# Relationship of the Third Heart Sound to Transmitral Flow Velocity Deceleration

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## Abstract

*Background *The third heart sound (S_{3}) occurs shortly after the early (E-wave) peak of the transmitral diastolic Doppler velocity profile (DVP). It is thought to be due to cardiohemic vibrations powered by rapid deceleration of transmitral blood flow. Although the presence, timing, and clinical correlates of the S_{3} have been extensively characterized, derivation and validation of a causal, mathematical relation between transmitral flow velocity and the S_{3} are lacking.

*Methods and Results *To characterize the kinematics and physiological mechanisms of S_{3} production, we modeled the cardiohemic system as a forced, damped, nonlinear harmonic oscillator. The forcing term used a closed-form mathematical expression for the deceleration portion of the DVP. We tested the hypothesis that our model’s predictions for amplitude, timing, and frequency of S_{3} accurately predict the transthoracic phonocardiogram, using the simultaneously recorded transmitral Doppler E wave as input, in three subject groups: those with audible pathological S_{3}, those with audible physiological S_{3}, and those with inaudible S_{3}.

*Conclusions *We found excellent agreement between model prediction and the observed data for all three subject groups. We conclude that, in the presence of a normal mitral valve, the kinematics of filling requires that all hearts have oscillations of the cardiohemic system during E-wave deceleration. However, the oscillations may not have high enough amplitude or frequency to be heard as an S_{3} unless there is sufficiently rapid fluid deceleration (of the Doppler E-wave contour) with sufficient cardiohemic coupling.

The third heart sound (S_{3}) is a soft, low-frequency sound in the range of 10 to 50 Hz.^{1} It occurs during deceleration of early rapid ventricular filling, after the early rapid filling (E-wave) peak of the transmitral Doppler echocardiogram. Although some studies claim that S_{3}s can occur before the E-wave peak,^{2} most studies concur that S_{3} occurs either coincident with or shortly after the E-wave peak.^{3} ^{4} ^{5} In characterizing physiological correlates of the S_{3}, investigators have found that a “more rapid than normal” deceleration of flow is more likely to produce an S_{3} than when the deceleration is “less rapid.”^{3} ^{4} It is believed that blood flow through the mitral valve, decelerating in the left ventricle, causes vibrations of the entire cardiohemic structure, that is, the ventricular wall, intraventricular blood pool, and surrounding structures. These vibrations can be heard as an S_{3}, provided the rate of deceleration of transmitral blood flow is high enough to provide sufficient power for sound generation.

Presence of an S_{3} is also associated with a steeper rise in the early filling portion of the left ventricular pressure tracing, which is related to decreased ventricular compliance.^{6} ^{7} This rapid, compliance-related pressure rise affects the frequency of S_{3} and is inscribed after minimum left ventricular diastolic pressure. It corresponds to a rapid rise of the reversed (left ventricular pressure> left atrial pressure) transmitral pressure gradient and a consequent rapid deceleration of transmitral blood flow.^{7} ^{8} The effect of the rapid development of this pressure gradient on the transmitral Doppler echocardiogram is a more rapid deceleration or steeper downslope of the E wave.

Clinical correlates of S_{3} are found in patients with certain ventricular disorders as well as in some healthy children and young athletes. A low-amplitude, low-frequency S_{3} that is not audible but can be seen on a phonocardiographic recording is commonly present in many physiologically normal young people but is rarely seen after the age of 35 years. In older patients, presence of an audible S_{3} may be indicative of pathological conditions such as mitral regurgitation, anemia, thyrotoxicosis, and left ventricular failure.^{9} The presence of an S_{3} in older adults with left ventricular failure can be an ominous prognostic sign. In young, healthy people with S_{3}s, the mechanism of S_{3} generation is slightly different than in older patients with heart disease. Studies have shown that in young subjects, the steeper rate of deceleration is caused by a higher maximum E-wave velocity while the duration of deceleration remains the same, whereas in subjects with heart disease the steeper rate of deceleration is explained by a normal E-wave amplitude with a shorter deceleration time.^{4} Regardless of which of these mechanisms is primarily responsible, it is this greater rate of deceleration that results in the rapid cessation of transmitral blood flow, which is responsible for the S_{3}.

