A Mathematical Model of Intervillous Blood Flow in the Human Placentone

(a) A cast of a large villous tree from the human term placenta (incomplete injection, reproduced from [8] with permission), showing differences in density of villi forming a central cavity (top). (b) A schematic (modified from [7]) of the placentone at term, enclosed between the basal (BP) and chorionic (CP) plates. Maternal blood enters the central cavity (CC) via the spiral artery (SA) and is drained through the decidual veins (DV) at the periphery. Note the anchoring stem villi (AV) surrounding the CC and connecting the CP with the BP. The dashed lines show the central cavity and notional outer boundary of the placentone. The intermediate and terminal villi are not shown.

A schematic placental circulatory unit confined between the plane S₁ and hemispherical surface S₂ of radius L. Maternal blood enters from a source (artery) at z=0 and exits through sinks (veins) at z=±z_v, as indicated by arrows: (a) a homogeneous intervillous space; (b) including a central cavity. In the cylindrical coordinate system, z measures distance along the axis in S₁ on which the spiral artery and decidual veins lie; r measures distance perpendicular to this axis. The flow is assumed axisymmetric, i.e. independent of the azimuthal angle θ measured in a plane perpendicular to the z-axis from the plane S₁.

(a,b) predicted streamlines (blue) and constant pressure lines (green) for steady flow in a homogeneous placentone plotted in a cross-section perpendicular to the z-axis holding the basal vessels, where r is a distance from the axis of symmetry. Arrows indicate the direction of flow. (c,d) Intervillous blood pressure along the z-axis (r≃0.005cm, P_ref=5mmHg, q₀=5ml/min). The decidual veins are located either near the centre ((a,c), h=0.45) or near the periphery ((b,d), h=0.9) of the placentone.[For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.].

Tracer dynamics in the placentone: (a) leading and trailing tracer fronts (blue dots) and the shell thickness profile (solid red, representing x-ray intensity) 15s following the appearance of the tracer from the spiral artery (shown in a sagittal section through the placentone; the decidual veins lie at z≃±2cm); (b,c) computed intensity of the tracer (grayscale) in the orthogonal projection on the basal plate 15 s (b) and 25 s (c) after the introduction of tracer to the model (white dots indicate the location of the spiral artery and decidual veins); (d) dependence of cross-sectional area of tracer bolus on time; (e) serial radioangiographic film of monkey uterus 18.5s following injection of Renografin (arrows indicate ring-like structures), reproduced from [1].[For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.].

Streamlines (white lines) and normalised concentration field (C/C₀, colours) in the placentone for different values of uptake parameter and relative position of the decidual veins (arrows): (a) small source-sink distance (Da=1, h=0.45); (b) veins near the periphery (Da=1, h=0.9); (c) low uptake rate (Da=0.25, h=0.9); (d) high uptake rate (Da=4, h=0.9). The corresponding values of the relative net uptake rate N_r are (a) 0.43, (b) 0.68, (c) 0.33, (d) 0.94, and of the absolute net uptake rate N_a/q₀C₀ are (a) 0.43, (b) 0.68, (c) 1.34, (d) 0.24.[For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.].

Flow streamlines (white lines) and concentration distribution (colours) in a hemisphere in the presence of the central cavity (assuming first-order uptake kinetics, h=0.9): (a) a small cavity of effective radius R≃0.6cm with Da≃0.43 outside the cavity and relative net uptake rate N_r≃0.45; (b) a large cavity of effective radius R≃1cm with Da≃0.39 and N_r≃0.37.