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Modeling the Variability of Shapes of a Human Placenta

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Modeling the Variability of Shapes of a Human Placenta
q
M. Yampolsky
a
,
*
, C.M. Salaﬁa
b
,
c
, O. Shlakhter
d
, D. Haas
e
, B. Eucker
f
, J. Thorp
f
a
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S2E4
b
Department of Psychiatry, New York University School of Medicine. 550 First Avenue, New York, NY 10016, United States
c
Department of Obstetrics and Gynecology, St Luke’s Roosevelt Hospital, New York, NY 10019, United States
d
Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S3G8
e
Department of Pathology, St Luke’s Roosevelt Hospital, New York, NY 10019, United States
f
Department of Obstetrics and Gynecology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States
a r t i c l e i n f o
Article history:
Accepted 16 June 2008
Keywords:
Placental growthPlacental shapePlacental vasculatureFractalDiffusion limited aggregation
a b s t r a c t
Background:
Placentas are generally round/oval in shape, but ‘‘irregular’’ shapes are common. In theCollaborative Perinatal Project data, irregular shapes were associated with lower birth weight forplacental weight, suggesting variably shaped placentas have altered function.
Methods:
(I) Using a 3D one-parameter model of placental vascular growth based on Diffusion LimitedAggregation (an accepted model for generating highly branched fractals), models were run witha branching density growth parameter either ﬁxed or perturbed at either 5–7% or 50% of model growth.(II) In a data set with detailed measures of 1207 placental perimeters, radial standard deviations of placental shapes were calculated from the umbilical cord insertion, and from the centroid of the shape (abiologically arbitrary point). These two were compared to the difference between the observed scalingexponent and the Kleiber scaling exponent (0.75), considered optimal for vascular fractal transportsystems. Spearman’s rank correlation considered
p
<
0.05 signiﬁcant.
Results:
(I) Unperturbed, random values of the growth parameter created round/oval fractal shapes.Perturbation at 5–7% of model growth created multilobate shapes, while perturbation at 50% of modelgrowth created ‘‘star-shaped’’ fractals. (II) The radial standard deviation of the perimeter from theumbilical cord (but not from the centroid) was associated with differences from the Kleiber exponent(
p
¼
0.006).
Conclusions:
A dynamical DLA model recapitulates multilobate and ‘‘star’’ placental shapes via changingfractal branching density. We suggest that (1) irregular placental outlines reﬂect deformation of theunderlying placental fractal vascular network, (2) such irregularities in placental outline indicatesub-optimal branching structure of the vascular tree, and (3) this accounts for the lower birth weightobserved in non-round/oval placentas in the Collaborative Perinatal Project.
2008 Elsevier Ltd. All rights reserved.
1. Introduction
Theplacentaistheprimaryfetalsourceofoxygenandnutrients.As such, it is a principal regulator of fetal growth and fetal health.Typical placentas will grow uniformly out from the umbilical cordinsertion, resulting in a round to oval disk with a centrally insertedcord. A variable maternal uteroplacental environment (the mater-nal ‘‘soil’’) affects macroscopic placental structure as a change inshape.Wherethematernal‘‘soil’’isnotreceptive,placentaswillnotgrow, or not robustly. Irregularities in disk outline, umbilical cordinsertion and in disk thickness are markers of fetal–placentalenvironmental pathology, denoting variable placental arborization,and as such, deformation of normal placental growth resulting inan abnormal placental structure. The microscopic growth of thehuman placenta involves repeated branching, analogous to theroots of a tree; its mature arborization pattern is complex (e.g.,[1–8]), so complex that it cannot be measured reliably even byexpert,dedicatedpediatricpathologists[9,10].Justasthepatternof roots reﬂects the underlying soil’s fertility and predicts the healthof plants that depend on those roots for sustenance, placentalarborization reﬂects the health of the maternal environment andimpacts on fetal health [11].The typical shape of a human placenta is well-understood [12],however, there are many possible deviations from it. Shapes ‘‘otherthan round’’ can be difﬁcult to classify; the Collaborative Perinatal
q
This work was partially supported by NSERC Discovery Grant (M. Yampolsky),by NARSAD Young Investigator Award (C. Salaﬁa), by K23 MidCareer DevelopmentAward NIMH K23MH06785 (C. Salaﬁa).
