The role of morphology in mathematical models of placental gas exchange

Diffusion-Controlled Reactions: An Overview

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

Encounter-based approach to the escape problem

We revise the encounter-based approach to imperfect diffusion-controlled reactions, which employs the statistics of encounters between a diffusing particle and the reactive region to implement surface reactions. We extend this approach to deal with a more general setting, in which the reactive region is surrounded by a reflecting boundary with an escape region. We derive a spectral expansion for the full propagator and investigate the behavior and probabilistic interpretations of the associated probability flux density. In particular, we obtain the joint probability density of the escape time and the number of encounters with the reactive region before escape, and the probability density of the first-crossing time of a prescribed number of encounters. We briefly discuss generalizations of the conventional Poissonian-type surface reaction mechanism described by Robin boundary condition and potential applications of this formalism in chemistry and biophysics.

Diffusion toward a nanoforest of absorbing pillars

Spiky coatings (also known as nanoforests or Fakir-like surfaces) have found many applications in chemical physics, material sciences, and biotechnology, such as superhydrophobic materials, filtration and sensing systems, and selective protein separation, to name but a few. In this paper, we provide a systematic study of steady-state diffusion toward a periodic array of absorbing cylindrical pillars protruding from a flat base. We approximate a periodic cell of this system by a circular tube containing a single pillar, derive an exact solution of the underlying Laplace equation, and deduce a simple yet exact representation for the total flux of particles onto the pillar. The dependence of this flux on the geometric parameters of the model is thoroughly analyzed. In particular, we investigate several asymptotic regimes, such as a thin pillar limit, a disk-like pillar, and an infinitely long pillar. Our study sheds light onto the trapping efficiency of spiky coatings and reveals the roles of pillar anisotropy and diffusional screening.

The Pivotal Role of the Placenta in Normal and Pathological Pregnancies: A Focus on Preeclampsia, Fetal Growth Restriction, and Maternal Chronic Venous Disease

The placenta is a central structure in pregnancy and has pleiotropic functions. This organ grows incredibly rapidly during this period, acting as a mastermind behind different fetal and maternal processes. The relevance of the placenta extends far beyond the pregnancy, being crucial for fetal programming before birth. Having integrative knowledge of this maternofetal structure helps significantly in understanding the development of pregnancy either in a proper or pathophysiological context. Thus, the aim of this review is to summarize the main features of the placenta, with a special focus on its early development, cytoarchitecture, immunology, and functions in non-pathological conditions. In contraposition, the role of the placenta is examined in preeclampsia, a worrisome hypertensive disorder of pregnancy, in order to describe the pathophysiological implications of the placenta in this disease. Likewise, dysfunction of the placenta in fetal growth restriction, a major consequence of preeclampsia, is also discussed, emphasizing the potential clinical strategies derived. Finally, the emerging role of the placenta in maternal chronic venous disease either as a causative agent or as a consequence of the disease is equally treated.

A Damaged-Informed Lung Ventilator Model for Ventilator Waveforms

Motivated by a desire to understand pulmonary physiology, scientists have developed physiological lung models of varying complexity. However, pathophysiology and interactions between human lungs and ventilators, e.g., ventilator-induced lung injury (VILI), present challenges for modeling efforts. This is because the real-world pressure and volume signals may be too complex for simple models to capture, and while complex models tend not to be estimable with clinical data, limiting clinical utility. To address this gap, in this manuscript we developed a new damaged-informed lung ventilator (DILV) model. This approach relies on mathematizing ventilator pressure and volume waveforms, including lung physiology, mechanical ventilation, and their interaction. The model begins with nominal waveforms and adds limited, clinically relevant, hypothesis-driven features to the waveform corresponding to pulmonary pathophysiology, patient-ventilator interaction, and ventilator settings. The DILV model parameters uniquely and reliably recapitulate these features while having enough flexibility to reproduce commonly observed variability in clinical (human) and laboratory (mouse) waveform data. We evaluate the proof-in-principle capabilities of our modeling approach by estimating 399 breaths collected for differently damaged lungs for tightly controlled measurements in mice and uncontrolled human intensive care unit data in the absence and presence of ventilator dyssynchrony. The cumulative value of mean squares error for the DILV model is, on average, ≈12 times less than the single compartment lung model for all the waveforms considered. Moreover, changes in the estimated parameters correctly correlate with known measures of lung physiology, including lung compliance as a baseline evaluation. Our long-term goal is to use the DILV model for clinical monitoring and research studies by providing high fidelity estimates of lung state and sources of VILI with an end goal of improving management of VILI and acute respiratory distress syndrome.

