Hilmi Demiray

A note on the elasticity of soft biological tissues

Abstract In this short note a simple possible form of the strain energy function for soft biological tissues is studied. The similarity of the result of an example problem to experimental results is encouraging.

WAVE PROPAGATION THROUGH A VISCOUS FLUID CONTAINED IN A PRESTRESSED THIN ELASTIC TUBE

In this work a theoretical analysis is presented for the wave propagation through a viscous incompressible fluid contained in a prestressed thin elastic tube. The fluid is assumed to be incompressible and Newtonian, whereas the tube material is considered to be incompressible, isotropic and elastic. Such an elastic tube is subjected to a mean pressure Pi and the axial force N (or, stretch ratio A). Assuming that the disturbance added on this initial deformation is small, the governing differential equations are obtained for the elastic tube and the fluid. A harmonic wave type of solution is sought for these field equations and the dispersion relation is obtained. Various special cases as well as the general case are throughly investigated and the present formulation is compared with previous works on the same subject. For a healthy person the mean pressure in an artery is approx. 100 mm Hg and the axial stretch is about 1.5. This mean pressure and the axial stretch create relatively high circumferential (hoop) and axial initial stresses. On the other hand, the pressure deviation in the course of periodic motion of heart is around f20 mm Hg. The dynamical deformation resulting from this pressure deviation may be assumed to be small as compared to initial deformation. Therefore, the theory of small deformations superimposed on initial static deformation may be utilized in analyzing the wave propagation in such a composite structure. Propagation of harmonic waves in an initially stressed (or unstressed) cylindrical tube filled with a viscous (or inviscid) fluid is a problem of interest since the time of Thomas Young who first obtained the speed of pulse waves in human arteries. The current literature on the subject is so rich that it is impossible to cite all the works here. The historical evolution of the subject may be found in the papers by Lambossy (l) and Skalak (2), and in the books by McDonald (3) and Fung (4). Significant contributions on wave motions of an elastic tube filled with a viscous fluid have been given by Morgan and Kiely (5), Womersley (6), and Atabek and Lew (7). All these researchers have assumed that the arterial wall is an isotropic and incompressible elastic material. However, Lawton (8) and Fen (9), in their experimental measurements, observed that the arterial wall material may be viscoelastic and anisotropic. These characterictics of blood vessels have partially been taken into account by Mirsky (lo), Atabek (ll), more recently by Rachev (12) and Kuiken (13). In all these works, either the effect of initial stresses have been neglected, or taken into account in ad hoc manner. Moreover, the elastic coefficients of the incremental stress have been treated to be constant. In reality these coefficients depend on the initial deformations. In this work, first, the equations of motion of the incremental deformation and the associated incremental stress are obtained for a thin elastic tube and an incompressible viscous fluid contained in it. For simplicity, the arterial wall is taken to be an isotropic and incompressible elastic material with a constitutive relation proposed by Demiray (14). Seeking a harmonic wave type of solution to the field equations of fluid and solid and using the boundary conditions, the dispersion relation is obtained and various special cases are discussed and the result is compared with previous studies on the same subject. Due to difficulty of the analysis of the general dispersion relation by analytical means, a numerical technique has been employed

Large deformation analysis of some basic problems in biophysics

This study is concerned with the application of a possible form of a strain-energy function suitable for soft biological tissues. Two problems considered here are related to simultaneous extension and inflation of cylindrical arteries, and inflation of the left ventricle under a given internal pressure. The values of the material constants are obtained via comparison of theoretical results with experimental findings. Some details concerning the wall stresses and the elastic stiffness are also given in the paper. For each case, it is seen that experiment and theory are in good agreement.

Solitary waves in prestressed elastic tubes

In this work, we studied the propagation of non-linear waves in a pre-stressed thin elastic tube filled with an inviscid fluid. In the analysis, analogous to the physiological conditions of the arteries, the tube is assumed to be subject to a uniform pressureP0 and a constant axial stretch ratio λz. In the course of blood flow it is assumed that a large dynamic displacement is superimposed on this static field. Furthermore, assuming that the displacement gradient in the axial direction is small, the non-linear equation of motion of the tube is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. The result is discussed for some elastic materials existing in the literature.

A layered cylindrical shell model for an aorta

Abstract In this work, utilizing the result of the experiment that we conducted on a dogs upper thoracic aorta, a two layered cylindrical shell model is presented. A series of bi-axial tests were carried out on the specimens taken from the media and adventitia of an aortic segment and the stress-stretch relations were obtained. Considering the morphological structures of media and adventitia, an orthotropic elastic model for media and an isotropic elastic model for adventitia is presented. By comparing theoretical results and experimental measurements, the values of material constants appearing in the model have been determined. Finally, using the elastic models introduced, the inflation and axial extension of a layered cylindrical aorta was studied.

A stress-strain relation for a rat abdominal aorta

Assuming the arterial wall is homogeneous, incompressible, isotropic and elastic, a stress-strain relation has been presented for a rats abdominal aorta. As an illustrating example, the problem of simultaneous inflation and the axial stretch of a cylindrical artery under physiological loading has been solved and then the material coefficients are determined by comparing theoretical results with the existing experiments. The result indicates that the maximum deviation between the theory and experiment for various pressure levels is 3.7% which seems to be a good approximation of theory to the experiments. The variation of circumferential stress and the incremental pressure modulus with inner pressure are also depicted in the work.

Small torsional oscillations of an initially twisted circular rubber cylinder

Abstract Small torsional oscillations of a circular, rubber cylinder initially subjected to a finite pure torsion are studied. The resulting equations are solved exactly for neo-Hookean materials and approximately for Mooney materials. It is shown that longitudinal oscillations are associated with torsional oscillations. For Mooney materials, first order longitudinal oscillations and the frequency of oscillations up to the second order are calculated.

Head-on collision of solitary waves in fluid-filled elastic tubes

Abstract In this work, treating the arteries as a thin walled, prestressed thin elastic tube and the blood as an inviscid fluid, we have studied the propagation of nonlinear waves, in the longwave approximation, through the use of extended PLK perturbation method. The results show that, up to O ( ϵ 2 ) , the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the collision. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly.

A note on the exact travelling wave solution to the KdV–Burgers equation

Abstract In the present note, by use of the hyperbolic tangent method, a progressive wave solution to the Korteweg–de Vries–Burgers (KdVB) equation is presented. The solution we introduced here is less restrictive and comprises some solutions existing in the current literature (see [Wave Motion 11 (1989) 559; Wave Motion 14 (1991) 369]).

Electromechanical properties and related models of bone tissues: A review

Abstract In this survey the mechanical and piezoelectrical properties of bone tissue and associated theoretical models have been reviewed and related references are supplied throughout. In doing this, we consider the connection of remodelling characteristics of bone tissue with electromechanical properties, by special emphasis on the feedback mechanism. In order to make the article self-sufficient, to some extent, the physiology of bone is summarized in Section 2. Examining the general picture of the research works in this area, the mechanical and electrical properties of bone tissues are studied separately. The mechanical properties of compact bone, with some contributing factors to these properties, and related theoretical models are presented in Sections 3 and 4, respectively. In Section 5, the piezoelectric properties of dry and wet bones, which are considered to be the main factor in remodelling processes, are surveyed in a chronological order. Finally, possible electromethanical models for dry and wet bones and their limitations are reported in Section 6.