Abstract.

The extrusion through a cylindrical nozzle of continuous and uniform filaments of cell-laden hydrogel is important to ensure accurate tissue constructs in 3D bioprinting. The hydrogel typically exhibits yield stress and shear-thinning properties, so we choose Carbopol as a model viscoplastic fluid to perform extrusion experiments. We study the gravity flow of an extruded filament and its breakup to explore how to control filament length and generate uniform filaments. Breakage occurs at an approximately constant distance from the nozzle for modest extrusion flow rates which typically produces short filament segments. In this limit, the experiment provides a ‘syringe test’ to simply measure the extensional yield stress [1]. We focus on larger extrusion flow rates for which we find that the breakup distance from the nozzle rapidly increases with both increasing flow rate and increasing yield stress. We find a similar trend for the length of the detached filament segment. We identify a threshold flow rate for each fluid, beyond which the breakup distance increases rapidly, and interpret this transition in terms of shifts in the balance of viscoplastic, capillary and inertial forces within the filament for low and high values of yield stress.
[1] A. Geffrault, H. Bessaies-Bey, N. Roussel & P. Coussot (2023) J. Rheol., 67 (2), 305–314.

Abstract.

Microcapsules with liquid cores encapsulated by thin membranes have many applications in science, medicine and industry. In this paper, we design a suspension of microcapsules which can flow and deform like red blood cells (RBCs), as a valuable tool to investigate microhaemodynamics. A reconfigurable and easy-to-assemble 3D nested glass capillary device is used to robustly fabricate water-oil-water double emulsions which are then converted into spherical microcapsules with hyperelastic membranes by cross-linking the polydimethylsiloxane (PDMS) layer coating the droplets. The resulting capsules are monodisperse to within 1% and can be made in a wide range of size and membrane thickness. We use osmosis to deflate by 36% initially spherical capsules of diameter 350 μm and a membrane thickness of 4% of their radius. Hence, we can match the reduced volume of RBCs but not their biconcave shape, since our capsules adopt a buckled shape. We compare the propagation of initially spherical and deflated capsules under constant volumetric flow in cylindrical capillaries of different confinements. We find that only deflated capsules deform broadly similarly to RBCs over a similar range of capillary numbers Ca - the ratio of viscous to elastic forces. Similarly to the RBCs, the microcapsules transition from a symmetric ’parachute’ to an asymmetric ’slipper’-like shape as Ca increases within the physiological range, demonstrating intriguing confinement-dependent dynamics. In addition to biomimetic RBC properties, high-throughput fabrication of tunable ultra-soft microcapsules could be further functionalized and find applications in other areas of science and engineering.

Abstract.

The flow and transport of solute molecules in the intricate structure of the placenta make the organ a fetal life-support system.

Abstract.

We investigate the transport of a solute past isolated sinks in a bounded domain when advection is dominant over diffusion, evaluating the effectiveness of homogenization approximations when sinks are distributed uniformly randomly in space. Corrections to such approximations can be non-local, non-smooth and non-Gaussian, depending on the physical parameters (a Péclet number Pe, assumed large, and a Damköhler number Da) and the compactness of the sinks. In one spatial dimension, solute distributions develop a staircase structure for large Pe, with corrections being better described with credible intervals than with traditional moments. In two and three dimensions, solute distributions are near-singular at each sink (and regularized by sink size), but their moments can be smooth as a result of ensemble averaging over variable sink locations. We approximate corrections to a homogenization approximation using a moment-expansion method, replacing the Green’s function by its free-space form, and test predictions against simulation. We show how, in two or three dimensions, the leading-order impact of disorder can be captured in a homogenization approximation for the ensemble mean concentration through a modification to Da that grows with diminishing sink size.

Abstract.

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the microhaemodynamics of a vascular network is underpinned by its interconnected structure, and certain structural alterations such as capillary dilation and blockage can substantially change blood flow patterns. However, for extravascular media with disordered microstructure (e.g. the porous intervillous space in the placenta), it remains unclear how the medium’s structure affects the haemodynamics. Here, we simulate cellular blood flow in simple models of canonical porous media representative of extravascular biological tissue, with corroborative microfluidic experiments performed for validation purposes. For the media considered here, we observe three main effects: first, the relative apparent viscosity of blood increases with the structural disorder of the medium; second, the presence of red blood cells (RBCs) dynamically alters the flow distribution in the medium; third, symmetry breaking introduced by moderate structural disorder can promote more homogeneous distribution of RBCs. Our findings contribute to a better understanding of the cell-scale haemodynamics that mediates the relationship linking the function of certain biological tissues to their microstructure.