We investigated the effects of fluid deceleration in the ventricle and its causal relation to oscillation of the cardiohemic system. We chose a kinematic model for the motion of the lumped system (the ventricle, its contents, and surrounding structures) displaced by inertial forces (blood deceleration). The lumped parameters include inertia, resistance, and stiffness. The system is allowed to oscillate in response to the external force derived from the inertia of fluid deceleration. We specifically avoided any attempt to anatomically model individual cardiohemic features or cardiac structures because we sought the response of the lumped system as a whole to the effects of fluid deceleration rather than the behavior of any individual structure. We derived a cause-and-effect rule in mathematical terms that all filling hearts must follow, predicted mathematically and verified by experimental in vivo data in humans, that oscillations (of the lumped-parameter cardiohemic system) must result from the application of a decelerating force.

## Methods

To investigate the mechanism of S_{3} production and its relation to the Doppler E wave, we treated the cardiohemic system in kinematic terms as a forced nonlinear harmonic oscillator having lumped parameters of inertia, resistance, and time-dependent stiffness. The oscillator’s forcing term, reflecting the role of blood deceleration as the source of S_{3}, was derived from the Doppler E-wave contour. We sought to predict the existence and attributes of S_{3} based solely on information from the model and the Doppler E wave. Parameters of the nonlinear harmonic oscillator include inertia *M*, resistance *C*, and time-dependent stiffness *K(t)*. The mathematical details of the model derivation are presented in the “Appendix”; however, important features are presented in this section. The equation of motion (Newton’s law) for the system is

where the dot denotes differentiation with respect to time, *u*(*t*) is the oscillator’s displacement, *G*(*t*) is the forcing term derived from the clinical E-wave contour, and *v* is transmitral flow velocity. We rewrite the forcing term by introducing a variable *Q*, a coupling constant that expresses the fraction of the applied force that contributes to the production of sound. The remainder of the total driving force is dissipated as heat, fluid turbulence, and motion of the system as a whole. Once values for *C* and *K*(*t*) are specified, solution of Equation 1 with initial conditions *u*(0)=0 and *du/dt*(0)=0 (indicating that the oscillator has no initial stored energy and is displaced in response only to the forcing term) predicts the amplitude, duration, and frequency of oscillation of the system. Fig 1⇓ shows a schematic of the oscillator for S_{3} generation.

We used established facts of cardiac physiology to estimate or derive the damping constant *C* and the stiffness parameter *K*(*t*). (These derivations are detailed in the “Appendix.”) The damping constant can be estimated by considering the (measured) typical duration of an S_{3} and choosing a numerical value for *C*, which damps oscillations on the same timescale. The time-dependent stiffness can be obtained by considering *dP/dV*, where *P* is the ventricular pressure and *V* is the ventricular volume. It is derived from the Doppler E wave by considering Bernoulli’s equation for nonsteady flow. The final equation of motion (see derivation in “Appendix”) is

where *M* has been set to unity to yield parameters per unit mass, and *v*, *v̇*, and *v̈* are obtained from differentiation of the Doppler E-wave velocity contour. *S* represents an effective surface area driven by the decelerating fluid, *Δx* is the spatial region of acceleration, ρ is the density of blood, and μ is an empirically determined proportionality constant indicating the fraction of fluid energy that contributes to a change in chamber pressure.

### Oscillatory Properties

In accordance with the properties of solutions to second-order differential equations, the solution of Equation 2, u(t), will oscillate if *C*^{2}−4*K*(*t*)<0. Since *K* is time dependent, there are time periods during the E wave when oscillations are allowed and other periods when oscillations are prohibited. When *v̇* is positive (the acceleration portion of the E wave), oscillations are prohibited. When *v̇* is negative (the deceleration portion of the E wave), *C*^{2}−4*K*(*t*) becomes negative, and oscillations, which damp out on a timescale ≈2/*C*, are permitted. Therefore, the model predicts damped oscillations of the cardiohemic system (heart sounds) shortly after the peak of the E wave, when the oscillatory parameters are such that the system can oscillate (the “underdamped” regime). Note that the frequency of the oscillations is time dependent because *K* is time dependent (see Equation 8 in “Appendix”).

