# Hemodynamic consequences of incomplete uterine spiral artery transformation in human pregnancy, with implications for placental dysfunction and preeclampsia

## Abstract

Normal human pregnancy requires a dramatic increase in uteroplacental blood flow, which is achieved by a transformation in the geometry of uterine spiral arteries, a key element in this blood supply system. The transformation is mediated by trophoblast invasion directed at converting a portion of the spiral artery into an open funnel, thereby greatly reducing resistance to flow. The converted portion lies within the depth of the decidua and part of the myometrium. Insufficient depth of trophoblast invasion in early pregnancy predisposes to inadequate perfusion of the developing placenta and fetus and may lead to preeclampsia, fetal growth restriction, and preterm delivery, sometimes referred to as the “Great Obstetrical Syndromes.” We examine the hemodynamic consequences of spiral artery transformation in human pregnancy and the relationship between the degree of transformation and the corresponding change in flow rate and resistance to flow. We identify two key variables in determining the hemodynamic change: the longitudinal converted fraction of the spiral artery and the relative downstream diameter of the open funnel. Our results indicate that there is a critical threshold in the value of the converted fraction required to achieve the marked increase in uteroplacental blood flow in normal pregnancy. This finding validates common clinical observations that the depth of trophoblast invasion reflects the “adequacy” of the increase in uteroplacental blood supply required in normal human pregnancy. Our results provide a quantitative measure of that adequacy and may serve as a future diagnostic marker for high-risk pregnancy.

**NEW & NOTEWORTHY** Human pregnancy requires dramatic increase in uteroplacental blood supply achieved by geometric transformation of uterine spiral arteries and facilitated by trophoblast invasion of these arteries to greatly reduce resistance to flow. Incomplete transformation has been associated with failed pregnancies, preeclampsia, and other pathologies, but a quantitative measure of “incompleteness” has been unavailable so far. We use a mathematical model to obtain a numerical threshold for this measure which may serve as a future diagnostic marker.

# INTRODUCTION

In early pregnancy, uteroplacental blood flow undergoes dramatic transformation. Between implantation of the fertilized conceptus and eight weeks gestation, extravillous trophoblasts migrate from the tips of chorionic villi to penetrate the uterine decidua and invade the spiral arteries (1–3), initially causing endovascular trophoblast plug formation and undetectable flow through these vessels (4, 5). Flow through the spiral arteries into the intervillous space is first detected between 8 and 10 wk gestation and becomes unimpeded by 10–12 wk gestation following marked transformation of these vessels (6).

The transformation of spiral arteries is depicted in the context of surrounding anatomical structures in Fig. 1. This transformation involves a major geometric remodeling of these vessels whereby an increasing fraction of the spiral artery is converted into an inverted funnel opening widely at its distal end into the intervillous space (4, 7, 8). Prior to that transformation, the artery is a tightly coiled vessel with a “reserve” of high resistance to flow, to be purged later in the course of pregnancy as the demand for blood supply rises (9). Furthermore, although spiral artery flow before this remodeling is under neurovascular control, this control is disabled by the trophoblast invasion and depletion of vascular smooth muscle, thus the spiral artery is no longer an effector organ of the sympathetic nervous system (10). Following remodeling, therefore, blood flow is determined primarily by the passive conditions imposed by the geometry of the spiral-plus-funnel combination. The fraction of spiral artery converted into an open funnel, and the relative diameter of the distal end of the funnel, are therefore critical determinants of the hemodynamic changes produced by spiral artery transformation.

Failure of trophoblasts to penetrate sufficiently deep into the decidua and myometrium in early gestation, and hence failure to achieve the full geometric transformation of spiral arteries, has been associated with the development of preeclampsia, fetal growth restriction, and preterm birth (11, 12). Furthermore, impaired perfusion of the developing placenta leads to hemodynamic effects in the maternal systemic circulation (13), including hypertension, exaggerated maternal hemodynamic response to catecholamines (14), and maternal renal injury (15). A recent paper by Brosens et al. (16) has indeed described the pathogenesis of preeclampsia as a severe pregnancy disorder characterized by incomplete transformation/remodeling of the uteroplacental spiral arteries.

In 1990, Redman (17) proposed a two-stage model of preeclampsia with impaired trophoblast-mediated remodeling of uterine spiral arteries in a first trimester preclinical phase, followed by a second trimester clinical phase, presenting as placental hypoxia leading to maternal preeclampsia and fetal growth restriction. This model is particularly applicable to early-onset preeclampsia (18). In the present study, the focus is on spiral artery remodeling, which is critical to early-onset preeclampsia.

