Abstract
Dimensional variables measured for study of left ventricular mechanics are subject to errors arising from difficulty in determining zerostress dimensions for use as a reference. Based on a method validated for measurements within individuals, we have devised an approach that facilitates comparison between individuals while minimizing random scatter. We define an exact mathematical index of strain, ln(h _{0}/h), using wall thickness (h) referenced to extrapolated wall thickness at zeroluminal volume (h _{0}). Noninvasive data from rabbits, pigs, and humans all yielded highly similar myocardial stress, ln(h _{0}/h), and work values. The stressln(h _{0}/h) relationship during afterload variation was constant among individual pigs with a twofold variation in ventricular mass. Stressln(h _{0}/h) data from our analysis displayed lower scatter than either pressurevolume data normalized to myocardial mass or stressln(h _{0}/h) data referenced to enddiastolic dimensions. A FrankStarlinglike curve with high correlation (r ^{2} = 0.96) was constructed from single points from different pigs, suggesting a low level of size and intersubject scatter. This method offers high precision for noninvasive characterization of ventricular and myocardial mechanics and for comparisons between subjects and between species.
 ventricular function
 wall stress
 ventricular geometry
recent reports have described a framework for noninvasive analysis of myocardial behavior based on mean wall stress and a strainlike quantity, the natural logarithm of reciprocal of wall thickness, ln(1/h) (2123). Wall thickness is used as a precise index of myocardial area under the assumption of the approximate constancy of myocardial volume. This approach has been shown to provide accurate estimates of myocardial work per unit volume of tissue (21, 22). Within individuals, it is sensitive to the effects of βblockers and other inotropes (21). It has the further advantage of being based on easily obtainable echocardiographic measurements.
The above investigations avoided the inclusion of a reference value for wall thickness. When stressstrain analysis is used to investigate constitutive relationships, such a reference measurement is generally made at a point of zero net stress, a point not obtainable under clinical conditions. Thus all analysis in these studies was in terms of increments in the variable ln(1/h) that indexed strain.
The lack of a reference, although practical for determining energetic and geometric changes, prevents direct comparison of values between different individuals. This lack also causes the units of the strainlike variable to be ambiguous. We investigated whether use of a calculated reference would provide a basis for comparison of values between individuals, as well as avoided the use of the ambiguous units of natural logarithm of reciprocal distance. Because myocardial mass has been shown to be effective in normalizing other descriptors of myocardial behavior (4, 15), we specifically examined references calculated as cuberoot functions of this quantity to produce a value with units of length rather than volume. The present paper discusses results obtained by using a reference that is the extrapolated value of wall thickness that would occur if the luminal volume could go to zero.
Glossary
 A
 Area of a truncated conical section of myocardium, cm^{2}
 D
 Instantaneous endocardial shortaxis of left ventricle, cm
 D_{m}
 Instantaneous midwall shortaxis of left ventricle, cm
 D_{m0}
 Midwall shortaxis of left ventricle at zeroluminal volume, cm
 ESPVR
 Endsystolic pressurevolume ratio
 FS
 Fractional shortening of the left ventricle
 h
 Instantaneous measured wall thickness, cm
 h_{0}
 Wall thickness at zeroluminal volume, cm
 L
 Endocardial longaxis of left ventricle, cm; mean sarcomere length, μm
 LV
 Left ventricle
 L_{0}
 Mean sarcomere length at zeroluminal volume, μm
 P
 Blood pressure in ventricular lumen, g/cm^{2}
 p, q,r, a,b
 Arbitrary intermediate quantities for solving a cubic equation
 T
 Mean of circumferential and meridional tension, g/cm
 V
 Volume of a truncated conical section of myocardium, cm^{3}
 V_{c}
 Cavity volume of the left ventricle, cm^{3}
 V_{lum}
 Left ventricular luminal volume, cm^{3}
 V_{myo}
 Left ventricular myocardial volume, cm^{3}
 V_{T}
 Left ventricular total volume (luminal + myocardial volume), cm^{3}
 V_{w}
 Left ventricular wall volume, cm^{3}
 V_{0}
 Extrapolated left ventricular luminal volume at zero pressure, cm^{3}
 VCF
 Velocity of circumferential fiber shortening, circumferences/s
 VCF_{c}
 Corrected velocity of circumferential fiber shortening, circumferences/s
 W
 Work done by a truncated conical section of myocardium, g ⋅ cm
 α
 Arbitrary proportion of myocardium
 ς
 Mean of circumferential and meridional wall stress or fiber stress, g/cm^{2}
THEORY
Definition of variables.
