The University of AucklandBioengineering Institute

Eindhoven University of TechnologyDepartment of Biomedical Engineering

Supervisors:

M.P. Nash (UoA)P.H.M. Bovendeerd (TU/e)

An anatomically accuratefinite element model of the

human left ventricle

Hanneke Gelderblom

BMTE 08.03December 2007

2

Abstract

Sudden cardiac death is one of the main causes of death in the Western world. In mostcases it is caused by ventricular fibrillation, a form of cardiac arrhythmia. To study themechanisms of cardiac arrhythmias, models which combine a description of cardiac me-chanics and electrophysiology are developed at the Auckland Bioengineering Institute andDepartment of Theoretical Biology of Utrecht University. Such models should contain anaccurate representation of the cardiac anatomy and are therefore based on data sets of thehuman heart. In this study a finite element model of the human left ventricle that includesanatomical features and fibre architecture is developed. The model was suitable for solvingthe following mechanics problems: passive inflation, isovolumic contraction and ejection ofthe heart. In order to make the model useful for coupled mechanics and electrophysiologysimulations it has to be further expanded and improved.

3

4

Acknowledgements

I would like to thank Peter Hunter for inviting me at the Auckland Bioengineering Instituteand arranging my internship project. Martyn Nash, my supervisor, for his enthusiasm andsupport during the project. Kumar Mithraratne and Glenn Ramsey for their assistancewith CMISS and Peter Schmiedeskamp for the Linux support. A big thanks to VickyWang, Helen Lam, Angela Lee and Shannon Li who have been very nice colleagues andhelped me whenever possible. Finally, I would like to thank Peter Bovendeerd for helpingme to arrange my visit to Auckland.

5

6

Contents

1 Introduction 111.1 The human heart data set . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 CMISS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Theory for solving finite elasticity problems 152.1 Cubic Hermite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Collapsed nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 A finite element model of the human left ventricle 193.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Geometric fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Improvements on the model . . . . . . . . . . . . . . . . . . . . . . 223.1.3 Fibre fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Ventricular geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Muscle fibre field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Limitations of the fibre representation by the model . . . . . . . . . 31

4 Simulating ventricular mechanics 334.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1.2 Solution fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.3 Displacement boundary conditions . . . . . . . . . . . . . . . . . . 354.1.4 Reference state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.5 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.6 Passive inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.7 Isovolumic contraction . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.8 Ejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.9 Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Discussion and conclusion 435.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8

List of Figures

1.1 The human heart surface data set . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 An apex element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 The left ventricular surface data . . . . . . . . . . . . . . . . . . . . . . . . 203.2 The initial unfitted model . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Fibre angle correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 The initial mesh surrounded by the data . . . . . . . . . . . . . . . . . . . 263.5 The fitted mesh surrounded by the data . . . . . . . . . . . . . . . . . . . 273.6 The fitted model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.7 Fibre orientations on the endocardium . . . . . . . . . . . . . . . . . . . . 293.8 Fibre orientations on the epicardium . . . . . . . . . . . . . . . . . . . . . 30

4.1 The cardiac cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Pressure volume curve of passive inflation . . . . . . . . . . . . . . . . . . 384.3 The initial model before start of passive inflation . . . . . . . . . . . . . . 394.4 The model after passive inflation . . . . . . . . . . . . . . . . . . . . . . . 394.5 The model after isovolumic contraction . . . . . . . . . . . . . . . . . . . . 404.6 The model after ejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9

10

Chapter 1

Introduction

Sudden cardiac death is one of the main causes of death in the Western world. In mostcases it is caused by ventricular fibrillation. Ventricle fibrillation is the most dangerousform of all cardiac arrhythmias, where severely disturbed and irregular conduction of theelectrical excitation wave disorganises the contractions of the ventricles to such an extentthat the heart cannot sustain blood pressure or maintain cardiac output [21, 13]. Thecoupling between electrical activation and mechanical contraction of the heart plays animportant role in cardiac arrhythmias. Disturbed activation patterns lead to uncoordi-nated cardiac contraction via excitation-contraction coupling and mechanical deformationhas been shown to alter the electrical properties of myocytes through mechanoelectric feed-back [17].

Mathematical models can be used to simulate cardiac behaviour and study quantities thatcannot be measured in clinical or experimental settings, such as mechanical stress in thebeating heart and electrical activity throughout the wall. Heart modelling groups at theBioengineering Institute of The University of Auckland and the Department of TheoreticalBiology of Utrecht University are working on a model which combines a description ofthe mechanical and electrical activity of the heart in order to study cardiac arrhythmias.Accurate description of ventricular anatomy is considered to be of great importance fora realistic modelling of the mechanics and electrophysiology of the beating heart. Themyocardium is electrically and mechanically anisotropic: the tissue stiffness is higher [16]and the electrical conductivity is faster [21] in the direction parallel to the muscle fibres.Therefore, a model describing the complete human ventricular geometry and muscle fibreorientation is necessary.

In this study, a finite element model of the human left ventricle which provides realisticdescriptions of both the geometry and muscle fibre orientation is created. This is accom-plished by fitting an initial ellipsoid finite element model to a data set of the human heart.The resulting model is used to simulate the mechanics of the heart during the cardiac cycle.

The origin of the human heart data set and the software used for creation of the model are

11

discussed in this remainder of the chapter. Chapter 2 provides a theoretical backgroundfor finite element modelling of the heart. The development of the model is described inChapter 3, followed by its use to simulate the cardiac cycle in Chapter 4. In Chapter 5 thewhole study is discussed and concluded.

1.1 The human heart data set

A data set containing a 3D description of the human ventricular anatomy and fibre orien-tation was obtained from the Department of Theoretical Biology, Utrecht University. Thedata set was used to model the electrical activity of the heart during ventricular fibrillationand other cardiac arrhythmias [21]. The data are obtained from an excised structurallynormal human heart, as described in detail by Hren [12]. Briefly, the heart was positionedas in the thorax and sectioned transaxially in 1 mm thick slices. These slices were digitisedinto 0.5 mm3 voxels and stacked on top of each other to form a 3D data set. The result-ing data set contained 1,693,010 data points. A data set containing endo- and epicardialsurface data was also provided and consisted of 258,571 data points (Fig. 1.1). The fillingstate of the heart at the moment the data were obtained was not mentioned, but it isassumed that the heart was in the zero load state (zero pressure in the cavities).

