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Validation of a novel noninvasive cardiac index of left ventricular contractility in patients

¹Department of Cardiology, National Heart Centre, ²Division of Engineering, Science, and Technology, University of New South Wales-Asia, and ³College of Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore; and Departments of ⁴Biomedical Engineering, ⁵Surgery, and ⁶Cellular and Integrative Physiology, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana

Submitted 25 May 2006 ; accepted in final form 15 January 2007

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
Although there are several excellent indexes of myocardial contractility, they require accurate measurement of pressure via left ventricular (LV) catheterization. Here we validate a novel noninvasive contractility index that is dependent only on lumen and wall volume of the LV chamber in patients with normal and compromised LV ejection fraction (LVEF). By analysis of the myocardial chamber as a thick-walled sphere, LV contractility index can be expressed as maximum rate of change of pressure-normalized stress (d*/dt_max, where * = /P and and P are circumferential stress and pressure, respectively). To validate this parameter, d*/dt_max was determined from contrast cine-ventriculography-assessed LV cavity and myocardial volumes and compared with LVEF, dP/dt_max, maximum active elastance (E_a,max), and single-beat end-systolic elastance [E_es(SB)] in 30 patients undergoing clinically indicated LV catheterization. Patients with different tertiles of LVEF exhibit statistically significant differences in d*/dt_max. There was a significant correlation between d*/dt_max and dP/dt_max (d*/dt_max = 0.0075dP/dt_max – 4.70, r = 0.88, P < 0.01), E_a,max (d*/dt_max = 1.20E_a,max + 1.40, r = 0.89, P < 0.01), and E_es(SB) [d*/dt_max = 1.60E_es(SB) + 1.20, r = 0.88, P < 0.01]. In 30 additional individuals, we determined sensitivity of the parameter to changes in preload (intravenous saline infusion, n = 10 subjects), afterload (sublingual glyceryl trinitrate, n = 10 subjects), and increased contractility (intravenous dobutamine, n = 10 patients). We confirmed that the index is not dependent on load but is sensitive to changes in contractility. In conclusion, d*/dt_max is equivalent to dP/dt_max, E_a,max, and E_es(SB) as an index of myocardial contractility and appears to be load independent. In contrast to other measures of contractility, d*/dt_max can be assessed with noninvasive cardiac imaging and, thereby, should have more routine clinical applicability.

cardiac mechanics; ventricular elastance; ventriculography; wall stress

An additional difficulty with LV dP/dt_max is that it is not preload independent (23). Conceivably, the LV pressure-volume relation and elastance reflect LV contractile function more accurately (22, 31, 32, 40, 42, 43). Suga and Sagawa (42) formalized this concept as the time-varying elastance of the ventricle by defining elastance (E) as follows: E(t) = P(t)/[V(t) – V_d], where P(t) and V(t) represent ventricular pressure and volume, respectively, which vary with time t, and V_d is the LV volume corresponding to zero LV pressure obtained by drawing a tangent to the pressure-volume curves at end ejection.

The end-systolic pressure-volume (ESPV) relation, which is the locus of pressure and volume points at end systole, has been shown to be insensitive to variations of the end-diastolic volume (preload) and the mean arterial pressure (afterload). The ESPV relation is usually a straight line with a slope of E_es (i.e., elastance at end systole). It has been found that E_es remains essentially constant if the preload and afterload are allowed to vary within the physiological range but is sensitive to inotropic agents and ischemia. Hence, E_es has been proposed as a "load-independent" index of contractility of the ventricle (32, 33, 37). The loading conditions do affect E_es, however, if ventricular pressure and volume are made to vary over a wider range (4, 25). Under these conditions, the ESPV relation becomes curvilinear, concave toward the volume axis, and load dependent. Hence, indexes such as E_es and the ESPV relation lose their clinical applicability (23). Elastance measures also require cardiac catheterization for measurement of pressure, which further reduces their clinical utility. An additional limitation of E_es is the difficulty of changing afterload and obtaining multiple pressure-volume data points in a given subject while maintaining a constant contractility. Hence, it is impractical to use E_es clinically for patient-specific LV catheterization-ventriculography data.

Recently, several approaches, generally referred to as the single-beat method [E_es(SB)], have been proposed for estimating E_es without loading interventions (37, 38, 44). Senzaki et al. (37) described a method for estimating E_es(SB) that is based on normalized time-varying elastance curves. In subsequent studies, this method was improved by adjustment of the normalized elastance curve to compensate for load dependencies and contractility (38).

