Probability distributions for measures of placental shape and morphology
Joshua S. Gill, Mischa P. Woods, Carolyn M. Salafia, Dimitri D. Vvedensky
PProbability distributions for measures of placental shapeand morphology
J. S. Gill a , M. P. Woods a , C. M. Salafia b , D. D. Vvedensky a a The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom b Placental Analytics LLC, 93 Colonial Avenue, Larchmont, New York 10538
AbstractGoal.
Weight at delivery is a standard cumulative measure of placentalgrowth. But weight is a crude summary of other placental characteristics,such as the size and shape of the chorionic plate and the location of the um-bilical cord insertion. Distributions of such measures across a cohort revealinformation about the developmental history of the chorionic plate that isunavailable from an analysis based solely on the mean and standard devia-tion.
Methods & Materials.
Various measures were determined from digitizedimages of chorionic plates obtained from the Pregnancy, Infection, and Nu-trition Study, a prospective cohort study of preterm birth in central NorthCarolina between 2002 and 2004. The centroids (the geometric centers) andumbilical cord insertions were taken directly from the images. The chori-onic plate outlines were obtained from an interpolation based on a Fourierseries, while eccentricity (of the best-fit ellipse), skewness, and kurtosis weredetermined from a shape analysis using the method of moments. The dis-tribution of each variable was compared against the normal, lognormal, andL´evy distributions.
Results.
We found only a single measure (eccentricity) with a normal dis-tribution. All other placental measures required lognormal or “heavy-tailed”distributions to account for moderate to extreme deviations from the mean,where relative likelihoods in the cohort far exceeded those of a normal dis-tribution.
Conclusions.
Normal and lognormal distributions result from the accumu-lated effects of a large number of independent additive (normal) or multi-plicative (lognormal) events. Thus, while most placentas appear to developby a series of small, regular, and independent steps, the presence of heavy-
Preprint submitted to Placenta October 29, 2018 a r X i v : . [ q - b i o . Q M ] S e p ailed distributions suggests that many show shape features which are moreconsistent with a large number of correlated steps or fewer, but substantiallylarger, independent steps. Keywords: placental measures, chorionic plate, shape, distributions
1. Introduction
The placenta is the interface across which all oxygen and nutrients areexchanged between mother and fetus. Understanding the development andfunction of the human placenta is crucial to gaining insight into the environ-ment of the developing fetus, whose health is thought to be an importantinfluence on childhood and lifelong health [1].The placenta is conventionally thought to develop uniformly outward fromthe site of the umbilical cord insertion, leading to an approximately circularshape. However, while circular placentas are infrequently observed, recentwork [2] has suggested that the “average” placental shape within a cohortis, in fact, close to circular, though there remains some debate on this is-sue [3, 4]. The ability of the chorionic late to extend laterally uniformlyoutward from the cord insertion is due, in part, to the suitability of thematernal uteroplacental environment. Any deficiencies in that environmentcan have adverse effects on placental and, by extension, fetal development.Consequently, the analysis of the deviations of mature placental chorionicsurface shapes from “regularity” (circular, or otherwise) can provide infor-mation about the uterine environment and possibly provide indicators aboutthe health of the child.The structure of the mature placenta is geometrically complex. The um-bilical cord is usually attached near the center of the placenta, but this is notalways the case; eccentric, marginal and velamentous cords inserted onto theextraplacental membranes are not rare [5]. From the point of the cord inser-tion onto the chorionic plate, the fetal chorionic vascular system branches andspreads laterally across the chorionic plate. At later stages in the branchingand extension of chorionic surface vessels, veins and arteries dive down intothe placenta and continue branching to contribute to disk thickness. Thechorionic surface outline of the delivered placenta is a culmination of lateralplacental vascular growth.A mature placenta can take many different shapes, from near-circular tomulti-lobed to star-shaped. There is little or no explanation as to why such2ariations of placental shapes exist, apart from “trophotropism” [6–8], anargument which says, in effect, that “the placenta grows where it can and doesnot grow where it cannot”. There is no data about whether shape variationsare associated with particular complications or subsequent health problems.One of the main goals of the present study is to understand the genesis,development, and evolution of mature placental chorionic surface shape fromthe distributions of various measures of placental shape. The shape of aplacental chorionic surface or, indeed, any two-dimensional object, can