The mechanical bidomain model is a new mathematical description of the elastic behavior of cardiac tissue. Its primary advantage over previous models is that it accounts for forces acting across the cell membrane arising from differences in the displacement of the intracellular and extracellular spaces. In this paper, I describe the development of the mechanical bidomain model. I emphasize new predictions of the model, such as the existence of boundary layers at the tissue surface where the membrane forces are large, and pressure differences between the intracellular and extracellular spaces. Although the theoretical analysis is quite mathematical, I highlight the types of experiments that could be used to test the model predictions. Finally, I present open questions about the mechanical bidomain model that may be productive future directions for research. 1. Introduction This paper focuses on a quantitative analysis of the biomechanical forces acting on the intracellular and extracellular spaces of cardiac tissue, and especially on the coupling of these two spaces across the cell membrane. Our understanding of these forces is rudimentary, yet they may play a vital role in tissue engineering and mechanobiology. This paper concentrates on the mechanical behavior of cardiac tissue, although the results may apply to many other tissues. Before presenting a mathematical model that describes membrane forces in cardiac tissue, let us consider the biological and biomedical problems that motivate this research. Chiquet [1] reviewed remodeling of the extracellular matrix and identified integrin proteins as playing a crucial role, because “they physically link the extracellular matrix to the cytoskeleton and hence are responsible for establishing a mechanical continuum by which forces are transmitted between the outside and the inside of cells.” Chiquet et al. [2] compare integrins and associated focal adhesion proteins to “molecular springs” coupling mechanical forces in the intracellular and extracellular spaces. Integrins may be important in tumor biology and cancer therapy [3]. The interaction of intracellular and extracellular forces plays a role in cardiac tissue remodeling. Kresh and Chopra [4] conclude that “ultimately, understanding how the highly interactive mechanical signaling can give rise to phenotypic changes is critical for targeting the underlying pathways that contribute to cardiac remodeling associated with various forms of dilated and hypertrophic myopathies, myocardial infarction, heart failure, and reverse remodeling.” Mechanotransduction
References
[1]
M. Chiquet, “Regulation of extracellular matrix gene expression by mechanical stress,” Matrix Biology, vol. 18, no. 5, pp. 417–426, 1999.
[2]
M. Chiquet, L. Gelman, R. Lutz, and S. Maier, “From mechanotransduction to extracellular matrix gene expression in fibroblasts,” Biochimica et Biophysica Acta, vol. 1793, no. 5, pp. 911–920, 2009.
[3]
C. Jean, P. Gravelle, J. J. Fournie, and G. Laurent, “Influence of stress on extracellular matrix and integrin biology,” Oncogene, vol. 30, no. 24, pp. 2697–2706, 2011.
[4]
J. Y. Kresh and A. Chopra, “Intercellular and extracellular mechanotransduction in cardiac myocytes,” Pflugers Archiv European Journal of Physiology, vol. 462, no. 1, pp. 75–87, 2011.
[5]
M. L. McCain and K. K. Parker, “Mechanotransduction: the role of mechanical stress, myocyte shape, and cytoskeletal architecture on cardiac function,” Pflugers Archiv European Journal of Physiology, vol. 462, no. 1, pp. 89–104, 2011.
[6]
F. Rodriguez, F. Langer, K. B. Harrington et al., “Alterations in transmural strains adjacent to ischemic myocardium during acute midcircumflex occlusion,” Journal of Thoracic and Cardiovascular Surgery, vol. 129, no. 4, pp. 791–803, 2005.
[7]
C. E. Sarraf, W. R. Otto, and M. Eastwood, “In vitro mesenchymal stem cell differentiation after mechanical stimulation,” Cell Proliferation, vol. 44, no. 1, pp. 99–108, 2011.
[8]
P. Kohl and F. Sachs, “Mechanoelectric feedback in cardiac cells,” Philosophical Transactions of the Royal Society A, vol. 359, no. 1783, pp. 1173–1185, 2001.
[9]
D. E. Hansen, C. S. Craig, and L. M. Hondeghem, “Stretch-induced arrhythmias in the isolated canine ventricle. Evidence for the importance of mechanoelectrical feedback,” Circulation, vol. 81, no. 3, pp. 1094–1105, 1990.
