Flow Rate through a Blood Vessel Deformed Due To a Uniform Pressure
ABSTRACT
In this paper, we present the mathematical equations that govern the deformation of an imbedded blood vessel under external uniform pressure taking into consideration the nonliner behavior of the soft tissue surrounding the vessel. We present a bifurcation analysis and give explicit formulas for the bifurcation points and the corresponding first order approximations for the\emph{non-trivial} solutions. We then show the results of a MATLAB program that integrates the equilibrium equations and calculates the blood flow rate through a deformed cross section for given values of the elasticity parameters and pressure. Finally, we provide (numerical) verification that the flow rate as a function of the elasticity parameters of the soft tissue surrounding the blood vessel is convex, and therefore validate the invertibility of our model.
In this paper, we present the mathematical equations that govern the deformation of an imbedded blood vessel under external uniform pressure taking into consideration the nonliner behavior of the soft tissue surrounding the vessel. We present a bifurcation analysis and give explicit formulas for the bifurcation points and the corresponding first order approximations for the\emph{non-trivial} solutions. We then show the results of a MATLAB program that integrates the equilibrium equations and calculates the blood flow rate through a deformed cross section for given values of the elasticity parameters and pressure. Finally, we provide (numerical) verification that the flow rate as a function of the elasticity parameters of the soft tissue surrounding the blood vessel is convex, and therefore validate the invertibility of our model.
Cite this paper
nullA. Cypher, J. Elgindi, H. Kouriachi, D. Peschman and R. Shotwell, "Flow Rate through a Blood Vessel Deformed Due To a Uniform Pressure," Journal of Biomaterials and Nanobiotechnology, Vol. 2 No. 4, 2011, pp. 369-377. doi: 10.4236/jbnb.2011.24046.
nullA. Cypher, J. Elgindi, H. Kouriachi, D. Peschman and R. Shotwell, "Flow Rate through a Blood Vessel Deformed Due To a Uniform Pressure," Journal of Biomaterials and Nanobiotechnology, Vol. 2 No. 4, 2011, pp. 369-377. doi: 10.4236/jbnb.2011.24046.
References
[1] C. Y. Wang, L. T. Watson and M. P. Kamat, “Buckling, Postbuckling, and Flow Rate Through a Tethered Elastic Cylinder Under External Pressure,” Journal of Applied Mechanics, Vol. 50, No. 1, pp. 13-18, 1983. doi:10.1115/1.3166981 [2] D. H. Bergel, “The Properties of Blood Vessels,” In: Biomechanics, Its Foundations and Objectives, Eds., Prentice-Hall, Englewood, 1972. [3] Y. C. Fung, “Biomechanics, Mechanical Properties of Living Tissues,” Springer-Verlag, New York, 1990. [4] A. C. Guyton and J. E. Hall, “Textbook of Medical Physiology,” Saunders Company, 1996. [5] A. H. Moreno, A. I. Katz, L. D. Gold and R. V. Reddy, “Mechanics of Distension of Dog Veins and Other Very Thin-Walled Tubular Sturctures,” Circulation Research, Vol. 27, No. 6, 1970, pp. 1069-1080. [6] K. Osterloch, P. Gaehtgens, and A. Pries, “Determination of Microvascular Flow Pattern Formation in Vivo,” American Journal of Botany, Vol. 278, No. 4, 2000, pp. H1142-H1152. [7] J. Stoer and R. Bulirsch, “Introduction to Numerical Analysis,” Springer-Verlag, New York, 1980. [8] M. B. M. Elgindi, D. H. Y. Yen and C. Y. Wang, “Deformation of a Thin-Walled Cylindrical Tube Submerged in a Liquid,” Journal of Fluids and Structure, Vol. 6, No. 3, 1992, pp. 353-370. doi:10.1016/0889-9746(92)90014-T
[1] C. Y. Wang, L. T. Watson and M. P. Kamat, “Buckling, Postbuckling, and Flow Rate Through a Tethered Elastic Cylinder Under External Pressure,” Journal of Applied Mechanics, Vol. 50, No. 1, pp. 13-18, 1983. doi:10.1115/1.3166981 [2] D. H. Bergel, “The Properties of Blood Vessels,” In: Biomechanics, Its Foundations and Objectives, Eds., Prentice-Hall, Englewood, 1972. [3] Y. C. Fung, “Biomechanics, Mechanical Properties of Living Tissues,” Springer-Verlag, New York, 1990. [4] A. C. Guyton and J. E. Hall, “Textbook of Medical Physiology,” Saunders Company, 1996. [5] A. H. Moreno, A. I. Katz, L. D. Gold and R. V. Reddy, “Mechanics of Distension of Dog Veins and Other Very Thin-Walled Tubular Sturctures,” Circulation Research, Vol. 27, No. 6, 1970, pp. 1069-1080. [6] K. Osterloch, P. Gaehtgens, and A. Pries, “Determination of Microvascular Flow Pattern Formation in Vivo,” American Journal of Botany, Vol. 278, No. 4, 2000, pp. H1142-H1152. [7] J. Stoer and R. Bulirsch, “Introduction to Numerical Analysis,” Springer-Verlag, New York, 1980. [8] M. B. M. Elgindi, D. H. Y. Yen and C. Y. Wang, “Deformation of a Thin-Walled Cylindrical Tube Submerged in a Liquid,” Journal of Fluids and Structure, Vol. 6, No. 3, 1992, pp. 353-370. doi:10.1016/0889-9746(92)90014-T