Abstract.

The placenta is a vital interface between the mother and her developing fetus. Micro-haemodynamics of the placenta, where the particulate nature of blood flow cannot be ignored, mediates the relationship between the organ's structure and its function. However, the placenta's complex architecture and its relation to pregnancy pathologies remain poorly understood. This review covers current challenges in characterising placental micro-haemodynamics. Recent progress in three-dimensional multiscale imaging has stimulated the development of image-based theoretical models, but existing approaches do not fully harness the available data, and new tools are needed for the assimilation of complex imaging datasets. Although the placenta at term is available for in vivo imaging or ex vivo experimentation, insight into placental micro-rheology is limited, necessitating the use of biomimetic models. Microfluidic approaches offer opportunities for well-controlled characterisation of micro-rheology in complex geometries, but challenges remain in the robust fabrication of these systems. Recent advances in high-performance simulations for suspension flows enable parametrisation of key physical processes at the micro-scale. Future progress can be made by optimising computational architecture and integrating micro-haemodynamics with solute transport. Both experimental and computational approaches require translation to the organ scale. New upscaling approaches will need to accommodate non-local interactions in microvascular network flows and address the lack of clear scale separation across the placental architecture. Together, recent advances in cross-disciplinary imaging and modelling over the last ten years have opened a pathway for an in silico human placenta, accelerating the development of precision obstetrics medicine in the next decade.

Abstract.

Multi-scale structural assessment of biological soft tissue is challenging but essential to gain insight into structure–function relationships of tissue/organ. Using the human placenta as an example, this study brings together sophisticated sample preparation protocols, advanced imaging and robust, validated machine-learning segmentation techniques to provide the first massively multi-scale and multi-domain information that enables detailed morphological and functional analyses of both maternal and fetal placental domains. Finally, we quantify the scale-dependent error in morphological metrics of heterogeneous placental tissue, estimating the minimal tissue scale needed in extracting meaningful biological data. The developed protocol is beneficial for high-throughput investigation of structure–function relationships in both normal and diseased placentas, allowing us to optimize therapeutic approaches for pathological pregnancies. In addition, the methodology presented is applicable in the characterization of tissue architecture and physiological behaviours of other complex organs with similarity to the placenta, where an exchange barrier possesses circulating vascular and avascular fluid spaces.

Abstract.

Multi-scale structural assessment of biological soft tissue is challenging but essential to gain insight into structure-function relationships of tissue/organ. Using the human placenta as an example, this study brings together sophisticated sample preparation protocols, advanced imaging, and robust, validated machine-learning segmentation techniques to provide the first massively multi-scale and multi-domain information that enables detailed morphological and functional analyses of both maternal and fetal placental domains. Finally, we quantify the scale-dependent error in morphological metrics of heterogeneous placental tissue, estimating the minimal tissue scale needed in extracting meaningful biological data. The developed protocol is beneficial for high-throughput investigation of structure-function relationships in both normal and diseased placentas, allowing us to optimise therapeutic approaches for pathological pregnancies. In addition, the methodology presented is applicable in characterisation of tissue architecture and physiological behaviours of other complex organs with similarity to the placenta, where an exchange barrier possesses circulating vascular and avascular fluid spaces.

Abstract.

We are writing to call for caution when interpreting the predictions of computational models at the organ-level in the context of tissue-scale effects. Having read recent work by Donovan and colleagues with great interest, we would like to raise a few points about the use of modelling to understand the mechanisms of therapeutic action of bronchial thermoplasty (BT) in asthma control.

Abstract.

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.

Abstract.

The placenta is a multifunctional 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 length scales. Models describe jets of maternal blood emerging from spiral arteries into a disordered and deformable porous medium, as well as 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

Abstract.

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.

Abstract.