becharacterized by area, perimeter, compactness (perimeter squared dividedby area), eccentricity (of a bounding ellipse), elongation and rectangularity(of a bounding box), etc. In addition, the chorionic plate outline can beanalyzed in terms of its “roughness” and “correlation”, both of which arestandard measures used in the statistical analysis of rough surfaces [9]. Theeccentricity and orientation of the best-fit ellipse, skewness, and kurtosis canbe calculated from the lower-order moments of the chorionic plate [10, 11],and the distance between the umbilical cord insertion and the centroid, whichprovides an indication about how the placenta developed with respect to theumbilical cord, is extracted directly from the images. The roughness andcorrelation function are based on a Fourier representation of each chorionicplate outline.Although an ideal placental shape is expected to be regular, if only tominimize the cost of maintaining its vascular network, deviations from reg-ularity can be quite pronounced, as noted above, and are not uncommon.This indicates that the lateral growth of the placental chorionic surface isnot typically a uniform process, but has an element of randomness in many,if not most pregnancies, that is, different regions of the placental chorionicsurface may develop at different random rates. The potentially importantcorollary to this is that the “fetal programming” hypothesis may be germaneto the majority of births, since few placentas are round with perfectly centralcords. Various measures can and have been extracted from digitized imagesof placentas [12] and plotted as distributions, but an analysis of their distri-butions has yet to reported. Our fundamental premise is that the form ofthese distributions can provide information about the statistical propertiesof these measures that encode the underlying developmental properties thatled to these distributions. Attaining a better understanding of the timing ofdevelopment of placental chorionic surface shape features, which may reflectearly perturbations of placental vascular growth [2] may clarify how risk ofthe wide range of diseases that have been associated with gestational pathol-3gy develops, or when in subsequent pregnancies of that mother surveillancemight be expected to be useful in identification of recurrence, since there isa low but finite risk of recurrence after preelampsia [13], preterm birth [14],fetal growth restriction [15], stillbirth [16], or even miscarriage [17, 18].Placental growth has been shown to be empirically modeled by growth ofa fractal by diffusion-limited aggregation [19]. From this basic observation,we can consider the notion of a random walk [20], where a “walker” takessmall sequential steps to the left or right, each chosen randomly with equalprobability. As the number of steps increases, the distribution of possibledistances from the walker’s initial position approaches a normal distribution.An alternative version of a random walk is based on independent random relative increments which, as the number of steps increases, leads to a log-normal distribution [21]. Finally, a random walk with step sizes that decay asa power law for large step lengths is known as a “L´evy flight”. The likelihoodof a large step is much greater than for a random walk, which has the effectof enhancing the rate of displacement compared to a random walk, and theresulting displacements follow the L´evy distribution [22].
2. Methods & Materials
The data set for our analysis is obtained from the digital images of pla-centas collected from the Pregnancy, Infection, and Nutrition Study, a cohortstudy of women recruited at mid-pregnancy from an academic health centerin central North Carolina. The study population and recruitment techniquesare described in detail elsewhere [23]. Beginning in March 2002, all womenrecruited into this study were requested to consent to a detailed placentalexamination. As of October 1, 2004, 1159 women (94.6%) consented to suchexamination and 1014 (87.4%) had placentas collected and photographed forimage analysis. Of these, 1008 (99%) were suitable for analysis.Placental gross examinations, histology reviews, and image analyses wereperformed at EarlyPath Clinical and Research Diagnostics, a New YorkState-licensed histopathology facility under the direct supervision of Dr. Car-olyn Salafia. The institutional review board from the University of NorthCarolina at Chapel Hill approved this protocol. The fetal surface of eachplacenta was wiped dry and placed on a clean surface, after which the ex-traplacental membranes and umbilical cord were trimmed from the placenta.4he fetal surface was photographed with the laboratory identificationnumber together with 3 cm of a plastic ruler in the field of view using astandard high-resolution digital camera with a minimum image size of 2.3megapixels. A trained observer captured the ( x, y ) coordinates that markedthe site of the umbilical cord insertion and a series of such points along theperimeter of the fetal surface. The perimeter coordinates were captured atintervals no greater than 1 cm, with additional coordinates if it appearedessential to accurately capturing the shape of the fetal surface.