At each data point three geometrical (x, y, z) coordinates and three coordinates defining avector pointing in the muscle fibre direction were prescribed. Because fibre orientations inthe human heart used for the geometrical data set were unknown, detailed fibre orientationdata from a canine heart were used. Anatomical landmarks on principle surfaces enclosingthe human heart were determined and, using the same landmarks on the canine heart sur-faces, a mapping between the two was obtained. Canine fibre orientations were assigned tothe human ventricular surfaces using this mapping. The intramural fibre orientation wasderived from the surface fibre directions and general knowledge of human intramural fibrerotation. The obtained fibre orientation field was validated by comparing it with recentlyobtained MR-diffusion tensor imaging data of the human heart [21].

1.2 CMISS

The numerical methods developed for creating the finite element model and solving me-chanics problems were implemented in CMISS. CMISS (Continuum Mechanics, Imageanalysis, Signal processing and System identification [1]) is a mathematical modelling en-vironment developed by researchers at the Auckland Bioengineering Institute. It can beused as a modelling tool for solving (nonlinear, time dependent) partial differential equa-tions over complex domains. CMISS consists of two software packages which are importantfor finite element modelling: cm and cmgui. Cm is written in Fortran and used for com-putational modelling. It is entirely command line driven, using a Perl interpreter. Cmguiis a graphical user interface and can be used for model visualisation and manipulation.

12

(a) Posterior view

(b) Base to apex view

Figure 1.1: Human heart surface data set.

13

14

Chapter 2

Theory for solving finite elasticityproblems

To study the mechanics of the beating heart, a heart model can be loaded with a (physio-logical) cavity pressure. The deformation of model under the applied load can be examinedby solving equilibrium equations to find the deformed state where the load is balanced byinternal wall stresses. The relation between deformation and stress in the material is givenby a constitutive relation and the material is characterised by a material law. The resultingequations and the domain they are solved on are too complex to allow analytical solvingand therefore finite element method is used. The domain is discretised into a number oftricubic Hermite elements, forming the finite element mesh, over which quantities of in-terest (geometric coordinates of a point, strain and stress) are continuously approximated.For each element, the equations are expressed in terms of known material properties andunknown nodal displacements. The resulting system of nonlinear equations is solved sub-ject to boundary conditions. This yields a set of deformed nodal coordinates from whichdeformation patterns are approximated using interpolation.

2.1 Cubic Hermite elements

The advantages of cubic hermite elements over linear elements are that the elements havemore degrees of freedom (DOFs), the element interpolation is cubic and there is C1 conti-nuity over the element boundaries. In this way smooth anatomically accurate surfaces canbe obtained with relatively few elements. Interpolation functions are used to interpolatecontinuous functions (for example describing the shape of a surface) within elements usingvalues of the function at the nodes. For cubic instead of linear interpolation, four insteadof two known element parameters are necessary (in 1D elements). In cubic Hermite ele-ments those two extra parameters are nodal derivatives, to ensure C1 continuity across theelement boundary. So instead of just the nodal coordinates, x, also the nodal derivatives,dxdξ

, are stored, with x the global coordinate vector and ξ the local element coordinate.

15

Eq.(2.1) gives the resulting element interpolation function for a 1D cubic Hermite element:

x(ξ) = Ψ01(ξ)x1 + Ψ1

1(ξ)dx

dξ|1 + Ψ0

2(ξ)x2 + Ψ12(ξ)

dx

dξ|2, (2.1)

with basis functions:

Ψ01(ξ) = 1− 3ξ2 + 2ξ3 Ψ1

1(ξ) = ξ(ξ − 1)2

Ψ02(ξ) = ξ2(3− 2ξ) Ψ1

2(ξ) = ξ2(ξ − 1).(2.2)

To make sure derivatives are continuous across element boundaries, derivative dxdξ|n at node

n has to be shared between adjacent elements. However, dxdξ|n is dependent on the local

element ξ-coordinate and will be different in two adjacent elements with the same globalnode if the elements have different arc lengths. Therefore, instead of storing the local

element nodal derivative, a global (arc length) derivative(

dxds

)∆(n,e)

is stored and eq.(2.3)

is used to determine dxdξ|n:

dx

dξ|n =

(dx

ds

)∆(n,e)

(ds

dξ

)e, (2.3)

with ∆(n, e) the global node number of local node n in element e and(

dsdξ

)e

an element

scale factor, denoted as Se, which scales the global arc length derivative to the local ξ-coordinate derivative and is a measure for the element size. This scale factor is requiredto be the same at each node regardless of the current element, so often the average of twoarclengths on either side of the node is used. The set of mesh parameters u hence containsthe nodal coordinates, arc length derivatives and scale factors.

In the 3D case, tricubic Hermite elements are used and the number of nodal degrees offreedom increases from two to eight per direction (x, y, z), containing the nodal coordinateand (cross)derivatives in ξ1-,ξ2- and ξ3-direction:

x,∂x

∂ξ1

,∂x

∂ξ2

,∂2x

∂ξ1∂ξ2

,∂x

∂ξ3

,∂2x

∂ξ2∂ξ3

,∂3x

∂ξ1∂ξ2∂ξ3

leading to a total number of 24 DOFs per node [4].

2.2 Collapsed nodes

To create an apex for the heart model presented in Chapter 3, collapsed nodes were used.Collapsing means placing one of the eight local nodes of an element at the same geometriclocation as another local node of the same element. To create the apex of the model,elements with two collapsed nodes will be used, which means they have six nodes with dif-ferent geometric locations instead of eight, resulting in elements with a shape as depicted

16

Figure 2.1: An apex element with 2 collapsed nodes.

in Fig. 2.1.

In CMISS, an element based normalised material coordinate system (ξ1, ξ2, ξ3 in 3D) isused. In the heart model, the ξ1 coordinate is directed circumferentially, the ξ2 longitu-dinally and ξ3 transmurally. Hence, the three directions are not necessarily orthogonalto each other. Because two nodes per apex element will be collapsed in circumferentialdirection, the ξ1-direction does not exist for these nodes. However, each element now hastwo different ξ2-directions starting from the same node. To make sure both lines will havea different shape, i.e. different values of the nodal derivatives in ξ2-direction, multipleversions of the node can be used. A version is an additional set of nodal parameters thatallows different lines in the same ξ-direction to originate from the same node.

If multiple versions of the same node are used and the model is subjected to, for ex-ample, a fitting procedure, one has to make sure that the geometric coordinates of thoseversions stay the same (i.e. you do not end up with different nodes), as well as some lines(for example the line in the ξ3-direction of the model, which connects the endo- with theepicardial apex) where no different versions are desired. This can be done with a mapping.If versions are mapped together during the fit, the corresponding rows in the set of systemequations describing the mesh will be deleted before solving the system, so all versions aresolved together and nodal parameters will get the same value.

