Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture

21 Pages·2014·6.09 MB·English

Journal of Basic & Applied Sciences, 2014, 10, 393-413 393
Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture
Tor A. Kwembe* and Ashley M. Sanders
Department of Mathematics and Statistical Sciences, Jackson State University, JSU Box 17610, Jackson, MS
39217, USA
Abstract: We have presented a mathematical model to study the evolution, growth and risk rupture of nontraumatic
aneurysms contained within a cylindrical region of blood vessels. Analytical and numerical solutions are studied. Results
affirmed that the intra-aneurysmal pressure and bloodstream flow account for the evolution and growth of aneurysms,
and we find that an aneurysm may rupture when the ratio of the lateral membrane contraction to longitudinal membrane
extension approaches one. Numerical properties of intra-aneurysmal pressure, impact fluid velocity, membrane
displacement and the deformed radius with respect to the Poisson ratio, membrane thickness and extensional rigidity are
studied. The importance of the findings is rested on the fact that they can be used to improve noninvasive means for
predicting aneurysm rupture, and treatment and management decisions after rupture.
Keywords: Elastodynamics, Filtration, Navier-Stokes, Numerical Solutions, Permeability, Poisson Ratio, Aneurysm.
1. INTRODUCTION bloodstream that acts parallel to the vessel wall [3, 9,
12]. The overall impact of these forces on the thinning
In this paper, we describe a mathematical model
of the aneurysm wall has not been suggested in the
which may lead to understanding the evolution, growth
literature. In this study, we have shown that when the
and rupture potential of nontraumatic aneurysms. The
lateral contraction of the membrane wall is in balance
model describes a quasi-static, non-convectional
with the longitudinal extension, an aneurysm may
acceleration, axi-symmetric Navier-Stokes equations in
rupture and that at the rupture point about 80% of the
cylindrical coordinates coupled with the Camenschi-
membrane wall thins out. The analytical and numerical
Fung [1, 2] type linear elastodynamic system of
results presented here affirms the results in the
equations with filtration. The profile of solutions of the
literature [3, 9, 10] that the intra-aneurysmal pressure
system of equations described may help to provide
and the bloodstream flow contribute to the evolution
insights in developing noninvasive means for detecting
and development of aneurysms.
when a nontraumatic aneurysm may rupture and
deciding the best treatment and management Modern neuroimaging techniques often detect
strategies of ruptured aneurysms. unruptured cerebral artery aneurysm, which is
estimated to be present in 3%-6% of the population [9,
The genesis of a nontraumatic aneurysm is
10]. Aggressive rupture preventing treatment is often
contingent on any condition that causes the walls of the
an option, but may lead to morbidity. The specific risk
blood vessel to weaken [3-8]. The most commonly
for rupture of a nontraumatic aneurysm is unknown and
investigated forms of aneurysms are the aortic,
risk assessments are based on general knowledge of
cerebral artery and intracranial aneurysm[3, 5-7, 9-11].
factors leading to subarachnoid hemorrhage deduced
The development of the cerebral aneurysm, for
from epidemiological studies [6, 9, 10]. Additionally,
example, is contingent on various physical factors
aneurysms of large size, proximal location, and small
associated with blood flow [3, 5, 10]. Studies suggest
neck, or fundus ratio are associated with increased risk
that the inertial forces of the bloodstream result in the
for rupture [5, 9, 10]. Thus, more reliable parameters to
local elevation of intravascular pressure and the flow
predict the risk of aneurysmal rupture are needed.
impact [9]. This means that, the impacting forces and
Intra-aneurysmal pressure gradients, bloodstream flow
the local pressure elevation at the aneurysm have a
profiles, membrane displacement profiles, membrane
large contribution to the development of cerebral
thickness, and Poisson ratio could provide additional
aneurysms. The other contingent factor is the wall
information regarding the risk of rupture. Moftakher, R.
shear stress, which is the viscous friction of the
et al. hypothesized in [13] that Phase Contrast with

vastly undersampled isotropic projection reconstruction
could accurately assess intra-aneurysmal pressure

*Address correspondence to this author at the Department of Mathematics and gradients in a canine aneurysmal model when
Statistical Sciences, Jackson State University, JSU Box 17610, Jackson, MS
39217, USA; Tel: [phone]; Fax: [phone]; compared with invasive measurements. Isaken, J. G. et
E-mail: [email protected]
al. developed in [9] a computational model for
AMS subject classification. 35Q92, 92C37, 65N99, 35Q30, 35Q92, 76Z05,
76Z99. simulation of fluid-structured interaction in cerebral

ISSN: 1814-8085 / E-ISSN: 1927-5129/14 © 2014 Lifescience Global
394 Journal of Basic & Applied Sciences, 2014 Volume 10 Kwembe and Sanders
aneurysms based on patient specific lesion geometry, which remain in place after the media has deflated
with emphasis on wall tension. (aneurysm has ruptured), into consideration. We have
considered the Poisson ratio in the anisotropic material
In the proposed model, we have introduced range 1 < (cid:1)0 < 2 in this study to indicate that the
additional parameters, the Poisson ratio and membrane extensional rigidity is weakened by
generating a substantial anisotropy in stiffness. This is
membrane wall thickness, as determining measures
discussed in section 2 in the activities leading to the
predicting the potential for an aneurysm to rupture. The
development of an aneurysm sac in the media layer of
mathematical model is constituted by equations (1) -
the membrane which is surrounded by the inner and
(14) of section 2. We used the Camenschi outer layers made of dominantly polyurethane material
dimensionless variable transforms and quasi-static which is anisotropic [15]. In section 4, we provided the
conditions [1] to reduce the problem to parameters that post rupture analysis and demonstrated how the
can be measured by noninvasive means. The membrane extensional rigidity deduced from the
membrane wall thickness and Poisson ratio can be
analytical solutions provide conditions for membrane
used to design a grading scale to measure the severity
enlargement to minimum and maximum stretches that
of aneurysm rupture.
depends on the intra-aneurysmal pressure gradients
and rates. It also reveals that a nontraumatic aneurysm
2. MATHEMATICAL MODEL
may rupture when the Poisson ratio, (cid:1), approaches
0
anisotropic material values. In section 3, we provide We will derive a model and then illustrate how it can
numerical analysis of the solutions based on be used to model the evolution, growth, and predict the
experimental data of parameters derived from the cited rupture potential of nontraumatic aneurysms. It is
literature. The analysis confirms that the profile of the known that inertial forces of the bloodstream results in
local elevation of intra-vascular pressure and the flow
deformed radius and the displacement components of
impacting force together with local pressure elevation
the membrane becomes discontinuous as the Poisson
at the aneurysm contributes to the development of
ratio approaches anisotropic material values.
aneurysms [10, 12]. It is also hypothesized that
Numerical analysis of intra-aneurysmal pressure, aneurysms rupture when the wall tension exceeds the
membrane displacement and thickness affirm that their strength of the wall tissue. We shall consider an axi-
profile before and after the rupture of an aneurysm are symmetric motion of equation in two media; the
consistent with in vitro and in vivo observations. In Newtonian fluid, coupled with the linear elastic
membrane. Due to the nature of the origin of the blood
particular, it shows that when 80% of the membrane
flow into the blood vessels, which is imposed by the
wall thins out then the aneurysm may rupture.
heart pumping blood into the circulatory system, we
shall consider non-convection acceleration, axi-
There are many hypotheses regarding aneurysm
symmetric, viscous, incompressible pressure driven
enlargement and rupture [3-10]. However, the roles of
Navier-Stokes equations in cylindrical coordinates
the Poisson ratio and the thinning of the membrane
coupled with the Camenschi-Fung type elastic
wall are not well understood. Shah and Humphrey
membrane equations [1, 2, 16]. We shall assume that
suggested in [14] that studies on the mechanics of
the aneurysm occurs within a cylindrical portion of the
saccular aneurysms should be focused on quasi-static
analyses that investigate the roles of lesion geometry vessel of length L. We let a be the undeformed
0
and material properties including growth and radius of the cylinder and a(z,t) be the deformed

