POLITECNICO DI MILANO
School of Industrial and Information Engineering
Master of Science Degree in Mathematical EngineeringComputational Science and Engineering
Master’s Thesis
Active and Passive Mechanics
of the Atria of the Human Heart:
Mathematical and Numerical
Modelling
Advisor Dr. Luca Dede’ Candidate Valeria Mutinelli Matr. 882982Indice
Indice ii
Elenco delle figure iv
Elenco delle tabelle vii
Acknowledgements viii
Abstract ix
Sommario x
Introduction 1
1 Basic concepts of cardiac anatomy and physiology 4
1.1 Heart function . . . 4
1.2 Histology . . . 6
1.3 Physiology of the atria . . . 7
1.4 Anatomy of the atria . . . 8
1.5 Fiber architecture in the atria . . . 12
2 Mathematical modeling of the heart 19 2.1 Existing models of the atria . . . 20
2.2 A review of continuum mechanics . . . 21
2.3 Passive mechanics . . . 25
2.3.1 The Holzapfel-Ogden orthotropic model . . . 26
2.3.2 A transversely isotropic model . . . 29
2.3.3 The model proposed by Jernigan et al. . . 29
2.3.4 The models proposed by Bellini and Di Martino . . . . 30
2.4 Nearly-incompressible formulation . . . 31
2.5 Active mechanics . . . 33
Indice
2.5.2 Active strain approach . . . 33
2.6 Fibers reconstruction . . . 34
2.7 The full mechanics problem . . . 36
3 Numerical approximation 38 3.1 Weak formulation . . . 38
3.2 Space discretization . . . 39
3.3 Time discretization . . . 40
3.4 The Newton-Raphson method . . . 41
4 Idealized left atrium study 44 4.1 Construction of the idealized geometry of the left atrium . . . 45
4.2 Fibers generation . . . 47
4.3 Setting the numerical simulations . . . 55
4.4 Mechanical activation . . . 57
4.4.1 Linear activation in time . . . 57
4.4.2 Quadratic activation in time . . . 58
4.5 Mesh refinement tests . . . 61
4.6 Influence of the fibers on the deformation . . . 67
Conclusions 74
A Numerical solutions of a benchmark problem 76
B Non-unicity of solutions for a benchmark problem 81
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1.1 Schematic view of the heart. . . 5
1.2 Representation of the layers composing the heart wall. . . 6
1.3 Working myocytes. . . 7
1.4 Schematic representation of a sarcomere. . . 8
1.5 Right atrium components. . . 9
1.6 Lateral perspective of the right atrium. . . 10
1.7 Left anterior perspective of the right and left atrium obtained by dissection through the atria. . . 11
1.8 Major fiber bundles in the human left atrium. . . 14
1.9 Endocardial view of fibers orientation in the pectinate muscles and the crista terminalis in the right atrium. . . 15
1.10 Schematic representation of common pattern in the posterior aspect of left atrial specimen. . . 16
1.11 View of the roof and posterior wall of the left atrium. . . 17
1.12 Transmural changes in fiber orientation across the atrial wall. 18 2.1 Sketch of the fiber disposition in the left ventricle. . . 28
4.1 Different views of the idealized geometry of the left atrium. . . 46
4.2 Partition of the boundary of the domain for the fibers gener-ation problem. . . 47
4.3 Visualization of g1(x) and of g2(x). . . 49
4.4 View of the fibers field on the dome of the left atrium. . . 50
4.5 View of the circular bundle around the mitral valve orifice. . . 50
4.6 Lateral perspective of the fiber field. . . 51
4.7 Oblique perspective of the fiber field. . . 51
4.8 Top view of the longidutinal bundle (left) and lateral view of the whole fiber field (right) obtained by setting different values of φy in Eq. (4.1) and Eq. (4.2). . . 52
4.9 Fiber field obtained by setting the functions g1(x) and g2(x) equal to 0 in Eq. (2.55). . . 53
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4.10 Visualization of g1(x) on the anterior wall of the left atrium
for different values of γ. . . 54 4.11 Different perspectives of the fiber field for different values of γ. 54 4.12 Linear and parabolic activation functions. . . 58 4.13 Time evolution of the volume reduction and of the ejection
fraction with linear activation. . . 59 4.14 Time evolution of the volume reduction and of the ejection
fraction using three different values of the activation constant and three different values of the bulk modulus. . . 60 4.15 Time evolution of the 3D deformation of the idealized atrium
with parabolic activation. . . 62 4.16 Different of the final 3D deformation of the idealized atrium
with parabolic activation. . . 63 4.17 Final 3D deformation of the idealized for three different values
of the activation constant. . . 63 4.18 Stress along the fibers direction for different values of the bulk
modulus and of the activation constant. . . 64 4.19 Module of the stress along the fibers direction for different
values of the bulk modulus and of the activation constant. . . 65 4.20 Frontal and inferior view of the three different meshes used for
the simulations. . . 66 4.21 Time evolution of the volume reduction and of the ejection
fraction in time with three different meshes. . . 67 4.22 Different perspectives of the final 3D deformation obtained in
three different meshes. . . 68 4.23 Comparison of the 3D deformation of the idealized atrium with
three different meshes. . . 69 4.24 Time evolution of the volume reduction and of the ejection
fraction for different values of φy. . . 71
4.25 Time evolution of the volume reduction and of the ejection fraction for different values of γ. . . 71 4.26 Final 3D deformation of the idealized atrium at time t =
300 ms for three different values of φy. . . 72
A.1 Meshes adopted in Test 1, Test 2, Test 3 and Test 4. . . 77 A.2 Exponential activation functions adopted in Test 1, Test 2,
Test 3, Test 4. . . 78 A.3 L2- and H1-norms of the error u − u∗ in Test 1 (a), Test 2
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A.4 On the left, values of the L2-norm of the error in the
displace-ment vs. the basis degree computed in Test 1, Test 2, Test 3 ; on the right an analogous plot for the H1-norm of the error. . 80
A.5 Magnitude of the displacement u∗ in Test 1 (left) and Test 3
(right). . . 80 B.1 Displacement at time t = 100 ms using finite element of order
r= 2 on a mesh made of 4 × 4 × 1 elements (on the left) and of 8 × 8 × 2 elements (on the right). . . 82 B.2 Time evolution of the displacement using finite element of
or-der r = 1 (on the left) and r = 2 (on the right) on a mesh made of 4 × 4 × 2 elements. . . 83 B.3 Time evolution of the displacement using finite element of
or-der r = 1 (on the left) and r = 2 (on the right) on a mesh made of 8 × 8 × 2 elements. . . 84
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2.1 Material constants of the Holzapfel-Ogden orthotropic con-stitutive law. . . 28 2.2 Material constants of the isotropic specialization of the
Holzapfel-Ogden constitutive law. . . 29 2.3 Material constants of the Mooney-Rivlin constitutive law
pro-posed byJernigan et al. . . 30 4.1 Values of the angles that identify the position of the four
Acknowledgements
I would like to express my graditude to my advisor Luca Dede’ for his support, his availability and his valuable advice to my work.
I would like to thank all my fellow students and in particular Francesco, Gabriele, Martina, Nicola, Nicol`o, Sabrina, Salvatore, Silvia, Simone, Sonia, Stefano and Veronica for having shared with me these years at the Politecnico di Milano. I would never have been able to support the demandings of these studies without their aid and friendship.
I wish to express a special thanks to my high school mates Alberto and Elisa, for the sincere affection they showed me from the first day we met, and to Ambra, Claudia and Maria Sole for showing me the importance of being different in friendship.
I have to express my very profound gratitude to all my family and expe-cially to my parents, for their economical and emotional support in my studies and beyond, and to my sister Livia, for the uniqueness of our relationship.
Finally, I would like to thank Andrea, who has always been a point of re-ference for me since the very first day at Mathematical Engineering. Without his irreplaceable help and his constant support I would have never been able to achieve the results that I have pursed over these years.
Abstract
The aim of this work is to develop a mathematical model for the atrial active mechanics and to simulate their contraction for idealized geometries.
