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Huml D and Beus-Huml M PDF Version of this article

Institute for Health Protection "Energoinvest", Sarajevo, B&H

The choise of the hemodynamic cube or elipsoid algorithm for the left ventricular dimensions and ejection function assessment is in the direct dependence on the cardiac apex movement and/or on the left ventricular length and transversal dimension ratio. The parameters of the global ejection function, an ejection fraction (EF) and a stroke volume (SV), respectively, alters at the same fractional shortening of the cavitary diameter depending on the cube or elipsoid hemodynamic conditions. The elipsoid diameters of the left ventricle are derived from "the general form of the third grade equation" by known values of the EF and the SV; the cube diastolic and systolic diameter are the cube root from the SV and EF ratio or the expression difference of this ratio and the SV.

The presented method is the marked contribution in the polycardiographic (phonomechanocardiographic) and/or the Dopple-cardiographic quantification, because it enables, besides the ejection function assessment, also left ventricular dimension approximation, the transversal and longitudinal ones, what was, up to the present time, in the domain of the sofisticated and expensive echocardiography and "invasive" cineventriculography.

Keywords: the polycardiography, the Doppler-cardiography, the left ventricular ejection function and dimensions, the cube and elipsoid hemodynamic algorithm

The left ventricular dimensions and ejection function assessment is important from diagnostical as well as prognostical point of view. These parameters can be measured repeatedly on a great number of chronic patients with heart disease only noninvasively. The supreme noninvasive method for this approximation is echocardiography as high sophisticated, but also an expensive one. However, among the noninvasive methods the polycardiography means the lowest cost, it is the least time consuming and the required instrumentation is avable widely. Since 1970, when Garrard and coworkers presented their results, the PEP/LEVT ratio was introduced as one of the noninvasive parameters to estimate the left ventricular ejection fraction 1. There was assessed in a population of older people up to the 60 years without left ventricular wall disinergy, also, a high specific and sensitive correlation of the PEP/LEVET and ejection fraction 2. In patients with the higher degree of the left ventricular wall disinergy, however, the PEP/LEVET ratio fairly frequently did not proved suitable for the "ventriculographic" ejection fraction assessment in the conditions at rest in distinction from the conditions at the interventions tests 3, 4, 5, 6. From starting point of view that an apexcardiography has a close relation to the left ventricular systolic wall movement it was found that this noninvasive method correlates closely and more significant with left ventricular ejection function than the PEP/LEVET ratio, particularly in patients with endsystolic apical anterior deflexion 7, 8, 9. It was showen that this method, known as Antani's one, was more accurate in patients with wall disinergy when the amplitude during aortic closure was corrected by reducing for the hight of endsystolic deflexion 9. On the other hand, if the cardiac apex endsystolic deflexion is paradoxical or significantly increased, then the left ventricular ejection function is overestimated according to ratio of the cubed endsystolic deviation, reduced for this pathologic deflexion and that one at the systole onset. If the ejection fraction according to cube algorithm is corrected with the elipsoid conversion coefficient in the cases with endsystolic deflexion of the cardiac apex movement, then one could obtain the high significant correlation with the PEP/LEVET ratio (the prersonal opservation). The aim of this study is to approximate the ejection fraction, stroke volume and the cavitary left ventricular dimensions by polycardiographic method with anticipation of the cube or elipsoid algorithms. These parameters up to the present time could be approximated only by the sofisticated expensive echocardiography and "invasive" cineventriculography.


In the study were included the 114 individuals, nonselected, healthy ones, and these with hypertensive and coronary heart disease, respectively, from which 50% have endsystolic deflexion od the cardiac apex movement, including also the paradoxical ictus movement. There is possible to anticipate the hemodynamic algorithm according to the x ray cardiac silhouette shape or by the inspection and palpation of the cardiac apex movement. The cavitary length (L) according to cube algorithm is approximatly or exactly correspond to the sum of two transversal diameters (L = 2 D). At the elipsoid algorithm the L is not the product or the sum of two D. The exaggerated and sustained anterior movement is a sign of the mild left ventricular hypertrophy. The double apical external movement, the presystolic and systolic one, in consequence of the increased arterial contribution to the left ventricular filling, ussually is going with the anterior wall displacemnt in the left parasternal region (also in an absance of the right ventricular hypertrophy!) with the delay to apical movement. One could reveal the apex endsystolic deflexion as the apical prolonged anterior movement with the faculatative wall retraction in left parasternal, sternal and right parasternal region. The criteria of lusitropy and inotropy dysfunction exists in the advanced "two hearts" disease, when one could again discover the pathologic apical endsystolic deflexion relatively to the prolonged anterior movement, and so parasternaly as anterior prolonged sysolic movements, too.

