The human placenta supports fetal development by mediating the exchange of gases, nutrients, drugs, and waste across a highly heterogeneous microscale interface formed by terminal villi. While placental transport has long been studied using simplified multiscale models, progress has been constrained by limited access to realistic microstructure geometries, uncertainty in barrier composition, and difficulties linking local geometries to organ-level measurements. Here, we review recent advances in placental imaging and mathematical modelling that are enabling better characterisation of microscale exchange. We outline how multimodal structural imaging, including X-ray phase-contrast approaches, microCT, optical microscopy and volume electron microscopy, can quantify villous and vascular architecture across scales. We discuss modelling frameworks informed by these data, including descriptions of maternal and fetal haemodynamics, image-based exchange-capacity concepts, and compartmental approaches that incorporate delivery limitations. We also discuss recent advances in the understanding of transporter-mediated transfer, and the emerging role of spatial 'omics in parametrising membrane function and regional heterogeneity. We conclude by identifying likely next steps and ways forward for image-based models of exchange in the human placenta.

The placenta is a fundamental organ for human reproduction, facilitating fetal growth via the exchange of oxygen, nutrients and waste products between mother and fetus, by means of a dense network of fetal villi bathed in maternal blood that flows through the intervillous space (IVS). However, despite its role in adverse pregnancy outcomes associated with impaired maternal-fetal transfer, the influence of placental structure on maternal haemodynamics and the delivery of oxygen and nutrients to the developing fetus is not well understood. This study employs computational fluid dynamics within physiologically informed 2D and 3D whole-organ-scale placental geometries, whose features are informed by recent ex vivo experimental and micro-CT data, to examine comprehensively the influence of placental anatomy on maternal flow and transport in the IVS. In particular, we consider in detail the impact of the number and placement of maternal decidual arteries and veins that supply and drain the IVS, the location and height of so-called septal walls that loosely separate the placenta into functional units (cotyledons) and sub-units (lobules), and the density of the fetal villous trees as reflected in the rate of uptake of dissolved solutes from the maternal blood and the resistance to flow. We first exploit the computational efficiency of simulation in a representative 2D geometry to study in detail the sensitivity of haemodynamic markers to these parameters. These results guide our 3D study which reveals that the flow, transport and oxygen uptake are strongly influenced by placental structure, and exposes the vein-to-artery ratio as a key indicator of placental efficiency, regulating a trade-off between a preferential maternal flow environment and fetal oxygen uptake. Conversely, the location and height of the septal walls, a feature that is not well studied, have minimal systematic impact on the macroscopic haemodynamic and transport measures considered here. We also introduce a reduced model, for which analytical progress can be made, and demonstrate its utility in exposing key drivers of maternal-fetal transport. ### Competing Interest Statement The authors have declared no competing interest.

Melanoma is a cancer of the melanocyte, known to have an ability to readily switch between different transcriptional cell states that convey different phenotypic properties (e.g. hyper-differentiated, neural crest-like). This ability is believed to underpin intratumour heterogeneity and plastic adaptation, which contributes to resistance to therapy and immune evasion of the tumour. Therefore, understanding the mechanisms underlying acquisition of transcriptional cell states and cell-state switching is crucial for the development of therapies. We model a minimal gene regulatory network comprising three key transcription factors, whose varying gene expression encodes different melanoma cell states, and use deterministic spatiotemporal differential-equation models to study gene-expression dynamics. We exploit an approximation, based on cooperative binding of transcription factors, in which the models are piecewise-linear. We classify stable states of the local model in a biologically relevant manner and, using a naïve model of intercellular communication, we explore how a population of cells can take on a shared characteristic through travelling waves of gene expression. We derive a condition determining which characteristic will become dominant, under sufficiently strong cell-cell signalling, which creates a partition of parameter space.

