A multiscale analysis of flow and transport in the human placenta

The human placenta is characterised by a unique circulatory arrangement, with numerous villous trees containing fetal vessels immersed in maternal blood. Placental tissue therefore manifests a multiscale structure balancing microscopic delivery of nutrients and macroscopic flow. The aims of this study are to examine the interaction between these scales and to understand the influence of placental organisation on the effectiveness of nutrient uptake, which can be compromised in pathologies like pre-eclampsia and diabetes. We first systematically analyse solute transport by a unidirectional flow past an array of microscopic sinks, taking up a dissolved nutrient or gas, for both regular and random sink distributions. We classify distinct asymptotic transport regimes, each characterised by the dominance of advective, diffusive or uptake effects at the macroscale, and analyse a set of simplified model problems to assess the accuracy of homogenization approximations as a function of governing parameters (Peclet and Damkohler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is determined by large spatial gradients at the scale of individual villi (depending on transport parameter values) and substantial fluctuations that can be correlated over lcngthscales comparable to the whole domain. In addition, we consider the nonlinear advective effects of solute-carriers, such as red blood cells carrying oxygen. Homogenization of the solute-carrier-facilitated transport introduces an effective Peclet number that depends on the slowly varying leading-order concentration, so that an asymptotic transport regime can be changed within the domain. At large Peclet and Damkohler numbers (typical for oxygen transport in the human placenta), nonlinear advection due to solute-carriers leads to a more uniform solute distribution than for a linear carrier-free transport, suggesting a "homogenizing" effect of red blood cells on placental oxygen transport. We then use image analysis and homogenization concepts to extract the effective transport properties (diffusivity and hydraulic resistance) from the microscopic images of histological sections of the normal human placenta. The resulting two-dimensional tensor quantities allow us to assess the anisotropy of placental tissue for solute transport. We also show how the pattern of villous centres of mass can be characterised using an integral correlation measure, and identify the minimum spatial scale over which the distribution of villous branches appears statistically homogeneous. Finally, we propose a mathematical model for maternal blood flow in a placental functional unit (a placentone), describing flow of maternal blood via Darcy's law and steady advective transport of a dissolved nutrient. An analytical 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. Predictions of the model agree with experimental radioangiographic studies of tracer dynamics in the intervillous space. The model supports the hypothesis that basal veins are located on the periphery of the placentone in order to optimise delivery of nutrients. We also explain the importance of dilatation of maternal spiral arteries and suggest the existence of an optimal volume fraction of villous tissue, which can both be involved in the placental dysfunction. Theoretical studies of this thesis thus constitute a step towards modelling-based diagnostics and treatment of placental disorders