Linköping Studies in Science and Technology, Dissertation No. 1081
Soft Tissue Mechanics with
Emphasis on Residual Stress Modeling
Tobias Olsson
Division of Mechanics
Institute of Technology, Linköping University
SE–581 83, Linköping, Sweden
Linköping, February 2007
Cover:
Illustration of how growth and remodeling can reduce the stress gradients.
The left picture is the stress due to a constant internal pressure, and the
rightmost figure shows the stress after growth and remodeling.
Printed by:
LiU–Tryck
Linköping University
SE–581 83 Linköping, Sweden
ISBN 978–91–85715–50–3
ISSN 0345–7524
Distributed by:
Institute of Technology, Linköping University
Department of Management and Engineering
SE–581 83, Linköping, Sweden
c 2007 Tobias Olsson
This document was prepared with LATEX, February 27, 2007
No part of this publication may be reproduced, stored in a retrieval system,
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Preface
This work has been carried out at the Division of Mechanics at Linköping
University. There are some people that I would like to acknowledge for supporting me through these years. First, my supervisor Prof. Anders Klarbring
(Linköping University) for giving me the opportunity to work at the division
and introducing me to the field of biomechanics. Second, I want to send
my best wishes to my colleagues, specially Dr. Jonas Stålhand, for fruitful
discussions. Third, for arranging my stay in Lisbon, I would like to thank
Prof. João A. C. Martins (I.S.T.). This gave me the chance to work with
challenging problems in soft tissues and taught me how important it is to
collaborate with colleagues in other adherent topics. Also, a special mention
goes to my colleague António Pinta da Costa (I.S.T.) for making my stay i
Lisbon the best possible. Finally, I would like to thank my family and friends
for supporting and believing in me during these years.
This work was partly supported by the Swedish Research Council.
Linköping, February 2007.
Tobias Olsson
iii
“Mistakes are the portals of discovery.”
—Joyce, James (1882–1941)—
iv
Abstract
This thesis concerns residual stress modeling in soft living tissues. The word
living means that the tissue interacts with surrounding organs and that it
can change its internal properties to optimize its function. From the first day
all tissues are under pressure, due, for example, to gravity, other surrounding
organs that utilize pressure on the specific tissue, and the pressure from the
blood that circulates within the body. This means that all organs grow and
change properties under load, and an unloaded configuration is never present
within the body. When a tissue is removed from the body, the obtained
unloaded state is not naturally stress free. This stress within an unloaded
body is called residual stress. It is believed that the residual stress helps
the tissue to optimize its function by homogenizing the transmural stress
distribution.
The thesis is composed of two parts: in the first part an introduction
to soft tissues and basic modeling is given and the second part consist of
a collection of five manuscripts. The first four papers show how residual
stress can be modeled. We also derive evolution equation for growth and
remodeling and show how residual stress develops under constant pressure.
The fifth paper deals with damage and viscosity in soft tissues.
v
To my mother and father, the best parents one could wish for.
vi
List of Papers
This dissertation consists of a short summary and a collection of five research
papers:
I
Anders Klarbring and Tobias Olsson, On Compatible Strain with Reference to Biomechanics, Zeitschrift für Angewandte Mathematik und
Mechanik, 85, 440–448, 2005.
II
Anders Klarbring, Tobias Olsson and Jonas Stålhand, Theory of Residual Stresses with Application to an Arterial Geometry, Submitted for
publication 2007.
III Tobias Olsson, Jonas Stålhand and Anders Klarbring, Modeling Initial Strain Distribution in Soft Tissues with Application to Arteries,
Biomechanics and Modeling in Mechanobiology, 5, 27–38, 2006.
IV Tobias Olsson and Anders Klarbring, Residual Stresses in Soft Tissues
as a Consequence of Growth and Remodeling, Submitted for publication 2007.
V
Tobias Olsson and João A. C. Martins, Modeling of Passive Behavior of
Soft Tissues Including Viscosity and Damage, III European Conference
on Computational Mechanics, C.A. Mota Soares et al. (Eds.), Lisbon,
5–9 June, 2006.
The author of this thesis has contributed to the development of the theory,
written substantial parts of the text and implemented all the numerical algorithms used in the papers.
vii
Contents
Preface
iii
Abstract
v
List of Papers
vii
PART I: INTRODUCTION
1 Background
1
2 Soft Tissues
2.1 Elastic Arteries . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Growth and Remodeling . . . . . . . . . . . . . . . . . . . . .
3
3
5
5
3 Mechanics
3.1 Growth and Remodeling . . . .
3.2 Damage . . . . . . . . . . . . .
3.2.1 Validation of the Model
3.3 Viscosity . . . . . . . . . . . . .
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9
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4 Future Work
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5 Abstract of Appended Papers
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PART II: APPENDED PAPERS
Paper I
33
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2
The basic theorem and its use . . . . . . . . . . . . . . . . . . 36
ix
CONTENTS
3
4
Applications . . . . . . . . . . . . . . . . .
