hemodynamic information and analysis of cardiac pumping

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Journal of Mechanics in Medicine and BiologyVol. 8, No. 1 (2008) 87–95c© World Scientific Publishing Company

HEMODYNAMIC INFORMATION AND ANALYSISOF CARDIAC PUMPING

T. K. HUNG∗,‡, S. R. BALASUBRAMANIAN∗, M. A. SIMON†,M. S. SUFFOLETTO†, H. S. BOROVETZ∗ and J. GORCSAN†

∗Department of Bioengineering, University of Pittsburgh749 Benedum Hall, Pittsburgh, PA 15261, USA†Division of Cardiology, University of Pittsburgh200 Lothrop Street, Pittsburgh, PA 15213, USA

‡[email protected]

Received 28 June 2007Accepted 20 August 2007

This study extracted hemodynamic information from echocardiogram to complement theleft ventricle ejection fraction and the stroke volume for cardiac evaluation and diagnosis.The dynamic characteristics of irregular wall motions can be analyzed by kinetic energyfluxes transferred from the left ventricle to the blood flow. A set of cardiac indices isdeveloped for quantification and classification of echocardiogram for clinical application.The pumping characteristics can be further quantified through the work done by pressureand viscous stresses of the ventricle.

Keywords: Hemodynamics; left ventricle; echocardiogram.

1. Introduction

Heart failure is the most frequent diagnosis in the United States. It accounts forfive million American adults, with an estimated annual cost of about US$30 billion.The lifetime risk of developing heart failure is 11% for men and 15% for women.1

Of those with heart failure, 50%–60% have systolic ventricular dysfunction (ejec-tion fraction <50%), the rest having diastolic dysfunction. One third of those withsystolic dysfunction have moderate to severe systolic dysfunction (ejection fraction<40%). The ejection fraction (EF), stroke volume, and left ventricle (LV) volume atend diastole are well used for assessing the cardiac condition/function of a patient.However, EF alone may not fully represent the dynamic pump performance of theLV, and stroke volume is difficult to obtain; this is in part due to the uncertaintyin measuring the three-dimensional configuration and motion of the left ventricle.The aim of the present study is to extract more hemodynamic information fromthe data stored in echocardiogram. A procedure for fluid dynamic quantification ofnormal and diseased ventricular pumping is formulated for this endeavor.

‡Corresponding author.

87

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88 T. K. Hung et al.

2. Theoretic Approach: Work-Energy Equation

In addition to employing EF, the dynamic pump performance of the LV can befurther assessed by using the work-energy equation of blood flow in the ventricle.The hemodynamic indices can be derived from an integral form of the work-energyequation2–5:

∫∫ρV 2

2

(u

∂x

∂n+ v

∂y

∂n+ w

∂z

∂n

)dS +

∫∫∫ (ρ

2∂V 2

∂t

)dxdydz

= −∫∫

p∗(

u∂x

∂n+ v

∂y

∂n+ w

∂z

∂n

)dS

+∫∫

µ

(∂u2

∂x

∂x

∂n+

∂v2

∂y

∂y

∂n+

∂w2

∂z

∂z

∂n

)dS

+∫∫

µ

(v

(∂v

∂x+

∂u

∂y

)+ w

(∂w

∂x+

∂u

∂z

))∂x

∂ndS

+∫∫

µ

(u

(∂v

∂x+

∂u

∂y

)+ w

(∂v

∂z+

∂w

∂y

))∂y

∂ndS

+∫∫

µ

(u

(∂w

∂x+

∂u

∂z

)+ v

(∂w

∂y+

∂v

∂z

))∂z

∂ndS

−∫∫∫

µ

(2

(∂u

∂x

)2

+ 2(

∂v

∂y

)2

+ 2(

∂w

∂z

)2

+(

∂u

∂y+

∂v

∂x

)2

+(

∂u

∂z+

∂w

∂x

)2

+(

∂w

∂y+

∂v

∂z

)2)

dxdydz, (1)

where ρ = blood density, V = velocity, (u, v, w) = velocity components in theCartesian coordinates (x, y, z), p∗ = pressure which includes the gravity effect,µ = dynamic viscosity of blood, n = outer normal direction on the wall, anddS = increment for surface integral along the LV cavity and the aortic root (referto the three-chamber view in Fig. 1). On the left-hand side of Eq. (1), the firstintegral represents the instantaneous kinetic energy flux along the ventricle walland the aortic root, while the second integral is the time rate of change of kineticenergy in the ventricle. The first two integrals on the right-hand side represent,respectively, the rates of work done by pressure and by normal viscous stresses;the third to fifth integrals are the rate of work done by shear stresses; and the lastintegral is the energy dissipation rate of blood in the ventricle. The first integral canbe calculated directly from the data recorded in echocardiogram. The rest of theintegrals cannot be evaluated until the velocity and pressure fields of the ventricularpumping are calculated from the Navier–Stokes equations. The characteristics ofenergy transfer from cardiac contraction to blood can be quantified; they can beused to identify the normal and abnormal ventricular contractions.

