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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
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Fig. 1.2 Schematic of distribution of atherosclerosis in the vascular system including coronary
arteries in category I (left upper panel). Reproduced from DeBakey et al. (1985) with permission
shown in Fig. 1.2. The various categories include regions of bifurcations, curvature,
and infra-renal regions. The common biomechanical characteristics of these regions
include transient flow reversal (i.e., flow disturbances, low fluid shear, oscillatory
shear index) and high intramural stresses at regions of curvature. Biomechanics is
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necessary to understand these phenomena and to devise therapies to mitigate and
treat atherosclerosis.
This chapter outlines a basic biomechanical approach for the understanding of
coronary vascular physiology and pathology. The geometry, material properties, and
boundary conditions in conjunction with the laws of mechanics allows a precise and
quantitative description of the problem, and associated method of solution (e.g., by
employing the finite element method, computational fluid dynamics method, and
fluid–solid interaction method). We shall describe each of the components of this
approach, which will set the stage for the study of specific problems of the coronary
circulation in the subsequent chapters.
1.2 Basic Terminology in Biomechanics
Table 1.1 summarizes some common terminology used in biomechanics. The
concept of stress and strain is intimately related to force and deformation. Forces
applied to fluids cause flow, while forces applied to solids cause strain or deformation (i.e., solids resist the stresses). When external forces are applied to a vessel, it
deforms to resist the forces. It is common to use distensibility and stiffness to
describe the deformation and the resistance to deformation, respectively. Definition
of these parameters for the blood vessels can be difficult since no single parameter
can describe the complex mechanical behavior of the blood vessels. To arrive at
useful approximations typically used in physiology, it is important to understand the
basic relations between stresses (i.e., force) and strains (i.e., deformation).
1.2.1 Stress
Stress is force per unit cross-sectional area (Table 1.1 and Fig. 1.3), i.e., force per
unit area of the material on the positive side (exterior) of a vector perpendicular to the
surface exerts on the negative side (interior). On any surface, the force may be
applied either perpendicular to the surface, such as the bolus pressure (normal stress)
exerted on the wall from the blood pressure or from the surrounding tissue (e.g.,
myocardium), or parallel to the surface, such as the force exerted by the fluid flow
(shear stress) on the wall. Normal stresses may be either compressive (e.g., forces on
coronary vessels from surrounding heart musc le) or tensile (e.g., forces on heart wall
from blood pressure). A force may be applied in any direction and can induce
stresses and strains in various directions.
At any given point in the body, the state of stress is described by a stress tensor
which consists of three normal stresses and six shear stresses (three are independent).
Tensors are geometric objects that are used as the language of continuum mechanics.
Both stress and strain are tensor quantities represented by a 3 3 matrix with nine
components in three-dimensional (3D) space (2 2 in 2D). Since both stress and
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Table 1.1 Common biomechanics terms
Term Definition
Stress Force per unit surface area that the part lying on the positive side of a surface
Strain Force applied to a solid causes deformation or strain. Consider a string with
Elastic modulus Proportionality constant between stress and strain in given direction. For
Isotropy Materials whose mechanical properties do not depend on directions are said to
Viscoelasticity Time dependence of the response to stress or strain. Stress relaxation, creep,
Preconditioning In mechanical testing of living tissues in vitro, the loading and unloading
Constitutive
equation
Zero-stress state Tissue configuration where no stress is present. For a tubular organ, the zero-
Plastic
deformation
element (the side on the positive side of the outer normal) exerts on the part
lying on the negative side. Stress is a tensor quantity with six independent
components. Three of the components are called normal stresses, and the
remaining three components are called shear stresses. A normal stress is
perpendicular to the surface while a shear stress is parallel to the surface.