We investigate gas injection into axisymmetric water-saturated porous channels with Gaussian and parabolic profiles, as idealized models of underground gas storage in dome-shaped anticlines. Exploiting the slenderness of each channel, we derive an evolution equation for the gas/liquid interface using a composite asymptotic approximation that accommodates large channel slopes and has a simplified small-slope form describing spreading in weakly curved channels. In the high gas-mobility limit, in contrast with flat planar channels, buoyancy influences the dynamics through different mechanisms in each geometry. For gas injected steadily into a Gaussian channel, buoyancy can continually affect the flow due to the attenuation of the gas velocity caused by axisymmetry. In parabolic channels, the increasing channel slope ensures that buoyancy eventually influences the flow, at a timescale depending on injection rate and fluid properties. Asymptotic analysis of the parabolic-channel flow reveals five temporal regimes, each with multiple spatial regions and a distinct spreading rate, reflecting the evolving spatiotemporal competition between injection and buoyancy. Initially, a thin film of gas spreads along the upper boundary; the channel slope and elongation of the film then generate a hydrostatic pressure gradient, which strengthens until buoyancy arrests the upper contact line and thickens the film. Beneath the film, liquid then drains until the interface flattens under buoyancy. Analytical solutions of reduced-order models capture interface evolution and contact-line motion through each regime and are validated against full numerical simulations. These results have implications for subsurface hydrogen and CO_2 storage, where a horizontal interface that advances vertically enhances both safety and storage efficiency.

The Golgi apparatus has an intricate spatial structure characterized by flattened membrane-bound compartments, known as cisternae. Cisternae house integral membrane enzymes that catalyse glycosylation, the addition of polymeric sugars to protein cargo, which is important for the trafficking and function of the products. The unusual and specific shape of Golgi cisternae is highly conserved across eukaryotic cells, suggesting significant influence in the correct functioning of the Golgi. Motivated by experimental evidence that disruption to Golgi morphology can lead to observable changes in secreted cargo mass distribution, we develop and analyse a mathematical model of polymerisation in a cisterna that combines chemical kinetics, spatial diffusion and adsorption and desorption between lumen and membrane. Exploiting the slender geometry, we derive a non-local non-linear advection-diffusion equation that predicts secreted cargo mass distribution as a function of cisternal shape. The model predicts a maximum cisternal thickness for which successful glycosylation is possible, demonstrates the existence of an optimal thickness for most efficient glycosylation, and suggests how kinetic and geometric factors may combine to promote or disrupt polymer production.

INTRODUCTION:Stereological estimates of villous membrane thickness and surface area are widely used to infer the diffusive exchange capacity of the human placenta. A key geometric determinant of exchange capacity can be expressed as an effective diffusive length scale. Here we combine virtual histological sections with computational modelling in realistic villous geometries to assess the accuracy of classical stereological estimates of this diffusive length scale. METHODS:Two terminal villi, reconstructed from three-dimensional imaging, were digitally sectioned to generate random two-dimensional geometries containing fetal capillaries and surrounding villous tissue. For each section, we simulated steady diffusive transport between the fetal-capillary and intervillous-space boundaries to obtain a physics-based diffusive length scale as a reference case. Using the same geometries, we applied standard line-intercept stereology to measure harmonic-mean barrier thickness and boundary-length densities, from which a stereological estimate of diffusive length scale was derived. RESULTS:Across both villi, stereology systematically overestimated the diffusive length scale by approximately 15-25%, depending on villus and section. We identified sources of this discrepancy, including interface curvature and assumptions underpinning the stereological correction factors, using idealised models of villous structure. CONCLUSION:These findings highlight the need for stereological approaches that account for curvature when interpreting placental structure-function relationships.

