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

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DOI: 10.1007/s13239-014-0176-8

- Cite this article as:
- Mossahebi, S. & Kovács, S.J. Cardiovasc Eng Tech (2014) 5: 25. doi:10.1007/s13239-014-0176-8

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## Abstract

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 (DT_{s}) and relaxation (DT_{r}) components of DT such that DT = DT_{s} + DT_{r}. For validation, pressure–volume (*P*–*V*) and E-wave data from 12 control (DT < 220 ms) and 12 delayed-relaxation (DT > 220 ms) subjects, 738 beats total, were analyzed. For each E-wave, DT_{s} and DT_{r} was compared to simultaneous, gold-standard, high fidelity (Millar catheter) determined, chamber stiffness (*K* = Δ*P*/Δ*V*) and chamber relaxation (time-constant of isovolumic relaxation—*τ*), respectively. For the group linear regression yielded DT_{s} = *α**K* + *β* (*R* = 0.82) with *α* = −0.38 and *β* = 0.20, and DT_{r} = *m**τ* + b (*R* = 0.94) with *m* = 2.88 and *b* = −0.12. We conclude that PDF-based E-wave analysis provides the DT_{s} and DT_{r} 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.

### Keywords

LV stiffnessLV relaxationDiastolic functionPDF formalismE-wave deceleration time### Nomenclature

- AT
E-wave acceleration time (ms)

- DF
Diastolic function

- DR
Delayed relaxation

- DT
E-wave deceleration time (ms)

- DT
_{r} Relaxation component of DT (ms)

- DT
_{s} Stiffness component of DT (ms)

*E*_{dur}E-wave duration

- IVR
Isovolumic relaxation

- IVRT
Isovolumic relaxation time (ms)

*K*Diastatic stiffness (mmHg/mL)

- LV
Left ventricle/ventricular

- LVEDV
Left ventricular end diastolic volume (mL)

- LVEF
Left ventricular ejection fraction

- LVEDP
Left ventricular end diastolic pressure (mmHg)

- NR
Normal relaxation

Parametrized Diastolic Filling

*P*–*V*Pressure–volume

*τ*Time constant of isovolumic relaxation (ms)

## Introduction

The clinical syndrome formerly referred to as “diastolic heart failure” is now called “heart failure with normal” or “heart failure with preserved ejection fraction”. It has been recognized as a major cause of cardiovascular morbidity and mortality and has reached epidemic proportions.15,20,24,36,48 Hence, the ability to quantitate diastolic function (DF) and the presence and severity of diastolic dysfunction is important. Among invasive DF indices, left ventricular (LV) chamber stiffness (Δ*P*/Δ*V*) and relaxation (*τ*) comprise the gold-standard.15,32,47,48 Conventionally, chamber stiffness has been computed from Δ*P*_{avg}/Δ*V*_{avg} using invasive methods.11,21,22,25,31,39 Although obtaining chamber stiffness, Δ*P*_{avg}/Δ*V*_{avg} itself usually involves an “absolute” measurement of LV pressure requiring catheterization, chamber stiffness, being the ratio of two derivatives, is a “relative” index and can be determined using “relative measurement” methodology, such as echocardiography, which is the preferred method of quantitative DF characterization. Hence, Doppler E-wave contours can only provide relative, rather than absolute, pressure information. It is known that model-based analysis of the inflow pattern, i.e., Doppler E-waves, generated by the atrioventricular pressure gradient (a relative measure), can accurately determine LV diastatic (passive) stiffness (also a relative measure).27

*et al.*23 used physiologic modeling to predict that E-wave DT is determined by stiffness alone. Their equation relating stiffness

*K*

_{LV}to DT was:

*ρ*is the density of blood,

*L*the effective mitral plug-flow length, and

*A*the mitral area.