Zhong et al. (46) introduced the formulation of an active LV elastance term, E_a,max, derived from LV pressure-volume data. E_a represents LV myocardial sarcomere activation; hence, the peak E_a (E_a,max) is hypothesized to represent LV contractile function. Nevertheless, the use of E_a,max as an intrinsic LV contractility index is tempered by the need for invasive LV pressure measurement and complex and time-consuming computations (see APPENDIX A).

The dP/dt_max is based on LV intracavity pressure, which is generated by an active myocardial stress. Hence, it is more appropriate to formulate an LV contractility index on the basis of LV wall stress (5, 9, 24). Numerous methods have been devised to measure LV wall stress (13) or to approximate it by use of geometric models (8, 19, 36). Measurement of transmural wall stress (with implantable strain gauges) is hampered, however, by the degree of coupling between the transducer and the ventricular wall (5).

We previously hypothesized that LV contractility is the capacity of the LV to develop intramyocardial stress to eject blood volume as rapidly as possible (45). On the basis of this concept, we developed an LV contractility index based on wall stress in a spherical LV model. The LV contractility index is formulated as follows: d(/P)/dt_max or d*/dt_max, where * = /P. In contrast to dP/dt_max, E_a,max, and E_es(SB), this novel index is derived solely from LV intracavity and myocardial wall volume data, which can be determined noninvasively. Since this approach obviates the need for invasive pressure measurement, it has the potential to become a clinical routine with noninvasive imaging of ventricular volume and myocardial mass.

In the present study, we describe a proof of concept for this novel stress-based LV contractility index by validating it against dP/dt_max, E_a,max, and E_es(SB) in 30 individuals with disparate ventricular function. Furthermore, we verified the load independence of the index under different preload and afterload conditions.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
Experiment I: comparison of d*/dt_max with dP/dt_max, E_a,max, and E_es(SB) determined from invasive cardiac catheterization. Thirty patients with diverse cardiac conditions undergoing clinically indicated invasive cardiac catheterization were enrolled in the study, which was approved by the Local Research Ethics Committee. Informed consent was obtained from each patient.

The catheterization laboratory employs a Philip Integris Allura 9 with the Dynamic Flat Detector (Philips Medical Systems, Best, The Netherlands) X-ray imaging system. Standard techniques used to advance catheters in retrograde fashion across the aortic valve allowed continuous measurement of LV chamber pressure. The LV pressure was then differentiated with respect to time for determination of maximal rate of increase during systole (LV dP/dt_max). The peak ECG R wave was used as an end-diastolic timing marker. Straight or angled six-side-hole pigtail catheters (5-F or 6-F) were used at the discretion of the operator. Each patient underwent rapid bolus infusion of 30–36 ml of nonionic radiopaque contrast solution into the LV through the pigtail catheter at 10–12 ml/s.

Ventriculograms were obtained in the standard 30° right anterior oblique projection with a 9-in. image intensifier at a speed of 50 frames per second. Ventriculography and hemodynamic data analyses were performed offline. Planimetry of single-plane LV endocardial and epicardial areas (aided by automated border detection) and LV long-axis length determination allowed calculation of frame-to-frame variation in LV intracavity volumes as well as LV myocardial volumes using the corresponding Viewstation (Philips Medical Systems; Fig. 1) (3, 30, 34). ECGs were recorded continuously with the ventriculograms. The images chosen for planimetry were from the first sinus beat with adequate opacification. LV ejection fraction (LVEF) was easily calculated from LV end-diastolic volume (LVEDV) and LV end-systolic volume (LVESV) as follows: LVEF = (LVEDV – LVESV)/LVEDV.

View larger version (71K): [in this window] [in a new window]   Fig. 1. ECG-triggered contrast cine left ventriculography at the right anterior oblique projection depicting opacified left ventricular (LV) cavity at the end-diastolic phase of the cardiac cycle. End-diastolic and end-systolic endocardial contours are outlined. An 4-cm segment of LV free wall epicardium at the junction of the apical and middle thirds is also depicted. Average LV wall thickness is obtained from the quotient of this segmental area and segmental length. From this, myocardial wall volume can be estimated (30, 34).

