  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.