*
Corresponding author. Tel.:
þ
1 416 978 4637; fax:
þ
1 416 978 4107.
E-mail address:
yampolsky.michael@gmail.com (M. Yampolsky).
Contents lists available at ScienceDirect
Placenta
journal homepage: www.elsevier.com/locate/placenta
0143-4004/$ – see front matter
2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.placenta.2008.06.005
Placenta 29 (2008) 790–797
Project used a variety of terms to attempt to describe such abnor-mal placental shapes but had to resort, after ‘‘bipartita’’ and‘‘tripartite’’ to terms such as ‘‘multiplex’’ to convey the complexityof placental shapes. Such subjective and imprecise terminology hasnot advanced our understanding of the genesis of such shapes, andhas limited our ability to analyze the relationship of abnormalplacental growth shapes to the health of the fetus and child.However,it is evident thatmanyatypicallyshapedplacentasarequite regular, and can be classiﬁed into several well-deﬁned geo-metrical patterns. This regularity suggests that there may bea common underlying pathological mechanism(s) responsible formuchofthevariabilityofobservedshapesofplacentas.Inthisworkwe propose such a mechanism: we derive the variability of pla-centas from a change in the branching structure of their vasculartrees. To demonstrate how this may occur, we introduce a dynamicmodel for the growth of the vascular tree of a placenta. This modelis based on a biologically realistic random growth process. We thenshow how a change of the parameters of the growth at a singleinstance leads to the appearance of the observed variability of placentas.
2. Materials and methods
2.1. Placental cohort
The
Pregnancy, Infection, and Nutrition Study
is a cohort study of pregnantwomenrecruited at mid pregnancyfroman academic health center in centralNorthCarolina. Our study population and recruitment techniques are described in detailelsewhere [13]. Beginning in March 2002, all women recruited into the
Pregnancy,Infection, and Nutrition Study
were requested to consent to a detailed placental ex-amination. As of October 1, 2004, 94.6% women consented to such examination. Of those women who consented, 87.4% had placentas collected and photographed forimage analysis. Of the 1207 consecutive placentas collected, two cases were ex-cluded because the trimmed placental weight was not recorded, and six cases weredelivered in fragments, such that measurement of chorionic plate landmarks wasnot possible. This left 1199 cases for analysis, 99% of the available placental sample.Placental gross examinations, histology review, and image analyses were per-formed at
EarlyPath Clinical and Research Diagnostics
, a New York State-licensedhistopathology facility under the direct supervision of Dr. Salaﬁa. The institutionalreview board from the University of North Carolina at Chapel Hill approved thisprotocol.The fetal surface of the placenta was wiped dry and placed on a clean surfaceafter which the extraplacental membranes and umbilical cord were trimmed fromthe placenta. The fetal surface was photographed with the Lab ID number and 3 cmof a plastic ruler in the ﬁeld of view using a standard high-resolution digital camera(minimumimagesize2.3megapixels).Atrainedobserver(D.H.)capturedseriesof
x
,
y
coordinates that marked the site of the umbilical cord insertion, the perimeter of the fetal surface, and the ‘‘vascular end points’’, the sites at which the chorionicvesselsdisappearedfromthefetalsurface.Theperimetercoordinateswerecapturedatintervalsofnomorethan1 cm,andmorecoordinateswerecapturedifitappearedessential to accuratelycapturing the shape of the fetal surface. The chorionic vesselsextend out from the umbilical cord insertion and, at varying intervals from the edgeof the fetal surface, divebeneath the surface sothat theyareno longervisible. Those‘‘vascular end points’’ mark the terminal differentiation of the chorionic vessels intofetal stems and the ﬁner structures of the placental functional units (P. Kaufman,MD, personal communication).
