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Gestational growth trajectories derived from a dynamic fetal–placental scaling law

Daniel Baller

Daniel Baller

Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996, USA

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Diana M. Thomas

Diana M. Thomas

Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996, USA

[email protected]

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Kevin Cummiskey

Kevin Cummiskey

Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996, USA

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Carl Bredlau

Carl Bredlau

Department of Computer Science, Montclair State University, Montclair, NJ 07043, USA

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Nadav Schwartz

Nadav Schwartz

Division of Maternal Fetal Medicine, 3400 Spruce Street, Philadelphia, PA 19104, USA

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Kelly Orzechowski

Kelly Orzechowski

Thomas Jefferson University Hospital, Philadelphia, PA 19107, USA

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Richard C. Miller

Richard C. Miller

Department of Obstetrics and Gynecology, St Barnabas Medical Center, Livingston, NJ 07039, USA

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Anthony Odibo

Anthony Odibo

Division of Maternal Fetal Medicine, University of South Florida, Tampa, FL 33620, USA

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Carolyn M. Salafia

Carolyn M. Salafia

Placental Analytics, New Rochelle, NY 10538, USA

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    Fetal trajectories characterizing growth rates in utero have relied primarily on goodness of fit rather than mechanistic properties exhibited in utero. Here, we use a validated fetal–placental allometric scaling law and a first principles differential equations model of placental volume growth to generate biologically meaningful fetal–placental growth curves. The growth curves form the foundation for understanding healthy versus at-risk fetal growth and for identifying the timing of key events in utero.

    1. Introduction

    In routine obstetric care, fetal growth measured from ultrasound technology is monitored against reference fetal growth curves to classify small for gestational age (SGA), appropriate for gestational age (AGA) and large for gestational age (LGA) infants. The terms SGA, AGA and LGA are defined by the location of ultrasound estimated fetal weight along time-dependent percentile curves [1]. At a given gestational age, below the 10th percentile is defined as SGA and above the 90th percentile is defined as LGA [1]. Any birth weight in between the 10th and 90th percentiles is referred to as AGA [1]. Both extremes of fetal growth (SGA and LGA) are at higher risk for long-term complications such as developmental delay [2,3] (SGA) and obesity-related comorbidities [4] (LGA). Fetal growth curves generally fit cross-sectional birth weight or ultrasound data obtained from a sufficiently large database [58]. Static growth curves have been invaluable to distinguish ‘healthy’ versus ‘at risk’ pregnancies, but they do not generate longitudinal growth trajectories from early pregnancy markers nor provide insight into the underlying physiology and dynamic interplay between fetal and placental growth patterns. Moreover, there is an ongoing debate on whether current growth curves should be ‘customized’ or adjusted for variables like parental height and weight, parity and race despite evidence that these factors impact what is considered healthy fetal weight for gestational age [9,10].

    Advances in ultrasound technology and quantitative placental analysis have resulted in detailed descriptors of fetal–placental landmarks [1115]; some aspects of delivered placental shape and fetal outcomes are already determined early in gestation [16]. Abnormal placental development has also been linked to health consequences well into adulthood [1720]. Taken together, these findings suggest that a mathematical model that mechanistically describes the symbiotic relationship between placental growth and fetal growth is valuable for both description and prediction.

    A dynamic model that incorporates the interaction between placenta and fetus was developed by Thomas et al. [21]. The model [21] included the theoretical impact of high versus low glycaemic maternal diet and levels of maternal exercise using an energy balance approach. The ideas in [21] were modified to simplify the placental volume model resulting in a model that can be fitted to individual data and generate the variety of observed individual placental growth curves [22]. Both models suffer from a foundational reliance on objective knowledge of maternal intake. While maternal intake can be objectively measured from multiple measures of body composition and energy expenditure from the doubly labelled water method [23], these methods are costly, require special equipment and personnel, and must be performed at a clinic. Self-reported energy intake, on the other hand, has been demonstrated deeply unreliable and cannot be used to derive scientific conclusions [24]. A fetal–placental model that does not rely on knowledge of maternal intake would eliminate this challenge.

    Here, we use a fetal–placental scaling law validated in over 20 000 pregnancies to develop a flexible dynamical systems model that predicts fetal and placental trajectories. The dynamic model parameters were estimated using a large database of longitudinally measured placental volumes from uncomplicated pregnancies and pregnancies complicated by SGA. The resulting model for healthy pregnancies was then compared to the model derived from pregnancies complicated by SGA.

    2. Methods

    2.1. Study design

    This study leverages a well-established fetal–placental scaling law and an existing dynamical systems model for placental volume to arrive at a differential equation model that generates fetal and placental growth curves. The system of equations was developed to answer the following three questions:


    What parameter ranges in the fetal–placental model yield different observed placental and fetal growth patterns?


    How can the fetal–placental model be used to identify pregnancies at risk for complications, namely SGA?


    What theoretical insights does the model yield that were not previously known?