Figure 1.3 shows an example of stresses induced in vessel wall in response to
pressures and flow
initial length L
in length by dimensionless ratios such as L/L
the absolute length from consideration. Elongation causes tensile (positive)
strain while shortening causes compressive (negative) strain. Figure 1.4
shows an example of strains induced in vessel wall in response to pressure and
axial force
example, Hooke’s law applies for a homogenous, isotropic, linearly elastic
material implying that in a given dimension a single elastic modulus describes
the stiffness, i.e., spring constant k. The mechanical behavior in soft biological
tissues is generally nonlinear and the elastic modulus is not constant but
depends on the load
be isotropic. Biological tissues are usually anisotropic, mainly due to their
heterogeneous, layered structure
and hysteresis are features of viscoelasticity
processes are repeated for a number of cycles until the stress–strain relation
becomes stabilized and repeatable results are obtained
A constitutive equation describes the material properties of a material; e.g.,
the stress– strain relatio n. A simple example for a spring is the equation of the
form F ¼ kx, where F is the force or stress and x is the displacement or strain
and k is the material constant
stress state is obtained by making radial cuts in a ring of tissue such that it
springs open into a sector. The difference in strain between the zero-stress
state and the no-load state where all external forces are absent is called
residual strain
Deformation that does not return to its initial state when the stress is removed
and stretched length L. Strain is useful to describe the change
o
or (L Lo)/Loas this eliminates
o
strain are symmetric tensors in the absence of external moments, the number of
independent components reduces to six in 3D (i.e., only three unique shear components). The rows correspond to the direction of outer normal to a surface, whereas
the columns correspond to the direction of force (Fung, 1994).
In a cylindrical tube (e.g., a blood vessel), radial, circumferential, and longitudinal components of stress can be defined in the respective directions. These are the
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Fig. 1.3 Schematic of blood vessel under pressure, flow, and external (e.g., intramyocardial
pressure, IMP for the heart) loadings. The isotropic pressures act in all direction to induce
circumferential (τ
endothelium
), axial (τz), and radial (τr) stresses. The blood flow induces shear stress on the
θ
normal components of stress in the wall of the cylinder (Fig. 1.3). There are also
three additional shear components. In tubular organs, the major tensile stress induced
by distension is in the circumferential direction (Dobrin, 1978). During luminal
pressure loading, the equilibrium condition requires the force in the vessel wall in the
circumferential direction to be balanced by the force in the vessel lumen contributed
by the inflation pressure. Under the assumption that the vessel geometry is cylindrical, it can be shown that the average circumferential wall stress is σ ¼ Pr/h, where P,
r, and h are the pressure, internal radius, and wall thickness, respectively. This
formula is commonly known as Laplace’s law (see Appendix 1 for derivation)
which is applicable for thin wall vessels, such as blood vessels. This equation
explains clinical phenomenon such as why aneurysms will continue to expand
once dilated, and why rupture occurs when segments are excessively distended,
i.e., as the radius increases, the stress or tension increases which leads to further
increase in radius and so on unti l the failure stress is reached. Another important
implication of this equation is that the wall stress is related to pressure and the radiusto-wall thickness ratio. It should be noted that the stress is averaged over the
thickness of the segment and does not describe the transmural distribution of stress
across the wall thickness, as in the case of a thick-walled vessel, i.e., a thick-walled
cylinder will bear the highest tensile stress on the inner surface. Furthermore,
residual strain (i.e., strain that remains in the tissue when all external loads are
removed) is often found in biological tissues as shown by a vessel segment opening
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Fig. 1.4 Schematic of
definition of strain in the
circumferential
(pressurized) and axial
(elongated) directions. The
circumferential deformation
can be defined as the stretch
ratio d/d
d
reference to undeformed
configuration) or (d d
(Eulerian if in reference to
deformed configuration),
etc. Similarly, the axial
stretch can be defined as L/
L
(L L
or strain (d do)/
o
(Lagrangian if in
o
or (L Lo)/Loor
o
)/L, etc.
o
)/d
o
into a sector when cut radially (Chap. 3), which is not considered in Laplace’s
equation.
1.2.2 Strain
Strain refers to stretch or deformation of a material and is usually expressed as a
fraction of the initial length (Lagrangian strain, ε), as defined in Table 1.1 (see
Fig. 1.4 for vessel as an example). It may also be defined in terms of a stretch ratio, λ
(length divided by initial length referred to as Lagrangian), which is useful if the
material is incompressible since the product of the stretch ratios in the three principal
directions (i.e., circumferential, axial, and radial) is equal to 1. Hence, if the stretch
ratios in two directions are known, the third stretch ratio can be computed. The
relation between Lagrangian strain (ε) and stretch ratio (λ)isε ¼ λ 1. Alterna-
tively, strain can be defined in reference to deformed state referred to as Eulerian. In
contrast to stress, stra in (Lagrangian or Eulerian) is dimensionless and the gradient in
circumferential strain is more uniform across the wall in tubular organs. The
dimensionless property of strain facilitates the comparison of various experiments.
The strain is dependent on the determination of the correct initial length which may
be uncertain since in blood vessels and other biological tissues (e.g., smooth muscle
cells of bladder), the resting length can accommodate a broad range of physiology.
