POLITECNICO DI MILANO
Facolt`
a di Ingegneria Industriale e dell’Informazione
Corso di Laurea in Ingegneria Matematica
A.A. 2013/2014
A Mathematical and Numerical Study of the Left
Ventricular Contraction based on the Reconstruction
of a Patient Specific Geometry
Candidate: Luca barbarotta
Matricola: 760674
Supervisor:
Prof. Alfio quarteroni
Assistent supervisors:
Ph.D Simone rossi
Ph.D Elena faggiano
Abstract
The heart is the most important organ of the cardio circulatory system. The development of new diagnostic methods and new treatments dedicated to the heart constitutes an important part of medical research. Nevertheless, the difficulties in data collection with non-invasive methods makes necessary the use of alternative methods to study certain diseases. For this reason, the computational methods applied in this area are subject of increasing interest. In particular, more and more accurate models are available for the description of cardiac mechanics. An important aspect that needs to be considered is the nonlinear anisotropic behavior of the myocardium. Such anisotropy is mainly due to the presence of muscular fibers. Recently, various studies have proved the importance of the presence of collagen in the muscular contraction which introduces another anisotropic direction. This effect must be included in the mathematical models that describe the elastic properties of the ventricle. During the last years different constitutive laws have been proposed which account for the orthotropy of the myocardium, considering both muscular fibers and collagen sheets. The heart is an active material that can change its configuration in absence of external loads. In fact, muscular fibers, if excited with an electric signal, contract. Hence in order to describe the behavior of the myocardium it is necessary to introduce the activation in the mechanical model. Two different approaches are known: the active stress and the active strain models. Actually, both models originate from the same thermodynamical principle but with different constitutive choice for the energy. In this thesis, we will consider only the active strain model because it introduces naturally an anisotropic activation. In particular, we will use at first a transversely isotropic activation. However, such model needs a contraction for the fibers far greater than the observed value in order to reproduce physiological values for the ventricular wall thickening. For an accurate description of the macroscopic deformation of the ventricle we will use an orthotropic activation model. Furthermore, several studies have evidenced that the distribution of the radial strains through the myocardium has a strong dishomogeneity between the epicardium and the endocardium. For this reason, we propose to modify the orthotropic activation model in order to include effects of dishomogeneous activation.
We compare the proposed model with both transversely isotropic activation and or-thotropic activation models on an idealized geometry, represented by a truncated ellipsoid. Then, we apply the proposed method on a real ventricle, reconstructed from medical images. To reconstruct the patient specific geometry, we used a region-based segmentation method based on the minimization of the region scalable fitting energy functional by means of the Split-Bregman approach. Despite the good performances of the segmentation model, we need a procedure for the image pre-processing in order to obtain the isolated ventricle.
The obtained results, on both ideal and patient-specific geometries, show that the method we proposed keeps the good performance of the orthotropic activation model. Moreover, it succeeds in describing the transmural dishomogeneity of the ventricular wall thickening and
Sommario
Il cuore `e l’organo pi`u importante del sistema cardio circolatorio. Lo sviluppo di nuovi strumenti diagnostici e di nuove cure dedicate al cuore occupa una parte importante della ricerca medica. Ci`o nonostante, le difficolt`a nella raccolta di dati con strumenti non inva-sivi rende necessario l’utilizzo di tecniche alternative per lo studio di alcune patologie. Per questo motivo i metodi computazionali applicati in questo ambito sono oggetto di crescente interesse.
In particolare, modelli sempre pi`u accurati sono disponibili per la descrizione della mec-canica cardiaca. Un aspetto di particolare importanza `e il comportamento non lineare ed anisotropo del miocardio. Tale anisotropia `e dovuta principalmente alla presenza di fibre muscolari. Recentemente, vari studi hanno dimostrato l’importanza della presenza di col-lagene nella contrazione muscolare. La conformazione del colcol-lagene introduce nel tessuto un’altra direzione di anisotropia. Questo effetto deve essere incluso nei modelli matematici che descrivono le propriet`a elastiche del ventricolo. In letteratura sono stati proposti diverse leggi costitutive che descrivono il tessuto cardiaco come ortotropo, tenendo in considerazione sia le fibre muscolari sia i fogli di collagene.
Il cuore `e un materiale attivo che pu`o cambiare la propria configurazione in assenza di carichi esterni. Infatti le fibre muscolari reagiscono ad uno stimolo elettrico contraendosi, inducendo cos`ı la contrazione del miocardio. Quindi per descrivere il comportamento del miocardio `e necessario introdurre l’attivazione nel modello meccanico. Due differenti approcci sono pre-senti in letteratura: il modello active stress ed il modello active strain. In realt`a `e possibile vedere che entrambi i modelli derivano dallo stesso principio termodinamico ma da diverse scelte della forma costitutiva dell’energia. In questa tesi, noi utilizzeremo l’approccio active strain perch´e introduce naturalmente un’attivazione anisotropa. In particolare utilizzeremo dapprima un’attivazione trasversalmente isotropa. Tuttavia tale approccio per riprodurre valori fisiologici per l’ispessimento della parete ventricolare necessit`a valori di contrazioni delle fibre molto maggiori di quelli osservati. Per una descrizione pi`u accurata delle de-formazioni macroscopiche del ventricolo utilizzaremo quindi un modello ortotropo. Inoltre, diversi lavori hanno evidenziato che l’andamento degli sforzi radiali attraverso il miocardio presenta una forte disomogeneit`a fra l’epicardio e l’endocardio. Per questa ragione in questo lavoro proponiamo di modificare il modello di attivazione ortotropa per includere effetti di attivazione disomogenea.
Confrontiamo il modello proposto con i modelli trasversalmente isotropi ed ortotropi di active strain su ua geometria idealizzata del ventricolo sinistro, rappresentata da un ellisoide troncato. Successivamente applichiamo il modello proposto nel caso di un ventricolo sinistro ricostruito da immagini mediche. Per ricostruire la geometria patient specific abbiamo usato un metodo di segmentazione region based basato sulla minimizzazione con metodo di Split-Bregman del funzionale di energia del region scalable fitting model (RFE). Malgrado le buone caratteristiche del modello di segmentazione, `e necessario un procedimento di preprocessing dell’immagine per poter ottenere il ventricolo isolato.
il metodo da noi proposto oltre a mantenere le buone prestazioni del modello di attivazione ortotropo, riesce a descrivere la disomogeneit`a dell’ispessimento della parete ventricolare e migliora la descrizione del meccanismo di attivazione meccanica ventricolare.