    To address Question 1, we first applied placental volume data from uncomplicated pregnancies to fit the placental volume model parameters. Using these values as a starting basis, the parameter values were perturbed to generate fetal growth curves to match published reference percentile curves to identify what combinations of placental parameters yield different fetal growth trajectories and birth outcomes of SGA, AGA and LGA.

    A comparison of placental volume model parameter values and placental growth trajectories from uncomplicated pregnancies versus pregnancies diagnosed as SGA was performed to answer Question 2.

    Finally, we conducted a perturbation analysis of the fetal growth differential equation model to address Question 3.

    2.2. Model development

    Two interconnected differential equations were developed. They describe placental volume growth and fetal growth, respectively. The fetal growth model leverages a validated fetal–placental scaling law. Model variables, parameters and assumptions are summarized in table 1.

    Table 1. List of model assumptions, model variables with their units and model parameters with their units.

    model assumptions
    assumption number assumption statement
    1 the early rate of placental volume growth is directly proportional to placental volume
    2 initial growth is limited by a saturation value beyond which the placental volume cannot increase
    3 the self-limiting component of the model is described by multiplying the term which exhibits exponential growth by a limiting factor, (1PVK)
    4 both the proportionality constant, r, and the carrying capacity, K, are independent of time
    5 the placental weight, PW(t), and fetal weight, FW(t), on gestational day t obey a power scaling law: PW(t)=α(t)FW(t)β
    6 the value of β is 3/4 throughout the pregnancy, while α(t) is a decreasing function that converges to 1 at term
    7 the placental volume is related to placental weight by PV=ρPW, where ρ is approximately 1 g ml−1
    variable and parameter descriptions
    variable/parameter description units
    PV(t) the placental volume on day t of gestation ml
    r The placental volume growth rate in early gestation is directly proportional to the current placental volume. The value of r is the proportionality constant 1/d
    K the carrying capacity of the placenta which is the absolute possible limiting volume the placenta cannot exceed ml
    P0 the volume of the placenta in the first trimester (approx. 84 days) ml
    PW(t) the placental weight on day t of gestation g
    FW(t) the fetal weight on day t of gestation g
    α(t) the time-varying proportionality constant in the power law: PW(t)=α(t)FW(t)β g(1/4)
    β the power of the scaling relationship in PW(t)=α(t)FW(t)β β=3/4
    ρ the density of fetal weight ml g−1
    time of inflection point the time point when the curve shifts from concave up to concave down in the S shape gestational day

    2.2.1. Model for placental volume growth

    We adopt a model that yields sigmoidal growth trajectories for placental volume similar to previous modelling studies [21,22]. The sigmoidal shape of placental volume growth has been experimentally established [2528] and suggests that during early pregnancy, placental volume increases exponentially. However, eventually, due to space restrictions and other growth-related mechanisms, the rate of placental volume growth declines to zero yielding a growth plateau.

    Referred to by mathematical biologist Alfred J. Lotka as the ‘law of population growth’ [29], sigmoidal placental growth can be expressed by the logistic growth model [30]

    where PV is the placental volume (ml) on gestational day t, r is the proportionality constant and K is the carrying capacity. Using the method of separation of variables [31], the closed-form solution to this model is
    given the initial condition, PV(84) = P0. The initial condition was selected at day 84 because that is the end of the first trimester by which time most women know that they are pregnant.

    The model in equation (2.1) assumes the following founded in dynamic logistic growth [30,32]:


    The early growth rate of the placental volume is directly proportional to placental volume, rPV , where 0 < r < 1 represents the proportionality constant. The validity of this assumption for placental growth is supported by the fact that early placental growth is primarily due to cell division, which has been well established to follow this assumed growth pattern [33].


    Increased placental volume over gestation is eventually limited by a saturation value beyond which the placental volume cannot increase [33]. This saturation value is referred to as the placental carrying capacity in millilitres and denoted by the value K. The carrying capacity is not the placental volume at term, but rather an upper bound which placental volume cannot increase beyond.


    Placental growth has a self-limiting property because the early growth term, rPV, is multiplied by a limiting factor (1(PV/K)). This limiting multiplier has the property that when PV is close to K, the factor is close to zero [29].


    Both the proportionality constant, r, and the carrying capacity, K, are independent of time.

    2.2.2. The fetal–placental metabolic power scaling law

    Power scaling laws describe a relationship between two quantities that may not change isometrically. For example, it has been well established that human body weight scales to height squared [34] or mathematically

    where α is referred to as the proportionality constant. The constant α differs from individual to individual, but the power of 2 remains fixed. In this case, the body weight scaling law implies that a 10% increase in height will result in more than a 10% increase in weight.

    It was hypothesized by Rubner in 1883 that the basal metabolic rate in mammals scales to body weight raised to the 2/3 power [35]. This model was reformulated in 1947 by Kleiber from analysis of experimental data to conclude that metabolic rate is proportional to body weight raised to the 3/4 power [36]. Since the derivation of what is today referred to as Kleiber's law, the value of the power is highly debated [37,38], but the fact that metabolic rate scales to body weight as a power law and that the power is a fraction between 0 and 1 is established [3941]. While there is debate on the exact value of the power, West et al. [42] have published the most widely accepted theoretical argument relying on a fractal analysis of branching capillaries supporting the value experimentally determined by Kleiber of 3/4.