We consider a transversely isotropic model for the passive mechanics of the atria to take into account the anisotropy of the tissue. Since the heart muscle experiences small volume changes during its contraction, we opt for a nearly-incompressible formulation. In order to account for the active mech-anics, we follow the active stress approach. Then, we approximate the math-ematical model with the finite element method to simulate the atrial contrac-tion and we construct an idealized geometry used to perform the simulacontrac-tions. To include the fiber structure we propose a method aimed at qualitatively reproducing the main fiber bundles.
By means of the numerical simulations we analyze the deformations, the stress along the fiber direction, the fraction of blood ejected during the con-traction and the volume change of the muscle. To the best of our knowledge, this work represents one of the first attempts to build a simplified framework for the modelling of atrial mechanics.
Sommario
Questo lavoro si propone di sviluppare un modello matematico per la mec-canica attiva degli atri del cuore umano e di simulare la loro contrazione su geometrie idealizzate.
Per la meccanica passiva consideriamo un modello trasversalmente iso-tropo al fine di tenere in considerazione l’anisotropia del tessuto. Poich`e il volume del muscolo cardiaco si riduce leggermente durante la contrazione, utilizziamo una formulazione quasi-incomprimibile. Per includere la mecca-nica attiva, invece, seguiamo un approccio active stress. Al fine di simulare la contrazione degli atri approssimiamo il modello matematico con il metodo degli elementi finiti, dopodich`e costruiamo una geometria idealizzata per l’a-trio sinistro per svolgere le simulazioni. Per includere la struttura delle fibre, che negli atri `e particolarmente complessa, proponiamo un metodo finalizzato a riprodurre qualitativamente i principali fasci di fibre della muscolatura.
Sulla base dei risulati delle simulazioni analizziamo le deformazioni ot-tenute, gli sforzi nella direzione delle fibre, la frazione di volume di sangue espulsa dal cuore durante la contrazione e la riduzione di volume del muscolo cardiaco. Al meglio delle nostre conoscenze, questo lavoro rappresenta uno dei primi tentativi di costruire un contesto semplificato per la modellizzazione della meccanica atriale.
Introduction
The heart has the fundamental role to receive deoxigenated blood from the circulatory system, send it to the lungs, where the blood is oxygenated, and pump it back into the systemic circulation (Klabunde 2011). Cardiovascu-lar diseases represent the most common cause of death in the world. In 2016, about 17.6 milion deaths were associated to cardiovascular diseases globally (Benjamin et al. 2016). Over the past 25 year, mathematical and computational modeling of the heart has gained increasing popularity in the research communities since numerical simulations can give an insight into the complex behaviour underlying each component of the the cardiovascular sys-tem. However, outstanding challenges are still posed by the modelling of the functionalities of the system, among these the mechanics of the heart muscle, the electrophysiology and the fluid dynamics of the blood (Quarteroni, Ded`e et al. 2019).
The heart mechanics is the main object of this work. In particular, our study is aimed at modeling the atrial mechanics and providing an idealized geometry of the left atrium together with an admissible reconstruction of the major bundles of muscle fibers to simulate the active atrial contraction.
Our study is motivated by the need of a better understanding of atrial mechanics due to the paucity of models devoted to the atrial mechanics in the literature. Indeed, the majority of the studies conducted so far on the heart mechanics is based on the characterization of the ventricles (e. g. Gerbi et al. 2018; Quarteroni 2015; Rossi 2014) and, in particular, of the left vent-ricle, for its fundamental function of pumping the oxygenated blood received from the left atrium to the whole body through the circulatory system. Most existing atrial models, on the other hand, are limited to the study of elec-trophysiology, for the role played by atrial fibrillation in electrophysiology disfunctions. Atrial fibrillation is the most common cardiac arrhythmia that can causes several damages to the heart function if associated to other cardiac diseases. Few models couple the electrophysiology with the atrial mechan-ics (Adeniran et al. 2015; Di Martino et al. 2011; Fritz et al. 2013; Jernigan et al. 2007; Satriano et al. 2013) but in these works the anisotropic mechanical
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behaviour of the atrial muscle is neglected or treated relying on the models developed for the ventricles. For these reasons, since the 90s several con-stitutive models of different derivation have been developed for the ventricles (among these the much celebrated Holzapfel and Ogden (2009) model; see all the references therein) while, to the best of our knowledge, only the recent works of Jernigan et al. (2007), Bellini and Di Martino (2012) and Bellini, Di Martino and Federico (2013) have been devoted to the characterization of the atrial mechanics.
A topic to which we devote particular attention in our model is the recon-struction of the fiber architecture. The heart muscle, called myocardium, is made of bundles of fibers which are formed by the alignment of cardiac muscle cells and that are responsible of the anisotropic behaviour of the muscle. The propagation of the electrical signal that initiates the contraction spreads mainly along in the fibers direction, causing their simultanous shortening which results in the heart contraction. Therefore, in developing a model of the atrium mechanics, their inclusion is fundamental (Ho and S´anchez-Quintana 2009). However, the manner in which the fiber architecture of the atria can be generated still remains an open question. First, at an anatomical level, the fiber disposition in the atria is particularly complex (Ho, Anderson et al. 2002; Ho, Cabrera, Tran et al. 2001; Ho and S´anchez-Quintana 2009; Ho, S´anchez-Quintana et al. 1999; Wang et al. 1995), therefore a complete detailed description of the fiber architecture is not a simple task. Indeed, sev-eral bundles of fibers with different orientations can be recognized throughout the atria, and these bundles intersect each other, causing abrupt changes in their disposition. Moreover, the small thickness of the atrial walls prevents certain types of investigation on the transmural variation of the fiber orient-ation (Pashakhanloo et al. 2016); up to now, to the best of our knowledge, there is still not full knowledge of the changes in the orientation of the fibers throughout the atrial walls, even if the development and improvement of some new technologies as the diffusion tensor magnetic resonance imaging (DT-MRI) with submillimeter resolution has brought new important information and, in the future, is expected to improve the comprehension of the whole three-dimensional fibrous structure of the atria (Pashakhanloo et al. 2016). Furthermore, at the present, a patient-specific estimation in vivo of the fiber architecture is still not feasible (Fastl et al. 2018). As a consequence of the complexity of the fiber structure and of the description of its transmural vari-ation, the development of an efficient algorithm aimed at reproducing in a quite accurate way the fiber architecture is a complex task (Fastl et al. 2018). In several methods proposed in literature, the generation of the fibers is based on the solution of a Laplace problem. This procedure has been succesfully applicated to the ventricles (Wong and Kuhl 2014) in which, however, the
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fibrous structure is simpler than that in the atria and the fiber orientation varies quite smoothly across the wall thickness. Since this procedure can generate smoothly varying vector fields, it is suitable to the case of the vent-ricles, while its extension to the case of the atria presents some difficulties, other than being more involved. In fact, due to the abrupt changes in the fiber orientation in the atria, manual interventions of correction of the vec-tor field generated by the procedure are often required, increasing inter- and intra-variability (Fastl et al. 2018).
In this work we build an idealized three-dimensional geometry of the left atrium, preferred to the right one for its shape and its strict relation to the left ventricle. While the geometry of the left ventricle has often been simplified as a prolate ellipsoid, an analogous idealization, to our knowledge, has not been done for the left atrium before now. To generate the fiber field we propose a method, straightforward and totally repeatable, which could be able to represent the main fiber bundles. To model the atrial passive mechanic we adopt a transversely isotropic specialization of the constitutive law proposed by Holzapfel and Ogden (2009) for the left ventricle. The modeling of the active mechanics is based on the active stress approach (G¨oktepe and Kuhl 2010; Panfilov et al. 2005; Smith et al. 2004).
The thesis is structured as follows. In Chapter 1 we overview some ba-sic concepts of cardiac anatomy and physiology, with a particular attention to the anatomy of the atria and to the atrial musculature. Chapter 2 is concerned with the mathematical modeling of the heart mechanics. The numerical approximation of the mechanics problem with the finite element method is given in Chapter 3. Chapter 4 focuses on the simulations on the left atrial idealized geometry. We first present the construction of the geo-metry, then we discuss the application of the method for the fiber generation presented in Chapter 2 and we show the results of some tests performed on the model. Finally, we draw the Conclusions and future developments of this work.
