Time-varying effective mitral valve area: prediction and validation using cardiac MRI and Doppler echocardiography in normal subjects

A mathematical model of pressure and flow waveforms in the aortic root

The differences in the pressure and flow waveforms in the aortic root have not been explained so far in a satisfactory mathematical way. It is a generally accepted idea that the existence of the reflected wave causes the differences in shapes of pressure and flow. In this paper, a mathematical model is proposed that explains the blood pressure and flow waveforms based on changes in left ventricular volume during blood ejection into the aorta. According to the model, a change in volume of the left ventricle during contraction can be mathematically presented with solutions of differential equations that describe the behavior of a second-order system. The proposed mathematical equations of pressure and flow waveforms are derived from left ventricular volume change and basic equations of fluid dynamics. The position of the reflected wave depends on the age and elasticity of arteries, and has an effect on the flow and pressure waveforms. The model is in acceptable agreement with the experimental data available.

What global diastolic function is, what it is not, and how to measure it

Despite Leonardo da Vinci's observation (circa 1511) that “the atria or filling chambers contract together while the pumping chambers or ventricles are relaxing and vice versa,” the dynamics of four-chamber heart function, and of diastolic function (DF) in particular, are not generally appreciated. We view DF from a global perspective, while characterizing it in terms of causality and clinical relevance. Our models derive from the insight that global DF is ultimately a result of forces generated by elastic recoil, modulated by cross-bridge relaxation, and load. The interaction between recoil and relaxation results in physical wall motion that generates pressure gradients that drive fluid flow, while epicardial wall motion is constrained by the pericardial sac. Traditional DF indexes (τ, E/E′, etc.) are not derived from causal mechanisms and are interpreted as approximating either stiffness or relaxation, but not both, thereby limiting the accuracy of DF quantification. Our derived kinematic models of isovolumic relaxation and suction-initiated filling are extensively validated, quantify the balance between stiffness and relaxation, and provide novel mechanistic physiological insight. For example, causality-based modeling provides load-independent indexes of DF and reveals that both stiffness and relaxation modify traditional DF indexes. The method has revealed that the in vivo left ventricular equilibrium volume occurs at diastasis, predicted novel relationships between filling and wall motion, and quantified causal relationships between ventricular and atrial function. In summary, by using governing physiological principles as a guide, we define what global DF is, what it is not, and how to measure it.

Measurements of changes in left ventricular volume, strain, and twist during isovolumic relaxation using MRI

Left ventricular (LV) active relaxation begins before aortic valve closure and is largely completed during isovolumic relaxation (IVR), before mitral valve opening. During IVR, despite closed mitral and aortic valves, indirect assessments of LV volume have suggested volume increases during this period. The aim of this study is to measure LV volume throughout IVR and to determine the sources of any volume changes. For 10 healthy individuals (26.0 ± 3.8 yr), magnetic resonance imaging was used to measure time courses of LV volume, principal myocardial strains (circumferential, longitudinal, radial), and LV twist. Mitral leaflet motion was observed using echocardiography. During IVR, LV volume measurements showed an apparent increase of 4.6 ± 1.5 ml (5.0 ± 2.0% of the early filling volume change), the LV untwisted by 4.5 ± 1.9° (36.6 ± 18.0% of peak systolic twist), and changes in circumferential, longitudinal, and radial strains were +0.87 ± 0.64%, +0.93 ± 0.57%, and −1.46 ± 1.66% (4.2 ± 3.3%, 5.9 ± 3.3%, and 5.3 ± 7.5% of peak systolic strains), respectively. The apparent changes in volume correlated ( P < 0.01) with changes in circumferential, longitudinal, and radial strains ( r = 0.86, 0.69, and −0.37, respectively) and untwisting ( r = 0.83). The closed mitral valve leaflets were observed to descend into the LV throughout IVR in all subjects in apical four- and three-chamber and parasternal long-axis views by 6.0 ± 3.3, 5.1 ± 2.4, and 2.1 ± 5.0 mm, respectively. In conclusion, LV relaxation during IVR is associated with changes in principal strains and untwisting, which are all correlated with an apparent increase in LV volume. Since closed mitral and aortic valves ensure true isovolumic conditions, the apparent volume change likely reflects expansion of the LV myocardium and the inward bowing of the closed mitral leaflets toward the LV interior.

