Resting Myocardial Properties

Since, by the Frank-Starling mechanism, end-diastolic volume directly affects systolic ventricular work, the mechanics of resting myocardium also have fundamental physiological significance. Most biome-chanical studies of passive myocardial properties have been conducted in isolated, arrested whole heart or tissue preparations. Passive cardiac muscle exhibits most of the mechanical properties characteristic of soft tissues in general [Fung, 1993]. In cyclic uniaxial loading and unloading, the stress-strain relationship is nonlinear with small but significant hysteresis. Depending on the preparation used, resting cardiac muscle typically requires from 2 to 10 repeated loading cycles to achieve a reproducible (preconditioned) response. Intact cardiac muscle experiences finite deformations during the normal cardiac cycle, with maximum Lagrangian strains (which are generally radial and endocardial) that may easily exceed 0.5 in magnitude. Hence, the classical linear theory of elasticity is quite inappropriate for resting myocardial mechanics. The hysteresis of the tissue is consistent with a viscoelastic response, which is undoubtedly related to the substantial water content of the myocardium (about 80% by mass). Changes in water content, such as edema, can cause substantial alterations in the passive stiffness and viscoelastic properties of myocardium. The viscoelasticity of passive cardiac muscle has been characterized in creep and relaxation studies of papillary muscle from cat and rabbit. In both species, the tensile stress in response to a step in strain relaxes 30 to 40% in the first 10 sec [Pinto and Patitucci, 1977; Pinto and Patitucci, 1980]. The relaxation curves exhibit a short exponential time constant (<0.02 sec) and a long one (about 1000 sec), and are largely independent of the strain magnitude, which supports the approximation that myocardial viscoelasticity is quasilinear. Myocardial creep under isotonic loading is 2 to 3% of the original length after 100 sec of isotonic loading and is also quasilinear with an exponential timecourse. There is also evidence that passive ventricular muscle exhibits other anelastic properties such as maximum strain-dependent "strain softening" [Emery et al., 1997a,b], a well-known property in elastomers first described by Mullins [1947].

Since the hysteresis of passive cardiac muscle is small and only weakly affected by changes in strain rate, the assumption of pseudoelasticity [Fung, 1993] is often appropriate. That is, the resting myocardium is considered to be a finite elastic material with different elastic properties in loading vs. unloading. Although various preparations have been used to study resting myocardial elasticity, the most detailed and complete information has come from biaxial and multiaxial tests of isolated sheets of cardiac tissue, mainly from the dog [Demer and Yin, 1983; Halperin et al., 1987; Humphrey et al., 1990]. These experiments have shown that the arrested myocardium exhibits significant anisotropy with substantially greater stiffness in the muscle fiber direction than transversely. In equibiaxial tests of muscle sheets cut from planes parallel to the ventricular wall, fiber stress was greater than the transverse stress (Fig. 11.7) by an average factor of close to 2.0 [Yin et al., 1987]. Moreover, as suggested by the structural organization of the myocardium described in Section 11.2, there may be also be significant anisotropy in the plane of the tissue transverse to the fiber axis.

The biaxial stress-strain properties of passive myocardium display some heterogeneity. Novak et al. [1994] measured regional variations of biaxial mechanics in the canine left ventricle. Specimens from the inner and outer thirds of the left ventricular free wall were stiffer than those from the midwall and

FIGURE 11.7 Representative stress-strain curves for passive rat myocardium computed using Eqs. (11.17) and (11.19). Fiber and crossfiber stress are shown for equibiaxial strain. (Courtesy of Dr. Jeffrey Omens.)

interventricular septum, but the degree of anisotropy was similar in each region. Significant species variations in myocardial stiffness have also been described. Using measurements of two-dimensional regional strain during left ventricular inflation in the isolated whole heart, a parameter optimization approach showed that canine cardiac tissue was several times stiffer than that of the rat, though the nonlinearity and anisotropy were similar [Omens et al., 1993]. Biaxial testing of the collagenous parietal pericardium and epicardium have shown that these tissues have distinctly different properties than the myocardium, being very compliant and isotropic at low biaxial strains (<0.1 to 0.15), but rapidly becoming very stiff and anisotropic as the strain is increased [Lee et al., 1987; Humphrey et al., 1990].

Various constitutive models have been proposed for the elasticity of passive cardiac tissues. Because of the large deformations and nonlinearity of these materials, the most useful framework has been provided by the pseudostrain-energy formulation for hyperelasticity. For a detailed review of the material properties of passive myocardium and approaches to constitutive modeling, see Chapters 1-6 of Glass et al. [1991]. In hyperelasticity, the components of the stress1 are obtained from the strain energy W as a function of the Lagrangian (Green's) strain ERS.

