Analytical theory of oxygen transport in thehuman placenta

A.S. Serov ∗ 1, C.M. Salafia2, M. Filoche1, and D.S. Grebenkov1

1Laboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, CNRS, 91128 PalaiseauCedex, France

2Placental Analytics LLC, 93 Colonial Avenue, Larchmont, New York 10538, USA

February 5, 2015

Abstract

We propose an analytical approach to solving thediffusion-convection equations governing oxygen trans-port in the human placenta. We show that only twogeometrical characteristics of a placental cross-section,villi density and the effective villi radius, are needed topredict fetal oxygen uptake. We also identify two combi-nations of physiological parameters that determine oxy-gen uptake in a given placenta: (i) the maximal oxygeninflow of a placentone if there were no tissue blockingthe flow, and (ii) the ratio of transit time of maternalblood through the intervillous space to oxygen extractiontime. We derive analytical formulas for fast and simplecalculation of oxygen uptake and provide two diagramsof efficiency of oxygen transport in an arbitrary placen-tal cross-section. We finally show that artificial perfu-sion experiments with no-hemoglobin blood tend to givea two-orders-of-magnitude underestimation of the in vivooxygen uptake and that the optimal geometry for suchsetup alters significantly. The theory allows one to adjustthe results of artificial placenta perfusion experiments toaccount for oxygen-hemoglobin dissociation. Combinedwith image analysis techniques, the presented model cangive an easy-to-use tool for prediction of the human pla-centa efficiency.

Keywords: Diffusion–convection; Stream-tube pla-centa model; Optimal villi density; Transport efficiency;Pathology diagnostics.

This article was published in the Journal of TheoreticalBiology (Serov et al., 2015b) and can be accessed by itsdoi: 10.1016/j.jtbi.2014.12.016.

∗Corresponding author. E-mail:alexander.serov@polytechnique.edu. Corresponding address:Laboratoire de Physique de la Matiere Condensee, Ecole Polytech-nique, CNRS, 91128 Palaiseau Cedex, France. Tel.: +33 1 69 3347 07, Fax: +33 1 69 33 47 99.

1 Introduction

The human placenta consists of maternal and fetalparts (Fig. 1a). The maternal part is a blood basinwhich is supplied by spiral arteries and drained by ma-ternal veins (Benirschke et al., 2006). The fetal part isa villous tree, inside which fetal blood goes from umbil-ical arteries to the umbilical vein through fetal capillar-ies. Maternal blood percolates through the same arbore-ous structure on the outside. Maternal blood and fetalblood do not mix, so the gas and nutrient exchange takesplace at the surface of the villous tree, sections of whichcan be observed in a typical histological 2D placentalslide (Fig. 1b). Modeling and understanding the relationbetween the geometrical structure of the exchange sur-face of the villous tree and the efficiency of the transportfunction of the placenta constitutes the central object ofour study.

Placenta models have been proposed previously (seediscussions in Aifantis, 1978, Battaglia and Meschia,1986, Chernyavsky et al., 2010, Gill et al., 2011). 1Dmodels dealt with oxygen transport at the scale of ei-ther one single villus or the whole placenta, in bothcases imposing a flat exchange surface between mater-nal and fetal blood (Bartels et al., 1962, Shapiro et al.,1967, Kirschbaum and Shapiro, 1969, Hill et al., 1972,1973, Longo et al., 1972a,b, Power et al., 1972a,b, Lard-ner, 1975, Wilbur et al., 1978, Groome, 1991); some 2Dmodels were used to study the co-orientation of mater-nal and fetal flows (Bartels et al., 1962, Metcalfe et al.,1964, Shapiro et al., 1967, Faber, 1969, Kirschbaumand Shapiro, 1969, Guilbeau et al., 1970, Moll, 1972,Schroder, 1982, Battaglia and Meschia, 1986); other mod-els represented the placenta as a porous medium (Erianet al., 1977, Schroder, 1982, Chernyavsky et al., 2010).A lumped element model was also proposed to calcu-late 1D placental diffusing capacity and to relate morpho-metric data to the efficiency of gas transport (see May-hew et al., 1984, 1986 and references therein). To our

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Figure 1. (a): Schematic representation of the human placenta (reproduced from Gray, 1918). Basal plate (maternal side)is at the top, chorionic plate (fetal side) is at the bottom. (b): A small fragment of a typical 2D slide of the human placenta.White space is intervillous space, normally filled with maternal blood, which has been washed away during the preparation ofthe slides (some residual red blood cells are still present). Red shapes are cross-sections of fetal villi. Redder regions insidecorrespond to fetal capillaries and the dark, violet dots at the perimeter are syncytiotrophoblast boundary layers. The sectionshave been taken in the direction from the basal plate to the chorionic plate, and are H&E stained

knowledge, the only 3D placenta model was introducedby Chernyavsky et al. (2010) to study how the positionof venous outlets and the existence of a central cavityinfluences oxygen transport in a hemispherical porous-medium placentone model.

However, none of these models uses fine geomet-rical structure of experimentally obtained placentalslides (Fig. 1b) as direct input. In a recent paperwe introduced a stream-tube placenta model (STPM;Fig. 2b), which is built upon histological placental cross-sections (Serov et al., 2015a) in contrast to previous pla-centa models. In this model, cross-sections of streamtubes of maternal blood flow (MBF) in the intervillousspace of the human placenta were reconstructed fromplacental cross-sections (Fig. 1b) and virtually extendedalong the third dimension. Although successive cross-sections of a stream tube obviously vary in the placenta,this variation cannot be reproduced from a single cross-section and was ignored in this model. Relevant phys-iological and geometrical parameters of the model wereestimated from available experimental data. Numericalsimulations of oxygen transport for identical circular villiwere then performed and showed that the model exhibitsan optimal villi density yielding maximal oxygen uptake.Deviations from these optimal characteristics with vari-ations of model parameters were estimated. The ob-tained optimal villi density (0.47 ± 0.06) corresponds tothat experimentally obtained in healthy human placen-tas (0.46± 0.06).

The present manuscript relies on the same STPM, butprovides the first approximate analytical theory of oxy-gen uptake in the human placenta based on histologicalcross-sections. The present work significantly developsthe results of the previous study by:

• allowing for a fast calculation of oxygen uptake forarbitrary placental cross-sections, while only circularvilli were considered before;

• demonstrating explicit dependence of oxygen uptakeon model parameters and their interrelation, whichcould not be obtained numerically;

• introducing two uptake efficiency indicators forwhich analytical formulas and diagrams are pro-vided;

• showing that accounting for oxygen–hemoglobin re-action is important for interpretation of artificialperfusion experiments (with no-Hb blood) and pro-viding a method of recalculation of the results ofsuch experiments to account for oxygen–hemoglobinreaction.

In the following, the construction of the analytical the-ory is preceded by a full description of the geometricalmodel, physical assumptions and parameters describingthe human placenta.

