Finite element stress analysis of left ventricular mechanics in the beating dog heart

Finite element modelling and simulations in cardiovascular mechanics and cardiology: A bibliography 1993–2004

The paper gives a bibliographical review of the finite element modelling and simulations in cardiovascular mechanics and cardiology from the theoretical as well as practical points of views. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1993 and 2004. At the end of this paper, more than 890 references are given dealing with subjects as: Cardiovascular soft tissue modelling; material properties; mechanisms of cardiovascular components; blood flow; artificial components; cardiac diseases examination; surgery; and other topics.

MRI-based finite-element analysis of left ventricular aneurysm

Tagged MRI and finite-element (FE) analysis are valuable tools in analyzing cardiac mechanics. To determine systolic material parameters in three-dimensional stress-strain relationships, we used tagged MRI to validate FE models of left ventricular (LV) aneurysm. Five sheep underwent anteroapical myocardial infarction (25% of LV mass) and 22 wk later underwent tagged MRI. Asymmetric FE models of the LV were formed to in vivo geometry from MRI and included aneurysm material properties measured with biaxial stretching, LV pressure measurements, and myofiber helix angles measured with diffusion tensor MRI. Systolic material parameters were determined that enabled FE models to reproduce midwall, systolic myocardial strains from tagged MRI (630 +/- 187 strain comparisons/animal). When contractile stress equal to 40% of the myofiber stress was added transverse to the muscle fiber, myocardial strain agreement improved by 27% between FE model predictions and experimental measurements (RMS error decreased from 0.074 +/- 0.016 to 0.054 +/- 0.011, P < 0.05). In infarct border zone (BZ), end-systolic midwall stress was elevated in both fiber (24.2 +/- 2.7 to 29.9 +/- 2.4 kPa, P < 0.01) and cross-fiber (5.5 +/- 0.7 to 11.7 +/- 1.3 kPa, P = 0.02) directions relative to noninfarct regions. Contrary to previous hypotheses but consistent with biaxial stretching experiments, active cross-fiber stress development is an integral part of LV systole; FE analysis with only uniaxial contracting stress is insufficient. Stress calculations from these validated models show 24% increase in fiber stress and 115% increase in cross-fiber stress at the BZ relative to remote regions, which may contribute to LV remodeling.

The relation between viable segments and left ventricular ejection fraction improvement

For patients with coronary artery disease and left ventricular dysfunction who undergo revascularization, it is important to estimate the left ventricular ejection fraction (LVEF) improvement after revascularization, as this is a strong indicator of the long-term outcome. Identification of viable segments from echocardiography has been considered a predictive sign of LVEF improvement. However, a quantitative relation between segmental function recovery and global ejection fraction improvement has not been established. There is a clinical need to determine parameters that are predictive to LVEF improvement. A cylindrical left ventricular model is proposed to establish the relation between segmental myocardial function and LVEF based on a 12-segment echocardiograph model. Model results show that LVEF improvement is directly related to the contraction ratio in normal segments and a weighted sum of the number of viable segments that recover to normal or hypokinetic, which is equal to a weighted sum of the change in wall motion scores. This new combined parameter is a better predictor of the amount of LVEF improvement than the total number of viable segments or preoperative ejection fraction. The predictive value of the model was illustrated in a group of four patients with coronary artery disease who underwent revascularization.

Akinetic myocardial infarcts must contain contracting myocytes: finite-element model study

Infarcted segments of myocardium demonstrate functional impairment ranging in severity from hypokinesis to dyskinesis. We sought to better define the contributions of passive material properties (stiffness) and active properties (contracting myocytes) to infarct thickening. Using a finite-element (FE) model, we tested the hypothesis that infarcted myocardium must contain contracting myocytes to be akinetic and not dyskinetic. A three-dimensional FE mesh of the left ventricle was developed with echocardiographs from a reperfused ovine anteroapical infarct. The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction, and an elastance model for active fiber stress was incorporated. The diastolic stiffness (C) and systolic material property (isometric tension at longest sarcomere length and peak intracellular calcium concentration, T(max)) of the uninfarcted remote myocardium were assumed to be normal (C = 0.876 kPa, T(max) = 135.7 kPa). Diastolic and systolic properties of the infarct necessary to produce akinesis, defined as an average radial strain between -0.01 and 0.01, were determined by assigning a range of diastolic stiffnesses and scaling infarct T(max) to represent the percentage of contracting myocytes between 0% and 100%. As C was increased to 11 times normal (C = 10 kPa) the percentage of T(max) necessary for akinesis increased from 20% to 50%. Without contracting myocytes, C = 250 kPa was necessary to achieve akinesis. If infarct stiffness is <285 times normal, contracting myocytes are required to prevent dyskinetic infarct wall motion.

