ATS
HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
QUICK SEARCH:   [advanced]


     

Ann Thorac Surg 2011;91:150-156. doi:10.1016/j.athoracsur.2010.06.132
© 2011 The Society of Thoracic Surgeons
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Right arrow Citation Map
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Add to Personal Folders
Right arrow Download to citation manager
Right arrow Author home page(s):
Amod Tendulkar
Arthur W. Wallace
Mark B. Ratcliffe
Right arrow Permission Requests
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Zhang, Z.
Right arrow Articles by Ratcliffe, M. B.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Zhang, Z.
Right arrow Articles by Ratcliffe, M. B.
Related Collections
Right arrow Cardiac - physiology
Right arrow Myocardial infarction

Original Articles: Adult Cardiac

Comparison of the Young-Laplace Law and Finite Element Based Calculation of Ventricular Wall Stress: Implications for Postinfarct and Surgical Ventricular Remodeling

Zhihong Zhang, MSa,e, Amod Tendulkar, MDa, Kay Sun, PhDa,e, David A. Saloner, PhDd,e, Arthur W. Wallace, MD, PhDc,e, Liang Ge, PhDa,b,e, Julius M. Guccione, PhDa,b,e, Mark B. Ratcliffe, MDa,b,e,*

a Department of Surgery, University of California, San Francisco, California
b Department of Bioengineering, University of California, San Francisco, California
c Department of Anesthesia, University of California, San Francisco, California
d Department of Radiology, University of California, San Francisco, California
e Veterans Affairs Medical Center, San Francisco, California

Accepted for publication June 29, 2010.

* Address correspondence to Dr Ratcliffe, Surgical Service (112), San Francisco Veterans Affairs Medical Center, 4150 Clement St, San Francisco, CA 94121 (Email: mark.ratcliffe{at}med.va.gov).


   Abstract
Top
 Abstract
Introduction
 Material and Methods
 Results
 Comment
 Acknowledgments
 References
 
Background: Both the Young-Laplace law and finite element (FE) based methods have been used to calculate left ventricular wall stress. We tested the hypothesis that the Young-Laplace law is able to reproduce results obtained with the FE method.

Methods: Magnetic resonance imaging scans with noninvasive tags were used to calculate three-dimensional myocardial strain in 5 sheep 16 weeks after anteroapical myocardial infarction, and in 1 of those sheep 6 weeks after a Dor procedure. Animal-specific FE models were created from the remaining 5 animals using magnetic resonance images obtained at early diastolic filling. The FE-based stress in the fiber, cross-fiber, and circumferential directions was calculated and compared to stress calculated with the assumption that wall thickness is very much less than the radius of curvature (Young-Laplace law), and without that assumption (modified Laplace).

Results: First, circumferential stress calculated with the modified Laplace law is closer to results obtained with the FE method than stress calculated with the Young-Laplace law. However, there are pronounced regional differences, with the largest difference between modified Laplace and FE occurring in the inner and outer layers of the infarct borderzone. Also, stress calculated with the modified Laplace is very different than stress in the fiber and cross-fiber direction calculated with FE. As a consequence, the modified Laplace law is inaccurate when used to calculate the effect of the Dor procedure on regional ventricular stress.

Conclusions: The FE method is necessary to determine stress in the left ventricle with postinfarct and surgical ventricular remodeling


   Introduction
Top
 Abstract
 Introduction
Material and Methods
 Results
 Comment
 Acknowledgments
 References
 
Regional coronary blood flow [1], myocardial oxygen consumption, hypertrophy, [2] and remodeling are all determined by ventricular wall stress. In addition, a reduction in wall stress has been used to justify a number of new cardiac operations including the Batista operation [3], the Acorn cardiac support device [4], and the Myocor Myosplint [5]. However, Huisman and coworkers [6] previously showed that it is impossible to directly measure regional in vivo myocardial stress because of tethering from surrounding myocardium [7] As a consequence, ventricular wall stress is usually calculated with force balance methods such as the Young-Laplace law [8, 9]. Unfortunately, localized shape change in and around the myocardial infarction (MI) and regional changes in systolic and diastolic material properties [10] make the left ventricle a mechanically complex structure. The accuracy of force balance based calculation of left ventricular (LV) stress may be, therefore, severely limited.

