Numerical simulation of failure response of vascular tissue due to deep penetration

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Numerical simulation of failure response of vascular tissue due to deep penetration

Caroline Forsell

Licentiate thesis no. 110, 2011 KTH School of Engineering Sciences

Department of Solid Mechanics Royal Institute of Technology SE-100 44 Stockholm Sweden

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TRITA HLF-0500 ISSN 1654-1472

ISRN KTH/HLF/R-11/02-SE

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The important thing is not to stop questioning. Curiosity has its own reason for existing.

Albert Einstein

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Preface

The work presented in this thesis has been carried out at the department of Solid Mechanics, Royal Institute of Technology (KTH), Stockholm between January 2008 and December 2010.

The work has been financially supported by the Project Grant No. 2007-4514 from the Swedish Research Council, witch is gratefully acknowledged.

First of all I would like to express my sincere gratitude to my supervisor T.Christian Gasser for giving me the opportunity to be a part of this project. I am thankful for all his support and encouragement and guidance during this project.

I would also like to thank Martin ¨Oberg for his help in the laboratory and Per Berg for his help with the cluster for the simulations.

I would also like to thank my colleagues and friends at KTH solid Mechanics for making this a great place to work especially thanks to my room mate Irene for our nice discussions private and professional.

Last but not least I would like to thank my family and friends for their support and love.

Especially thanks to my mum Yvonne for inspiring me to start with research and to my love in life Alejandro for his encouragement and love.

Stockholm, January 2011 Caroline Forsell

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List of appended papers

Paper A: Numerical simulation of the failure of ventricular tissue due to deep penetration:

The impact of constitutive properties Caroline Forsell and T.Christian Gasser Journal of Biomechanics 44 (2011) 45–51.

Paper B: On the numerical implementation of invariant-based viscoelastic formulations at finite strains. An anisotropic model for the passive myocardium.

T.Christian Gasser and Caroline Forsell

Report 499, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of Technology, Stockholm, Sweden

In addition to the appended papers, the work has resulted in the following conference contri- butions:

Numerical Simulation of Deep Penetration of Ventricular Tissue C. Forsell, T.C. Gasser, P. Gudmundson and G. Dohr

WCCM8 and ECCOMAS,Venice, Italy, June 30 - July 5, 2008.

Modeling of vascular failure with application to myocardial failure due to deep penetration

C. Forsell and T.C. Gasser

10th US National Congress on Computational Mechanics, Columbus, Ohio, US, July 16-19, 2009.

Modeling of myocardial splitting due to deep penetration C. Forsell and T.C. Gasser

ECCMR 2009 - 6th European Conference on Constitutive Models for Rubber, Dresden, Ger- many, 7 - 10 September 2009.

Experimental and modeling of myocardial splitting C. Forsell and T.C. Gasser

ECCM 2010 - IV European Conference on Computational Mechanics, Paris, France, May 16-21, 2010.

The impact of constitutive properties on myocardial tissue perforation C. Forsell and T.C. Gasser

WCB 2010 - 6th World Congress on Biomechanics, Singapore, Singapore, August 1-6, 2010.

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Introduction

Heart structure

The heart is a rhythmic contracting organ that consists of a thick muscular vascular system.

It has a size, for humans, of about 12x9x6 cm and it contains four chambers; the right and left atrium and the left and right ventricle, see Figure 1. The heart pumps blood through the body carrying nutrients, wastes and oxygen. It consists of three layers; endocardium, myocardium and epicardium, epicardium is the most external and endocardium the most internal layer, Bloom and Fawcett (1970).

The myocardium is the thickest layer and consist of myocardial fibers bound to connective tissue, myocytes and fiberoblasts, Humphrey (2002). The mycocyters are embedded with an extracellular matrix consisting of type I and III collagen, and the myocytes are arranged in parallel muscle fibers, Humphrey (2002). Elastic fibers are forming large networks in the atria of the myocardium, Bloom and Fawcett (1970) and a predominant fiber direction can be seen. This direction is nearly parallel to the epicardial surface (it differs around 3 − 5 degrees), Humphrey (2002). In Vitro tests suggest that the myocardium is anisotropic, non linear, nearly elastic and possibly heterogeneous in regions, Okamoto et al. (2000).

