Patient-specific predictions of aneurysm growth and remodeling in the ascending thoracic aorta using the homogenized constrained mixture model
aa r X i v : . [ phy s i c s . m e d - ph ] D ec Noname manuscript No. (will be inserted by the editor)
Patient–specific predictions of aneurysm growth andremodeling in the ascending thoracic aorta using thehomogenized constrained mixture model
S. Jamaleddin Mousavi · SolmazFarzaneh · Stéphane Avril the date of receipt and acceptance should be inserted later
Abstract
In its permanent quest of mechanobiological homeostasis, our vascula-ture significantly adapts across multiple length and time scales in various physi-ological and pathological conditions. Computational modeling of vascular growthand remodeling (G&R) has significantly improved our insights of the mechanobio-logical processes of diseases such as hypertension or aneurysms. However, patient–specific computational modeling of ascending thoracic aortic aneurysm (ATAA)evolution, based on finite-element models (FEM), remains a challenging scientificproblem with rare contributions, despite the major significance of this topic ofresearch. Challenges are related to complex boundary conditions and geometriescombined with layer-specific G&R responses. To address these challenges, in thecurrent paper, we employed the constrained mixture model (CMM) to model thearterial wall as a mixture of different constituents such as elastin, collagen fiberfamilies and smooth muscle cells (SMCs). Implemented in Abaqus as a UMAT,this first patient–specific CMM-based FEM of G&R in human ATAA was firstvalidated for canonical problems such as single–layer thick–wall cylindrical andbi–layer thick–wall toric arterial geometries. Then it was used to predict ATAAevolution for a patient–specific aortic geometry, showing that the typical shapeof an ATAA can be simply produced by elastin proteolysis localized in regions ofderanged hemodymanics. The results indicate a transfer of stress to the adventitiaby elastin loss and continuous adaptation of the stress distribution due to changeof ATAA shape. Moreover, stress redistribution leads to collagen deposition wherethe maximum elastin mass is lost, which in turn leads to stiffening of the arterialwall. As future work, the predictions of this G&R framework will be validatedon datasets of patient–specific ATAA geometries followed up over a significantnumber of years.
Keywords finite–elements · constrained mixture theory · anisotropic behaviour · zero–pressure configuration · residual stresses S. Jamaleddin Mousavi · Solmaz Farzaneh · S. Avril ∗ Mines Saint-Étienne, Univ Lyon, Univ Jean Monnet, INSERM, U 1059 Sainbiose, Centre CIS,F - 42023 Saint-Étienne France.Phone: 0477420188, Fax: +33477420000, ∗ Corresponding author E-mail: [email protected] S. Jamaleddin Mousavi et al.
List of symbols
In folowing i ∈ (cid:8) e , c j , m (cid:9) and k ∈ { c j , m } a k The unit vector pointing direction of the k th fiber C i el Elastic right Cauchy-Green deformation tensor of the i th constituent C i el Modified elastic right Cauchy-Green deformation tensor of the i th con-stituent D max Maximum damage of elastin ˙ D i g Generic rate function of i th constituent F The total deformation gradient of the mixture F i tot Total deformation gradient of the i th constituent F i el Elastic deformation gradient of the i th constituent F i gr Total inelastic (G&R) deformation gradient of the i th constituent F i g Deformation gradient of the i th constituent due to growth F i r Deformation gradient of the i th constituent due to remodelling G i h Deposition stretch tensor of the i th constituent J Jacobian of the mixture I i First invariant of the right Cauchy-Green deformation tensor for the i thconstituent I i Fourth invariant of the right Cauchy-Green deformation tensor for the i thconstituent k c j σ Gain or growth parameter of collagen fiber families k k Fung-type material coefficient the k th constituent k k Fung-type material coefficient the k th constituent L dam The spatial damage spread parameter of elastin S Second Piola–Kirchhoff stress T i The average turnover time of the i th constituent t dam The temporal damage spread parameter of elastin W The specific strain energy density function of the mixture W i The strain energy of the i th individual constituents X Material point in a reference configuration x Material point in a deformed or current configuration α c j each direction of collagen fiber families µ e Neo Hookean material coefficient of elastin κ Bulk modulus of elastin σ i Current stress of extant i th constituent σ c j h Average stress of i th constituent at homeostasis λ e z Axial elastin deposition stretch value λ e θ Circumferential elastin deposition stretch value λ k Deposition stretch value of k th constituent in fiber direction Ω Reference configuration Ω ( t ) Deformed or current configuration ̺ i Mass densities of of the i th constituent before G&R ̺ it Mass densities of the i th constituent at time t ˙ ̺ e ( t ) The rate of mass degradation of the elastin atient–specific predictions of aneurysm growth and remodeling 3 ˙ ̺ c j adv ( t ) The rate of mass degradation or deposition in the adventitia for collagenfibers ˙ ̺ c j med ( t ) The rate of mass degradation or deposition in the media for collagen fibers
Growth and remodeling (G&R) are fundamental mechanobiological processes innormal tissue development and in various pathological conditions. It is suggestedthat G&R in tissues may be mediated by mechanical stresses. For example, cardiachypertrophy and normal cardiac growth develop in response to increased hemo-dynamic loading and altered systolic and diastolic wall stresses [27]. Sustainedhypertension is also associated with changes such as increased wall thickness inlarge arteries [36]. This adaptation ability of soft tissues is related to the exis-tence of a mechanical homeostasis across multiple length and time scales in thevasculature. At the tissue scale, this manifests through continuous mass changesof the components of the extracellular matrix (ECM) such as collagen, elastin andproteoglycans [14, 39].In the current paper, we are interested in continuum finite-element formula-tions to simulate G&R in arteries. The first model of mechano–regulated soft tissuegrowth was presented by Rodriguez et al [52] in the mid–1990s, incorporating theassociated growth by multiplicative decomposition of the total deformation gra-dient into an elastic and inelastic part. Thereafter, this conceptual simplicity hasbeen widely used by others. Comellas et al [12] introduced an original constitutivemodel to study remodeling of damaged tissue within the framework of continuumdamage mechanics and open-system thermodynamics. The total damage rate wascalculated as the sum of a healing rate and a mechanical damage rate. In or-der to couple biochemical and biomechanical damage, the healing rate was drivenby mechanical stimuli and subjected to simple metabolic constraints. Althoughtheir model was based on the mixture theory, it did not account for the evolvingprestretch of each constituent.Although many theories of G&R have modelled the tissue as a homogenized(single-constituent) solid continuum [33,34], the constrained mixture model (CMM)has been increasingly employed by a number of authors [3,8,14,20,24,43,45,56,59,64,67] to simulate G&R in arteries, including non–homogenized [20,56,59] and ho-mogenized [8, 14, 45] approaches. For example, Valentín et al [59, 60] established anonlinear finite element model (FEM) based on the non–homogenized constrainedmixture theory (CMT) of G&R to facilitate numerical analyses of various casesof arterial adaptation and maladaptation. Watton et al [64] presented the firstmathematical framework to study G&R in two-layered cylindrical membranes. Intheir framework the natural configurations of each individual constituent were up-dated at each time step and thus cell-mediated G&R effects could be handled [38].They introduced collagen fibre recruitment and collagen fibre density in the strainenergy density functions. Baek et al [3] made another important contribution tostudy the growth of intracranial cerebral aneurysms with a constrained mixturethin-walled model, permitting to account for evolving strain energy density func-tions. Although both adopted a CMM (deforming different constituents altogetherunder a total mixture deformation gradient but having different natural referenceconfigurations) the mathematical foundations were slightly different. The former
