Nonlinear isochrones in murine left ventricular pressure-volume loops: how well does the time-varying elastance concept hold?

The linear time-varying elastance theory is frequently used to describe the change in ventricular stiffness during the cardiac cycle. The concept assumes that all isochrones (i.e., curves that connect pressure-volume data occurring at the same time) are linear and have a common volume intercept. Of specific interest is the steepest isochrone, the end-systolic pressure-volume relationship (ESPVR), of which the slope serves as an index for cardiac contractile function. Pressure-volume measurements, achieved with a combined pressure-conductance catheter in the left ventricle of 13 open-chest anesthetized mice, showed a marked curvilinearity of the isochrones. We therefore analyzed the shape of the isochrones by using six regression algorithms (two linear, two quadratic, and two logarithmic, each with a fixed or time-varying intercept) and discussed the consequences for the elastance concept. Our main observations were 1) the volume intercept varies considerably with time; 2) isochrones are equally well described by using quadratic or logarithmic regression; 3) linear regression with a fixed intercept shows poor correlation (R(2) < 0.75) during isovolumic relaxation and early filling; and 4) logarithmic regression is superior in estimating the fixed volume intercept of the ESPVR. In conclusion, the linear time-varying elastance fails to provide a sufficiently robust model to account for changes in pressure and volume during the cardiac cycle in the mouse ventricle. A new framework accounting for the nonlinear shape of the isochrones needs to be developed.     

Frequency analysis of time-varying elastance model of the left ventricle

Zadeh's transfer function method for linear time-variable systems is used to apply frequency-domain analysis to a periodically time-varying elastance model of the left ventricle. Left ventricular pressure computed from the system function of the time-varying elastance and the phasors of aortic flow shows a typical waveform of the measured ventricular pressure.     

Dynamic left ventricular elastance: a model for integrating cardiac muscle contraction into ventricular pressure-volume relationships.

To integrate myocardial contractile processes into left ventricular (LV) function, a mathematical model was built. Muscle fiber force was set equal to the product of stiffness and elastic distortion of stiffness elements, i.e., force-bearing cross bridges (XB). Stiffness dynamics arose from recruitment of XB according to the kinetics of myofilament activation and fiber-length changes. Elastic distortion dynamics arose from XB cycling and the rate-of-change of fiber length. Muscle fiber stiffness and distortion dynamics were transformed into LV chamber elastance and volumetric distortion dynamics. LV pressure equaled the product of chamber elastance and volumetric distortion, just as muscle-fiber force equaled the product of muscle-fiber stiffness and lineal elastic distortion. Model validation was in terms of its ability to reproduce cycle-time-dependent LV pressure response, DeltaP(t), to incremental step-like volume changes, DeltaV, in the isolated rat heart. All DeltaP(t), regardless of the time in the cycle at which DeltaP(t) was elicited, consisted of three phases: phase 1, concurrent with the leading edge of DeltaV; phase 2, a brief transient recovery from phase 1; and phase 3, sustained for the duration of systole. Each phase varied with the time in the cycle at which DeltaP(t) was elicited. When the model was fit to the data, cooperative activation was required to sustain systole for longer periods than was possible with Ca(2+) activation alone. The model successfully reproduced all major features of the measured DeltaP(t) responses, and thus serves as a credible indicator of the role of underlying contractile processes in LV function.     

Global cardiac function: mechano-energetico-informatics

This review on the global cardiac function covers cardiac mechanics, energetics, and informatics that I have developed with my collaborators over the last 30 years in Japan and USA. We first established E(max) (end-systolic maximum elastance or pressure/volume ratio) as a new index of ventricular contractility using canine hearts. We then expanded the E(max) concept to PVA (systolic pressure-volume area consisting of external mechanical work and mechanical potential energy) as an innovative measure of total mechanical energy of ventricular contraction and discovered it to be a reliable determinant of ventricular energetics or O(2) consumption (V(O(2))). We have discovered that E(max) shifts the V(O(2))-PVA relation and the E(max) dependency (O(2) cost of E(max)) varies among different pathophysiological hearts. We also searched for the basis of E(max) in crossbridge behavior information contained in an X-ray diffraction of papillary muscle. Recently, we established a new integrative analysis to estimate total Ca(2+) recruited for excitation-contraction coupling in a beating heart using the E(max)-PVA-V(O(2)) information. These global, mechano-energetico-informatic approaches seem to facilitate better understanding of cardiac function, as required in the present post-genomic era when more physiomic knowledge is required not only in cardiac function but also in all other physiologic functions.     