    The sole mechanism for fetal–metabolic exchange is through the placenta and therefore conjecturing a power law between fetal and placental weight is natural where placental weight is substituted as a proxy for metabolic rate. This allometric model of placental and fetal weight was first posed by Aherne in 1966 [43]. In 2009, Salafia et al. experimentally verified that placental weight scales to birth weight to the 3/4 power by log-transforming the equation

    and applying linear regression to estimate the coefficients of the linear model above [11]. Remarkably, the value for β was close to 3/4 which matched the value for Klieber's laws in mammals [11]. Because the branching placental villi represent an analogue for the human circulatory system, a fractal analysis identical to the one employed by West [42] provided theoretical justification for the value of β = 3/4 [44].

    In addition to the experimental and theoretical justification for β = 3/4, the fetal–placental scaling law was experimentally verified in mice at various stages of gestation [45] and in twin human pregnancies [46]. The dynamic fetal growth model relies on this power scaling law.

    2.2.3. The model for fetal growth

    To advance dynamic fetal growth model development, we begin by listing a series of assumptions that underly the fetal differential equation model:


    The placental and fetal weight obey the power law PW(t) = α(t)FW(t)β during gestation, where PW(t) represents the grams of placental weight on gestational day t, and FW(t) represents grams of fetal weight on gestational day t. This assumption is based on the experimental and theoretical analysis performed in [11,14,4446] where a power law relationship was verified in human and mouse pregnancies independent of gestational age at term.


    The value of β is 3/4 throughout the pregnancy, while α(t) is a decreasing function that converges to 1 at term. The assumption that α(t) = 1 at term is supported by the validation in over 20 000 pregnancies performed in [44]. The assumption that β = 3/4 throughout pregnancy is supported by the decades of studies, both experimental and theoretical, that conclude the metabolic scaling exponent is 3/4 in adults [42,4749] and that the exponent is 3/4 in human singleton pregnancies [11,14,44]. The value of 3/4 has also been experimental validated in mice placental studies at different gestational ages [45] and in twin pregnancies [46].

    Cross-sectional placental weights and fetal weights were published in [50] from pregnancies as early as eight weeks. From these data, we computed the ratio of placental weight and fetal weight raised to the 3/4 power to obtain an estimate for α as a function of gestational age. Plotting α as a function of gestational age, t (figure 1), suggests that α(t) is an exponentially decaying function of gestational age that decreases to 1 at term.


    The placental volume is related to placental weight by PV = ρPW , where ρ is approximately 1 g ml−1. This assumption is simplifying and while there is evidence the density is close to 1 [51], the estimate varies in the literature [52].

    Figure 1.

    Figure 1. Plot of α versus gestational day calculated from the ratio placental weight/fetal weight3/4 using data from [50].

    Using assumptions A5–A7, we derive the fetal growth differential equation model through the following steps.

    Step 1: Let ρ represent the density of the placenta (according to assumption A6, we assume ρ = 1 for numerical calculations) and substitute PW = ρPV into the fetal–placental scaling law


    Step 2: Calculate the derivative of PV in terms of FW


    Step 3: Substitute the scaling law expression, PV=(α(t)/ρ)FWβ, into the placental volume differential equation


    Step 4: Equate the expression in Step 2 with the expression in Step 3 and solve for W/dt


    After input of the parameters r, β, ρ, K and α(t), the solution to this model generates a predicted fetal growth curve, FW(t).

    For theoretical analysis, the functional form for α(t) is not necessary; however, for numerical simulation, a functional form is required. Under the assumption that β = 3/4 and that placental tissue density is approximately 1, we calculated the value of α for the given gestational day, t, using cross-sectional data that included both placental weight and fetal weight [50] from the eighth week of pregnancy forward from the equation α(t)=PW(t)/FW(t)3/4. Direct inspection revealed that the data appeared to follow a power curve. As a result, for simulation purposes, we fit the data to a power curve (figure 1)


    The solution to the fetal growth differential equation, equation (2.2), cannot be expressed in closed form, so all simulations presented here were developed numerically using equation (2.3) for α(t).

    2.3. Placental volume model parameter estimation

    2.3.1. Subjects

    Five different studies were used to fit placental model parameters in normal uncomplicated pregnancies and then compared to placental model parameters in pregnancies diagnosed as SGA. Four of the studies included ultrasound measured placental volume in uncomplicated pregnancies: New York University (NYU), University of Pennsylvania (UPENN), Washington University St Louis (WUSL) and St Barnabas Medical Health Center (St Barnabas). Two additional studies that included at term placental weights were applied to compare the best fit healthy trajectory to placental volume growth curves in SGA pregnancies: the Pregnancy, Infection and Nutrition (PIN) study and the University of Connecticut (UCONN) study.