The deformation of the heart wall, as well as all soft tissue, is complex due to that the material is inhomogeneous nonlinear, anisotropic, elastic and has viscous behavior, Abolhassani et al.

(2007). We are also dealing with large deformations and this can cause geometric nonlinear- ities, Zienkiewicz and Taylor (1989).

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Numerical simulation of failure response of vascular tissue due to deep penetration

A pulse generator

A pulse generator (defibrillator or pacemaker) can be used to regulate cardiac arrhythmias. A pacemaker is needed if the heart beats to slow because this can cause damage to the cardiac system. If there is a dysfunction in the heart electrical system the pacemaker can listen to the heart activity and send electrical signals, when necessary. The pacemaker consists of a box that is put under the skin close to the clavicle that is connected to leads that conduct the electrical pulses to the heart. The leads are put inside the heart chamber close to the wall, St.Jude (2009), see Figure 1. If the patients heart beat too fast, or irregularly, a cardiverter defibrillator can be needed. This devise stimulates the heart to contract and relax properly, St.Jude (2006).

Figure 1: Illustration of how the pacemaker leads are put close to the heart wall and of the four chambers of the heart St.Jude (2008).

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Lead perforation

A lead perforation has occurred if the lead from the pulse generator has penetrated the heart wall. Lead perforation can be acute, subacute, or delayed, depending on the time from the operation to the event, Alla et al. (2010). Lead perforation is a rare, but serious, complication of pacemaker implantation with numerous case reports and case series known, 0.1%−0.8% for pacemaker leads and 0.6 − 5.2% for impartable defibrillator leads, Khan et al. (2005). Factors that have been reported to cause lead perforation are: patient characteristics, concomitant therapies such as steroids and anticoagulants, implant techniques, and design characteristics of the lead, Carlson et al. (2008). A perforation-resistant lead design requires detailed under- standing regarding the mechanical failure of the ventricular wall, which however, is not yet available in the literature, Khan et al. (2005).

Constitutive models

Constitutive modeling of vascular tissue is an active field of research and numerous models have been constructed. In the biological cells, tissues and individual organisms are orga- nized from molecules and atoms. In these systems it is convenient to consider a material as a continuum. A continuum is an isomorphism or a real number system in a three di- mensional Euclidean space. In this approach you can consider the mechanical properties of the system by a constitutive relation, Fung (1981). Constitutive relations is a mathematical relation that describes how a material behaves under loading under certain conditions. The differentiation of a strain energy function describes the relationship between stress and strain.

Common models that have been used were models for rubber like materials, thus describing the anisotropy; Mooney - Rivling, Ogden and Neo-hookean model, Humphrey and Delange (2004).

For vascular tissue we have for example a description for arterial wall where two layers are described, Mooney- Rivling is used for the innermost layer and Money-Rivling model plus a double helically arranged fibers four the outer layer, Holzapfel et al. (2000). For the ventricular wall, Hunter et al. (1998), Humphrey et al. (1990a), Yang and Taber (1991), have been

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reported, describing different strain energy functions. The most comprehensive (reliable) models integrate data from biaxial in-vitro testing, an approach also supposed to describe the passive properties of myocardial tissue, Humphrey et al. (1990a,b).

Vascular tissue is a multiphasic material with a large amount of not particularly mobile fluid, which makes it incompressible under many loading conditions. A particular challenge of the ventricular wall arises where as the passive heart wall can be regarded as incompressible.

Vascular tissue show different types of inelastic phenomena, where specifically viscoelastic- related phenomena are experimentally observed, Humphrey (2002). Different modeling assumptions accounting for inelastic phenomena of vascular tissue have been suggested, where specifically viscoelastic modeling was fostered, Fung (1981).