S. Jamaleddin Mousavi et al. employed a rate-based approach whilst the latter used an integral approach. Therate-based approach was shown to be more efficient and as accurate as the inte-gral one [25]. Including disease progression and evolving geometries, CMT-basedmodels were able to predict changes in fiber orientations and quantities, degra-dation of elastin and loss of smooth muscle cells (SMCs). The same concept wasimplemented by Famaey et al [20] in Abaqus [30] to predict adaptation of a pul-monary autograft over an extended period. There are a number of other appliedcontributions considering, for example: – two dimensional (2D) non–homogenized CMM for arterial G&R proposed byBaek et al [3], for cerebral aneurysms and extended by Valentín et al [55,57,58]for cerebral arteries – the evolving geometry, structure, and mechanical properties of a representativestraight cylindrical artery subjected to changes in mean blood pressure and flowin 3D [41].Braeu et al [8] and Cyron et al [14] introduced the homogenized CMM frame-work for G&R using an informal temporal averaging approach. Lin et al. [45]combined homogenization and the CMT to simulate the dilatation of abdominalaortic aneurysms. Their methodology is computationally less expensive than non–homogenized CMM and it can yet capture important aspects of G&R such as massturnover in arterial walls. Unlike non–homogenized CMMs in which one must dealwith myriads of evolving configurations, the homogenized CMM is based on a sin-gle time–independent reference configuration for each material species and eachpoint with a time–dependent inelastic local deformation of G&R. Recently, La-torre and Humphrey [44] introduced a new rate-based CMM formulation suitablefor studying mechanobiological equilibrium and stability of soft tissues exposedto transient or sustained changes, permitting direct resolution of G&R problemswith a static approach.Although prior work in the CMM framework have significantly improved ourinsights of arterial wall G&R, they have been mostly limited to canonical prob-lems in arterial mechanics such as 2D [3, 55, 57, 58] or simplified 3D cases, us-ing membrane [14] or single–layer thick–wall axisymmetric [8, 45] approximations.Therefore, the framework still requires to be extended to more realistic and di-verse analyses including patient–specific arterial geometries. To this end, severalproblems still need to be addressed within the CMM framework such as layerspecificity, irregular boundary conditions and complex deformations. These prob-lems can become extremely challenging in the case of ascending thoracic aorticaneurysms (ATAA) due to the simulateneous and region specific evolution of ge-ometry, material properties [22], and hemodynamic loads [13, 38].In the present work, the objective is to set up the first nonlinear FEM basedon the homogenized CMT to simulate G&R in patient–specific ATAA. After itsimplementation, the FEM is first validated on an idealized single–layer thick–wallcylinder. In a second stage, the model is illustrated for a canonical problem inarterial mechanics: G&R of a toric bilayer thick–wall arterial geometry. Then itis used to predict ATAA evolution for a patient–specific aortic geometry, showingthat the typical shape of an ATAA can be obtained simply with a proteolysis ofelastin localized in regions of deranged hemodymanics. atient–specific predictions of aneurysm growth and remodeling 5 χ : Ω be the general mapping in a R domain. Ω is considered as the invivo (for example healthy) configuration of a blood vessel before any specific G&Rstarts. The total deformation gradient of a mixture of n different constituents (e.g.elastin, collagen fiber families, or SMCs), F , between a material point, X ∈ Ω ,from a reference configuration, Ω (0) = Ω , and a position, x = χ ( X , t ) ∈ Ω , ina deformed or current configuration, Ω ( t ) = Ω t , at time t can be defined as F ( X , t ) = ∂ x ∂ X (1)Reference volume elements d V ∈ Ω are mapped to volume elements d v = J d V ∈ Ω t with the Jacobian J = | F | > 0.According to the CMT, we assume that all constituents in the mixture deformtogether under the total deformation gradient F in the stressed field while eachconstituent has a different "total" deformation gradient resulting from its owndeposition stretch. Therefore, assuming that G i h ( i ∈ { e , c j , m } , where superscriptse, c j and m represent respectively the elastin, the constituent made of each of the n possible collagen fiber families and the SMCs, all these constituents makingthe mixture) is the deposition stretch of the i th constituent with respect to thereference homeostatic configuration [10, 49, 50], the "total" deformation gradientof the i th constituent can be calculated as F i tot = FG i h (2)where the deposition stretch tensor of elastin may be written such as G eh =diag[ λ e θ λ e z , λ e θ , λ e z ] to ensure incompressibility, and the deposition stretch tensorof collagen families and SMCs may be written such as G kh = λ k a k ⊗ a k + √ λ k ( I − a k ⊗ a k ) , k ∈ { c j , m } . λ e θ and λ e z are the deposition stretches of elastin in thecircumferential and longitudinal directions, respectively, and λ k is the depositionstretch of the k th constituent in the fiber direction with a unit vector a k .The local stress–free state may vary between each constituent and even betweenthe differential mass depositions of these constituents at different time increments.Thus, for each differential mass increment of the i th constituent deposited at time τ , the total deformation gradient of each constituent in Eq. 2 may be rewrittenby a multiplicative decomposition into an elastic F i el and inelastic (namely G&R)part F i gr as F i tot = F i el F i gr (3)It is noteworthy that due to continuous G&R, the inelastic G&R deformation gra-dient includes the changes between the local stress-free configurations of differentmass increments resulting from deposition in a different configuration and at a dif-ferent time. This process is schematically shown in Fig. xx. Dynamic effects suchas inertia or viscoelasticity can usually be neglected during G&R as they occur atslow time scales [8]. S. Jamaleddin Mousavi et al. W = ̺ e t (cid:0) W e ( I e ) + U ( J eel ) (cid:1) + n X j =1 ̺ c j t W c j ( I c j ) + ̺ m t W m ( I m ) (4)where ̺ it and W i ( i ∈ { c j , m } ) refer respectively to the mass densities and strainenergy of the individual constituents based on the first ( I i ), fourth ( I i ) invariantsand Jacobian ( J ).Strain energy density of elastin is described by a Neo–Hookean function inwhich incompressibility is enforced by a penalty function of the Jacobian [10,31,49]as W e ( I e ) = µ e I e − (5a) U e ( J eel ) = κ ( J eel − (5b)where µ e and κ are respectively a material parameter and the bulk modulus(stress–like dimensions), and I e = tr ( C eel ) (6a) C eel = F eel( T ) F eel (6b) F eel = 1 J eel / F eel (6c) J eel = det ( F eel ) > (6d)Noting that det ( F eel ) = 1 .The passive strain energy density of SMCs and collagen families are are de-scribed using an exponential expression respectively as [10, 49, 51, 53] W c j ( I c j ) = k c j k c j h e k c j ( I c j − − i (7)and W m ( I m ) = k m k m h e k m ( I m − − i (8) k c j and k m are stress–like material parameters while k c j and k m are dimensionlessmaterial parameters. These parameters can take different values when fibers are atient–specific predictions of aneurysm growth and remodeling 7 under compression or tension [7, 50]. Noting that the fourth invariant and rightCauchy–Green stretch tensor can respectively be written such as I k = 1 k F kgr a k k C k el : a k ⊗ a k (9a) C k el = F k el ( T ) F k el (9b)where k ∈ { c j , m } and a k = F kgr a k k F kgr a k k .For every 3D hexahedral or tetrahedral finite element across the geometry ofthe artery, the same strain energy density function is assumed, however differentmaterial properties and mass densities of the individual constituents may be usedfor each layer (media and adventitia).Referring to Eq. 4 leads to the expression of the second Piola-Kirchhoff stresstensor as: S = ̺ e t ( S e + J p C − ) + n X i =1 ̺ c j t S c j + ̺ m t S m (10)where S e = 2 ∂W e ∂ C and S j = 2 ∂W j ∂ C are the second Piola-Kirchhoff stress of corre-sponding constituents of the mixture, ( i ∈ { c j , m } ), and p = d U e d J el , the hydrostaticpressure. The Cauchy stress tensor is derived from the second Piola-Kirchoff stressas σ = J − FSF T (11) Mass turnover and inelastic G&R deformation gradient