An alternative approach to quantifying ventricular wall stress and stiffness is mathematical modeling based on the conservation laws of continuum mechanics, the most versatile of which is the finite element (FE) method [11]. The FE method used in this study for continuum analysis of the heart includes several features that are uncommon in conventional FE methods. First, the constitutive relationship is nonlinear and anisotropic [12, 13], with direction based directly on measured three-dimensional myofiber angle distributions [14, 15]. Next, the models undergo large or finite deformation with a difference in dimensions between end diastole and end systole that is greater than 10%. More recent models now also include the transmural heterogeneity of cellular excitation-contraction coupling mechanisms [16].

We calculated stress with the assumption that wall thickness is very much less than the radius of curvature with large deformation FE methods (Young-Laplace law) and without the assumption (modified Laplace) using data collected from sheep after anteroapical MI and after Dor procedure. We tested the hypothesis that stress in the circumferential direction calculated with Laplace law is equal to values obtained with the FE method. Since stress in the cross-fiber direction is likely to cause volume overload hypertrophy and stress in the fiber direction is likely to cause pressure overload hypertrophy, we also compared stress calculated using force balance with the magnitude of stress in the fiber and cross fiber directions calculated with the FE method.


   Material and Methods
Top
 Abstract
 Introduction
 Material and Methods
Results
 Comment
 Acknowledgments
 References
 
Animals used in this study were treated in compliance with the "Guide for the Care and Use of Laboratory Animals" prepared by the Institute of Laboratory Animal Resources, National Research Council (published by the National Academy Press, revised 1996).

Magnetic resonance (MRI) images with noninvasive tags were used to calculate three-dimensional myocardial strain in 5 sheep 16 weeks after anteroapical MI and in 1 of those sheep 6 weeks after Dor procedure (Fig 1) [17]. These experimental results were previously reported [17, 18].


Figure 1
View larger version (12K):
[in this window]
[in a new window]

  		Fig 1. (A) Long-axis magnetic resonance (MR) images from sheep with anteroapical myocardial infarction (MI). (B) Short-axis MR image. Both images were obtained 16 weeks after MI. Both images have noninvasive tissue tags, and both images were obtained at end systole. (MI = dyskinetic infarct; SI = septal infarct.)

(A) Reprinted from Zhang, et al, J Thorac Cardiovasc Surg 2007;134:1017–24 [17], with permission from the American Association for Thoracic Surgery.

 
Force Balance Calculations
The Young-Laplace law
The endocardial contour was divided into 32 points. As seen in Figure 2, three adjacent points on the endocardial contour were identified and used to determine the radius of curvature [19]. Two line segments were created by connecting point 1 and point 2, point 2 and point 3, where two bisectors to these two line segments were then constructed. The radius of curvature for point 2 was calculated from the distance from point 2 to the bisector's intersection. Note that short axis contours near the apex (infarct area) were not used because the image plane in that area cuts the LV wall obliquely.


Figure 2
View larger version (3K):
[in this window]
[in a new window]

  		Fig 2. Method of measuring endocardial radius of curvature: r is the radius of curvature, and h is the wall thickness.

 
The derivation of the Young-Laplace law is as follows:



Formula 0001

(1)
where P is intracavitary pressure, r is the endocardial radius of curvature, and h is the wall thickness. Equation 1 simplifies to:



Formula 0002

(2)
if h<<r Equation 2 is reduced to the Young-Laplace law:



Formula 0003

(3)

Modified Laplace law
Throughout the paper, we refer to Equation 3 as "Young-Laplace" and to Equation 2 as "modified Laplace."

Finite Element Model
Analysis of borderzone function using FE models has been previously reported [20].

Finite element method
Animal-specific FE models were created from MRI images at the early diastole, which was considered as the initial unloaded reference state [21]. Representative surface and FE meshes are seen in Figure 3. Myofiber angles of –37 degrees, 23 degrees, and 83 degrees were assigned at the epicardium, midwall, and endocardium, respectively, in the remote and borderzone regions [22]. At the aneurysm region, fiber angles were set to 0 degrees [23]. Circumferential displacement of basal epicardial nodes was constrained.


Figure 3
View larger version (12K):
[in this window]
[in a new window]

  		Fig 3. Finite element model of the left ventricle with anteroapical myocardial infarction. (A) Animal specific contours were generated from magnetic resonance imaging. (B) Solid mesh was broken into three regions (green = remote; brown = borderzone; blue = infarct).