View larger version (10K): [in this window] [in a new window]   Fig. 2. Top: LV pressure (P) and volume (V) throughout the cardiac cycle [time (in seconds) from start of isovolumic contraction] in 1 representative subject (subject x). Measurements have been corrected for geometric distortion from the respective recording systems. Bottom: pressure-volume relation in subject x. Ascending numbers indicate frame number (each frame lasts 20 ms) from beginning of isovolumic contraction (frame 1). Frames 1–5, isovolumic contraction; frames 5–17, ejection; frames 17–21, isovolumic relaxation; frames 21–36, ventricular filling.

Experiment II: load independence of echocardiographically derived d*/dt_max.

There were three groups of volunteers: 10 healthy subjects were given sublingual glyceryl trinitrate (500–1,000 µg) to decrease afterload, with a target decrease in systolic blood pressure of 10% (group A), 10 healthy subjects received intravenous normal saline solution (500 ml) over 20–30 min to increase preload (group B), and 10 consecutive patients were scheduled to undergo clinically indicated dobutamine stress echocardiography (group C). In all subjects, echocardiography was performed at baseline, as well as after alterations of afterload, preload, and myocardial contractility. In group C, echocardiographic measurements were made during intravenous dobutamine infusion at 5.0 µg·kg^–1·min^–1, a dose that increases myocardial contractility but does not have significant cardiac chronotropic or volume effects.

All echocardiograms were acquired using Vivid 7 (General Electric). M-mode measurements of the LV were obtained, and LV mass was calculated using standardized methodology (6, 18). Myocardial volume was calculated by dividing LV mass by myocardial density (assumed to be 1.05 g/ml). Furthermore, two-dimensional apical four- and two-chamber views of the LV were acquired, and end-diastolic and end-systolic endocardial contours were manually outlined. The corresponding LVEDV and LVESV were then automatically determined using Simpson's biplane method. Pulse-wave echocardiographic Doppler interrogation of the LV outflow tract (LVOT) allowed, in the absence of significant mitral regurgitation or aortic valve dysfunction, calculation of the maximal LV volume rate (dV/dt_max) during ejection: dV/dt_max = V_peak * AVA, where V_peak is the peak velocity sampled at the LVOT and AVA is the aortic valve area (AVA = D²/4, where D is the LVOT diameter measured in the 2-dimensional parasternal long-axis image of the heart). Substitution of values of myocardial volume and dV/dt_max into Eq. 7 (see below) yields d*/dt_max.

E_es(SB) estimation using bilinearly approximated time-varying elastance. The algorithm is based on the elastance curve derived from the pressure-volume loop (38). It is primarily focused on the shape of the curve, with the assumption of a dependence on contractility and loading conditions. The time-varying elastance is approximated by a bilinear function: one for the isovolumic contraction phase and another for the ejection phase. E_es(SB) is then given by

(1)

where P_ad is pressure at the end of isovolumic contraction, P_ed is end-diastolic pressure, P_es is end-systolic pressure, PEP is preejection period, ET is ejection time, is the ratio of the slope in the ejection phase to the slope in the isovolumic contraction phase, and SV is stroke volume. The bivariate model of Shishido et al. (38) was used to estimate : = –0.210 + 1.348EF + 0.682PEP/(PEP + ET), where EF is LV ejection fraction.

Derivation of a novel wall stress-dependent LV contractility index. For simplicity, we approximate the LV as a thick-walled spherical shell consisting of incompressible, homogeneous, isotropic, elastic material. The inner and outer radii of the shell are denoted r_i and r_e, respectively. The epicardial (outer) surface is considered load free, whereas the endocardial surface is subjected to LV intracavity pressure P(t), where t is time. The circumferential wall stress () can be expressed at any transmural radial position in the wall (r) as

(2)

(see APPENDIX B for derivation). In this model, the maximum wall stress occurs at the endocardium, r = r_i, and is given by

(3)

The geometric relation between wall volume (V_m), LV cavity volume (V), r_i, and r_e can be expressed as

(4)

If we combine Eqs. 3 and 4, we obtain the desired result

(5)

By normalizing wall stress to LV intracavity pressure (P), an index of LV contractile function may result as

(6)

This notion is based on the premise that LV wall stress (due to LV myocardial sarcomere contraction) is responsible for the development of LV pressure. Hence, it is more rational to base LV contractility index on LV wall stress per pressure, rather than on stress or pressure alone.