Bioengineering Approaches for Placental Research

Research into the human placenta's complex functioning is complicated by a lack of suitable physiological in vivo models. Two complementary approaches have emerged recently to address these gaps in understanding, computational in silico techniques, including multi-scale modeling of placental blood flow and oxygen transport, and cellular in vitro approaches, including organoids, tissue engineering, and organ-on-a-chip models. Following a brief introduction to the placenta's structure and function and its influence on the substantial clinical problem of preterm birth, these different bioengineering approaches are reviewed. The cellular techniques allow for investigation of early first-trimester implantation and placental development, including critical biological processes such as trophoblast invasion and trophoblast fusion, that are otherwise very difficult to study. Similarly, computational models of the placenta and the pregnant pelvis at later-term gestation allow for investigations relevant to complications that occur when the placenta has fully developed. To fully understand clinical conditions associated with the placenta, including those with roots in early processes but that only manifest clinically at full-term, a holistic approach to the study of this fascinating, temporary but critical organ is required.

Pseudo-Darwinian evolution of physical flows in complex networks

The evolution of complex transport networks is investigated under three strategies of link removal: random, intentional attack and “Pseudo-Darwinian” strategy. At each evolution step and regarding the selected strategy, one removes either a randomly chosen link, or the link carrying the strongest flux, or the link with the weakest flux, respectively. We study how the network structure and the total flux between randomly chosen source and drain nodes evolve. We discover a universal power-law decrease of the total flux, followed by an abrupt transport collapse. The time of collapse is shown to be determined by the average number of links per node in the initial network, highlighting the importance of this network property for ensuring safe and robust transport against random failures, intentional attacks and maintenance cost optimizations.

Paradigm Shift in Diffusion-Mediated Surface Phenomena

Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane permeation, and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this Letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-particle interactions in terms of stopping conditions, we obtain in a unified way major diffusion-reaction characteristics such as the propagator, the survival probability, the first-passage time distribution, and the reaction rate. This general formalism allows us to describe new surface reaction mechanisms such as for instance surface reactivity depending on the number of encounters with the diffusing particle that can model the effects of catalyst fooling or membrane degradation. The disentanglement of the geometric structure of the medium from surface reactivity opens far-reaching perspectives for modeling, optimization, and control of diffusion-mediated surface phenomena.

Diffusion-influenced reactions on non-spherical partially absorbing axisymmetric surfaces

The calculation of the diffusion-controlled reaction rate for partially absorbing, non-spherical boundaries presents a formidable problem of broad relevance. In this paper we take the reference case of a spherical boundary and work out a perturbative approach to get a simple analytical formula for the first-order correction to the diffusive flux onto a non-spherical partially absorbing surface of revolution. To assess the range of validity of this formula, we derive exact and approximate expressions for the reaction rate in the case of partially absorbing prolate and oblate spheroids. We also present numerical solutions by a finite-element method that extend the validity analysis beyond spheroidal shapes. Our perturbative solution provides a handy way to quantify the effect of non-sphericity on the rate of capture in the general case of partial surface reactivity.

Quantifying the impact of tissue metabolism on solute transport in feto-placental microvascular networks

The primary exchange units in the human placenta are terminal villi, in which fetal capillary networks are surrounded by a thin layer of villous tissue, separating fetal from maternal blood. To understand how the complex spatial structure of villi influences their function, we use an image-based theoretical model to study the effect of tissue metabolism on the transport of solutes from maternal blood into the fetal circulation. For solute that is taken up under first-order kinetics, we show that the transition between flow-limited and diffusion-limited transport depends on two new dimensionless parameters defined in terms of key geometric quantities, with strong solute uptake promoting flow-limited transport conditions. We present a simple algebraic approximation for solute uptake rate as a function of flow conditions, metabolic rate and villous geometry. For oxygen, accounting for nonlinear kinetics using physiological parameter values, our model predicts that villous metabolism does not significantly impact oxygen transfer to fetal blood, although the partitioning of fluxes between the villous tissue and the capillary network depends strongly on the flow regime.

