Chris Smith

The Placenta - a Mathematical Model

Figure 1 - The placenta provides an interface between mother and baby, allowing exchange of materials useful for growth and development, gas exchange, and removal of waste products.

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).

Figure 2 - The structure of a villus tree. The yellow background represents the lining of the uterus and maternal blood.

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 tree-like

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 pre-eclampsia 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.

Figure 3 - The position and appearance of the placenta in situ, and after delivery.

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 two-dimensional

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.

Figure 4 - Simplified mathematical model of the placenta examining two-dimensional blood flow within a compartment filled with a porous medium.

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 2-3cm 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 central-eastern 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 theologist-turned mathematician Pierre-Simon 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.

Figure 5 - Darcy's Law and Forcheimer's equation show the relationship between pressure (p) and velocity (u), and can be rewritten as the LaPlace Equation, below.

Solving problems involving complicated two-dimensional geometries

can be challenging, but the process can be simplified by using a

technique called conformal mapping. This involves converting

the two-dimensional 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.

Figure 6 - Modelling the blood flowing through the placenta from the central artery to the two adjacent veins on each side. The analytical solutions, shown on the left, agree very well with the numerical solutions, shown on the right.

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.

Figure 7 - Fluid inertia (Reynold's number - Rek) increases flow penetration into the compartment (y axis)

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.

A developing baby suspended within the amnion and linked to the placenta (left) via the umbilical cord. Image © ADAM

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.

 

- August 2005

About the Author

Chris Smith (the other one) is a maths PhD student in the Dept. of Applied Mathematics at Strathclyde University. Chris is also the president of the Strathclyde University Christian Union, and plays the keyboard for a ceilidh band called "Jiggered".