In CMT–based models, G&R is a conceptual phenomenon during which simul-taneous degradation and deposition of different constituents continuously occur.This mass turnover is a stress mediated process during which extant mass is con-tinuously degraded and new mass is deposited into the extant matrix by a stressmediated rate [8,14,20]. In this work, in two-layer arterial models, mass turnover ofcollagen families is mediated by SMC stresses in the media and by collagen stressesin the adventitia (for the latter, it is assumed that fibroblasts of the adventitiawould be sensitive to the stresses of collagen, both in intensity and directionality).Therefore, the rate of mass degradation or deposition in the media and in theadventitia can be respectively calculated as ˙ ̺ c j med ( t ) = ̺ c j t k c j σ σ m − σ mh σ mh + ˙ D c j g (12a) ˙ ̺ c j adv ( t ) = ̺ c j t k c j σ σ c j − σ c j h σ c j h + ˙ D c j g (12b)where ̺ c j t = ̺ c j ( t ) are mass densities of collagen families at time t , k c j σ stands forcollagen growth (gain) parameter, σ mh and σ c j h are average SMCs and collagen fiberstresses at homeostasis, σ m and σ c j denote the current stress of extant collagenfibers and SMCs.Moreover, it is assumed that elastin can be only subjected to degradation, ifany, and its mass loss cannot be compensated by new elastin deposition. ˙ ̺ e ( t ) = ˙ D eg (13) S. Jamaleddin Mousavi et al. ˙ D c j g and ˙ D eg , so called generic rate function, is used to describe additional deposi-tion or degradation due to any damage in collagen and elastin, respectively. Thoseare not stress mediated but can be driven by other factors like chemical degra-dation processes or mechanical fatigue. Besides, no mass turnover is assumed forSMCs.Even when there is a mass balance between mass degradation and mass pro-duction ( ̺ it = 0 ), the traction–free state changes as new mass is deposited witha prestress which is not usually identical to the current stress at which the ex-isting mass is removed. This results in changes of the current average stress and,in turn, of the traction–free state of a constituent. Therefore, some change of themicrostructure of the tissue, so-called remodeling, should accompany this massbalance. However, the local traction–free configuration of a constituent will bechanged also by growth when there is no mass balance between mass degradationand production ( ̺ it = 0 ). Thus in addition to this turnover–based remodeling,which is a volume preserving process, the mass turnover is generally associatedwith a local change of the volume by growth which accommodates the mass ina certain region of the body. Consequently, the traction–free configuration of acertain constituent should be amended by both remodeling– and growth–relatedinelastic local changes of the microstructure and volume. To this end, we takeadvantage of the homogenized CMT–based G&R model presented by Cyron etal [8, 14]. Therefore, multiplicative decomposition of the inelastic G&R deforma-tion gradient of the i th constituent deposited at different times reads F i gr = F i r F i g (14)where F i g and F i r are inelastic deformation gradients due to G&R, respectively.The former is related to any change in the mass per unit reference volume andthe latter captures changes in the microstructure due to mass turnover. Therefore,having the net mass production rate based on [8, 14], the evolution of the inelasticremodeling deformation gradient of the i th constituent at time t is calculated bysolving the following system of equations (cid:20) ˙ ̺ it ̺ it + 1 T i (cid:21) h S i − S i pre i = (cid:20) ∂ S i ∂ C i el : ( C i el L i r ) (cid:21) (15)where S is the second Piola–Kirchhoff stress and subscript "pre" denotes depositionprestress while L i r = ˙ F i r F i r − is the remodeling velocity gradient. T i is the periodwithin which a mass increment is degraded and replaced by a new mass increment,named the average turnover time.It is assumed that elastin is not produced any longer during adulthood, iteven undergoes a slow degradation with a half–life time of several decades [8, 15].Therefore, elastin growth can be basically calculated based on its degradation rate( ˙ D eg ) which in turn depends on elastin half–life time (some decades). This impliesthat the remodeling velocity gradient is zero, then the remodeling gradient is theidentity ( L er = , ˙F er = and F er = I ).Any change in the mass of each constituent in a region of the arterial wallgenerally induces a local change of wall volume which can be captured by an in-elastic deformation gradient namely the growth deformation gradient. The inelasticgrowth deformation gradient relates the change of shape and size of a differentialvolume element to the degraded or deposited mass in that element. Basically it atient–specific predictions of aneurysm growth and remodeling 9 is the geometrical and micromechanical features of the underlying growth pro-cess that dictates the local deformation gradient due to a certain mass change.The degradation or deposition of each constituent induces element deformationsat each time increment that can be captured by an inelastic deformation gradi-ent rate for each constituent. Based on the homogenized CMT, Braeu et al [8]suggested that all constituents experience the same inelastic growth deformationgradient: F i g = F g . Therefore, the total inelastic growth deformation gradient rateequals the sum of the growth–related deformation gradient rates of each individualconstituent and can be obtained by ˙ F g = n X i =1 ˙ ̺ it ̺ tott h F i g-T : a i g ⊗ a i g i a i g ⊗ a i g (16)where unit vector a i g stands for growth direction of individual constituents which,for example, can represent an anisotropic growth in arterial wall thickness direc-tion. ̺ tott = P ni =1 ̺ it denotes total volumic mass at each time. It is noteworthy thatSMCs do not experience growth according to Eq. 16, which means that no massturnover is assumed for them. However, because the second term on the left-handside of Eq. 15 is never null, SMCs continuously undergo remodeling, leading to acontinuous update of their reference configuration.2.3 Finite–Element implementationThe proposed model was implemented within the commercial FE software Abaqus[30] through a coupled user material subroutine (UMAT) [21]. A 3D structuralmesh made of hexahedral elements was reconstructed across the wall of the artery.The mesh was structural, which means that the edge of each element was locallyaligned with the material directions of the artery: radial, circumferential and axial.For non–perfectly cylindrical geometries, the radial direction is defined as theoutward normal direction to the luminal surface, the axial direction is defined asthe direction parallel to the luminal centerline in the direction of the blood flow,and the circumferential direction is perpendicular to the two previously defineddirections. It is assumed that each element is a mixture of elastin, collagen andSMCs with mass density varying regionally.The deformation of the artery is computed for every time step correspondingto one month of real time. It is obtained by feeding equilibrium equations with theconstitutive equations introduced previously, and solving the resulting nonlinearequations using the Newton Raphson method. G& R deformations tensors areobtained at each time step based on stresses assessed at the previous step. Onlythe initial time step is assumed to satisfy homeostatic conditions. Three different models were considered:1. The first case was an thick–wall cylindrical artery responding to localizedelastin loss. It was initially solved by Braeu et al. [8] using the homogenized
