Post-infarct evolution of ventricular and myocardial function.

Adverse ventricular remodeling following acute myocardial infarction (MI) may induce ventricular dilation, fibrosis, and loss of global contractile function, possibly resulting in heart failure (HF). Understanding the relation between the time-dependent changes in material properties of the myocardium and the contractile function of the heart may further our understanding of the development of HF post-MI and guide the development of novel therapies. A finite element model of cardiac mechanics was used to model MI in a thick-walled truncated ellipsoidal geometry. Infarct core and border zone comprised 9.6 and 8.1% of the LV wall volume, respectively. Acute MI was modeled by inhibiting active stress generation. Chronic MI was modeled by the additional effect of infarct material stiffening, wall thinning and fiber reorientation. In acute MI, stroke work decreased by 25%. In the infarct core, fiber stress was reduced but fiber strain was increased, depending on the degree of infarct stiffening. Fiber work density was equal to zero. Healthy tissue adjacent to the infarct showed decreased work density depending on the degree of infarct stiffness and the orientation of the myofibers with respect to the infarct region. Thinning of the wall partially restored this loss in work density while the effects of fiber reorientation were minimal. We found that the relative loss in pump function in the infarcted heart exceeds the relative loss in healthy myocardial tissue due to impaired mechanical function in healthy tissue adjacent to the infarct. Infarct stiffening, wall thinning and fiber reorientation did not affect pump function but did affect the distribution of work density in tissue adjacent to the infarct.

A method for the quantification of the pressure dependent 3D collagen configuration in the arterial adventitia.

Collagen plays an important role in the response of the arterial wall to mechanical loading and presumably has a load-bearing function preventing overdistension. Collagen configuration is important for understanding this role, in particular in mathematical models of arterial wall mechanics. In this study a new method is presented to image and quantify this configuration. Collagen in the arterial adventitia is stained with CNA35, and imaged in situ at high resolution with confocal microscopy at luminal pressures from 0 to 140mm Hg. The images are processed with a new automatic approach, utilizing techniques intended for MRI-DTI data. Collagen configuration is quantified through three parameters: the waviness, the transmural angle and the helical angle. The method is demonstrated for the case of carotid arteries of the white New Zealand rabbit. The waviness indicated a gradual straightening between 40 and 80mm Hg. The transmural angle was about zero indicating that the fibers stayed within an axial-circumferential plane at all pressures. The helical angle was characterized by a symmetrical distribution around the axial direction, indicating a double symmetrical helix. The method is the first to combine high resolution imaging with a new automatic image processing approach to quantify the 3D configuration of collagen in the adventitia as a function of pressure.

Towards model-based analysis of cardiac MR tagging data: relation between left ventricular shear strain and myofiber orientation.

Many cardiac pathologies are reflected in abnormal myocardial deformation, accessible through magnetic resonance tagging (MRT). Interpretation of the MRT data is difficult, since the relation between pathology and deformation is not straightforward. Mathematical models of cardiac mechanics could be used to translate measured abnormalities into the underlying pathology, but, so far, they even fail to correctly simulate myocardial deformation in the healthy heart. In this study we investigated to what extent (1) our previously published three-dimensional finite element model of cardiac mechanics [Kerckhoffs, R.C.P., Bovendeerd, P.H.M., Kotte, J.C.S., Prinzen, F.W., Smits, K., Arts, T., 2003. Homogeneity of cardiac contraction despite physiological asynchrony of depolarization: a model study. Ann. Biomed. Eng. 31, 536-547] can simulate measured cardiac deformation, and (2) discrepancies between strains in model and experiment are related to the choice of the myofiber orientation in the model. To this end, we measured midwall circumferential strain E(cc) and circumferential-radial shear strain E(cr) in three healthy subjects using MRT. E(cc) as computed in the model agreed well with measured E(cc). Computed E(cr) differed significantly from measured E(cr). The time course of E(cr) was found to be very sensitive to the choice of the myofiber orientation, in particular to the choice of the transverse angle. Discrepancies between circumferential-radial shear strain in model and experiment were reduced strongly by increasing the transverse angle in the original model by 25%.

Modeling the relation between cardiac pump function and myofiber mechanics.