remodeling. The model presented here has taken the
radius describing the curvature of the aneurysm. We let
material properties, growth and remodeling into
P(z,t) be the intra-aneurysmal pressure with P(t) and
account. Remodeling of the membrane wall in 1
responses to tears is incorporated in the filtration P(t) the end pressures at z=0 and z= L respectively
2
coefficient and retain the normal balance in the micro-
(See Figure 1).
vascular mechanism in supplying tissues or the
surrounding fluids with nutrients and clearing waste (cid:1)
If we let (cid:1)=((cid:1),u) be the velocity vector of the fluid
products. However, the model does not take into
flowing through the portion of the vessel containing the
consideration the healing process of the membrane
layers. Another interesting finding is that when the aneurysms where (cid:1)(z,r,t) and u(z,r,t) are the
(cid:1)
Poisson ratio is in the range 0 < (cid:1) < 1, and (cid:1) > 2, the transversal and longitudinal components, and (cid:1)=((cid:2),(cid:3))
0 0
aneurysm may stretch or shrink but not rupture. In fact, is the displacement vector of the membrane. (cid:1)(z,t)

most experimental studies choose values of Poisson
and (cid:1)(z,t) are the transversal and longitudinal
ratio in the isotropic material range 0 < (cid:1) < 0.5 and do
0
components of the membrane displacement
not factor the filtration process and anisotropic material
respectively as illustrated in Figure 1 and furthermore,
composition of the membrane’s inner and outer layers,
Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture Journal of Basic & Applied Sciences, 2014 Volume 10 395

Figure 1: Model of a deformed membrane contained within a cylindrical portion of a blood vessel.
if t and t are the stress tensor components in the A normal artery wall consist of three layers. The
rz rr
innermost endothelial layer is called the intima, the
fluid then, we have the Camenschi-Fung type system
middle layer consisting of smooth muscle is called the
of equations [1, 2]:
media and the other layer consisting of connecting
tissues is called the adventitia [2]. The deformed
(cid:4)(cid:2) 1 (cid:4)P μ(cid:5)(cid:4)2(cid:2) 1(cid:4)(cid:2) (cid:4)2(cid:2) (cid:2)(cid:8)
=(cid:1) + (cid:6) + + (cid:1) (cid:9); (1) curvature of the blood vessel or aneurysmal sac is
(cid:4)t (cid:3)(cid:4)r (cid:3)(cid:7)(cid:4)r2 r (cid:4)r (cid:4)z2 r2(cid:10) composed of only the intima and adventitia. The
material composition of the intima and adventitia is
(cid:3)u 1 (cid:3)P μ(cid:4)(cid:3)2u 1(cid:3)u (cid:3)2u(cid:7) made up of dominantly anisotropic polyurethane. The
=(cid:1) + (cid:5) + + (cid:8); (2) intima, based on in vitro and in vivo observations [17,
(cid:3)t (cid:2)(cid:3)r (cid:2)(cid:6)(cid:3)r2 r (cid:3)r (cid:3)z2(cid:9) 18] remains normal but subintimal cellular proliferation
also occurs when aneurysm developed. The internal
(cid:2)(cid:1) (cid:1) (cid:2)u elastic membrane responsible for the thinning of the
+ + =0 (3)
(cid:2)r r (cid:2)z artery wall is either reduced in size or absent causing
the media to retract to the junction of the aneurysm
neck with the parent blood vessel. This development
(cid:5)2(cid:4) (cid:6)(cid:1) (cid:5)(cid:3) (cid:5)2(cid:4)(cid:9)
(cid:2) h = D 0 + +t r=a ; (4) transforms the membrane within the aneurysm region
m (cid:5)t2 (cid:8)(cid:7)a (cid:5)z (cid:5)z2(cid:11)(cid:10) rz 0 from pseudoelastic isotropic to anisotropic media.
0
Thus, the deformation transforms the curvature of the
(cid:6)2(cid:4) (cid:7)(cid:2) (cid:6)(cid:5) (cid:4)(cid:10) weakened portion of the blood vessel into an
(cid:3) h =(cid:1)D(cid:8) 0 + (cid:11)(cid:1)t r=a ; (5) aneurysmal sac and hence tolerating the extension of
m (cid:6)t2 (cid:9)a0 (cid:6)z a02(cid:12) rr 0 the membrane and transverse shear, curving and
pressure difference. We shall consider the blood
(cid:4)(cid:3)u (cid:3)(cid:2)(cid:7) vessels as permeating deformable shell filled with a
trz =(cid:1)μ(cid:6)(cid:5)(cid:3)r + (cid:3)z(cid:9)(cid:8) r=a0 (6) pulsating incompressible fluid and constrained by the
surrounding tissue and fluid. The extensional rigidity D
of the membrane depend on its stiffness and the
(cid:3) (cid:2)u(cid:6) deformation grows in the direction of the space created
t = P(cid:1)2μ r=a (7)
rr (cid:5)(cid:4) (cid:2)r(cid:8)(cid:7) 0 by the surrounding tissue due to impacting forces on