Contents
Abstract 1
Sommario 3
1 Mathematical models of cardiac mechanics 12
1.1 Heart physiology . . . 12
1.2 Introduction to finite elasticity . . . 13
1.2.1 Introduction to finite elasticity . . . 13
1.2.2 Anisotropy and Holzapfel–Ogden constitutive law . . . 15
1.2.3 Nearly-incompressible formulation . . . 16
1.3 Active mechanics . . . 19
1.3.1 Active Strain formulation . . . 20
1.3.2 Transversely isotropic activation . . . 21
1.3.3 Orthotropic activation . . . 21
1.3.4 Transmurally non homogeneous orthotropic activation . . . 23
2 Numerical discretization 24 2.1 Weak formulation . . . 24
2.2 Linearization . . . 27
2.2.1 The Newton-Rapshon method . . . 27
2.2.2 Tangent problem . . . 27
2.3 Discrete problem . . . 30
3 Geometry reconstruction 33 3.1 Introduction to Image Segmentation . . . 33
3.2 The Region Scalable Fitting Energy Functional . . . 34
3.3 The Split-Bregman Method . . . 35
3.4 Minimize RSFE using Split-Bregman Method . . . 38
3.5 Numerical Implementation . . . 41
3.6 Customizations and Additional Tools . . . 47
3.7 Application to Medical Image . . . 49
3.7.1 Preprocessing and Segmentation . . . 49
4 Idealized left ventricle study 53 4.1 Fibers and sheets fields computation . . . 53
4.2 Preload and contraction . . . 54
4.3 Simulation settings . . . 57
4.4 Study on ventricular wall thickening . . . 60
4.4.1 Transversely isotropic activation test . . . 60
4.4.3 Transmurally non homogeneous orthotropic activation test . . . 69
5 Patient–specific left ventricle study 78 5.1 Left ventricle mesh generation . . . 78
5.2 Recovery of a stress free configuration . . . 80
5.3 Implementation of the transmurally non homogeneous orthotropic activation model for a patient–specific ventricle . . . 82
5.4 Activation comparison . . . 83
5.4.1 Orthotropic activation . . . 85
5.4.2 Transmurally linear orthotropic activation k0epi= 0, 75 . . . 91
Conclusions and Perspectives 97
List of Figures
1.1 Multiplicative decomposition of the total deformation. The deformation of gradient F that takes B0 to B2 is split in two subsequent deformation with
gradients Fa and Fe respectively. . . 20
3.1 Illustration of the pre-processing steps. . . 51 3.2 Left ventricular segmentation. . . 52 4.2 Different views for the fibers field of the idealized ventricle. In the first two
images at the top it is showed a view of the streamlines related to the fibers vector field viewed form the base and from the apex of the ventricle. . . 57 4.1 Idealized ventricular geometry from two points of view where we show the
boundary partition. . . 58 4.3 The levels used in the computation of thickness. The first and the last level
individuate the apex and base plan where the rotation and torsion is com-puted . . . 59 4.4 Transversely isotropic activation model. The deformation of the ideal
ven-tricle at different moments of activation. 4.4a, 4.4b are two instants of the isochoric contraction phase while 4.4c, 4.4d are two instants of the ejection phase. The displacements are shown in cm. . . 61 4.5 Transversely isotropic activation model. The fibers elongation of the ideal
ventricle at different moments of activation. . . 62 4.6 Transversely isotropic activation model. Pressure-volume diagram. On
X-axis there is volume (cm3) while on Y-axis the pressure (mmHg). The values written in the graph are the activation state corresponding to the deformation of the images 4.4. The diastolic volume is 95,794 cm3, the final volume is
48,469 cm3, the ejection fraction is 49,40% . . . 62 4.7 Transversely isotropic activation model. Left ventricular length variation. In
the Y-axis it is shown the per cent variation of the length with respect to the diastolic length. . . 63 4.8 Transversely isotropic activation model. Wall thickening at each activation
step. The four plots represent on the Y-axis the percent thickening variation with respect to the diastolic configuration in each level. . . 63 4.9 Transversely isotropic activation model. Transmural wall thickening - radial
strain (adimensional). In abscissa it is shown the curvilinear coordinate of the line on which we computed the strain, the smallest value coincides with the endocardium, the greatest coincides with the epicardium . . . 64 4.10 Orthotropic activation model. The deformation of the ideal ventricle at
dif-ferent moments of activation. 4.10a, 4.10b are two instants of the isochoric contraction phase while 4.10c, 4.10d are two instants of the ejection phase. The displacements are shown in cm. . . 65
4.11 Orthotropic activation model. The fibers elongation of the ideal ventricle at different moments of activation. . . 66 4.12 Orthotropic activation model. Pressure-volume diagram. On X-axis there is
volume (cm3) while on Y-axis the pressure (mmHg). The values written in
the graph are the activation state corresponding to the deformation of the images4.10. The diastolic volume is 95,794 cm3, the final volume 47,669 cm3, the final ejection fraction is 50,24%. . . 66 4.13 Orthotropic activation model. Left ventricular length variation. In the Y-axis
it is shown the percent variation of the length with respect to the diastolic length. . . 67 4.14 Orthotropic activation model. Wall thickening at each activation step. The
four plots represent on the Y-axis the percent thickening variation with re-spect to the diastolic configuration in each level. . . 67 4.15 Orthotropic activation model. Transmural wall thickening - radial strain
(adimensional). In abscissa it is shown the curvilinear coordinate of the line on which we computed the strain, the smallest value coincides with the endocardium, the greatest coincides with the epicardium . . . 68 4.16 Linear orthotropic activation with kepi = 0. The deformation of the ideal
ventricle at different moments of activation. . . 69 4.17 Linear orthotropic activation with kepi= 0. The fibers elongation of the ideal
ventricle at different moments of activation. . . 70 4.18 Linear orthotropic activation with kepi= 0. Pressure-volume diagram. On
X-axis there is volume (cm3) while on Y-axis the pressure (mmHg). The values
written in the graph are the activation state corresponding to the deformation of the images4.16. The diastolic volume is 95,794 cm3, the final volume 50,42
cm3, the final ejection fraction is 47,36%. . . . 71
4.19 Linear orthotropic activation with kepi= 0. Left ventricular length variation.
In the Y-axis it is shown the percent variation of the length with respect to the diastolic length. . . 71 4.20 Linear orthotropic activation with kepi= 0. Wall thickening at each activation
step. The four plots represent on the Y-axis the percent thickening variation with respect to the diastolic configuration in each level. . . 72 4.21 Linear orthotropic activation with kepi = 0. Transmural wall thickening
-radial strain (adimensional). In abscissa it is shown the curvilinear coordinate of the line on which we computed the strain, the smallest value coincides with the endocardium, the greatest coincides with the epicardium . . . 73 4.22 Linear orthotropic activation with kepi= 0.75. The deformation of the ideal
ventricle at different moments of activation. . . 74 4.23 Linear orthotropic activation with kepi = 0.75. The fibers elongation of the
ideal ventricle at different moments of activation. . . 75 4.24 Linear orthotropic activation with kepi = 0.75. Pressure-volume diagram.
On X-axis there is volume (cm3) while on Y-axis the pressure (mmHg). The values written in the graph are the activation state corresponding to the deformation of the images4.22. The diastolic volume is 95,794 cm3, the final
volume 49,8 cm3, the final ejection fraction is 48,01%. . . 75 4.25 Linear orthotropic activation with kepi = 0.75. Left ventricular length
varia-tion. In the Y-axis it is shown the percent variation of the length with respect to the diastolic length. . . 76
4.26 Linear orthotropic activation with kepi = 0.75. Wall thickening at each
ac-tivation step. The four plots represent on the Y-axis the percent thickening variation with respect to the diastolic configuration in each level. . . 76 4.27 Linear orthotropic activation with kepi= 0.75. Transmural wall thickening
-radial strain (adimensional). In abscissa it is shown the curvilinear coordinate of the line on which we computed the strain, the smallest value coincides with the endocardium, the greatest coincides with the epicardium . . . 77 5.1 Three different views of the diastolic mesh. The five different colors
corre-sponds to five different tags. . . 80 5.2 Three different views for the fibers field of the reconstructed ventricle. The
orientation is expressed in degrees and is computed with respect of the cen-terline (0.338, 0.385, 0.859). . . 80 5.3 . . . 84 5.4 Two views of the stress free configuration (blue) compared with the diastolic
configuration (green). . . 85 5.5 Orthotropic activation model. Deformation viewed from the side at six
dif-ferent step of activation, respectively from top left to bottom right, γf: 0,
-0.05, -0.1, -0.125, -0.13, -0.135. The red ventricle is the systolic configuration. 86 5.6 Orthotropic activation model. Deformation viewed from the base at six
dif-ferent step of activation, respectively from top left to bottom right, γf: 0,
-0.05, -0.1, -0.125, -0.13, -0.135. The red ventricle is the systolic configuration. 87 5.7 Orthotropic activation model. Deformation viewed from the septum at six
different step of activation, respectively from top left to bottom right, γf: 0,
-0.05, -0.1, -0.125, -0.13, -0.135. The red ventricle is the systolic configuration. 88 5.8 Orthotropic activation model. Pressure-Volume diagram. The number near
the stars are the values of γf. The vertical line corresponds to the volume of
the target systolic configuration. . . 89 5.9 Orthotropic activation model. Ventricular shortening. The numbers on the
curve are the values of γf. . . 89
5.10 Orthotropic activation model. Radial strain computed through the my-ocardium. The left side of the abscissa correspond to the endocardium, the right side to the epicardium. . . 90 5.11 Linear orthotropic activation with kepi= 0.75. Deformation viewed from the
side at six different step of activation, respectively from top left to bottom right, γf: 0, -0.05, -0.1, -0.125, -0.14, -0.15. The red ventricle is the systolic
configuration. . . 92 5.12 Linear orthotropic activation with kepi= 0.75. Deformation viewed from the
base at six different step of activation, respectively from top left to bottom right, γf: 0, -0.05, -0.1, -0.125, -0.14, -0.15. The red ventricle is the systolic
configuration. . . 93 5.13 Linear orthotropic activation with kepi= 0.75. Deformation viewed from the
septum at six different step of activation, respectively from top left to bottom right, γf: 0, -0.05, -0.1, -0.125, -0.14, -0.15. The red ventricle is the systolic
configuration. . . 94 5.14 Linear orthotropic activation with kepi = 0.75. Pressure-Volume diagram.
