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70 Chapter 4
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where Ji: ¼
, J4, J5and to set up the constitutive law for a locally transversally anisotropic
J
2
vJ
, aF: ¼ Fa. In laboratory, one performs four independent tests to find J1,
vI
i
material.
Some biological tissues (e.g., artery wall with collagen fibers) are strengthened by two
families of fibers given by unit vectors a, a
0
in the undeformed reference configuration.
Elastic potential
JðFÞ¼JðI
; I2; I4; I5; I6; I7; I8Þ¼J
1
has three additional invariants I
¼ a0$ðCa0Þ, I7¼ a0$C2a
6
ðI1; I2ÞþJ
iso
ðI4; I5; I6; I7; I8Þ
aniso
0
, and I
¼ a$ðCa0Þða $a0Þ,
8
and application of Eq. (4.39) yields constitutive relation:
S ¼pI þ 2J
þ2J
4aF5aF
0
a
þ2J
5a
6
F
B þ 2J
1
2
B B
I
1
þ 2J5ðaF5BaFþ BaF5aFÞ
0
þ 2J
F
0
5Ba
a
7
F
The sum of any isotropic incompressible elastic potential J
0
F
2
þ Ba
0
F
5a
0
F
þ J
8
a
iso
F
5a
0
þ a
F
and J
aniso
0
5a
:
F
F
can be used in
Eq. (4.43) to define the Cauchy stress tensor. An example of the elastic potential (Eq.
4.41) for the artery wall was suggested in [152]:
J
iso
¼
m
ðI
2
3Þ; J
1
aniso
¼
m
2m
2
3
n
exphm
ðI4 1Þ
3
with matrix elasticity parameter m and fiber elasticity parameters m
J
is active only if fibers are stretched, I4> 1orI6> 1.
aniso
i
2
þexphm
i2o
ðI6 1Þ
3
and m3. Note that
2
2
(4.43)
;
Now we are ready to consider several popular choices of the isotropic elastic potentials,
which are motivated by different assumptions on the medium behavior.
4.3.2 Models of hyperelastic materials
The GreeneLagrange strain tensor is given by
Tensor E measures the deformation relative to the reference (undeformed) configuration. It
is defined completely by C and reduces to the linear strain tensor
small deformations. Using the definition of E, one can rewrite Eq. (4.36) as:
E ¼
1
ðC IÞ¼
2
S ¼
1
Vu þVu
2
1
F
J
T
þVuTVu:
1
ðVu þVuTÞ in case of
2
vJ
T
: (4.44)
F
vE
Equations of fluid dynamics and elasticity 71
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The Saint VenanteKirchhoff model of a compressible isotropic material is given by
J
SVK
¼
l
2
ð
tr E
2
Þ
þ mtr E2: (4.45)
The Saint VenanteKirchhoff material is the simplest compressible model with the Hooke’s
constitutive law applicable for finite (but not necessarily small) deformations:
SVK
1
¼
Fðlðtr EÞI þ2mEÞF
J
S
T
: (4.46)
The linear elasticity model follows from Eqs. (4.44)e(4.46) under the assumption of
infinitesimal deformation, whenkuk 1 andkVuk 1. In this case, the second-order
term in E can be neglected and the GreeneLagrange strain tensor becomes linear:
1
¼
L
Vu þVu
2
E
T
:
An isotropic linear material is defined by the elastic potential:
J
L;iso
¼
l
ðtr E
2
Þ2þ mtr E
L
2
; (4.47)
L
where the Lame´parameters l and m are defined from the experimentally observed
Poisson’s coefficient n and Young’s modulus E:
l ¼
ð1 þ nÞð1 2nÞ
En
; m ¼
E
2ð1 þ nÞ
:
Eqs. (4.44) and (4.47) lead to the constitutive law of linear elasticity:
S
¼
L;iso
Under the assumption of linear elasticity, the strain is infinitesimal, and one has
Jz1 þtr E
, FzI, and FTzI. After simplifications based on these approximations, Eq.
