What maths can reveal about the function of the organ to which we all owe our existence...Chris Smith
The placenta forms on the uterus wall during pregnancy and, via the umbilical cord, links the baby to its mother. The word placenta means 'cake' in Latin and, appropriately enough, it's job is to nourish the developing baby by transferring nutrients from the mother's bloodstream into the fetal circulation, and removing waste products. The placenta is also responsible for gas exchange, including delivering oxygen to the baby and picking up carbon dioxide, controlling water balance and pH, and producing hormones, such as progesterone, which are involved in maintaining pregnancy. These roles are summarised in figure 1. The placenta is produced by the growing baby and develops throughout pregnancy. By the time the baby is born it resembles a large disc, 20cm across and 3cm thick, weighing about 500g (figure 3, below).
The side of the placenta closest to the baby, known as the fetal side, has the umbilical cord sticking out of it. The opposite face, in contact with the wall of the uterus and known as the maternal side, is readily identified by its distinctive lobes (see figure 3, below). Between the fetal and maternal sides is the "intervillous space" which is divided into compartments by protrusions (called "septa") from the maternal side. The intervillous space is filled with the mother's blood which circulates around treelike structures called villi that are produced by the fetus and contain fetal blood vessels. There is at least one villous tree in each compartment (figure 2, left). This arrangement brings the blood of the developing baby and the blood of the mother very close together without the two actually mixing, allowing nutrients and waste products to be exchanged. Fresh maternal blood continuously flows into the intervillous space from uterine spiral arteries, and drains away through uterine veins, which carry off the fetal waste products for the mother to metabolise. The purpose of the villi is to create a sufficiently large surface area for the exchange process to occur efficiently. Indeed, the placenta has a surface area of at least 14m², and contains over 50km of capillaries (tiny blood vessels). WHY MODEL THE PLACENTA? Clearly, for the baby to develop normally, the placenta needs to function efficiently. However, complications occasionally arise such as when the placenta develops in the wrong position in the uterus (called placenta previa), or separates prematurely from the uterine wall (known as placental abruption). Also, diseases such as diabetes and preeclampsia can affect the maternal and fetal blood flow rates, blood pressure, and villous tree architecture. It's also been shown that when a pregnant mother smokes, the placenta responds by 'closing' the villous trees to limit the delivery of toxins to the baby. But this also has the effect of reducing the normal transfer of essential nutrients and oxygen, which is why smoking during pregnancy can have such serious implications for the health and growth of the developing baby.
We're investigating maternal blood flow through the placenta so that we can better understand how it achieves optimal delivery of oxygen and nutrients, and efficient retrieval of waste and carbon dioxide. It's not easy to view the flow of blood in the placenta, as the group in Nottingham that we are collaborating with have found, but it might be possible to use maths to help us to work out what happens when blood spurts into the placenta from a spiral artery, and drains away into a uterine vein. Obviously, any mathematical model involves simplifying the biological system considerably, and making a number of assumptions, but hopefully it will improve our overall understanding of how the placenta works. MODELLING THE PLACENTA MATHEMATICALLY The first step involved in creating our model was to imagine a steady flow of maternal blood through a single twodimensional compartment filled with a uniform, porous medium (see figure 4, below), which represents a greatly simplified version of a villous tree. The compartment in our model has a central artery flanked symmetrically to either side by two veins.
