mass as described in the text. Note that all models depicted here describe the mass of the entire maternal body plus the products of conception (including the fetus, placenta, and amniotic fluid).

https://doi.org/10.1371/journal.pone.0215906.g001

In Table 1, we provide an itemized comparison of the various publications in the preceding list.

Download:

Table 1. Itemized comparison of selected publications that contain one or more formulae related to human gestation and pregnancy.

https://doi.org/10.1371/journal.pone.0215906.t001

We developed the current manuscript en route to producing a PBPK model for a human mother and fetus because we were not able to find in the literature useful models for all the time-varying quantities of interest that we needed to construct such a PBPK model. This manuscript expands on previous work by constructing and selecting empirical models for anatomical and physiological changes in a human mother and fetus that either (1) have not been described mathematically in previous similar collections of human pregnancy models (i.e., models for blood flow rates to maternal adipose tissue, gut, liver, and thyroid, and to the fetal lung, gut, kidney, brain, liver, and thyroid, and through the fetal ductus arteriosus and the foramen ovale); (2) have been described mathematically, but by models that appear to give unreasonably high or low values when compared with data (i.e., the Luecke et al. [15] models for maternal plasma volume, amniotic fluid volume, and fetal lung mass, the Gaohua et al. [29] model for maternal adipose mass, the Dallmann et al. [3] model for maternal glomerular filtration, and the Abduljalil et al. [33] model for fetal kidney mass); or (3) have been described mathematically, but using model parameterization and selection methods that are either not statistically rigorous (e.g., the models of Abduljalil et al. [28] and Zhang et al. [32]) or not described at all (e.g., the models of Luecke et al. [15] and Gaohua et al. [29]). Wherever possible, we have endeavored to compare our models to those presented elsewhere. Also, we include herein mathematical descriptions for “rest of body” volumes and blood flow rates that incorporate mass balance principles, whereas the aforementioned published collections (with the notable exception of Zhang et al. [32]) do not. These efforts have resulted in a comprehensive collection of empirical models for important time-varying quantities that one should consider when constructing a PBPK model of human gestation and pregnancy.

Methods

Data sets

We used both curated (multiple-source) data sets and original (single-source) data sets to calibrate empirical models for various anatomical and physiological quantities that vary during gestation. In particular, we relied heavily on composite data sets published by Abduljalil et al. [28] and Abduljalil et al. [33]; for those quantities not described in these two sources (e.g., fetal blood flow rates), we located data from other published studies. Also, for cases in which data were only available in a graphical form, we used the WebPlotDigitizer data extraction tool [34] to convert the graphically-presented data into numerical data. The sources for the specific data sets used to calibrate models for each quantity of interest are identified in Table 2, and additional details are given in the Results section.

Download:

Table 2. Data sources, preferred models, and relevant figures for various quantities of interest.

https://doi.org/10.1371/journal.pone.0215906.t002

Some data sets are based on gestational age, or time since the last menstruation, whereas others are based on fetal age, or time since fertilization of the ovum. For purposes data analysis and model comparison, we assumed that gestational age equals fetal age plus two weeks [42] to convert data and models based on fetal age to time scales based on gestational age.

Models

We examined four basic types of models to describe changes in masses, volumes, percentages (such as hematocrits) and rates (such as blood flow and glomerular filtration rates) that occur during the singleton pregnancy of an “average” healthy woman. In particular, we considered polynomial models (up to degree 3), Gompertz models, logistic models, and allometric power law models. For quantities expected to have initial values (i.e., values at the beginning of gestation) of zero (such as fetal body mass), we examined pure Gompertz and logistic growth models as candidate models, whereas for quantities expected to have initial values substantially greater than zero (such as maternal body mass) we examined modified Gompertz and logistic growth models that include an additional parameter describing a “baseline” value for the quantity of interest. In these models, the additional parameter is an additive constant that does not influence the rate of growth. For some of the quantities considered, power law models were used to relate the quantity to a body mass (i.e., the maternal or fetal mass). When the quantity in question had an initial value substantially greater than zero but the related body mass did not (as for fetal body mass), we examined modified power law models that include an additional parameter describing a baseline value. Formulae and references for all models we considered are provided in Table 3.