2.3 Equilibrium equations

By applying a pressure load on the endocardial surface, a heart model will be deformeduntil a new equilibrium state is reached. To find this deformed state, equilibrium equationsare solved using finite element method:

1. Conservation of mass for an incompressible material with density ρ, undeformed vol-ume V0, deformed volume V and Jacobian J of the transformation from undeformed

17

to deformed state:∫V0

ρ0dV0 =

∫V

ρdV =

∫V0

ρJdV0 → J = 1. (2.4)

2. Conservation of linear momentum in absence of body forces:

∂

∂XM

(TMN

∂xj

∂XN

)= 0, (2.5)

with T being the second Piola-Kirchhoff stress tensor, X the coordinates in theundeformed and x in the deformed body.

3. Conservation of angular momentum, which is satisfied by requiring the second Piola-Kirchhoff stress tensor to be symmetric:

TMN = TNM . (2.6)

The second Piola-Kirchhoff stress tensor is used rather than the Cauchy stress, because itrefers all stresses back to a know reference state instead of the unknown deformed state.

2.4 Constitutive relations

The relation between deformation and stress in an incompressible material is given by aconstitutive relation:

TMN =∂W

∂EMN

− J∂XM

∂xi

pδij

∂XN

∂xj

, . (2.7)

Strain energy function W is a function of strain E. The passive myocardium can bemodelled as an incompressible hyperelastic material, described by a transversely isotropicmaterial law. In this study, a material law adapted from Guccione [10] is used:

W = C1eQ,

Q = 2C2(Eff + Ess + Enn) + C3E2ff + C4(E

2ss + E2

nn + 2E2sn) + 2C5(E

2fs + E2

fn),(2.8)

with the strain energy function being a function of the material constants C1-C5 and thefibre strain components.

18

Chapter 3

A finite element model of the humanleft ventricle

To create a anatomically accurate model of the left ventricle, an initial model is fitted tothe human heart data set. First, an ellipsoid finite element mesh is created and the facesof this mesh are fitted to the surface data points. Second, a muscle fibre field is fitted tothe fibre orientation data set.

3.1 Methods

3.1.1 Geometric fitting

A new data set, containing only left ventricular surface data information, was created bydeleting the data belonging to the right ventricle using cmgui. This caused some holes inthe data set at the locations where both ventricles are connected. The endocardial andepicardial data had to be separated, in order to allow fitting of both surfaces individually.Some data in the basal region of the left ventricle were removed to obtain a clear partition-ing of both surfaces. In order to represent the data set in the same (rectangular cartesian)coordinate system as the initial model, the data were converted into a cardiac coordinatesystem with the x-axis pointing from base to apex, the y-axis from left to right ventricleand the z-axis orthogonal to both, using MATLAB (Fig. 3.1).

An initial mesh consisting of sixteen tricubic Hermite elements was created using a cardiacprolate spheroidal coordinate system (λ,µ,θ). Prolate coordinates are related to rectangularcartesian coordinates by:

x = d cosh(λ) cos(µ)y = d sinh(λ) sin(µ) cos(θ)z = d sinh(λ) sin(µ) sin(θ)

(3.1)

where d is the focus position on the x-axis. The mesh consisted of four elements in cir-cumferential (θ) and longitudinal (µ) direction and one element in radial (λ) direction. In

19

(a) Anterior view (b) Base to apex view

Figure 3.1: The left ventricular surface data (endocardial surface in red, epicardial surfacein blue) in a cardiac coordinate system.

the longitudinal direction, µ was varied from 0 (at the apex) to 120, with µ=90 at theequatorial plane. In circumferential direction, θ was varied from 0 to 360. Suitable valuesfor λ and d of the ellipsoid were calculated in order to embed the initial mesh in the dataset. The focal point was set so that the radial coordinate of the apex assumed a value ofunity [18] (eq.(3.2)). The λ-values were obtained by determining the y-values of the endo-and epicardial surfaces on the line x = 0, z = 0 (θ = 0, µ = 90; the equatorial plane)(eq.(3.3)).

d =xapex

cosh(1)(3.2)

λendo = sinh−1(yendo

d)

λepi = sinh−1(yepi

d)

(3.3)

To form the apex of the left ventricle, two nodes of the epi- and endocardial surface hadto be set to λ = 1 and λ = 0.9, respectively. This implies collapsing of two nodes of thenormally eight nodal elements to form an apex. Four versions of the ξ2-direction for bothapical nodes were used to create a good apex shape. After generation of the initial meshusing prolate coordinates, the model is converted into rectangular Cartesian coordinates,because of convenience for the mechanics simulations later on. Fig. 3.2 shows the resultinginitial mesh. The cm input files containing the mesh topology were generated with Perl.

20

Figure 3.2: The initial unfitted model.

In order to create a geometrically accurate finite element representation of the left ven-tricle, the initial mesh is fitted to the data. The performed fitting is a face fitting, whichmeans that the transmural (ξ3) direction is not involved in the fit. The endocardial andepicardial data are fitted to respectively the endocardial (ξ3 = 0) and epicardial(ξ3 = 1)faces of the model. The goal of the fitting procedure is to find the nodal positions andderivatives (the mesh degrees of freedom) that result in the best mesh approximation of thedata. For each data point zd an orthogonal projection zp onto the mesh can be found. TheEuclidean distance between zd and zp is the error that has to be minimised. Point zp canbe described as a function of the initial mesh parameters using the element interpolationformula eq.(3.4) (for simplicity the formula for fitting of a 1D cubic Hermite element isgiven here, see Chapter 2)[4].

zp(ξd) =∑n=1,2

[Ψ0

n(ξd)xn + Ψ1n(ξd)

(dx

ds

)n.Sn

]. (3.4)

The ξ-coordinates of point zp are found by cm using a non-linear iterative procedure: givena starting position ξ of the projection of zd onto the mesh, an error function can be set upfor the distance between this point and data point zd. The ξ-coordinates which minimisethis function, ξd, are the coordinates of the orthogonal projection of zd onto the mesh. Fora given projection of the data points onto the mesh (i.e. ξd is held constant) the objectivefunction to be minimised in the fit is formed by the sum of squares of the individual errors[4]:

F(u) =∑

d=1,D

‖zp(ξd)− zd‖2, (3.5)

21

where u is a vector of mesh parameters. For each node, the parameters that minimise thisobjective function can be found by differentiating eq.(3.5) with respect to each of the meshdegrees of freedom and setting these derivatives to zero. A set of equations describingthe entire mesh can be found by assembling a global stiffness matrix from all individualelement matrices (eq.(3.6)).The assembling is done to satisfy the continuity of the nodalvalues and nodal derivatives at the nodes common to different elements.

