Constitutive Relations and Finite Deformations of...

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Constitutive Relations and FiniteDeformations of Passive Cardiac Tissue II:

Stress Analysis in the Left VentricleJay D. Humphrey and Frank C.P. Yin

We present a new approach for estimation of transmural distributions of stress and strain in theequatorial region of a passive left ventricle. We employ a thick-walled cylindrical geometry,assume that myocardium is incompressible, and use a three-dimensional constitutive relationthat yields a material symmetry consistent with observed transmural variations in muscle fiberorientations. Moreover, we consider finite deformations including inflation, extension, twist,and transmural shearing and suggest a new method for determination of the requisitedeformation parameters directly from experimental strain data. We show representativetransmural distributions of stress and strain, and perform a parametric study to illustratediffering predictions of stress induced by varying boundary conditions, muscle fiber orienta-tions, or modes of deformation. Our analysis can be used to guide and check future predictionsof cardiac stresses, and to guide experimentalists by suggesting the accuracy of measurementsessential for stress analysis in the heart. (Circulation Research 1989;65:805-817)

Quantification of the mechanical stresses inthe heart remains a subject of intense inter-est. Mechanical stresses can, for example,

affect coronary blood flow, myocardial oxygen con-sumption, overall cardiac function, and the rate andextent of hypertrophy in certain pathological condi-tions. These stresses cannot be measured directly inthe intact heart,1-2 but must be inferred from biome-chanical analyses. Thus, most reported inferences ofstresses in the heart have been obtained from eitherfinitc-element-based calculations or from closed-form analytical solutions.3-6

Finite-element methods are numerical approxima-tion techniques that can, in theory, account for thegeometric irregularities, complex boundary condi-tions, large deformations, solid-fluid interactions,material heterogeneities, and nonlinear anisotropicmaterial behavior inherent to the heart. Thisapproach may, in fact, be the only practical tool foraccurate analysis of regional stresses in the intact

From the Department of Mechanical Engineering (J.D.H.),The University of Maryland, and the Departments of Medicineand Physiology (F.C.P.Y.), The Johns Hopkins Medical Institu-tions, Baltimore, Maryland.

Supported in part by Grant HL 33621-02 from the NationalHeart, Lung, and Blood Institute and by a Minta Martin grant.F.C.P.Y. is an Established Investigator of the American HeartAssociation.

Address for correspondence: Frank C.P. Yin, MD, PhD,Cardiology Division, Carnegie 565, The Johns Hopkins Hospital,600 North Wolfe Street, Baltimore, MD 21205.

Received October 15, 1987; accepted March 9, 1989.

heart.3 At present, however, realistic predictions ofthe three-dimensional state of stress in the heart arelimited by a lack of sufficient experimental data,particularly information on the material propertiesof myocardium. Consequently, results obtained usingfinite-element methods must be interpreted withcaution.

Because of the lack of complete data and thesometimes prohibitive cost and effort in performanceof finite-element calculations, closed-form analyticalsolutions can continue to provide valuable insightinto cardiac mechanics, as long as one views thepredictions within the framework of the underlyingassumptions. Such solutions presented before 1980generally considered the heart to be homogeneous,however, and did not account for the known distri-butions of muscle fibers throughout the walls.3"5

Thus, the utility of the associated predictions wasquestionable. Recent investigators, using a thick-walled cylindrical geometry to model the left ventri-cle, have attempted to take fiber angles into accountbut have assumed that myocardium consists solelyof "slippery" muscle fibers embedded in an inviscidfluid.7-14 Hence, it was tacitly assumed that myocar-dium can bear loads only in the direction of themuscle fibers. This assumption, however, is incon-sistent with recent experimental data on noncontract-ing tissue15; therefore, the usefulness of predictionsbased on these cylindrical models of passive leftventricular mechanics is also questionable.

In this paper, we present a new analyticalapproach for calculation of transmural distributions

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806 Circulation Research Vol 65, No 3, September 1989

of stress and strain in the equatorial region of anoncontracting left ventricle. Like other investiga-tors, we base our analysis on a thick-walled cylin-drical geometry, a reasonable representation of thedistribution of muscle fibers in the left ventricle,1216

incompressibility of the tissue, and large deforma-tions including inflation, extension, and twist. Incontrast with previous studies, however, we use arecent constitutive description of myocardial prop-erties derived from biaxial data17 and account fortransmural shearing strains, which were recentlyidentified in canine hearts.18 Our new constitutiverelation treats myocardium as transversely isotro-pic with respect to the muscle fiber directions andaccounts for both the "fiber" and the previouslyneglected "cross-fiber" stiffness13 of the tissue.Finally, whereas the parameters that define thedeformation are often inferred from assumed load-ing conditions,811-13 we calculate these parametersdirectly from available experimental data18 and bysatisfaction of the equations of equilibrium.

We show transmural distributions of stress andstrain in the equatorial region of a representativepassive left ventricle. We also examine changes inthe predicted stresses induced by different distribu-tions of muscle fibers, varying amounts and types ofdeformation, and differing boundary conditions.Because of the current lack of sufficient experimen-tal data, however, our predictions of stress distri-butions, like all others, should be viewed cautiouslywith regard to physiological implications. Nonethe-less, our analysis is reasonably general, and can berefined to include new information as it becomesavailable. Finally, our analysis can be used to guideand check future calculations of cardiac stresses,and to guide experimentalists by suggesting therequisite accuracy of measurements that are neededfor more complete and realistic assessments ofstress in the heart.