The placenta is crucial for life. It is an ephemeral but complex organ acting as the barrier interface between maternal and fetal circulations, providing exchange of gases, nutrients, hormones, waste products and immunoglobulins. Many gaps exist in our understanding of the detailed placental structure and function, particularly in relation to oxygen handling and transfer in healthy and pathological states in utero. Measurements to understand oxygen transfer in vivo in the human are limited, with no general agreement on the most appropriate methods. An invasive method for measuring partial pressure of oxygen in the intervillous space through needle electrode insertion at the time of Caesarean sections has been reported. This allows for direct measurements in vivo whilst maintaining near normal placental conditions; however, there are practical and ethical implications in using this method for determination of placental oxygenation. Furthermore, oxygen levels are likely to be highly heterogeneous within the placenta. Emerging non-invasive techniques, such as MRI, and ex vivo research are capable of enhancing and improving current imaging methodology for placental villous structure and increase the precision of oxygen measurement within placental compartments. These techniques, in combination with mathematical modelling, have stimulated novel cross-disciplinary approaches that could advance our understanding of placental oxygenation and its metabolism in normal and pathological pregnancies, improving clinical treatment options and ultimately outcomes for the patient.

Abstract.

Bronchial thermoplasty is a treatment for asthma. It is currently unclear whether its histopathological impact is sufficiently explained by the proportion of airway wall that is exposed to temperatures necessary to affect cell survival. Airway smooth muscle and bronchial epithelial cells were exposed to media (37–70°C) for 10 s to mimic thermoplasty. In silico we developed a mathematical model of airway heat distribution post-thermoplasty. In vivo we determined airway smooth muscle mass and epithelial integrity pre- and post-thermoplasty in 14 patients with severe asthma. In vitro airway smooth muscle and epithelial cell number decreased significantly following the addition of media heated to ≥65°C. In silico simulations showed a heterogeneous heat distribution that was amplified in larger airways, with <10% of the airway wall heated to >60°C in airways with an inner radius of ∼4 mm. In vivo at 6 weeks post-thermoplasty, there was an improvement in asthma control (measured via Asthma Control Questionnaire-6; mean difference 0.7, 95% CI 0.1–1.3; p=0.03), airway smooth muscle mass decreased (absolute median reduction 5%, interquartile range (IQR) 0–10; p=0.03) and epithelial integrity increased (14%, IQR 6–29; p=0.007). Neither of the latter two outcomes was related to improved asthma control. Integrated in vitro and in silico modelling suggest that the reduction in airway smooth muscle post-thermoplasty cannot be fully explained by acute heating, and nor did this reduction confer a greater improvement in asthma control.

Abstract.

Workshops are an important part of the IFPA annual meeting as they allow for discussion of specialized topics. At IFPA meeting 2017 there were four themed workshops, all of which are summarized in this report. These workshops discussed new knowledge and technological innovations in the following areas of research: 1) placental bed; 2) 3D structural modeling; 3) clinical placentology; 4) treatment of placental dysfunction.

Abstract.

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.

Abstract.

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.

Abstract.

Despite a large amount of in-vitro data, the dynamics of airway smooth muscle (ASM) mass increase in the airways of patients with asthma is not well understood. Here, we present a novel mathematical model that describes qualitatively the growth dynamics of ASM cells over short and long terms in the normal and inflammatory environments typically observed in asthma. The degree of ASM accumulation can be explained by an increase in the rate at which ASM cells switch between non-proliferative and proliferative states, driven by episodic inflammatory events. Our model explores the idea that remodelling due to ASM hyperplasia increases with the frequency and magnitude of these inflammatory events, relative to certain sensitivity thresholds. It highlights the importance of inflammation resolution speed by showing that when resolution is slow, even a series of small exacerbation events can result in significant remodelling, which persists after the inflammatory episodes. In addition, we demonstrate how the uncertainty in long-term outcome may be quantified and used to design an optimal low-risk individual anti-proliferative treatment strategy. The model shows that the rate of clearance of ASM proliferation and recruitment factors after an acute inflammatory event is a potentially important, and hitherto unrecognised, target for anti-remodelling therapy in asthma. It also suggests new ways of quantifying inflammation severity that could improve prediction of the extent of ASM accumulation. This ASM growth model should prove useful for designing new experiments or as a building block of more detailed multi-cellular tissue-level models.

Abstract.