Complexity of the geometry and structure of the heart hampers easy modeling of cardiac mechanics. The modeling can however be simplified considerably when using the hypothesis that in the normal heart myofiber structure and geometry adapt, until load is evenly distributed. A simple and realistic relationship is found between the hemodynamic variables cavity pressure and volume, and myofiber load parameters stress and strain. The most important geometric parameter in the latter relation is the ratio of cavity volume to wall volume, while actual geometry appears practically irrelevant. Applying the found relationship, a realistic maximum is set to left ventricular pressure after chronic pressure load. Pressures exceeding this level are likely to cause decompensation and heart failure. Furthermore, model is presented to simulate left and right ventricular pump function with left-right interaction.

Influence of endocardial-epicardial crossover of muscle fibers on left ventricular wall mechanics.

The influence of variations of fiber direction on the distribution of stress and strain in the left ventricular wall was investigated using a finite element model to simulate the mechanics of the left ventricle. The commonly modelled helix fiber angle was defined as the angle between the local circumferential direction and the projection of the fiber path on the plane perpendicular to the local radial direction. In the present study, an additional angle, the transverse fiber angle, was used to model the continuous course of the muscle fibers between the inner and the outer layers of the ventricular wall. This angle was defined as the angle between the circumferential direction and the projection of the fiber path on the plane perpendicular to the local longitudinal direction. First, a reference simulation of left ventricular mechanics during a cardiac cycle was performed, in which the transverse angle was set to zero. Next, we performed two simulations in which the spatial distribution of either the transverse or the helix angle was varied with respect to the reference situation, the spatially averaged variations being about 3 and 14 degrees, respectively. The changes in fiber orientation hardly affected the pressure-volume relation of the ventricle, but significantly affected the spatial distribution of active muscle fiber stress (up to 50% change) and sarcomere length (up to 0.1 micron change). In the basal and apical region of the wall, shear deformation in the circumferential-radial plane was significantly reduced by introduction of a nonzero transverse angle. Thus, the loading of the passive tissue may be reduced by the endocardial-epicardial crossover of the muscle fibers.

Dependence of local left ventricular wall mechanics on myocardial fiber orientation: a model study.

The dependence of local left ventricular (LV) mechanics on myocardial muscle fiber orientation was investigated using a finite element model. In the model we have considered anisotropy of the active and passive components of myocardial tissue, dependence of active stress on time, strain and strain rate, activation sequence of the LV wall and aortic afterload. Muscle fiber orientation in the LV wall is quantified by the helix fiber angle, defined as the angle between the muscle fiber direction and the local circumferential direction. In a first simulation, a transmural variation of the helix fiber angle from +60 degrees at the endocardium through 0 degrees in the midwall layers to -60 degrees at the epicardium was assumed. In this simulation, at the equatorial level maximum active muscle fiber stress was found to vary from about 110 kPa in the subendocardial layers through about 30 kPa in the midwall layers to about 40 kPa in the subepicardial layers. Next, in a series of simulations, muscle fiber orientation was iteratively adapted until the spatial distribution of active muscle fiber stress was fairly homogeneous. Using a transmural course of the helix fiber angle of +60 degrees at the endocardium, +15 degrees in the midwall layers and -60 degrees at the epicardium, at the equatorial level maximum active muscle fiber stress varied from 52 kPa to 55 kPa, indicating a remarkable reduction of the stress range. Moreover, the change of muscle fiber strain with time was more similar in different parts of the LV wall than in the first simulation. It is concluded that (1) the distribution of active muscle fiber stress and muscle fiber strain across the LV wall is very sensitive to the transmural distribution of the helix fiber angle and (2) a physiological transmural distribution of the helix fiber angle can be found, at which active muscle fiber stress and muscle fiber strain are distributed approximately homogeneously across the LV wall.

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.

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact.

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.

Optimization of left ventricular fibre orientation of the normal heart for homogeneous sarcomere length during ejection.