The number near the stars are the values of γf. The vertical line corresponds
5.15 Linear orthotropic activation with kepi = 0.75. Ventricular shortening. The
numbers on the curve are the values of γf. . . 95
5.16 Linear orthotropic activation with kepi = 0.75. Radial strains computed
through the myocardium. The left side of the abscissa correspond to the endocardium, the right side to the epicardium. . . 96
List of Tables
1.1 Coefficients of the Holzapfel–Ogden orthotropic constitutive law for the
pas-sive behavior of the myocardium [10]. . . 17
3.1 Algorithm’s parameters adopted to segment the ventricular cavity. See3.1d. 51 3.2 Algorithm’s parameters adopted to segment the left ventricular myocardium. See 3.2. . . 51
4.1 Physiological values of the end phase pressure for each phase. . . 55
4.2 Values of the parameters we used for the fixed point iteration scheme and for the windkessel model. . . 56
4.3 Pressure used for the phase selection in our tests. . . 57
4.4 Values of µ that define the levels . . . 59
5.1 Pressure used for the phase selection in our tests. . . 83
5.2 Coefficients used for the robin boundary conditions. . . 84
5.3 Orthotropic activation model. Thickness comparison with the real systolic geometry about at 50% of ejection fraction . . . 87 5.4 Transmurally linear orthotropic activation model - kepi0 = 0.75. Thickness
Chapter 1
Mathematical models of cardiac
mechanics
In this chapter, we introduce the mathematical models we use to describe cardiac mechanics. In the first part, we briefly describe the anatomy of the heart and we explain the complex processes taking place during a single heartbeat.
In the second part, we introduce the finite elasticity framework, which is essential to describe the large deformations occurring in the heart. We assume the tissue is perfectly elastic such that its behavior is fully described by a phenomenological strain-energy function which accounts for material anisotropy. Eventually, we present the specific models for active and passive mechanics of the myocardium.
1.1
Heart physiology
Heart pumps blood throughout the vessels in order to bring oxygen, provisions and other substances, to all the cells of our body. It is surrounded by the pericardium, a tough double layered sac bounded to the diaphragm with the purpose of protecting it from shocks and sustaining it [19]. The inner layer of pericardium is divided in two other layers inter-spaced by a viscous fluid lubricating the two surfaces and allowing motion inside the pericardium. The heart is a hollow muscular organ, divided into four cavity: two atria and two ventricles. The right atrium gathers the deoxygenated venous blood from inferior and superior vena cava. In order to enrich the blood with oxygen the right ventricle pumps it to the lungs through the pulmonary artery. The left atrium receives the oxygenated blood coming from the lungs from left and right pulmonary veins. Eventually, the left ventricle pumps it to the aorta from which it reaches all the peripheral areas of the body.
The myocardium is the muscle responsible for the contraction of the atria and ventricles. The contractile cells of the myocardium are called cardiomyocytes and they constitutes 70% of the total heart mass [20]. Cardiomyocytes, or fibres, are arranged in a helical fashion and their orientation varies transmurally vary from epicardium to endocardium. The cardiomyocites are immersed in a collagen bone. Groups of three-four fibers are firmly tied up in laminar sheets which instead are weakly bound together. It is because of this structure that the tissue presents different properties in the cross-fibers directions [23, 14].
The heart cycle can be mainly classified in two phase: the passive phase, called diastole, and the active phase, called systole. During diastole the blood pressure inflates the ventricle making the fibers stretch. During systole the fibers contract making the blood flows out the ventricle.
1.2
Introduction to finite elasticity
During the systolic contraction the volume of the ventricular chamber is reduced up to 60%, the wall thickness can increase more than 40% and the ventricle shortens about 20-30% with respect to the diastolic configuration. In order to deal with such large deformations we will make use of finite elasticity.
1.2.1
Introduction to finite elasticity
In finite elasticity theory we distinguish the reference configuration of the body Ω0 from the
actual configuration Ω. We denote with X the position vector in the reference configuration Ω0, while the vector x, represents the position vector in the actual configuration.
The actual configuration describe the configuration of the body after deformation. A deformation is a map from the reference to the actual configuration:
ϕ : Ω0 → Ω
x = ϕ(X)
We suppose that ϕ is a diffeomorphism from Ω0 to Ω, and its derivative
F(X) := ∂ϕ
∂X, [Fij] = ∂ϕi
∂Xj
, i, j ∈ {1, 2, 3},
is the deformation gradient tensor. We define J as the determinant of F. It represents the ratio of the volume of the deformed configuration over the volume of the reference configuration. In fact, given three vector in the reference configuration: a, b and c, the determinant of F can be expressed as:
det(F) = (Fa ∧ Fb) · Fc (a ∧ b) · c .
We recall that (a ∧ b) · c is the volume of the parallelepiped with axes along a, b and c. Another condition we must set on the deformation map is that
J = det(F) > 0. (1.2.1)
Hence, if it becomes zero the deformation violates the mass conservation law since the body collapse to a point. For continuity arguments F cannot be negative, starting from a positive determinant it must pass from det(F)=0 to become negative.
The previous characterization of the determinant of F allows us to write the equation for the mass conservation:
ρ0 = det(F)ρ.
The polar decomposition theorem states that given a non singular deformation F there exist an orthogonal tensor R and two symmetric tensors U, V such that:
F = RU = VR,
where U and V represent the pure deformation in the reference and actual configuration respectively. Given a deformation it is always possible to decompose F into a rigid rotation and a pure deformation. We define the left and right Cauchy–Green and Green strain tensors as follows:
C := FTF, B := FFT, and E := 1
respectively. From the polar decomposition theorem we deduce that C = U2 and B = V2.
The Cauchy theorem states that the stress state of a body can be described by a tensor, the Cauchy stress tensor T. The Cauchy stress tensor is defined in the actual configuration. The application of the stress tensor on a normal vector characterizing a surface gives the stress acting on that surface
t(x, vn) = T(x)ν.
The energy balance equation reads d dt Z Ω ρv2 dv = Z Ω ρb · v dv + Z ∂Ω t · v da = Z Ω ρb · v dv + Z ∂Ω Tvn· v da = Z Ω ρb · v dv + Z Ω div T · v dv. Applying the additional hypothesis of quasi-static deformation, we obtain
Z Ω ρb · v dv + Z Ω div T · v dv = 0.
This equation is valid in every inertial system and then we find that for all v: Z
Ω
(ρb + div T) · v dv = 0
⇒ − div T = ρb, (1.2.2)
where b is a volume force, t is the stress and ν is the surface normal. Equation (1.2.2) is expressed in the actual configuration which is typically unknown. In order to write equation (1.2.2) in the reference configuration we must use the Piola transformation:
Z ∂Ω v · ν da = Z ∂Ω0 J F−1v · N dA, (1.2.3)
where ν and N are the surface normal in the actual and reference configuration respectively. Thanks to the Piola transformation, we can write the first and second Piola–Kirchhoff tensors:
P := J TF−T, S := F−1P = J F−1TF−T. (1.2.4) In the reference configuration, equation (1.2.2) becomes:
− Div P = ρ0b0, in Ω0, (1.2.5)
where ρ0 is the density of the body in the reference configuration while b0 represents an
external bulk force per unit volume. Note that all terms in (1.2.5) are in material coordinates and that Div =P
i∂/∂Xi.
In order to properly solve equation (1.2.5), we must provide a set of boundary conditions. At this purpose we partition the boundary of Ω0 as follows ∂Ω0 = ΓD∪ ΓN ∪ ΓR on which
different type of condition could be imposed. Where we indicated with ΓD, ΓN and ΓR the
portions of boundary on which we impose dirichlet, natural and robin conditions.
A Dirichlet condition is used to impose the displacement on a region of the boundary. This condition is needed in order to get uniqueness of the solution, otherwise the solutions would be infinite which differentiate for a rigid translation. A natural condition, allows to prescribe
a stress on the boundary, such as an external superficial load. The linear combination of the two previous condition provides the Robin condition which introduces a linear relation between the stress and the displacement. In others words this condition works like a spring: the greater is the strain the greater is the stress that opposes the motion.
We use a dirichlet condition on the base of the ventricle to prescribe a null orthogonal displacement in order anchor it to aorta and to the left atrium. We set the natural condition at endocardium, so that we can impose the stress that blood exerts on the ventricular wall. The epicardium is the portion of border where we set the robin condition. We use a robin condition at epicardium to describe the interaction between heart and other surrounding tissues. − Div P(u) =0 in Ω0 u =uD on ΓD Pν =t on ΓN Pν + αu =0 on ΓR. (1.2.6)
Note that we neglected the external force ρ0b0, due to gravity, because its contribution is
negligible.