    2.3.2. The New York University study

    The NYU study [53,54] was a prospective cohort study performed at the obstetric and gynaecologic ultrasound unit of the New York University Medical Center. Pregnant women (N = 95) were enrolled between 11 and 14 weeks gestation and an 11–14 week three-dimensional (3D) volume sweep of the placenta was obtained using a transabdominal probe, Voluson E8; GE Healthcare, Milwaukee, WI, USA. Placental volume was estimated using 4D View software (GE Healthcare, Kretztechnik, Zipf, Austria). Placental weight was also retained at term.

    Only the data from pregnancies without complications were retained for parameter fitting. The NYU study protocol was approved by the New York University Institutional Review Board, and written informed consent was obtained from all study participants.

    2.3.3. The University of Pennsylvania study

    The UPENN study was a prospective cohort study of pregnant women with a singleton pregnancy. Pregnant women were recruited and two ultrasounds were performed, one in the first trimester and one in the second trimester at the Hospital of the University of Pennsylvania. Complete records from N = 551 participants were retained. A 3D volume sweep of the placenta was obtained transabdominally (4–8 MHz probe, GE Voluson Expert, GE Healthcare, Wisconsin, USA) and stored for later evaluation. The stored placenta volume sets were evaluated using 4D View software (GE, Austria) and placental volume was retained.

    Data from pregnancies below the 10th percentile were retained to fit the parameters of the model in the SGA case.

    The UPENN study protocol was approved by the University of Pennsylvania Review Board, and written informed consent was obtained from all study participants.

    2.3.4. The Washington University St Louis study

    Placental volume was obtained in N = 712 pregnant women at WUSL Division of Ultrasound and Genetics between 11 and 14 weeks of pregnancy [16]. Only data from 538 uncomplicated pregnancies were retained. The 3D volume images were obtained using Voluson 730 Expert ultrasound machines (GE Medical Systems, Milwaukee, WI, USA) equipped with a 4–8 MHz transducer. The placental images were stored and similar to the NYU and UPENN studies, placental volume was estimated later using 4D View computer software (GE Medical Systems).

    The WUSL study was approved by the Institutional Review Board at Washington University in St Louis and all participants provided consent.

    2.3.5. The St Barnabas Medical Health Center (St Barnabas) study

    Pregnant women who answered posted advertisements were recruited from two private practice obstetrical offices in northern New Jersey. Subjects were eligible for the study if they were between 18 and 35 years old and less than 12 weeks pregnant at enrolment confirmed by first-trimester ultrasonography. Women were excluded for: (i) history of smoking and/or drug abuse, (ii) a history of prior gestational diabetes or preeclampsia, (iii) medical comorbidities (i.e. chronic hypertension, diabetes, asthma, etc.), (iv) known uterine anomalies or fibroids, and (v) subsequent diagnosis of gestational diabetes or preeclampsia. Twenty women responded; 13 were qualified to participate, and 1 was diagnosed with gestational diabetes after enrolment. Data from the remaining 12 women were retained for parameter estimation.

    Maternal height and weight were recorded at 12, 17, 22, 27 and 32 weeks gestation. For placental volume measurement, the entire placenta was identified by 2D ultrasonography. Placental volume scans were acquired using Voluson E8 ultrasound machines (GE Medical Systems) with a 4–8 MHz transducer. Volumes were calculated from manual contours performed by the same physician. Three scans were obtained at each time point, and averaged.

    The St Barnabas study was approved by the St Barnabas Medical Health Center institutional review board and all participants gave consent.

    2.3.6. Pregnancy, Infection and Nutrition study

    Placental measurements in the PIN study conducted at the University of North Carolina (N = 1260) were obtained from delivered placentas at term [13]. The placentas of participants were weighed both before and after trimming the umbilical cord to within 1 cm of its insertion and extraplacental membranes were trimmed from the chorionic plate perimeter. Only the data from pregnancies without complications were retained for parameter fitting and analysis.

    The institutional review board from the University of North Carolina at Chapel Hill approved this protocol and all participants provided consent to take part in the study.

    2.3.7. University of Connecticut

    The data from this study (N = 461) were obtained from delivered pregnancies between April 1988 and March 1994 at the University of Connecticut, John Dempsey Hospital that underwent pathologic examination [55]. All evaluated cases were singleton delivery, live birth, gestational age between 22 and 32 weeks and no maternal history of chronic hypertension, diabetes mellitus or placenta praevia. The final dataset was composed of 431 preterm placentas (22–32 weeks' gestation), and non-hypertensive abruption placentae (n = 27) defined as antepartum vaginal bleeding followed by preterm delivery. Data from SGA pregnancies were retained to fit model parameters in pregnancies complicated by SGA.

    This study was approved by the John Dempsey Hospital institutional review board and all participants consented to the study.

    2.4. Data treatment

    All datasets were prepared for analysis using the software package R [56]. The R Base Package was used for all data cleaning and manipulation. We first removed observations from the UPENN dataset with measurements of gestational days greater than 365 and missing measurements of gestational day or placenta volume, which accounted for 23.4% of observations. Additionally, observations were removed from the NYU dataset for missing placenta weights at term. This removed 33.7% of the observations from the NYU dataset. The PIN dataset used for model fitting was restricted to pregnancies that did not experience any complications. Gestational days for the WUSL study were calculated in weeks and were converted to days for analysis. We then constructed a dataset consisting of 2352 measurements of gestational days and placenta volumes from uncomplicated pregnancies using the UPENN, NYU, PIN, St Barnabas and WUSL datasets.