 
Loading
The inner endocardial wall was loaded to the measured in vivo end-diastolic and end-systolic LV pressures.

Constitutive relationship
Passive [13] and active myocardial [24] constituitive relationships have been previously described. The active myocardial material property law was implemented using a user-defined material subroutine in LS-DYNA (Livermore Software Technology Corporation, Livermore, CA). Diastolic [25] and systolic [24] material variables have been previously reported.

Material property optimization
The method of myocardial material property optimization was previously described by Sun and colleagues [26]. Specifically, the commercial FE optimization software, LS-OPT [27], was used to find the optimal value of T max for each region.

Simulations were performed on a small Linux cluster (seven nodes; each node with two AMD Opteron 240 processors and 1 gB memory).

Statistical Analysis
All values are expressed as mean ± SD and compared by repeated measures analysis using a mixed model to test for both fixed and random effects (Systat Software, Chicago, IL). The statistical model was as follows:


Formula 0004

(0004)
where Method, Region, and Layer are dummy variables, and where Method is either Laplace or FE, Region is either BZ or remote, and Layer is either endocardium, midwall, or epicardium. The statistical significance of individual group comparisons was tested using the Student t test with the Bonferoni correction for multiple comparisons. Significance was set at p less than 0.05.


   Results
Top
 Abstract
 Introduction
 Material and Methods
 Results
Comment
 Acknowledgments
 References
 
Computational time was recorded while performing the FE method on a single FE model. Time for FE simulation was 385 s.

Average Circumferential Stress
Table 1 shows average LV wall stress in the circumferential direction calculated using the Young-Laplace law, the modified Laplace law, and the FE method. The average wall thickness to radius of curvature ratio was 0.22 ± 0.022 at end diastole and 0.24 ± 0.030 at end systole. As a consequence, circumferential stress calculated with the Young-Laplace law and the modified Laplace law is significantly different.


View this table:
[in this window]
[in a new window]

  		Table 1 Average Stress in Circumferential Direction at End Diastole and End Systole
 
Furthermore, circumferential stress calculated with the modified Laplace is closer to results obtained with the FE method than stress calculated with the Young-Laplace law. Specifically, average modified Laplace stress was 26% higher than FE-based calculation of circumferential stress at end diastole (p = 0.006) but was not different at end systole.

Regional Variation
There were significant regional differences. Stress calculated with the modified Laplace was higher than circumferential stress calculated with the FE method in the remote (37%, p = 0.048) but not borderzone regions (Fig 4A), and higher than circumferential stress calculated with the FE method in the endocardial layer (93%, p < 0.001) at end diastole (Fig 4B).


Figure 4
View larger version (3K):
[in this window]
[in a new window]

  		Fig 4. Stress calculated with the Laplace and finite element (FE) methods at end diastole by (A) region and (B) layer of the left ventricle wall. Note that both the Young-Laplace law and the modified Laplace law overestimate stress in the circumferential direction and cross-fiber direction in the remote myocardium at end diastole. (Black bars = FE fiber; dark gray bars = FE cross fiber; light gray bars = FE circumferential; white bars = LaPlace; hatched bars = modified LaPlace.)

 
Furthermore, stress calculated with the modified Laplace law was higher than circumferential stress calculated with the FE method in the borderzone (modified Laplace 52%, p = 0.048) but not in the remote myocardium (Fig 5A), and higher than circumferential stress calculated with the FE method in the endocardial layer (93%, p < 0.001) at end systole (Fig 5B).


Figure 5
View larger version (3K):
[in this window]
[in a new window]

  		Fig 5. Stress calculated with the Laplace and the finite element (FE) methods at end systole by (A) region and (B) layer of the left ventricle wall. Note that the Young-Laplace law underestimates stress in the fiber direction at end systole. The Young-Laplace law is not different than circumferential stress in the remote myocardium but fails to account for transmural variation in stress. Statistically significant comparisons with cross-fiber stress are not marked. See the text for those details. (White bar = LaPlace; hatched bar = modified LaPlace; black bars = FE fiber; dark gray bars = FE cross fiber; light gray bars = FE circumferential; Endo = endocardium; Epi = epicardium; Mid = midwall.)

