Heterogeneous growth-induced prestrain in the heart

M. Genet a,b,1,n, M.K. Rausch c, L.C. Lee a,f, S. Choy g, X. Zhao g, G.S. Kassab g,h,i, S. Kozerke b,J.M. Guccione a, E. Kuhl c,d,e

a Department of Surgery, School of Medicine, University of California at San Francisco, USAb Institute for Biomedical Engineering, University and ETH Zürich, Switzerlandc Department of Mechanical Engineering, Stanford University, CA, USAd Department of Bioengineering, Stanford University, CA, USAe Department of Cardiothoracic Surgery, Stanford University, CA, USAf Department of Mechanical Engineering, Michigan State University, MI, USAg Department of Biomedical Engineering, Indiana University–Purdue University Indianapolis, USAh Department of Cellular and Integrative Physiology, Indiana University–Purdue University Indianapolis, USAi Department of Surgery, Indiana University–Purdue University Indianapolis, USA

a r t i c l e i n f o

Article history:Accepted 10 March 2015

Keywords:Residual stressPrestrainOpening anglePatient-specific modelingFinite element methodFinite strain

a b s t r a c t

Even when entirely unloaded, biological structures are not stress-free, as shown by Y.C. Fung's seminalopening angle experiment on arteries and the left ventricle. As a result of this prestrain, subject-specificgeometries extracted from medical imaging do not represent an unloaded reference configurationnecessary for mechanical analysis, even if the structure is externally unloaded. Here we propose a newcomputational method to create physiological residual stress fields in subject-specific left ventriculargeometries using the continuum theory of fictitious configurations combined with a fixed-pointiteration. We also reproduced the opening angle experiment on four swine models, to characterizethe range of normal opening angle values. The proposed method generates residual stress fields whichcan reliably reproduce the range of opening angles between 8.771.8 and 16.6713.7 as measuredexperimentally. We demonstrate that including the effects of prestrain reduces the left ventricularstiffness by up to 40%, thus facilitating the ventricular filling, which has a significant impact on cardiacfunction. This method can improve the fidelity of subject-specific models to improve our understandingof cardiac diseases and to optimize treatment options.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Biological tissues inherently contain some prestrain, or equiva-lently residual stress (Vaishnav and Vossoughi, 1987; Fung, 1993;Humphrey, 2002). This has been evidenced in the opening angleexperiments, performed originally on the artery in Chuong andFung (1986), and then on the left ventricle in Omens and Fung(1990). The exact origin of the prestrain is still an open question: itmay be induced by inhomogeneous tissue growth, or the constantexchange of matter within the tissues and associated remodeling,cell death, and renewal, etc. (Fung, 1993; Humphrey, 2002).Moreover, prestrain potentially exists across multiple spatialscales, including the sub-cellular (Kumar et al., 2006; Deguchiet

et al., 2006), cellular, and structural levels (Fung, 1993; Humphrey,2002; Guo et al., 2007).

It is important to distinguish between these different types ofprestrain. Indeed, sub-cellular prestrain surely affects the responseof individual cells, but is confined to the cells and cannot becharacterized on the macroscopic level (Kumar et al., 2006;Deguchi et al., 2006; Webster et al., 2014). Conversely, structural,or global, residual stress (imagine a stress-free straight beam gluedinto a circular shape for which a single cut would release all theresidual stress at once) is most probably not seen in vivo. Insupport of this, while original experiments showed that the firstcut released most of the residual stress (Liu and Fung, 1989; Hanand Fung, 1996), later experiments revealed that subsequent cutsalways release additional residual stress (Greenwald et al., 1997).Hence, we investigate the prestrain generated on the cellular scale,a scale where living tissues are composed of many differentcomponents in frictional contact and constant relative motion.Cellular prestrain is analogous to the one present in engineering

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Journal of Biomechanics

http://dx.doi.org/10.1016/j.jbiomech.2015.03.0120021-9290/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author at: Institute for Biomedical Engineering, Gloriastrasse 35,8092 Zurich, Switzerland.

E-mail address: genet@biomed.ee.ethz.ch (M. Genet).1 Marie-Curie International Outgoing Fellow

Journal of Biomechanics 48 (2015) 2080–2089

materials (e.g., between grains in metals) or between componentsin composites, which is usually induced by the process ordifferences in coefficients of thermal expansion. It is presenteverywhere in living systems and it can have a tremendous impacton their macroscopic behavior.