CMM and the purpose was to use these previous results for validating ourmodel.2. The second case was a thick–wall toric artery responding to localized elastinloss. The toric model was previously used by Alford and Taber [2] to studyG&R in the aortic arch.3. The third case was a thick–wall patient–specific artery responding to localizedelastin loss.Previous work with the homogenized CMM considered a single layer to modelthe artery. Similar single–layer models were used in the first case using materialproperties taken from [8] and reported in Table 2. In other cases, in order to havea more realistic model of G&R, we considered two–layer thick–wall arteries, withdifferent material properties for the media and the adventitia. Additional materialparameters used in the two–layer thick–wall model were calibrated with data ofour group [17, 50], they are reported in Table 3.3.1 Application to a single–layer thick–wall cylindrical artery responding toelastin lossAn idealized single–layer thick–wall cylindrical artery with r = 10 mm and h =1 . mm was considered. It was assumed that this geometry, which was set as thereference configuration, was related to a reference pressure of 13.3 kPa and wasat homeostasis. The deposition stretch of elastin permitting to obtain mechanicalequilibrium in the reference configuration was solved using the algorithm presentedin [49]. Following [8], elastin was degraded with the following rate: ˙ D eg ( X , t ) = − ̺ e ( X , t ) T e − D max t dam ̺ e ( X , e − . X L dam ) − tt dam (17)where L dam and t dam are the spatial and the temporal damage spread parameters,respectively, while D max is maximum damage. X is the material position in theaxial direction of the cylinder. Due to symmetrical geometry one–fourth of thecylinder was modeled using symmetric boundary conditions. The axial directionwas defined such as ≤ X ≤ L . The first term in Eq 17 denotes a normal elastinloss by age while the second one is related to a sudden and abnormal local damagestarting at t = 0 with maximum value at the center of the cylinder ( X = 0 )and fading at X = L . The 3D results obtained with our model on this caseare compared with the corresponding 3D results of [8] for six different growthparameters, k c j σ .3.2 Application to a two–layer thick–wall toric artery responding to elastin lossWe employed the model on a torus shown in Fig. xx with Rr = 4 . Its thickness andinner radius were assumed identical to the ones of the cylindrical artery definedin section 3.1 ( r = 10 mm and h = 1 . mm). Due to symmetry only a quarter ofthe torus was modeled using symmetric boundary conditions.Again, it was assumed that this geometry, which was set as the reference con-figuration, was related to a reference pressure of 13.3 kPa and was at homeostasis. atient–specific predictions of aneurysm growth and remodeling 11 The deposition stretch of elastin permitting to obtain mechanical equilibrium inthe reference configuration was solved using the algorithm presented in [46, 49].Then, an elastin degradation rate with temporal and spatial damage was as-sumed as ˙ D eg ( X , t ) = − ̺ e ( X , t ) T e − D max t dam ̺ e ( X , e − . θθ dam ) − tt dam (18)where θ dam is the spatial damage spread parameters. ≤ θ ≤ is the materialposition varying as shown in Fig. xx, indicating maximal and minimal elastin lossat θ = 0 ◦ and θ = 90 ◦ , respectively.Material parameters reported in Table 2 were employed considering that themedia comprised 97% of total elastin, 100% of total SMCs, and 15% of total axialand diagonal collagen fibers while the adventitia comprised 3% of total elastin, 85%of total axial and diagonal collagen, and 100% of total circumferential collagen [6].3.3 Application to a two–layer thick–wall patient–specific human ATAAresponding to elastin lossTo demonstrate the applicability of the model to predict patient–specific wallG&R, the model was employed onto the geometry of a real human ATAA. AnATAA specimen and the preoperative CT scan of the patient were obtained afterinformed consent from a donor undergoing elective surgery for ATAA repair atCHU–SE (Saint-Etienne, France). The lumen of the aneurysm was clearly visible Table 2
Material parameters employed for a single–layer thick–wall cylindrical artery and atwo–layer thick–wall toric artery [8]. α c , α c , α c and α c are axial, circumferential and twodiagonal directions of collagen fiber families, respectively. Symbol Values α c j , j = 1 , , ..., π and ± π µ e
72 [J/kg] κ
720 [J/kg] k c j
568 [J/kg] k c j k m k m ̺ e ] ̺ c ] ̺ c ] ̺ c = ̺ c ] ̺ m ] λ e z λ c j λ m T e
101 [years] T c j
101 [days] T m
101 [days] L dam
10 [mm] t dam
40 [days] D max in the DICOM file, but detection of the aneurysm surface was not possible auto-matically. A non–automatic segmentation of the CT image slices was performedusing MIMICS (v. 10.01, Materialise NV) to reconstruct the ATAA geometry.The reconstructed geometry was meshed with 7700 hexahedral elements. A wallthickness of 2.38 mm was defined evenly in the reference configuration, yieldingan average thickness of 2.67 mm at zero pressure, which corresponded to themeasured thickness on the supplied sample [22]. Material parameters (reportedin Table 3) such as deposition stretch of collagen and exponents were taken fromliterature [6, 10] and others were calibrated with data of our group [17]. Note that97% of total elastin, 100% of total SMC, and 15% of total axial and diagonalcollagen fibers were assigned to the media. Conversely, 3% of total elastin, 85% oftotal axial and diagonal collagen, and 100% of total circumferential collagen wereassigned to the adventitia [6,49,50]. The geometry obtained from the CT scan wasassigned as the reference configuration. It was subjected to a luminal pressure of80 mmHg (diastole). An axial deposition stretch of 1.3 was defined for the elastinand the deposition stretches of collagen and SMC components were set to 1.1. Thespatially varying circumferential deposition stretch of elastin was determined toensure equilibrium with the luminal pressure using our iterative algorithm [49].Both ends of the ATAA model were fixed in axial and circumferential directions,allowing only radial displacements.4D flow magnetic resonance imaging (MRI) with full volumetric coverage ofATAAs can reveal complex aortic 3D blood flow patterns, such as flow jets, vor-tices, and helical flow [13,32]. For the same patient, 4D flow MRI datasets were alsoacquired, revealing a jet flow impingement against the aortic wall around the bulgeregion (downstream the area of maximum dilatation) as shown in Fig xx. Guzzardiet al. [29] found that regions with largest WSS underwent greater elastin degra-dation associated with vessel wall remodeling in comparison with the adjacentregions with normal WSS. Consequently, based on these findings we considered alocalized elastin degradation and simulated its effects on ATAA G&R. The localelastin degradation rate was written such as ˙ D eg ( X , t ) = − ̺ e ( X , t ) T e − D max t dam ̺ e ( X , e − tt dam (19)Three different values of t dam (as listed in Table 