Road, Manchester M13 9WL, UK January 5, 2015 (Serov et al.,2015) and can be accessed by its doi: 10.1016/j.jtbi.2014.09.022. 1 Introduction The human placenta is the sole organ of

Optimal villi density for maximal oxygen uptake in the human placenta

A.S. Serov ∗ a, C.M. Salafiab, P. Brownbillc,d, D.S. Grebenkova, and M. Filochea

aLaboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, FrancebPlacental Analytics LLC, 93 Colonial Avenue, Larchmont, New York 10538, USA

cMaternal and Fetal Health Research Centre, Institute of Human Development, University of Manchester, OxfordRoad, Manchester M13 9WL, UK

dSt. Mary’s Hospital, Central Manchester University Hospitals NHS Foundation Trust, Manchester Academic HealthScience Centre, Manchester M13 9WL, UK

January 5, 2015

Abstract

We present a stream-tube model of oxygen exchange insidea human placenta functional unit (a placentone). The ef-fect of villi density on oxygen transfer efficiency is assessedby numerically solving the diffusion-convection equation in a2D+1D geometry for a wide range of villi densities. For eachset of physiological parameters, we observe the existence of anoptimal villi density providing a maximal oxygen uptake as atrade-off between the incoming oxygen flow and the absorbingvillus surface. The predicted optimal villi density 0.47± 0.06is compatible to previous experimental measurements. Sev-eral other ways to experimentally validate the model are alsoproposed. The proposed stream-tube model can serve as a ba-sis for analyzing the efficiency of human placentas, detectingpossible pathologies and diagnosing placental health risks fornewborns by using routine histology sections collected afterbirth.

Keywords: human placenta model; oxygen transfer;diffusion-convection.

This article was published in the Journal of TheoreticalBiology (Serov et al., 2015) and can be accessed by its doi:10.1016/j.jtbi.2014.09.022.

1 Introduction

The human placenta is the sole organ of exchange betweenthe mother and the growing fetus. It consists of two principalcomponents: (i) fetal chorionic villi, and (ii) the intervillousblood basin in which maternal blood flows (Fig. 1). The pla-centa fetal villi have a tree-like structure and are immersedin the maternal blood basin anchoring to the basal plate andsupporting the placenta shape (Benirschke et al., 2006). Fetalarterial blood goes from the umbilical arteries to the capillar-ies inside the villi and returns to the umbilical vein, while ma-ternal blood percolates outside this arboreous structure. The

∗Corresponding author.E-mail: [email protected] address: Laboratoire de Physique de la Matiere Con-densee, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France.Tel.: +33 1 69 33 47 07, Fax: +33 1 69 33 47 99

exchange of oxygen and nutrients takes place at the surfaceof the villous tree. We aim to understand how the geometryof the villous tree affects the placenta exchange function.

Figure 1. Structure of the human placenta (reproduced fromGray, 1918).

The normal development of the baby largely depends onthe ability of the placenta to efficiently transfer oxygen, nu-trients and other substances. Both placenta weight andplacenta-fetus weight ratio are associated with the newbornshealth (Hutcheon et al., 2012, Teng et al., 2012). Moreover,“fetal origins of adult health” research has identified links be-tween placenta size and risk of heart disease in adults (Barker,1995).

Routinely stained 2D histology sections of the placenta col-lected after birth (Fig. 2) may provide valuable insights intoits function during pregnancy. Figure 2 illustrates our basicidea: random sampling of a normal placenta (Fig. 2B) con-tains intervillous space (IVS) and villous tree sections in “nor-mal” proportions (which allow efficient function), whereaspre-eclamptic (Fig. 2A, disproportionally large IVS, rare villi)and diabetic (Fig. 2C, denser and larger villi) cases exhibit avery different geometry.

Assessing the relation between the placenta structure andits function is problematic because (i) it is unethical to manip-ulate the human placenta in vivo; (ii) histological and physi-ological features commonly co-vary (e.g. changes in villi dis-

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Figure 2. Typical 2D placenta cross-sections: (A) pre-eclamptic placenta (rarefied villi, reduced exchange surface); (B) normal placenta;(C) diabetic placenta (dense villi, reduced surface accessibility). White space is intervillous space (IVS), normally filled with maternalblood, which has been washed away during the preparation of the slides (some residual red blood cells are still present). The dark shapesare cross-sections of fetal villi. The sections are H&E stained and have been taken in the direction from the basal (maternal) to thechorionic (fetal) plate.

tribution alter the blood flow); (iii) quantitative histologicalanalysis of the placenta structure can only be performed post-partum when maternal blood flow (MBF) and fetal blood flowhave ceased. Mathematical modeling and numerical simula-tions are in this case valuable tools to gain deeper insightsinto the in vivo functioning of the placenta.

Mathematical models of the human and animal placentafunction have been proposed for at least 60 years (see dis-cussions in Aifantis, 1978, Battaglia and Meschia, 1986,Chernyavsky et al., 2010, Gill et al., 2011). Previous mod-els mainly focused either on a single villus scale or on thewhole placenta scale (different kinds of the flat wall exchangermodel: Bartels et al., 1962, Gill et al., 2011, Groome, 1991, Hillet al., 1973, Kirschbaum and Shapiro, 1969, Lardner, 1975,Longo et al., 1972, Shapiro et al., 1967, Wilbur et al., 1978);several studies dealt with flow patterns (co-orientation ofmaternal and fetal flows: Bartels et al., 1962, Battaglia andMeschia, 1986, Faber, 1969, Guilbeau et al., 1970, Kirschbaumand Shapiro, 1969, Metcalfe et al., 1964, Moll, 1972, Schroder,1982, Shapiro et al., 1967); other works represented the pla-centa as a porous medium (Chernyavsky et al., 2010, Erianet al., 1977, Schmid-Schonbein, 1988), restricted, with one ex-ception (Chernyavsky et al., 2010), to one or two dimensions.Some efforts have been devoted to understanding the relationbetween morphometric data and gas transfer in 1D in termsof diffusing capacity (see Mayhew et al., 1986 and referencestherein).

Better understanding of the transfer function of the pla-centa requires a model of the 3D geometry of the organ ona larger scale than a few villi, predictions of which could becompared to experimental data. We use a simplified engineer-ing representation of the complex system in order to revealthe most relevant transport mechanisms.

The driving questions of our placenta modeling are:

• What are the relevant parameters (geometrical and phys-iological) that govern oxygen transfer efficiency?

• For a given set of physiological parameters (such as MBFvelocity or oxygen content of blood), is there an “opti-mal” villous tree geometry which maximizes the oxygenuptake?

In the following, we present a simplified stream-tube pla-centa model (STPM) of oxygen exchange. The mathemat-ical equations governing oxygen transfer in the placentone

are then numerically solved for different values of geometricaland physiological parameters. The results are compared topublished histomorphometric measurements near term (Ah-erne and Dunnill, 1966, Lee and Mayhew, 1995, Mayhew andJairam, 2000, Mayhew et al., 1993, Nelson et al., 2009).

2 Mathematical model

2.1 Outline

Our 3D STPM is inspired by the observation that the hu-man placenta resembles “a closed cubical room supportedby cylindrical pillars running from floor to ceiling” (Leeand Mayhew, 1995). The model consists of a large cylin-der (Fig. 3A) representing a stream tube along which ma-ternal blood flows (Fig. 3C). This cylinder contains multiplesmaller parallel cylinders which represent fetal villi (terminaland mature intermediate), filled with fetal blood. In this pa-per, we consider these small cylinders to be of identical sizes,while their lateral spacings are randomly distributed (withno cylinders overlap). This technical assumption can be eas-ily relaxed in order to treat routinely stained 2D histologicalsections of the placenta.

