Journal of Theoretical Biology 227 (2004) 437–443
A phenomenological model for atherosclerotic plaque growthand rupture
T.I. Zohdi
a,
*, G.A. Holzapfel
b
, S.A. Berger
a
a
Department of Mechanical Engineering, University of California, 6195 Etcheverry Hall, Berkeley, CA 947201740, USA
b
Institute for Structural AnalysisComputational Biomechanics, Graz University of Technology, 8010 Graz, Schiesstattgasse 14B, Austria
Received 18 September 2003; received in revised form 19 November 2003; accepted 20 November 2003
Abstract
The objective of this communication is to develop a computerbased framework for the overall coupled phenomena leading togrowth and rupture of atherosclerotic plaques. The modeling is purposely simpliﬁed to expose the dominant phenomenologicalcontrolling mechanisms, and their coupled interaction. The main ingredients of the present simpliﬁed modeling approach, describingthe events that occur due to the presence and oxidation of excess lowdensity lipoprotein (LDL) in the intima, are: (i) adhesion of monocytes to the endothelial surface, which is controlled by the intensity of the blood ﬂow and the adhesion molecules stimulated bythe excess LDL, (ii) penetration of the monocytes into the intima and subsequent inﬂammation of the tissue, and (iii) rupture of theplaque accompanied with some degree of thrombus formation or even subsequent occlusive thrombosis. The set of resulting coupledequations, each modeling entirely different physical events, is solved using an iterative staggering scheme, which allows the equationsto be solved in a computationally convenient decoupled fashion. Theoretical convergence properties of the scheme are given as afunction of physical parameters involved. A numerical example is given to illustrate the modeling approach and an a prioriprediction for time to rupture as a function of arterial geometry, diameter of the monocyte, adhesion stress, bulk modulus of theruptured wall material, blood viscosity, ﬂow rate and mass density of the monocytes.
r
2003 Elsevier Ltd. All rights reserved.
Keywords:
Atherosclerotic plaque; Vulnerable plaque; Plaque growth; Plaque rupture
1. Introduction
Atherosclerosis is a vascular disease associated withthe accumulation of lipids leading to invasion of leukocytes (monocytes) and smooth muscle cells intothe intima, a process which may proceed to theformation of atheroma. Biomechanical and biochemicalmechanisms are involved in the development of thelesions characteristic of atherosclerotic plaque. Myocardial infarction and stroke can result from plaque ruptureand subsequent release of highly thrombogenic materialand lipids into the blood stream.Lesions with high risk of rupture are termed
vulnerable
, (see, for example, Fuster, 2002). These lesions,culprits in the sudden lifethreatening cardiovascularevents, are associated with, among other features,micromorphological characteristics, such as plaque capthickness (Richardson et al., 1989; Loree et al., 1992),
and lipid core size (Davies et al., 1993). The study
suggested that silent subclinical plaque rupture occursfrequently in patients with atherosclerosis. The size andbulk of lesions is likely to increase as a result of repeatedsequences of rupture and repair. This may explain whymyocardial infarction often occurs in asymptomaticpatients. No adequate diagnostic strategy for the identiﬁcation of vulnerable plaques is available yet. Investigators are striving to determine why one plaque is vulnerableand lifethreatening while another one is resistant andinnocuous. Elucidation of the mechanisms and factorsinvolved in plaque initiation, growth, and developmentcould lead to strategies to stabilize atherosclerotic plaquesand to prevent plaque rupture through lesionspeciﬁcinterventions (see, for example, Libby and Aikawa, 2002).Certain aspects of the overall growth process are wellunderstood. Essentially, excess lowdensity lipoprotein
ARTICLE IN PRESS
*Corresponding author.