To address cardiovascular diseases, one approach is the use ofpersonalized models for quantitative diagnostic, prognostic, andtreatment optimization (Lee et al., 2014b). The current bottlenecksin personalized cardiac modeling include the assessment ofmyofiber architecture (Toussaint et al., 2013; Harmer et al.,2013), non-invasive pressure measurement, tissue propertiesorthotropy and regionality (Lee et al., 2011; Xi et al., 2011;Marchesseau et al., 2013). Another important limitation is ourunderstanding of prestrain and residual stress. Indeed, as a resultof prestrain, subject-specific geometries of biological structuresextracted from medical imaging do not represent the referenceconfiguration required for mechanical analysis, even if the imagescorrespond to an externally unloaded state. To identify the trulyunstressed reference configuration, we have to solve an inverseproblem that is highly non-linear.

In this paper, we propose a computational method to introducephysiological residual stress fields in subject-specific left ventri-cular geometries. The proposed method is embedded within thecontinuum theory of fictitious configurations and uses a fixed-point iteration on the geometry itself. Wang et al. (2013b) recentlyproposed an alternative approach, also using fixed-point iterationson the geometry to recover a relaxed configuration. However, theyuse Shams et al. (2011)'s method to introduce residual stress in theinvariant-based Holpzapfel model (Holzapfel and Ogden, 2009)through additional pseudo-invariants. This implies that (i) themethod is limited to invariant-based formulations, (ii) residualstress is introduced as an ad hoc parameter to fit experimentaldata (Costa et al., 1997), and (iii) the resulting residual stress fieldis not strictly speaking divergence free, i.e., auto-balanced. Here,the residual stress is generated mechanistically by the underlyingbiological process, heterogeneous growth, and is naturally auto-balanced.

2. Materials

2.1. Opening angle experiment

2.1.1. Animal preparationWe performed all animal experiments in accordance with national and local

ethical guidelines, including the Principles of Laboratory Animal Care, the Guide for

the Care and Use of Laboratory Animals (National Research Council (US) Committeefor the Update of the Guide for the Care and Use of Laboratory Animals, 2011) andthe National Association for Biomedical Research (Cardon et al., 2012), and anapproved Indiana University Purdue University Indianapolis IACUC protocolregarding the use of animals in research. Four Yorkshire pigs (55.173.0 kg bodyweight) of either sex were used in this study. The pigs were fasted overnight andsurgical anesthesia was induced with TKX (Telazol 10 mg/kg, Ketamine 5 mg/kg,Xylazine 5 mg/kg) and maintained with 1–2% Isoflurane-balance O2 during acute,non-aseptic surgery. An introducer sheath was placed percutaneously in the rightjugular vein for administration of drugs. The chest was opened through a mid-linesternotomy and the heart was arrested with an IV bolus injection of saturated KClsolution (20 mL) under deep anesthesia and proper anticoagulation with Heparin(100 IU/kg body weight).

2.1.2. Heart preparationThe heart was excised and placed in cardioplegic solution. We cannulated and

perfused the left anterior descending, the left circumflex, and the right coronaryarteries with a solution (Lin and Yin, 1998) of the following composition in mmol:NaCl, 127; KH2PO4, 1.3; MgSO4, 0.6; NaHCO3, 25; KCl, 2.3; CaCl2, 2.5; dextrose, 11.2;BDM (2,3 butanedione monoxime, Sigma, St. Louis, MO), 30. After perfusion, theaorta and pulmonary arteries, the right and left atria, and the right ventricular freewall were carefully cut, keeping an intact left ventricle in the shape of a cone.

2.1.3. Experimental procedureAfter the first basal slice was discarded, three 1-cm-thick slices were cut from

the left ventricle (basal, equatorial, and apical) and placed in petri dishes containingperfusion solution (see Fig. 3). Two cameras were set up to photograph the cross-sectional surfaces of the slice from the top and the bottom surfaces simultaneously.A baseline picture was taken before a radial cut was made in the middle of the leftventricular free wall. Immediately after the cut was made, continuous pictureswere taken every five minutes for a period of 30 min. The opening angle wasmeasured with a program for image processing and analysis, ImageJ 1.44p(National Institutes of Health).

2.2. Mechanics of prestrained biological systems

2.2.1. Continuum modelWe characterize prestrain using the concept of fictitious configurations and

introduce a stress-free reference configuration, a residually stressed but mechani-cally unloaded configuration, and residually stressed mechanically loaded config-uration (Rausch and Kuhl, 2013).