Download:

Table 3. Types of models used to describe changes in masses, volumes, percentages, and physiological rates during pregnancy and gestation.

https://doi.org/10.1371/journal.pone.0215906.t003

Model calibration

In considering any particular model and any given quantity of interest (e.g., maternal body mass), we used data to identify the model parameters {θ₀, θ₁, …} that allow one to estimate the quantity of interest as a function y of an independent variable t (e.g., a gestational age or body mass) and said parameters (cf. Table 3). The data sets utilized included means, standard deviations, and sample sizes paired with times (gestational ages) or body masses. For data sets in which standard deviations and samples sizes were unavailable, we assumed a 20% coefficient of variation and a sample size of one for each data point. We used a standard maximum likelihood approach to obtain a maximum likelihood estimate (MLE) for the model parameters. Further details are provided in the Supporting Information (S1 File).

Model selection

After obtaining the MLE for each model for a given data set and quantity of interest, we chose the most parsimonious model by applying the Akaike information criterion (AIC) [47]. We computed the AIC score of each model as where k is the number of parameters in the model, is the MLE, represents the data, and denotes the log-likelihood function. The model with the lowest AIC score was then selected as the most parsimonious model. For those cases in which the most parsimonious model gave negative values for the quantity of interest at any point in the domain of applicability (e.g., for gestational ages from 0 to 42 weeks), the model with the next lowest AIC and that did not produce negative values was identified as the “preferred” model.

We remark that we have not attempted to statistically compare our models to other published models. This is because, in many cases, the other models were developed using different data sets. In selecting models, we compared our candidate models to one another (but not to any other published models) using robust statistical methods designed to assess parsimony and agreement with a specific data set; in each case, we then proposed a preferred model. We have provided representations of models published by other authors in our various figures for purposes of visual comparison only. Thus, readers can see in which cases our models appear to be very similar or very dissimilar to other published models.

Composite models

For some important anatomical and physiological quantities, raw data expressed as values vs. gestational ages were not available. For such quantities (e.g., maternal blood flow to adipose tissue and fetal blood flow through the foramen ovale), we constructed composite models. That is, in each such case, we used information about relationships between the quantity of interest and other modeled quantities, possibly along with proportionality constants or relative percentages, to construct a model for the quantity of interest.

Programming details

For all analyses described herein, we used Python 3.6.4 with the NumPy (version 1.13.3), SciPy (version 0.19.1), Matplotlib (version 2.1.2), and PIL (version 5.0.0) packages. Scripts are available in the Supporting Information (S2 File).

Naming conventions

We have used mathematical symbols to denote various quantities of interest throughout this manuscript. For example, represents the volume of the adipose tissue in the mother. In general, a superscript on such a symbol will be “m” in the case a maternal quantity and “f” in the case of a fetal quantity. When present, the subscript on such a symbol is typically a four-letter code and it indicates a particular physiological compartment. A list of compartment codes is provided in Table 4. For symbols representing blood flow rates through temporary blood vessels or routes in the fetus, the subscript is an upper-case two-letter code indicating the specific blood vessel or route. In particular, “DA”, “DV”, and “FO” indicate the ductus arteriosus, ductus venosus, and foramen ovale, respectively.

Download:

Table 4. Four-letter codes used to represent compartments in a human mother or fetus.

https://doi.org/10.1371/journal.pone.0215906.t004

Results

For each quantity of interest discussed in this section, we describe: (1) the data we used for model calibration, (2) the model selected by us after considering both parsimony and plausibility, and (3) previously published models for the same quantity of interest. Our recommended, or “preferred”, models for all quantities of interest are identified by equation number in Table 2.

Maternal mass

To identify an optimal model for the mass of an average woman throughout gestation, we used data curated by Abduljalil et al. [28] to calibrate linear, quadratic, and cubic functions (i.e., polynomial functions of degree 1, 2, and 3), as well as modified Gompertz and logistic functions. The cubic model, which had the lowest AIC, gives the mass (kg) of an average pregnant woman as (1) where t is the gestational age (weeks) of her fetus. Table 5 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. The cubic model of Eq 1 is shown in Fig 1. It is important to note that “maternal mass” includes the mass of the fetus, the placenta, and the amniotic fluid.