OBJECTIVES: Despite a large amount of in vitro experimental data, the dynamics of airway smooth muscle (ASM) hyperplasia in the airways of patients with asthma is not well understood. Here, we present a novel mathematical model that describes the growth dynamics of ASM cells over short and long time-scales in both normal and inflammatory environments typically observed in asthma. METHODS: Each ASM cell is assumed as being in a proliferative or non-proliferative state. Switching between the states is modulated by a series of inflammatory events that affect the inflammatory status. We use a combination of analytical and computational techniques to explore possible ASM growth scenarios. MAIN RESULTS: Mild and severe ASM hyperplasia can be explained by an increase in the rate at which ASM cells switch between non-proliferative and proliferative states, driven by episodic inflammatory events. Our model embodies the idea that remodelling due to ASM hyperplasia increases with magnitude and frequency of acute inflammatory events. It also highlights the importance of the rate of inflammation clearance by showing that when resolution of inflammation is slow, even a series of small exacerbation events can result in significant remodelling, which persists after the inflammatory episodes. Finally, we demonstrate how the uncertainty in long-term outcome can be quantified and used to design an optimal low-risk individual anti-proliferative treatment strategy. CONCLUSIONS: The model shows that the rate of clearance of ASM proliferation and recruitment factors after an acute inflammatory event is a potentially important, and hitherto unrecognised, target for anti-remodelling therapy in asthma. It also suggests new ways of quantifying the inflammation severity (e.g. using airway eosinophil density) that could improve the prediction of airway remodelling. This ASM growth model may prove useful for designing new experiments or as a building block of more detailed multi-cellular tissue-level models.

Abstract.

The transport of a scalar quantity in a disordered medium is a common problem in science and engineering. To understand the interplay between deterministic transport dynamics and stochasticity of the underlying microstructure, we analyse a simple model for unidirectional advection-diffusion-reaction over a random array of point sinks. The homogenized concentration distribution over a periodic array provides a leading-order approximation for a wide range of ergodic stationary random sink distributions of comparable mean density. However the fluctuations about this state depend strongly on the statistical properties of the array, the relative sizes of the scale-separation parameter (the ratio of mean inter-sink distance to domain size) and the physical parameters (expressed as dimensionless Péclet and Damköhler numbers). Using a combination of Monte-Carlo simulation and asymptotic analysis, we characterize the spatial variability and correlation statistics of the transported quantity and show how underlying regularity of the microstructure, particularly at low Péclet numbers, ensures a much smaller fluctuation magnitude than in the case of a uniformly random microstructure. Even when sink locations are almost uncorrelated to each other, we find that the concentration fluctuations correlate strongly over lengthscales comparable to the whole domain. Thus boundary conditions can determine the spatial profile of both the averaged leading-order distribution of the transported quantity and its fluctuations.

Abstract.

The growth of a pollen tube, a protuberance of the germinating pollen grain, is crucial for plant reproduction. This growth is extremely rapid and involves targeted intracellular cargo-transport, and the expansion of a pollen tube as a high-pressure vessel strongly depends on the mechanical properties of cell wall [1]. Although the intracellular streaming in plant and animal cells is a common phenomenon [2,3], the understanding of intracellular flow dependence on cell geometry and the role of streaming in transport of solutes and suspended particles is incomplete [4]. In addition, a growing pollen tube responds to a variety of environmental cues, and previous studies showed orientation response of pollen tubes growing in a medium of varying stiffness [5]. In this study, we address the response to the mechanical environment and some aspects of intracellular streaming in pollen tubes, as a representative model of cellular growth. To probe the impact of mechanical stresses - distributed over the pollen tube surface - on the growth of the cell, we have employed controlled perfusion in a microchannel. We have found some indication that the pollen tube can respond to the applied Hele-Shaw flow by reversibly slowing down the growth rate and aligning parallel to the flow streamlines. Due to high internal pressure relative to the created external viscous stress, the growth of the cell thus likely involves a combination of passive and active response mechanisms regulating its adaptation to the mechanical environment. We have also modelled drag-induced cytosol flow inside a pollen tube, using a superposition of distributed Stokeslets and higher-order multipoles. Our model indicates that the relative distance between counter-forcing cytoskeletal filaments as well as the forcing ratio can change both the direction and magnitude of net cytosol motion, which could be of significance for advective intracellular transport. The study was supported by the David Crighton Fund. [1] Rojas, E. R.; Hotton, S. & Dumais, J. (2011) Chemically mediated mechanical expansion of the pollen tube cell wall. Biophys. J. 101:1844-1853. [2] Pickard, W. F. (2003) The role of cytoplasmic streaming in symplastic transport.Plant, Cell & Environment 26:1-15. [3] Kamiya, N. (1981) Physical and chemical basis of cytoplasmic streaming. Annu. Rev. Plant. Physiol. 32:205-236. [4] Verchot-Lubicz, J. & Goldstein, R. (2010) Cytoplasmic streaming enables the distribution of molecules and vesicles in large plant cells. Protoplasma 240:99-107. [5] Gossot, O. & Geitmann, A. (2007) Pollen tube growth: coping with mechanical obstacles involves the cytoskeleton. Planta 226:405-416.

