Content uploaded by Zoltán Molnár
Author content
All content in this area was uploaded by Zoltán Molnár on Aug 13, 2014
Content may be subject to copyright.
Report no. 04/03
A Hydro-elastic Model of Hydrocephalus
A. Smillie1, I. Sobey2& Z. Molnar3
We combine elements of poroelasticity and of fluid mechanics to construct a
mathematical model of the human brain and ventricular system. The model is
used to study hydrocephalus, a pathological condition in which the normal flow of
the cerebrospinal fluid is disturbed, causing the brain to become deformed. Our
model extends recent work in this area by including flow through the aqueduct,
by incorporating boundary conditions which we believe more accurately represent
the anatomy of the brain and by including time dependence. This enables us to
construct a quantitative model of the onset, development and treatment of this con-
dition. We formulate and solve the governing equations and boundary conditions
for this model and give results which are relevant to clinical observations.
1Present address: Imperial College, London
2Corresponding author 3Department of Human Anatomy, Oxford
Oxford University Computing Laboratory
Numerical Analysis Group
Wolfson Building
Parks Road
Oxford, England OX1 3QD February, 2004
2
1 Introduction
Hydrocephalus is an illness in which abnormal flow of cerebrospinal fluid (CSF) through the
cerebral ventricular system causes the brain to become deformed. The disease itself is well
known, due both to its relatively high incidence and to its debilitating and often fatal effects
(Drake and Sainte-Rose, 1995). Despite the level of awareness and the progress which has been
made in recent decades in understanding the condition, a treatment which is both reliable and
widely applicable remains elusive, see Drake et al. (1998).
CSF is produced mainly in the choroid plexuses; long, convoluted strands of vascularized
tissue located in the the lateral, third and fourth ventricles. The mechanism by which fluid
is produced and secreted is rather complex, but the production rate of around 500 ml/day
in human adults is well known and is reported to be independent of external influences such
as intraventricular pressure (Bradbury, 1993). Given that the total volume of the cerebral
ventricular system is around 150 ml we may suppose that the CSF is renewed about three
times daily.
In a healthy brain, CSF flows from production sites in the choroid plexuses of the lateral
and third ventricles through a single narrow cerebral aqueduct and into a fourth ventricle. It
then moves through another series of narrow passageways known as the median and lateral
apertures into the pontine cistern and cisterna magna near the base of the skull, from where
it passes into the subarachnoid space between the brain and the dura matter. A little fluid is
also believed to flow down the subarachnoid space around the spinal cord and then flow back
up into the cranial subarachnoid space, though it seems difficult to explain how this flow could
be maintained in terms of fluid mechanics.
CSF absorption occurs in the arachnoid villi, small granulations of the arachnoid which
protrude into the dura matter. The barrier between the CSF and the blood in these granula-
tions is thin, enabling CSF to pass into the bloodstream where it is absorbed. In contrast with
CSF production, the rate of absorption is pressure dependent, specifically depending upon
the difference between the intraventricular pressure and the superior sagittal venous pressure
(Bradbury, 1993). The structure of the arachnoid villi is such that even in the unlikely event
of venous blood pressure exceeding the intraventricular pressure no flow will take place from
the blood into the CSF system, so that the villi effectively act as a one-way valve for removal
of CSF.
The build up of fluid associated with hydrocephalus may in theory be caused by overproduc-
tion of CSF in the ventricles, under-absorption in the subarachnoid space or some obstruction
of the CSF pathways. With the exception of a few rare cases it is the latter of these which is
the cause of the illness.
Obstruction of the CSF pathways can happen at any point in the ventricular system, but
the long, narrow aqueduct of Sylvius which runs between the third and fourth ventricles is
reported as being the most frequent site for a blockage to occur (Weller et al., 1993). There is
in the literature the descriptors, connecting and non-connecting hydrocephalus to distinguish
cases where the aqueduct is open or blocked but since we are interested in the whole range
of states of the aqueduct we shall not use these here. The potential causes of a blockage in
the flow pathways are many and varied. The most frequent cause of congenital and infantile
hydrocephalus is a malformation in one or more parts of the ventricular system, for example
stenosis of the aqueduct or membranous occlusion of the foramen of Monro. A blockage
can also be the result of a blood clot entering the CSF system and occluding the aqueduct.
Hydrocephalus has also been known to occur as a result of occluded flow pathways in the
3
subarachnoid space itself, though this is relatively rare.
A consequence of the disease is normally oedema of the parenchyma, particularly the white
matter adjacent to the ventricles. It is this oedema, together with the deformation of the brain
itself that is the cause of most of the long term tissue damage associated with hydrocephalus,
which can result in a range of symptoms including headaches, gait disorder, intellectual im-
pairment and ultimately death.
A traditional approach to modelling hydrocephalus is a lumped parameter model where
the contents of the skull are represented by a series of interconnected compartments through
which fluid is exchanged. All of the resistance to flow between these compartments is lumped
at these interfaces. The formulation of such models leads to a system of coupled differential
equations for the evolution in time of the fluid pressure in each compartment. The solution
of that system gives a relationship between the intracranial pressure and the volume of the
ventricles, see for instance Sivaloganathan et al. (1998). Such a pressure-volume relationship is
useful to clinicians in the diagnosis and treatment of hydrocephalus but the chief weakness of
the approach is that no spatial variation is permitted in any of the physical parameters, hence
it is not possible to describe the stress and strain distributions in the brain tissue or to make
predictions regarding the distribution of fluid in hydrocephalus. Such significant limitations
mean that over the last decade attention has shifted away from lumped parameter models to
the formulation of a spatially more realistic models for the hydrocephalic brain.
A number of authors have proposed mechanical models of hydrocephalus based on the
theory of poroelasticity. It is hoped that such models will give a better understanding of
the condition and hence better treatment. These existing poroelastic models consider hydro-
cephalus in the final diseased state and do not consider the transition from the healthy to
the pathological condition of the brain when there is still be flow through the aqueduct. Nor
have such models included the transient effects associated with shunting, the most widely used
treatment for hydrocephalus. In this paper we construct a model of the brain and ventricular
system which is sufficiently complex to reproduce the behaviour of the hydrocephalic brain yet
simple enough to be mathematically tractable, and use the model to analyse the onset and
treatment of the condition. A review of the general area of application of poroelascticty to the
brain may be found in Tenti et al. (2000).
The use of a mechanical model with a more realistic spherical geometry was first proposed
in Hakim et al. (1976), as was the concept of the brain as a spongelike material. The governing
equations formulated by Hakim et al do not however incorporate this spongelike behaviour
and so are unable to simulate the build up of fluid in brain tissue (oedema) observed in cases
of hydrocephalus.
A crucial step forward was made by Nagashima et al. (1987), who used the consolidation
theory developed by Biot (1941) to model the brain as a porous, linearly elastic solid. This
enabled them to formulate governing equations which modelled both the stress and strain
distribution and the pressure of distribution of fluid (CSF) through the brain. These authors
used a finite element method to solve their model numerically for an anatomically realistic
geometry. While this approach yielded results which were in qualitative agreement with clinical
observations, the quantitative accuracy of their results was limited by their use of inaccurate
values of some the material parameters and by the somewhat naive boundary conditions used
in solving the governing equations.
Recent papers by Kaczmarek et al. (1997), Tenti et al. (1998) and Levine (1999) have
attempted to resolve these difficulties with varying degrees of success. In Kaczmarek et al.
4
(1997) and Tenti et al. (1998) a cylindrical geometry is used in order to facilitate the analytic
solution of the governing equations. Such analytic solutions are desirable as they give a deeper
insight into the behaviour of pressure and stress through the brain, but the use of such a
geometry seems somewhat unreasonable, especially when it comes to specifying boundary
conditions at the ends of the cylindrical ‘brain’. Levine uses a spherically symmetric geometry
similar to that in Hakim et al. (1976) to construct analytic solutions and Stastna et al. (1998)
extended the model in Tenti et al. (1998) to include some transient effects.
All of the authors model the brain as a poroelastic solid undergoing small strains, Kacz-
marek et al and Tenti et al use data from a range of medical and anatomical studies to find
the values of for example, the Young’s modulus and permeability of the brain tissue. Levine
modified previous poroelastic models by attempting to incorporate the absorbtion of fluid in
the brain tissue and the effect of the venous bed in the skull. This approach necessitated the
use of more physical parameters and the results of Levine and Tenti et al are qualitative in
nature. The model of Kaczmarek et al. (1997) was able provide predictions for flow and stress
in the brain tissue.
In the next section we set out the results from poroelastcity and fluid flow which we use
in our model. Then we describe the assumptions regarding geometry, material properties and
time dependance which we make in order to complete a model of hydrocephalus. We then
present solutions to the model and consider the physical parameters which we shall require
in order to make quantitative predictions about the behaviour of the brain and CSF. These
solutions are discussed in terms of their usefulness relative to some of the other models available
and their consistency with clinical data.