Spiral artery remodeling has typically been considered as a binary of two states: a non-pregnant state of low-flow spiral arteries with muscular walls under sympathetic neural control, and a pregnant state of high-flow spiral artery plus inverted funnel combination lined by trophoblasts and not under sympathetic neural control. In this study, we examine the hemodynamic consequences of a full range of spiral artery plus inverted funnel combinations between these two binary extremes, with the aim of identifying the limits of spiral artery remodeling required to provide the marked increase in uteroplacental perfusion required in normal pregnancy.

Two recent studies of uteroplacental blood flow (19, 20) have provided important information on the nature of blood flow exiting the spiral arteries and on the impact of the emerging “jet flow” on the development of the intervillous space.

Scant ultrasound-based data have also focused on jet properties of flow into the intervillous space (5, 21). The focus of the present study is on the upstream precursor to that jet flow, that is, on flow rate and resistance to flow within the spiral artery at different degrees of its geometric transformation. Furthermore, because of the scarcity of physiological data, the present study is designed to provide insight into a range of possible scenarios rather than make assumptions based on isolated and possibly spurious data points.

An accepted premise in the literature (11–18) is that preeclampsia and fetal growth restriction are associated with impaired uteroplacental perfusion resulting from inadequate depth of trophoblast invasion leading to inadequate trophoblast-mediated spiral artery remodeling. What is unknown is whether there is a critical threshold depth for trophoblast invasion to yield adequate spiral artery transformation to meet the needs of normal pregnancy, and what that threshold is in quantitative terms. Present technology cannot provide this data because of access and imaging limitations. In this study, we use a mathematical model to quantify uteroplacental perfusion in terms of blood flow, and quantify the depth of trophoblast invasion in terms of the conversion fraction of spiral arteries. Using this model, we provide a relationship between these two variables and demonstrate that there is a critical quantifiable threshold depth of trophoblast invasion needed to provide the increased uteroplacental perfusion for normal pregnancy. Furthermore, we show that this threshold changes remarkably little over a wide range of scenarios examined.

# METHODS

## Key Geometric Variables of Spiral Artery Transformation

We consider a spiral artery emanating from a radial artery and terminating at the intervillous space. In the analysis to follow, the unwound length of the spiral artery is denoted by *L* and the straight line depth of tissue it traverses is denoted by *H*. For the purpose of comparison with flow in a hypothetical straight vessel that has the same length as *H*, we have scaled *H* to 1.0 arbitrary units. The use of arbitrary units here is again necessitated by the lack of physiological data and the desire to avoid calculations based on isolated physiological data points. The use of arbitrary units also allows us to widen the scope of the results to enable potential comparison with a range of different physiological data as they become available.

Prior to transformation by invading trophoblasts, the spiral artery is considered to be complete in its full coiling form, starting with a coiling radius of *R = R*_{0} and terminating with a coiling radius *R = *0. In the course of transformation, a distal portion of the spiral artery, denoted by *h*, is converted into an inverted open funnel as illustrated schematically in Fig. 2. The converted fraction of the spiral artery is to be denoted by *cf* and expressed as a fraction of 1.0, and the unconverted fraction is then *(H − h)*/*H*, or *1 − cf*, as shown in Fig. 2.

In a study of the consequences of conversion of spiral arteries for uteroplacental blood flow during human pregnancy by Burton et al. (22), the unconverted portion of spiral artery was considered to be a 1.0-cm long straight artery segment. In the present study, we retain the spirality of the unconverted portion of the spiral artery in order to more accurately represent the effects of “depth” of trophoblast invasion. In an earlier study (9), we have shown that the calculated resistance to flow in a coiled vessel is determined predominantly by the unwound length of the vessel (*L*) and significantly less so by the curvature effects of the coiling of the vessel. The difference between the length *L* and depth *H* is clearly critical in determining the effects of the trophoblast invasion as spiral artery transformation progresses.

## Underlying Assumptions

Flow in a spiral artery is subject to two extraneous effects compared with flow in a straight artery. First, the added vessel length per depth of tissue being penetrated and, second, the added complexity of the flow pattern within the vessel because of the coiling curvature of the vessel (9). In both cases, the net result is added resistance to flow compared with that in a straight vessel under the same pressure drop. In a previous study where these two effects were compared (9), we showed that the curvature effects were negligible compared with the length effects, except at the distal end of the spiral where the coiling radius may diminish to the point of producing very high curvature effects. Since in the spiral-funnel combination to be considered in the present study the distal portion of the spiral artery is replaced by an inverted funnel, only the length effects will be considered in the analysis to follow.

