Maternal, placental blood flow: a model with velocity-dependent permeability

Impact of tissue porosity and asymmetry on the oxygen uptake of the human placenta: A numerical study

INTRODUCTION

This study proposes a computational fluid dynamics model of a human placenta's independent exchange unit (placentome) to assess the effect that the inner villi distribution and decidual veins (DVs) location and number, have on the oxygen uptake.

METHODS

The internal placentome porosity distribution was altered in symmetric morphology, while asymmetry was introduced by varying the location and number of DVs. The DV asymmetry was introduced by either displacing them circumferentially, thereby changing the angle between them, or by adding DVs in the inlet cross-section. The results were analyzed by the changes in the normalized oxygen mass fraction and the oxygen uptake.

RESULTS

Oxygenated blood was shown to be delivered deeper into the placentome when the area of non-homogeneous porosity was larger. The largest oxygen uptake was achieved in the asymmetric model with the smallest angle distance between the DVs, where a 10% decrease relative to the farthest case was obtained. Placing DVs adjacent to the spiral artery opening enhanced the drainage of oxygenated blood.

DISCUSSION

This study demonstrates the importance of the local porosity distribution for the proper perfusion of the intervillous space and proposes a novel approach to improve our understanding of the role of the DVs in placental oxygen uptake.

Blood flow and transport in the human placenta

The placenta is a multi-functional organ that exchanges blood gases and nutrients between a mother and her developing fetus. In humans, fetal blood flows through intricate networks of vessels confined within villous trees, the branches of which are bathed in pools of maternal blood. Fluid mechanics and transport processes play a central role in understanding how these elaborate structures contribute to the function of the placenta, and how their disorganization may lead to disease. Recent advances in imaging and computation have spurred significant advances in simulations of fetal and maternal flows within the placenta, across a range of lengthscales. Models describe jets of maternal blood emerging from spiral arteries into a disordered and deformable porous medium, and solute uptake by fetal blood flowing through elaborate three-dimensional capillary networks. We survey recent developments and emerging challenges in modeling flow and transport in this complex organ.

Advances in Human Placental Biomechanics

Pregnancy complications are a major clinical concern due to the related maternal and fetal morbidity. Many are caused through defective placentation, but research into placental function is difficult, principally because of the ethical limitations associated with the in-vivo organ and the difficulty of extrapolating animal models. Perfused by two separate circulations, the maternal and fetal bloodstreams, the placenta has a unique structure and performs multiple complex functions. Three-dimensional imaging and computational modelling are becoming popular tools to investigate the morphology and physiology of this organ. These techniques bear the potential for better understanding the aetiology and development of placental pathologies, however, their full potential is yet to be exploited. This review aims to summarize the recent insights into placental structure and function by employing these novel techniques.

Association of Placental Jets and Mega-Jets With Reduced Villous Density

Spiral arteries (SAs) lie at the interface between the uterus and placenta, and supply nutrients to the placental surface. Maternal blood circulation is separated from the fetal circulation by structures called villous trees. SAs are transformed in early pregnancy from tightly coiled vessels to large high-capacity channels, which is believed to facilitate an increased maternal blood flow throughout pregnancy with minimal increase in velocity, preventing damage to delicate villous trees. Significant maternal blood flow velocities have been theorized in the space surrounding the villi (the intervillous space, IVS), particularly when SA conversion is inadequate, but have only recently been visualized reliably using pulsed wave Doppler ultrasonography. Here, we present a computational model of blood flow from SA openings, allowing prediction of IVS properties based on jet length. We show that jets of flow observed by ultrasound are likely correlated with increased IVS porosity near the SA mouth and propose that observed mega-jets (flow penetrating more than half the placental thickness) are only possible when SAs open to regions of the placenta with very sparse villous structures. We postulate that IVS tissue density must decrease at the SA mouth through gestation, supporting the hypothesis that blood flow from SAs influences villous tree development.

Feeding Your Baby In Utero: How the Uteroplacental Circulation Impacts Pregnancy

The utero-placental circulation links the maternal and fetal circulations during pregnancy, ensuring adequate gas and nutrient exchange, and consequently fetal growth. However, our understanding of this circulatory system remains incomplete. Here, we discuss how the utero-placental circulation is established, how it changes dynamically during pregnancy, and how this may impact on pregnancy success, highlighting how we may address knowledge gaps through advances in imaging and computational modeling approaches.