Ku = f, (3.6)

with K the stiffness matrix and f the zero vector. This results in a linear system ofequations if the scale factors S are kept constant during the fit. Because both S and thedata projections are kept constant, the resulting mesh may not represent the smallest error.Therefore, the fitting is done in more iterations and after each iteration the scale factor anddata projections are recalculated. The error after each iteration is the root mean square(RMS) error, which represents the accuracy of the fit. The complete fitting algorithm isas follows:

1. Define an initial mesh and calculate the initial scale factors

2. Calculate the initial data point projections

3. Repeat until converged:

(a) Fit the mesh

(b) Update the scale factors

(c) Calculate the data point projections onto the new mesh

(d) Calculate RMS error

During the fit, the derivatives of the apex nodes in ξ1-direction had to be fixed to zero,since the ξ1-direction does not exist for those nodes. If these derivatives are released duringthe fit, cm will assign values to them, resulting in very odd shapes of the apex elements.The derivatives in ξ3-direction had to fixed as well, since this direction is not part of thefitting problem.

3.1.2 Improvements on the model

After fitting the initial model faces to the surface data, it was found that in the basalregion, the endocardial surface is pointing out of the epicardial surface due to a lack ofdata points. To solve this problem, the initial mesh was manipulated by dragging the nodalpoints of the basal plane closer to the surface data. To improve the accuracy of the fit,the mesh was refined in the ξ1-direction. The new mesh consisted of one element in radialdirection, four in longitudinal direction and eight in circumferential direction. The apexnodes in this new mesh have eight versions each, and during the fit the nodal derivativein ξ2-direction of version 1 was mapped to version 5, 2 to 6,3 to 7 and 4 to 8 to create

22

smoother apex elements to improve numerical stability when solving mechanics later on.Because the ξ3-direction is not part of the fitting problem, curved lines in this directionprovide no useful geometry information. In order to get straight lines, the nodal derivativesin ξ3-direction were recalculated based on the straight line lengths between the nodes afterthe fitting. To control the arc curvature, arc length and the surface curvature of the meshfaces during the fit and penalise excessive curvature, a Sobelov smoothing term can beadded to the objective function (eq.(3.7))[4]:

F(u) =∑

d=1,D

‖zp(ξd)− zd‖2 + G(u) (3.7)

G(u) =

∫Ω

α(‖ ∂u

∂ξ1

‖2 + ‖ ∂u

∂ξ2

‖2)

+ β(‖∂2u

∂ξ21

‖2 + 2‖ ∂2u

∂ξ1∂ξ2

‖2 + ‖∂2u

∂ξ22

‖2)

dξ (3.8)

The first two terms of eq.(3.8) measure element arc lengths, the third and fifth termmeasure arc curvature and the fourth, cross derivative term measures surface curvature.Acceptable values for the smoothing factors had to be found by parameter variation. Theepicardial surface required additional smoothing to correct for the rough shape and lack ofdata points at the septum, where the right ventricle was removed. Part of the endocardialsurface data described the papillary muscle region. These data had a great influence on theshape of the fitted endocardial faces of the mesh. The current model does not representthe papillary muscles and therefore this part of the data set was removed using cmgui, sothat it could not influence in the fit. With the new mesh and data set the fitting procedurewas repeated.

3.1.3 Fibre fitting

Fibre fitting is a 3D fitting, which means the complete volume data set will be used and theξ3-direction of the model is also involved in the fit. After removal of the right ventricle datawith cmgui, the volume data set consisted of 1,380,946 data points. At each data point, aunit vector representing the fibre orientation (the fibre vector) was specified. Because thenumber of data points was much larger than the number of degrees of freedom in the meshthat could be fitted, the data set was resampled. Furthermore, only data points containedby the model were used for fibre fitting. A Perl script was used to resample the data andexclude the data points that were not contained by the model. The remaining data setconsisted of 10,904 data points. The fibre vectors were rotated to convert them into thecardiac coordinate system. Two angles defining the fibre orientation could be deducedfrom the fibre vectors: one elevation angle with respect to the horizontal sections of themodel, with 0 meaning that the fibre is in the horizontal plane, and one azimuth anglewith respect to the global x-axis [21]. For cardiac muscle fibres, 2 angles are of importance:the fibre angle, defined as the angle between the fibre vector and the circumferential direc-tion of the heart, and the sheet angle, defining the orientation of the sheets that bundlethe muscle fibres. Unfortunately no information about sheet angles was provided, so theycould not be included in the model. Furthermore it was assumed that muscle fibres are

23

aligned with the local surface tangent planes, i.e. have no imbrication angle, leaving onlythe fibre angle as a parameter to fit.

To be able to determine the fibre angles from the given fibre vectors, it was necessaryto assume that circumferential direction of the heart is aligned with the yz-plane in thecardiac coordinate system. This allowed the fibre angle to be calculated as the angle α ofthe fibre vector with the yz-plane using:

α = arcsin( −x√

(x2 + y2 + z2)

)(3.9)

The angle between yz-plane and a fibre vector pointing in negative x-direction is definedas positive. Note that the fibre angle is only unique between −π

2and +π

2. In CMISS, the

fibre angle is defined as the angle with respect to the local circumferential direction (ξ1) inthe model. In order to represent the cardiac geometry accurately, the ξ1-direction of themodel was not enforced to be in the yz-plane. Therefore, the calculated fibre angles hadto be corrected for the angle between the yz-plane and the local ξ1 direction before fitting.This was done with a cm subroutine that works as follows:

1. For each data point the ξ-coordinates in the corresponding element in the model aredetermined.

2. For each element a vector g1, lying in the yz-plane at the intersection with the ξ1ξ2-plane, and a vector g2, lying in the ξ1ξ2-plane and aligned with the ξ1-direction, areconstructed (see Fig. 3.3):

g1 = a∂xk

∂ξ1

ik + b∂xk

∂ξ2

ik, (3.10)

where xk = (x, y, z) and ik are unit vectors along the three rectangular cartesian

axes. a = −∂x∂ξ2∂x∂ξ1

and b = 1 to ensure g1(1) = 0, i.e. g1 lies in the yz-plane.

g2 =∂xk

∂ξ1

(3.11)

3. The angle between g1 and g2 is the correction angle, which represents the anglebetween the ξ1-axis and its projection on the yz-plane:

θ = arccos(g1,g2), (3.12)

where g1 and g2 have been normalised to unit length.

4. If ∂x∂ξ1

is negative, so the g2 makes a positive angle with the yz-plane, π is subtractedfrom θ to form the correction angle.

5. Correction angle is added to the fibre angles of the data points lying in that element.

24

Figure 3.3: Fibre angle correction for non-circumferential ξ1 direction (adapted from: [20]).