Email addresses:
zohdi@newton.berkeley.edu (T.I. Zohdi),gh@biomech.tugraz.ac.at (G.A. Holzapfel), saberger@me.berkeley.edu (S.A. Berger).00225193/$see front matter
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2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2003.11.025
(LDL) particles accumulate in the intima inducing aseries of biochemical events, which cause endothelialcells to produce cell adhesion molecules, which latchonto blood monocytes. In the intima, the monocytesmature into active macrophages, which then ingestmodiﬁed lipoprotein particles, ending up as fatladenfoam cells. Smooth muscle cells in the intima divide, andother smooth muscle cells migrate into the intima fromthe media attracted by cytokines. Smooth muscle cellsthen elaborate extracellular matrix, promoting extracellular matrix accumulation in the
growing
atherosclerotic plaque. In later stages, a (thin) ﬁbrous cap,composed primarily of extracellular matrix proteinssuch as elastin and collagen (Virmani et al., 2000), may
form over a large lipid pool. This is the typicalmorphological structure of vulnerable plaques associated with rupture. Later, for example, inﬂammatorysubstances secreted by the tissue can cause the thinﬁbrous cap to become highly stressed, and possiblyrupture, releasing material which can lead to potentiallylethal blood clots. Excellent overviews of the currentthinking in the medical community pertaining to thegrowth and rupture of atherosclerotic plaques areprovided in Shah (1997), van der Wal and Becker
(1999), Chyu and Shah (2001) and Libby (2001a, b),
among others.Corresponding laboratory tests to investigate thebiological processes mentioned above are extremelydifﬁcult, even though the individual events are relativelystraightforward to describe from the mechanical pointof view. Physical and numerical modeling of individualportions of this process have been undertaken bynumerous researchers, one example is the modeling of the highly nonlinear deformation mechanisms and stressdistributions in healthy and diseased arteries underdifferent loading conditions, see, for example, thestudies (Holzapfel et al., 2000, 2002a, b) by Holzapfel
and coworkers, or the more general overview given byHumphrey (2002) and Holzapfel and Ogden (2003),
among others. However, while there are numerousresearches detailing speciﬁc clinical events involved inthe growth and rupture of atherosclerotic plaques, to theknowledge of the authors, there appears to be anabsence of works which focuses attention on developinga comprehensive mathematical model, which bothaddresses all of the events simultaneously and providesrobust solutions.It was the objective of this research to develop aconstitutive and computational framework, which assembles very simple models governing essentiallydifferent physical events, in order to describe thecomplex coupled events of atherosclerotic plaquegrowth and rupture. Since modeling of these coupledevents is rather complex, and there being no frameworkavailable which could be used as a basis, the presentwork aims to provide this larger perspective on theproblem. In Section 2 we provide the physical setting,while in Section 3 we present computational aspectsrequired to solve the coupled system of nonlinearequations. The concluding remarks mainly focus attention on the limitations of the presented approach, andattempt to motivate several possible improvements.
2. Phenomenological mathematical idealizations
We start by developing a phenomenological model todescribe how the crosssection of the artery,
A
¼
p
r
2
in
(Fig. 1), changes in response to the presence of an excessof lipids in the intima and the subsequent growth of thetissue due to inﬂammation associated with the adhesionof monocytes onto the intima wall and penetrationtherein. The inner radius of the artery is denoted by
r
in
:
We assume that the blood contains lipids and monocytes in suspension that are convected along by theblood via drag forces. We consider an incompressible,steady, 1D ﬂow proﬁle, with a constant ﬂow rate
Q
0
¼
v
0
A
0
¼
R
A
v
d
A
¼
Q
;
through an artery, givenby
ð
q
>
0
Þ
v
¼
v
max
1
rr
in
q
:
ð
1
Þ
Because the ﬂow rate is constant
ð
Q
0
¼
Q
Þ
this implies
v
max
¼
Q
0
A
ð
1
½
2
=
q
þ
2
Þ
:
ð
2
Þ
The velocity of a monocyte particle of diameter
d
(seeFig. 1), which is convected with the ﬂuid near the
ARTICLE IN PRESS
FLOWBLOODARTERIALLDL
r
in out
r
INTIMA
INTIMAFaFddmvtrestresLDL
MONOCYTE
Fig. 1. Schematic for the model problem.