2.2. Computer software
Numerical simulations of vascular trees were carried out using
dla-3d-placenta
,a Unix-based, ANSI C, 3-dimensional, diffusion-limited aggregation simulationpackage. In its development, we have used Mark Stock’s
dla-nd
arbitrary-di-mensional diffusion-limited aggregationsimulator,a freesoftwaredeveloped underthe terms of the GNU General Public License as published by Free Software Foun-dation. For DLA cluster visualization we have used
PovRay
: a freeware ray tracingprogram available for a variety of computer platforms.
2.3. Random number generators
Random number generators used in this study include:
drand48, an ANSI C system-supplied double-precision linear congruentialuniform number generator, (generates sequence of integers
I
1
,
I
2
,
I
3
,
.
, eachbetween 0 and
M
by the recurrence relation
I
n
þ
1
¼
aI
n
þ
c
(mod
M
)).
ran1(long *idum), uniform random number generator of Park and Millerwith Bays-Durham shufﬂe and added safeguards [14].
ran2(long *idum), long period random number generator of L’Ecuyer withBays-Durham shufﬂe and added safeguards [14].
2.4. Seeing stars (I)
The typical shape of a placenta is round, with the umbilical cord insertionroughlyatthecenter[12],asillustratedinFig.1(leftcolumn).Wealsocalculatedthe
mean
shapeofaplacentalsurfaceinourdataset.Theinsertionpointoftheumbilicalcordwasplacedatthesrcin,andthepointontheperimeterclosesttotheraptureof the amniotic sac was positioned on the negative vertical axis, for consistency. Theperimeters were rescaled to a common size. The points in the perimeter were thenaveraged inside a sector of 18
, thus obtaining 20 radial markers for each placenta.Each of the markers was then averaged over the whole data set, thereby giving 20mean placental radii from the umbilical insertion point spaced at 18
angular in-tervals.Theywerefurtheraveragedtoobtainthemeanaverageplacentalradius.The20 mean radii were within 5% of the average placental radius, with the standarddeviation from the average equal to 1.8%. Thus, the mean shape is round, witha centrally inserted umbilical cord. We aim to explain some of the variability fromthe round/oval shape, concentratingon tworegular shapes which appear prevalent:
Star-shapedplacentas(about5%ofallcases).Atypicalstarhasbetween5and7prominent spikes. Variability of the radius of a star, as measured from theumbilical cord, is typically within 20–30% (Fig. 1, middle column).
Several pronounced lobes. These placentas also account for about 5% of thetotal cases. The number of lobes is typically 2 or 3 (Fig. 1, right column).
We propose to explain this observed variability by a deviation in the growth of the vascular treeof the placenta. In the next sectionweintroducea dynamic growthmodel for a vascular network. We then demonstrate how the model explains theobserved shapes, and present evidence to support our explanation.
2.4.1. A note on the existing models of vascular trees
Numerous
static
models of vascular trees exist in the literature (e.g. see Ref. [15]and references therein). They are usually based on various optimization algorithmsforﬁllingthespatialstructureofanorgan.Suchanapproachis,however,completelyunsuitable for our purposes. We aim to demonstrate how a deviation in the processof the
dynamical
growth of the vascular network affects the macroscopic shape of a placenta. We thus present a
dynamic
model of growth of a vascular tree. Localmodels of angiogenesis exist in previous literature (e.g. see Ref. [16]), however, theydo not take into account the global vascular tree structure and only describea growth of a single ‘‘strand’’ of the tree – which is again unsuitable for us.