3.1
The rotationally symmetric cylinder
3.2
The rotationally symmetric sphere
Conclusions . . . . . . . . . . . . . . . . .
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39
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Paper II
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
General theory of a residually stressed body . . . . . . . . .
2.1
Geometry . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Balance and constitutive laws . . . . . . . . . . . . .
2.3
Existence of a stress free compatible reference configuration . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Arterial geometry . . . . . . . . . . . . . . . . . . . . . . . .
3.1
An identification problem . . . . . . . . . . . . . . .
3.2
Compatible stress free reference configuration . . . .
4
Riemannian manifold . . . . . . . . . . . . . . . . . . . . . .
4.1
The tensor m as a metric on B0 . . . . . . . . . . .
4.2
Determinants, volume elements and densities . . . . .
5
Boundary value problems . . . . . . . . . . . . . . . . . . . .
5.1
Boundary value problem on B0 with metric m . . .
5.2
Boundary value problem on B0 with metric G . . . .
5.3
Boundary value problem on B with metric γ . . . .
5.4
Incompressibility . . . . . . . . . . . . . . . . . . . .
5.5
Comparison of formulations . . . . . . . . . . . . . .
1
Appendix – Piola Identity . . . . . . . . . . . . . . . . . . .
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79
Paper III
1
Introduction . . . . . . . . . . . . . . . . . .
2
General theory . . . . . . . . . . . . . . . .
2.1
Geometry . . . . . . . . . . . . . . .
2.2
Constitutive law . . . . . . . . . . .
2.3
Equilibrium . . . . . . . . . . . . . .
2.4
General identification problem . . . .
3
The rotationally symmetric case . . . . . . .
3.1
Geometry . . . . . . . . . . . . . . .
3.2
Equilibrium and boundary conditions
3.3
Identification problem . . . . . . . .
4
A numerical example . . . . . . . . . . . . .
4.1
A specific constitutive law . . . . . .
4.2
Results . . . . . . . . . . . . . . . . .
5
Discussion . . . . . . . . . . . . . . . . . . .
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CONTENTS
6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Paper IV
1
Introduction . . . . . . . . . . . . . . . . .
2
General Theory . . . . . . . . . . . . . . .
2.1
Kinematics . . . . . . . . . . . . .
2.2
Balance Equations . . . . . . . . .
2.3
Constitutive Equations . . . . . . .
3
Arterial Application . . . . . . . . . . . .
3.1
Geometry . . . . . . . . . . . . . .
3.2
Equilibrium . . . . . . . . . . . . .
3.3
Growth and Remodeling Equations
3.4
Strain Energy . . . . . . . . . . . .
3.5
Remodeling of the Collagen Fibers
4
Numerical Solution . . . . . . . . . . . . .
5
Application . . . . . . . . . . . . . . . . .
6
Discussion . . . . . . . . . . . . . . . . . .
7
Conclusions . . . . . . . . . . . . . . . . .
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115
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136
Paper V
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2
Mathematical Framework . . . . . . . . . . . . . . . . .
2.1
Kinematics . . . . . . . . . . . . . . . . . . . . .
2.2
Constitutive Relations . . . . . . . . . . . . . . .
2.3
Damage Evolution . . . . . . . . . . . . . . . . .
2.4
The non–Equilibrium Stresses . . . . . . . . . . .
3
The Elasticity Stiffness Tensor . . . . . . . . . . . . . . .
3.1
The Derivative of the 2nd Piola–Kirchhoff Stress .
4
Illustrative Examples . . . . . . . . . . . . . . . . . . . .
5
Discussion and Conclusions . . . . . . . . . . . . . . . .
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143
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xi
Part I
Introduction
1
Background
Biomechanics can be defined as the development and application of mechanics
to solve problems in biology. The main point is that we first observe the
biology and then try to develop a model that represents that behavior. It
may not be possible to determine the beginning of biomechanics; one of the
pioneers was Leonardo da Vinci (1452–1519). He discovered how the valves in
veins made the blood flow in only one direction, from the veins to the heart
and to the arteries. That, together with the assumption of conservation
of mass, led to the conclusion that blood must return from the arteries to
the veins, and therefore circulate within the body. But the development
of biomechanics was slow, due to that the particle mechanics derived by
Galileo and Newton was not suitable to describe the continuous blood flow.
The development of a theory appropriate to the mechanics of a continuous
media, continuum mechanics, began in the early 18th century with Euler and
later with Navier, Cauchy and others. In the late 19th century, experiments
on different specimens indicated that soft tissues do not obey Hooke’s law,
that is their constitutive behavior is not linear. With the non–linearity of
soft tissues, further understanding was delayed until the middle of the 20th
century, when the theory of finite deformations was developed.