the vessel wall. The deformation of the vessel is not
volume preserving but the biological material properties
defined in the region 0<r(cid:1)a (cid:1)a(z,t);0<z<L;t(cid:2)0
0 of the membrane are more or less the same
where (cid:1) is the mass density of the fluid, μ is the
everywhere and transition from isotropic to anisotropic
dynamic viscosity, (cid:1) is the mass density of the media as aneurysm evolved. It is also biologically
m
known that the interior of a cell is anisotropic due to
membrane, h is the thickness of the membrane wall,
intracellular organelles. This characteristic may be
Eh
D= is the extensional rigidity of the membrane, responsible for the development of two or more
1(cid:1)(cid:2)2 aneurysms on the same artery. Multiple aneurysms are
0
due to defects in the arterial wall and may be
E is the Young’s modulus, and (cid:1) is the Poisson ratio.
0
396 Journal of Basic & Applied Sciences, 2014 Volume 10 Kwembe and Sanders
congenital [19]. Nevertheless, we shall consider the We will now justify (10). In a normal vascular
membrane as a pseudoelastic isotropic material in the system of functions, there is a free exchange of
biological sense [2] that the material properties of the nutrients, water, electrolytes and microphage between
membrane remain the same throughout the the intra-vascular and extra-vascular components of
deformation process becoming anisotropic only during the blood vessels. Several mechanisms are
the development of aneurysm sac. This feature holds responsible for this critical function of the vascular
only for nontraumatic aneurysms. The theory of system. Physiologists, including Michel [12, 25, 26]
isotropic elasticity allows the Poisson ratio in the range investigated the mechanism by which plasma and its
(cid:1)1(cid:3)(cid:2) (cid:3)0.5 for an object with surface with no solutes cross the vascular barrier. They discovered that
0
capillaries are the vascular segment responsible for
constraint. Physically, this means that for the material
molecular exchange in normal tissues and that gases,
to be stable, the stiffness must be positive. That is, it is
water and microphages cross the capillary endothelial
required that both the bulk and shear modulus be
cell barrier freely, but the passage of larger molecules
greater than or equal to zero [20, 21]. On the other
such as plasma proteins are tightly restricted. They
hand, isotropic objects that are constrained at the
also discovered that several mechanisms are involved
surface can have Poisson ratios outside the above
in this exchange. The most important, though, are the
range and be stable [20, 21]. Since the blood vessels
bulk flow and diffusion. The rate of change in either
are constrained by the surrounding tissue and fluid, we
direction is determined by physical factors such as
shall consider values of Poisson ratio to include values
hydrostatic pressure, osmotic pressure, and the
outside the isotropic range (cid:1)1(cid:3)(cid:2) (cid:3)0.5. This
0 physical nature of the barrier separating the blood and
consideration also includes the regime of anisotropic the interstitium of the tissue. That is, the permeability of
deformation, when the constituted material of the the membrane wall. While the diffusion process is
membrane within the aneurysm region is made up of deemed the most important mechanism in this
the intima and adventitia, since the concept of Poisson exchange, the diffusion coefficient in the Fick equation
ratio can be extended to anisotropic materials with [27] depends on molecular size [28]. It is important for
values outside the isotropic range (cid:1)1(cid:3)(cid:2) (cid:3)0.5 [15, 20- the exchange of small molecules which is driven by
0
molecular concentration gradient across vascular
23].
endothelium defined by the Fick equation
We shall also assume that the natural vascular J =k(C (cid:1)C ). C (cid:1)C , is the concentration difference.
0 (cid:2) i (cid:2) i
process of exchange of nutrients and waste from the Sample and Golovin employed this condition in [29, 30]
evolution and development of aneurysms are by to study the dynamics of a double-lipid bilayer
membrane by coupling intermembrane separation and
filtration. Thus, the boundary conditions expressing the
the lipid chemical composition of a two-component
adherence of the fluid to the membrane wall and the
membrane and dependence on the membrane
fluid filtration through the membrane are:
curvatures. They focused on the thermodynamical
equilibrium in [29] and non-equilibrium in [30] of fluxes
(cid:1)(z,a(z,t),t)= (cid:3)(cid:2)+ (cid:1)n,f (8) across the membrane. In this derivation, we are
(cid:3)t 2 considering the impacting forces on the membrane,
(cid:4)(cid:3)a(cid:7)
1+ rather than the concentration of the fluid content of the
(cid:5) (cid:8)
(cid:6)(cid:3)z(cid:9) blood vessels and hence admit the exchange of large

molecular fluxes, such as plasma proteins.
(cid:4)(cid:3)a(cid:7) Consequently, filtration is much more important than
(cid:3)(cid:2) (cid:1)n,f (cid:6)(cid:5)(cid:3)z(cid:9)(cid:8) diffusion for flux of large molecules such as plasma
u(z,a(z,t),t)= + (9) proteins and is governed by the Starling equation [12,
(cid:3)t 2* 13, 25, 26]:
(cid:4)(cid:3)a(cid:7)
1+
(cid:5) (cid:8)
(cid:6)(cid:3)z(cid:9)
k*
(cid:2) = (cid:5)(P(z,t)(cid:1)P)(cid:1)(cid:4)((cid:3)(cid:1)(cid:3) )(cid:7),
nf μh(cid:6) e e (cid:8)

