Statistical Geometry and Topology of the Human Placenta 201 Boundary Conditions Statisticalfor GeometryThree and Topology Placental of the Human Placenta Flow Fields 201 That May Predict and Cause Intra-Uterine Growth Retardation C.G. Ball1, D. Grynspan2, A. Gruslin3 1. Department of Pathology and Laboratory Medicine, University of Ottawa / Ottawa Hospital, Ottawa, Ontario, Canada 2. Department of Pathology and Laboratory Medicine, University of Ottawa / Children’s Hospital of Eastern Ontario, Ottawa, Ontario, 3. Departments of Obstetrics and GynecologyStatistical Geometry and Topology of Cellular of the Human PlacentaMolecular Medicine, University 197 of Ottawa / Ottawa Hospital, Ottawa, Ontario, Canada

Introduction The Spiral Arteries

Altered end-diastolic flow in the umbilical artery is a reliable predictor of Variation in the size and shape of the spiral arteries is associated with the intrauterine growth retardation (IUGR). It appears that an earlier and more physiological state of the uterus (non-gravid, post-partum, gravid, in pre- sensitive prediction can be made when chorionic vessel flow patterns are eclampsia, etc.). In the diseased state, the causality of these morphological characterised; we hypothesize that this results from remodelling of the villous changes in the spiral arteries is not entirely known (altered trophoblastic invasion circulatory bed. However, neither the true extent nor the causative mechanism has been implicated). Nonetheless, it is postulated variations in spiral arterial of these altered flow patterns is known. In addition to the villous tree, the flow bring about other changes in the health of the uterus, fetus, and mother lacunar blood flows and the spiral arteries may also be prime suspects. (such as villous tree changes, IUGR, pre-eclampsia, and so on). Furthermore, Unfortunately, direct observation of these physical flows is restricted. the shear interactions between the blood flow and arterial walls may result in Preliminary enquiry indicates that each system may be compared to a unique additional alteration of the geometry of the spiral arteries. physical flow problem. The villous tree as a complicated flow-in-pipes system; Accordingly, it would be beneficial to precisely know the flow in the spiral arteries the lacunar pools as a porous medium flow; the spiral arteries as a wall shear in order to accurately determine uteroplacental flow, and to assess whether stress interactions problem. incipient changes in spiral arterial flow incite further local and regional pathologic changes. The Chorionic Vascular Tree Fig 1. Highlighted vasculature of theFig. chorionic 7 Six tree networksplate: a) corresponding Fig 2. to Variation the vasculature of ofthe human chorionic placentas, obtained according Fig. 5 (a) A digital photograph of the chorionic plate ofto the the placenta procedure with in handSect. tracing4.1,withtheindicatedbirthweights(Fig. which 7 Six tree networks correspondinglower panels to the)togetherwiththeir vasculature of human placentas, obtained according entropy profiles (upper panelsto). Forvasculature, the procedureeach placenta, in Sect.six the computational network4.1,withtheindicatedbirthweights( is shown in gray and the corre-lower panels)togetherwiththeir highlightsPhotograph the visible surface of vasculatureplacenta (arterieswith arterial are traced tree, here). (b)b) andand (c )c) show the hand traced Our Goals The chorionic vascular tree may be approximated to a system of flow-in-pipes. sponding triangulation shown adjacententropyarterial profilesin red tree (upper networks panels). from For each human placenta, the network is shown in gray and the corre- arterial andarterial venous and vasculature venous respectively vasculature, prior to d) computer computational aided processing. (d)Computersponding triangulation shown adjacent in red In engineering systems, these problems are normally solved empirically. generated graph showing both the arterial (red)andvenousvasculature placenta, with fetal birth weights approximation of both vascular trees (from 2). G.J. Burton et al. / Placenta 30 (2009) 473–482 1. Determine479 the extent of feasible simplification of the chorionic vascular tree. These solutions approximate losses and gains in a flow system by accounting Bearing in mind the limitations(from 2.) of a small sample size, as well as the potential for measurement errors of theBearing placental in vasculature, mind the limitations Fig. 8 shows of severala small2. features sampleDetermine of size, as an well idealised as the potential geometryResults for the