2 The model

2.1 Model assumptions

Maternal blood arrives into the intervillous space of thehuman placenta by spiral arteries (Fig.2a). It then per-colates through the branching structure of a tree of fetalvilli and leaves the intervillous space by decidual veins.The total pattern of the MBF can be virtually subdividedinto small regions (stream tubes), each following the flowand extending from the central cavity to a decidual vein.

2

chorionic plate

basal plate

central cavity

spiral artery

stream tubes

decidual veins

axis of the model

(a)

u

Fetalvilli

Intervillousspace

(b)

Figure 2. (a) Scheme of placental blood flow and location of stream tubes in the placenta. The dashed line schematicallyoutlines the central cavity. Curved arrows in the right part show maternal blood losing oxygen while going from the centralcavity to decidual veins percolating through the villous tree (shaded region). Curved lines on the left schematically show streamtubes of blood flow. Our model corresponds to one such stream tube unfolded; small straight arrows show the entrance pointsof the model. The exchange is not modeled in the central cavity, but only after it, in the MBF pathway. The concept thatspiral arteries open into the IVS near the central cavity corresponds to current physiological views (Benirschke et al., 2006,Chernyavsky et al., 2010). (b) Our geometrical model of one unfolded stream tube of maternal blood in a human placentone.The cross-section of the stream tube as well as that of the fetal villi inside can be either determined from 2D histological slidesor chosen arbitrary. Red arrows show the flow of maternal blood

Each stream tube comes into contact with numerous fe-tal villi, at the surface of which mass exchange betweenmaternal and fetal blood takes place.

We model one such stream tube unfolded as a cylinderof an arbitrary cross-section containing multiple parallelcylinders of arbitrary cross-sections and sizes, which rep-resent fetal villi (Fig. 2b). The shapes and locations ofthe villi can be taken from a histological slide. Since weaim to base the STPM on histological slides (Fig. 1b)which provide only one stream-tube section withoutany information about the change of this section alongthe MBF, we further postulate that the same shapes andlocations of villi are conserved along the stream tube.This is obviously an oversimplification of the irregular 3Dstructure of the placenta, but it is the most straightfor-ward assumption given the lack of complete 3D geomet-rical data spatially resolving all villi.

The model relies on several other assumptions:

1. Fetal blood is considered as a perfect oxygen sink.

2. MBF is considered to be laminar with slip conditionsat all boundaries (no liquid-wall friction), so that thevelocity profile in any cross-section is flat.

3. The oxygen–hemoglobin dissociation curve is lin-earized in the physiological range of partial pressureof oxygen (0–60 mmHg) observed in the human pla-centa.

4. Oxygen uptake occurs at the feto-maternal interface,i.e. at the boundaries of the small cylinders, and isdirectly proportional to the interface permeabilityand to oxygen concentration on the maternal side ofthe interface.

5. In a cross-section perpendicular to the MBF, oxygenis only redistributed by diffusion.

6. Erythrocytes are uniformly distributed in the ma-ternal blood.

7. Oxygen bound to hemoglobin does not diffuse; onlyoxygen dissolved in the blood plasma does.

8. Oxygen uptake is stationary.

The validity of these assumptions is thoroughly discussedin Serov et al. (2015a). The model also includes geometri-cal and biological parameters, which are listed in Table 1.It will further be shown that fetal oxygen uptake is de-termined by two parameter combinations which naturallyappear in the development of the theory.

3 Mathematical formulation

3.1 Time scales of the system

We identify three different physical transport processesin the placenta, each of which operates on a charac-teristic time scale: hydrodynamic blood flow throughthe IVS characterized by an average velocity u andtransit time τtr; diffusion of oxygen with characteris-tic time τD; and equilibration between oxygen bound tohemoglobin and oxygen dissolved in the blood plasmawith characteristic time τp. This last thermodynamicequilibrium is described as equal partial pressure of oxy-gen in both states. The three times can be estimated asfollows:

• τtr, the transit time of blood through the IVS, isof the order of 27 s from the results of angiographic

3

Table 1. Parameters of the human placenta used in the STPM (see Serov et al., 2015a for the method of calculation of thesevalues)

Parameter Symbol Mean ± SD

Maximal Hb-bound oxygen concentration at 100 % Hb saturation, mol/m3 cmax 7.30 ± 0.11Oxygen-hemoglobin dissociation constant B 94 ± 2Concentration of oxygen dissolved 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 re 41 ± 3Permeability of the effective materno-fetal interface, 10−4 m/s w 2.8 ± 1.1Placentone radius, 10−2 m R 1.6 ± 0.4Velocity of the maternal blood flow, 10−4 m/s u 6Stream tube length, 10−2 m L 1.6

studies at term (Burchell, 1967, Serov et al., 2015a).

• τD, reflecting oxygen diffusion over a length δ inthe IVS is τD ∼ δ2/D, where D is oxygen diffu-sivity in the blood plasma (Table 1). Either fromcalculations (Mayhew and Jairam, 2000), or directlyfrom normal placental sections (Fig. 1b), the meanwidth of an IVS pore can be estimated as δ ∼ 80µm,yielding τD ∼ 4 s.

• τp, an equilibration time scale, which includescharacteristic diffusion time for oxygen to reachHb molecules inside a red blood cell (∼ 10 ms, seeFoucquier et al., 2013) and typical time of oxygen–hemoglobin dissociation (∼ 20 ms for the slowestprocess, see Yamaguchi et al., 1985). Together thesetimes sum up to τp ∼ 30 ms.

These three characteristic times are related as fol-lows: τp � τD . τtr. This relation suggests that oxygen-hemoglobin dissociation can be considered instantaneousas compared to diffusion and convection; the latter two,by contrast, should be treated simultaneously.

3.2 Equilibrium between bound and dissolved oxygen

The very fast oxygen–hemoglobin reaction can be ac-counted for by assuming that the concentration of oxygendissolved in the blood plasma (cpl) and that of oxygenbound to hemoglobin (cbnd) instantaneously mirror eachother’s changes. Mathematically both concentrations canbe related by equating oxygen partial pressures in thesetwo forms.

Oxygen dissolved in plasma. Because of the low sol-ubility of oxygen in blood, the partial pressure of thedissolved oxygen (pO2) can be related to its concentra-tion (cpl) using Henry’s law:

pO2=khnρbl· cpl, (1)

where ρbl ≈ 1000 kg/m3 is the density of blood and thecoefficient khn can be estimated from the fact that a con-centration cpl ≈ 0.13 mol/m3 of the dissolved oxygen cor-responds to oxygen content of 3 ml O2/l blood or partialpressure of 13 kPa at normal conditions (Law and Buk-wirwa, 1999), yielding khn ∼ 7.5 · 105 mmHg · kg/mol forthe oxygen dissolved in blood.

Hemoglobin-bound oxygen. The partial pressure ofthe hemoglobin-bound oxygen depends on its concentra-tion through Hill equation:

cbnd = cmax S(pO2), S(pO2

) ≡ (khlpO2)α

1 + (khlpO2)α, (2)

where cmax is the oxygen content of maternal blood atfull saturation; khl ≈ 0.04 mmHg−1 and α ≈ 2.65 are co-efficients of Hill equation, obtained by fitting the experi-mental curve of Severinghaus (1979, see Fig. 3).