where (cid:1) is the filtration velocity normal to the
nf
membrane wall, and we supposed that it is governed where k*, the filtration coefficient, is a property of the
by the Darcy-Starling Law [14, 24] given as: membrane wall and a measure of the permeability of
the membrane to water, P(cid:1)P and (cid:2)(cid:1)(cid:2) are
e e
k*
(cid:2) = (P(z,t)(cid:1)P) (10) hydrostatic and osmotic pressure differences,
n,f μh e respectively, between the plasma and the interstitium,
(cid:1) is the osmotic reflection coefficient and varies from
where k* is the permeability coefficient and P is the zero to one. High values of (cid:1) indicate little plasma-
e
constant exterior pressure. protein escape [28]. When (P(z,t)(cid:1)P)(cid:1)(cid:3)((cid:2)(cid:1)(cid:2) ) is
e e
Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture Journal of Basic & Applied Sciences, 2014 Volume 10 397
positive, filtration takes place and when it is negative all forms of aneurysms within a cylindrical portion of the
reabsorption takes place. The amount of fluid filtered or vessel.
reabsorbed per unit time, the filtration velocity or flux is
determined by the permeability of the membrane and Assumption 2 in (14) leads to a quasi-static
by the surface area available for the exchange. problem. That is, initial conditions are not needed to
solve and analyze the problems of aneurysm evolution
Vascular permeability is essential for the health of and development. We now non-dimensionalize
equations (1) - (13) using (14) and the following
normal tissues and it is also an important characteristic
Camenschi [1] dimensionless variable transformations
of many disease state in which it is greatly increased
[12, 25, 26, 28]. Since aneurysm is caused by the
L u a
weakening of the membrane which results in high t= t0; z= Lz0; r=a r0;(cid:1)= * 0(cid:1)0;u=uu0;
u 0 L *
levels of plasma-protein escape activity, we shall take *
the osmotic reflection coefficient to be zero and,
μu L
therefore, arrive at the Darcy-Starling filtration velocity P= * P0;(cid:3)=(cid:1)L(cid:3)0; (cid:2)=(cid:1)a(cid:2)0; a(z,t)=a a0(z0,t0)
given in equation (10). That is, when an aneurysm a2 0 0
0
evolved, the reabsorption process is stopped.
where (cid:1)<1 is a dimensionless parameter given as
The values of the pressure at the ends of the
aneurysm region are given by: Pa μu L
(cid:1)= * 0 = * .
D a D
P(0,0,t)= P(t); P(L,0,t)= P(t). (11) 0
1 2
The expression for (cid:1) and condition (14) are natural
We further assume that, the end at z=0, the
scaling obtained from the nondimensional analysis
beginning of the aneurysm region is fixed and the end
process and agreed with those of fluid flow in
of the aneurysm region z= L is free of stress, we
permeable media and Camenschi [1]. The superscript
have:
index "0" is used for dimensionless quantities. Upon
dropping the index after transformation, equations (1) -
(cid:3) =0; D(cid:6)(cid:5)(cid:3)+(cid:1) (cid:2)(cid:9) =0;t(cid:4)0 (12) (13) reduces to the following dimensionless equivalent
z=0 (cid:8)(cid:7) (cid:5)z 0 a (cid:11)(cid:10) z=L equations:
0
The deformed radius is given by (cid:3)P
=0; 0<r(cid:1)1(cid:1)a(z,t); t(cid:2)0 (15)
(cid:3)r
a(z,t)=a +(cid:1)(z,t); 0<z<L; t(cid:2)0. (13)
0
(cid:3)P 1 (cid:3) (cid:4) (cid:3)u(cid:7)
= r ; 0<r(cid:1)1(cid:1)a(z,t); 0<z<1;t(cid:2)0 (16)
We further make the following assumptions: (1) the (cid:5) (cid:8)
(cid:3)z r (cid:3)r(cid:6) (cid:3)r(cid:9)
length of the aneurysm region must be much larger
than the undeformed radius of the blood vessel and (2):
1 (cid:3) (cid:3)u
(r(cid:1))+ =0; 0<r(cid:2)a (cid:2)a(z,t); 0<z<1 (17)
r (cid:3)r (cid:3)z 0
(cid:2)a (cid:5)2 (cid:1)u a (cid:1) hu2
(cid:3) 0(cid:6) (cid:1)1; * 0 (cid:1)1; m * (cid:1)1. (14)
(cid:4) L(cid:7) μ D (cid:5)(cid:2) (cid:5)2(cid:3) (cid:5)u
(cid:1) + = r=1; 0(cid:4)z(cid:4)1 (18)
0 (cid:5)z (cid:5)z2 (cid:5)r
where u is the characteristic longitudinal speed.
*
(cid:6)(cid:3)
(cid:1) +(cid:2)= P(z,t); 0(cid:4)z(cid:4)1; t(cid:5)0 (19)
Thus, equations (1) - (14) constitute the 0 (cid:6)z
mathematical model describing the evolution and
development of aneurysms contained within a along with the boundary conditions:
cylindrical portion of the blood vessel. If any form of
(cid:8)(cid:5)
aneurysm occurs within the cylindrical portion of the (cid:4)(z,r,t) =(cid:3) +(cid:2)(P(cid:1)P); 0(cid:6)z(cid:6)1; t(cid:7)0 (20)
arteries or blood vessels, in general, then the deformed r=a(z,t) (cid:8)t e
radius a(z,t) defines the geometry of the aneurysm.
k*L2
Since, as Figure 1 shows, the extension, or stretch, of where (cid:1)= .
a(z,t) is measured through the neck of the aneurysm, ha03
or the opening into the aneurysm (fundus aspect ratio),
from the bloodstream vessel. Consequently, the model (cid:5)(cid:2)
u(z,r,t) =(cid:1) ; t(cid:4)0; 0(cid:3)z(cid:3)1 (21)
given here can be used to study the characteristics of r=a(z,t) (cid:5)t
398 Journal of Basic & Applied Sciences, 2014 Volume 10 Kwembe and Sanders
(cid:1) =0;t(cid:2)0 (22) aneurysm is in the weakening of the membrane at
z=0
r=a . Therefore, (28) and (29) constitute the
0
components that measure the dynamical changes of
(cid:7)(cid:6)(cid:4) (cid:10)
(cid:1) +(cid:2)(cid:3) =0;t(cid:5)0 (23) the aneurysm wall and the deformed radius given in
(cid:9)(cid:8) (cid:6)z 0 (cid:12)(cid:11) z=1
(25) in dimensionless representation, measures the
size of the aneurysm radially and whose change with
P(0,t)= P(t); P(1,t)= P(t); t(cid:1)0 (24) respect to P(z,t) may not be linear.
1 2
a(z,t)=1+(cid:1)(cid:2)(z,t); 0(cid:3)z(cid:3)1; t(cid:4)0. (25) The dimensional equivalent of the aneurysm size is

