Parametrized Diastolic Filling Research Articles

E-wave asymmetry elucidates diastolic ventricular stiffness-relaxation coupling: model-based prediction with in vivo validation.

Load, chamber stiffness, and relaxation are the three established determinants of global diastolic function (DF). Coupling of systolic stiffness and isovolumic relaxation has been hypothesized; however, diastolic stiffness-relaxation coupling (DSRC) remains unknown. The parametrized diastolic filling (PDF) formalism, a validated DF model incorporates DSRC. PDF model-predicted DSRC was validated by analysis of 159 Doppler E-waves from a published data set (22 healthy volunteers undergoing bicycle exercise). E-waves at varying (46-120 bpm) heart rates (HR) demonstrated variation in acceleration time (AT), deceleration time (DT), and E-wave peak velocity. AT, DT, and Epeak were converted into PDF parameters: stiffness ([Formula: see text]), relaxation ([Formula: see text]), and load (xo) using published numerical methods. Univariate linear regression showed that over a twofold increase in HR, AT, and DT decrease ([Formula: see text] = -0.44; P < 0.001 and r = -0.42; P < 0.001, respectively), while, DT/AT remains constant (r = -0.04; P = 0.67). Similarly, [Formula: see text] increases with HR (r = 0.55; P < 0.001), while [Formula: see text] has no significant correlation with HR (r = 0.08; P = 0.32). However, the dimensionless DSRC parameter ψ = c2/4k shows no significant correlation with HR (r = -0.03; P = 0.7). Furthermore, ψ is uniquely determined by DT/AT rather than AT or DT independently. Constancy of ψ in spite of a twofold increase in HR establishes that stiffness (k) and relaxation (c) are coupled and manifest via a HR-invariant parameter of E-wave asymmetry and should not be considered independent of each other. The manifestation of DSRC through E-wave asymmetry via ψ underscores the value of DT/AT as a physiological, mechanism-derived index of DF.NEW & NOTEWORTHY: Although diastolic stiffness and relaxation are considered independent chamber properties, the cardio-hemic inertial oscillation that generates E-waves obeys Newton's law. E-waves vary with heart rate requiring simultaneous change in stiffness and relaxation. By retrospective analysis of human heart-rate varying transmitral Doppler-data, we show that diastolic stiffness and relaxation are coupled and that the coupling manifests through E-wave asymmetry, quantified through a parametrized diastolic filling model-derived dimensionless parameter, which only depends on deceleration time and acceleration time, readily obtainable via standard echocardiography.

Alternative diastolic function models of ventricular longitudinal filling velocity are mathematically identical.

The spatiotemporal features of normal in vivo cardiac motion are well established. Longitudinal velocity has become a focus of diastolic function (DF) characterization, particularly the tissue Doppler e'-wave, manifesting in early diastole when the left ventricle (LV) is a mechanical suction pump (dP/dV < 0). To characterize DF and elucidate mechanistic features, several models have been proposed and have been previously compared algebraically, numerically, and in their ability to fit physiological velocity data. We analyze two previously noncompared models of early rapid-filling lengthening velocity (Doppler e'-wave): parametrized diastolic filling (PDF) and force balance model (FBM). Our initial numerical experiments sampled FBM-generated e'(t) contours as input to determine PDF model predicted fit. The resulting exact numerical agreement [standard error of regression (SER) = 9.06 × 10-16] was not anticipated. Therefore, we analyzed all published FBM-generated e'(t) contours and observed identical agreement. We re-expressed FBM's algebraic expressions for e'(t) and observed for the first time that model-based predictions for lengthening velocity by the FBM and the PDF model are mathematically identical: e'(t) = γe-αtsinh(βt), thereby providing exact algebraic relations between the three PDF parameters and the six FBM parameters. Previous pioneering experiments have independently established the unique determinants of e'(t) to be LV relaxation, restoring forces (stiffness), and load. In light of the exact intermodel agreement, we conclude that the three PDF parameters, relaxation, stiffness (restoring forces), and load, are unique determinants of DF and e'(t). Thus, we show that only the PDF formalism can compute the three unique, independent, physiological determinants of long-axis LV myocardial velocity from e'(t).NEW & NOTEWORTHY We show that two separate, independently derived physiological (kinematic) models predict mathematically identical expressions for LV-lengthening velocity (Doppler e'-wave), indicating that damped harmonic oscillatory motion is a physiologically accurate model of diastolic function. Although both models predict the same "overdamped" velocity contour, only one model solves the "inverse problem" and generates unique, lumped parameters of relaxation, stiffness (restoring force), and load from the e'-wave.

