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. 2000 Nov;79(5):2369–2381. doi: 10.1016/S0006-3495(00)76482-4

A cell-based constitutive relation for bio-artificial tissues.

G I Zahalak 1, J E Wagenseil 1, T Wakatsuki 1, E L Elson 1
PMCID: PMC1301124  PMID: 11053116

Abstract

By using a combination of continuum and statistical mechanics we derive an integral constitutive relation for bio-artificial tissue models consisting of a monodisperse population of cells in a uniform collagenous matrix. This constitutive relation quantitatively models the dependence of tissue stress on deformation history, and makes explicit the separate contribution of cells and matrix to the mechanical behavior of the composite tissue. Thus microscopic cell mechanical properties can be deduced via this theory from measurements of macroscopic tissue properties. A central feature of the constitutive relation is the appearance of "anisotropy tensors" that embody the effects of cell orientation on tissue mechanics. The theory assumes that the tissues are stable over the observation time, and does not in its present form allow for cell migration, reorientation, or internal remodeling. We have compared the predictions of the theory to uniaxial relaxation tests on fibroblast-populated collagen matrices (FPMs) and find that the experimental results generally support the theory and yield values of fibroblast contractile force and stiffness roughly an order of magnitude smaller than, and viscosity comparable to, the corresponding properties of active skeletal muscle. The method used here to derive the tissue constitutive equation permits more sophisticated cell models to be used in developing more accurate representations of tissue properties.