Fig. 1 illustrates the three configurations, which can also be used to modelprestrain induced by other physical mechanisms such as friction (Hild, 1998;Fagiano et al., 2014). Under the small deformation hypothesis, this decompositioninduces an additive decomposition of the strain itself: the total elastic strain is thesum of prestrain and mechanically-induced strain. Under finite deformations, thisdecomposition induces a multiplicative decomposition of the deformation gradi-ent: The total elastic tensor Fe is the product of a prestrain tensor Fp and themechanically-induced deformation gradient F :

Fe ¼ F � Fp with F ¼ Fe � F p�1 ¼∇φ ð1Þ

Inherent to this approach, only the mechanically-induced deformation gradientF ¼∇φ is a gradient of a vector field, while Fe and Fp are generally incompatible.We can then adopt the classical finite deformation theory for nearly incompressiblebodies (Holzapfel, 2000), however, now based on the total elastic tensor Fe insteadof the purely mechanically-induced deformation gradient F . This implies that wecan express the strain energy potential, which is generally a function of F and Fp , asa function of Fe only:

ψ F ; Fp� �

¼ψ e Fe� �

ð2Þ

The most widely used method to impose incompressibility or quasi-incompressibilityconsists of further decomposing the elastic tensor into volumetric and isochoricparts:

Fe ¼ Fv � Fe ð3Þ

with the volumetric contribution Fv ¼ Je� �1=31 with Je ¼ det Fe

� �and the isochoric

contribution Fe ¼ Je� ��1=3Fe so that Je ¼ det Fe

� �¼ 1. As a result of this decom-

position, the independent variables are now the elastic Jacobian Je and the isochoricelastic Cauchy–Green tensor Ee :

Je ¼ det Fe� �

and Ee ¼ 12

Fet � Fe �1

� �ð4Þ

We can then split the elastic strain energy potential into volumetric and deviatoricparts:

ψ e Je ; Ee� �

¼U Je� �þψ e Ee

� �ð5Þ

For the volumetric part of the strain energy function (Auricchio et al., 2013), we could

Fig. 1. Kinematics of prestrained biological systems. Prestrain maps the stress-freereference configuration onto the residually stressed but mechanically unloadedconfiguration through the prestrain tensor Fp . Mechanical loading maps theresidually stressed but mechanically unloaded configuration into the mechanicallyloaded configuration through the deformation gradient F . The total elasticdeformation Fe is a result of prestrain Fp and mechanical deformation F . Onlythe mechanical deformation F ¼∇φ is the gradient of a vector field, while theprestrain tensor Fp and the elastic tensor Fe are incompatible. Only the elastictensor Fe generates stress.

M. Genet et al. / Journal of Biomechanics 48 (2015) 2080–2089 2081

suggest the following expression:

U Je� �¼ 1

D0

12

ðJeÞ2�1� �

� ln Je� �� �

ð6Þ

For the deviatoric part, we use a transversely isotropic Fung (1993) law adaptedfor myocardial tissue (Guccione et al., 1991):

ψ e Ee� �

¼ 12C0 exp Ee : B0 : Ee

!�1

!ð7Þ

where B0 is a fourth-order constitutive tensor, which takes a diagonal format in

Voigt notation (Mehrabadi and Cowin, 1990; François, 1995):

cB0 ¼Diag Bff ;Bss ;Bnn ;2Bfs ;2Bfn ;2Bsn� � ð8Þ

Its individual parameters Bff, Bss, Bnn, Bfs, Bfn, and Bsn weigh the influence of thenormal fiber, sheet, and normal strains and of the corresponding shear compo-nents. Throughout this paper, we use the normal human myocardial propertiescharacterized in Genet et al. (2014) and summarized in Table 1. Note that it is well-known that the Fung law is not poly-convex (Holzapfel et al., 2000; Holzapfel andOgden, 2009), making the proposed model not strictly speaking thermodynami-cally consistent. However, it is still widely used for the myocardium; for this paperwe solve our model implicitly and have not encountered any material instability.Furthermore, the framework presented here for growth-induced residual stress isindependent of the choice of the strain energy potential, and a convex potentialcould be used instead.

These equations can, in principle, allow us to calculate the in vivo stress state ofthe heart. Unfortunately, however, neither the initial nor the grown configuration isknown. Only the residually stressed mechanically unloaded configuration and themechanically loaded configuration can be imaged in vivo. In what follows, wepropose a fixed-point iteration to computationally retrieve the initial configurationfrom the in vivo-acquired configuration. The core assumption of this algorithm isthat residual stresses are a result of biological growth.

2.2.2. Computational modelThe constitutive laws have been implemented as a user subroutine into

Abaqus/Standard. The subroutine is written in Cþþ , and uses the LMTþþ libraryfor linear algebra (Leclerc, 2010; Genet, 2010). The implementation is straightfor-ward as outlined in detail in Rausch and Kuhl (2013).