MethodsPreliminary Considerations

We will first address several issues that are fun-damental to our analysis, including the applicabilityof the continuum hypothesis and the choice ofconstitutive relations, geometry, deformations, andboundary conditions.

Continuum approach. Some investigators, usingmeasurements of gross forces and deformations,have attempted to calculate stresses and extensionsin individual muscle fibers or sarcomeres.7-8>1113>19

To accurately determine these stresses and exten-sions, however, one must know the precise detailsof the microstructure, including the number, orien-tation, distribution, diameter, and properties of thetissue constituents and their interactions. Since thisinformation is not available, there is no way todiscriminate between the loads borne by individualmuscle fibers and those borne by neighboring tissue

components; hence, estimates of stress in individualmuscle or collagen fibers are likely to be misleading.

We, like many others, employ an approach that isuseful when one considers gross mechanical behav-ior of a material, namely, the relationship betweenlocally averaged stresses and strains at any pointwithin a body. This continuum approach is reason-able when the size of the constituents that make upthe material and the distances between them areorders of magnitude smaller than the dimensions ofinterest. In the canine heart, for example, the leftventricle is roughly 1-2 cm thick, whereas thediameters of muscle fibers and intramuscular colla-gen and the interfiber distances are on the order of0.1-3 /mi.20 Hence, for the usual considerations of,for example, transmural distributions of stress, thecontinuum hypothesis seems reasonable.

Constitutive relations. Equations that character-ize a material and its response to applied loads arecalled constitutive relations since they describegross behavior due to the internal constitution ofthe material. Recent biaxial experimental data revealthat potassium-arrested myocardium can supportloads both transverse to and in the direction of themuscle fibers, and that the tissue behavior is non-linear and anisotropic.15-21 Thus, existing character-izations of myocardium as a fiber-reinforced, fluid-filled structure,8-10-13 or as an isotropic material,22-26

are inappropriate descriptors of the tissue behavior.Consequently, we employ a constitutive relation

derived directly from biaxial data and convenientlyexpressed in terms of a pseudostrain-energy func-tion, W(I,,a), as17

W=c[ei(I'-3)-l]+A[efl(a-1)2-l] (1)

where c, b, A, and a are pseudoelastic materialparameters derived from multiaxial data, and I, anda, a stretch ratio in the direction of a muscle fiber,are coordinate invariant measures of the deforma-tion. Stress-strain relations are obtained from Equa-tion 1 by taking derivatives of W(I,,a) with respectto the appropriate measures of the deformation (see"Appendix").

Geometry. A thick-walled sphere does not closelyapproximate the actual geometry of the heart or leftventricle. Moreover, the deformation of a sphereinto another sphere induced by uniform pressureson its surfaces implicitly requires either isotropic ortransversely isotropic (with respect to radial direc-tion) material symmetry. Thus, spherical modelsare not very useful for estimation of stresses andstrains in the heart even though the solutions arestraightforward.22-24-27 Conversely, a prolate spher-oid resembles the geometry of the left ventricle, butintroduces substantial mathematical complexity, par-ticularly when large deformations are considered.Thus, simplifying assumptions are often intro-duced, such as treatment of the ventricular wall asa series of uncoupled thin layers described byLaplace's relation,28-29 a conjecture that remainsquestionable.18



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Humphrey and Yin Stress Analysis in Passive Myocardium 807

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FIGURE 1. Schematic drawing of an ideal-ized left ventricle (panel A) wherein thincross sections in the equatorial region areidealized as a thick-walled cylindrical annu-lus with the six independent componentsof Cauchy stress (force/current area) shownon a differential element of wall (panel B)and uniformly applied pressures denotedon inner and outer surfaces (panel C).

Alternatively, a thick-walled cylindrical geome-try is mathematically tractable even when one con-siders certain complex finite deformations of anincompressible, anisotropic, nonlinear material.Thus, like many investigators, we employ a cylin-drical geometry to study the passive left ventricle.We emphasize, however, that a cylindrical geome-try can, at best, approximate the left ventricle onlyin a small annular region (Figures 1A and IB; seealso Figure 2 in Streeter16) and not over its entirety.28

Although left ventricular cross sections are admit-tedly not circular near the equator, this approxima-tion seems warranted if one is not interested indetailing the stresses and strains in the trabeculaeand other interstices near the endocardial surface.Finally, consideration of the equatorial region ascylindrical is consistent with finite-element predic-tions that the variations in stress are primarily radialin the equatorial plane, but are both radial and axialnearer the apex and base.30-31

Muscle fiber orientations. The data of Streeter16

reveal that the orientations of the muscle fibers varynonlinearly from the endocardium to the epicar-dium in the equatorial region of the canine leftventricle. Moreover, the muscle fibers appear tofollow regular helices on constant radial surfaces.Thus, Tozeren12 suggested the following equation to

represent these reported distributions of musclefiber orientations, <!>, as

0(R)=(D0{[2R-(R0+Ri)]/[R0-Rl]}3 (2)

where <$>o is the value of <£ at the epicardium (e.g.,-60° to -90°) and R, and R,, denote unloadedendocardial and epicardial radii, respectively.Although this equation only approximates the actualmuscle fiber orientations in the heart, we employ itto obtain representative predictions of transmuralstress. Finally, a linear distribution of 4>(R) resultsfrom Equation 2 if the exponent on the term in {. . .}is changed from 3 to I.11-12

