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14 1 Biomechanics
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important determinant of blood flow because the period of the cardiac cycle is
considerable smaller than the time constant of coronary blood flow (Spaan, 1985).
Constitutive equations describe the relation between stress applied to a material
and the associated deformation. They are a function of the molecular structure and
forces in the material. Constitutive equations must be established by experiments
under thermodynamic constraints and they are mathematical approximation, i.e.,
they represent important features, ignoring most details. A well-known phenomenological approach to elasticity of bodies capable of large deformation is to postulate
the form of constitutive equation or strain energy function (Green & Adkins, 1960).
The strain–energy density may be expressed as an exponential, logarithmic, or
polynomial function in terms of strain components or principle stretches, and the
material parameters are determined experimentally. Most of these models are phenomenological, which mainly represent mathematical curve fits of the experimental
data. The major shortcoming of phenomenological models is that the material
constants have no direct physical meaning, and hence do not facilitate an understanding of the connection between tissue architecture and the mechanical behavior
of the tissue (e.g., response to load, remodeling, growth, disease). Recently, there has
been great progress towards incorporation of structural information into the constitutive relations (Chen et al., 2013; Chen, Liu, Zhao, Lanir, & Kassab, 2011).
Microstructure-based constitutive models provide more accurate predictions of the
overall mechanical responses of tissues than phenomenological approaches
(Chap. 4).
Structure-based models (Chap. 4) are essential for the analysis of tissue mechanical response and understanding of the individual role of each of tissue constituents
in health and disease. Since several vascular pathologies are related to the degradation of tissue fibers, the prediction of the onset of vascular diseases can only be done
by using a model which adequately incorporates the influence of each fiber type.
Furthermore, the determination of microstructural stress requires a constitutive
model based on the ultrastructure and the corresponding material properties. Complex microstructure and strong nonlinear mechanical behaviors of soft tissue, however, present significant challenges in constitutive modeling. Despite the complexity,
the current trend in biomechanics is to move from phenomenological to
micromechanical models in order to predict the overal l nonlinear and microstructural
responses of inhomogeneous soft tissues (Chen & Kassab, 2016; Chen, Zhao, Lu, &
Kassab, 2013). New developments in imaging and biochemistry will continue to
provide more details of the constitutive properties of soft tissues and continue to
advance the development of structure-based models. The determination of the
relative importance of the contribution made by the various microstructural components can be assessed by a sensitivity analysis. The various microstructural and
mechanical parameters can be varied over a large range (model inputs) and their
effect (model outputs) on the parameters of interest (e.g., fiber or cell stress or strain)
can be assessed. This approach can guide the focus of experiments on those
microstructural a nd mechanical parameters of greatest sensitivity, which is further
elaborated in Chaps. 3 and 4.
1.7 Boundary Conditions 15
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1.6 Laws of Mechanics
Fluid (e.g., blood, urine, bile, air) flow must obey the conservation laws (Fung,
1997). The laws for fluid flow are conservation of mass (continuity) and momentum
equations that include velocity, pressure, strain rate tensor, density, dynamic viscosity, and body force typically in the form of gravity which lead to the Navier–Stokes
equation (Appendix 3). The Navier–Stokes (NS) equation that describes the motion
of an incompressible fluid such as the blood states that the inertial force ¼ body
force pressure force + viscous or diffusion force. Inertial force consists of transient
and convective terms where the transient term is time dependent while convective
term changes with space or geometry. Since the NS equation is very complex to
solve in its entirety, approximation can be made to solve only those terms that
dominate the flow under certain hemodynamic conditions. For example, consider
two flow fields with the following: (1) With geometrically similar boundaries (i.e.,
geometric similarity), (2) With the same Reynolds number (i.e., dynamic similarity),
and (3) With the same boundary conditions in the initial conditions expressed in
nondimensional quantities. Then the solution for the NS equation will be the same.
This analysis emerges from nondimensionalization of the NS which results in Reynolds and Womersley numbers as noted above. This allows us to compare the terms of
the NS and ignore those that are relatively negligible to obtain a simpler equation.
