zbMATH — the first resource for mathematics

A three-dimensional computational method for blood flow in the heart. I: Immersed elastic fibers in a viscous incompressible fluid. (English) Zbl 0668.76159
The heart wall is modelled as a viscous incompressible fluid that contains elastic fibres. The fluid has mass and volume whereas the fibre is assumed to transmit stress only and does not have mass or volume. The appropriate three dimensional time dependent equations are established. These consist of the Navier-Stokes equations in the fluid, equations for the fibre and a set of linking equations. A suitable numerical method is developed for solving the equations and this is then implemented on a large computer. There is a detailed discussion of this process including such topics as vectorization and the use of the central memory of the computer. As a preliminary to considering the behaviour of the heart the equations and method of solution are used to solve the problem of the vibration of a fibre wound toroidal tube. Numerical results are presented in graphical form. This is an impressive paper covering a wide spread of mathematical techniques with direct application to a practical problem.
Reviewer: G.Eason

76Z05 Physiological flows
92Cxx Physiological, cellular and medical topics
74L15 Biomechanical solid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs
Full Text: DOI
[1] Thomas, C.E., Amer. J. anatomy, 101, 17, (1957)
[2] Streeter, D.D.; Powers, W.E.; Ross, M.A.; Torrent-Guasp, F., Three-dimensional fiber orientation in the Mammalian left ventricular wall, (), 73
[3] Sagawa, K.; Suga, H.; Nakayama, K., Instantaneous pressure-volume ratio of the ventricle versus instantaneous force-length relation of papillary muscle, (), 99
[4] Peskin, C.S., J. comput. phys., 25, 220, (1977)
[5] Peskin, C.S.; McQueen, D.M., J. comput. phys., 37, 113, (1980)
[6] McQueen, D.M.; Peskin, C.S.; Yellin, E.L., Amer. J. physiol., 242, H1095, (1982)
[7] McQueen, D.M.; Peskin, C.S., J. thorac. cardiovasc. surg., 86, 126, (1983)
[8] McQueen, D.M.; Peskin, C.S., Scand. J. thorac. cardiovasc. surg., 19, 139, (1985)
[9] Meisner, J.S.; McQueen, D.M.; Ishida, Y.; Vetter, H.O.; Bortolotti, U.; Strom, J.A.; Frater, R.W.M.; Peskin, C.S.; Yellin, E.L., Amer. J. physiol., 249, H604, (1985)
[10] Chorin, A.J., Math. comput., 22, 745, (1968)
[11] Chorin, A.J., Math. comput., 23, 341, (1969)
[12] Greenberg, S.; McQueen, D.M.; Peskin, C.S., Three-dimensional fluid dynamics in a twodimensional amount of central memory, (), 185
[13] Fischer, D.; Golub, G.; Hald, O.; Leiva, C.; Widlund, O., Math. comput., 28, 349, (1974)
[14] O’Leary, D.P.; Widlund, O., Math. comput., 33, 849, (1979)
[15] Gottlieb, A., An overview of the NYU ultracomputer project, ()
[16] Horowitz, E.J., J. comut. phys., 68, 56, (1987)
[17] Nishiguchi, A.; Orii, S.; Yabe, T., J. comput. phys., 61, 519, (1985)
[18] Peskin, C.S., Commun. pure appl. math., 42, 79, (1989)
[19] Chadwick, R.S., Biophys. J., 39, 279, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.