Effect of the Geometry of the Left Ventricle

Transcription

1 EF AD LV GEOMETRY/Dumesnil and Shoucri 91 Rearranging equation 2 and solving for the activity yields =C u * T S e-ud (I - eut) (3) If the activity within the left ventricle is treated as a point source at the center of the ventricle, the calculated activity A' from the count rate C is given by C A' Sud (4) where d = D + 1/2 T, the depth of the center of the left ventricle. Given a count rate C from the left ventricle, the error in correcting C for attenuation using equation 4 instead of equation 3 is given by A A' (ut/2) = sinh (5) (ut/2) where sinh (ut/2) = 1/2 (e(ut/2) - e -(ut/2)). As u and T go to 0, the error term goes to 1. For u and T greater than 0, the error term is less than 1, and the use of equation 4 will result in too large a calculated activity. However, at technetium-99m energy in water, the error is under 1% for T less than 3 cm, under 5% for T less than 8 cm, and under 10% for T less than 10 cm. For comparison, the average thickness of a prolate spheroid viewed perpendicular to its major axis is two-thirds of the minor-axis length. For a 5-cm-thick ventricle, the error would thus be approximately 1%. Effect of the Geometry of the Left Ventricle on the Calculation of Ejection Fraction JEA G. DUMESIL, M.D., AD RACHAD M. SHOUCRI, PH.D. SUMMARY We used a cylindrical model for the left ventricle contracting both radially and longitudinally to show how the ejection fraction is related to different variables that describe the ventricular geometry. The relations during ventricular contraction between wall thickening, midwall radius shortening and longitudinal shortening are used to derive precise formulas for the calculation of ventricular blood flow velocity and ejection fraction. Validation of the model is given by results from the literature and by a study of 40 patients. The formulas derived can be used to validate angiographic measurements or to assess more precisely blood flow velocity and ejection fraction from the M-mode echocardiogram. Ejection fraction is determined not only by the extent of myocardial shortening, but also by the relationship of ventricular wall thickness to ventricular cavity size. I A RECET STUDY,1 a mathematical model was used to demonstrate the relations between the geometric variables most frequently used to describe left ventricular contraction. It was shown that for a ventricle contracting both radially and longitudinally, ventricular longitudinal axis shortening, midwall radius shortening and wall thickening are directly related and that the value for one of these variables can be derived from the knowledge of the other two. It was also demonstrated that for a similar extent of ventricular contraction, the change in the internal radius of the ventricle during systole is directly influenced by the shape of the ventricle, as expressed by the midwall radius-to-wall thickness ratio (R/h) of the ventricle. In this paper, we examine how the relationship between myocardial shortening and the ejection fraction is influenced by left ventricular geometry. The implications are important because ejection fraction is one of the main guidelines used to assess left ventricular performance in the clinical situation. We also compared the results from our model with those of From the Quebec Heart Institute, Laval University, Quebec, Canada. Supported by grant MA-5766, Medical Research Council of Canada. Address for correspondence: Jean G. Dumesnil, M.D., 2725, chemin Sainte-Foy, Quebec, GIV 4G5 Canada. Received December 10, 1980; revision accepted April 24, Circulation 65, o. 1, other methods of calculating ejection fraction from the M-mode echocardiogram. Methods Mathematical Model The left ventricle is represented by a uniform, thickwalled cylinder that contracts both radially and longitudinally (fig. 1). The inner volume is given as V = r a2 L (1) and the given as wall of the ventricle, assumed constant, is Vw = 2 r RhL (2) where a, b = inner and outer radii of the cylinder, L = length of the cylinder, h = b - a = wall thickness and R= b~2 a= midwall radius. Differentiation of equa- 2t tion 1 gives 1 dv _ 2 da 1 dl V dt a dt L dt (3) and differentiation of equation 2 gives 1 dr 1 dh 1 dl R dt h dt L dt _ O (4) From equation 1, the ejection fraction for the cylindrical model can be given as

2 92 CIRCULATIO VOL 65, o 1, JAUARY 1982 For the cylindrical model, we also previously demonstrated' that Aa/a, AR/R, AL/L and R/h are related Aa/a ( R/h1/2) (AR ) FIGURE 1. The left ventricle is represented by thick-walled cylinder contracting radially and longitudinally. b = external radius; a = internal radius; R = midwall radius; h = wall thickness; L = length. The volume of the wall of the cylinder (Vw = 2 ir RhL) is assumed to be constant during contraction. A L In echocardiographic studies, the cube of the ventricular internal diameter is often used to calculate ventricular volume and the ejection fraction. By the cube volume, the ejection fraction is given as 'A V Dd3 Ds3( (5) t VJ8 Dds ~~~~~(6) ~V Dd3 where Dd = diastolic diameter and Ds = systolic