Chapter 1
Basic concepts of cardiac
anatomy and physiology
In this chapter we recall some fundamental elements of cardiac anatomy that are essential to develop the mathematical model of the heart. Since the aim of this work is to study a model for the mechanics of the atria, particular attention will be devoted to the anatomy of these chambers.
In the first section we describe briefly the heart structure and the cardiac cycle; then we mention the cardiac cells responsible of the mechanical and electrical behaviour of the heart in the second section. The third and fourth sections focus on the physiology and the anatomy of the atria and lastly, the fifth section provides a detailed description of the atrial musculature.
1.1
Heart function
The heart is a hollow muscular organ of size comparable to a closed human fist, located in the middle compartment of the chest between the lungs. Its function is to receive deoxigenated blood from the circulatory system, send it to the lungs, where the blood is enriched with oxygen, and pump the oxygenated blood back into the systemic circulation.
The heart consists of four chambers: the right atrium (RA), the right ventricle (RV), the left atrium (LA) and the left ventricle (LV), as depicted in Figure 1.1. The right atrium collects venous blood from the main circulatory system via the inferior and superior caval vein (IVC/SVC) and from the heart muscle via the coronary sinus (CS), which runs in the left atrioventricular groove on the posterior side of the heart and collects blood coming from the coronary veins. Then, the right atrium passes the blood to the right ventricle through the triscuspid valve (TV). The blood is subsequently pumped from
1.1 - Heart function
Figure 1.1: Schematic view of the heart. Blue and red arrows represent the direc-tion of the blood flow; blue arrows follow the direcdirec-tion of the deoxigenated blood, while red arrows show the path of the oxygenated blood. The chambers and the vessels coloured in blue are those which collect deoxygenated blood, while those in red receive oxygenated blood. The white curved lines represent the valves. Image taken from Franzone et al. (2014).
the left ventricle into the pulmonary artery through the pulmonary valve, and carried to the lungs. The left atrium collects oxygenated blood coming from the lung circulatory system, usually through four pulmonary veins. Then, it passes the blood through the mitral valve (MV) to the left ventricle, which pumps the blood through the aortic valve into the aorta, that carries the blood into the systemic circuit and to the heart itself via the the coronary arteries, which originate from the ascending part of the aortic arch. The phase of the cardiac cycle in which the ventricles contract is called systole, while the part in which the two chambers relax is called diastole.
The function of the atria is generally to collect, while the function of the ventricles is to pump. The atria are electrically isolated from the ventricles by fibro-fatty tissues at the valve plane.
The valves allow blood flows only in one direction, and their opening and closure is induced by pressure gradients.
1.2 - Histology
Figure 1.2: Representation of the three layers composing the heart wall.
A schematic representation of the four chambers, the main vessels and the valves is shown in Figure 1.1.
1.2
Histology
The heart wall is composed of three layer: the epicardium, the myocardium and the endocardium. The epicardium is an external membrane of thick-ness of the order of 100µm; the middle layer, the myocardium, is the heart muscle; the endocardium is a serous membrane with a thickness of 100µm that adheres to the inner wall of the myocardium and serves as an interface between the muscle and the blood. A schematic representation of the three layers is depicted in Figure 1.2.
The myocardium constitutes the major part of the heart wall. Myocardial tissue is constituted by muscle cells called cardiomyocytes and connective tissue. Several types of myocytes can be distinguished in the adult human heart. Among them, we cite the working myocytes, responsible of the muscle mechanical contraction, and the nodal cells, responsible of pacemaker activity in the sino-atrial node and of ventricular conduction delay in the atrio-ventricular node. For a more detailed description we refer to Katz (2010).
Working cardiomyocytes have an elongated shape (Figure 1.3a); in gen-eral, they have a shorter width of about 10-40µm and a longer dimension around 50-200µm, which is the direction of the mechanical force generation axis (Iaizzo 2009). The mechanical force produced by these cells is generated by fundamental contractile units called sarcomeres; these structures are made
1.3 - Physiology of the atria
(a) Sketch of working myocytes. (b) Myocardial tissue.
Figure 1.3: Working myocytes. Figure 1.3a: working myocytes are elongated cells connected one to each other through intercalar discs. Figure 1.3b: connections of working myocytes (in pink) form a macroscopic organization bundles of fibres with the same general orientation; the nucleus of the cells are coloured in purple.
of sliding filaments that, with their shortening and elongation, produce the contraction and relaxation of the whole heart muscle. Working cardiomyo-cytes are in general composed of one nucleus and possess branches that link them with the neighbouring myocytes.
At the extremities of the cardiomyocytes, structures called intercalar discs provide mechanical linkages between the cells. Inside the intercalar discs, structures called gap junctions, characterized by low electrical resistance, allow a fast and synchronous propagation of the electrical signal. Thanks to these linkages, cardiomyocytes form long chains of cells with the same general orientation (see Figure 1.3b) and these chains are often referred to as fibers.
The myocardium is an involuntary muscle and the nervous system can modulate, but not initiate, its contraction, whose rate consists of about 80 contractions/min (Iaizzo 2009).
1.3
Physiology of the atria
The atria perform three basic mechanical functions: reservoir, conduit and active conctractile pump.
The reservoir phase occurs at the closure of the atrioventricular valves, when the blood coming from the pulmonary veins in the left atrium and from the superior and inferior caval vein in the right atrium flows into the two chambers during the ventricular systole.
The conduit phase identifies the passage of the blood from the atria to the ventricles during early diastole, after the opening of the atrioventricular valves, when the blood flow is driven only by the pressure gradients between
1.4 - Anatomy of the atria
Figure 1.4: Schematic representation of a sarcomere. Sliding filaments (Titin, Actin, Myosin) induce the shortening of the contractile unity.
the two chambers.
The phase of the active contractile pump denotes the active contraction of the atria that occurs in the late diastole, to facilitate the passage of the blood into the ventricles.
1.4
Anatomy of the atria
Right atrium
The right atrium is conventionally divided in four components: the septum, the appendage (or auricule), the venous component and the vestibule (Ho, Anderson et al. 2002), as depicted in Figure 1.5.
The septum is the portion of the wall common to both the atria. The septal surface of the right atrium is marked by the valve of the fossa ovalis, which is the remnant of a thin fibrous sheet that, during fetal development, covered the foramen ovale, that allowed blood flow between right and left atria.
The appendage is situated anterior and laterally, and it is the most prom-inent external feature of the right atrium. In the interior of the right atrium,
1.4 - Anatomy of the atria
Figure 1.5: Perspective of the right atrium opened by a cut through its append-age parallel to the atrioventricular junction, showing the four components of the chamber (septum, appendage, vestibule and venous component). A portion of the wall of appendage is reflected upwards. Image taken from Anderson et al. (2010).
a thick portion of heart muscle called crista terminalis marks the union between the right atrium and the appendage. An array of pectinate muscles arises from the crista terminalis and spread through the appendage, covering the lateral and inferior wall of the right atrium. These muscles have a char-acteristic comb-tooth shape, as their name suggests. The terminal grove, or sulcus terminalis, corresponds externally to the crista terminalis.
The venous component is characterized by smooth walls. While in the entrace of the superior caval vein there is not any valve, the entrance of the inferior caval vein is guarded by the eustachian valve. Sometimes sleeves of myocardium surround the entrance of the superior caval vein.
The vestibule, a smooth muscular rim, surrounds the tricuspid valve. For a sketch of all these structures we refer to Figure 1.6.
Left atrium
As well as the right atrium, the left atrium is divided in four components: septum, appendage, venous component and vestibule (Ho, Anderson et al. 2002), as depicted in Figure 1.7.