Multicellular tissues, such as the epithelium coating a developing embryo, often combine complex tissue shapes with heterogeneity in the spatial arrangement of individual cells. Discrete approximations, such as the cell vertex model, can accommodate these geometric features, but techniques for analysis of such models are underdeveloped. Here, we express differential operators defined on a network representing a monolayer of confluent cells in a framework inspired by discrete exterior calculus, considering scalar fields defined over cell vertices and centres and vector fields defined over cell edges. We achieve this by defining Hodge stars, wedge products and musical isomorphisms that are appropriate for a disordered monolayer for which cell edges and links between cell centres are not orthogonal, as is generic for epithelia. We use this framework to evaluate the harmonic vector field arising in an ablated planar monolayer, demonstrating an approximate 1/r scaling of the upper bound of the field’s amplitude, where r is the distance from the ablation. Using a vertex model that incorporates osmotic effects, we then calculate the mechanical response of a monolayer in a jammed state to ablation. Perturbation displacements exhibit long-range coherence, monopolar and quadrupolar features, and an approximate 1/r near-hole upper-bound scaling, implicating the harmonic field. The upper bounds on perturbation stress amplitudes scale approximately like 1/r^2 , a feature relevant to long-range mechanical signalling.

Underground gas storage is a critical technology in global efforts to mitigate climate change. In particular, hydrogen storage offers a promising solution for integrating renewable energy into the power grid. When injected into the subsurface, hydrogen's low viscosity compared with the resident brine causes a bubble of hydrogen trapped beneath caprock to spread rapidly into an aquifer through release of a thin gas layer above the brine, complicating recovery. In long aquifers, the large viscous pressure drop between source and outlet induces significant pressure variations, potentially leading to substantial density changes in the injected gas. To examine the role of gas compressibility in the spreading dynamics, we use long-wave theory to derive coupled nonlinear evolution equations for the gas pressure and gas/liquid interface height, focusing on the limit of long domains, weak gas compressibility and low gas/liquid viscosity ratio. Simulations are supplemented with a comprehensive asymptotic analysis of parameter regimes. Unlike the near-incompressible limit, in which gas spreading rates are dictated by the source strength and viscosity ratio, and compressive effects are transient, we show how compression of the main gas bubble can generate dynamic pressure changes that are coupled to those in the thin gas layer that spreads over the liquid, with compressive effects having a sustained influence along the layer. This coupling allows compressibility to reduce spreading rates and gas pressures. We characterise this behaviour via a set of low-order models that reveal dominant scalings, highlighting the role of compressibility in mediating the evolution of the gas layer.

The umbilical cord plays a critical role in delivering nutrients and oxygen from the placenta to the fetus through the umbilical vein, while the two umbilical arteries carry deoxygenated blood with waste products back to the placenta. Although solute exchange in the placenta has been extensively studied, exchange within the cord tissue has not been investigated. Here, we explore the hypothesis that the coiled structure of the umbilical cord could strengthen diffusive coupling between the arteries and the vein, resulting in a functional shunt. We calculate the diffusion of solutes, such as oxygen, and heat in the umbilical cord to quantify how this shunt is affected by vascular configuration within the cord. We demonstrate that the shunt is enhanced by coiling and vessel proximity. Furthermore, our model predicts that typical vascular configurations of the human cord tend to minimize shunting, which could otherwise disrupt thermal regulation of the fetus. We also show that the exchange, amplified by coiling, can provide additional oxygen supply to the cord tissue surrounding the umbilical vessels.

We use a three-dimensional formulation of the cell vertex model to describe the mechanical properties of a confluent planar monolayer of prismatic cells. Treating cell height as a degree of freedom, we reduce the model to a two-dimensional form. We show how bulk effects, associated with cell volume and total surface area, lead to coupling between energy variations arising from changes in the cell apical area and the apical perimeter, a feature missing from standard implementations of the two-dimensional vertex model. The model identifies five independent mechanisms by which cells can lose in-plane rigidity, relating to variations in total cell surface area, the strength of lateral adhesion, and constrictive forces at the apical cortex. The model distinguishes bulk from in-plane stresses, and it identifies two primary measures of cell shear stress. In the rigid regime, the model shows how lateral crowding in a disordered isolated monolayer can lead to cell elongation towards the monolayer center. We examine the loss of in-plane rigidity in a disordered monolayer and connect isolated patches of stiffness that persist during the rigidity transition to the spectrum of a Laplacian matrix. This approach enables bulk mechanical effects in an epithelium to be captured within a two-dimensional framework.