A multiscale model of placental oxygen exchange: The effect of villous tree structure on exchange efficiency

The placenta is critical to fetal health during pregnancy as it supplies oxygen and nutrients to maintain life. It has a complex structure, and alterations to this structure across spatial scales are associated with several pregnancy complications, including intrauterine growth restriction (IUGR). The relationship between placental structure and its efficiency as an oxygen exchanger is not well understood in normal or pathological pregnancies. Here we present a computational framework that predicts oxygen transport in the placenta which accounts for blood and oxygen transport in the space around a placental functional unit (the villous tree). The model includes the well-defined branching structure of the largest villous tree branches, as well as a smoothed representation of the small terminal villi that comprise the placenta's gas exchange interfaces. The model demonstrates that oxygen exchange is sensitive to villous tree geometry, including the villous branch length and volume, which are seen to change in IUGR. This is because, to be an efficient exchanger, the architecture of the villous tree must provide a balance between maximising the surface area available for exchange, and the opposing condition of allowing sufficient maternal blood flow to penetrate into the space surrounding the tree. The model also predicts an optimum oxygen exchange when the branch angle is 24 °, as villous branches and TBs are spread out sufficiently to channel maternal blood flow deep into the placental tissue for oxygen exchange without being shunted directly into the DVs. Without concurrent change in the branch length and angles, the model predicts that the number of branching generations has a small influence on oxygen exchange. The modelling framework is presented in 2D for simplicity but is extendible to 3D or to incorporate the high-resolution imaging data that is currently evolving to better quantify placental structure.

Computational Fluid Dynamic Simulations of Maternal Circulation: Wall Shear Stress in the Human Placenta and Its Biological Implications

Introduction

In the human placenta the maternal blood circulates in the intervillous space (IVS). The syncytiotrophoblast (STB) is in direct contact with maternal blood. The wall shear stress (WSS) exerted by the maternal blood flow on the STB has not been evaluated. Our objective was to determine the physiological WSS exerted on the surface of the STB during the third trimester of pregnancy.

Material and Methods

To gain insight into the shear stress levels that the STB is expected to experience in vivo, we have formulated three different computational models of varying levels of complexity that reflect different physical representations of the IVS. Computations of the flow fields in all models were performed using the CFD module of the finite element code COMSOL Multiphysics 4.4. The mean velocity of maternal blood in the IVS during the third trimester was measured in vivo with dynamic MRI (0.94±0.14 mm.s^-1). To investigate if the in silico results are consistent with physiological observations, we studied the cytoadhesion of human parasitized (Plasmodium falciparum) erythrocytes to primary human STB cultures, in flow conditions with different WSS values.

Results

The WSS applied to the STB is highly heterogeneous in the IVS. The estimated average values are relatively low (0.5±0.2 to 2.3±1.1 dyn.cm^-2). The increase of WSS from 0.15 to 5 dyn.cm^-2 was associated with a significant decrease of infected erythrocyte cytoadhesion. No cytoadhesion of infected erythrocytes was observed above 5 dyn.cm^-2 applied for one hour.

Conclusion

Our study provides for the first time a WSS estimation in the maternal placental circulation. In spite of high maternal blood flow rates, the average WSS applied at the surface of the chorionic villi is low (<5 dyn.cm^-2). These results provide the basis for future physiologically-relevant in vitro studies of the biological effects of WSS on the STB.

The role of morphology in mathematical models of placental gas exchange

The performance of the placenta as a gas exchanger has a direct impact on the future health of the newborn. To provide accurate estimates of respiratory gas exchange rates, placenta models need to account for both the physiology of exchange and the organ morphology. While the former has been extensively studied, accounting for the latter is still a challenge. The geometrical complexity of placental structure requires use of carefully crafted approximations. We present here the state of the art of respiratory gas exchange placenta modeling and demonstrate the influence of the morphology description on model predictions. Advantages and shortcomings of various classes of models are discussed, and experimental techniques that may be used for model validation are summarized. Several directions for future development are suggested.