The site of the atrial septum is represented by the flap valve of the oval foramen in right wall of the left atrium. The left atrial appendage extends anteriorly over the atrioventricular (or coronary) sulcus and it can have
1.4 - Anatomy of the atria
Figure 1.6: Lateral perspective of the right atrium. The lateral wall of the right atrium occupied by the pectinate muscles has benn partially removed. SA node indicates the atrio-ventricular node; SVC, superior caval vein; IVC inferior caval vein.
many shapes among individuals. The walls of the left atrial appendage are pectinate, while the walls of the left atrium are smooth, and the division between these rough and smooth walls is at the mouth of the appendage.
The pulmonary veins arose from the superior and posterior walls of the left atrium, an area that is usually referred as the dome of the left atrium, with the left veins located more superior than the right veins. The pulmon-ary veins pass immediately behind the superior caval vein. Typically there are four pulmonary veins, but their number can vary among individuals (Ho, S´anchez-Quintana et al. 1999 and Nathan and Eliakim 1966). Sometimes, two veins of one, or both, sides are united prior to their entry to the at-rium. Typically, sleeves of myocardium surround the proximal portion of the pulmonary veins, but the lengths of the sleeves vary from heart to heart and from vein to vein (Ho, S´anchez-Quintana et al. 1999 and Nathan and Eliakim 1966). Nathan and Eliakim (1966) found that myocardial sleeves are better developed around the superior pulmonary veins than around the inferior veins. The function of the cardiac muscle in the pulmonary venous wall has not been investigated thoroughly but some studies reported that myocardial sleeves are longest in the superior pulmonary veins, correspond-ing to the highest frequency of ectopic focuses (Ho, S´anchez-Quintana et al. 1999, Haissaguerre et al. 1998, Saito et al. 2000). Related posteriorly and
1.4 - Anatomy of the atria
Figure 1.7: Left anterior perspective of the right and left atrium obtained by dissection through the atria. Image taken from Ho, Cabrera and S´anchez-Quintana (2012).
inferiorly to the left atrium is the coronary sinus which occupies the left atrioventricular groove.
The vestibular component is the smooth circumferential area surrounding the orifice of the mitral valve.
Left atrial volume and wall thickness
Atrial volume and atrial thickness can vary very much from patient to patient, depending, for example, on the gender, the weight, the age and the history of cardiac diseases. The atrial volume determines the maximum volume of blood that can be stored in the atrial cavity. Lang et al. (2006) reported a reference value of 22 − 52 mL of the left atrial volume in women and a value of 18 − 58 mL in men in the absence of cardiac diseases. This values can grow strongly in presence of pathologies, leading to volumes grater than 73 mL for women and 79 mL for men. Similar values were reported by Pritchett et al. (2003). Structural remodeling caused by atrial fibrillation (AF) commonly increases left atrial volume (Schotten et al. 2003).
Several studies have been done to measure the local wall thickness of the atria, and a partial list of literature regarding wall thickness is reported in Kr¨uger (2013, p. 16). All these studies reported that walls of the atria are non uniform in thickness. For example, Nathan and Eliakim (1966) reported mean values ranging from 0.8 to 1.8 mm of the atrial wall thickness between
1.5 - Fiber architecture in the atria
the pulmonary veins; in Ho, S´anchez-Quintana et al. (1999), the authors reported a mean thickness of 3.3 mm in the anterior of the left atrium, of 4.1 mm in the posterior, of 4.5 mm in the superior, of 3.9 mm in the lateral wall and of 2.3 mm in the vestibule. For a more detailed description of the variation of the atrial wall thickness, we refer to Ho, Cabrera and S´anchez-Quintana (2012).
Again, atrial fibrillation can induce changes in wall thickness in the medium-long term.
1.5
Fiber architecture in the atria
It is well-estabilished that the understanding of the architecture of atrial musculature is fundamental for the comprension of the activation and the contraction of the heart, and it has to be taken into account when developing computer models (Ho, Cabrera, Tran et al. 2001). The reconstruction of a precise patient-specific map of the fibers in the atria still remains a challenge due to the complexity of the disposition of the bundles and the small thickness of the atrial walls, which hinders certain from of investigation as, for example, dissection (Pashakhanloo et al. 2016).
In this section, a brief state of the art regarding fiber architecture is summarized. Then, the main features of the fibers arrangement as reported in literature are discussed.
We adopt here the terminology defined by Ho, S´anchez-Quintana et al. (1999): the term fibers describes “the macroscopic appearance of the strands of myocytes that impart a grain-like appearance to the atrial wall”; in the left atrium fibers are circumferential when “they run parallel to the plane of the mitral orifice” and longitudinal “when they are arranged at right angles to the orifice”; the term bundle refers to “a collection of fibers with the same general orientation” and a layer “indicates a change of fiber orientation from the more superficial to the deeper fibers”.
State of the art
The first works of morphologists that mentioned atrial musculature are those of Keith (1907), Keith and Flack (1907) and Papez (1920). The first work that deeply investigated the main fiber bundles in the left human atrium is probably that of Nathan and Eliakim (1966), with a particular attention to the junction between the atrium and the pulmonary veins. In the work of Wang et al. (1995), 9 normal postmortem human hearts were analysed. By dissection of atrial muscles with macrophotography, the gross
arrange-1.5 - Fiber architecture in the atria
ment of muscular bundles on sub-epicardium and super-endocardium was examined, confirming a circumferential arrangement at the base of the atria and a longitudinal arrangement predominating in the parietal walls. With the same methodology adopted in Wang et al. (1995), the authors in Ho, S´anchez-Quintana et al. (1999) reviewed the gross structure of the left atrium, highlighting the complexity of the musculature made of overlapping bundles of fibers. By charting the layers of the atrial musculature on transparent sheets and by stacking them, they were able to map the major transmural changes in the fiber orientation. Ho, Anderson et al. in 2002 revisited the gross morphology of the fiber arrangement of both the atria to provide a morphological basis for atrial conduction and they also provided a compar-ison of the heart structure between humans and animals commonly used in experimental laboratory. In the following years other studies have investig-ated more deeply some specific structure or sites of the atria. Among these studies we mention those of Cabrera et al. (2008) and S´anchez-Quintana et al. (2002). In Markides et al. (2003), the relation between patterns of activation of the human left atrium and atrial myocardial architecture has been investigated.
An unprecedented level of information on human myoarchitecture was obtained by Pashakhanloo et al. (2016) with the first submillimeter diffusion tensor magnetic resonance imaging (DT-MRI) acquisition of fiber architec-ture in human atria. In their work, entire intact human heart specimens ex vivo were imaged. The submillimeter resolution allowed quantitative meas-urements of the transmural variation of fiber orientation and a semiautomatic clustering was performed in order to group the set of bundles common to all specimens.
Main atrial bundles
We limit ourselves to mention here only the principal bundles of fibers in-dividuated in the right and left atrium. For a more detailed description we refer to the articles previously cited.
The main bundle on the posterior wall of the left atrium runs longitud-inally along the dome (Figure 1.8, bundle c and Figure 1.11) and is recog-nised in literature as the septopulmonary bundle (Papez 1920). Even if the longitudinal orientation seems to be the most common in the atrial dome, other patterns are not unusual, as documented in Nathan and Eliakim (1966) and Pashakhanloo et al. (2016). Nathan and Eliakim defined four types of predominant patterns: vertical, horizontal, oblique and mixed, reported in Figure 1.10.
1.5 - Fiber architecture in the atria
Figure 1.8: Major fiber bundles in the human left atrium (LA) found by Pashakhanloo et al. using a cluster algorithm on MRI data and tractography. The color map reveals the transmureal position of the fibers between the endo-cardial layer (yellow) and the epiendo-cardial layer (red). A–C, Posterior. D and E, Anterior view. F, Septum. CS indicates crista terminalis; LAA, left atrial append-age; LIPV, left inferior pulmonary vein; LSPV, left superior pulmonary vein; MV, mitral valve; OF, oval fossa; RAA, right atrial appendage; RIPV, right inferior pulmonary vein; RSPV, right superior pulmonary vein; and SVC, superior caval vein. Image taken from Pashakhanloo et al. (2016).
