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The purpose of this paper is to present standard results on the effect of nonlinearities on the computed pulsatile flow in a cylindrical distensible tube as a first stage in the calculation of flow in a tapered tube and blood flow in arteries. The calculations are made using the pressure-radius relationship of a rubber tube with no longitudinal motion and for a linearised relationship. The one-dimensional equations of motion are solved by the method of finite differences. The values of skin friction that are incorporated are determined from the vorticity and continuity equations for a rigid tube and a correction made to the current diameter at each time step. The accuracy of the results is assessed and the effect of varying parameters investigated. The method is applied to a segment of an infinite tube for which the linear analytical solution is available. The characteristics of the velocity wave calculated from an input pressure wave are presented as departures from the linear theory values of these characteristics, the wave speed, flux and transmission factor per wavelength. Computations are made at values of non-dimensional frequency (Stokes number α) of about 3 and 10. It is concluded that as far as physiological application is concerned (i.e. small amplitude and long wavelength) the results of linear theory are a very good first approximation for the cylindrical tube. At α=10, the relative departure of wave speed is about 0·5 times the relative diameter amplitude ( $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ amplitude/mean diameter) when the pressure-radius relation is linear and the pressure and velocity waves have the same characteristics. At α=3 the corresponding wave speed departure is about 0·1 $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ . The relative departure of the flux is less than 0·05 $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ at α=3 and about 0·5 $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ at α=10. The transmission coefficient has a relative departure of less than 0·05 $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ at α=10 and its relative increase at α=3 is about 0·3 $${{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}$$ . [ABSTRACT FROM AUTHOR]

The one-dimensional equations are used in the calculation of blood flow in arteries. The majority of the treatments use the method of characteristics and because of the nature of the method it is necessary to use a simplified value of the skin friction. A commonly used simplification is to assume the zero frequency value of the skin friction. The effect of the use of this approximation is compared with results using the full linear theory value. It is shown that the phase difference between the skin friction and the flux has an appreciable effect on the velocity wave calculated from a given pressure wave. [ABSTRACT FROM AUTHOR]

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Copyright of Medical & Biological Engineering & Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The displacement relationships describing the deformation of an elastic vessel under excess internal pressure which are derived from different theories of elasticity are compared. The main result of the comparison is that theories which take account of the thickness of the wall of the vessel produce a significantly better representation than those theories which treat the wall as a membrane. The classical and statistical theories of thick-walled tubes result in complicated pressure-radius relationships. It is shown that there is little difference between the results of the more exact theories and those for a thin membrane corrected by means of a simple thickness factor. A review of the different theories is necessary to decide which pressure-displacement relationship to apply as an approximation for the elastic properties of arteries. An indication is given of the manner in which the relationship is used in numerical computations. In Part 2 the experimental determination of the pressure-radius relationship for a rubber tube is described. The results are in agreement with the conclusions of the comparison of theoretical treatments in Part 1. [ABSTRACT FROM AUTHOR]