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh-Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.

Echocardiographic Assessment of the Right Ventricular Stress-Velocity Relationship Under Normal and Chronic Overload Conditions

The effects of chronic volume or pressure overload on the velocity of right ventricular ejection have not been previously well defined. We hypothesized that, as formerly shown for the left ventricle, there would be a direct relationship between the velocity of ejection and an estimate of systolic wall stress.

METHODS

Echocardiograms of asymptomatic patients, not on cardiac medications, with either an isolated secundum atrial septal defect > or = 5 mm in diameter or isolated pulmonic stenosis with a peak instantaneous pressure gradient > or = 20 mmHg, were reviewed. Forty-one patients with an atrial septal defect and 34 with pulmonary stenosis met criteria, and were compared to age-matched normal controls. Total subjects were 127 with ages ranging from 1 day to 54 years. Right ventricular monoplane ejection fraction, ejection time corrected for heart rate (ETc), mean normalized systolic ejection rate (MNSERc) and meridianal peak-systolic wall stress (WSps) were measured.

RESULTS

Compared to controls, ejection fractions were not significantly different, but WSps averaged 81% and 110% higher, ETc 8% and 9% longer, and MNSERc 5% and 9% slower in the atrial septal defect and pulmonary stenosis groups, respectively. Among all subjects WSps had a significant linear correlation with ETc (r = 0.61, P < 0.01), MNSERc (r =-0.46, P < 0.01), and ejection fraction (r =-0.19, P < 0.05).

CONCLUSIONS

Increases in WSps cause an incremental slowing of MNSERc in the right ventricle, with a relationship that is linear over a wide range of normal and abnormal loading conditions.

Haemodynamics and mechanics following partial left ventriculectomy: a computer modeling analysis

Mechanics following partial left ventriculectomy is still poorly understood. A computational cylindrical model of the left ventricle was developed, based on the myocardial fibre behaviour for the evaluation of the mechanical and haemodynamical effects of the operation. A healthy left ventricle with physiological geometry and function and a dilated hypokinetic heart were investigated. Haemodynamic and mechanical data were obtained at baseline and compared with those obtained at different degrees of volume reduction. Data included: ejection fraction (EF); stroke volume (SV); end-systolic and end-diastolic pressure-volume relationships (ESPVR and EDPVR), and efficiency. EF increases following volume reduction in both simulation but, concurrently, SV shows modest improvement (dilated ventricle) or reduction (healthy ventricle) at progressive degrees of resection. The ESPVR and EDPVR slope increases and shifts leftward with the resection extent, but the increase of the ESPVR slope is more pronounced in dilated ventricle. Efficiency is improved in the dilated heart after resections, while does not improve when the healthy-heart volume is reduced. The simulation of partial left ventriculectomy suggests an improvement of systolic performance, counterbalanced by increased diastolic stiffness following inverse remodelling. Efficiency of simulated dilated ventricles is enhanced by volume reduction, suggesting a favourable effect of reduction of the metabolic demand of the failing heart.

Myosplint decreases wall stress without depressing function in the failing heart: a finite element model study

BACKGROUND

The Myocor Myosplint is a transcavitary tensioning device designed to change left ventricular (LV) shape and reduce wall stress. Regional wall stress cannot be measured in the intact heart and LV function after surgical remodeling is often confounded by inotropic agents and mitral repair. We used a realistic mathematical (finite element) model of the dilated human LV to test the hypothesis that Myosplint decreased regional ventricular fiber stress and improved LV function.