Two widely used descriptors of material deformation are LaGrangian strain, (x −x _{0})/x _{0}, and natural strain, ln(x/x _{0}). The natural strainlike quantity used by Nakano et al. (2123) is not referenced and thus is not directly comparable in magnitude between different hearts. One logical reference dimension,h _{0}, would be the wall thickness at zero external load, producing the variable ln(h _{0}/h). However, the roughly tangential approach of the myocardial stressstrain relationship to the zero external stress axis (10, 39) results in large random errors in any measurement of this reference point. Consequently, we sought a reference that was more precisely determinable despite the fact that it would not correspond to zero net stress.
The net effect of a reference thickness taken at a new point is to cause a constant additive shift in all ln(h _{0}/h) values parallel to the ln(h _{0}/h) axis. This is due to the identity ln(h _{0}/h) = ln(h _{0}) − ln(h). A newh _{0} is simply a different additive constant that does not change the shape of empirical stressln(h _{0}/h) contours.
Because of the demonstrated usefulness of myocardial mass (and the directly proportional myocardial volume) in normalizing descriptors of myocardial mechanics (4, 15), we sought a reference value directly related to this quantity. The line plotted in Fig.1 is the relationship between wall thickness and luminal dimension based on the approach of Hugenholtz et al. (17), which models the myocardial wall as an ellipsoid of constant wall volume. This assumption has been shown to provide an accurate prediction of wall thickness over the physiological range of ventricular contraction (17). Extrapolation beyond the physiological range to zeroluminal diameter provides a clearly defined calculated value that is precisely representative of the myocardial volume while having the proper units (length) for use as a reference quantity. Although this value is not obtainable under physiological conditions, we felt that its clear definition and precise relationship to myocardial mass strongly suggested its appropriateness. This approach is also analogous to the use of allometric equations in which reference variables (e.g., height, weight, or body surface area) are taken to a whole or fractional power so as to provide more appropriate dimensional units (11, 16).
Tangential wallstress relation to radial thickness.
Description of the mathematical relationship between (radial) wall thickness and (tangential) wall stress has been published previously by others (21, 22). Briefly, approximation of the myocardium as incompressible allows substitution of a variable (thickness) that lies along the transmural dimension for the mean of the dimensional variables (arc lengths or areas) that lie in the plane of the wall. Contraction of the myocardium is described in terms of perimeter tension and area strain of a truncated conical region of the ventricular wall. Work, W, done by this region is the integral of perimeter tension, T, with respect to change in area,A: W =
For the present extension of this approach, we took the integration between the calculated reference of extrapolated thickness at zeroluminal volume,h _{0}, and instantaneous thickness, h, measured from Mmode echocardiograms. The general result is W = ς ln(h _{0}/h), where ln(h _{0}/h) is an exact mathematical index of natural strain in the circumferential direction, ln(length/reference length). In that their product is work, ς and ln(h _{0}/h) form a pair of intensive and extensive mechanical variables analogous to pressure and volume or force and length. However, the work expressed is per unit volume of material rather than total work.
The reference value,h _{0}, is in the numerator rather than the denominator. When the index of strain is defined this way, its value increases in parallel with luminal diameter and volume, i.e., as the wall becomes thinner. Work calculated by using a more conventional definition, ln(h/h _{0}), would have a minus sign during ejection of blood. We chose to avoid this potential source of confusion by keeping the inverted form resulting from the above derivation. The zero point of this index of strain corresponds to the zero value of luminal volume, resulting in a qualitative similarity between pressurevolume plots and stressln(h _{0}/h) plots.
Previous work (21) has demonstrated shifts in data within individuals in response to βblockers and other inotropes. The addition of a reference dimension does not alter this sensitivity. Because all data from one individual will be referenced to the same value based on the myocardial mass (h _{0}), all data will be shifted identical distances with the result that prereference shifts will remain. Relative position is only changed between data points obtained from hearts of different mass.