The solid (e.g., heart, skeletal muscle, bones, bladde r, intestines) mechanics
analysis is based on the governing equations of equilibrium and momentum
(Fung, 1990), i.e., Newton’s laws of mechanics (Appendix 3). These equations
include stresses in the various directions, the surface traction vector, and the acceleration of a material point. For a full formulation, the constitutive equation that
relates stress to strain must be specified. Unlike the constitutive equation for a
Newtonian fluid that describes blood, the constitutive equations for biological solids
tend to be more complex (Chap. 4). Problems that involve both fluid and solid
mechanics (fluid–structure interaction) require the specifications of both laws of
fluids and solids and their interactions, as outlined in Appendix 3.
1.7 Boundary Conditions
The solution of any biomechanical problem (fluids, solids, fluid–solid interaction,
mass transport, heat transfer, multi-physics) requires the prescription of boundary and
initial conditions. Boundary conditions incorporate the neighborhood of the organ,
including the surface interaction between the organ of interest and neighboring tissue
or organs internal or external to the object of interest. The boundary conditions
imposed on the boundary of the object of interest can be in the form of displacement
(stress per area) or traction (force per area). If the problem of interest depends on time
(not steady state), then there is need for prescription of initial condition, i.e., if the
body is at rest initially, then the velocity may be set to zero at t ¼ 0.
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1.8 Boundary Value Problems
Boundary value problems (BVPs) are solved under specific boundary and initial
conditions. The mechanics of fluid and solid in the cardiovascular system are so
complicated that they are rarely solvable analytically (Fung, 1997). Most of the
analytical solutions are carried out, based on simplified geometries (e.g., cylinders,
spheres, ellipses). Experiments are used to measure quantities such as geometry
(by using imaging techniques such as angiography, computed tomography, magnetic
resonance imaging, intravascular ultrasound, and optical coherence tomography),
strain (by measuring change in geometry using imaging techniques), material properties (by using imaging in conjunction with elastography), and flow (by using
single-photon emission tomography, positron emission tomography, etc.). Solid
wall stresses, however, cannot be measured but only calculated from computational
finite element models.
For blood flow in a vessel, computational fluid dynamics (CFD) methods are used
in conjunction with the experimental waveforms (as boundary conditions) to determine the blood flow disturbances (e.g., flow separation, secondary flow, stagnation
point flow, reversed flow, and/or turbulence) due to convective inertia. Since the
spatial complexities of blood flow in the cardiovascular system cannot be fully
visualized with current imaging methods, theory and computational modeling are
necessary.
For stress analysis in a vessel wall, the BVPs are solved by applying the equation
of radial equilibrium under quasi-static conditions. The solutions of these BVPs
provide the transmural distribution of stress components. The constitutive equation
and BVPs are intimately related. No BVP analysis can be done without the constitutive equation (Chaps. 5–8). Conversely, the material parameters of the constitutive
equation cannot be determined without the solution of a BVP (Chap. 4). We shall
consider several BVPs in Chaps. 5–8, for fluid and solid problems in coronary
circulation.