Deformations. Attempts to rigorously quantifythe complex deformations experienced by the hearthave been reported only recently.18-32-34 These datareveal that large in-plane (i.e., in constant radialsurfaces) and transmural shearing strains accom-pany finite extensional strains throughout the car-diac cycle.18 Transmural shearing strains corre-spond to the tn and tr, stresses, and the in-planeshear strain corresponds to tte (Figure IB). Whereasthe presence of transmural shearing strains hasbeen neglected in previous cylindrical models of theleft ventricle, we consider a deformation that isconsistent with the existence of all six reportedcomponents of Green strain.18



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Stress boundary conditions. Uniform pressuresare normally assumed to act on the inner and outersurfaces of the heart (Figure 1C). The pressure atthe inner surface, P|, is the intraventricular cavitypressure, whereas the external pressure, Po, isusually assumed to be zero.8-11-13 Neglect of Po issuspect, however, in light of increasing data andspeculation on the mechanical role of the parietaland visceral pericardia.35-37 Moreover, because theleft ventricle consists of the interventricular septumand the free wall, both pericardial and right ventric-ular pressures should be considered as left ventric-ular stress boundary conditions. Inclusion of theprecise stress boundary conditions, however, mustawait additional experimental data and three-dimensional finite-element analyses. Here, we con-sider that the right ventricular pressure and pericar-dial influence are of the same order of magnitudeduring diastole, and can be approximated by asingle nonzero value of PO.

AnalysisBased on the above considerations, we examine a

finite inflation, extension, torsion, and transmuralshearing of a thick-walled, incompressible cylindri-cal annulus composed of a material described byEquation 1 with muscle fiber orientations defined byEquation 2. The Appendix contains a detaileddescription of the associated kinematics, constitu-tive considerations, satisfaction of equilibrium, andmethod of solution.

General outline. Briefly, we formulate the prob-lem in terms of five deformation quantities: r,, theinner radius in a loaded, or pressurized, configura-tion; A, a uniform axial extension; iff, a twist perunloaded axial length; and o»(R) and w(R), radiallydependent parts of the displacement in the circum-ferential and axial directions, respectively. Thevalues of ri5 A, and t/» are calculated directly fromexperimental strain data. The radial location of anypoint in the wall, in a loaded configuration, is thencalculated from the incompressibility constraint andknowledge of r,, A, and the inner and outer radii inan unloaded configuration, R; and R,,. Finally, forthe calculation of stress and strain, we only needthe derivatives of w(R) and w(R) with respect to theunloaded radius (i.e., d&VdR denoted as a>' anddw/dR denoted as w'). We determine a>' and w'from experimental strains at a single location in thewall and from satisfaction of two of the threeequilibrium equations. The final equilibrium equa-tion yields that portion of the stress associated withthe incompressibility of the tissue.

Preliminary calculations. Using Equations A-23athrough A-23e in the Appendix, we calculated rep-resentative deformation parameters from the dia-stolic strain data of Waldman et al18 (i.e., the databetween 0.4 and 0.57 seconds in Figure 3 and 4 ofthat study. Waldman et al18 calculated these strainsfrom the motions of radially directed tetrahedronsformed by lead markers embedded in the heart wall;

thus, their measurements were averaged radially.Consequently, for data from the subendocardialregion, we assume that their measurements reflectstrains midway through the inner third of the wall(i.e., R=R,+[Ro-Ri]/6), where R, = 1.5 cm andR<,=2.55 cm.38 With these assumptions and correc-tion for our reference state being the beginning ofdiastasis (Waldman used end diastole) and ourcoordinate system being R,G,Z (Waldman used0,Z,R), we found that r,=<1.77 cm, ^ 1.7°/cm, andA=«1.015. Interestingly, we also calculated A fromEquation A-3 using the heart dimensions in Olson etal and obtained A~1.02. Finally/we let w'~0.1rad/cm and w'««0.2 at R=R;=1.5 cm. The remainingvalues of <o' and w' were determined separately foreach radial location by use of the method outlined inthe Appendix (Equations A-24 through A-28).

Due to a lack of complete data, we assumed thatthe material parameters in Equation 1 do not varyacross the wall, and employed the following values:c=0.115 kPa, 6=9.665, A=0.082 kPa, and a=61.52(where kPa=l,000 N/m2).17 Moreover, we letPo=0.98 kPa, <t>o=-90°, and <I>(R) vary cubically(Equation 2). Collectively, these parameters wereemployed to generate numerical results for an equa-torial region in a "representative" canine left ven-tricle. Finally, we also examined changes in thepredicted stresses that were induced by assuming alinear muscle fiber distribution11-12 instead of a morerealistic cubic one (Equation 2); neglecting possibleexternal surface pressures on the heart8-11-13; neglect-ing transmural shearing strains8-11-13; and neglectingpossible twisting of the ventricle.

ResultsRepresentative Stresses and Strains

All six components of the Green strain variedmonotonically as a function of radius as prescribedby Equations A-l through A-5 (Figure 2). Becausethe circumferential (Eee) and radial (ERR) strains areinfluenced primarily by the term (r/R)2, their radialgradients near the inner wall can change dramati-cally (not shown) with even small changes in (r/R).Thus, there is a need for accurate measurements ofventricular geometry (e.g., Olson et al38) rather thanindirect estimates (e.g., Feit8). Due to the magnitudesof the deformations, infinitesimal strain assumptionsare clearly inappropriate.