2.4.2. Modeling growth at the tips, or a cloud of blind ﬂies
Growth of blood vessels is recognized to depend on the concentration gradientsof appropriate growth factors. Cells situated at the tip of vascular sprouts sense andnavigate the environment, while cells in the sprout stalks proliferate and formavascularlumen[17–19].Howdowemodelthenewgrowthofavasculartreeatthetips of the branches? For simplicity, let us discretize both time and the units of growth. For a three-dimensional tree we should then ask where its tips are, to de-termine where the growth is likely to occur in the next unit of time. While this doesnot appear to be a simple task even for a treewith a few branches, there is a general(and easily automated) method for doing this.Forillustration,consideratree.Atsomedistancefromthetree,releaseacloudof blind ﬂies. Flying randomly, without any sense of direction, our ﬂies will only stopwhentheycomeincontactwiththetree.Atthispoint,aﬂywouldcloseitswings,andsit at the spotwhere the contact occurred. It is a remarkable mathematical fact that:
It is one of the tips of the tree, where a ﬂy is most likely to land
(see Fig. 2).We can thus identify the tips as the places with the most ﬂies, and the relativenumber of ﬂies which have landed at a given extremity will correspond to thelikelihood that the next unit of growth will happen there.Toformulatethisin morepreciseterms,let usselectasa unitof growth(which wewill call a
particle
), a three-dimensional ball of a small diameter
d
. At each moment of time our tree grows by one particle. At the initial time
t
¼
1, we start with a tree
C
1
consisting of a single particle. Thus, at time
n
, our tree is a
cluster C
n
consisting of
n
particles.Togrow it byoneunit,considera largesphere
S
around
C
n
(muchlargerthanthe size of the cluster itself). Randomly select a point
w
in
S
, as the initial position of ablindﬂy.Theﬂywillperforma
randomwalk
(orarandomﬂightinthiscase)srcinatingfrom
w
. That is, it will randomly perform a sequence of moves up/down, left/right,forward/backward. Recordingits spatial positions, we will see a sequence of points
w
0
¼
w
;
w
1
.
;
w
t
1
;
w
t
.
where
w
t
is produced from
w
t
1
by adding an increment
s
t
. The increment alwayshas the same size, comparable to
d
(to ﬁx the ideas, we can take it equal to
d
). Thedirection of the increment has to be aligned with one of the three coordinate axes
M. Yampolsky et al. / Placenta 29 (2008) 790–797
791
(up/down, left/right, forward/backward) and randomly chosen from the sixpossibilities.If at some point of time
t
the point
w
t
is at a distance of less than 2
d
from thecluster
C
n
, then we attach a new particle to this position, and thus obtain a biggercluster
C
n
þ
1
. If before this happens, the particle
w
t
drifts outside of the large sphere2
S
, then we discard it, and select a new initial point
w
.Such randomgrowth, ﬁrst introduced by Witten and Sander in 1981 ([20,21]), isknown to physicists under the somewhat unwieldy name
Diffusion Limited Aggre- gation
, or
DLA
for short.
2.4.3. Visualizing a DLA tree as a vascular network
A tree
C
n
constructed in this way consists of
n
three-dimensional balls, all withthe same small diameter
d
. To visualize it consistently with the structure of a vas-cularnetwork,weneedtomakesomeof thebranchesthicker,andtheotherthinner.We ﬁrst identify its branch structure in the following way. We saythat a particle
x
isan immediate descendant of a particle
y
, if
x
is attached to
y
, and
y
is older than
x
.Further,
x
n
is a descendant of
x
¼
x
0
if there is a chain of DLA particles
x
0
;
x
1
;
.
;
x
n
1
;
x
n
such that
x
i
is an immediate descendant of
x
i
1
for
0
<
i
n
1. Now we can assignavisualization size
v
toeach DLAparticle, consistently with theprinciple that biggervessels grow ﬁrst. Speciﬁcally, let the weight
m
of a particle
x
be the total number of its descendants. Thus, older particles, which are located closer to the root (the veryﬁrst particle
C
1
) of the DLA tree, will have a larger weight. We then assign
v
¼
e
am
with a suitably chosen parameter
a
to visualize the model vascular tree. Changingthe diameter of each DLA particle to its visualization size
v
, we obtain a modelvascular treewith the thickness of branchesvarying consistently with the structure.
Fig. 1.
Placental shapes. Left column: normal round to oval shapes, middle column: ‘‘stars’’, right column: multilobate shapes.