METHODS

A finite element model was used to simulate the effects of Myosplint on the LV stroke volume/end-diastolic pressure (Starling) relationship and regional distributions of stress in the local muscle fiber direction (fiber stress) for a wide range of diastolic and end-systolic material properties. The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction. An elastance model for active fiber stress was incorporated in an axisymmetric geometric model of the globally dilated LV wall.

RESULTS

Both diastolic compliance and end-systolic elastance shifted to the left on the pressure-volume diagram. LV end-diastolic volume and end-systolic volumes were reduced by 7.6% and 8.6%, respectively. Mean end-diastolic and end-systolic fiber stress was decreased by 24% and 16%, respectively. Although the effect of Myosplint on the Starling relationship was not significant, there were trends toward an improvement in this relationship at low diastolic stiffness, C, high peak intracellular calcium concentration, Ca(0), and high arterial elastance, E(A). Of note, the effect of C was twice that of Ca(0) and E(A). Diastolic function would, therefore, be expected to be the prime determinant of success with Myosplint.

CONCLUSIONS

Myosplint reduces fiber stress without a decrement in the Starling relationship. Myosplint should be much more effective than partial ventriculectomy as a surgical therapy for patients with dilated cardiomyopathy and end-stage congestive heart failure.

Intracellular calcium handling heterogeneities in intact guinea pig hearts

Regional heterogeneities of ventricular repolarizing currents and their role in arrhythmogenesis have received much attention; however, relatively little is known regarding heterogeneities of intracellular calcium handling. Because repolarization properties and contractile function are heterogeneous from base to apex of the intact heart, we hypothesize that calcium handling is also heterogeneous from base to apex. To test this hypothesis, we developed a novel ratiometric optical mapping system capable of measuring calcium fluorescence of indo-1 at two separate wavelengths from 256 sites simultaneously. With the use of intact Langendorff-perfused guinea pig hearts, ratiometric calcium transients were recorded under normal conditions and during administration of known inotropic agents. Ratiometric calcium transients were insensitive to changes in excitation light intensity and fluorescence over time. Under control conditions, calcium transient amplitude near the apex was significantly larger (60%, P < 0.01) compared with the base. In contrast, calcium transient duration was significantly longer (7.5%, P < 0.03) near the base compared with the apex. During isoproterenol (0.05 microM) and verapamil (2.5 microM) administration, ratiometric calcium transients accurately reflected changes in contractile function, and, the direction of base-to-apex heterogeneities remained unchanged compared with control. Ratiometric optical mapping techniques can be used to accurately quantify heterogeneities of calcium handling in the intact heart. Significant heterogeneities of calcium release and sequestration exist from base to apex of the intact heart. These heterogeneities are consistent with base-to-apex heterogeneities of contraction observed in the intact heart and may play a role in arrhythmogenesis under abnormal conditions.

Modeling Total Heart Function

Computational models of the electrical and mechanical function of the heart are reviewed. These models attempt to explain the integrated function of the heart in terms of ventricular anatomy, the structure and material properties of myocardial tissue, the membrane ion channels, and calcium handling and myofilament mechanics of cardiac myocytes. The models have established the computational framework for linking the structure and function of cardiac cells and tissue to the integrated behavior of the intact heart, but many more aspects of physiological function, including metabolic and signal transduction pathways, need to be included before significant progress can be made in understanding many disease processes.

A Finite Element Model of Left Ventricular Cellular Transplantation in Dilated Cardiomyopathy

Surgical therapies for heart failure that reduce left ventricular (LV) size have failed to improve LV function. Recent reports describe the direct injection of myocytes into the LV wall and suggest that myocyte transplantation improves regional contractile ability and improves LV function. Using a previously described finite element model, we simulated myocyte transplantation in the failing LV and tested the hypothesis that myocyte transplantation improves LV function (Starling relationship). An elastance model for active fiber stress was incorporated in an axisymmetric geometric model of the dilated, poorly contractile LV (dilated cardiomyopathy [DCM]). The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction. Systolic material properties were depressed by assigning a peak intracellular calcium concentration (Ca2+) of 1.8 micromoL (normal value: 4.2 micromoL). Six different simulations of myocyte transplantation were performed in which transplanted areas were assigned a peak intracellular calcium concentration (Ca2+) of 4.2 micromoL. The pattern of myocyte engraftment was varied (transmural versus subepicardial; confluent versus heterogeneous), as was the amount of the LV free wall that was transplanted (17% vs 33%). Models were created and loaded with a range of physiologic LV pressures. Simulated myocyte transplantation increased the slope of the end-systolic elastance curve, improved the Starling relationship, and improved stroke volume and ejection fraction compared with DCM. This study demonstrated an improvement in LV function after myocyte transplantation.