Our model relies on the following assumptions:

1. The large cylinder corresponds to an idealized unfoldedstream tube of maternal blood, extending from the bound-ary of the central cavity to a decidual vein (Fig. 3C). Ev-ery portion of arterial blood coming into placenta has itsown stream tube, along which blood exchanges oxygenwith fetal villi and gradually becomes venous. The con-torted axis of a stream tube follows the maternal bloodflow. The length of a tube can vary from one region toanother, but is expected to be of the order of placentaldisk thickness (see Appendix A for more details);

2. The stream-tube has the same cross-section along itsaxis. In fact, we aim to base our STPM on histolog-ical slides (like those shown in Fig. 2), which give usonly one section of a stream tube without any informa-tion about how this section changes along it. We thensuppose that the same integral characteristics, such asthe density or the perimeter of villi, are preserved alongthe same tube. This is obviously an oversimplification ofthe irregular 3D structure of the placenta, but it is themost straightforward assumption given the lack of the

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Figure 3. (A) The stream-tube model of oxygen exchange in a human placenta functional unit (a placentone). Arrows show the flow ofmaternal blood. All the calculations are presented for a cylindrical stream-tube shape. (B) An alternative (square) geometry of a streamtube. As discussed in Appendix A, the shape of the external boundary can be chosen arbitrary. (C) A scheme of the placentone andlocation of the stream tubes in the placenta. The dashed line schematically outlines the central cavity. Curved arrows on the right showmaternal blood loosing oxygen while going from the central cavity to decidual veins. Curved lines on the left schematically show streamtubes of blood flow. The STPM corresponds to one such tube unfolded; small straight arrows show the entrance point of the model. Theexchange is not modeled in the central cavity, but only after it, in the MBF pathway. The concept that spiral arteries open into the IVSnear the central cavity corresponds to the current physiological views (Benirschke et al., 2006, Chernyavsky et al., 2010).

complete 3D geometrical data with all spatially resolvedterminal and mature intermediate villi;

3. In any cross-section, oxygen is only redistributed by dif-fusion. The gradient of oxygen dissolved in the bloodplasma, not the gradient of oxygen bound to hemoglobin,determines the rate of oxygen diffusion in blood (Bar-tels et al., 1962). As oxygen has low solubility in blood,linear solubility (Henry’s law, Prausnitz et al., 1998) isassumed;

4. Fetal blood is considered to act as a perfect oxygen sink.This assumption means that we are studying oxygen par-tial pressure differences between the maternal and fetalblood. Additionally, it supposes that fetal circulation isefficient enough to rapidly carry away all oxygen thatis transferred from maternal blood. If villi are consid-ered to be perfect sinks, there is no difference betweenconcurrent, counter-current blood flow patterns or anyother organization of the flow in this specific geometry.The features of the fetal blood flow can therefore be dis-regarded, except that fetal blood flow rate is assumed tobe sufficient to maintain steep oxygen gradients for diffu-sion. In other words, the STPM separates the IVS geom-etry from capillary geometry inside the villi. Resistanceto oxygen transfer across the villus membrane, intravil-lous space and fetal capillary membranes are modeled byan effective feto-maternal interface of finite permeabilitycalculated from the placental diffusing capacity;

5. Oxygen uptake occurs at the feto-maternal interface, i.e.at the boundaries of the small cylinders in Fig. 3A. Itis directly proportional to the interface permeability andto the oxygen concentration on the maternal side of theinterface. This statement is a reformulation of Fick’s lawof diffusion, in which the rate of oxygen transfer acrossthe interface is proportional to the difference of partialpressures (pO2) on both sides of the interface;

6. Maternal blood flow is considered to be laminar with noliquid-walls friction (slip boundary conditions at all sur-

faces), so that the velocity profile in any cross-sectionis flat. This assumption is supported by calculationsobtained for capillary-tissue cylinders in brain (Reneauet al., 1967). The brain model has been used to com-pare the effect of non-slip boundary conditions (and thusa non-flat velocity profile) versus slip boundary condi-tions (and thus a flat velocity profile) on the distributionof pO2 in a Krogh’s type cylindrical tissue layer. Thedifference between the two cases in pO2 distribution wasless than 10 %. This strong assumption is discussed laterin the text;

7. Oxygen-hemoglobin dissociation curve is linearized in or-der to simplify the resolution of equations;

8. Erythrocytes are uniformly distributed in the IVS;

9. The solution is stationary, i.e. placental oxygen flow re-mains constant over time;

10. Blood flow matching does not occur, i.e. there is no influ-ence (feedback) of oxygen uptake on the parameters ofthe model (such as MBF velocity, see Talbert and Se-bire, 2004). No redundancy of some of the fetal cylinderswhen maternal IVS blood flow is reduced is taken intoaccount.

2.2 Solution of the model

2.2.1 Characteristic time scales of the transfer pro-cesses in the placenta

We identify three different physical transfer processes in theplacenta, each of which operates on a characteristic time scale:hydrodynamic blood flow through the IVS characterized byan average velocity u and transit time τtr; diffusion of oxygenwith characteristic time τD; and equilibration between oxy-gen bound to hemoglobin and oxygen dissolved in the bloodplasma with characteristic time τHb. This last thermody-namic equilibrium is described as equal partial pressure ofoxygen in both states. The three times can be estimated asfollows:

3

Table 1. Parameters of the human placenta used in the calcula-tions. Details of the parameters calculation can be found in Ap-pendix A. No experimental error estimations were available for uand L.

Parameter Symbol Mean ± SD

Maximal Hb-bound oxygen concentra-tion at 100 % Hb saturation, mol/m3

cmax 7.30 ± 0.11

Oxygen-hemoglobin dissociation con-stant

B 94 ± 2

Concentration of oxygen dis-solved in blood at the entrance tothe IVS, 10−2 mol/m3

c0 6.7 ± 0.2

Oxygen diffusivity in blood, 10−9 m2/s D 1.7 ± 0.5Effective villi radius, 10−6 m r 41 ± 3Permeability of the effective materno-fetal interface, 10−4 m/s

w 2.8 ± 1.1

Placentone radius, 10−2 m R 1.6 ± 0.4Radius of stream tube used in numeri-cal calculations, 10−4 m

Rnum 6

Velocity of the maternal bloodflow, 10−4 m/s

u 6

Stream tube length, 10−2 m L 1.6

• τtr, the transit time of blood through the IVS, is of theorder of 27 s from the results of angiographic studies atterm (Burchell, 1967, see Appendix A);

• τD, reflecting oxygen diffusion over a length δ in the IVSis τD ∼ δ2/D, where D = (1.7±0.5)·10−9 m2/s is oxygendiffusivity in the blood plasma (Moore et al., 2000). Ei-ther from calculations (Mayhew and Jairam, 2000), ordirectly from normal placenta sections (Fig. 2), meanwidth of an IVS pore can be estimated as δ ∼ 80µm,yielding τD ∼ 4 s;

• τHb, an equilibration time scale, which includes the char-acteristic diffusion time for oxygen to reach Hb moleculesinside a red blood cell (∼ 10 ms Foucquier et al., 2013)and a typical time of oxygen-hemoglobin dissociation (∼20 ms for the slowest process, see Yamaguchi et al., 1985).Together these times sum to τHb ∼ 30 ms.

These three characteristic times are related as follows: τHb �τD . τtr. This relation suggests that oxygen-hemoglobin dis-sociation can be considered instantaneous as compared to dif-fusion and convection; the latter two, by contrast, should betreated simultaneously. Further details of the solution of themodel can be found in Appendix B.

2.2.2 Parameters of the model

The parameters necessary for the construction of the modelare listed in Table 1, while their calculation is reported inAppendix A. Errors in the experimental parameters were es-timated in cases when such information was available from thesource; no error assumptions were made in other cases. Rela-tively small stream-tube radius Rnum has been used to speedup the computations, but all results have been rescaled to theplacentone radius R by multiplying the uptake by R2/R2

num.Note that besides the parameters traditionally used for

modeling 1D exchange in the human placenta, namely: in-coming (c0) and maximal oxygen concentrations (cmax), dif-fusivity of oxygen in blood (D), solubility of oxygen inblood (khn), oxygen-hemoglobin dissociation coefficient (β60),density of blood (ρbl), length of the blood flow path (L), a

N=1Φ=0.005

N=54Φ=0.25

N=80Φ=0.37

N=160Φ=0.75

Figure 4. A range of villi densities for which oxygen uptake hasbeen calculated. The number of villi and the corresponding villidensity is displayed above each case. Simulations have been carriedout for all geometries in the range N = 1–160.

characteristic size of the exchange region (R) and the localpermeability of the feto-maternal interface (w) or the inte-gral diffusing capacity of the organ (Dp), the construction ofthe STPM required several additional parameters to accountfor a 2D placental section structure (compare, for example,with Power et al., 1972). These are: (i) the linear maternalblood flow velocity in the exchange region (u, in m/s), sincethe usual integral flow (in ml/min) is not enough; (ii) thevolumetric density (φ) of small fetal villi in a given region;and (iii) the effective radius of fetal villi (r), which accountsfor the non-circular shape of fetal villi by providing the rightvolume-to-surface ratio.