Quantifying Diastolic Function: From E-Waves as Triangles to Physiologic Contours via the 'Geometric Method'.

Conventional echocardiographic diastolic function (DF) assessment approximates transmitral flow velocity contours (Doppler E-waves) as triangles, with peak (Epeak), acceleration time (AT), and deceleration time (DT) as indexes. These metrics have limited value because they are unable to characterize the underlying physiology. The parametrized diastolic filling (PDF) formalism provides a physiologic, kinematic mechanism based characterization of DF by extracting chamber stiffness (k), relaxation (c), and load (x o ) from E-wave contours. We derive the mathematical relationship between the PDF parameters and Epeak, AT, DT and thereby introduce the geometric method (GM) that computes the PDF parameters using Epeak, AT, and DT as input. Numerical experiments validated GM by analysis of 208 E-waves from 31 datasets spanning the full range of clinical diastolic function. GM yielded indistinguishableaverage parameter values per subject vs. the gold-standard PDF method(k: R2=0.94, c:R2=0.95, x o :R2=0.95, p<0.01 all parameters). Additionally, inter-rater reliability for GM-determined parameters was excellent (k:ICC=0.956c:ICC=0.944, x o :ICC=0.993). Results indicate that E-wave symmetry (AT/DT) may comprise a new index of DF. By employing indexes (Epeak, AT, DT) that are already in standard clinical use the GM capitalizes on the power of the PDF method to quantify DF in terms of physiologic chamber properties.

Quantification of Global Diastolic Function by Kinematic Modeling-based Analysis of Transmitral Flow via the Parametrized Diastolic Filling Formalism

Quantitative cardiac function assessment remains a challenge for physiologists and clinicians.Although historically invasive methods have comprised the only means available, the development of noninvasive imaging modalities (echocardiography, MRI, CT) having high temporal and spatial resolution provide a new window for quantitative diastolic function assessment. Echocardiography is the agreed upon standard for diastolic function assessment, but indexes in current clinical use merely utilize selected features of chamber dimension (M-mode) or blood/tissue motion (Doppler) waveforms without incorporating the physiologic causal determinants of the motion itself.The recognition that all left ventricles (LV) initiate filling by serving as mechanical suction pumps allows global diastolic function to be assessed based on laws of motion that apply to all chambers. What differentiates one heart from another are the parameters of the equation of motion that governs filling. Accordingly, development of the Parametrized Diastolic Filling (PDF) formalismhas shown that the entire range of clinically observed early transmitral flow (Doppler E-wave) patterns are extremely well fit by the laws of damped oscillatory motion.This permits analysis of individual E-waves in accordance with a causal mechanism (recoil-initiated suction) that yields three (numerically) unique lumped parameters whose physiologic analogues are chamber stiffness (k), viscoelasticity/relaxation (c), and load (xo).The recording of transmitral flow (Doppler E-waves) is standard practice in clinical cardiology and, therefore, the echocardiographic recording method is only briefly reviewed. Our focus is on determination of the PDF parameters from routinely recorded E-wave data.As the highlighted results indicate, once the PDF parameters have been obtained from a suitable number of load varying E-waves, the investigator is free to use the parameters or construct indexes from the parameters (such as stored energy 1/2kxo(2), maximum A-V pressure gradient kxo, load independent index of diastolic function, etc.) and select the aspect of physiology or pathophysiology to be quantified.

Diastolic Function in Normal Sinus Rhythm vs. Chronic Atrial Fibrillation: Comparison by Fractionation of E-wave Deceleration Time into Stiffness and Relaxation Components.

Although the electrophysiologic derangement responsible for atrial fibrillation (AF) has been elucidated, how AF remodels the ventricular chamber and affects diastolic function (DF) has not been fully characterized. The previously validated Parametrized Diastolic Filling (PDF) formalism models suction-initiated filling kinematically and generates error-minimized fits to E-wave contours using unique load (xo), relaxation (c), and stiffness (k) parameters. It predicts that E-wave deceleration time (DT) is a function of both stiffness and relaxation. Ascribing DTs to stiffness and DTr to relaxation such that DT=DTs+DTr is legitimate because of causality and their predicted and observed high correlation (r=0.82 and r=0.94) with simultaneous (diastatic) chamber stiffness (dP/dV) and isovolumic relaxation (tau), respectively. We analyzed simultaneous echocardiography-cardiac catheterization data and compared 16 age matched, chronic AF subjects to 16, normal sinus rhythm (NSR) subjects (650 beats). All subjects had diastatic intervals. Conventional DF parameters (DT, AT, Epeak, Edur, E-VTI, E/E') and E-wave derived PDF parameters (c, k, DTs, DTr) were compared. Total DT and DTs, DTr in AF were shorter than in NSR (p<0.005), chamber stiffness, (k) in AF was higher than in NSR (p<0.001). For NSR, 75% of DT was due to stiffness and 25% was due to relaxation whereas for AF 81% of DT was due to stiffness and 19% was due to relaxation (p<0.005). We conclude that compared to NSR, increased chamber stiffness is one measurable consequence of chamber remodeling in chronic, rate controlled AF. A larger fraction of E-wave DT in AF is due to stiffness compared to NSR. By trending individual subjects, this method can elucidate and characterize the beneficial or adverse long-term effects on chamber remodeling due to alternative therapies in terms of chamber stiffness and relaxation.