After the correction, the fibre angle is the angle of the fibre vector with respect to the localξ1-direction in the model.

An initial fibre field with zero fibre angles and fibre angle derivatives was set up. Ineach node one fibre angle and 7 fibre angle derivatives are specified, since cubic hermitefibre elements were used. For each data point the ξ-coordinates, ξd, were determined. Themodel fibre angle at each point was then determined using the same interpolation functionsas for the geometry (eq.(3.13) holds for 1D elements):

α(ξd) =∑n=1,2

[Ψ0

n(ξd)αn + Ψ1n(ξd)

(dα

ds

)n.Sn

](3.13)

The error between the data fibre angle (corrected to be with respect to the local ξ1-direction) and the model angle at each location is minimised by adapting the nodal fibreangles and derivatives, analogue to the geometric fitting. Since the geometry does notchange during the fit, the data projections and scale factors do not change, resulting in alinear set of equations that can be solved in one step.

25

(a) Endocardium (b) Epicardium

Figure 3.4: The initial mesh surrounded by the data.

3.2 Results

3.2.1 Ventricular geometry

Fig. 3.4 shows the refined initial mesh with adapted basal plane surrounded by the surfacedata. The endocardial face of the mesh has 798 degrees of freedom and is fitted to 34,177data points (after removal of the papillary muscle region). The epicardial face (same num-ber of DOFs) is fitted to 103,319 data points. The results after two iterations are depictedin Fig. 3.5 and 3.6.

iteration RMS [mm] smoothing factorsendocardium epicardium endocardium epicardium

0 9.17 8.04 - -1 1.36 1.43 α = 1.0, β = 1.5 α = 3, β = 3.52 1.14 1.20 α = 0.5, β = 0.75 α = 1.5, β = 1.753 1.07 1.10 α = 0.25, β = 0.375 α = 0.75, β = 0.8754 1.02 1.05 α = 0.125, β = 0.188 α = 0.375, β = 0.438

Table 3.1: Used smoothing factors and resulting RMS values for the geometric fit.

26

(a) Endocardium (b) Epicardium

Figure 3.5: The fitted mesh after 2 iterations surrounded by the data.

The RMS error of the fit decreases with each iteration (see Table 3.1), indicating anincreasing accuracy of the fit. The fitted mesh after two iterations will be used for me-chanics simulations later on, because the elements in the basal plane region are taller inξ3-direction than after three or four iterations, which will lead to better convergence.

3.2.2 Muscle fibre field

1560 Model degrees of freedom were fitted to 10,904 data point fibre angles. The initialRMS error was 33.8 and resulting RMS error after fitting was 10.2 overall and varied from1.83 to 34.4 in individual elements. High RMS errors were found in the septal region,low RMS errors in the left ventricular free wall.

Figure 3.7 and 3.8 show the variation of fibre orientation over the endo- and epicardialsurfaces. Because the fibre direction is only unique between −π

2and +π

2, fibre orientations

are depicted as cylinders rather than as vectors. Through the left ventricular wall, thefibre angle varied from 13 (endocardium) to −55 (epicardium) at the anterior site, 72

to −32 at the posterior site, 62 to −57 in the free wall and 14 to −37.3 in the septum(angles with respect to the local element circumferential direction).

27

Figure 3.6: The fitted model.

28

(a) LV free wall, septum

(b) Anterior, posterior view

Figure 3.7: Fibre orientations on the endocardium.

29

(a) LV free wall, septum

(b) Anterior, posterior view

Figure 3.8: Fibre orientations on the epicardium.

30

3.3 Discussion

3.3.1 Limitations of the fibre representation by the model

In literature, fibre angle variations from +60 at the subendocardium to −60 at thesubepicardium are reported [6]. The model fibre angles in the left ventricular free wall arecomparable with these observations. In the septum region, the left ventricular epicardiumalso represents the right ventricular endocardium, and therefore fibre angles in the mid wallare smaller than at the epicardial surface. The complex fibre orientation in the septumalso caused higher RMS errors and a more distorted fibre field in that region. The verysmall fibre angles at the anterior endocardial surface (around +13) were caused by theremoval of papillary muscle data before the geometric fitting, causing the model to inac-curately represent the endocardial surface of the data. Because only fibre angles of datapoints contained by the model were used in the fibre fitting, mid wall fibre orientationswere fitted to endocardial model regions, since the endocardial data are not embedded inthe model.

Because the fitted model did not embed the complete data set, some data points werepositioned outside the model. Those data points were not taken into account for the fibrefitting, and therefore epi- and endocardial surface fibre angles might not have been fitted.The fibre angle changes rapidly near the surface, so excluding some of these data reducesthe accuracy of fibre representation. It is possible to correct this using the projections ofvectors of data points outside the model onto the surface, as described by Nielsen [18].Because the fibre angle changes so rapidly in transmural direction, refinement of the meshin transmural direction could also improve the accuracy of the fit.

Muscle fibres were assumed to lie solely in planes spanned by the local circumferentialand longitudinal direction. According to imbrication angles reported in literature this as-sumption is valid for the equator region of the heart, but invalid for the apical and basalregion, where mid wall imbrication angles of −12 (apex) and +9 (base) are found [6].This simplification may significantly influence the local ventricular wall mechanics [3, 22].The current model did also not include information about fibre sheet orientations. Thisis a major limitation, since the mechanical properties of the myocardium are different insheet, sheet normal and fibre direction [16] and will influence the mechanics simulationswith the model later on.

31

32

Chapter 4

Simulating ventricular mechanics

This chapter presents the use of the finite element model to simulate mechanics during thecardiac cycle. The cardiac cycle consists of four different phases, as depicted in figure 4.1.Isovolumic contraction, the first phase of the systole, begins when active stress is generatedby contraction of the muscle fibres (point A in Fig. 4.1). Increasing pressure in the leftventricular cavity first closes the mitral valve, after which the pressure rises further untilthe aortic pressure is exceeded and the aortic valve opens. During isovolumic contractionthe cavity pressure rises at a constant volume. When the aortic valve opens, the ejectionphase begins (point B in Fig. 4.1). The ventricular volume reduces, but the ventricularpressure continues rising at first (the rapid ejection) because blood is flowing into the aortafaster than it is flowing out through its branches. About 70% of the emptying occurs inthis phase. During the second part of ejection (reduced ejection) the pressure in the leftventricular cavity drops gradually, until the third phase, isovolumic relaxation and thusthe diastole, starts (point C in Fig. 4.1). During this phase, both the aortic and themitral valve are closed and the ventricle relaxes at a constant volume, until the pressurefalls below the pressure in the left atrium, the mitral valve opens and the filling phasestarts (point D in Fig. 4.1). During the first part of the filling blood flows rapidly acrossthe mitral valve into the further relaxing ventricle. This rapid filling is followed by a slowfilling (diastasis), during which the pressure rises very gradually. Atrial contraction thenforces an additional 20-30% of the total filling volume into the left ventricle [13, 11].