1.9 Solutions of Boundary Value Problems
1.9.1 Computational Fluid Dynamics
Since there is a casual relation between local hemodynamics and endothelial function, the computational fluid dynamics (CFD) method to solve the NS equation has
emerged as a powerful tool to study flow patterns in blood vessels (Berger & Jou,
2002; Buchanan Jr., Kleinstreuer, Truskey, & Lei, 1999; Farmakis, Souli s,
Giannoglou, Zioupos, & Louridas, 2004; He & Ku, 1996; Kleinstreuer et al.,
2001; Ku, 1997; Lei, Kleinstreuer, & Truskey, 1996; Perktold, Resch, & Florian,
1991; Perktold, Resch, & Peter, 1991; Ramaswamy et al., 2004; Sankaranarayanan,
Ghista, Poh, Seng, & Kassab, 2006; Stroud, Berger, & Saloner, 2002; Taylor,
1.9 Solutions of Boundary Value Problems 17
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Hughes, & Zarins, 1998; Zeng, Ding, Friedman, & Ethier, 2003). The CFD model
has been used to describe the flow patterns in different anatomical geometries
(Perktold, Resch, & Florian, 1991; Perktold, Resch, & Peter, 1991; Ramaswamy
et al., 2004; Sankaranarayanan et al., 2006; Santamarina, Weydahl, Sigel, & Moore,
1998; Stroud et al., 2002; Taylor et al., 1998; Weydahl & Moore, 2001; Zeng et al.,
2003). Newtonian (stress is linearly proportional to strain rate, e.g., air, water,
plasma) and non-Newtonian (e.g., blood, synovial fluid) unsteady fluid flow has
been compared in normal carotid arteries by using the finite element method (FEM)
(Perktold, Resch, & Florian, 1991; Perktold, Resch, & Peter, 1991). Kleinstreuer and
his colleagues (Buchanan Jr. et al., 1999; Kleinstreuer et al., 2001; Lei et al., 1996;
Perktold, Resch, & Florian, 1991; Perktold, Resch, & Peter, 1991) studied the
relationship between non-uniform hemodynamics at the rabbit aorto-celiac junction.
Berger and his associates (Bao, Lu, & Frangos, 2001; Stroud et al., 2002) investigated the blood flow with a turbulence model in stenotic vessels. Ku’s group (Ku,
1997) solved the pulsatile flow in the human left coronary artery including the left
common coronary artery (LCCA), left anterior descending (LAD), and left circumflex arteries (LCx). Ramaswamy et al. (2004 ) performed the numerical simulation to
study the effects of motion of the coronary artery on the unsteady fluid dynamics.
The CFD method is a numerical model to treat a continuous fluid in a discretized
fashion. The fundamental basis of the model in a single-phase blood flow is the
partial differential equations (PDEs) or integro-differential equations of continuity
and Navier–Stokes (which arises from Newton’s second law of viscous fluid
motion), which can be discretized at specific locations in space and time, approximated by a system of algebraic equations, and then solved on the computer. The
method of a numerical solution can be summarized as follows:
1. Select an appropriate mathematical model and boundary conditions.
2. Select a suitable discretization method.
3. Select the correct coordinate system and basis vectors.
4. Create the numerical grid.
5. Solve the algebraic equations.
6. Set the convergence criteria for the iterative method.
The accuracy of the numerical solutions depends on the selected discretization
method. There are many discretization methods, the most popular approaches of
which are finite difference (FD), fi
(FE) methods. The FD method is the most classical and straightforward approach
for numerical solution of partial differential equations (PDE) in simple geometries.
In the FD met hod, the PDE is converted into a set of FD equations at each grid point
that can be solved subject to the appropriate boundary conditions. Taylor series
expansion or polynomial fitting is generally used to obtain an approximation to the
first and second derivatives of the variables with respect to the coordinates. The FD
method is very efficient on structured meshes (simple geometry), but difficult to
implement in complex geometries (Chap. 2).
nite volume (FV), and finite element
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In comparison with the FD method, the FV method enforces the integral conservation law in small control volumes defined by the computational mesh. In the FV
method, the variable values are calculated at the central node of each control volume
(CV). Interpolation is used to express variable values at the CV surface, and the
appropriate quadrature formulae are used to simulate the surface and volume integrals. The advantage of the FV method is that it is suitable for complex geometries,
but the disadvantage is that the higher order FV method is more difficult to develop
in three-dimensional (3D) geometries.
The FE method has many common features with FV method. The FE procedure
begins with the division of the continuum region into number of simply shaped
regions called elements. Within each element, the variables are interpolated by
functions of compatible order, in terms of values to be determined at a set of nodal
points. For the purpose of developing the equations for these nodal point unknowns,
an individual elem ent may be separated from the assembled system. The FE method
can address any arbitrary geometry, but it is time-consuming to solve the assembled
large sparse matrices. Since the FE method is similar to the FV method, only the FE
method is described in the following subsections and the details of the FV method
can be found in a standard textbook (Patankar, 1980).