2.2.3 Numerical simulation

The diffusion-convection equation (B.5) is solved by a finiteelements method in MatlabR© for various 2D villi distribu-tions (Fig. 4). Villi distributions are generated in the rangeof villi densities φ from 0.005 to 0.75. Villi density is defined asthe ratio of small villi cross-sectional area (Ssm.v) to the totalarea of the stream tube cross-section (Stot): φ = Ssm.v/Stot.These densities correspond to a range of villi numbers Nfrom 1 to 160 for a stream-tube of the radius Rnum. Theupper boundary of the range has been chosen close to themaximal packing density of circles that one can achieve with-out resorting to special packing algorithms (Specht, 2009).

3 Results

Figure 5A shows the dependence of oxygen uptake of a sin-gle placentone on the length of the stream tube. One cansee that the villi density which provides the maximal oxygenuptake, depends on the stream tube length, all other param-eters remaining fixed. In other words, the most efficient ge-ometry (the one which gives the highest uptake for a fixedlength, MBF velocity and incoming oxygen concentration) isdifferent for different stream tube lengths. The villi densitycorresponding to the maximal uptake will be called “opti-mal”. The fact that there is an optimal villi density can beclearly seen in Fig. 5B, which shows the variation of oxygenuptake with villi density for a fixed stream tube length.

Both the maximal oxygen uptake and the corresponding op-timal villi density depend on the average MBF velocity andon the stream-tube length L, which are the parameters themost susceptible to variations in different placentas and forwhich no error estimations were available (see Appendix A).If empirical distribution of MBF velocities in the IVS of thehuman placenta were known, with the help of Fig. 6A, onecould estimate the optimal villi density that would result in the

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Figure 5. Oxygen uptake of a single placentone as a function of several geometrical parameters. (A) Oxygen uptake as a function of thestream-tube length L for fixed villi densities φ. Various symbols represent the villi densities of Fig. 4. (B) Oxygen uptake as a functionof villi density φ for three lengths z: L/3, 2L/3, and L. The peak uptake moves to smaller villi densities when the length z increases.Oxygen uptake has been calculated for Rnum and rescaled to placentone radius R by multiplying by R2/R2

num. Transit time of maternalblood through the placentone (τtr = z/u), blood inflow per placentone (uπR2(1 − φ)) and oxygen uptake in cm3/min are shown on theadditional axes to simplify comparison of the results with physiological data.

maximal uptake in any given region. Conversely, if one knowsthe average villi density in a placenta region (e.g., from his-tological slides) and if optimal placenta function is assumed,one can estimate the MBF velocity in that region. The cor-responding maximal oxygen uptake can be determined fromFig. 6C. The dependence of the optimal characteristics on thestream-tube length is shown in Figs 6B, 6D.

Our calculations show that for a healthy placenta (with theparameters given in Table 1), the optimal villi density is φ0 =0.47 ± 0.06 and the corresponding maximal oxygen uptakeis Fmax = (1.2±0.6) ·10−6 mol/s ≈ 1.6±0.8 cm3/min. Detailsof the provided error estimations can be found in Appendix C.It is worth nothing that to the precision of the calculations,no dependence of the results on the diffusivity D of oxygenwas observed. Note also that since no experimental error datawas available for L and u, these variables were not includedin error calculation. The confidence intervals for φ0 and Fmax

are hence likely to be larger than the provided values.

4 Discussion

We have developed the STPM of oxygen exchange in thehuman placenta that relates oxygen uptake to cross-sectionhistomorphometry obtained from routinely prepared placentaslides. Our goal was to provide a model capable of identify-ing cases of suboptimal oxygen transfer from maternal to fetalblood by histological analysis of a placenta post-partum. Itwas shown that there is always an optimal villi density, atwhich oxygen uptake is maximal, all other parameters beingfixed.

The optimal density is determined by a competition of twofactors. On one hand, larger villi density leads to larger ex-change surface and thus larger uptake. On the other hand,larger villi density implies that less blood is coming into theplacentone per unit of time, hence a lower uptake rate. Fora given set of the placenta-specific parameters (Table 1), ourmodel allows one to calculate the optimal villi density for agiven placenta or placenta region.

4.1 Comparison to experimental data

Our calculated optimal density can be compared to exper-imental data. The villi density φ can be determined from

published histomorphometrical data according to the for-mula φ = Ssm.v/(Ssm.v + SIVS), where Ssm.v is the cross-sectional area of terminal and mature intermediate (small)villi, and SIVS is the cross-sectional area of the IVS. In theliterature, Ssm.v and SIVS are traditionally obtained as sur-face area fractions of a 2D section and then multiplied by thevolume of the placenta to yield volumes of small villi (Vsm.v)and IVS (VIVS). Villi density can be calculated by the sameformula from these volumes: φ = Vsm.v/(Vsm.v + VIVS) (Ta-ble 2). In cases when the fraction of terminal and matureintermediate villi in the total villi volume was not avail-able, we have used the average fraction of small villi volumeof α ≈ 0.665 in the total volume of the villi (Sen et al., 1979).Note that these experimental data refer to placentas at termsince a morphometrical study of a placenta can be only per-formed post-partum.

Table 2. Villi density (φ) as calculated from histomorphometricaldata for small villi (Vsm.v) and IVS (VIVS) volumes of placentas atterm obtained from normal pregnancies.

Vsm.v,cm3

VIVS,cm3

φ Source

130 170 0.43 Mayhew and Jairam (2000)220 220 0.50 Nelson et al. (2009)190 170 0.53 Aherne and Dunnill (1966)110 180 0.38 Lee and Mayhew (1995)150 170 0.47 Mayhew et al. (1993)

The optimal villi density 0.47±0.06 obtained in our model isconsistent with the experimental data which yield φ = 0.46±0.06 (mean ± SD). The quantitative agreement of the model’spredictions with physiological densities allows us to speculatethat normal placentas are optimal oxygen exchangers, whilepathological placentas (e.g. diabetic or pre-eclamptic) behavesub-optimally. Although such an argument is plausible, fur-ther analysis is required. Evidence from sheep specificallydemonstrates a large reserve capacity for fetal oxygen sup-ply, suggesting that variation in villi density and MBF ve-locity might be a normal tolerated biological process (Carter,1989). A similar behavior has been demonstrated earlier inhuman lungs, where lung geometry is not optimal under nor-mal conditions, but its reserve capacity is used during exer-cises (Sapoval et al., 2002).

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Figure 6. (A), (C): Dependence of the optimal villi density (A) and the maximal oxygen uptake (C) on the MBF velocity at a fixed length Lfor a single placentone. (B), (D): Dependence of the optimal villi density (B) and the maximal oxygen uptake (D) on the stream-tubelength L for a single placentone. Dashed lines show the expected average stream tube length L = 1.6 cm and MBF velocity u = 6 ·10−4 m/s.Transit time of maternal blood through the placentone (τtr = L/u), blood inflow per placentone (uπR2(1 − φ)) and oxygen uptakein cm3/min are shown on additional axes to simplify comparison of the results with physiological data.

The experimental morphometrical analysis of the villi den-sity in placenta sections post-partum may not represent pre-cisely the in vivo situation due to the cease of maternal andfetal blood flows and to the fixation procedure. Because ofthese effects, the histological villi density seen post-partummay overestimate the in vivo villi density. However, for themoment, there is no generally acknowledged method whichwould allow to correct for these changes.

4.2 Comparison to the porous medium model

Our optimal villi density can be compared to predictions ofthe porous medium model (PMM, Chernyavsky et al., 2010).Chernyavsky et al. have obtained a value of the optimal frac-tion of “villous material” φc close to 0.3. This value has to betranslated into φ obtained in the STPM to allow comparison.Defining φc as a fraction of villi volume in the total volumeof IVS and villi, and φ as a fraction of small villi volume inthe total volume encompassing IVS and small villi:

φc = Vvil/(Vvil + VIVS), φ = Vsm.v/(Vsm.v + VIVS),

one obtains the following translation formula: φ = α/(α −1 + 1/φc) ≈ 0.22, where again α ≡ Vsm.v/Vvil ≈ 0.665 is theaverage fraction of small villi in the total villi volume (Senet al., 1979). Comparison of this value to the average ex-perimental value φ = 0.46± 0.06 demonstrates a twofold un-derestimation of the observed villi density in the PMM. Theauthors themselves admit that this value corresponds ratherto high-altitude or pathological pre-eclamptic placentas than

to healthy ones. A possible reason of this mismatch is thatin the PMM the optimal villi density appears as a trade-offbetween blood flow resistance and uniform uptake capacity ofan unspecified passively transported substance. We proposea different explanation presenting the optimal villi density asa trade-off between the absorbing surface of fetal villi and theincoming flow of oxygen. In this claim, we underline the roleof the absorbing surface (which is hard to define in the PMM),but also suggest the importance of oxygen transport as com-pared to the passive transport of an unspecified substance.