Optimizing ventricular fibers: uniform strain or stress, but not ATP consumption, leads to high efficiency

The aim of this study was to investigate the influence of fiber orientation in the left ventricular (LV) wall on the ejection fraction, efficiency, and heterogeneity of the distributions of developed fiber stress, strain and ATP consumption. A finite element model of LV mechanics was used with active properties of the cardiac muscle described by the Huxley-type cross-bridge model. The computed variances of sarcomere length (SL(var)), developed stress (DS(var)), and ATP consumption (ATP(var)) have several minima at different transmural courses of helix fiber angle. We identified only one region in the used design space with high ejection fraction, high efficiency of the LV and relatively small SL(var), DS(var), and ATP(var). This region corresponds to the physiological distribution of the helix fiber angle in the LV wall. Transmural fiber angle can be predicted by minimizing SL(var) and DS(var), but not ATP(var). If ATP(var) was minimized, then the transverse fiber angle was considerably underestimated. The results suggest that ATP consumption distribution is not regulating the fiber orientation in the heart.

Wall stress misrepresents afterload in children and young adults with abnormal left ventricular geometry

Wall stress, although commonly used as an index of afterload, fails to take into account forces generated within the wall of the left ventricle (LV) that oppose systolic fiber shortening. Wall stress may, therefore, misrepresent fiber stress, the force resisting fiber shortening, particularly in the presence of an abnormal LV thickness-to-dimension ratio (h/D). M-mode LV echocardiograms were obtained from 207 patients with a wide range of values for LV mass and/or h/D. Diagnoses were valvar aortic stenosis, coarctation repair, anthracycline treated, and severe aortic and/or mitral regurgitation. End-systolic wall stress (WS(es)) and fiber stress (FS(es)) were expressed as age-corrected Z scores relative to a normal population. The difference between WS(es) and FS(es) was extreme when h/D was elevated or reduced [WS(es) Z score - FS(es) Z score = 0.14 x (h/D)(-1.47) - 2.13; r = 0.78, P < 0.001], with WS(es) underestimating FS(es) when h/D was increased and overestimating FS(es) when h/D was decreased. Analyses of myocardial mechanics based on wall stress have limited validity in patients with abnormal ventricular geometry.

Quantitative prediction of improvement in cardiac function after revascularization with MR imaging and modeling: initial results

PURPOSE

To evaluate a model that can be used quantitatively to predict changes in postrevascularization left ventricular function based on classification of myocardial tissue as hibernating, scarred, or normal with cine magnetic resonance (MR) imaging.

MATERIALS AND METHODS

Eleven patients with chronic left ventricular dysfunction were studied before and after revascularization with cine MR imaging. Regional myocardial contractility and wall thickness were used in the model to predict postrevascularization ejection fraction (EF). The actual EF from the postrevascularization MR images was compared with the EF from the prerevascularization images predicted with the model by using regression analysis and Bland-Altman analysis.

RESULTS

Correlation between the actual EF after revascularization and the EF predicted by using the model yielded an R value of 0.98, with a standard error of 1.3 EF percentage points. Predicting changes in function in a myocardial segment was less successful because only 55% of segments classified as hibernating actually improved resting function after revascularization. In nonimproved segments, 78% were either adjacent to infarcted segments or had nontransmural wall thinning.

CONCLUSION

A simple mathematical model combined with functional information provided by MR imaging was used to predict improvements in global EF resulting from revascularization.