diameter. By dividing equation 1 by equation 2 and rearranging the right-hand side of the equation, we obtain the formula V (R/h - 1/2)2 VW 2(R/h) Because Vw is assumed to be constant, another formula derived from the cylindrical model for calculating the ejection fraction is given as {A -V(-d (V)- (8) ~(8 where the suffixes d and s refer to end-diastole and end-systole and V is given by equation 7. In equation 7, V is the cavity volume of the left ventricle. Equations 5 and 8 can be calculated from M-mode echocardiographic measurements in the routine position. In the case of equation 5, A L/L can be calculated, as previously shown,' from the formula ( 1 AR ) 1 IL ( + jz5h - (9) (' 2(42) ( R/h-l/2 ) 1 (1 + AR/R)(1 + AL/L) (10) Clinical Data As an application of equations 3 and 4, we have used the results published by Gould et al.2 (figs. 2 and 3). They measured the maximal normalized velocities of midwall radius shortening ([1/R] [dr/dt]), longitudinal axis shortening ([1/L] [dl/dt]), and wall thickening ([I/h] [dh/dt]) directly from angiograms in nine groups with different diseases (total of 122 patients). These variables were measured independently of one another without assuming any mathematical relationship between them. For equation 3, we have assumed in our calculation that (1/a) - (da/dt) (l/r) (dr/dt) because the results for (1/a) (da/dt) were not given by Gould et al.2 We also oversimplified by assuming that the maximal values for the normalized variables occur simultaneously, which may not be so. As an application of equations 5 and 6, we used the results of Lewis and Sandler3 (table 1). These authors measured independently the relative changes in ventricular internal radius (Aa/a) and ventricular length (A L/L) during systole in 22 patients with different diseases without assuming the relations given by equations 5 and 8. Finally, we reviewed the M-mode echocardiograms and the angiograms of 40 patients studied in our institution. ine had chest pain of unknown origin and normal coronary arteries ("normal"), two had mitral valve prolapse, one patient had congestive cardiomyopathy, 12 patients had aortic stenosis, five had idiopathic hypertrophic subaortic stenosis, one patient had discrete subaortic stenosis, six patients had aortic insufficiency, and four had mitral regurgitation. As previously described,' the M-mode echocardiograms were recorded in the routine position, just below the mitral valve where the interventricular septum and the left ventricular posterior wall are best seen with some parts of both mitral leaflets or chordae. The internal dimension of the left ventricle was measured in both diastole and systole as the distance between the left side of the interventricular septum and the anterior border of the posterior free wall. The internal radius of the ventricle in the short axis was considered to be half the value of the internal dimension of the ventricle in that position. We also measured the thickness of the ventricular posterior free wall in both diastole and systole as the distance between the posterior border of the epicardial echo to the top of the endocardial echo. From these measurements, we then calculated the

3 EF AD LV GEOMETRY/Dumesnil and Shoucri 93 TABLE 1. Comparison of the Ejection Fraction Obtained by Three Methods Case Aa/a* AL/L* AV/V* AV/VC AV/VS AV/VR p < 0.05 < S *Cineangiographic values from Lewis and Sandler.3 The p value indicates the significance of the statistical comparison of each method with the cineangiographic values using the paired t test. Abbreviations: Aa/a = relative change in internal radius; AL/L = relative change in ventricular length; AV/V = ejection fraction; c = cylindrical model (equation 5); s = cube formula (equation 6); R = cylindrical model (equation 7). relative changes in left ventricular internal radius Ava/a, left ventricular midwall radius AR/R, left ventricular posterior wall thickness Ah/h and the ventricular midwall radius-to-wall thickness (R/h) ratio in diastole. Relative ventricular longitudinal axis shortening, AL/L, was calculated from equation 9 and compared with angiographic values. The ejection fraction was calculated from equation 5. We also calculated, for each patient, what the ejection fraction would have been if the R/h ratio had been either 2.0 or 4.0 instead of the actual value, and AR/R and AL/L had remained the same. The latter calculations were done using equations 5, 9 and 10. The purpose of these calculations is to make evident the influence of the geometry of the ventricle on the ejection fraction. In appendix A is a comparison of the ejection fraction calculated from the cylindrical model with other methods used to calculate the ejection fraction from the M-mode echocardiogram. A simple linear regression equation or the paired t test was used to make statistical comparisons. Statistical significance was at the p < 0.05 level. Results In figure 2, the results for (1/h) (dh/dt) as calculated from equation 4 are compared with the actual measurements made by Gould et al.2 The correlation between the two results is excellent (r = 0.99) and strongly