T.I. Zohdi et al. / Journal of Theoretical Biology 227 (2004) 437–443
438
endothelial surface, is approximated by the velocity
v
c
of its center
v
c
¼
def
v
max
1
r
in
ð
d
=
2
Þ
r
in
q
:
ð
3
Þ
To determine the critical velocity of the monocyte,
v
say, below which a particle will become arrested at thewall surface, we write an impulsemomentum balance(see Fig. 1)
mv
þ
Z
t
res
0
X
F
d
t
E
mv
þ
F
d
t
res
F
a
t
res
¼
0
;
ð
4
Þ
where
m
is the mass of the monocyte, and
F
d
is the dragforce imposed on the particle by the ﬂuid, which, for lowReynolds numbers is
F
d
E
3
pm
v
d
(law of Stokes), with
m
being the absolute viscosity of the ﬂuid. We assumethat the adhesive force
F
a
is proportional to the crosssectional area of the monocyte, and proportional to theadhesion stress
t
a
;
i.e. adhesive force per unit area,produced on the intima surface upon contact. We write
F
a
E
g
1
t
a
p
d
2
=
4
;
where
g
1
is a constant of proportionality, which scales the contact area to the crosssectionalarea of the particle, and
t
res
is the contact time,commonly referred to in the literature as the ‘residencetime’ (see, for example, Libby, 2001a, b). In addition, we
assume that
v
E
g
2
d
=
t
res
;
where
g
2
is another constant of proportionality, which scales the contact time to thecritical velocity of the monocyte. The mass of themonocyte is
m
¼
r
m
p
d
3
=
6
;
where
r
m
is the referencemass density of the monocyte particle. Substitution intothe impulsemomentum balance yields an explicitequation for the critical velocity of the monocyte, i.e.
ð
v
Þ
2
þ
18
g
2
m
v
d
r
m
3
g
1
g
2
t
a
2
r
m
¼
0
ð
5
Þ
which yields
v
¼
9
g
2
m
d
r
m
7
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
81
g
22
m
2
d
2
r
2
l
þ
3
g
1
g
2
t
a
2
r
m
s
;
ð
6
Þ
where the positive root is the physically correct one. It isassumed that the concentration of the monocytesarrested at the wall, denoted as
c
w
;
is proportional tothe difference between the velocity of the monocytes
v
c
and the critical velocity
v
:
Thus,
c
w
p
max 0
;
1
v
c
v
¼
def
Z
:
ð
7
Þ
As the velocity increases, particles are less likely to adhere,until a critical velocity is met, where no particles adhere.Essentially, one can interpret
Z
as a distribution function,which indicates the likelihood of a particle adhering to theintima surface as a function of the nearwall velocity.The assumption is now made, that the growingthickness of the intima, denoted as
a
;
is related to theconcentration of monocytes by
’
a
¼
F
ð
Z
;
y
;
material parameters
Þ
:
ð
8
Þ
This phenomenological model accounts for the intimalthickening due to the presence of the monocytes andsubsequent reactions due to monocytes and macrophages, etc. Speciﬁcally, this takes the form of asomewhat standard growth law, commonly used inanalyses of scale growth in pipe ﬂow (see, for example,Fontana, 1986; Shackelford, 2000)
’
a
¼
K
Z
;
ð
9
Þ
where
K
is a growth rate constant. If
Z
were timeinvariant, then the growth would be ‘linear’ of the form
a
ð
t
Þ¼
K
Z
t
þ
a
ð
0
Þ
:
ð
10
Þ
However, this observation can only be taken qualitatively, because
Z
is a function of
a
and
t
;
due to the factthat
v
is a function of
r
in
ð
t
Þ¼
r
out
a
ð
t
Þ
(see Fig. 1). InSection 3, we provide an algorithm to solve this coupledsystem.In order to describe the rupture of the thin ﬁbrouscap, which is coupled with the growth, we use a simpleconstitutive approach, and split the stored energy
C
intopurely isochoric
ð
%
C
Þ
and volumetric
ð
U
Þ
parts, i.e.
C
¼
%
C
þ
k
2
ð
J
1
Þ
2
ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ}
def
¼
U
;
ð
11
Þ
where
k
is the bulk modulus of the material, and
J
¼
det
F
is the Jacobian determinant of the deformationgradient
F
¼r
X
x
;
whereby
u
¼
x
X
is the displacement vector, and
X
and
x
indicate referential positionand spatial position of a point relative to a ﬁxed srcin.For the associated kinematics, see, for example,Holzapfel (2000).With
C
¼
F
T
F
denoting the right Cauchy–Greentensor, it is straightforward to derive the stress relation.In particular, the Cauchy stress tensor
r
is obtainedfrom
C
through the relation
r
¼
J
1
F
ð
2
@
C
=@
C
Þ
F
T
;
(Holzapfel, 2000), and by considering the particularconstitutive assumption (11) we get
r
¼
r
þ
p
1
;
where
r
¼
J
1
F
ð
2
@
%
C
=@
C
Þ
F
T
denotes the purely isochoriccontribution, and
p
1
the purely volumetric contributionto the stresses, with
p
¼
@
U
@
J
¼
k
ð
J
1
Þ
:
ð
12
Þ
Note that
p
¼
tr
r
=
3 can be identiﬁed as the hydrostaticpressure. Further, we propose that rupture in the thinﬁbrous cap occurs when the pressure
p
in the capexceeds some critical value
p
at a continuum particle,
p
X
p
say.