In contrast with the transmural distribution ofstrains, only three of the components of stressvaried monotonically as a function of radius (Figure3). In particular, the radial stress, t^, varies from-Pi to -P o , and the transmural shearing stresses, uand trff, are inversely proportional to r and r ,respectively, each as required by equilibrium. Con-versely, the circumferential (tw), axial (t^), andin-plane shearing (t9z) stresses had complex trans-mural distributions. The peaks in t^, tei, and t^ inthe inner third of the wall result primarily from theanisotropy introduced by the muscle fiber direc-



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UNDEFORUED RADIUS (cm)

FIGURE 2. Representative transmural Green strains areplotted versus undeformed radius, and are based on thefollowing values of the parameters: R,=1.5 cm, RO=2.55cm, rt=1.77 cm, ip=1.7°/cm, u)'=0.1 rad/cm at Rh

A=1.015, w'=0.2atRh <J>o=-90°, c=0.115kPa, b=9.665,A-0.082 kPa, 3=61.52, Po=0.98 kPa, and <t>(R) variescubicalty (Equation 2).

tions. As shown in the Appendix, these three com-ponents of stress can be used to calculate the stressin the direction of the muscle fibers.

It is noteworthy that these general characteristicsof the Cauchy stress hold for any transverselyisotropic material defined by a W(I,,a) with chang-ing muscle fiber directions lying in constant radialsurfaces (e.g., N in Equations A-7 and A-8) and,thus, are not restricted to the specific form of W(I,,a)in Equation 1, or the particular values of the param-eters. Therefore, as these figures illustrate, the com-ponents of the Cauchy stress cannot be simplyinferred from knowledge of the associated strains.

Factors Influencing the Stress DistributionsFigures 4 and 5 illustrate components of the

Cauchy stress based on the representative parame-ters above (Figures 2 and 3), except that we sepa-rately assumed a) a linear distribution of musclefibers, Equation 2 with the {. . .} term raised to the

first power; b) no external pressure, Po-0; c) notwist, 1/̂ =0; d) no RZ shear, w'=0; or e) no R0shear, w'=0. The figures also show the correspond-ing component of stress from Figure 3 for compar-ison. As can be seen, the radial stress becomes lessnegative, at almost all values of r, when each ofthese changes in parameters is invoked (Figure 4A).Thus, a difference in cavity pressure on the order of15-20% can result, for example, from inclusion orneglect of known amounts18 of transmural shearingstrains. Moreover, neglect of an external pressureof 0.98 kPa has a large effect on the magnitude of t^at all r values, even though the transmural pressure(P|—Po) varies only minimally.

The transmural shearing stresses, trt and tra, areidentically zero if to'=0 or w'=0, and are signifi-cantly influenced (on the order of 25%) by thepresence or absence of w' and a/, respectively(Figures 4B and 4C). The presence or absence of a1.7°/cm twist does not significantly affect the distri-bution of these components of stress, however.Finally, as expected (Equation A-12), alteration ofmuscle fiber orientations or external pressure doesnot influence the transmural shearing stresses.

The magnitude of the circumferential stress, t^,decreases at almost all values of r when the twistand transmural shearing strains are neglected (Fig-ure 5A). Conversely, neglect of Po increases thiscomponent of stress at all values of r (Figure 5D).Marked differences in both the magnitude and char-acter of the stress distribution arise due to linearversus cubic muscle fiber distributions (Figure 5D);that is, the peak value of stress both diminishes andmoves toward the midwall by assumption of a linearmuscle fiber distribution.11-12 Results for the axial(t^) and in-plane shearing (t8z) stresses are similar,except that Po does not affect t9z (Figures 5B, 5C,5E, and 5F). Nonetheless, note the dramatic differ-ences (some on the order of 50%) in t^ induced byinclusion or neglect of Po, w', and a/.

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1.9 2.1 2.3 2.5

DEFORMED RADIUS (cm)

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FIGURE 3. Representative transmuralCauchy stresses (kPa) are plotted versusdeformed radius, and are based on thefollowing values of the parameters: R-,=l. 5cm, RO=2.55 cm, r,=1.77cm, t//—L7"/cm,ay'=0.1 rad/cm at R,, A=1.015, w'=0.2 atRlt Qo=-90°, c=0.U5 kPa, b=9.665,A=0.082kPa, ^=61.52, Po=0.98kPa, and

varies cubicalty (Equation 2).



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- 4EN DO MID

DEFORMED RADIUS

FIGURE 4. Calculated transmural distributions of trn t^, and tn using parameters noted in Figure 2, except that weseparately prescribe either a linear muscle fiber distribution (dotted line), no twist (ip=0, long dashed line), no RG shear(<o'=0, dash-dot line), no RZ shear (w'—O, short dashed line), or no external pressure (Po—0, labeled solid line in panelA). For purposes of comparison, the appropriate component of stress from Figure 3 is shown in each panel as anunmarked solid line. Symbols associated with each curve denote the quantity that has been neglected or changed (eg.,<t>, dotted line, denotes the curve based on a linear, instead of cubic, muscle fiber distribution). Curves are denotedsimilarly in each panel.

Finally, the stretch ratio in the direction ofthe muscle fiber (Equation A-7) is illustratedin Figure 6. The effects of twist and the distri-bution of muscle fibers are pronounced. Al-

though the displacements, <o(R) and w(R), donot affect a, (see also Equation A-9), theyclearly affect the stresses, as seen in Fig-ures 3-5.

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FIGURE 5. Calculated transmural distributions oft,,, t)z, and tzz. Format is the same as in Figure 4 except that thecurves are denoted differently in panels A, B, and C (tp=O is long dashed line, w'=0 is short dashed line, and w'=0 isdotted line) and panels D, E, and F (no external pressure is dotted line, linear fiber angle is long dashed line).



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FIGURE 6. Calculated stretch ratio, a, with same formatas in Figure 4.