The umbilical cord exhibits striking structural diversity across placental mammals; however, the consequences of such variation on physiological function are poorly understood. Combining comparative anatomy across 130 mammal species with ancestral-trait reconstruction, multimodal imaging, and physics-based transport modelling, we show that the human umbilical cord structure is uncommon in other mammals and that the ancestral umbilical cord was unspiralled, with three vessels and an allantoic duct. Across species, vessel number and cord length correlate with birth weight: four-vessel cords and the persistence of an allantoic duct occur in species with heavier neonates, while two-vessel cords are confined to small muroid rodents. Mathematical modelling predicts that oxygen delivery remains near maximal across all cord architectures, but placental heat removal depends strongly on umbilical flow, and moderately on cord length and coiling. Scaling of umbilical flow with birth weight implies that larger species approach near-maximal heat-removal capacity, while smaller species are less efficient. Finally, three-dimensional imaging of the human cord reveals a triple-helix configuration of umbilical vessels, rather than arteries simply twisting around a central vein. This, and other observed cross-sectional vessel configurations in mammals and atypical human cords, aligns with theoretically predicted arrangements that minimize inter-vessel shunting, supporting an adaptive role for cord geometry in thermal regulation function. Together, these findings reframe the umbilical cord as a heat-exchange system and link its structural diversity to evolutionary history, neonatal mass, and maternal investment, with implications for fetal resilience to environmental stress. ### Competing Interest Statement The authors have declared no competing interest. BBSRC, BB/Y005953/1 EPSRC, EP/T02593X/1 A*STAR

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared with the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk–surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

Tissue crowding represents a critical challenge to epithelial tissues, which often respond via the irreversible process of live cell extrusion. We report that apical size reduction via macropinocytosis serves as a malleable and less destructive form of tissue remodeling that can alleviate the need for cell loss. We find that macropinocytosis is triggered by tissue crowding via mechanosensory signaling, leading to substantial internalization of apical membrane. This drives a reduction in apical surface which alleviates crowding. We report that this mechanism regulates the long-term organization of the developing epithelium and controls the timing of proliferation-induced cell extrusion. Additionally, we observe a wave of macropinocytosis in response to acute external compression. In both scenarios, inhibiting macropinocytosis induces a dramatic increase in cell extrusion suggesting cooperation between cell extrusion and macropinocytosis in response to both developmental and external compression. Our findings implicate macropinocytosis as an important regulator of dynamic epithelial remodeling.

Experiments have shown that surfactant introduced to a liquid-filled maze can find the solution path. We reveal how the maze-solving dynamics arise from interactions between the added surfactant and endogenous surfactant present at the liquid surface. We simulate the dynamics using a nonlinear model solved with a discrete mimetic scheme on a graph. Endogenous surfactant transforms local spreading into a non-local problem with an omniscient view of the maze geometry, key to the maze-solving dynamics. Our results offer insight into surfactant-driven transport in complex networks such as lung airways.

Liquid films coating vertical cylinders can form annular liquid collars which translate downwards under gravity. We investigate the dynamics of a thin viscoplastic liquid film coating the interior or exterior of a vertical cylindrical tube, quantifying how the yield stress modifies both the Rayleigh-Plateau instability leading to collar formation and the translation of collars down the tube. We use thin-film theory to derive an evolution equation for the layer thickness, which we solve numerically to examine the nonlinear dynamics. Instability and collar formation occur when gravity is sufficiently strong to make the fluid yield initially. We use matched asymptotics to derive a model describing the quasi-steady translation of a slender liquid collar when the Bond number is small. The structure of the asymptotic solution for a viscoplastic collar shares some features with the Newtonian version, but there are several novel asymptotic regions that emerge at the two ends of the collar. The global force balance, which determines the collar's speed, is modified by a leading-order contribution from viscous drag in the collar when the liquid is viscoplastic. We use the asymptotic model to describe slow changes in collar volume when the film thicknesses ahead of, and behind, the collar are unequal. When the film thickness ahead of the collar is less than a critical value that we determine, viscoplastic collars adjust their volume and reach a steadily-translating state. This contrasts with the Newtonian problem, where the only state in which steady translation occurs is unstable to small changes in the film thickness.