Placental structure in gestational diabetes mellitus

The placenta is a transitory organ, located between the mother and the foetus, which supports intrauterine life. This organ has nutritional, endocrine and immunologic functions to support foetal development. Several factors are related to the correct functioning of the placenta including foetal and maternal blood flow, appropriate nutrients, expression and function of receptors and transporters, and the morphology of the placenta itself. Placental morphology is crucial for understanding the pathophysiology of the organ as represents the physical structure where nutrient exchange occurs. In pathologies of pregnancy such as diabetes mellitus in humans and animal models, several changes in the placental morphology occur, related mainly with placental size, hypervascularization, higher branching capillaries of the villi and increased glycogen deposits among others. Gestational diabetes mellitus is associated with modifications in the structure of the human placenta including changes in the surface area and volume, as well as histological changes including an increased volume of intervillous space and terminal villi, syncytiotrophoblast number, fibrinoid areas, and glycogen deposits. These modifications may result in functional changes in this organ thus limiting the wellbeing of the developing foetus. This review gives an overview of recurrent morphological changes at macroscopic and histological levels seen in the placenta from gestational diabetes in humans and animal models. This article is part of a Special Issue entitled: Membrane Transporters and Receptors in Pregnancy Metabolic Complications edited by Luis Sobrevia.

Placental oxygen transfer reduces hypoxia-reoxygenation swings in fetal blood in a sheep model of gestational sleep apnea

Obstructive sleep apnea (OSA), characterized by events of hypoxia-reoxygenation, is highly prevalent in pregnancy, negatively affecting the gestation process and particularly the fetus. Whether the consequences of OSA for the fetus and offspring are mainly caused by systemic alterations in the mother or by a direct effect of intermittent hypoxia in the fetus is unknown. In fact, how apnea-induced hypoxemic swings in OSA are transmitted across the placenta remains to be investigated. The aim of this study was to test the hypothesis, based on a theoretical background on the damping effect of oxygen transfer in the placenta, that oxygen partial pressure (Po₂) swings resulting from obstructive apneas mimicking OSA are mitigated in the fetal circulation. To this end, four anesthetized ewes close to term pregnancy were subjected to obstructive apneas consisting of 25-s airway obstructions. Real-time Po₂ was measured in the maternal carotid artery and in the umbilical vein with fast-response fiber-optic oxygen sensors. The amplitudes of Po₂ swings in the umbilical vein were considerably smaller [3.1 ± 1.0 vs. 21.0 ± 6.1 mmHg (mean ± SE); P < 0.05]. Corresponding estimated swings in fetal and maternal oxyhemoglobin saturation tracked Po₂ swings. This study provides novel insights into fetal oxygenation in a model of gestational OSA and highlights the importance of further understanding the impact of sleep-disordered breathing on fetal and offspring development.NEW & NOTEWORTHY This study in an airway obstruction sheep model of gestational sleep apnea provides novel data on how swings in oxygen partial pressure (Po₂) translate from maternal to fetal blood. Real-time simultaneous measurement of Po₂ in maternal artery and in umbilical vein shows that placenta transfer attenuates the magnitude of oxygenation swings. These data prompt further investigation of the extent to which maternal apneas could induce similar direct oxidative stress in fetal and maternal tissues.

Physical and geometric determinants of transport in fetoplacental microvascular networks

Across mammalian species, solute exchange takes place in complex microvascular networks. In the human placenta, the primary exchange units are terminal villi that contain disordered networks of fetal capillaries and are surrounded externally by maternal blood. We show how the irregular internal structure of a terminal villus determines its exchange capacity for diverse solutes. Distilling geometric features into three parameters, obtained from image analysis and computational fluid dynamics, we capture archetypal features of the structure-function relationship of terminal villi using a simple algebraic approximation, revealing transitions between flow- and diffusion-limited transport at vessel and network levels. Our theory accommodates countercurrent effects, incorporates nonlinear blood rheology, and offers an efficient method for testing network robustness. Our results show how physical estimates of solute transport, based on carefully defined geometrical statistics, provide a viable method for linking placental structure and function and offer a framework for assessing transport in other microvascular systems.

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.

