Soft Tissue Mechanics with Emphasis on Residual Stress Modeling

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Link¨oping 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, Link¨oping University SE–581 83, Link¨oping, Sweden

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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

Link¨oping University

SE–581 83 Link¨oping, Sweden ISBN 978–91–85715–50–3 ISSN 0345–7524

Distributed by:

Institute of Technology, Link¨oping University Department of Management and Engineering SE–581 83, Link¨oping, Sweden

c

2007 Tobias Olsson

This document was prepared with LA_{TEX, February 27, 2007}

No part of this publication may be reproduced, stored in a retrieval system, or be transmitted, in any form or by any means, electronic, mechanical, pho-tocopying, recording, or otherwise, without prior permission of the author.

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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 sup-porting 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 St˚alhand, 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.

Link¨oping, February 2007.

Tobias Olsson

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“Mistakes are the portals of discovery.”

—Joyce, James (1882–1941)—

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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.

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To my mother and father, the best parents one could wish for.

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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 Ref-erence to Biomechanics, Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 85, 440–448, 2005.

II Anders Klarbring, Tobias Olsson and Jonas St˚alhand, Theory of Resid-ual Stresses with Application to an Arterial Geometry, Submitted for publication 2007.

III Tobias Olsson, Jonas St˚alhand and Anders Klarbring, Modeling Ini-tial 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 publica-tion 2007.

V Tobias Olsson and Jo˜ao 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 al-gorithms used in the papers.

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Part I

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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 stiff-1

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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 partic-ular 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.

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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 ex-ample, 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

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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

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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

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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 infarc-tion, 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) de-scribing the evolution.

As more and more sophisticated models appear that predict the residual 6

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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.

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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 = (X1_{, X}2_{, X}3_{) denote the coordinates in the reference configuration}

B₀_{and x = (x}1_{, x}2_{, x}3_{) 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 B0to 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,

re-spectively. 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

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CHAPTER 3. MECHANICS

F

T B0 T B

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 con-venient 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 trans-formation to the undeformed configuration is called a Piola transtrans-formation. 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 advanta-geous 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 ∂B tds + Z B ρb dv = 0.

Writing the surface integral as a volume integral by Stoke’s theorem (di-vergence theorem) using (1) and localizing, we obtain the local form of the equilibrium equation:

div σ + ρb = 0. (4)

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3.1. GROWTH AND REMODELING

Using the Piola transformation (2) we can show that the equilibrium equation transforms into

Div P + ρ0b= 0,

where Div is the divergence with respect to the reference coordinates X and ρ0= (det F )−1ρ 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

∂F F

T_.

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 B0onto the T B

to be the composition of an elastic part Feand a growth part G. That is,

F = FeG. An illustration is presented in Figure 4. Note that the growth

tensor maps vectors in T B0to vectors in T B0.

We also introduce a strain energy W as an isotropic scalar valued function dependent only on the elastic deformation Fe and remodeling (material)

parameters mα_{. The energy per unit volume in the reference configuration}

B₀_{is defined as} Ψ =

Z

B0

(det G)W(Fe, mα) dV0. (5)

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 ˙F, but we are also 11

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F

T B0

T B0 T B

G Fe

Figure 4: The deformation, growth, and the corresponding tangent spaces. interested in growth and remodeling. Introducing the growth rate ˙Gand the remodeling rates ˙mα_{we can define the internal power as}

Pi= − Z B0 P: ˙F+ Y : ˙G+ Mα_m_˙ α dV0, (6)

where P , Y and Mα_{are forces to balance the corresponding rates. Note that}

Y and Mα_{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,

˙

Ψ + 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 (det G)∂W ∂mα− Mα ˙ mα_{+ (det G)G}−₁ W − Y − FT eP ˙G≤ 0. (9) 12

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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 ma-terial is treated as a generalized standard mama-terial (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 ∂ ˙G = F T eP− (det G)G −_T W + Y (10) ∂ϕ ∂ ˙mα = Mα− (det G) ∂W ∂mα, (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 2G˙ : ˙G+ c1 2m˙ α_m_˙ α, 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 − FeP ∗ , (13) where P∗

is a given homeostatic stress. The equations (10) and (11) together with (12) and (13) become

c0G˙ = FTe(P − P ∗

) (14)

c1m˙α= Mα− (det G)∂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

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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 Wd _{and is written}

Wd_{= 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 ∂δ∗˙δ

∗

≤ 0, (17)

together with the evolution of the growth and the remodeling, the dissipa-tion 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). How-ever, we will not use that approach, instead we explicitly define the damage 14

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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

g(δ∗

) = 1 − e

γ(δ∗−_δ+₎

1 − eγ(δ−−_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+_.

1.1 1.12 1.14 1.16 1.18 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ* g( δ *) γ >0 γ <0 γ=0

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.

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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 modifi-cation 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, c1and c2. The fibrous part is constructed as an exponential

function with two parameters, c3 and c4. Most tissues include a high

per-centage 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(δ ∗

1)(Ψ0(J) + Ψ1(I1, I2)) + g2(δ ∗

2)Ψ2(I4) (19)

where I1, I2 and I4 are deformation invariants (see Paper V), and g1 and

g2 are damage functions with the corresponding damage variables δ∗1 and

δ∗

2, respectively. The material parameters are included in the volumetric,

isotropic, and anisotropic parts and the explicit expression for those are Ψ0= c0( J2_{− 1} 2 − ln J) (20) Ψ1= c1(I1− 3) + c2(I2− 3) (21) Ψ2= c3 c4 (ec4(I4−1)_{+ c} 4(I4− 1) − 1) (22)

where c0 is a penalization constant, c1 and c2 describe the elasticity of the

non fibrous part of the tissue and c3is 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 δ−

1 and δ −

2 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).

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3.3. VISCOSITY 1 1.1 1.2 1.3 1.4 1.5 1.6 0 1 2 3 4 5 6 Strain [−]

Nominal stress [MPa]

Elastic Parameters: c 1 = 6.877e+4 c 2 = 1.209e+4 c 3 = 6.078e+4 c 4 = 4.355 Damage Paramters: γ₁ = 7.300e−2 γ₂ = −4.704e−1 δ₁− = 1.316 (exp.) δ₁+ = 1.631 δ₂− = 1.250 (exp.) δ₂+ = 1.631

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= Svol+ Se+ H, (23)

where Svolis the stress solely due to volumetric changes, Seis 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

˙ H+1

τH= β ˙Se (24)

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

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integrated and the result can be written using convolution: H= H0e−T /τ₊Z T

0

˙

Seβe(t−T )/τ dt (25)

where H0 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: Hn+1= Hn_e−∆t/τ_{+ βτ}1 − e −_∆t/τ ∆t S n+1 e − Sne , (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.

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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.

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CHAPTER 4. FUTURE WORK

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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 as-sumed 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 biome-chanics, 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,

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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 formu-lated 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 B0with a

Eu-clidean 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 residu-ally 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 ini-tial 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

Conse-quence 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

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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 indepen-dent 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.

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CHAPTER 5. ABSTRACT OF APPENDED PAPERS

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