Analogous to dP/dt_max, we propose an LV contractility index as the maximal rate of change of pressure-normalized wall stress

(7)

Obviously, this index is independent of pressure by definition.

Data analysis. Mathematical modeling and curve fitting were performed using Matlab 6.5 software. Statistical analysis was performed using Medcalc statistical software. All parametric data are means ± SD. Statistical significance was defined at P < 0.05. In experiment I, the patients were divided into tertiles according to their LVEF for analysis. For multiple-group comparisons, ANOVA was followed by Student-Newman-Keuls test for pairwise comparisons if ANOVA revealed significant intergroup differences. Linear regression analysis was employed to quantify the correlation between d*/dt_max and other indexes of LV contractility [dP/dt_max, E_a,max, and E_es(SB)]. In experiment II, paired Student's t-test was used to compare parameters measured at baseline and after physiological manipulation.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
Correlation of d*/dt_max with dP/dt_max, E_a,max, and E_es(SB). Thirty volunteers [mean 58.1 (range 48–77) yr of age, 13:2 male-to-female ratio] with diverse cardiac conditions were recruited for experiment I. Sample LV pressure-volume data from one patient are shown in Fig. 2. LVEF and dP/dt_max were computed directly from these traces. E_a at various times was also computed from the pressure-volume loops (Table 1), as described in APPENDIX A. E_a,max was extrapolated from the peak of the E_a-time curve (Fig. 3). E_es(SB) was also computed from the pressure-volume loop using Eq. 1.

View larger version (8K): [in this window] [in a new window]   Fig. 3. Active elastance (Ea) values during isovolumic contraction and relaxation vs. time from start of isovolumic contraction in subject x. Curve fit to Eq. A2 yields Ea,0 = 1.48 mmHg/ml, C = 0.1555 s, ZC = 1.631, d = 0.28 s, R = 2.267 s, and Ea,17 = 1.1 mmHg/ml (root mean square error = 0.026 mmHg/ml).

Figure 4 summarizes the correlation between d*/dt_max, dP/dt_max, and E_a,max, as well as E_es(SB). Linear regression analysis revealed good correlation between d*/dt_max and dP/dt_max, E_a,max, and E_es(SB), with significant correlation coefficients of 0.9 in each case: d*/dt_max = 0.0075dP/dt_max – 4.70 (r =0.88, P < 0.01), d*/dt_max = 1.20E_a,max + 1.40 (r = 0.89, P < 0.01), and d*/dt_max = 1.60E_es(SB) + 1.20 (r =0.88, P < 0.01). In contrast, the correlation between d*/dt_max and LVEF is less strong (r = 0.71), as is the correlation between E_es(SB) and LVEF (r = 0.78), underscoring the lack of specificity of LVEF as an index of myocardial contractility.

View larger version (8K): [in this window] [in a new window]   Fig. 4. Linear regression analysis demonstrates good correlation between d*/dtmax and dP/dtmax (d*/dtmax = 0.0075dP/dtmax – 4.70, r =0.88; A), between d*/dtmax and Ea,max (d*/dtmax = 1.20Ea,max + 1.40, r =0.89; B), and between d*/dtmax and Ees(SB) [d*/dtmax = 1.60Ees(SB) + 1.20, r =0.88; C].

Load independence of d*/dt_max.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

During LV systole, LV wall stress is generated intrinsically by sarcomere contraction and results in the development of extrinsic LV pressure. LV wall stress is dependent on wall thickness, LV geometry, and chamber pressure and sarcomere contraction. Hence, it is rational to quantify the LV wall stress as an intrinsic measure of myocardial contractility. For example, Hood et al. (12) evaluated stress as an "index of the ‘appropriateness’ of the anatomic and functional response to any given pressure or volume load," but they were unable to provide a formulation for the normalized LV stress. A similar notion has been suggested by several investigators (5, 12, 24). Here, we present the first validated formulation of a stress-dependent contractility index d*/dt_max.

The LV wall stress also has a bearing on the myocardial oxygen supply-and-demand, which potentially influences contractile function. As wall stress increases, strain energy and myocardial oxygen demand increase in tandem (41). Increased wall stress can also reduce myocardial oxygen supply by compression of intramural arterial blood flow, which augments flow resistance (11). Hence, LV wall stress is only one determinant of LV myocardial perfusion. Other factors include pressure in the coronary vessels and coronary vasculature and LV myocardial porosity. The latter is in part influenced by LV wall stress.