2 Mathematical Model
In this section we set out some basic results from poroelasticity and fluid mechanics which
underly our model of the brain. As already indicated, the model can be applied to a healthy
or a damaged cerebrospinal fluid flow. We will assume that CSF is produced at a constant
rate and once produced can (a) remain in the ventricle, causing the ventricle to enlarge, (b)
flow through the aqueduct or (c) flow through the porous matrix of brain tissue.
2.1 Poroelastic equations
An isotropic elastic solid subject to a stress field σij such that it undergoes a small strain
deformation with displacements uiin the coordinate directions with a resultant strain tensor
ǫij satisfies the constitutive relationship from Hooke’s Law,
ǫij =1
Eh(1 + ν)σij −νσk kδij i,(2.1)
where δij is the Kronecker delta. The elastic constants Eand νrepresent the Young’s modulus
and Poisson’s ratio of the material respectively. The volumetric strain, or dilation, is defined
by
ε=ǫkk =∂uk
∂xk
.(2.2)
The poroelasticity model introduced by Biot (1941) generalized the above equations to
model a solid-fluid mixture by introducing a new variable, the fluid pressure, p. Equation (2.1)
5
became
ǫij =1
Eh(1 + ν)σij −νσk kδij i+p
3Hδij (2.3)
where Hwas a new physical parameter which was regarded as a measure of the mixture’s
compressibility for a change in fluid pressure.
In order to completely describe the condition of the body we require an additional parameter
ζ, the increment of fluid content. A positive value for ζindicates that fluid has been added by
the application of the stress field, a negative value indicates that fluid has been removed. By
considering the isotropy of the body and assuming the existence of a potential energy of the
mixture it can be shown (see, for example, Wang (2002)) that ζis given by
ζ=1
Hσkk +p
G.(2.4)
Gis another new physical constant which describes the change in fluid content for a given
change in fluid pressure.
We now invert equation (2.3) to give
σij =E
1 + νǫij +νE
(1 + ν)(1 −2ν)εδij −αpδij ,(2.5)
where the Biot-Willis parameter αis another poroelastic constant defined as
α=1
3
E
(1 −2ν)H.
The expression for the increment of fluid content (2.4) can be rewritten in terms of the
strain
ζ=αε +(1 −αβ)α
Kβ p. (2.6)
In this expression Kis the bulk modulus of the body,
K=E
3(1 −2ν),
and we denote Skempton’s coefficient as β,
β=G
H,
where βis essentially a measure of how the applied stress is distributed between the solid
matrix and the fluid. It tends to one for saturated mixtures where the load is supported
entirely by the fluid and zero for gas filled pores where the stress is transferred through the
solid. By setting p= 0 in (2.6) we can see that αmay be interpreted as the ratio of volume of
fluid displaced to the volumetric strain under drained conditions.
Note that in poroelasticity theory there exist a range of different material constants (such
as G,H,αand β) which characterize the behaviour of the fluid-solid mixture, see Wang (2002)
for an exhaustive list. There are however only two distinct constants which, together with two
elastic constants representing the average elastic properties of the solid matrix, fully describe
the material properties of the body. This is analogous to the case of isotropic linear elasticity
where only two constants are ever required to completely specify the elastic properties of a
6
material but several elastic moduli are defined and used in practice. For consistency and
simplicity we shall use E,ν,αand βas the four independent material constants. We do,
however, observe that two more constants, the undrained Poisson’s ratio νuand the undrained
Young’s modulus Eu, can be defined as
νu=3ν+αβ(1 −2ν)
3−αβ(1 −2ν)and Eu=(1 −2νu)
(1 −2ν)
E
1−αβ .(2.7)
Our reason for making these additional definitions will become evident when we consider the
values of the material parameters to be used in our model.
We also observe that the total stress tensor of the mixture, σij , may be regarded as an
additive mixture of the fluid pressure pand the ‘effective stress’ in the solid matrix σ′
ij,
σij =σ′
ij −αpδij (2.8)
The negative sign on the pressure follows from the convention in solid mechanics that pressures
are positive and compressive stresses negative. This idea of the separation of the total stress
into solid and fluid components will be useful in the application of the boundary conditions.
2.2 Fluid flow equations
In order to completely model the behaviour of a fluid-solid mixture we also require an expression
for the movement of fluid through the solid matrix. We use D’Arcy’s Law for the flow of fluid
through a porous medium, which for spherically symmetric flow with radius r, is
W=−k
µ
∂p
∂r .(2.9)
Here Wdenotes the flow relative to the solid per unit area, or filtration velocity, µis the
viscosity of the fluid and kis the permeability of the body, which will in general be strain
dependent. Klachnar and Tarbell (1987) proposed the modelling the the permeability of arterial
tissue as an exponentially increasing function of the strain. Kaczmarek et al. (1997) suggested
a small strain linear approximation between the inverse permeability and the strain. A model
of either form introduces a material parameter which has unknown value, so in this work we
keep the permeability constant and accept that one of the important elements of poroelasticity,
linkage between the permeability and strain, must await better measurements of the physical
properties of the parenchyma.
We shall also use the well known formula for steady flow of a Newtonian fluid through a
rigid cylindrical pipe, or Poiseuille flow, so that
Qa=πd4
128µL(p1−p2).(2.10)
where Qais the volume of fluid flowing per unit time, dis the diameter of the pipe, Lis the
length of the pipe and p1and p2are the pressures at either end.
2.3 Geometry
We use a spherical geometry based on the first mechanical model for a realistic brain geometry
proposed in Hakim et al. (1976). The brain is modelled as being composed of two concentric,
7
r=A r=B
r=C r=D
Skull
Grey matter
White matter
Aqueduct
Inner Ventricles
Subarachnoid space
(CSF removal)
(CSF production)
Figure 1: Schematic model of brain, showing inner ventricles, aqueduct, grey and white matter,
subarachnoid space (assumed to have negligible thickness) and skull
porous, linearly elastic thick shells with outer radii Band C, which represent the white and
grey matter respectively. Each layer may have different mechanical properties, reflecting the
different properties of each type of tissue. The ventricles are modelled as a spherical cavity
of radius A, located at the centre of the brain, while the Dura Matter, skull and scalp are
represented by a single spherical layer of impermeable solid, outer radius D, enclosing the
system. A narrow cylindrical channel (representing the aqueduct) of diameter druns from the
central ventricles to the interface between the grey matter and the skull (the subarachnoid
space). We suppose that since the volume of this channel will be very small relative to the
volume of the brain it will have no effect on the solid mechanical properties of the surrounding
tissue.
The central cavity is filled with fluid (CSF) of viscosity µwhich is produced at a constant
rate Qp. This fluid flows from the ventricles through the aqueduct and the porous tissue of the
parenchyma into the subarachnoid space, where it is absorbed into the blood. This absorption
is assumed to be proportional to the pressure difference between the blood and the CSF in
the subarachnoid space. We expect that in the normal physiological state the vast majority
of fluid transfer will occur through the aqueduct, as is observed in vivo. Should this flow
be constrained as a result of stenosis of the aqueduct (so that the effective diameter of the
aqueduct becomes small) we expect a much greater degree of flow through the brain itself, as
happens in patients with hydrocephalus. In the case where a shunt is used to divert CSF into
the bloodstream we suppose that fluid is removed directly from the ventricles at a rate QS.
The schematic geometry of our model is illustrated in figure 1, and while this represents
a significant simplification of the make up of a real brain but we believe that it captures the
key geometric and mechanical properties necessary for our purposes. In particular, there is
some justification for the use of a spherical model of the ventricles (which in a healthy brain
8
are in fact narrow, ‘C-shaped’ cavities) by their approximately spherical configuration which
is observed in hydrocephalus.
2.4 Material properties
Perhaps the most important decision to be made in modelling the material properties of the
brain tissue involves the choice of constitutive equation for the solid matrix. It is well known
that biological soft tissues rarely obey Hooke’s law but instead exhibit a mechanically nonlinear
stress-strain relationship, see Sahay (1984) and Fung (1993). Some progress has recently been
made in the formulation of such a non-linear model for the brain based on the theory of
hyperelasticity (see Sahay et al. (1992), Miller and Chinzei (1997)). Sahay and Kothiyal
(1984) even modelled the intracranial pressure-volume relationship in this fashion. They were,
however, unable to reproduce the behaviour of the the brain under pressure, most likely because
their model was unable incorporate the porous nature of the parenchyma. Indeed it seems that
at present no theoretical basis for modelling a mechanically nonlinear poroelastic material
exists. We therefore follow all of the previous authors in the field by using a Hookean form for
the stress-strain behaviour of the brain, see equations (2.1). Since the white and grey matter
may be in general be expected to exhibit different material properties we denote the Poisson’s
ratios and Young’s moduli of each as νwand νgand Ewand Eg. As indicated above, the white
and grey matter are taken to have constant permeability, kwand kgrespectively.
The outer layer consisting of the dura matter, skull and scalp is taken to be homogenous
with elasticity constants νsand Esand zero permeability. While in reality each of the three
components of this layer will exhibit distinct material properties, we justify our assumption
of homogeneity by suggesting that only the comparatively very rigid skull is likely to be of
mechanical significance. Since the grey matter and the outer layer of tissue are in contact
in our model we suppose that the radial displacement and stress will be continuous at the
interface (r=C). At the interface between the white and grey matter (r=B) we will need
to impose the condition that the displacement, radial stress, fluid pressure and flow rate are
all continuous.