3) were studied. atient–specific predictions of aneurysm growth and remodeling 13 are shown in Figs. xx. Elastin loss naturally leads to higher stresses in the othercomponents of the arterial wall and subsequently higher deposition of new collagenfibers.4.2 Response of a two–layer thick–wall toric artery to localized elastin lossThe effect of elastin loss during 15 years in a two–layer toric artery was consideredfor k c j σ = . T c j and k c j σ = . T c j . The change of the thickness and diameter due todegradation of the elastin are shown in Figs. xx and Figs. xx, respectively. The di-latation and thickness were never stable for small growth parameters ( k c j σ = . T c j ).Conversely, for relatively large growth parameters ( k c j σ = . T c j ), the thickness anddiameter became stable after about five years of transient growth period. The wallwas basically thickened on the outer curvature side, mainly in the media. There-fore, the response of a toric artery to elastin loss is unstable for small growthparameters while it recovers its stability, after some enlargement, for relativelylarge growth parameters. In addition, colormaps of the maximum principal stressand the collagen mass density for large and small growth parameters (Figs. xx)show that elastin loss continuously causes higher stresses and collagen depositionin the media. However, the balance between arterial dilatation and collagen depo-sition leads to higher collagen production for small gain parameters. This in turnends with higher stresses in the arch with small gain parameters. Table 3
Material parameters employed for two–layer patient–specific human ATAA modeladapted from [50]. α c , α c , α c and α c are axial, circumferential and two diagonal directionsof collagen fiber families, respectively. Symbol Values α c j , j = 1 , , ..., π and ± π µ e
82 [J/kg] κ µ e [J/kg] k c j , c = k m,c
15 [J/kg] k c j , c = k m,c k c j , t
105 [J/kg] k c j , t k m,t
10 [J/kg] k m,t ̺ e
250 [kg/m ] ̺ c j
460 [kg/m ] ̺ m
280 [kg/m ] λ e z λ c j λ m T e
101 [years] T c j
101 [days] T m
101 [days] t dam
20, 40 and 80 [days] D max t dam results in anincrease of maximum principal stresses in the arterial wall. It is induced by therelated increase of elastin degradation rate. In Fig. xx-b, d and f, the distributionof collagen mass density for different t dam shows that most of the collagen is de-posited in the media where elastin has been lost (recall that ∼
97% of the elastinis in media), causing finally a thickening of the arterial wall (Fig. xx). It is note-worthy that increase of t dam accelerates collagen deposition and consequently wallthickening. Moreover, we studied the sensitivity of ATA dilatation to the collagengrowth parameter, k c j σ . As shown in Fig. xx, larger growth parameters stabilizeATA dilatation induced by elastin loss. However, for relatively small growth pa-rameters, k c j σ = . T c j , as shown in Figs. xx-a, xx-b and xx-a, the ATA undergoesan excessive dilatation (the maximum ATA diameter increases continuously from ∼
42 mm to ∼
64 mm after ∼
180 months). As the newly deposited collagen has tocompensate for the elastin loss to maintain the homeostatic state, this induces theadaptation response. Conversely, increasing the growth parameter leads to a stablegrowth of ATA after 31 months (the maximum diameter of ATA after elastin lossstops increasing after ∼
31 months, enlarging from ∼
42 mm to ∼
47 mm). For allcases, whatever the growth parameter, remodeling induced by collagen depositionalways causes ATA wall thickening, mainly in the media (see Fig. xx).
A robust computational model based on the homogenized CMT was presented andits potential was shown to predict ATAA evolution for a patient–specific aorticgeometry, showing that the typical shape of an ATAA can be obtained simplywith a proteolysis of elastin localized in regions of deranged hemodymanics. Themost interesting result is that although elastin degradation occurs locally in theATAA at the location of WSS peak, the whole ATAA globally undergoes G&Rdue to redistribution of stresses distribution, leading to ATAA dilatation.A general advantage of the model presented here is that it was developed to ac-count for in situ prestrain (and therefore prestress). It permits to run FE analysisof G&R in soft biological tissues without requiring a zero–pressure configuration.This appears to be especially beneficial when a patient–specific geometry is re-constructed using CT scans or MRI data acquired in a pressurized configuration.Using this methodology, the prestress is calculated based on the prestrain, definedin terms of fiber prestretches (deposition stretches), assuming a hyperelastic elas-tic material behavior. Therefore, a drawback of this methodology is that we mayencounter some instability in the resolution if we enforce a particular depositionstretch for each constituent. Large distortions of elements may also occur withsmall variations of deposition stretchs and lead to the divergence of the resolution.This indicates that an arbitrary deposition stretch cannot be always imposed onan arbitrary reference configuration. atient–specific predictions of aneurysm growth and remodeling 15
For all geometries given herein for large gain parameters, k σ , the blood vesselrecovered a new stable state after a transient period of dilation and enlargement.In contrast, for small gain parameters it underwent unbounded dilation, experi-encing mechanobiological instability. Dilatation due to weakening of the arterialwall by elastin loss is physically consistent with previous findings [8, 14]. However,one can find 3D FE implementation of G&R in which the arterial radius decreasesafter elastin degradation [19, 28, 59]. This can be explained by the implementationof volumetric growth. [19, 28, 59] defined implicitly the growth directions usingthe volumetric deviatoric contributions of the deformation gradient and imposingincompressibility constraints. Only isotropic growth can be modeled with theirapproach, elastin degradation consequently causing a decrease of total tissue vol-ume. Therefore, in their model the arterial wall shrinks in all spatial directions,including the circumferential direction. Eriksson [18] attempted to overcome thisproblem by introducing the concept of constant and adaptive individual densitygrowth in which an elastin loss does not cause a contraction of the arterial walldue to loss of mass. Nevertheless, it is still controversial whether elastin loss wouldlocally lead to arterial shrinking in the thickness direction or if it would reallyinduce a change in the mass density of the tissue.Basically, two major approaches were so far proposed for numerical modelingof soft tissue G&R. Rodriguez et al. [52] introduced a kinematic growth theoryby multiplicative decomposition of the deformation gradient into an elastic andan inelastic growth contributions. The elastic part ensured geometric compati-bility and mechanical equilibrium while the inelastic growth part contained thelocal changes of mass and volume. Although their model was computationally effi-cient and conceptually simple it was intrinsically unable to compute the separateG&R of structurally different constituents. This limitation was fixed by CMT–based model introduced by [39] in which the in vivo situation can be realisticallymimicked using the concept of deposition stretches. The computational cost of theclassical CMT–based models is higher than that of simple kinematic growth theory.Homogenized CMT–based models introduced by Cyron et al. [14] combines the ad-vantages of both classical CMT–based models and simple kinematic growth modelsto overcome the drawback of each model. The results obtained by homogenizedCMT–based models are similar to the classical ones but with low computationalcost. Focusing on the example of simple membrane–like [14] and thick–wall [8]cylindrical vessels, they showed that homogenized CMT–based models are