2.3. Mechanics of growth-induced, prestrained biological systems

2.3.1. Continuum modelThe origin of prestrain is still an open question today. Here, we investigate the

hypothesis that the prestrain is induced by heterogeneous growth. In order tomodel the growth of biological tissues, we once again adopt the concept offictitious configurations and decompose the total deformation multiplicatively intoelastic, prestrain, and growth parts (Rodriguez et al., 1994; Ambrosi et al., 2011;Kuhl, 2014). This decomposition is a very general tool, and can also be applied forinstance to model tissue activation (Rossi et al., 2012; Pezzuto et al., 2014)

Fig. 2 shows the different configurations and transformations. Similar to themechanics of prestrained biological systems in Section 2.2, the elastic tensor Fe canbe multiplicatively decomposed into the prestrain Fp and the mechanically-induced deformation F 0 . Yet, we can also decompose the elastic tensor Fe intothe inverse growth Fg and the deformation gradient F :

Fe ¼ F 0 � Fp ¼ F � Fg �1 with F ¼ F 0 � Fp � Fg ¼∇φ ð9Þ

Note that in the special case where F ¼ F 0 , we have Fp ¼ Fg �1. However, in practicewe only work with Fg , which obeys the evolution equation defined in the nextparagraph, and Fe , which contains both the prestrain Fp and load-induceddeformation F 0 , and is used to compute stress through the elastic strain energypotential. Fig. 2 also illustrates the heterogeneous growth-induced prestrain mapFp between the incompatible grownconfiguration and the residually-stressedcompatible configuration.

Here, in order to focus on growth-induced residual stress and not on thegrowth itself, we will use a simple isotropic and strain-driven growth law. Indeed,

even though it is now established that maladaptive hypertrophic processes arelargely anisotropic (Taber, 1995; Frey and Olson, 2003), physiological growth mustinvolve both longitudinal and transverse growth, and isotropic growth is thesimplest growth model that can generate residual stresses. Moreover, even thoughthe hormonal mechanisms underlying tissue growth are complex and beyond thescope of this study, it is now accepted that hormonal signals are strongly regulatedby mechanical signals, and that mechanotransduction, especially stretch-sensitiv-ity, is one of the fundamental processes underlying tissue growth (Maillet et al.,2013). Note that elastic strain and stress are coupled through the constitutiverelationship and not mutually independent, so a stress-driven law could be used aswell. To characterize the kinematics of growth we can thus parameterize thegrowth tensor Fg in terms of a single scalar-valued growth multiplier ϑg:

Fg ¼ ϑg1 ð10Þ

To characterize the kinetics of growth, we assume that growth evolves in timeaccording to the following evolution law:

_ϑg ¼ 1

τtr Ee� �

ð11Þ

where τ is a time constant for growth. Since it is the only time constant of theproblem, we simply set it to 1, and consider all other time-dependent quantities interms of it. This implies that the driving force for growth is the trace of the elasticdeformation Ee .

2.3.2. Computational modelSimilar to the elasticity part in Section 2.2, we implemented the growth model

as a user subroutine into the finite element package Abaqus/Standard. Since growthis driven by the elastic strains Ee , which are in turn affected by growth asFe ¼ F � Fg �1, we iteratively determine the growth multiplier ϑg at the integrationpoint level at every global iteration step. Rather than using a consistent Newton–Raphson procedure, which requires a consistent algorithmic linearization (Goktepeet al., 2010a,b), we adopt a local fixed-point iteration, supplemented by Aitken'srelaxation (Genet et al., 2013). For the time integration, we use an implicit mid-point rule. This results in the following algorithm to determine the local growth

Table 1Normal human myocardial properties (Genet et al., 2014) used in this paper for the ventricular simulations.

D0 (kPa�1) C0 (kPa)

10�3 0.115Bff()

Bss()

Bnn()

Bfs()

Bfn()

Bsn ()

14.4 5.76 5.76 5.04 5.04 2.88

Fig. 2. Kinematics of growth-induced, prestrained biological systems. Growth turnsthe initial, stress-free and compatible configuration into the grown, stress-free butincompatible configuration. Prestrain maps the grown, stress-free configurationonto the residually stressed but mechanically unloaded configuration through theprestrain tensor Fp . Mechanical loading maps the residually stressed but mechani-cally unloaded configuration into the mechanically loaded configuration throughthe deformation gradient F 0 . The total elastic deformation Fe is a result of prestrainFp and mechanical deformation F 0 . The full deformation F is a result of growth Fg ,prestrain Fp , and mechanical deformation F 0 . Only the full deformation F ¼∇φ andmechanically-induced deformation F 0 ¼∇φ0 are gradients of a vector field, whilethe growth tensor Fg , the prestrain tensor Fp , and the elastic tensor Fe areincompatible. Note that Fig. 2 becomes identical to Fig. 1 if Fg ¼ 1 .