given in (13). The velocity and membrane components
and the aneurysm size are completely determined if the
We note here that the dimensionless system of
equations constitute only the intra-vascular pressure, pressure functions P(z,t),P(t) and P(t) are found.
1 2
the components of the fluid velocity and membrane
For the pressures P(t) at the beginning of the
1
displacement, the Poisson ratio (cid:1) and the filtration
0 aneurysm region z=0 and P(t) at the end of the
coefficient k* as the inertial forces impacting the aneurysm region z= L, we w ill2 use the Milnor [3, 31,
membrane wall. Since in simulation, (cid:1) will always be
32] pressure waveform
taken to be less than one, the mass densities and fluid
viscosity do not play a major role in the evolution and
P(t)= P +(cid:2)N (A cos(n(cid:1)t)+B sin(n(cid:1)t)),
development of an aneurysm and in its potential to m n=1 n n
rupture. In dimensionless analysis, the evolution of an
aneurysm will always depend on the size of the where P is the mean blood pressure, A and B are
m n n
transversal displacement of the membrane and wall
Fourier coefficients for N harmonics, and (cid:1) is the
thickness, h. circular frequency. In the case of Brachial Artery
aneurysms, P(t) and P(t) can be taken as the
We also note that the dimensionless equations of 1 2
the non-convection acceleration Navier-Stokes Brachial Artery Peripheral Pulse Pressures measured
equations are the same as with the Camenschi’s non-invasively with any of the modern conventional
convection acceleration, quasi-static equations. Thus, sphygmomanometry. P(t) is the diastolic pressure
1
making the mathematical model given here a slow flow,
quasi-static problem. The solutions to the system of wave form for N <10, and P2(t) is the systolic
equations (15) - (24) as with Camenschi in [1] are pressure waveform for N (cid:1)10.
obtained by direct integration under the conditions that
(cid:1)(z,r,t) and u(z,r,t) remain bounded at r=0 and So, P= P(z,t) is the intra-aneurysmal pressure and
noting from equation (15) that P = P(z, r, t) only. from (15), (16), (19), (20) and (26) - (28) it satisfies the
Hence, we have the following expression for the fluid nonlinear partial differential equation
velocity and membrane displacement components. The
3
argument (z, r, t) is omitted in P, (cid:1), and (cid:1) in some 1 (cid:12)(cid:15) (cid:9) 2(cid:1)(cid:4) (cid:4) (cid:19)(cid:22)(cid:15) (cid:5)2P(z,t)
(cid:13)1+(cid:3)(cid:10) 0 P(z,t)+ 0 P(t)(cid:20)(cid:23)
cases for printing convenience. 16(cid:14)(cid:15) (cid:11)(cid:10)2(1(cid:1)v02) 2(1(cid:1)v02) 2 (cid:21)(cid:20)(cid:24)(cid:15) (cid:5)z2
(cid:3)(2(cid:1)(cid:4))(cid:12)(cid:15) (cid:9) 2(cid:1)(cid:4) (cid:4) (cid:19)(cid:22)(cid:15)2(cid:6)(cid:5)P(z,t)(cid:16)2
(cid:3)(z,r,t)=(cid:7)(cid:9)(cid:8)(cid:1)1r63+8r(1+(cid:2)(cid:4))2(cid:10)(cid:12)(cid:11)(cid:6)(cid:6)2zP2 +(cid:2)4r(1+(cid:2)(cid:4))(cid:6)(cid:6)(cid:4)z (cid:6)(cid:6)Pz (cid:1)(cid:2)2r(cid:6)(cid:6)z2(cid:5)(cid:6)t (26) +(cid:3)8((cid:12)(cid:15)1(cid:1)v020)(cid:9)(cid:13)(cid:14)(cid:15)21+(cid:1)(cid:4)(cid:3)(cid:11)(cid:10)(cid:10)2(1(cid:1)v002)P(z,(cid:4)t)+2(1(cid:1)0v(cid:19)02)1P(cid:1)2(2t(cid:4))(cid:21)(cid:20)(cid:20)(cid:23)(cid:24)(cid:15) (cid:8)(cid:7) 2(cid:1)(cid:5)z(cid:4) (cid:22)(cid:15)(cid:18)(cid:17)(cid:5)P(z,t))
(cid:1) (cid:13)1+(cid:3)(cid:10) 0 P(z,t)+ 0 P(t)(cid:20) 0 + 0 (cid:23)
1 (cid:5)P(z,t) (cid:5)(cid:4)(z,t) 2(cid:14)(cid:15) (cid:11)(cid:10)2(1(cid:1)v02) 2(1(cid:1)v02) 2 (cid:21)(cid:20)2(1(cid:1)v02) (1(cid:1)v02)(cid:24)(cid:15) (cid:5)t
u(z,r,t)= [r2(cid:1)(1+(cid:2)(cid:3)(z,t))2] +(cid:2) (27)
4 (cid:5)z (cid:5)t =(cid:1) (cid:3) (cid:12)(cid:13)(cid:15)1+(cid:3)(cid:9)(cid:10) 2(cid:1)(cid:4)0 P(z,t)+ (cid:4)0 P(t)(cid:19)(cid:20)(cid:1)2(cid:4)(cid:22)(cid:23)(cid:15)P'(t)
4(1(cid:1)v02)(cid:14)(cid:15) (cid:11)(cid:10)2(1(cid:1)v02) 2(1(cid:1)v02) 2 (cid:21)(cid:20) 0(cid:24)(cid:15) 2
2(cid:1)(cid:2) (cid:2) +(cid:2)(P(z,t)(cid:1)P)
(cid:3)(z,t)= 0 P(z,t)+ 0 P(t) (28) e
2(1(cid:1)(cid:2)2) 2(1(cid:1)(cid:2)2) 2 (30)
0 0
We shall employ the perturbation method of [1, 33]
