Case 8
Compact arteriovenous malformations are pathological structures
in the vascular system of the brain that can lead to strokes; in
contrast to most other kinds of stroke, these usually affect the young.
Although the normal brain vasculature is complex, it is also rather
orderly, with low order branching and with the arteries and veins only
connected via capillaries. An AVM, on the other hand, is tangled, highly
branched, and involves direct connection from arteries to veins. As a
result, blood pours through the system unpredictably. Potential
consequences include hæmorrhage, when the flow breaches the vessel
walls; shunt, where the blood flows through the brain area too
quickly to properly deliver oxygen and glucose and pick up waste
products; and steal, where the increased flow through the AVM
diverts blood away from other areas, leading to chronic ischæmia.
Modelling the flow of blood through such a system, and predicting its
behaviour, is a tricky business and the methods immature. The governing
equations, as so often for fluid dynamics problems, are Navier-Stokes, a
complicated partial differential system that can be expressed in a
number of ways; here's one:1
The equations are well characterised and very widely used, in everything
from aeroplane design to climate change studies, but they can only be
solved analytically in a few very simple cases that do not, on the
whole, correspond to realistic problems. They are more commonly
estimated numerically, but this doesn't scale well to complex
interconnecting topologies such as those found in an AVM.
The system operates on two main length scales: an O(1) scale coming from
the vessel shape (the viscous term is significant close to the walls,
but the core can be considered inviscid) and an O(Re) scale for the
influence of fluid further downstream (for which the core must treated
as viscous). In the simplest case -- a long, steady flow through an
unbranching vessel of constant cross-section, with no transverse
velocity -- there is a pressure gradient from end to end, and the flow
has a parabolic profile.2
A soon as you add branching to the picture, even a single 1-2 branch, it
significantly complicates the short range effects. To simplify, we
assume that the Reynolds number is large (typically ~300 for blood flows
in the brain), so that the 1/Re term is small and can be ignored;
and that the du/dt term is small as well.3 Thus we need deal only with the steady
state.
We can derive a recurrence relation between the pressures and
displacements in the mother vessel and the two daughters:
which in this simple case can be solved analytically (for the unknown
u values). However, as we add more branches, things get rapidly
out of hand: the length scales reduce, and the recurrence relations
across the generations -- although still "easy" to construct --
become increasingly non-linear, analytically intractable, and in many
cases may have multiple solutions.
Numerical simulations can be run of some more complex systems, although
again there are fairly low limits to what can be managed with any kind
of computational stability. With multi-way branches, unexpected effects
can arise; for example, even with a higher pressure upstream of the
branching than on each of the downstream branches, situations arise
where the flow in some of the daughters goes upstream. With
reconnections -- so that the system becomes a network -- things
get even worse.
1 This version is for an incompressible fluid, and is not the most common formulation. It is non-dimensional and parameterised only by the Reynolds number, Re, which describes the relative importance of viscous and inertial forces. u is a Cartesian distance vector, p is pressure and t is time.
2 That is, the blood flows more quickly in the centre of the vessel and more slowly at the edges, with a (quadratic) smooth change across the width.
3 The justification for the latter assumption escapes me.
1 This version is for an incompressible fluid, and is not the most common formulation. It is non-dimensional and parameterised only by the Reynolds number, Re, which describes the relative importance of viscous and inertial forces. u is a Cartesian distance vector, p is pressure and t is time.
2 That is, the blood flows more quickly in the centre of the vessel and more slowly at the edges, with a (quadratic) smooth change across the width.
3 The justification for the latter assumption escapes me.