Evaluation of diastolic function with Doppler echocardiography: the PDF formalism

1987 ◽  
Vol 252 (1) ◽  
pp. H178-H187 ◽  
Author(s):  
S. J. Kovacs ◽  
B. Barzilai ◽  
J. E. Perez
Keyword(s):  
Harmonic Oscillator ◽  
Diastolic Function ◽  
Doppler Echocardiography ◽  
Diastolic Filling ◽  
Doppler Velocity ◽  
Elastic Recoil ◽  
Initial Displacement ◽  
Suction Pump ◽  
Forcing Term ◽  
Damped Harmonic Oscillator

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.

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

2020 ◽  
Vol 318 (5) ◽  
pp. H1059-H1067 ◽  
Author(s):  
Druv Bhagavan ◽  
William M. Padovano ◽  
Sándor J. Kovács
Keyword(s):  
Diastolic Function ◽  
Longitudinal Velocity ◽  
Force Balance ◽  
Balance Model ◽  
Diastolic Filling ◽  
Restoring Force ◽  
Suction Pump ◽  
Restoring Forces ◽  
Pdf Model

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−α tsinh(β t), 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.

scholarly journals DIASTOLIC FUNCTION: MEASURING LEFT VENTRICULAR UNTWISTING AS A DAMPED HARMONIC OSCILLATOR

2021 ◽  
Vol 77 (18) ◽  
pp. 583
Author(s):  
Forrest Gamble ◽  
Rifqi Aufan ◽  
Oleg F. Sharifov ◽  
Lamario J. Williams ◽  
Shane Reighard ◽  
...  
Keyword(s):  
Harmonic Oscillator ◽  
Diastolic Function ◽  
Left Ventricular ◽  
Damped Harmonic Oscillator

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

Path integral for the damped harmonic oscillator coupled to its dual

1994 ◽  
Vol 35 (3) ◽  
pp. 1185-1191 ◽  
Author(s):  
L. Chetouani ◽  
L. Guechi ◽  
T. F. Hammann ◽  
M. Letlout
Keyword(s):  
Harmonic Oscillator ◽  
Path Integral ◽  
Damped Harmonic Oscillator
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