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Hemodynamic Information and Analysis of Cardiac Pumping 89

The first integral in Eq. (1) is valuable for quantitative analysis of ventricularpumping. For a two-dimensional echocardiogram, it can be expressed as

∫ρV 2

2

(u

∂x

∂n+ v

∂y

∂n

)d` =

∫

wall

ρV 2

2(~V • ~n)d` +

∫

o

ρV 2

2(~V • ~n)d`, (2)

where d` represents the line increment. The first term on the right is the instanta-neous rate of kinetic energy transferred from the wall contraction to blood flow inthe ventricle. The second integral represents the kinetic energy outflux across theaortic root (indicated by the subscript “o” of the integral). As the LV flow is con-vergent to the aortic root, the instantaneous velocity Vo(t) = Vo can be considereduniform at the aortic root. Thus, the above expression becomes

∫

wall

ρV 2

2(~V • ~n)dl +

∫

o

ρV 2

2(~V • ~n)dl = −

∫

wall

ρV 2

2(VN)d` +

∫

o

ρV 2o

2(Vo)d`

= −∫

wall

ρV 2

2(VN)d` +

ρV 3o D

2, (3)

where D is the aortic root diameter and ~V • ~n = −VN (the minus sign is for aninward normal velocity VN). Substituting this expression into the two-dimensionalform of Eq. (1) yields

ρV 3o D

2+

∫∫ (ρ

2∂V 2

∂t

)dxdy

=∫

wall

ρV 2

2(VN)d`−

∫p∗(VN)d` + µ

∫ (∂u2

∂x

∂x

∂n+

∂v2

∂y

∂y

∂n

)d`

+µ

∫ (v

(∂v

∂x+

∂u

∂y

)∂x

∂n+ u

(∂v

∂x+

∂u

∂y

)∂y

∂n

)d`

−µ

∫∫ (2

(∂u

∂x

)2

+ 2(

∂v

∂y

)2

+(

∂u

∂y+

∂v

∂x

)2)

dxdy. (4)

The continuity equation for a two-dimensional ventricle section requires

Vo(t)D −∫

wall

~V • ~nd` = 0. (5)

Thus, the instantaneous velocity Vo(t) at the aortic root can be calculated from thevelocity data of the LV motion:

Vo(t) =1D

∫

wall

~V • ~nd`. (6)

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3. Kinetic Energy Wall Flux

The instantaneous rate of kinetic energy transferred from the LV wall to blood flowcan be calculated from a line integral of kinetic energy (ρV 2/2) along the boundary(wall):

KEW =∫

wall

ρV 2

2(VN)d`. (7)

The quantity KEW is referred to hereafter as the kinetic energy wall flux calcu-lated from an instantaneous velocity distribution [see Fig. 1(a)] of a two-chamberview of echocardiogram. The rationale for using the work-energy equation is basedon its intrinsic characteristics for energy transfer from LV contraction to blood flow.Figure 1(b) demonstrates time variations of velocity magnitudes at four points ofthe LV motion6; they are plotted as positive values (for inward velocity) for sys-tole. Figure 1(c) shows the time variation of KEW during systole. As indicated in

(a) (b)

(c)

Fig. 1. Patient A. (a) Endocardial wall velocity vector at 0.2 second. (b) Variation of wall velocitywith time at the mid-left, basal left, mid-right, and basal right segments in (a). (c) Kinetic energywall flux (KEW) transferred from the LV wall to the blood during systole.

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Eq. (5), the pattern and magnitude of KEW depend not only on the velocity V ,but also on its normal component VN. The tangential velocity component does notcontribute to kinetic energy transfer. In the proposed study, we will concentrate onthe systolic phase of LV motion. Integration of KEW with time for systole givesthe total kinetic energy flux (TKEW):

TKEW =∫ S

0

(∫

wall

ρV 2

2(VN)d`

)dt, (8)

where the superscript S represents the duration of systole. Both the pattern[Fig. 1(c)] and the total kinetic energy flux provide hemodynamic information ofthe echocardiogram.

The kinetic energy flux is only part of the energy transmitted from the LV cham-ber to blood. Most of the energy is transmitted through the work done by ventricularpressures. Figure 2 shows similar results for the same patient (patient A).

(a) (b)

(c)

Fig. 2. Patient A. (a) Endocardial wall velocity vector at 0.2 second. (b) Variation of wall velocitywith time at the four points indicated in (a). (c) Kinetic energy wall flux (KEW) during systole.