L
(4.48) becomes the standard Hooke’s law:
Another popular model for isotropic biological tissues is the neo-Hookean model of a
compressible isotropic material [122]:
which gives rise to the neo-Hookean constitutive relation:
1
Fðlðtr E
J
ÞI þ2mELÞFT¼
L
NH
¼
m
2
J
1
Flðdiv uÞI þmVu þVu
J
S
¼lðtr ELÞI þ 2mEL: (4.49)
L;iso
T
T
F
: (4.48)
3 þdJ212ðd þ1ÞðJ 1Þ; (4.50)
I
1
S
¼mB þ mðdJ d 1ÞI: (4.51)
NH
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The neo-Hookean model is used in surgical simulators for modeling of mechanical
behavior of kidney and liver [159]. However, the model cannot describe stiffening of
soft tissue at very large deformations. An often adopted rule of thumb is to use the
neo-Hookean model at range of deformation up to 100%.
Now, we consider several practical incompressible models. We already mentioned that, for
incompressible materials, the constitutive law contains the extra term pI.
If deformations are small, the incompressibility condition J ¼ 1 can be written in the form
of the constrain div u ¼ 0ortrE
¼ 0. Therefore, for the linear incompressible elastic
L
model, the elastic potential (Eq. 4.47) and the constitutive law (Eq. 4.48) reduce to
J
L;inc
¼mtr E
2
; S
L
¼ 2mEL pI; (4.52)
L;inc
whereas, for the neo-Hookean model (finite deformations), the elastic potential and
constitutive relation are
m
J
NH;inc
¼
ð
I
2
3Þ; S
1
¼ mB pI: (4.53)
NH;inc
Incompressible models applicable at large deformations are the Gent model [153] with
parameter J
and the Yeoh model with C
[1:
m
mJ
m
¼
J
G;inc
¼ m=2 and parameters C2; .; Cn[210],
1
J
Y;inc
ln1
2
n
X
¼
CiðI1 3Þi: (4.55)
i¼1
I1 3
J
m
; (4.54)
The Gent model can be used for soft tissues containing reinforcing fibers such as arterial
walls. The Yeoh model was shown to be adequate for a wide range of deformations [176].
The incompressible Gent and Yeoh constitutive laws are
n
S
G;inc
mJ
¼
Jm I1þ 3
m
B pI; S
Y;inc
¼ 2
X
iCiðI1 3Þ
i¼1
i1
B pI: (4.56)
These equations are readily reduced to Eq. (4.53) when J
4.4 Fluidestructure interaction
In cardiovascular simulations, one is often interested in numerical simulation of blood
dynamics and tissues deformation at once. Such a problem involves mutual interaction of
a fluid (blood) and an elastic structure (tissue): Fluid flow depends on the structure
/N and n ¼ 1.
m
Equations of fluid dynamics and elasticity 73
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displacement, whereas structure motion is influenced by the fluid dynamics. In
mathematical modeling, such setup is known as a fluidestructure interaction (FSI)
problem. Examples of FSI problems arising in computational hemodynamics include blood
flow in a complaint elastic vessel and flow passing heart valves or vascular stent.
In the previous sections, we separately considered equations governing the fluid dynamics
and the motion of elastic materials. For a complete description of an FSI problem, one
needs to prescribe coupling conditions on the fluidestructure interface and the evolution
of the fluid domain due to the motion of the structure. In the following, we discuss
interface conditions and equations constituting a standard FSI problem.
Trying to understand or simulate numerically an FSI problem, one should keep in mind
that equations of fluid dynamics are commonly given in the Eulerian coordinates, whereas,
for solids motion, one usually uses Lagrangian description.