A computational procedure for prediction of structural effects of edge-to-edge repair on mitral valve

Edge-to-edge technique is a surgical procedure for the correction of mitral valve leaflets prolapse by suturing the edge of the prolapsed leaflet to the free edge of the opposing one. Suture presence modifies valve mechanical behavior and orifice flow area in the diastolic phase, when the valve opens and blood flows into the ventricle. In the present work, in order to support identification of potentially critical conditions, a computational procedure is described to evaluate the effects of changing suture length and position in combination with valve size and shape. The procedure is based on finite element method analyses applied to a range of different mitral valves, investigating for each configuration the influence of repair on functional parameters, such as mitral valve orifice area and transvalvular pressure gradient, and on structural parameters, such as stress in the leaflets and stitch tension. This kind of prediction would ideally require a coupled fluid-structural analysis, where the interactions between blood flows and mitral apparatus deformation are simultaneously considered. In the present study, however, an alternative approach is proposed, in which results obtained by purely structural finite element analyses are elaborated and interpreted taking into account the Bernoulli type equations available in literature to describe blood flow through mitral orifice. In this way, the effects of each parameter in terms of orifice flow area, suture loads, and leaflets stresses can be expressed as functions of atrioventricular pressure gradient and then correlated to blood flow rate. Results obtained by using this procedure for different configurations are finally discussed.

E-wave deceleration time may not provide an accurate determination of LV chamber stiffness if LV relaxation/viscoelasticity is unknown

Average left ventricular (LV) chamber stiffness (ΔPavg/ΔVavg) is an important diastolic function index. An E-wave-based determination of ΔPavg/ΔVavg (Little WC, Ohno M, Kitzman DW, Thomas JD, Cheng CP. Circulation 92: 1933–1939, 1995) predicted that deceleration time (DT) determines stiffness as follows: ΔPavg/ΔVavg = N(π/DT)2 (where N is constant), which implies that if the DTs of two LVs are indistinguishable, their stiffness is indistinguishable as well. We observed that LVs with indistinguishable DTs may have markedly different ΔPavg/ΔVavg values determined by simultaneous echocardiography-catheterization. To elucidate the mechanism by which LVs with indistinguishable DTs manifest distinguishable chamber stiffness, we use a validated, kinematic E-wave model (Kovács SJ, Barzilai B, Perez JE. Am J Physiol Heart Circ Physiol 252: H178–H187, 1987) with stiffness ( k) and relaxation/viscoelasticity ( c) parameters. Because the predicted linear relation between k and ΔPavg/ΔVavg has been validated, we reexpress the DT-stiffness (ΔPavg/ΔVavg) relation of Little et al. as follows: DT k ≈ [Formula: see text]. Using the kinematic model, we derive the general DT-chamber stiffness/viscoelasticity relation as follows: DT k, c = [Formula: see text](where c and k are determined directly from the E-wave), which reduces to DT k when c ≪ k. Validation involved analysis of 400 E-waves by determination of five-beat averaged k and c from 80 subjects undergoing simultaneous echocardiography-catheterization. Clinical E-wave DTs were compared with model-predicted DT k and DT k, c. Clinical DT was better predicted by stiffness and relaxation/viscoelasticity ( r2 = 0.84, DT vs. DT k, c) jointly rather than by stiffness alone ( r2 = 0.60, DT vs. DT k). Thus LVs can have indistinguishable DTs but significantly different ΔPavg/ΔVavg if chamber relaxation/viscoelasticity differs. We conclude that DT is a function of both chamber stiffness and chamber relaxation viscoelasticity. Quantitative diastolic function assessment warrants consideration of simultaneous stiffness and relaxation/viscoelastic effects.