UNLABELLED: During the ejection phase of the cardiac cycle, left ventricular muscle fibres shorten while generating force. It was hypothesized that fibres are oriented in the wall such that the amount of shortening is the same for all fibres. We evaluated this hypothesis for the equatorial region of the left ventricle. In a finite element model of left ventricular wall mechanics fibre orientation was quantified by a helix angle which varied linearly from the inner to the outer wall. Fibre length was characterized by sarcomere length, set at 1.95 microns everywhere in the passive state of 0 transmural pressure. For a cavity pressure of 15 kPa, considered representative for ejection, inhomogeneity in mechanical loading was expressed by the variance of the sarcomere length. The variance was minimized by adapting the transmural course of fibre angle. First, only the slope was optimized and in a second optimization this was done for both slope and intercept. Optimal helix fibre angles were 69.6 degrees endocardially, 0 degree at the middle of the wall and -69.6 degrees epicardially for the first optimization and 78.2 degrees, 20.7 degrees and, -36.7 degrees respectively for the second. Sarcomere length changed from 1.95 to 1.975 +/- 0.012 and 1.981 +/- 0.004 microns (mean +/- SD) respectively. CONCLUSION: After optimization calculated helix fibre angles were in the physiological range. Describing the transmural course of fibre angle with slope and intercept significantly improved homogeneity in mechanical load.

Determination of muscle fibre orientation using Diffusion-Weighted MRI.

Biomechanical studies have shown that the distribution of stress and strain in biological tissue is strongly dependent on fibre orientation. Therefore, to analyze the local mechanical load, accurate data on muscle fibre orientation are needed. Traditional techniques to determine fibre orientation are inherently invasive. Here we used Diffusion Weighted MRI to non-invasively determine, in each image voxel of 0.23 x 0.23 mm, the diffusion tensor of water in the cat semimembranosus muscle. The direction corresponding to the largest eigenvector of this tensor was calculated. This direction was found to correspond qualitatively to the muscular fibre direction, as determined by visual inspection.

A model for arterial adaptation combining microstructural collagen remodeling and 3D tissue growth.

Long-term adaptation of soft tissues is realized through growth and remodeling (G&R). Mathematical models are powerful tools in testing hypotheses on G&R and supporting the design and interpretation of experiments. Most theoretical G&R studies concentrate on description of either growth or remodeling. Our model combines concepts of remodeling of collagen recruitment stretch and orientation suggested by other authors with a novel model of general 3D growth. We translate a growth-induced volume change into a change in shape due to the interaction of the growing tissue with its environment. Our G&R model is implemented in a finite element package in 3D, but applied to two rotationally symmetric cases, i.e., the adaptation towards the homeostatic state of the human aorta and the development of a fusiform aneurysm. Starting from a guessed non-homeostatic state, the model is able to reproduce a homeostatic state of an artery with realistic parameters. We investigate the sensitivity of this state to settings of initial parameters. In addition, we simulate G&R of a fusiform aneurysm, initiated by a localized degradation of the matrix of the healthy artery. The aneurysm stabilizes in size soon after the degradation stops.

Electromechanics of paced left ventricle simulated by straightforward mathematical model: comparison with experiments.

Intraventricular synchrony of cardiac activation is important for efficient pump function. Ventricular pacing restores the beating frequency but induces more asynchronous depolarization and more inhomogeneous contraction than in the normal heart. We investigated whether the increased inhomogeneity in the left ventricle can be described by a relatively simple mathematical model of cardiac electromechanics, containing normal mechanical and impulse conduction properties. Simulations of a normal heartbeat and of pacing at the right ventricular apex (RVA) were performed. All properties in the two simulations were equal, except for the depolarization sequence. Simulation results of RVA pacing on local depolarization time and systolic midwall circumferential strain were compared with those measured in dogs, using an epicardial sock electrode and MRI tagging, respectively. We used the same methods for data processing for simulation and experiment. Model and experiment agreed in the following aspects. 1) Ventricular pacing decreased systolic pressure and ejection fraction relative to natural sinus rhythm. 2) Shortening during ejection and stroke work declined in early depolarized regions and increased in late depolarized regions. 3) The relation between epicardial depolarization time and systolic midwall circumferential strain was linear and similar for the simulation (slope = -3.80 +/- 0.28 s(-1), R2 = 0.87) and the experiments [slopes for 3 animals -2.62 +/- 0.43 s(-1) (R2 = 0.59), -2.97 +/- 0.38 s(-1) (R2 = 0.69), and -4.44 +/- 0.51 s(-1) (R2 = 0.76)]. We conclude that our model of electromechanics is suitable to simulate ventricular pacing and that the apparently complex events observed during pacing are caused by well-known basic physiological processes.