Download:

Table 5. Maternal mass models (mass in kg vs. gestational age in weeks).

https://doi.org/10.1371/journal.pone.0215906.t005

Abduljalil et al. [28] and Gaohua et al. [29] both selected quadratic models to give the mass (in kg) of a pregnant woman as a function of gestational age t (in weeks). In both cases, the authors used the same maternal mass data used by us [28] in order to obtain their models. These two models are also shown in Fig 1.

Even without considering the starting mass of a woman, there is tremendous variability in mass gain trends of women during pregnancy. For example, some women have gained 23 kg or more at term, while others have actually experienced a reduction in total body mass [49]. Hytten and Leitch [48] depicted data summarizing the mean mass gain during pregnancy of 2868 normotensive primigravidae (i.e., women who were pregnant for the first time and who maintained blood pressure in the normal range throughout their pregnancies). Unfortunately, the actual data are not provided by Hytten and Leitch [48], so we were unable to calibrate models to that data as discussed in the Methods section. To compare the trend shown by Hytten and Leitch [48] with our model and the models of Abduljalil et al. [28] and Gaohua et al. [29], we captured data points from their graphical depiction of the data trend curve, calibrated a Gompertz growth curve to fit those points, then added an initial mass of 61.1 kg to match the initial mass of the Abduljalil et al. [28] model. The resulting curve is shown for purposes of comparison in Fig 1.

Fetal mass

To identify an optimal model for the volume of an average human fetus throughout gestation, we used data curated by Abduljalil et al. [28]. The Gompertz model, which had the lowest AIC, gives the volume (mL) of an average human fetus as (2) where t is the gestational age (weeks). Since the average density of a human fetus is approximately 1 g/mL throughout gestation [50], W^f(t) also represents the mass (g) of the fetus at a given gestational age. Table 6 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. Fig 2 shows the Gompertz model of Eq 2 and three published models for human fetal mass [3, 27, 28].

Download:

Table 6. Fetal mass models (mass in g vs. gestational age in weeks).

https://doi.org/10.1371/journal.pone.0215906.t006

Download:

Fig 2. Volume (mL) or mass (g) of a human fetus vs. gestational age.

The Gompertz model (solid line) given by Eq 2 was selected as the most parsimonious model in our analysis. The models of Abduljalil et al. [28] and Wosilait et al. [27] were also Gompertz models, though the model parameters used by those authors were different. Dallmann et al. [3] used a log-logistic model for fetal volume. The model of Abduljalil et al. [28] was calibrated using the same curated data set [28] used by us, while the models of Wosilait et al. [27] and Dallmann et al. [3] were calibrated using different data sets.

https://doi.org/10.1371/journal.pone.0215906.g002

Like us, Abduljalil et al. [28] and Wosilait et al. [27] both selected Gompertz models to give the volume (mL) of a human fetus as a function of gestational age t (weeks). The model of Abduljalil et al. [28] was calibrated using the same curated data set [28] used by us, but we were unable to reproduce the Gompertz model parameters reported by them. Nevertheless, as illustrated in Fig 2, our Gompertz model and the Abduljalil et al. [28] model predict similar fetal volumes for most gestational ages. The model of Wosilait et al. [27], also depicted in Fig 2, was calibrated using a different data set; thus, it is not surprising that they obtained different Gompertz model parameters and different predicted fetal volumes. Dallmann et al. [3] used a log-logistic model for fetal volume that was calibrated with yet another curated data set.

Maternal compartment volumes

While volumes of many maternal organs and tissues remain approximately constant throughout pregnancy, several of these undergo dramatic changes. In particular, the volumes of adipose tissue and blood components (plasma and red blood cells) generally increase as pregnancy progresses. In addition, the “products of conception”, which include the placenta and the amniotic fluid, increase in size as pregnancy progresses and therefore contribute to maternal mass gain [48].

Adipose tissue.