Rheological models are often used to describe viscoelastic behavior of materials. Some common used models are Maxwell model, the Voigt model and the Kelvin model. These models consists of linear springs and dashpots in different combination. A linear spring produce an instant deformation proportional to the load while a dashpot produce a velocity proportional to the load at any instant, Fung (1981). A Maxwell body consist of a linear spring and a linear dashpot. This means that the force in the spring relate linearly to its extension and the force in the dashpot relates linearly to its rate of extension. The total force in the Maxwell element is the total force from the dashpot and the spring, Fung (1981).

Failure mechanisms of vascular tissue

Deep penetration of soft solids by a punch is for example piercing of mammal skin or failure of rubber tyres by penetration of a foreign body such as a nail, Shergold and Fleck (2004).

Most articles about deep penetration seem to deal with skin penetration. Shergold and Fleck (2005) has used a phenomenological model where both flat-bottom punches and sharp-tipped punches were studied. It could be seen that in the flat-bottom punch a mode-II ring crack formed and propagates a head of the tip of the penetrator. The pressure for the flat-bottomed case was two to three times higher than for the sharp-tipped case (the toughness has to be

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equal in the flat-bottom and the sharp-tipped case), Shergold and Fleck (2004). The sharp tip on the other hand was found to form a planar mode-I crack, ahead of the tip where the crack faces wedged open by the advancing punch, Shergold and Fleck (2005). Literature on skin injection and literature about rubber penetrations indicate that sharp-tip punches crack and after the penetrator has been removed it is followed by substantial reversible deformation.

Not many studies investigate the differences in geometry of crack depending on the geometry of the penetrator, Shergold and Fleck (2005). Experiments have later been done and found that flat bottom punches create a mode-I failure, Gasser et al. (2009). See Figure 2 for picture of the penetrator site mode-I failure and flat bottom punches.

Figure 2: Electron microscopical image taken at the entrance of the punch illustration mode-I failure (splitting) Gasser et al. (2009)

The diameter, of the needle also effects the force. Studies reported an increasing of the force with larger diameter and also when the needle tip changed from triangular to bevel and from bevel to cone, Okamura et al. (2004).

Objective

The aim of this study is to identify, quantify and model failure in vascular tissue under different mechanical loading situations towards optimizing pacemaker leads. Analytical work aims at describing vascular tissue failure within the concept of finite strain continuum mechanics, which is guided/validated by tailored experimental testing. Numerical descriptions of devel-

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oped failure models are derived and implemented in Finite Element (FE) environments to address more challenging analysis and facilitate inverse material parameter estimation.

Objective 1: A non-linear FE model was developed that integrates constitutive data pub- lished in the literature with information from in-vitro tensile testing. One main objective is to investigate the impact of different constitutive parameters on the simulated results. Ob- jective 2: A conceptional framework for finite strain photoelasticity thought to be suitable to capture salient features of a class of passive soft biological tissues like the myocardium was developed. A linearization around the current configuration coincides with earlier suggested viscoelastic models and a linearized viscoelastic model has been implemented for a particular anisotropic constitutive model for the passive myocardium.

Experimental study design

The present study investigated the elastic and inelastic properties of myocardial tissue in cross-fiber direction. Experimental design and results are detailed in Section paper 1 and a summary is shown in Figure 3.

Figure 3: Whisker box plots (Boxes: upper and lower quartile. Bars: max/min values) rep- resenting the cross-fibre strength of myocardial tissue. The number of investigated test specimens is denoted by the number below the plots.

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Numerical modeling

Computational models have been developed and detailed in paper 1 and paper 2. The models aim at investigating the elastic and inelastic response of myocardial tissue with respect to pacing lead contact. To this end the Finite element packages FEAP (University of California at Berkley) and ABAQUS (Dassault Systems) were used and material models through the user-model-interfaces implemented.