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A tensile stress imposed on a material leads to elongation (positive strain), while a
compressive stress leads to shortening (negative strain). Like stress, strain is a tensor
with corresponding components. In the wall of a cylinder, radial, circumferential,
and longitudinal strains can be induced in the respective directions as well as three
shear strains. In vitro, strains are often computed from measurements of changes in
distance between markers located on the surface or embedded in the tissue. In intact
organs, the change in radius or circumference can be used as a measure of strain, as
will be discussed later. In in vitro experiments, the segments are often free to
lengthen, but in vivo the organs may be tethered to the surrounding tissues
(Chap. 3). For example, for coronary vessels, an important observation is that the
vessel is always under considerable axial stress in vivo. When removed from the
body, blood vessels shorten by up to 60% with a corresponding increase in the
diameter. This large residual axial strain has important implications on the stress
distribution, as will be shown in Chap. 8.
Stress and strain are related through material properties of the object. For a linear
elastic Hookean material, the proportionality constant between stress and strain is
called the Young’ s modulus (Table 1.1). Young’s modulus is a measure of the
stiffness of a material (i.e., the stiffer the material the larger the Young’s modulus).
For such a material, the constitutive equation can be simplified to Hooke’s law (see
Appendix 2). In soft biological tissues (e.g., blood vessels, heart), however, the
relation between stress and strain is nonlinear with large strain (i.e., finite deformation). The nonlinear (typically exponential-like) mechanical behavior in biological
tissues facilitates stretch in the physiological pressure range and prevents overstretch
and damage to the tissue at higher stress levels. Overstretch can induce plastic
deformation (Table 1.1) whereby the tissue can no longer return to its original
state when unstressed. In the elastic regime, it is possible to linearize the stress–
strain relation to compute an incremental elastic modulus (Chap. 4).
The blood vessel wall has complex 3D structure that has different material
properties in different directions (see Chap. 3). This important feature is called
anisotropy (compared with isotropy where the material properties are the same in
all directions, Table 1.1), and it implies that a large set of material parameters must
be specified in order to completely describe the mechanical behavior. The constitutive equation relates stress and strain through a set of material parameters or
constants. Appendix 2 illustrates the constitutive equation for an isotropic, Hookean,
linearly elastic solid.
1.2.3 Compliance, Stiffness, Distensibility, and Young’s
Modulus
The elastic response of a blood vessel can be expressed in terms of compliance,
distensibility, stiffness, or elastic modulus. Compliance is defined as the change in
luminal dimension (diameter, cross-sectional area [CSA] or volume) divided by the
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corresponding change in pressure (i.e., ΔD/ΔP, ΔCSA/ΔP or ΔV/ΔP, respectively).
Stiffness is the reciprocal of compliance, while distensibility is normalized compliance. The compliance can be measured under static loading or dynamic loading.
Dynamic loading is also referred to as the dynamic compliance or capacitance. If the
loading history is relatively slow (e.g., slow pressure loading), then measurements
correspond to those of a static compliance. It should be noted that the compliance
merely expresses the differences in luminal dimensions between pressure steps.
Hence, the compliance does not account for the actual degree of stretch that occurs
under luminal pressure loading, the variations in the unstressed luminal diameter, or
the wall thickness.
The pressure elastic modulus (E
change in pressure per change in strain (i.e., E
) is a measure of stiffness and is defined as
p
¼ ΔP/(ΔD/D)). This parameter can
p
be used to compare vessels with different pressures. It is more advantageous than
compliance because it considers the degree of stretch, but still does not account for
the wall thickness (i.e., it represents pressure, not stress).
In some cases, it may be advantageous to express the vessel wall distensibility
(rather than stiffness) in terms of cross-sectional area and transmural pressure.
Distensibility is defined as the ratio of fractional change of cross-sectional area
(CSA) to the change in transmural pressure (P
where CSA
is the reference cross-sectional area. This parameter can be directly
0
) as (1/CSA0)(ΔCSA/ΔPtm),
tm
computed from the slope of pressure–CSA curve. Despite this, it may be useful to
convert such a measurement into an incremental Young’s modulus for circumferential extensions of the vessel wall, considered as a uniform cylinder with homogenous, isotropic walls. The use of an incremental Young’s modulus is made necessary
by the nonlinearity of the relation between circumferential stress and cross-sectional
area (see Chap. 4 for more detail). Hence, a single elastic parameter can be employed
by considering small departures from a mean, pre-stressed, in vivo state, and
linearizing the stress–strain curves. This may be useful if the amplitude of the
pressure is small. For an increase in pressure within a thin-walled isotropic vessel
whose length is held constant, Young’s modulus (E) is related to the distensibility.