The two models offer two different points of view on mod-eling the human placenta, which we discuss below.

1. Representing villi as a uniform and isotropic porousmedium in the PMM ignores the placenta geometry. Theonly geometrical parameter left (volume fraction of the“villous material”, analogous to villi density) masks in-dependent roles played by the absorbing surface as wellas by villi shapes and sizes. With no uptake surface,the PMM requires assuming a uniform uptake propor-tional to the IVS volume fraction. At the same time,experimental results show a considerable difference inthe absorbing surface between healthy and pathologicalcases (Mayhew and Jairam, 2000).

Although the full 3D structure of the human placentaat the microscopic scale of individual villi is not avail-able from the experiment, its 2D sections can be ob-tained (Fig. 2). Both the PMM and STPM can relyon these slides in terms of villi density, but the presentmodel also accounts for the absorbing surface by using

6

an effective villi radius (see Appendix A) and can take awhole 2D section as input data (the last option requiresdevelopment of image analysis techniques). In the PMM,the analysis cannot go beyond the villi density;

2. Simple uniform uptake kinetics (ignoring interaction withhemoglobin) used in the PMM can only account for pas-sive transport of certain metabolites, but it is unable todescribe exchange of respiratory gases as it ignores Hb-binding (for instance, about 99 % of oxygen comes intothe placenta in the bound form). At the same time,the STPM can be used for oxygen and carbon dioxidetransport as well as for passive substances transport.

3. The use of slip boundary conditions yielding a flat veloc-ity profile in the STPM needs further discussion. Ideally,a full model of the placenta transport function would firstrequire the full high-resolution 3D placenta structure tocompute the velocity field by solving the Navier-Stokesequation with non-slip boundary conditions at the villoustree surface. After that, oxygen uptake could be calcu-lated using convection-diffusion equations with the ob-tained velocity profile. Since modern experimental tech-niques can neither acquire the placenta structure with asufficient spatial resolution for such calculations, nor pro-vide direct experimental measurements of the blood ve-locity field, one must resort to simplifications. One wayis to ignore the geometrical structure of the organ andsubstitute it by a uniform isotropic porous medium as inthe PMM. We chose an alternative way using geometricalinformation available from 2D histological slides.

From the hydrodynamic point of view, the non-slipboundary conditions would be the most natural for mod-eling blood flow in a real high-resolution 3D placentastructure. Since the blood flow path in a real placentageometry is expected to be irregular, no developed ve-locity profile is expected along the flow, and differentsections of the placenta may present different velocityprofiles. These real velocity profiles, however, can befar from that of a developed flow with non-slip boundaryconditions. As a consequence, the advantage of choosingthe non-slip boundary conditions over the slip ones inour already simplified structure is not so clear. In otherwords, although slip and non-slip boundary conditionsmay lead to different quantitative predictions, it is diffi-cult to say which prediction will be closer to the results ina real 3D placenta geometry. This statement can be ver-ified either by modeling the blood flow in a full placentageometry (unavailable by now), or by confronting modelspredictions to physiological observations. From the latterpoint of view, our model with slip boundary conditionspredicts the optimal villi density which is comparable tothe experimentally observed one. A comparison with pre-dictions from the same simplified structure with non-slipboundary conditions presents an interesting perspective.Note that the blood velocity used in our model shouldbe understood as an average velocity across the streamtube or a placenta region.

On the contrary, PMM implicitly takes into account thenon-slip boundary conditions (friction at the boundary)by using Darcy’s law. However, the PMM has similar lim-itations as it obtains a developed blood flow which may

be far from the real non-developed one. Moreover, quali-tative distribution of the flow in the placentone obtainedin the PMM remains approximate because: (i) Darcy’slaw has never been proved valid for the human placentadue to the lack of experimental data; (ii) Darcy’s law wasargued to ignore the inertia of the flow and can possiblymiss important characteristics of the flow in the organ:“given the architecture of a placental circulatory unit,the [maternal blood] jet penetration essential to good uti-lization of the full villous tree for mass transfer can occuronly with appreciable fluid inertia” (Erian et al., 1977).The validity of the last statement may however depend onthe location of the maternal vessels (Chernyavsky et al.,2010). Our approach operates with an average velocityof the maternal blood in the organ which includes theinertia implicitly;

4. The advantage of the PMM is that the results are ob-tained analytically in a simple form.

4.3 Other ways of comparison

Several other ways of validation of the presented model can besuggested, all of them requiring experimental measurementsof some quantitative criteria of oxygen transfer efficiency andhistomorphometry of placental sections:

1. For each pregnancy, medical doctors possess approximatequantitative criteria of newborns’ health, such as thebirth-weight to placenta-weight ratio. If such informationwere obtained together with histological sections of theplacenta in each case, a correlation could be studied be-tween the observed villi density, the optimal villi densitypredicted by the model and the health of the baby. Suchexperiment, however, requires the development of imageanalysis techniques that would allow for automatic seg-mentation and morphometric measurements to be per-formed on large histological placenta sections;

2. In the primate placenta (the structure of which is similarto that of the human placenta), more information canprobably be obtained from the experiment than it is eth-ical in the human placenta. Thus, (i) if one measured theaverage linear MBF velocity in the IVS of a placentone orin the spiral arteries supplying it, and (ii) if morphomet-rical measurements were obtained after birth for the sameplacentone, then the relation between the blood velocityand the observed villi density could be compared withFig. 6A. If additionally blood oxygenation were measuredin the spiral arteries and the decidual veins, the depen-dence of oxygen uptake of a placentone on the MBF ve-locity could be compared to Fig. 6C. A systematic studyof several placentones of the same or different placentascould further improve validation of the model;

3. Artificial perfusion experiments could also provide datacomparable to the results of the model. In such setups,one can estimate the oxygen uptake by the fetal circula-tion of a given placenta, while having control over per-fusion parameters. After the perfusion, histomorphome-trical measurements can be performed. The fact that inthe artificial perfusion experiments no-hemoglobin bloodis normally used does not hamper the comparison, as

7

such situation can be simulated in our model by allow-ing B = 1. Experimental measurements of villi densityand of oxygen uptake as a function of the perfusion pa-rameters can then be compared to the predictions of themodel.

These suggestions represent possible directions of further de-velopment of experimental techniques.

4.4 Model assumptions and other remarks

The assumption of fetal villi being a perfect sink erases differ-ences between concurrent and counter-current organizationsof flows since this difference essentially arises from the non-uniformity of oxygen concentration in the fetal blood. Thepresented STPM can describe oxygen transfer in both theseflow orientations under this assumption. As for the cross-current (multivillous) blood flow organization, the model ge-ometry can be changed to account for the angle α between thematernal and fetal flows directions. We have not incorporatedsuch calculations into our model, because (i) there does notseem to be any fixed co-orientation of the maternal and fetalblood flows in the human placenta, and (ii) the experimen-tally observed villi density is already a result of sectioning atan unknown (random) angle to fetal villi. If the value of thisangle were known, the effect of the sectioning angle could beapproximately corrected by multiplying the villi density andthe villi perimeter used in the model by the factor 1/ cos(α)before comparison to values observed in experimental slides.

Finally, it should be noted that the curves in Fig. 5 donot reach the expected limit of 1 for the optimal villi den-sity, but stop at 0.75. This is explained by the circular villishapes used in the calculations. The maximal packing den-sity for the used radii ratio of Rnum/r ≈ 14.63 is known tobe φ ≈ 0.83 or N = 177 cylinders (Specht, 2009). Withoutusing the optimal packing algorithms, we have carried outcalculations up to the density φ = 0.75, which we believe tobe sufficient to represent densely packed regions of placentaslides (cf. Figs 2, 4). The latter value is the limit of the curvesseen in Figs 5.