Heterogeneous continuous-time random walks

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

Geometric and hemodynamic characterization of uterine spiral arteries: The concept of resistance reserve

BACKGROUND

The coiled geometry of spiral arteries in the human uteroplacental circulation is a hemodynamic enigma because of added length of a spiral artery compared with that of a straight artery, as well as added complexity of the flow within the vessel because of the coiling curvature.

METHODS

We examined the geometric and hemodynamic characteristics of mathematically defined helical and spiral arteries and compared these with the corresponding characteristics of a straight artery traversing the same depth of tissue, with the aim of gaining some insight into the possible role of spiral geometry in uteroplacental perfusion.

RESULTS

The results indicate that the added length of a spiral artery provides the uteroplacental circulation with a reserve of high resistance to flow. The effect of coiling geometry on the flow within the artery is the development of churning vortices in planes normal (perpendicular) to the main flow direction.

CONCLUSIONS

In the early stages of pregnancy the reserve of high resistance is intact, thus keeping blood supply low. As pregnancy progresses, the reserve is gradually purged by trophoblast invasion and transformation of the distal portion of the spiral artery into an open funnel, thus providing the required high blood supply. The development of churning vortices within the spiral artery support earlier suggestions in the literature that the "spurts" of maternal blood emerging from these arteries may play a role in shaping the anatomy of the villous trees among placental lobules.

Analytical model of the feto-placental vascular system: consideration of placental oxygen transport

The placenta is a transient vascular organ that enables nutrients and blood gases to be exchanged between fetal and maternal circulations. Herein, the structure and oxygen diffusion across the trophoblast membrane between the fetal and maternal red blood cells in the feto-placental vasculature system in both human and mouse placentas are presented together as a functional unit. Previous models have claimed that the most efficient fetal blood flow relies upon structures containing a number of 'conductive' symmetrical branches, offering a path of minimal resistance that maximizes blood flow to the terminal villi, where oxygen diffusion occurs. However, most of these models have disregarded the actual descriptions of the exchange at the level of the intermediate and terminal villi. We are proposing a 'mixed model' whereby both 'conductive' and 'terminal' villi are presumed to be present at the end of single (in human) or multiple (in mouse) pregnancies. We predict an optimal number of 18 and 22 bifurcation levels in the human and the mouse placentas, respectively. Wherever possible, we have compared our model's predictions with experimental results reported in the literature and found close agreement between them.

Diffusion and perfusion quantified by Magnetic Resonance Imaging are markers of human placenta development in normal pregnancy

PURPOSE

To investigate the potential of bi-exponential model of diffusion-weighted (DW) signal decay to quantify diffusion and perfusion changes in human placenta of normal pregnancies due to its development.

METHODS

26 normal pregnancies at 19-37 weeks of gestation underwent Magnetic Resonance Imaging (MRI) examination at 1.5 T. DW Spin-Echo Echo Planar Imaging with diffusion gradients applied along 3 no-coplanar directions at seven different b-values (0,50,100,150,400,700,1000 s/mm²) was used. Apparent diffusion coefficient (ADC), pseudodiffusion (D*) and perfusion fraction (f) were extracted in selected placenta regions: umbilical (U-ROI), central (C-ROI) and peripheral (P-ROI). The relation between ADC, D*, f and mother age, gestational age (GA), Body-Mass Index (BMI), basal Glycaemia (bG), were evaluated. Pearson correlation with Bonferroni correction was used.

RESULTS

A significant negative correlation was found between ADC and GA, for GA≥30w in P-ROI, while no-dependence of ADC on GA was observed in GA range 19-29 weeks. A positive linear correlation was found between f and GA in the C-ROI and between f and GA in P-ROI for GA≥30 week. No significant correlations were found between ADC, D*, f and age, BMI, bG.

CONCLUSION

ADC measurements in P-ROI of normal placenta reflects tissue changes occurring in the third trimester of gestation. Specifically, ADC decreases with GA increase. Besides, f increases with the GA increase in the C-ROI and during the third trimester of pregnancy in the P-ROI. These results suggest the potential of diffusion and perfusion parameters extracted by using a biexponential model to provide information about placenta changes during its development.