fetal villous tree within the the effects of standard piping elements (i.e. elbows, tees, pumps, that placental development is even linked to adultthe health. entropy The profiles placenta of 13 grows placentasfor measurement and divided errors into categories of the placental of low (2–3vasculature, kg),uteroplacental normal Fig. 8 shows several blood features pool. of (3–4 kg) and high (4–5 kg) infant birth weights. In common with the placentas in contractions/expansions, etc.). However, the implied assumptions of such andevelops alongside the fetus (Fig. 1). The surface of the placenta attachedthe to entropy the profiles of 13 placentas divided into3. categoriesDetermine of lowthe (2–3uteroplacental kg), normal blood flow velocity field, using a scaled endometrium (lining) of the uterine wall is calledFig. the7,theentropiesoftheentirevasculatures[Figs. basal plate and the(3–4 surface kg) and high (4–5 kg)8(a), infant (b)] birth show weights. a much smaller In common with the placentas in engineering system may not be justifiable in the variable flow setting of the Fig. 7,theentropiesoftheentirevasculatures[Figs.experimental8(a), (b)] show model. a much smaller nearest to the fetus the chorionic plate. Between the two is a complex arborized chorionic vascular tree. Among the most difficult challenges are: vascular network through which oxygen, nutrient and waste exchange takes place. 4. Determine idealised geometries for the spiral arteries in normal gravid and G.J. Burton et al. / Placenta 30 (2009) 473–482 475 non-gravid uterus, with a goal of computing shear-wall interactions and their • determination of an idealised geometry for the vascular tree, The vascular system of the placenta consists of two arteries and one vein that form a possible effect on spiral arterial development. • understanding of the fluid dynamic effects of the pulsatile flow treelike network in the chorion with its origin placed at the umbilical cord insertion. Beyond the chorionic plate, the veins and arteries dive into the placenta with fur- • determination of an acceptable approximation for blood viscosity ther branching. In our analysis we focus on the vascular structure on the chorionic References • understanding the wall effect of the elastic blood vessels plate, which we regard as a 2D tree network. The analysis distinguishes between the 1. Ball CB, Grynspan D, Gruslin A. (2013) Porous media flow models for maternal arterial and venous vasculature. placental circulation. Annual ConclusionsMeeting of the International Federation of Placenta Utero-Placental Flow Associations, Whistler, British Columbia. 2. Bernischke K., Kaufmann P, Baergen, RN Pathology of the human placenta. 5th ed.

Numerous approximations of uteroplacental blood flow and solute transport4.1 Tracing the VasculatureFig. 6. Diagrammatic representationFig (not 4. to scale)Diagrammatic of the effects of spiral artery representation conversion on the inflow of of maternal normal blood into theand intervillous space and on lobuleSpringer; architecture 2006. predicted by modelling. Dilation of the distal segment in normal pregnancies will reduce the velocity of incoming blood, and the residual momentum will carry the blood into the have been proposed as a means to uncover underlying pathophysiologic central cavity (CC) of the lobule,pathologic from where it will disperse alteration evenly through of the villousthe tree.villous Transit time tree to the uterinein response vein is estimated to be in the order3. of 25–30Burton s, allowing GJ, Woods AW, Jauniaux E, Kingdom JCP. (2009) Rheological and adequate time for oxygen exchange. The pressure of the maternal blood, indicated in mmHg by the figures in blue, will drop across the non-dilated segment of the spiral artery, the processes (of IUGR and other entities). In these efforts, analytical solutions or dimensions of which are given alongside.altered In pathological spiral pregnancies, arterial where in-flow no or very limited (as conversion predicted occurs, the maternal by 3.) blood will enter the intervillous spacephysiological at speeds of consequences of conversion of the maternal spiral arteries for ExtractingFig 3. the A: network Large informationstem villi1–2 m/s. of from The the high the Reynolds placenta number predicts is a turbulent time-consuming flow, indicated by the multi- circular arrows. We suggest that the high momentum ruptures anchoring villi (asterisked) and step process. Figure 5 illustratesdisplaces the steps others to involvedform echogenic cysticin extracting lesions (ECL) lined the by thrombus placental (brown). vas- The transit time will be reduced, so that oxygen exchange is impaired and blooduteroplacental leaves in the blood flow during human pregnancy. Placenta. 