To illustrate the properties of the model and its solution, Fig 2⇓ shows the velocity contour for a hypothetical E wave, the derivative of the velocity contour (proportional to the forcing function in Equation 1), and the model-predicted oscillation, *u*(*t*). Note that the oscillation starts shortly after the peak of the E wave, after the curve of *v̇* dips below zero.

### Numerical Methods

A Macintosh Quadra 950 computer and labview software were used to solve Equation 2 numerically for *u*(*t*) using Euler’s method and a time step of 1 millisecond. Solutions were displayed graphically in labview. We conducted numerical experiments to determine the effects of varying the parameters *C*, *Q*, μ/*S*, and *Δx*, as well as varying the shape of the Doppler E wave.

### Data Acquisition

After appropriate informed consent, in accordance with Jewish Hospital and Washington University Medical Center guidelines, simultaneous transmitral Doppler echocardiographic and phonocardiographic recordings were obtained from patients and volunteers with and without audible S_{3}s. The presence of an audible S_{3} was initially determined by an experienced clinical cardiologist using a stethoscope and independently verified by at least two out of three independent, blinded observers.

An Acuson 128XP echocardiography machine, with a model S2194 transducer and a model 24601 microphone, was used to perform the examinations. For each subject, the microphone was attached to the chest over the apical impulse, using a built-in suction cup.^{10} A transmitral, pulsed Doppler study was performed by a certified echocardiography technician using the apical four-chamber view.^{11} To capture the data on our computer, the forward-channel audio Doppler and phonocardiogram outputs from the echocardiography machine were connected to a 12-bit, multichannel, analog-to-digital converter resident in the Quadra 950. The filters on the Acuson were set to allow the highest possible band width for each signal (125 Hz for the audio Doppler high-pass filter and 30 to 800 Hz for the phonocardiogram band-pass filter). These signals were digitized at 10 kHz to capture the full band width of the audio Doppler signal. The data were written to the computer’s hard disk for off-line analysis. Data collection epochs were 30 second each.

### Comparison of Model With Clinical Data

To carry out a robust validation process, we tested our hypothesis (that our model’s predictions for amplitude, timing, and frequency of S_{3} accurately predict the transthoracic phonocardiogram) by recording simultaneous phonocardiographic and transmitral Doppler velocity profiles in three distinct physiological categories: (1) audible pathological S_{3}s (three adult subjects), (2) audible physiological S_{3}s (three young subjects), and (3) subjects without audible S_{3}s (five young subjects). We performed a graphic comparison of the phonocardiographic recordings to the model-predicted oscillations, using the derivative of the clinical E-wave contour as the driving force for the S_{3} model.

We obtained a closed-form, mathematical expression for the contour of the clinical Doppler E wave by using a previously validated model of diastolic filling, the parameterized diastolic filling (PDF) formalism.^{12} The model’s derivation, properties of its solution, and other details can be found in Reference 12. In brief, the model uses a simple harmonic oscillator as a paradigm for the kinematics of filling and generates a unique, invertible, closed-form expression for the entire contour of the Doppler E and A waves. Model inversion using the clinical Doppler velocity profile images (which were reconstructed from the digital acoustic Doppler signal) provide unique E-wave model parameters (*c*, *k*, and *x*_{o}) and associated standard deviations.^{13} An illustrative example of a digitally acquired and reconstructed clinical Doppler E-wave image with the corresponding PDF model–predicted velocity contour superimposed is shown in Fig 3⇓.

The E-wave contour, determined by the PDF parameters *c*, *k*, and *x*_{o} obtained in the above manner, was used in the S_{3} model to calculate the predicted oscillation (the solution of Equation 2). The terms *S*/μ, *Q*, and *Δx* relate the model to the observed data in parametric terms. The coupling constant *S*/μ was chosen as 1%>*S*/μ>0.1% to obtain oscillations in the 10- to 50-Hz frequency range for the S_{3}. The damping constant, *C*, was chosen so that the oscillation damped in a time ≈2/*C*=0.02 to 0.05 seconds. It was typically in the range of 40 to 80 g/s. *Q* is a scaling factor in the solution, and its value was kept constant throughout the study. *Δx* was typically chosen to be in the range 0.01<*Δx*<1.0 cm so that oscillations commence at the appropriate point in time relative to the peak of the Doppler E wave.