4.4.1 Interface conditions
Assume that the fluid and solid motion takes place in a time-dependent domain UðtÞ3ℝ
3
that is partitioned into subdomain UfðtÞ occupied by the fluid and subdomain UsðtÞ
fs
occupied by the solid medium (structure). Let G
ðtÞ :¼ vUfðtÞXvUsðtÞ be the interface
where the interaction of the fluid and solid material happens. The initial configuration will
be further used to define the reference domains in the Lagrangian framework,
b
U
¼Ufð0Þ;bUs¼ Usð0Þ;bGfs¼ Gfsð0Þ:
f
The first condition, which is commonly assumed on the fluidestructure interface, is the
continuity of fluid and structure velocities, i.e., the fluid does not penetrate through the
structure, and fluid particles adhere to it (no slip). Mathematically, this condition can be
written as
v
f¼vs
on GfsðtÞ:
Since the solid motion is often defined in terms of displacements with respect to the
s
reference configuration, u
ðbx; t Þ, it is common to write the continuity of velocity interface
condition in the form:
f¼us
v
where xðbx; tÞ¼bx þ u is the Lagrangian mapping frombU
1
+x
t
on GfsðtÞ or vf+x ¼ u
s
onbGfs; (4.57)
t
to UsðtÞ (Fig. 4.2).
s
74 Chapter 4
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Figure 4.2
Reference (initial) and deformed (current) configurations.
The second condition results from the balance of contact forces on GfsðtÞ. Equating
surface forces acting on an elementary area of G
fs
ðtÞ from the solid and fluid sides leads to
the balance of the normal stresses:
f
n ¼ssn on GfsðtÞ;
s
f
where s
fs
G
wherebn is the normal vector tobG
, ssare fluid and surface Cauchy stress tensors, and n is the normal vector to
ðtÞ. Again switching from physical to reference coordinates, we can equivalently write
ð
s
+ xÞF
f
T
b
n ¼S
.
fs
F
s
T
b
n onbG
; (4.58)
fs
4.4.2 Fluid domain motion
In an FSI problem, fluid equations are solved in a time-dependent domain. There is,
however, an important difference to the situation discussed in Section 4.2. The evolution
f
of U
ðtÞ is not given a priori any more, and finding it is a part of the FSI problem.
Assuming for the sake of clarity that only G
time, we see that the evolution of fluid domain is completely determined by the motion of
the elastic structure and that the velocity ofbG
on the interface. To formulate this observation mathematically, we recall that the evolution
f
of U
ðtÞ can be described through the mapping from the reference domain to the physical
domain occupied by the fluid, x
f
:bUf/UfðtÞ. Thus, we have the condition:
part of the boundary vUfðtÞ depends on
fs
ðtÞ is equal to the velocity of solid particles
fs
This is, of course, equivalent to x
the interface. This mapping is Lagrangian and is defined by the displacement field u as
s
x
¼bx þ usin the solid partbUs. In the fluid part, however, the mapping does not have to
follow material trajectories, and one has an option to choose any sufficiently smooth
f
x
¼bx þ usonbGfs: (4.59)
f
¼ xsonbGfs, i.e., the mapping x is continuous over
Equations of fluid dynamics and elasticity 75
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mapping xf:bUf/UfðtÞ that satisfies Eq. (4.59) and xf¼bx on vbUfbGfs(other parts of
the fluid domain boundary are static). In fact, one needs to define a smooth extension of
the displacement field from the solid to the flow reference domains:
f
u
: ¼Extðu
s
Þ
; (4.60)
inbU
f
which is continuous over the interface (a consequence of Eq. (4.59)) and such that
f
¼0onvbUfbGfs: (4.61)
u
f
Furthermore, one may set x
mapping x
f
; one can opt for the Eulerian formulation of the fluid equations in the physical
ðbx; tÞ¼bx þuf. Now, when UfðtÞ is defined through the
domain or, as alternative, one can use quasi-Lagrangian or arbitrary LagrangianeEulerian
formulation of the fluid problem.
The extension in Eq. (4.60) can be defined in different ways. For numerical stability, one
is interested in an extension with possibly small gradients and such that its Jacobian inbU
is strictly positive and well separated from zero. Finding a suitable extension in the case of
not small deformations may be a challenging task. One popular extension method consists
in finding u
f
as a solution to an auxiliary elliptic PDE inbUfwith boundary conditions:
f
u
¼usonbG
fs
(4.62)
and Eq. (4.61). One standard choice of PDE is the linear elasticity equation (see, e.g.,
[188]):
divl
ðdiv uÞI þm
m
Vu þVu
m
with space-dependent auxiliary Lame parameters l
T
¼ 0inbU
, mm. However, other options including
m
; (4.63)
f
nonlinear elasticity and biharmonic equations are available (see, for example, [188,194,207]).