The effect of a myocardial infarction on the normalized time-varying elastance curve.

It has been suggested that the shape of the normalized time-varying elastance curve [E(n)(t(n))] is conserved in different cardiac pathologies. We hypothesize, however, that the E(n)(t(n)) differs quantitatively after myocardial infarction (MI). Sprague-Dawley rats (n = 9) were anesthetized, and the left anterior descending coronary artery was ligated to provoke the MI. A sham-operated control group (CTRL) (n = 10) was treated without the MI. Two months later, a conductance catheter was inserted into the left ventricle (LV). The LV pressure and volume were measured and the E(n)(t(n)) derived. Slopes of E(n)(t(n)) during the preejection period (alpha(PEP)), ejection period (alpha(EP)), and their ratio (beta = alpha(EP)/alpha(PEP)) were calculated, together with the characteristic decay time during isovolumic relaxation (tau) and the normalized elastance at end diastole (E(min)(n)). MI provoked significant LV chamber dilatation, thus a loss in cardiac output (-33%), ejection fraction (-40%), and stroke volume (-30%) (P < 0.05). Also, it caused significant calcium increase (17-fold), fibrosis (2-fold), and LV hypertrophy. End-systolic elastance dropped from 0.66 +/- 0.31 mmHg/microl (CTRL) to 0.34 +/- 0.11 mmHg/microl (MI) (P < 0.05). Normalized elastance was significantly reduced in the MI group during the preejection, ejection, and diastolic periods (P < 0.05). The slope of E(n)(t(n)) during the alpha(PEP) and beta were significantly altered after MI (P < 0.05). Furthermore, tau and end-diastolic E(min)(n) were both significantly augmented in the MI group. We conclude that the E(n)(t(n)) differs quantitatively in all phases of the heart cycle, between normal and hearts post-MI. This should be considered when utilizing the single-beat concept.     

Optimal control aspects of left ventricular ejection dynamics

A model of the ejecting left ventricle is developed in which ventricular elastance as a function of time is optimized with respect to a simple performance index selected on an energetic basis. The model correctly predicts a number of well known experimental findings concerning the effects of preload and afterload conditions and varying system parameters on left ventricular pressure and elastance waveforms and on the ejection period. The results characterize ventricular systolic elastance as dependent on both end-diastolic volume and mean aortic pressure.     

Model for left ventricular contraction combining the force length velocity relationship with the time varying elastance theory

A model for the contraction of the left ventricle (LV) is developed for a spheroidal geometry. The classical force-length-velocity relationship for a single muscle fiber is assumed. The linear maximum pressure volume relationship (maximum elastance), a measure of muscle contractility, is further extended into a time-varying function. This is achieved by utilizing a mechanical activation function, assumed as half a sinusoidal wave, to describe the time-dependent isometric stress for the activated cardiac muscle. This, in turn, results in the time-varying elastance function and represents the instantaneous activity of the muscle contractile proteins. The model is tested for a set of boundary conditions that determine preload, afterload, and the inherent properties of the muscle, i.e., the contractility. The computed results of the isovolumic contraction, auxotonic contraction, and isovolumic relaxation are in agreement with the expected behavior of the LV. The relations between the simulated variations on preload, afterload, and contractility, and the set of performance indexes of the LV, are presented and discussed.     

Clinical use of pressure-dimension and stress-shortening relations in systole and diastole

The concept of time-varying elastance is an important one and has clinical relevance for both systole and diastole. The application of this concept to systole has permitted the development of end-systolic pressure-volume (dimension) and stress-shortening analysis as a method of assessing myocardial contractility in patients at risk for congestive cardiomyopathy. The application of the time-varying elastance concept to diastole is particularly relevant to new findings concerning abnormal relaxation in hypertrophic cardiomyopathy and angina pectoris.     