A Fourier series (Appendix A) is used to interpolate between the dis-crete points captured along the perimeter of the chorionic plate (Sec. 2.1),resulting in a smooth outline. A Fourier series is a sum of trigonometricfunctions (sines and cosines) whose coefficients measure the deviation of theoutline from circularity. If only small deviations are present, then the firstfew terms in this series are sufficient. But more terms are required to capturean outline that has rapidly-varying features, such as lobes and protrusions.This interpolation is used to calculate moments of the region surrounded bythe outline (Appendix B). With increasing order, these moments providesuccessively more detailed information about the shape and morphology ofthe chorionic plate. Both the Fourier coefficients of the outline and the mo-ments of the region surrounded by the outline are used to calculate measuresof the shape and morphology of the chorionic plate. Table 1 summarizes themeasures and their formulas.The area bounded by the chorionic outline provides a cumulative measureof the development of the placenta at delivery. No information is providedabout the shape or morphology of the chorionic plate – this is containedin higher moments. The skewness measures the asymmetry with respectto the mean of projections of the image onto the x - and y -axes, viewed asdistributions. Similarly, the kurtosis measures the peakedness or flatness ofthese projections relative to that of a normal distribution, whose kurtosishas the value 3. Thus, a positive (resp., negative) kurtosis means that thedistribution is more (resp., less) peaked than a normal distribution. Eachchorionic plate has also been represented by a ellipse, whose eccentricity andorientation are determined by the zeroth and first moments.The chorionic plate outline provides complementary information to themoment analysis. The two measures we use are the roughness and the cor-relation function. The roughness, defined in (A.5), is an average over the5 able 1: Measures of the chorionic plate that are calculated in this paper. Against thename of each measure is its symbol, definition, and a formula expressed in terms of mo-ments µ ij of the region bounded by the chorionic plate outline (Appendix B) or theFourier coefficients a n and b n of the outline (Appendix A). The fundamental mathemat-ical definitions of these measures, from which the formulas in this table are derived, aregiven in Appendices A and B.Name Symbol Definition FormulaArea A Area within chorionic µ plate outlineCentroid ( x c , y c ) Geometric center of (cid:18) µ µ , µ µ (cid:19) area within chorionicplate outlineEccentricity e Eccentricity of (cid:114) − b a bounding ellipseSkewness ( S x , S y ) Asymmetry of image (cid:32) µ µ / , µ µ / (cid:33) projections onto x -and y -axesKurtosis ( K x , K y ) Peakedness relative to (cid:18) µ µ − , µ µ − (cid:19) normal distribution ofimage projections onto x - and y -axesRoughness W Standard deviation of (cid:34) N (cid:88) n =1 (cid:0) a n + b n (cid:1)(cid:35) / chorionic outline fromthe average radiusCorrelation C ( s ) Standard deviation of (cid:34) N (cid:88) n =1 (cid:0) a n + b n (cid:1) sin (cid:18) πsnL (cid:19)(cid:35) / Function chorionic outline as afunction of separation chorionic plate outline of root-mean-squared deviations from an average ra-dius. Thus, roughness measures the “width” of the deviations of the outlinefrom a circle. A small roughness indicates a narrow width, which corresponds6o an approximately circular outline, while a large width results from largerdeviations from circularity, such as those of lobed or star-shaped outlines.A related measure of the irregularity of the outline is the correlationfunction, defined in (A.7) as the standard deviation of differences betweenall pairs of radii on the chorionic plate outline at a fixed separation. Whereasthe roughness measures the deviation from circularity by summing individualpoints along the outline, yielding a number , the correlation function involvesdifferences between radii at a fixed separations along the outline, which isexpressed as a function of this separation. Thus, the correlation functionis measure of roughness that is spatially resolved along the chorionic plateoutline. In this paper, however, we will not discuss the spatial dependenceof the correlation function, but focus on its average properties.
When calculated for all of the placentas in our cohort, the measures com-piled in Table 1 yield ranges of values that can be represented as distribu-tions, that is, the relative frequencies of occurrences of the outcomes of themeasures. These distributions embody information about the developmentalcharacteristics of placentas, which can be identified by comparing them withdistributions that are associated with particular types of processes. The dis-tribution functions that we use in this paper are summarized below, withdetails provided in Appendix C.The most common probability distribution is the Gaussian, or normal ,distribution. The probability density of this distribution is completely char-acterized by its mean µ and standard deviation σ . Normal distributionsare so common because of the central limit theorem, which states that suchdistributions are the cumulative result of a large number of additive randomevents [20]. A related distribution is the lognormal , which is the probabilityof a variable whose logarithm is normally distributed [21]. The lognormal is askewed distribution, which occurs when averages are low, variances compar-atively large, and values of the quantity being measured cannot be negative.This distribution is the cumulative result of a large number of multiplicative random events.A distribution that is qualitatively different from the normal and log-normal distributions is the symmetric L´evy distribution [22]. The maindistinguishing characteristic of L´evy distributions is that the probability ofextreme variations decays like a power of that variation, as indicated in (C.7).Hence, the occurrence of such variations is far more likely than for a normal7 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) x (cm) y ( c m ) Figure 1: Two examples of interpolations of chorionic plate outlines that have been de-termined by the method described in Appendix A. The origin of the data points foreach outline has been shifted to its centroid. The open circles mark the original datapoints and the broken curve shows the interpolation for an outline with a single-valuedradius function. The corresponding umbilical cord insertion is indicated by the interioropen circle. The closed circles mark the original data points and the solid curve showsthe interpolation for an outline with a multi-valued radius function. The correspondingumbilical cord insertion is indicated by the interior closed circle. distribution, which decays exponentially . For this reason, L´evy distributionsare called “heavy tailed.” L´evy distributions arise from additive randomevents which may involve quite large changes. In contrast, the events thatresult in the normal and lognormal distributions are comparatively small.