The effect of conversion of a distal portion of the spiral artery into an inverted funnel is twofold. First, the coiling length of the converted spiral artery segment is replaced by a shorter straight vessel segment and, second, the caliber of that vessel segment increases from the vessel diameter of the spiral artery (*d*_{1}) at the proximal end to *d*_{2} (*>d*_{1}) at the distal end (Fig. 2). In both cases, the result is a considerable reduction in resistance to flow, hence an increase in flow rate into the intervillous space.

There are two additional factors that may affect the resistance to flow in the transformed spiral artery. Flow in a diverging conduit is subject to “boundary layer separation” (23) whereby the flow separates from the walls of the conduit and disturbed recirculating flow regions develop near the wall as illustrated schematically in Fig. 3. In studies of flow in diverging ducts (diffusers) in engineering systems (24, 25), it is found that the onset of separation depends on the opening angle of the diverging conduit and on the Reynolds number (Re) (26)

*ρ*and

*µ*are density and viscosity of the fluid, respectively, $\overline{u}$ is the mean velocity of the flow entering the diverging conduit, and ${d}_{1}$ is the conduit diameter at that entrance. Higher opening angles and low Re values promote flow separation, but the exact values of these two variables or combinations of values at which separation occurs, as reported by different authors, vary considerably because the onset of separation depends also on the experimental conditions or theoretical model on which these values are based (24, 25). The engineering literature on this subject therefore does not offer definitive results. As an approximate guideline, some studies have suggested that separation is not likely to occur at angles less than 7° and most likely to occur at angles greater than 15°. However, the validity/usefulness of these limits has been questioned because the onset of flow separation depends also on Re number and on the experimental setup (24).

These uncertainties are compounded even further in the biological situation at hand because neither the opening angle nor Re are fixed, nor indeed known with any accuracy. They vary in the course of pregnancy because the funnel diameter ratio *d*_{2}/*d*_{1} and conversion fraction *cf* change in the course of spiral artery remodeling.

In the face of these limitations, it is clear that determining the absolute state of hemodynamics at different degrees of spiral artery transformation is unattainable. In the analysis to follow, therefore, we consider a wide range of values of *cf* and *d*_{2}*/d*_{1} and compare the relative states of hemodynamics at different values of these variables and, in this comparison, we focus on only the primary effects of the diverging funnel, omitting the secondary effects of flow separation.

Thus, the results we present will be approximate, the degree of approximation involved is not easily determined because flow separation has two opposite effects, one is that of removing friction at the funnel wall and the other is that of adding energy expenditure within the disturbed separated region near the wall. On balance, the net result may in fact be small. Also, the hemodynamic state of most interest is at the end state of spiral artery transformation because this is the state at which flow rate into the intervillous space reaches its highest level. In this state, both *cf* and *d*_{2}/*d*_{1} have reached their highest values and, consequently, the opening angle of the funnel is at its smallest value (Fig. 3) whereas Re is at its highest value because of the high-flow rate. The combination of high Re and small opening funnel angle diminish even further the onset or extent of flow separation.

Notwithstanding these considerations, we acknowledge that the phenomenon of flow separation within the transformed spiral artery may have a different role to play, namely that of shaping the placental lobules and forming the central cavities, as suggested in an earlier paper by Reynolds (27). The subject is clearly beyond the scope of the present study.

We assume, finally, blood to be an incompressible viscous Newtonian fluid for the purpose of this study. This assumption becomes questionable only when the diameter of a blood vessel becomes comparable with the size of red blood cells and other blood cells (26, 28). In the current situation, the typical diameter of a spiral artery is 0.45 mm (=450 µm) whereas the typical diameter of a red blood cell is 10 µm. The assumption of a Newtonian fluid is clearly valid. It is widely accepted that blood, despite its corpuscular composition, is incompressible. More accurately, it can be treated as incompressible in the equations governing the flow within the typical range of velocities in blood flow (26, 28).

## Flow Rate in the Transformed Spiral Artery

As a model of transformed spiral artery at different stages of transformation, we consider a spiral artery of penetration depth *H* of which a distal fraction *cf* × *H* has been converted into an inverted funnel. If the resistance to flow in the untransformed portion of the spiral artery is denoted by Ω_{s} and the resistance to flow in the funnel is denoted by Ω_{f}, the flow rate *Q*_{sf} in the spiral-funnel combination, since the two elements (the unconverted portion of the spiral artery and the funnel) are in series, is given by is given by the following equation (26):

*2*)

*P*

_{0}and

*P*

_{2}are the pressures at the entry to the spiral artery and at the exit from the funnel, respectively, as indicated in Fig. 2.