1
We now consider a problem with the simplekinematics, as shown in Fig. 2. Thereby, a continuumparticle on the endothelial surface, with referential
ARTICLE IN PRESS
1
The pressure criterion is a logical choice, since it is believed thatrupture occurs when the thin ﬁbrous plaque cap, which is essentially athin membrane, bursts, in a similar manner as an overpumped balloon.The material in the intima is then released, leading to a stroke.
T.I. Zohdi et al. / Journal of Theoretical Biology 227 (2004) 437–443
439
position
X
(with referential coordinates
ð
X
1
;
X
2
;
X
3
Þ
), isdisplaced to the current position
x
(with spatialcoordinates
ð
x
1
;
x
2
;
x
3
Þ
) so that
u
1
¼
u
2
¼
0 and
u
3
¼
D
a
ð
t
Þ
X
3
;
with
D
a
ð
t
Þ ¼
a
ð
t
Þ
a
ð
t
¼
0
Þ
;
are the Cartesiancomponents of the displacement vector
u
:
Hence, thematrix representation
½
F
of the deformation gradient
F
and the volume ratio
J
reduces to
½
F
¼
1 0 00 1 00 0
D
a
ð
t
Þ þ
1
264375
;
J
¼
det
½
F
¼
D
a
ð
t
Þ þ
1
;
ð
13
Þ
which leads, with Eq. (12), to
p
¼
k
D
a
ð
t
Þ
:
ð
14
Þ
Remark.
Although the deformation used is relativelysimple, it captures the essence of the swelling of theintima. More complicated deformations would be anoverkill since they would be still coupled to the simpleﬂuid model. Furthermore, since the pressure is the onlyquantity sought after, this deformation is adequate.However, if a more sophisticated rupture criteria wouldbe used, for example involving shear stresses, then thedeformation analysis would have to be more detailed.
3. A staggered solution scheme for growth and rupture
To solve the coupled system of equations in theprevious section, an implicit ﬁxedpoint recursion isused to update the nonlinear velocitygrowth couplingwithin each time step. Given the critical velocity
v
through (6), the procedure at a given time
t
and with thetime increment
D
t
is as follows:
ð
I
Þ
Solve for fluid velocity
ð
fix
a
i
ð
t
Þ ¼
r
out
r
i in
ð
t
ÞÞ
:
v
c
;
i
þ
1
ð
t
Þ ¼
v
i
max
1
r
i in
ð
t
Þ ð
d
=
2
Þ
r
i in
ð
t
Þ
q
)
update :
Z
i
þ
1
ð
t
Þ ¼
max 0
;
1
v
c
;
i
þ
1
ð
t
Þ
v
;
ð
II
Þ
Solve for growth
ð
fix
v
c
;
i
þ
1
ð
t
ÞÞ
:
’
a
i
þ
1
ð
t
Þ ¼
K
Z
i
þ
1
)
a
i
þ
1
ð
t
Þ ¼
K
Z
i
þ
1
D
t
þ
a
ð
t
D
t
Þ)
update :
r
i
þ
1
in
ð
t
Þ ¼
r
out
a
i
þ
1
ð
t
Þ
;
ð
III
Þ
Repeat until :
jj
r
i
þ
1
in
ð
t
Þ
r
i in
ð
t
Þjj
p
TOL
:
ð
15
Þ
The index of iteration at time
t
is denoted by
i
¼
1
;
y
;
N
:
The ﬁrst step starts at ﬁxed thickness
a
of the intima. The ﬂuid mechanical problem is solved for thevelocity
v
c
of the center of the monocyte, and thereafterthe particle/wall adhesion check is performed—thedistribution function
Z
is computed. In a second step,at ﬁxed velocity
v
c
;
the evolution equation for intimalgrowth is solved for
a
;
and thereafter
r
in
is computed.This twostep algorithm is continued as long as
jj
r
i
þ
1
in
ð
t
Þ
r
i in
ð
t
Þjj
p
TOL
;
where
TOL
is a small tolerance value.After the process has converged at
t
;
the failure criterion
p
ð
t
Þ
X
p
is checked. A schematic ﬂowchart is presented inFig. 3. Increased accuracy is acquired for smaller timeincrements
D
t
in the growth law.