3. Results
Figure 1 shows typical fits to the data points of chorionic plate outlinesobtained with the method described in Appendix A. Two types of outlinesare shown: one with a single-valued and one with a multi-valued radius. Asingle-valued radius means that a line emanating from the centroid intersectsevery point on the perimeter only once, while a multi-valued radius functionmay intersect the perimeter more than once. In the latter case, the perimeter8olds back on itself, and the corresponding chorionic plate has lobes or someother irregular shape. Note the irregular spacing of the points along theperimeter, as described in Sec. 2.1. The outline with the single-valued radiushas a regular shape, so relatively few data points are needed. However,the outline with the multi-valued radius has intervals where more points areneeded to describe regions of greater curvature, which can occur for a smallprotrusion or, as in this case, a large morphological entity such as a lobe.This is reflected in the number of terms that must be included in the Fourierseries to produce an accurate interpolation. The series for the outline withthe single-valued radius required fewer terms than that for the outline withthe multi-valued radius because regions of larger curvature mean that morerapidly varying trigonometric functions must be included in the interpolation.
The distribution of areas A bounded by the outlines of the chorionic platesare shown as a histogram in Fig. 2. These histograms were constructed byfirst defining normalized areas as the original areas A divided by the averagearea A av of the cohort. These data points are grouped into contiguous “bins”of width 0.1, a choice dictated by the balance between the inherent statisticalfluctuations in such a limited sample against the smoothness of the resultingrelative frequency profile. Choosing a width of 0.05 produced a somewhatnoisier distribution but did not substantially alter any of the fits. The relativefrequencies f are obtained by dividing the fraction of the total number of datapoints within each bin by the bin width, so the shaded area in the histogramin Fig. 2 is equal to 1. This way of plotting histograms, which eliminates theunits of the quantities being plotted, allows distributions of different measuresto be compared directly, as well as providing the conceptual convenience ofhaving the mean at 1.Superimposed on the area histogram are the normal (a) and lognormal(b) distributions with mean and standard deviation determined from thedata, in the latter case using (C.3) and (C.4), and an optimized fit to aL´evy distribution (c), which yielded the parameters α = 1 . ± .
02 and γ = 0 . ± .
04 in (C.5). This fit was obtained by the least squares method,wherein a L´evy distribution was calculated at the center of each bin, and thesum of the squares of the differences between these values and those of thebins was minimized by varying α and γ . Figures 2(b,d,f) show the same dis-tributions plotted on a logarithmic scale for the frequencies (but maintainingthe same linear scale for the areas). Such plots are used to accentuate the9 av f (a) av (cid:60) (cid:60) (cid:60) l n f (b) av f (c) av (cid:60) (cid:60) (cid:60) l n f (d) av f (e) av (cid:60) (cid:60) (cid:60) l n f (f) Figure 2: Histogram of chorionic plate areas shown as the relative frequencies of binsof normalized areas. These are compared with (a,b) the normal distribution, (c,d) thelognormal distribution, and (e,f) and an optimized L´evy distribution, each of which isshown by a solid curve. The histogram and distributions are plotted on a logarithmicscale for the relative frequencies, indicated as points, in (b,d,f). extreme variations of data (the “tails” of the relative frequencies) to assesshow various distributions account for this regime. Note that, according to(C.7), the fit in Fig. 2(c), yields a probability for large deviations from themean decreases as (
A/A av ) − . . Measures of the chorionic plate outline that provide information aboutshape are the roughness of the perimeter(A.5) and the integrated correlation10 av f (a) av (cid:60) (cid:60) (cid:60) l n f (b) av f (c) av (cid:60) (cid:60) (cid:60) l n f (d) Figure 3: Histograms of the roughness (a,b) and the integrated correlation function in(A.9) (c,d). Linear plots are shown in (a) and (c) and the corresponding plots with alogarithmic frequency scale in (b) and (d) with the frequencies associated with each binindicated by points. Each of the histograms is compared with a lognormal distribution,which is indicated by the solid curve. function (A.9). Figure 3(a,c) shows the histograms of these quantities, witheach normalized as in Fig. 2, i.e. each measure is divided by its average overthe cohort and the frequencies are defined such that the sum of the shadedregions is equal to 1. The bin width for each histogram was again taken as0.1. Plotted with each histogram is a lognormal distribution whose meanand standard deviation were determined from the data by using (C.3) and(C.4). Figure 3(b,d) shows the histograms and corresponding lognormal dis-tributions plotted on a logarithmic scale for the frequencies. The tails ofthese distributions extend to much larger values than the area distributionsin Fig. 2, so the semi-logarithmic plots provide correspondingly more infor-mation about the distribution. These histograms are significantly skewed, soonly the lognormal distribution is appropriate, as both the normal and L´evydistributions are symmetric. 11 av f (a) av (cid:60) (cid:60) l n f (b) Figure 4: (a) Histogram of the distances between the centroid and the umbilical cordinsertion compared with the lognormal distribution. (b) Semi-logarithmic plot of thehistogram and distributions in (a), with the bin frequencies represented by points.