Recent reports provide some separate support for the usefulness of referencing to zerovolume dimensions (2, 3, 10). These studies found that, during ejection, the instantaneous mean sarcomere length,L, is accurately predicted by a function including calculated sarcomere length at zeroluminal volume,L _{0}, and the ratio of luminal and wall volumes, V_{lum}and V_{w}:L =L _{0} [1 + (3V_{lum})/V_{w}]^{1/3}. Thus the sarcomere length can be referenced to zerovolume length in a useful predictive equation:L/L _{0}= [1 + (3V_{lum})/V_{w}]^{1/3}.
METHODS
Human studies.
Patient data were collected retrospectively from records of the Division of Pediatric Cardiology at our institution. Data were selected from patients who underwent 1) successful ablation of substrate of intermittent supraventricular tachycardia (WolffParkinsonWhite or atrioventricular node reentrant tachycardia) in a heart without shunts, incompetent or stenosed valves, or restrictions to flow in the great vessels; and2) same or nextday postcatheterization echocardiographic study. Mmode echocardiographic measurements were taken perpendicular to the long axis of the ventricle at the tops of the papillary muscles with the cursor directed through the septum to a point between the papillary muscles. Measurements taken from the Mmode recordings were enddiastolic wall thickness (mean of posterior wall and intraventricular septum), enddiastolic luminal diameter, and endsystolic luminal diameter. Because blood pressures were not available as part of the echocardiographic studies, systolic and diastolic cuff pressures recorded during catheterization were used. Sixteen patients were included (ages 6–32 yr, mean 14 yr). Calculated ventricular masses (including intraventricular septum; seeCalculations below) ranged from 37 to 124 g.
Animal models.
All animals were treated and cared for in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals (National Research Council, Washington, DC, 1996).
Swine studies.
Ten Yorkshire pigs (25–30 kg) of either sex were part of studies of congestive heart failure in which animals were subjected to rapid atrial pacing. For the present study, data were collected from animals that had yet to receive pacing, after 2 wk of recovery from surgery. The surgical techniques of pacemaker and aortic access port implantation have been described previously (32, 34).
Pigs were presedated with diazepam (10 mg) and brought to the laboratory in a sling. For recording of aortic pressure tracings, a vascular access port was entered by using a 12gauge Huber needle attached to a salinefilled tube and a Statham 23ID (Gould, Oxnard, CA) pressure transducer. Further sedation was achieved with 5 to 10mg boluses of midazolam given through the vascular access port. Surface electrocardiographic (ECG) electrodes were attached to each limb, and standard ECG leads I, II, and III were displayed on a physiological recorder (EVR16, Electronics for Medicine, Overland Park, KS). Twodimensional and Mmode echocardiographic studies of left ventricle (LV) function were performed (2.25MHz transducer, ATL Ultramark VI, Bothell, WA) from the right parasternal approach. LV dimensions were measured at end systole and end diastole as described for patients. Calculated LV myocardial masses (seeCalculations below) ranged from 23 to 64 g.
Six pigs were given variable doses of phenylephrine to increase afterload. Phenylephrine infusion started at a rate of 1.3 μg ⋅ kg^{−1} ⋅ min^{−1}and increased in such a manner so as to obtain four to seven different simultaneous Mmode tracings and blood pressures. This allowed determination of four to seven different paired values of afterload and fractional shortening for each pig.
Rabbit studies.
Data were obtained from three shamoperated rabbits (New Zealand White, 3.5–4.5 kg, 5 mo of age) that served as controls in a study of pacinginduced congestive heart failure. Procedures for instrumentation in this model have been published previously (33). The animals were administered 4 mg/kg sc of enrofloxacin (Baytril, Miles, Shawnee Mission, KS) twice per day on postoperative days 1 and 2. In 4 wk of time, twodimensional and Mmode echocardiography (5MHz transducer, ATL Ultramark VI) were used to image the LV from the right parasternal approach (33). The same ventricular dimension measurements were taken in rabbits as in pigs and patients. Before being brought to the laboratory, rabbits were sedated with 10 mg of intramuscular midazolam (Versed, HoffmanLaRoche, Nutley, NJ) and placed in a custom sling, and an ECG was established.
Pressure was obtained at the time of the terminal experiment with a fluidfilled catheter advanced into the LV via the right carotid artery. The catheter was connected to a previously balanced and calibrated Statham 23ID pressure transducer (Gould). LV masses obtained at autopsy ranged from 3.4 to 4.7 g wet wt.