For the mass of the adipose (fat) tissue of an average woman throughout pregnancy, we used data curated by Abduljalil et al. [28] to calibrate various models. In calibrating the polynomial models and the modified Gompertz and logistic models, we used the total maternal fat mass (mean and standard deviation) vs. gestational age data exactly as tabulated by Abduljalil et al. [28]. In an alternative approach, we followed the working assumption of Luecke et al. [15] that maternal fat mass gain correlates with fetal mass; that is, we considered allometric relationships between these two quantities. Using Eq 2, we calculated a fetal mass for each gestational age data point that was tabulated [28] for total maternal fat mass. We then calibrated modified versions of the power law and Luecke power law models (cf. Table 3) using data for the total maternal fat mass (mean and standard deviation) vs. (calculated) fetal mass. Of all models considered, the linear model had the lowest AIC. This model gives the mass (kg) of the adipose tissue of an average pregnant woman as (3) where t is the gestational age (weeks). Since the mean density of human adipose tissue is 0.950 kg/L [51], the volume of the maternal adipose tissue (L) can be computed as (4)

Table 7 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. The linear model of Eq 3 and several other models for maternal fat mass are shown in Fig 3.

Download:

Table 7. Maternal fat mass models (mass in kg vs. fetal mass in kg for power law models, mass in kg vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t007

Download:

Fig 3. Adipose tissue mass of a human mother of vs. gestational age.

The linear model (solid line) given by Eq 3 was selected as the most parsimonious model in our analysis. The models of Abduljalil et al. [28] and Gaohua et al. [29], both quadratic, were calibrated using the same curated data set [28] used by us. The latter of these models was modified as described in the text. The model of Luecke et al. [15] was calibrated using different data. It predicts maternal fat mass as a function of total fetal mass, and was interpreted as described in the text. Dallmann et al. [3] also selected a linear model, but they calibrated their model with different data.

https://doi.org/10.1371/journal.pone.0215906.g003

Abduljalil et al. [28] and Gaohua et al. [29] each selected quadratic models to give the mass (kg) of adipose tissue in a pregnant woman as a function of gestational age t (weeks). In both cases, the authors used the same maternal total fat mass data used by us [28] to obtain their models. Gaohua et al. [29] claim that their model gives the total fat mass (kg) vs. gestational age, but the masses predicted for various gestational ages are about 50% larger than the mean data values upon which the model is based. In an effort to explain this discrepancy, we hypothesized that the Gaohua et al. [29] quadratic model for maternal fat mass is actually a model for the percentage of the mother’s body mass that is fat. Thus, to obtain fat mass values for comparison, we divided the values predicted by their model by 100 and then multiplied the resulting values by corresponding values from the maternal mass vs. gestational age model that these authors provide in the same manuscript [29]. That is, in Fig 3, the model labeled “Gaohua et al. (2012) Model” differs from the model described by those authors.

Luecke et al. [15] constructed a model for maternal fat mass gain, so to compare its predictions with those of the other models, we added an initial mass of 17.14 kg (the mean value for non-pregnant women from the data set curated by Abduljalil et al. [28]) to the predicted mass gain values. The model of Luecke et al. [15], modified as just described to account for an initial maternal fat mass, is shown in Fig 3. For earlier gestational ages, this model predicts maternal fat masses considerably lower than the data values curated by Abduljalil et al. [28]. This may be because the model is a power law based on fetal mass, which does not increase substantially (from an initial mass of about zero) until about 15 weeks, whereas the data suggest that maternal fat mass does begin to increase substantially between conception and 13 weeks.

Note that the total fat mass predicted (by our preferred linear model) for a woman at conception, , represents about 27.9% of the total body mass, W^m(0) = 61.103 kg. This represents a considerable deviation from the figure of 37.5% reported in Table 8 for adipose body mass percentage for women, but note that that number represents an average for all women [35]. The individuals that have been included in studies of pregnancy [3, 28], appear to have a lower mean body fat percentage than the population of all women, or even all women of childbearing age [52].

Download:

Table 8. Percent of total body mass and density (references listed) for various human organs and tissues.

https://doi.org/10.1371/journal.pone.0215906.t008

Plasma.