DiscussionImplications of the Results

Armed with the present or easily generated addi-tional numerical calculations, it is tempting to sug-gest physiological implications. Although this couldbe done here, we prefer to resist this temptation andsimply note the following: The deformation [i.e.,r/R, i/f, w(R), w(R), and A] can influence the stressesin a highly nonlinear fashion. Similarly, differentfunctional forms of the constitutive relations, dif-fering values of the material parameters and bound-ary conditions, and various descriptions of musclefiber orientations can markedly change the pre-dicted stresses. Thus, it is difficult, even with thepresent results, to discern the effects of these param-eters categorically. Perhaps most importantly, how-ever, we find that even slight variations in manydifferent parameters can significantly alter not onlythe magnitudes but also the character of the trans-mural distributions of stress and strain. For exam-ple, it may not seem necessary, or feasible, toexperimentally determine the "diastolic twist" inthe heart more accurately than ±17cm. As shownin Figure 7, however, the circumferential stress,based on the aforementioned representative param-eters with i/f= 1.07cm or </>= 3.07cm, can differ sig-

nificantly (-30%) over this small range of twist. Inlight of these observations, we emphasize that infer-ences of the distribution of stress in an actual leftventricle must be made judiciously, and be based onaccurate experimental data.

Need for Additional DataRegional variations in myocardial properties have

been suggested based on limited data13 and ontheoretical arguments,35-39-40 but have not been ade-quately investigated. Since it is easy to show thattransmural variations in material properties cansignificantly affect predicted distributions of stress,there is an obvious need to obtain regional data.Similarly, as pointed out by Fung,41 it is likely thatresidual stresses exist in the heart. Residual stressesare those stresses that remain after removal of allexternal loads. Since residual stresses can signifi-cantly influence the actual distribution of stress inorgans,41 experimental and theoretical characteriza-tions of the residual stress phenomena must comeforth before we can quantify the true state of stressin the heart.

Accurate experimental data on the deformationsexperienced by the heart are fundamental to theperformance and validation of stress analyses.Hence, reliable in vivo and in situ strain data are ofparamount importance. We employed the data ofWaldman and colleagues,18 which appear to be thebest data presently available, although likely incom-plete. Due to the potentially large radial gradients instrain, particularly in the subendocardium, it isimperative that the appropriate error analyses beperformed in the future to ensure confidence in thistype of data. Furthermore, the possible presence ofresidual strains must be explored and documented,and the most appropriate state to which the strainsshould be referred must be identified.

Finally, a nonzero external pressure can signifi-cantly affect the normal stresses. Since the pericar-dia (visceral and parietal) may contribute substan-tially to a nonzero value of Po on the free wall, themechanics of the pericardia must be better under-stood. Moreover, the influence of other thoracic

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DEFORMED RADIUS

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FIGURE 7. Calculated values of f« based on repre-sentative parameters used in Figures 2 and 3, exceptthat ip=r/cm or ifr^

2.8



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structures (e.g., great vessel attachments) like thatof hilar support on the distribution of stress in thelungs should be investigated.42

Utility of Present AnalysisIn spite of the lack of necessary data, theory and

experiment must continue to progress in harmony.Our current analysis extends previous attempts tomodel the heart with a cylindrical geometry in thatwe present a general solution valid for any consti-tutive description of myocardium in terms of atransversely isotropic pseudostrain-energy function,W(Il5a), including arbitrary distributions of musclefiber angles that lie in constant radial surfaces. Ratherthan restricting the myocardial properties to be homo-geneous, we can easily include radially dependentmaterial parameters. We also consider a generaldeformation that is consistent with experimentalobservations that all six components of the Greenstrain are significant in the canine heart (i.e., we donot neglect transmural shearing strains as in prioranalyses). Moreover, we provide an approachwherein the deformation parameters can be deter-mined directly from experimental strain data andsatisfaction of the equilibrium equations. This is incontrast with previous approaches, wherein an incom-plete set of deformation parameters is inferred fromassumed loading conditions.8-11>12 Until the issues ofresidual stress and the coupling between systolic anddiastolic properties are clarified, these assumed load-ing conditions will remain questionable.

We also hope that the present analysis, andrefinements thereof, can be used by experimental-ists to guide their data acquisition. For example, ifone desires to predict stresses to within a 10%accuracy given experimental strains, the presentanalysis can be used to examine possible require-ments on the necessary accuracy of the strainmeasurement (e.g., see Figure 7).

Another utility of our analysis is the ease ofinclusion of new information. For example, Chuongand Fung43 suggested that residual strains in arteriescan be quantified by consideration of the following"residual deformation." Similar to Equation A-l inthe Appendix, consider material particles in a stress-free configuration (p,#,£) to go to (R,9,Z) in anunloaded, but residually stressed, configuration,namely

R=R(P), e= (3)

where 0O is an opening angle that can be measuredwhen a longitudinal cut is introduced into an arteryto relieve residual stresses, and A is an axial exten-sion ratio that can be measured by observation ofthe axial shortening or lengthening associated withrelief of the residual stresses.

The developments of Chuong and Fung43 can beeasily incorporated into our analysis. Thus, if Equa-tion 3, or comparable relations, are shown to ade-quately describe the residual strains in the equato-rial region of the heart, the present analysis can be

modified to include residual stresses. The effects ofresidual stress were not considered here, however,due to a lack of sufficient information.