    Next, we constructed a dataset of growth-restricted pregnancies using observations of SGA pregnancies from the UCONN dataset, and observations from pregnancies in the below the 10th percentile category (SGA) in the UPENN dataset. This resulted in 136 measurements of gestational days and placenta volumes from growth-restricted pregnancies.

    2.5. Statistical analysis

    The main question that drove this analysis was: is there a difference in the r (proportionality constant) and K (carrying capacity) parameters for uncomplicated and growth-restricted pregnancies resulting in different placental growth curves?

    2.5.1. Estimating parameters, P0, r and K, for the placental volume model using uncomplicated and growth-restricted pregnancy data

    A nonlinear least-squares approach was used for model fitting using the statistical program R [56]. The R package ‘stats’ was used to fit equation (2.1) to the uncomplicated dataset providing estimated r, K and P0 (initial volume) parameters for the uncomplicated pregnancy model. The same process was used to fit equation (2.1) to the growth-restricted dataset providing estimated r, K and P0 parameters for the growth-restricted pregnancy model.

    2.5.2. Calculating the gestational day at the point of the inflection

    The inflection point mathematically reflects the exact switching point in time when early exponential growth transitions to logarithmic growth, and represents the maximal placental growth rate in ml d−1 [31]. The timing of the inflection point was calculated directly from the differential equation by taking the second derivative of equation (2.1), setting it equal to zero and algebraically solving


    Setting PV=K/2 on the left-hand side of the closed-form solution yields


    Solving for gestational day gives us


    2.5.3. Comparison of model placental volume trajectories for normal versus intrauterine growth restriction pregnancies

    The differences in the parameters were assessed by analysing both one-dimensional confidence intervals and the joint confidence regions for the parameters of each model. The R package ‘nlstools’ [57] was used to create confidence intervals and regions for each set of estimated parameters from the nonlinear regression model fit of both the uncomplicated pregnancies and growth-restricted pregnancies.

    2.5.4. Perturbation analysis of the fetal growth model

    Many complications of pregnancy manifest initial signs and symptoms by the end of the second trimester. Unfortunately, it is often too late to turn the course of the pregnancy towards positive outcomes. Thus, it is important to identify as early as possible quantitative indicators of potential problems. A perturbation analysis separates a differential equation into a component that is exactly solvable and components that are small perturbations of this exactly solvable component. The exactly solvable component approximates the differential equation during early time or in our case, early gestation.

    The carrying capacity is large relative to placental volume in early pregnancy and thus ε = 1/K is small. Rewriting equation (2.2) in terms of ε

    and considering a perturbation expansion in terms of ε yields
    where FW0(t) is the solution to the exactly solvable component and FW(t), FW1(t), FWN(t) are the functions that comprise the perturbed component.

    Substituting the summation into equation (2.4) and equating coefficients generates the following equation for the leading order terms:


    The above equation can be explicitly solved for FW0(t)


    2.6. Web-based calculator

    A calculator that simulates the placental volume model, the fetal growth model and provides fetal–placental ratios after input of the parameter values was programmed in the RShiny application for placement on the Web. The calculator requires user input of fetal weight and placental volume at 84 days, r, K and β. The time varying α(t) was set using equation (2.3). The calculator output is the resulting growth trajectories for fetal weight (g), placental volume (ml) and fetal–placental ratio.

    3. Results

    3.1. Subjects

    Summary characteristics of the reference dataset applied for parameter estimation appear in table 2.

    Table 2. Summary characteristics reported as mean ± s.d. for studies used to fit parameter values in the dynamic placental volume model. Observations with missing or unreasonable data were removed prior to calculating summary statistics. Data for maternal weight in the UCONN study were unavailable.

    study maternal age (years) maternal weight (kg) birth weight (g)
    NYU 33.11 ± 4.32 (63) 61.95 ± 11.58 (36) 3311.67 ± 455.84 (63)
    UPENN 30.57 ± 5.72 (519) 72.68 ± 19.43 (521) 3332.85 ± 530.32 (517)
    WUSL 32.20 ± 5.38 (538) 69.71 ± 17.53 (523) 3368.20 ± 576.45 (534)
    St Barnabas 31.54 ± 2.91 (11) 64.99 ± 17.80 (11) 3327.27 ± 586.67 (11)
    UCONN SGA 27.56 ± 5.93 (36) 932.76 ± 323.51 (38)
    PIN 29.08 ± 5.47 (884) 66.81 ± 15.89 (881) 3417.26 ± 438.28 (882)
    UPENN SGA 29.59 ± 5.44 (54) 68.54 ± 19.90 (54) 2467.72 ± 413.75 (54)

    3.1.1. Comparison of model placental volume trajectories for normal versus small for gestational age pregnancies

    The values of r and K in uncomplicated pregnancies were r = 0.0344 and K = 449.2152. The value for P0 for uncomplicated pregnancies was 56.7 ml.