As the placenta grows outwards from a central point, the position ofthe umbilical cord insertion relative to the centroid of the placenta, whichis a measure of the centrality of this point, provides information about theisotropy of placental development. If the umbilical cord insertion is closeto the centroid, then the placenta has, on average, grown outwards moresymmetrically than if the cord insertion is displaced appreciably from thecentroid. This does not imply that the chorionic plate is circular in this case,just that lateral growth was not skewed in any direction. The histogramof the distances between the centroid and the umbilical cord insertion isshow in Fig. 4, plotted with the distances divided by their average, withfrequencies that sum to 1. The data have been grouped into bins of width 0.1.Superimposed on the histograms are the lognormal distribution whose meanand standard deviation are determined from the data by using (C.3) and(C.4). Note that, in common with the histograms in Fig. 3, the histogramof the distances is highly skewed, with a long tail, so only the lognormaldistribution is appropriate.
The moment expansion method described in Appendix B has been usedto calculate the best-fit ellipse for the chorionic plate of each placenta inthe cohort. This includes the semi-major and semi-minor axes and the ori-entation angle. Figure 5 shows a selection of placentas together with theirbest-fit ellipses. Most apparent from this figure is that some outlines fit theirellipse quite well. These correspond to chorionic plates with regular shapes.For chorionic plates with irregular shapes that have pronounced lobes and12 (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) (cid:60) (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) (cid:60) (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) (cid:60) (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) (cid:60) (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) (cid:60) (cid:60)
10 0 10 20x (cm) (cid:60) (cid:60) y ( c m ) Figure 5: Chorionic plates (shown shaded) and best-fit ellipses (solid curves) for a selectionof placentas from our cohort. α = 2 and γ = 0 .
012 for the eccentricity, α = 1 . γ = 0 .
009 for the skewness, and α = 1 .
75 and γ = 0 . x - and y -projections of each quantity.
4. Discussion
There are large variations in the characteristics of mature placental shapes.How do these variations arise? The uterine environment plays a part, for ex-ample, in cases where “trophotropism” suggests that the placenta can differ-entially grow, effectively migrating to a more suitable location in the uterus.However, there may also be manifestations of randomness within placen-tal growth. The chorionic plate outline of the mature placenta reflects, inessence, a summary of the effects of all factors that can impact the lateralexpansion of the chorionic surface. Identifying whether a placental measurefollows one of the distributions in Appendix C or another distribution isimportant for assessing the statistical properties of lateral growth or the pro-cesses underlying growth. The results we described in the preceding sectionare summarized in Table 2. 14 f (a) f (b) (cid:60) (cid:60) f (c) (cid:60) (cid:60) f (d) (cid:60) (cid:60) (cid:60) (cid:60) (cid:60) f (e) (cid:60) (cid:60) (cid:60) (cid:60) (cid:60) f (f) Figure 6: Histograms of the eccentricities of the best-fit ellipses compared with (a) a normaland (b) an optimized L´evy distribution. Histograms of the skewness of the chorionicplate, projected onto the x - and y -axes (shown as shaded and unshaded bins, respectively)compared with the (c) normal and (d) an optimized L´evy distribution. Histograms of thekurtosis of the chorionic plate, projected onto the x - and y -axes (shown as shaded andunshaded bins, respectively) compared with the (e) normal and (f) an optimized L´evydistribution. The fits in (c)–(f) were obtained by averaging over the x - and y -projectionsof each quantity. The Gaussian and lognormal distributions provide good accounts of thegross shape of the histogram of chorionic plate areas, but underestimate thepeak near the mean [Fig. 2(a)]. The larger positive deviations from the mean15 able 2: Summary of best-fit distributions for the measures in Table 1, as well as thedistance between the centroid and the umbilical cord insertion.