[10]
N. Trayanova, W. Li, J. Eason, and P. Kohl, “Effect of stretch-activated channels on defibrillation efficacy,” Heart Rhythm, vol. 1, no. 1, pp. 67–77, 2004.
[11]
W. Li, V. Gurev, A. D. McCulloch, and N. A. Trayanova, “The role of mechanoelectric feedback in vulnerability to electric shock,” Progress in Biophysics and Molecular Biology, vol. 97, no. 2-3, pp. 461–478, 2008.
[12]
R. Kn？ll, M. Hoshijima, H. M. Hoffman et al., “The cardiac mechanical stretch sensor machinery involves a Z disc complex that is defective in a subset of human dilated cardiomyopathy,” Cell, vol. 111, no. 7, pp. 943–955, 2002.
[13]
W. H. Zimmermann, I. Melnychenko, and T. Eschenhagen, “Engineered heart tissue for regeneration of diseased hearts,” Biomaterials, vol. 25, no. 9, pp. 1639–1647, 2004.
[14]
H. Naito, I. Melnychenko, M. Didié et al., “Optimizing engineered heart tissue for therapeutic applications as surrogate heart muscle,” Circulation, vol. 114, no. 1, pp. I-72–I-78, 2006.
[15]
A. D. Bach, J. Stern-Straeter, J. P. Beier, H. Bannasch, and G. B. Stark, “Engineering of muscle tissue,” Clinics in Plastic Surgery, vol. 30, no. 4, pp. 589–599, 2003.
[16]
E. A. Silva and D. J. Mooney, “Synthetic extracellular matrices for tissue engineering and regeneration,” Current Topics in Developmental Biology, vol. 64, pp. 181–205, 2004.
[17]
C. A. Powell, B. L. Smiley, J. Mills, and H. H. Vandenburgh, “Mechanical stimulation improves tissue-engineered human skeletal muscle,” American Journal of Physiology, vol. 283, no. 5, pp. C1557–C1565, 2002.
[18]
R. G. Katare, M. Ando, Y. Kakinuma, and T. Sato, “Engineered heart tissue: a novel tool to study the ischemic changes of the heart in vitro,” PLoS ONE, vol. 5, no. 2, Article ID e9275, 2010.
[19]
S. Puwal and B. J. Roth, “Mechanical bidomain model of cardiac tissue,” Physical Review E, vol. 82, no. 4, Article ID 041904, 5 pages, 2010.
[20]
L. Tung, A bidomain model for describing ischemic myocardial dc potentials [Ph.D. thesis], Massachusetts Institute of Technology, Cambridge, Mass, USA, 1978.
[21]
B. J. Roth, “How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle,” Journal of Mathematical Biology, vol. 30, no. 6, pp. 633–646, 1992.
[22]
C. S. Henriquez, “Simulating the electrical behavior of cardiac tissue using the bidomain model,” Critical Reviews in Biomedical Engineering, vol. 21, no. 1, pp. 1–77, 1993.
[23]
I. R. Efimov, R. A. Gray, and B. J. Roth, “Virtual electrodes and deexcitation: new insights into fibrillation induction and defibrillation,” Journal of Cardiovascular Electrophysiology, vol. 11, no. 3, pp. 339–353, 2000.
[24]
N. Trayanova and G. Plank, “Bidomain model of defibrillation,” in Cardiac Bioelectric Therapy: Mechanisms and Practical Implications, I. R. Efimov, M. Kroll, and P. Tchou, Eds., pp. 85–109, Springer, New York, NY, USA, 2009.
[25]
J. Ohayon and R. S. Chadwick, “Effects of collagen microstructure on the mechanics of the left ventricle,” Biophysical Journal, vol. 54, no. 6, pp. 1077–1088, 1988.
[26]
R. S. Chadwick, “Mechanics of the left ventricle,” Biophysical Journal, vol. 39, no. 3, pp. 279–288, 1982.
[27]
B. J. Roth, “Boundary layers in the mechanical bidomain model,” in Proceedings of the Biomedical Engineering Society Annual Meeting (BMES), Atlanta, Ga, USA, October 2012.
[28]
B. J. Roth, “Boundary layers and the distribution of membrane forces predicted by the mechanical bidomain model,” submitted.