    Ninety-five per cent confidence intervals for these parameters were CIr = (0.0327, 0.0362) and CIK = (444.6464, 453.7839). The value of r and K in the pregnancies diagnosed as SGA were r = 0.0474 and K = 260.7875. Ninety-five per cent confidence intervals for these parameters were CIr = (0.0367, 0.0581) and CIK = (241.2979, 280.2770). The value for P0 for SGA complicated pregnancies was 48.2 ml. The one-dimensional confidence intervals for the parameters r and K do not overlap and indicate a difference in the parameters for the two models.

    The R package ‘nlstools’ was used to create projections of the 95% joint confidence region of r and K according to Beale's criterion shown in figure 2. To create these intervals, parameter values for r and K were randomly selected from a hypercube centred on their least-squares estimate and kept if the corresponding residual sum of squares satisfies Beale's criterion in the following equation:

    Figure 2.

    Figure 2. Joint confidence regions of placental parameters r and K for uncomplicated and growth-restricted pregnancies. The two regions do not overlap and are greatly separated by the K parameter. Variability in the r and K parameters is much greater for the growth-restricted model leading to a larger joint confidence region. (Online version in colour.)

    In the above equation, n is the number of observations and p is the number of parameters in the model. This process is repeated until 2000 acceptable parameter sets have been generated [56,57]. The two joint confidence regions for uncomplicated and growth-restricted pregnancies do not overlap and are greatly separated by the K parameter.

    It is important to note that while the estimated value for r is larger for the SGA pregnancies, it does not imply that the overall placental growth rate is larger since K was estimated substantially smaller. In the SGA pregnancies, the overall placental growth rate, dPV/dt, was dominated by K and as a result is smaller for most of gestation when compared with uncomplicated pregnancies.

    3.1.2. Estimation of gestational age at the inflection point

    The R package ‘stats’ [56] was used to estimate the inflection point for each model based on the parameters obtained from the nonlinear least-squares fitted model. The inflection point for uncomplicated pregnancies occurred at 140.17 days of gestation. Pregnancies identified as SGA inflected earlier at approximately 115.32 days of gestation.

    3.1.3. Perturbation analysis of the fetal growth model

    The solution, equation (2.6), represents an estimate of the early fetal growth trajectory and is explanatory of the fetal–placental balance. For example, if α(t) < αN(t), where αN(t) represents the time-varying coefficient in an uncomplicated pregnancy, then by the scaling law, the fetal–placental ratio is smaller than expected. The trajectory in the solution to equation (2.6) indicates that early fetal growth in this case will be greater than expected. These phenomena are more commonly exhibited at term by pregnancies diagnosed with gestational diabetes mellitus (GDM) [58]. On the other hand, if α(t) > αN(t), then the placenta is inefficient. Placental size is larger than expected for normal growth. In this case, fetal growth is smaller than expected. This could encompass both preeclampsia and SGA since the birth weight of pregnancies diagnosed with preeclampsia are often classified as SGA [52]. While not surprising that the model describes quantifications consistently observed at term, it is important that these calculations can be done in utero. The value for α(t) at any time t can be estimated in vivo through a placental volume and fetal weight measure obtained by ultrasound technology.

    3.1.4. Web-based calculator

    The calculator allows users to input model parameters and observe the resulting simulations of fetal growth and placental volume over the period of pregnancy. The calculator was made available at the URL:

    4. Discussion

    Current fetal growth curves are derived by fitting curves to cross-sectional data and do not provide underlying explanations for why and how fetal growth trajectories slow, plateau or accelerate. We advance the field by developing the first dynamical model that couples changes in in vivo placental measures with fetal weight using a validated placental–fetal scaling law. The final model generates an entire fetal growth curve from measurable early gestational in vivo placental volume measures. The model can test hypotheses involving fetal–placental growth without costly experiments, including the testing of risk detection paradigms for pregnancy complications. Our analysis demonstrates that deviations from predicted placental growth curves were related to pregnancy complications of SGA.

    4.1. Application of the fetal–placental growth model to detect pregnancy complications

    The utility of the model is its capacity to detect pregnancy complications and risk. There are several methods outlined here in the study that can be used to identify risk. Each of these methods require calculating the values for r and K using at least two placental volume scans between 11 and 17 weeks. The values can be solved for by inputting the gestational day of the measurement for t and the initial placental volume for P0 and setting the closed-form solution PV(t) equal to the measured placental volume. The two measurements will result in two equations which can be simultaneously solved to find r and K. Once r, K and P0 are known, the personalized placental volume trajectory can be generated.