Plasma is one of the two major constituents of blood. We used the curated data of Abduljalil et al. [28] to calibrate various models for plasma volume in a human mother during pregnancy. The modified logistic model, which we found to be the best of the candidate models based on AIC, gives the plasma volume (L) of the mother as (5) where t is the gestational age (weeks). Table 9 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. We remark that the modified Luecke power law model yielded a lower AIC (365.07 vs. 365.13), but the AIC difference is too small to recommend one model over the other. Furthermore, the modified Luecke power law relates maternal plasma volume to fetal mass whereas the modified logistic model relates maternal plasma volume directly to gestational age; thus, the latter model has the advantage of not requiring an (intermediate) estimate of fetal mass at each time point of interest. Fig 4 shows the modified logistic model of Eq 5 and several other models for maternal plasma volume. (Fig 4 does not show the modified Luecke power law model, but values predicted by that model and the modified logistic model differ by less than 2% throughout the period from 0 to 42 gestational weeks.)

Download:

Table 9. Maternal plasma volume models (volume in L vs. fetal mass in kg for power law models, volume in L vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t009

Download:

Fig 4. Maternal plasma volume vs. gestational age.

The modified logistic model (solid line) given by Eq 5 was selected as the most parsimonious model in our analysis. The models of Abduljalil et al. [28] and Gaohua et al. [29], both cubic polynomials, were calibrated using the same curated data set [28] used by us. The model of Luecke et al. [15] was calibrated using different data. It assumes an initial maternal plasma volume of 2.6 L (or an initial plasma mass of 2.6 kg) and predicts an increase in maternal plasma volume (or mass) as a function of total fetal mass. Dallmann et al. [3] calibrated their cubic model with different data.

https://doi.org/10.1371/journal.pone.0215906.g004

We remark that some longitudinal studies of human subjects have shown that plasma volume typically decreases slightly at the end of pregnancy [55–58]. When applied to the aggregated data set [28] our model selection process led us to choose a model for plasma volume (Eq 5) that does not predict such decrease, though some of the alternative models presented in the literature do [3, 28, 29].

Red blood cells.

Red blood cells (RBCs) make up the other major component of human blood. We used curated data from Abduljalil et al. [28] to calibrate various models for RBC volume in a human mother during pregnancy. We selected the modified logistic model given by (6) as the most parsimonious of the candidate models. Here, denotes the volume (L) of the RBCs at gestation age t (weeks). Table 10 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values.

Download:

Table 10. Maternal RBC volume models (volume in L vs. fetal mass in kg for power law models, volume in L vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t010

Through analysis of data sets describing maternal plasma volume, RBC volume, and hematocrit [28], we independently obtained models for each of these quantities (cf. Eqs 5, 6 and 24). However, as one might expect when independently constructing models of interrelated quantities, the models that arose are not perfectly consistent. Because hematocrit represents the volume percentage of RBCs in whole blood, and because whole blood is mostly made up of plasma and RBCs (with only a small fraction made up of white blood cells and platelets), we can estimate the volume (L) of RBCs in maternal blood as (7) where H^m(t) and represent the maternal hematocrit and plasma volume (L) at gestation age t (weeks). Thus, if one uses the models for maternal plasma volume, RBC volume, and hematocrit given by Eqs 5, 7 and 24, respectively, the model predictions will be consistent with one another. The modified logistic model of Eq 6, the alternate hematocrit-based model of Eq 7, and three published models for maternal RBC volume [3, 28, 29] are shown in Fig 5.

Download:

Fig 5. Maternal RBC volume vs. gestational age.

The modified logistic model (solid line) given by Eq 6 was selected as the most parsimonious model in our analysis, but the hematocrit-based model (second in legend) of Eq 7 ensures consistency with models for plasma volume (Eq 5) and hematocrit (Eq 24). The models of Abduljalil et al. [28] and Gaohua et al. [29], both of which are linear models, were calibrated using the same curated data set [28] used by us. Dallmann et al. [3] did not create a model for maternal RBC volume, so the model attributed to them here is algebraically derived from their models for plasma volume and hematocrit.

https://doi.org/10.1371/journal.pone.0215906.g005

Placenta.