The central artery, where maternal blood flows in, is represented mathematically by a source of strength of 2m, meaning that fluid enters the compartment with a speed 2m/r, where r is the distance from the source. In other words, the speed of the fluid at any point depends upon how far away it is from the artery. Similarly, the two veins are represented as sinks of strength m. (At these points, fluid is 'sucked out' of the compartment uniformly at a speed of m/r, where r is the distance from the sink.) The veins in our model are symmetrically positioned around the central artery, but could be situated anywhere from next to the septum, to adjacent to the artery. The height of the compartment in our model is also adjustable but is typically about 23cm in reality. The next step was to use a formula developed by a nineteenth century Frenchman, Henry Philibert Gaspard Darcy, who was was an Inspecteur Général de Ponts et Chaussées (Inspector General of Bridges and Roadways). In 1856 he published a report on the public fountains and sewer systems of Dijon, in centraleastern France, which included a description of observations he'd made and experiments he'd performed that led him to come up with an equation now known as Darcy's Law (see figure 5, below), which is a cornerstone of the study of hydrology and how liquids flow through porous media. To simplify matters, Darcy's Law for an incompressible fluid, which relates two variables  pressure (p) and velocity (u), can be rewritten as Laplace's Equation (see figure 5, below), named after French theologistturned mathematician PierreSimon de Laplace, which deals with just a single variable at a time  either pressure, or the flow streamlines (related to the velocity), which is much easier to use.
Solving problems involving complicated twodimensional geometries can be challenging, but the process can be simplified by using a technique called conformal mapping. This involves converting the twodimensional geometries into simpler forms, which are easier to solve, and then returning to the original geometry again. The same trick can be used, for example, to predict the flow of air around an aeroplane wing. The problem is converted into one involving flow around a circular cylinder, so that each point on the wing is mapped, in other words corresponds, to a point on the cylinder. The resulting problem is much easier to solve. Similarly, applying this technique to our placental model provided exact formulae for the blood flow that we could plot, enabling us to check the results later. This method allowed us to see what the flow would look like if the placental compartment were infinitely wide, or infinitely deep  in other words a box with no sides, or no top. While these aren't situations you'd ever want in practice, this approach allowed us to check the results against the numerical work we had already done. Using FEMLAB, a powerful, interactive computing environment for modelling and solving scientific and engineering problems, we found that the 'topless' model was in excellent agreement with the numerical calculations for the complete compartment, as highlighted by figure 6, below.
This analysis is fine as far as it goes, but suffers from one serious shortcoming, namely that it is true only if the blood flows relatively slowly. In practise, blood spurts into the placenta very quickly, so it's also important to consider the inertia, or momentum, of the blood entering the compartment. The more inertia the blood has as it enters the placenta, the less likely it is to slow down when it flows round the villi, and so the further it is likely to penetrate into the compartment. So, in order to include the effects of fluid inertia in the model, and to find out if including these effects makes any significant difference to the penetration of the blood into the placenta, we used the Forchheimer Equation (see figure 5, above), which dates from 1901 and is an extension of Darcy's Law to include fluid inertia effects.
Next, we combined all the parameters in our model (such as viscosity µ, permeability k and density ρf) into a single value called the Reynolds number, denoted by Rek, which is a measure of the inertia of the flow. Reynolds Numbers are often used in mathematical modelling to describe the ratio of inertial forces to viscous forces. In order to calculate realistic values of Rek for the flow in the placenta we needed to find out the values of all the parameters in our model. Some of these are rather difficult to ascertain, but our best estimate is that Rek lies somewhere between between 1 and 10. We used our model to examine the effect of varying the Reynolds Number (Rek), and found that values in the physically realistic range result in significantly greater penetration into the compartment. But more importantly, we found that increasing Rek beyond the range of realistic values does not increase the penetration any further; in other words, in a real placenta, blood flows in at the optimum speed to achieve maximum penetration without wasting energy. Of course, the role of the placenta is to exchange substances, such as oxygen and carbon dioxide, between the mother and the baby. So we're now using the flow patterns that we have discovered to learn more about how this exchange works, using an equation that describes the convection and diffusion of oxygen in maternal blood, coupled to a simple expression for the diffusion of oxygen through the villi. Hopefully we'll be able to report on more developments on this mathematically exciting, and medically important problem in the near future.
ACKNOWLEDGEMENTS My PhD supervisors are Prof. Stephen Wilson and Dr. Brian Duffy of the University of Strathclyde, Glasgow, and the work is funded by the Carnegie Trust for the Universities of Scotland. References
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