Abstract.

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.

Abstract.

INTRODUCTION. The placenta is an essential element of the life-support system for the developing fetus, enabling nutrients and waste to be exchanged between the fetal and maternal circulations. Unlike most microcirculatory networks, in which blood is confined to narrow capillaries, maternal blood in the human placenta flows freely in the space between the branches of villous trees, within which are fetal vessels. Placental development is vulnerable to maternal diseases such as pre-eclampsia and diabetes, and damage to the fetus due to placental insufficiency can have immediate conseqeunces as well as effects that last into adulthood. It is therefore valuable to understand how blood flow and nutrient exchange are influenced by the complex geometric arrangement of villous branches. METHODS. The functional placental unit is a 'placentone,' which comprises a spiral artery in the uterine wall (the decidua), a single villous tree and nearby decidual veins. In a first approximation, the movement of maternal blood from the spiral artery into the intervillous space can be modelled as a flow in a homogeneous porous medium, using Darcy's law [1] or its variants [2] to describe velocity and pressure fields averaged over many villous branches; an advection/diffusion/reaction equation may also be used to model the macroscopic transport of nutrients such as glucose or oxygen. Formally, both models can be derived using a homogenization approximation that assumes a spatially periodic or uniformly random microstructure. Numerous refinements to this relatively crude approximation may be considered. (i) The rheology of blood in networks of capillaries (treated as bifurcating cylindrical tubes) is reasonably well characterised. Nonlinearities in viscosity and hematocrit transport properties can lead to complex hematocrit dynamics even in simple networks [3,4]; we examine this using coupled nonlinear hyperbolic PDEs distributed over networks. In contrast, blood rheology in the tortuous intervillous space is much less well understood, as are the implications for hematocrit distributions in the placenta. (ii) Villous branches have a complex spatial distribution that requires careful statistical characterisation. We apply two measures to histological images (sampling villous area fraction as a function of window size; and estimating Ripley's K-function for spatial point processes) to establish the minimum distance over which the distribution of villous branches appears statistically homogeneous. (iii) Given the statistical properties of the distribution of villous branches, we assess the accuracy of the leading-order homogenization approximation for nutrient transport. We do so using a simplified model problem (steady solute transport by a unidirectional flow past a distribution of point sinks) which incorporates the influence of two key transport parameters (Péclet and Damköhler numbers, Pe and Da). RESULTS. Modelling maternal blood flow in a placentone using Darcy's law reveals the importance of the calibre of the spiral artery and decidual veins in determining the overall flow resistance of the unit [1]. This may be offset by the presence of a cavity in the villous tree near the exit of the spiral artery. This leading-order model also suggests how the effciency of nutrient transport can be optimized at a suitable villous-branch volume fraction. Analysis of histological data shows that the villous branches appear to have a statistically homogeneous distribution over sufficiently large distances, but a so-called 'hard-core' distribution over shorter distances. For physiologically relevant parameter values, and using periodic and stochastic sink distributions in the model solute transport problem, we find that the difference between the leading-order homogenization approximation and the exact nutrient distribution (the 'homogenization residue') is characterized by large spatial gradients at the scale of individual villi: the leading-order approximation converges only weakly to the true concentration distribution. Furthermore, using both Monte Carlo simulations and a multiple-scales analysis, we find that the residue exhibits substantial spatial fluctuations that can be correlated over lengthscales comparable to the whole domain. In contrast to transport problems with periodic microstructure, disorder in the spatial distribution of villous branches substantially reduces the size of the regime in (Pe,Da)-space in which the homogenization approximation is valid. Simulations accounting for nonlinear rheology in simple capillary networks point to the prevelance of temporal fluctuations in hematocrit; such effects can be expected to further enrich transport phenomena in the placenta. CONCLUSIONS. Our results highlight the importance of quantifying errors due to spatial and temporal disorder in multiscale approximations of transport processes in physiological systems. Leading-order homogenization approximations provide valuable insights of overall function but fail to resolve fine-scale structures which may be of physiological significance. REFERENCES. [1] Chernyavsky, I.L., Jensen, O.E., Leach, L.: A mathematical model of intervillous blood flow in the human placentone. Placenta 31:69-76, 2010. [2] Erian, F.F., Corrsin, S., Davis, S.H.: Maternal placental blood flow: a model with velocity-dependent permeability. J. Biomech. 10:807-814, 1977. [3] Kiani, M.F., Pries, A.R., Hsu, L.L., Sarelius, I.R., Cokelet, G.R.: Fluctuations in microvascular blood flow parameters caused by hemodynamics mechanisms. Am. J. Physiol. 266:65-73, 1994. [4] Pop, S.R., Richardson, G.W., Waters, S.L., Jensen, O.E.: Shock formation and nonlinear dispersion in a microvascular capillary network. Math. Med. Biol. 24:379-400, 2007.