M. Genet et al. / Journal of Biomechanics 48 (2015) 2080–20892082

multiplier ϑg:

given : ϑgt ; F tþ1

;Δt

initialize growth rate and iteration counter : _ϑgtþ1=2 ¼ 0;niter ¼ 0

loopupdate iteration counter : niter ¼ niterþ1

update growth : ϑgtþ1 ¼ϑg

t þ _ϑgtþ1=2 Δt

calculate elastic deformation : Eetþ1

¼ 12

1ðϑg

tþ 1 Þ2F t

tþ1� F

tþ1�1

� �

evaluate residual :if ðniter41Þ then rold ¼ r

r¼ _ϑgtþ1=2�

1τtr

EetþEe

tþ 12

� �8><>:check convergence : if rotoleranceð Þ then exit loop

calculate relaxation :if niter ¼ 0ð Þ or r ¼ rold

� �then α¼ 1

else α¼ �α rold=ðr�roldÞ

(update growth rate : _ϑ

gtþ1=2 ¼ _ϑ

gtþ1=2�α r

������������������������������end loop

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:For a tolerance of tol¼10�6, the algorithm converges rapidly in only a fewiterations.

2.3.3. Example of growth-induced, prestrained left ventricleThroughout this paper, we will illustrate our proposed method on a patient-

specific left ventricular model of a healthy human (Genet et al., 2014). This modelcharacterizes healthy human ventricular mechanics. Briefly, the geometry wasextracted from magnetic resonance images by manual segmentation, and triangu-lated using linear hexahedrons. A generic myofiber orientation field, with linearlyvarying helix angle from þ601 at the endocardium to �601 at the epicardium(Streeter and Bassett, 1966; Streeter et al., 1969), was assigned to the mesh. We usea standard displacement finite element method, with fully implicit time stepping.In terms of boundary conditions, we constrain normal displacement of the basalnodes, and apply pressure loading at the endocardial faces.

To introduce residual stresses in a subject-specific geometry, we solve thefollowing problem: we load the left ventricle to a given ventricular pressure P , thenkeep the pressure constant for a time t to allow the ventricle to grow, and thengradually remove the pressure. Growth duration t is usually expressed as a functionof the growth time constant τ, which is the only time constant of the problem. As aresult of heterogeneous growth, the final configuration is unloaded, but not stressfree, i.e., residually stressed. Our goal is now to iteratively determine the referenceconfiguration for which the in vivo grown geometry for given a given P and time tmatches our subject-specific imaging data. The following section describes how weidentify the reference configuration using fixed-point iterations.

2.4. Computation of stress-free reference configuration of biological systems

2.4.1. Fixed-point iterationAs discussed in Section 2.3 and illustrated in Fig. 2, the mechanically unloaded

configuration from medical imaging is not a true reference configuration in thecontinuum mechanics sense since it is residually stressed. The stress free referenceconfiguration is unknown and has to be determined iteratively. A commonapproach to this problem is to solve the so-called inverse elastic problem(Govindjee and Mihalic, 1996, 1998). Yet, this approach requires non-standardformulations and solvers on the continuum and discrete levels. Here we suggest analternative fixed-point-based approach, which requires only modular modificationsto the original forward problem. Fixed-point iterations have originally beenproposed by Sellier (2011), and have recently been adopted for in the context ofcardiac mechanics by Wang et al. (2013a,b) and Krishnamurthy et al. (2013).However, previous approaches have only used the fixed-point iteration to identifythe reference configuration associated with a loaded in vivo configuration. Here, weuse it to find the reference configuration associated with an unloaded in vivo

configuration. We assume that this unloaded structure contains heterogeneousgrowth-induced residual stresses.

Let X!

ref denote the nodal positions of the residually stressed, mechanicallyunloaded configuration obtained, e.g., from in vivo imaging. The objective of thefixed-point iteration is to find the nodal positions X

!of the stress-free, incompa-

tible configuration that will deform into X!

ref through the loading-growth-unloading scenario described in Section 2.3. Similar to the fixed-point iterationsolver used for the growth law in Section 2.3.2, we adopt Aitken's (1950) relaxationto improve convergence. For time integration, we use here an implicit backwardEuler scheme. The algorithm to iteratively determine the nodal positions X

!in the

stress-free configuration looks as follows:

given : X!

ref ; X!

init

initialize configuration and iteration counter : X!¼ X

!init;niter ¼ 0

loopupdate iteration counter : niter ¼ niterþ1

for given X!

; P and t ;

solve growth problem : X!

def

evaluate residual : if niter41ð Þ then R!old ¼ R

!R!¼ X

!def � X

!ref

ncheck convergence : if max R

!� otolerance� �

then exit loop

calculate relaxation :

if niter ¼ 0ð or R!¼ R

!old� �

then α¼ 1

else α¼ �αR!old

� R!� R

!old� �

R!� R

!old� �

� R!� R

!old� �

8>>>>>>><>>>>>>>:

1CCCCCCCAupdate configuration : X

!¼ X!�α R

!