The prediction was experimentally validated (*r*^{2} = 0.88) in conscious dogs by invasively determining LV stiffness (Δ*P*_{avg}/Δ*V*_{avg}).23 An alternative kinematic modeling based analysis that incorporates the mechanical suction-pump feature of the physiology [the Parametrized Diastolic Filling (PDF) formalism] showed19 that the PDF parameter *k* (the analog of stiffness) is the algebraic equivalent of *K*_{LV}. For E-wave contours well fit by the “underdamped” oscillatory regime of motion, the relationship between the PDF stiffness parameter *k* and Little’s expression for stiffness *K*_{LV} is given by *k* = 1.16[*A*/(*ρL*)] *K*_{LV} + 41, *r*^{2} = 0.92.

Although Little *et al.*23 proposed that DT is determined by chamber stiffness (*K*_{LV}) alone, Shmuylovich *et al.*38 showed that two subjects can have indistinguishable E-wave DTs, but can have significantly different catheterization determined (gold standard) chamber stiffness (*dP*/*dV*). They showed that DT is actually jointly determined by both stiffness (PDF stiffness parameter *k*) and relaxation (PDF relaxation parameter *c*).38

LV relaxation is conventionally characterized by the time constant (*τ*) of isovolumic relaxation (IVR),43 where *τ* is the e-folding time (the time interval during which the pressure falls by a factor of 1/e) assuming pressure, after peak –*dP/dt* to mitral valve opening, declines exponentially. The interval from aortic valve closure to mitral valve opening, the isovolumic relaxation time (IVRT), non-invasive echocardiographic measurement, is another commonly used, but less, specific surrogate.41 Chamber stiffness, the slope (Δ*P*/Δ*V*) of the end-diastolic pressure–volume relationship, is usually determined from multiple beats. Diastatic (passive) stiffness is the slope of the diastatic pressure–volume relationship, inscribed by the locus of load varying *P*–*V* points achieved at the end of each diastatic interval after E-wave termination, after the chamber has fully relaxed.6,21,22,34,44 During diastasis, LV and left atrial pressures are equal, the pressure gradient across the mitral valve is zero,6 there is no transmitral flow, hence the resultant forces generated by and acting on the ventricle are balanced (but not zero).35 Accordingly, diastasis comprises the static equilibrium state of the passive LV. In engineering terms, the volume at diastasis is the resting (equilibrium) volume relative to which the chamber oscillates.

## Materials and Methods

### Patient Selection

*E*/

*A*(ratio of

*E*

_{peak}and

*A*

_{peak}(peak of late filling)) < 1 and a DT > 220 ms (12 subjects), normal relaxation (NR) pattern was defined as

*E*/

*A*> 1, DT < 220 ms and normal Doppler tissue velocity (

*E*′ > 8 cm/s) (12 subjects). Both groups included the range of LV end diastolic pressure (LVEDP) representative of a patient population encountered clinically. Among the 12 normal deceleration time (DT) datasets, 8 had normal end-diastolic pressure (LVEDP < 14 mmHg), 2 had 15 mmHg < LVEDP < 20 mmHg and 2 had elevated LVEDP (>21 mmHg). The distribution of LVEDPs in the 12 DR group datasets were: 4 with LVEDP < 14, 4 with 15 < LVEDP < 20 mmHg and 4 with LVEDP > 21. A total of 738 cardiac cycles (31 beats/subject) of simultaneous echocardiographic-high fidelity hemodynamic (conductance catheter) data were analyzed. The clinical descriptors of the 24 subjects and their hemodynamic and echocardiographic indices are shown in Table 1.