Finally, it is almost certain that once the requisiteexperimental data become available, full three-dimensional finite-element analyses will be neces-sary for the estimation of cardiac stresses. Due tothe inherent nonlinearities in cardiac mechanics,however, convergence to the correct solution byfinite elements cannot be guaranteed, and thus thepredictions must be checked against known solu-tions. Our analysis, albeit limited to cylindricalgeometry, includes many of these nonlinearitiesand, therefore, could be used to check sophisticatedfinite-element predictions. Since our analysis maybe of use to experimentalists and numerical analystsalike, we will gladly supply interested investigatorswith our IBM AT FORTRAN programs.

In summary, we reviewed many assumptionscommonly employed in the analysis of stress in thepassive heart. We concur with Moriarty39 that finitedeformations, the finite thickness of the wall, andnonlinear stress-strain relations must be included inany analysis of regional stresses and strains. Addi-tionally, we suggest that a) until sufficient ultrastruc-tural data are available, the continuum hypothesisremains a useful approximation; b) any myocardialconstitutive relation must describe available multi-axial data (i.e., isotropic and fiber-reinforced fluidmodels are not appropriate); c) experimentally mea-sured transmural shearing strains should beaccounted for; and d) a cylindrical geometry isperhaps the most useful analytically although itrepresents, at best, only select regions within theleft ventricle.

Consequently, we presented an analysis of afinite deformation of a thick-walled cylindrical annu-lus composed of a nonlinear, anisotropic, incom-pressible material. We employed a new pseudoelas-tic constitutive relation derived from experimentalbiaxial data on potassium-arrested myocardium,included a reasonable description of the transmuraldistribution of muscle fibers, and considered finitedeformations including inflation, extension, twist,and transmural shearing. Illustrative predictions oftransmural variations in stress and strain were pre-sented based on available experimental data. Theseresults are not to be viewed as final, however, dueto the lack of complete data.

In contrast with finite-element analyses, our solu-tion has the disadvantage of being restricted to acylindrical geometry; this limitation does not appearto be severe in the equatorial region of the leftventricle, however. Advantages of our solutionover finite elements include the cost- and time-efficient performance of parametric studies, andcertain explicit results. For example, the transmuralshearing stresses must vary inversely with r or r2 asrequired by equilibrium. Finally, our analysis canserve to check finite-element solutions.



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Due to the sensitivity of the stresses to thevarious parameters, it is essential to base analysesof stresses on complete and reliable data. In partic-ular, there is a need to determine the following:regional stress-strain relations for normal and dis-eased myocardium in the passive and active states,the properties and contributions of the pericardia,regional strain data throughout the cardiac cycle,the possible effects of trabeculations on endocardialstress concentrations, residual stresses, inertialeffects, regional boundary conditions, and so forth.Theory and experiment must proceed hand in hand.Until the requisite data are available, regardless ofour computational sophistication, predictions of stressin the heart will remain only speculative and theassociated physiological interpretations suspect.

AppendixKinematics

We consider finite deformations of a thick-walled, incompressible cylindrical annulus includ-ing inflation, extension, twist, and transmural shear-ing. Thus, we consider material particles in anunloaded configuration (R,6,Z) to go to (r,0,z) afterloading, namely

r=r(R), 6=e+ipZ+w{R), z=AZ+w(R) (A-l)

where iff is a twist-per-unit unloaded length, w is aradially dependent portion of the circumferentialdisplacement, A is an extension ratio denoting uni-form changes in axial lengths per unloaded lengths,and w is a radially dependent part of the axialdisplacement. Analyses of similar deformations ofcylindrical tubes are found in Green and Adkins44

and Truesdell and Noll.45

The physical components46 of the deformationgradient, F, are determined from the motion (Equa-tion A-l), and are conveniently written in matrixform as

and repeated indices imply summation, for exam-ple, trC=CMM=Cu+C22+Cj3. In matrix form, B m is

dr/BR ar/Rae dr/dZ

Td0/dR TddfRde Tdd/dZ

dz/dR dz/RdO

r' 0 0

rco' r/R rip

w' 0 A

(A-2)

wherein the prime denotes an ordinary derivativewith respect to the unloaded radius R, as, forexample, r' denotes dr/dR. Incompressibility (detF=l) requires that (r')(Ar/R)=l, which, when inte-grated, yields either the functional dependence of ron R, or the overall incompressibility relation

i^-r.Htf-Rfl/A or r02-ri

2=(R02-R,2)/A (A-3)

Subscripts i and o denote inner and outer surfaces,respectively.

The physical components of the left and rightCauchy-Green deformation tensors (B, C) are,Bmn=FmMFnM and CMN=F;nMFmN where m, n, M, andN=1,2,3. Wherever possible, lowercase and upper-case indices denote quantities associated with theloaded and unloaded configurations, respectively,

(r')2 r'(rw') r'w'

r'(rw') (r/R)2+(r<o')2+(r.//)2

•w' (w')2+(A)2

and CMN is

(r')2+(ra/)2+(W)2 (r/R)ro/ (To>')Tip+Aw'

(r/R)ro)' (r/R)2

(ro)')ri/H-Aw' (r/R)ri/

(A-4)

(ri/f)2+(A)2

(A-5)

Thus, both B and C are symmetric. Waldman et al18

use the Green strain measure, E=(C —1)/2, which iseasily calculated from C.