More than twenty years ago, laboratory work began to develop and more
and more realistic models began to appear. Even today there are a great
many challenging problems in soft tissue modeling that need more study, for
example residual stress, growth, remodeling and inelasticity. Residual stress
is the stress that is contained in an unloaded tissue. Most tissues grow within
the body and during a lifetime the mechanical properties may change due to
the wellness of that particular person. It may be that some parameters that
describe the tissue are dependent on the person’s age and former illnesses.
Remodeling means that the tissue can change its constitutive behavior due
to some deficiency, for example the heart may remodel itself after a cardiac
infarction and arteries can develop aneurysms due to degradation of the stiff1
CHAPTER 1. BACKGROUND
ness in the arterial wall. Inelastic effects such as damage and viscosity are
also important for understanding the total behavior of the tissue, in particular damage propagation during hypertension.
Cardiovascular diseases are a major health problem. In the United States
of America, for example, they accounted for almost 39% of all deaths in
2001 (American Heart Association, 2003). That is almost 3 times as many as
those caused by cancer, and the British Heart Foundation Health Promotion
Research Group (2000) reports that the percentage is the same in Europe.
In this dissertation different models that include residual stress, growth,
remodeling and damage are developed. We give examples of how to find
the residual stress by optimizing specific material parameters against real
experimental data. We develop a model that lets the tissue grow under load.
If we define a stress free unloaded tissue and then apply pressure and let the
tissue grow and remodel for some time, and then remove the pressure, we
will see that the tissue is no longer stress free when unloaded. In that way
we can say that the residual stress develops during growth and remodeling.
When an artery grows it can also develop artifacts such as aneurysms and it is
believed that they grow during a phase when the wall is weakening due to the
degradation of some constituents. This type of degradation of the elasticity
of the wall is called remodeling, and a theory for this is also presented. If an
aneurysm grows in such a way that it can rupture, it is very critical for the
survival of the patient. Therefore, it is of importance to understand how the
damage propagates through the wall, due to some disease or hypertension,
and where an aneurysm may appear.
2
2
Soft Tissues
Soft tissues are complex materials. Although each type of tissue has unique
behaviors, there are many properties that are common. All organs live in a
pressurized environment which is complex to simulate and most tissues are
comprised of different layers, were each layer has specific properties. For example, large elastic arteries consist mainly of three layers each with different
material properties. Furthermore, visco–elastic and anisotropic behaviors are
not uncommon. If the environment changes, for example due to some kind
of disease, the tissue can grow and remodel itself to optimize its function in
this new environment. All tissues are under pressure in their normal state.
Therefore, is it difficult to obtain accurate data from experiments done in
vitro (in a laboratory rather than in the tissue’s natural setting).
From a mechanical point of view, it is also difficult to find a suitable
configuration that is stress free. It is important to be able to conclude that
stress in the reference set is known or zero. In standard continuum mechanics
it is often assumed that the reference configuration is stress free, but in soft
tissue mechanics it is not generally true that an unloaded configuration is
stress free; we call the stress within an unloaded body residual stress. To be
able to make accurate stress estimations it is important to find this residual
stress or to find a configuration where the body is stress free. One way to
find a stress free configuration is to cut the tissue into many pieces where
each part is assumed to be stress free. This procedure can be described
locally by a deformation, but will in general end up with a non–compatible
configuration, see for example Klarbring et al. (2007).
2.1
Elastic Arteries
Most elastic arteries consist of three layers; the intima (tunica intima), the
media (tunica media) and the adventitia (tunica adventia), see Figure 1. In
3
CHAPTER 2. SOFT TISSUES
healthy young people the innermost layer, the intima, consists of mainly
one layer of endothelial cells. Due to the thinness of the innermost layer,
it is often assumed that the intima does not contribute to the mechanical
properties of the wall. However, the endothelial cells have an indirect effect
on the mechanical properties since they are sensitive to shear stress and can
stimulate the tissue to grow. Even though the intima has no mechanical
properties in young people, it thickens with age and may contribute to the
mechanical properties in older people.
The media is the largest part of the wall (about 67%), and consists of
a complex three dimensional network of smooth muscle cells and a mix of
collagen and elastin fibers (Sonneson et al., 1994). The smooth muscle cells
are concentrically arranged through the arterial wall and its active properties
help to regulate the stiffness of the artery and the the blood flow. It is
often assumed that the media is responsible for the elastic properties of the
artery. The post–natal generation of elastin is mainly developed during the
first two weeks of life and the turnover time is very long. Elastin’s half–life
is best measured in years (Dubick et al., 1981; Humphrey, 2001) whereas
for collagen it is best measured in days (Reinhart et al., 1978; Humphrey,
2001). Due to the slow adaptation of elastin it is believed that it may be
responsible for the development of residual stresses in arteries during post–
natal growth. The media is separated from the intima and adventitia by two
Figure 1: Schematic of the different layers in an elastic artery. The innermost
layer (black) is the intima, the middle layer (grey) is the media, and the
outermost layer (light grey) is the adventitia.
elastic membranes. The mix of smooth muscle cells and the fibers constitute
a helix with a small twist (pitch) (Holzapfel et al., 2000). This arrangement
gives the media great strength and the ability to resist loads in both axial
and circumferential directions.