suggests that there is indeed a direct relationship between the maximal normalized velocities of wall thickening, midwall radius shortening and longitudinal axis shortening during ventricular contraction. However, the results are mean values in nine groups of subjects and the correlation for individual results might not have been as good. Figure 3 is a comparison of the results for the normalized ejection velocity (1/V) (dv/dt) as calculated from equation 3, with the values measured directly by Gould et al.2 Again, there is a very good correlation between the two results (r = 0.99). The calculated values for (1/V) (dt/dt) are all smaller than the measured values; this is to be expected because we have assumed in our calculation that (1/a) (da/dt) (I/R) (dr/dt), but actually, (1/R) (dr/dt) is always smaller than (1/a) (da/dt).' The relationship of (1/a) (da/dt) and (1/R) (dr/dt) is also influenced by the R/h ratio of the ventricle so that it is not necessarily the same in each case. evertheless, the results support the validity of equation 3. From this equation, the maximal normalized ejection velocity of the ventricle can be directly calculated from measurements of the maximal normalized velocities of internal radius shortening and longitudinal axis shortening of the ventricle.

4 94 CIRCULATIO VOL 65, o 1, JAUARY 1982 K K 1.5k 1.0O 0.5 dh OBSERVED y= 1.019x r= ( dh )CALCULATED FIGURE 2. Comparison ofthe maximal velocity ofventricular systolic wall thickening (I/h) (dh/dt) as measured angiographically by Gould et al.2 in nine groups ofpatients (ordinate) with the results obtained using equation 4 (abcissa). See text. Table 1 is a comparison of the calculations of the ejection fraction AV/V made by Lewis and Sandler3 (column 3) to the results obtained when equations 5, 6 and 8 are used (columns 4, 5 and 6). There is a very good correlation between column 3 (cineangiography) and column 4 (equation 5) (y = 0.98 x , r = 0.99) and between column 3 and column 5 (cube formula) (y = 1.08 x , r = 0.99). However, the cube formula significantly overestimates the ejection fraction compared with angiography (A = 0.06 ± 0.03, p < 0.001), whereas there is only a slight overestimation (A = 0.01 ± 0.02, p < 0.05) when equation 5 is used. Equation 8 (column 6) also shows a good correlation (y = 1.03x , r = 0.96) with the angiographic results, and does not significantly over- or underestimate (A = 0.01 ± 0.05, S) the angiographic results. Table 2 shows the results from the 40 patients studied in our laboratory. Ventricular longitudinal axis shortening calculated from echocardiography in these patients ranged from There is good agreement between the values measured from the angiograms and the results derived from equation 9 (r = 0.89) and there is no significant difference in results between the two methods (A = , S). The R/h ratio of the ventricle in these patients ranged from Column 7 of table 2 gives the actual measurements for the ejection fraction and columns 8 and 9 show what the ejection fraction becomes when the R/h ratio of the ventricle is artificially assumed to be either 2.0 or 4.0 and AR/R and AL/L remain identical. The difference in values between columns 8 and 9 is about 20%. In columns 7, 8 and 9, the ejection fraction is calculated assuming identical values for circumferential (AR/R) and longitudinal (AL/L) shortening and the differences that we observe from one column to the other are uniquely due to a change in ventricular geometry (R/h ratio). Discussion The results of the present study further demonstrate that a cylindrical model can be used to derive the relationships between some of the variables used to describe ventricular contraction. The measurements made by Gould et al.2 and Lewis and Sandler3 were done without knowledge of the equations given by the cylindrical model. Yet, the results of both studies are highly consistent with the present model. The relations based on the cylindrical model may be particularly useful in echocardiographic studies because all of the variables can be derived from the M-mode echocardiogram in the routine position. Hence, it may be possible to obtain noninvasive evaluations of maximal normalized ejection velocity, a variable that is sensitive to changes in ventricular function.' However, further studies correlating the M-mode echocardiogram with blood flow velocity measurements will be necessary. In angiographic studies, these equations may be useful for validating the direct measurements performed on the angiograms. For instance, a discrepancy between the formulas and the actual measurements could indicate an error of measurement. The data definitely show that these equations are applicable to the symmetrically contracting ventricle. Gould et al.2 studied two groups of patients with regional ventricular dysfunctions, and their results are i F 2.0F F V d)observed y=0.975x+4 r = ~~~~~~~~~~~~~~~~..- I (1 dv CALCULATED V d t FIGURE 3. Comparison of the maximal blood flow velocity as measured by Gould et al.2 in nine groups of patients (ordinate) with the results obtained using equation 3 (abcissa). See text.