1.5 - Fiber architecture in the atria
Figure 1.9: Endocardial view of fibers orientation in the pectinate muscles and the crista terminalis in the right atrium (RA), found by Pashakhanloo et al. using a cluster algorithm on MRI data and tractography. The fibers follow the tra-beculated structure. The color map reveals the transmural position of the fibers between the endocardial layer (yellow) and the epicardial layer (red). Image taken from Pashakhanloo et al. (2016).
is crossed by a set of fibers that runs circumferentially around the base of the left atrium, as depicted in Figure 1.11.
On the sleeves of myocardium around the pulmonary veins, bundles of fibers run cirumferentially to the orifices (Figure 1.8A, bundle a1,
Fig-ure 1.8C, bundle a2, Figure 1.10 and Figure 1.11).
An interatrial bundle, known as the Buchmann’s bundle, originates from the superior caval vein and encircles the inferior part of the anterior wall of the left atrium until the left atrium appendage, where it bifurcates into two bundles that split around the origin of the appendage (1.8E, bundle l). Fibers coming from the pulmonary veins and from the posterior wall of the left ventricle encircle the fossa ovalis (Figure 1.8F).
In the right atrium wall, the orientation of the fibers generally follows the trabeculated structure of the endocardial wall (Pashakhanloo et al. 2016), as depicted in Figure 1.9.
Concering the transmural changes in fiber orientation, differences in the fiber distribution between the endocardial layer and the epicardial layer have been documented in several works, e. g. in that of Ho, S´anchez-Quintana et al. (1999). However, it was only in the work of Pashakhanloo et al. (2016) that the distribution across the whole wall has been documented in a quantit-ative manner, thanks to the use of DT-MRI and clustering algorithms instead of simple dissection. The work of Pashakhanloo et al. revealed a regional
het-1.5 - Fiber architecture in the atria
(a) Predominant vertical pattern. (b) Predominant horizontal pattern.
(c) Predominant oblique pattern. (d) Predominant mixed pattern.
Figure 1.10: Schematic representation of common pattern in the posterior aspect of left atrial specimen. Image taken from Nathan and Eliakim (1966).
1.5 - Fiber architecture in the atria
Figure 1.11: View of the roof and posterior wall of the left atrium with transil-lumination. LS indicates left superior pulmonary vein; LI, left inferior pulmon-ary vein; ICV, inferior caval vein. Image taken from Ho, Cabrera and S´ anchez-Quintana (2012).
erogeneity of the fiber distribution from endocardium to epicardium, showing that in some regions, as in the lateral wall and on the dome of the left at-rium, the orientation of the fibers does not change from endocardium to epicardium, while abrupt variations are present in other regions, where two layers of orthogonal fibers intersect each other in the middle of the atrial wall, revealing a bilayer structure (Figure 1.12).
1.5 - Fiber architecture in the atria
Figure 1.12: Transmural changes in fiber orientation across the atrial wall (pos-terior view) found by Pashakhanloo et al. A, histograms and profiles against the thickness of the atrial wall of the local fiber orientation in two points (A and B) of the posterior left atrial wall reveal a local bilayer structure, with perpen-dicular bundles of fibers. B, transmural dispersion of fiber orientation. LIPV indicates left inferior pulmonary vein; LSPV, left superior pulmonary vein; RIPV, right inferior pulmonary vein; RSPV, right superior pulmonary vein. Image taken from Pashakhanloo et al. (2016).
Chapter 2
Mathematical modeling of the
heart
Over the course of the cardiac cycle the myocardium undergoes large de-formations, therefore the finite elasticity theory is an appropriate framework for the mathematical description of heart mechanics (Quarteroni, Ded`e et al. 2019). The myocardium is classified as a hyperelastic fiber-reinforced ma-terial (Holzapfel and Ogden 2009), due to the ability of the heart muscle to support a periodic load without experiencing a permanent deformation and the presence of myocytes which group into oriented muscle bundles that strengthen the material response when a load is applied along their direction. Living tissue has the ability to actively deform (Ambrosi, Arioli et al. 2011). Therefore, in the study of a mathematical model for the myocaridum, it is necessary to model the passive mechanics, which is the passive elastic accomodation of tissue subjected to an external force, and the active mech-anics, which stands for the ability of the pacemaker cells to actively generate the electric pulse that drives the mechanical contraction of the muscle.
Since the propagation of the electrical signal and the contraction of the myocytes occur along the fiber direction, an accurate reconstruction of the fiber orientation is necessary when modelling the mechanics as well as the electrophysiology of the heart.
In this chapter we discuss all these aspects. In Sec. 2.1, we mention the ex-isting models for the atria. In Sec. 2.2, a brief review of continuum mechanics recalls some fundamental concepts of kinematics, stress, balance principles, and hyperelasticity necessary to introduce the mathematical model of the atria. Sec. 2.3, 2.4 and Sec. 2.5 are devoted to the passive and active mech-anics. In Sec. 2.6 we mention the methods commonly used for the generation of the fibers in the atria, highlighting the differences with the case of the ventricles. Finally, we formulate the full mechanics problem in Sec. 2.7.
2.1 - Existing models of the atria
We remark that, in our treatment, we will often mention the case of the ventricles; indeed, the first mathematical models of the heart were developed for these chambers, in particular for the left ventricle. Therefore, the study of the models of the electromechanics of the ventricles have been a point of reference for the development of those of the atria.
2.1
Existing models of the atria
Due to the importance of atrial fibrillation, the most common arrhythmia which increases the risk of developing other cardiovascular diseases (Fastl et al. 2018), most of the computational models developed for the atria are devoted to electrophysiology. Some of these are 3D models (e. g. Dokos, Cloherty et al. 2007; Harrild and Henriquez 2000; Seemann et al. 2006) based on anatomically realistic atrial geometries; nevertheless, due to the small thickness of the atrial walls, these models are computationally expensive, therefore shell models, which consider the atria as 2D membranes, have often been preferred (e. g. Al Abed et al. 2009; Chapelle et al. 2013; Rotter et al. 2007; Virag et al. 2002).
Very few models couple the electrophysiology with the mechanics. In some of these the mechanical behaviour of the atria was assumed to be iso-tropic and the anisotropy induced by the presence of the fibers was regarded only for the propagation of the electrical pulse in solving the problem of electrophysiology (Di Martino et al. 2011; Jernigan et al. 2007; Satriano et al. 2013); others treat the atrial myocardium as transversely isotropic (Ad-eniran et al. 2015; Fritz et al. 2013) adopting the Guccione constitutive law developed for passive ventricular myocardium (Guccione et al. 1991).
To the best of our knowledge, the only works devoted to the characteriz-ation of the mechanics of the atria on the base of experimental data on atrial tissue are those of Bellini and Di Martino (2012), Bellini, Di Martino and Federico (2013) and Jernigan et al. (2007).
The model of Jernigan et al. (2007) was based on experiments on porcine atrial tissue specimens; passive left atrial tissue was characterized with a Mooney-Rivlin hyperelastic model and the material constants of the model were derived by uniaxial tensile tests. The adoption of an isotropic model was justified by the observation that stress vs. strain relationships did not significantly depend on the fibre direction. To the best of our knowledge, the model of Jernigan et al. (2007) was the first model to provide specific material parameters for the atria based on experimental data.
Opposite results were obtained by Bellini and Di Martino (2012) with planar biaxial tests performed on tissue specimens from pigs; in their findings,
2.2 - A review of continuum mechanics
the atrial tissue showed an orthotropic behaviour. Moreover, while Jernigan et al. did not observed any variation of the mechanical response throughout the left atrium, changes between the anterior and the posterior of the left atrium were observed by Bellini and Di Martino. A further contribution of Bellini and Di Martino (2012) was to fit experimental data with two dif-ferent anisotropic constitutive relashionships, namely a four-parameter Fung model and a microstructurally-motivated model, providing average material parameters for the two constitutive laws.
The work of Bellini, Di Martino and Federico (2013) was the first devoted to the mechanical characterization of tissues from human atria and based on experimental measurements collected with planar biaxial tests. In this case a model based on a Fung-type elastic strain energy potential was chosen and material parameters obtained by fitting the data from human atrial tissue specimens were provided.