 
Comparison With Fiber and Cross-Fiber Stress Calculated With the FE Method
The magnitude of stress calculated with the modified Laplace law was very different than stress in the fiber and cross-fiber direction. Average fiber stress with the FE method was 3.119 ± 1.784 kPa at end diastole and 38.618 ± 10.653 kPa at end systole. Average cross-fiber stress with the FE method was 1.935 ± 1.112 kPa at end diastole and 8.800 ± 7.102 kPa at end systole. Average modified Laplace stress was 22% higher than FE-based calculation of cross-fiber stress at end diastole (p < 0.001) and 121% higher at end systole (p < 0.001).

Once again, there were significant regional differences. Stress calculated with the modified Laplace law was higher than cross-fiber stress calculated with the FE method in the remote region (22%, p = 0.012) at end diastole (Fig 4A). Stress calculated with both the modified Laplace law was substantially lower than fiber stress calculated with the FE method in both remote (modified Laplace 64%, p < 0.001) and borderzone regions (modified Laplace 35%, p = 0.012) at end systole (Fig 5A). Finally, stress calculated with the modified Laplace law was lower than fiber stress at all layers (p < 0.001) at end systole (Fig 5B).

Change in Stress After Dor Procedure
Stress at end systole calculated with the Young-Laplace law, modified Laplace law, and FE method in the infarct borderzone before and after Dor procedure is seen in Table 2. The reduction in average borderzone stress calculated with the Young-Laplace and modified Laplace law was –23.5% and –26.2%, respectively. However, the change in regional stress calculated with the FE method was quite different, with a substantial increase in stress in the inner layer ({Delta} in circumferential stress +196.7%; {Delta} in fiber stress +15%) and a more pronounced decrease in the outer layer ({Delta} in circumferential stress –70.4%; {Delta} in fiber stress –49.7%).


View this table:
[in this window]
[in a new window]

  		Table 2 Stress at End Systole Calculated With Young-Laplace Law, Modified Laplace Law, and Finite Element (FE) in Infarct Borderzone Before and After Dor Procedure
 

   Comment
Top
 Abstract
 Introduction
 Material and Methods
 Results
 Comment
Acknowledgments
 References
 
The principal findings of our study are (1) circumferential stress calculated with the modified Laplace is closer to results obtained with the FE method than stress calculated with the Young-Laplace law; (2) there are pronounced regional differences between stress calculated with the modified Laplace law and the FE method; and (3) the magnitude of stress calculated with the modified Laplace law is very different than stress in the fiber and cross-fiber directions calculated with the FE method.

Finite Element Method as the Gold Standard
The decision to use the large deformation FE method as the reference or gold standard is reasonable given that the FE models were optimized using measured myocardial strain in addition to end-diastolic and end-systolic volumes. A comparison of Laplace's law with the FE method in a physical model constructed from materials with known material properties, such as silicone [28] or silicone with embedded elastic fibers, and one that has simplified geometry would be interesting, and we propose to carry out this study in the future. However, even when completed, physical models of that sort will lack the ability to simulate active contraction.

Limitations of the Young-Laplace Law
There are a number of inaccuracies and limitations associated with the application of Laplace type calculations of LV wall stress [29] An inaccuracy specific to the Young-Laplace law is the restriction that the wall thickness (h) be very much less than the radius of curvature (r). However, the h/r ratio in this study was 0.22 ± 0.022 at end diastole and 0.24 ± 0.030 at end systole. This suggests that Equation 2 (modified Laplace), which does not have the h << r restriction, is more reasonable. In fact, we found in all cases that stress calculated with Equation 2 was closer to FE-based calculation of circumferential stress than stress calculated with the Young-Laplace law.

Stress Variation Across the LV Wall
When a thick-walled pressure vessel is inflated, most of the deformation in the circumferential and longitudinal directions occurs at the inner surface and decreases monotonically toward the outer surface. Thus, we would expect circumferential and longitudinal stress to vary transmurally in a similar manner, although the actual transmural variation in these stress components will depend on the relationship between stress and strain (constitutive relation) of the myocardium. For instance, Guccione and colleagues [30] previously measured fiber stress in the left ventricle of a normal dog using a finite deformation FE method similar to that used in the current study [30]. The investigators found the transmural gradient in fiber stress was 3.3 kPa at the base and 4.6 kPa at the apex at end diastole [30]. The transmural gradients were small near the LV base during systole and were as high as 43 kPa between the midventricle and apex [30]. In the present study, transmural gradients in fiber stress were smaller but still significant with a gradient of 1.55 kPa in the remote myocardium at end diastole and 7.23 kPa at end systole. The Young-Laplace law is based on a global force balance, which ignores myocardial material properties. Thus, the Young-Laplace law can be used to estimate only average stress across the full wall thickness in the circumferential and longitudinal directions.