5 _ TABLE 2. Effect of Ventricular Geometry on the Calculation of the Ejection Fraction in 40 Patients AL/L.V/V AV/V Diagnosis AR/R Ah/h Aa/a Echo Angio R/h AV/V (R/h = 2) MVP MVP CM DS Al MR MR MR MR Mean ± SD ±0.05 ± ± p S < V/V (R/h = 4) ± 0.10 < The p value is the level of significance for statistical comparison of AL/L by angio and echo, and for comparison of zv/v in columns 8 and 9 to actual measurement (column 7) using the paired t test. Abbreviations: AR/R = relative change in ventricular midwall radius; Ah/h = relative change in ventricular wall thickness; z\a/a = relative change in ventricular internal radius; AL/L = relative change in ventricular length; R/h = ventricular midwall radius-to-wall thickness ratio; AV/V = ejection fraction; = normal; MVP = mitral valve prolapse; CM =- congestive cardiomyopathy; = idiopathic hypertrophic subaortic stenosis; DS = discrete subaortic stenosis; = aortic stenosis; = aortic insufficiency; MR = mitral regurgitation. EF AD LV GEOMETRY/Dumesnil and Shoucri

6 96 CIRCULATIO VOL 65, o 1, JAUARY 1982 consistent with the equations given by the cylindrical model; but Gould et al.2 measured ventricular wall thickness by the method of Hugenholtz et al.,5 which is an average measurement of wall thickness change throughout the ventricle during systole. However, differences in regional wall thickness changes are frequently found in patients with coronary artery disease,6 and in these cases, regional wall thickness measurements should probably not be used to make the calculations. In table 1, the ejection fraction was calculated by using three formulas. The results from equation 5 (column 4) compare very well to the results given by Lewis and Sandler9 (column 5) based on an ellipsoidal model. This finding further demonstrates the equivalence between the cylindrical and ellipsoidal models.' The results in column 5 were obtained with the cube formula used in echocardiographic studies. This formula assumes that the extent of ventricular longitudinal shortening during systole is equivalent to the extent of circumferential shortening and that the relationship between these two variables remains constant. The present data and our previous results,' however, show that the extent of ventricular longitudinal shortening is normally less than the extent of internal radius shortening and that the relationship between the two variables may not be the same in each patient. The ejection fraction calculated from equation 6 is therefore usually higher than the actual measurements. Equation 5 is probably more appropriate and it is applicable to echocardiography because Aa/a can be measured directly and AL/L can be calculated by using equation 9. The results in column 6 of table 1 are also derived from the cylindrical model, based on equation 8. In this case, the calculation is based on the change of the midwall radius-to-wall thickness ratio, which incorporates both circumferential and longitudinal shortening. The results are also consistent with the actual measurements, and the two values vary probably because the R/h ratio is a very sensitive measurement and because a slight change in its values can significantly affect the results for the ejection fraction. Therefore, very reliable measurements are necessary to use this formula. When applied to the present results, it nevertheless demonstrates the validity of the cylindrical model. The advantage of using equations 5 and 8 is that no prior assumption of any relation between the inner radius and the length is required. Table 2 shows that the ejection fraction varies in relation to the R/h ratio of the ventricle. The results in columns 7, 8 and 9 were calculated on the assumption that there is no change in AvR/R and AL/L and that the differences we observed are due uniquely to a change in the R/h ratio of the ventricle. The R/h ratio is the relation between ventricular cavity size and ventricular wall thickness, whereas ventricular midwall radius shortening (zr/r) and ventricular longitudinal shortening (AL/L) are directly related to the extent of shortening within the ventricular wall during contraction. Considering the range of values in different clinical situations, the ejection