Distensibility ¼ (1 α
2
)D/(Eh), where h is the wall thickness, D is vessel diameter,
and α is Poisson’s ratio defined as negative of the ratio of transverse to axial strain.
The Poisson’s ratio is equal to 0.5 if the material is incompressible (see Appendix 2).
This result follows from the classical elasticity of shells and is referenced by Bergel
(1972) in a more general form for thick-walled tubes.
1.2.4 Viscoelasticity
Biological tis sues reveal properties of both elastic solid and viscous fluid. Thus, the
stress depends not only on the applied strain as in a solid, but also on the rate of strain
as in a viscous fluid. In other words, the mechanical response of the tissue is time
dependent in that the stress–strain response does not occur instantly. When the
material is suddenly stretched and the strain is maintained constant, the
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corresponding stresses induced in the wall decrease with time. This phenomenon is
called stress relaxation. If the material is suddenly stressed and the stress is
maintained constant, the material will continue to deform. This phenomenon is called
creep. If the material is subjected to a cyclic loading, the stress–strain relationship in
the loading process is somewhat different from that in the unloading process, and the
phenomenon is called hysteresis (Fung, 1993). Stress relaxation, creep, and hysteresis are features of viscoelasticity. Often, the viscoelastic behavior is described in
terms of time-dependent models, e.g., the Maxwell model describes the mechanical
behavior of the tissue material by using a spring and a dashpot in series.
1.3 Approach
As proposed by Fung (1983), four basic prerequisites to the solution of any problem
in biomechanics are as follows:
1. Identification of the geometry or structure of the system, including anatomical,
morphological, histological, and microstructural studies
2. Determination of the materials of the system and delineation of their mechanical
properties, involving the study of chemistry, mechanical testing, and constitutive
equation formulation
3. Analysis of basic constitutive laws governing the system, including outlining the
field equations depending on the number of assumptions invoked
4. Prescription of initial and boundary conditions, which are required to constrain
the solutions to problems of physiological or clinical significance
In solving biomechanical problems, the ideal approach is to minimize the number
of ad hoc assumptions (i.e., know Prerequisites 1 and 2) and to allow only the most
basic principles as axioms (Prerequisite 3), such as static and dynamic equilibrium
based on Newton’s law of mechanics, the balance laws of mass, momentum, and
energy; the second law of thermodynamics, and so on. The final requirement of
initial and boundary conditions depends on the starting point and the neighborhood
of the specific problem at hand, respectively. Collectively this approach leads to
what can be termed as well-posed bou ndary value problems (BVPs). In this way,
biomechanics provides a mathematical framework for integration of geometry and
material constitutive properties to predict function, and hence provides a link
between structure and function.
1.4 Structure and Geometry
The prescription of geometry or morphometry (measurement of form or shape) is
necessary for the formulation of any boundary value problem (BVP). Since a BVP is
a problem which has values assigned on the physical boundary of the domain in
which the problem is speci fied, the importance of geometry or form is obvious.
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Specifically, there are several reasons for specifications of geometry of a blood
vessel or other fluid flow organ:
1. A mathematical model must obey geometric similarity which requires knowledge
of the dimensions of the organ.
2. A mathematical analysis must also obey the rule of dynamic simil arity which
reduces to the simulation of two dimensionless parameters, (a) the Reynolds
number, N
dimension, and ρ and μ are the density and viscosity of blood, respectively), and
(b) Womersley numbe r, N
of pulsatile flow). The Reynolds number (N
¼ ρUD/μ where U is the mean flow velocity, D is the lumen
R(NR
W(NW
¼ D/2(ρω/μ)
1/2
where ω is the circular frequency
) represents a ratio of inertial to
R
viscous forces, i.e., the flow in the highly inertial flow in the heart and aorta has
high N
Womersley number (N
i.e., N
3. For a steady laminar flow, the Poiseuille’s formula can be employed to determine
the flow rate Q ¼ ΔP πR
while the viscous flow in the mic rocirculation has low NR. Similarly, the
R
) is the ratio of transient inertial forces to viscous forces,
W
is high in aorta and small in capillaries.
W
4
/8 μL, in terms of the pressure drop, radius R, viscosity
μ, and vessel length L.
4. In an unsteady flow, the characteristic impedance is the ratio ρc/A, where ρ is the
density of blood, c is the speed of flexural waves in the blood vessel, and A is the
cross-sectional area (proportional to the square of diameter) of the vessel.