## Results

Numerical experiments show that, shortly after the peak of the Doppler E wave, the solution of Equation 2 exhibits damped oscillation with frequency given (by Equation 7) in the “Appendix.” Fig 2⇑ shows an E wave, the forcing function, and the corresponding model-predicted oscillation. The oscillations in Fig 2⇑ are a general consequence of the kinematics of the system, that is, as long as the system has mass, appropriate magnitude for damping, and stiffness and is subjected to an external driving force (fluid deceleration), it must oscillate. What varies from case to case is the amplitude, frequency, and time of onset of the oscillations. Hence, this is general kinematic law that all filling hearts must follow.

### Dependence of Model-Predicted S_{3} on Parameters

The time at which the oscillation begins depends primarily on the parameter *Δx* and to a lesser extent on *C*. For increasing values of *Δx*, the onset of the model-predicted S_{3} moves further beyond the peak of the E wave. *Q* is a scaling factor related to the amplitude of the heart sound, while *S*/μ affects the oscillation frequency (see “Appendix”).

We conducted numerical experiments to observe how the magnitude of the model-predicted S_{3} solution depends on the shape of the Doppler E wave by varying the E-wave PDF parameters *c*, *k*, and *x*_{o}. For a fixed value of *Q*, the magnitude of the model-predicted S_{3} solution increases with increasing the E-wave parameter values for *k* and *x*_{o} (higher amplitude and narrower E wave) because these parameter values correspond to higher deceleration rates. The amplitude decreases with increasing values of the E-wave parameter *c* (broader E wave) because this corresponds to slower E-wave deceleration rates.^{12} ^{13} Thus, the model predicts higher-amplitude oscillations for those patients who have higher-amplitude phonocardiographic S_{3} recordings (or audible S_{3}s) because these patients tend to have steeper rates of E-wave deceleration.^{3} ^{4}

The amplitude of the phonocardiographically recorded S_{3} is dependent not only on the mechanics of filling but also on the coupling of the sound vibrations to the chest wall and other nonquantifiable factors such as the coupling between the chest wall and the heart sound microphone. In phonocardiographic recordings, we adjusted the amplifier gain to obtain the highest possible resolution for the digital phonocardiographic signal. As a consequence, no calibrated or absolute determination of S_{3} amplitude on a subject-to-subject basis was possible. For a given subject, the coupling constant *Q* (verified by numerical experiments) is a constant multiplier of the S_{3} amplitude. Its variability on a subject-to-subject basis indicates that the magnitude of the coupling of fluid deceleration to the cardiohemic system is variable among subjects.

### Comparison of Model With Clinical Data

Graphic comparisons between the model’s predictions and the measured S_{3} were conducted for all subjects. We did not perform a quantitative amplitude comparison between the model-predicted oscillation and the clinical S_{3} data because of our inability to obtain a calibrated or absolute measure of recorded S_{3} amplitude. The differences between patients with respect to coupling between the sound and the chest wall also prevented reliable amplitude comparison. We compared timing, duration, and frequency in graphic form (See Figs 4 through 6⇓⇓⇓). Further model-to-data comparison will not provide additional information. Our model-predicted oscillation, when graphically compared with the clinical data, shows excellent agreement with respect to timing, duration, frequency, and relative amplitude (E waves with steeper decelerations have higher-amplitude S_{3}s than E waves with slow deceleration).

#### Audible and Phonocardiographically Present Pathological S_{3}

Fig 4⇑ illustrates simultaneously recorded clinical Doppler velocity profiles (DVPs) and phonocardiograms in subjects with pathological S_{3}s. In addition, the velocity contour for the E wave from the kinematic model^{12} and the corresponding nonlinear oscillator-predicted sound in two patients having audible S_{3}s are shown. Fig 4A⇑ shows data from a 63-year-old male patient with acute myocardial infarction; Fig 4B⇑ shows data from a 25-year-old diabetic female patient. Note the close agreement (time of onset, duration, and frequency of oscillation) between the phonocardiogram and the model prediction. Also note the variation in the magnitude of the baseline noise typical of different subjects.

#### Audible and Phonocardiographically Present Physiological S_{3}

Fig 5⇑ is the same format as Fig 4⇑, in subjects with physiological S_{3}s. Fig 5A⇑ shows data from a 44-year-old female volunteer; Fig 5B⇑ shows data from a 23-year-old male athlete.