3.1. General convergence criteria
The convergence to a ﬁxedpoint solution, as described in scheme (15), is also controlled by the timeincrement
D
t
:
Thus, during the simulations, if the timestep solution does not converge within a certain numberof iterations, the time step is reduced.
2
In abstract terms,consider
D
ð
r
in
ð
t
ÞÞ ¼
F
;
where
r
in
ð
t
Þ
is to be solved for attime
t
:
It is convenient to write an operator split
D
ð
r
in
ð
t
ÞÞ
F
¼
G
ð
r
in
ð
t
ÞÞ
r
in
ð
t
Þ þ
z
¼
0
;
ð
16
Þ
where
z
is the remainder term in the operator split. Astraightforward, well established, type of iterativescheme is
r
i in
ð
t
Þ ¼
G
ð
r
i
1
in
ð
t
ÞÞ þ
z
;
ð
17
Þ
where
i
¼
1
;
y
;
N
is the index of iteration at time
t
:
Theconvergence of such a scheme is dependent on the
ARTICLE IN PRESS
X
3
X
1
X
2
∆
a(t)
=
a(t)
−
a(t=0)DEFORMEDUNDEFORMED
(x)(X)
Fig. 2. Continuum particle on the endothelial surface with referentialposition
X
displaced to the current position
x
;
with
u
1
¼
u
2
¼
0 and
u
3
¼
D
a
ð
t
Þ
X
3
:
ADHESION CHECKPARTICLE/WALLFLUID MECHANICS PARTICLE FLOWINTIMA GROWTH IF NOT CONVERGEDIF CONVERGEDWALL RUPTURECHECK
INCREMENTTIME
Fig. 3. Schematic ﬂowchart.
2
In addition to convergence issues, for accurate numerical solutionssuch approaches require small time steps, primarily because thestaggering error accumulates with each successive increment.
T.I. Zohdi et al. / Journal of Theoretical Biology 227 (2004) 437–443
440
behavior of
G
:
Namely, a sufﬁcient condition forconvergence is that
G
is a contraction mapping for all
r
i in
ð
t
Þ
:
Accordingly, we deﬁne the error as e
i
ð
t
Þ ¼
r
i in
ð
t
Þ
r
in
ð
t
Þ
:
A necessary restriction for convergence is iterativeselfconsistency, i.e. the exact solution must be represented by the scheme
G
ð
r
in
ð
t
ÞÞ þ
z
¼
r
in
ð
t
Þ
:
Enforcing thisrestriction, a sufﬁcient condition for convergence is theexistence of a contraction mapping
j
e
i
ð
t
Þj ¼ j
r
i in
ð
t
Þ
r
in
ð
t
Þj¼ j
G
ð
r
i
1
in
ð
t
ÞÞ
G
ð
r
in
ð
t
ÞÞj
p
l
j
r
i
1
in
ð
t
Þ
r
in
ð
t
Þj
;
ð
18
Þ
where
l
denotes the ﬁxedpoint constant. If 0
p
l
o
1 foreach iteration
i
;
then e
i
ð
t
Þ

0 for any arbitrary startingvalue
r
i
¼
0
in
ð
t
Þ
as
i

N
:
The type of contraction conditiondiscussed is sufﬁcient, but not necessary, for convergence. For more details see, for example, Ostrowski(1966), Ortega and Rockoff (1966), Kitchen (1966),
Ames (1977) or Axelsson (1994).