We assume the myocardium to act as a hyperelastic material, that is we assume that there exists a function W : Ω0 × Lin+ → R, called strain energy function, which describes
the energetic state of the body in the configuration determined by F. W is such that: P = ∂W
∂F , or, equivalently S = 2 ∂W
∂C , (1.2.7)
W must be objective, that is, for any rotation Q applied to the reference system the energy must not change
W(X, F) = W(X, QF).
Considering the relation between C and U provided by the polar decomposition theorem, C does not account for rigid rotation of F. Then in order to automatically consider the objectivity it is convenient to describe the energy function as a function of C.
1.2.2
Anisotropy and Holzapfel–Ogden constitutive law
The symmetry group describing the material symmetries is a subset of the rotation tensors set. The rotations included are those who do not change the stress state of the body.
G =nQ ∈ O(3) : T(FQ) = T(F), ∀F ∈ Lin+o,
or equivalently, in energetic terms:
G =nQ ∈ O(3) : W(FQ) = W(F), ∀F ∈ Lin+o,
where O(3) is the group of the orthogonal tensors and Lin+ is the group of tensor with positive determinant.
The symmetry group allow us to introduce an equivalence relation among deformations in the mechanical constitutive relation. Its meaning is: the stress tensor T does not change if we rigidly rotate the body with an element of its symmetry group before the deformation. If a material behaves in the same way whatever rotation we apply on it, then it is said to be isotropic. As we already seen in section1.1, the cardiac muscle is made of fibers and sheets. The myocardial stiffness is greater along the fibers direction and softer in the sheet-normal
direction. Hence the material is not isotropic because the stress depends on the direction of the normal with which it has been computed. Indeed, if we fix a point of the body and compute the stress alongside the fiber direction it would be different with respect to the stress computed in the same point but alongside the sheet direction. The symmetry group therefore must be a subgroup of the proper orthogonal tensors. In particular for an orthotropic material the symmetry group will be:
Gorth =
n
Q ∈ O(3) : Q(m ⊗ m)QT = m ⊗ m, for m ∈ {f◦, s◦, n◦}
o
, (1.2.8)
where we indicated with f◦, s◦ and n◦ the direction of anisotropy, which for the myocardium
represent the fibers, sheets and normal to the fibers-sheets plane directions respectively. In order to account for the presence of fibers and sheets, we need to prescribe a function W(C) invariant with respect of (1.2.8). In practice, we must provide an irreducible set of invariants so that we can write the strain anergy as a function of those invariants [26].
In [18], the authors provide a strain–energy function describing myocardial tissue de-pending on the following set of invariants:
isotropic: I1 = tr C, I3 = det C,
anisotropic: I4,f = f · f , I4,s= s · s, I8,fs =f · s.
The isotropic invariants are those usually involved in the isotropic part of the strain energy [17, 28]. I4,f, I4,s and I8,fs are responsible for the anisotropic response of the material.
In particular, I4,f and I4,s are the quadratic modulus of the deformed fibers and sheets
respectively. The invariant I8,ab represents, instead, the projection of the fibers onto the
sheets direction. It is related to the angle spanned by the vectors f and s, supposing that f◦ and s◦ are orthogonal. The Holzapfel–Ogden model for incompressible material assumes
that the strain–energy density function takes the form: W(C) = W(I1, I4,f◦, I4,s◦, I8f◦s◦)
= W1(I1) + W4,f◦(I4,f◦) + W4,s◦(I4,s◦) + W8,f◦s◦(I8,f◦s◦).
(1.2.9) The specific form of each term in (1.2.9) is of exponential type, in order to accommodate the typical response of a biological tissue, where the greater is the strain, the greater is the apparent stiffness: W1(I1) = a 2b h eb(I1−3)− 1i, W4,f◦(I4,f◦) = af 2bf h ebf(I4,f◦−1)2 − 1 i , W4,s◦(I4,s◦) = as 2bs h ebs(I4,s◦−1)2 − 1 i , W8,f◦s◦(I8,f◦s◦) = afs 2bfs h ebfsI8,f◦s◦2 − 1i. (1.2.10)
The values of the coefficients of (1.2.10) are showed in Table 1.1.
1.2.3
Nearly-incompressible formulation
Biological tissue is mostly made of water. The water flow is often treated as incompressible. However in the ventricle perfused blood is responsible for large volume variations [46].
a (kPa) b af (kPa) bf as (kPa) bS af s (kPa) bf s
Orthotropic
Holzapfel–Ogden 0.059 8.023 18.472 16.026 2.481 11.120 0.216 11.436 Table 1.1: Coefficients of the Holzapfel–Ogden orthotropic constitutive law for the passive behavior of the myocardium [10].
Neglecting the role of perfused blood, the volume variations are very moderate. For this reason we propose to adopt a nearly-compressible formulation.
The nearly-incompressible formulation can be obtained from an incompressible model introducing a multiplicative decomposition of the deformation gradient tensor [13]:
F = FFvol, (1.2.11)
that is, the local deformation is the composition of purely volumetric deformation, followed by an isochoric one. In particular, we impose that det F = 1 and Fvol = J . Within this
context, the two deformations F and Fvol contribute additively to the strain–energy [13],
W(C) = Wiso(I1, I4,f, I4,s, I8,f s) + Wvol(I3), (1.2.12)
where the invariants of the deviatoric component of the strain–energy are computed with respect of Ciso:
C := FTF = J−23C
and are defined as:
I1 = tr C = J− 2 3I1, I4,f = f · Cf = J− 2 3I4,f, I4,s= s · Cs = J− 2 3I 4,s, I8,fs= 1 2 h f · Cs + s · Cfi= J−23I8,fs.
The term Wisoaccounts for the isochoric deformation and it must be described by the strain–
energy of an incompressible material. For this reason it depends only on the deviatoric component of the deformation. We chose to describe the deviatoric strain-energy term with the Holzapfel–Ogden model showed in section1.2.2.
Wiso(C) = W(C) = W1(I1) + W4,f◦(I4,f◦) + W4,s◦(I4,s◦) + W8,f◦s◦(I8,f◦s◦) = a 2be b(I1−3)+ af 2bf ebf(I4,f◦−1)2 − 1+ as 2bs ebs(I4,s◦−1)2 − 1 + afs 2bfs ebfs(I8,f◦s◦)2 − 1. (1.2.13)
The second term of the right hand side of (1.2.12) is the volumetric energy, and it increases when J 6= 1. This means that the deformation becomes more energetically expensive the more the deformation is far from being isochoric. Several forms for Wvol can be found in the
literature [31]: as a rule of thumb, a good choice would be a function bounded from below, convex and whose derivative is zero for J = 1. We use the following form for Wvol [31]:
Wvol(J ) =
k 4(J
2− 1 − 2 ln J). (1.2.14)
It is worthwhile to notice that such expression is unbounded for J approaching zero or +∞ and has a physical meaning: collapsing down to a point the body or dilating it indefinitely would require an infinte energy. The parameter k in (1.2.14) is the bulk modulus. It is a pe-nalization factor that allows to enforce incompressibility. Increasing the value of k will make non isochoric deformation energetically more expensive and penalizing the volume variation of a deformation. For very small volume variation we could encounter some difficulties. In fact a large value for the bulk modulus could lead to locking phenomena. The penalization in this case is too strong and it prevents the body from moving.
The first Piola–Kirchhoff can be obtained applying the chain rule to the derivatives:
P = X k∈{1,4f,4s,8f s} ∂Wk ∂Ik ∂Ik ∂F + ∂Wvol ∂I3 ∂I3 ∂J ∂J ∂F, (1.2.15)
where the derivatives of the strain–energy are ∂W1 ∂I1 = a 2expb I1− 3 , ∂W4,f ∂I4,f = af I4,f − 1 exp h bf I4,f − 1 2i ∂W4,s ∂I4,s = as I4,s− 1 exp h bs I4,s− 1 2i , ∂W8,f s ∂I8,f s = af sI8,f sexpbf s I8,f s2 ∂Wvol ∂J = k 2 J2− 1 J ,
and the derivative of the invariants are ∂I1 ∂F = ∂ J−23I1 ∂F = J −2 3∂I1 ∂F + ∂J−23 ∂F I1 = 2J −2 3F − 2 3J −2 3I1F−T, ∂I4,f ∂F = ∂J−23I4,f ∂F = J −2 3∂I4,f ∂F + ∂J−23 ∂F I4,f = 2J −2 3f ⊗ f◦−2 3J −2 3I4,fF−T, ∂I4,s ∂F = ∂J−23I4,s ∂F = J −23∂I4,s ∂F + ∂J−23 ∂F I4,s = 2J −23 s ⊗ s◦− 2 3J −23I 4,sF−T, ∂I8,f s ∂F = ∂J−23I8,f s ∂F = J −2 3∂I8,f s ∂F + ∂J−23 ∂F I8,f s = J −2 3 (f ⊗ s◦+ s ⊗ f◦) − 2 3J −2 3I 8,f sF−T, ∂J ∂F = ∂ det F ∂F = J F −T .