The outermost layer, the adventitia (about 33%), consists mainly of cells
that produce collagen and elastin. The collagen fibers are arranged in a
helical structure and reinforce the wall. The adventitia is not as stiff at
4
2.2. RESIDUAL STRESS
low pressures as it is at high pressures. This is due to the fact that at low
pressures the collagen fibers are undulated and do not reach its full length,
but when the pressure increases the collagen fibers straighten and begin to
carry load. When all collagen fibers are stretched the adventitia becomes
almost rigid and prevents the artery from rupture.
2.2
Residual Stress
As already mentioned, residual stress is the stress that is left within the
body when all external forces are removed. One of the first discoveries that
unloaded arteries are not stress free was made by Bergel (1960). He wrote:
“When an artery is split open longitudinally it will unroll itself. . .
This surely indicates some degree of stress even when there is no
distending pressure.”
Twenty years later, Chuong and Fung (1986) performed experiments with
arteries from rabbits. They found that when the arteries were cut along
their symmetry axis the arteries opened up. Later, they performed the same
type of experiments with left ventricles and they found the same type of
behavior as with the arteries. This indicates that stress exists in unloaded
soft tissues (not only arteries). It is believed that this residual stress reduces
the stress gradients in the pressurized environment and gives the tissue a more
homogenized stress distribution. In arteries it mainly reduces the tangential
stress at the inner wall, which otherwise would be high. The residual stress
is believed to develop during growth in a pressurized environment. Figure 2
shows how the tangential residual stress is develops from a stress free state.
The left picture shows a hypothetical embryo that is by definition stress free.
That embryo is then pressurized with a luminal pressure at 13.3 kPa and to
homogenize the stresses the growth process begins. After some time, when
the growth has stabilized, the pressure is removed and the picture to the
right with the residual stress is obtained.
2.3
Growth and Remodeling
The mass of a living tissue both increases and decreases with time. The
change of mass is often referred to as growth. Since the tissue occupies a
part of a pressurized body, the growth must take place under the influence of
pressure. During growth the tissue develops residual stresses, which means
that the residual stress field is dependent on time. It may be possible to model
5
CHAPTER 2. SOFT TISSUES
Figure 2: Schematic of the initial (unloaded) stress free configuration (left)
and the corresponding grown configuration (unloaded) with the tangential
component of the residual stress (right).
the residual stress due to growth by assuming a stress free configuration at
some reference time and then letting a strain–like tensor (growth tensor)
evolve with time. This tensor should be defined locally at every material
point, so each part of the tissue can grow independently of other parts. This
unfortunately also implies that a grown unstressed body is not necessarily
compatible. The residual stress is then believed to occur as a result of the
elastic deformation required from the incompatible grown configuration to a
compatible physical configuration, see Figure 4 for a simple sketch and Klisch
et al. (2001), Rodriguez et al. (1994), Skalak et al. (1996), and Taber and
Eggers (1996).
When using continuum mechanics to describe growth one must add that
the mass changes with time, instead of remaining constant. Mass change in
living tissues arises primarily from a change of volume whereas the density
is almost conserved.
The remodeling of soft tissues is also important for their behavior. As
examples of remodeling we have: changes in the heart wall after an infarction, remodeling of the arterial wall stiffness due to illness or changes in
the surrounding environment, changes of the angle between the fibers, and
changes in material constants. Since growth and remodeling are both time–
dependent, we need equations or differential equations (evolution laws) describing the evolution.
As more and more sophisticated models appear that predict the residual
6
2.3. GROWTH AND REMODELING
stress, growth and remodeling, the ability to make a correct stress calculation
will increase. The possibility to estimate the stress may help in decisions
involving surgery and increase the ability to predict ruptures, diseases etc.
7
CHAPTER 2. SOFT TISSUES
8
3
Mechanics
The first step in mechanical modeling is to define a set that represents the
body in a particular configuration. We define a set B0 , as a sub–domain
of the physical space. Further, let χ define a time–dependent deformation
from B0 onto a deformed configuration B that represents the body under a
particular load. We write this as
χ:
B0 × R → B.
Let X = (X 1 , X 2 , X 3 ) denote the coordinates in the reference configuration
B0 and x = (x1 , x2 , x3 ) the coordinates in the physical configuration B. The
deformation in then given by
x = χ(X, t).
Given a deformation, the deformation gradient is calculated as the derivative
of the deformation χ:
F =
∂χ(X, t)
∂x
=
.