f
4.4.3 Complete system of equations and energy balance
Once we defined a (auxiliary) displacement field ufin the fluid domain, it is convenient to
write out the FSI system of PDEs using the continuous displacement and velocity fields
defined in the reference domain:
In this section, we shall omit hats in notations of Lagrangian velocities inbU. Velocity is
continuous due to the interface condition (4.57) and u is continuous by construction of the
u ¼
(
s
u
inbUs;
f
inbUf;
u
v ¼
s
v
inbUs;
f
inbUf:
v
(
76 Chapter 4
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extension operator. In the reference solid domain (but not in fluid domain), one, of course,
has the relation:
¼v inbUs: (4.64)
u
t
Hence, v is the Lagrangian velocity for the solid and quasi-Lagrangian for the fluid. The
deformation gradient is F ¼ I þ Vu, and its determinant J :¼ detðFÞ is also defined
globally inbU.
Denote by r
and rfthe densities of solid and fluid. The momentum Eqs. (4.29) and (4.33)
s
for the fluid and the solid in the reference subdomains can be combined:
vv
vt
¼
8
>
>
<
>
>
Þ1divJsf+x
ðJr
:
f
1
divJSsF
r
s
f
F
T
T
þbf inbU
ðVvÞF
1
;
s
v
vu
vt
þbf inbU
(4.65)
:
f
If the fluid is incompressible, then we also have the mass conservation equation:
divJF
and an additional variable, the fluid pressure p
1
v¼0inbUf; (4.66)
f
. For the complete FSI system, one
complements Eqs. (4.65) and (4.66) with Eq. (4.64), extension rule (Eq. 4.60), interface
conditions (4.57), (4.58), and (4.62), as well as suitable initial and boundary conditions.
It is instructive to find the total energy balance of the FSI system. For the brevity, we
consider the homogeneous boundary conditions in Eqs. (4.23) and (4.24) for fluid and the
free-stress boundary condition for the solid oncvU
bGfs. Using integration by parts, one
s
checks the identity:
Z
ððw$VuÞvÞþ
b
U
f
Z
1
¼
2
b
U
f
1
ððdiv wÞuvÞdx
2
ððw$VuÞvÞððw$VvÞuÞdx þ
Z
1
2
vbU
f
(4.67)
ðn$wÞuv ds:
Multiplying the first equality in Eq. (4.65) by r
the reference domain, and employing Eq. (4.67) give
v, the second one by Jrfv, integrating over
s
Equations of fluid dynamics and elasticity 77
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j
2
jvj
dx
1
C
A
f
2
dx þ
F
r
T
f
2
r
f
2
Z
vJ
jvj
vt
b
U
f
: Vv dx
Z
v$njvj2ds ¼
G
out
2
dx
Z
b
f$v d x:
b
U
0
dt
B
@
Z
r
jvj
s
b
U
s
12d
Z
þ
JSsFT: Vv dx þ
b
U
s
Z
r
f
2
b
U
f
divJF
2
dx þ r
1
v
f
b
U
Z
Z
b
U
vu
vt
Jjv
f
f
Jsf+x
Now note that the Reynolds transport equality (4.7) applied to f ¼ J brings us to the
identity:
vJ
þdivJF
vt
1
v
vu
vt
¼ 0inbU
: (4.68)
f
This identity leads to some cancellations and we get
12d
þ
0
B
@
dt
Z
Jsf+x
b
U
f
Z
2
r
dx þ r
jvj
s
b
U
s
f
T
F
Z
Jjv
f
b
U
f
: Vv dx þ
2
dx
j
Z
r
f
2
G
1
Z
C
þ
A
b
U
s
v$njvj2ds ¼
out
JSsFT: Vv dx
Z
b
f$v d x:
b
U
The third term corresponds to the variation of the elastic energy. To see this, we
can rewrite it using the elastic potential J. From the principle of the mechanical work
(Eq. 4.35),wehaveJ
ðFÞ¼ðJSFTÞ : Ft. With the help of this equality, we get
t
¼
Z
JSsFT: Vv dx ¼
b
U
s
Z
JSsFT:
b
U
s
vF
vt
dx ¼
Z
b
U
s
vJðFÞ
Z
JSsFT: V
b
U
s
dx ¼
vt
vu
d
dt
vt
dx
Z
b
U
s
JðFÞdx:
78 Chapter 4
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Now we need the Piola identity divJF
1
¼ 0, which implies the following equality:
Using the notationbDðvÞ¼
divJF
1
VvF
2
1
v¼JðVvÞ: F
1þFT
ðVvÞ
T
inbUf: (4.69)
T
for the rate of deformation tensor in the