(cid:4)(z,t)= 1(cid:1)2(cid:2)0 (cid:5)zP((cid:3),t)d(cid:3)(cid:1) P2(t) z (29) by developing the pressure function P(z,t) in a power
2(1(cid:1)(cid:2)02) 0 2(1(cid:1)(cid:2)02) series with respect to the small parameter (cid:1) and
neglecting the O((cid:1)2) terms. That is, we assume that
We know that the influence of the stress tensor
components t and t in the fluid have been
rz rr (cid:2) (cid:1)nP(z,t)
incorporated in solving for the components of the P(z,t)=(cid:3) n (31)
n!
membrane displacement. Hence, the genesis of the n=0
Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture Journal of Basic & Applied Sciences, 2014 Volume 10 399
Consequently, we have the following system of P <0 or upward if P >0 and the morphology can
zz zz
partial differential equations in P(z,t) and P(z,t):
0 1 also be up in some direction and down in others. We
shall examine these characteristics with respect to the
1 (cid:3)2P(z,t) values of the Poisson ratio (cid:1) . Theorem 2.1
0 =(cid:2)[P(z,t)(cid:1)P] (32) 0
16 (cid:3)z2 0 e establishes conditions that allow for a minimum and
maximum stretch of the deformed radius and hence
aneurysmal curvature. Theorem 2.2 establishes
1 (cid:4)2P(z,t) 1 (cid:8) 2(cid:1)(cid:3) 3(cid:3)P(t)(cid:14)(cid:4)2P(z,t)
1 (cid:1)(cid:2)P(z,t)= (cid:1) (cid:9)3 0 P(z,t)+ 0 2 (cid:15) 0 conditions on the Poisson ratio conditioned for
16 (cid:4)z2 1 16(cid:10)(cid:9) 2(1(cid:1)(cid:3)02) 0 2(1(cid:1)(cid:3)02)(cid:16)(cid:15) (cid:4)z2 aneurysm rupture.
(2(cid:1)(cid:3))(cid:5)(cid:4)P(z,t)(cid:11)2 1(cid:8)1(cid:1)2(cid:3) (cid:4)P(z,t) P'(t)(cid:14)
(cid:1)8(1(cid:1)(cid:3)002)(cid:7)(cid:6) 0(cid:4)z (cid:13)(cid:12) +4(cid:10)(cid:9)(cid:9)1(cid:1)(cid:3)020 0(cid:4)t (cid:1)1(cid:1)2(cid:3)02(cid:16)(cid:15)(cid:15) THEOREM 2.1. Let P be twice differentiable with
2(cid:1)(cid:3) (cid:4)P(z,t) (cid:3)P'(t) (cid:1)2P(z,t) (cid:1)2P(z,t)
+ 0 0 + 0 2 respect to z and t and = such that
2(1(cid:1)(cid:3)02) (cid:4)t 2(1(cid:1)(cid:3)02) (cid:1)z(cid:1)t (cid:1)t(cid:1)z
(33)
2
(cid:3)2P(z,t) (cid:3)2P(z,t) (cid:2) (cid:3)2P(z,t) (cid:5)(cid:3)2P(z,t)(cid:8)
The solution of equations (32) - (33) are solved by (cid:4) + 0 P"(t)> ,
direct integration, using the boundary conditions in (cid:3)z2 (cid:3)t2 2(cid:1)(cid:2) (cid:3)z2 2 (cid:7)(cid:6) (cid:3)z(cid:3)t (cid:10)(cid:9)
0
(24). The perturbation solution is then
P(z,t)= P(z,t)+(cid:1)P(z,t). The expressions for P(z,t) and (cid:1) (cid:3)[0,1)(cid:1)(1,2)(cid:1)(2,(cid:2)). If a(z,t) given in (34) is
0 1 0 0
and P(z,t) are given in Appendix A. The approximate twice differentiable in z and t then the aneurysm wall
1
aneurysm wall development using (25) and (28) is attains a minimum extension at some point z(cid:1)(0,1) for
given by the equation all t(cid:1)0 provided
(cid:4) 2(cid:1)(cid:3) (cid:3) P(t) (cid:7) (cid:2)f((cid:3) ,P(z,t),P)P"(t)+g((cid:3) )(P(z,t)(cid:1)P)>0
a(z,t)=1+(cid:2)(cid:5) 0 P(z,t)+ 0 2 (cid:8) (34) 0 2 2 0 e
(cid:6)(cid:5)2(1(cid:1)(cid:3)02) 2(1(cid:1)(cid:3)02)(cid:9)(cid:8) Similarly, if
Thus, the evolution and development of the
(cid:2)f((cid:3) ,P(z,t),P)P"(t)+g((cid:3) )(P(z,t)(cid:1)P)<0
aneurysm size is dependent on the following impact 0 2 2 0 e
forces and parameters: (i) The Poisson ratio, (cid:1) , which
0 then the aneurysm wall attains a maximum extension
we henceforth defined as the ratio of the lateral or
where
transversal contraction to longitudinal extension, (ii)
The pressures at the beginning of the aneurysm region
z=0;P1(t) and at the end of the aneurysm region f(v0,P(z,t),P2)=(1(cid:1)v02)1(2(cid:1)v0)(cid:2)(cid:3)(cid:5)(cid:4)(cid:5)2(21(cid:1)(cid:1)vv002)P(z,t)+2(1v(cid:1)0v02)P2(t)(cid:6)(cid:7)(cid:5)(cid:8)(cid:5)
z=1,P(t); The intra-aneurysmal pressure P(z,t),
2
and
which in turn depends on (cid:1) , P(t), P(t), the rates
0 1 2
P'(t) of P(t) and P'(t) of P(t) and the filtration 2(cid:2)
1 1 2 2 g(v )=
velocity coefficient (cid:1). The filtration velocity coefficient 0 v2(cid:1)v +1
0 0
(cid:1) accounts for the thinning of the aneurysm wall and
contributes to the way the intra-aneurysmal pressure, Furthermore, the aneurysm curvature is up in one
P(z,t), is moderated. It also accounts for the direction and down in the other if

