Biomathematics Seminar: Transport in the human placenta: homogenizing advection-diffusion-uptake in a disordered medium
12 March 2013 14:00 in Department of Mathematical Sciences
Homogenized descriptions of heterogeneous media have long attracted attention since the first analytical estimates of bulk electrical conductivity by Maxwell and Rayleigh. An interesting and important biomedical application is found in solute transport in the human placenta, an interface between the mother and the foetus, which is characterised by a unique multi-scale arrangement of densely packed fetal tissues, bathed by maternal blood that delivers nutrients and removes waste products. While maternal placental blood flow can be described using established models of transport in porous media, estimating the accuracy of these models poses a challenging problem. To understand the interplay between deterministic transport dynamics and stochasticity of the microstructure, we analyse a simple model for advection-diffusion-uptake over a random array of point sinks, characterised by the Peclet number Pe (relating the strength of advection to diffusion), Damkohler number Da (relating the rate of uptake to diffusion) and the statistics of the sink distribution. Imaging of normal placental tissue and spatial statistics are used to motivate the choice of sink distributions. We systematically map transport regimes in (Da,Pe)-parameter space and identify regions where random fluctuations or large micro-scale gradients of concentration have a significant impact on the averaged description. Our analysis shows that even when the sink locations in a medium are uncorrelated, the difference between the exact and homogenized concentration profile can exhibit a strong long-range correlation. The global boundary conditions can thus determine not only the spatial profile of averaged solute distribution but also the degree of uncertainty in this prediction.
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