Clinical descriptors including hemodynamic and echocardiographic indexes

Clinical descriptors | NR group | DR group | Significance |
---|---|---|---|

| 12 | 12 | N.A. |

Age (years) | 56 ± 11 | 67 ± 11 | 0.02 |

Gender (male/female) | 7/5 | 9/3 | N.A. |

Heart rate (bpm) | 62 ± 10 | 58 ± 4 | 0.20 |

LVEF (%) | 71 ± 7 | 72 ± 1 | 0.89 |

LVEDP (mmHg) | 16 ± 5 | 18 ± 3 | 0.37 |

LVEDV (mL) | 129 ± 25 | 149 ± 43 | 0.17 |

DT (ms) | 185 ± 21 | 252 ± 24 | <0.0001 |

DT | 45 ± 13 | 98 ± 13 | <0.0001 |

DT | 140 ± 11 | 154 ± 16 | <0.03 |

| 24 ± 6 | 39 ± 3 | <0.0001 |

| 76 ± 6 | 61 ± 3 | <0.0001 |

| 1.16 ± 0.19 | 0.84 ± 0.13 | <0.0005 |

IVRT (ms) | 75 ± 6 | 95 ± 9 | <0.0001 |

| 58 ± 6 | 75 ± 5 | <0.0001 |

### Data Acquisition

Our simultaneous high-fidelity, *P*–*V* and echocardiographic transmitral flow data recording method has been previously detailed.1,3,18,19,21,28 Briefly, LV pressure and volume were acquired using a micromanometric conductance catheter (SPC-560, SPC-562, or SSD-1043, Millar Instruments, Houston, TX) at the commencement of elective cardiac catheterization, prior to the administration of iodinated contrast agents. Pressure signals from the transducers were fed into a clinical amplifier system (Quinton Diagnostics, Bothell, WA, and General Electric). Conductance catheterization signals were fed into a custom personal computer *via* a standard interface (Sigma-5, CD Leycom). Conductance volume data were recorded in five channels. Data from low-noise channels providing physiological readings were selected, suitably averaged and calibrated using absolute volumes obtained by calibrated ventriculography during the same procedure.

### Doppler E-Wave Analysis

For each subject, approximately 1–2 min of continuous transmitral flow data were recorded in the pulsed-wave Doppler mode. Echocardiographic data acquisition is performed in accordance with published American Society of Echocardiography30 criteria. Briefly, immediately before catheterization, patients are imaged in a supine position using a Philips (Andover, MA) iE33 system. In accordance with convention, the apical 4-chamber view was used for Doppler E-wave recording with the sample volume located at the leaflet tips. An average of 31 beats per subject of simultaneous echocardiographic-hemodynamic data were analyzed (738 cardiac cycles total for the 24 subjects). DT was measured manually using standard criteria9 as the base of the triangle approximating the deceleration portion of the E wave. Each E-wave was also analyzed *via* PDF formalism (see Appendix) to yield mathematically unique PDF parameters for each E-wave (stiffness parameter (*k*), chamber viscoelasticity/relaxation parameter (*c*), load parameter (*x*_{o})).16,17,21

Because DT has been shown to explicitly depend on both stiffness and relaxation,38 in this work we provide the method that decomposes E-wave DT into its stiffness (DT_{s}) and relaxation (DT_{r}) components. Accordingly DT = DT_{s} + DT_{r}. The decomposition utilizes (PDF) analysis of Doppler E-waves. For validation, we determine the relationship between DT_{s} and DT_{r} and conventional and gold-standard (simultaneous) invasive DF parameters of stiffness (slope of diastatic pressure–volume relationship) and relaxation (*τ*, IVRT). The diastatic pressure–volume relationship is obtained by a linear (or exponential) fit to diastatic load-varying *P*–*V* data. Since previous work45 has shown that a linear or exponential fit to the same diastatic *P*–*V* data yields a similar measure of goodness of fit, a linear fit was used.

### Determination of Diastatic Stiffness from *P*–*V* Data

Hemodynamics were determined from the high-fidelity Millar LV *P*–*V* data from each beat. The method used to compute volumes has been previously detailed.21,27,28,45 Quantitative ventriculography was used to determine end-systolic and end-diastolic volumes which defined (calibrated) the systolic and diastolic volume limits of conductance catheter recorded continuous volume signal. After calibration of conductance volume, LV pressure and volume at diastasis were measured beat-by-beat using a custom MATLAB program. Although relaxation is often fully complete at the end of the E-wave, when diastasis begins, to assure full relaxation and achievement of the passive state of the LV we analyzed data at the end of diastasis, i.e., at ECG P-wave onset. We selected cardiac cycles having diastatic intervals during which pressure as a function of time was essentially constant or varied by <2 mmHg during all of diastasis. At sufficiently low heart rates (HRs), end-diastasis points were defined by ECG P-wave onset.28,45,46 In our analyzed subjects the average HR was 62 ± 10 bpm for the NR group and 58 ± 4 bpm for the DR group. As previously,28,45,46 for each subject diastatic *P*–*V* data points were fit by linear regression, from which diastatic chamber stiffness was determined as the slope (*K*) of diastatic pressure–volume relationship. Micromanometric conductance catheter P measurement precision is < 0.1 (mmHg).