Airway constriction and blockage in obstructive lung diseases cause ventilation heterogeneity and create barriers to effective drug deposition. Established computational particle-deposition models have not accounted for these impacts of disease. We present a new particle-deposition model that calculates ventilation based on the resistance of each airway, such that ventilation responds to airway constriction. The model incorporates distal airway constrictions representative of cystic fibrosis, allowing us to investigate the resulting impact on patterns of deposition. Unlike previous models, our model predicts how constrictions affect deposition in airways throughout the lungs, not just in the constricted airways. Deposition is reduced in airways directly distal and proximal to constrictions. When constrictions are clustered together, central-airways deposition can increase significantly in regions away from constrictions, but distal-airways deposition in those regions remains largely unchanged. We use our model to calculate lung clearance index (LCI), a clinical measure of ventilation heterogeneity, after applying constrictions of varying severities in one lobe. We find an increase in LCI coinciding with significantly reduced deposition in the affected lobe. Our results show how the model provides a framework for development of computational tools that capture the impacts of airway disease, which could significantly affect predictions of regional dosing.

This report relates to a study group hosted by the EPSRC funded network, Integrating data-driven BIOphysical models into REspiratory MEdicine (BIOREME), and supported by The Insigneo Institute and The Knowledge Transfer Network. The BIOREME network hosts events, including this study group, to bring together multi-disciplinary researchers, clinicians, companies and charities to catalyse research in the applications of mathematical modelling for respiratory medicine. The goal of this study group was to provide an interface between companies, clinicians, and mathematicians to develop mathematical tools to the problems presented. The study group was held at The University of Sheffield on the 17 - 20 April 2023 and was attended by 24 researchers from 13 different institutions. This report relates to a challenge presented by Arete Medical Technologies relating to impulse oscillometry (IOS), whereby a short pressure oscillation is imposed at a person's mouth during normal breathing, usually by a loudspeaker. The resulting pressure and flow rate changes can be used to the impedance of the airways, which in turn can provide proxy measurements for (patho)physiological changes in the small airways. Disentangling the signal so that airway mechanics can be measured accurately (and device properties/environmental effects can be accounted for) remains an open challenge that has the potential to significantly improve the device and its translation to clinic. In this report, several approaches to this problem, and the wider problem of interpreting oscillometry resuts are explored.

Cell division orientation, which is key to cell fate and shaping tissues, is influenced by mechanical forces. For example, in uniaxially stretched tissue most divisions orient along the stretch axis. This mechanosensitive division orientation can be achieved by distinct mechanisms involving a direct response to force or an indirect response to force via cell shape changes. Cells also generally align divisions along their long axis of interphase shape. Since stretching a tissue elongates cells, uncoupling whether divisions orient in response to force or to cell shape remains challenging. Here, we utilised the Xenopus laevis embryonic animal cap tissue, to which a reproducible external stretch can be applied, to understand mechanosensitive division orientation. We focused on nuclear mitotic apparatus protein (NuMA), which is key to spindle orientation. We show that NuMA is dynamically localised to the cell cortex earlier during mitosis in uniaxially stretched tissues. This temporal cortical recruitment of NuMA coincides with the onset and subsequent stretch-induced amplification of spindle oscillations. Additionally, we show that knockdown of NuMA reduces spindle oscillations and disrupts division orientation according to stretch and cell shape. We also show that divisions dynamically align with cell shape under tissue stretch and that this involves a direct response to force via NuMA localisation, rather than an indirect response to force via cell shape changes. Overall, using live tissue imaging, mathematical modelling, and tissue mechanics our results indicate that NuMA fine tunes spindle dynamics and ensures the accurate alignment of divisions with cell shape and anisotropic tension. ### Competing Interest Statement The authors have declared no competing interest.