Finally, the first Piola-Kirchhoff tensor reads P = a expb I1− 3 J− 2 3 F − I1 3 F −T + 2J−23af I4,f − 1 exp h bf I4,f − 1 2i f ⊗ f◦− I4,f 3 F −T ! + 2J−23as I4,s− 1 exp h bs I4,s− 1 2i s ⊗ s◦− I4,s 3 F −T ! + 2J−23a f sI8,f sexpbf s I8,f s2 " f ⊗ s◦+ s ⊗ f◦ 2 − I8,f s 3 F −T # +k 2 J 2− 1 F−T.
1.3
Active mechanics
Active materials have the ability of changing their own conguration without the need of an external load. Living matter is one of the most noticeable examples of active materials. For instance, muscles contract after being electrically stimulated. In this context, the description of the correct active behavior of the cardiac muscle represents an exciting challenge in the field of mechanobiology.
In cardiac mechanics, the contraction of the muscle is due to the presence of cardiomyocytes which react to electric excitation. Cardiomyocytes are spindly shaped cells organized in collagen sheets and their stiffness in the longitudinal direction is higher than the cross-sectional stiffness [42]. We need to define a reference configuration for the fibers and sheets microstructure to describe the muscular anisotropy.
The anisotropy is a fundamental feature in the mechanical description of the myocardium. Fibers and sheets when elongated contribute differently to the deformation due to the dif-ferent stiffness in the respective directions. Furthermore, during activation, fibers undergo an isochoric deformation shortening along their axis and thickening in the orthogonal di-rections. Although the anisotropy description could be different for the passive and active behavior, it plays an important role in both cases.
A common approach in the description of the active mechanics is to add an active term to the stress tensor. This active term depends on the deformation and on some quantities synthesizing the biochemical state of the cell. Because of the active term addition this approach is called active stress.
This approach describes the passive behavior of the material as we did in section1.2.2, then add an active stress component from constitutive considerations. For further details about the active stress approach see [10, 35].
We use a thermodynamically consistent active strain approach [35] which we briefly expose in the next section.
1.3.1
Active Strain formulation
An alternative approach, the so–called active strain, consists in a multiplicative decompo-sition of the deformation gradient tensor
F = FeFa
in an elastic deformation Fe and an active distortion Fa.
The idea is the following: an inelastic process, dictated by the biochemistry, locally changes the length and the shape of the fibres; then, an elastic deformation accommodates the active strain distortion Fain order to preserve the compatibility. The physiological basis
of the approach resides in the contractile units of the myocytes: the sarcomeres shorten because of the sliding filaments of the actin–myosin molecular motor, and this shortening is encoded by Fa, and the fictitious intermediate placement determined by Fa is the new
reference configuration for the elastic deformation.
As explained in [2], the strain energy only accounts for the elastic deformation, conse-quently the Piola stress tensor computed in the usual fashion is:
Pe =
∂W(Fe)
∂Fe
. (1.3.1)
Pe, due to the multiplicative decomposition of the deformation gradient, works on the
con-figuration B1 of figure 1.1 and produce stress on B2. Since our reference configuration is B0,
Pe must be pulled back to that configuration by mean of the Piola transformation (1.2.3):
P = det(Fa)PeF−Ta (1.3.2)
F
Fa Fe
B0
B
2
B
1Figure 1.1: Multiplicative decomposition of the total deformation. The deformation of gradient F that takes B0 to B2 is split in two subsequent deformation with gradients Fa and
Fe respectively.
The Cauchy stress tensors is:
T = (det Fe)−1
∂W(Fe)
∂Fe
FTe.
The active strain model needs a constitutive law for the description of Fa. In the next
sections we will show two possible choice: the transversely isotropic activation model and the orthotropic activation model.
1.3.2
Transversely isotropic activation
Since fibers contract along their axis a natural choice for Fa could be considering a
trans-versely isotropic activation model. This model is based on the hypothesis that myocites do not change their volume significantly during contraction [3]. Then an isochoric active deformation must be provided.
The form of the active deformation gradient is then:
Fa = γ1f◦⊗ f◦+ γ2(I − f◦⊗ f◦),
where the contraction γ1occurs along fibers direction f◦while γ2account for the deformation
in the orthogonal directions in order to preserve the volume. At this purpose introducing three orthogonal axes f◦, s◦ and n◦ we compute the determinant of Fa in order to determine
the relation between γ1 and γ2 that permits to preserve volume:
1 = det(Fa) = (Faf◦∧ Fas◦) · Fan◦ (f◦∧ s◦) · n◦ = (γ1f◦∧ γ2s◦) · γ2n◦ n◦· n◦ = γ1γ2n◦· γ2n◦ = γ1γ22, (1.3.3) ⇒ γ2 = 1 √ γ1 .
Taking γ = γ1 the final form of the active deformation gradient is:
Fa = γf◦⊗ f◦+
1 √
γ(I − f◦⊗ f◦), (1.3.4)
1.3.3
Orthotropic activation
An alternative to the transversely isotropic activation model consist in considering an or-thotropic activation. This kind of description allows us to obtain an additional parameter that we can exploit to describe the thickening occurring during myocardial contraction. Myocardium indeed thickens up to 40% during contraction [32], and such thickening it is supposed to be due to the sliding of the collagen sheets. In other words, contracted fibers enlarge in the directions orthogonal to their axis, and this enlargement forces the fibers to rearrange inside the sheetlets increasing myocardial wall thickness. In [35], the trans-versely isotropic activation model has been compared with the orthotropic one. The authors showed that the transversely isotropic activation model is not capable to reproduce such large thickening. Because, if a single myocyte shortens about 6%, in the orthogonal direction it would enlarge about 3%. They conclude that only the description of the isochoric fibers deformation at the microscale is not sufficient to reproduce the wall thickening occurring to the contracted myocardium. In order to include the hypothesis of sliding sheetles, in [35] the authors proposed to introduce a macroscopic rearrangement mechanics that occurs simultaneously with a microscopical rearrangement.
Following what they did in [35], we decompose the deformation gradient as follows:
where Feis the elastic deformation gradient, Fs the deformation gradient at the intermediate
scale and Fm the microscopic deformation gradient. Fm has the following expression
Fm = I + ξff◦⊗ f◦+ ξss◦⊗ s◦+ ξnn◦⊗ n◦,
where f◦, s◦, n◦ are the unitary vectors along fibers and sheets directions and along the
normal vector to the plane on which lay fibers and sheets. ξf is the microscopic deformation
occurring along fibers axis, ξs and ξn are the microscopic deformation along the s◦ and n◦.
We assume that fibers shortening is not influenced by the sliding process, therefore the sliding deformation gradient can be written as
Fs= I + ςss◦⊗ s◦+ ςnn◦⊗ n◦.
with ςs and ςnrepresenting the deformation occurring in the intermediate scale along s◦ and
n◦ respectively. This hypothesis implies that the order of the multiplication of Fs and Fm
is irrelevant, then we obtain the usual active strain decomposition: F = FeFsFm = FeFmFs= FeFa,
where Fa has the following form
Fa = I + γff◦ ⊗ f◦+ γss◦⊗ s◦+ γnn◦⊗ n◦.
with γf = ξf, γi = ςi+ ξi+ ςiξi for i ∈ {s, n}.
At this stage we include in the description of the middle–scale deformation a model used to link the microscopic deformation with that happening at the macro–scale. At this purpose we choose, as in [35], a linear relation among ςn and ξn:
ςn = k0ξn. (1.3.6)
During contraction the fiber expand in the cross–fiber direction, hence ξn is positive, while
the sliding process at the intermediate scale make the sheets thickens along s◦. Since the
active deformation must be isochoric, ςn must be negative, and hence also k0.
The total active cross–fiber deformation then is:
γn = ςn+ ξn+ ςnξn= (1 + k0)ξn+ k0ξn2. (1.3.7)
Using volume conservation at the micro scale, and symmetry in the cross–fiber strain ξn
can be brought back from ξf:
det Fm = (1 + ξf)(1 + ξs)(1 + ξn) = 1,
=⇒ ξn= ξs=
1 p1 + ξf
− 1. (1.3.8)
Finally, exploiting volume conservation on the total active strain tensor and (1.3.8), Fa
depends only on the microscopic deformation in the fiber direction: det Fa = (1 + γf)(1 + γs)(1 + γn) = 1 =⇒ γf = ξf γn = k0ξn(ξf) γs = 1 (1 + γf)(1 + γn) − 1
1.3.4
Transmurally non homogeneous orthotropic activation
In [35] the authors stated that their orthotropic activation model is capable of getting a sub-stantial wall thickening, but they also express that it fails to reproduce the wall thickening dishomogeneity occurring to the myocardium. Hence, they conclude that this phenomenon must be better investigate. We propose then a modification of the orthotropic active strain model with the aim of succeeding in modeling such phenomenon.