### Determination of Time-Constant of Isovolumic Relaxation from Pressure Data

*τ*is the time-constant of IVR, and

*P*

_{∞}is the pressure asymptote.

As previously described8 the pressure phase plane (*dP*/*dt* vs. *P*) was used to determine *τ* (conventional invasive relaxation index) for each beat in each subject.

### Graphical Determination of Stiffness and Relaxation Components of E-Wave DT

*E*

_{dur}), acceleration time (AT), and DT are measured as usual from Doppler echo data, by approximating E-wave shape as a triangle (Fig. 1). PDF parameters (

*k*,

*c*, and

*x*

_{o}) can be obtained for each E-wave (defined as the original E-wave)

*via*PDF analysis (see Appendix). The effect of DR on an ideal (generated by recoil only) E-wave is to decrease its peak amplitude and lengthen its DT. Accordingly, DT

_{r}is determined by setting PDF relaxation parameter zero (

*c*= 0) and generating an ideal contour

*via*the PDF formalism, using the same

*x*

_{o}and

*k*as the original E-wave. Therefore, the model predicted ideal E-wave has the same stiffness and initial load (displacement from the equilibrium) as the actual E-wave being analyzed, but recoils (oscillates) without resistance. Subtracting the ideal E-wave duration from actual PDF fit total DT yields DT

_{r}(see Fig. 1). Therefore, E-wave DT becomes DT = DT

_{s}+ DT

_{r}.

By determining DT_{s} and DT_{r} of each E-wave, the total DT can be normalized and fractionated as the fraction due to stiffness (*S* = DT_{s}/DT) and the fraction of DT due to relaxation (*R* = DT_{r}/DT) for each cardiac cycle such that *S* + *R* = 1.

### Algebraic Determination of Stiffness and Relaxation Components of E-Wave DT

*c*= 0), \( \omega = \sqrt k \) and the duration of the ideal E-wave using the PDF model is:

*c*≠ 0) and ideal E-wave (

*c*= 0) for the “underdamped” (

*c*

^{2}< 4

*k*) regime of oscillation from the PDF model are

*π*/

*ω*and \( \pi /\sqrt k. \) Therefore, the relaxation component of DT (DT

_{r}) defined by the difference between actual and ideal E-wave durations is:

## Results

_{s}(

*R*

^{2}= 0.63) and DT

_{r}(

*R*

^{2}= 0.89) in 738 analyzed cardiac cycles from 24 subjects. DT

_{s}did not significantly correlate with DT

_{r}(

*R*

^{2}= 0.30) across all subjects. Table 2 shows the average DT components in all subjects (12 NR and 12 DR datasets).