We have validated a new LV contractility index, d*/dt_max, based on the maximum rate of development of LV wall stress with respect to LV pressure. From the right-hand side of Eq. 7, this index is also seen to represent the maximal flow rate from the ventricle (cardiac output) normalized to myocardial volume (or mass). The cardiac output has been used as a measure of myocardial contractility in rats, provided that the influences of afterload are taken into account (7). Furthermore, normalization of ventricular measurements to the myocardial volume or mass has been used to standardize pressure-volume relations (2, 10). The present model formalizes these features analytically.

The peak rates of myocardial stress (d/dt) and LV pressure (dP/dt) development typically occur during the isovolumic contraction phase (Fig. 5, B and D, respectively). The rate of development of pressure-normalized wall stress (* = /P) during isovolumic contraction, however, is nearly zero because of the small changes of LV geometry during isovolumic contraction (Fig. 5F). In Eq. 7, the peak rate of normalized wall stress d*/dt (left-hand side of Eq. 7) occurs during the ejection phase (Fig. 5F), consistent with the peak cardiac output during the ejection phase (right-hand side of Eq. 7). Hence, there is synchronization between the novel contractility index and the rate of ejection, as suggested by Eq. 7.

View larger version (16K): [in this window] [in a new window]   Fig. 5. Plot of wall stress (; A), rate of wall stress (d/dt; B), LV pressure (P; C), rate of LV pressure (dP/dt; D), pressure-normalized wall stress (* = /P; E), and rate of * (d*/dt; F) vs. time. Data were obtained from 1 typical subject.

In the 30 additional subjects/patients studied in experiment II, d*/dt_max is insensitive to preload changes induced by intravenous saline infusion and afterload changes induced by glyceryl trinitrate but is sensitive to changes in contractility provoked by an infusion of dobutamine. The use of nitroglycerin to vary afterload results in a decline in preload (a decrease in EDV, albeit not statistically significant). In future studies, the afterload resistance or impedance must be measured to ensure the desired effect on afterload; i.e., the index needs to be further tested in the setting of acute changes in vascular impedance load.

Traditional axisymmetric models of LV determine wall stress at the equator of a thick-walled sphere. For the formulation of the new index, the LV was modeled as a pressurized thick-walled sphere. Similar geometric assumptions had been made by investigators studying LV pressure-volume dynamics in small animal models (14, 26). This was done to reduce the mathematical complexity of calculation, which is especially pertinent in clinical cases. Regardless of the simplified LV spherical model, the present results indicate that d*/dt_max can distinguish between impaired and normal LVEF and correlates well with dP/dt_max, E_a,max, and E_es(SB). Future studies must consider more realistic geometries, such as an ellipsoid or patient-specific finite-element model of the heart. Those improvements in model must be weighed against the incurred computational cost.

Angiocardiography is frequently used for determination of heart dimensions in humans. Numerous studies have shown that, when used properly, such measurements are reliable and reasonably accurate (3, 20, 34, 35). There is no doubt that MR imaging techniques are more accurate and reproducible. The development and refinement of three-dimensional cine-MR imaging techniques have enabled reconstruction of the ventricle by the interleaving of multiple long- and short-axis images. Similarly, three-dimensional echocardiography has also developed to a level of sophistication that permits expeditious and accurate determination of LV volumes and mass (27, 28). These imaging modalities can thus provide very accurate geometric models of the heart that can yield reliable estimates of myocardial contractility.

In conclusion, we validate a novel index of LV contractility that can discriminate between patients with different LVEF values. This index can be determined noninvasively from LV cavity and wall volume measurements and can become a routine clinical adjunct to echocardiographic or cardiac MR imaging for assessment of LV contractility and function. Our data demonstrate that d*/dt_max correlates well with dP/dt_max, E_a,max, and E_es(SB), is independent of changes in preload and afterload, and is sensitive to changes in myocardial contractility.

	APPENDIX A

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
Calculation of E_a from LV pressure and volume data. The governing differential equation relating LV intracavity pressure (P) and volume (V) is given by Zhong et al. (46)

(A1)

where M is the inertia term and E is the sum of LV passive and active elastance (E_p and E_a). E_p governs the pressure response to volume change and can be expressed as a monoexponential function, and E_a characterizes intrinsic LV stiffness attributable to activation of the myocardial sarcomere during active contraction and relaxation. The relaxation can also be deemed to be an active process requiring active decoupling of Ca²⁺ from myocardial contractile proteins.