Analytical theory of oxygen transport in the human placenta

We propose an analytical approach to solving the diffusion-convection equations governing oxygen transport in the human placenta. We show that only two geometrical characteristics of a placental cross-section, villi density and the effective villi radius, are needed to predict fetal oxygen uptake. We also identify two combinations of physiological parameters that determine oxygen uptake in a given placenta: (i) the maximal oxygen inflow of a placentone if there were no tissue blocking the flow and (ii) the ratio of transit time of maternal blood through the intervillous space to oxygen extraction time. We derive analytical formulas for fast and simple calculation of oxygen uptake and provide two diagrams of efficiency of oxygen transport in an arbitrary placental cross-section. We finally show that artificial perfusion experiments with no-hemoglobin blood tend to give a two-orders-of-magnitude underestimation of the in vivo oxygen uptake and that the optimal geometry for such setup alters significantly. The theory allows one to adjust the results of artificial placenta perfusion experiments to account for oxygen-hemoglobin dissociation. Combined with image analysis techniques, the presented model can give an easy-to-use tool for prediction of the human placenta efficiency.

HOMOTOPY SIMULATION OF TWO-PHASE THERMO-HEMODYNAMIC FILTRATION IN A HIGH PERMEABILITY BLOOD PURIFICATION DEVICE

A two-phase thermo-hydrodynamic model is presented for transport in the vertical chamber of a porous media blood filtration device. A non-Darcy drag force formulation was employed. The Marble–Drew fluid–particle suspension model was used to simulate the plasma phase and the suspension (erythrocyte) particle phase. The non-dimensional transport equations were solved using a semi-computational procedure known as the homotopy analysis method (HAM). With the judicious use of the auxiliary parameter ℏ, HAM affords a powerful mechanism to adjust and control the convergence region of solution series. This method provides an efficient approximate analytical solution with high accuracy, minimal calculation and avoidance of physically unrealistic assumptions. Detailed computations are presented for the effects of Grashof number (Gr), momentum inverse Stokes number (Skm), Darcy number (Da), Forchheimer number (Fs), particle loading parameter (PL), buoyancy parameter (B) and temperature inverse Stokes number (SkT) on the dimensionless fluid phase velocity (U), dimensionless particle phase velocity (Up), dimensionless fluid phase temperature (Φ) and the dimensionless temperature of particle phase (Φp). A Prandtl number of 25 was used to simulate blood at room temperature. Excellent correlation was obtained between the HAM and numerical shooting quadrature solutions. The results indicated that there is a strong decrease in fluid phase velocities with increasing Darcian (first order) drag and second-order Forchheimer drag, and a weaker reduction in particle phase velocity field. Applications of the study include porous media bio-filtration devices and dialysis simulations.

Transport in the placenta: homogenizing haemodynamics in a disordered medium

The placenta is an essential component of the life-support system for the developing foetus, enabling nutrients and waste to be exchanged between the foetal and maternal circulations. Maternal blood flows between the densely packed branches of villous trees, within which are foetal vessels. Here, we explore some of the challenges in modelling maternal haemodynamic transport using homogenization approaches. We first show how two measures can be used to estimate the minimum distance over which the distribution of villous branches appears statistically homogeneous. We then analyse a simplified model problem (solute transport by a unidirectional flow past a distribution of point sinks) to assess the accuracy of homogenization approximations as a function of governing parameters (Péclet and Damköhler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is characterized by large spatial gradients at the scale of individual villi and substantial fluctuations that can be correlated over lengthscales comparable to the whole domain. This study highlights the importance of quantifying errors owing to spatial disorder in multi-scale approximations of physiological systems.

A mathematical model of intervillous blood flow in the human placentone

We present a mathematical model for maternal blood flow in a placental circulatory unit (a placentone), describing flow of maternal blood via Darcy's law and steady advective transport of a dissolved nutrient. The method of images and computational integration along streamlines are employed to find flow and solute concentration distributions, which are illustrated for a range of governing system parameters. The model shows how the calibre of the basal vessels can be a dominant determinant of the maternal blood flow rate through the placentone, given a driving pressure difference between the spiral arteries and decidual veins. The model supports the hypothesis that basal veins are located on the periphery of the placentone in order to optimise delivery of nutrients and suggests the existence of an optimal volume fraction of villous tissue.