Equilibrium relation between cpl and cbnd can then beobtained by substituting Eq. (1) into Eq. (2):

cbnd = cmax S

(khnρbl

cpl

). (3)

3.3 Diffusive-convective transport of oxygen

Diffusive-convective transport of oxygen is governed bythe mass conservation law for the total concentration ofoxygen in a volume of blood:

∂(cpl + cbnd)

∂t+ div~jctot = 0, (4)

where ~jctot is the total flux of oxygen, transported bothby diffusion and convection for the dissolved form andonly by convection (RBCs being too large objects) forthe bound form:

~jctot = −D~∇cpl + ~u(cpl + cbnd), (5)

where ~u denotes the velocity of the MBF and

−→∇ ≡

{∂

∂x;∂

∂y;∂

∂z

}.

4

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 3. Oxygen–hemoglobin dissociation curve. In thefigure: (i) dots are experimental data at normal conditionsas obtained by Severinghaus (1979); (ii) solid curve showsa fit of these data with Hill equation, the coefficients be-ing α = 2.65, khl = 0.04 mmHg−1; (iii) straight dashed lineis a linear approximation of the curve in the 0–60 mmHgregion as discussed in Section 3.4, slope of the line be-ing β60 ≈ 0.017 mmHg−1

Omitting the time derivative in the stationary regime,substituting Eq. (5) into Eq. (4) and choosing z as thedirection of the MBF, we obtain

∆cpl =u

D

∂(cpl + cbnd)

∂z,

where

∆ ≡ ∂2

∂x2+

∂2

∂y2+

∂2

∂z2

is the Laplace operator. Using the relation (3) betweenthe dissolved and bound oxygen concentrations we thenderive an equation for the unknown cpl only:

∆cpl =u

D

∂

∂z

(cpl + cmax S

(khnρbl

cpl

))=

u

D

(1 +

cmaxkhnρbl

S′(khnρbl

cpl

))∂cpl∂z

.

(6)

This equation is non-linear as cpl appears also in the ar-gument of the derivative of Hill saturation function S′.In a first approximation, S can be linearized by assum-ing S′ to be constant in the range of partial pressures ofoxygen encountered in the human placenta.

3.4 Linearization of Hill equation

The idea of linearization is simple: to replace the sigmoidsaturation function (2) with a linear function of pO2 . Al-though it is natural to make the line pass through theorigin, the slope of the line may be chosen differently de-pending on the range of partial pressures in which weapproximate the curve (Fig. 3). Data found in the liter-ature indicate that maternal blood in the IVS of the hu-man placenta has pO2 of about 60 mmHg (Rodesch et al.,

1992, Jauniaux et al., 2000, Challier and Uzan, 2003).We further suppose that this partial pressure is the max-imal value in the IVS and hence delimits the range of theneeded linear approximation. Fitting the experimentalcurve of Severinghaus (1979) in the region 0–60 mmHgwith a straight line passing through zero we obtain a lin-ear approximation

S(pO2) ≈ β60 pO2

, S′(pO2) = β60 ≈ 0.017 mmHg−1,

displayed in Fig. 3.This approximation leads to the following relation be-

tween ctot, cpl and cbnd: ctot ≡ cpl + cbnd = cplB, or

cpl =cbndB − 1

=cmax

B − 1S(pO2

),

where B ≡ 1 +cmaxβ60khn

ρbl.

We emphasize here that ignoring oxygen-hemoglobin in-teraction would be equivalent to setting B = 1, whichwould lead to a hundred-fold underestimation of this con-stant (Table 1).

From Eq. (5), a linearized version of the correspondingtotal oxygen flux is then

~jctot = −D~∇cpl + ~ucplB. (7)

Finally the partial differential equation (6) becomes

∆cpl =uB

D

∂cpl∂z

. (8)

3.5 Boundary conditions

Boundary conditions should be imposed on Eq. (8):

• the boundary of the large cylinder represents theouter boundary of a stream tube. Assuming thereis no exchange of oxygen between different streamtubes, we consider zero flux on its wall:

∂cpl∂n

= 0,

where ∂/∂n is the normal derivative directed outsidethe IVS;

• uptake at the effective feto-maternal interface is pro-portional to the concentration of oxygen dissolved inthe maternal blood plasma:

D∂cpl∂n

+ wcpl = 0,

where w is the permeability of the interface whichaccounts for the resistance of IVS–villus and villus–capillary membranes as well as for diffusion in theconnective tissue separating the two membranes;

5

• the total concentration of oxygen in blood is uniformand constant at the entrance of the stream tube (z =0):

cpl(x, y, z = 0) = c0, ∀(x, y) ∈ SIVS,

where c0 is oxygen concentration in the incomingblood plasma and SIVS is the part of the stream-tube cross-section occupied by the IVS.

3.6 Conversion to a 2D eigenvalue equation

To solve Eq. (8), we separate the coordinate z along thestream-tube axis from the coordinates x and y in thetransverse cross-section. The general solution of Eq. (8)then takes the following form:

cpl(x, y, z) = c0

∞∑j=1

ajvj(x, y) e−µjz, (9)

where {µj} are decay rates in the z-direction and {aj} areweights of cpl in the orthonormal eigenbasis {vj(x, y)} ofthe Laplace operator ∆xy in the transverse cross-section.{vj(x, y)} satisfy the following equations:

(∆xy + Λj) vj = 0,

∂vj∂n

= 0 on stream-tube boundary,(∂

∂n+w

D

)vj = 0 on villi boundaries,∑

j

ajvj = 1 in the z = 0 plane,

(10)

(11)

(12)

(13)

where ∆xy ≡∂2

∂x2+

∂2

∂y2, Λj ≡ µ2

j + µjuB

D. (14)

Eigenvalues {µj}, eigenfunctions {vj(x, y)} andweights {aj} are determined by Eqs (10)–(14) fora given cross-section. In particular, from Eq. (13) itfollows that aj ≡

∫SIVS

vjdS, when eigenfunctions are

L2-normalized (∫SIVS

v2j = 1).

3.7 General expression for oxygen uptake

According to the mass conservation law, oxygen uptakeup to length L is equal to the difference between oxygenflow coming into the system at z = 0 and oxygen flowleaving the system at z = L. Using Eqs (7) and (9) onecan derive an explicit dependence of oxygen uptake onthe stream-tube length:

F (L) =

∫SIVS

(~jctot · ~nz

)∣∣∣z=0

dS −∫

SIVS

(~jctot · ~nz

)∣∣∣z=L

dS

= c0

∞∑j=1

a2j (Dµj + uB)(1− e−µjL

), (15)

where the definition of {aj} has been used. This is anexact expression for oxygen uptake, into which the ge-ometrical structure of the placental cross-section entersthrough the spectral characteristics {µj} and {aj}. Ourgoal now is to simplify this expression and to identify themost relevant geometrical and physiological parametersthat determine oxygen uptake.