fraction can vary by as much as 20% on the basis of a change in the R/h ratio. Because the ejection fraction is one of the main guidelines used to assess ventricular contraction in the clinical situation, this finding has important implications. For example, in a patient with concentric ventricular hypertrophy due to aortic stenosis or systemic hypertension (small R/h value), the ejection fraction is higher for the same AR/R and zvl/l than in a normal subject, whereas it is lower in a patient with ventricular dilatation and normal wall thickness (high R/h value). Considering the results for the ejection fraction, one should therefore consider that it is determined not only by the extent of shortening within the ventricular wall, but also by the specific shape of the ventricle. An interesting implication for the heart muscle mechanics is that the same amount of muscular fiber shortening as measured by AR/R and AL/L will result in different values of the ejection fraction zv/v, depending on the geometry of the ventricular cavity as characterized by the ratio R/h. In conclusion, this study further demonstrates the relation between the geometric variables used to describe ventricular contraction and shows that a simple expression of their relationship can be obtained by using a cylindrical model of the ventricle. This model can be used to validate measurements and to calculate variables that cannot be measured directly, as in M- mode echocardiography. Myocardial shortening as well as ventricular geometry have a significant influence on ventricular ejection fraction. References I. Dumesnil JG, Shoucri RM, Laurenceau JL, Turcot J: A mathematical model of the dynamic geometry of the intact left ventricle and its application to clinical data. Circulation 59: 1024, Gould KL, Kennedy MJ, Frimer M, Pollack GH, Dodge HT: Analysis of wall dynamics and directional components of left ventricular contraction in man. Am J Cardiol 38: 322, Lewis RP, Sandler H: Relationship between changes in left ventricular dimensions and the ejection fraction in man. Circulation 44: 548, Peterson KL, Skloven D, Ludbrook P, Uther JB, Ross J Jr: Comparison of isovolumetric and ejection phase indices of myocardial performance in man. Circulation 49: 1088, Hugenholtz PG, Kaplan K, Hall E: Determination of left ventricular wall thickness by angiocardiography. Am Heart J 78: 513, Dumesnil JG, Ritman EL, Frye RL, Gau GT, Rutherford BD, Davis GD: Quantitative determination of regional left ventricular wall dynamics by roentgen videometry. Circulation 50: 700, Teichholz LE, Kreulen T, Herman MV, Gorlin R: Problems in echocardiographic volume determination; echocardiographicangiographic correlations in the presence or absence of asynergy. Am J Cardiol 37: 7, Fortuin J, Wood WP, Sherman ME, Craige E: Determination of left ventricular volumes by ultrasound. Circulation 44: 575, Gault JH, Ross J, Braunwald E: Contractile state of the left ventricle in man: instantaneous tension-velocity-length relations in patients with and without disease of the left ventricular myocardium. Circulation 22: 451, Kronik G, Slany J, M8sslacher H: Comparative value of eight M-mode echocardiographic formulas for determining left ventricular stroke volume - a correlative study with thermodilution and left ventricular single-plane angiography. Circulation 60: 1308, 1979

7 EF AD LV GEOMETRY/Dumesnil and Shoucri 97 TABLE Al. Ejection Fraction from the M-mode Echocardiogram by Four Different Methods Ejection fraction LVIDd h Cylindrical Teichholz Cube Fortuin Patients (mm) (mm) Aza/a AL/L model et al.7 formula et al.8 ormal subjects Mean ±SD ±3 ±0.7 ±0.04 ±0.04 ±0.05 ±0.05 ±0.05 ±0.05 Miscellaneous (see table 2) Mean ±SD ±10 ±2.9 ±0.14 ± ± ± ± ± Aortic stenosis Mean ±SD ±6 ±2.8 ±0.09 ±0.07 ±0.09 ±0.10 ±0.10 ± 0.12 Valvular regurgitation : Mean ±SD ±8 ±1.5 ±0.06 ±0.03 ±0.06 ±0.08 ±0.08 ±0.08 Total Mean ±SD ±11 ±2.8 ±0.10 ±0.08 ±0.09 ±0.10 ± 0.10 ±0.12 p S < < Abbreviations: See table 2; LVIDd - left ventricular internal diameter in diastole; h = posterior wall thickness. The p value indicates the level of significance for statistical comparison of ejection fraction by cylindrical model with other methods using paired t test.

