Mathematical modelling

1
Placental blood flow
M. D. Finn1, L. Leach2, P. A. Gowland3 and O. E.
Jensen1 School of Mathematical Sciences1, School
of Biomedical Sciences2, School of Physics and
Astronomy3, University of Nottingham,
University Park, Nottingham NG7 2RD, UK
Figure 4 Advection-diffusion
simulations with solute absorption by the villous
tree. Solute-rich blood (indicated in red) enters
through the spiral artery. Solute is absorbed by
the villi causing the concentration to fall as
the blood filters back to the veins. The
left-hand plot corresponds to low Reynolds number
flow where the lower villi receive more solute.
The right-hand plot corresponds to a larger
Reynolds number with solute delivered deep into
the lobule. Magnetic resonance imaging Numerical
simulations of the flow are to be compared with
magnetic resonance imaging of the velocity field
inside artificially perfused, delivered
placentas. Using a 0.5 Tesla body scanner and
specially adapted perfusion apparatus we have
made preliminary scans at a resolution of around
3mm. Using a 3 Tesla coil we hope to improve this
resolution to 1mm, which will provide us with
images that can be compared to our numerics. We
are also considering other methods for measuring
the flow, such as laser Doppler imaging.
Mathematical modelling Blood flow is simulated
computationally using the Lattice-Boltzmann (LB)
method in an idealised, two-dimensional placental
lobule. In the LB method, the velocity field is
discretised onto a square lattice, and written in
terms of a number of densities representing the
motion of particles in the different lattice
directions 5. Time-stepping is performed by
streaming the densities between adjacent lattice
points and allowing for particle collisions which
dissipate energy. It can be shown that the LB
method produces solutions to the Navier-Stokes
equation that are second-order accurate in space
and in time 5. An illustration of our LB
program is shown in Figure 2, displaying a model
square lobule. The top represents the chorion,
where the main villous branches enter. The base
represents the decidua, with a maternal artery
located at the centre, and veins at both ends. A
pulsatile inlet flow is prescribed at the artery,
and flow is free to exit the box at the veins at
a prescribed pressure. The villous tree is
modelled as a series of line segments, with a
density reflecting that which would be
anticipated if one were to take a cross-section
through a real placental lobule. Although LB uses
an underlying regular square lattice, the no-slip
boundary conditions on the irregular villous tree
are easily implemented using the interpolation
scheme of Bouzidi et al. 6. Fig
ure 2 Predictions of Lattice-Boltzmann
simulations, implemented in Java. The main window
shows the lobule tree geometry, streamlines from
the central artery to two veins, and the colours
represent the pressure field. The colour plots in
the small window show the flow speed (top) and
vorticity (bottom). Using LB allows us to study
transient dynamics of the pulsatile flow at a
range of Reynolds numbers. The simplicity of the
LB method means that time-stepping can be
performed fast enough on a modern desktop
computer for the evolution of the flow to be
visualised, which provides significant insight
into various flow phenomena. Validation studies
have been performed on our code. Our preliminary
results show how inertia promotes mixing and
transport of blood deep within the lobule. Figure
3 shows how an increase in the arterial jet
Reynolds number increases the flow of blood
through villi higher in the lobule. Figure 4
shows results of advection-diffusion simulations
with solute absorption by the villi at low and
high Reynolds numbers.
Introduction The placenta forms during early
pregnancy in the uterine wall and performs the
role of the fetal lungs, gut and kidneys, by
exchanging gases, nutrients and waste between
maternal and fetal blood 1. We study blood flow
and solute transport in a placental subunit
(lobule), where maternal arteries periodically
squirt blood into a cavity in which floats a
compliant villous tree of fetal vasculature. The
maternal and fetal blood flow rate and pressure,
and the architecture of the villous tree, may be
altered by pathologies such as preeclampsia or
diabetes. By developing a model of the maternal
blood/villous tree interactions, we hope
ultimately to understand how pathological
placental development impacts upon the efficiency
of solute transfer between mother and
baby. Placental physiology An illustration of a
fully developed placenta is shown in Figure 1
2. At delivery, the placenta is around 20cm in
diameter, 3cm thick and has a mass of around
0.5kg 1, 3. The placenta is divided into
around forty isolated lobes by thin membranes
called septa. Each lobe contains at least one
villous tree of fetal vasculature. A single
villous tree, and the surrounding region of
maternal blood is called a lobule, typically
around 2cm in diameter. Fetal blood in the villi
is separated from the bath of maternal blood by a
thin permeable membrane. Transport of solutes
across the membrane occurs by diffusion, active
transport, and pinocytosis. Maternal blood is
periodically squirted into the lobules by spiral
arteries in the uterine wall. Typically there are
one or two spiral arteries per lobule. In places,
the terminal branches of the villous tree are
very densely packed, so maternal blood flow has
been previously be modelled as flow in a porous
medium. However the density of villi varies over
orders of magnitude in the lobule. For instance,
immediately above the spiral artery there are
typically very few villi, allowing inertia to
carry the blood jet further into the lobule.
Attempts have been made to adapt porous medium
models to allow for inertia by having a velocity
dependent permeability 4. We are developing a
general computer code to simulate advection and
diffusion in a lobule to capture the effects of
inertia and pulsatility more accurately. We are
also interested in the effect that compliance of
the fetal vasculature might have upon solute
transfer efficiency. Figure 1
Illustration of a placenta (reproduced from 2).
The magnified view shows a placental lobe which
divides into two lobules, each containing a
villous tree of fetal vasculature. Spiral
arteries emerging from the decidua periodically
squirt maternal blood into each lobule, which
passes around the outside of the compliant
villous tree before draining back into the
decidua. Solutes are exchanged between maternal
and fetal blood across the villous tree membrane.
Figure 3 Plot of the mean horizontal blood speed
against height into the lobule. At low arterial
jet Reynolds number most of the blood passes
directly from artery to vein. As the Reynolds
number increases, blood penetrates up the centre
of the lobule and through more of the villi.

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• This work is funded by the Medical Research
  Council (MRC).