We assumed that the strains at the endocardium must be necessarily greater than those occurring at epicardium because of the greater arm of the epicardial fibers. Therefore, we thought that also the thickening due to sheetlets sliding must be affected by the different forces exerted by the fibers.
We modeled this behavior of the myocardium by means of a transmurally orthotropic acti-vation model where we exploited the good performance in terms of wall thickening of the orthotropic activation model and we will try to enhance it by modulating the thickening by means of a non constant value for k0.
Eventually, we must identify the best form of the scalar field k0, but initially we will consider a linear function joining the extremal values, kendo and kepi, we set for the endocardium and
for the epicardium:
k0(λ) = k0 kendo λ − λepi λepi− λendo + kepi λ − λendo λepi− λendo ,
where we indicated with lambda a curvilinear coordinate through the wall and with k0 the orthotropic activation model parameter. At the beginning we will set the extremal values kendo, kepi equal to one and zero respectively and then we will vary them.
In chapter 4 we will apply the models herein exposed, in particular we will take for granted the passive mechanics model and we will vary the activation model. Therefore we will consider an hyperelastic nearly–incompressible material with an exponential constitutive law provided by the Holzapfel and Ogden model, while we will use the three different active strain model herein presented: beside the transversely isotropic and orthotropic activation models we will use the transmurally orthotropic activation model that we proposed.
We will apply this framework at first to an idealized ventricle and then to a reconstructed real left ventricle in order compare the different activation model and to assess the capacity of our model to represent the dishomogeneity of the wall thickening
Chapter 2
Numerical discretization
In order to represents the complex phenomena occurring during the cardiac cycle we must solve the mathematical model showed in chapter 1. Its resolution is carried out by means of a numerical discretization method. In particular we choose to use a Galerkin method. We use the Galerkin method to compute the weak solution of (1.2.6). This method applies to linear partial differential equations and hence a linearization is required.
We first rewrite the problem in its weak formulation providing some appropriate functional spaces. Then, the Newton-Raphson method is used to solve the non linear equation. The solution of the nonlinear problem is achieved if the sequence of solutions of the linearized problem is convergent.
After linearization, we study the well posedness of the problem.
Eventually, we choose the finite dimensional space on which the solution is projected. At this purpose, after we partition the domain we choose a finite element space by providing a space basis. Then, we project the weak formulation of the linearized problem onto the finite dimensional basis obtaining its discretized version: an algebraic system of linear equations.
2.1
Weak formulation
Recall the model used to describe cardiac mechanics is given by the elastostatic equation − Div P(u) = 0 in Ω0 u = uD on ΓD Pν = t on ΓN Pν + αu = 0 on ΓR. (2.1.1)
We want to find the weak solution of (2.1.1). We must look for the solution in a proper functional space X. At this purpose we choose X to be the Sobolev space [HΓ1D(Ω0)]D =
{v ∈ [H1(Ω
0)]D : v|ΓD = 0}.
We obtain the weak formulation multiplying equation (2.1.1) by a test function v ∈ X and integrating it over the domain Ω0.
find u ∈ X: Z
Ω0
− Div P(u) · vdV = 0 ∀v ∈ X. (2.1.2)
Integrating by parts (2.1.2) and rearranging the boundary terms, supposing for the sake of exposition an homogeneous Dirichlet condition, the equation becomes:
find u ∈ X: Z Ω0 P(u) : ∇0vdV − Z ΓR αu · vdA = Z ΓN t · vdA ∀v ∈ X. (2.1.3)
The nearly incompressible formulation and the description of the active behaviors are in-cluded in P. We obtain the expression of P by firstly additively split the strain energy in order to include the nearly incompressible formulation
W = Wisochoric I1, I4,f, I4,s, I8,f s + Wvolume I3 , (2.1.4)
then, splitting the first term of the right hand side of (2.1.4) for the introduction of active strain model Wisochoric = W1 IE1, IE4,f, IE4,s, γf + W4,f IE4,f, γf + W4,s IE4,s, γf + W8,f s IE8,f s, γf , where the apex E indicates that the invariant are computed in the intermediate configuration introduced by the active strain model. By deriving the resulting strain energy function, we obtain the first Piola–Kirchoff stress tensor
P = P + Pvol= P1+ P4,f + P4,s+ P8,f s+ Pvol = X k∈{1,4f,4s,8f s} ∂Wk ∂IEk X j∈{1,4f,4s,8f s} ∂IEk ∂Ij ∂Ij ∂F + ∂Wvol ∂I3 ∂I3 ∂J ∂J ∂F, (2.1.5)
where the apexes E indicate that the invariants must be computed in the intermediate configuration of the multiplicative decomposition of the deformation gradient of the active strain model. Hence IE1, IE4,f, IE4,s, IE8,f s indicate respectively tr(Ce) = tr(FeTFe), Cef◦ · f◦,
Ces◦· s◦ and Cef◦· s◦. Their expressions are
IE1 = 1 − γn(γn+ 2) (γn+ 1)2 I1 + γn(γn+ 2) (γn+ 1)2 − γf(γf + 2) (γf + 1)2 I4,f + γn(γn+ 2) (γn+ 1)2 −γs(γs+ 2) (γs+ 1)2 I4,s, IE4,f = I4,f (γf + 1)2 , IE4,s= I4,s (γs+ 1)2 , IE8,f s = I8,f s (γf + 1)(γs+ 1) .
In order to write the final form of P we must provide the expression of the derived terms in (2.1.5) ∂W1 ∂IE1 = a 2exp h bIE1 − 3i, ∂W4,f ∂IE4,f = af IE4,f − 1exp bf IE4,f − 12 ∂W4,s ∂IE4,s = as IE4,s− 1exp bs IE4,s− 12 , ∂W8,f s ∂IE8,f s = af sI E 8,f sexp h bf s IE8,f s2i ∂Wvol ∂J = k 2 J2− 1 J ,
∂IE1 ∂I1 = 1 − γn(γn+ 2) (γn+ 1)2 , ∂I E 1 ∂I4,f = γn(γn+ 2) (γn+ 1)2 − γf(γf + 2) (γf + 1)2 , ∂IE1 ∂I4,s = γn(γn+ 2) (γn+ 1)2 −γs(γs+ 2) (γs+ 1)2 , ∂I E 4,f ∂I4,f = 1 (γf + 1)2 , ∂IE4,s ∂I4,s = 1 (γs+ 1)2 , ∂I E 8,f s ∂I8,f s = 1 (γf + 1)(γs+ 1) , where the omitted terms are zero, and:
∂I1 ∂F = ∂J−23I1 ∂F = J −2 3∂I1 ∂F + ∂J−23 ∂F I1 = 2J −2 3F − 2 3J −2 3I1F−T, ∂I4,f ∂F = ∂J−23I4,f ∂F = J −2 3∂I4,f ∂F + ∂J−23 ∂F I4,f = 2J −2 3f ⊗ f◦−2 3J −2 3I 4,fF−T, ∂I4,s ∂F = ∂J−23I4,s ∂F = J −2 3∂I4,s ∂F + ∂J−23 ∂F I4,s = 2J −2 3s ⊗ s◦− 2 3J −2 3I 4,sF−T, ∂I8,f s ∂F = ∂J−23I8,f s ∂F = J −2 3∂I8,f s ∂F + ∂J−23 ∂F I8,f s = J −2 3 (f ⊗ s◦+ s ⊗ f◦) − 2 3J −2 3I8,f sF−T, ∂J ∂F = ∂ det F ∂F = J F −T.
Finally, the first Piola-Kirchhoff tensor reads P = 1 −γn(γn+ 2) (γn+ 1)2 a exp h b IE1 − 3iJ−23 F − I1 3 F −T + 2J−23 ( γn(γn+ 2) (γn+ 1)2 − γf(γf + 2) (γf + 1)2 a 2exp h b IE1 − 3i+ 1 (γf + 1)2 af IE4,f − 1exp bf IE4,f − 12 ) f ⊗ f◦− I4,f 3 F −T ! + 2J−23 ( γn(γn+ 2) (γn+ 1)2 − γs(γs+ 2) (γs+ 1)2 a 2exp h bIE1 − 3i+ 1 (γs+ 1)2 as IE4,s− 1exp bs IE4,s− 12 ) s ⊗ s◦− I4,s 3 F −T ! + 2J−23 1 (γf + 1)(γs+ 1) af sI E 8,f sexp h bf s IE8,f s2i " f ⊗ s◦+ s ⊗ f◦ 2 − I8,f s 3 F −T # +k 2 J 2− 1 F−T .