∂X
∂X
The deformation gradient maps tangent vectors on B0 to tangent vectors on
B, this is written
F :
T B0 → T B,
where T B and T B0 are the union of all tangent spaces to B and B0 , respectively. For an illustration, see Figure 3.
Now introduce a normal vector n(x) defined on the boundary of physical
configuration B. From Cauchy’s theorem (see, for example Gurtin, 1981)
we know that there exists a tensor σ such that σ times the normal vector is
9
CHAPTER 3. MECHANICS
F
TB
T B0
Figure 3: The deformation gradient and the corresponding tangent spaces.
equal to the traction t applied on the surface at the point x. This is usually
written as
t(n) = σn.
(1)
The tensor σ is more well known as the Cauchy stress tensor. This stress
measure is represented by force per deformed area. Sometimes it is more convenient to define a stress tensor that represent the force per undeformed area.
To define such a tensor we use the deformation gradient and its determinant:
P = (det F )σF −T ,
(2)
where P is known as the first Piola–Kirchhoff stress tensor and the transformation to the undeformed configuration is called a Piola transformation.
For later use we also define the second Piola–Kirchhoff stress tensor as
S = (det F )F −1 σF −T = F −1 P .
(3)
The physical meaning of the second Piola–Kirchhoff stress is more vague
than, for example the Cauchy or the first Piola–Kirchhoff stress respectively.
Sometimes, for example when modeling viscosity, it can be more advantageous to use this kind of stress measure.
Now assume the body is in equilibrium. The forces acting on the body
are of two kinds: traction forces t on the body surface and body forces ρb,
where ρ is the density of the physical body B. The total force acting on the
body in equilibrium is equal to zero and is written
Z
Z
t ds +
ρb dv = 0.
∂B
B
Writing the surface integral as a volume integral by Stoke’s theorem (divergence theorem) using (1) and localizing, we obtain the local form of the
equilibrium equation:
div σ + ρb = 0.
10
(4)
3.1. GROWTH AND REMODELING
Using the Piola transformation (2) we can show that the equilibrium equation
transforms into
Div P + ρ0 b = 0,
where Div is the divergence with respect to the reference coordinates X and
ρ0 = (det F )−1 ρ is the reference density.
If we assume the existence of a strain energy function W dependent on the
elastic deformation gradient F , we can show from the dissipation inequality
(see equation (7) below) that the stress P can be calculated as the derivative
of the strain energy. That is,
P =
∂W
,
∂F
and with (2) we obtain the Cauchy stress as
σ = (det F )−1
∂W T
F .
∂F
A material where a strain energy can be defined is called hyper–elastic.
Hyper–elastic materials are a very large class of materials and this is the
most common way (by a large margin) to model elastic phenomena.
3.1
Growth and Remodeling
To determine what kind of equations may drive growth and remodeling we
again use the dissipation inequality. This law says, in words, that the sum
of the internal power and the change of energy is always less than or equal
to zero. We take the total deformation gradient F from T B0 onto the T B
to be the composition of an elastic part F e and a growth part G. That is,
F = F e G. An illustration is presented in Figure 4. Note that the growth
tensor maps vectors in T B0 to vectors in T B0 .
We also introduce a strain energy W as an isotropic scalar valued function
dependent only on the elastic deformation F e and remodeling (material)
parameters mα . The energy per unit volume in the reference configuration
B0 is defined as
Z
(5)
Ψ=
(det G)W(F e , mα ) dV0 .
B0
To define the internal power we need to choose what kind of velocity fields
we want to use. Of course we have the deformation rate Ḟ , but we are also
11
CHAPTER 3. MECHANICS
F
T B0
TB
G
Fe
T B0
Figure 4: The deformation, growth, and the corresponding tangent spaces.
interested in growth and remodeling. Introducing the growth rate Ġ and the
remodeling rates ṁα we can define the internal power as
Z Pi = −
(6)
P : Ḟ + Y : Ġ + M α ṁα dV0 ,
B0
where P , Y and M α are forces to balance the corresponding rates. Note that
Y and M α are material forces (configurational forces), for example they are
related to the change in the structure or properties of the body. The colon
( : ) should be interpreted as a double contraction, and, as is customary, a
repeated index means summation over that index. Now, if we require that
the dissipation inequality,
Ψ̇ + Pi ≤ 0,
(7)
must hold for all subsets of B0 , it can be localized. Standard arguments now
give that the force P can be explicitly expressed as
P = (det G)
∂W
∂F
(8)
and we obtain the reduced dissipation inequality
∂W
−1
T
α
−
M
(det G)
α ṁ + (det G)G W − Y − F e P Ġ ≤ 0. (9)
∂mα
12
3.2. DAMAGE
We note that the force P in (8) is in fact the first Piola–Kirchhoff stress
tensor. To be able to satisfy the reduced dissipation inequality (9) the material is treated as a generalized standard material (Halphen and Nguyen,
1975; Moreau, 1974) and a convex potential function ϕ is introduced. The
resulting evolution equations are given by the following system of ordinary
differential equations
∂ϕ
= F Te P − (det G)G−T W + Y
∂ Ġ
∂ϕ
∂W
= Mα − (det G)
,
∂ ṁα
∂mα
(10)
(11)
The simplest possible potential that can be chosen and at the same time
guarantees that the thermodynamics are satisfied is the quadratic function
ϕ=
c0
c1
Ġ : Ġ + ṁα ṁα ,
2
2
c0 , c1 > 0.