5 Conclusions

We presented a 2D+1D stream-tube placenta model of oxygentransfer in the human placenta. Our model incorporates onlythe most significant geometrical and hemodynamic features,enabling it to remain simple while yielding practical results.

Modeling oxygen transport in the human placenta tradi-tionally involved the following parameters: incoming (c0)and maximal oxygen concentrations (cmax), diffusivity ofoxygen in blood (D), solubility of oxygen in blood (khn),oxygen-hemoglobin dissociation coefficient (β60), density ofblood (ρbl), length of the blood flow path (L), a characteristicsize of the exchange region (R) and the local permeability ofthe feto-maternal interface (w) or the integral diffusing capac-ity of the organ (Dp). The construction of the STPM showedthat several additional parameters are required to accountfor a 2D placental section structure: (i) the linear maternalblood flow velocity in the exchange region (u, in m/s), sincethe usual integral flow (in ml/min) is not enough; (ii) the vol-umetric density (φ) of small fetal villi in a given region; and(iii) the effective radius of fetal villi (r), which accounts for the

non-circular shape of fetal villi by providing the right volume-to-surface ratio. Although we have indirectly estimated theseparameters from the available experimental data, their directmeasurement in the human placenta is a promising directionof further development of experimental techniques.

In spite of its simplicity, the model predicts the existence ofan optimal villi density for each set of model parameters as atrade-off between the incoming oxygen flux and the absorbingvillus surface. The predicted optimal villi density 0.47±0.06 iscompatible with experimentally observed values. Dependenceof the optimal characteristics on the length of the stream tubeand the velocity of the MBF was investigated.

In a perspective, one can check the effect of relaxing someof the model assumptions. One direction of the future workconsists in relaxing the slip boundary conditions and com-paring the results of the two approaches, which are expectedto describe different limiting cases of the real in vivo flow.One can also study the effect of villi shapes and sizes distri-butions on the predicted optimal villi density and maximaluptake. Besides, the placenta is a living organ with its ownoxygen consumption rate, which may be considered propor-tional to the weight of its tissues. Once this effect is taken intoaccount, the amount of oxygen transferred at the same villidensity from mother to the fetus should decrease, shifting theposition of the optimal villi density peak to the left (Fig. 5B).Another direction is to apply the STPM to 2D histologicalslides in order to relate geometrical structure of placenta re-gions to their oxygen uptake efficiency and hence to be ableto distinguish between healthy and pathological placentas.

6 Acknowledgements

This study was funded by the International Relations Depart-ment of Ecole Polytechnique as a part of the PhD projectof A.S. Serov, by Placental Analytics LLC, NY, by ANRSAMOVAR project n◦ 2010-BLAN-1119-05 and by ANRproject ANR-13-JSV5-0006-01. The funding sources did notinfluence directly any stage of the research project.

7 Conflicts of interest

The authors declare to have no potential conflicts of interest.

8 Contributions of authors

All authors have contributed to all stages of the work exceptfor numerical calculations performed by A.S. Serov. All au-thors have approved the final version of the article.

Appendices

A Parameters of the model

Maximal oxygen concentration in the maternalblood (cmax) cmax is the maximal oxygen concentration,which a unit of blood volume would contain at a 100 %saturation only due to binding to hemoglobin molecules. Tocalculate cmax we need to multiply the following quantities:

• concentration of hemoglobin in blood.

Pregnant women hemoglobin level at termis 0.1194± 0.0007 g Hb/ml blood (Wills et al., 1947);

8

• hemoglobin binding capacity. Adult personhemoglobin binding capacity is 1.37 ± 0.02 ml O2/g Hb(Dijkhuizen et al., 1977);

• amount of substance in 1 l of oxygen. As Avo-gadro’s law states, a liter of any gas taken at normalconditions contains 1/22.4 mol of substance.

Using these data we obtain

cmax = 0.1194g Hb

ml blood· 1.37

ml O2

g Hb· 1

22.4

mol

l O2

= 7.30± 0.11 mol/m3

Oxygen-hemoglobin dissociation constant (B)Oxygen-hemoglobin dissociation constant B representsthe interaction of oxygen with hemoglobin in Eq. (B.5), whenthe Hill equation has been linearized.

Substituting cmax = 7.30 ± 0.11 mol/m3, β60 = 0.0170 ±0.0003 mmHg−1 (slope of the linearized Hill equation, see Sec-tion B.3), khn ≈ 7.5 · 105 mmHg · kg/mol (see Section B.1)and ρbl ≈ 103 kg/m3 (as close to the density of water) intoEq. (B.5), we obtain B = 94± 2.

Oxygen concentration at the entrance to the IVS (c0)c0 is the concentration of oxygen dissolved in the blood plasmaof maternal blood entering the system. To calculate the con-centration c0, we need to know oxygen saturation of maternalblood which enters the placenta. We will then be able tocalculate c0 by the formula which follows from the Hill equa-tion (B.1):

c0 = cpl(z = 0) =1

B − 1· S(pO2) · cmax.

After 12-th week postmenstruation, pO2 was found tobe 90 mmHg in the uterine artery, 61 mmHg in the IVSand 47 mmHg in the decidual vein (Challier and Uzan, 2003,Jauniaux et al., 2000, Rodesch et al., 1992). Taking the IVSvalue of 61 mmHg (no error estimation available) togetherwith the hemoglobin saturation curve (Severinghaus, 1979) weget hemoglobin saturation S(61 mmHg) ≈ 85 %. Supposingthat it is equal to the hemoglobin saturation at the entranceto the IVS, we then obtain c0 = (6.7± 0.2) · 10−2 mol/m3.

Oxygen diffusivity in blood plasma (D) Oxygen diffu-sivity in the blood plasma has been measured under differentconditions: D = (1.7 ± 0.5) · 10−9 m2/s (Moore et al., 2000,Wise and Houghton, 1969).

Effective villi radius (r) Histological slides of the humanplacenta show that the cross-sections of fetal villi are not cir-cular. Traditionally, the radius of the villi is estimated byaveraging the size of a cross-section in all directions. Thisyields a villi radius 25–30µm for terminal and mature inter-mediate villi, which contribute the most to the uptake (Senet al., 1979). However, such calculation ignores the fact thatthe real perimeter of a villus is larger than that of a circlewith the same radius. At the same time, the absorbing sur-face of villi is important for transport processes as, in the firstapproximation, oxygen uptake is proportional to its area.

To let our model appropriately account for both cross-sectional area (Ssm.v) and the absorbing perimeter (Psm.v)of the small villi, an effective villi radius (r) is introduced

to give the same perimeter-to-cross-sectional-area ratio (ε =Psm.v/Ssm.v, also called surface density) as in histologicalslides. In our model, ε = 2πrN/(πr2N) = 2/r, and we thendefine the effective villi radius as r = 2/ε so as to give theright perimeter-to-area ratio for the villi cross-sections. Thevalue of ε is taken from histomorphometrical studies.

Experimental values of the perimeter-to-area ratio ε andradii calculated from them are shown in Table A.1. Thearea (Ssm.v) and the perimeter (Psm.v) of the small villi cross-sections are usually presented in terms of small villi vol-ume (Vsm.v) and small villi surface area (Asm.v) respectivelyby scaling them to the size of the whole placenta. In spite ofthis recalculation procedure, the same formula remains validfor ε: ε = 2Asm.v/Vsm.v. One then gets r = 41 ± 3µm forthe effective villi radius (a value slightly larger than the aver-age villi radius as it accounts for the correct perimeter-to-arearatio of villi cross-sections).

