Method of testing very soft biological tissues in compression

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1 ARTICLE IN PRESS Journal of Biomecanics 38 (2005) Metod of testing very soft biological tissues in compression Karol Miller* Scool of Mecanical Engineering, Te University of Western Australia, 35 Stirling Higway, Crawley/Pert WA 6009, Australia Accepted 20 February 2004 Abstract Mecanical properties of very soft tissues, suc as brain, liver, kidney and prostate ave recently joined te mainstream researc topics in biomecanics. Tis as appened in spite of te fact tat tese tissues do not bear mecanical loads. Te interest in te biomecanics of very soft tissues as been motivated by te developments in computer-integrated and robot-aided surgery in particular, te emergence of automatic surgical tools and robots as well as advances in virtual reality tecniques. Mecanical testing of very soft tissues provides a formidable callenge for an experimenter. Very soft tissues are usually tested in compression using an unconfined compression set-up, wic requires ascertaining tat friction between sample faces and stress strain macine platens is close to zero. In tis paper a more reliable metod of testing is proposed. In te proposed metod top and bottom faces of a cylindrical specimen wit low aspect ratio are rigidly attaced to te platens of te stress strain macine (e.g. using surgical glue). Tis arrangement allows using a no-slipboundary condition in te analysis of te results. Even toug te state of deformation in te sample cannot be treated as ortogonal te relationsips between total cange of eigt (measured) and strain are obtained. Two important results are derived: (i) deformed sape of a cylindrical sample subjected to uniaxial compression is independent on te form of constitutive law, (ii) vertical extension in te plane of symmetry l z is proportional to te total cange of eigt for strains as large as 30%. Te importance and relevance of tese results to testing procedures in biomecanics are igligted. r 2004 Elsevier Ltd. All rigts reserved. Keywords: Soft tissue; Mecanical properties; Matematical modelling; Compression experiment 1. Introduction Mecanical properties of very soft tissues, suc as brain, liver, kidney and prostate ave recently joined te main directions of researc in biomecanics. Tis as appened in spite of te fact tat tese tissues do not bear mecanical loads. Te interest in te biomecanics of very soft tissues as been motivated by te developments in computer-integrated and robot-aided surgery in particular, te emergence of automatic surgical tools and robots as well as advances in virtual reality tecniques. Mecanical testing of very soft tissues provides a formidable callenge for an experimenter. Very soft tissues are usually tested in compression using an unconfined compression set-up, Fig. 1 (Estes and McElaney, 1970; Miller and Cinzei, 1997). Wile conducting te experiment care must be taken to minimise friction between te sample and macine platens. Only wen friction can be assumed to be zero *Tel.: ; fax: address: kmiller@mec.uwa.edu.au (K. Miller). can one assume tat te sample expands uniformly during te compression and, terefore, tat te state of deformation witin te sample is ortogonal, Eq. (1). F ¼ l 1=2 z l 1=2 z l z 3 7 5; ð1þ were F is a deformation gradient; l z is a stretc in vertical direction. In Eq. (1) incompressibility and isotropy of te sample material are assumed. If te assumption of friction being negligible is violated, Eq. (1) is invalid and measurement results misleading. Current biomecanics literature recognises tis problem, and addresses it by proposing reliable (but complicated) ways to ensure te friction is close to zero, see e.g. (Miller and Cinzei, 1997; Nasseri et al., 2003). To strengten te argument, I present finite element simulations of unconfined compression experiment wit coefficient of friction between te sample and macine platens ranging from 0 to 0.1, conducted using commercial program ABAQUS (ABAQUS, 1998). Te /$ - see front matter r 2004 Elsevier Ltd. All rigts reserved. doi: /j.jbiomec

2 154 ARTICLE IN PRESS K. Miller / Journal of Biomecanics 38 (2005) Z Z Impermeable platen Load Measured compressive force Tissue 2*H Tissue R R (a) Fig. 1. Layout of unconfined compression experiment set-up wit coordinate axes. z Table 1 Reaction force at 20% compression of cylindrical sample (30 mm diameter, 10 mm eigt) for coefficients of friction between 0 and 0.1 Coefficient of friction m= m= m= Reaction force (N) mes was taken from (Miller, 1999): 30 m 10 m 2 cylinder modelled by 480 CAX4RH four-node, axisymmetric elements. For simplicity, te Mooney Rivlin material (Mooney, 1940) was cosen, wit material constants C 1 =C 2 =200 Pa. Tis rougly corresponds to brain mecanical properties at very slow loading. Because of tissue incompressibility te ybrid elements (wit pressure as additional variable) were cosen. Te compression plates were taken to be circular, 50 mm in diameter (i.e. muc larger tan specimen), and rigid. Te results presented in Table 1 sow tat even low friction as substantial effect of producing sear stresses, wic leads to increasing te measured reaction force, and consequently te overestimation of tissue s stiffness. Te numerical result for zero friction agrees exactly wit te analytical solution. Analytical solutions for cases wit non-zero friction do not exist. In tis paper a more reliable metod of testing is considered. In te proposed metod top and bottom faces of a cylindrical specimen wit low aspect ratio are rigidly attaced to te platens of te stress strain macine (e.g. using surgical glue), Fig. 2. Tis arrangement allows using a no-slipboundary condition in te analysis of te results. Te no-slip boundary condition set-upas been used previously in my work on soft tissue properties in tension (Miller, 2001; Miller and Cinzei, 2002). It was sown tat te no-slip boundary condition, wit te application of surgical glue as described in (Miller and Cinzei, 2002) or coarse sand paper as Lynne Bilston used for er sear experiments (Bilston et al., 2001) is reliable. Surgical glues do not infiltrate te tissue. To conduct te experiment is very easy: one prepares a cylindrical sample, applies surgical glue to tissue faces (or attac Measured compressive force (b) 2* Deformed sample Fig. 2. Sketc of te experimental set-up wit no-slip boundary conditions; sample eigt 2 and vertical force are measured; (a) undeformed sample, (b) deformed sample. coarse sand paper to macine platens) and conducts te compression. It is te analysis of measurement results tat is difficult. Te manuscript addresses tis problem. Even toug te state of deformation in te sample cannot be treated as ortogonal te relationsips between total cange of eigt and strain (needed for te constitutive law identification) are derived below in te way analogous to tat used for extension. Two novel (toug analogous to similar results for extension) results are obtained: (i) Te sape of te compressed cylindrical sample of isotropic, incompressible tissue does not depend on te constitutive law describing te properties of te tissue. (ii) Te vertical extension in te plane of symmetry (middle of te sample eigt) l z is proportional to te total cange of eigt for nominal strains as large as 30%. It is claimed tat tese two results allow te analysis of uniaxial compression experiments wit no-slip boundary conditions performed on cylindrical samples wit low aspect ratio in analogous way to tat routinely used in te unconfined compression wit no-friction boundary conditions. 2. Compression experiment set-up Typically, in experiments on brain tissue cylindrical samples of diameter B30 mm and eigt B10 mm are r

