In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

https://hal.archives-ouvertes.fr/hal-01297725/document

A new constitutive framework for arterial wall mechanics and a comparative study of material models

http://biomechanics.stanford.edu/me338/me338_project04.pdf

Stress-dependent finite growth in soft elastic tissues

It is now becoming apparent that dynamic changes occur within the interstitium that directly contribute to adverse myocardial remodeling following myocardial infarction (MI), with hypertensive hear...

https://www.physiology.org/doi/pdf/10.1152/physrev.00012.2007

Myocardial Matrix Remodeling and the Matrix Metalloproteinases: Influence on Cardiac Form and Function

The regional function of the left ventricle can be visualized in real-time using the new strain rate imaging method. Deformation or strain of a tissue segment occurs over time during the cardiac cycle. The rate of this deformation, the strain rate, is equivalent to the velocity gradient, and can be estimated using the tissue Doppler technique. We present the strain rate as color-coded 2-dimensional cine-loops and color M-modes showing the strain rate component along the ultrasound beam axis. We tested the method in 6 healthy subjects and 6 patients with myocardial infarction. In the healthy hearts, a spatially homogeneous distribution of the strain rate was found. In the infarcted hearts, all the infarcted areas in this study showed up as hypokinetic or akinetic, demonstrating that this method may be used for imaging of regional dysfunction. Shortcomings of the method are discussed, as are some possible future applications of the method. (J Am Soc Echocardiogr 1998;11:1013-9.)

Real-time strain rate imaging of the left ventricle by ultrasound

A method for direct therapeutic treatment of myocardial tissue in a localized region of a heart having a pathological condition. The method includes identifying a target region of the myocardium and applying material directly and substantially only to at least a portion of the myocardial tissue of the target region. The material applied results in a physically modification the mechanical properties, including stiffness, of said tissue. Various devices and modes of practicing the method are disclosed for stiffening, restraining and constraining myocardial tissue for the treatment of conditions including myocardial infarction or mitral valve regurgitation.

Prevention of myocardial infarction induced ventricular expansion and remodeling

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

Passive material properties of intact ventricular myocardium determined from a cylindrical model.

To examine transmural finite deformation in the wall of the canine left ventricle, closely spaced columns of lead beads were implanted at a single site on the left ventricular free wall. The three-dimensional coordinates of these myocardial markers were obtained with high-speed biplane cineradiography. Any four noncoplanar markers forming small tetrahedral volumes (less than or equal to 0.1 cc) were used to calculate finite normal and shear strains with respect to a cardiac coordinate system at end diastole. Due to the symmetry of the finite strain tensor, the algebraic eigenvalue problem could be solved to compute principal strains and the directions of the principal axes of deformation with respect to the reference coordinates. An examination of the principal strains in a number of tetrahedra in five animals indicates that deformation increases with depth beneath the epicardium. For example, the transmural variation of principal shortening strain averages -0.014 +/- 0.009 per 10% increment in thickness from epicardium to endocardium. Furthermore, shortening and thickening strains at midwall and deeper are too large (0.10 to 0.40) to be described accurately by infinitesimal theory. These strains are often accompanied by substantial in-plane and transverse shears which are not predicted by typical membrane or shell theories, indicating that these theories must be applied with caution when computing indices of regional ventricular performance. The directions of the principal axes of shortening vary substantially less than the fiber direction varies across the wall (20 degrees - 40 degrees compared with 100 degrees - 140 degrees for fiber direction), supporting the concept that there are substantial interactions between neighboring fibers in the left ventricular wall.

Transmural myocardial deformation in the canine left ventricle. Normal in vivo three-dimensional finite strains.