    The first method is to generate the fetal–placental growth curves using baseline values as described above and calculate the point of inflection for the placental volume curve. Our analysis demonstrated that in uncomplicated pregnancies, this point of inflection should occur at approximately 140.17 days of gestation. Pregnancies complicated by SGA had inflection points earlier in gestation at approximately 115.32 days of gestation, indicating an accelerated timing for a key gestational time point. To interpret the physiology behind the earlier inflection point, it is valuable to consider this timing as the maximal growth rate of the placenta. Placental volume/weight growth rates are related to fetal growth rates with the maximal growth rate of the placenta followed by the maximal growth rate of the fetus [59]. If the maximal growth rate of the placenta occurs early, the maximum growth rate of the fetus will also be early, and this is concordant with inadequate fetal growth rates or SGA.

    The second method is to compare the resulting predicted placental growth trajectory after input of r, K and P0 to the trajectory generated in the uncomplicated case in figure 3. From here, we can evaluate if the two curves deviate. Our analysis demonstrated that the curves generated for uncomplicated pregnancies and pregnancies complicated by SGA had statistically different parameter values.

    Figure 3.

    Figure 3. Scatter plot of placental volume in millilitres versus gestational day plotted for uncomplicated pregnancies (NYU, UPENN, WUSTL, St Barnabas and PIN), represented by triangles. The values of r and K for the placental growth curve model were fitted using the uncomplicated pregnancy data and the model using these values is represented by the black line. The point of inflection is reflected as a solid red circle and occurs at 140.15 days of gestation. Placental volumes of pregnancies complicated by SGA (UCONN SGA and UPENN SGA) represented by squares are also plotted. The values of r and K for the placental growth curve model were fitted using the SGA pregnancy data and the model using these values is represented by the orange curve. The point of inflection is denoted by a solid black circle and occurs at 115.32 days of gestation. (Online version in colour.)

    A third method of detecting appropriate fetal growth patterns is to generate a full fetal growth curve from first-trimester initial values in order to compare the resulting trajectory to known percentile curves. It is not known yet whether changing diet or activity to alter the value of r and K will turn a pregnancy to an optimal growth trajectory; however, this can be tested in pregnancy lifestyle interventions.

    Finally, assembling the curve for α(t) to observe deviations from αN(t) opens an opportunity to determine the existence of placental inefficiencies early in gestation.

    4.2. Comparison to the current practice applying fetal weight percentile curves

    The fetal growth curves presented here differ from existing fetal weight percentile curves in three significant ways. First, a fetal growth curve from equation (2.2) represents a customized longitudinal growth trajectory of an individual fetus that can be generated by solely using placental volume data from the first trimester. This differs from a curve that arises from percentiles. For example, a 50th percentile fetal weight curve is generated by measuring a cohort of fetal weights at the 20th week of gestation and breaking these measured weights into percentiles. The first point would be the 50th percentile fetal weight from this cohort at the 20th week. Then a different cohort of fetal weights at the 21st week is estimated, and a second point is generated at the 50th percentile weight. These two points are then connected. This process is advanced until we have assembled a curve. The points on the curve do not represent a single trajectory of an individual fetus, but rather connected cross-sectional data. We can think of this fundamental difference as a longitudinal trajectory versus a cross-sectional curve. Second, volumes by ultrasound are estimated using voxels or by breaking a structure into cubes and summing the volume of the cubes. The placenta is a solid, often disc-like structure [60] which renders it more readily available for volume calculations as early as the 10th week of gestation. Fetal weight by ultrasound is first estimated at approximately week 20, largely because the fetus is not as easy to break into voxels until it is large enough [61]. Therefore, tracking fetal growth by ultrasound requires waiting until the 20th week. On the other hand, the inflection point and the deviation between the generated trajectory arising from equation (2.1) can be evaluated earlier in the first trimester. Third, the placental volume curve generated by equation (2.1) has readily available quantities associated with the curve such as the value of r, K and the timing of the inflection point that originate from calculus. These numbers can be evaluated for whether they are in a healthy range similar to a blood pressure measurement. The percentile curves do not have a functional form like the solution to equation (2.1) and as a result do not have similar single valued quantities that can be immediately evaluated against benchmark values. We note that the values for r and K derived here were from six datasets of placental volume while the percentile curves for fetal growth are derived from a completely different population and different quantity, namely fetal weights.

    Despite these differences, the differential equation models presented here should not supplant the current practice of using percentile curves to track fetal growth. There is a large and long-standing body of literature linked to percentile curves that has successfully identified complications in fetal growth [62,63]. Rather, the model information presented here has higher utility if used in tandem with the existing standard of monitoring fetal growth against percentile curves.

    4.3. Limitations

    While our initial results are promising, there are several limitations that need to be considered. These limitations involve current unknowns which potentially can be determined if data are gathered in future ultrasound studies.

    4.3.1. Functional form of α(t)

    We relied on cross-sectional data [50] to assume that α(t) is a decreasing function that converges to 1 at term. We also used the same data to fit a curve to α(t) for numerical simulation. We do not know whether the functional form of α(t) in an individual pregnancy longitudinally follows the cross-sectional curve which notably was derived from terminated pregnancies [50]. These assumptions can be strengthened and tested using current cross-sectional data similar to [50] and frequent longitudinal ultrasound data of placental volume and fetal weight from uncomplicated pregnancies.