The isovolumic relaxation to early rapid filling relation: kinematic model based prediction with in vivo validation.

Although catheterization is the gold standard, Doppler echocardiography is the preferred diastolic function (DF) characterization method. The physiology of diastole requires continuity of left ventricular pressure (LVP)‐generating forces before and after mitral valve opening (MVO). Correlations between isovolumic relaxation (IVR) indexes such as tau (time‐constant of IVR) and noninvasive, Doppler E‐wave‐derived metrics, such as peak A‐V gradient or deceleration time (DT), have been established. However, what has been missing is the model‐predicted causal link that connects isovolumic relaxation (IVR) to suction‐initiated filling (E‐wave). The physiology requires that model‐predicted terminal force of IVR (F t IVR) and model‐predicted initial force of early rapid filling (F i E‐wave) after MVO be correlated. For validation, simultaneous (conductance catheter) P‐V and E‐wave data from 20 subjects (mean age 57 years, 13 men) having normal LV ejection fraction (LVEF>50%) and a physiologic range of LV end‐diastolic pressure (LVEDP) were analyzed. For each cardiac cycle, the previously validated kinematic (Chung) model for isovolumic pressure decay and the Parametrized Diastolic Filling (PDF) kinematic model for the subsequent E‐wave provided F t IVR and F i E‐wave respectively. For all 20 subjects (15 beats/subject, 308 beats), linear regression yielded F t IVR = α F i E‐wave + b (R = 0.80), where α = 1.62 and b = 1.32. We conclude that model‐based analysis of IVR and of the E‐wave elucidates DF mechanisms common to both. The observed in vivo relationship provides novel insight into diastole itself and the model‐based causal mechanistic relationship that couples IVR to early rapid filling.

Kinematic Modeling Based Decomposition of Transmitral Flow (Doppler E-Wave) Deceleration Time into Stiffness and Relaxation Components

The mechanical suction-pump feature of the left ventricle aspirates atrial blood and generates a rapid rise and fall in transmitral flow (Doppler E-wave). Initially, E-wave deceleration time (DT), a routine index of clinical diastolic function, was thought to be determined only by chamber stiffness. Kinematic modeling of filling, in analogy to damped oscillatory motion [Parametrized Diastolic Filling (PDF) formalism], has been extensively validated and accurately predicts clinically observed E-wave contours while, revealing that DT is actually an algebraic function of both stiffness (PDF parameter k) and relaxation (PDF parameter c). We hypothesize that kinematic modeling based E-wave analysis accurately predicts the stiffness (DTs) and relaxation (DTr) components of DT such that DT = DTs + DTr. For validation, pressure–volume (P–V) and E-wave data from 12 control (DT 220 ms) subjects, 738 beats total, were analyzed. For each E-wave, DTs and DTr was compared to simultaneous, gold-standard, high fidelity (Millar catheter) determined, chamber stiffness (K = ΔP/ΔV) and chamber relaxation (time-constant of isovolumic relaxation—τ), respectively. For the group linear regression yielded DTs = αK + β (R = 0.82) with α = −0.38 and β = 0.20, and DTr = mτ + b (R = 0.94) with m = 2.88 and b = −0.12. We conclude that PDF-based E-wave analysis provides the DTs and DTr components of DT with simultaneous chamber stiffness (K) and relaxation (τ) respectively, as primary determinants. This kinematic modeling based method of E-wave analysis is immediately translatable clinically and can assess the effects of pathology and pharmacotherapy as causal determinants of DT.