To determine the relation between local myofiber anatomy and local deformation in the wall of the left ventricle, both three-dimensional transmural deformation and myofiber orientation were examined in the anterior free wall of seven canine left ventricles. Deformation was measured by imaging columns of implanted radiopaque markers with high-speed, biplane cineradiography (16 mm, 120 frames/sec). Hearts were fixed at end diastole and sectioned parallel to the local epicardial tangent plane to determine the transmural distribution of fiber directions at the site of strain measurement. The principal direction of deformation associated with the greatest shortening was compared with the local fiber direction in the outer (21 +/- 8% of the wall thickness from the epicardium) and inner (65 +/- 9%) halves of the wall. Although the fiber direction varied substantially with depth from the epicardium, the principal direction did not. In the outer half of the wall, fiber direction averaged -8 +/- 24 degrees, while the principal direction averaged -33 +/- 24 degrees from circumferential (counterclockwise angles are positive). In the inner half, fiber direction averaged 69 +/- 10 degrees, while the principal direction averaged -22 +/- 21 degrees. Therefore, while fiber and principal directions were not substantially different in the outer half, the greatest shortening occurred orthogonally to the fiber direction in the inner half. Normal and shear strains measured in a cardiac coordinate system (circumferential, longitudinal, and radial coordinates) were rotated (transformed) to "fiber" coordinates in both halves of the wall. In the outer half, normal strains observed in the fiber (-0.09 +/- 0.04) and cross-fiber (-0.04 +/- 0.04) directions were not significantly different (paired t test, p less than 0.05). In the inner half, more than twice as much strain occurred in the cross-fiber (-0.17 +/- 0.03) than in the fiber direction (-0.06 +/- 0.06). Moreover, the only shear strain that remained substantial after transformation was transverse shear in the plane of the fiber and radial coordinates. These results suggest that both reorientation and cross-sectional shape changes of myofibers or the interstitium may contribute to the large wall thickenings observed during contraction, particularly in the inner half of the ventricular wall.

Relation between transmural deformation and local myofiber direction in canine left ventricle.

Models of contracting ventricular myocardium were used to study the effects of different assumptions concerning active tension development on the distributions of stress and strain in the equatorial region of the intact left ventricle during systole. Three models of cardiac muscle contraction were incorporated in a cylindrical model for passive left ventricular mechanics developed previously [Guccione et al. ASME Journal of Biomechanical Engineering, Vol. 113, pp. 42-55 (1991)]. Systolic sarcomere length and fiber stresses predicted by a general "deactivation" model of cardiac contraction [Guccione and McCulloch, ASME Journal of Biomechanical Engineering, Vol. 115, pp. 72-81 (1993)] were compared with those computed using two less complex models of active fiber stress: In a time-varying "elastance" model, isometric tension development was computed from a function of peak intracellular calcium concentration, time after contraction onset and sarcomere length; a "Hill" model was formulated by scaling this isometric tension using the force-velocity relation derived from the deactivation model. For the same calcium ion concentration, the sarcomeres in the deactivation model shortened approximately 0.1 microns less throughout the wall at end-systole than those in the other models. Thus, muscle fibers in the intact ventricle are subjected to rapid length changes that cause deactivation during the ejection phase of a normal cardiac cycle. The deactivation model predicted rather uniform transmural profiles of fiber stress and cross-fiber stress distributions that were almost identical to those of the radial component. These three components were indistinguishable from the principal stresses. Transmural strain distributions predicted at end-systole by the deactivation model agreed closely with experimental measurements from the anterior free wall of the canine left ventricle.

Mechanics of Active Contraction in Cardiac Muscle: Part II—Cylindrical Models of the Systolic Left Ventricle

A three-dimensional finite element method for nonlinear finite elasticity is presented using prolate spheroidal coordinates. For a thick-walled ellipsoidal model of passive anisotropic left ventricle, a high-order (cubic Hermite) mesh with 3 elements gave accurate continuous stresses and strains, with a 69 percent savings in degrees of freedom (dof) versus a 70-element standard low-order model. A custom mixed-order model offered 55 percent savings in dof and 39 percent savings in solution time compared with the low-order model. A nonsymmetric 3D model of the passive canine LV was solved using 16 high-order elements. Continuous nonhomogeneous stresses and strains were obtained within 1 hour on a laboratory workstation, with an estimated solution time of less than 4 hours to model end-systole. This method represents the first practical opportunity to solve large-scale anatomically detailed models for cardiac stress analysis.

L. K. Waldman

Papers

Passive material properties of intact ventricular myocardium determined from a cylindrical model.

Transmural myocardial deformation in the canine left ventricle. Normal in vivo three-dimensional finite strains.

Relation between transmural deformation and local myofiber direction in canine left ventricle.

Mechanics of Active Contraction in Cardiac Muscle: Part II—Cylindrical Models of the Systolic Left Ventricle

A Three-Dimensional Finite Element Method for Large Elastic Deformations of Ventricular Myocardium: II—Prolate Spheroidal Coordinates