2.5 Quasi-steady approximation
In formulating the governing equations for our model we shall consider only the quasi-steady
behaviour of the system as the evolution of hydrocephalus occurs on a timescale of days and
weeks (Hakim et al., 1976), inertial terms in the governing equations which represent the
propagation of waves through the tissue are unlikely to affect the process of the brain settling
into a hydrocephalic state. This conclusion is supported by the work of Stastna et al. (1998)
who showed that the retention of such terms led to waves propagating on a time scale of order
10−2s, much too short a time to affect the onset of the condition.
Thus by assuming changes happen slowly, we introduce time dependence to the behavior of
the ventricle walls, the displacement of which will be of critical importance in the comparison
of our results with clinical observations. We model the time rate of change of the volume of
the ventricular cavity as being equal to the difference between the production rate of CSF and
the total drainage rate through the aqueduct, parenchyma and any shunting device present,
see equation (4.2). This will allow us to construct a phase plot of the behaviour of the ventricle
wall, and hence to investigate the stability or otherwise of our steady state solutions. We will
also be be able to analyse how some variation in the material parameters affects the position
9
and stability of the steady state.
2.6 Onset and Treatments
We shall analyse the onset of hydrocephalus in three ways. Stenosis of the aqueduct, the
main cause of the illness, can be modelled by reducing the value of the aqueduct diameter, d.
We further simulate a) hydrocephalus caused by intraventricular infection or haemorrhage by
varying the viscosity and b) hydrocephalus from impaired absorption in the subarachnoid space.
We also wish to consider the effect of treatments, in particular shunting, on the mechanical
behavior of the brain.
Using a phenomenonological model for the quantitative properties of shunts based on the
results recorded by Czosnyka et al. (1997) we suppose that the flow rate through the shunt,
QS, driven by a pressure difference between the ventricle and the blood, ∆p=pw(A, t)−pbp,
takes the form
QS=S1∆pfor ∆p > 0,
0 for ∆p≤0,(2.11)
for ‘ball on spring’ devices and
QS=S2∆p+S3∆p2for ∆p > 0,
0 for ∆p≤0,(2.12)
for shunts with silicone diaphragm valves. Here S1-S3are physical parameters which will
depend on the material properties of the shunt, pw(A, t) is the CSF pressure in the ventricles
and pbp is the blood pressure.
Other methods of treating hydrocephalus can also be studied using the model. A surgical
procedure to widen the aqueduct (the removal of a tumour, say) could be represented by
increasing dafter it had been constricted for some time. We shall use the quasi-steady model to
study the effect of a lumbar puncture by instantaneously reducing the volume of the ventricles
and observing how, or indeed if, they return to a steady configuration.
3 Hydro-elastic system of equations
3.1 Fluid
The continuity equation for spherical flow with velocity Vof an incompressible fluid in a
biphasic medium is ∂ζ
∂t +1
r2
∂
∂r r2V+Qab = 0 (3.1)
where flux can be decomposed in terms of the fluid velocity Wand the matrix velocity utas
V=W+ut.
Applying our assumptions of quasi-steady state and no CSF absorption in the brain, we have
that the equilibrium equation for the fluid is simply
∂W
∂r +2W
r= 0.
Now we apply D’Arcy’s Law, (2.9), to express the filtration velocity in terms of the pressure
and then split the domain into the white matter A≤r < B and grey matter B≤r < C, so
10
that the governing equations for the fluid pressure in the white matter, pw, and grey matter,
pg, are
∂2pw
∂r2+2
r
∂pw
∂r = 0 A≤r≤B(3.2)
and ∂2pg
∂r2+2
r
∂pg
∂r = 0 B≤r≤C. (3.3)
We do not include expressions for the pressure in the region C≤r≤Dsince no fluid is present
in that part of the domain.
3.2 Solid matrix equations
The governing equations for the solid phase are derived from the poroelastic equations for
spherically symmetric deformation again assuming quasi-steady conditions.
The strains and the dilation is given in terms of the radial displacement, u, by
ǫrr =∂u
∂r , ǫθθ =ǫφφ =u
r, ǫrθ =ǫθφ =ǫφr = 0,
ε=ǫrr +ǫθθ +ǫφφ =∂u
∂r + 2 u
r.
Substituting these expressions in equation (2.5) leads to expressions for the stresses,
σrr =E1−ν
(1 + ν)(1 −2ν)∂u
∂r +2Eν
(1 + ν)(1 −2ν)
u
r−αp,
σθθ =σφφ =Eν
(1 + ν)(1 −2ν)
∂u
∂r +E
(1 + ν)(1 −2ν)
u
r−αp, (3.4)
σrθ =σθφ =σφr = 0.
Since there are no body forces and using the quasi-steady approximation, the stress is
divergence free, giving
∂σrr
∂r +1
r(2σrr −σθθ −σφφ) = 0.
We now substitute expressions (3.4) into the above equation to find the governing equations
for the displacement of the white matter uw,
∂2uw
∂r2+2
r
∂uw
∂r −2uw
r2=E∗
w
∂pw
∂r A≤r≤B, (3.5)
and of the grey matter ug,
∂2ug
∂r2+2
r
∂ug
∂r −2ug
r2=E∗
g
∂pg
∂r B≤r≤C, (3.6)
where
E∗
w=α(1 + νw)(1 −2νw)
Ew(1 −νw)and E∗
g=α(1 + νg)(1 −2νg)
Eg(1 −νg).
11
We can simply perform the above analysis with p= 0 to find the governing equation for
the displacement of the impermeable outer layer of tissue (the skull), us,
∂2us
∂r2+2
r
∂us
∂r −2us
r2= 0 C≤r≤D. (3.7)
We now have a set of governing equations (3.2)-(3.3) and (3.5)-(3.7) which describe the be-
haviour of the independent variables for pressure and displacement.
4 Boundary Conditions
We have one second order ordinary differential equation each governing the fluid pressure
and the solid displacement in the white and grey matter and another for the displacement of
the skull, giving a total of five second order equations. We therefore require ten boundary
conditions in order to completely solve the system; four on the pressure/fluid velocity and
six on the displacement/solid stress. We apply two boundary conditions at the ventricle wall
(r=A), four at the interface between the white and grey matter (r=B), three at the interface
of the grey matter and the skull (r=C) and one at the outer surface of the skull (r=D).
4.1 Ventricles
The boundary condition for the pressure in the ventricles, pw(A, t), is the most complicated
and perhaps the most important which we shall apply. It is here where we incorporate the flow
of fluid through the aqueduct and shunting into the model. The deformation of the ventricles
is also the clearest clinical sign of hydrocephalus, so the size of the radial deformation of the
ventricle wall, uw(A, t), will be critical in evaluating the model.
If we are considering a steady state situation, variables are independent of time and since
the volume of CSF being produced must be equal to the volume of fluid flowing out,
1
z}| {
πd4
128µL[pw(A)−pg(C)] −
2
z}| {
4πA2kw
µ
∂pw
∂r r=A+
3
z}| {
S(pw(A)) =
4
z}|{
Qp.(4.1)
The first term models the flow of fluid through the aqueduct, Qa, driven by the pressure
difference between the ventricles and the subarachnoid space [pw(A)−pg(C)], and dand L
represent an effective diameter and length of the aqueduct respectively, see (2.10). If the
aqueduct is being forced into collapse then it will not remain cylindrical but its length will
not change much so we continue to use a Poiseuille flow approximation but interpret das the
diameter of a cylindrical tube with the same flow rate for a given pressure drop. The second
term models the flow of fluid across the ventricle wall into the parenchyma using the velocity
of the flow from D’Arcy’s law (2.9). The third term is the flow rate through any shunt and
the fourth term, is the production rate of fluid, assumed constant. If now we invoke a quasi-
steady approximation and allow the ventricle radius to vary slowly we incorporate this time
dependence by introducing a term ˙
V, the rate of change of the ventricular volume with respect
to time. Thus we have
˙
V=Qp−πd4
128µL[pw(A, t)−pg(C, t)] + 4πA2kw
µ
∂pw
∂r r=A−S(pw(A, t)).(4.2)
12
Now expressing the ventricular volume in terms of the initial radius Aand the deformation of
the ventricle, uw(A, t),
˙
V=d
dt 4
3π(A+uw(A, t))3.