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Selected References

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  1. Barocas V. H., Moon A. G., Tranquillo R. T. The fibroblast-populated collagen microsphere assay of cell traction force--Part 2: Measurement of the cell traction parameter. J Biomech Eng. 1995 May;117(2):161–170. doi: 10.1115/1.2795998. [DOI] [PubMed] [Google Scholar]
  2. Barocas V. H., Tranquillo R. T. An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance. J Biomech Eng. 1997 May;119(2):137–145. doi: 10.1115/1.2796072. [DOI] [PubMed] [Google Scholar]
  3. Bausch A. R., Möller W., Sackmann E. Measurement of local viscoelasticity and forces in living cells by magnetic tweezers. Biophys J. 1999 Jan;76(1 Pt 1):573–579. doi: 10.1016/S0006-3495(99)77225-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Bell E., Ivarsson B., Merrill C. Production of a tissue-like structure by contraction of collagen lattices by human fibroblasts of different proliferative potential in vitro. Proc Natl Acad Sci U S A. 1979 Mar;76(3):1274–1278. doi: 10.1073/pnas.76.3.1274. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Darby I., Skalli O., Gabbiani G. Alpha-smooth muscle actin is transiently expressed by myofibroblasts during experimental wound healing. Lab Invest. 1990 Jul;63(1):21–29. [PubMed] [Google Scholar]
  6. Eschenhagen T., Fink C., Remmers U., Scholz H., Wattchow J., Weil J., Zimmermann W., Dohmen H. H., Schäfer H., Bishopric N. Three-dimensional reconstitution of embryonic cardiomyocytes in a collagen matrix: a new heart muscle model system. FASEB J. 1997 Jul;11(8):683–694. doi: 10.1096/fasebj.11.8.9240969. [DOI] [PubMed] [Google Scholar]
  7. Evans E., Yeung A. Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration. Biophys J. 1989 Jul;56(1):151–160. doi: 10.1016/S0006-3495(89)82660-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Ford L. E., Huxley A. F., Simmons R. M. The relation between stiffness and filament overlap in stimulated frog muscle fibres. J Physiol. 1981 Feb;311:219–249. doi: 10.1113/jphysiol.1981.sp013582. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Glerum J. J., Van Mastrigt R., Van Koeveringe A. J. Mechanical properties of mammalian single smooth muscle cells. III. Passive properties of pig detrusor and human a terme uterus cells. J Muscle Res Cell Motil. 1990 Oct;11(5):453–462. doi: 10.1007/BF01739765. [DOI] [PubMed] [Google Scholar]
  10. Harris D. E., Warshaw D. M. Length vs. active force relationship in single isolated smooth muscle cells. Am J Physiol. 1991 May;260(5 Pt 1):C1104–C1112. doi: 10.1152/ajpcell.1991.260.5.C1104. [DOI] [PubMed] [Google Scholar]
  11. Intengan H. D., Deng L. Y., Li J. S., Schiffrin E. L. Mechanics and composition of human subcutaneous resistance arteries in essential hypertension. Hypertension. 1999 Jan;33(1 Pt 2):569–574. doi: 10.1161/01.hyp.33.1.569. [DOI] [PubMed] [Google Scholar]
  12. Katz B. The relation between force and speed in muscular contraction. J Physiol. 1939 Jun 14;96(1):45–64. doi: 10.1113/jphysiol.1939.sp003756. [DOI] [PMC free article] [PubMed] [Google Scholar]
  13. Kolodney M. S., Elson E. L. Contraction due to microtubule disruption is associated with increased phosphorylation of myosin regulatory light chain. Proc Natl Acad Sci U S A. 1995 Oct 24;92(22):10252–10256. doi: 10.1073/pnas.92.22.10252. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Kolodney M. S., Elson E. L. Correlation of myosin light chain phosphorylation with isometric contraction of fibroblasts. J Biol Chem. 1993 Nov 15;268(32):23850–23855. [PubMed] [Google Scholar]
  15. Kolodney M. S., Wysolmerski R. B. Isometric contraction by fibroblasts and endothelial cells in tissue culture: a quantitative study. J Cell Biol. 1992 Apr;117(1):73–82. doi: 10.1083/jcb.117.1.73. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Murray J. D., Oster G. F. Cell traction models for generating pattern and form in morphogenesis. J Math Biol. 1984;19(3):265–279. doi: 10.1007/BF00277099. [DOI] [PubMed] [Google Scholar]
  17. Murray J. D., Oster G. F., Harris A. K. A mechanical model for mesenchymal morphogenesis. J Math Biol. 1983;17(1):125–129. doi: 10.1007/BF00276117. [DOI] [PubMed] [Google Scholar]
  18. Odell G. M., Oster G., Alberch P., Burnside B. The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Dev Biol. 1981 Jul 30;85(2):446–462. doi: 10.1016/0012-1606(81)90276-1. [DOI] [PubMed] [Google Scholar]
  19. Sung K. L., Dong C., Schmid-Schönbein G. W., Chien S., Skalak R. Leukocyte relaxation properties. Biophys J. 1988 Aug;54(2):331–336. doi: 10.1016/S0006-3495(88)82963-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  20. Tranquillo R. T., Murray J. D. Continuum model of fibroblast-driven wound contraction: inflammation-mediation. J Theor Biol. 1992 Sep 21;158(2):135–172. doi: 10.1016/s0022-5193(05)80715-5. [DOI] [PubMed] [Google Scholar]
  21. Tranquillo R. T., Murray J. D. Mechanistic model of wound contraction. J Surg Res. 1993 Aug;55(2):233–247. doi: 10.1006/jsre.1993.1135. [DOI] [PubMed] [Google Scholar]
  22. Tranquillo R. T. Self-organization of tissue-equivalents: the nature and role of contact guidance. Biochem Soc Symp. 1999;65:27–42. [PubMed] [Google Scholar]
  23. Wakatsuki T., Kolodney M. S., Zahalak G. I., Elson E. L. Cell mechanics studied by a reconstituted model tissue. Biophys J. 2000 Nov;79(5):2353–2368. doi: 10.1016/S0006-3495(00)76481-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Zahalak G. I., McConnaughey W. B., Elson E. L. Determination of cellular mechanical properties by cell poking, with an application to leukocytes. J Biomech Eng. 1990 Aug;112(3):283–294. doi: 10.1115/1.2891186. [DOI] [PubMed] [Google Scholar]

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