(12)
In arteries it is believed that the transmural stress distribution is almost
constant. In Figure 5 it is shown how we can use the growth evolution to
drive the stress to a more homogenized state. The growth is initiated after a
luminal pressure at 13.3 kPa is applied. The tissue is then allowed to grow.
The almost constant stress distribution is obtained by choosing a driving
force Y as
Y = (det G)G−1 W − F e P ∗ ,
(13)
where P ∗ is a given homeostatic stress. The equations (10) and (11) together
with (12) and (13) become
c0 Ġ = F Te (P − P ∗ )
c1 ṁα = Mα − (det G)
(14)
∂W
,
∂mα
(15)
The evolution law (14) is such that the tissue grows (increases and decreases
its mass) until the stress P is equal to the given homeostatic stress P ∗ .
3.2
Damage
As discussed in previous sections, the zero stress state for soft tissues is not
the same as the unloaded state and residual stresses are important for the
tissue to function in its natural environment. But when the tissue is stretched
way beyond its normal working range, the residual stress is small compared
13
CHAPTER 3. MECHANICS
Figure 5: Illustration of how growth can influence the stress distribution.
The tissue is initiated with a inner pressure at 13.3 kPa and the tissue is
then allowed to change its mass to homogenizes the stress distribution.
with the stress within this overstretched tissue, see for example Fung (1993).
In this section we neglect the residual stresses and model the degradation of
a tissue due solely to very large strains.
When modeling damage one usually multiplies the strain energy by a
smooth function g with a range in the interval 1 to 0, where 1 signifies
undamaged material and 0 total failure or rupture, see Lemaitre (1992). The
damaged strain energy is denoted W d and is written
W d = g(δ ∗)W
(16)
where δ ∗ is some kind of strain measure and W is the undamaged strain
energy. For example, when modeling damage of fibers, δ ∗ may be the stretch
of the fibers. If we use the damaged strain energy in the dissipation inequality,
we see by using (5) and (7) that if we require that
∂g ∗
δ̇ ≤ 0,
∂δ ∗
(17)
together with the evolution of the growth and the remodeling, the dissipation inequality is always satisfied. This means that if the strain measure δ ∗
increases, the damage function g(δ ∗ ) must decrease and vice versa.
The damage function g(δ ∗ ) can be described by an evolution law in the
same way as growth and remodeling, see equations (14) and (15). However, we will not use that approach, instead we explicitly define the damage
14
3.2. DAMAGE
function as a monotonically decreasing function with a few parameters. In
Natali et al. (2003) and Natali et al. (2005) it is suggested and shown that for
a transversely isotropic material, e.g., tendons and ligaments, the following
functional form gives satisfactory results
+
1 − eγ(δ −δ )
1 − eγ(δ− −δ+ )
∗
g(δ ∗) =
(18)
where γ is a parameter that sets the characteristic of the damage and, δ + and
δ − describe at what strain the tissue fails and when the damage is initiated.
This means that for strains less than δ − , the tissue is undamaged (g = 1),
and for strains equal to δ + , the tissue ruptures (g = 0). Figure 6 showns how
the damage function depends on the parameter γ. For small absolute values
the damage is close to linear and for large positive values the damage is very
slow at the beginning, but as the strain δ ∗ tends to the limit δ + , the damage
function goes rapidly to zero and the tissue fails. For negative values of γ
the damage propagates rapidly close to the initial strain δ − and slows down
when the strain gets close to the rupture state δ + .
1
γ >0
0.9
0.8
0.7
g(δ*)
0.6
γ=0
0.5
0.4
0.3
0.2
0.1
0
1.1
γ <0
1.12
1.14
δ*
1.16
1.18
1.2
Figure 6: A plot of how the damage function g(δ ∗ ) depends on the parameter
γ. The initial strain is δ − = 1.1 and the maximum strain is taken to be
δ + = 1.2
3.2.1
Validation of the Model
This section is taken from an upcoming conference paper. It is a continuation
of Paper V and for simplicity the notation is as therein, and may differ from
that is used in the rest of this introduction.
15
CHAPTER 3. MECHANICS
To test the damage theory against experimental data, an extension test
was performed on a porcine ligament and the resulting stress–strain curve was
used to find the material and damage parameters. Ligaments are treated as
a transversely isotropic material since their anisotropic behavior is mainly
due to one fiber family, aligned along the ligament.