Measure Best Fit CommentsArea L´evy Normal distribution for smallvalues, L´evy distribution formoderate to large valuesRoughness lognormal power law (“heavy”) tailCorrelation lognormal power law (“heavy”) tailFunctionCentroid– lognormal poor fit near peak of histogramUmbilical CordDistanceEccentricity normal optimized L´evy distributionalso yields normal distributionSkewness L´evy average over x - and y - directionsKurtosis L´evy average over x - and y - directions of the area are better described by the lognormal distribution, but the L´evydistribution provides a discernibly better overall fit to the entire histogramthan either the normal or lognormal distributions [Fig. 2(c)]. In particu-lar, the L´evy distribution gives a much better account of the peak near themean and at large positive deviations from the mean, where the decay ismuch slower than for the normal distribution. This can be seen directly inFig. 2(b,d), where we plot the logarithm of the frequencies in Figs. 2(a,c)against the normalized area. In this coordinate system, the normal distribu-tion appears as an inverted parabola, as follows from (C.1). Figure 2(b,d)clearly shows that the lognormal distribution provides a good fit for moderate16ositive deviations from the mean, but that the L´evy distribution providesa much better fit at all large positive deviations from the mean. The nor-mal distribution, however, provides a better description of the data at valuesbelow the mean.To appreciate the consequences of the better fit provided by the L´evydistribution, we return to the notion of a random walk [20], where a “walker”takes small sequential steps to the left or right, each chosen randomly withequal probability. As previously noted, as the number of steps increases, thedistribution of possible distances from the walkers initial position approachesa normal distribution. L´evy distributions arise from random walks with stepsizes chosen from a distribution for which step sizes decay as a power law forlarge step lengths. Hence, the likelihood of a large step is much greater thanfor a random walk. This has the effect of enhancing the rate of displacementdisplacement compared to a random walk. The fits in Fig. 2 thereby suggestthat placentas whose chorionic plate area is much smaller than the mean,which follow a normal distribution, developed by a series of small independentrandom steps. Placentas with a chorionic area that is much larger than themean, however, developed by large steps, or a series of smaller correlatedsteps. In either case, the growth of placentas with a large chorionic area ismanifestly inconsistent with normal behavior. The skewed histograms in Fig. 3 mean that symmetric distributions arenot appropriate, so we have focused on the lognormal distribution. Thisdistribution provides a reasonable fit to the data, though there are evidentdiscrepancies, especially for small values of the correlation function. How-ever, as the semi-logarithmic plots in Fig. 3(b,d,f) show, while the lognormaldistribution accurately accounts for moderate positive deviations from themean, extreme deviations (the tails) show systematic differences from thisdistribution.An analysis of this regime is carried out in Fig. ?? , where we show log-log plots of the data in Figs. 2(a,c,e) together with a linear fit to the tailsof each distribution. Bearing in mind that there are fewer placentas forextreme values, so the scatter in the data is correspondingly greater than forsmaller values, the linear fits provide an acceptable account of these tails.The significance of this becomes apparent when we refer to the discussion inAppendix C and, in particular, the power law behavior of the tails of theL´evy distribution in (C.6). The linear fits in Fig. 7 show that the tails of17 .4 0.6 0.8 1 2 3 4ln(W / W av )10 (cid:60) (cid:60) (cid:60) l n f (a) av )10 (cid:60) (cid:60) (cid:60) l n f (b) Figure 7: Log-log plots of the histograms in Fig. 3(a,c), with the bin frequencies representedas dots. The lines in each panel represent an optimized linear fit to the tails of thedistributions, with slopes of − .
47 and − .
49, respectively. The quality of these fitssuggest that the tails of the corresponding distributions have a power law decay, as in(C.6). these histograms are indeed consistent with a power law decay. Although thisis indicative of the wild fluctuations associated with L´evy distributions, wehave not been able to fit a L´evy distribution to the entire range of the data.Nevertheless, the analysis in Fig. 7 is very suggestive. But we conclude thisdiscussion with a word of caution. Linear fits to log-log plots typically rely onseveral decades (i.e. powers of ten on a log-log plot) of data to enable a firmconclusion to be drawn about the existence of power law tails. Our analysisis based on less than half a decade, which is the nature of the measures weare using, so our conclusions must be tempered accordingly.