Calculations.
For determination of wall thickness at other than end diastole, we used the approach validated by Hugenholtz et al. (17) with luminal volume values determined as in Zile et al. (40). The latter method assumes constant myocardial volume and is based on the modeling of the LV as a halfellipsoid of circular cross section with a constant ratio of epicardial long and short axes of 1.2 (40). Wall thickness at the apex is assumed to be 0.4 times that at the base (40). Accordingly, using1) averaged enddiastolic wall thickness, 2) enddiastolic luminal dimension, and 3) endsystolic luminal dimension, we calculated endsystolic (endejection) and zeroluminal wall thicknesses (Eqs. EA5 and EA6 , respectively, in the ). Startejection wall thickness was taken as equal to enddiastolic thickness.
Wall stresses (mean fiber stress) at start and end ejection were calculated based on enddiastolic and endsystolic arterial pressures, respectively, with the use of the formula of Regen (25). Endsystolic pressure was estimated as equal to the calculated mean aortic pressure (28, 29)
LV luminal and myocardial volumes, including free wall and septum, were calculated based on the model of Hugenholtz et al. (17). We used the Zile geometric model (40) for the determination of the long axis of the ventricle. We recognize that use of this model may have resulted in a bias in determined volume and mass values. However, because we were concerned only with the relative sizes of the ventricles, the use of a consistent formula was felt to be a valid approach. Myocardial mass was calculated as myocardial volume times 1.05.
The area under ejection trajectories was calculated as mean stress times the difference in ln(h _{0}/h) values at start and end ejection. The mean stress was calculated as the simple mean of the startejection and endejection stresses.
Ejection time was measured from Mmode recordings as the time between start of contraction and end of contraction.
Data are presented as means ± SD. Statistical difference between regression lines was tested by using confidence ellipses (24). A Bonferroni correction for multiple tests was used in tests for significance by analysis of variance (35). Spearman’s rank correlation was used to test for ordering within clusters (31).
RESULTS
Size independence of stress, ln(h_{0}/h), and work variables.
Figure 2 shows a plot of mean values and 95% confidence intervals of stress and ln(h _{0}/h) at beginning and end systole obtained from rabbits, pigs, and patients. Estimated ventricular masses had means (and ranges) of 4 (35), 40 (23–64), and 78 (37–124) g for rabbits, pigs, and humans, respectively. On the basis of the form of the Laplace law, the similarity of the stress values might be expected, both within and between species, due to the similarities in cardiac anatomy and blood pressures (102/76, 114/80, and 109/62 mmHg for rabbits, pigs, and humans, respectively). Again, because of the form of the Laplace law, one may expect that an index of strain would vary only a small amount because the proportional anatomy of these ventricles is similar both within and between species. Only differences in inotropic state would be expected to result in variations of ln(h _{0}/h) as it does for ln(1/h). The plot indeed displays the independence from ventricular size of not only stress but also ln(h _{0}/h). The variation among species is on the same order as that within species as indicated by the error bars. Multiple oneway analysis of variance of stress and ln(h _{0}/h) values at start and end ejection detected no statistically significant differences among these points.
Figure 3 shows the same plot as in Fig. 2with the addition of stressln(h _{0}/h) values that were calculated from measurements extracted from cardiac literature. The species studied were rats (18, 19), lambs and sheep (1), and humans (9). These results, although based on data from completely independent studies, agree closely with present results. This pattern suggests that these quantities are insensitive to interobserver variation, and that interspecific differences in these quantities, if present, are small for these species. Endsystolic values, which would be expected to vary less with likely differences in loading between experimental populations, show closer similarity.
By using start and endejection values, ejection work was calculated by approximating the ejection trajectory in the stressln(h _{0}/h) plane as a straight line. Validity of this approximation for normally ejecting ventricles in dogs and humans is supported by published studies of stressstrain loops (22, 37). On the basis of the Laplace law relating stress to pressure and proportional geometry, the similarity in ventricular geometry and pressure waveforms among the species included in this study should result in similar ejection trajectories. The area under this line segment represents the work per unit myocardium during ejection. The values are similar, despite a roughly 150fold range in myocardial mass (Table1). Observed variations are on the order of what would be expected within single individuals because of preload and afterload variations. The values are also similar to those reported by Nakano et al. (Ref. 22; 40–96 g ⋅ cm^{−1} ⋅ cm^{3}) for patients for the full cardiac cycle.