2.2
A review of continuum mechanics
Consider a continuum B, namely a body whose mass and volume are con-tinuous functions of continuum particles. The continuum B is supposed to be moving in time and space, from an initial configuration Ω0, called
ref-erence (or undeformed ), corresponding to the initial time t = 0, to a new region Ω, called current (or deformed ) configuration, at a subsequent time t >0. Therefore, during its motion, the continuum B occupies a continuous sequence of geometrical regions denoted by Ω0, . . . ,Ω, and every particle of
B corresponds to a geometrical point in Ω0, . . . ,Ω. The position vector X
identifies the position of a particle P ∈ B in the reference configuration, while the position vector x identifies the position of P in the current configuration. The motion of the body B is described by the map χ such that
x = χ(X, t)
for all X ∈ Ω0 and for all times t > 0. The motion χ carries points X ∈ Ω0
to points x ∈ Ω, and is assume to be uniquely invertible. The vector field
U(X, t) = x(X, t) − X (2.1)
represents the displacement field of a particle P occupying a position X ∈ Ω0
at time t = 0 and moving to the position x ∈ Ω at time t > 0. The displacement field in the spatial description is denoted by u and is defined as
2.2 - A review of continuum mechanics
By means of the motion χ it comes straightforward that U(X, t) = u(x, t)
(see Holzapfel 2002); therefore, with an abuse of notation, we will always refer to u even when dealing with the reference configuration. The behaviour of a motion in the neighborhood of a point is characterized by the quantity F called the deformation gradient, defined as
F(X, t)..= ∂χ(X, t)
∂X .
The tensor
C..= FT
F
is called the right Cauchy-Green strain tensor or Green deformation tensor. By definition, it is symmetric and positive definite (detC = (detF)2 >0) for
each X ∈ Ω0. We denote with J the determinant of F:
J(X, t)..= det(F(X, t))
and since F is invertible, it must be J(X, t) 6= 0. Moreover, from the defin-ition of F ad J, it comes that, if dV and dv denote infinitesimal volume elements in the reference and in the current configuration respectively, then
dv= J(X, t)dV (2.3)
and deformations such that J(X, t) < 0 are not allowed due to the impenet-rability of matter.
We denote with ρ0(X) and with ρ(x, t) the mass density of the particle
P ∈ B occupying the position X in the material configuration and the posi-tion x = χ(X, t) in the current configuraposi-tion. The conservaposi-tion of mass (or continuity mass equation) is represented (see Holzapfel 2002) by the formula
ρ0(X) = J(X, t)ρ(χ(X, t), t)
that holds for every X ∈ Ω0.
The velocity field of the continuum body B with a given motion χ(X, t) is defined as
V(X, t)..= ∂χ(X, t)
∂t (2.4)
and the spatial description of the velocity is obtained by means of the motion χ:
2.2 - A review of continuum mechanics
In order to recall the concept of stress in continuum mechanics, we consider a continuum body occupying a region Ω at time t, subjected to forces acting on its surface (external forces) or within the interior of the body (internal forces). Consider two arbitrary infinitesimal surface elements dS and ds lying on an imaginary section of the body in the material and current configuration respectively. We claim that there exist two traction vectors t and T such that the infinitesimal resultant force df acting on the surface elements is given by
df = tds = TdS
where in principle t = t(x, t, n) and T = T(X, t, N), n and N being the outer normal to the surface elements ds and dS respectively. By the Cauchy’s stress theorem there exist unique second-order tensor fields σ and P such that
t(x, t, n) = σ(x, t)n, (2.6)
T(X, t, N) = P(X, t)N, (2.7)
i.e. the traction vectors depend linearly on the normals to the surface ele-ments. The tensor σ is called the Cauchy (or true) stress tensor, while P is called first Piola-Kirchhoff (or nominal ) stress tensor. It can be proven that σ is symmetric. The transformation between the two spatial tensors is given by the Nanson’s formula:
P = JσF−T.
We now introduce the balance of linear momentum, the fundamental equation that, at any time, provides the displacement of every point of a continuum body that is moving subjected to some force. In a closed system, the total linear momentum L is defined as:
L(t) = Z
Ω
ρ(x)v(x, t) dv. (2.8)
The balance of linear momentum is postulated as: ˙L(t) = D
Dt Z
Ω
ρ(x)v(x, t) dv = F(t), (2.9)
where F(t) is the resultant force acting on the body B. If we denote with b = b(x, t) the body forces acting on B and with t = t(x, t, n) the surface forces which the body is subjected to, then the resultant force F(t) acting on B is given by: F(t) = Z ∂Ω t ds + Z Ω b dv. (2.10)
2.2 - A review of continuum mechanics
Therefore, from (2.9) and (2.10) the balance of linear momentum in the spatial description reads:
D Dt Z Ω ρv dv = Z ∂Ω t ds + Z Ω b dv. (2.11)
From Eq. (2.6) and the divergence theorem it follows that: Z ∂Ω t(x, t, n) ds = Z ∂Ω σ(x, t)n ds = Z Ω ∇ · σ(x, t) dv (2.12) Moreover, it can be proven that:
D Dt Z Ω ρv dv = Z Ω ρ˙v dv (2.13)
Therefore, from Eq. (2.11), (2.12), and (2.13), it follows the Cauchy’s first equation of motion:
Z
Ω
(ρ ˙v − ∇ · σ − b) dv = 0 (2.14)
Since the domain Ω is arbitrary, the equation (2.14) can be expressed in the differential form:
ρ˙v − ∇ · σ − b = 0 ∀x ∈ Ω, ∀t > 0 (2.15) Since in many applications the current configuration in unknown, equations (2.14) and (2.15) are often expressed in the material description, where they take the form (for the proof see Holzapfel 2002):
Z
Ω0
(ρ0V − ∇˙ 0· P + −B) dV = 0 (2.16)
ρ0V − ∇˙ 0· P − B = 0 ∀X ∈ Ω, ∀t > 0. (2.17)
The mechanics of myocardium will be determined by solving the momentum balance equation (2.17), provided that appropriate boundary, initial condi-tions and a constitutive law for the frist Piola-Kirchoff P tensor are given. We postpone the treatment of the boundary and the initial conditions, since they depend on the specific application. Instead, for what concerns the con-stitutive law for the tensor P, we now introduce some basic concepts of the theory of hyperelasticity since many models for passive myocardium have been developed in this context.
2.3 - Passive mechanics
A material is said to be hyperelastic if there exist a function W depending on the deformation gradient F called Helmholtz free-energy function such that
P = ∂W(F)
∂F . (2.18)
Since the amount of energy generated in a body by the motion χ(x, t) should not change after a translation or rotation of the body (i.e. a rigid motion), the Helmholtz free-energy must satisfy
W(F) = W(QF) (2.19)
for all the proper orthogonal tensors Q. If W satisfies (2.19), then it is said to be objective. Eq. (2.19) can be also stated in an alternative form, relying on the right Cauchy-Green tensor. In this regard, we recall the polar decomposition theorem, which states that, for every non singular tensor F, there exist a unique decomposition
F = RU = vR, (2.20)
where R is a proper orthogonal tensor with det(R) = 1 and U and v are positive definite symmetric tensors which measure the local stretching along their mutually orthogonal eigenvectors in the reference and current config-uration respectively. With the particular choice Q = RT in Eq. (2.19), one
has that
W(F) = W(RTF) = W(RTRU) = W(U), (2.21)
which shows that the Helmholtz free-energy depends only on the stretching part U of the deformation gradient F. Since C = FTF = UTRTRU = U2,
equation (2.19) can be stated as
W(F) = W(C). (2.22)
Therefore, expressing the Helmholtz free-energy as a function of C is a ne-cessary (thanks to the uniqueness of the polar decomposition) and sufficient condition for the objectivity of W.