A 2D finite element model of the interventricular septum under normal and abnormal loading

The interventricular septum is the structure that separates the left and right ventricles of the heart. Under normal loading conditions, it is concave to the left ventricle, but under abnormal loading the septum flattens and occasionally inverts. In the past, the septum has frequently been modelled as integral to the left ventricle with the effects of pressure from the right ventricle being ignored. Under abnormal loading, the septum has been described as behaving equivalent to a "flapping sail". There has been no consideration of structural behaviour under these conditions. A 2-D plane stress FE model of the septum was used to investigate the difference in structural behaviour of the septum during diastole between normal and abnormal loading. The biaxial stress patterns that develop are distinctively disparate. Under normal loading, the septum behaves much like a thick-walled cylinder subject to internal and external pressure, with the resulting stresses being circumferential tension and radial compression, both varying with radius. These stresses are very low throughout most of diastole. However, under abnormal loading, the septum behaves in an arch-like fashion, with high compressive stresses almost circumferential in direction, combined with radial compression. We conclude that right ventricular pressures cause bending effects in the wall of the heart, and that under abnormal loading, the compressive stresses that develop in the septum may lead to an understanding of certain, previously unexplained, pathological conditions.

Regional septal dysfunction in a three-dimensional computational model of focal myofiber disarray

MLC2v/ras transgenic mice display a phenotype characteristic of hypertrophic cardiomyopathy, with septal hypertrophy and focal myocyte disarray. Experimental measurements of septal wall mechanics in ras transgenic mice have previously shown that regions of myocyte disarray have reduced principal systolic shortening, torsional systolic shear, and sarcomere length. To investigate the mechanisms of this regional dysfunction, a three-dimensional prolate spheroidal finite-element model was used to simulate filling and ejection in the hypertrophied mouse left ventricle with septal disarray. Focally disarrayed septal myocardium was modeled by randomly distributed three-dimensional regions of altered material properties based on measured statistical distributions of muscle fiber angular dispersion. Material properties in disarrayed regions were modeled by decreased systolic anisotropy derived from increased fiber angle dispersion and decreased systolic tension development associated with reduced sarcomere lengths. Compared with measurements in ras transgenic mice, the model showed similar heterogeneity of septal systolic strain with the largest reductions in principal shortening and torsional shear in regions of greatest disarray. Average systolic principal shortening on the right ventricular septal surface of the model was -0.114 for normal regions and -0.065 for disarrayed regions; for torsional shear, these values were 0.047 and 0.019, respectively. These model results suggest that regional dysfunction in ras transgenic mice may be explained in part by the observed structural defects, including myofiber dispersion and reduced sarcomere length, which contributed about equally to predicted dysfunction in the disarrayed myocardium.

Biomechanical 3-D finite element modeling of the human breast using MRI data

Breast tissue deformation modeling has recently gained considerable interest in various medical applications. A biomechanical model of the breast is presented using a finite element (FE) formulation. Emphasis is given to the modeling of breast tissue deformation which takes place in breast imaging procedures. The first step in implementing the FE modeling (FEM) procedure is mesh generation. For objects with irregular and complex geometries such as the breast, this step is one of the most difficult and tedious tasks. For FE mesh generation, two automated methods are presented which process MRI breast images to create a patient-specific mesh. The main components of the breast are adipose, fibroglandular and skin tissues. For modeling the adipose and fibroglandular tissues, we used eight noded hexahedral elements with hyperelastic properties, while for the skin, we chose four noded hyperelastic membrane elements. For model validation, an MR image of an agarose phantom was acquired and corresponding FE meshes were created. Based on assigned elasticity parameters, a numerical experiment was performed using the FE meshes, and good results were obtained. The model was also applied to a breast image registration problem of a volunteer's breast. Although qualitatively reasonable, further work is required to validate the results quantitatively.

Mechanism underlying mechanical dysfunction in the border zone of left ventricular aneurysm: a finite element model study

BACKGROUND

The global left ventricular dysfunction characteristic of left ventricular aneurysm is associated with muscle fiber stretching in the adjacent noninfarcted (border zone) region during isovolumic systole. The mechanism of this regional dysfunction is poorly understood.