*P*

_{1}is an intermediate pressure whose value depends primarily on the extent of trophoblast invasion

*cf*. The flow rate through the spiral-funnel combination is driven by the pressure difference:

*P*

_{0}−

*P*

_{2}.

For the purpose of comparison and scaling, we consider flow in a straight artery traversing the same depth of penetration *H* and under the same pressure difference *P*_{0} − *P*_{2}. If the flow rate is denoted by *Q*_{0} and the resistance to flow is denoted by Ω_{0}, then

*3*)

The resistance Ω_{0} in this case is known (26), namely

*4*)

## Spiral Resistance to Flow

We consider a spiral artery in three-dimensional coordinates *x*,*y,z*, its path being described by the spiral curve as follows:

*5*)

*T*is the sweeping angle from the first point on the spiral (

*T*= 0), such that when

*T*= 2

*π*the spiral will have completed a full loop (revolution) and advanced one pitch in the negative

*z*direction to

*z*=

*H*−

*h*. The normal projection of this curve onto the

*xy*plane is a circle of diminishing radius, with an initial radius

*R*

_{0}, decreasing by an amount

*r*in each revolution (9).

Following the discussion of curvature effects and length effects in a spiral artery in Underlying Assumptions, the resistance to flow of the untransformed segment of the spiral artery will be based on length effects only. If the length of the untransformed segment of the spiral artery is *L*_{s}, the resistance to flow in that segment, using *Eq. 4*, is given as follows:

*6*)

If the number of loops completed by the full spiral artery, before transformation, is *n*, the sweeping angle covered by the full spiral is *T _{n}* =

*n*× 2

*π*, and the sweeping angle covered by only the untransformed fraction of the spiral is given by the following:

*7*)

*8*)

The differentials d*x*, d*y*, d*z* are determined from *Eq. 5*, and the integral is evaluated numerically.

## Funnel Resistance to Flow

In the analysis to follow, it is assumed that the flow within the inverted funnel remains attached to the funnel wall as discussed in Underlying Assumptions. Although this may not be the case in the physiological setting, depending on the values of *cf* and *d*_{2}/*d*_{1}, as well as on the value of Re (24, 25), the resistance to flow being calculated on this basis will serve as an upper bound estimate of the actual resistance in the physiological setting, again, as discussed in Underlying Assumptions.

We define the inverted funnel as a truncated cone with diameters *d*_{1}, *d*_{2}, with *d*_{2} *> d*_{1} (Fig. 2) and length *L*_{f} where

*9*)

The flow rate *Q* through the funnel is governed by the following equation (26):

*10*)

where *µ* is blood viscosity as before and d(*z*) is the diameter of the funnel at a cross-section where the pressure gradient is $\frac{\text{d}p}{\text{d}z}$, noting that here *Q* is negative because the flow is occurring in the negative *z*-direction whereas $\frac{\text{d}p}{\text{d}z}$ is positive since pressure is increasing in the positive *z*-direction, and thus both sides of *Eq. 10* are negative.

The resistance to flow through the funnel is defined as follows:

*11*)

where *P*_{1} is the pressure at the spiral-funnel junction (Fig. 2), noting that flow *Q* is in the negative *z*-direction (hence the minus sign). In order to find this pressure, *Eq. 10* is integrated over the length of the funnel form *z *=* *0 where the pressure is *P*_{2} to $z={L}_{\text{f}}$ where the pressure is *P*_{1}

*12*)

Writing

*13*)

*14*)

*Eq. 12*becomes

*15*)

The integral on the right is evaluated in a straightforward manner and, omitting the details, the result is

*16*)

*17*)

Substituting these results in *Eq. 11*, we find the resistance to flow through the funnel

*18*)

As a test, we note that if *d*_{2} = *d*_{1}, hence $\u03f5=0$, the funnel becomes a straight artery of length *L*_{f}, and the resistance to flow becomes the same as that in *Eq. 4* for a straight artery, but here with length *L*_{f} instead of *H*.