remodeling and growth of the membrane in response to
2
the tension exerted on it by the forces of fluid flow and (cid:2)2P(z,t) (cid:2)2P(z,t) v (cid:2)2P(z,t) (cid:4)(cid:2)2P(z,t)(cid:7)
(cid:3) + 0 P"(t)<
stress tensors. (cid:2)z2 (cid:2)t2 2(cid:1)v (cid:2)z2 2 (cid:6)(cid:5) (cid:2)z(cid:2)t (cid:9)(cid:8)
0
We will now examine the morphology of the
Proof. a(z,t) is differentiable in z and t since P(z,t) is
aneurysm curvature by establishing conditions
and from equation (34), we see that the equilibrium
determining the extent to which the deformed radius
a(z,t) can stretch. As seen in (34), a(z,t) depends on (cid:1)P
points occurs when =0 and
P(z,t) and so the extrema of a(z,t) occurs at the (cid:1)z

same location within the aneurysm region as those of (cid:2)P =(cid:1) v0 P'(t);v (cid:3)2. Upon substituting the
P(z,t). That is, the morphology of the aneurysmal wall (cid:2)t 2(cid:1)v 2 0
0
curves the same way in all directions; downward if equilibrium points into the pressure equation in (30) we
400 Journal of Basic & Applied Sciences, 2014 Volume 10 Kwembe and Sanders
(cid:1)2P (cid:1)2P and (cid:1)(z,t,v ). Consequently, they blow up as v
see that >0 for a minimum extension and <0 0 0
(cid:1)z2 (cid:1)z2 approaches one. This implies that the aneurysm may
2 rupture at some point z ;0<z <1 and time t(cid:1)t for
(cid:2)2a (cid:2)2a (cid:4) (cid:2)2a(cid:7) 0 0 0
for a maximum extension and (cid:3) (cid:1) >0.
(cid:2)z2 (cid:2)t2 (cid:6)(cid:5)(cid:2)z(cid:2)t(cid:9)(cid:8) some fixed t0. However, for v0 (cid:1)1, the transversal and
longitudinal components of the membrane
2
Furthermore, (cid:2)(cid:2)2zP2 (cid:3)(cid:2)(cid:2)2tP2 +2(cid:1)v0v0 (cid:2)(cid:2)2zP2 P2"(t)<(cid:4)(cid:6)(cid:5)(cid:2)(cid:2)z2(cid:2)Pt(cid:7)(cid:9)(cid:8) , dCiosnpslaecqeumeenntlyt , trheem aanine urybsomun wdaeldl remfoar insfi xsetadb le (az0t ,at0n)y.
(cid:2)2a (cid:2)2a (cid:4) (cid:2)2a(cid:7)2 fixed point (z0,t0);0(cid:1)z0 (cid:1)1;t(cid:2)t0, when v0 (cid:1)1.
gives that (cid:3) (cid:1) <0 and the theorem
(cid:5) (cid:8)
(cid:2)z2 (cid:2)t2 (cid:6)(cid:2)z(cid:2)t(cid:9)
2.1. The Impact of Filtration and Fluid Flux
holds.
In this section, we examine the impact of filtration
THEOREM 2.2. Let a(z,t,v ) be given in (34) and
and the total fluid flux on the aneurysm wall. In
0
P(z,t)= P(z,t)+(cid:1)P(z,t). Then, there exists a dimensionless parameters, the filtration velocity v is
0 1 n,f
functional M(z,t,v ,P(t),P'(t),P(t),P'(t)) such that for given as
0 1 1 2 2
z and t fixed, we have
0 0 v =(cid:2)(P(z,t)(cid:1)P) (35)
n,f e
(1(cid:1)v )2a(z ,t ,v ) (cid:2)1 M(z ,t ,v ,P(t ),P'(t ),P(t ),P'(t )) which is completely defined by the perturbation solution
0 0 0 0 4 0 0 0 1 0 1 0 2 0 2 0 for P(z,t). The total flux crossing the undeformed

and
radius, denoted by Q , is given as
n,f
lim (1(cid:1)v )2a(z ,t ,v ) (cid:2)N
v0(cid:3)1 0 0 0 0 Q =(cid:3)(cid:3) v d(cid:1) (36)
n,f (cid:2) n,f
where N > 0 is a constant. Furthermore, there exists
2
(cid:6)(cid:5)a(cid:9)
N(cid:1) = N(cid:1)(z,t,v ,P(t),P'(t),P(t),P'(t)) that is finite and where d(cid:3)=a(z,t) 1+ d(cid:2)dz;0(cid:4)(cid:2)(cid:4)2(cid:1);0(cid:4)z(cid:4)L,
0 1 1 2 2 (cid:8)(cid:7)(cid:5)z(cid:11)(cid:10)
positive at each fixed point (z ,t ) given that 0 < z < 1
0 0 and (cid:1) is the whole surface of the aneurysm region. In
and t > 0 such that
dimensionless parameters, Q is given as
n,f
lim(1(cid:1)v )2a(z ,t ,v ) (cid:2)N(cid:1)(z ,t ,v ,P(t ),P'(t ),P(t ),P'(t )).
v0(cid:3)1 0 0 0 0 0 0 0 1 0 1 0 2 0 2 0 Q =(cid:2)(cid:3)1a(z,t)[P(z,t)(cid:1)P]dz.
n,f 0 e
Similarly, there exists
N = N (z,t,v ,P(t),P'(t),P(t),P'(t)) and The profile of the impact of the filtration flux is given
1 1 0 1 1 2 2
in subsection 3.3, when O((cid:1)2) terms of the series
N = N (z,t,v ,P(t),P'(t),P(t),P'(t)) which are finite solution are neglected. On the other hand, the total
2 2 0 1 1 2 2
fluid flux, Q(z,t) in any cross-section of the aneurysm
and positive at each fixed point (z ,t ) such that
0 0 given in dimensionless parameters is
lim (1(cid:1)v )2(cid:2)(z ,t ,v ) (cid:3)N and a(z,t)
v0(cid:4)1 0 0 0 0 1 Q(z,t)=2(cid:1)(cid:2)0 ru(z,r,t)dr (37)
lim (1(cid:1)v )2(cid:2)(z ,t ,v ) (cid:3)N for all points
v0(cid:4)1 0 0 0 0 2 which is completely defined by (27), and (34). The
(z,t)(cid:3)(0,1)(cid:2)0,(cid:1)). profile is given in subsection 3.3.
Proof. The proof is a direct consequence of (28), The filtration velocity as with the pressure, deformed
(29), (34) and the perturbation expression for the intra- radius and components of the membrane displacement
aneurysmal pressure P(z,t). depends on the Poisson ratio v . Numerical analysis of
0
the filtration rate shows that the flux blows up as v
We note that Theorem 2.2 says that v =1 is a 0
0 approaches one. A similar conclusion is reached for the
singularity of the deformed radius a(z,t,v ) and the case of total fluid flux Q(z,t).
0
components of the membrane displacement (cid:1)(z,t,v )
0
Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture Journal of Basic & Applied Sciences, 2014 Volume 10 401
So for any fixed point (z ,t );0<z <1;t (cid:1)0, if the figure in the numerical solutions. The surviving
0 0 0 0 parameters are size of the aneurysm region, L,
Poisson ratio approaches one, both Q and Q(z,t)
n,f permeability and filtration coefficient (cid:1), undeformed
blows up, indicating the potential for rupture. radius a , membrane thickness, h, the Poisson ratio
0
v , and the time scale. All of which can be measured
3. NUMERICAL METHODS AND PARAMETERS 0
by noninvasive means. Intra-aneurysmal pressure
Human aneurysms are formed in different sizes and P(z,t), fluid flow patterns, membrane displacement