METHODS

An anteroapical transmural myocardial infarct was created by coronary arterial ligation in an adult Dorset sheep and was allowed to mature into left ventricular aneurysm for 10 weeks. The animal was imaged subsequently using magnetic resonance imaging with simultaneous recording of intraventricular pressures. A realistic mathematical model of the three-dimensional ovine left ventricle with an anteroapical aneurysm was constructed from multiple short-axis and long-axis magnetic resonance imaging slices at the beginning of diastolic filling.

RESULTS

Three model simulations are presented: (1) normal border zone contractility and normal aneurysmal material properties; (2) greatly reduced border zone contractility (by 50%) and normal aneurysmal material properties; and (3) greatly reduced border zone contractility (by 50%) and stiffened aneurysmal material properties (by 1000%). Only the latter two simulations were able to reproduce experimentally observed stretching of border zone fibers during isovolumic systole.

CONCLUSIONS

The mechanism underlying mechanical dysfunction in the border zone region of left ventricular aneurysm is primarily the result of myocardial contractile dysfunction rather than increased wall stress in this region.

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.

Effects of mitral valve replacement on regional left ventricular systolic strain

BACKGROUND

Mitral valve replacement (MVR) with chordal excision impairs left ventricular (LV) systolic function, but the responsible mechanisms remain incompletely characterized. Loss of normal annular-papillary continuity also adversely affects LV torsional deformation, possibly due to changes in myocardial fiber contraction pattern.

METHODS

Twenty-seven dogs underwent insertion of LV myocardial markers and a sham procedure (cardiopulmonary bypass, no MVR, n = 6), conventional MVR with chordae tendineae excision (n = 7), or chordal-sparing MVR with reattachment of the anterior leaflet chordae to the anterior annulus (n = 7) or to the posterior annulus (n = 7). In the anterior, lateral, posterior, and septal LV regions, linear chords were constructed from each region's central marker to its surrounding markers. Percent systolic shortening (regional LV strain) was calculated for each chord, and the chords were assigned to one of four angular groups: I, left-handed oblique (subepicardial fiber direction); II, circumferential (midwall); III, right-handed oblique (subendocardial); or IV, longitudinal. Regional LV strain data were compared before and after MVR.

RESULTS

Sham and anterior chordal-sparing MVR had minimal effects on regional LV strain. With posterior chordal-sparing MVR: anteriorly, left-oblique (I) strain fell (31%, p<0.05), as did circumferential (II) and right-oblique (III) strains (by 49% and 51%, respectively; p<0.01). Laterally, left-oblique (I) strain fell by 36% (p<0.05), as did longitudinal (IV) strain (54% decline, p<0.01). Conventional MVR with chordal excision disrupted regional fiber shortening diffusely, affecting oblique fibers (I and III) in the anterior and septal regions and impairing longitudinal (IV) strain in all regions (45% to 68% fall, p<0.05).

CONCLUSIONS

Sham and anterior chordal-sparing MVR did not substantially alter regional LV strain; however, loss of normal anatomic valvular-ventricular integrity (conventional MVR) or posterior chordal-sparing MVR resulted in pronounced alterations in LV strain, most notably in the longitudinal and oblique fiber directions. These findings demonstrate that the deleterious effects of chordal excision are associated with perturbed internal myocardial systolic deformation, which suggests that chordal disruption distorts myofiber architecture or regional systolic loading.

The effects of cross-fiber deformation on axial fiber stress in myocardium

We incorporated a three-dimensional generalization of the Huxley cross-bridge theory in a finite element model of ventricular mechanics to examine the effect of nonaxial deformations on active stress in myocardium. According to this new theory, which assumes that macroscopic tissue deformations are transmitted to the myofilament lattice, lateral myofilament spacing affects the axial fiber stress. We calculated stresses and deformations at end-systole under the assumption of strictly isometric conditions. Our results suggest that at the end of ejection, nonaxial deformations may significantly reduce active axial fiber stress in the inner half of the wall of the normal left ventricle (18-35 percent at endocardium, depending on location with respect to apex and base). Moreover, this effect is greater in the case of a compliant ischemic region produced by occlusion of the left anterior descending or circumflex coronary artery (26-54 percent at endocardium). On the other hand, stiffening of the remote and ischemic regions (in the case of a two-week-old infarct) lessens the effect of nonaxial deformation on active stress at all locations (9-32 percent endocardial reductions). These calculated effects are sufficiently large to suggest that the influence of nonaxial deformation on active fiber stress may be important, and should be considered in future studies of cardiac mechanics.