#### Inaudible and Phonocardiographically Absent S_{3}

Fig 6⇑ illustrates an entire diastolic interval consisting of E and A waves for a healthy 20-year-old subject with no phonocardiographic evidence of an S_{3}. Although the model predicts an oscillation for this data, its amplitude is 100 dB lower (a factor of 10^{5}) than the predicted amplitude of the S_{3} shown in subjects in whom the phonocardiogram recorded an S_{3}. Therefore, for the slow E-wave deceleration seen in this subject, the predicted oscillation does not have high enough amplitude to be recorded or to be audible as a heart sound. (The oscillation in the phonocardiographic tracing seen during the A wave on the DVP is a fourth heart sound.)

#### Inaudible and Phonocardiographically Present S_{3}

Fig 7⇓ illustrates low-amplitude S_{3}s seen on the phonocardiographic recording of a young (age, 21 years), healthy volunteer with no audible S_{3}. Several seconds of the phonocardiogram are shown in Fig 7A⇓. The phonocardiogram was filtered using a low-pass filter with cutoff frequency of 100 Hz to minimize background noise. The log power was computed, averaged with a sliding window of 0.05 seconds, and the result displayed in Fig 7B⇓. Although the presence of an S_{3} may be difficult to discern from baseline noise in the raw data by the untrained observer (Fig 7A⇓), the presence and periodic nature of S_{3} is clearly discernible in the log power display in the graph. Fig 7C⇓ illustrates the Doppler E wave, the predicted contour,^{12} and the nonlinear oscillator-predicted S_{3} for the beat of the phonocardiogram in Fig 7A⇓ marked with the arrow.

## Discussion

Our model predicts that damped cardiohemic oscillations occur as a result of fluid deceleration and expresses this relation mathematically. We specifically avoided attempts at anatomic modeling in our derivation in the sense of modeling vibrations of individual anatomic cardiohemic structures because we sought the kinematic behavior of the system as a whole. The parameters *C* and *K(t)* represent the lumped (kinematic) properties of the cardiohemic system. Our motivation was to mathematically express, predict, and verify, by comparing model prediction with in vivo data, that S_{3} generation is causally related to and kinematically explained by the deceleration of transmitral flow.

The timing, presence, and clinical correlates of the S_{3} have been studied extensively. It has been shown that S_{3}s are associated with E waves that have a “more rapid than normal” deceleration.^{3} ^{4} It has been suggested that S_{3} oscillation is initiated when the rate of ventricular filling exceeds ventricular diastolic distensibility, causing a sudden deceleration of the E wave.^{14} Its source can be envisioned as being due to loss of fluid kinetic energy that drives the oscillation. No prior attempts have been made to kinematically derive a causal mathematical relation between transmitral flow deceleration and the genesis of S_{3}.

The first studies of the third heart sound were done in the late 19th century by Potain,^{15} who attributed the S_{3} to sudden cessation of distention (stiffening) of the ventricle in early diastole. Researchers in the past have attributed the S_{3} to valvular causes,^{16} ^{17} ^{18} but the advent of M-mode ultrasound indicated that this was not the case because S_{3} occurs as the mitral valve is moving toward partial closure (E-wave deceleration) from its fully open position (E-wave peak). Other causes have been suggested, such as a sudden intrinsic limitation of longitudinal expansion of the left ventricular wall during early diastolic filling.^{19} The current accepted view is that S_{3} is caused by compliance-related rapid deceleration of transmitral flow and associated vibration of the entire cardiac–blood pool (cardiohemic) system. The more rapid the deceleration, the more likely an S_{3} will be present.^{2} ^{3} It is known that in patients after bypass surgery, a decrease in deceleration time of the Doppler E wave is associated with increased diastolic chamber stiffness, that is, decreased compliance.^{20}

Since our model uses blood flow deceleration as the forcing term for the nonlinear harmonic oscillator, higher-amplitude solutions are associated with higher forcing amplitudes, that is, higher rates of deceleration of the E wave. This is in good agreement with previous data. Higher E-wave deceleration rates also result in higher oscillation frequencies (see “Appendix”). Therefore, the steeper the downslope of the E wave, the more likely the oscillation (sound) is to be of sufficiently high frequency and amplitude to be heard through a stethoscope as a third heart sound.