2.3
Passive mechanics
We now focus on the specific expression that the energy function W assumes in the case of materials made of myocardial tissue. In Sec. 2.1 we reported the only existing models for the characterization of the atrial mechanics: in our opinion, the model of Jernigan et al. (2007) is too reductive in considering
2.3 - Passive mechanics
the atrial myocardium as isotropic, while the model of Bellini, Di Martino and Federico (2013), of complex derivation, requires a non trivial definition of three mutually orthogonal directions of anisotropy that change throughout the atria; consequentely we adopted a transversely isotropic specialization of the orthotropic model proposed by Holzapfel and Ogden (2009), which is the reference model in cardiac mechanics and has been developed for the mechanics of the left ventricle.
We first introduce the orthotropic constitutive model of Holzapfel and Ogden (2009) and then, we derive its isotropic specialization; finally, for the sake of completeness, we think that it is worthwhile to report also the models of Bellini and Di Martino (2012) and Jernigan et al. (2007).
2.3.1
The Holzapfel-Ogden orthotropic model
The assumption underlying the model proposed by Holzapfel and Ogden (2009) is that the myocardium of the left ventricle is a orthotropic, thick-walled, nonlinear elastic and incompressible material. The constitutive law was developed on the base of the use of invariants related to the myocardium structure; for more details on the invariant theory of fibre-reinforced material, see Spencer et al. (1984).
In order to define the orthotropic materials, we first introduce the notion of simmetry group. A simmetry group describes all the material symmetries of a body, i.e. all the possible rotations of the body that do not change its stress state. In the continumm mechanics theory, the simmetry group S of a body is defined as S =Q ∈ O(3) : σ(FQ) = σ(F) ∀F ∈ Lin+ (2.23) or, equivalently, S =Q ∈ O(3) : W(FQ) = W(F) ∀F ∈ Lin+ (2.24) where Lin..=F : R3
→ R3 linear application, det(F) 6= 0
(2.25)
Lin+ ..= {F ∈ Lin : det(F) > 0} (2.26)
O(3) =Q ∈ Lin : QTQ = I . (2.27)
A hyperelastic material is said to be orthotropic if the simmetry group of the Helmholtz free-energy W is
2.3 - Passive mechanics
where f0,s0,n0 are the three mutually orthogonal directions. This means
that the stress state of the body resulting after a certain deformation does not change if the body is rotated along one of these three directions before being deformed.
In the case of the ventricular myocardium, fibers are organized in layers (or sheets) with different orientation across the ventricular wall. Fibers in the same layer have the same orientation, which varies between +50◦ to
+70◦ in the sub-epicardial region and −50◦ to −70◦ in the sub-endocardial
region with respect to the circumferential direction of the left ventricle. In the definition (2.28), f0 represents the fiber direction, s0 the sheet (or
cross-fiber) direction and n0 the sheet-normal direction, defined to be orthogonal
to the other two. A schematic representation of the fiber disposition in the left ventricle is reported in Figure 2.1.
In order to express the energy function W in terms of C, as stated by Eq. (2.22), several invariants of the right Cauchy-Green strain tensor can be taken into account. On the base of experimental and mathematical con-siderations, Holzapfel and Ogden included in their model only the following ones: I1 = tr(C) (2.29) I3 = det(C) (2.30) I4,f = f0· Cf0 (2.31) I4,s = s0· Cs0 (2.32) I8,fs = f0· Cs0, (2.33)
where I1 and I3 are isotropic invariants, associated with the non-collagenous
and non-musculare matrix, while I4,f, I4,s and I8,fs are orthotropic invariants
associated to the fiber and cross-fiber directions in the reference configur-ation. By fitting the shear test data of Dokos, Smaill et al. (2002), the following energy function was proposed:
W = a 2be b(I1−3)− 1 + af 2bf h ebfhI4,f−1i2 − 1 i + as 2bs h ebshI4,s−1i2 − 1i+ af s 2bf s h ebf sI8,fs2 − 1 i , (2.34)
where hxi denotes the positive part of x. The parameters a, b, af, bf, as,
bs, af s and bf s are positive material constants whose values are reported in
Table 2.1. The constants a, af, asand af s have the dimension of a stress and
represent the stiffness of the physical structures they are associated with. In particular, one has that af > as > an > a.
Notice that the third invariant is assumed to be equal to one, since the material is assumed to be incompressible.
2.3 - Passive mechanics
Figure 2.1: Schematic representation of the fiber disposition in the left ventricle. (a), idealized left ventricle and a tissue specimen from the equator; (b), sketch of the fiber orientation across the thickness of the specimen; (c), fiber orientation in five different sections of the tissue specimen; (d), representation of the local coordinate system of orthogonal axes; (e), cube of layered tissue. Image taken from Holzapfel and Ogden 2009.
Table 2.1: Material parameters for the energy function (2.34) for the left ventricle model used by Holzapfel and Ogden (2009) to fit the shear data of Dokos, Smaill et al. (2002) from porcine hearts.
a b af bf as bs af s bf s
(kPa) (-) (kPa) (-) (kPa) (-) (kPa) (-)
2.3 - Passive mechanics
Table 2.2: Material parameters for the energy function (2.35) for the left ventricle model used by Holzapfel and Ogden (2009) to fit the biaxial tension data of F. C. P. Yin et al. (1987) from canine hearts.
a b af bf
(kPa) (-) (kPa) (-) 2.280 9.726 1.685 15.779
2.3.2
A transversely isotropic model
The transversely isotropic specialization of the orthotropic model of Holzapfel-Ogden is obtained by setting as and af s equal to 0 in equation (2.34). In
this way, only the contribution of the isotropic extra-cellular matrix and of the fibers in taken into account in the mechanical behaviour of the tissue. This model was used by Holzapfel and Ogden (2009) to fit the biaxial experi-mental data of F. C. P. Yin et al. (1987) on ventricular myocardium. Even if the myocardium, on the base of the shear data of Dokos, Smaill et al. (2002), proved to be an orthotropic material, the transversely isotropic specialization of the model proved to be sufficient to fit the biaxial data alone. Therefore, we adopted this model as a first simplified approach for the mechanics of the atria, but we remind that an orthotropic model, as suggested by Bellini, Di Martino and Federico (2013), may be more accurate.
The energy function that we adopted for the passive mechanics of the atria is then: W = a 2be b(I1−3)− 1 + af 2bf h ebfhI4,f−1i2 − 1 i (2.35) and the values of the material constants are summarized in Table 2.2.
2.3.3
The model proposed by Jernigan et al.
The energy function proposed by Jernigan et al. (2007) was previously used in applications for coronary arteries and heart valves. The left atrial tissue was thus characterized as a Mooney-Rivlin hyperelastic material, therefore isotropic and incompressible. The energy function proposed was:
W = c10(I1− 3) + c01(I2− 3) + c20(I1− 3)2
+ c11(I1− 3)(I2 − 3) + c02(I1− 3)2,
2.3 - Passive mechanics
Table 2.3: Material parameters of the energy function (2.36) for the left atrial model used by Jernigan et al. (2007) to fit uniaxial test data from porcine hearts.
c10 c01 c20 c11 c02
(kPa) (kPa) (kPa) (kPa) (kPa)
−58.4 63.4 16.0 × 103 −35.3 × 103 19.7 × 103
where I1 and I2 are the first and the second invariant of the right
Cauchy-Green strain tensor. The material parameters c10, c01, c20, c11, c02 where
ob-tained by fitting uniaxial test data from porcine hearts. Their values are reported in Table 2.3.
2.3.4
The models proposed by Bellini and Di Martino
In the work of Bellini and Di Martino (2012), two anisotropic constitutive laws were validated on the base of biaxial tests performed on atrial tissue specimens from pigs. The specimens were taken from both left and right atrium. The local orientation of the fiber bundles were prior investigated on rabbits hearts, revealing two orthogonal families of fibers. In their find-ings, atrial tissue exhibited orthotropic behaviour. They choosed to adopt a microstructurally-motivated model of the from of the orthotropic model pro-posed by Holzapfel-Ogden (Holzapfel and Ogden 2009) reported in Sec. 2.3.1:
W = ciso(I1− 3) + k1 2k2 h ek2hI4,f−1i2 − 1 i + k3 2k4 h ek4hI4,s−1i2 − 1 i (2.37) and a Fung-type strain energy function:
W(Q) = 1 2cF(e Q− 1) (2.38) with Q= a1E112 + a2E222 + 2a3E11E22, (2.39) where E2
11 and E222 are the components of the Green-Lagrange strain tensor
defined as E = 1
2(C − I) and I is the second-order identity tensor. Subscript
1 was consistently assigned to the orientation of the fibers, associated with the maximum stiffness. Values of the material parameters ciso, k1, k2, k3,
2.4 - Nearly-incompressible formulation
in the left and in the right atrium; for their values we refer to Bellini and Di Martino (2012).