4 Approximate analytical solution

4.1 Form of the approximation

A quick analysis of Eq. (15) shows that F (L) is a smoothmonotonous curve which is linear at small lengths and ex-ponentially saturates at large lengths. In a first approx-imation, Eq. (15) can then be replaced by an expressionwhich has the same behavior at these limits:

F ap(L) = A(1− e−αL), (16)

where A and α are two parameters: A is oxygen uptakeat L → ∞ (equal to the total incoming oxygen flow),and α is the mean decay rate of oxygen concentrationwith stream-tube length. We will now relate A and α tothe parameters of the model.

4.2 Uptake at the infinite length

Large lengths are characterized by saturation when allincoming oxygen is transferred to the fetal blood.

An exact expression for the saturation limit can beobtained from Eq. (15) as L→∞:

F∞ = c0uBSIVS + c0D

∞∑j=1

a2jµj , (17)

where the identity∑∞j=1 a

2j = SIVS, resulting from

Eq. (13), was used.Note that the flow in Eq. (17) includes two contribu-

tions: convective flow (the first term) and diffusive flowalong the z-axis (the second term). It turns out that thesecond term is much smaller than the first one, so thatEq. (17) can be simplified by omitting the diffusive term:

F∞ ≈ A = c0uBSIVS. (18)

This approximation is justified by the following argu-ments:

1. Relative roles of convection and diffusion in a hy-drodynamic problem are described by the Pecletnumber, which is the ratio of characteristic times ofdiffusive and convective transport to the same dis-tance δ: Pe = uδ/D. Large Peclet numbers (Pe� 1)indicate predominant convective transport, whilstsmall values signify prevalence of the diffusive trans-port.

6

For the human placenta, the ratio u/D is of the orderof 105 m−1 (see Table 1), which yields that Pe � 1for lengths δ � 10µm. As the characteristic lengthof the stream tube is L0 ∼ 1.6 cm � 10µm, weconclude that diffusion along the stream tube canbe omitted as compared to convection. At the sametime, diffusion in the cross-section cannot be ignoredas it is the only in-plane mechanism of oxygen trans-port.

2. Since 99 % of oxygen is bound to hemoglobin in redblood cells (RBC), and RBCs are too large to dif-fuse, the error from ignoring the diffusive transportterm in F∞ does not exceed 1 % in terms of oxygencontent.

Mathematically, the simplification we have used can bewritten as

µj � uB/D, (19)

so that Eq. (14) becomes µj ≈ ΛjD/(uB). It should benoted that statement (19) does not contradict with thefact that {µj} grow to infinity with eigenvalue number j.In fact, each µj contributes to the final expression witha weight aj , which diminishes with j. Eq. (19) shouldbe then understood as only valid for all eigenvalues thathave significant contributions aj .

Oxygen uptake (15) can then be approximated as

F (L) ≈ c0uBSIVS

1−∞∑j=1

a2jSIVS

exp

(− D

uBΛjL

) .

(20)

4.3 Average concentration decay rate

Using Eq. (18) and comparing Eq. (20) with the approxi-mate form of oxygen uptake (16), one obtains the follow-ing definition of the average concentration decay rate α:

α(L) ≡ − 1

Lln

∞∑j=1

a2jSIVS

exp

(− D

uBΛjL

) . (21)

In this formula, α depends explicitly on L and implicitlyon the cross-sectional geometry. Our goal now is to ex-tract the main part of this implicit dependence. For thispurpose, we integrate Eq. (10) over the IVS in the cross-section (SIVS). We further apply the divergence theo-rem (Arfken et al., 2005) to transform the integral overthe IVS to an integral over its boundary (Ptot, which in-cludes the perimeter P of the absorbing boundary of thevilli and that of the outer boundary of the stream tube

in a cross-section):

Λj = −∫Ptot

∂vj/∂n dP∫SIVS

vj dS

=w

D

∫Pvj dP∫

SIVSvj dS

=

(wP

DSIVS

)q2j , (22)

where boundary conditions (11) and (12) were used to re-move the contribution from the non-absorbing boundaryand

q2j ≡1P

∫Pvj dP

1SIVS

∫SIVS

vj dS

is the dimensionless ratio of the mean value of the eigen-function vj over the villi boundary to its mean value inthe IVS.

In the first approximation, the coefficient wP/(DSIVS)in Eq. (22) describes the dependence of Λj on the cross-sectional geometry. Introducing qj and the dimensionlesslength `(L) ≡ wP

uBSIVSL into Eq. (21), we transform it into

α =wP

uBSIVSκ(L), (23)

where κ(L) ≡ − 1

`(L)ln

∞∑j=1

a2jSIVS

exp(−`(L)q2j )

is a dimensionless coefficient depending on L and thecross-sectional geometry and containing fine details ofa given villi distribution and shapes, which are ig-nored by the integral parameters P and SIVS. Fig-ure 4a shows the dependence of κ(L) on villi den-sity (φ ≡ 1 − SIVS/Stot = Svil/Stot, the part of thecross-section occupied by fetal villi) as calculated nu-merically for a stream-tube of circular cross-section filledwith circular villi. One can see that in a first approx-imation, κ(L) can be considered as independent of thecross-sectional geometry in a wide range of biologicallyrelevant stream-tube lengths. For each L, the value of κcan be determined from Fig. 4b. For the average stream-tube length (L0 = 1.6 cm, Table 1), κ ≈ 0.35. UsingEqs (18) and (23), the approximate oxygen uptake (16)can then be rewritten as

F ap(L) = c0uBSIVS

(1− exp

(− wPL

uBSIVSκ(L)

)). (24)

4.4 Dimensionless geometrical parameters

In Eq. (24), both geometrical parameters P and SIVS

depend on the size of the analyzed region. To facili-tate physical analysis of the approximate solution andits comparison to experimental data, we identify two ge-ometrical characteristics of a placental cross-section thatare independent of the size of the region:

7

æ

æ

ææ æ æ æ

æ æ æ

æ

à à à àà

àà

à à

à

à

ì ì ìì

ìì

ìì ì

ì

ìò ò

òò

òò ò

òò

ò

ò

ô

ô

ôô ô ô

ô ô

ô ô

ô

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

Villi density HΦL

ΚHLL

æ L = L0 � 10

à L = L0 � 3ì L = 2 L0 � 3ò L = L0

ô L = 10 L0

(a)

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææææææææææ

-2.5 -2.0 -1.5 -1.00.2

0.3

0.4

0.5

0.6

0.7

log10 L, m

ΚHLL

(b)

Figure 4. Dependence of κ(L) on the stream-tube cross-sectional geometry for circular villi in a circular stream tube (Fig. 8).(a): Dependence of κ(L) on villi density φ ≡ Svil/Stot in a large range of stream-tube lengths. Here Svil is the area of the cross-section occupied by fetal villi, Stot is the total area of a cross-section, and L0 is the average stream-tube length (see Table 1).One can see that in the first approximation, κ(L) may be assumed independent of villi configuration. (b): Dependence of κ(L)averaged over φ on the stream-tube length L. Dashed vertical line denotes the average stream-tube length L0

• The fraction of the cross-section occupied by fetalvilli, which we define as the ratio of the total area ofvilli in a cross-section to the total area of the cross-section: φ ≡ Svil/Stot.