Permeability of the effective feto-maternal inter-face (w) The permeability of the feto-maternal interface isa local transfer characteristic, which has been studied beforeunder the form of oxygen diffusing capacity (Dp). We showbelow how the two characteristics are related and how onecan be recalculated into the other. Following Bartels et al.(1962), Mayhew et al. (1984), we define Dp as a volume ofgas (measured at normal conditions) transferred per minuteper average oxygen partial pressure difference between thematernal and fetal blood: Dp = VO2/(t(pmat − pfet)), wherethe horizontal bar denotes averaging over the feto-maternalinterface surface. Oxygen diffusing capacity has beencalculated for the placenta as a whole, but the detailsof calculations (namely, rescaling) demonstrate its localnature (Aherne and Dunnill, 1966, Laga et al., 1973, Mayhewet al., 1984, Teasdale, 1982, 1983). The following points needto be clarified:

1. Our model assumption that fetal blood is a perfect sinkimplies that the partial pressure of oxygen (pO2) appear-ing in our model is not its absolute value in the ma-ternal blood, but the difference in oxygen partial pres-sures on both sides of the membrane: pO2 = pmat − pfet.This partial pressure can be recalculated into the con-centration of oxygen dissolved in the blood plasma (cpl)with Henry’s law (Appendix B): pO2 = cplkhn/ρbl. Thevolume of transferred oxygen VO2 measured at nor-mal conditions can be recalculated into amount of sub-stance with the help of Avogadro’s law: VO2 = νO2Vm,where Vm = 22.4 l/mol. One then obtains

Dp =νO2

t

Vmρblkhncpl

orνO2

t=DpkhncplVmρbl

, (A.1)

where the horizontal bar denotes averaging over the feto-maternal interface surface;

2. Using the boundary conditions froom Appendix B, wecan write oxygen uptake as F = −

∫Asm.v

D ∂cpl/∂n dS =

w∫Asm.v

c dS = wcplAsm.v, where Asm.v is the absorbingsurface of fetal villi along the blood flow. The perme-ability w can then be rewritten as w = F/(Asm.vcpl).By definition, oxygen uptake F is the amount of oxy-gen transferred to the fetus (νO2) divided by the time of

9

Table A.1. Estimations of the effective villi radius r. Average small villi fraction in all villi volume (66.5 %) and surface area (77.3 %) (Senet al., 1979) were used for calculations, when these data were not directly available from the experiment.

Mayhew andJairam (2000)

Nelson et al.(2009)

Aherne andDunnill (1966)

Lee and Mayhew(1995)

Mayhew et al.(1993)

Small villi volume (Vsm.v), cm3 130 220 190 110 150Small villi surface area (Asm.v), m2 7 11 9 5 7Perimeter-to-area ratio (ε), 104 m−1 5.4 5.0 4.7 4.5 4.7Effective villi radius (r), µm 37 40 43 44 42

exchange t, so for w one gets

w =νO2

t

1

cplAsm.v. (A.2)

Substituting Eq. (A.1) into Eq. (A.2), one relates the perme-ability w to the experimental diffusive capacity:

w =Dpkhn

VmρblAsm.v.

The mean value of the diffusing capacity is reported tobe Dp = 3.9 ± 1.0 cm3/(min ·mmHg); as calculated fromstructural and physical properties of the organ (Aherne andDunnill, 1966, Laga et al., 1973, Mayhew et al., 1984, Teas-dale, 1982, 1983). The absorbing surface of villi Asm.v =7.8 ± 2.3 m2 is that of small villi (terminal and mature in-termediate) as they correspond to the main site of exchange(Table A.1). Using these data, we finally get w = (2.8± 1.1) ·10−4 m/s. Note also that calculations of Mayhew et al. (1993)show that oxygen diffusive capacity in diabetic pregnancies,although slightly higher, is of the same order of magnitude asin healthy pregnancies.

Radius of the placentone at term (at least 37 com-pleted weeks of pregnancy) (R) A normal placentanumbers 50 ± 10 (Benirschke et al., 2006, p. 161) pla-centones, whereas the average radius of the placental diskis 11 ± 2 cm (Salafia et al., 2012). Division of the disk areaby the number of functional units yields R = 1.6± 0.4 cm. Aplacentone of this radius contains about 1.5 · 105 fetal villi ofradius r. Numerical resolution of the problem for such a largenumber of villi would be too time-consuming. At the sametime, oxygen uptake depends rather on villi density than onthe number of villi. We have thus performed the calculationsfor a smaller radius Rnum = 6 ·10−4 m, and then rescaled oxy-gen uptake to the real size R of the placentone by multiplyingit by R/R2

num.Note that the analysis of the characteristic time scales of the

transfer processes in the human placenta shows that duringthe time spent in a placentone by each volume of blood, thecharacteristic distance of oxygen diffusion in a cross-sectionis of an IVS pore size (about 80µm, see Sect. 2.2.1). Thatmeans that the solution of the diffusion-convection equationin any part of the stream-tube volume that is farther fromits boundary than that distance, will be independent of theshape of the boundary, and hence, any shape of the externalboundary of a placentone can be chosen.

Maternal blood flow velocity (u) Information on thelinear maternal blood flow in the IVS of the human placentain vivo is scarce. To our knowledge, no direct measurementshave ever been conducted. Among the indirect estimations,the following approaches are worth attention:

Table A.2. Radioactive dye diffusion in the human IVS atterm (original data from Burchell, 1967)

Diffusion diameter (2a), cm 6 3.5Dye observation time (tobs), s 30Incoming blood flow per 1 placentone (q), ml/min 24Transit time of blood through the IVS (τtr), s 28Mean velocity of blood in the IVS (u), m/s 6 · 10−4

1. The modeling by Burton et al. (2009) concludes thatblood velocity at the ends of spiral arteries falls downto 0.1 m/s due to dilation of spiral arteries towards term.MBF velocity in the IVS must drop even further;

2. The order of magnitude of the MBF velocity can also beestimated as follows: the total incoming flow of bloodcan be divided by the surface of the whole placentacross-section occupied by the IVS. The total incomingflow is known to be of the order of 600 ml/min (Assaliet al., 1953, Bartels et al., 1962, Browne and Veall, 1953),or 450 ml/min if one considers that 25 % of the flow passesthrough myometrial shunts (Bartels et al., 1962); cross-section area of the placental disk is π(11 cm)2 ≈ 380 cm2;and the percentage of the cross-section occupied bythe IVS is around 35 % (Aherne and Dunnill, 1966, Ba-con et al., 1986, Laga et al., 1973, Mayhew et al., 1986).There is no contradiction between the villi density ofaround 0.5 (main text, Table 2) and the value 0.35, asthe former is the fraction of small villi in a cross-sectioncontaining small villi and IVS only (all other compo-nents subtracted), whereas the latter also accounts forlarge villi and non-IVS components of a placenta cross-section. Finally, for the estimation of the average MBFwe get: u ∼ 450 ml/min/(380 cm2 · 0.35) ∼ 5 · 10−4 m/s;

3. A more accurate estimation of the blood velocity canbe obtained from the results of the experimental studyby Burchell (1967), in which angiographic photographs ofthe human uterus at different stages of pregnancy havebeen obtained by X-rays with a radioactive dye. Thetime during which the radioactive dye was observed in theplacenta (tobs) and the corresponding diffusion diameterof the dye spot (2a) at term were reported (Table A.2).The diffusion diameter is defined as the mean maximaldiameter of the zone of propagation of the radioactivematerial, and the dye observation time (tobs) is the timeelapsed since the appearance of the radioactive dye tillits disappearance from the IVS.

Data in Table A.2 shows that the diameter of the diffu-sion space where the radioactive dye was observed (upto 3.5 cm) is of the order of the placentone diame-ter (2a ≈ 2.5 cm), which justifies the conclusion of the au-

10

thor that in this study, blood propagation in placentoneshas been observed. The dye volume Vdye was reported tobe 50 cm3 in all the experiments. For estimations, we ne-glect the effect of its further dissolution and of increaseof the dye volume and assume that the dye volume isequally distributed between ncot = 50 placentones of theplacenta at term (Benirschke et al., 2006).

From the presented data, one can estimate the totalblood flow in one placentone (q, in ml/min), the IVSblood flow velocity (u) and the time (τtr) that takes ablood volume to pass a placentone. For this purpose, weassume that the IVS has the shape of a hemisphere of ra-diusR, and blood flows uniformly from its center in radialdirections to the hemisphere boundary (this descriptionis confirmed by the “doughnut” shape of dye diffusionclouds reported in the study and corresponds to the cur-rent views, see Benirschke et al., 2006, Chernyavsky et al.,2010). In this case, q = (2πa3/3 + Vdye/ncot)/tobs andthe transit time of maternal blood through the placen-tone is τtr = 2πa3/(3q). The mean velocity of blood inthe placentone can be defined as u = a/τtr.

The results of the calculations of q, u and τtr are pre-sented in the lower part of Table A.2. The average IVSblood flow velocity is estimated to be of the orderof 6 · 10−4 m/s.