Characterization of the normal cardiac myofiber field in goat measured with MR-diffusion tensor imaging.

Cardiac myofiber orientation is a crucial determinant of the distribution of myocardial wall stress. Myofiber orientation is commonly quantified by helix and transverse angles. Accuracy of reported helix angles is limited. Reported transverse angle data are incomplete. We measured cardiac myofiber orientation postmortem in five healthy goat hearts using magnetic resonance-diffusion tensor imaging. A novel local wall-bound coordinate system was derived from the characteristics of the fiber field. The transmural course of the helix angle corresponded to data reported in literature. The mean midwall transverse angle ranged from -12 +/- 4 degrees near the apex to +9.0 +/- 4 degrees near the base of the left ventricle, which is in agreement with the course predicted by Rijcken et al. (18) using a uniform load hypothesis. The divergence of the myofiber field was computed, which is a measure for the extent to which wall stress is transmitted through the myofiber alone. It appeared to be <0.07 mm(-1) throughout the myocardial walls except for the fusion sites between the left and right ventricles and the insertion sites of the papillary muscles.

The large shear strain dynamic behaviour of in-vitro porcine brain tissue and a silicone gel model material.

The large strain dynamic behaviour of brain tissue and silicone gel, a brain substitute material used in mechanical head models, was compared. The non-linear shear strain behaviour was characterised using stress relaxation experiments. Brain tissue showed significant shear softening for strains above 1% (approximately 30% softening for shear strains up to 20%) while the time relaxation behaviour was nearly strain independent. Silicone gel behaved as a linear viscoelastic solid for all strains tested (up to 50%) and frequencies up to 461 Hz. As a result, the large strain time dependent behaviour of both materials could be derived for frequencies up to 1000 Hz from small strain oscillatory experiments and application of Time Temperature Superpositioning. It was concluded that silicone gel material parameters are in the same range as those of brain tissue. Nevertheless the brain tissue response will not be captured exactly due to increased viscous damping at high frequencies and the absence of shear softening in the silicone gel. For trend studies and benchmarking of numerical models the gel can be a good model material.

Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall.

Pumping power as delivered by the heart is generated by the cells in the myocardial wall. In the present model study global left-ventricular pump function as expressed in terms of cavity pressure and volume is related to local wall tissue function as expressed in terms of myocardial fiber stress and strain. On the basis of earlier studies in our laboratory, it may be concluded that in the normal left ventricle muscle fiber stress and strain are homogeneously distributed. So, fiber stress and strain may be approximated by single values, being valid for the whole wall. When assuming rotational symmetry and homogeneity of mechanical load in the wall, the dimensionless ratio of muscle fiber stress (sigma f) to left-ventricular pressure (Plv) appears to depend mainly on the dimensionless ratio of cavity volume (Vlv) to wall volume (Vw) and is quite independent of other geometric parameters. A good (+/- 10%) and simple approximation of this relation is sigma f/Plv = 1 + 3 Vlv/Vw. Natural fiber strain is defined by ef = In (lf/lf,ref), where lf,ref indicates fiber length (lf) in a reference situation. Using the principle of conservation of energy for a change in ef, it holds delta ef = (1/3)delta In (1 + 3Vlv/Vw).

Effect of ventricular contraction, pressure, and wall stretch on vessels at different locations in the wall.

A cylindrical model of the heart was used to calculate the influence of ventricular filling and (isovolumic and isobaric) contraction on the cross-sectional area and resistance of a subendocardial and subepicardial maximally dilated arteriole and venule. Contraction is defined as the difference between static diastole and static systole. Furthermore, a small piece of rectangular myocardium containing the vessel was modeled to distinguish between the individual contributions of contractility (i.e., myocardial elastic properties), ventricular pressure, and local circumferential stretch to the changes in vascular area and resistance during contraction. Calculations were performed assuming the muscle fibers ran in either an apex-to-base or a circumferential direction. The results were similar for the two directions. Assuming constant, physiological arteriolar and venular pressures of 45 and 10 mmHg, respectively, coronary blood vessels were predicted not to collapse during ventricular contraction. Moreover, vascular area reduction was found to be larger for the arteriole (approximately 50%) than for the venule (approximately 30%) during both isovolumic and isobaric contractions. Consequently, arteriolar resistance was found to increase more than venular resistance (approximately 340 and 120%, respectively). Subendocardial area reductions were found to be somewhat smaller than subepicardial area reductions for the venule (by approximately 10%) but not for the arteriole. Contractility was found to be the main contributor to the changes in vascular area and resistance in the subepicardium but to contribute by < 50% to the changes in the subendocardium. Because pressure does, but stretch does not, contribute to the area change during isovolumic contraction and the reverse is true during isobaric contraction, it was concluded that although changes in vascular area and resistance may be similar for different contractions, the causes for these changes are very different.