The first principal strain invariant (I,=trC=trB) is

I,=(r')2+(ra>')2+(r/R)2+(r$2+(w')2+(A)2 (A-6)

Thus, I, depends on all of the deformation parame-ters in Equation A-2 (r, <\>, A, a>, and w'). It will alsoprove useful to calculate the stretch ratio, a, in thedirection of a muscle fiber. Thus, we consider17

«2=N-C-N=CMNNMNN (A-7)

where N is a unit vector in the direction of a musclefiber in the unloaded configuration. For example,we can assume12 that N has components

NA=[N,,N2,N3]=[0, cos<D(R), sin$(R)] (A-8)

where <I>(R) describes the transmural distribution ofmuscle fibers through the heart wall (e.g., Equation2 in the text, although other descriptions may bephysiologically more realistic). Regardless, a2 is

o2={(r/R)2 cos2*(R)-l-2(r/R)r^ sind>(R) cos#(R)

+ [(r,A)2+(A)2] sin2*(R)} (A-9)

Thus, the stretch ratio in the direction of a musclefiber is independent of w(R) and (o(R), but dependson <£>(R), r, t/r, and A.

Constitutive Relation

Calculation of stresses from the deformationrequires knowledge of a constitutive (stress-strain)relation for the tissue. For the present purposes, weconsider noncontracting myocardium to be trans-versely isotropic with respect to the directions ofthe muscle fibers17-4748; that is, we assume thatmyocardium responds equally to loads applied inany direction excluding the muscle fiber directionwherein a different force-deformation behaviorexists. A general form of a transversely isotropicpseudostrain-energy function for myocardium canbe written as a function of I, and a17

W=W(I,,a) (A-10)



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A specific form of W(Iwa) is provided in the text(Equation 1). The Cauchy stress (force/current area)for a material described by Equation A-10 is givenby17

t=-pI+2WlB + (Wa/a)F-N<g)N-FT (A-ll)

where t is the Cauchy stress, p is a Lagrangemultiplier enforcing incompressibility, I is the iden-tity tensor, Wi = dW/dI,, Wa= <3W/da, ® denotestensor product, and the superscript T denotes trans-pose. Thus, from Equations A-2, A-4, A-10, andA-ll, the nonzero physical components46 of theCauchy stress are

ttr=-p(r)+2W1(r')2 (A-12a)

tM=-p(r)+2W1[(r/R)2+(r.A)2+(ra/)2] +(WâHa ' -Aîn^R)] (A-12b)

t2Z=-p(r)+2W,[(w')2+(A)2] +

(Wa/a)[A2sin2<&(R)] (A-12c)

trfi=2W,(rW) (A-12d)

tn=2W,(r'w') (A-12e)

{[sin<I)(R)Ar/R][Rîn$(R)+cos<I)(R)]} (A-12f)

where t=tT; 4>(R) can be any transmural distributionof muscle fibers, and similarly W, and Wa can becalculated for any constitutive relation expressedby W(li,a). For example, for Equation 1 in the text

W,=6ceb(I'-3) and Wa=2a(a-l)Aea(<r"l)2

Since Wt and Wa may depend on Il3 and since Ijdepends on a>(R) and w(R), each of the componentsof stress can depend on w and w, components of thedeformation previously neglected in cylindrical mod-els of the heart. Finally, the components of stressand strain are a function of only one spatial vari-able, r, thereby keeping the analysis tractable.

Stress TransformationsThe component of Cauchy stress in the direction

of the muscle fibers is easily determined from theusual transformation relations (tmn*=QmiQnjtij, whereQral are the directions cosines for the transformationand the * denotes a transformed quantity). Thus,the stress in the direction of a muscle fiber is

(A-12g)

wherein the deformed muscle fiber angle (<f>) is

with n=F-N/a and n,=[0, cos^(r), sin^r)].4* Thus,the stress in the direction of a muscle fiber dependsonly on three components of the Cauchy stress:circumferential, axial, and in-plane shear. Finally,remember that the stress in the muscle fiber direc-tion is a continuum quantity reflecting locally aver-

aged loads borne by muscle, collagen, and otherconstituents.

Equilibrium and Boundary ConditionsIn the absence of body forces, the equilibrium

equations (div t=0) are44-46

dtn/dr+(trT-tM)/r=O (A-14a)

dtl9/dr+2tr9/r=0 (A-14b)

dtJdr+tn/r=0 (A-14c)

which arc easily integrated, uncoupled ordinarydifferential equations. For example, Equation A-14b can be written as

[d(r2trt)/dr]/r2=0 (A-15a)

from which we obtain

1^,=constant=H (A-15b)

Similarly, Equation A-14c can be written as

[d(rtrz)/dr]/r=0 (A-16a)

and thus

rtI7=constant=G (A-16b)

Equilibrium requires, therefore, that t^ and tra areinversely proportional to r2 and r, respectively, bothindependent of the particular form of W(I,,a).

If we assume that the inner and outer surfaces ofthe cylinder experience uniform pressures (Figure1C), then the radial stress boundary conditions are

a r , ) = - P , and t^ro)"-?o (A-17)

Integration of Equation A-14a, with the aid ofEquation A-17, yields the expression for the "trans-mural pressure"

P,-Po= f {• - -}drJr.

(A-18)

or the Lagrange multiplier in Equations A-ll andA-12

r

p(r)=2W1(r')2+Pi-jr|{- • -}dr (A-19)

where the integrands are

{. . .}={2W1[(r/R)2+(r(/02+(ra/)2-(r')2] +

(Wa/a)[a2-A2sin2*(R)]}/r (A-20)

These equations are easily integrated numericallyvia a Romberg method.49 The correctness of ournumerical algorithms was verified by comparison ofthe solution of a finite inflation of a thick-walledMooney-Rivlin cylinder46 with a finite-element solu-tion by a commercial program (ABAQUS).