1.9.2 Finite Element Method
Mathematical modeling of the cardiovascular system using the FE method is necessary given the complex geometry of the organs (heart, disease vessels, etc.) and
complex boundary conditions (Guccione, Kassab, & Ratcliffe, 2010). Advances in
computational sciences have made the FE method both more powerful and more user
friendly. The FE method is a computer-aided mathematical technique for obtaining
the approximate numerical solution to the physical phenomena subject to initial and
boundary conditions in both fluid and solid mechanics. The FE method originally
arose from applications in solid mechanics (elas ticity, plasticity, statics, and dynamics). To date, applications have been expanded to the broad field of engineering
sciences, such as heat transfer (conduction, conve ction, and radiation), fluid mechanics (inviscid or viscous, compressible or incompressible), acoustics, electromagnetics, and many others.
The basic idea of the FE method is summarized as follows: (a) Domain of the
problem is partitioned into smaller regions, called elements; (b) In each element, the
governing equations are transformed into algebraic equations called the element
equations; (c) Terms in the element equations are numerically evaluated for each
element in the mesh; (d) Resulting numbers are assembled into a much larger set of
algebraic equations called the system equations; (e) System equations are solved by
using the numerical technique on a computer; and (f) Final operation displays the
solutions in tabular, graphical, or pictorial form.
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1.9.3 Fluid–Structure Interaction
Problems in fluid–structure interaction (FSI) involve both fluid and solid mechanics.
Nature has an abundance of problems in FSI involving both fluid and solid molecules phenomena. Some examples in the cardiovascular system include intraventricular flow, valve opening and closure, ventricular ejection, and coronary
vessel/myocardial interaction in coronary circulation, flow past cilia, and many
others. FSI problems are generally complex because the structures are usually freely
moving with large deformation. Typically, the Navier – Stokes equations (conservation of mass and momentum) are formulated in Eulerian coordinates (observer
focuses on speci fic location in the space through which the fluid flows over time),
and the solid motion equations are formulated in Lagrangian (observer follows a
material point as it deforms). The two sets of partial differential equations are
coupled on the moving interfaces, which separate the fluid and solid component
(Appendix 3). The coupling is not known a priori and must be solved as part of the
problem, i.e., the FSI is a free moving boundary problem. The FSI models have an
additional unknown variable (the time-dependent interface position) and are more
challenging than a corresponding fixed boundary problem. Thus, analytical solutions
to FSI problems are rare and a computational approach is usually the only option.
There are a variety of methods for FSI, including the immersed boundary
(IB) (Peskin, 1972, 1977, 2002), the Arbitrary Lagrangian Eulerian (ALE) (Donea,
Giuliani, & Halleux, 1982; Formaggia & Nobile, 1999; Hughes, Liu, & Zimmermann, 1981), and the fictitious domain methods (Glowinski, Pan, Hesla, Joseph, &
Periaux, 2001; Glowinski, Pa n, & Periaux, 1994). The ALE and IB method have
been widely used in computational studies of the cardiovascular system and are
briefly described below.
1.9.4 ALE Formulation for Fluid–Structure Interaction
The arbitrary Lagrangian–Eulerian method is an effective way to treat FSI problems.
In the Lagrangian approach, an observer follows an individual fluid element while in
the Eulerian, the observer focuses on specific location in the space through which the
fluid flows as time passes. Instead of using either a single Lagrangian approach or a
single Eulerian approach, the ALE describes the motion of fluid in a moving
reference frame which has a velocity distribution that is based on the sole constraint
that the velocity on the fluid–solid boundary must equal to that of the boundary. The
velocity of the reference frame is usually neither the fluid particle velocity such as in
a pure Lagrangian description, nor zero in a pure Eulerian description. The boundary
conditions for the inflow boundaries can be measured experimentally. The boundary
condition for the surface traction can be prescr ibed at the outflow boundaries.
The Navier–Stoke equati ons for the fluid, and the equilibrium equations for the
solid are coupled on the fluid–solid interface via kinematic and dynamic conditions.