���������������������������������

1CCCCCCCCCCCCCCCCCCCCCCCCCCCAend loop

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:Here X

!init are the initial nodal position of the stress-free reference configura-

tion, and X!

def are the deformed nodal position after the loading-growth-unloadingprocedure. This implies that X

!init could, but does not have to, be equal to X

!ref .

Similar to the algorithm in Section 2.3.2, the fixed-point iteration converges rapidlyin a few iterations for a tolerance of 10�1 mm.

2.4.2. Computational modelWe implemented the fixed-point iteration as a python script, which solves the

growth problem inside the inner loop using Abaqus/Standard and then post-processes the nodal positions.

3. Results and discussion

3.1. Opening angle experiment

Fig. 3 illustrates our opening angle experiment. It shows anisolated heart slice before (left) and after (right) cutting the sliceradially in the middle of the left ventricular free wall.

Table 2 summarizes the opening angles of the basal, equatorial,and apical slices of all four hearts. Average opening angles increasefrom base to apex (from 8.71 to 16.61) with an increasing standarddeviation (from 71.81 to 713.71). The variability in opening anglemay be attributed to a combination of geometric and structuraldifferences between the two control slices, and the location of theradial cut. Nevertheless, opening angles are significantly lowerthan values found for rat hearts (451) (Omens and Fung, 1990),

Fig. 3. Opening angle experiment. Isolated heart slice before (left) and after (right) radial cutting in the middle of the left ventricular free wall.

M. Genet et al. / Journal of Biomechanics 48 (2015) 2080–2089 2083

hence potentially revealing a size-effect in residual stressdevelopment.

3.2. Generating growth-induced prestrain

Fig. 4 illustrates our protocol to generate prestrain throughheterogeneous growth in a patient specific left ventricular geo-metry (Genet et al., 2014). For the sake of illustration, we used anominal ventricular pressure P of 1 mmHg, and a nominal growthduration t of one growth time constant τ. During the strain-drivengrowth step, the myofiber stress field becomes more and morehomogeneous. While homogeneous isotropic growth does notgenerate residual stress, homogeneous anisotropic growth(Rausch et al., 2011; Kerckhoffs, 2012) and heterogeneous isotropicgrowth (Goktepe et al., 2010a,b) do. Thus, the final configuration isresidually stressed: the sub-endocardial region is in compression,the sub-epicardial region in tension.

This is in contrast to some models in the literature (Kroon et al.,2007, 2009), which hypothesize that the growth-induced residualstresses are relieved over time, such that the grown configurationis always assumed to be stress free. Our opening angle experimentindicates though that the inner layer of the ventricle, similar toarteries, is in compression and the outer layer is in tension. This

result is compatible with strain-driven growth-induced prestrain,the underlying hypothesis in this paper.

We have used the trace of the elastic Green–Lagrange straintensor as the thermodynamical force to drive growth. Otherauthors have used the fiber stretch (Lee et al., 2014a); however,the fiber stretch is largest in the mid-wall and not in the sub-endocardial region (Choi et al., 2010; Lee et al., 2014a), and theassociated residual stress fields are not physiological. Others haveused the Kirchhoff (Klepach et al., 2012) or Mandel (Goktepe et al.,2010a,b) stresses, which are more “isotropic” quantities. Actually,in the case of isotropic growth and compressible elasticity, theMandel stress is the energy conjugate of the growth state variable;however, in the case of quasi-incompressible elasticity consideredhere, the energy conjugate of the isotropic growth state variable isbasically the isostatic pressure, which does not seem to be areasonable driving force. In other words, the distribution ofresidual stresses constrains the functional form of the growthlaw. This is why, after systematically testing a collection of otherdriving forces for growth, we decided to select the trace of theelastic deformation.

As for growth kinetic equation itself, for the lack of appropriateexperiments, we use a simple linear evolution law. More complexevolution laws have been proposed, including thresholds andlimiting values (Kuhl, 2014), and even multiple domains (Lee et al.,2014a) to predict both growth and reverse growth. The growthlaw we have selected here is the simplest law to predict hetero-geneous growth.

3.3. Growth-induced prestrain in patient-specific left ventricle

Fig. 5 illustrates the fixed-point iteration method to solve forthe reference configuration associated with a given amount ofprestrain, once again generated with a nominal ventricular

Table 2Opening angle experimental data. All values are reported in degrees.