The simulation of mechanics starts with a passive inflation of the left ventricular cavity, toreach the end diastolic volume and pressure in the cavity. This is followed by isovolumiccontraction, rapid and reduced ejection. Due to lack of time, the simulation had to beterminated at the end of the ejection phase.

33

Figure 4.1: A representation of the cardiac cycle (adapted from Glass [9]). A: Start ofisovolumic contraction B: Start of ejection C: Start of isovolumic relaxation D: Start offilling.

4.1 Methods

4.1.1 Material properties

Because there was no information available about fibre sheet orientations from the dataset, it was assumed that cardiac tissue is transversely isotropic material: it has one axisof symmetry, aligned with the average fibre direction. Passive tissue properties are thendescribed by the constitutive law given in Chapter 2 (eq.(2.8)). The material parametersC1, C2, C3, C4 and C5 were obtained from estimations made by Schmid et al. [19]. Theyestimated material parameters from experimental data of six pig hearts for the orthotropicmaterial law found by Costa [5]:

W = 12a(eQ − 1),

Q = bffE2ff + 2bfn

(12(Efn + Enf )

)2

+ 2bfs

(12(Efs + Esf )

)2

+ bnnE2nn

+2bns

(12(Ens + Esn)

)2

+ bssE2ss.

(4.1)

34

The parameters as determined for one of these hearts were used for the current materiallaw: a = 0.19, bff = 30.5, bfn = 12.1, bfs = 13.9, bnn = 17.8, bns = 9.86 and bss = 13.6.Because the currently used material law is transversely isotropic, material parameters insheet and sheet normal directions were averaged to get equal material properties in thosedirections:

C1 =a

2= 0.095, C2 = 0, C3 = bff = 30.5, C4 =

bnn + bss + bns

3= 13.75,

C5 =bfn + bfs

2= 13.0

The material properties are assumed to be homogeneous throughout the whole left ven-tricular wall.

4.1.2 Solution fields

There are four fields of unknowns to solve the equilibrium equations presented in Chap-ter 2 for: the geometric coordinates x, y, z and the hydrostatic pressure p induced by theincompressibility constraint. In this finite element model myocardial stresses are derivedfrom the hydrostatic pressure and the strain tensor, of which the latter depends on thederivatives of the geometric coordinates. To obtain C0 continuity of stress over the ele-ment boundaries, C1 continuity of the geometric coordinates is necessary. The geometriccoordinates are interpolated using tricubic Hermite basis functions to ensure this. Thehydrostatic pressure is interpolated using trilinear Lagrange interpolation functions andhas therefore C0 continuity over the element boundaries.

4.1.3 Displacement boundary conditions

To prevent rigid body rotation of the model, nodal positions and derivatives at the basewere fixed. It was found that for the sake of numerical convergence, it was better to fixboth the endo- and epicardial nodes of the base, to prevent the elements in the basal ringfrom becoming too small during deformation of the model. The fixation of the basal nodeshas a great influence on the mechanics in that region. A refinement of the elements inthe basal region was made to create a small layer of basal elements were the displacementboundary conditions could be applied. The remaining number of degrees of freedom of themodel is given in Table 4.1.

4.1.4 Reference state

In large deformation mechanics, material deformation and derived stresses and strains arecalculated with respect to a well defined reference configuration, for which the materialstrain components are assumed to be zero. Such a reference state does not exist for theintact heart, but it is assumed that the residual stresses and strains are negligible if theheart is in its zero load state (zero pressure in the cavities). It is assumed that the dataset on which the model is based, is obtained from a heart that was in this state.

35

number of nodes DOFsstandard 64 1536fixed 16 0collapsed with versions 2 60total 82 1596

Table 4.1: Calculation of the number of degrees of freedom in the model.

4.1.5 Solution methods

Deformation of the left ventricle model is governed by the equilibrium equations and con-stitutive relations presented in Chapter 2. A system of nonlinear equations is assembled inCMISS using finite element method. These equations are rearranged into a set of residuals(with zeros on the right hand side) which have to minimised with respect to the set of so-lutions variables (array u) and linearised using Taylor expansion. The solution u for whichthe residuals are zero is approximated using the Newton Raphson iterative method. Theresulting set of linear equations can then be solved for each iteration using a direct solver(the LU decomposition method was used here). Convergence is achieved when both theratio of constraint to unconstraint residuals and the sum of solution vector increments forthe current Newton iteration are below a prescribed error tolerance. Constrained residualsare residuals for which essential boundary conditions are prescribed (the solution is know),while for the unconstrained residuals the solution has to be determined.

4.1.6 Passive inflation

Assuming the heart had initially been in the zero load state, the first step of the simulationwas filling of the heart until the end of the diastolic phase was reached. To simulate this,the pressure in the left ventricular cavity was increased from 0 to 1 kPa, by applying aload on the endocardial surface of the model in the form of a pressure boundary condition.Since this is a nonlinear hyperelastic problem, applying the complete load on the boundaryat once is not possible, because it would deform the model too much and would not lead toa converged solution. Therefore, the load was applied gradually, by increasing the pressureboundary condition in 10 steps, 0.1 kPa at a time, to 1 kPa. For each step, the referencestate is still the zero load state, but the initial guess for the nonlinear equation will be thelast converged solution. Each step brings the model closer to the final deformed state.

4.1.7 Isovolumic contraction

The diastolic phase ends when an electrical wave propagates through the myocardium toexcite the muscle fibres, causing them to contract and generate force. This process wassimulated by turning on an ‘activation parameter’, which is related to the intracellulaircalcium concentration, determining the level of generated active stress. At the moment

36

the activation parameter is turned on, all muscle cells are activated simultaneously. It wasassumed that the cardiac muscle fibres can generate stress in the direction of the musclefibre only. This means that just one term had to be added to the passive 3D stress tensorto model active behaviour of the myocardium. The total second Piola-Kirchhoff stresstensor now becomes [16]:

TMN =∂W

∂EMN

+ J∂XM

∂xi

(σA

ijδi1δ

j1 − pδi

j

)∂XN

∂xj

, (4.2)

with σA the active tension generated by the muscle fibres. For the current modelling,this tension is made dependent on the level of the activation parameter and muscle fibreextension ratio only (for detailed description refer to [15]). The activation parameteris incrementally increased during the simulation. For each level of activation the leftventricular cavity pressure (i.e. the pressure boundary condition at the endocardium) isdetermined such, that it balances the increasing fibre stress to keep a constant cavityvolume. A mesh of the left ventricular cavity was created to evaluate the volume aftereach activation increment. If the volume had changed, the cavity pressure was increaseduntil the end diastolic volume was reached again. The end of the isovolumic contractionwas defined to be at a cavity pressure of 10 kPa.