An E_a term was previously formulated as (46)

(A2)

where t represents time from the start of isovolumic contraction, E_a,0 is the E_a coefficient, _C is the time coefficient of the rate of elastance rise during LV contraction, _R is the time coefficient of the rate of elastance fall during LV relaxation, and d is a time constant. The exponents Z_C and Z_R were introduced to smooth out the E_a curve during isovolumic contraction and relaxation. It was proposed that E_a,max may be adopted as an intrinsic measure of LV contractility. The parameter d is a time constant during the ejection phase; the term u(t – d) is the unit step function: u(t – d) = 0 for t < d. During isovolumic contraction, when u(t – d) = 0, Eq. A2 reduces to

(A3)

During isovolumic contraction and isovolumic relaxation, LV volume and E_p are constant. With dV = 0 and d = 0, Eq. A1 becomes VdE = dP, which can be discretized for any time i during the isovolumic contraction as

(A4)

and, for time i during the isovolumic relaxation, as

(A5)

where V_i and P_i are LV volume and pressure measured at time i, E_p,es is passive elastance at end diastole, and E_p,es is passive elastance at end systole.

Applying Eqs. A4 and A5 to study measured LV pressure and volume data allows calculation of E_a during isovolumic contraction and relaxation for various patients (Table 1). In each patient, E_a,0, _C, and Z_C in Eq. A3 can be determined from a curve fit of LV pressure and volume data during isovolumic contraction. Substituting the values of these parameters into the more general E_a of Eq. A2, the remaining parameters d, _R, and Z_R can be determined similarly to fit LV pressure and volume data during isovolumic relaxation. The resultant smooth curve fit function of Eq. A2 represents E_a at all times and enables determination of E_a,max (Fig. 2).

	APPENDIX B

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
Analysis of wall stress in a thick-walled spherical LV model based on the theory of elasticity (39). The LV was modeled as a pressurized thick-walled spherical shell. If u_r represents the radial displacement of a spherical surface element in the LV wall at radius r, then, at the outer surface of radius r + r, the corresponding radial displacement can be expressed as u_r + (u_r/r)dr. Strain in the radial direction (_r) is then given by

(B1)

Spherical symmetry implies that strains in the orthogonal circumferential axes ( and ) are identical. Therefore, strain in the circumferential direction () is

(B2)

Strains along the three orthogonal axes are

(B3)

Similarly, = . For equilibrium in the radial direction, we have

(B4)

Since = , d_r = (_r/r)r, and Eq. B4 reduces to

(B5)

On the basis of Hooke's law, we can express the constitutive equation as

(B6)

where E is the elastic modulus and v is the Poisson's ratio. Similarly, from Eq. B2

(B7)

Differentiating Eq. B7 with respect to r, we obtain

(B8)

Equations B6 and B8 can be combined as

(B9)

If we combine (Eq. A5) into Eq. A9, we obtain

(B10)

If we substitute r²_r = y, Eq. B10 becomes

(B11)

This yields a homogeneous linear equation with the solution

(B12)

where A and B are constants. From Eqs. B5 and B12, we find

(B13)

For a thick-walled spherical shell subjected to internal pressure (P_i) and zero external pressure, the boundary conditions are _r = –P_i when r = r_i and _r = 0 when r = r_e. From Eq. B12, we obtain

(B14)

Hence, we can determine the constants of integration as

(B15)

From Eqs. B12–B14, expressions for radial and circumferential stresses are

(B16)

(B17)

The present formulation will focus on the circumferential stress, which is significantly larger in magnitude than and opposite in sign (tensile) from the radial (compressive) stress.

	GRANTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 
This study was supported by Singapore A*STAR Research Grant 0221010023 and National Heart, Lung, and Blood Institute Grant 2 R01 HL-055554-08.

	ACKNOWLEDGMENTS

 
We are grateful to Drs. Lim Soo-Teik and K. Gunasegaran (National Heart Centre) for professional and technical assistance in cine ventriculography and stress echocardiography.

	FOOTNOTES

 

Address for reprint requests and other correspondence: G. S. Kassab, Dept. of Biomedical Engineering, Indiana-Purdue Univ., Indianapolis, IN 46202 (e-mail: gkassab{at}iupui.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B GRANTS REFERENCES

 

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