We used the curated data of Abduljalil et al. [28] to calibrate various models for human placenta volume. The cubic growth model given by (8) was selected as the most parsimonious model for placenta volume (mL) at gestational age t (weeks). Table 11 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. Fig 6 shows the cubic growth model of Eq 8 and three published models for placenta volume [3, 15, 28].

Download:

Table 11. Placenta volume models (volume in mL vs. fetal mass in kg for power law models, volume in mL vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t011

Download:

Fig 6. Placenta volume vs. gestational age.

The cubic growth model (solid line) given by Eq 8 was selected as the most parsimonious model in our analysis. The model of Abduljalil et al. [28], also a cubic polynomial model, was calibrated using the same curated data set [28] used by us. The model of Luecke et al. [15] was calibrated using different data and assumes a relationship between placenta volume and fetal mass. Dallmann et al. [3] calibrated their cubic model with different data.

https://doi.org/10.1371/journal.pone.0215906.g006

It is worth noting that Eq 8 yields negative volumes for the placenta for gestational ages less than about 1.97 weeks; thus, this equation should not be used to estimate placental volume during that time frame. In any case, conception does not occur until a gestational age of about 2 weeks, so neither the placenta nor the embryo, itself, come into existence until after that time. At a fetal age of about 4 days, or a gestational age of about 18 days, the cells in the periphery of the blastocyst (i.e., the early embryo) become distinguishable as a trophoblast; this trophoblast is a precursor to the fetal component of the placenta [42]. We propose that Eq 8 be used to estimate the volume of the placenta after 2 weeks, but it should not be considered accurate until 9 weeks (the time of the first data point of Abduljalil et al. [28]).

Amniotic fluid.

We used the curated data of Abduljalil et al. [28] to calibrate various models for human amniotic fluid volume. The logistic model given by (9) was selected as the most parsimonious model for amniotic fluid volume (mL) at gestational age t (weeks). Table 12 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. The logistic model of Eq 9 and three published models for amniotic fluid volume [3, 15, 28] are shown in Fig 7.

Download:

Table 12. Amniotic fluid volume models (volume in mL vs. fetal mass in kg for power law models, volume in mL vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t012

Download:

Fig 7. Amniotic fluid volume vs. gestational age.

The logistic model (solid line) given by Eq 9 was selected as the most parsimonious model in our analysis. The model of Abduljalil et al. [28], which is a fifth degree polynomial model, was calibrated using the same curated data set [28] used by us. The model of Luecke et al. [15] was calibrated using different data and assumes a relationship between amniotic fluid volume and fetal mass. That model is shown here both as originally stated (in the publication) and after correcting a presumed error (to obtain the “Adjusted Model”) as described in the text. Dallmann et al. [3] calibrated their fourth degree polynomial model with different data.

https://doi.org/10.1371/journal.pone.0215906.g007

Eq 9 yields nonnegative values for amniotic fluid volume on the time domain of interest (t > 0); however, as with the placenta volume model, it is worth noting that conception does not occur until a gestational age of about 2 weeks and so none of the products of conception (including amniotic fluid) exist until after that time. To be clear, Eq 9 should not be used to estimate the volume of the amniotic fluid prior to 2 weeks, and should not be considered accurate until 9 weeks (the time of the first data point of Abduljalil et al. [28]).

In Fig 7, we present two versions of the amniotic fluid volume model of Luecke et al. [15]. The first version shown (“Luecke et al. (1994) Model”) represents a Luecke power law model (cf. Table 3) with coefficient values as shown in Table 3 of Luecke et al. [15]. Since that model seems to under-predict amniotic fluid volumes by an order of magnitude, we hypothesized that the first coefficient printed in their table (θ₀ = 0.002941) may have been off by a factor of 10. We therefore increased that coefficient by a factor of 10 (θ₀ = 0.02941) to obtain the second model version (“Luecke et al. (1994) Adjusted Model”) shown in Fig 7.

Other specific compartments.