M. Yampolsky et al. / Placenta 29 (2008) 790–797
792
2.4.4. Growing denser trees
Our model has a single parameter 0
<
k
1 which will be completely crucial toour study. It is the probability that a DLA particle will stick to the cluster when itcollides with it. Think again of a blind ﬂy which has encountered one of the tips of the tree. If
k
s
1, thenwith non-zeroprobability 1
k
the ﬂy will continue its travels.Mostlikely,itwillendupsittingatsomenearbypointofthetree,butnotnecessarilyat the tip. The effect of this is to add some diffusion to DLA growth. The smaller thevalue of
k
is, the more ‘‘hairy’’ the resulting tree will become.Frommodelingconsiderations,itisalsousefultonotethatthephysicalsize
d
(asopposed to the visualization size) of a DLA particle can be changed during thegrowth of the cluster. This is a quick and dirty way of changing the branchingstructure.A smaller particlewill also tendtodiffuse along the tips of the DLAclusterformed by larger particles, hence a reduction of
d
will create more branching.
2.4.5. Growing a human placenta with DLA
To model the growth of a placental vascular tree, we surround our growth witha spherical constraint, modeling the walls of the uterus. The constraint absorbs anyDLAparticlewhich hits it fromthe interior, thus ensuring that no growth penetratesthe constraint. The initial cluster marks the insertion of the umbilical cord. It isplaced near the ‘‘south pole’’ of the constraint.To ensure that the initial growth is directed towards the constraint (theumbilical cord does not grow away from the wall of the uterus), it is helpful toselect the initial cluster
C
1
consisting not of one, but of several DLA particles. Thenumber of particles in
C
1
is still insigniﬁcant, compared with the overall size of the DLA growth (in our experiments, it is between 0.01 and 1% of the total). Wehave experimented both with a deterministically deﬁned
C
1
and with a shortinitial cluster given by an opportune DLA growth, with identical results. The totalnumber of DLA particles in a grown model vascular tree is between 150 and 200thousand.Once the parameters, such as the position and the shape of the initial clusterhave been chosen, the model was tested by varying the random number generator(RNG) used to grow DLA. We have tested the model in two ways: by changing the‘‘seed’’ of a given RNG (this changes the random number sequence but not thealgorithmwhich produces it); and bychanging the RNG itself. The results were veryrobust (Fig. 3).
2.5. Seeing stars (II)
To produce models of placentas of irregular shapes we have varied thebranching parameter
k
of the growth at various stages of the development of themodel placenta. A single instance of the change in the value of
k
over the course of growth of the model is required to produce the observed variability (see Fig. 4):
Decreasing the branching parameter
k
(by a multiple 0.01) at an early stage of the growth (after 5–7% of DLA particles have attached) leads to a picture withseveral(typicallytwoorthree)largelobes.Qualitatively,thelargevesselsgrowapart early, and with an increased degree of branching, the tree does notproduce enough ‘‘medium size’’ branches to uniformly ﬁll a circular shape(Fig. 4, middle column).
Well-deﬁned stars with 5–7 pronounced arms correspond to a change of branching at a late stage of growth (typically after 50% of DLA particles haveattached). We have achieved this result by decreasing
k
, and alternatively, bydecreasing the physical size of theDLA particle(Fig. 4, right column).Changingthe physicalsize of a DLAparticle turns out tobe a moreconvenient parameterhere (the size was typically decreased by 20%).As further comparison of model shapes, we calculate the mean distance of asmallvesseltothepointofinsertionof theumbilicalcordin asectorof5
,whichisthen rotated in increments of 2.5
. In each of the 5
sectors, we then mark the pointon the bisector, whose distance to the vertex is equal to the mean distance for thisparticular sector. Connecting these points by line segments, we obtain a diagram,tracing the shape of the model placenta. The diagrams in the bottom row of Fig. 4correspond to the model placentas above them, in the third row of the same ﬁgure.A further evidence for the mechanism of star formation we are describing isfoundinthevariabilityofthicknessofstar-shapedplacentas.Achangeinthedensityof the vascular tree is going to correspond to changing thickness of the placenta.Thus in a ‘‘star’’, the arms will be thicker than the areas between them. In a verticalslice, this would produce a wavy pattern, schematically shown in Fig. 5 (top).In Fig. 5 wedemonstrate vertical slices of placentas taken at 1 cm intervals. Firstis a typical placenta, with slices exhibiting uniform thickness. Next are pictures of slices of stars, with a wavy pattern clearly visible.