Energetics of ventricular contraction as traced in the pressure-volume diagram

The pressure-volume (P-V) relationship of the canine left ventricle can reasonably be simulated by a time-varying elastance model. In this model the total mechanical energy generated by a contraction can be determined theoretically from the change in the elastance. Applying this theory to the actual left ventricle, we have found that the area in the P-V diagram circumscribed by the end-systolic P-V relation line, the end-diastolic P-V relation curve, and the systolic segment of the P-V trajectory is equivalent to the total mechanical energy generated by ventricular contraction. We call this area the systolic P-V area (PVA). We have studied experimentally the correlation between the PVA and myocardial oxygen consumption (VO2) in the canine left ventricle. VO2 was linearly correlated with PVA regardless of the contraction mode and loading conditions in a given left ventricle. The VO2-PVA relation parallel shifted upward with positive inotropic agents. This shift comprised a significant increase in VO2 component for the unloaded contraction. We therefore consider that further analyses of the VO2-PVA relationship will greatly promote our understanding of cardiac energetics.     

The relative value of strain and strain rate for defining intrinsic myocardial function

It is well accepted that strain and strain rate deformation parameters are not only a measure of intrinsic myocardial contractility but are also influenced by changes in cardiac load and structure. To date, no information is available on the relative importance of these confounders. This study was designed to investigate how strain and strain rate, measured by Doppler echocardiography, relate to the individual factors that determine cardiac performance. Echocardiographic and conductance measurements were simultaneously performed in mice in which individual determinants of cardiac performance were mechanically and/or pharmacologically modulated. A multivariable analysis was performed with radial and circumferential strains and peak systolic radial and circumferential strain rates as dependent parameters and preload recruitable stroke work (PRSW), arterial elastance (E(a)), end-diastolic pressure, and left ventricular myocardial volume (LVMV) as independent factors representing myocardial contractility, afterload, preload, and myocardial volume, respectively. Radial strain was most influenced by E(a) (β = -0.58, R(2) = 0.34), whereas circumferential strain was strongly associated with E(a) and moderately with LVMV (β = 0.79 and -0.52, respectively, R(2) = 0.54). Radial strain rate was related to both PRSW and LVMV (β = 0.79 and -0.62, respectively, R(2) = 0.50), whereas circumferential strain rate showed a prominent correlation only with PRSW (β = -0.61, R(2) = 0.51). In conclusion, strain (both radial and circumferential) is not a good surrogate measure of intrinsic myocardial contractility unless the strong confounding influence of afterload is considered. Strain rate is a more robust measure of contractility that is less influenced by changes in cardiac load and structure. Thus, peak systolic strain rate is the more relevant parameter to assess myocardial contractile function noninvasively.     

Positive inotropic and negative lusitropic effects of endothelin receptor agonism in vivo

The endothelin (ET) system is involved in the regulation of myocardial function in health as well as in several diseases, such as congestive heart failure, myocardial infarction, and septic myocardial depression. Conflicting results have been reported regarding the acute contractile properties of ET-1. We therefore investigated the effects of intracoronary infusions of ET-1 and of the selective ET(B) receptor-selective agonist sarafotoxin 6c with increasing doses in anesthetized pigs. Myocardial effects were measured through analysis of the left ventricular pressure-volume relationship. ET-1 elicited increases in the myocardial contractile status (end-systolic elastance value of 0.94 +/- 0.11 to 1.48 +/- 0.23 and preload recruitable stroke work value of 68.7 +/- 4.7 to 83.4 +/- 7.2) that appear to be mediated through ET(A) receptors, whereas impairment in left ventricular isovolumic relaxation (tau = 41.5 +/- 1.4 to 58.1 +/- 5.0 and t(1/2) = 23.0 +/- 0.7 to 30.9 +/- 2.6, where tau is the time constant for pressure decay and t(1/2) is the half-time for pressure decay) was ET(B) receptor dependent. In addition, intravenous administration of ET-1 impaired ventricular relaxation but had no effect on contractility. Intracoronary sarafotoxin 6c administration caused impairments in left ventricular relaxation (tau from 43.3 +/- 1.8 to 54.4 +/- 3.4) as well as coronary vasoconstriction. In conclusion, ET-1 elicits positive inotropic and negative lusitropic myocardial effects in a pig model, possibly resulting from ET(A) and ET(B) receptor activation, respectively.     