The response of endsystolic stress and ln(h _{0}/h) to afterload variation from graded doses of phenylephrine is shown in Fig. 4. Endsystolic points for each pig fall along straight lines with little scatter (r ^{2} = 0.94–0.99). These lines cluster tightly without ordering according to the approximately twofold variation in ventricular mass (23–40 g, mean 31 g) [Spearman’s rank correlation between mean endsystolic ln(h _{0}/h) and myocardial mass not significantly different from zero]. None of the regression lines shows a statistically significant difference from the others.
Figure 5 combines Fig. 4 with results calculated from published dimension values from six openchest dogs undergoing afterload variation because of aortic constriction (23). The values were obtained from plotted data with the use of a calibrated digitizer tablet. Based on reverse calculation from values of derived quantities in tables, the masses of these canine ventricles ranged from 19 to 67 g with a mean of 41 g. As an indicator of the location of the data set from each dog, we calculated the meanxposition within each animal’s set of points. In the original analysis, these mean ln(h _{0}/h) values had a total range of 0.496. The range of the present results was reduced by ∼50% to 0.26 with a mean position very close to, and not statistically different from, that of the swine results. Changes in both mass and species resulted in little or no position change in the stressln(h _{0}/h) domain.
The values of both slope and intercept from the recalculated dog results range from below to above those of the pig data. The variation in parameter values is much greater in the recalculated dog data than in the pig data. This variation might be due to a roundoff error introduced when published data were reanalyzed, or it may represent real variations in the slopes and intercepts that are characteristic of the species. As would be expected because of the complete overlap of ranges, no statistically significant difference was found between the population means of the canine and swine regression parameters (slope and intercept) with the use of simultaneousttests of slopes and intercepts with Bonferroni correction.
Preliminary investigation of the stressln(h _{0}/h) relationship under constant afterload was based on data from two literature sources in which the endsystolic pressurevolume ratio (ESPVR) (36) was investigated (15, 39) in isolated, crosscirculated hearts from rabbit (5 g) and dog (131 g), which contracted isovolumically and displayed quite different ESPVRs. In contrast to the afterload variation results discussed in Size independence of stress, ln(h_{0}/h), and work variables, these data display clearly nonlinear ESPVRs. This is probably due to the lower range of preloads that is attainable with this type of preparation. The higher preloads observed under afterload variation apparently span only a linear portion of the total curve.
As can be seen in Fig. 6, the stressln(h _{0}/h) curves are almost superimposable, despite the species and size differences. If the data are presented in the ln(1/h) format of Nakano et al. (2123), i.e., without referencing, the meanxpositions of the data sets are −0.62 and 0.60. After referencing, the mean positions are 0.56 and 0.51.
Precision of stress and ln(h_{0}/h) variables.
As opposed to the direct measurement of volume carried out for pressurevolume results (4, 15), noninvasive approaches require the calculation of volumes from linear measurements. Such calculated volumes will have a magnified level of scatter (lower statistical precision or sensitivity) because of the cubing of measured values. To demonstrate this effect, the stressln(h _{0}/h) values from one of the phenylephrine infusion experiments in Fig. 4 is reproduced in Fig.7 A, whereas Fig. 7 B displays values of pressure and luminal volume over mass that have been calculated from the same raw data. The value of the coefficient of determination,r ^{2}, is the fraction of total variance in the data that is accounted for by the regression relationship (24) and thus is a direct indicator of the precision of the data. Ther ^{2} value was higher for the present analysis (0.95 vs. 0.72), indicating a lower level of scatter. As shown in Table 2, this pattern of r ^{2}values was true for data from all six pigs from the phenylephrineinfusion experiments (Fig. 4).
Other noninvasive methods of ventricular performance assessment, such as FS (7), velocity of circumferential fiber shortening (VCF) (5), and corrected VCF (VCF_{c}) (8), use end diastole as a reference state. This is also true for some invasive methods (20). However, natural variation in enddiastolic distension from beat to beat would be expected to result in additional random scatter not accounted for by an underlying stressln(h _{0}/h) relationship. Figure 7 C displays values of stress paired with ln(h _{0}/h), calculated from the same raw data set as Fig.7 A but by using enddiastolic thickness as a reference rather than thickness at zeroluminal volume. As expected, the value ofr ^{2} was higher (0.95 vs. 0.87) when the reference was the zeroluminal thickness. Again, this pattern was true for all six pigs from the phenylephrineinfusion experiments.