Basal slice Equatorial slice Apical slice

Heart 1 9.3 8.7 9.8Heart 2 9.8 19.9 16.5Heart 3 9.7 9.0 4.2Heart 4 6.1 14.5 35.7

Mean7SD 8.771.8 13.075.3 16.6713.7

Fig. 4. Protocol to generate heterogeneous growth-induced residual stress in a patient specific left ventricular geometry. The ventricle is loaded to a ventricular pressure of P ,(a)–(b), then allowed to grow for a duration t , (b)–(c), and then unloaded, (c)–(d). The strain-driven growth drives the growing configuration into a state with a morehomogeneous myofiber stress and reduces regional stress variations (c). The final configuration is unloaded, but not stress free, with the sub-endocardial region incompression, and the sub-epicardial region in tension (d).

M. Genet et al. / Journal of Biomechanics 48 (2015) 2080–20892084

pressure of 1 mmHg and a nominal growth duration of one growthtime constant. For each iteration (i.e., each column), we show thereference stress-free configuration (top row) and the final (afterloading, growth, and unloading) configuration (bottom row).During the fixed-point iterations, we compute the error betweenthe reference configuration (obtained from medical imaging) andthe deformed configuration (after loading-growth-unloading). Thevery last geometry (last column, bottom row) is identical to thevery first (first column, top row; this geometry is the in vivounloaded geometry extracted from medical imaging), but it con-tains heterogeneous growth-induced residual stress.

Fig. 6 shows, for multiple growth durations ranging from 1 to6 growth time constants, the reference configuration computedusing the fixed-point iteration method (top row) as well as theprestrained configuration. The reference configuration is alwaysstress-free, but the geometry varies. On the contrary, the prestrained

configuration has always the same geometry (the one extractedfrom medical imaging data), but different levels of residual stress.

By assuming that the early-diastolic ventricular geometryextracted from medical imaging is unloaded, we implicitly neglectedthe ventricular pressure at the beginning of diastole. However, thishypothesis is not necessary for using the fixed-point iterations. Inprinciple, it is possible to prescribe any loading at the end of theforward step of the inverse problem, thus taking into account themechanical load applied to the tissues.

3.4. Virtual opening angle experiment

After generating residual stresses in the subject-specific leftventricular geometry, we slice the virtual heart longitudinally andthen cut it open. The result is illustrated in Fig. 7, for a nominalventricular pressure of 1 mmHg, and a nominal growth duration of

Fig. 5. Fixed-point iteration method to compute the prestrained, mechanically unloaded reference configuration (last column, top row) so that the prestrained, mechanicallyloaded configuration (last column, bottom row) matches the in vivo geometry extracted from magnetic resonance images (first column, top row) but contains auto-balancedresidual stresses. Colors represent residual fiber stress, in kPa. (For interpretation of the references to color in this figure caption, the reader is referred to the web version ofthis paper.)

Fig. 6. Prestrained configurations for varying prestrain levels. Prestrain is generated by heterogeneous growth. The reference configurations are computed using a fixed-point iteration, so that the prestrained configurations (bottom row) match the in vivo geometry extracted from magnetic resonance images (top row). Colors representresidual fiber stress, in kPa. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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six growth time constants. During this virtual opening angleexperiment, the apical and basal parts of the mesh are fixed, andonly the slice is allowed to deform. To reproduce the experimentalconditions, the nodes belonging to the upper and lower faces canonly move in the sliding plane. To remove rigid body motions, thenodes of one face of the cut can only move in the cutting plane.

Fig. 8 shows the relationship between the maximum principalresidual strain in the slice and the opening angle. Different levelsof residual strain are generated using different growth durations,

ranging from 1 to 6 growth time constants, while a 1 mmHgventricular pressure is always used. The black line and the graybox mark the average and standard deviation from the physicalopening angle experiment on the mid-level slice in Section 3.1.The graph confirms that the proposed method is capable ofreproducing physiologically realistic opening angle ranges.

It is not necessary to shrink the ventricle to an infinitesimallysmall size to then induce enough residual stress through growth togenerate a realistic opening angle. This means that part of theresidual stress induced by growth during organogenesis is released,as hypothesized in Kroon et al. (2007, 2009). Our proposed methodthus allows us to distinguish between the residual stress that isreleased, and the residual stress that remains within the tissues.Moreover, the fact that some of the residual stress is released justifiesthe original assumption that there exists a stress-free compatibleconfiguration, which is described in Fig. 2, and computed using thefixed-point algorithm described Section 2.4.