• The effective villi radius, which we define as re ≡2Svil/P ≡ 2φSIVS/(P (1 − φ)). In morphometricstudies, the inverse parameter 2/re = P/Svil isknown as “villi surface density”. For circular villiof radius r, re ≡ r. The mean value of re for thehuman placenta can be found in Table 1.

Substitution of these definitions into Eq. (24) gives

ζ(γ, φ) ≡ F ap(γ, φ)

F0

= (1− φ)

(1− exp

(−γ(re, L)

φ

1− φ

)), (25)

where γ(re, L) ≡ 2wκ

uBreL, F0 ≡ c0uBStot. (26)

The physical meaning of F0 follows from its definition: itis the maximal oxygen flow entering a stream tube thatwould be achieved if no villi obstructed the IVS of thestream-tube. The incoming flow in the presence of villiis F∞ ≡ F0(1− φ). Note that γ includes information onthe average villus shape through the parameter re, andthat F0 is independent of the cross-sectional geometry.Normalized oxygen uptake ζ(γ, φ) ≡ F ap/F0 (which wewill call oxygen extraction efficiency) is plotted in Fig. 5a.The physical meaning of γ is discussed later.

4.5 Optimal cross-sectional geometry

Looking at Fig. 5a and expression (25) for oxygen uptake,it is natural to ask what are the “optimal” values of theparameters φ and γ that maximize oxygen uptake at agiven stream-tube length. Figure 5a clearly shows twotrends:

• for any fixed value of φ, larger γ provides larger up-take;

• for any fixed value of γ, there exists some interme-diate value of φ: 0 < φopt(γ) < 1 that maximizesoxygen uptake. This optimal value φopt(γ) tends todiminish with γ.

Note that from the definition (26) it follows that in termsof cross-sectional geometrical parameters, an increaseof γ corresponds to a decrease of the effective radius re.Figure 5a can then be interpreted in the following way:it is more efficient to have many small villi than fewerbig villi occupying the same area. This prediction can beunderstood if one considers the fact that small villi havemore absorbing surface per unit of cross-sectional area.

However, in the human placenta, re cannot be infinitelysmall (and hence γ infinitely large). Indeed, villi possessan internal structure (e.g., fetal blood vessels) to trans-port the absorbed oxygen to the fetus. The decrease of rebelow some value is likely making the villi less efficientin transporting the already absorbed oxygen. This argu-ment is supported by an experimental observation thatin the terminal and mature intermediate villi of the hu-man placenta (the smallest villi), blood vessels normallyoccupy the main part of the internal volume. The meanradius of these smallest villi is r ≈ 25–30µm (see Ta-ble 28.7 in Benirschke et al., 2006) and is not reportedto significantly vary within the same placenta or betweendifferent placentas (Benirschke et al., 2006). At the sametime, villi density may exhibit significant spatial fluctua-tions within the same placenta as well as between differ-ent placentas (Bacon et al., 1986).

It is then reasonable to reformulate the initial ques-tion of optimization of the cross-sectional geometry aswhich villi density provides the highest oxygen uptakefor a given effective villi radius re. Mathematically, it isthe question of finding the maximum of F against φ fora fixed γ(re).

4.6 Optimal villi density

Under the constraint of fixed γ, Eq. (25) implies the ex-istence of maximal uptake at a certain villi density. Thereasoning is the following:

8

++

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-2 -1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

log10 Γ

Φ

ΖHΓ,ΦL º Fap�F0

(a)

++

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-2 -1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

log10 Γ

Φ

Η HΓ, ΦL º Fap�Fmaxap

(b)

Figure 5. Diagrams of oxygen extraction efficiency ζ(γ, φ) ≡ F ap/F0 (a) and villi density efficiency η(γ, φ) = F ap/F apmax (b)

as functions of villi density φ and the dimensionless parameter γ. The plus symbol marks the parameters γ ≈ 1.4 (Sect. 6.1.1)and φ ≈ 0.46 (Serov et al., 2015a) expected on the average in a healthy human placenta (κ = 0.35)

• F (φ = 0) = 0, because the condition φ = 0 meansno feto-maternal interface and hence no uptake.

• F (φ = 1) = 0, because fetal vessels occupy the en-tire cross-section of the stream tube and the incom-ing MBF is zero as it does not have space to flow.

• F > 0 for 0 < φ < 1, which corresponds to the factthat the placenta transfers oxygen from mother tofetus for intermediate villi densities. Hence, therealways exists a maximal oxygen uptake Fmax(L) ata certain villi density 0 < φopt(L) < 1 for any γ(re).

Here we have used F and not F ap symbol for oxygenuptake to underline that these arguments are general andare valid not only for the approximate flow, but for theexact flow as well.

The optimal villi density can then be obtained by solv-

ing the equation ∂F ap(φ,L)∂φ

∣∣∣φ=φopt

= 0,

or exp

(γ

φopt1− φopt

)= 1 +

γ

1− φopt. (27)

One can note that Eq. (27) is an explicit equationfor φopt as a function of γ only. As a consequence, itdoes not require any eigenvalues calculation.

The substitution x ≡ γφopt/(1−φopt) reduces Eq. (27)to the form γ = g(x), whereg(x) ≡ ex−x − 1, and its solution can be representedas x = g−1(γ). Although the inverse function g−1(γ)does not have an explicit representation, its form can beeasily calculated once and then the tabulated values can

be used in practice. Returning to the definition of x, oneobtains φopt as a function of γ:

φopt(γ) =1

1 +γ

g−1(γ)

. (28)

The function φopt(γ) (Fig. 6a) can be used to calcu-late an optimal villi density for a placental region if γis known for this region. Substitution of the last resultinto Eq. (25) gives the corresponding maximal uptake:

F apmax(γ)

F0= γ

1− exp(−g−1(γ))

γ + g−1(γ). (29)

From the asymptotic behavior of g(x) at small andlarge x, we obtain the following asymptotic formulasfor φopt and F ap

max/F0:

φopt(γ) '

1

1 +√γ/2

, γ � 1

ln(γ)/γ, γ � 1

(30)

F apmax(γ)

F0'

1− e−

√2γ

1 +√

2/γ, γ � 1

1− 1/γ

1 + ln(γ)/γ, γ � 1

(31)

Figure 6 shows that these asymptotics accurately ap-proximate φopt(γ) and F ap

max(γ)/F0 not only in the limitsof γ � 1 and γ � 1, but for all γ. For instance, it canbe calculated that if small-γ asymptotic is used for γ 6 3and large-γ asymptotic is used for γ > 3, the maximal

9

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è small-Γ

asymptotics

ò large-Γ

asymptotics

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

log10 Γ

Φopt

(a)

èèèèèèèèèèèèèèèèèèèè

è

è

è

è

è

èè

ò

ò

ò

ò

ò

ò

òòòòòòòòòòòòòò

è small-Γ

asymptotics

ò large-Γ

asymptotics

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

log10 Γ

Fm

ax

ap�F

0

(b)

Figure 6. Analytical predictions of the optimal villi density (a) and of the normalized maximal uptake (b) as functions of γ(solid lines). Small γ asymptotics are shown by circles; large γ asymptotics are shown by triangles. Dashed lines marks γ ≈ 1.4observed in a healthy human placenta (see Sect. 6.1.1).

relative error of the second formula of Eq. (31) is lessthan 4 %. In other words, we obtained simple explicitapproximations for the optimal villi density φopt and thenormalized maximal oxygen uptake F ap

max/F0 as functionsof a single parameter γ.