Thus in all six sets of regression results, the present approach produced the highestr ^{2} value and highest precision. With the use of a sign test (31), this result was tested against a null hypothesis of ther ^{2} value of the present approach not being the highest (i.e., one or both of the other approaches having equal or higherr ^{2}). The test showed significance with a P value of 0.016.
Ejection energetics and enddiastolic ln(h_{0}/h).
If indeed the tested method has a high precision, it should allow the discernment of underlying patterns that are obscured by random scatter in other methods. Accordingly, a final area of investigation was the relationship of ejection work per unit volume of myocardium to enddiastolic ln(h _{0}/h). This relationship is analogous to a FrankStarling curve relating stroke volume to enddiastolic luminal volume. Singlepoint data, each based on Mmode and pressure measurements from a single time point, from 10 healthy control pigs of similar age are plotted in Fig.8. Linear regression analysis demonstrated a high correlation (r ^{2} = 0.96) between ejection work and enddiastolic ln(h _{0}/h). The probability of this relationship of points occurring by chance, when no actual relationship exists, is <0.0001 (P value of the regression). As would be expected based on the FrankStarling relationship, work increased with diastolic distension [high values of ln(h _{0}/h)]. Remarkably, the present result has been obtained by using independent baseline measurements from 10 different animals. Although the contractilities, and hence FrankStarling curves, of these animals might be expected to be similar, it would be predicted that measurements from separate animals would be confounded by individual variations in size and physiology, thus lowering correlation.
Regression with the use of endsystolic points, points not expected to follow a FrankStarling relationship, showed no correlation (r ^{2} = 0.02,P < 0.71), suggesting that the enddiastolic correlation was not due simply to correlation between work and overall ventricular mass. Regression was also performed by using data obtained from the same animals after 3 wk of rapid atrial pacing used to induce heart failure. The heartfailure animals would be expected to have a large range of (reduced) contractilities (32) and thus would not be expected to share similar FrankStarling relationships. The regression failed to show any correlation (r ^{2} = 0.05,P < 0.56).
We also investigated the behavior of calculated ejection power (ejection work/ejection time) with enddiastolic ln(h _{0}/h) in the same set of control pigs (Fig. 9). We again found a high correlation (r ^{2} = 0.89) with enddiastolic ln(h _{0}/h), but not endsystolic ln(h _{0}/h) (r ^{2} = 0.01,P < 0.78) or enddiastolic ln(h _{0}/h), after induction of heart failure by rapid atrial pacing (r ^{2} = 0.01,P < 0.76).
DISCUSSION
The method of Nakano et al. (2123) offers the advantages of being noninvasive and physics based. It has been demonstrated to provide accurate estimates of myocardial work and to be sensitive to myocardial inotropic state. The present study has demonstrated the additional precision achieved by referencing the index of strain, ln(1/h) to calculated thickness at zeroluminal volume,h _{0}. Measured values of stress, ln(h _{0}/h), and work per unit volume of myocardium from differentsized individuals of one species are very similar when zerovolume referencing is used. Furthermore, the similarity of myocardial energetic and mechanical properties among different mammalian species, widely demonstrated at tissue and biochemical levels, can be directly observed in vivo by using this noninvasive approach.
Despite the noninvasive nature of the method, patterns of stress and ln(h _{0}/h) under varying afterload were found to have high correlations and remarkable similarity among ventricles of differing size in pigs. Position shifts in plots of stress vs. strain index, observed in previous studies employing the unreferenced strain index ln(1/h) (21), thus have significance for absolute as well as relative value. The absolute location of a stressln(h _{0}/h) curve could potentially indicate an abnormal myocardial state without reference to a control value within the same animal. Initially variant or abnormal individuals could be identified and analyzed accordingly.