[29]
V. M. Punal and B. J. Roth, “A perturbation solution of the mechanical bidomain model,” Biomechanics and Modeling in Mechanobiology, vol. 11, no. 7, pp. 995–1000, 2012.
[30]
B. J. Roth and J. P. Wikswo Jr., “A bidomain model for the extracellular potential and magnetic field of cardiac tissue,” IEEE Transactions on Biomedical Engineering, vol. 33, no. 4, pp. 467–469, 1986.
[31]
N. G. Sepulveda, B. J. Roth, and J. P. Wikswo Jr., “Current injection into a two-dimensional anisotropic bidomain,” Biophysical Journal, vol. 55, no. 5, pp. 987–999, 1989.
[32]
N. A. Trayanova, B. J. Roth, and L. J. Malden, “The response of a spherical heart to a uniform electric field: a bidomain analysis of cardiac stimulation,” IEEE Transactions on Biomedical Engineering, vol. 40, no. 9, pp. 899–908, 1993.
[33]
B. J. Roth, “Approximate analytical solutions to the bidomain equations with unequal anisotropy ratios,” Physical Review E, vol. 55, no. 2, pp. 1819–1826, 1997.
[34]
M. van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, NY, USA, 1964.
[35]
R. S. Johnson, Singular Perturbation Theory—Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, NY, USA, 2005.
[36]
H. Schlichting and K. Gersten, Boundary-Layer Theory, Springer, New York, NY, USA, 8th edition, 2000.
[37]
J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill, New York, NY, USA, 1972.
[38]
J. M. Guccione and A. D. McCulloch, “Finite element modeling of ventricular mechanics,” in Theory of Heart, L. Glass, P. Hunter, and A. McCulloch, Eds., pp. 121–144, Springer, New York, NY, USA, 1991.
[39]
Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York, NY, USA, 1981.
[40]
L. Glass, P. Hunter, and A. McCulloch, Theory of Heart, Springer, New York, NY, USA, 1991.
[41]
J. C. Neu and W. Krassowska, “Homogenization of syncytial tissues,” Critical Reviews in Biomedical Engineering, vol. 21, no. 2, pp. 137–199, 1993.
[42]
M. Dembo and F. Harlow, “Cell motion, contractile networks, and the physics of interpenetrating reactive flow,” Biophysical Journal, vol. 50, no. 1, pp. 109–121, 1986.
[43]
V. H. Barocas and R. T. Tranquillo, “An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance,” Journal of Biomechanical Engineering, vol. 119, no. 2, pp. 137–145, 1997.
[44]
V. C. Mow, M. H. Holmes, and W. M. Lai, “Fluid transport and mechanical properties of articular cartilage: a review,” Journal of Biomechanics, vol. 17, no. 5, pp. 377–394, 1984.
[45]
A. F. Mak, W. M. Lai, and V. C. Mow, “Biphasic indentation of articular cartilage-I. Theoretical analysis,” Journal of Biomechanics, vol. 20, no. 7, pp. 703–714, 1987.
[46]
S. Puwal and B. J. Roth, “Elastic wave propagation in the mechanical bidomain model of cardiac tissue,” in Proceedings of the Fall Meeting of the Ohio Region Section of the American Physical Society, Muncie, Ind, USA, October 2011.
[47]
L. Tung and A. G. Kléber, “Virtual sources associated with linear and curved strands of cardiac cells,” American Journal of Physiology, vol. 279, no. 4, pp. H1579–H1590, 2000.
[48]
N. Bursac, Y. Loo, K. Leong, and L. Tung, “Novel anisotropic engineered cardiac tissues: studies of electrical propagation,” Biochemical and Biophysical Research Communications, vol. 361, no. 4, pp. 847–853, 2007.
[49]
D. S. Rosenbaum and J. Jalife, Optical Mapping of Cardiac Excitation and Arrhythmias, Futura, Armonk, NY, USA, 2001.
[50]
B. J. Roth, “Artifacts, assumptions, and ambiguity: pitfalls in comparing experimental results to numerical simulations when studying electrical stimulation of the heart,” Chaos, vol. 12, no. 3, pp. 973–981, 2002.
[51]
D. L. Butler, S. A. Goldstein, R. E. Guldberg et al., “The impact of biomechanics in tissue engineering and regenerative medicine,” Tissue Engineering B, vol. 15, no. 4, pp. 477–484, 2009.