# RESULTS

In what follows, we present values of the flow rate through the transformed spiral artery (spiral + funnel combination) with a range of values for the depth of spiral artery remodeling *cf* and with a range of values for the relative diameter of the distal end of the funnel *d*_{2}/*d*_{1}. The flow in the spiral-funnel combination, like the flow in a tube, depends only on the pressure difference driving flow and on shear stress and any other impediments in the way. It does not depend on the position of the tube in space. These flow rate values, therefore, are independent of the anatomical position of the spiral-plus-funnel combination relative to the decidua, and no assumption is made about the fraction of spiral artery remaining spiral over gestation.

Change of flow rate with change in the spiral converted fraction *cf*, for a specific value of *d*_{2}/*d*_{1}, is shown in Fig. 4. Of particular note is the value of *cf* at which the flow rate increases above the straight vessel flow rate value. We shall refer to this value of *cf* as a “threshold.” The effect of changing the value of *d*_{2}/*d*_{1} on the position of the threshold is shown in Fig. 5. It is noted that the threshold value of *cf* remains high (*>*0.8) even with very high values of *d*_{2}/*d*_{1}. This is because as the opening angle of the inverted funnel becomes large, the funnel resistance becomes negligible compared with that of the unconverted portion of the spiral artery, thus higher values of *d*_{2}/*d*_{1} (*>*5.0 or so) offer little advantage as seen in Fig. 5. Corresponding results in terms of resistance to flow are shown in Fig. 6.

An indication of the relative importance of the two key variables in spiral artery transformation, namely *cf* and *d*_{2}/*d*_{1}, is shown in Fig. 7. The figure offers yet a different view of the dependence of the flow rate threshold on these two variables. As stated earlier in **Underlying Assumptions**, in the course of spiral artery transformation, changes in *cf* and in *d*_{2}/*d*_{1} may occur in tandem and in a manner for which we have no physiological data on which to base specific flow rate calculations. Therefore, in the absence of such data, in Fig. 8 we present a range of possible combinations of *d*_{2}/*d*_{1} and *cf*, with an indication of whether the corresponding value of the flow rate in each case is higher or lower than the flow rate in a straight vessel.

# DISCUSSION

The main aim of this study was to determine the geometric and hemodynamic consequences of spiral artery transformation in the course of human pregnancy and, once determined, to use these as markers to estimate the change in blood supply to the intervillous space which these vessels provide at different degrees of their transformation and then assess how blood supply to the intervillous space is impacted in the event of an incomplete spiral artery transformation.

As a model of spiral artery transformation, we considered a distal segment of the spiral artery to be remodeled into an inverted funnel, opening into the intervillous space. This remodeling leads to an increase in flow rate because the high resistance of a spiral artery segment is replaced by the low resistance of an open funnel, with the pressure drop being unchanged. We assessed two key variables for their effects on flow rate into the intervillous space, the fraction of spiral artery converted into an open funnel *cf* and the ratio of distal to proximal diameters of the funnel *d*_{2}/*d*_{1}. Our results indicate that *cf* is the more important of the two variables.

In comparing the flow rates at different degrees of spiral artery transformation, we found a useful reference to be the flow rate in a straight vessel which has the same diameter as the untransformed spiral artery, penetrates the same depth of tissue, and is under the same pressure drop. Using this reference, we identified a threshold value of *cf* (≈0.85) where the flow rate in the transformed spiral-funnel artery rises above the corresponding straight artery flow rate and continues to rise precipitously with increasing values of *cf* above this threshold (Fig. 4).

It is important to emphasize that we use the flow rate in a straight line vessel as only a “measure,” a “yard stick,” for comparison with flow in spiral-plus-funnel combinations. We do not assume that the straight vessel flow rate is required for successful pregnancy. We make only the conservative assumption that at least that amount of flow rate must be exceeded in the course of successful pregnancy. The value of *cf* at which the flow rate in a spiral-funnel combination crosses the straight vessel flow rate is used only because it is the point at which the flow rate begins to rise precipitously. We make no assumption about how much of that rise is needed for a successful pregnancy.

The flow rate through the spiral-funnel combination shown in the figures is scaled by the flow rate in a straight vessel of the same depth of penetration as the spiral-funnel combination and is driven by the same pressure difference as the spiral-funnel combination, namely *P*_{0} − *P*_{2}. Thus, if maternal pressure *P*_{0} changes, the scaled flow rate will again represent how the flow in the spiral-funnel combination is compared with the flow rate in a straight vessel of the same depth of penetration and driven by the same pressure difference. Our model thus allows for maternal hypertension, which is an important component of preeclampsia.