Results obtained from published data in a different species (dog), collected completely independently from the present work, also demonstrated the increased clustering of results within a species that is achieved by the referencing toh _{0}. The mean positions of the clustered data points were similar between the species as were the mean slopes and intercepts of regression lines. Again, this allows analysis in terms of absolute as well as relative position of the stress vs. strain index curves. Whereas the dog curves showed a greater variation in parameters (mean abcissa, slope, and intercept), the means of these parameters were not different from those of the swine data. This difference in variance of the location parameter would not be detectable under systems in which location is not adjusted according to myocardial mass, such as when ln(1/h) is used.
We have demonstrated the lower random scatter (higher precision) that results from the present approach, compared with approaches based on calculated volumes or on enddiastolic references. Random scatter was also low enough to allow the construction of a FrankStarlinglike curve by using single points from different animals. The known underlying relationship was discernible despite variations between individuals.
Referencing to zerovolume thickness may seem problematic because such dimensions are not physiological. This thickness, however, is a characterization, or parameter, of the basic geometry of the ventricle rather than a physiological measurement. It would be mathematically equivalent to our present method to use a dimension occurring at a different luminal volume, as long as that luminal volume were a constant proportion of myocardial volume. For example, calculating a reference thickness at luminal volume equal to myocardial volume, rather than at zero lumen, would lower all presented ln(h _{0}/h) values by a constant value, 1.056. Whereas most values would shift to negative, all correlations among the data and similarities between data sets would remain unchanged. Although a choice among mathematically equivalent reference points cannot avoid being somewhat arbitrary, we feel that the choice of a point at zero volume avoids possible confusion engendered by negative values of ln(h _{0}/h) as well as resulting in the coincidence of zero ln(h _{0}/h) and zeroluminal volume.
Comparison to massbased normalizations.
Normalization of ventricular measurements to the myocardial mass has been used to standardize pressurevolume relations (4, 15). The present method is analogous to such an approach because wall thickness at zeroluminal volume is directly related to the total myocardial volume. However, normalizations that are based on mass utilize a ratio involving two different quantities (luminal volume and wall mass) in contrast to the ratio of two values of the same quantity (wall thickness) as in the present approach. The combination of values from different geometric structures cannot be used as an index of strain in analysis, thus complicating possible comparisons to stressstrain behavior (curve shapes) in other situations, such as isolated myocardial strips and trabeculae (6, 14).
Because, due to its noninvasiveness, the present method relies on distance measurements rather than direct measurement of volume, only calculated volumes can be obtained. The present approach avoids this demonstrated source of scatter (Fig. 7) by measuring and calculating linear quantities only.
Results comparable with the present study are theoretically obtainable when midwall measurements are normalized or referenced to calculated values at zeroluminal volume rather than enddiastolic dimensions, as have been used previously (12, 13, 20, 27, 30). The mathematical relationship between wallthickness ratios and midwalldiameter ratios is not linear, however (see the ); therefore, the shapes of the relationships with stress, work, and power may also be different. The use of wall thickness and luminal diameter in the present investigation is based on the ease of obtaining these measurements through echocardiographic imaging.
Physical meaning of variables.
The present study investigated stress, ln(h _{0}/h), work, and power. These variables are fundamental physical quantities expressed in fundamental physical units. Accordingly, these noninvasively determined values may be directly compared with values obtained from experiments that used other fundamental quantities, such as force and distance, pressure and volume, or generated heat. This situation contrasts with that of other noninvasively obtained quantities, such as FS (7), VCF (5), and VCF_{c} (8), that can only be compared with other measurements of the same indexes.
The work of Nakano et al. (21) using ln(1/h) suggests that the relationship of that variable to mean tangential stress can be an index of contractility. Comparison of the present work with their results would suggest that the stressln(h _{0}/h) relationship could be used in a similar fashion. We did not explicitly manipulate contractility, however, and, therefore, have not addressed this question. Accordingly, our main conclusion is that the confounding effects of size and species on measurements of mechanical and energetic variables appear to be substantially reduced or eliminated by our approach.
Limitations of the present work.
We chose to use a particular geometric model of the LV that, as with any geometric model, contains some inaccuracy. The consistent use of this model for all calculations should result in equal bias for all results and thus allow the valid comparison of values.
Because it is a material property, ln(h _{0}/h) should be only slightly affected by the shape of the chosen geometric model. Furthermore, because the stress variable used here is the mean of equatorial and meridional stress, changes in shape have little effect on its final value (2). We tested these predictions and found <1% change in calculated values of stress or ln(h _{0}/h) when the geometric model was varied in the ratio of the long axis to the short axis from 1 to 1.4. Thus our particular choice of longaxis/shortaxis ratio should have a negligible effect on the accuracy of the results.