30:473-482. mathematical models are applied to approximate physical processes in the villous tree, B) Higheruterine magnification vein with a higher oxygen concentration than normal. Trophoblastic microparticulate debris (dotted) may be dislodged from the villous surface, leading to maternal Conclusions culature from a 2D digital photographendothelial cellof activation. the chorionic Finally, the retention plate of of smooth the muscle placenta. cells (SMC) These around the spiral artery will increase the risk of spontaneous vasoconstriction and ischaemia– uteroplacental flow bed. These models, while intricate, do not account for the of peripheral branchesreperfusion of the injury. villous 4. Chernyavsky IL, Leach L, Dryden IL, Jensen OE. (2011) Transport in the placenta: steps are: presumed variations in intervillous flow mechanics that may be seen. tree (from 1). Homogenizing haemodynamics in a disordered medium. Phil. Trans. R. Soc. A. maternal posture on umbilical arterial waveforms. In the supine uteroplacental perfusion cannot be performed in the369:4162—4182. human, it is position the pregnant uterus compresses the inferior vena cava, notable that patients with high levels of endogenous release The utero-placental flow in a single cotyledon (or placentone) is generated by leading to a build-up in pressure and a reduction in intervillous through phaeochromocytoma have an elevated fetal5. lossSeong rate in the R-K, Getreue P, Li Y, Girardi T, Salafia CM, Vvedensky DD. (2013) Statistical Fig 5. Diagrammatic representationblood flow [47,48]. At the same time there is a reversible increase in second and third trimesters of pregnancy, which may be due to the ‘sources’ and ‘sinks’ of spiral arteries and decidual veins. In light of the the umbilical arterial resistance, indicating acute compression of excessive constriction of these segments [50]. Geometry and Topology of the Human Placenta, Advances in Applied Mathematics, of spiral arteries in thethe non-gravid fetal placental vessels [49]. It is essential therefore that the For much of pregnancy intermittent placental perfusion may be symmetric homogeneity of the villous tree, it may be reasonable to construct transmembrane pressure differential is positive in a fetal to of little consequence, as the maternal blood retainedModeling, within the and Computational Science. 66:187-208 uterus, in pregnancy, maternalin pre- direction if the fetal capillaries are to remain distended. intervillous space will provide a supply of oxygen6. and Taylor nutrients toCA, Humphrey JD. (2009) Open problems in computational vascular an idealised model of the placentone and determine utero-placental flow As can be seen in Fig. 5b most of the maternal pressure drop occurs tide the placental tissues over until the circulation restarts. eclampsia, in post-partumacross the (from non-dilated 3). part of the artery, so that the pressure in the However, towards term, when fetal and placental oxygenbiomechanics: extrac- Hemodynamics and arterial wall mechanics. Comp. Methods. in App. experimentally. terminal dilated section of the artery is equivalent to that in the tion are at their maximum, even temporary cessation in the intervillous space. circulation is likely to lead to a dip in the oxygen concentrationMech. Eng. 198(45-46):3514-3523. Fig. 2. Diagrammatic representation of uterine and placental vasculature (red shading arterial; blue shading venous) in the non-pregnant, pregnant and immediate post- ¼ ¼ partum state. Normal pregnancy is characterized by the formation of large arterio-venouswithin shunts that thepersist intervillous in the immediate post-partum space. period.This By will contrast be pregnancies exacerbated by the fact 4.3. Constancy of bloodcomplicated flow by severe preeclampsia are characterized by minimal arterio-venous shunts, andthat thus the narrower volume uterine arteries. of maternal Extravillous cytotrophoblastblood acting invasion as in a normal reservoir around each pregnancy (diamonds) extends beyond the decidua into the inner myometrium resulting invillus the formation will of funnels decrease at the discharging as the tips size of the of spiral the arteries. intervillous Contrast with pores diminishes The effects on ratesevere preeclampsia. and pressure (Preparedof by Ms. inflow Leslie Proctor, are MSc.) mediated by the [51]. Fluctuating oxygen concentrations is a powerful inducer of terminal