Substituting into (4.2) we have a first order non-linear differential equation for uw(A, t)≡
uA(t),
duA
dt =1
4π(uA(t) + A)2Qp−πd4
128νL [pw(A, t)−pg(C, t)]
+4πA2kw
µ
∂pw
∂r r=A−S(pw(A, t)).(4.3)
The second boundary condition to be applied here refers to the solid matrix. Since the
brain is untethered at the ventricle wall we assume that the radial stress in the solid,
σ′
rr =σrr +αp,
is zero at its boundary. Hence our second boundary condition is
Ew1−νw
(1 + νw)(1 −2νw)∂uw
∂r r=A+2Ewνw
(1 + νw)(1 −2νw)
uw(A, t)
A= 0.(4.4)
4.2 Interface of White and Grey Matter
The four boundary conditions to be applied at r=Ball come from the continuity of physical
quantities in our model; the displacement, radial stress, fluid pressure and filtration velocity
must all be continuous. Hence we match displacements
uw(B, t) = ug(B , t) (4.5)
and stresses
Ew1−νw
(1 + νw)(1 −2νw)∂uw
∂r r=B+2Ewνw
(1 + νw)(1 −2νw)
uw(B, t)
B−αpw(B, t)
=Eg1−νg
(1 + νg)(1 −2νg)∂ug
∂r r=B+2Egνg
(1 + νg)(1 −2νg)
ug(B, t)
B−αpg(B, t) (4.6)
in the solid. Note that since the αp terms will cancel from each side of the equation it does
not matter whether we choose to equate the effective stresses σ′
rr (B) or total stresses σrr(B).
We also match fluid pressures
pw(B, t) = pg(B , t) (4.7)
and filtration velocities
−kw
µ
∂pw
∂r r=B=−kg
µ
∂pg
∂r r=B(4.8)
across the interface.
13
4.3 Subarachnoid Space
The first two boundary conditions at r=Care a result of the continuity of the behaviour of
the solid between the brain and the skull. Thus we apply continuity of displacements
ug(C, t) = us(C, t) (4.9)
and stresses
Eg1−νg
(1 + νg)(1 −2νg)dug
dr r=C+2Egνg
(1 + νg)(1 −2νg)
ug(C)
C−αpw(C, t)
=Es1−νs
(1 + νs)(1 −2νs)dus
dr r=C+2Esνs
(1 + νs)(1 −2νs)
us(C, t)
C(4.10)
is a similar way to those at the interface of the white and grey matter.
The third boundary condition relates to the absorption of fluid which occurs in the sub-
arachnoid space. Modelling the absorption as proportional to the pressure difference between
the CSF and the blood stream, we have that
pg(C, t)−pbp
Rµ =πd4
128µL[pw(A, t)−pg(C, t)] −4πC2kg
µ
∂pg
∂r r=C.(4.11)
The term on the left hand side of this equation refers to the flow of fluid through the
arachnoid villi, which is driven by the pressure difference (pg(C)−pbp) and where Ris a
parameter which models the resistivity of the villi to flow. Note that since flow cannot occur
from the blood into the subarachnoid space we implicity assume that in all cases pg(C)≥pbp.
4.4 Skull
The last boundary condition applies to the outside of the skull, r=D. Here we simply assume
that the solid is untethered and hence stress free so that
Es1−νs
(1 + νs)(1 −2νs)dus
dr r=D+2Esνs
(1 + νs)(1 −2νs)
us(D, t)
D= 0.(4.12)
The total stress and effective stress are identical here since there is no fluid pressure, hence
there is no need to make any distinction between them in applying this boundary condition.
4.5 Solution method
The formulation we have derived provides us with a set of ten boundary conditions, (4.1) or
(4.2) and (4.4)-(4.12) with which to solve the governing equations (3.2)-(3.3) and (3.5)-(3.7).
The spherical geometry allows the pressure and displacement to be determined in terms of
simple powers or the radial distance and unknown constants. In the steady case the solutions,
when substituted into the boundary conditions, these reduce to solving a 10×10 linear system
for the unknown constants. This has been done using Maple to solve the linear system ana-
lytically. In the unsteady case, the nine boundary conditions (4.4)-(4.12) are applied leaving
just the ventricle displacement undetermined. Equation (4.3) can then be solved numerically
using MATLAB to provide the time evolution of the deformation of the ventricle wall and
corresponding stress and displacement within the grey and white matter, for full details see
Smillie (2003). In addition to following the time evolution, we also use a phase plot of uAvs.
duA
dt to determine the stability of a steady state solution to the model.
14
5 Parameter Estimation
The evaluation of the material parameters is often a non-trivial problem in biomechanics since
the usual engineering tests used to measure the physical properties of a material are often
difficult to apply to soft biological tissue. There is also the question of different material
behaviour in vivo and in vitro, with experimentation in the former case being ethically as well
as practically problematic. In the following section we shall therefore attempt to estimate
numerical values for all of the parameters in our model, but some will inevitably be rough
approximations.
5.1 Geometry of the Brain
Since the spherically symmetric geometry of the brain in the model is an idealization we must
calculate values for A, B, C, D, L and dwhich are in some sense equivalent to those in the real
geometry. We use values for the radii of the ventricles, white matter and grey matter which
approximately correspond to their location in the adult male brain Kaczmarek et al. (1997).
Hence we have that the ventricles are of radius A= 3 ×10−2m, the interface of the white and
grey matter is at B= 7 ×10−2m and the the brain itself has radius C= 10 ×10−2m. Taking
that the skull to be of thickness 0.2 cm (Drossos et al., 2000), the outer layer of the model will
be of radius D= 10.2×10−2m.
The diameter of a healthy cerebral aqueduct varies along its length and between individuals,
we use the average value reported in Bickers and Adams (1949), d= 4 ×10−3m. Since we have
assumed that the aqueduct is straight and runs from the ventricles to the subarachnoid space
we suppose that it is of length L=C−A= 7 ×10−2m.
5.2 Poroelastic constants
The poroelastic constants are perhaps the most difficult class of parameters for which to
find numerical values. Kaczmarek et al. (1997) use a value of E= 1 ×104N/m2for the
Young’s modulus, taken from Metz et al. (1970). As was noted in Levine (1999) however,
this represents the instantaneous, or undrained, value of the coefficient, the drained value is
likely to be somewhat smaller. It is also unclear whether the value of the Poisson’s ratio used,
ν= 0.35, was measured under drained or undrained conditions.
In formulating a model of the brain all of the previous authors have assumed that the brain
perfectly saturated so that α=β= 1. In order to enable us to convert from undrained to
drained elastic moduli we instead suppose that the brain is almost but not perfectly saturated
with fluid, and use the values given in Wang (2002) for such a mixture,
α= 1 and β= 0.99.
Now, if the value of νquoted above refers to drained conditions then, using equation (2.7), we
have that the undrained Poisson’s ratio is
νu= 0.4983,
that is, the mixture is virtually incompressible under undrained conditions. This may explain
why Nagashima et al erroneously used a value of ν= 0.4999 in their model of the hydrocephalic
15
brain. Given an undrained Young’s modulus of Eu= 1 ×104N/m2and inverting the second
of equations (2.7) we have a value of the drained Young’s modulus
E=1−2ν
1−2νu
Eu(1 −αβ) = 0.9010 ×104N/m2,
only a little lower than the value used by Kaczmarek et al.
However, if the value of the Poisson’s ratio used in Kaczmarek et al. (1997) was measured
under undrained conditions (so that νu= 0.35), we have that the drained value of the Poisson’s
ratio is
ν=3νu−αβ(1 + νu)
3−2αβ(1 + νu)=−0.8761,
a value of νwhich is unusual, but not physically inadmissable. Materials with negative Pois-
son’s ratio exist and are thought to occur in other places in the body, see for example Lakes
(1993), Lakes (2002) so while the suggestion of a negative Poisson’s ration for brain tissue is
highly unusual, it is not an impossible suggestion. Again assuming that Eu= 1 ×104N/m2,
we then have a Young’s modulus of
E= 0.9177 ×103N/m2,
an order of magnitude lower than that used by Kaczmarek et al. One of the main weaknesses of
their model was the unrealistically high levels of intraventricular pressure (∼40kPa) required to
maintain displacements of the ventricle wall consistent with those observed in hydrocephalus.
A significantly lower value of Emight resolve this discrepancy, therefore we shall consider
these values of the elastic constants as being potentially viable despite the unusual form of ν.
Since the value of νhas to be something of an open question we present results for both of the
above scenarios (ν= 0.35 and νu= 0.35) and compare the results.
The above results were measured using a section of brain composed mainly of white matter.
In the absence of any quantitative data regarding different elastic properties of the grey matter
we shall assume that Ew=Eg=Eand νw=νg=νwhen computing our results, though in
reality it is likely that the grey matter will be a little more rigid. For the skull we use values
of
Es= 1 ×109N/m2and νs= 0.3,
from van Rietbergen et al. (1995).
In order to fully characterize the poroelastic behaviour of the brain we require numerical
values for the permeabilities kwand kg. Kaczmarek et al calculated the permeability of the
brain using the results of Reulen et al. (1977), however the value given as ‘permeability’ in
their paper is actually the permeability scaled with fluid viscosity. Based on their calculations
the appropriate values for the permeabilities of the white and grey matter are
kw= 1.426 ×10−14m2and kg= 1.426 ×10−16m2respectively.
5.3 Fluid Flow and Drainage
The physical properties of the CSF system itself are rather better documented. Since CSF has
physical properties similar to water we assume that it has dynamic viscosity (Fay, 1994)
µ= 8.91 ×10−4Ns/m2.