able toreproduce realistically both pathological growth responding to an elastin loss (asobserved in aneurysms) and adaptive growth in healthy vessels due to hyperten-sion. The prominent privilege of CMT–based models is the inherent incorporationof anisotropic volumetric growth in the thickness direction of arterial wall (provedby experimental observations of [48]). Moreover, recently Lin et al. [45] combinedhomogenization and CMT to capture G&R of the abdominal aorta and to con-sider dilatation of abdominal aortic aneurysms under loading. They focused on atransversely isotropic mixture subjected to uniform extension in the direction ofcollagen fibers assuming that they are embedded in an isotropic elastin matrix,ignoring the role of SMCs. Considering a very special case of isotropic growth,their model can successfully predict the continuous enlargement of an abdominalaortic aneurysm by combined effects of elastin degradation, loss of extant collagenand production of new collagen, as well as fiber remodeling. As anisotropic growth may stabilize the arterial wall under perturbations moreefficiently than the isotropic growth [8] so that the ability of the homogenizedCMT–based model implemented herein can be considered an ideal tool to realis-tically study the patient–specific geometries undergoing G&R in response to anunexpected degradation of elastin. Following [6, 8, 10, 63, 64], it was assumed thatpatient–specific G&R resulted from specific temporal and spatial distributionsof elastin degradation. We considered multiple temporal damage parameters forelastin degradation leading to different aneurysm growth rate. Although the globalshape of the aneurysm resembles, the thickening and collagen production rates aredifferent for different cases. Different temporal damage constants showed signifi-cant effects on the expansion rate where the higher t dam delivers the higher G&Rrate. Although we simply employed temporal damage parameter for elastin degra-dation, elastin degradation during ATAA growth involves multiple biological andmechanical parameters including abnormal distribution of wall shear stress [29]and circumferential stress [39]. The formation of intraluminal thrombus is specificto AAA [61, 62]. It may stimulate proteolytic effects but this was not consideredhere as thrombus are very rare in ATAAs.In the patient–specific study it also appears that collagen deposition tends tocompensate the elastin loss. It is worth noting that as aneurysm grows, principalstress may not increase necessarily in a damaged location. This is observed in AAAgrowth as well [68]. Moreover, although elastin was degraded locally, dilatation ofthe ATA was spread across a larger area due to stress redistribution.The in vivo images were obtained when the artery was under pressure so thatthe stress–free or zero pressure configuration was not basically available. Hence,for hyperelastic models such as Holzapfel–type models [31], approaches such asinverse elastostatic methods [69] or Lagrangian-Eulerian formulations [26] are re-quired to estimate the stress–free state of in vivo geometries obtained from medicalimages. One of the advantages of current CMT-based model is that G&R analy-sis of a patient-specific model can be directly performed on the in vivo geometryreconstructed from medical images obtained under pressure, without needing tocompute the stress–free geometry.Salient features of the response of arterial walls to altered hemodynamics[4, 5, 11, 55] were captured by 2D and 3D CMT-based models [9, 39, 41]. Despitethe major interest of this prior work on CMT-based models, two novelties can behighlighted in our work: application of CMT-based models to patient-specific ge-ometries and integration of layer-specific properties (media and adventitia). Futurework will focus on fully coupling the present model with CFD analyses [13] to studythe effects on aortic G&R of different hemodynamic metrics, such as helicity, wallshear stress (WSS), time averaged WSS (TAWSS), oscillatory shear index (OSI)or relative residence time (RRT). Such fluid-solid-growth simulations have alreadybeen developed by different authors for cerebral [65] or abdominal [1, 28, 47, 54]aneurysms and we will extend them to ATAA to provide additional insight intothe evolution of these aneurysms.This altogether indicates that the present model has the potential for clinicalapplications to predict G&R of patient–specific geometries if a realistic rate ofelastin loss and collagen growth parameter are available.There are still several limitations and technical challenges associated with cur-rent model: atient–specific predictions of aneurysm growth and remodeling 17 – The active role of SMCs is not considered in the present model, despite itsmajor role in mechanosensing [40]. – Theory of G&R is based on a key assumption, the existence of mechanicalhomeostasis [37,42]. It is difficult to have the assumption of a homeostatic statesatisfied at every point of the arterial wall. For an idealized model, such as idealthick–wall cylinders, the in vivo material properties are typically assumed tobe uniform across the domain. When a patient–specific geometry is used for aclinical study, it will be essential to prescribe the distribution of material andstructural parameters such as thickness and fiber orientations consistent with in vivo data. Therefore, considering the arterial wall with a uniform thicknesscan be considered as additional limitations of the current work. – Another difficulty associated with patient–specific models is estimating theconstitutive parameters of the model for different patients. Here, these param-eters were estimated by curve fitting from the ex vivo bulge inflation data ofan ATAA segment excised after the surgical intervention of the same patient.However, in clinical applications, it will be needed to identifying noninvasivelythe in vivo material properties of ATAAs [22, 23].
In summary, in this manuscript, a robust computational model based on the ho-mogenized CMT was presented and its potential was shown for patient–specificpredictions of growth and remodeling of aneurysmal human aortas in response tolocalized elastin loss. As future application, the predictions of this G&R frame-work will be validated on datasets of patient–specific ATAA geometries followedup over a significant number of years.
The authors are grateful to the European Research Council for grant ERC-2014-CoG BIOLOCHANICS. The authors would also like to thank Nele Famaey (KULeuven, Belgium), Christian J. Cyron (TU Hamburg, Germany) and Fabian A.Braeu (TU München, Germany) for inspiring discussions related to this work.
There is no conflict of interest.
References
1. P. Di Achille, G. Tellides, and J.D. Humphrey. Hemodynamics-driven deposition of in-traluminal thrombus in abdominal aortic aneurysms.
Int J Numer Method Biomed Eng ,33(5):e2828, 2017.2. P.W. Alford and L.A. Taber. Computational study of growth and remodelling in the aorticarch.
Comput Methods Biomech Biomed Engin , 11(5):525–38, 2008.3. S. Baek, K.R. Rajagopal, and J.D. Humphrey. A theoretical model of enlarging intracranialfusiform aneurysms.
J Biomech Eng , 128(1):142–9, 2006.8 S. Jamaleddin Mousavi et al.4. S. Baek, K.R. Rajagopal, and J.D. Humphrey. A theoretical model of enlarging intracranialfusiform aneurysms.
J Biomech Eng , 128(1):142–9, 2006.5. S. Baek, A. Valentín, and J.D. Humphrey. Biochemomechanics of cerebral vasospasmand its resolution: II. constitutive relations and model simulations.
Ann Biomed Eng. ,35:1498–509, 2007.6. C. Bellini, J. Ferruzzi, S. Roccabianca, E.S. Di Martino, and J.D. Humphrey. A microstruc-turally motivated model of arterial wall mechanics with mechanobiological implications.
Ann. Biomed. Eng. , 42(3):488–502, 2014.7. M.R. Bersi, C. Bellini, P. Di Achille, J.D. Humphrey, K. Genovese, and S. Avril. Novelmethodology for characterizing regional variations in the material properties of murineaortas.