dilations of the spiral arteries, yet the vessels undergo placental oxidative stress [52], and so any mismatch in maternal conversion as far as the inner third of the myometrium where the supply and feto-placental demand may have significant physio- vessel does not dilateand penetrate to the thesame vessel extent. walls from As their discussed outside. In earlier, normal preg- we terminallogical part consequences. of a spiral artery at term having a maximum diameter speculate that thenancy, goal the of interstitial this deeper trophoblast invasion cells invade is to as deep remove as the theinner of 2.4Physiological mm (Fig. 1c), which conversion represents may an approximately therefore be 4-fold a means by which the smooth muscle fromthird the of the highly myometrium, contractile where they segment progressively of transform the spiral into increaseconflicting in the requirementsdiameter of the vessel for at vasoreactivity the myometrial–endo- to support menstrua- artery in the junctionalimmotile zone giant (Fig. cells 1 (b).Fig. Doing 2). Both so endovascular will reduce and the interstitial risk metrialtion, andboundary vaso-inactivity and within the endometrium. to support Why placental only the perfusion,final can be met of spontaneous vasoconstriction,invasion are associated and with so the physiological extent of conversion intermittent of the sectionwithin of a convertedsingle vessel. artery is involved in this exaggerated dila- perfusion that mightspiral otherwise arteries, although occur, the molecular as evinced mechanisms by the involved rhesus are tation is not clear, but it may be sculpted by the extent of the monkey. Ensuringstill an unclear uninterruptible[9,11]. During this circulation process the arteries to the loose placenta the smooth endovascular4.4. Volume invasion of intervillous of the vessels in blood early pregnancy. flow Initially the must be of the utmostmuscleimportance, in their walls and since their elastic removal lamina, of and the assegment a result it is invasion is so extensive that the tips of the arteries are effectively that normally preventscommonly excessive stated that the menstrual vessels dilate blood and are loss converted might into pluggedUterine by the trophoblast artery cells blood[14,20] flow, and so thereincreases is little, ifdramatically any, during reasonably be expectedflaccid conduits. to predispose The extent the of woman conversion to varies an increased across the maternalgestation blood in flow all into mammalian the placenta (Fig. species 3). whether or not trophoblast risk of post-partumplacental haemorrhage. bed, and is greatest Although in the central experiments region where tropho- to invasionEqually, it should occurs, be remembered and so that is trophoblast most likely invasion an reaches endocrine mediated examine the effectsblast invasion of administration is most extensive [19] of. catecholamines on neitherevent. the radialThe arteries, dilatation nor the arcuate of the and uterine human arteries. uterine None- artery initially The final outcome was elegantly demonstrated by Harris and theless, all these undergo profound dilation during pregnancy, partic- Ramsey, who performed three-dimensional reconstructions of ularly beneath the implantation site [21,22].Fromuniqueradiographic a number of spiral arteries from hysterectomy specimens at various studies Burchell observed that the diameter of the uterine artery stages of pregnancy [16]. They observed that the arteries undergo doubles by 6.5 weeks of pregnancy. He also observed another unique a generalised, but non-uniform, dilation as pregnancy advances, property of the uterine circulation during pregnancy, namely that the with considerable variation in size between arteries within the diameter of the vessels increases, rather than decreases, as they same specimen, and even at different points along individual approach their target organ. Thus, by mid-pregnancy the diameter of arteries. Most importantly, they found that the terminal coils of the the arcuate arteries exceeds that of the uterine vessels, and by term spiral arteries are enormously dilated, often reaching 2–3 mm in some are twice the diameter [22].Thisprogressivedilationisreflected diameter. This transition is quite abrupt, so that the dilated in sonographic measurements demonstrating that the peak systolic segments form a funnel-shaped chamber that opens through the velocityand pulsatility index are higher in the uterine arterycompared basal plate, often with a slit-like orifice. Thus, they depicted the to the arcuate artery (Fig. 3) [23].