16
The production rate of CSF in the ventricles is reported to be (Bradbury, 1993)
Qp= 5.833 ×10−9m3/s,
which we take to be independent of intraventricular pressure. We may now compare the relative
importance of the aqueduct and the porous brain in draining fluid from the ventricles. Taking
a typical length scale of Land pressure scale of Pand dividing term 1 of equation (4.1) by
term 2 gives
πd4P/128µL
4πA2kwP/µL ≈1750.
Thus, in a healthy brain, flow through the aqueduct accounts for virtually all of the transfer
of CSF through the ventricular system. Only in a pathological state, for example when d→0,
will a significant proportion of flow occur through the parenchyma. We therefore have that a
typical flow speed in the aqueduct is
U=4Qp
πd2= 1.16 ×10−4m/s.
Then, given a kinematic viscosity µ/ρ = 8.91 ×10−7m2/s, we have that the Reynolds number
for flow in the aqueduct is
Re = ρUL
µ≈10−1.
To evaluate R, the resistance of the arachnoid villi, we suppose that the resistance is
such that the rate of outflow is exactly equal to the CSF production rate Qpfor the normal
physiological value of p(A)−pbp = 441 N/m2. Hence
R=[p(A)−pbp]norm
µQp
= 8.490 ×1013m−3.
We assume a typical value for venous blood pressure of 100 mmHg or pbp = 1.333 ×104N/m2.
In order to determine values for S1,S2and S3, the parameters governing flow through
a shunt, we use a least squares fit to the data in Czosnyka et al. (1997) to give that S1=
1.253 ×10−10m5/Ns, S2= 3.031 ×10−11m5/Ns and S3= 3.766 ×10−14m7/N2s.
6 Results
We now consider solutions of the model to test against clinical observations and to examine the
behaviour of the response of the brain to a variation in these parameters. The major indicators
we consider are the displacement of the ventricle wall, uw(A, t), and the intraventricular pres-
sure, pw(A, t), since these are the key clinical signs of hydrocephalus. The ventricular aspect
ratio,
Γ = C+ug(C, t)
A+uw(A, t),
was introduced by Hakim et al. (1976) as an alternative measure of the extent of hydrocephalus,
we shall use it when making a comparison with the results of those authors. We shall also use
the notion of the magnitude of the shear stress
|τ(r, t)|=|σr(r, t)−σθ(r, t)|
2
17
as measure of tissue damage, see Holbourn (1943). A summary of parameter values is given in
table 1.
A3×10−2mE(Case 1) 9.010 ×103N/m2
B7×10−2mE(Case 2) 9.177 ×102N/m2
C10 ×10−2mν(Case 1) 0.35
D10.2 ×10−2mν(Case 2) -0.8761
L7×10−2mEs1×109N/m2
d4×10−3mνs0.3
α1Qp5.833 ×10−9m3/s
β0.99 µ8.91 ×10−4Ns/m2
kw1.426 ×10−14 m2R8.490 ×1013 m−3
kg1.426 ×10−16 m2pbp 1.330 ×104N/m2
Table 1. Values of physical constants used to calculate model solutions.
6.1 Severe Hydrocephalus
We begin by solving the model for the case of complete occlusion of the aqueduct, d= 0, and
no shunt, QS= 0. In this case it is to be expected that the intraventricular pressure will be
significantly above normal physiological values, since all of the CSF produced in the ventricles
must be driven through the parenchyma, a much more resistant pathway than the aqueduct.
This pressure rise will induce a deformation in the parenchyma, the defining symptom of
hydrocephalus.
Setting d= 0 is also a useful starting point since the results from this configuration of the
model can be compared not only with clinical observations of severe hydrocephalus, but also
with the other mathematical models of this condition (for example Kaczmarek et al. (1997)
and Tenti et al. (1998)).
6.1.1 Flow and Pressure Distribution of CSF
In figure 2 we show the excess in pore pressure over the venous blood pressure. The intra-
ventricular pressure, pw(A) is around 13 kPa above b.p. This value is rather higher than that
typically observed clinically (CSF pressures in excess of 2 kPa above b.p. are usually consid-
ered abnormal) but is likely to be within the upper end of recorded hydrocephalic ventricular
pressures. It is certainly closer to physiological levels than the intraventricular pressures in
excess of 40 kPa predicted by the model of Kaczmarek et al.
The pressure falls off relatively slowly through the white matter (A≤r≤B) then drops
much more sharply in the grey matter region of the brain (B≤r≤C), due to the lower
permeability here. The step change in the gradient of fluid pressure at the interface is not
physiological, its presence is due to our assumption that the brain can be divided into distinct
regions of white and grey matter. In reality both tissue types are found throughout the
parenchyma, with grey matter becoming more prevalent towards the periphery. A model
in which the permeability is a monotonically decreasing function of radius may resolve this
difficulty, though would of course lead to a different form of the solutions.
In steady state the volume of fluid flowing out through the arachnoid villi and into the
bloodstream is constant, regardless of whether the fluid has arrived in the subarachnoid space
via the aqueduct or the parenchyma. This means that in our model, as a consequence of
18
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
2000
4000
6000
8000
10000
12000
14000
r (m)
p(r) (Pa)
Figure 2: Fluid pressure in the white and grey matter
equation (4.11), the pressure in the subarachnoid space pg(C) will be always be relatively close
to blood pressure. We were unable to find any empirical data on the fluid pressure in the
subarachnoid space in hydrocephalic patients (probably due to the difficulty of performing
such measurements), so it is difficult to asses the validity of this result.
Figure 3 shows the filtration velocity decreasing as the square of the radius. This is expected
since no fluid is absorbed as it passes through the brain, hence the volume of fluid flowing
through any given spherical surface should be constant. The filtration velocity is much higher
in this case than the typical flow speed calculated in section 5.3. As the volume of CSF has
risen, the residence time for the CSF will be greater but the time spent in the parenchyma will
be less and such a change may have further consequences for the brain if the CSF plays a role
in transferring nutrients from the ventricles to the subarchnoid space through the parenchyma.
6.1.2 Displacements and Stresses through the Brain
The elasticity constants Eand νare critical in determining the behaviour of the solid compo-
nents of the brain. As explained above, we have calculated solutions for two values of Poisson’s
ratio, one positive (case 1) and one which is negative (case 2), and consequentially of uncertain
applicability.
The radial displacement of the brain in both cases is shown in figure 4, where case 1 refers
to (ν= 0.35, E = 9.010 ×103N/m2), while case 2 has (ν=−0.8761, E = 9.177 ×102N/m2).
The displacement of the ventricle wall uw(A), which we shall consider to be the result most
indicative of the extent of hydrocephalus, is in both cases around 0.012 m, while the outer
surface of the brain experiences virtually no deformation, a consequence of the relatively rigid
nature of the skull. This gives a ventricular aspect ratio of
C+ug(C)
A+uw(A)= 2.38,
which is close to the value of 2 given in Hakim et al. (1976) for a typical case of adult hydro-
cephalus.
19
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
1
2
3
4
5
6x 10−7
r (m)
W(r) (m/s)
Figure 3: Filtration velocity (superficial) in the white and grey matter
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
0.005
0.01
0.015
0.02
0.025
r (m)
u(r) (m)
ν = −0.8761
ν = 0.35
Figure 4: Radial displacement of tissue in white and grey matter for the two cases: ν= 0.35
(—–) and ν=−0.8761 (- - -)
20
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
−2.7
−2.6
−2.5
−2.4
−2.3
−2.2
−2.1
−2 x 104
r (m)
σr (Pa)
ν = 0.35
ν = −0.8781
Figure 5: Radial stress for the two cases: ν= 0.35 (—–) and ν=−0.8761 (- - -).
While the displacements of the outer and inner boundaries of the skull are similar for
both sets of elastic constants the deformation through the brain is very different. In case 1 the
deformation of the white matter is relatively constant, while in case 2 the deformation becomes
larger away from the ventricles. This latter type of behaviour is not physiological, suggesting
that the elastic parameters in this case are unlikely to be valid. The slight increase in the
displacement as rapproaches Bin case 1 is also probably not physiological, this is again a
consequence of our assumption that the brain can be divided into two distinct regions of white
and grey matter. The results for the displacement in Case 1 are similar to those in Kaczmarek
et al. (1997).
The radial stress distributions are shown in figure 5. The negativity of the total stresses
in both cases 1 and 2 indicates that the stresses are compressive in nature. The stress falls
quickly through the skull to the stress free outer surface of the head, again a consequence of
the relative rigidity of the skull. For clarity we have omitted this from the figure since such a
large change makes the stress distribution through the brain less clear.
The total stress at the ventricle wall is equal and opposite to the fluid pressure (includ-
ing blood pressure) at this point. Since the solid parenchyma is untethered at the ventricle
wall (4.4), it is the the pressure gradient in the fluid alone which induces the compression
of the brain. In contrast to this, the magnitude of the total stress at the outer edge of the
parenchyma is much greater then the fluid pressure and the stress there is largely due to the
skull constraining the brain.
Since the solid exhibits unphysiological behaviour in case 2 we believe that the elastic
parameters used in that case are not physiological, with the value of -0.8761 for the Poisson’s
ratio being the most likely cause of such unrealistic results. For the remainder of this report
we shall therefore assume that
E= 9.010 ×103N/m2and ν= 0.35.