Importance of Fiber and Cross-Fiber Stress
Another limitation of the Young-Laplace and modified Laplace laws is that they cannot be used to calculate stress in the local muscle fiber or cross-fiber direction as stress in the fiber and cross-fiber directions are probably the causes of hypertrophy. There is increasing evidence that stress in the cross-fiber direction causes eccentric or volume overload type hypertrophy [31, 32]. Although evidence is less clear, it is probable that end-systolic fiber stress causes concentric or pressure overload type hypertrophy.

Implications for the Dor Procedure
Regional differences between stress calculated the Young-Laplace law and FE methods occurs especially in the inner and outer layers of the infarct borderzone. When coupled with inability of Laplace type methods to measure fiber and cross-fiber stress, this leads the Young-Laplace and modified Laplace laws to miscalculate the change in stress in the infarct borderzone after the Dor procedure. Since the rationale for the Dor procedure in specific and surgical ventricular remodeling in general is reduction in wall stress [33], accurate knowledge of stress in the fiber and cross-fiber directions is of obvious importance. Furthermore, in our opinion, improvement in borderzone contractility is the primary therapeutic target of the Dor procedure [20].

Utility of Finite Deformation Type FE Model of the Left Ventricle
The FE method can be used to simulate the effect of surgical ventriculoplasty [34] and passive constraint on the left ventricle. For instance, Dang and colleagues [34] previously used a FE model of the finite deformation type to calculate the effect of surgical remodeling on stroke volume and mean fiber stress. Force balance methods such as the Young-Laplace law can only calculate stress from an existing left ventricle image obtained during animal experiments or from patients. Thus, it cannot be used as a predictive theoretical tool. In our opinion, this is the greatest limitation of the Young-Laplace law.

Conclusions
Circumferential stress calculated with the modified Laplace law was superior to the Young-Laplace law, although stress calculated with the modified Laplace law remained statistically different (higher) than FE-based calculation of circumferential stress at end diastole.

Conversely, the magnitude of stress calculated with the modified Laplace law is very different than stress in the fiber and cross-fiber directions calculated with the FE method. Also, there are pronounced regional differences, with the largest differences between the modified Laplace and FE methods occurring in the inner and outer layers of the infarct borderzone. As a consequence, the modified Laplace law is inaccurate when used to calculate the effect of the Dor procedure on regional ventricular stress. The FE method is necessary to determine stress in the left ventricle with postinfarct and surgical ventricular remodeling.


   Acknowledgments
Top
 Abstract
 Introduction
 Material and Methods
 Results
 Comment
 Acknowledgments
References
 
This study was supported by National Institutes of Health grant R01-HL-77921 (Dr. Guccione), and R01-HL-63348 (Dr. Ratcliffe). This support is gratefully acknowledged.


   References
Top
 Abstract
 Introduction
 Material and Methods
 Results
 Comment
 Acknowledgments
 References
 