Wave intensity analysis of left ventricular filling: application of windkessel theory

We extend our recently published windkessel-wave interpretation of vascular function to the wave intensity analysis (WIA) of left ventricular (LV) filling dynamics by separating the pressure changes due to the windkessel from those due to traveling waves. With the use of LV compliance, the change in pressure due solely to LV volume changes (windkessel pressure) can be isolated. Inasmuch as the pressure measured in the cardiovascular system is the sum of its windkessel and wave components (excess pressure), it can be substituted into WIA, yielding the isolated wave effects on LV filling. Our study of six open-chest dogs demonstrated that once the windkessel effects are removed from WIA, the energy of diastolic suction is 2.6 times greater than we previously calculated. Volume-related changes in pressure (i.e., the windkessel or reservoir effect) must be considered first when wave motion is analyzed.

Cardiac dysfunction in heart failure with normal ejection fraction: MRI measurements

Cardiovascular magnetic resonance is a non-invasive 3-dimensional imaging technique which can provide morphologic and functional information as well as tissue characterization without the use of ionizing radiation or nephrotoxic contrast agents. It has a high accuracy and reproducibility and is optimally suited to quantify structural and functional abnormalities and to follow a patient over time. In the setting of heart failure with normal ejection fraction it can be used as an alternative to echocardiography in those patients with suboptimal image quality but it can also provide unique information for the differential diagnosis and the underlying physiopathology of this syndrome.

Wave intensity analysis of left atrial mechanics and energetics in anesthetized dogs

The left atrium (LA) acts as a booster pump during late diastole, generating the Doppler transmitral A wave and contributing incrementally to left ventricular (LV) filling. However, after volume loading and in certain disease states, LA contraction fills the LV less effectively, and retrograde flow (i.e., the Doppler Ar wave) into the pulmonary veins increases. The purpose of this study was to provide an energetic analysis of LA contraction to clarify the mechanisms responsible for changes in forward and backward flow. Wave intensity analysis was performed at the mitral valve and a pulmonary vein orifice. As operative LV stiffness increased with progressive volume loading, the reflection coefficient (i.e., energy of reflected wave/energy of incident wave) also increased. This reflected wave decelerated the forward movement of blood through the mitral valve and was transmitted through the LA, accelerating retrograde blood flow in the pulmonary veins. Although total LA work increased with volume loading, the forward hydraulic work decreased and backward hydraulic work increased. Thus wave reflection due to increased LV stiffness accounts for the decrease in the A wave and the increase in the Ar wave measured by Doppler.

Frequency-based analysis of the early rapid filling pressure-flow relation elucidates diastolic efficiency mechanisms

Stiffness- and relaxation-based diastolic function (DF) assessment can characterize the presence, severity, and mechanism of dysfunction. Although frequency-based characterization of arterial function is routine (input impedance, characteristic impedance, arterial wave reflection), DF assessment via frequency-based methods incorporating optimization/efficiency criteria is lacking. By definition, optimal filling maximizes (E wave) volume and minimizes "loss" at constant stored elastic strain energy (which initiates mechanical, recoil-driven filling). In thermodynamic terms, optimal filling delivers all oscillatory power (rate of work) at the lowest harmonic. To assess early rapid filling optimization, simultaneous micromanometric left ventricular pressure and echocardiographic transmitral flow (Doppler E wave) were Fourier analyzed in 31 subjects. A validated kinematic filling model provided closed-form expressions for E wave contours and model parameters. Relaxation-based DF impairment is indicated by prolonged E wave deceleration time (DT). Optimization was assessed via regression between the dimensionless ratio of 2nd (Q2) and 3rd flow harmonics (Q3) to the lowest harmonic (Q1), i.e., (Q2/Q1) or (Q3/Q1) vs. DT or c, the filling model's viscosity/damping (energy loss) parameter. Results show that DT prolongation or increased c generated increased oscillatory power at higher harmonics (Q2/Q1 = 0.00091DT + 0.09837, r = 0.70; Q3/Q1 = 0.00053DT + 0.02747, r = 0.60; Q2/Q1 = 0.00614c + 0.15527, r = 0.91; Q3/Q1 = 0.00396c + 0.05373, r = 0.87). Because ideal filling is achieved when all oscillatory power is delivered at the lowest harmonic, the observed increase in power at higher harmonics is a measure of filling inefficiency. We conclude that frequency-based analysis facilitates assessment of filling efficiency and elucidates the mechanism by which diastolic dysfunction associated with prolonged DT impairs optimal filling.