3.5. Influence of the prestrain on the ventricular mechanicalresponse

An important consequence of prestrain is the influence of theresidual stress on the mechanical response of the heart. Fig. 9shows the passive pressure–volume relationship for differentlevels of prestrain associated with different opening angles. Onceagain, the different levels of prestrain are generated using differentgrowth durations, here ranging from 1 to 6 growth time constants,and the ventricular pressure during growth is kept constant1 mmHg. Because of the nature of residual stresses in growingcircular cross sections, with compressive stresses at the sub-

Fig. 7. Virtual opening angle experiment. The prestrained configuration is slicedand then cut. The slice naturally springs open, similar to the physical experimentillustrated in Fig. 3. Cutting the slice relieves some, but not all residual stress. Colorsrepresent residual fiber stress, in kPa. (For interpretation of the references to colorin this figure caption, the reader is referred to the web version of this paper.)

Fig. 8. Opening angles for varying prestrain levels. The green curve summarizes the computationally simulated opening angles for a pressure of P ¼ 1 mmHg. The black lineand gray box mark the experimentally measured opening angle of 1375.31 for the equatorial slice. Computationally simulated opening angles lie within the range ofexperimentally measured opening angles. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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endocardial region and tensile stresses at the sub-epicardialregion, prestrain tends to make the ventricle more compliant.For instance, for a maximal principal prestrain of 15.34%, a fiberresidual stress varying from �0.12 kPa to þ0.055 kPa, and anopening angle of 17.11, i.e., close to the average experimental value,the ventricular pressure after a passive filling corresponding to anunloaded ventricular volume (i.e., a 50% ejection fraction) isreduced by 25.8% compared to the prestrain-free ventricle. As aconsequence of this increased compliance, the ventricle requiresless energy to fill. In other words, besides the well-known effect ofhomogenizing the ventricular wall stress (Omens and Fung, 1990),residual stresses could be a mechanism to directly improveventricular diastolic function.

4. Summary and perspectives

In this paper, we investigated the hypothesis that residualstress in the beating heart is induced by growth of the organ.We used the finite growth framework, together with a fixed-pointiteration on the geometry itself, to introduce residual stress in asubject-specific left ventricular geometry. We showed that theresidual stress fields induced by heterogeneous growth are com-patible with experimental results, both qualitatively and quantita-tively. Indeed, our method can reproduce the classical openingangle experiment, by Chuong and Fung (1986) and Omens andFung (1990). It can successfully predict the physiological range ofopening angles. We showed that the prestrain can have a sig-nificant impact on the passive mechanical response of the ven-tricle, making it much more compliant.

The present study could be complemented at multiple levels. Forinstance, different growth kinematics such as longitudinal andtransverse growth (Goktepe et al., 2010a,b) could be compared interms of the opening angle they generate. Moreover, it would beinteresting to explore regional variations along the longitudinal axisand compare prestrain in different slices. Furthermore, we plan toquantify the out-of-plane displacement, both experimentally andcomputationally, to provide additional information about the local

residual stress field and the degree of anisotropy. Finally, we coulduse a fully convex elastic strain energy potential (Holzapfel andOgden, 2009), making our framework thermodynamically consistent.

Some more general perspectives follow from this work. First, inaddition to studying the influence of the prestrain on the passiveventricular response alone, we will characterize the sensitivity ofthe active response, and, eventually, of the entire pressure–volumerelationship, for different prestrain levels. Another importantfollow up study will be to incorporate the algorithm introducedhere within a personalized left ventricular model (Genet et al.,2014), and compute personalized passive and active cardiacmechanical properties by taking into account physiological pre-strain levels. Finally, we will study the interactions betweenphysiological residual stress and residual stress induced by theinjection of biomaterials in diseased myocardium as a potentialtreatments (Lee et al., 2013, 2014c), and their respective influenceon ventricular mechanics. We believe that prestrain could be thekey to explain various discrepancies in the literature betweenin vivo and ex vivo measurements and computational simulations.Including the effects of prestrain has the potential to makesubject-specific modeling more reliable and more applicable totoday's cardiac health problems.

Conflict of interest statement

None.

Acknowledgments

This work was supported by a Marie-Curie international out-going fellowship within the 7th European Community FrameworkProgram (M. Genet); NIH Grants R01-HL-077921 (J.M. Guccione),R01-HL-118627 (J.M. Guccione and G.S. Kassab) and U01-HL-119578 (G.S. Kassab, J.M. Guccione, E. Kuhl); NSF grants 0952021and 1233054 (E. Kuhl).

Fig. 9. Passive pressure–volume response of the left ventricle for varying prestrain levels. Increasing the prestrain level increases the ventricular compliance: for an openingangle of 17.11, i.e., close to the average experimental value, the ventricular pressure after passive filling corresponding to a 50% ejection fraction is reduced by 25.8% comparedto the prestrain-free case. This shows that residual stresses could be a mechanism to directly improve ventricular diastolic function.

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