21
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
r (m)
ζ(r)
Figure 6: Increment of fluid content, ν= 0.35
6.1.3 Effects on the Brain
Most of the damage to the tissue of the brain, and hence most of the symptoms of hydro-
cephalus, occur as the result of a combination of oedema and shear stresses in the solid. The
increment of fluid content may be regarded as a measure of the former, see figure 6. Notice how
the increase in fluid content is largely confined to the white matter and is most pronounced in
the regions adjacent to the ventricle wall. This is in agreement with the clinical observations
of oedema in hydrocephalus. In the areas of grey matter close the the subarachnoid space
ζbecomes negative, indicating that in this region fluid has been squeezed out of the brain.
Given that most of the compression of the parenchyma occurs in this region this seems like
reasonable behaviour, and is supported by clinical evidence from CT scans of hydrocephalic
patients (Kaczmarek et al., 1997).
Note that since we have assumed that the brain is almost completely saturated with fluid,
hence α= 1 and β= 0.99, we have from equation (2.6) that the increment of fluid content,
ζ(r), is approximately the same as the dilation, ε(r). This indicates that close to the ventricles
the solid matrix expands, while in the peripheral regions it is compressed against the skull.
The magnitude of the shear stresses are plotted in figure 7. These indicate that in our
model the most significant tissue damage takes place in the white matter adjacent to the
ventricles and in the grey matter near the skull. As normal there is a step change in the
nature of the solution at the interface of the white and grey matter, in this case the minimum
amount of damage takes place here. The increased level of tissue damage at the periphery of
the brain is in contrast to the results of Levine (1999), who found that shear stress decreased
monotonically from a maximum at the ventricle wall. The high degree of compression of this
region of the brain observed in hydrocephalus would suggest however that significant levels of
damage may occur here, and hence our results may indicate a real physiological effect, rather
than a mathematical anomaly caused by the simplified geometry employed in formulating the
model.
22
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
500
1000
1500
2000
2500
3000
r (m)
τ(r) (Pa)
Figure 7: Shear stress, ν= 0.35
6.2 Onset of Hydrocephalus
One of the main advantages of our model is that it allows simulation of some of the causes of
hydrocephalus by varying material parameters. First, we shall consider the effect of stenosis
of the aqueduct, the most frequent cause of the condition, by reducing the value of d. Second,
we model impaired absorption in the subdural space by increasing Rand third, we consider
variation in the viscosity of the CSF.
6.2.1 Stenosis of the Aqueduct
The dependence of the intraventricular pressure and ventricle wall displacement on the width
of the aqueduct is shown in figure 8. It is clear that the system is resistant to even a relatively
large decrease in dwith little discernible effect so long as d > 0.8×10−3m, a fifth of its
physiological value. Once ddrops below that value the effect on the brain is pronounced,
with a large increase in intraventricular pressure, pw(A), and wall displacement, uw(A) as the
effective aqueduct radius, d, approaches zero. Note that the pressure and the displacement tend
to the equilibrium values determined for complete blockage of the aqueduct. The sensitivity
of the model to changes in the diameter of the aqueduct below 0.8×10−3m is a result of the
d4term which appears in boundary conditions (4.1) and (4.11). This reinforces the view that
complete occlusion of the cerebral aqueduct is the main cause of hydrocephalus.
We may model the effect of aqueduct stenosis on the time evolution of the system using
the ventricular pressure boundary condition (4.2). In particular the stability of steady state
solutions is illustrated in figure 9 by a phase plane diagram of uw(A) against duw(A)
dt for a
range of values of d. The phase diagram is shown for three cases; complete occlusion of the
aqueduct, d= 0m, major occlusion of the aqueduct, d= 0.25 ×10−3m, and minor occlusion of
the aqueduct, d= 1 ×10−3m. The steady state for the complete obstruction of the aqueduct is
located at uw(A) = 0.012m, corresponding to the deformation state described in the previous
section, while for the aqueduct open the steady state is located at the origin, corresponding to
the undeformed configuration of the brain (the phase plot for d= 4 ×10−3m, its physiological
23
0 1 2 3 4
x 10−3
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8 x 104
d (m)
pw(A) (Pa)
0 1 2 3 4
x 10−3
−2
0
2
4
6
8
10
12 x 10−3
d (m)
uw(A) (m)
Figure 8: Effect of varying the aqueduct radius,d, on the ventricular pressure, pW(A), and wall
displacement, uW(A)
value, is very similar to that for d= 1 ×10−3m, only with an even steeper gradient). When
d= 0.25 ×10−3m the deformation state is located between these two extremes.
All of the steady states are stable in nature. This has important consequence for the
behaviour of the brain in both healthy and pathological conditions. In the healthy state
the ventricles will return to their undeformed configuration (that is uw(A) = 0) even if they
are instantaneously subject to a large deformation, due to blow to the head for example.
In the pathological state the ventricles will return to a deformed configuration even if their
volume is temporarily reduced by treatment. Unless either dcan be increased, by surgery to
remove a tumour for example, or a permanent alternative drainage pathway can be constructed,
for example by shunting, fluid will continue to be driven through the parenchyma with the
consequent rise in intraventricular pressure and deformation of the brain.
The relatively steep gradient of the phase plot in the case of the open aqueduct indicates
that the brain will quickly return to steady state after being perturbed, while the shallower
gradients when d= 1×10−3m and d= 0 mean that this configuration will recover more slowly.
We are able to plot the time evolution of hydrocephalus by solving the differential equaiton
for the deformation of the ventricles (4.3) numerically, the results are shown in figure 10. In
formulating these solutions we assumed that the aqueduct was completely blocked and used
the initial condition
uw(A, 0) = 0,
that is that the brain is initially in its undeformed state. Both the displacement of the ventricles
and the intraventricular pressure approach their steady state values asymptotically. The time
scale over which the onset of hydrocephalus occurs, 2.5×105s≈3 days, is within the range of
‘days and weeks’ given by Hakim et al. (1976) as typical for the development of the condition.
6.2.2 Impaired absorption of CSF
We now consider potential alternatives to stenosis of the aqueduct as a cause of hydrocephalus.
One such alternative is impaired absorption of fluid in the subarachnoid space, which in our
24
−0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03
−4
−3
−2
−1
0
1
2
3
4
5
6x 10−6
uw(A) (m)
duw(A)/dt (m/s)
d=0
d=0.00025
d=0.001
Figure 9: Phase plots for the wall displacement, uw(A, t) showing a stable fixed point (circled)
for various aqueduct diameter: d= 0 (—–), d= 0.00025 (- - -), d= 0.001 (···)
0 0.5 1 1.5 2 2.5
x 105
0
0.005
0.01
0.015
t (s)
uw(A) (m)
0 0.5 1 1.5 2 2.5
x 105
1
1.5
2
2.5
3x 104
t (s)
pw(A) (Pa)
Figure 10: Time evolution of the ventricular pressure, pW(A, t), and wall displacement,
uW(A, t) for severe hydrocephalus following a sudden blockage of the aqueduct
25
0 5 10
x 1015
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6x 104
R (m−3)
pw(A) (Pa)
0 5 10
x 1015
5.23
5.2305
5.231
5.2315
x 10−8
R (m−3)
uw(A) (m)
Figure 11: Effects of increasing the absorption resistance,R, on the ventricular pressure, pW(A),
and wall displacement, uW(A)
model should correspond to an increase in R, the resitivity of the arachnoid villi. Figure 11
shows the effect on the intraventricular pressure and displacement of increasing Rby up to
two orders of magnitude. Both plots show a linear dependence on the resistance, but the
change in the magnitude of the pressure is much greater than the infinitesimal increase in the
displacement. Given that it is a pressure increase in the ventricles which causes hydrocephalus
such an effect seems difficult to explain. However it is not the fluid pressure which is the cause
of the deformation but the pressure gradient, evident from the presence of only the derivative of
pin equations (3.5) and (3.6). An increase in Rmeans that the fluid pressure rises throughout
the brain and ventricular system and so does not induce a deformation in the parenchymal
matrix. Thus, in our model, an increase in the resistance of the arachnoid villi will not cause
hydrocephalic damage to the brain tissue, although the rise in fluid pressure may have other
detrimental effects.
6.2.3 Increased viscosity of CSF
An other potential cause of hydrocephalus reported clinically is an increase in the viscosity
of the CSF, for instance due to haemorrhage in the ventricles. Since blood has a higher
viscosity than CSF we might imagine that intraventricular bleeding would lead to a thickening
of the fluid in the system and hence to an increase in pressure and a deformation in the
brain. Should some protein enter the CSF system, perhaps as the result of an intraventricular
infection or a malformation of the choroid plexus, the increase in viscosity would be likely
to be even more pronounced. We have modelled this by varying the value of the viscosity
from µ= 8.91 ×10−4Ns/m2, its physiological value, to µ= 8.91 ×10−2Ns/m2, two orders of
magnitude greater.