3.2. Qualitative behavior
The ﬁxedpoint constant
l
may be determined
explicitly
in terms of the material parameters and ﬂowrates involved. This is achieved by collapsing all of theequations into a single one involving the primaryvariable
r
in
ð
t
Þ
(for convenience, we will omit subsequently the index of iteration at time
t
). By referring to(15), during the staggering process, we ﬁnd, by deﬁning
a
0
¼
a
ð
t
D
t
Þ
and
g
ð
r
in
ð
t
ÞÞ ¼
v
c
ð
t
Þ
=
v
;
that
a
ð
t
Þ ¼
K
Z
D
t
þ
a
0
:
This implies
r
in
ð
t
Þ ¼
r
out
K
Z
D
t
a
0
¼
G
ð
r
in
ð
t
ÞÞ þ
z
;
ð
19
Þ
in which the deﬁnitions
G
ð
r
in
ð
t
ÞÞ ¼
K
D
tg
ð
r
in
ð
t
ÞÞ
;
z
¼
r
out
a
0
K
D
t
ð
20
Þ
are to be used. In this case, with Eqs. (1), (2) and(6) weﬁnd that
G
ð
r
in
ð
t
ÞÞ
¼
K
D
t Q
0
=
ð
p
r
2
in
ð
t
Þð
1
ð
2
=
q
þ
2
ÞÞÞ½
1
ð
r
in
ð
t
Þ
d
=
2
=
r
in
ð
t
ÞÞ
q
ð
9
g
2
m
=
d
r
m
Þ þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
81
g
22
m
2
=
d
2
r
2
l
Þ þ ð
3
g
1
g
2
t
a
=
2
r
m
Þ
q
:
ð
21
Þ
Therefore, we have the following observations:
*
As either
r
m
;
d
;
Q
0
;
K
or
D
t
increase, then
G
ð
r
in
ð
t
ÞÞ
increases, thus impairing convergence,
*
As either
t
a
;
m
or
q
decrease, then
G
ð
r
in
ð
t
ÞÞ
increases,thus impairing convergence.Eq. (21) allows an explicit expression for the necessarytime step
D
t
to achieve convergence. We obtain
K
D
t Q
0
=
ð
p
r
2
in
ð
t
Þð
1
2
=
ð
q
þ
2
ÞÞÞ½
1
ð
r
in
ð
t
Þ ð
d
=
2
Þ
=
r
in
ð
t
ÞÞ
q
9
g
2
m
=
d
r
m
þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
81
g
22
m
2
=
d
2
r
2
l
Þ þ ð
3
g
1
g
2
t
a
=
2
r
m
Þ
q
o
TOL
o
1
ð
22
Þ
which leads to
D
t
o
TOL
ðð
9
g
2
m
=
d
r
m
Þ þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
81
g
22
m
2
=
d
2
r
2
l
Þ þ ð
3
g
1
g
2
t
a
=
2
r
m
Þ
q
Þ
KQ
0
=
ð
p
r
2
in
ð
t
Þð
1
2
=
ð
q
þ
2
ÞÞÞ½
1
ð
r
in
ð
t
Þ ð
d
=
2
Þ
=
r
in
ð
t
ÞÞ
q
:
ð
23
Þ
3.3. Approximate critical time to rupture
Here we approximate the critical time,
t
say, atwhich rupture occurs. This approximation is based onthe premise that rupture depends solely on some criticalvalue
p
of the hydrostatic pressure at a continuumparticle.For the rupture criterion to be met, we have
p
¼
p
;
where the particularized form for the hydrostaticpressure is given (14), i.e.
p
¼
k
D
a
ð
t
Þ
:
To ﬁnd the criticalgrowth we invert this expression, which gives
D
a
ð
t
Þ ¼
p
k
:
ð
24
Þ
By considering the case where
Z
is slowly varying, thus
Z
E
ð
v
v
c
Þ
=
v
;
this leads to an explicit expression for
t
:
From (10) we ﬁnd with
D
a
ð
t
Þ ¼
a
ð
t
Þ
a
ð
0
Þ
that
a
ð
t
Þ
E
K
Z
t
þ
a
ð
0
Þ )
t
¼
D
a
ð
t
Þ
K
Z
:
ð
25
Þ
Using (24) we then get the critical time
t
¼
p
k
K
Z
ð
26
Þ
at which rupture occurs.
3.4. Numerical example
In the following, a speciﬁc numerical example isbrieﬂy presented with the goal of showing the evolutionof certain quantities with time obtained from thesolution of the coupled velocitygrowth model, asdescribed in this section. In particular, the geometry of the artery and the material parameters used are shownin Table 1 (Berger, 1996). The only constant that was
not obtained from the literature was
K
:
This constantwas estimated from known approximate times (
E
15–20years) to rupture. Therefore, one can consider such asimulation as the result of an inverse problem where
K
was sought. The constants of proportionality,
g
1
and
g
2
;
were set to unity and a quadratic velocity proﬁle wasused (
q
¼
2).Fig. 4 shows the evolution of the inner radius
r
in
of theartery, the velocity
v
c
of the center of the monocyte, andthe pressure
p
with time
t
:
Under the conditionssimulated the atherosclerotic plaque would rupture in
t
¼
16 years. During this process, the inner radiusdecreases to approximately twothirds of its original
ARTICLE IN PRESS
T.I. Zohdi et al. / Journal of Theoretical Biology 227 (2004) 437–443
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