2.5.1. Further evidence that variable shapes reﬂect the architecture of the vascular fractal: deviations from the 3/4 rule
The
3/4 rule
(or
Kleiber’s Law
) is a famous allometric scaling law, which postu-lates that the basal metabolic rate
B
and the body mass
M
are related as
B
w
M
3
=
4
:
The appearance of a power
3/4
is explained by the fractal structure of thevascular tree (e.g. see Ref. [22]). The fact that the same power law is observedin many different organisms suggests a universality of vascular architecture.Some explanations of this universality (such as the model in Ref. [22]) haveappeared in the literature.Wehaveveriﬁedthisscalinglawforhumanfetuses,usingtheplacentalweightasaproxyforthebasalmetabolicrate([23]).Thisresultsinanallostericscalingequationlog
½
placental weight
¼
log
a
þ
b
log
½
birth weight
;
(1)inwhich
b
represents the unknownpower. The calculated value of
b
was
0.78
0.02
in an excellent agreement with the rule.OurDLAmodelispremisedontheshapeoftheplacentareﬂectinganunderlyingvascular fractal. It is thus reasonable to expect that a deviation from the typicalround shape of a placenta with a central insertion of the umbilical cord will becorrelated with abnormal placental vascular architecture. The latter should result ina deviation from the normal value of
b
z
3/4.To test this prediction, we have employed several measures of deviation froma round placental shape. Firstly, we have calculated the standard deviation of theradius, measured from the point of insertion of the umbilical cord. For a roundplacenta with a central insertion, this value is
0
. However, this measure will notdifferentiate between a regular ellipse and a star. We have used another simplemeasure, the
roughness
, calculated as the perimeter of the shape divided by theperimeterofthesmallestconvexhullthatcontainedtheshape.Itisequalto
1
foranyconvex shape, such asa circle. Descriptivestatistics determinedthe percentagewithvalues for both parameters within
10%
of the values for a round circle (
0, 1
re-spectively), and for the subsequent deciles of deviation.In Table 1 the results of Spearman’s rank correlations are presented for ourdata set. The variable
OutSdRCrd
is the deviation of the radius from the point of insertion,
OutRough
is the roughness. Both of them are signiﬁcantly correlatedwith the variable
D
b
which is the difference between the computed value of
b
from the scaling Eq. (1), and the ‘‘ideal’’ value
0.75
. The inverse (–) association of the radial deviation and
D
b
means that as the deviation increases the differencebetween
b
observed and
0.75
is greater.Ourinterpretationisconﬁrmedbythelackofcorrelationofthevariable
OutSdR
(thestandard deviation of the radius calculated from the centroid of the placental shape)with the deviation from
0.75
. This is as we predict, since the centroid of the placentalshape is an arbitrarygeometrical center that does not directly relate to the underlyingvasculararchitecture that ramiﬁes out from the umbilical cord insertion site.
3. Conclusion
We have presented a mechanism which accounts for two of the most common patterns of abnormal placental shapes
Fig. 2.
A tree with eight symmetric branches, and the positions where the ﬁrst 100‘‘blind ﬂies’’ land on it, generated by a numerical simulation (marked with dots).
M. Yampolsky et al. / Placenta 29 (2008) 790–797
793
Fig. 3.
Some examples of model vascular trees. On the left is an XY-plane projection, on the right is a YZ-projection. In the YZ-plane the spherical constraint is apparent.
M. Yampolsky et al. / Placenta 29 (2008) 790–797
794

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