Cardiac mechanoenergetics replicated by cross-bridge model.

The aim of this work is to reproduce the experimentally measured linear dependence of cardiac muscle oxygen consumption on stress-strain area using a model, composed of a three-state Huxley-type model for cross-bridge interaction and a phenomenological model of Ca2+-induced activation. By selecting particular cross-bridge cycling rate constants and modifying the cross-bridge activation model, we replicated the linear dependence between oxygen consumption and stress-strain area together with other important mechanical properties of cardiac muscle such as developed stress dependence on the sarcomere length and force-velocity relationship. The model predicts that (1) the amount of the "passenger" cross bridges, i.e., cross bridges that detach without hydrolyzing ATP molecule, is relatively small and (2) ATP consumption rate profile within a beat and the amount of the passenger cross bridges depend on the contraction protocol.

Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection.

The strain of muscle fibers in the heart is likely to be distributed uniformly over the cardiac walls during the ejection period of the cardiac cycle. Mathematical models of left ventricular (LV) wall mechanics have shown that the distribution of fiber strain during ejection is sensitive to the orientation of muscle fibers in the wall. In the present study, we tested the hypothesis that fiber orientation in the LV wall is such that fiber strain during ejection is as homogeneous as possible. A finite-element model of LV wall mechanics was set up to compute the distribution of fiber strain at the beginning (BE) and end (EE) of the ejection period of the cardiac cycle, with respect to a middiastolic reference state. The distribution of fiber orientation over the LV wall, quantified by three parameters, was systematically varied to minimize regional differences in fiber shortening during ejection and in the average of fiber strain at BE and EE. A well-defined optimum in the distribution of fiber orientation was found which was not significantly different from anatomical measurements. After optimization, the average of fiber strain at BE and EE was 0.025 +/-0.011 (mean+/-standard deviation) and the difference in fiber strain during ejection was 0.214+/-0.018. The results indicate that the LV structure is designed for maximum homogeneity of fiber strain during ejection.

Subepicardial fiber strain and stress as related to left ventricular pressure and volume.

In a mathematical model of the mechanics of the left ventricle (LV) by Arts et al. (1), assuming uniformity of fiber stress (sigma f) and fiber strain (delta epsilon f) in the wall during the ejection phase, fiber stress and fiber strain were related to LV cavity pressure (Plv), LV cavity volume (Vlv) and wall volume (Vw) by the following pair of equations: sigma f = Plv (1 + 3 Vlv/Vw) and delta epsilon f = 1/3 delta ln (1 + 3 Vlv/Vw). The ratio of Vlv to Vw appeared to be the most important geometric parameter, whereas the actual LV shape was of minor importance. The relationships on fiber strain and stress were evaluated experimentally in six anesthetized open-chest dogs during normal and elevated (volume loading) end-diastolic LV pressure. Subepicardial fiber strain was measured simultaneously in 16 adjacent regions of the LV anterior wall, using optical markers that were attached to the epicardial surface and recorded on video. Changes in Vlv were measured by use of four inductive coils sutured to the LV in a tetrahedric configuration. Vw was measured postmortem. During control as well as hypervolemia the following results were found. At the anterior free wall of the LV, the slope of the estimated linear relationship between measured and calculated fiber strain was 1.017 +/- 0.168 (means +/- SD), which is not significantly different from unity. Calculated fiber stress corresponded qualitatively and quantitatively with experimental results reported on isolated cardiac muscle. Calculated subepicardial contractile work per unit of tissue volume was not significantly different from global pump work as normalized to Vw. These findings support the assumption of homogeneity of muscle fiber strain and stress in the left ventricular wall during the ejection phase. Furthermore, average values of fiber stress and strain can be estimated on the basis of measured left ventricular pressure and volume.

A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction.

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.