2.2
Linearization
As we anticipated, in order to solve the elastostatic problem we will make use of the Galerkin method.
Clearly, the Galerkin method cannot be used to solve (2.1.3) because the its first term is not a bilinear form.
The nonlinearity of the problem is only partially due to the exponential constitutive rela-tion. There is also a geometrical nonlinearity due to the large deformation occuring to the body. In fact, even if we choose a linear relation, the large deformation requires the Piola transformation, which is non linear, to materialize the deformed body. An alternative to directly resolve (2.1.3), can be the resolution of the corresponding linearized problem. At this purpose we can exploit a method originally born for the research of roots in non linear function and then adapted also in more general problems: the Newton-Raphson method.
2.2.1
The Newton-Rapshon method
We will make use of the Newton-Raphson method in order to solve the nonlinear elasto-static problem. However a more general formulation of the method is needed to deal with functional defined over Banach spaces.
Let V and W be Banach spaces, and U ⊂ V an open subset of V . A functional F : U → W is said Fr´echet differentiable at x if there exists a bounded linear operator DF (x) : V → W such that: lim khkV→0 kF (x + h) − F (x) − DF (x)[h]kW khkV = 0 ∀h : x + h ∈ U, (2.2.1) and the bounded linear operator DF (x) is the Fr´echet differential.
Exploiting the Fr´echet differential, the problem:
F (v) = 0, (2.2.2)
can be approximated at first order:
F (v) + DF (v)[δv] = 0. (2.2.3)
Taking δvk = vk+1 − vk and taking advantage of the linearity of the differential operator,
(2.2.3) lead us directly to the usual formulation of the Newton’s method:
DF (vk)[vk+1] = DF (vk)[vk] − F (vk). (2.2.4) The apexes on v in (2.2.4) highlight that, in order to get the solution of a single nonlinear problem, we must follow the sequence of solutions of the linearized problem. Hence, equation (2.2.4) must be solved until convergence.
However, the method is not assured to converge, but if it converges the convergence will be at least linear (Newton-Kantorovich theorem [29]). The lack of convergence assurance impose us to use an incremental procedure both in the external loads imposition and in the fiber activation step.
2.2.2
Tangent problem
Given an initial guess u, using the Newton’s method the weak formulation reads:
where: a(δu, v; u) = Z Ω0 ∂P ∂F(u)[∇0δu] : ∇0vdV − Z ΓR αδu · vdA (2.2.6) R(v; u) = − Z Ω0 P(u) : ∇0vdV + Z ΓR αu · vdA (2.2.7) L(v) = Z ΓN t · vdA. (2.2.8)
In order to compute the differential of the stress tensor, we apply again the chain rule: ∂P ∂F[∇0δu] = ∂ ∂F X k∈{1,4f,4s,8f s} ∂Wk ∂IE k X j∈{1,4f,4s,8f s} ∂IE k ∂Ij ∂Ij ∂F + ∂Wvol ∂J ∂J ∂F : ∇0δu = X k∈{1,4f,4s,8f s} ∂2W k ∂IE k∂IkE X j∈{1,4f,4s,8f s} ∂IE k ∂Ij ∂Ij ∂F X j∈{1,4f,4s,8f s} ∂IE k ∂Ij ∂Ij ∂F : ∇0δu + X k∈{1,4f,4s,8f s} ∂Wk ∂IE k X j∈{1,4f,4s,8f s} ∂IE k ∂Ij ∂2I j ∂F∂F : ∇0δu + ∂ ∂J ∂Wvol ∂J ∂J ∂F ∂J ∂F : ∇0δu, = ∂2W 1 ∂IE 1 ∂I1E ∂IE 1 ∂I1 ∂I1 ∂F + ∂IE 1 ∂I4,f ∂I4,f ∂F + ∂IE 1 ∂I4,s ∂I4,s ∂F ∂IE 1 ∂I1 ∂I1 ∂F + ∂I1E ∂I4,f ∂I4,f ∂F + ∂I1E ∂I4,s ∂I4,s ∂F : ∇0δu + " ∂2W4,f ∂IE 4,f∂I4,fE ∂I4,fE ∂I4,f ∂I4,f ∂F !# ∂I4,fE ∂I4,f ∂I4,f ∂F ! : ∇0δu + " ∂2W 4,s ∂I4,sE ∂I4,sE ∂IE 4,s ∂I4,s ∂I4,s ∂F !# ∂IE 4,s ∂I4,s ∂I4,s ∂F ! : ∇0δu + " ∂2W 8,f s ∂IE 8,f s∂I8,f sE ∂I8,f sE ∂I8,f s ∂I8,f s ∂F !# ∂I8,f sE ∂I8,f s ∂I8,f s ∂F ! : ∇0δu + ∂W1 ∂IE 1 ∂IE 1 ∂I1 ∂I1 ∂F∂F + ∂IE 1 ∂I4,f ∂I4,f ∂F∂F + ∂IE 1 ∂I4,s ∂I4,s ∂F∂F : ∇0δu + " ∂W4,f ∂IE 4,f ∂IE 4,f ∂I4,f ∂I4,f ∂F∂F !# : ∇0δu + " ∂W4,s ∂IE 4,s ∂I4,sE ∂I4,s ∂I4,s ∂F∂F !# : ∇0δu
+ " ∂W8,f s ∂IE 8,f s ∂I8,f sE ∂I8,f s ∂I8,f s ∂F∂F !# : ∇0δu + ∂Wvol ∂J + J ∂2W vol ∂J2 F−T : ∇0δu JF−T + J ∂Wvol ∂J −F −T∇ 0δuF−T .
Where the second derivatives of the strain–energy function are: ∂2W 1 ∂IE1∂IE1 = ab 2 exp b IE1 − 3, ∂2W4,f ∂IE4,f∂IE4,f = 2afbf IE4,f − 12exp bf IE4,f − 12, ∂2W4,s ∂IE4,s∂IE4,s = 2asbs IE4,s− 12exp bs IE4,s− 12, ∂2W8,f s ∂IE8,f s∂IE8,f s = 2af sbf sI E 8,f s2exp bf s IE8,f s2, ∂2Wvol ∂J ∂J = k 2 1 − 1 J2 ,
while the second derivatives of the invariants are ∂2I1 ∂F∂F[∇0δu] = ∂2J−23F − 2 3J −2 3I1F−T ∂F = 2J−23 h −2 3F −T : ∇0δu F − I1 3 F −T + ∇0δu − 2 3(F : ∇0δu) F −T +I1 3 F −T (∇0δu)T F−T i , ∂2I 4,f ∂F∂F [∇0δu] = ∂ 2J−23 h f ⊗ f◦− I4,f 3 F −Ti ∂F [∇0δu] = −4 3J −2 3F−T : ∇0δu f ⊗ f◦− I4,f 3 F −T + 2J−23∇0δu f◦⊗ f◦ −2 3J −2 3F−T (f ⊗ f◦ : ∇0δu) + 2J− 2 3I4,f 3 F−T (∇0δu) T F−T
∂2I 4,s ∂F∂F [∇0δu] = ∂2J−23 h s ⊗ s◦− I4,s 3 F −Ti ∂F [∇0δu] = −4 3J −23F−T : ∇ 0δu s ⊗ s◦− I4,s 3 F −T + 2J−23∇0δu s◦⊗ s◦ − 2 3J −23F−T(s ⊗ s ◦ : ∇0δu) + 2J− 2 3I4,s 3 F−T(∇0δu)T F−T ∂2I 8,f s ∂F∂F [∇0δu] = ∂2J−23 f ⊗s◦+s⊗f◦ 2 − I8,f s 3 F −T ∂F [∇0δu] = −4 3J −23F−T : ∇ 0δu f ⊗ s◦+ s ⊗ f◦ 2 − I8,f s 3 F −T + 2J−23∇0δu (f◦ ⊗ s◦+ s◦⊗ f◦) − 2 3J −2 3 f ⊗ s◦+ s ⊗ f◦ 2 : ∇0δu F−T + 2 3J −2 3I8,f sF−T (∇0δu) F−T , ∂2J ∂F∂F[∇0δu] = ∂ J F−T ∂F [∇0δu] = J F −T : ∇0δu F−T − JF−T (∇0δu) T F−T.
