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.