How to test very soft biological tissues in extension?

doi:10.1016/S0021-9290(00)00236-0

Journal of Biomechanics

Volume 34, Issue 5, May 2001, Pages 651-657

https://doi.org/10.1016/S0021-9290(00)00236-0 Get rights and content

Abstract

Mechanical properties of very soft tissues, such as brain, liver and kidney, until recently have largely escaped the attention of researchers because these tissues do not bear mechanical loads. However, developments in Computer–Integrated and Robot–Aided Surgery — in particular, the emergence of automatic surgical tools and robots — as well as advances in Virtual Reality techniques, require closer examination of the mechanical properties of very soft tissues and, ultimately, the construction of corresponding, realistic mathematical models. A body of knowledge about mechanical properties of very soft tissues, assembled in recent years, has been almost exclusively based on the results of compression, indentation and impact tests. There are no results of tensile tests available. This state of affairs, in the author's opinion, is caused by the lack of analytical solution relating a measured quantity — machine head displacement — to strain in simple extension experiments of cylindrical samples with low aspect ratio. In the paper this important solution is presented. The theoretical solution obtained is valid for isotropic, incompressible materials for moderate deformations (<30%) when it can be assumed that planes initially perpendicular to the direction of applied extension remain plane. Two astonishing results are obtained: (i) deformed shape of a cylindrical sample subjected to uniaxial extension is independent on the form of constitutive law, (ii) vertical extension in the plane of symmetry λ_z is proportional to the total change of height for strains as large as 30%. The importance and relevance of these results to testing procedures in Biomechanics is highlighted.

Introduction

Mechanical properties of living tissues form a central subject in Biomechanics. Recent developments in robotics technology, especially the emergence of automatic surgical tools and robots (e.g. Brett et al., 1995) as well as advances in virtual reality techniques (Burdea, 1996) provide immediate and important applications to research into the mechanical properties of very soft tissues, such as brain, liver and kidney. These tissues have been largely neglected before because they do not bear mechanical loads. Mathematical models of brain tissue mechanical properties may find application, for example, in a surgical robot control system where the prediction of deformation is needed (Miller and Chinzei (1995), Miller and Chinzei (2000)); surgical operation planning and surgeon training systems based on the virtual reality techniques (Burdea, 1996 and references cited therein) where force feedback is needed; and in registration (Lavallée, 1995) where knowledge of local deformation is required.

There is a wealth of information available in the literature about the mechanical properties of brain tissue in-vitro (Ommaya, 1968; Estes and McElhaney, 1970; Galford and McElhaney, 1970; Pamidi and Advani, 1978; Bilston et al., 1997; Donnelly and Medige, 1997; Miller and Chinzei, 1997). Some information on liver and kidney tissue mechanical properties has also been published (e.g. Melvin et al., 1973; Farshad et al (1998), Farshad et al (1999); Miller, 2000). However, the experimental results available in literature are limited to compression, indentation and impact tests and to a lesser extent, torsional tests. To the best of the author's knowledge, there are no results in the literature concerning very soft tissue properties in tension. Why is that? The author believes that, besides technical problems with conducting extension tests on brain and other soft tissues, the main reason is that the analytical relation between the tensile stress machine head displacement and strain is not known for cylindrical samples with a low aspect ratio.

In this paper I develop the rigorous mathematical description of the deformation in the uniaxial extension experiment and derive the relation between the machine head displacement and strain needed for the constitutive law identification. It is claimed that the results obtained allow the analysis of uniaxial extension experiments performed on low aspect ratio cylindrical samples of soft biological tissues in an analogous way to that routinely used in the unconfined compression.

Section snippets

Extension experiment set-up

Typically, in experiments on brain tissue cylindrical samples of diameter ∼30 mm and height ∼10 mm are used (Miller and Chinzei, 1997). Steel pipe (30 mm diameter) with sharp edges is used to cut the samples. The faces of the cylindrical brain specimens are smoothed manually, using a surgical scalpel.

Uniaxial tension of brain (or other very soft tissue) can be performed in a testing stand sketched in Fig. 1. This particular geometry is dictated by the difficulties in attaching faces of cylindrical

Theoretical analysis of extension experiment

In extension experiment the kinematics of the deformation is complex, prohibiting the existence of the exact analytical solution for the deformation of the sample for any realistic constitutive law chosen to describe tissue mechanical properties. This, in the author’ opinion, is one of the reasons why there are no extension results for soft tissues published in the biomechanics literature. However, with a few reasonable assumptions an approximate solution can be found.

I consider a circular

Results

By differentiating Eq. (14) with respect to Z, Eq. (16) with respect to R, and equating the results one obtains a single ordinary differential equation for the shape of the deformed sample $f (Z)$ : $∂ W ∂ I_{1} d d Z f″ f + ∂ W ∂ I_{2} d^{3} d Z^{3} 12 f^{2} =0.$ Unfortunately, the solutions of Eq. (17) can be found only in the implicit form: $z+ EllipticE ArcSin f [z] f [0],− C_{2} f [0]^{2} C_{1} f [0] f[0]^{2} −f[z]^{2} f[0]^{2} (C_{1} +C_{2}) Const 1(−f[0]^{2} +f[z]^{2} C_{1} +C_{2} f[z]^{2} C_{1} +C_{2} f[z]^{2} C_{1} =Const2$ $−z+ EllipticE ArcSin f [z] f [0],− C_{2} f[0]^{2} C_{1} f [0] f [0]^{2} −f [z]^{2} f [0]^{2} (C_{1} +C_{2}) Const 1 (−f [0]^{2} +f [z]^{2}) C_{1}$

Discussion and conclusions

Two theoretical results presented in this paper have important implications for testing in biomechanics of very soft tissues. As shown above, in a uniaxial extension experiment, in the plane of symmetry Z=z=0 (see Fig. 2) the orthogonal state of deformation can be assumed. This state of deformation can be described, as in the case of the unconfined compression experiment (Miller and Chinzei, 1997), by a diagonal deformation gradient: $F (Z=0)= f(0) 000 f(0) 000 g′(0) = λ_{z}^{−1/2} 000 λ_{z}^{−1/2} 000 λ_{z}$ Therefore, the

Acknowledgements

The financial support of the Australian Research Council, Japanese Science and Technology Agency and Japanese Agency for Industrial Science and Technology is gratefully acknowledged. The author would like to thank Dr Lynne Bilston and Professor Nhan Phan-Thien for very helpful discussions during the author's visit at the University of Sydney.

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How to test very soft biological tissues in extension?

Abstract

Introduction

Section snippets

Extension experiment set-up

Theoretical analysis of extension experiment

Results

Discussion and conclusions

Acknowledgements

Biorheology

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Force and Touch feedback for Virtual Reality

Compression of swine brain tissueexperiment in-vitro

Journal of Mechanical Engineering Laberatory

Shear properties of human brain tissue

Transactions of ASME, Journal of Biomechanical Engineering