The lognormal distribution in Fig. 4 provides a good account of the pro-file of the histogram of distances between the centroid and the umbilicalcord insertion – only the main peak of the histogram is overestimated byapproximately 10%. Even more significant is the fit in Fig. 6(b), whichshows that the lognormal distribution provides an excellent account of thetail of the histogram. Hence, we conclude that the distribution associatedwith this quantity, when measured in mature placentas across a cohort, isthe cumulative result of small multiplicative random steps. This, in turn,leads to two further considerations. Consider first the fact that a vasacu-logenic zone is already evident at the 5th week of development [24]. Thus,in the early stages of development, we expect that the centroid of the de-veloping chorionic plate and umbilical cord insertion are strongly correlated.However, as further development occurs, random factors diminish this corre-18ation, eventually producing the uncorrelated behavior seen in Fig. 4. Why isthe distribution lognormal, rather than normal? Chemical reactions and, byextension, biochemical reactions, are inherently multiplicative processes [21]because concentrations of particular species must be simultaneously presentat a specific location for development to occur. The amount of each quantity,which varies across the placenta, determines the extent to which developmentoccurs. The comparisons in Fig. 4 suggests that these spatial variations arerandom.
The most striking result in Fig. 6 is how well the normal distributionaccounts for the eccentricities of the best-fit ellipses across the cohort. Thisis confirmed by the optimized L´evy distribution, which has ( α = 2) and astandard deviation of √ γ = 0 . σ = 0 .
5. Summary
Placental weight is a standard measure of placental development and isoften used as a primary indicator of fetal health. But weight is just one waythat factors affect the developmental history of a placenta. Other measureshave been presented before [12] and are revisited here in light of their dis-tributions. Working from interpolations between data points obtained fromdigitized images of the cohort described in Ref. [23], we have calculated sev-eral measures of chorionic plate morphology, including its area, the roughnessand correlation function of the outline, the distance between the centroid andthe umbilical cord insertion, and several shape parameters.Our focus here is determining the extent to which the distributions ofplacental measures are described by normal or lognormal distributions, inother words, the extent to which the fluctuations of these measures resultfrom the sum or product, respectively, of relatively independent factors. In19act, we found that normal distributions provide an accurate account only ofthe distribution of the eccentricities of the best-fit ellipses. Taken together,the results presented here demonstrate how an analysis of a cohort can revealfundamental aspects in the development of placentas. The deviations fromnormal or log-normal behavior, in particular, provide the most direct indica-tion of the presence of correlations in the development of the placenta. Largedeviations from mean behavior are not simply the result of mild independentfluctuations, as normal or lognormal distributions would imply, but embodythe wild fluctuations that lead to power law decay. While we have focusedentirely on geometric and morphological features in this paper, other charac-teristics of the chorionic plate would also benefit from our analysis, especiallythose which take account of vasculature. Such studies are in progress andwill be reported in a future publication.20 ppendix A. Fourier Series for the Chorionic Plate Outline
The chorionic plate outline is represented by a set of points with coordi-nates ( x k , y k ), for k = 1 , . . . , N obtained from the digitized images (Sec. 2.1).To eliminate any bias in the data, we first calculate the coordinates ( x c , y c )of the centroid by taking the average of each perimeter coordinate: x c = 1 N N (cid:88) k =1 x k , y c = 1 N N (cid:88) k =1 y k . (A.1)The centroid is taken as the origin of coordinates for the points along thechorionic outline. The radius r is specified in terms of the length s along theperimeter, which has length L . The Fourier series for r ( s ) is r ( s ) = r av + N (cid:88) n =1 (cid:20) a n cos (cid:18) πsnL (cid:19) + b n sin (cid:18) πsnL (cid:19)(cid:21) , (A.2)where the Fourier coefficients are a n = 2 L N (cid:88) k =1 r k cos (cid:18) πs k nL (cid:19) , (A.3) b n = 2 L N (cid:88) k =1 r k sin (cid:18) πs k nL (cid:19) , (A.4)in which r k is the radius of the k th data point at a distance s k along theperimeter. The corresponding series for the coordinates ( x ( s ) , y ( s )) of theperimeter are of the same form as (A.2), but with x k and y k in turn replacing r k in (A.3) and (A.4).The interpolation of the chorionic plate outline can be used to calculateseveral measures associated with the deviations of this outline from