8 98 CIRCULATIO Appendix A For the 40 patients studied by echocardiography, we compared the results of the cylindrical model to those of other methods used to calculate the ejection fraction from the M-mode echocardiogram (Teichholz et al.,7 cube formula and Fortuin et al.8) (table Al). When the cylindrical model is applied to the M-mode echocardiogram, equations 5 and 8 will yield identical results for the ejection fraction because AL/L is not measured directly, but is calculated from equation 9 using AR/R and Ah/h. Overall, the results from the cylindrical model are closest to the results from the method of Teichholz et al.7 (A = 0.00 ± 0.05, S), whereas the cube formula agrees more closely with the method of Fortuin et al.8 (A = 0.01 i± 0.05, S). Compared with either to the cylindrical model or the method of Teichholz et al., the cube formula and the method of Fortuin et al.8 consistently and significantly (p < 0.001) overestimate the ejection fraction. In individual patients, the main discrepancies between the cylindrical model and the method of Teichholz et al.7 are in patients where ventricular long-axis shortening is either high (cases 7, 10, 31 and 39) or low (cases 17, 19, 22, 25 and 27) relative to ventricular internal radius shortening. In our normal subjects, ventricular longitudinal shortening averaged ± 0.04 and ventricular internal radius shortening 0.35 i 0.04 (mean values for the whole group are ± 0.08 and ± 0.10, respectively). Similar values have been reported previously.l Kronik et al.10 showed that compared with other methods, such as VOL 65, o 1, JAUARY 1982 the cube formula and the method of Fortuin et al.,8 the method of Teichholz et al.7 best correlated with thermodilution and angiographic measurements of stroke volume and ejection fraction. Both the cube formula and the method of Fortuin et al.8 assume more or less that ventricular long-axis shortening during systole is equivalent to ventricular internal radius shortening, and both formulas are derived from measurements of ventricular internal radius shortening. In fact, ventricular long-axis shortening is most often less than ventricular internal radius shortening, and this probably explains the overestimates of ejection fraction observed with these methods. The formula of Teichholz et al.7 is based on the observation that the ventricular long axis-to-minor axis ratio increases with a decrease in ventricular size, and it thus takes into account that the extent of ventricular long-axis shortening during systole is usually less than the extent of ventricular minor-axis shortening. However, the formula of Teichholz et al.7 is based on a relation determined empirically between the length L and the inner diameter D of the left ventricle, and the individual data points used to derive this empirical relation are considerably scattered and, as such, suggest that the relationship between ventricular long-axis shortening and short-axis shortening may not be the same in each patient. Moreover, it has been shown (table 2) that in patients with aortic stenosis, AL/L may be selectively decreased. The advantage of the cylindrical model in this respect is that no assumption is made of any relation between the longitudinal axis L and the inner diameter D of the ventricle. Longitudinal axis shortening in the present approach is determined in terms of AR/R and Ah/h by using equation 9, and we have shown (table 2) that the values of AL/L calculated from echocardiography correlate well with angiographic measurements. Thus, it is not surprising that the main discrepancies between the method of Teichholz et al.7 and the cylindrical model were observed in cases where the relationship between ventricular longitudinal shortening and short-axis shortening does not hold. In such cases, the results from the cylindrical model are probably more accurate. However, for these measurements to be valid, ventricular wall thickness must be measured precisely, which is not always possible. Also, posterior wall thickening is assumed to be representative of wall thickening throughout the ventricle and, as with the other methods, our model will probably not be valid in subjects with regional ventricular asynergy.