Any given PBPK model may include compartments that correspond to specific organs and tissues in the mother that do not vary significantly during a normal pregnancy. To obtain volumes for such compartments that coincide with the time-varying volumes already described, we examined reference values for total body masses of “typical” women and the masses of various organs and tissues in such women as reported by the ICRP [35]. We used these to compute the percent of total body mass accounted for by various tissues. These percentages, along with densities associated with various organs and tissues, are displayed in Table 8. We calculated the static compartment volumes for a typical pregnant woman by multiplying the non-pregnant body mass (W^m(0) = 61.103 kg) by the female mass percentage of the relevant tissue or organ and dividing by the relevant density. The volumes obtained for maternal brain, thyroid, kidneys, gut, liver, and lungs are shown in Table 13.

Download:

Table 13. Volumes of (some) maternal compartments that do not change during pregnancy.

https://doi.org/10.1371/journal.pone.0215906.t013

Rest of body.

We used the principle of mass balance to obtain a formula for the volume of a “rest of body” compartment comprising all mass in the pregnant female body that has not been accounted for in one of the specific compartments already described. Assuming that the fat-free mass of the mother has an average density of 1.1 g/mL throughout pregnancy [59], the total volume (L) of the maternal fat-free mass is (10)

Here, we have assumed that the products of conception appear at gestational age 2 weeks, and that the densities of the placenta and amniotic fluid are 1.02 g/mL [53] and 1.01 g/mL [54], respectively (cf. Table 8). (We further assumed that W^m, W^f, , and are described by Eqs 1, 2, 3, 8 and 9, respectively.) Also, the total volume (L) of all the specific maternal compartments, excluding adipose tissue, is (11)

(We assumed that the quantities on the right-hand side of this equation are described by Eqs 5 and 7 and the values listed in Table 13.) Thus, the volume of the maternal rest of body compartment (L) is (12) where t is the gestational age (weeks). As shown in Fig 8, Eq 12 results in volumes for the maternal rest of body compartment that fluctuate between approximately 31 and 34 L during pregnancy.

Download:

Fig 8. Volume of maternal “rest of body” compartment vs. gestational age.

The formula for the depicted model, which is based on mass balance, is provided as Eq 12.

https://doi.org/10.1371/journal.pone.0215906.g008

Maternal blood flow rates

During pregnancy, a woman experiences an increase in cardiac output as well as significant changes in the relative blood flow rates to various organs and tissues [35]. We first provide an expression for the maternal cardiac output, which corresponds to the quantity we denote (the blood flow rate into the mother's arterial blood compartment). Then, we describe flow rates to the other maternal compartments as proportions of the cardiac output. As shown in Table 2.44 from Section 12.2.7 of the ICRP [35] report, the percentage of cardiac output directed to any particular organ or tissue can change as pregnancy progresses. For blood flow rates for which time course data were not available, we assumed that the change from the “non-pregnant” percentage to the near-term “pregnant” percentage (as reported in that table) occurs in a linear fashion between 0 and 40 weeks of gestational age; i.e., the percentage is a linear function of the gestational age. Hereafter, we refer to such models as “linear transition models”.

Cardiac output.

We used the curated data of Abduljalil et al. [28] to calibrate various models for maternal cardiac output. We assume this to be both the total flow rate into the maternal arterial blood compartment (hence the use here of the symbol ) and the total flow rate into the maternal venous blood compartment. We selected the cubic model given by (13) as the most parsimonious for maternal cardiac output (L/h) at gestational age t (weeks). Table 14 shows the maximum likelihood estimates of the parameter values for all models considered along with the associated log-likelihood and AIC values. Fig 9 shows the cubic model of Eq 13 along with several other models for maternal cardiac output.

Download:

Table 14. Maternal cardiac output models (flow rate in L/h vs. maternal mass in kg for power law models, flow rate in L/h vs. gestational age in weeks for all other models).

https://doi.org/10.1371/journal.pone.0215906.t014

Download:

Fig 9. Maternal cardiac output vs. gestational age.

The cubic model (solid line) given by Eq 13 was selected as the most parsimonious model in our analysis. The models of Abduljalil et al. [28] and Gaohua et al. [29], both of which are quadratic models, were calibrated using the same curated data set [28] used by us. Dallmann et al. [3] calibrated their model with different data.

https://doi.org/10.1371/journal.pone.0215906.g009