4.7 Villi density efficiency

Basing on the optimal villi density and the maximal up-take introduced in the previous section, one can definea quantitative measure of optimality of villi density in agiven (not optimal) geometry.

If the given geometry is characterized by the param-eters (γ, φ), its villi density efficiency can be defined asthe ratio of oxygen uptake in this particular geometryto the maximal value, which can be obtained with thesame γ (Fig. 6b):

η(γ, φ) ≡ F ap(γ, φ)

F apmax(γ)

=(1− φ)

(1− exp

(−γ φ

1−φ

))(1− φopt(γ))

(1− exp

(−γ φopt(γ)

1−φopt(γ)

)) . (32)

Figure 5b presents the villi density efficiency η(γ, φ) in aphysiological range of (γ, φ) and at L = L0.

Following the comment after Eq. (31), Eq. (32) can berewritten as

η(γ, φ) '

(1− φ)(

1− exp(−γ φ

1−φ

))(1 +

√2/γ

1− e−√2γ

, γ 6 3

(1− φ)(

1− exp(−γ φ

1−φ

))(1 + ln(γ)/γ)

1− 1/γ, γ > 3

with a maximal relative error of 4 %. Note that thelast equation does not require calculation of φopt and isan explicit function of γ and φ.

Note finally that the optimality indicators ζ and η playdifferent roles. Oxygen extraction efficiency ζ (Fig. 5a)

indicates the fraction of the maximal possible incomingoxygen flow F0 that is absorbed by a given cross-sectionalgeometry. The higher is the value of ζ, the higher is theabsolute value of fetal oxygen uptake. At the same time,villi density efficiency η (Fig. 5b) shows how far the villidensity of a given cross-section is from its optimal valuefor a fixed γ(re). The higher is the value of η, the closeris fetal oxygen uptake to the maximal value for the givenvilli radius re.

5 Results

Figure 7 shows that fetal oxygen uptake predicted bythe analytical Eq. (25) agrees well with numerically cal-culated results (Serov et al., 2015a) in wide ranges ofstream-tube lengths (L) and villi densities (φ). Fig-ure 7a demonstrates the existence of maximal oxygenuptake corresponding to an optimal villi density for eachstream-tube length. The value of κ(L) was determinedfrom Fig. 4b for each considered length L. These resultswere calculated for the same geometries as in our earliernumerical simulation (Fig. 8). We emphasize that nu-merical simulations with identical circular villi are shownonly for the purpose of validation. The proposed analyt-ical theory, which uses only villi density and the effectivevilli radius (re) as geometrical information, is applicableto villi of arbitrary shapes and sizes. The agreement ofanalytical curves with numerical points shows that foruniform villi distributions, knowing villi density and theeffective villi radius is enough to predict the oxygen up-take.

Variations of the parameter γ in analytical ex-pressions for optimal villi density and maximal up-take (Eqs (28), (29)) can be interpreted in terms ofchanges of individual parameters of the model, other pa-rameters being fixed. For example, optimal villi densityand maximal uptake can be plotted as functions of MBFvelocity (Figs 9a, 9c) or stream-tube length (Figs 9b

10

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ààà

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ììì

æ L=L0

à L=2L0�3

ì L=L0�3

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

12

0 5 10 15 20 25

0.0

0.3

0.5

0.8

1.0

1.3

1.5

Villi density

Oxygen

upta

ke,10-

7m

ol�

s

Blood inflow per placentone, cm3�min

Oxygen

upta

ke,cm

3�m

in

(a)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææææææææææææææææææ

à

à

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à

àààààà à

à àà à

à à àà à à à

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ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òò

æ N=80, Φ=0.37

à N=160, Φ=0.75

ì N=54, Φ=0.25

ò N=1, Φ=0.005

0.0 0.5 1.0 1.5

0

2

4

6

8

10

0 5 10 15 20 25

0.0

0.3

0.5

0.8

1.0

1.3

1.5

Stream-tube length, cm

Oxygen

upta

ke,10-

7m

ol�s

Transit time of blood HΤtrL, s

Oxygen

upta

ke,cm

3�m

in

(b)

Figure 7. Oxygen uptake of a single placentone as a function of geometrical parameters. Solid lines correspond to theanalytical approximation; symbols reproduce the results of numerical simulations of Serov et al. (2015a). (a) Oxygen uptakeas a function of villi density φ for three lengths L: L0/3, 2L0/3, and L0. For each of these lengths, the corresponding valueof κ was determined from Fig. 4b: κ ≈ {0.46, 0.39, 0.35} for L = {L0/3, 2L0/3, L0} respectively. MBF velocity u = 0.6 mm/swas used. It can be observed that peak uptake moves to smaller villi densities for larger stream-tube lengths L. Note that inthe analytical theory oxygen uptake is calculated directly for the radius R, whereas in the numerical simulation oxygen uptakeis calculated for Rnum and then rescaled to the placentone radius R by multiplying by R2/R2

num. Note also that the numericalcurves do not go beyond the villi density φ ≈ 0.75 because there exists a maximal packing density of circles in a large circle, andnumerical results cannot be calculated beyond that density (see Specht, 2009). The analytical theory, on the contrary, does notrely on particular shapes or distributions of villi, but operates only with villi density and the effective villi radius, thus allowingthe results to be calculated for villi densities beyond this limit (although in the region of φ > 0.75 villi cannot be circular, thesame re is maintained). (b) Oxygen uptake as a function of stream-tube length L for a fixed villi density φ. Small deviationsof the theory from the numerical results seen in the figure are explained by the fixed κ = κ(L0) ≈ 0.35 used for all lengths.Various symbols represent villi densities of Fig. 8

and 9d). An agreement between the plotted curves andnumerical results of Serov et al. (2015a) can be observed.

All four plots in Fig. 9 feature a dashed black curve rep-resenting a fictitious case of blood having no hemoglobinbut transporting only oxygen dissolved in the bloodplasma. Mathematically, this case is described byoxygen-hemoglobin dissociation parameter B = 1, whichis about 100 times smaller than that for blood with Hb.As predicted by Eq. (28), the no-Hb curves for optimalvilli density have the same shape but are shifted by twoorders of magnitude as compared to those for normalblood.