Since the Fung model revealed to provided a better prediction of the Green-Lagrange strain, the work of Bellini, Di Martino and Federico (2013) later provided a novel structural model based on a Fung-type elastic strain energy potential, validated by fitting experimental data from biaxial planar tests on specimens from human hearts atrial tissue. Since the model is quite complex, we refer to Bellini, Di Martino and Federico (2013) for its derivation. However, we remark that this was the first time that the human heart atrial tissue was characterized from a mechanical point of view.
2.4
Nearly-incompressible formulation
In the constitutive law of Holzapfel and Ogden (2009) the myocardium is regarded as incompressible. However, F. C. Yin et al. (1996) showed that during contraction or passive stretching, the increase of the tissue stiffness causes a decrease of the volume of blood in the vessels that perfuse the myocardium. Moreover, changes in myocardium has been observed during ventricle systole (Ashikaga et al. 2008; Cheng et al. 2005). In Cheng et al. 2005; F. C. Yin et al. 1996, the authors estimated a volumetric change of the ventricular myocardium during the caridac cycle between the 2% and the 15% (Cheng et al. 2005; F. C. Yin et al. 1996). Even if in literature we did not find data related to the atrial myocardium, we may expect similar values. Therefore, we move to a nearly-incompressible formulation, which has been widely adopted in several other works (e. g. Barbarotta et al. 2018; Eriksson et al. 2013; Gerbi et al. 2018). This approach is also advantageous for the finite element approximation (Holzapfel 2002).
This formulation is based on a multiplicative decomposition of the de-formation gradient, originally proposed by Flory (1961) and then applied in isothermal finite elasticity. In this context, the deformation gradient F is split into a so-called volumetric and isochoric part:
F = ¯FFvol (2.40) where ¯ F = J−13F (2.41) Fvol = J 1 3I (2.42)
The isochoric part, denoted with ¯F, accounts for the volume-preserving part of the deformation (indeed it is immediate that det(¯F) = 1), while the
volu-2.4 - Nearly-incompressible formulation
metric part, denoted with Fvol, takes into account volume changes.
Accord-ingly, we can define a modified right Cauchy-Green tensor ¯
C = ¯FTF
and its invariant
¯I1 = tr( ¯C) = J−
2 3I
1.
The multiplicative decomposition of the deformation gradient reflects into an additive decomposition of the energy function W:
W = Wiso+ Wvol. (2.43)
Since the Holzapfel-Ogden constitutive law accounts for the isochoric part of the deformation, Wiso is the energy function of the Holzapfel-Ogden model,
rewritten as a function of the invariants of ¯C (see Eriksson et al. (2013)). However, Sansour (2008) proved that, for fiber-reinforced materials, the volumetric-isochoric split should be applied to the matrix part only in or-der not to violate certain physical requirements. Therefore, the final form of the energy function W is:
W = Wiso(¯I1,I4,f) + Wvol(J)
= W1(¯I1) + W4,f(I4,f) + Wvol(J). (2.44) where W1(¯I1) = a 2b h eb(¯I1−3)− 1 i , (2.45) W4,f(I4,f) = af 2bf h ebfhI4,f−1i2 − 1 i . (2.46)
The volumetric part of the energy Wvol, instead, is a penalization term.
Generally, it is required that Wvol(J) is a convex function bounded from
below whose slope is null in J = 0, so that it becomes unbounded when J → 0 or J → ∞, conditions that correspond, respectively, to the collapse of the body into a single point and to an indefinite dilation. In this work the following form has been adopted (Simo and Taylor 1982):
Wvol(J) = k 4 (J − 1) 2 + log(J)2 (2.47) where k is called bulk modulus and penalizes volume changes.
For a list of other alternative choices we refer to Hartmann and Neff (2003).
2.5 - Active mechanics
2.5
Active mechanics
Active mechanics models the contraction of the cardiac muscle due to the electric pulse generated by the pacemaker cells of the heart. To model the active mechanics, two main approaches have been developed in literature: the active stress (Smith et al. 2004, Panfilov et al. 2005, G¨oktepe and Kuhl 2010) and the active strain models (Cherubini et al. 2008, Ambrosi, Arioli et al. 2011). For advantages and disadvantages of the two approaches we refer to Ambrosi, Arioli et al. (2011) and Ambrosi and Pezzuto (2012)), while we refer to Rossi et al. (2014) for computational results.
2.5.1
Active stress approach
In the active stress model, the first Piola-Kirchhoff stress tensor P is decom-posed into the sum of two tensors: the first, denoted with Pe, accounts for
the stress induced by pure passive deformation; the second, denoted with Pa, models the stress caused by the active contraction of the cardiac tissue.
Therefore
P = Pe+ Pa. (2.48)
Pe is the first Piola-Kirchhoff tensor used to model the passive mechanics:
Pe =
∂W
∂F, (2.49)
while Pa has to be constitutively specified. A common choice, that has been
adopted also in this work, is:
Pa= αFf ⊗ f0, (2.50)
where f ..= Ff0 and αF is a parameter with the dimension of a stress that
comes from the electrophysiology and has to be properly modeled.
2.5.2
Active strain approach
In the active strain approach, the gradient deformation tensor F is split into an elastic deformation Fe and an active deformation Fa through a
multiplic-ative decomposition:
F = FeFa. (2.51)
At a physical level, the idea underlying this decomposition is that the elastic deformation does not occur in the reference configuration, but in an inter-mediate configuration where the fibers have been shorten as a consequence of the propagation of the electrical pulse. Therefore, the deformation Fa
2.6 - Fibers reconstruction
represents the shortening of the sarcomeres in the myocytes that initiate the contraction, while Fe represents the subsequent deformation induced by
the elastic accomodation of the tissue in response to the propagation of the electrical pulse.
Again, the energy function W accounts only for the elastic deformation; therefore, the stress tensor in the intermediate configuration, denoted with Pe, is computed as
Pe=
∂W(Fe)
∂Fe
(2.52) and the first Piola-Kirchhoff tensor is recovered by pulling back equation (2.52) in the reference configuration:
P = det(Fa)PeF−Ta . (2.53)
The deformation Fa has to be constitutively provided and several choices
exist in literature; we refer to Gerbi (2018) for a list of common models.
2.6
Fibers reconstruction
Due to the microscopical structure of the myocardium, electrical pulses travel faster along the fiber direction, and the shortening of all the fibers causes the cardiac contraction. Therefore, any mechanical model of the heart should account for a precise reconstruction of fiber orientation; however, this is still a challenging problem in cardiac mechanics. First of all, an accurate ana-tomical description of the fiber bundles in needed; then, a suitable algorithm that can reconcile precision in the fiber description, low computational cost, and absence of inter- and intra-variability is to be found. Moreover, the task is still more challenging if a patient-specific reconstruction is required, since no reliable estimate of the fiber architecture can be provided in-vivo with the current technology (Fastl et al. 2018).
In the literature, a great effort as been done to find a tool for the gen-eration of the fibers in the ventricles. In these chambers, the fiber structure is more regular with respect to the one in the atria and can be easily de-scribed as a helicoidal pattern, as shown briefly in Sec. 2.3.1. Moreover, the larger thickness of the walls of the ventricles allowed to study more easily the change in the fiber orientation across the ventricular walls, which var-ies quite smoothly across the thickness. For these reason, there exist many models that can reconstruct in a quite accurate way the fiber architecture in these chambers, taking into account also the transmural variation of their orientation. As an example, we cite the method proposed in Wong and Kuhl (2014), that relies on the Poisson interpolation. This algorithm is quick and