In Paper V we have chosen to use a strain energy that is a slight modification of the model presented in Natali et al. (2005) and Natali et al. (2003),
it is composed of an isotropic part and another that is anisotropic. The
isotropic contribution is given by a standard Mooney–Rivlin material with
two parameters, c1 and c2 . The fibrous part is constructed as an exponential
function with two parameters, c3 and c4 . Most tissues include a high percentage of water and can therefore be treated as almost incompressible. The
incompressibility condition is obtained by penalization. To do this we add
a volumetric part to the strain energy, and it is customary to use one that
tends to infinity (+∞) as the volume ratio J = det F approaches zero (see
Paper V). The total strain energy is given by
Ψ = g1 (δ1∗ )(Ψ0 (J) + Ψ1 (I 1 , I 2 )) + g2 (δ2∗ )Ψ2 (I 4 )
(19)
where I 1 , I 2 and I 4 are deformation invariants (see Paper V), and g1 and
g2 are damage functions with the corresponding damage variables δ1∗ and
δ2∗ , respectively. The material parameters are included in the volumetric,
isotropic, and anisotropic parts and the explicit expression for those are
J2 − 1
− ln J)
2
Ψ1 = c1 (I 1 − 3) + c2 (I 2 − 3)
c3
Ψ2 = (ec4 (I 4 −1) + c4 (I 4 − 1) − 1)
c4
Ψ 0 = c0 (
(20)
(21)
(22)
where c0 is a penalization constant, c1 and c2 describe the elasticity of the
non fibrous part of the tissue and c3 is the resistance in the undulated fibers,
and c4 represents the strength of the stretched fibers. The parameters are
obtained by using a Nelder–Mead method (using the fminsearch function in
MATLAB) for solving a non–linear minimization problem in the least square
sense.
The damage is here modeled by two damage functions one for the strain
energies Ψ0 and Ψ1 and another for Ψ2 , see equation (19). This means
that we need to determine 6 damage parameters (10 parameters in total).
For simplicity we have taken the initial damage values δ1− and δ2− from the
experimental data set. This gives a total of 8 parameters. The result of the
parameter fit is given in Figure 7 and it shows that the model is capable of
capturing the behavior of this kind of tissue (ligament).
16
3.3. VISCOSITY
6
5
Elastic Parameters:
c = 6.877e+4
1
c = 1.209e+4
2
c = 6.078e+4
3
c = 4.355
Nominal stress [MPa]
4
4
3
Damage Paramters:
γ = 7.300e−2
1
γ2 = −4.704e−1
δ − = 1.316 (exp.)
1
δ + = 1.631
1
δ − = 1.250 (exp.)
2
δ + = 1.631
2
2
1
0
1
1.1
1.2
1.3
Strain [−]
1.4
1.5
1.6
Figure 7: Result after optimization of the damage model used on a porcine
ligament. Two damage functions are used: one for the isotropic part and one
for the anisotropic part respectively.
3.3
Viscosity
A viscous material is such that the current stress depends on the evolution
of the deformation. The stress is rate–dependent. The total stress is usually
divided into a volumetric part, an elastic part, and a viscous part. When
modeling viscosity it is preferable to use the second Piola–Kirchhoff stress
measure. The total stress is written:
S = S vol + S e + H,
(23)
where S vol is the stress solely due to volumetric changes, S e is the stress from
the pure elastic process, and H is a second Piola–Kirchhoff–like stress tensor
describing the viscous stress (non–equilibrium stress). The viscous effects
are assumed to follow the behavior of a generalized Maxwell element (see for
example Simo, 1987; Holzapfel and Gasser, 2001) and the non–elastic stress
H is described by the following ordinary differential equation
1
Ḣ + H = β Ṡ e
(24)
τ
where τ is the relaxation time and β is a free energy factor associated with
the relaxation time (see, Holzapfel and Gasser, 2001). Equation (24) is easily
17
CHAPTER 3. MECHANICS
integrated and the result can be written using convolution:
Z T
H = H 0 e−T /τ +
Ṡ e βe(t−T )/τ dt
(25)
0
where H 0 is the non–elastic stress at time t = 0, here taken to be zero
(no initial viscosity). To solve (25) we use the following recursive algorithm,
derived from the midpoint rule (Simo, 1987), that has proved to converge
rapidly:
H n+1 = H n e−∆t/τ + βτ
1 − e−∆t/τ
− S ne ,
S n+1
e
∆t
(26)
where n + 1 is the current iteration and ∆t is the time increment.
To obtain the true stress (Cauchy stress) we solve equation (7) for σ and
the Cauchy stress is given by
σ = (det F )F −1 SF −T .