On the occasion of the left ventricular dimension calculation by the two algorithms one could respect the procedure order. The ejection fraction and stroke volume must be determined primarly by both of two algorithms. The cube pattern of ejection fraction is the simple ratio of the cubed SO and EO displacement substracted from one; according to the elipsoid algorithm one must perform the correction by the conversion coefficient (Figure 1)

F i g u r e 1.

The both of these two procedurers are incorporated into the formula for an ejection fraction according to the PEP/LEVET ratio:

EF = 1.365 e -2.35 (PEP/LEVET) ; N = 114 ; R = 0.98 (1)

The formula is related to the individuals with and without left ventricular wall disinergy. The correlation of the ejection fraction according to the ratio of PEP and LVET and/or the cardiac apex shortening of EO and SO deviations by an anticipation of two algorithms, with the Sabah's first time derivative of the Doppler- cardiographic velocity curve 10 is on the level of functional connection. For the approximation of the borderline ejection fraction od 0.50 the method sensitivity and negative predictive value is 96%; the method specifity and postive predictive value is 95%, respectively. The stroke volume is determined according to the LVET, the pulse pressure (PP) and the value of diastolic arterial pressure (TAd), depending of an actual normotension:
SV = 9.356 e -0.008 TAd * LVET * PP, (2)

or depending on the hypertension:
SV = 3.962 e -0.001 TAd * LVET * PP. (3)

The method reproducibility was discussed earlier 11, 12. The left ventricular ejection fraction could be estimated also by hemodynamic algorithm from the standard electrocardiogram by extrapolating the PEP and the LVET from the JT and the QT interval 11, 12.

In primary cube algorithm one must anticipate cube ejection fraction pattern (EFc) with the endsystolic SO and the maximal EO deviation of the cardiac apex movement without endsystolic curve deflexion as one could see in figure 1, at panel on bottom:
EFc = 1 - (SO3 / EO3) (4)

The cube systolic and diastolic left ventricular dimension is derived directly by the known value of the SV and EF:



Dd = Ö

(SV / EF)        (5)



Ds = Ö

(SV / EF) - SV     (6)

The stroke volume in the hemodynamic interphases of cube algorithm is the difference of actual cubed diastolic and systolic diameter or the product of cube diastolic diameter and "cube" ejection fraction: SVc = Dd3 - Ds3 = (1 - (Dd3 / Dd3) Dd3     (7)

In primary elipsoid algorithm one could anticipate the elipsoid ejection fraction (EFe) where the SO is a deviation during the aortic closure at endsystole or during the second heart sound (S2). The EO is the standardised inicial deviation of the cardiac apex movement, which must not be also the maximal one at the eliposid algorithm. If the apical movement is paradoxical one because of the left ventricular wall disinergy in ischemic heart disease, when the protosystolic wave deviation is lesser than endsystolic one, or when it stays the endsystolic deflexion with higher protosystolic wave deviation, the SO is corrected by reduction for the hight of this pathologic deflexion in the majority of cases, and the EF is "elipsoid" one, e.g. it is corrected by the conversion coefficient (Figure 1., the upper and midle panel):

EFe = (1 - (SO3/EO3)((EO+2.4)/(SO+2.4)))          (8)

The authenthic left ventricular dimensions in the elipsoid pattern by a real values of the SV and the EF is derived from the general form of the third grade equation:

p> ax3 + bx2 + cx + d = 0

a = 7 EF, b = 0, c = SV and d = 2.4 SV for the diastolic diameter; for the systolic diameter a = 7, b = 0, c = -(EDV-SV) and d = 2.4(EDV - SV).