We have defined all terms of (2.2.5). In the next section we will apply the Galerkin’s method in order to obtain the discrete version of the original problem.
2.3
Discrete problem
In this section we discretize the variational problem (2.2.5) by means of the Galerkin method. First, we project the infinite dimensional Hilbert space X onto a finite dimensional subset Xh ⊂ X depending on a positive parameter h.
A partition Ph of the domain Ω0 is such that: Ph = ∪j=1nel Kj and given Ki and Kj in Ph
then Ki∪ Kj = ∂Ki∪ ∂Kj for every couple of i and j, where K indicates a generic element
and with nel we indicated the total number of elements. Then, we can define the finite
element space Xh, as:
Xh ⊂ C0(Ω0)
The variational problem becomes:
find δuh ∈ Xh : a(δuh, vh; uh) = R(vh; uh) + L(vh) ∀vh ∈ Xh, (2.3.1)
a(δuh, vh; uh) = Z Ω0 ∂P ∂F(uh)[∇0δuh] : ∇0vhdV − Z ΓR αδuh· vhdA (2.3.2) R(vh; uh) = − Z Ω0 P(uh) : ∇0vhdV + Z ΓR αuh· vhdA (2.3.3) L(vh) = Z ΓN t · vhdA. (2.3.4)
• the functional space where the problem is set must be an Hilbert space; • ellipticity and continuity of the bilinear form;
• continuity of the linear functional;
In the present cotext, the above hypothesis are satisfied, as it has been shown in [34], hence the resolution with the Galerkin method will lead to the unique weak solution.
Let {φi}ni=1b be a vectorial basis for Xh, where nb is the dimension of the basis. Since {φi}ni=1b
is a basis for Xh, we need (2.3.1) to be tested for all basis function, as every function of Xh
is a linear combination of {φi}ni=1b . Then, we require that
find δuh ∈ Xh : a(δuh, φi; uh) = R(φi; uh) + L(φi) ∀i ∈ {1, ..., nb} (2.3.5)
the unknown δuh can be expressed as linear combination of the basis function δuh =
Pnb
j=1δujφj. Including it in the previous expression and taking advantage of the
linear-ity of a(·, ·), then (2.3.5) becomes:
a nb X j=1 δujφj, φi; uh ! = R(φi; uh) + L(φi) ∀i ∈ {1, ..., nb}, nb X j=1 δuja (φj, φi; uh) = R(φi; uh) + L(φi) ∀i ∈ {1, ..., nb}, nb X j=1
δujAi,j(uh) = ri(uh) + li ∀i ∈ {1, ..., nb},
A(uh)δu = r(uh) + l, (2.3.6)
where Ai,j(uh), ri(uh) and li are
Ai,j(uh) = Z Ω0 ∂P ∂F(uh)[∇0φj] : ∇0φidV − Z ΓR αδφj · φidA (2.3.7) ri(uh) = − Z Ω0 P(uh) : ∇0φidV + Z ΓR αuh· φidA (2.3.8) li = Z ΓN t · φidA. (2.3.9)
Note that, since δuh = uk+1h − ukh, equation (2.3.6) provides actually a sequence of solutions
uk+1h = ukh+ A(ukh)−1r(ukh) + l .
If the sequence is convergent, then it converges to the unique solution of the problem. In Algorithm 1, we show the procedure used to compute the elastostatic solution.
Algorithm 1 Resolution of the elastostatic problem
1: u0 ← initial guess f or the displacement . initialize displacement
2: τ ← tolerance . set the Newton’s method tolerance
3: while kδuk+1− δukk > τ do
4: update A(uk) . update the system matrix
5: update r(uk) . update the r.h.s depending on previous solution
6: Solve A(uk)δuk+1 = r(uk) + l . solve the linear system
Chapter 3
Geometry reconstruction
This chapter is dedicated to the description of the segmentation method used for patient-specific reconstruction of the ventricle. Among the variety of segmentation techniques ex-isting in literature, we choose to use the Split-Bregman minimization of the Region-Scalable Fitting Energy (RSFE) method as proposed by Yang et al. [45] because of its properties to work with inhomogeneous intensity images. In a previous work with M. Fedele and F. Cremonesi [12], we implemented this algorithm in a C++ library enriching it with some ad-ditional tools to make it more suitable to process medical images. After a brief introduction to the concept of image segmentation (section3.1), we focus on the mathematical derivation of the algorithm (sections 3.2, 3.3 and 3.4) and we provide a description of our numerical implementation (section3.5). Finally we talk about our customizations focused on medical applications (section 3.6) showing an example of pre-processing and segmentation in the last section (section 3.7). We refer to the project report [12] for more details about code implementation.
3.1
Introduction to Image Segmentation
In computer vision, image segmentation is the process of partitioning a digital image into multiple segments (sets of pixels). The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyse. More strictly, image segmentation is the process of assigning a label to every pixel in an image such that pixels with the same label share certain visual characteristics or computed properties (such as colour, intensity, or texture). Hence, the result of image segmentation is a set of subregions such that each region is significantly different with respect to the char-acteristics of interest. It is typically used to locate objects and boundaries inside images. Concerning 3D medical images like X-ray Computed Tomography (CT) or Magnetic Res-onance Imaging (MRI), the most common way of looking at them is scrolling through the 2D slices in which they are divided. Generating contours inside a 3D medical image using segmentation allows instead the 3D reconstructions of any human organ.
The most used methods in image segmentation are the active contours models [7,11,21,
8,6], which can be divided in two typical approaches: edge-based methods and region-based methods.
Edge-based methods [8, 21, 4] use local edge information to move the active contour to-ward image boundaries. The ability of these methods relies principally on the definition of a good edge detector and their major limitation is the need of strong edges between objects. Region-based methods [6, 45] use certain region descriptor such as for example mean intensity to guide the active contour. Limits of the majority of region-based methods
are due to their assumption of intensity homogeneity inside the regions to be segmented. In fact, intensity inhomogeneity often occurs in real images and in particular in medical images from different modalities.
In our work we implement the Split-Bregman algorithm for the minimization of the Region-Scalable Fitting Energy model (SB-RSFE) [45] because of its ability to work also on regions with intensity inhomogeneities.
In this thesis the treated images are always three dimensional grey scale images. From a mathematical standpoint, a 3D image can be identified with a domain Ω ⊂ R3 (i.e. the
image domain) and a function u0(x) standing for the image intensity distribution over Ω.
3.2
The Region Scalable Fitting Energy Functional
The model is based on the RSFE functional minimization [24] and was created with the aim to overcome the limit of the Piecewise Constant (PC) Chan-Vese model [6].
The idea behind the Chan-Vese model is to divide the image in two regions and approxi-mate the intensity of the image inside each region with a constant value representing the average of the intensity inside the region. The regions are obtained minimizing a functional derived from Mumford-Shah functional [6]. This method is conceived in order to segment images consisting of homogeneous regions and fails to provide the correct segmentation in inhomogeneous images as shown in [24].
The RSFE model can be seen as an improvement of this model. In fact the idea behind is similar but it substitutes the constant values with functions that approximate the image intensity in a local region through the use of a Gaussian kernel. The name comes from this: the method is considered scalable because through the use of the Gaussian kernel with a scale parameter it is possible to decide the size of the stencils on which the average of the intensity around a pixel is calculated.
We notice that the RSFE model with a large scale parameter tends to the PC model; from here we can sense the importance of the scale parameter in RSFE for the segmentation of inhomogeneous images.
The aim of the segmentation algorithm is to divide the image in two parts: the object Ω1 ⊂ Ω and the background Ω2 ⊂ Ω with Ω1 ∩ Ω2 = 0 and Ω1∪ Ω2 = Ω. We notice that
the interface between the object and the background is the 3D surface or the 2D contour representing the segmented region; we call contour this interface. The RSFE method is based on a level set formulation. This means that the solution of the method (i.e. the searched contour) is described as the α-level iso-surface S(t) = {x ∈ Ω : φ(x, t) = α} of a scalar function φ(x, t) : Ω × R+→ [a
0, b0], where α = (a0+ b0)/2.
The result of the segmentation is obtained minimizing the subsequent functional called the region scalable fitting energy F :
F (φ, f1(x), f2(x)) = Z Ω " 2 X i=1 λi Z Ω Kσ(x − y) u0(y) − fi(x) 2 Miε(y)dy # dx + ν Z Ω ∇Mε 1(x) dx + µ Z Ω 1 2 ∇φ(x) − 1 2 dx, (3.2.1)
The first term in 3.2.1 is the error we make by approximating the image intensity u0(x)