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The solid and fluid models can be coupled by the fluid nodal positions on the FSI
interfaces, which are determined by the kinematic conditions. The displacements of
the other fluid nodes are determined to preserve the initial mesh quality. The ALE
modified governing equations for fluid flow are then solved. For the dynamic case,
the fluid stresses are integrated along the fluid–solid interface and applied on the
corresponding solid nodes.
The computational procedure is iterative. For each time step, the fluid model is
solved first, with the imposed boundary condition. The fluid pressure and shear
stress fields are then passed onto the solid model, and the displacements and stresses
of the solid are solved. With the solid model updated, onto the next time step, the
fluid variables are solved again. This process continues until the solutions converge
(Bathe, 1995; Donea et al., 1982).
1.9.5 Immersed Boundary (IB) Method
The immersed boundary (IB) method is a practical and effective method for mathematically formulating and numerically solving problems involving interactions
between an elastic structure and an incompressible viscous fluid. The immersed
elastic structure can be either passive (e.g., a flapping flag) or active (e.g., a
swimming eel); it can also be neutrally buoyant (e.g., a swimming sperm) or
heavier/lighter than the surrounding fluid (e.g., a flying insect). The IB method
was formulated by Peskin (1972, 1977) for studying the flow patterns around
human heart valves in the 1970s which has since become a general method for
investigating flexible-structure–viscous-fluid interaction problems. It has been
applied to both natural and prosthetic cardiac valves (McQueen & Peskin, 1983,
1985; McQueen, Peskin, & Yellin, 1982; Peskin, 1972), and to the modeling of the
whole heart (Ko vacs, McQueen, & Peskin, 2001 ; McQueen & Peskin, 2000;
Vigmond, Clements, McQueen, & Peskin, 2008).
A variety of different versions of the IB method exist (Peskin, 2002). The
underlying philosophy is that the entire system (viscous fluid + elastic body) is
treated as an incompressible composite material, and an Eulerian description is used
to describe its dynamics as a whole. In addition, a Lagrangian description is used to
depict the motion of the immersed elastic body where the connection between
Eulerian and Lagrangian variables are realized by the Dirac delta function
(a generalized function,ordistribution that is zero everywhere except at zero, with
an integral of one). In general, the incompressible viscous Naviver–Stokes equations
with additional forcing term from the immersed boundary (and variable density
resulting from the mass of the immersed body if it is not neutrally buoyant) are used
to govern the whole system. The governing equations are discretized on a fixed
Eulerian uniform grid, while the equations of the elastic structure are discretized on a
collection of Lagrangian moving points that do not necessarily coincide with the
fixed Eulerian mesh points. It should be noted that the shape and position of the
immersed boundary is not known in advance and must be determined from the
Appendix 1: Derivation of Circumferential Stress (Laplace’s Law)... 21
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solution. Thus, the Navier–Stokes solver does not require the shape and location of
the moving elastic body. The influence of the latter is considered by spreading the
mass and force to the nearby fluid. The motion of IB is updated by the surrounding
fluid velocity via interpolation because of the no-slip conditions at the boundary. The
IB formulation is a nonlinear system of integral-differential equations, which may be
numerically solved by different versions of the IB method among which are the Fast
Fourier Transform version and the multi-grid version (Chorin, 1968, 1969; Peskin,
1977, 2002; Peskin & McQ ueen, 1993).
Appendix 1: Derivation of Circumferential Stress (Laplace’s
Law) and Longitudinal Stress in a Vessel
If a body is in equilibrium, every part of it is in equilibrium. To determine the internal
reaction stresses, one can cut free certain parts of the body and examine their
conditions of an equilibrium. Consider a cylindrical vessel subjected to an internal
pressure P
The blood pressure induces stress in the vessel wall. Under equilibrium conditions,
the force in the vessel wall in the circumferential direction 2τ
by the force in the vessel lumen contributed by the pressure 2Lr
in Fig. 1.5b. Hence, the equilibrium equation in the circumfer ential direction is
given by:
, as shown in Fig. 1.5a below, where cuts are made in different planes.
i
ri)L, is balanced
θ(ro
, as shown
iPi
τ
where τ
¼ Piri= ro r
θ
is circumferential stress, riis internal radius, and rois outer radius of vessel.