shapes, and they exhibit a variety of material behavior and the dynamics of the deformed radius are
[6, 8, 10, 13]. As indicated earlier much of the work expressed in terms of the surviving parameters. As can
geared towards understanding the development and be seen, the numerical results given here indicate that
evolution of an aneurysm has focused mostly on the the Poisson ratio, intra-aneurysmal pressure gradients,
fluid content dynamics, shape and size. The current
flow profiles and membrane displacement could
interests are to determine more reliable parameters to
provide additional information regarding the rupture
predict the potential for aneurysmal rupture. So, we will
potential of aneurysms.
use the data collected in the literature [6, 8-10, 13, 31,
34] to fit the mathematical model developed in section Towards that end, we use the MATLAB® platform
2. We have considered an aneurysm within a version 7.10, 207B, 208B to numerically and
cylindrical portion of a blood vessel of length L=2cm, graphically understand the roles the bloodstream
a membrane of thickness h=0.25cm, and an impact forces, intra-aneurysmal pressure elevations
and Poisson ratio play in the rupture potential of
undeformed radius of a =0.5cm. We also considered
0 aneurysms contained within cylindrical portions of
a permeability coefficient of k* =2.5(cid:2)10(cid:1)7 and the blood vessels. The method begins with the
k*L2 dimensionless solutions of the fluid velocity
Darcy-Starling coefficient (cid:1)= . For the end components v(z,r,t), u(z,r,t); the membrane
ha3
0
displacement components (cid:1)(z,t), (cid:1)(z,t); the deformed
pressure P(t) and P(t), we shall considered the
1 2 radius a(z,t); the intra-aneurysmal pressure, P(z,t);
Ferguson [3, 6, 31] pressure formula described by the
Fourier series the filtration velocity, v ; the rate of filtration Q , and
nf nf
the cross-sectional fluid flux Q(z,t). We have

P(t)= P +(cid:2)N [A cos(n(cid:1)t)+B sin(n(cid:1)t)];i=1,2. developed MATLAB® codes of these solutions to
i m n=1 n n
produce numerical data and graphical representation at
Milnor [3, 6, 31, 32] suggested that N =10 is the circular frequencies (cid:2)=2(cid:1)f where

sufficient to describe systemic blood pressure in
f =0.2,0.5,1,1.2,5,8,10 to simulate in vitro tests. It is
general with N <10 in the diastole vasculature. Here,
known that most actual laboratory tests are performed
we shall take P(t) to represent the diastolic pressures
1 at values of f at 0.1 or 0.2 [6, 31]. The impact of the

and P(t) the systolic pressures. All pressure Poisson ratio on the comparative relationship of the
2
measurements considered here are in mmHg. We deformed radius, a(z,t), with intra-aneurysmal

shall take Pm =65.7mmHg, see [6], the coefficients of pressure, P(z,t), and with the local elevations of the
the Fourier series given in row vector form are: intra-aneurysmal pressure is expressed for the Poisson
ratio ranges of 0<v <1,1<v (cid:1)2, and v >2.
0 0 0
A =[(cid:1)7.13 (cid:1)3.08 (cid:1)0.130 (cid:1)0.346 0.0662 …]
n
3.1. Results and Analysis
B =[4.64 (cid:1)1.18 (cid:1)0.564 (cid:1)0.346 (cid:1)0.120 …]
n Computations revealed discontinuity in the profiles
of the impact fluid velocity and its components, the
and we have taken P =70mmHg where we have used
e membrane displacement and its components, the intra-
the Shah and Humphrey data in [6]. aneurysmal pressure, the deformed radius, the rate of
filtration and the flux on any cross-section of the
Since the heart beats about 72 times per minute, we membrane within the aneurysm region when the
2(cid:1) 60 Poisson ratio approaches 1. We shall interpret the
shall take (cid:2) = 72, or (cid:1)=8rad/s for a regular heart points of discontinuity in the solution profiles as
beat. As deduced in section 2, the mathematical model indicators for aneurysm rupture. The dynamic profiles
developed is a quasi-static form of a slow flow. of the bloodstream impacting forces, intra-aneurysmal
Consequently in dimensionless parameters, the fluid pressure elevations and the deformed radius exhibit
characteristics such as density and viscosity do not similar characteristics for f =0.1,0.2,0.5,1,1.2,and5.
402 Journal of Basic & Applied Sciences, 2014 Volume 10 Kwembe and Sanders
However, we shall present the results for the cases discontinuity and then drops gradually after the
f =1.2 corresponding to the regular heart beat of 72 discontinuity and then oscillates and die out. In Figures

10 and 11, the intra-aneurysmal pressure, P(z,t)
beats per minute, and f =0.2, the experimental value

shows local pressure elevations before the
at irregular heart beat of 12 beats per minute. In each
discontinuity. Following the point of discontinuity, the
case, we considered two scenarios, the case where a
pressure profile drops and then elevates and then
fixed point (z ,t ) is considered within the aneurysm
0 0 oscillates and die out. The transversal component of
and one where all points (z,t) are considered within the membrane displacement shows the same

the aneurysm region. In both cases, we see that for characteristics with the deformed radius, a(z,t). But
f =0.2 and f =1.2, time changes of the simulations of this is not surprising, since a(z,t)=1+(cid:1)(cid:2)(z,t). The

the intra-aneurysmal pressure, P(z,t) are in synchrony longitudinal component of the membrane displacement

with the membrane displacement components, its elevates slightly followed by a drop leading into the
magnitude and the deformed radius (See Figures 2, 3, point of discontinuity, then delays afterwards slightly
12, 13 and 13). For the figures presented here, we and elevates, then oscillates to vanish. The profiles of
maintained P(t) and P(t) at the same oscillating the impact fluid velocity in Figure 10 and 11 shows
1 2 elevated values leading into the point of discontinuity
values. Even when one end is maintained at higher
and oscillates to zero afterward. Similarly, the profile of
pulsating pressure values there were no significant
the magnitude of the membrane displacement elevates
differences, in regards to rupture potential indicators, in
rapidly leading to the point of discontinuity and then
the characteristics of the profiles of the intra-
oscillates to zero afterward. There are more oscillations
aneurysmal pressure, membrane displacement,
following the point of discontinuity for the case of a
deformed radius, filtration velocity, filtration rate and
regular heart beat than the case of irregular heart beat.
fluid flux.
3.2. Impact of the Poisson Ratio on the
In Figures 2 and 3, the profile of the deformed
Relationship of a(z,t) with P(z,t)
radius a(z,t) is flat before and after the discontinuity.

However, in Figure 10 which is associated with the
The impact of the Poisson ratio on the relationship
case of a regular heart beat, it shows local elevations
between the deformed radius and intra-aneurysmal
before the discontinuity at the time v0 =1 and then it pressure can be seen from the linear relationship
oscillates with time afterwards to vanish. In Figure 11 between a(z,t) and P(z,t) given in equation (34) for

corresponding to the case of irregular heart beat, the
values of v (cid:1)(0,1).
profile of the deformed radius increases before the 0

Figure 2: Membrane displacement, deformed radius, pressure vs time, f = 1.2 , z and v varies.
0