Echocardiography-derived left ventricular end-systolic regional wall stress and matrix remodeling after experimental myocardial infarction

OBJECTIVES

We tested the hypothesis that regional end-systolic left ventricular (ESLV) wall stress is associated with extracellular matrix remodeling activity after myocardial infarction (MI).

BACKGROUND

Increased left ventricular (LV) wall stress is a stimulus for LV enlargement, and echocardiography can be used to estimate regional wall stress. A powerful validation of a noninvasive method of estimating wall stress would be predicting cellular responses after a MI.

METHODS

Echocardiographic images were obtained in rats 1, 7, 14 or 21 days after coronary ligation (n = 11) or sham surgery (n = 5). End-systolic left ventricular wall stress was calculated by finite element analysis in three regions (infarcted, noninfarcted and border) from short-axis images. Matrix metalloproteinase-9 (MMP-9) and macrophage density were determined by immunohistochemistry, and positive cells were counted in high power fields (hpf).

RESULTS

Average ESLV wall stress was higher in rats with MI when compared to shams irrespective of time point (p < 0.01), and ESLV wall stress in the infarcted regions increased with time (25.1 +/- 5.9 vs. 69.9 +/- 4.4 kdyn/cm2, day 1 vs. 21; p < 0.01). Matrix metalloproteinase-9 expression was higher in infarcted and border regions when compared to noninfarcted regions (22.1 vs. 25.7 vs. 0.10 cells/hpf, respectively; p < 0.01). Over all regions, ESLV wall stress was associated with MMP-9 (r = 0.76; p < 0.001), macrophage density (r = 0.72; p < 0.001) and collagen content (r = 0.67; p < 0.001). End-systolic left ventricular wall stress was significantly higher when MMP-9 positive cell density was greater than 10 cells/hpf (45+/-20 vs. 14+/-10 kdyn/cm2; p < 0.001).

CONCLUSIONS

Regional increases in ESLV wall stress determined by echocardiography-based structural analysis are associated with extracellular matrix degradation activity.

Estimation of regional left ventricular wall stresses in intact canine hearts

Left ventricular (LV) wall stress is an important element in the assessment of LV systolic function; however, a reproducible technique to determine instantaneous local or regional wall stress has not been developed. Fourteen dogs underwent placement of twenty-six myocardial markers into the ventricle and septum. One week later, marker images were obtained using high-speed biplane videofluoroscopy under awake, sedated, atrially paced baseline conditions and after inotropic stimulation (calcium). With a model taking into account LV pressure, regional wall thickness, and meridional and circumferential regional radii of curvature, we computed average midwall stress for each of nine LV sites. Regional end-systolic and maximal LV wall stress were heterogeneous and dependent on latitude (increasing from apex to base, P < 0.001) and specific wall (anterior > lateral and posterior wall stresses; P = 0. 002). Multivariate ANOVA demonstrated only a trend (P = 0.056) toward increased LV stress after calcium infusion; subsequent univariate analysis isolated significant increases in end-systolic LV wall stress with increased inotropic state at all sites except the equatorial regions. The model used in this analysis incorporates local geometric factors and provides a reasonable estimate of regional LV wall stress compared with previous studies. LV wall stress is heterogeneous and dependent on the particular LV site of interest. Variation in wall stress may be caused by anatomic differences and/or extrinsic interactions between LV sites, i.e., influences of the papillary muscles and the interventricular septum.

Three-dimensional fetal echocardiography

The complex anatomy and dynamics of the heart make it a challenging organ to image. The fetal heart is particularly difficult because it is located deep within the mother's abdomen and direct access to electrocardiographic information is difficult. Thus more complex imaging and analysis methods are necessary to obtain information regarding fetal cardiac anatomy and function. This information can be used for medical diagnosis, model development and theoretical validation. The objective of this article is to provide scientists and engineers with an overview of three-dimensional fetal echocardiography.

Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy

The three-dimensional geometry and anisotropic properties of the heart give rise to nonhomogeneous distributions of stress, strain, electrical activation and repolarization. In this article we review the ventricular geometry and myofiber architecture of the heart, and the experimental and modeling studies of three-dimensional cardiac mechanics and electrophysiology. The development of a three-dimensional finite element model of the rabbit ventricular geometry and fiber architecture is described in detail. Finally, we review the experimental results, from the level of the cell to the intact organ, which motivate the development of coupled three-dimensional models of cardiac electromechanics and mechanoelectric feedback.

Estimates of regional work in the canine left ventricle

Assessment of the magnitude of regional myocardial work requires knowledge of regional fiber stress and fiber shortening. The theoretical development and experimental validation of a method is presented which used values of estimated active and passive fiber stress according to a fluid-fiber model, and measured fiber strain values. This enables the construction of regional stress-strain diagrams, a regional analog of the pressure-volume area model by Suga and co-investigators, which can be linked to regional oxygen consumption. In the left ventricle, either normally or asynchronously activated, the method yields reliable data on strain and active and passive fiber stress. The relation between estimated regional work and myocardial oxygen demand is in quantitative agreement with previously reported relations between global oxygen demand and measured pressure-volume area. During coronary artery occlusion, however, these values were less reliable, which might be due to inaqdequate knowledge of the (passive) material properties of the myocardium.

Tissue remodeling with micro-structurally based material laws

Cardiomyocytes and the extracellular collagen matrix which holds them together respond to changes in their mechanical environment by adapting their orientation, size and composition. We examine local mechanical feedback mechanisms affecting the fiber orientation, sheet orientation and passive fiber direction stiffness, using an axisymmetric finite element model of the left ventricle (LV), with material constitutive laws based on the fibrous-sheet microstructure of myocardium.

Optimization of cardiac fiber orientation for homogeneous fiber strain at beginning of ejection

Mathematical models of left ventricular (LV) wall mechanics show that fiber stress depends heavily on the choice of muscle fiber orientation in the wall. This finding brought us to the hypothesis that fiber orientation may be such that mechanical load in the wall is homogeneous. Aim of this study was to use the hypothesis to compute a distribution of fiber orientation within the wall. In a finite element model of LV wall mechanics, fiber stresses and strains were calculated at beginning of ejection (BE). Local fiber orientation was quantified by helix (HA) and transverse (TA) fiber angles using a coordinate system with local r-, c-, and l-directions perpendicular to the wall, along the circumference and along the meridian, respectively. The angle between the c-direction and the projection of the fiber direction on the cl-plane (HA) varied linearly with transmural position in the wall. The angle between the c-direction and the projection of the fiber direction on the cr-plane (TA) was zero at the epicardial and endocardial surfaces. Midwall TA increased with distance from the equator. Fiber orientation was optimized so that fiber strains at BE were as homogeneous as possible. By optimization with TA = 0 degree, HA was found to vary from 81.0 degrees at the endocardium to -35.8 degrees at the epicardium. Inclusion of TA in the optimization changed these angles to respectively 90.1 degrees and -48.2 degrees while maximum TA was 15.3 degrees. Then the standard deviation of fiber strain (epsilon f) at BE decreased from +/- 12.5% of mean epsilon f to +/- 9.5%. The root mean square (RMS) difference between computed HA and experimental data reported in literature was 15.0 degrees compared to an RMS difference of 11.6 degrees for a linear regression line through the latter data.

ECG response of the human body subjected to vibrations

In this paper, the ECG response of the human body subjected to vibrations is investigated. Measurements relied on recording the ECG and then computing the normalized difference of the ECG power spectrum density. Ten subjects aged 20-22 years old were exposed for 15 min to vertical vibrations in the frequency range 5-30 Hz. Results show either depression or elevation of the ST segment indicating heart muscle fatigue. The power spectrum density normalized difference also shows that the maximum difference takes place at 8 Hz vibrations frequency which is thought to be around the resonance frequency of the heart.