6 Discussion

6.1 Parameters γ and F0

6.1.1 Values

Taking πR2 as the total area of the cross-section, pa-rameters from Table 1 and κ(L0) ≈ 0.35 for the averagestream-tube length L0 ≈ 1.6 cm (Fig. 4b), from Eq. (26)one can estimate the values of γ and F0 which character-ize a “healthy” regime of our placenta model: γ ≈ 1.4,F0 ≈ 3 · 10−6 mol/s. The obtained average value of γtogether with the average villi density φ ≈ 0.46 (Serovet al., 2015a) are marked by crosses in the diagrams inFig. 5. One can see that the theory predicts that an

average placenta extracts around 35 % of the maximalpossible incoming oxygen flow F0 (Fig. 5a), and that thisvalue is close to the maximal one for the given effectivevilli radius (Fig. 5b). Although 35 % seems to be a lowvalue, note that oxygen extraction efficiency of 100 % isnever achievable since in the presence of villi only a partof the flow unobstructed by villi (F0) can be transferredto the fetus.

To have predictive power, γ and φ need to be mea-sured for different healthy as well as pathological placen-tas over the whole exchange region. Such measurementsrequire development of image analysis techniques, whichcould automatically determine these characteristics forhistological placental slides. Such measurements havenot yet been performed and present an important per-spective to this study. At the same time, because of thelack of experimental information about several other pa-rameters (namely, u, w, L) in each studied placenta, cor-relations of changes of γ and φ with changes of fetal de-velopment characteristics (such as birth-weight, placentaweight or their ratio) are expected to be of more practi-cal use than absolute values of γ and φ. Note finally thatthe optimal geometry and maximal uptake may changefor non-slip boundary conditions; further studies are re-quired to clarify this point (for discussion see Serov et al.,2015a).

11

N=1Φ=0.005

N=54Φ=0.25

N=80Φ=0.37

N=160Φ=0.75

Figure 8. Villi distributions for which oxygen uptake was calculated in Serov et al. (2015a). Number of villi (N) and thecorresponding villi density (φ) are displayed above each case. The analytical theory was applied to the corresponding φ with regiven in Table 1. Maternal blood flows in the white space in the direction perpendicular to the cross-sections.

6.1.2 Parameter γ

The two parameters γ and F0 play different roles. Ac-cording to Eq. (27), γ alone determines the optimal villidensity, while F0 together with γ determines the maximaloxygen uptake (Eq. (29)).

A clear physical interpretation of γ can be obtained byrewriting (26) as

γ =L/u

Bre/(2wκ)=τtrτe,

where τtr ≡ L/u is the transit time of maternalblood through the placenta (while it flows along astream tube of length L with an average velocity u)and τe ≡ Bre/(2wκ) is oxygen extraction time of aplacental cross-section. As a consequence, γ can beunderstood as a quantitative measure of balance be-tween two oxygen transport mechanisms: the longitudi-nal convective flow and the transverse diffusion. In otherwords, γ describes the level of adaptation of the geometryof the cross-section and uptake parameters to the incom-ing MBF. Large values of γ (γ � 1) mean that oxygen isquickly transferred to the fetal circulation at the begin-ning of the stream-tube and is rapidly depleted, so thatpoor in oxygen maternal blood flows through the remain-ing part. Thus, this remaining part does not functionefficiently. Small values of γ (γ � 1) mean that ma-ternal blood passes too quickly through the placenta ascompared to the oxygen extraction time, so that a con-siderable part of the incoming oxygen flow may not betransferred. One can then speculate that transport ofoxygen is the most efficient in the placentas, for which γis of the order of 1. γ ≈ 1.4 calculated from the modelparameters suggests that a healthy placenta may indeedfunction optimally.

6.2 The analytical theory

The advantages of the analytical solution over the nu-merical one are numerous:

1. Oxygen uptake can be estimated for a histologicalcross-section of arbitrary geometry.

2. The villi density φ and the effective villi radius re areshown to be the only geometrical parameters neces-

sary to predict oxygen uptake of a rather uniformvilli distribution in a placental cross-section (seeFigs 7, 9). These two parameters allow for a sim-ple application of the theory to distributions of villiof arbitrary shapes. The validity of the theory inthe case of strongly non-uniform villi distributionsremains to be investigated.

Finer details of villi distributions which produce dif-ferences between numerical and analytical results inFigs 7 and 9, are “stored” in the coefficient κ. Thiscoefficient encompasses not only the details of villidistributions, but also determines the strength oftheir influence on oxygen uptake at a given length L.In other words, it quantitatively describes the factthat in each geometry, different regions of the IVSare not equivalent due to random distribution of villi,and that with length L, oxygen in some regions isexhausted faster than in other regions. However, thedependence of κ and α on L is rather weak as canbe seen in Fig. 7b, in which F ap(L) is plotted for alllengths with the same κ = κ(L0) ≈ 0.35. In the firstapproximation, κ can hence be considered indepen-dent of L.

3. The efficiency of oxygen transport in a given pla-cental cross-section can be estimated by means ofoxygen extraction efficiency ζ and villi density effi-ciency η plotted in Fig. 5. For these two quantities,simple analytical formulas and diagrams are pro-vided, which allow for comparison of different pla-centas or placental regions once the parameters φand γ are calculated for them. To have predictivepower, the efficiency estimations provided by themodel need to be studied for correlations with inde-pendent indicators of placental exchange efficiency,such as placenta shape, placenta weight, placenta-fetus birth-weight ratio (Misra et al., 2010, Hutcheonet al., 2012) or pulsatility indices determined byDoppler velocimetry in the umbilical cord or mater-nal vessels (Todros et al., 1999, Madazli et al., 2003),which were demonstrated to vary between normaland pre-eclamptic pregnancies or pregnancies com-plicated by fetal growth restriction.

4. The analytical theory suggests that oxygen uptake

12

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æ æ ææ æ

æ æ

æ æ ææ æ æ

æ æ æ æ

-3.8 -3.6 -3.4 -3.2 -3.0 -2.8 -2.60.0

0.2

0.4

0.6

0.8

1.00.81.01.21.41.61.82.0

Log10 Blood velocity, m�s

Optim

alvilli

den

sity

Log10 Transit time of blood HΤtrL, s

(a)

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0

5

10

15

20

25

Log10 Stream-tube length, m

Opt

imal

villi

dens

ity

Log10 Transit time of blood HΤtrL, s

Blo

odin

flow

,cm

3 �min

(b)

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1.0

1.5

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3.00.81.01.21.41.61.82.0

0

1

2

3

4

Log10 Blood velocity, m�s

Max

imal

upta

ke,

10-

6m

ol�s

Log10 Transit time of blood HΤtrL, s

Oxygen

upta

ke,

cm3�m

in

(c)

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0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

Log10 Stream-tube length, mM

axim

alupta

ke,

10-

6m

ol�s

Log10 Transit time of blood HΤtrL, s

Max

imal

upta

ke,

cm3�m

in

(d)