Abstract.

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.

Abstract.

Mathematical formulation of the human placental circulation and metabolic exchange was started more than 30 years ago by Longo et al. [L.D. Longo, H. Bartels (eds), Nat. Inst. Child Health Human Dev., 1972]. Nevertheless, it is still not clear how placental structure and geometry, in particular decidual vasculature distribution, affects intrauterine haemodynamics and solute concentration patterns, which could result in placental insufficiency. The aims of this study were to examine the influence of the relative positions of decidual vessels and maternal blood flow rates in a single placentone on the effectiveness of nutrient uptake, where the placentone is defined as a placental circulatory unit formed by a single villous tree and its corresponding maternal vessels. Our model considers steady filtration of an incompressible viscous liquid in a porous medium, and assumes that advective transport of solutes dominates molecular diffusion. Blood flow velocity and pressure fields are calculated analytically using the theory of images in a hemispherical domain. The concentration field is described by a solute transport equation incorporating different types of uptake kinetics. We investigated the dependence of net nutrient uptake rate on the system parameters. We showed that a ratio of solute consumption rate to maternal blood flow rate imposes a restriction on homogeneous spatial solute distribution. The model suggests that a large distance between the decidual artery and veins provides more effective metabolic exchange, supporting the hypothesis of basal veins' location on the periphery of the placentone. Supported by Marie Curie EST Network MMBNOTT.

Abstract.

Local regulation of blood flow and blood pressure in the cardiovascular system is provided by a complex network of various mechanical and biochemical factors. Fluid shear stress is one of the important regulators mediated via the endothelium-derived relaxing factor (EDRF) that is nitric oxide. This mechanism involves biochemical reactions in the arterial wall. The autoregulation process is managed by the vascular tone and gives a negative feedback to the shear stress change. A new mathematical model for the autoregulation of the arterial lumen under constant transmural pressure is presented. The model takes account of EDRF, the multi-layer structure of an arterial wall, its nonlinear viscoelastic properties, and diffusion-kinetic processes. The exact stationary distributions of NO, Ca²⁺ and phosphorylated myosin in an arterial wall are found. The condition of linear stability for equilibrium state is obtained analytically. Numerical simulation for the autoregulation process is presented. It is shown that there is limit cycle bifurcation at some wall viscosity values outside linear stability range. The transition of the system to a new equilibrium state with a delay in response to the change in a blood flow rate is demonstrated. Slow oscillations of about 0.01 Hz managed by muscle tone are observed. The results are in good agreement with the experiments of Snow, et al. and Melkumyants, et al. carried out in canine and feline arteries.

Abstract.

For describing the autoregulation of the blood flow in an artery under constant transmural pressure a mathematical model that takes into account the multilayer structure of the arterial wall, the diffusion and kinetic processes in the wall, and the nonlinear viscoelastic properties of the wall material is proposed. The limiting case of a thin-walled artery is studied analytically. The arterial-wall viscosity range on which the equilibrium state of the system is stable is estimated. Accurate stationary distributions of the nitric oxide, calcium and myosin concentrations in the arterial wall are found. Numerical simulation of the autoregulation process demonstrated the possibility of arterial adaptation to radius perturbations, the existence of slow oscillations, and system transition to a new equilibrium state with change in blood flow level. The results are in good agreement with the experimental data.

Abstract.

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.