    4.3.2. Variance and error in initial value estimates

    For our numerical analysis, we applied PV(84)=56.70ml determined from parameter estimation which was close to 54.80 ml reported in the literature [64]. We also used FW(84)=20g from the same sources [64]. The value for initial fetal weight was obtained from cross-sectional data from pregnancies that terminated. If the pregnancy was terminated, by default, it was complicated. Moreover, the variance in the initial fetal weight and how it differs in complicated versus uncomplicated pregnancies are not known. Technically, these values can be obtained using ultrasound data; however, even these estimates of placental volume and fetal weight hold variance and measurement error which contribute to model predictions. Full knowledge of the degree of variance needs to be understood in the context of model accuracy to identify pregnancy risks.

    4.3.3. Additional pregnancy complications

    We focused our comparisons to pregnancies complicated by SGA because our dataset included enough pregnancies to justify such a comparison. Whether the model can be used to identify other complications like GDM is not known, yet very important. If detected early, the course of a pre-GDM environment can be turned around through improved diet and activity. Analysis of the model and parameters for complications like GDM are important future considerations.

    4.3.4. The need for data that include placental imaging and pregnancy complications

    While the dataset of placental volumes from both uncomplicated and SGA pregnancies assembled here represents, to our knowledge, the largest existing database for analysis, there remains a lack of measurements in the field. This is because (i) the placenta is not routinely imaged and (ii) complicated pregnancies are usually eliminated from the study protocol [65]. There is growing rigorous evidence that placental growth and morphology are predictive of pregnancy complications but to leverage this evidence, a placental image database needs to amass measurements from both healthy and complicated pregnancies. Additionally, more diverse data that span different maternal ages, races and parity will be important for understanding how parameter values like r and K are a function of these covariates. It is known that race and ethnicity results in different fetal growth percentile curves [9,10] so adding to our database using a wider range of subjects is imperative. Moreover, an independent dataset would allow for validation of the placental volume model simulation and the estimated values for r and K derived here.

    Imaging the placenta with ultrasound is initially challenging since it requires the imager to partially segment the image manually and requires repeated measures to obtain a quality placental image. However, with more experience, our team became adept at acquiring placental volumetric data within 5 min. Currently, engineering teams are developing automated algorithms that will eliminate this placental imaging burden for investigators [66]. A more serious limitation is that there are presently no ultrasound standards for normal in vivo placental size. Estimated placental biometry and volume during pregnancy are correlated with their measurements at postnatal assessment, but they are not equivalent [67] with in vivo measurements of surface area tending to underestimate ex vivo [67]. However, placental thickness and placental volume measurements overestimate ex vivo measurements. These differences reflect the collapse of intervillous space due to draining of maternal intervillous blood after birth. Azpurua et al. [68] described that placental weight could be accurately predicted by 2D ultrasound with volumetric calculation. Second, intra- and inter-observer variability are much more important to in vivo sonographic measurements than ex vivo, real-life measurements [67]. More recently, a new technique was established for estimating placental volume from 3D ultrasound scans through a semi-automated technique [66]. While these investigators showed good inter-rater reliability for in vivo placental volume, they made no comparison to post-delivery measures. It is not possible to reconstruct the dimensions of the intervillous space in the post-delivery placenta, nor can we control to guarantee equal emptying of the intervillous space post birth. We anticipate that these differences in pre- to post-measurements cannot be easily resolved; however, our work suggests that the variability is not so great as to preclude the development of a useful predictive model.

    Despite these limitations, the developed fetal–placental growth equations provide new insight into the coupled relationship of fetal–placental growth and its capacity to detect risk for pregnancy complications.


    All individual studies were approved by the home institution's respective IRB and this study was approved by the Western Institutional Review Board's (WIRB) WIRB Work Order no. 1-883897-1.

    Data accessibility

    The data applied in this project is available as part of the electronic supplementary material.

    Authors' contributions

    D.M.T. and C.M.S. conceived the study. D.M.T. developed the dynamic models and performed the perturbation analysis. D.B. cleaned and prepared all datasets and D.B. and K.C. performed the statistical analysis. N.S., K.O., R.C.M., A.O., R.S. and C.M.S. explained the measurements in the data and reviewed model findings. D.B. prepared a full draft of the manuscript. D.B. and C.B. developed the Web-based app. All authors reviewed and approved manuscript drafts.

    Competing interests

    D.M.T. and C.M.S. report the patent ‘System and Method for Predicting Fetal and Maternal Health Risks’, which was filed within the USA and internationally. The US application was filed on 30 March 2015, and assigned serial no. 61/973,565, and the international application was filed on 30 March 2015, and assigned no. PCT/US2015/023257.


    D.M.T. and C.M.S. were funded by NIH (grant no. R43 HD066952).


    The authors would like to thank Dr James F. Clapp III. This project originated from Dr Clapp's groundbreaking work on exercise, the placenta and pregnancy. Without his mentorship, our work would not exist.


    Electronic supplementary material is available online at

    Published by the Royal Society. All rights reserved.