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The total kinetic energy (TKEW) can be complemented by a dimensionlessindex defined by a ratio (KEA) of the total kinetic energy transfer (from the wallto blood flow on this two-dimensional echocardiogram) to that when the normalvelocity [in the denominator of Eq. (9)] is based on the absolute value |VN|:

KEA =

∫ S

0

∫wall

ρV 2

2 VNdldt∫ S

0

∫wall

ρV 2

2 |VN|dldt. (9)

In the absence of dyssynchrony, the index KEA is equal to unity. Its deviationfrom 1 would indicate the seriousness of dyssynchrony. Since the magnitude of thenormal velocity VN is also important to TKEW, another dimensionless parameter(ratio) is defined as

KEI =

∫ S

0

∫wall

ρV 2

2 VNdldt∫ S

0

∫wall

ρV 2

2 |V |dldt. (10)

The denominator is based on an idealized (hypothetical) wall velocity which isnormal to the wall (VN = V ). The index KEI reflects the effect of direction of thewall velocity on kinetic energy transfer from the LV motion to blood flow. KEI isalways less than 1. This ratio (index KEI) indicates the direction of wall velocity foreffective pumping. In nature, the value of VN is expected to be somewhat close toV for an optimal energy transfer from the ventricular wall to blood flow. Figure 3shows the results for another plane of patient A. While the velocity plots [referto Fig. 3(b)] do not deviate much from those shown in Figs. 1(b) and 2(b), theKEW shown in Fig. 3(c) is much lower than those in Figs. 1(c) and 2(c). TheTKEW is 18.4 erg, which is much lower than 30.3 erg and 30.2 erg shown in Figs. 1and 2, respectively. The reduction in TKEW is caused by smaller values of VN.This ineffective transfer of kinetic energy is identified and quantified by KEI =0.36, which is equal to 0.7 and 0.6 in Figs. 1 and 2, respectively. The comparisondemonstrates a useful quantification of the LV contraction by the total kineticenergy wall flux TKEW and the index KEI.

4. Dyssynchronized Echocardiogram

Echocardiograms for patient B were also analyzed. As demonstrated in Fig. 4(b),the velocity variations for an abnormal LV with dyssynchronized wall motion areevident. The kinetic energy transfer during systole is very low in Fig. 4(c). Theinstantaneous kinetic energy wall flux is KEW = SKE + DKE, in which the valueof SKE (synchronized kinetic wall flux) is positive while the value of DKE (dysyn-chronized kinetic wall flux) is negative due to abnormal dilation during systole.Clearly, the pumping effectiveness is drastically reduced and is well quantified byTKEW = 7.9 erg (in comparison with 30.3 erg and 30.2 erg in Figs. 1 and 2,respectively, for patient A). As shown in the table in Fig. 4, the dyssynchronizedindex KEA is 0.7. The value KEA = 1 represents that there is no incidence of car-diac dyssynchrony. The dimensional index TKEW (in erg) quantifies the strength

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(a) (b)

(c)

Fig. 3. Patient A. (a) Endocardial wall velocity vector at 0.33 second. (b) Variation of wall velocitywith time at the four points indicated in (a). (c) Kinetic energy wall flux (KEW) during systole.

of pumping. The dimensionless index KEI indicates the effectiveness of velocitydirection on cardiac pumping.

Integration of the kinetic energy wall flux (KEW) over the period of systoleyields the total kinetic energy wall flux TKEW [see Eq. (2)]. For the same strokevolume, the value of TKEW will decrease with the lower wall velocity of an enlargedventricle, as is the case for the ejection fraction (EF). The index EF provides alinear relationship between the stroke volume and the dilated ventricle volume.The total kinetic energy of the wall (TKEW) reflects a nonlinear relationship,making it feasible to correlate TKEW with various types of diseased ventricularmotion. However, the total energy transfer from the LV contraction to blood flowalso requires an evaluation of work done by pressure and viscous stresses. The totalenergy pumped to blood by the left ventricle is

TEW =∫ S

0

∫

wall

(ρV 2

2+ p∗

)VNdldt, (11)

where p∗ is the instantaneous pressure on the ventricle wall.

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(a) (b)

(c)

Fig. 4. Patient B. (a) Endocardial wall velocity vector at 0.2 second. (b) Variation of wall velocitywith time at the four points indicated in (a). (c) Kinetic energy wall flux (KEW) during systole.DKE indicates the dyssynchronized part of kinetic energy wall flux, while the synchronized partof KEW is represented by SKE.

5. Conclusion

The present study has demonstrated that the data recorded in echocardiogramcan be well used for quantification of cardiac pumping characteristics associatedwith cardiac functions and diagnosis. The dimensional quantities — TKEW andTEW — and the dimensionless indices — EF, KEA, and KEI — will be usefulfor correlating with the conditions of ineffective cardiac pumping. In our secondphase of the study, we will develop an efficient computational analysis to calculatepressures and velocity distributions for systole. A patient-specific computation willbe established to characterize cardiac pumping of echocardiograms.

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hemodynamic information and analysis of cardiac pumping

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