θ
ðÞ ð1:1Þ
i
The longitudinal stress in a vessel wall can be determined based on the equilibrium of forces in the longitudinal direction. The product of the longitudinal stress
and the cross-sectional area of the vessel wall is the force that balances the total
longitudinal force acting on the vessel as shown in Fig. 1.5c. The longitudinal force
in the vessel wall τ
2
2
z
π r
o
is balanced only by the pressure component P
r
i
2
πr
as
i
i
the external pressure is assumed to be zero. Thus, the desired equation follows:
2
2
¼ Pir
τ
z
where τ
r
o
is the longitudinal stress. If the wall thickness-radius ratio is small, so that
z
¼ ri¼ r and ro ri¼ h, then these equations are simplified to
τ
¼ Pir=h, τz¼ Pir=2h: ð1:3Þ
θ
i
= r
2
r
o
i
ð1:2Þ
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Radial
Longitudinal
r
i
(a)
Circumferential
r
τ
θ
P
i
r
o
τ
θ
(b)
τ
z
z
τ
z
(c)
Fig. 1.5 A pressurized cylindrical tube. (a)Aninfinitesimal element of the longitudinal and
circumferential cylindrical tube showing the radial, longitudinal and circumferential directions.
(b) A free-body diagram of half of the tube cut parallel to the central axis. (c) A free-body diagram
of the tube cut perpendicular to the central axis. Reproduced from Gregersen and Kassab (1996)
with permission
Appendix 2: Constitutive Equation of a Homogeneous,
Isotropic, and Linear Elastic Solid (Hooke’s Law)
The constitutive equations of a solid that consists of a homogeneous, isotropic,
linearly elastic material contain only two material constants given by Hook e’s law:
12μλ
¼
ε
ij
3λ þ 2μ
where i and j are indices ranging from integers 1 to 3. The ith index denotes the
component in the ith direction whereas the jth index denotes the surface perpendicular to the jth direction. The repetition of an index in a term denotes a summation
with respect to that index over its range.
Several special cases will be considered:
Special Case 1
A uniaxial state of stress with the non-zero stress-component τ
the x
-direction. From Eq. (1.4), the following holds:
1
τ
ijδij
þ τ
ij
corresponding to
11
ð1:4Þ
Appendix 2: Constitutive Equation of a Homogeneous, Isotropic, and... 23
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ε11¼
λ þ μ
μ 3λ þ 2μðÞ
τ
, ε22¼ ε33¼
11
λ
2μ 3λ þ 2μðÞ
τ
11
ð1:5Þ
The coefficient between stress and strain can be expressed as:
E ¼ μ
3λ þ 2μðÞ
λ þ μ
ð1:6Þ
which is called the elastic modulus or Young’s modulus. This module can be readily
measured in a uniaxial tension test. The ratio ε
v ¼
λ
2 λ þ μðÞ
¼ε33/ε11is given by:
22/ε11
ð1:7Þ
which is known as Poisson’s ratio. Poisson’s ratio is a measure of the lateral
contraction (extension) produced by an axial tension (compression).
Special Case 2
Consider a state of plane stress in pure shear in which the only non-zero component
of the stress tensor is τ
¼ τ216¼ 0. The corresponding non-zero strain component is
12
given by
1
ε
12
τ
¼
12
2μ
ð1:8Þ
The ratio of shearing stress τ
and the corresponding change 2ε12of an initially
12
right material angle is known as the shear modulus. In the engineering literature, the
symbol G is widely used to describe the shear modulus.
Special Case 3
Finally, consider a hydrostatic state of stress, given by:
τ
¼pδ
ij
ij
ð1:9Þ
If this result is combined wi th Eq. (1.4), the following holds:
1
ε
¼
ij
3λ þ 2μ
pδ
ij
ð1:10Þ
Setting i ¼ j, and summing on j, one finds:
where ε
p ¼ K ε
¼ ε11+ ε22+ ε33and proportionality constant given by:
ii
ii
ð1:11Þ
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