The three estimates (u1 < 0.1 m/s, u2 ≈ 5 · 10−4 m/s,u3 ≈ 6 · 10−4 m/s) are compatible, and for calculations weuse the value u = 6 · 10−4 m/s obtained from the third, themost precise method. We emphasize that the MBF veloc-ity may significantly differ between placental regions, and thevalue u = 6 · 10−4 m/s should be understood as a mean valuefor the whole placenta.

Cylinder length (L) The length of the cylinder L is fixedto provide the time of passage (τtr = 27 s) as calculatedfrom Burchell (1967) (Table A.2) if the velocity of the flowu ≈ 6 · 10−4 m/s: L ≈ 1.6 cm (no error estimation available).Note that this value is comparable to, but smaller than the ba-sic estimate as a doubled placenta thickness (for the idealizedcase when blood goes from the basal plate to the chorionicplate and then back) because: (i) a placentone may not oc-cupy all the thickness of the placenta; (ii) the central cavityis likely to have less resistance to the flow and larger bloodvelocities, and should hence probably not be considered as apart of the exchange region; and (iii) blood flow in the ex-change region is not directed “vertically” between the basaland the chorionic plates, but rather from the boundary of thecentral cavity to decidual veins (see Fig. 3C).

B Mathematical formulation

B.1 Interaction between bound and dissolvedoxygen

The very fast oxygen-hemoglobin reaction can be accountedfor by assuming that the concentrations of oxygen dissolvedin the blood plasma (cpl) and of oxygen bound to hemoglo-bin (cbnd) instantaneously mirror each other’s changes. Math-ematically, both concentrations can be related by equatingoxygen partial pressures in these two forms.

Because of a low solubility, the partial pressure of the dis-solved oxygen can be related to its concentration by usingHenry’s law of ideal solution: pO2 = cplkhn/ρbl, where ρbl ≈1000 kg/m3 is the density of blood, and the coefficient khncan be estimated from the fact that a concentration cpl ≈0.13 mol/m3 of dissolved oxygen corresponds to an oxygencontent of 3 ml O2/l blood or a partial pressure of oxy-gen of 13 kPa at normal conditions (Law and Bukwirwa,1999). No error estimations are provided for these data.These estimates give the Henry’s law coefficient: khn ∼ 7.5 ·105 mmHg · kg/mol for oxygen dissolved in blood.

The partial pressure of hemoglobin-bound oxygen dependson its concentration through oxygen-hemoglobin dissociationcurve (the Hill equation):

cbnd = cmaxS(pO2), S(pO2) ≡ (khlpO2)α

1 + (khlpO2)α, (B.1)

with khl ≈ 0.04 mmHg−1 and α ≈ 2.65, that we obtain by fit-ting the experimental curve of Severinghaus (1979), and cmax

is the oxygen content of maternal blood at full saturation.Equilibrium relation between cpl and cbnd can then be ob-

tained by substituting Henry’s law into the Hill equation:

cbnd = cmax · S(cplkhnρbl

). (B.2)

B.2 Diffusive-convective transfer

Diffusive-convective transfer is based on the mass conserva-tion law for the total concentration of oxygen in a volume ofblood:

∂(cpl + cbnd)

∂t+ div~jctot = 0, (B.3)

where ~jctot = −D~∇cpl + ~u(cpl + cbnd) is the total flux of oxy-gen, transferred both by diffusion and convection for the dis-solved form and only by convection (RBCs being too largeobjects) for the bound form; and ~u denotes the velocityof the MBF. Omitting the time derivatives in the station-ary regime, substituting the expression for oxygen flux intoEq. (B.3) and assuming z being the direction of the MBF, weobtain

∆cpl =u

D

∂(cpl + cbnd)

∂z.

Using the relation (B.2) between the dissolved and boundoxygen concentrations, we obtain an equation which containsonly oxygen concentration in blood plasma:

∆cpl =u

D

∂

∂z

(cpl + cmaxS

(khnρbl

cpl

)). (B.4)

This equation is nonlinear as cpl appears also as an argumentof the Hill saturation function S(pO2) (Fig. B.1).

B.3 Linearization of the Hill equation

Data found in the literature show that the maternal blood atthe entrance to the IVS of the human placenta has a pO2

of about 60 mmHg (see calculation of c0 in Appendix A).It is natural to suppose that this pressure is the maxi-mal value in the exchange region. From Fig. B.1, onecan see that in the range [0, 60] mmHg the shape of thedissociation curve can be approximated by a straight line.

11

æææææææææ

æ

æ

æ

æ

æ

æææææææææææææææææ

æ æ æ æ æ æ æ æ

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

Partial pressure of oxygen HpO2L, mmHg

Satu

ratio

nofh

emog

lobi

nHSL

Figure B.1. Oxygen-hemoglobin dissociation curve. On the fig-ure: (i) the dots are experimental data at normal conditions asobtained by Severinghaus (1979); (ii) the curved line shows ourfitting of these data by the Hill equation with the coefficients be-ing α = 2.65, k = 0.04 mmHg−1; (iii) the straight dashed line isour linear approximation in the [0, 60] mmHg region, slope of theline being β60 ≈ 0.0170 ± 0.0003 mmHg−1 (linear fit ± standarderror).

In that case, the 3D equation (B.4) can be converted toa 2D+1D equation which allows for faster computations, whilekeeping the main features of the solution. Such linearizedequation can be also studied analytically. Fitting the ex-perimental curve in the region [0, 60] mmHg by a straightline passing through zero we obtain a linear approxima-tion S(pO2) ≈ β60 pO2 , β60 ≈ 0.017 mmHg−1, with the helpof which the diffusion-convection equation (B.4) transformsinto

∆cpl =u

DB∂cpl∂z

, (B.5)

where B ≡ ctotcpl

= 1 +cmaxβ60khn

ρbl.

The total oxygen uptake is determined as an integral of thenormal component of the total flux of oxygen

~jctot = −D~∇cpl + ~ucplB (B.6)

across the total surface of the fetal villi. Note that the de-scribed linearization procedure is similar to that used for theintroduction of an effective oxygen solubility in blood (Bartelsand Moll, 1964, Opitz and Thews, 1952).

B.4 Boundary conditions

Equation (B.5) should be completed with boundary condi-tions derived from the model assumptions:

• there is no uptake at the boundary of the placentone (ofthe large external cylinder): ∂cpl/∂n = 0, where ∂/∂n isa normal derivative directed outside the IVS;

• oxygen uptake at the feto-maternal interface is propor-tional to the concentration of oxygen dissolved in theIVS: (∂/∂n + w/D)cpl = 0, where w is the permeabilityof the effective feto-maternal interface;

• the concentration of oxygen dissolved in the maternalblood plasma is uniform at the entrance of the streamtube (at z = 0 plane): cpl(x, y, z = 0) = c0, ∀(x, y) ∈SIVS, where SIVS is the maternal part of a horizontalcross-section.

B.5 Numerical solution and validation

Equation (B.5) is then solved numerically in thePDE ToolboxR© in MatlabR© with the help of spectraldecomposition over Laplacian eigenfunctions {vj} in a 2Dsection (Davies, 1996):

cpl = c0∑j

ajvj e−µjL,

F (L) = c0∑j

a2j (Dµj + uB)(1− e−µjL),

where oxygen uptake F (L) at the stream-tube length L iscalculated from the law of conservation of mass. The mesh ofthe 2D geometry of the IVS necessary for an implementationof a finite elements method is created with Voronoı-Delaunayalgorithm (Fig. B.2A). Figures B.2B–B.2D illustrate oxygenconcentration cpl distribution in a 2D cross-section calculatedat three different stream-tube lengths.

Convergence of the results has been tested by varying thenumber of eigenfunctions found and the number of meshpoints. Figure B.3 shows the dependence of the uptake atlength L on the number of eigenfunctions. One can see thata good convergence is achieved with around 3000 eigenfunc-tions for all geometries and for all three lengths shown in thearticle (z = L/3, z = 2L/3 and z = L).

