Igor L. Chernyavsky

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 Darcys 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.

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.

The Role of Inflammation Resolution Speed in Airway Smooth Muscle Mass Accumulation in Asthma: Insight from a Theoretical Model

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.

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.

Human placental oxygenation in late gestation: experimental and theoretical approaches

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.

In vitro, in silico and in vivo study challenges the impact of bronchial thermoplasty on acute airway smooth muscle mass loss

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. Bronchial thermoplasty treatment for asthma has unexpected possible mechanisms of action http://ow.ly/ZcuE30jsaSa

IFPA meeting 2017 workshop report: clinical placentology, 3D structure-based modeling of placental function, placental bed, and treating placental dysfunction

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.