The analysis we present in this study is based on steady flow because the analysis of steady flow in a given vascular network is a necessary prerequisite for the subsequent analysis of pulsatile flow in that bed (26, 28). The effects of pulsatility are secondary and can only be considered once the primary effects of steady flow have been established. In the present study, our focus is on the primary effects of changing geometry in the course of pregnancy on the steady part of the flow. Furthermore, it is not clear whether there is any appreciable pulsatile component to flow in the inverted funnel. The pulsatility of blood flow in the cardiovascular system diminishes as the flow progresses down the hierarchy of the vascular tree. It is expected, and we have assumed, that pulsatility of the flow in the spiral-funnel combination is negligibly small. Again, no physiological data are currently available to address this.

These results provide a theoretical lower bound for the fraction of spiral artery that must be converted in order to provide the required marked increase in blood flow into the intervillous space in the course of normal pregnancy. Trophoblast invasion which does not meet this lower bound of spiral artery conversion fraction may predispose to inadequate blood supply to the intervillous space and, in turn, to pathological consequences associated with suboptimal perfusion, including preeclampsia, fetal growth restriction, and preterm birth (11, 12, 29).

The extent of trophoblast invasion in the first trimester is a critical determinant of normal pregnancy (29). Trophoblast invasion of the decidua and remodeling of spiral arteries is predominantly completed in the early second trimester, and fetal growth restriction typically presents later in pregnancy, as placental and fetal demands increase. In normal pregnancy, trophoblast invasion affects the entire depth of the decidua and the inner one third of the myometrium (22, 30).

Low-oxygen tension in the placental implantation site in the first weeks of gestation plays an important role in the initiation and depth of trophoblast invasion (31). Before the tenth week of gestation, normal oxygen tension in the intervillous space is less than 20 mmHg (6, 32). During the first weeks of pregnancy, plugs of endovascular trophoblasts completely occlude spiral artery blood flow—these plugs dissipate between 8 and 10 wk gestation to allow uninterrupted flow to the intervillous space by 10–12 wk gestation (5). Indeed, the presence of detectable blood flow into the intervillous space by ultrasound before the 8th week of pregnancy is a strong predictor for subsequent pregnancy complications, including spontaneous abortion and preeclampsia (8).

There are limited data in the literature regarding the measured dimensions (length, radius, coiling, etc.) of spiral arteries and their transformation in normal and abnormal pregnancy. Specimens from cesarean hysterectomies performed for cervical cancer, placenta accreta, uterine atony, among other conditions may provide data on depth of trophoblast invasion, but these specimens are by definition not obtained from normal pregnancies. Emerging noninvasive technologies such as ultrasound-directed photoacoustic imaging may in the future provide such data in normal pregnancy. The threshold value of *cf* found in this study may provide a useful context for such data and for the extent of trophoblast invasion as assessed by 3D histological analysis of spiral artery sections of basal plate specimens from in situ placentas attached to decidua. If the depth of trophoblast invasion can be reliably measured using these or other emerging technologies, then our threshold of *cf* may provide a diagnostic aid to predict which patients will go on to develop preeclampsia and fetal growth restriction.

In conclusion, the lack of physiological data on this subject has so far stood in the way of progress in our understanding of the physiology of pregnancy and preeclampsia, and this lack of data is not going to change in the near future. The spiral arteries are not accessible to direct invasive flow measurements, whereas current and emerging imaging technologies are unable to provide these data noninvasively. The present study and the results we present should therefore be viewed primarily as a novel way of making some progress in the face of these realities. To compensate for the lack of physiological data, we used a wide range of values for the unknown variables involved and were able to demonstrate that there is a critical, quantifiable, threshold depth of trophoblast invasion needed to provide the increased uteroplacental perfusion for normal pregnancy and that this critical threshold changes remarkably little over the wide range of values studied.

# GRANTS

This work was supported in part by Natural Sciences and Engineering Research Council of Canada Grant R1721 (to M. Zamir), in part by Israel Science Foundation Grant 1817/13 (to Y. Ginosar), and in part by The Foundation for Barnes-Jewish Hospital, Washington University School of Medicine.

# DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

# AUTHOR CONTRIBUTIONS

M.Z., D.M.N., and Y.G. conceived and designed research; M.Z. performed modeling; M.Z. analyzed data; M.Z., D.M.N., and Y.G. interpreted results of modeling; M.Z. and Y.G. prepared figures; M.Z., D.M.N., and Y.G. drafted manuscript; M.Z., D.M.N., and Y.G. edited and revised manuscript; M.Z., D.M.N., and Y.G. approved final version of manuscript.

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