The concentration cpl(r) and oxygen uptake F (L) for thecase of one fetal villus have been compared to the exact ana-lytical solution (Lundberg et al., 1963):

vj(r) = Aj

(J0(√

Λjr)−J1(√

ΛjR)

Y1

(√ΛjR

)Y0

(√Λjr

)),

µj =

√B2u2 + 4D2Λj −Bu

2D,

Aj =

(2π

∫ R

r0

(J0(√

Λjr)

−J1(√

ΛjR)

Y1

(√ΛjR

)Y0

(√Λjr

))2

rdr

)−1/2

,

aj = 2π

∫ R

r0

Aj

(J0(√

Λjr)−J1(√

ΛjR)

Y1

(√ΛjR

)Y0

(√Λjr

))rdr,

and {Λj} are defined as roots of the equation(wDJ0(√

Λjr0)

+√

ΛjJ1(√

Λjr0))

Y1

(√ΛjR

)−(wDY0

(√Λjr0

)+√

ΛjY1

(√Λjr0

))J1(√

ΛjR)

= 0,

Jm(r) and Ym(r) are the m-th order Bessel functions of thefirst and second kind respectively, r0 and R are the radii ofthe small and large cylinders respectively and r is the radialcoordinate. Figure B.4 shows that the relative error betweenthe numerical and analytical oxygen uptake for the case of onevillus is less than 5 % for all physiologically relevant stream-tube lengths.

C Error estimation

The uncertainties of measurement of the parameters of themodel can be used to estimate the error in determinationof the optimal villi density φ0 and the maximal oxygen up-take Fmax. If a variable f depends on several independent

12

(A) (B) (C) (D)

Figure B.2. (A) A typical calculation mesh for N = 80 villi. Plates (B), (C), (D) show oxygen concentration distribution at stream-tubelengths z = L/3, z = 2L/3 and z = L respectively for the same geometry. For this illustration, the incoming oxygen concentration issupposed to be 1, and the color axis represents oxygen concentration in the range [0, 1]. All the figures are plotted with 4500 eigenfunctions.

æ

æ

æ

æ

ææ

ææææææ æ

æ æ ææ æ æ æ

æ æ æ æ æ æ æ æ ææ

à

à

ààà

à

à

àààà à à

à à à àà à à à

à à à à à à àà à

ì

ì

ì

ì

ììì ì ì

ì ì ì ìì ì ì ì ì ì

ì ì ì ì ì ì ì ì ìì ì

ò

ò

ò

ò

ò

òòò ò

ò òò ò

ò òò ò ò

ò ò ò ò ò òò ò ò ò ò ò

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

Eigenfunctions number HnL

FHL�3

;nL�FHL�3

;n m

axL

æ N=1

à N=54

ì N=107

ò N=160

(A)

æ

æ

æ

æ

æææ æ

æ ææ æ æ

æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ

à

à

àààààà à

à à àà à à à à à à à à

à à à à à à à à à

ì

ì

ì

ì

ì ìì ì ì ì ì ì

ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì

ò

ò

ò

ò

òò ò

ò òò ò

ò òò ò

ò ò ò òò ò ò ò ò ò ò ò

ò ò ò

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

Eigenfunctions number HnL

FHL

;nL�FHL

;n m

axL

æ N=1

à N=54

ì N=107

ò N=160

(B)

Figure B.3. Convergence of oxygen uptake F (z;n) at stream-tube lengths z = L/3 (A) and z = L (B) with the number of eigenfunctions nfor different numbers of villi N . The uptake for each geometry is normalized to the value F (z;nmax) obtained for the maximal calculatednumber of eigenvalues nmax = 4500. One can see that the convergence is slower at smaller lengths.

variables xi with known standard deviations δxi, it is knownthat its standard deviation δf has the following form (Taylor,1996):

(δf)2 =∑i

(∂f

∂xiδxi

)2

.

This formula can be applied to estimate the standard devi-ations of φ0 and Fmax due to uncertainties of measurementof the parameters of the model, for which error estimationsare known (B, D, r, R, w, c0). Variations of the parame-

æ

æ

æ

æ

ææææææææææææææææææææææææææ

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

0 5 10 15 20 25

0.0

0.5

1.0

1.5

2.0

Stream tube length, cm

Oxygen

upta

ke,

10-

8m

ol�s

Time of blood passage HΤuL, s

Oxygen

upta

ke,

10-

2cm

3�m

in

Figure B.4. Comparison of oxygen uptake for one villus calcu-lated numerically with 4500 geometrical eigenvalues (points) andanalytically (solid line).

ters cmax, β60, khn are included in the variation of B. Forthe parameters u and L no error estimations are available.Variations δφ0 and δFmax are then:

(δφ0)2 =

(∂φ0

∂BδB

)2

+

(∂φ0

∂DδD

)2

+

(∂φ0

∂rδr

)2

+

(∂φ0

∂RδR

)2

+

(∂φ0

∂wδw

)2

+

(∂φ0

∂c0δc0

)2

,

(δFmax)2 =

(∂Fmax

∂BδB

)2

+

(∂Fmax

∂DδD

)2

+

(∂Fmax

∂rδr

)2

+

(∂Fmax

∂RδR

)2

+

(∂Fmax

∂wδw

)2

+

(∂Fmax

∂c0δc0

)2

.

Since analytical formulas for the dependence of φ0 and Fmax

on the parameters of the model are not available, the par-tial derivatives appearing in the variations δφ0 and δFmax

were calculated numerically. For each variable independently,small deviations were assumed and estimations for the partialderivatives were obtained:

• for φ0: ∂φ0∂B

= −1.6 · 10−3 , ∂φ0∂D

≈ 0 s/m2, ∂φ0∂r

=

−3800 m−1, ∂φ0∂R

≈ 0 m−1, ∂φ0∂w

= 550 s/m, ∂φ0∂c0

≈0 m3/mol;

• for Fmax: ∂Fmax∂B

= 6.8 · 10−9 mol/s, ∂Fmax∂D

≈ 0 mol/m2,∂Fmax∂r

= −1.2 · 10−2 mol/(m · s), ∂Fmax∂R

= 1.4 ·10−4 mol/(m · s), ∂Fmax

∂w= 1.7 · 10−3 mol/m,

∂Fmax∂c0

= 1.7 · 10−5 m3/s.

13

Note that at the numerical precision, φ0 and Fmax appear tobe independent of the diffusivity D of oxygen. Such behaviorcannot be universal for all values of parameters (since D = 0,for example, would obviously give no oxygen uptake), butseems to be approximately valid for the normal placenta pa-rameters (Table 1). Substituting the standard deviation ofthe parameters: δB = 2, δr = 3 · 10−6 m, δR = 4 · 10−3 m,δw = 1.1 · 10−4 m/s, δc0 = 2 · 10−3 mol/m3 (Table 1), weobtain δφ0 = 0.06 and δFmax = 6 · 10−7 mol/s.

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http://dx.doi.org/10.1046/j.1469-7580.2000.19720263.x

http://dx.doi.org/10.1046/j.1469-7580.2000.19720263.x





http://dx.doi.org/10.1016/S0143-4004(86)80003-0


http://dx.doi.org/10.1007/BF02374479

http://dx.doi.org/10.1007/BF02374479





http://dx.doi.org/10.1002/(SICI)1522-2594(200002)43:2<295::AID-MRM18>3.0.CO;2-2

http://dx.doi.org/10.1002/(SICI)1522-2594(200002)43:2<295::AID-MRM18>3.0.CO;2-2



http://dx.doi.org/10.2337/db09-0739

http://dx.doi.org/10.2337/db09-0739



http://dx.doi.org/10.1007/BF02119166

http://dx.doi.org/10.1007/BF02119166






http://dx.doi.org/10.1073/pnas.122352499


http://dx.doi.org/10.1007/978-1-4615-8109-3_2

http://dx.doi.org/10.1007/978-1-4615-8109-3_2



http://dx.doi.org/10.1007/BF00234853








http://packomania.com

http://dx.doi.org/10.1016/j.mehy.2003.10.025

http://dx.doi.org/10.1016/j.mehy.2003.10.025




http://dx.doi.org/10.1016/S0143-4004(83)80012-5

http://dx.doi.org/10.1016/S0143-4004(83)80012-5



http://dx.doi.org/10.3109/14767058.2012.671871

http://dx.doi.org/10.3109/14767058.2012.671871





http://dx.doi.org/10.1079/BJN19470024

http://dx.doi.org/10.1016/S0006-3495(69)86367-8

http://dx.doi.org/10.1016/S0006-3495(69)86367-8




Documents

Road, Manchester M13 9WL, UK January 5, 2015 (Serov et al.,2015) and can be accessed by its doi: 10.1016/j.jtbi.2014.09.022. 1 Introduction The human placenta is the sole organ of