Given the substantial changes in the more proximal uterine vasculature during pregnancy, the unique conversion of the distal portions of the spiral arteries must have additional physiological advantages other than to add in a small way to the overall reduction in uteroplacental and systemic vascular resistance. In an attempt to answer this question, we have modelled flow in the spiral arteries at term, and will consider the impact of conversion on four aspects of flow into the intervillous space that are critical for placental exchange, namely the rate, pressure, constancy and volume of flow. We will then consider the impact on uterine arterial vascular resistance.
The spiral arteries are modelled as being 10 mm in length, with a pressure drop of 80 mmHg along their length. We have taken the total flow supplied to the uterus to be 750 ml/min 
, although as discussed previously it is now uncertain whether all this volume of maternal blood is delivered into the intervillous space. The volume flow rate Q
through each artery is governed by the local Poiseuille flow relation
) is the local radius of the artery, with x
the downstream distance, p
the pressure and μ
the viscosity of the blood. Here, we have assumed that the change of the radius of the artery over a distance comparable to the radius is small compared to the radius so that, to leading order, the flow is parallel to the downstream direction x
If the radius is constant along the length of the artery then the pressure gradient is constant, and in this case a illustrates the flow rate as a function of the radius. Curves are given for blood viscosities of 3 mPa s (upper curve) and 6 mPa s (lower curve), which span the measurements provided for the end of pregnancy 
. We infer from this figure that for an artery with a radius in the order of 0.2–0.25 mm (diameter 0.4–0.5 mm), as depicted by Harris and Ramsey (c) 
, the flow rate will be about 0.2–0.4 ml/s. With smaller arteries, the flow rate decreases rapidly given the dependence of the flow on the fourth power of the radius. The mean flow speed in the artery is in the order of 1–2 m/s, given the flow rate and cross-sectional area, as indicated in b.
Fig. 4 a) Graphs showing the relationship between the flow volume and the radius of a spiral artery, assuming a length of 1 cm and a pressure drop of 80 mmHg. The upper curves correspond to a viscosity of 3 mPa s and the lower (more ...)
This simple calculation suggests that in the absence of any dilation at the end of the artery a very high speed jet, of 1–2 m/s depending on the viscosity and the exact radius, would enter the intervillous space, and this would have considerable momentum. Such a flow rate would imply a Reynolds number of the flow of about 20–80, and so the viscous stress will dominate the flow resistance.
However, if there is a region of dilation in the artery near the opening into the intervillous space, then the flow speed will decrease, and the overall flow for a given pressure gradient will increase. For example, in the simplified case that there is a linearly divergent section of the artery in the region a
, where x
is the terminal end of the artery, with the radius increasing according to r
) in this terminal zone, from a fixed radius r0
upstream to a final radius r0
), then for a given pressure drop across the artery, the speed of the flow entering the intervillous space decreases substantially as the increase in radius becomes larger, as shown in the blue line in a. The flow has been calculated from the modified relation for the pressure drop as a function of the flow rate Q
Fig. 5 a) Graphs showing the relationship between speed of flow on exit from the artery (solid lines) and Reynolds number (dashed lines) with the radius of the artery. The upper curves correspond to a viscosity of 3 mPa s and the lower curves (more ...)
In turn, the Reynolds number of the jet, given by Re = ur/ν
is the kinematic viscosity of the blood, also decreases as it enters the intervillous space as the amount that the distal end of the artery diverges increases. In a, we assume the divergence occurs in the last 3 mm of the artery, with the radius increasing from an upstream value of 0.25–1.2 mm at the mouth as depicted by Harris and Ramsey 
. The effect of the dilation is to reduce the flow speed substantially down to values in the region of 10 cm/s from the upstream values of 2–3 m/s, owing to the larger flow area.
It is notable that the volume of flow remains similar in both these cases, with the non-diverging artery of radius 0.25 mm supplying a flow of 0.27 ml/s, and that with the dilated end providing 0.37 ml/s in the present calculations. If we assume a flow of 0.2–0.4 ml/s and a total uterine blood flow of 750 ml/min, then we estimate that 30–60 spiral arteries are required to deliver that flow. This is towards the lower end of the range given by Boyd and Hamilton 
, but approximates closely to the estimate of 40–50 of Reynolds 
based on the number of lobules observed. As Boyd and Hamilton comment, counts based on openings through the basal plate alone may overestimate the number of functional arteries. Remodelling during pregnancy can leave some segments of the arteries redundant.
Finally, it is of interest to note how the pressure varies along the spiral artery. It is seen that the pressure losses occur primarily upstream of the dilation zone, and in this distal segment the pressure is comparable to that of the intervillous space (b). This equates with the pressures measured by Moll et al. in the terminal parts of the spiral arteries in the rhesus monkey just before they enter into the intervillous space