Factors influencing left-ventricular stiffness

The aim of the study was to investigate the relative contributions of geometrical and material factors to overall left-ventricular cavity stiffness. Left-ventricular cavity shapes were reconstructed using a computer and the variation of myocardial elastic modulus was calculated, by the finite element method, through the passive phase of diastole when rising volume coincided with rising pressure. Geometric data were obtained from biplane cineangiography, with micromanometer pressure measurements, for ten patients with left ventricular disease. Dimensional analysis was applied to the initial and derived data from which the influences of myocardial compliance, wall thickness-to-long dimension ratio, and aspect ratio (long-to-short axes) were determined. The ratio between the volume elasticity and the myocardial modulus of elasticity, the normalized stiffness ratio (NSR), is proposed as a useful index of left ventricular mechanical behaviour in diastole. The volume elasticity of the chamber is dependent not only upon the myocardium elastic modulus and the wall thickness ratio, but also on the shape of the chambe. Changes in the thickness/radius ratio of the ventricle have less effect upon its distention than those in the long dimension/radius ratio. The left ventricle becomes more spherical in shpae through diastole and hence becomes stiffer by this geometric mechanism.     

Dynamic geometry of the intact left ventricle

Knowledge of left ventricular chamber dynamics is central to our understanding of cardiac physiology. The complicated changes in left ventricular geometry observed in the dog during various phases of the cardiac cycle can be represented as distinct linear relationships between chamber eccentricity and intracavitary volume during diastole and ejection, and probably represent structural properties of the ventricular wall. Chamber geometry of the left ventricle is a major determinant of overall myocardial function. The slope of the radius of curvature (r) to wall thickness (h) relationship is a geometric constant that determines the mural force at any given transmural pressure. Chronic pressure and volume overload produce changes in this geometric relationship as a result of increased mural force resisting ejection. The adaptive mechanism of ventricular hypertrophy in this setting alters the r/h ratio and returns systolic mural force toward normal. Coronary occlusion induces acute changes in regional geometry characterized by holosystolic wall bulging and systolic wall thinning, which shift the r/h relationship upward and to the left. The geometric alteration during ischemia probably increases systolic mural force and could adversely affect myocardial function. Recent studies with patients have shown the r/h ratio to be of value in distinguishing between reversible and irreversible impairment of myocardial performance. Because most myocardial diseases produce major alterations in the structure of the ventricular wall, analysis of dynamic chamber geometry may prove of prognostic value in assessing patients with cardiac disorders.     

The time-varying elastic properties of the left ventricular muscle

We combine two techniques in order to discuss the time-varying elastic properties of the left ventricular muscle. An analytic model for the shape and forces in the left ventricle is combined with the Fourier series representations of certain of the ventricular dimensions and pressure to derive expressions for the stress and strain in the left ventricle. The strain is thus a function of the elastic material properties, which are then expressed as functions of time by using Fourier series. The only data needed for a numerical study using these techniques are closed-chest determinations of the ventricular dimensions and the ventricular pressure.     

Acute elevation of triglycerides increases left ventricular contractility and alters ventricular-vascular interaction

Acute elevation of circulating lipids, such as the postprandial state, contributes to increased cardiovascular risk. However, the effect of acutely elevated triglycerides on arterial and left ventricular function is not completely understood. We aimed to assess whether an acute increase in triglycerides affects ventricular-vascular interaction. Fifteen healthy men (age, 49 ± 8 yr) underwent blinded, randomized infusion of saline and intravenous fat emulsion to acutely raise plasma triglycerides. All subjects underwent both randomization trials, in random order on two separate days. Ventricular-vascular interaction measures were recorded by tonometry (central blood pressure) and echocardiography (left ventricular volumes, strain, and strain rate) at baseline and after 1 h infusion. Net ventricular-vascular interaction was defined by the effective arterial elastance (E(A))-to-left ventricular end-systolic elastance (E(LV)) ratio (E(A)/E(LV)). When compared with saline, the infusion of intravenous fat emulsion increased triglycerides and free fatty acids (ΔP < 0.001 for both) and improved left ventricular contractility (ΔE(LV), end-systolic volume and strain rate; P < 0.05 for all). However, arterial function was unchanged (ΔE(A), brachial and central blood pressure; P > 0.05 for all). Overall, E(A)/E(LV) was decreased by an infusion of intravenous fat emulsion (P = 0.004) but not saline (P > 0.05, P = 0.001 for Δ between trials). We conclude that intravenous fat emulsion and acute elevation of blood lipids (including triglycerides and free fatty acids) alter ventricular-vascular interaction by increasing left ventricular contractility without affecting arterial load. These findings may have implications for cardiovascular responses to parenteral nutrition.     