The myocardium is generally assumed to be an incompressible material, which is a good approximation in the isolated tissue, although in the intact heart significant redistribution of tissue volume is sometimes associated with phasic changes in regional coronary blood volume. Incompressibility is included as a kinematic constraint in the finite elasticity analysis, which introduces a new pressure variable that is added as a Lagrange multiplier in the strain energy. The examples that follow are various strain-energy functions, with representative parameter values (for W in kPa, i.e., mJ-ml-1), that have been suggested for cardiac tissues. For the two-dimensional properties of canine myocardium, Yin and colleagues [1987] obtained reasonable fits to experimental data with an exponential function,

35 Eji2 +20 E22

1In a hyperelastic material, the second Piola Kirchhoff stress tensor is given by P = —

RS 2

where En is the fiber strain and E22 is the crossfiber in-plane strain. Humphrey and Yin [1987] proposed a three-dimensional form for W as the sum of an isotropic exponential function of the first principal invariant I1 of the right Cauchy-Green deformation tensor and another exponential function of the fiber stretch ratio :

The isotropic part of this expression has also been used to model the myocardium of the embryonic chick heart during the ventricular looping stages, with coefficients of 0.02 kPa during diastole and 0.78 kPa at end-systole, and exponent parameters of 1.1 and 0.85, respectively [Lin and Taber, 1994]. Another, related transversely isotropic strain-energy function was used by Guccione et al. [1991] and Omens et al. [1991] to model material properties in the isolated mature rat and dog hearts:

where, in the dog,

Q = 26.7e121 + 2.°(e222 + 4 + e223 + 4 )+14.7 (4 + 4 + 4 + 4) (H.17)

In Eqs. (11.17) and (11.18), normal and shear strain components involving the radial (X3) axis are included. Humphrey and colleagues [1990] determined a new polynomial form directly from biaxial tests. Novak et al. [1994] gave representative coefficients for canine myocardium from three layers of the left ventricular free wall. The outer third follows as:

W = 4.8(4 -1)2 + 3.4(4 -1)3 + °.77(i! - 3) - 6.1(lt - 3)F -1) + 6.24 - 3) (11.19) The midwall region follows as:

W = 5.3(Af -1)2 + 7.5(4 -1)3 + 0.43(l! - 3)- 7.7(lj - 3)P -1) + 5.6(l - 3) (11.20) The inner layer of the wall follows as

W = 0.51(4 -1)2 + 27.6(4 -1)3 + 0.74(l! - 3) - 7.3(lj - 3) -1) + 7.°4 - 3) (11.21)

A power law strain-energy function expressed in terms of circumferential, longitudinal, and transmural extension ratios (4 4, and 4 was used [Gupta et al., 1994] to describe the biaxial properties of sheep myocardium 2 weeks after experimental myocardial infarction, in the scarred infarct region:

32 30 31

and in the remote, non-infarcted tissue:

22 26 24

Finally, based on the observation that resting stiffness rises steeply at strains that extend coiled collagen fibers to the limit of uncoiling, Hunter and colleagues have proposed a pole-zero constitutive relation in which the stresses rise asymptotically as the strain approaches a limiting elastic strain [Hunter et al., 1998].

The strain in the constitutive equation, must generally be referred to the stress-free state of the tissue. However, the unloaded state of the passive left ventricle is not stress free; residual stress exists in the intact, unloaded myocardium, as shown by Omens and Fung [1990]. Cross-sectional equatorial rings from potassium-arrested rat hearts spring open elastically when the left ventricular wall is resected radially. The average opening angle of the resulting curved arc is 45 ± 10° in the rat. Subsequent radial cuts produce no further change. Hence, a slice with one radial cut is considered to be stress free, and there is a nonuniform distribution of residual strain across the intact wall, being compressive at the endocardium and tensile at the epicardium, with some regional differences. Stress analyses of the diastolic left ventricle show that residual stress acts to minimize the endocardial stress concentrations that would otherwise be associated with diastolic loading [Guccione et al., 1991]. An important physiological consequence of residual stress is that sarcomere length is nonuniform in the unloaded resting heart. Rodriguez et al. [1993] showed that sarcomere length is about 0.13 |im greater at epicardium than endocardium in the unloaded rat heart, and this gradient vanishes when residual stress is relieved. Three-dimensional studies have also revealed the presence of substantial transverse residual shear strains [Costa et al., 1997]. Residual stress and strain may have an important relationship to cardiac growth and remodeling. Theoretical studies have shown that residual stress in tissues can arise from growth fields that are kinematically incompatible [Skalak et al., 1982; Rodriguez et al., 1994].

Although ventricular pressures and volumes are valuable for assessing the global pumping performance of the heart, myocardial stress and strain distributions are need to characterize regional ventricular function, especially in pathological conditions, such as myocardial ischemia and infarction, where profound localized changes may occur. The measurement of stress in the intact myocardium involves resolving the local forces acting on defined planes in the heart wall. Attempts to measure local forces [Feigl et al., 1967; Huisman et al., 1980] have had limited success because of the large deformations of the myocardium and the uncertain nature of the mechanical coupling between the transducer elements and the tissue. Efforts to measure intramyocardial pressures using miniature implanted transducers have been more successful but have also raised controversy over the extent to which they accurately represent changes in interstitial fluid pressure. In all cases, these methods provide an incomplete description of three-dimensional wall stress distributions. Therefore, the most common approach for estimating myocardial stress distributions is the use of mathematical models based on the laws of continuum mechanics [Hunter and Smaill, 1989]. Although there is not room to review these analyses here, the important elements of such models are the geometry and structure, boundary conditions and material properties, described in the foregoing sections. An excellent review of ventricular wall stress analysis is given by Yin [1981]. The most versatile and powerful method for ventricular stress analysis is the finite element method, which has been used in cardiac mechanics for over 20 years [Yin, 1985]. However, models must also be validated with experimental measurements. Since the measurement of myocardial stresses is not yet reliable, the best experimental data for model validation are measurements of strains in the ventricular wall.

The earliest myocardial strain gauges were mercury-in-rubber transducers sutured to the epicardium. Today, local segment length changes are routinely measured with various forms of the piezoelectric crystal

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