DT components in all 24 subjects

NR | DR | |||
---|---|---|---|---|

DT | DT | DT | DT | |

Subject 1 | 53 ± 4 | 139 ± 6 | 88 ± 12 | 155 ± 17 |

Subject 2 | 54 ± 7 | 130 ± 9 | 94 ± 11 | 148 ± 10 |

Subject 3 | 24 ± 3 | 127 ± 10 | 112 ± 19 | 192 ± 18 |

Subject 4 | 53 ± 8 | 141 ± 12 | 92 ± 12 | 140 ± 17 |

Subject 5 | 56 ± 5 | 158 ± 13 | 111 ± 18 | 165 ± 13 |

Subject 6 | 22 ± 2 | 142 ± 7 | 77 ± 11 | 144 ± 14 |

Subject 7 | 51 ± 10 | 137 ± 12 | 95 ± 13 | 171 ± 15 |

Subject 8 | 39 ± 6 | 125 ± 5 | 89 ± 11 | 160 ± 15 |

Subject 9 | 44 ± 6 | 139 ± 9 | 121 ± 13 | 147 ± 15 |

Subject 10 | 33 ± 5 | 131 ± 5 | 92 ± 12 | 132 ± 13 |

Subject 11 | 56 ± 7 | 153 ± 5 | 112 ± 14 | 141 ± 14 |

Subject 12 | 57 ± 7 | 152 ± 11 | 97 ± 12 | 154 ± 12 |

### Stiffness Component of DT and Diastatic Stiffness

_{s}and diastatic stiffness derived from

*P*–

*V*data (

*K)*were correlated (DT

_{s}= −0.38

*K*+ 0.20,

*R*

^{2}= 0.67) (Fig. 3). The negative slope in DT

_{s}vs.

*K*correlation was expected from the inverse relation between DT (and stiffness component of DT) and chamber stiffness.

### Relaxation Component of DT and Relaxation Indexes

### Fractionation of DT in terms of Stiffness and Relaxation Components in Normal and DR

*p*< 0.0001). Figure 6 shows the fraction of DT accounted for by stiffness (

*S*) in the DR group is significantly less than in the NR group (

*p*< 0.0001), and the fraction of DT due to the relaxation (

*R*) in the DR group is significantly higher than in the NR group (

*p*< 0.0001).

### Interobserver Variability and Bland–Altman Analysis

As in previous work,2 interobserver variability in applying the PDF formalism for E-wave analysis of the current data was ≤8%. Two months after the initial analysis, we carried out an inter-observer variability study where datasets were reanalyzed in random order. Bland–Altman analysis shows that PDF parameters, AT, and DT have very good agreement between observers. Less than 5% of all measurements reside outside 1.96 SD of the percentage difference, in keeping with the criteria of Bland and Altman, representing 95% confidence intervals in the results.

## Discussion

*k*), viscoelasticity/relaxation (

*c*) and load parameters (

*x*

_{o}) for each E-wave. Chamber stiffness11,21,22,25,31,39 is defined by the slope Δ

*P*/Δ

*V*of the

*P*–

*V*relation. Lisauskas

*et al.*has demonstrated the expected high correlation between PDF parameter

*k*and Δ

*P*

_{avg}/Δ

*V*

_{avg}in a large dataset. Little

*et al.*23 have proposed that chamber stiffness (

*K*

_{LV}) is related to E-wave DT as:

*K*

_{LV}was shown to correlate with stiffness determined from the LV end-diastolic pressure–volume relationship.

Considering the physiology in kinematic modeling terms that incorporates the suction-pump attribute of the LV, Shmuylovich *et al.*38 have shown that DT is jointly (algebraically) determined by stiffness (PDF parameter *k*) and relaxation (PDF parameter *c*). Importantly, Shmuylovich *et al.* have also shown that two subjects with *indistinguishable* E-wave determined DTs, mitral valve areas, and chamber volumes (LVEDV) can have *distinguishable* catheterization-determined values of chamber stiffness, because of differences in the viscoelastic/relaxation parameter (PDF parameter *c*) in the two subjects.

Relaxation can be characterized by the time constant (*τ*) or the logistic time constant (*τ*_{L}), from cardiac catheterization data, or by IVRT and DT from echocardiography. The concordance of delayed-relaxation (DT > 220 ms) and associated prolonged *τ* indicates that impaired relaxation is a feature of diastolic dysfunction.15,32,47,48 The PDF chamber relaxation/viscosity parameter *c* has been shown: (1) to have a significant linear correlation with 1/*τ,*3 and with the “pressure recovery ratio”, directly determined from the LV waveform after mitral valve opening,46 and (2) differentiate diabetic from non-diabetic hearts in animals7 and in humans.37

Because constrictive-restrictive E-wave patterns inscribe tall and narrow E-waves with short DT, the E-wave fits generate higher (compared to normal) PDF parameter *k* values, indicating increased stiffness, relative to normal DT patterns. In contrast, PDF fits to DR patterns (long DT) generate higher *c* values, indicating DR.