(quasi)-Lagrangian coordinates, we get with the help of Eqs. (4.28), (4.66), and (4.69):
Z
b
U
f
Here and in the remainder,
Jsf+ x
$
F
f
T
: Vv dx ¼2m
F
stands for the Frobenius norm. Therefore, the final energy
Z
b
DðvÞ
2
dx:
F
J
f
b
U
f
equality takes the form:
Z
b
f$v d x;
b
U
1
C
A
12d
dt
þ2m
0
B
@
f
Z
b
U
Z
r
s
b
U
s
JjbDðvÞj
f
v
2
dx þ r
2
F
dx þ
dx þ
Z
JðFÞdx
b
U
s
Z
2
Jjv
b
U
f
Z
r
f
2
G
out
j
v$njvj2ds ¼
f
i.e., the variation of the total system energy is balanced by the fluid viscous dissipation,
and the energy rate at the open boundary and the work of external forces.
In case of Saint VenanteKirchhoff material (Eq. 4.45), the elastic energy is
Z
JðFÞdx ¼
b
U
s
Z
1
l
2
b
U
s
trðE
s
2
Þ
þ 2msjEj
2
dx:
F
(4.70)
In case of incompressible neo-Hookean material (Eq. 4.53), the elastic energy is
Z
JðFÞdx ¼
b
U
s
Z
1
2
2
m
s
b
U
s
jFj
3dx:
F
CHAPTER 5
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3D vascular and heart hemodynamics
5.1 Introduction
In Chapter 4, we reviewed some fundamental laws and equations of fluid dynamics and
the physics of deformable media. Building on this basis, this chapter introduces
mechanical models of (different parts of) a human cardiovascular system. We further apply
computational methods to solve these models numerically. Numerical simulation of a
complete cardiovascular system would include interaction of elastic vessels, deformable
tissues, and blood flows on a range of spatial scales from aortic to capillary flows. Such
direct computations look unfeasible even with modern supercomputers. To address this
computational challenge, model order reduction techniques can be used, and we discuss
some of them in Chapters 6 and 7. At the same time, it is often required to simulate in full
detail local phenomena, such as a blood flow in the heart chambers or the interaction of
the blood flow with a vascular stent. In this case, full-scale 3D models should be used to
set up a numerical experiment. Such models and methods are the subject of this chapter.
We first discuss blood flow modeling in the case of noncompliant boundaries, which is a
reasonable approximation for flows when the vessel walls lack elasticity due to a
pathology or no sufficient information about their elastic properties is available.
Furthermore, we continue with fluidestructure models for the blood flows in compliant
vessels and blood flow in the heart. We complete the chapter with a more detailed
discussion about numerical procedures used to compute for solutions of these models.
5.2 Simulation of blood flow in vessel with nondeformable walls
5.2.1 Mathematical model
The setup of a mathematical model requires definitions of equations, a domain where the
equations are posed, as well as boundary and initial conditions. We start with the domain
definition. The simulation domain U is represented by a 3D tube with rigid walls, which
may have one or several bifurcations (see Fig. 5.1). The rigid wall defines a part of the
boundary G
vessel outlet G
sections.
Personalized Computational Hemodynamics. https://doi.org/10.1016/B978-0-12-815653-7.00005-1
Copyright © 2020 Elsevier Inc. All rights reserved.
. The vessel inlet Ginis represented by a cross section of the tube. The
wall
out
or several outlets, G
k
; k ¼ 1; .; N
out
79
are also given by the tube cross
out
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