5. The mean circumferential Cauchy stress, σ (force per deformed area), in the vessel
wall is given by σ ¼
PD
where P is the blood pressure and h is the wall thickness.
,
2h
Hence, the geometry (e.g., diameter, length, wall thickness, and curvature) must
be properly quantified for a realistic biomechanical analysis of function.
Since a detailed biomechanical analysis requires data on the structure and geometry of an organ, tissue, or cell, developments in biomechanics overlap with
advances in anatomical imaging. Structural imaging is necessary for measuring
and quantifying organs, tissues, cellular structures, and molecular structures,
which serve as a basis for the construction of biomechanical and integrative models.
Imaging modalities include magnetic resonance imaging (MRI), computerized
tomography (CT), positron emission tomography (PET), and ultrasound (US) at
the organ level; micro-CT and optical coherence tomography (OCT) and intravascular US (IVUS) at the tissue level; confocal and interference microscopy, multiphoton microscopy (MPM), and electron tomography (ET) at the cellular level; and
X-ray crystallography at the molecular level. These imaging techniques provide the
structure and geometry that is essential to perform a biomechanical analysis.
1.5 Material Properties
Biological tissues are subject to the same conservation laws of mass, momen tum,
and energy as inanimate objects. What distinguishes biological tissues from inanimate materials are their unique constitutive equations. Soft biological tissues are
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typically nonlinear, non-isotropic, viscoelastic, and hyperelastic materials (Zhang,
Chen, & Kassab, 2007). Modeling the mechanical properties can be formulated in
many ways, varying in degree of generality and complexity.
The pressure–diameter relation has been extremely popular among cardiovascular physiologists since it plays an important role in the pressure–flow relation of
blood flow through an organ. Indeed, it can be shown that the nonlinearity of the
pressure–flow relation stems from the distensibility of the vasculature (Kassab,
2001). Furthermore, the compliance of the vessels, as derived from a pressure–
diameter relationship, can be shown to be proportional to the diameter-wall thickness
ratio of the vessel and inversely proportional to the Young’s modulus of the vessel
wall material as described above. For the foregoing reasons, the distensibility of the
vessels has been the subject of a vast number of studies (For coronary vessels, see
(Douglas & Greenfield, 1970; Giezeman, VanBavel, Grimbergen, & Spaan, 1994;
Gregg, Green, & Wiggers, 1935; Kuo, Chilian, & Davis, 1991 ; Kuo, Davis, &
Chilian, 1988; Manor, Beyar, Shofti, & Sideman, 1994; Nakayama, Osol, &
Halpern, 1988; Patel & Janicki, 1970; Reneman & Arts, 1985; Tomoike, Ootsubo,
Sakai, Kikuchi, & Nakamura, 1981)). Although the mechanical properties of a blood
vessel are important determinants of the pressure–flow relationship, the speed of
pulse waves in the vessels, the stress distribution in the vessel wall, and the
phenomena of mass transport through the arterial wall (Fung, 1990), a complete
and systematic set of data on coronary blood vessel elasticity is not available for any
species.
The mechanical properties of blood vessels are derived from microstructural
constituents (e.g., collagen and elastin fibers, smooth muscle cells, and ground
substances) of the wall. The literature on blood vessels includes numerous references
to their material components (see review by Fung (1990)). The macroscopic effective strain–stress relationship of the vessel wall is associated with the geometrical
features and mechanical properties of elastin, collagen fibers, cells, and ground
substance. Specifically, fibers can have variable densities and topologies such as
orientation, length, width, and degree of undulation. For example, coronary blood
vessels have three layers (i.e., intima, media, and adventitia) from the lumen to the
external surface, which have mechanical properties that are differentiated by the
respective arrangement of collagen and elastin fibers, and cells. The deformation or
stress of each component (such as a single fiber) depends on its own stiffness,
geometry, and its interaction with other fibers. Thus, an accurate description of
geometrical and mechanical properties of microstructure is essential for a
microstructure-based constitutive model. The topic will be discussed thoroughly in
Chap. 4.
The mechanical properties of blood vessels depend not only on the intrinsic
properties of the blood vessel wall but also on the properties of neighboring tissue.
For example, the intramural coronary blood vessels are embedded in the myocardium, where the interaction of blood pressure, vessel elasticity, smooth muscle tone,
and tissue stress lead to a complex time-dependent interaction between blood flow
and muscle contraction. For example, the transient muscle–vessel interaction is an
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