Relationship of pulmonary vein flow to left ventricular short-axis epicardial displacement in diastole: model-based prediction with in vivo validation

Previous studies in healthy humans have established that the (approximately 850 ml) volume enclosed by the pericardial sac is nearly constant over the cardiac cycle, exhibiting a transient approximately 5% decrease (approximately 40 ml) from end diastole to end systole. This volume decrease manifests as a "crescent" at the ventricular free wall level when short-axis MRI images of the epicardial surface acquired at end systole and end diastole are superimposed. On the basis of the (near) constant-volume property of the four-chambered heart, the volume decrease ("crescent effect") must be restored during subsequent early diastolic filling via the left atrial conduit volume. Therefore, volume conservation-based modeling predicts that pulmonary venous (PV) Doppler D-wave volume must be causally related to the radial displacement of the epicardium (Delta) (i.e., magnitude of "crescent effect" in the radial direction). We measured Delta from M-mode echocardiographic images and measured D-wave velocity-time integral (VTI) from Doppler PV flow of the right superior PV in 11 subjects with catheterization-determined normal physiology. In accordance with model prediction, high correlation was observed between Delta and D-wave VTI (r=0.86) and early D-wave VTI measured to peak D-wave velocity (r=0.84). Furthermore, selected subjects with various pathological conditions had values of Delta that differed significantly. These observations demonstrate the volume conservation-based causal relationship between radial pericardial displacement of the left ventricle and the PV D-wave-generated filling volume in healthy subjects as well as the potential role of the M-mode echo-derived radial epicardial displacement index Delta as a regional (radial) parameter of diastolic function.

Consequences of increasing heart rate on deceleration time, the velocity-time integral, and E/A

The ascendancy of diastolic heart failure to "epidemic" proportions has increased the use of and reliance on Doppler echocardiography as a source for diagnosis and as the preferred method for determining indexes of diastolic function (DF). Current indexes are primarily derived from shape-based features of Doppler E and A waves, such as their amplitudes, slopes, durations, and areas. Load dependence and pathologic correlates of these indexes have been considered, but DF indexes are not routinely corrected for heart rate (HR). To determine the dependence of selected Doppler-derived indexes of DF on HR, transmitral Doppler flow velocities and electrocardiograms were simultaneously recorded during supine bicycle exercise in 21 young, healthy volunteers. Standard E- and A-wave shape-based indexes (acceleration time, deceleration time [DT], peak E, peak A) were measured using triangle approximation. Velocity-time integrals (VTIs) were calculated by trapezoidal and triangular approximations. A-wave peak velocity (A) was measured conventionally, relative to baseline, and also using 2 alternative methods: A*, measured relative to the E@A velocity, and Ac, relative to the E-wave deceleration value at peak A-wave velocity. E/A was calculated conventionally and by using A* and Ac. The results showed that DF indexes derived from individual E waves are essentially HR independent. DT showed a mere 20% decrease for a 100% increase in HR. A triangular approximation for the E-wave VTI and the corrected E/Ac were found to be nearly HR independent. In conclusion, on the basis of the established continuity of cardiac output as a function of increasing HR and the observed data, Doppler-derived indexes of DF (DT, VTIs, E/Ac) can be treated as essentially HR independent only if the VTI and A-wave peak are corrected for HR as described.

A concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction

Numerical modeling of the human cardiovascular system has always been an active research direction since the 19th century. In the past, various simulation models of different complexities were proposed for different research purposes. In this paper, an improved numerical model to study the dynamic function of the human circulation system is proposed. In the development of the mathematical model, the heart chambers are described with a variable elastance model. The systemic and pulmonary loops are described based on the resistance-compliance-inertia concept by considering local effects of flow friction, elasticity of blood vessels and inertia of blood in different segments of the blood vessels. As an advancement from previous models, heart valve dynamics and atrioventricular interaction, including atrial contraction and motion of the annulus fibrosus, are specifically modeled. With these improvements the developed model can predict several important features that were missing in previous numerical models, including regurgitant flow on heart valve closure, the value of E/A velocity ratio in mitral flow, the motion of the annulus fibrosus (called the KG diaphragm pumping action), etc. These features have important clinical meaning and their changes are often related to cardiovascular diseases. Successful simulation of these features enhances the accuracy of simulations of cardiovascular dynamics, and helps in clinical studies of cardiac function.

Prediction and assessment of the time-varying effective pulmonary vein area via cardiac MRI and Doppler echocardiography

Accurately estimating left atrial (LA) volume with Doppler echocardiography remains challenging. Using angiography for validation, Marino et al. (Marino P, Prioli AM, Destro G, LoSchiavo I, Golia G, and Zardini P. Am Heart J 127: 886–898, 1994) determined LA volume throughout the cardiac cycle by integrating the velocity-time integrals of Doppler transmitral and pulmonary venous flow, assuming constant mitral valve and pulmonary vein areas. However, this LA volume determination method has never been compared with three-dimensional LA volume data from cardiac MRI, the gold standard for cardiac chamber volume measurement. Previously, we determined that the effective mitral valve area is not constant but varies as a function of time. Therefore, we sought to determine whether the effective pulmonary vein area (EPVA) might be time varying as well and also assessed Marino's method for estimating LA volume. We imaged 10 normal subjects using cardiac MRI and concomitant transthoracic Doppler echocardiography. LA and left ventricular (LV) volumes were measured by MRI, transmitral and pulmonary vein flows were measured by Doppler echocardiography, and time dependence was synchronized via the electrocardiogram. LA volume, estimated using Marino's method, was compared with the MRI measurements. Differences were observed, and the discrepancy between the echocardiographic and MRI methods was used to predict EPVA as a function of time. EPVA was also directly measured from short-axis MRI images and was found to be time varying in concordance with predicted values. We conclude that because EPVA and LA volume time dependence are in phase, LA filling in systole and LV filling in diastole are both facilitated. Application to subjects in select pathophysiological states is in progress.

Duration of diastole and its phases as a function of heart rate during supine bicycle exercise

The duration of diastole can be defined in terms of mechanical events. Mechanical diastolic duration (MDD) is comprised by the phases of early rapid filling (E wave), diastasis, and late atrial filling (A wave). The effect of heart rate (HR) on diastolic duration is predictable from kinematic modeling and known cellular physiology. To determine the dependence of MDD of each phase and the velocity time integral (VTI) on HR, simultaneous transmitral Doppler flow velocities and ECG were recorded during supine bicycle exercise in healthy volunteers. Durations, peak values, and VTI using triangular approximation for E- and A-wave shape were measured. MDD, defined as the interval from the start of the E wave to end of the A wave, was fit as an algebraic function of HR by MDD = BMDD + MLMDD·HR + MIMDD/HR, derivable from first principles, where BMDD is a constant, and MLMDD and MIMDD are the constant coefficients of the linear and inverse HR dependent terms. Excellent correlation was observed ( r2 = 0.98). E- and A-wave durations were found to be very nearly independent of HR: 100% increase in HR generated only an 18% decrease in E-wave duration and 16% decrease in A-wave duration. VTI was similarly very nearly independent of HR. Diastasis duration closely tracked MDD as a function of HR. We conclude that the elimination of diastasis and merging of E and A waves of nearly fixed durations primarily govern changes in MDD. These observations support the perspective that E- and A-wave durations are primarily governed by the rules of mechanical oscillation that are minimally HR dependent.