J. Biomech. Eng. , 138(7):doi: 10.1115/1.4033674, 2016.8. F.A. Braeu, A. Seitz, R.C. Aydin, and C.J. Cyron. Homogenized constrained mixturemodels for anisotropic volumetric growth and remodeling.
Biomech Model Mechanobiol ,16(3):889–906, 2017.9. L. Cardamone and J.D. Humphrey. Arterial growth and remodelling is driven by hemody-namics.
In: Ambrosi D., Quarteroni A., Rozza G. (eds) Modeling of Physiological Flows.MS&A âĂŤ Modeling, Simulation and Applications , 5 Springer, Milano, 2012.10. L. Cardamone, A. Valentin, J.F. Eberth, and J.D. Humphrey. Origin of axial prestretchand residual stress in arteries.
Biomech. Model. Mechanobiol. , 8:431–46, 2009.11. L. Cardamone, A. Valentín, J.F. Eberth, and J.D. Humphrey. Modelling carotid arteryadaptations to dynamic alterations in pressure and flow over the cardiac cycle.
Math MedBiol. , 27(4):343–71, 2010.12. E. Comellas, T.C. Gasser, F.J. Bellomo, and S. Oller. A homeostatic-driven turnoverremodelling constitutive model for healing in soft tissues.
J R Soc Interface , 13(116):doi:10.1098/rsif.2015.1081, 2016.13. F. Condemi, S. Campisi, M. Viallon, T. Troalen, G. Xuexin, A.J. Barker, M. Markl,P. Croisille, O. Trabelsi, C. Cavinato, A. Duprey, and S. Avril. Fluid- and biomechanicalanalysis of ascending thoracic aorta aneurysm with concomitant aortic insufficiency.
AnnBiomed Eng , 45(12):2921–32, 2017.14. C.J. Cyron, R.C. Aydin, and J.D. Humphrey. A homogenized constrained mixture (andmechanical analog) model for growth and remodeling of soft tissue.
Biomech ModelMechanobiol , 15(6):1389–1403, 2016.15. C.J. Cyron and J.D. Humphrey. Growth and remodeling of loadbearing biological softtissues.
Meccanica , 52(3):645–64, 2016.16. C.J. Cyron, J.S. Wilson, and J.D. Humphrey. Mechanobiological stability: a new paradigmto understand the enlargement of aneurysms?
J R Soc Interface , 11(100):20140680, 2014.17. F.M. Davis, Y. Luo, S. Avril, A. Duprey, and J. Lu. Local mechanical properties of humanascending thoracic aneurysms.
J Mech Behav Biomed Mater , 61:235–49, 2016.18. T.S.E. Eriksson. Modelling volumetric growth in a thickwalled fibre reinforced artery.
JMech Phys Solids , 73:134–150, 2014.19. T.S.E. Eriksson, P.N. Watton, X.Y. Luo, and Y. Ventikose. Modelling volumetric growthin a thickwalled fibre reinforced artery.
J Mech Phys Solids , 73:134–50, 2014.20. N. Famaey, J. Vastmans, H. Fehervary, L. Maes, E. Vanderveken, F. Rega, S.J. Mousavi,and S. Avril. Numerical simulation of arterial remodeling in pulmonary autografts.
ZAngew Math Mech , pages 1–19, 2018.21. S. Farzaneh, O. Paseta, and M.J. Gómez-Benito. Multi-scale finite element model of growthplate damage during the development of slipped capital femoral epiphysis.
Biomech ModelMechanobiol , 14(2):371–85, 2015.22. S. Farzaneh, O. Trabelsi, and S. Avril. Inverse identification of local stiffness across ascend-ing thoracic aortic aneurysms.
Biomech Model Mechanobiol , 10.1007/s10237-018-1073-0,2018.23. S. Farzaneh, O. Trabelsi, B. Chavent, and S. Avril. Identifying local arterial stiffnessto assess the risk of rupture of ascending thoracic aortic aneurysms.
Ann Biomed Eng ,47(4):1038–50, 2019.24. C.A. Figueroa, S. Baek, C.A. Taylor, and J.D. Humphrey. Acomputational frameworkfor fluid-solid-growthmodeling in cardiovascular simulations.
Comput Methods Appl MechEng , 198(45–46):3583–602, 2009.25. Ch. Gasser and A. Grytsan. Biomechanical modeling the adaptation of soft biologicaltissue.
Current Opinion in Biomedical Engineering , 1:71–77, 2017.26. M.W. Gee, C. Forster, and W.A. Wall. A computational strategy for prestressing patient-specific biomechanical problems under finite deformation.
Int J Numer Meth BiomedEngng , 26:52–72, 2010.atient–specific predictions of aneurysm growth and remodeling 1927. W. Grossman. Cardiac hypertrophy: useful adaptation or pathologic process?
Am J Med ,69(4):576–84, 1980.28. A. Grytsan, P.N. Watton, and G.A. Holzapfel. Athick-walled fluid-solidgrowth modelof abdominal aortic aneurysm evolution: application to a patient-specific geometry.
JBiomech Eng , 137(3):031008, 2015.29. D.G. Guzzardi, A.J. Barker, P. van Ooij, S.C. Malaisrie, J.J. Puthumana, D.D. Belke,H.E. Mewhort, D.A. Svystonyuk, S. Kang, S. Verma, J. Collins, J. Carr, R.O. Bonow,M. Markl, J.D. Thomas, P.M. McCarthy, and P.W. Fedak. Valve-related hemodynamicsmediate human bicuspid aortopathy: Insights from wall shear stress mapping.
J Am CollCardiol , 66(8):892–900, 2015.30. Hibbit, Karlson, and Sorensen.
Abaqus-Theory manual , 6.11-3 edition, 2011.31. A.G. Holzapfel, C.T. Gasser, and R.W. Ogden. A new constitutive framework for arterialwall mechanics and a comparative study of material models.
Journal of Elasticity , 61:1–48,2000.32. T.A. Hope, M. Markl, L. Wigström, M.T. Alley, D.C. Miller, and R.J. Herfkens. Compar-ison of flow patterns in ascending aortic aneurysms and volunteers using four-dimensionalmagnetic resonance velocity mapping.
J Magn Reson Imaging , 26(6):1471–9, 2007.33. H.S. Hosseini, K.E. Garcia, and L.A. Taber. A new hypothesis for foregut and hearttube formation based on differential growth and actomyosin contraction.
Development ,144(13):2381–91, 2017.34. H.S. Hosseini and L.A. Taber. How mechanical forces shape the developing eye.
Progressin biophysics and molecular biology , pages 1–12, 2018.35. J.D. Humphrey. Mechanics of arterial wall: Review and directions.
Crit. Rev. Biomed.Eng. , 23(1-2):1–162, 1995.36. J.D. Humphrey. Mechanisms of arterial remodeling in hypertension: coupled roles of wallshear and intramural stress.
Hypertension , 52(2):195–200, 2008.37. J.D. Humphrey. Vascular adaptation and mechanical homeostatsis at tissue, cellular, andsub-cellular levels.