Figure 9. (a), (c): Dependence of the optimal villi density (a) and the maximal oxygen uptake (c) on the MBF velocity ata fixed length L0 (Table 1) for a single placentone. (b), (d): Dependence of the optimal villi density (b) and the maximaloxygen uptake (d) on stream-tube length at a fixed MBF velocity u (Table 1) for a single placentone. Solid curves representanalytical results, while filled circles correspond to numerical simulations for circular villi (Fig. 8). Dashed curves show analyticalresults in blood with no hemoglobin (B = 1, as in artificial perfusion experiments). Straight dashed lines indicate the expectedaverage MBF velocity u and stream-tube length L0 (Table 1). Step growth of numerical results seen in Figs (a), (b) is explainedby discrete changes of villi density in the numerical simulations due to discrete changes in the number of villi in the cross-section

in the human placenta is rather robust to changes ofvilli density. Indeed, the diagram in Fig. 5a showsthat placental villi density can vary by about 10 %around the optimal value with the villi density effi-ciency η staying in the 90–100 % interval. Far fromthe optimal villi density, η tends to decrease faster.

5. One can analyze the consequences of neglectingoxygen-hemoglobin reaction on the predictions ofoxygen uptake and the optimal villi density of aplacental region. Moreover, the theory gives amethod of recalculation of the results obtained forno-Hb blood in artificial placenta perfusion exper-iments into those for normal blood. Imagine thatat the end of an artificial perfusion experimentwith no-Hb blood, one obtains the total oxygen in-flow Fin into the placenta, fetal oxygen uptake Fand the average villi density φ from histomorphom-etry of the same placenta (note that Fin and Fdiffer from Fin and F which would have been ob-tained for normal blood). From these data one can

calculate F0 = Fin/(1 − φ) (see the discussion ofEq. (26)) and then γ as a root of Eq. (25) (with F ap

replaced by F ). These values can be recalculated

for blood containing Hb: γ = γ/B and F0 = F0B,where B ≈ 94 (Table 1), and can be substituted intoEq. (25) to give oxygen uptake F in the same pla-centa for blood containing Hb. One can see that oxy-gen uptake in a no-Hb perfusion experiment gives onthe average a hundred-times underestimation of thereal uptake. Finally, the values of γ and φ for thegiven placenta can be compared with the diagramin Fig. 5b to determine how far the geometry of theregion is from the optimal one. Note that this re-calculation introduces a small error as in no-Hb casethe diffusive part of the total flow, which is omittedin Eq. (17), becomes important;

6. The computation time is reduced since calculationsof eigenfunctions and eigenvalues of the diffusionequation are not required. Note that due to longcomputation time, numerical simulations of Serovet al. (2015a) had to be performed on a smaller pla-centone radius Rnum and then rescaled to the ra-dius R by multiplying oxygen uptake by R2/R2

num.This constraint does not apply to the analytical the-ory. In particular, good agreement between both ap-proaches justifies the rescaling of results performed

13

(a) (b)

Figure 10. Illustration of the difference between the total villous perimeter and the effective absorbing villous perimeter. (a)Example of an isolated group of villi. In the shaded villi group, some parts of absorbing villi boundaries are inefficient as theyare screened by other villi from the outside of the isolated group, where the main reservoir of oxygen is supposed to be. Theeffective absorbing perimeter of the group is close to the perimeter outlined by the dashed line. This perimeter of the IVSsurrounding the villi can be several times smaller than the total perimeter of the villi in the shaded group. Note also thatadding a new villus into such group (the villus outside the shaded group) does not increase the effective absorbing perimeterproportionally to the increase of the total villous perimeter. In the example in the figure, the new perimeter (the dotted contour)has approximately the same length as the old one. (b) Illustration of the same concept in a histological slide of the humanplacenta. Two dashed contours show the effective absorbing perimeters of two groups of villi, which are considerably smallerthan the total perimeters of villi in the groups. Discussion of similar screening concepts can be found in Sapoval et al. (2002),Felici et al. (2005), Gill et al. (2011)

in the numerical calculations.

Note finally that the derivation of Eq. (26) implies that,strictly speaking, P is not the total perimeter of the villi,but the effective absorbing perimeter of the villi (i.e. onlyits part that is directly in contact with the IVS). In thecase of well-separated villi, there is no difference betweenthe two definitions. However, it is not always the casein placental cross-sections. For instance, in Fig. 10b onecan see several isolated groups of villi, inside which villilie so close to each other, that there is virtually no IVSleft between them. Parts of the villous boundary whichare not in contact with large parts of the IVS are thenscreened from participating in oxygen uptake, and henceshould not be accounted for in the effective absorbingperimeter of the villi. This remark can be understood byconsidering the fact that oxygen diffuses in the IVS, andonly parts of the villous boundary that are in contactwith the IVS will participate in the uptake. In the caseof well separated singular villi, the entire perimeter isabsorbing. A schematic description of this situation isshown in Fig. 10a. Note finally that in our model thescreening effect is implicitly taken into account by thediffusion equation (10).

7 Conclusions

In the present work, an analytical solution to thediffusion-convection equation governing oxygen transportin the human placenta has been developed. Oxygen up-take was calculated for an arbitrary cross-sectional geom-etry of the stream tubes of maternal blood. It was shown

that for a rather uniform spatial distribution of villi in aplacental cross-section, only two geometrical characteris-tics, villi density φ and the effective villi radius re, areneeded to predict fetal oxygen uptake.

It was also demonstrated that all the parameters of themodel do not influence oxygen uptake independently, butinstead form two combinations: (i) the maximal oxygeninflow of one placentone F0, and (ii) the ratio γ of thetransit time of maternal blood through the IVS and theoxygen extraction time. These two parameters togetherwith villi density determine oxygen uptake. Analyticalformulas and diagrams were obtained which allow foroxygen uptake calculation and quantitative estimationof the efficiency of oxygen transport of a given placentalregion based on measurements of φ and re.

Finally, a fictitious case of blood containing nohemoglobin was analyzed to study oxygen transport inartificial placenta perfusion experiments. It was demon-strated that artificial perfusion experiments with nohemoglobin tend to give a two-orders-of-magnitude un-derestimation of the in vivo oxygen uptake. A methodof recalculation of the results of artificial perfusion ex-periments to account for oxygen-hemoglobin dissociationwas proposed.

Once combined with image analysis techniques, theproposed analytical theory can be the mathematicalground for a future tool of fast diagnostics of placentaefficiency based on histological placental slides.

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8 Acknowledgements

The authors thank Dr Paul Brownbill for fruitful discus-sions.

This study was funded by the International RelationsDepartment of Ecole Polytechnique as a part of the Ph.D.project of A.S. Serov, by Placental Analytics LLC, NY,by SAMOVAR project of the Agence Nationale de laRecherche n◦ 2010-BLAN-1119-05 and by Agence Na-tionale de la Recherche project ANR-13-JSV5-0006-01.

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