  1. Jan KM. Distribution of myocardial stress and its influence on coronary blood flow J Biomechanics 1985;18:815-820.[Medline]
  2. Grossman W. Cardiac hypertrophy: useful adaptation or pathologic process? Am J Med 1980;69:576-584.[Medline]
  3. Batista RJ, Verde J, Nery P, et al. Partial left ventriculectomy to treat end-stage heart disease Ann Thorac Surg 1997;64:634-638.[Abstract/Free Full Text]
  4. Mann DL, Acker MA, Jessup M, et al. Rationale, design, and methods for a pivotal randomized clinical trial for the assessment of a cardiac support device in patients with New York Heart Association class III-IV heart failure J Card Fail 2004;10:185-192.[Medline]
  5. McCarthy PM, Takagaki M, Ochiai Y, et al. Device-based change in left ventricular shape: a new concept for the treatment of dilated cardiomyopathy J Thorac Cardiovasc Surg 2001;122:482-490.[Abstract/Free Full Text]
  6. Huisman RM, Elzinga G, Westerhof N, Sipkema P. Measurement of left ventricular wall stress Cardiovasc Res 1980;14:142-153.[Abstract/Free Full Text]
  7. Yin FCP. Ventricular wall stress Circ Res 1986;49:829-842.
  8. Fung YC. Biomechanics: mechanical properties of living tissuesNew York: Springer-Verlag; 1981.
  9. Woods RH. A few applications of a physical theorem to membranes in the human body in a state of tension J Anat Physiol 1892;26:362-370.[Medline]
  10. Gupta KB, Ratcliffe MB, Fallert MA, Edmunds LHJ, Bogen DK. Changes in passive mechanical stiffness of myocardial tissue with aneurysm formation Circulation 1994;89:2315-2326.[Abstract/Free Full Text]
  11. McCulloch A, Waldman L, Rogers J, Guccione J. Large-scale finite element analysis of the beating heart Crit Rev Biomed Eng 1992;20:427-449.[Medline]
  12. Guccione JM, McCulloch AD. Mechanics of active contraction in cardiac muscle: part I—constitutive relations for fiber stress that describe deactivation J Biomech Eng 1993;115:72-81.[Medline]
  13. Guccione JM, McCulloch AD, Waldman LK. Passive material properties of intact ventricular myocardium determined from a cylindrical model J Biomech Eng 1991;113:42-55.[Medline]
  14. Streeter DD, Spotnitz HM, Patel DP, Ross JRJ, Sonnenblick EH. Fiber orientation in the canine left ventricle during diastole and systole Circ Res 1969;24:339-347.[Abstract/Free Full Text]
  15. Walker JC, Guccione JM, Jiang Y, et al. Helical myofiber orientation after myocardial infarction and left ventricular surgical restoration in sheep J Thorac Cardiovasc Surg 2005;129:382-390.[Abstract/Free Full Text]
  16. Saucerman JJ, Brunton LL, Michailova AP, McCulloch AD. Modeling beta-adrenergic control of cardiac myocyte contractility in silico J Biol Chem 2003;278:47997-48003.[Abstract/Free Full Text]
  17. Zhang P, Guccione JM, Nicholas SI, et al. Endoventricular patch plasty for dyskinetic anteroapical left ventricular aneurysm increases systolic circumferential shortening in sheep J Thorac Cardiovasc Surg 2007;134:1017-1024.[Abstract/Free Full Text]
  18. Zhang P, Guccione JM, Nicholas SI, et al. Left ventricular volume and function after endoventricular patch plasty for dyskinetic anteroapical left ventricular aneurysm in sheep J Thorac Cardiovasc Surg 2005;130:1032-1038.[Abstract/Free Full Text]
  19. Jackson BM, Gorman JH, Salgo IS, et al. Border zone geometry increases wall stress after myocardial infarction: contrast echocardiographic assessment Am J Physiol Heart Circ Physiol 2003;284:H475-H479.[Abstract/Free Full Text]
  20. Sun K, Zhang Z, Suzuki T, et al. Dor procedure for dyskinetic anteroapical myocardial infarction fails to improve contractility in the border zone J Thorac Cardiovasc Surg 2010;1401:233-239.
  21. Moustakidis P, Maniar HS, Cupps BP, et al. Altered left ventricular geometry changes the border zone temporal distribution of stress in an experimental model of left ventricular aneurysm: a finite element model study Circulation 2002;106(Suppl 1):168-175.
  22. Omens JH, May KD, McCulloch AD. Transmural distribution of three-dimensional strain in the isolated arrested canine left ventricle Am J Physiol 1991;261:H918-H928.[Medline]
  23. Moonly SM. Experimental and computational analysis of left ventricular aneurysm mechanics [PhD thesis]San Francisco, CA: Department of Bioengineering, University of California, San Francisco with University of California, Berkeley; 2003.
  24. Guccione JM, Waldman LKL, McCulloch AD. Mechanics of active contraction in cardiac muscle: part II—cylindrical models of the systolic left ventricle J Biomech Eng 1993;115:82-90.[Medline]
  25. Walker JC, Ratcliffe MB, Zhang P, et al. MRI-based finite-element analysis of left ventricular aneurysm Am J Physiol Heart Circ Physiol 2005;289:H692-H700.[Abstract/Free Full Text]
  26. Sun K, Stander N, Jhun C-S, et al. A computationally efficient formal optimization of regional myocardial contractility in a sheep with left ventricular aneurysm J Biomech Eng 2009;131:111001.[Medline]
  27. Stander N, Roux W, Eggleston T, Craig K. LS-OPT User's Manual Version 3.2, 2007Livermore, CA: Livermore Software Technology Corporation; 2008.
  28. Dokos S, LeGrice IJ, Smaill BH, Kar J, Young AA. A triaxial-measurement shear-test device for soft biological tissues J Biomech Eng 2000;122:471-478.[Medline]
  29. Moriarty TF. The law of Laplace. Its limitations as a relation for diastolic pressure, volume, or wall stress of the left ventricle. Circ Res 1980;46:321-331.[Free Full Text]
  30. Guccione JM, Costa KD, McCulloch AD. Finite element stress analysis of left ventricular mechanics in the beating dog heart J Biomech 1995;28:1167-1177.[Medline]
  31. Gopalan SM, Flaim C, Bhatia SN, et al. Anisotropic stretch-induced hypertrophy in neonatal ventricular myocytes micropatterned on deformable elastomers Biotechnol Bioeng 2003;81:578-587.[Medline]
  32. Senyo SE, Koshman YE, Russell B. Stimulus interval, rate and direction differentially regulate phosphorylation for mechanotransduction in neonatal cardiac myocytes FEBS Lett 2007;581:4241-4247.[Medline]
  33. Burkhoff D, Wechsler AS. Surgical ventricular remodeling: a balancing act on systolic and diastolic properties J Thorac Cardiovasc Surg 2006;132:459-463.[Free Full Text]
  34. Dang AB, Guccione JM, Zhang P, et al. Effect of ventricular size and patch stiffness in surgical anterior ventricular restoration: a finite element model study Ann Thorac Surg 2005;79:185-193.[Abstract/Free Full Text]