Cell Biochemistry and Biophysics , 50:53–78, 2008.38. J.D. Humphrey and G.A. Holzapfel. Mechanics, mechanobiology, and modeling of humanabdominal aorta and aneurysms.
J Biomech , 45(5):805–84, 2012.39. J.D. Humphrey and K.R. Rajagopal. A constrained mixture model for growth and remod-eling of soft tissues.
Math Models Methods Appl Sci , 12(03):407–30, 2002.40. J.D. Humphrey, M.A. Schwartz, G. Tellides, and D.M. Milewicz. Role of mechanotransduc-tion in vascular biology: focus on thoracic aortic aneurysms and dissections.
CirculationResearch , 116(8):1448–1461, 2015.41. I. Kar˘saj, J. Sorić, and J.D. Humphrey. A 3-d framework for arterial growth and remod-eling in response to altered hemodynamics.
Int J Eng Sci , 48(11):1357–72, 2010.42. G.S. Kassab. Mechanical homeostasis of cardiovascular tissue.
In: Artmann, G.M., Chien,S. (Eds.), Bioengineering in Cell and Tissue Research. Springer , pages 371–391, 2008.43. M. Latorre and J.D. Humphrey. Critical roles of time-scales in soft tissue growth andremodeling.
APL BIOENGINEERING , 2:026108, 2018.44. M. Latorre and J.D. Humphrey. Mechanobiological stability of biological soft tissues.
Journal of the Mechanics and Physics of Solids , 125:298–325, 2018.45. W.J. Lin, M.D. Iafrati, R.A. Peattie, and L. Dorfmann. Growth and remodeling withapplication to abdominal aortic aneurysms.
J Eng Math , 109(1):113–137, 2017.46. L. Maes, H. Fehervary, J. Vastmans, S.J. Mousavi, S. Avril, and N. Famaey. Constrainedmixture modeling affects material parameter identification from planar biaxial tests.
JMech Behav Biomed Mater , 95:124–35, 2019.47. A.L. Marsden and J.A. Feinstein. Computational modeling and engineering in pediatricand congenital heart disease.
Current opinion in pediatrics , 27(5):587, 2015.48. T. Matsumoto and K. Hayashi. Response of arterial wall to hypertension and residualstress.
In: Hayashi K., Kamiya A., Ono K. (eds) Biomechanics. Springer, Tokyo , pages93–119, 1996.49. S.J. Mousavi and S. Avril. Patient-specific stress analyses in the ascending thoracic aortausing a finite-element implementation of the constrained mixture theory.
Biomech ModelMechanobiol , s10237:doi: 10.1007/s10237–017–0918–2, 2017.50. S.J. Mousavi, S. Farzaneh, and S. Avril. Computational predictions of damage propagationpreceding dissection of ascending thoracic aortic aneurysms.
Int J Numer Method BiomedEng , 34(4):e2944, 2018.51. F. Riveros, S. Chandra, E.A. Finol, T.C. Gasser, and J.F. Rodriguez. A pull-back algo-rithm to determine the unloaded vascular geometry in anisotropic hyperelastic aaa passivemechanics.
Ann Biomed Eng , 41(4):694–708, 2013.0 S. Jamaleddin Mousavi et al.52. E.K. Rodriguez and A. Hoger. Stress-dependent finite growth in soft elastic tissues.
JBiomech , 27(4):455–67, 1994.53. J.F. Rodriguez, C. Ruiz, M. Doblaré, and G. Holzapfel. Mechanical stresses in abdominalaortic aneurysms: influence of diameter, asymmetry, and material anisotropy.
ASME J.Biomech. , 130(2):021023, 2008.54. A. Sheidaei, S.C. Hunley, S. Zeinali-Davarani, L.G. Raguin, and S. Baek. Simulation ofabdominal aortic aneurysm growth with updating hemodynamic loads using a realisticgeometry.
Medical engineering & physics , 33(1):80–88, 2011.55. A. Valentín, L. Cardamone, S. Baek, and J.D. Humphrey. Complementary vasoactivityand matrix remodelling in arterial adaptations to altered flow and pressure.
J R SocInterface , 6(32):293–306, 2009.56. A. Valentín and G.A. Holzapfel. Constrained mixture models as tools for testing competinghypotheses in arterial biomechanics: A brief survey.
Mech Res Comm , 42:126–33, 2012.57. A. Valentín and J.D. Humphrey. Evaluation of fundamental hypotheses underlying con-strained mixture models of arterial growth and remodelling.
Philos Transact A Math PhysEng Sci , 367:3585–606, 2009.58. A. Valentín and J.D. Humphrey. Parameter sensitivity study of a constrained mixturemodel of arterial growth and remodeling.
J Biomech Eng , 131:101006, 2009.59. A. Valentín, J.D. Humphrey, and G. Holzapfel. A finite element-based constrained mixtureimplementation for arterial growth, remodeling, and adaptation: Theory and numericalverification.
Int J Numer Method Biomed Eng , 29(8):822–49, 2013.60. A. Valentín, J.D. Humphrey, and G.A. Holzapfel. A multi-layered computational modelof coupled elastin degradation, vasoactive dysfunction, and collagenous stiffening in aorticaging.
Ann Biomed Eng , 39(7):2027–45, 2011.61. D.A. Vorp and J.P. Vande Geest. Biomechanical determinants of abdominal aorticaneurysm rupture.
Arterioscler Thromb Vasc Biol , 25(8):1558–66, 2005.62. D.A. Vorp, P.C. Lee, D.H.J. Wang, M.S. Makaroun, E.M. Nemoto, S. Ogawa, and M.W.Webster. Association of intraluminal thrombus in abdominal aortic aneurysm with localhypoxia and wall weakening.
J Vasc Surg , 34(2):291–99, 2001.63. P.N. Watton and N.A. Hill. Evolving mechanical properties of a model of abdominal aorticaneurysm.
Biomech Model Mechanobiol , 8(1):25–42, 2009.64. P.N. Watton, N.A. Hill, and M. Heil. A mathematical model for the growth of the ab-dominal aortic aneurysm.
Biomech Model Mechanobiol , 3(2):98–113., 2004.65. P.N. Watton, N.B. Raberger, G.A. Holzapfel, and Y. Ventikos. Coupling the hemody-namic environment to the evolution of cerebral aneurysms: computational framework andnumerical examples.
J Biomech Eng , 131(10):doi: 10.1115/1.3192141, 2009.66. J.S. Wilson, S. Baek, and J.D. Humphrey. Parametric study of effects of collagen turnoveron the natural history of abdominal aortic aneurysms.
Proc R Soc A , 469(2150):20120556,2013.67. S. Zeinali-Davarani and S. Baek. Medical image-based simulation of abdominal aorticaneurysm growth.
Mechanics Research Communications , 42:107–17, 2012.68. S. Zeinali-Davarani, A. Sheidaei, and S. Baek. A finite element model of stress-mediatedvascular adaptation: application to abdominal aortic aneurysms.
Comput MethodsBiomech Biomed Engin , 14(9):803–17, 2011.69. X. Zhou, M.L. Raghavan, R.E. Harbaugh, and J. Lu. Specific wall stress analysis incerebral aneurysms using inverse shell model.