Result

Numerical predictions due the pacing lead contact are derived that included failure due to deep penetration, see Figures 4 and 5.

Figure 4: Numerically-predicted crack-tip deformation of myocardial tissue penetration. Re- sults are achieved with Elastic Set I, Prony Set I and Fracture Set II (see Paper A). Images illustrate a view on the symmetry plane (see Fig.6, Paper A) at Peak load (left) and after the drop of the penetration force (right). Color code denotes the Cauchy stress in MPa in direction perpendicular to the punch and parallel to the symmetry plane.

Discussion

The applicability of the modeling assumptions and derived results are discussed, where particularly their limitations are underlined, see section 4, paper 1 and section 6, paper 2.

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1 1.05 1.1 1.15 1.2 1.25

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Stretch

Punch Force [N]

Anisotropic model Isotropic model

(a) (b) (c) (d)

Figure 5: Elastic response of a punch pushing down on a disk of porcine myocardial tissue.

(a) Punch force displacement response using the anisotropic (eq.(32); solid line) and the isotropic (eq.(33);dashed line) constitutive description for myocardial tissue. (b) and (c) Cauchy stress in fiber and cross-fiber directions as predicted by the anisotropic constitutive model (32). (d) Cauchy stress in fiber and cross-fiber directions as predicted by the isotropic constitutive model (33).(see Paper B for references)

Conclusion

A mixed experimental numerical approach is required to explore elastic and inelastic properties, which in turn may help to design penetration resistant pacing lead tips. Numerical modeling plays a key role in that process as it is helpful to interpret and further understand data derived from experimental investigations.

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Summary of appended papers

Paper A: Numerical simulation of the failure of ventricular tissue due to deep penetration:

The impact of constitutive properties

In this paper the myocardial failure due to deep penetration by an advancing rigid punch was investigated. A non-linear FE model was developed that integrates constitutive data pub- lished in the literature with information from in-vitro tensile testing in cross-fibre direction of porcine myocardial tissue. The Finite element model considered the non-linear, isotropic and visco-elastic properties of the myocardium. The Failure model is phenomenologically described by a Traction Separation Law. In-vitro penetration testing of porcine myocardium was used to validate the Finite Element model, and a particular objective of the study was to investigate the impact of different constitutive parameters on the simulated results.

Paper B: On the numerical implementation of invariant-based viscoelastic formulations at finite strains. A model for the passive myocardium.

In this paper a conceptional framework for finite strain photoelasticity thought to be suitable to capture salient features of a class of passive soft biological tissues like the myocardium was developed. A superposition of an Elastic Body and a Maxwell Body. The Elastic Body defines the viscoelastic continuum, both related to two independent reference configurations.

The reference configuration of the Maxwell Body stores the history of the deformation and it moves in space at it is described (apart form rigid body rotation) by a rate equation in strain space. At thermodynamical equilibrium the reference configuration of the Maxwell Body coincides with the current configuration of the continuum. The Helmholtz free energy is expressed as a function of two independent strain variables, which once defined entirely render the viscoelastic body. Its lineralization around the current configuration coincides with earlier suggested viscoelastic models although it differs to reported viscoelastic consents for finite strain. The linearized viscoelastic model has been implemented for a particular anisotropic constitutive model for the passive myocardium. Material responds due to pushing

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a rigid punch into the myocardium was studied. Non negative dissipation of the model is sat- isfied and an update equation of the over-stress that allows for variable time steps was used.

Results between anisotropic and isotropic descriptions of the myocardium differ significantly which justified the implementation of an anisotropic model for the myocardium.

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BIBLIOGRAPHY

Zienkiewicz, O. C., Taylor, R. L., 1989. The Finite Element Method. Basic Formulation and Linear Problems, 4th Edition. Vol. 1. McGraw-Hill, London.

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Numerical simulation of failure response of vascular tissue due to deep penetration