Noninvasive assessment of left ventricular and myocardial contractility in middle-aged men and women: disparate evolution above the age of 50?

End-systolic elastance (E(es)) is a frequently used index of left ventricular (LV) contractility. However, because of its inherent dependence on LV geometry, E(es) cannot be used to compare myocardial contractile state between ventricles with different geometries, which is the case in any cross-sectional study. Various normalization methods for E(es) have been proposed in the literature, but a standardized method is still lacking. In this study, we introduced a novel alternative normalization technique and compared it with three previously suggested methods. We tested all normalization methods to assess the age- and sex-related differences in myocardial contractility in a large population sample of 2,184 middle-aged (ages, 35-55 yr) untreated subjects free from overt cardiovascular disease. Ventricular contractility E(es) was determined using a previously validated noninvasive single-beat method, based on two-dimensional echocardiographic and brachial blood pressure measurements. Myocardial contractility was estimated as 1) E(es).end-diastolic volume (EDV); 2) E(es).LV mass (LVM); 3) 0.433.E(es).LVM/relative wall thickness (RWT), based on a theoretical LV model; and 4) 0.0941.E(es).LVM(0.455).RWT(-0.159), a novel semiempirical expression derived in this study. Because of the difference in their underlying assumptions, the various myocardial contractility indexes do not provide consistent information with respect to sex differences. Despite these discrepancies, it was found that myocardial contractility in women appears to be better preserved after the age of 50 yr compared with that in men. The physiological mechanisms behind this potentially clinically important phenomenon at population level require further investigation.     

Measuring right ventricular function in the normal and hypertensive mouse hearts using admittance-derived pressure-volume loops

Mice are a widely used animal model for investigating cardiovascular disease. Novel technologies have been used to quantify left ventricular function in this species, but techniques appropriate for determining right ventricular (RV) function are less well demonstrated. Detecting RV dysfunction is critical to assessing the progression of pulmonary vascular diseases such as pulmonary hypertension. We used an admittance catheter to measure pressure-volume loops in anesthetized, open-chested mice before and during vena cava occlusion. Mice exposed to chronic hypoxia for 10 days, which causes hypoxia-induced pulmonary hypertension (HPH), were compared with control (CTL) mice. HPH resulted in a 27.9% increase in RV mass (P < 0.005), a 67.5% increase in RV systolic pressure (P < 0.005), and a 61.2% decrease in cardiac output (P < 0.05). Preload recruitable stroke work (PRSW) and slope of the maximum derivative of pressure (dP/dt(max))-end-diastolic volume (EDV) relationship increased with HPH (P < 0.05). Although HPH increased effective arterial elastance (E(a)) over fivefold (from 2.7 ± 1.2 to 16.4 ± 2.5 mmHg/μl), only a mild increase in the ventricular end-systolic elastance (E(es)) was observed. As a result, a dramatic decrease in the efficiency of ventricular-vascular coupling occurred (E(es)/E(a) decreased from 0.71 ± 0.27 to 0.35 ± 0.17; P < 0.005). Changes in cardiac reserve were evaluated by dobutamine infusion. In CTL mice, dobutamine significantly enhanced E(es) and dP/dt(max)-EDV but also increased E(a), causing a decrease in E(es)/E(a). In HPH mice, slight but nonsignificant decreases in E(es), PRSW, dP/dt(max)-EDV, and E(a) were observed. Thus 10 days of HPH resulted in RV hypertrophy, ventricular-vascular decoupling, and a mild decrease in RV contractile reserve. This study demonstrates the feasibility of obtaining RV pressure-volume measurements in mice. These measurements provide insight into ventricular-vascular interactions healthy and diseased states.