4.1.8 Ejection

Ejection starts as soon as the model cavity pressure reaches 10 kPa. The increase incavity pressure during the rapid ejection phase is modelled by increasing the pressureboundary conditions at the endocardium incrementally until 15 kPa is reached. To geta realistic volume change at each pressure increment, the desired volume at the currentpressure is obtained from a look up table and the activation parameter (so the activetissue stress) is increased until this volume is reached. Reduced ejection is simulatedby incrementally decreasing the pressure boundary condition until 14 kPa at constantactivation level, reducing the cavity volume.

4.1.9 Numerical verification

Although a low number of elements is desirable to reduce computation time, it is impor-tant to have a mesh with enough degrees of freedom to accurately represent the actualdeformations. A too low number of degrees of freedom will induce artificial stiffness in themodel, because it cannot be deformed in the way it should be. To check if the solutionsfound during passive inflation were converged, i.e. would not change if the mesh wouldbe further refined, a convergence analysis was performed. The initial mesh was refinedseveral times in each direction. The displacement of the nodes in the left ventricular wallduring passive inflation was compared between the refinements. The RMS error betweentwo refinements per direction (x, y, z) is defined as the square root of the mean squareddifference in nodal displacements during passive inflation.

37

Figure 4.2: Left ventricular cavity volumes and pressures during passive inflation.

4.2 Results

The results of the simulation can be seen in a short movie [8] and Fig. 4.3 to 4.6. Duringpassive inflation the left ventricular cavity was inflated from 41.3 to 66.9 ml (see Fig. 4.2)and the apex to base length (axial length) of the model increased from 80.6 to 86 mm.During isovolumic contraction the lengthening continued to an axial length of 92.6 mm,and also a twist of the apex appeared. During rapid ejection the axial length shortened to87.7 mm and the volume decreased to 33.4 ml. A further decrease to 27.1 ml took placein the reduced ejection phase, resulting in an ejection fraction of 59%, with 84% of theejection occurring in the rapid ejection phase.

From the convergence analysis, it was found that refinement in the apex region did notlead to converged solutions. Therefore, the refinements were made in the base and midwall regions only (so the eight apex elements were not refined). The model was refined one,two and three times in longitudinal, ξ2, direction and one and two times in transmural, ξ3

direction. More refinements in ξ3 direction led to convergence problems, due to high aspectratios of elements. It was not possible to refine in all directions at once, because this wascomputationally too expensive. Refining in ξ2 direction did reduce error a little bit, butthe error between zero and one refinements was already quite small, as for the ξ3 direction(see Fig. 4.7). Due to lack of time, no refinements in circumferential, ξ1, direction weremade, but it is expected that the number of DOFs in this direction was already sufficient.

38

Figure 4.3: The initial model before start of passive inflation, septum (left) and anteriorview (right).

Figure 4.4: The model after passive inflation, septum (left) and anterior view (right).

39

Figure 4.5: The model after isovolumic contraction, septum (left) and anterior view (right).

Figure 4.6: The model after ejection, septum (left) and anterior view (right).

40

Figure 4.7: The RMS error between refinements in ξ2 and ξ3 direction.

4.3 Discussion

According to the convergence analysis the solutions are not very sensitive to further dis-cretisation of the mesh. The initial model thus contained sufficient degrees of freedom toaccurately represent deformations occurring during passive inflation and therefore, presum-ably, during the whole cardiac cycle. The cavity volume of the initial model was smaller(41.6 ml) than the cavity volume of a human heart which is at the beginning of the fillingphase in vivo (60-70 ml) [13]. The cadaver heart the data set was obtained from mighthave been contracted due to rigor mortis, causing a smaller cavity volume. Furthermore,an in vivo heart is never completely empty at the beginning of the filling phase. Theejection fraction and percentage of cavity emptying during rapid ejection are in agreementwith the expected values. The shape change of the model during isovolumic contraction isnot physiological, as an axial length shortening and circumference thickening is expected,giving a more rounded contour to the cavity [13], instead of more elongated. The twistof the apex is probably due to the fact that the modelling of fibre orientation around theapex was not accurate. The imbrication angle, which may significantly influence local wallmechanics at apex and base [6] was neglected, as well as the presence of fibre sheets. Theapical twist started as soon as active fibre stress was generated. To prevent this movementof the apex, one could think of turning off the active stress in this region or incorporat-ing imbrication angles. A possible explanation for the model elongation is the absence ofpapillary muscles and the pericardial sac to restrict lengthening of the heart [15]. Also thepresence of a right ventricle will limit the movement of the left ventricle, and will imposepressure boundary conditions on the septum.

41

Ejection was modelled in a rather artificial way, with the cavity volume (and thus ejectionfraction) being a prescribed instead of a solution parameter. Increasing the activation pa-rameter to obtain a certain cavity volume during rapid ejection, and keeping the activationparameter constant during reduced ejection has no physiological meaning or explanation.Therefore, it would be better to include a (Windkessel) model of the systemic circulation,to determine the volume change during ejection depending on the afterload of the heart. A3-element Windkessel model of the systemic circulation consists of an aortic characteristicimpedance in series with an arterial compliance, representing the arterial storage of blood,which is parallel to a peripheral resistance, modelling the pressure drop in the capillaries.Aortic flow can be expressed as a function of the Windkessel model parameters and thepressure difference over the characteristic impedance, and should be equal to the rate ofchange of the left ventricular cavity volume if hemodynamic coupling is achieved. If thisis not the case, the aortic flow is adapted, leading to a new pressure in the cavity and thusa new change in cavity volume after solving the equilibrium equations. This procedure isrepeated for each point in time until hemodynamic coupling is achieved. Using this proce-dure would also lead to a better way to determine the end of the ejection phase: reversionof the aortic flow instead of exceeding a certain predefined pressure level [2].

Finally it is important to note that the material laws used for active and passive mate-rial behaviour are both phenomenological, so no physiological measures are incorporated.The material properties were assumed to be homogeneous throughout the whole heart andduring systole, the whole heart was activated at once, which is not the case in reality.