There may be a varying degree of inaccuracy of the geometric model, depending on the size and, especially, species origin of the heart. This potential source of error cannot be eliminated from the present study. Considering the similarity of values among the species investigated in the present work, it would seem that this error is not of great magnitude.
There may be a high percentage of error in measurements of small values of wall thickness. Despite the fact that many of the data points were single measurements of wall thickness, little scatter was observable in plots of endsystolic stress and ln(h _{0}/h) under varying afterload. The high correlations obtained in these experiments would suggest that this source of measurement error had little effect on the results.
The systolic trajectory in the stressln(h _{0}/h) relationship was approximated as a straight line, as suggested for results in dogs and humans. It is not known how consistently this pattern is found within these species or in the other species that are used in the present study. The trajectory may instead be concave up or down. Thus it must be stated that the work and power estimates made in this study may contain an unknown amount of error or bias. As stated inresults, we have assumed that the similarity in ventricular geometry and pressure waveforms among the species included in this study would result in similar ejection trajectories because of the law of Laplace. Analysis in terms of this principle suggested that changes in pressure within a physiologically realistic range (for normal individuals) during ejection would produce only small departures from linearity of this trajectory. These changes would result in <20% increase in the ejection work estimate.
The present stressln(h _{0}/h) data could not be used directly to derive a constitutive or phenomenological (predictive) equation. The advantages gained in ease and precision of measurement would seem to make this limitation an acceptable tradeoff.
It can be argued that the stress values used here are hypothetical because they are not measured directly (2). Pressure waves within the lumen and anisotropy within the wall certainly cause variations in stress among different depths and regions of the myocardium that are beyond the analysis provided by the present approach.
The stress values are not hypothetical, however, when the mean fiber stress over the full thickness of the myocardium in the geometric model employed here is discussed. The overall balance of mean pressure and mean stress in ellipsoidal shapes has been demonstrated to be almost completely independent of the ratios of diameters (2, 26). This fact and the fundamental law of balance of stresses in a plane (38) require that measured pressures and wallluminal geometries uniquely and accurately predict values of mean fiber stress.
In summary, we have developed and tested a formalism for LV chamber and myocardial mechanics that includes referencing of wallthickness measurements to calculated thickness at zeroluminal volume. This approach allows increased direct comparison between mechanical and energetic measurements by 1) reducing scatter, 2) eliminating size dependence, and 3) employing variables that are expressed in basic physical units. Underlying relationships are less obscured by random variations in measurements. The fact that the approach involves only noninvasive measurements makes it practical for use in longitudinal studies and clinical applications.
Footnotes

Address for reprint requests and other correspondence: S. Denslow, The Children’s Heart Center of South Carolina, 165 Ashley Ave., PO Box 250915, Charleston, SC 29425 (Email: denslows{at}musc.edu).

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 Copyright © 1999 the American Physiological Society
Appendix
A cubic equation was used to calculate wallthickness values from the measured values of 1) enddiastolic values of wall thickness and luminal diameter and2) endsystolic luminal diameter. It was based on the geometric model used by Zile et al. (40) and the approach used by Hugenholtz et al. (17). LV geometry was assumed to be a halfellipsoidal shell. The proportionality of the epicardial long and short axes of this solid of revolution was assumed to be 1.2, and the wall thickness at the apex was assumed to be 0.4 times that at the base. In the following, L is the endocardial long axis, D is the endocardial shortaxis, h is wall thickness at the base, V_{T} is total ventricular volume (luminal + wall), and V_{c} is cavity (luminal) volume. The constant myocardial volume is V_{myo}.
The basic equations describing this model are
An enddiastolic midwall encloses some proportion, α, of the total myocardial mass, V_{myo}. The diameter enclosing this fraction of mass at an arbitrary point in contraction isD _{m}, whereas the diameter of the lumen is D. At zeroluminal volume, this fraction of mass would form a sphere with diameter D _{m0}.
The total volume of the lumen plus myocardium can be expressed in two ways
At zeroluminal volume, the midwall diameter still encloses the same proportion, α, of myocardium, but no lumen, whereas the wall thickness encloses the full myocardial volume