18
4
Future Work
The development of realistic models for soft tissues has just begun and much
more work needs to be done in this field. So far, most models treat the tissues
as consisting of passive materials, but recent research has shown that the
cells that make up the tissue are highly active and can regulate growth and
remodeling of the tissue. For example, the endothelial cells in the intima react
on the concentration of Ca2+ ions and can send signals through the tissue
to stimulate growth. If we can find out how the cells react on mechanical
and chemical stimuli, we may be able to understand the complex behavior
of cardiovascular diseases, for example aneurysms. This challenging field is a
union of many different fields. To be successful, a close collaboration between
biologists, chemists, physicians and mechanicans is needed.
Another interesting area is to build a finite element model of a portion
of an artery and simulate the blood flow. By using the blood shear stress as
boundary conditions on the endothelial cells, we may be able to simulate a
realistic growth and remodeling response.
The damage theory presented in this thesis is not stable when the stress
reaches the point where the tangent stiffness is zero (the maximum stress
or turning point on the curve). Obtaining a more stable method that can
simulate rupture should be of interest.
19
CHAPTER 4. FUTURE WORK
20
5
Abstract of Appended Papers
I
On Compatible Strain with Reference to
Biomechanics
In previous studies, residual stresses and strains in soft tissues have been
experimentally investigated by cutting the material into pieces that are assumed to become stress free. The present paper gives a theoretical basis
for such a procedure, based on a classical theorem of continuum mechanics.
As applications of the theory we study rotationally symmetric cylinders and
spheres. A computer algebra system is used to state and solve differential
equations that define compatible strain distributions. A mapping previously
used in constructing a mathematical theory for the mechanical behavior of
arteries is recovered as a corollary of the theory, but is found not to be unique.
It is also found, for a certain residual strain distribution, that a sphere can
be cut from pole to pole to form a stress and strain free configuration.
II
Theory of Residual Stresses with Application
to an Arterial Geometry
This paper presents a theory of residual stresses, with applications to biomechanics, especially to arteries. For a hyper–elastic material, we use an initial
local deformation tensor K as a descriptor of residual strain. This tensor, in
general, is not the gradient of a global deformation, and a stress free reference
configuration, denoted B, therefore, becomes incompatible. Any compatible
reference configuration B0 will, in general, be residually stressed. However,
21
CHAPTER 5. ABSTRACT OF APPENDED PAPERS
when a certain curvature tensor vanishes, it does actually exist a compatible
and stress free configuration, and we show that the traditional treatment of
residual stresses in arteries, using the opening–angle method, relates to such
a situation.
Boundary value problems of non-linear elasticity are preferably formulated on a fixed integration domain. For residually stressed bodies, three
such formulations naturally appear: (i) a formulation relating to B0 with a
non-Euclidean metric structure; (ii) a formulation relating to B0 with a Euclidean metric structure; and (iii) a formulation relating to the incompatible
configuration B. We state these formulations, show that (i) and (ii) coincide
in the incompressible case, and that an extra term appears in a formulation
on B, due to the incompatibility.
III
Modeling Initial Strain Distribution in Soft
Tissues with Application to Arteries
A general theory for computing and identifying the stress field in a residually stressed tissue is presented in this paper. The theory is based on the
assumption that a stress free state is obtained by letting each point deform
independently of its adjacent points. This local unloading represents an initial strain, and can be described by a tangent map. When experimental data
is at hand in a specific situation the initial strain field may be identified by
stating a non linear minimization problem where this data is fitted to its
corresponding model response. To illustrate the potential of such a method
for identifying initial strain fields, the application to an in vivo pressure–
radius measurement for a human aorta is presented. The result shows that
the initial strain is inconsistent with the strain obtained with the opening–
angle–method. This indicates in this case that the opening-angle-method has
a too restrictive residual strain parametrization.
IV
Residual Stresses in Soft Tissue as a Consequence of Growth and Remodeling
We develop a thermodynamically consistent model for growth and remodeling
in elastic arteries. The model is specialized to a cylindrical geometry, strain
22
energy of the Holzapfel–Gasser–Ogden type and remodeling of the collagen
fiber angle. A numerical method for calculating the evolution of the adaption
process is developed. For a particular choice of the thermodynamic forces of
growth and remodeling (configurational forces), it is shown that an almost
homogeneous transmural axial and tangential stress distribution is obtained.
Residual stresses develop during this adaption process and these stresses
resemble what is found by the widely used opening-angle model.
V
Modeling of Passive Behavior of Soft
Tissues Including Viscosity and Damage
This article describes a continuum damage model for anisotropic soft tissues.
The model is developed with the underlying framework of hyper-elasticity.
As usual, the corresponding strain energy is additively split into a volumetric
part and a volume–preserving part. the damage of the tissue involves both
isotropic and anisotropic contributions. The viscous properties of the tissue
are modeled by a generalized linear standard solid with a finite number of
Maxwell elements, which allows for the approximation of frequency independent responses. The results are obtained with the commercial FE software
ABAQUS and are in agreement with other studies done by different authors
in the field.
23
CHAPTER 5. ABSTRACT OF APPENDED PAPERS
24
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