One particularly interesting feature of our model is the prediction that all E waves that decelerate will generate some degree of oscillation, that is, all cases should have vibrations of the cardiohemic system associated with fluid deceleration. (In cases of mitral stenosis where very low E-wave deceleration rate may be present, the usual S_{3} oscillations beyond the peak of the E wave are not expected. The “opening snap” of mitral stenosis, however, merits further investigation as an example that could be modeled and explained by similar kinematics.) However, these oscillations may not have high enough amplitude or sufficiently high frequency to be distinguished from background noise or may fall below the threshold of human hearing (audibility), which is logarithmically related to the amplitude and frequency of the signal.^{21} S_{3} is commonly present in normal young people as a low-frequency, usually subaudible vibration that can sometimes be seen on the phonocardiogram. Our model’s predictions agree with these semiquantitative observations (see Fig 7⇑). Of five young volunteers without audible S_{3} by multiple independent auscultators, four had visible phonocardiographic (physiological) S_{3} oscillation.

For an S_{3} to be audible clinically, the acoustic energy reaching the thoracic surface must be above a certain threshold. The frequency must be high enough to fall in the audible range (>20 to 40 Hz) and there must be sufficient coupling between the sound (oscillations) and the chest wall. One limitation of our study was our inability to quantitate the individual differences among subjects with respect to this coupling. The phonocardiographic device provided with our echocardiography machine was such that we were unable to obtain a calibrated or absolute measure of recorded S_{3} amplitude from subject to subject. Other factors that contributed to this problem were varying amounts of background noise, the nature of the contact between the heart sound recording transducer and the chest wall, and attenuation of the sound through the chest. For example, the fluid inside the heart will only support longitudinal waves, since a (nearly ideal) fluid will not support transverse (shear) waves. In the chest wall, however, longitudinal waves will be quickly damped out, and most of the energy will be transmitted in the form of shear waves. A potential solution of this problem and a further test of our model’s predictions would be to obtain appropriately calibrated intracardiac heart sound recordings in conjunction with Doppler echocardiography and phonocardiography.

When comparing clinical data with model-predicted features in subjects with and without S_{3}s, we obtained excellent agreement with the model’s predictions. Those who had S_{3}s had higher model-predicted amplitudes than those who had no visible phonocardiographic oscillations (keeping the parameter *Q* constant in the model). The differences in amplitude between those who had S_{3}s and those who had no phonocardiographic oscillation varied by several orders of magnitude in most cases. For example, the subject in Fig 6⇑ had no S_{3} on the phonocardiographic recording. The model-predicted amplitude for this patient (a 60-year-old with left bundle branch block) was 100 dB lower (lower by a factor of 10^{5}) than the model-predicted amplitude for the subjects in Figs 4⇑ and 7⇑ who either had audible S_{3}s (Fig 4⇑) or visible phonocardiographic oscillations (Fig 6⇑). Note that the deceleration of the E wave in Fig 5⇑ is not as steep as the deceleration in the figures with S_{3}s. Since higher rates of deceleration are associated with S_{3}s and higher deceleration rates cause higher model-predicted oscillation amplitudes, agreement seen in this respect is expected.

The model-predicted amplitude for those with audible S_{3}s tended to be higher by a factor of 20 to 40 dB than those without audible S_{3}s but who had some visible phonocardiographic oscillation. These subjects had steeper E-wave decelerations and higher amplitude model-predicted oscillations than those with no visible phonocardiographic oscillations.

The results of our study indicate that a causal relation between transmitral flow and the S_{3} can be expressed in mathematical terms and kinematically explains why all cases must have oscillations of the cardiohemic system associated with the Doppler E wave. The force transmitted to the ventricle and the blood pool and surrounding structures by the deceleration of transmitral flow generates oscillations. The greater the deceleration, the greater the force that is applied during the period of deceleration and the higher the amplitude and frequency of the resulting oscillation. If the amplitude is high enough and the frequency is in the audible range, the oscillation will be heard as an S_{3}, provided the coupling between the cardiohemic system and the chest wall facilitates transmission to the surface of the chest.