The effect of such an increase on the intraventricular pressure and displacement is shown
in figure 12. Both the pressure and deformation exhibit an apparently linear dependence on
viscosity, but as in the previous case, the effect on the pressure is far greater. This is because in
our model the main consequence of an increase in the viscosity is impairment of the absorbtion
26
0 0.05 0.1
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6x 104
µ (Ns/m2)
pw(A) (Pa)
0 0.05 0.1
0
1
2
3
4
5
6x 10−6
µ (Ns/m2)
uw(A) (m)
Figure 12: Effects of increasing the CSF viscosity, µ, on the ventricular pressure, pW(A), and
wall displacement, uW(A)
of fluid through the arachnoid villi, flow through the cerebral aqueduct is also impaired but
the resistivity of this pathway is relatively low, so even a large increase in viscosity has little
effect on the pressure gradient. Thus it seems that, based on our model, an increase in the
viscosity of the CSF is unlikely to be a major cause of hydrocephalus. The only mention
of hydrocephalus due to a change in fluid mechanical properties of the CSF in the medical
literature refers to blood entering the ventricular system and a clot blocking the flow pathway,
see Bradbury (1993). In terms of our model this would better correspond to the previous case
of d→0 which, as we showed above, does lead to the symptoms of hydrocephalus, than to
increased viscosity.
6.3 Treatments
Finally we model some of the treatments used in cases of hydrocephalus. The most effective,
and hence the most widely used, treatment is shunting and so we shall consider it in some
detail, but we begin with lumbar puncture.
6.3.1 Lumbar Puncture
In lumbar puncture some volume of CSF is removed from the patient, usually via the spinal
cord, in order to relieve fluid pressure in the skull. In terms of our model we regard this as a
step decrease in the intraventricular pressure, and hence in the displacement of the ventricle
wall, without any permanent change in any of the material parameters. Clinically such a
procedure has not proved to be successful in the long term, and is generally used only in severe
cases where an immediate reduction of CSF pressure is required or in conjunction with another
procedure such as shunting. The time dependent version of the model would appear to explain
these observations; since all of the hydrocephalic steady states are stable (see Figure 5.8) any
perturbation in uw(A) or pw(A) which is not accompanied by some change in the underlying
material parameters will eventually decay, and uw(A) and pw(A) will eventually return to their
27
−2 0 2 4 6 8 10
x 10−4
−7
−6
−5
−4
−3
−2
−1
0
1
2x 10−5
uw(A) (m)
duw(A)/dt (m/s)
Silicone diaphragm
shunting
Ball−on−spring
shunting
Figure 13: Phase plots of the ventricular wall displacement, uw(A) with shunting showing a
single stable fixed point for ball on spring shunt (- - -) and two fixed points (one stable and
one unstable) for silicone diaphragm shunt (—–)
steady state pathological values. We do not include a plot of this behaviour since it would
have a form the same as Figure 5.9, which shows the pressure and deformation moving to
equilibrium.
6.3.2 Shunting
Shunt insertion is a major surgical procedure which entails implanting a tube with a valve
leading from the ventricles into some point in the the body, normally an artery of the abdomen,
where CSF is allowed to drain into the bloodstream. The intention of such a procedure is
to create an alternative pathway for CSF drainage, and hence to relieve hydrocephalus. In
addition to the difficulties normally associated with surgery on the brain there is the question
of how much fluid to drain and how to control the rate of drainage, too much can lead to the
collapse of the ventricles, so called ‘slit ventricle syndrome’, while too little will be insufficient
to cure the illness. The main types of shunt currently used are ball-on-spring valves and silicone
diaphragm valves which we model using equations (2.11) and (2.12) respectively. Note that
in all cases in this section we consider the case of severe hydrocephalus where the aqueduct is
blocked and d= 0.
Phase plots of the behaviour of the ventricle wall with shunt inserted are shown in figure
13, corresponding to a ball-on-spring shunt (linear pressure flow relationship) and to a silicone
diaphragm shunt (quadratic pressure flow relationship). In each case there is a stable steady
state close to uw(A) = 0, indicating that shunting may be expected to be successful in reducing
the deformation and returning the ventricles to their physiological size. Notice however that
for a quadratic pressure flow relationship there is a second, unstable steady state at uw(A)
28
0 50 100 150 200 250
1
1.5
2
2.5
3x 104
t (s)
pw(A) (Pa)
0 50 100 150 200 250
−5
0
5
10
15 x 10−3
t (s)
uw(A) (m)
Figure 14: Time evolution of ventricular pressure, pW(A, t) and displacement, uW(A, t) fol-
lowing introduction of ball-on-spring shunt for three initial configurations, (- - - ) severe hy-
drocephalus, (—–) normal brain, (···) small negative perturbation
just less than zero. This indicates that should the radius of ventricles be perturbed below this
second equilibrium point, by a blow to the head or the patient coughing say, the ventricular
cavity will collapse, leading to slit ventricle syndrome. No such effect occurs with a linear
pressure flow shunt, since there is only one, stable, steady state.
This contrasting behaviour is illustrated in figures 14 and 15, where we plot numerical
solutions to equation (4.3) for the case of ball-on-spring and silicone diaphragm valves respec-
tively. The figures show both the time evolution of the displacement of the ventricles and the
intraventricular pressure for initial conditions of
uw(A, 0) = 1.2×10−2m (hydrocephalus),
uw(A, 0) = 0 (normal),
uw(A),0 = −0.2×10−2m (negative perturbation).
In the case of a linear pressure-flow shunt, the brain returns to its healthy undeformed
state for each initial condition. The time scale over which this occurs, 150 s, seems relatively
short, though we do not have any clinical data with which to make a direct comparison. Our
method of approximating the the material parameter which describes the shunt resistance, S1,
was fairly crude which may be the reason the model predicts such a short time for resolution
of hydrocephalus. If the timescale for shunting to be effective is indeed rather longer than 150
s our results would explain why lumbar puncture is often used in conjunction with shunting
in the treatment of hydrocephalus, the lumbar puncture quickly returns the ventricles to their
undeformed configuration while the shunt prevents the intraventricular pressure from rising,
hence maintaining the brain in its undeformed state.
The time evolution of the system with a silicone diaphragm shunt inserted is more com-
plicated. For the first two initial conditions, corresponding to a hydrocephalic brain and a
healthy brain, the behaviour of the deformation and the pressure is similar to the previous
case, both return to a healthy state within a timescale of 100 s. In the case of a small negative
29
0 50 100 150 200
0
1
2
3x 104
t (s)
pw(A) (Pa)
0 50 100 150 200
0
5
10
x 10−3
t (s)
uw(A) (m)
Figure 15: Time evolution of ventricular pressure, pW(A, t) and displacement, uW(A, t) fol-
lowing introduction of a silicone diaphragm shunt for three initial configurations, (- - - ) severe
hydrocephalus, (—–) normal brain, (···) small negative perturbationsilicone diaphragm shunt-
ing
perturbation to uw(A) the results are markedly different, both the radial displacement and
intraventricular pressure fall farther below their normal values, until some kind of singularity
appears at t≈110 s. At this point both drop very sharply and it is likely that a number of
assumptions made in formulating the model, such as the small strain approximation, will cease
to be valid. For this reason we do not attempt to model the behaviour of the system past this
point, the crucial observation is that in the case of a valve exhibiting quadratic pressure volume
relationship, shunting may introduce the possibility of collapse of the ventricular cavity.
7 Conclusions
The three main areas in which we have made significant refinements to existing models are: the
specification of an anatomically realistic set of boundary conditions for the system, a review
of the parameters to be used in describing the poroelastic properties of the brain tissue and a
quasi-steady model for the time dependent behaviour of the system which gives the evolution
of hydrocephalus or the evolution of clinical treatment.
An important step forward in modelling the anatomy of the brain comes with our inclusion
of the cerebral aqueduct flow in addition to poroelastic flow through the parenchyma. This
enables us to simulate the behaviour of the brain and CSF pathways in their normal, non-
pathological state.
A two layered structure of the brain is useful since we can incorporate the differing material
properties of the white and grey matter, though due to a lack of experimental data we are
limited to prescribing different values for the permeability only. Our approach here is essentially
the same as that of Kaczmarek et al. (1997), and the results for pressure and fluid content
through the white and grey matter were similar to the findings of those authors.
The boundary conditions for the subarachnoid space are more sophisticated than in existing
30
models, where the authors have simply taken the fluid pressure to be fixed and the solid to either
rigid (Nagashima et al. (1987), Kaczmarek et al. (1997) and Levine (1999)) or unconstrained
Tenti et al. (1998). Our boundary conditions better enable us to consider the effects of a
change in the resistivity of the arachnoid villi or the viscosity of the CSF and to model the
deformation induced in the skull. For material parameters corresponding to an adult skull the
resultant displacement is very small, so in some sense our boundary conditions are similar to
those of a perfectly rigid outer layer of tissue. However, if values were known for the material
properties of an infant’s skull, our model could be used to analyse the magnitude of expected
deformation due to congenital hydrocephalus and the effect of treatments such as compressive
head wrapping.