The real solution of formula depends from the discriminant "D " (D > 0, D = 0, D < 0). If we wish to solve the equation instead "x" we must introduce a new variable "y":

y = x + (b/3a) ; y3 + 3 py + 2q = 0

D = q2 + p3

2q = (2b3 / 27a3) - (bc / 3a2) + (d/a)

3p = ((3ac - b2)/(3 a2))

The last deduced formula for the left ventricular dimension is:

3 _________________       3






D = y = Ö

-q + Ö

q2 + p3            

Ö-q - Ö

q2 + p3          (9)

The stroke volume in hemodynamic interphases of the elipsoid algorithm one could extrapolated according to formulas:
SVe = (7 Dd3/(Dd+2.4)) (1-7Ds3/(Ds+2.4))/(7Dd3/(Dd+2.4)))
= (7 Dd3/(Dd+2.4)) - (7 Dd3/(Dd+2.4))           (10)

The length of long axis in left ventricle at the cube algorithm is approximatly two transversal diameters:
Lc = 2 D (11)
At elipsoid algorithm long axis is extrapolated from the formula:
Le = (7 D) / ((2.4 + D) (p /6)) (12)


There were presented the nomogram for the left ventricular systolic and diastolic dimension assessment by hemodynamic cube and elipoid algorithm (Tables 1, 2, 3, 4, respectively).

T a b l e 1.

T a b l e 2.

T a b l e 3.

T a b l e 4.

Afterwards, the results of the original polycardiographic method for the ejection fraction assessment and the relationship of our method ("the method 3") and the methods according to Garrard and coauthors ("the method 1") 1 and according to Schuster and coauthors ("the method 2") 2 are listed also like as nomograms (Tables 5 and 6).

T a b l e 5.

T a b l e 6.

The diastolic left ventricular dimension of 5.7 cm and more according to the cube algorithm in males is defined by the regression:
(PEP/LVET) >= 21.197 SV -0.8715 # EF =< 0.0054 SV 1.0016

For the diastolic left ventricular dimension of 5.2 cm and more in females are valid the following formulas according to the same algorithm:
(PEP/LVET) >= 17.360 SV -0.8769 # EF =< 0.0068 SV 1.0082

The left ventricular systolic dimension of 3.5 cm and more, according to cube algorithm is going with the following correlations:
(PEP/LVET) >= 2.5080 SV -0.4740 # EF =< 0.1080 SV 0.4057

The borderline normal values of left ventricular chamber dimensions are in concoradance with the recommended standards of the Cardiology Association, 1985 (The CA Commission for Noninvasive Diagnostic) or according to Devereux 13, 14.

The elipsoid algorithm itself suggest on pathologic relationship in transversal and longitudinal dimensions of the left ventricle. The power law regressions, also, point at the diastolic dimensions of 6.4 cm and 5.7 cm, respectively, and more (cube "pendent" is 5.9 cm and 5.4, respectively, and more):

(PEP/LVET) >= 28.465 SV -0.9066 # EF =< 0.0049 SV 0.9936
(PEP/LVET) >= 13.215 SV -0.7879 # EF =< 0.0068 SV 1.0082

The systolic dimension of 4 cm and more according to elipsoid algorithm (and the same (!) cube "pendent") could be calculate on the basis of the power law regression formula:
(PEP/LVET) >= 1.8061 SV -0.3507 # EF =< 0.1350 SV 0.3184


The presented method is the marked contribution in the polycardiographic quantification, because it enables, besides the ejection function assessment, in addition to the left ventricular dimension approximation, the transversal and longitudinal ones, respectively, what was up to the present time in the domain of the sofisticated and expensive echocardiography and "invasive" cineventriculography. The significancy is so far greater since the left ventricular dimensions sould be derived from the Doppler- cardiograms or the standard electrocardiograms.


The author express thanks to dipl eng Maja Dimitrijevic on the most useful advices from the mathematis field and particularly on the derivation of general form of the third grade equation.


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