circu-larity. The roughness W of this outline is defined as the root-mean-squareddeviations from its average radius r av : W = (cid:26) L (cid:90) L (cid:2) r ( s ) − r av (cid:3) ds (cid:27) / , (A.5)in which r ( s ) is the distance from the centroid to the chorionic plate outline ata point s along the outline and L is the length of the outline. The roughness21s expressed in terms of the coefficients in (A.3) and (A.4) as W = (cid:20) ∞ (cid:88) n =1 (cid:0) a n + b n (cid:1)(cid:21) / . (A.6)The correlation function C ( s ), defined as C ( s ) = (cid:26) L (cid:90) L (cid:2) r ( s + t ) − r ( t ) (cid:3) dt (cid:27) / , (A.7)is the standard deviation of pairs of radii on the chorionic plate outline as afunction of their separation, is expressed in terms of the coefficients in (A.3)and (A.4) as C ( s ) = (cid:20) ∞ (cid:88) n =1 (cid:0) a n + b n (cid:1) sin (cid:18) πsnL (cid:19)(cid:21) / . (A.8)The relation between the correlation function and the roughness can be ob-tained directly from (A.6) and (A.8): (cid:90) L C ( s ) ds = L N (cid:88) n =1 (cid:0) a n + b n (cid:1) = 2 LW , (A.9)so the correlation function corresponds to a roughness that is spatially re-solved along the chorionic plate outline. Appendix B. Moments of Chorionic Plate Shape
An alternative to the contour-based analysis of chorionic plate shape usingthe Fourier series in Appendix A is the area-based approach of moments. Wedefine a function f ( x, y ) that takes the value 1 within the chorionic plate areaand the value 0 outside this area. The ( p, q )th moment µ p,q of the enclosedarea is defined as µ p.q = (cid:90) (cid:90) x p y q f ( x, y ) dx dy , (B.1)where p, q = 0 , , , · · · . If all of the moments are calculated, then the originalshape can be restored. In practice, only lower-order moments, for which p + q ≤ µ , determines the area A of the chorionicplate, A = µ , = (cid:90) (cid:90) f ( x, y ) dx dy , (B.2)and the coordinates ( x c .y c ) of the centroid are expressed in terms of µ , thefirst-order moments µ , and µ , as x c = 1 A (cid:90) (cid:90) x f ( x, y ) dx dy = µ , µ , , (B.3) y c = 1 A (cid:90) (cid:90) y f ( x, y ) dx dy = µ , µ , . (B.4)The zeroth- and second-order moments determine the best-fit ellipse.This ellipse is centered at the centroid of the chorionic plate and its semi-major and semi-minor axes a and b , respectively, are the perpendicular linesthat pass through the centroid for which the second-order central momentsabout these lines are maximum and minimum, respectively. The semi-majorand semi-minor axes are given by [10] a = √ (cid:26) µ , + µ , + (cid:2) ( µ , − µ , ) + 4 µ , (cid:3) / µ , (cid:27) / , (B.5) b = √ (cid:26) µ , + µ , − (cid:2) ( µ , − µ , ) + 4 µ , (cid:3) / µ , (cid:27) / , (B.6)where the tilt angle φ of the ellipse, measured counterclockwise with respectto the original coordinate axes, is [10] φ = 12 tan − (cid:18) µ , µ , − µ , (cid:19) . (B.7)The convention is that φ is the angle between the x -axis and the semi-majoraxis, where, by definition, a ≥ b . The eccentricity e of the best-fit ellipse isgiven by the usual formula: e = (cid:114) − b a . (B.8)Higher-order moments include quantities such as skewness and kurtosisof x and y projections of the placental shape (for example, the x -projectionis the image obtained by summing over all pixels in the x -direction). Expres-sions for these quantities are compiled in Table 1.23 ppendix C. Probability Density Functions The probability density function p ( x ) of a continuous random variablerepresents the relative likelihood that the random variable occurs at a givenpoint x . The probability density function is nonnegative, and its integralover all possible values of x is equal to one. The probability density of thenormal distribution is p ( x ; µ, σ ) = 1 √ πσ exp (cid:20) − ( x − µ ) σ (cid:21) , (C.1)in which µ is the mean σ the standard deviation. The corresponding quantityfor the lognormal distribution is p ( x ; µ, σ ) = 1 x √ πσ exp (cid:20) − (ln x − µ ) σ (cid:21) . (C.2)where µ is the mean and σ the standard deviation for ln x . These are relatedto the mean µ (cid:48) and variance σ (cid:48) of a random variable that is log-normallydistributed by µ = ln( µ (cid:48) ) − ln (cid:18) σ (cid:48) µ (cid:48) (cid:19) , (C.3) σ = ln (cid:18) σ (cid:48) µ (cid:48) (cid:19) . (C.4)The probability density function of the L´evy distribution p ( x ; α, γ ) = 1 π (cid:90) ∞ e − γk α cos( kx ) dk , (C.5)where 0 < α ≤ γ > α = 1 and α = 2, with the latter yielding thenormal distribution in the form p ( x ; 2 , γ ) = 1 √ πγ exp (cid:18) − x γ (cid:19) , (C.6)which is a normal distribution with µ = 0 and σ = 2 γ . In all other casesthe L´evy distribution must be evaluated numerically.One of the most important characteristics of L´evy distributions is thatthe probability density of extreme variations of a random variable follows apower law: p ( x ; α, γ ) → | x | − α − as | x | → ∞ . (C.7)24 eferences [1] Barker DJ. Fetal origins of coronary heart disease. 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