We have considered the question of which values to use for the material parameters that
appear in the governing equations. In particular we found what we believe to be suitable values
for the Young’s modulus and Poisson’s ratio of the white and grey matter by reinterpreting
the calculations made in Kaczmarek et al. (1997) using the concept of drained and undrained
elastic constants in poroelasticty. We also computed values for the permeability of the white
and grey matter and the Reynolds number for flow in the aqueduct.
The mathematical analysis of the development and treatment of hydrocephalus which we
performed has not been attempted previously, we therefore believe that such an analysis rep-
resents a significant step towards a model of the illness which is both physically realistic and
clinically useful. Our results for the onset of hydrocephalus due to stenosis of the aqueduct
appear to be in agreement with clinical observations, both in terms of the stability of the
hydrocephalic steady state and the time scale for its development. The results for some of
the more unusual aetiologies, such as an increase in the viscosity of the CSF or impaired ab-
sorption in the subarachnoid space, were less successful in replicating clinical observations of
hydrocephalus. This may however be due to our use of an inappropriate method of simulating
of these aetiologies, rather than a weakness in our model overall.
Our results for the effect of shunting appear to be of some relevance, indicating that while
shunting should in general be effective in reducing the intraventricular pressure which is the
cause of hydrocephalus, certain types of shunt are likely to be more susceptible than others
to over drainage of the ventricles. Since our model of shunting is somewhat crude it is the
qualitative nature of these results, rather than the precise values for the location of the steady
states which arise and the (short) timescale for the relief of the condition, which we believe
to be of most interest. We also found that while lumbar puncture alone is ineffective in
treating hydrocephalus, when combined with shunting it can quickly and permanently relieve
the condition.
There are a number of extensions to this model which need to be considered. The very
simple spherical geometry could be enhanced by using more sophisticated numerical solutions.
Levine (1999) proposed a refinement to the governing equation for the fluid so that it includes
the effect of trans-parenchymal absorption of CSF, that some fluid is absorbed by capillaries in
the brain before it reaches the subarachnoid space. This leads to a new term in equations (3.2)
and (3.3), which results in solutions in the form of hyperbolic sine and cosine functions. It may
also be possible to incorporate the variable permeability model of Klachnar and Tarbell (1987)
where again full numerical solutions would be necessary but as we have pointed out, such
a vital extension requires better knowledge of the physical characteristics of the parenchyma.
Since brain tissue is unlikely to display a linear stress-strain response, non-linear elastic theory,
especially hyperelasticity, which has recently been applied with some success in other areas of
31
biomechanics (Humphrey, 2003), may be a way of formulating a more accurate constitutive
relationship for the white and grey matter. Incorporating the porous nature of brain tissue
into this type of non-linear theory is likely to present a mathematically challenging problem
which, if resolved, may have a wide range of applications in biomechanics.
The linear and quadratic models used for ball-on-spring and silicone diaphragm shunts
respectively are purely phenomenonological in nature and the line fitting method of parameter
estimation is likely to be approximate at best. This means that our results regarding the brain
deformation and timescale for the effectiveness of shunting may need to be re-evaluated, but
the qualitative nature of our findings should still be significant. We chose to use this approach
since the only existing mathematical models of shunting in the literature (Buchheit et al. (1982)
and Portnoy (1982)) are of an elementary form and we did not have sufficient time to construct
a detailed model of our own. A pressure-flow relationship S=S(∆p), based on a study of the
hydrodynamic properties of shunts and validated experimentally, would therefore be of great
use in the modelling of the treatment of hydrocephalus.
References
Bickers D. and Adams R. (1949). Hereditary Stenosis of the Aqueduct of Sylvius as a Cause
of Congenital Hydrocephalus.Brain,72:246–262.
Biot M. (1941). General Theory of Three Dimensional Consolidation.Journal of Applied
Physics,12:55–164.
Bradbury M. (1993). Anatomy and Physiology of CSF . In Schurr P. and Polkey C., editors,
Hydrocephalus.
Buchheit F., Maitrot D., Healy J., and S. G. (1982). How to choose the best valve. In Choux
M., editor, Shunts and Problems in Shunts.
Czosnyka M., Czosnyka Z., Whithouse H., and Pickard J. (1997). Hydrodynamic Properties
of Hydrocephalus Shunts: UK Shunt Evaluation Laboratory.Journal of Neurology, Neuro-
surgery and Psychiatry,62:43–50.
Drake J., Kestle J., and Milner R. (1998). Randomized Clinical Trials of Cerbrospinal Fluid
Shunt Design in Pediatric Hydrocephalus.Neurosurgery,43:294–305.
Drake J. and Sainte-Rose C. (1995). The Shunt Book. Blackwell Science.
Drossos A., Santomaa V., and Kuster N. (2000). The Dependence of Electromagnetic En-
ergy Absorbtion upon Human Head Tissue Composition in the range 300-3000 MHz .IEEE
Transactions on Microwave Theory and Techniques,48:1988–1995.
Fay J. (1994). Introduction to Fluid Mechanics. MIT Press.
Fung Y.C. (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag.
Hakim S., Venegas J.G., and Burton J. (1976). The Physics of the Cranial Cavity, Hydro-
cephalus, and Normal Pressure Hydrocephalus: Mechanical Interpretation and Mathmatical
Model.Surgical Neurology,5:187–210.
32
Holbourn A. (1943). Mechanics of Head Injuries.The Lancet,2:403–410.
Humphrey J. (2003). Continuum Biomechanics of Soft Biological Tissues.Proceeding of the
Royal Society, London,A459:3–46.
Kaczmarek M., Subramaniam R., and Neff S. (1997). The Hydromechanics of Hydrocephalus:
Steady State Solution for Cylindrical Geometry.Bulletin of Mathematical Biology,59:295–
323.
Klachnar M. and Tarbell J. (1987). Modelling Water Flow through Arterial Tissue.Bulletin
of Mathematical Biology,49:651–669.
Lakes R. (1993). Materials with structural hierarchy.Nature,361:511–515.
Lakes R. (2002). Making and characterizing negative Poisson’s ratio material.International
Journal of Mechanics Engineering Education,30:50–58.
Levine D. (1999). The Pathogenesis of Normal Pressure Hydrocephalus.Bulletin of Mathe-
matical Biology,61:875–916.
Metz H., McElhaney J., and Ommaya A. (1970). A Comparison of the Elasticity of Live, Dead
and Fixed Brain Tissue.Journal of Biomechanics,3:453–458.
Miller K. and Chinzei K. (1997). Constitutive Modelling of Brain Tissue: Experiment and
Theory.Journal of Biomechanics,30:115–1121.
Nagashima T., Tamaki N., Matsumoto S., Horwitz B., and Seguchi Y. (1987). Biomechanics
of Hydrocephalus: A New Theoretical Model.Neurosurgery,21:898–904.
Portnoy H. (1982). Hydrodynamics of shunts. In Choux M., editor, Shunts and Problems in
Shunts.
Reulen H., Graham R., Spatz M., and Klatzo I. (1977). Role of Pressure Gradients and Bulk
Flow in the Dynamics of Vasogenic Brain Edema.Journal of Neurosurgery,46:24–35.
Sahay K. and Kothiyal K. (1984). A Nonlinear Hyperelastic Model of the Pressure Volume
Function of Cranial Components.International Journal of Neuroscience,23:301–309.
Sahay K., Mehrotra R., Sachdeva U., and Banerji A. (1992). Elastomechanical Characterization
of Brain Tissues.Journal of Biomechanics,25:319–326.
Sahay K.B. (1984). On the choice of Strain Energy Function for Mechanical Characterisation
of Soft Biological Tissues.Engineering in Medicine,13:11–14.
Sivaloganathan S., Drake J., and Tenti G. (1998). Mathematical Pressure Volume Models of
the Cerebrospinal Fluid.Applied Mathematics and Computation,94:243–266.
Smillie A. (2003). A Biomechanical Model of the Pathogenesis and treatment of Hydrocephalus.
Master’s thesis, Oxford University.
Stastna M., Tenti G., Sivaloganathan S., and J. D. (1998). Brain Biomechanics: Consolidation
Theory of Hydrocephalus. Variable Permeability and Transient Effects.Canadian Applied
Mathematics Quarterly,7:93–110.
33
Tenti G., Drake J., and Sivaloganathan S. (2000). Brain biomechanics: Mathematical Modelling
of Hydrocephalus.Neurological Research,22:19–24.
Tenti G., Sivaloganathan S., and Drake J. (1998). Brain Biomechanics: Steady-state Consolo-
dation Theory of Hydrocephalus.Canadian Applied Mathematics Quarterly,7:111–125.
van Rietbergen H., Weinars B., Huiskes R., and Odgaard A. (1995). A New Method to deter-
mine Bone Elastic Properties and Loading using Micromechanical Finite Element Models.
Journal of Biomechanics,28:69–81.
Wang H. (2002). Theory of Linear Poroelasticity: with Applications to Geomechanics and
Hyrdogeology. Princeton University Press.
Weller R., Kida S., and Harding B. (1993). Aetiology and Pathology of Hydrocephalus. In
Schurr P. and Polkey C., editors, Hydrocephalus.