This article has been cited by other articles:


Home page
 Ann. Thorac. Surg.Home page
R. Carrick, L. Ge, L. C. Lee, Z. Zhang, R. Mishra, L. Axel, J. M. Guccione, E. A. Grossi, and M. B. Ratcliffe
Patient-Specific Finite Element-Based Analysis of Ventricular Myofiber Stress After Coapsys: Importance of Residual Stress
Ann. Thorac. Surg., June 1, 2012; 93(6): 1964 - 1971.
[Abstract] [Full Text] [PDF]

Home page
 Ann. Thorac. Surg.Home page
V. M. Wong, J. F. Wenk, Z. Zhang, G. Cheng, G. Acevedo-Bolton, M. Burger, D. A. Saloner, A. W. Wallace, J. M. Guccione, M. B. Ratcliffe, et al.
The Effect of Mitral Annuloplasty Shape in Ischemic Mitral Regurgitation: A Finite Element Simulation
Ann. Thorac. Surg., March 1, 2012; 93(3): 776 - 782.
[Abstract] [Full Text] [PDF]

Home page
 Ann. Thorac. Surg.Home page
M. Soleimani, M. Khazalpour, G. Cheng, Z. Zhang, G. Acevedo-Bolton, D. A. Saloner, R. Mishra, A. W. Wallace, J. M. Guccione, L. Ge, et al.
Moderate Mitral Regurgitation Accelerates Left Ventricular Remodeling After Posterolateral Myocardial Infarction
Ann. Thorac. Surg., November 1, 2011; 92(5): 1614 - 1620.
[Abstract] [Full Text] [PDF]

This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Right arrow Citation Map
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Add to Personal Folders
Right arrow Download to citation manager
Right arrow Author home page(s):
Amod Tendulkar
Arthur W. Wallace
Mark B. Ratcliffe
Right arrow Permission Requests
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Zhang, Z.
Right arrow Articles by Ratcliffe, M. B.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Zhang, Z.
Right arrow Articles by Ratcliffe, M. B.
Related Collections
Right arrow Cardiac - physiology
Right arrow Myocardial infarction

HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
ANN THORAC SURG ASIAN CARDIOVASC THORAC ANN EUR J CARDIOTHORAC SURG
J THORAC CARDIOVASC SURG ICVTS ALL CTSNet JOURNALS