Automated method for characterization of diastolic transmitral doppler velocity contours: Early rapid filling

Doppler echocardiographic studies of transmitral flow have become a routine clinical tool for the assessment and characterization of ventricular diastolic (filling) function. We have previously derived a parametrized diastolic filling (PDF) formalism for the purpose of diastolic function assessment using Doppler echocardiography. The model accommodates the mechanical “suction” feature of early diastolic filling of the heart by using a simple harmonic oscillator (SHO) as a paradigm for the kinematics of filling. PDF model predictions of transmitral flow velocity have shown excellent agreement with human echocardiography Doppler contours (temporal profiles) when a visual, transparency overlay method of model fit to clinical Doppler contour comparison was used. The determination of PDF model parameters from the clinical Doppler contour is equivalent to the solution of the “inverse problem” of diastole. Previously, this determination consisted of a manual, iterative method of graphical overlay, in which model predicted contours were visually compared with the echocardiography machine generated Doppler contour using transparencies. To automate the process of model parameter estimation ( i.e., solution of the “inverse problem”) for the early or “rapid filling” phase of diastole (known in cardiology as the E-wave of the clinical Doppler velocity profile [Dvp]) we recorded the acoustic pulsed Doppler signal using the forward channel of a commercial echocardiography machine. The Doppler spectrogram for a particular E-wave was recreated using short-time Fourier transform processing. The maximum velocity envelope (MVE) was extracted from the spectrogram. The PDF model was fit to the E-wave MVE using a Levenberg-Marquardt (iterative) algorithm by the requirement that the mean-square error between the clinical data (MVE) and the model be minimized. Because the model is linear, all of the PDF parameters for the Doppler E-wave can be uniquely determined. We show that: (1) solution of the “inverse problem of diastole” is possible; (2) clinical Doppler E-wave contours can be accurately reproduced and quantified using the PDF formalism and its parameters; and (3) our proposed, automated method of PDF parameter determination for the E-wave is robust.

Response of blood flow entering the heart to dissynchronous ventricular recoil

To account for mechanical suction at the initiation of the cardiac filling process, we have previously modeled transmitral blood flow driven by synchronous ventricular recoil in terms of a one-dimensional simple harmonic oscillator (SHO) in the Parametrized Diastolic Filling (PDF) formalism. PDF model predictions of blood flow velocity have shown excellent agreement with the diastolic Doppler velocity profile (DVP), the echocardiographically measured blood flow velocity of filling. However, dissynchrony of the cardiac recoil process, defined by the early or late recoil of one region relative to the rest of the ventricle, is known to exist in-vitro and has been observed in-vivo. To model dissynchronous recoil and mathematically explore the range of model predicted blood flow velocity responses, we have developed a new model. Our two-dimensional model utilizes two second order constant coefficient linear differential equations, whose solutions combine to form v( t), the new model's prediction of blood flow velocity. The differential equations are solved as initial value problems with zero initial velocities, unknown initial displacements and unknown damping and spring constants. For a given data set (DVP) we solve the ‘inverse problem’ and determine initial oscillator displacements and parameters iteratively by graphical comparison of v( t) to the DVP contour. We discuss the kinematic features of this refined model, its testable predictions, and provide evidence for its validity by comparison of v( t) to the clinical DVP in a case of known dissynchrony. We comment on the benefits and limitations of this modeling approach.

Evaluation of diastolic function with Doppler echocardiography: the PDF formalism.

A new parametrized diastolic filling (PDF) formalism for evaluation of holodiastolic (left and right) ventricular function via Doppler echocardiography is presented. It is motivated by the empiric observation that during diastole the heart behaves as a suction pump whose dynamics, in certain respects, are those of a damped harmonic oscillator. An expression for elastic recoil (suction) initiated ventricular diastolic fluid inflow velocity v(t) is obtained by differentiation from the solution x(t) of the linear differential equation that describes the motion of a forced, damped harmonic oscillator. It is solved for "over-damped" motion, for zero initial velocity and initial displacement = xo cm. An explicit forcing term F(t) = Fosin(omega t) is included to account for late diastolic (atrial) filling. The quantitative parameters of the model include inertia (mass; m), viscosity (damping constant; c), source of stored energy for suction (spring constant; k), and its initial displacement xo, the amplitude and frequency of the (atrial) forcing term Fo, omega. The mathematical behavior of the solution v(t) and its dependence on the parameters xo, c, and k, which characterize the contour of the Doppler velocity profile (DVP), is discussed. When clinical examples of normal and abnormal transmitral DVPs are compared with v(t) calculated using the harmonic oscillator model, excellent agreement [DVP-v(t)]/v(t) approximately 0.05 is obtained throughout diastole. Thus the model allows accurate qualitative and quantitative characterization of global ventricular diastolic behavior by noninvasive means in a variety of normal and abnormal stiffness-compliance states. In addition, it may serve as a prototype for a class of mathematical models that can encompass the essential dynamic elements of ventricular diastolic function that couple to flow and further enhance the role of the heart as a suction pump.