42

Chapter 5

Discussion and conclusion

5.1 Discussion

The overarching aim of the heart modelling group at the Auckland Bioengineering Insti-tute and the Department of Theoretical Biology of Utrecht University is to develop a heartmodel which combines a description of mechanical and electrical activity of the heart. Theventricular geometry and muscle fibre orientations have a great influence on the mechanicsand electrophysiology of the heart. Therefore, a model which combines both should con-tain an accurate representation of the cardiac anatomy. The data set used in this researchprovides a description of human ventricular geometry and fibre orientations. The group inUtrecht already developed a finite difference model based on this data set to solve equa-tions describing the excitable behaviour of the heart. The Auckland group is working onfinite element models to simulate cardiac mechanics. The used equations describing activefibre contraction contain a certain ’activation parameter’, which determines the amountof active tension generated by the cardiac muscle fibres. The model of the Utrecht groupcan provide the distribution and level of muscle activation through the cardiac wall, whichcould then be used as an input for the Auckland model to solve the mechanics. The result-ing deformations, affecting the electrical properties of myocytes through mechanoelectricfeedback, can than be fed back into the Utrecht model, that solves the next step. Thiswill provide a more realistic modelling of the active behaviour of the heart during systole.A better way to model ejection could be through the use of a Windkessel model for thesystemic circulation, to simulate hemodynamic coupling during the period the aortic valveis open and determine the end of the ejection phase based on a reversion of aortic flowwhich causes the aortic valve to close.

To make the currently created model suitable for the general purpose of the Aucklandand Utrecht groups, it has to be expanded to include a right ventricle and papillary mus-cle for a more accurate representation of ventricular anatomy. The modelling of passivebehaviour of the heart could be improved by including fibre sheet orientations and usingan orthotropic material law. A sensitivity study can be performed to see how sensitive the

43

results are to variations in fibre angle and what the influence is of including imbricationangles (see [7, 22]). This could show how important it is to incorporate patient specificfibre orientations rather than a general distribution.

5.2 Conclusion

The goal of this project was to develop a finite element model of the human left ventriclethat accurately represents anatomical features and fibre architecture, based on the dataset of the Utrecht group. The created model accurately represents the geometry of the leftventricular wall, but lacks detailed information about the basal region, aortic and mitralvalve and the papillary muscles. The model does contain information about muscle fibreorientation, but lacks information about the sheets that embed the fibres. The imbricationangle was neglected, resulting in a less realistic representation of the fibres in the apex andbase region. It was shown that the model is suitable for solving simple mechanics problems(passive inflation, isovolumic contraction and ejection of the left ventricle), but in order tomake the model useful for coupled mechanics and electrophysiology simulations it has tobe further expanded and improved.

44

Bibliography

[1] Auckland Bioengineering Institute. www.cmiss.org.

[2] PHM Bovendeerd. The mechanics of the normal and ischemic left ventricle during thecardiac cycle. PhD thesis, Rijksuniversiteit Limburg, Maastricht, The Netherlands,1990.

[3] PHM Bovendeerd, JMRJ Huyghe, T Arts, DH van Campen, and RS Reneman. In-fluence of endocardial-epicardial crossover of muscle fibers on left ventricular wallmechanics. J Biomech, 27:941–951, 1994.

[4] CP Bradley, AJ Pullan, and PJ Hunter. Geometric modelling of the human torsousing cubic hermite elements. Ann Biomed Eng, 25:96–111, 1997.

[5] KD Costa, JW Holmes, and AD McCulloch. Modelling cardiac mechanical propertiesin three dimensions. Phil Trans Royal Soc Lond A, 359:1233–1250, 2001.

[6] L Geerts, PHM Bovendeerd, K Nicolay, and T Arts. Characterization of normalcardiac myofiber field in goat measured with MR-diffusion tensor imaging. Am JPhysiol Heart Circ Physiol, 283:H139–H145, 2002.

[7] L Geerts, RCP Kerckhoffs, PHM Bovendeerd, and T Arts. Towards patient specificmodels of cardiac mechanics: a sensitivity study. In Functional imaging and modelingof the heart, pages 81–90, 2003.

[8] H Gelderblom. heart inf ivc eje.mpg, 2007. unpublished.

[9] L Glass, PJ Hunter, and AD McCulloch. Theory of the heart. ISBN 0-387-97483-0.New York: Springer-Verlag, 1991.

[10] JM Guccione, AD McCulloch, and LK Waldman. Passive material properties of intactventricular myocardium determined from a cylindrical model. J Biomech Eng, 113:43–55, 1991.

[11] AC Guyton and EJ Hall. Textbook of medical physiology. ISBN 0-8089-2187-8. W.B.Saunders Company, 2000.

45

[12] R Hren. A realistic model of the human ventricular myocardium:application to thestudy of ectopic activation. PhD thesis, Dalhousie University, Halifax, Canada, 1996.

[13] AM Katz. Physiology of the heart. ISBN 0-7817-5501-8. London: Lippincott Williamsand Wilkins, 2006.

[14] IJ LeGrice, PJ Hunter, and BH Smaill. Laminar structure of the heart: a mathematicalmodel. Am J Physiol Heart Circ Physiol, 41:H2466–H2476, 1997.

[15] MP Nash. Mechanics and material properties of the heart using an anatomicallyaccurate mathematical model. PhD thesis, The University of Auckland, Auckland,New Zealand, 1998.

[16] MP Nash and PJ Hunter. Computational mechanics of the heart. J Elasticity, 61:113–141, 2000.

[17] MP Nash and AV Panfilov. Electromechanical model of excitable tissue to studyre-entrant cardiac arrhythmias. Prog Biophys Mol Bio, 85:501–522, 2004.

[18] PMF Nielsen, IJ LeGrice, BH Smaill, and PJ Hunter. Mathematical model of geometryand fibrous structure of the heart. Am J Physiol Heart Circ Physiol, 260:H1365–H1378,1991.

[19] H Schmid, PO O’Callaghan, MP Nash, W Lin, IJ LeGrice, BH Smaill, AA Young, andPJ Hunter. Myocardial material parameter estimation. Biomech Model Mechanobiol,2007. published online.

[20] C Stevens. An anatomically-based computational study of cardiac mechanics and my-ocardial infarction. PhD thesis, The University of Auckland, Auckland, New Zealand,2002.

[21] KHWJ Ten Tusscher, R Hren, and AV Panfilov. Organization of ventricular fibrillationin the human heart. Circ Res, 100:e87–e101, 2007.

[22] SWJ Ubbink, PHM Bovendeerd, T Delhaas, T Arts, and FN van de Vosse. Towardsmodel-based analysis of cardiac MR tagging data: Relation between left ventricularshear strain and myofiber orientation. Med Image Analysis, 10:632–641, 2006.

46