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).     

Cardiac mechanoenergetics in silico

The aim of this thesis is to investigate the link between biochemical intracellular processes and mechanical contraction of the cardiac muscle. First, the regulation of intracellular energy fluxes between mitochondria and myofibrils is studied. It is shown, that the experimentally observed metabolic stability of the cardiac muscle is reproducible by a simple feedback regulation mechanism, i.e., ATP consumption in myofibrils and ATP production in mitochondria are balanced by the changes of the high energy phosphate concentrations. Second, an important property of energy transformation from biochemical form to mechanical work in the cardiac muscle, the linear relationship between the oxygen consumption and the stress-strain area, is replicated by a cross-bridge model. Third, by using the developed cross-bridge model, the correlation between ejection fraction of the left ventricle and heterogeneity of sarcomere strain, developed stress and ATP consumption in the left ventricular wall is established. Fourth, an experimentally observed linear relationship between oxygen consumption and the pressure-volume area can be predicted theoretically from a linear relationship between the oxygen consumption and the stress-strain area. Summing up, it is shown how the macrovariables of a cardiac muscle are interwoven with intracellular physiological processes into a whole.     

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.     

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.     

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.     

Structure and torsion in the normal and situs inversus totalis cardiac left ventricle. II. Modeling cardiac adaptation to mechanical load

Mathematical models provide a suitable platform to test hypotheses on the relation between local mechanical stimuli and responses to cardiac structure and geometry. In the present model study, we tested hypothesized mechanical stimuli and responses in cardiac adaptation to mechanical load on their ability to estimate a realistic myocardial structure of the normal and situs inversus totalis (SIT) left ventricle (LV). In a cylindrical model of the LV, 1) mass was adapted in response to myofiber strain at the beginning of ejection and to global contractility (average systolic pressure), 2) cavity volume was adapted in response to fiber strain during ejection, and 3) myofiber orientations were adapted in response to myofiber strain during ejection and local misalignment between neighboring tissue parts. The model was able to generate a realistic normal LV geometry and structure. In addition, the model was also able to simulate the instigating situation in the rare SIT LV with opposite torsion and transmural courses in myofiber direction between the apex and base [Delhaas et al. (6)]. These results substantiate the importance of mechanical load in the formation and maintenance of cardiac structure and geometry. Furthermore, in the model, adapted myocardial architecture was found to be insensitive to fiber misalignment in the transmural direction, i.e., myofiber strain during ejection was sufficient to generate a realistic transmural variation in myofiber orientation. In addition, the model estimates that, despite differences in structure, global pump work and the mass of the normal and SIT LV are similar.     

Structure and torsion of the normal and situs inversus totalis cardiac left ventricle. I. Experimental data in humans

In 1926, the famous American pediatric cardiologist, Dr. Helen B. Taussig, observed that in situs inversus totalis (SIT) main gross anatomical structures and the deep muscle bundles of the ventricles were a mirror image of the normal structure, while the direction of the superficial muscle bundles remained unchanged (H. B. Taussig, Bull Johns Hopkins Hosp 39: 199-202, 1926). She and we wondered about the implication of this observation for left ventricular (LV) deformation in SIT. We used magnetic resonance tagging to obtain information on LV deformation, rotation, and torsion from a series of tagged images in five evenly distributed, parallel, short-axis sections of the heart of nine controls and eight persons with SIT without other structural (cardiac) defect. In the controls, during ejection, the apex rotated counterclockwise with respect to the base, when looking from the apex. Furthermore, the base-to-apex gradient in rotation (torsion) was negative and similar at all longitudinal levels of the LV. In SIT hearts, torsion was positive near the base, indicating mirrored myofiber orientations compared with the normal LV. Contrary to expectations, torsion in the apical regions of SIT LVs was as in normal ones, reflecting a normal internal myocardial architecture. The transition zone with zero torsion, found between the apex and base, suggests that the heart structure in SIT is essentially different from that in the normal heart. This provides a unique possibility to study regulatory mechanisms for myocardial fiber orientation and mechanical load, which has been dealt with in the companion paper by Kroon et al.     

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.     

Theoretical models for coronary vascular biomechanics: progress & challenges

A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.