## Appendix A

### Mathematical Details

#### Forcing Term

The forcing term *G*(*t*) in Equation 1 is obtained directly from the clinically recorded E-wave contour. The force driving the oscillation is provided by the rate of change of momentum, *dp/dt*, of the decelerating blood. Since momentum *p=Mv*, where *v* is the blood inflow velocity, and *M*=1, the force driving cardiohemic oscillation (S_{3}) is the time derivative of the E-wave Doppler velocity contour. The physiological cause of the E-wave deceleration is the decrease of the atrioventricular pressure gradient, which generates the E wave.^{8}

#### Damping Constant

The magnitude of the damping constant *C* can be estimated by considering the (measured) typical duration of an S_{3}. An exponential decay term (e^{−ct/2}) having decay constant ≈2/*C* determines the envelope of oscillation of the solution of Equation 1. Hence, the appropriate value for *C* is that which damps the solution on the same timescale as a typical S_{3}.

#### Stiffness

The kinematic equivalent of the system’s stiffness can be obtained by considering *dP/dV*, where *P* is the ventricular pressure and *V* is the ventricular volume. This quantity is a function of time: *dP/dV*=(*dP/dt*)/(*dV/dt*). An expression for *dP/dt* can be obtained by differentiating Bernoulli’s equation for nonsteady flow. Bernoulli’s equation is written as

where *v* is blood velocity, *P* is pressure, and the subscripts 1 and 2 refer to the atrium and the ventricle, respectively. The one-dimensional distance along a streamline is *dx*, ρ is the density of blood, and μ is an empirically determined proportionality constant indicating the fraction of fluid energy that contributes to a change in chamber pressure.

To obtain *dP*_{2}*/dt*, we make several simplifying approximations: (1) the velocity in the atrium is less than the velocity in the ventricle during the E wave (*v*_{1}^{2}/*v*_{2}^{2}<<1), (2) the rate of change of pressure in the atrium is less than the rate of change of pressure in the ventricle during the E wave ([*dp*_{1}/*dt*]/[*dp*_{2}/*dt*]<<1), and (3) *dv/dt* is constant over the spatial region of acceleration, *Δx* (Reference 22). Using these approximations and differentiating, Equation 3 yields

where *Δx* is the spatial region of acceleration. The subscript 2 has been dropped for simplicity. Filling volume *V* is given by *V*=*A∫vdt*, hence *dV/dt=Av*, where *A* is the mitral valve area and *v* is transmitral blood flow velocity, given by the E-wave contour. The resulting expression for *dP/dV* during the E wave becomes

From Equation 5 for *dP/dV* (pressure per unit volume) in the ventricle, the time-dependent “spring constant” or stiffness, *K* (force per unit length), can be found by conversion to its one-dimensional kinematic analogue. This yields

where *S* is an “effective surface area” driven by the decelerating blood. Thus μ/*S* can be viewed as a coupling constant per unit surface area of the endocardium. The resulting kinematic equation for S_{3} generation then becomes

where *M* has been set to unity and *v*, *v̇*, and *v̈* are obtained from differentiation of the Doppler E-wave velocity contour. The oscillations will have characteristic frequencies given by

Note that the frequency is a function of time.

## Acknowledgments

This study was supported in part by the Whitaker Foundation, the Cardiovascular Biophysics Laboratory Research Fund, and Delos Research Fellowship (A.L.M.), Washington University Department of Physics. The authors would like to thank Jose Rosado, MD, Peggy Brown, Tinoa Terry, Galina Dub, and the staff of the Cardiovascular Biophysics Laboratory for their assistance in echocardiographic and phonocardiographic data recording.

- Received December 27, 1994.
- Accepted January 19, 1995.

- Copyright © 1995 by American Heart Association

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- Relationship of the Third Heart Sound to Transmitral Flow Velocity DecelerationAbigail L. Manson, Scott P. Nudelman, Michael T. Hagley, Andrew F. Hall and Sándor J. KovácsCirculation. 1995;92:388-394, published online before print August 1, 1995http://dx.doi.org/10.1161/01.CIR.92.3.388
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- Relationship of the Third Heart Sound to Transmitral Flow Velocity DecelerationAbigail L. Manson, Scott P. Nudelman, Michael T. Hagley, Andrew F. Hall and Sándor J. KovácsCirculation. 1995;92:388-394, published online before print August 1, 1995http://dx.doi.org/10.1161/01.CIR.92.3.388

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