In Vivo Quantification of Placental Insufficiency by BOLD MRI: A Human Study

Fetal health is critically dependent on placental function, especially placental transport of oxygen from mother to fetus. When fetal growth is compromised, placental insufficiency must be distinguished from modest genetic growth potential. If placental insufficiency is present, the physician must trade off the risk of prolonged fetal exposure to placental insufficiency against the risks of preterm delivery. Current ultrasound methods to evaluate the placenta are indirect and insensitive. We propose to use Blood-Oxygenation-Level-Dependent (BOLD) MRI with maternal hyperoxia to quantitatively assess mismatch in placental function in seven monozygotic twin pairs naturally matched for genetic growth potential. In-utero BOLD MRI time series were acquired at 29 to 34 weeks gestational age. Maps of oxygen Time-To-Plateau (TTP) were obtained in the placentas by voxel-wise fitting of the time series. Fetal brain and liver volumes were measured based on structural MR images. After delivery, birth weights were obtained and placental pathological evaluations were performed. Mean placental TTP negatively correlated with fetal liver and brain volumes at the time of MRI as well as with birth weights. Mean placental TTP positively correlated with placental pathology. This study demonstrates the potential of BOLD MRI with maternal hyperoxia to quantify regional placental function in vivo.

Image-Based Modeling of Blood Flow and Oxygen Transfer in Feto-Placental Capillaries

During pregnancy, oxygen diffuses from maternal to fetal blood through villous trees in the placenta. In this paper, we simulate blood flow and oxygen transfer in feto-placental capillaries by converting three-dimensional representations of villous and capillary surfaces, reconstructed from confocal laser scanning microscopy, to finite-element meshes, and calculating values of vascular flow resistance and total oxygen transfer. The relationship between the total oxygen transfer rate and the pressure drop through the capillary is shown to be captured across a wide range of pressure drops by physical scaling laws and an upper bound on the oxygen transfer rate. A regression equation is introduced that can be used to estimate the oxygen transfer in a capillary using the vascular resistance. Two techniques for quantifying the effects of statistical variability, experimental uncertainty and pathological placental structure on the calculated properties are then introduced. First, scaling arguments are used to quantify the sensitivity of the model to uncertainties in the geometry and the parameters. Second, the effects of localized dilations in fetal capillaries are investigated using an idealized axisymmetric model, to quantify the possible effect of pathological placental structure on oxygen transfer. The model predicts how, for a fixed pressure drop through a capillary, oxygen transfer is maximized by an optimal width of the dilation. The results could explain the prevalence of fetal hypoxia in cases of delayed villous maturation, a pathology characterized by a lack of the vasculo-syncytial membranes often seen in conjunction with localized capillary dilations.

An international network (PlaNet) to evaluate a human placental testing platform for chemicals safety testing in pregnancy

The human placenta is a critical life-support system that nourishes and protects a rapidly growing fetus; a unique organ, species specific in structure and function. We consider the pressing challenge of providing additional advice on the safety of prescription medicines and environmental exposures in pregnancy and how ex vivo and in vitro human placental models might be advanced to reproducible human placental test systems (HPTSs), refining a weight of evidence to the guidance given around compound risk assessment during pregnancy. The placental pharmacokinetics of xenobiotic transfer, dysregulated placental function in pregnancy-related pathologies and influx/efflux transporter polymorphisms are a few caveats that could be addressed by HPTSs, not the specific focus of current mammalian reproductive toxicology systems. An international consortium, "PlaNet", will bridge academia, industry and regulators to consider screen ability and standardisation issues surrounding these models, with proven reproducibility for introduction into industrial and clinical practice.

Three-dimensional modeling of human placental terminal villi

INTRODUCTION

Placental transport is the main factor affecting the health and development of the fetus. Due to the placenta's geometrical and mathematical complexity, the structure-function relations of placental terminal villi have not been successfully modeled. Hence, a novel modeling approach is proposed.

METHODS

Computational models of four different specimens were generated from the three-dimensional reconstruction of confocal laser scanning microscopic image stacks. To evaluate the capabilities of the proposed methodology, stationary oxygen diffusion transport was calculated in the terminal villus volumes.

RESULTS

The reconstructions automatically provided the spatial arrangement of the fetal capillaries inside the terminal villi. The surface and volume ratios between the fetal capillaries and the villus were also calculated, and the effects of model parameters on the placental diffusive capacity were assessed by parametric analysis.

DISCUSSION

The potential of three-dimensional reconstructions combined with finite element analysis as a research tool for the human placenta was tested. The methodology herein could serve in the future as a simulation platform for complicated in vivo and in vitro scenarios.