If one wishes to calculate the externally appliedloads necessary to maintain this deformation (Equa-tion A-l), then one might consider, for example, theaxial load (L) and moment (M) equations44-45

PoTrro2 (A-21a)



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and

(r t>dr (A-21b)

Equations A-21a and A-21b were used in threestudies8-1' •'2 to infer values of the deformation param-eters (A, i/») by the assumption that L=0 and M=0were balanced by the appropriate integrals overpassive stresses. If L and M are actually zero,however, it is likely that they will be balanced bythe total ^ and t,8 stresses, including contributionsfrom residual, passive, and perhaps active (associ-ated with diastolic tone) stresses. Thus, unless thetotal state of stress is used, deformation parametersinferred from assumed gross balance relations aresuspect. Since the total stress state is not known atpresent, we prefer to calculate the values of thedeformation parameters directly from experimentalstrain data.

Method of SolutionComponents of the Green strain, E, arc obtained

from Equation A-4 and areERR=[(r')2+(ra/)2+(w')2-l]/2 (A-22a)

Eee=[(r/R)2-l]/2 (A-22b)

Ezz=[(r^)2+(A)2-l]/2 (A-22c)

ERe=[(ro/)(r/R)]/2 (A-22d)

Eez=[(r/R)(r,/0]/2 (A-22f)

Thus, if the components of the Green strain areknown at a single radial location in the wall of theheart, say R, then we can calculate the deformationparameters (r, (/», A, co', and w') in Equation A-2directly. That is, from Equations A-22b, A-22f,A-22c, A-22d, and A-22e, respectively, we have

r=RV(l+2Eee) (A-23a)

A=V(l+2Ezz-(r^)2)

o>'=(R/r2)(2ERe)

(A-23b)

(A-23c)

(A-23d)

(A-23e)

Equation A-22a can be used to check the assump-tion of incompressibility.

Since the values of ip and A in Equation A-l donot vary with radial location, Equations A-23b andA-23c yield values that are valid at all values of R.Similarly, r' is easily evaluated at any radial loca-tion by use of the incompressibility constraint r' = (R/rA). Conversely, co' and w' vary with radial loca-tion, and although they are known at one point inthe wall via Equations A-23d and A-23e, they areyet undetermined functions of radius. We suggestone possible approach to determine w'(R) and w'(R).

Once we know the deformation parameters at asingle radial location (Equations A-23a through A-

23e), we can then determine the integration con-stants, G and H, from Equations A-12, A-15, andA-16 as

and

= [2W,(r'w'r)]

= [2W,(r'w'r3)]

(A-24a)

(A-24b)

where | R denotes "evaluated at" the R valuecorresponding to the strains used in Equation A-24.With G and H known, Equations A-24a and A-24bcan be used to solve for all values of co' and w'.

Although a is independent of w' and co', I\ dependsnonlinearly on both w' and co'. Thus, unless W! isindependent of I, (e.g., NeoHookean rubber whereW=c[I)-3] and c is a material parameter), EquationA-24 represents two coupled nonlinear algebraicequations for w' and co' (when both are nonidenti-cally equal to zero). These equations are decoupledand solved easily, however, by consideration of thefollowing ratio of transmural shearing stresses

trAz=(r«/)/(w')=(H/Gr) (A-25)

which is valid for all W(Ii,a). We find to' in terms ofw', therefore, as

( f\ /TT//^ \ / /A ^Z"\

r&i ) = (H/Gr)w (A-26)Substituting (H/Gr)w' into I, (Equation A-6) for TO>'and letting x denote w', we obtain

I1=(r')2+(r/R)2+(r^)2+(A)2+[l+(H/Gr)2]x2 (A-27)Thus, Equation A-24a becomes an uncoupled non-linear algebraic equation in terms of w' (or x), andcan be written as a function of x

f(x)=2W,(rr')x-G=0 (A-28a)

Moreover,

dtfdx=2(rr')[W,-l-x(dW1/dx)] (A-28b)

and, thus, we can determine x (or w') via a straight-forward root-finding algorithm. We chose theNewton-Raphson method49 to determine values of xby iteratively improving on an initial guess for x(say x,), namely

x,+1=x,-[f(x,)]/[df(x,)/dx] (A-28c)

where xn is the previous value (77=1 is the initialguess) and xn+1 is the current value (77=1,2,. . .).Iterations continue until the current value of x^+1converges to within the desired accuracy. ThisNewton-Raphson method works well when reason-able initial "guesses" for x are available. Here, wecan suggest judicious initial guesses for x since weknow one value of x from the strain data (EquationA-23e). Note, too, that for Equation 1 in the text

dW,/dx=2xb[l+(H/Gr)2]W,

After w' is known at each radial location, wedetermine co' from Equation A-26. If either w' or co'is identically zero at all radial locations, however,then either G or H is zero, Equations A-24a and



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A-24b become uncoupled, and the nontrivial equa-tion can be solved directly via a Newton-Raphsonmethod.

Given w' and a>' at all values of r, one can thensolve Equations A-18 and A-19 and thus computethe stresses and strains at any radial location.Hence, this approach yields "analytical" solutions,except for the numerical integrations and root findingfor w' and a>'. Finally, w(R) and «(R) are difficult todetermine directly since the necessary equationsare coupled and nonlinear, but can be reconstructedfrom knowledge of w'(R) and a)'(R). Only the deriv-atives of w(R) and OJ(R) are needed for calculationof stress and strain, however, and there is no needto reconstruct the actual displacements.

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J D Humphrey and F C Yinthe left ventricle.

Constitutive relations and finite deformations of passive cardiac tissue II: stress analysis in

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