In the current study we analyzed simultaneous LV *P*–*V* and transmitral flow (echo) data and decomposed E-wave DT is to stiffness (DT_{s}) and relaxation (DT_{r}) components. As expected DT_{s} was highly correlated with (simultaneous) invasively determined (passive) diastatic chamber stiffness.45 Similarly, very strong correlation was observed between DT_{r} and the time-constant of IVR (*τ*) from simultaneous high fidelity pressure data and between IVRT determined by echocardiography.

Our study provides a novel methodologic approach employing rigorous causal analytical and modeling methods, that, for the first time, fractionates total DT into its stiffness and relaxation components.

### The Load Dependence of DT_{s} and DT_{r}

Because all conventional indexes of DF are load-dependent we assessed the correlation between total DT, DT_{s}, DT_{r} and load. Although mitral valve opening pressure is the ideal index of load, it was not available, hence we employed LVEDP as the load surrogate, since LVEDP and mitral valve opening pressure are known to be closely correlated.14,26,29,33 The results, for the group as a whole are that DT vs. LVEDP (*R*^{2} < 0.17), DT_{s} vs. LVEDP (*R*^{2} < 0.13), and DT_{r} vs. LVEDP (*R*^{2} < 0.20) indicating that DT, DT_{s}, DT_{r} are very weakly load-dependent as expected.

### The Heart Rate Dependence of DT_{s} and DT_{r}

The HR dependence of the duration of diastole and its phases (E-wave, diastasis and A-wave) have been previously detailed.4 Importantly, for a 100% increase in HR, E-wave duration diminishes by 15%; hence we expect that DT, DT_{s}, DT_{r} would only be weakly HR dependent. Our results, for the group as a whole, indicate that DT vs. HR (*R*^{2} < 0.21), DT_{s} vs. HR (*R*^{2} < 0.16), and DT_{r} vs. HR (*R*^{2} < 0.20) justify this conclusion.

## Limitations

### Conductance Volume

The conductance catheter method of volume determination has known limitations related to noise, saturation and calibration that we have previously acknowledged.21,27,28,45 In this study, the channels which provided physiologically consistent *P*–*V* loops were selected and averaged. However, since there was no significant volume signal drift during recording, any systematic offset related to calibration of the volume channels did not affect the result when the limits of conductance volume were calibrated *via* quantitative ventriculography.

### Sample Size

Although the number of subjects (*n* = 24) is modest, and may be viewed as a minor limitation, the total number of cardiac cycles analyzed (*n* = 738) and the very high *R*^{2} values observed, mitigates the sample size limitation to an acceptable degree.

## Conclusions

We used the PDF formalism to decompose E-wave DT into its stiffness and relaxation components and utilized *in vivo*, human, simultaneous *P*–*V* and transmitral echocardiographic data to validate model prediction. We showed that DT_{s} is primarily determined by the diastatic (passive) chamber stiffness (*K*), and DT_{r} is determined by relaxation (*τ*). This method is general and can be used to decompose any E-wave into its stiffness and relaxation components. It therefore facilitates rigorous noninvasive assessment of the differential effects of pathophysiology and of alternative therapies as determinants of DT and its components.

## Acknowledgments

This work was supported in part by the Alan A. and Edith L Wolff Charitable Trust, St. Louis, and the Barnes-Jewish Hospital Foundation. Sina Mossahebi was supported in part by a teaching assistantship from the Physics Department, Washington University College of Arts and Sciences. We thank sonographer Peggy Brown for expert echocardiographic data acquisition, and the staff of Barnes Jewish Hospital Cardiovascular Procedure Center’s Cardiac Catheterization Laboratory for their assistance.

### Conflict of interest

The authors have no conflicts of interest to disclose with the reported study.

### Human Subjects

Prior to data acquisition, subjects provided signed, institutional review board (IRB) approved informed consent for participation in accordance with Washington University Human Research Protection Office (HRPO) criteria.

### Animal Studies

This work did not include any animal studies.