3 ARTICLE IN PRESS K. Miller / Journal of Biomecanics 38 (2005) used (Miller and Cinzei, 1997; Miller and Cinzei, 2002). Steel pipe (30 mm diameter) wit sarp edges is used to cut te samples. Te faces of te cylindrical brain specimens are smooted manually, using a surgical scalpel. Uniaxial compression of brain (or oter very soft tissue) can be performed in a testing stand sketced in Fig. 2. Te testing apparatus sould be able to move te macine ead witin large range of velocities (to simulate strain rates typical to impact, surgical or quasi-static conditions) and measure accurately small (fractions of a Newton) vertical forces. 3. Teoretical analysis of compression experiment In compression experiment described above te kinematics of te deformation is complex, proibiting te existence of exact analytical relations between te measured force and stress, and between te measured total cange of eigt and strain in te sample for any realistic constitutive law cosen to describe tissue mecanical properties. Tis, in my opinion, is one of te reasons wy suc a set-upas not been used before. However, wit a few reasonable assumptions an approximate solution can be found. Te matematical description of te experiment and te analysis of te state of deformation in te sample are similar to tose presented for extension in (Miller, 2001). I consider a circular cylinder bonded between two rigid end-plates (Fig. 2). Te disc of radius R and eigt 2H in undeformed state is compressed to te final eigt 2 by uniform surface forces applied normal to te endplates. For te coordinate system in te unstrained state we take Cartesian coordinates fx; Y; Zg: Te Cartesian coordinate system fx; y; zg for deformed body is taken to coincide wit te system fx; Y; Zg: In te analysis te following assumptions are employed: 3.1. Incompressibility Very soft tissues are most often assumed to be incompressible (see e.g. Pamidi and Advani, 1978; Wals and Scettini, 1984; Saay et al., 1992; Mendis et al., 1995; Miller and Cinzei, 1997; Farsad et al., 1999; Miller, 1999, 2000, 2002) Isotropy Very soft tissues do not bear mecanical loads and do not exibit directional structure (provided tat a large enoug sample is considered: for brain we used samples of 30 mm diameter and 10 mm eigt). Terefore, tey may be assumed to be initially isotropic (see e.g. Pamidi and Advani, 1978; Wals and Scettini, 1984; Saay et al., 1992; Mendis et al., 1995; Miller and Cinzei, 1997; Farsad et al., 1999; Miller, 1999, 2000; Bilston et al., 2001, Miller and Cinzei, 2002). Prange and Margulies (2002) report anisotropic properties of brain tissue. However, teir sample sizes were 1 mm wide. At suc a small lengt scale a fibrous nature of most tissues will come into play and directional properties will be detected. Experimental tecnique discussed ere aims at identifying average properties at te lengt scale of approx. 1 cm. At suc lengt scales most very soft tissues can be safely assumed not to exibit directional variation of mecanical properties Te planes perpendicular to te direction of te applied force remain plane From te above assumptions it follows (Miller, 2001) tat in te plane of symmetry, Z=0, off-diagonal components of te deformation gradient vanis. Tis is a very important observation te deformation in te plane of symmetry is ortogonal. FðX; Y; 0Þ ¼ l 1=2 z l 1=2 z l z 3 7 5; ð2þ were l z is a stretc in vertical direction (Fig. 2). Following te procedure described in detail in Miller (2001), one obtains te following implicit equation for te deformed sape of te compressed sample: Elliptic E arcsin f ½zŠ ; C 2f ½0Š 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ½0Š 2 þ f ½zŠ 2 f ½0Š f ½0Š C 1 f ½0Š 2 z ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s const 2; u tðc 1 þ C 2 Þconst 1ð f ½0Š 2 þ f ½zŠ 2 Þ C 1 þ C 2 f ½zŠ 2 C 1 þ C 2 f ½zŠ 2 C 1 were f ðzþ ¼r=R is te sape of te side of te compressed sample. C 1 and C 2 are integration constants. Elliptic E denotes te elliptic integral of te second kind. It is difficult to calculate integration constants and convert Eq. (3) into an explicit formula for f ðzþ: However, one can obtain important relationsips for two extremes of te material beaviour: * Neo Hookean material=> W=C 1 (I 1 3); m/2=c 1 ; * Extreme Mooney material=> W=C 2 (I 2 3); m/2=c 2 were W is a potential function, I 1 and I 2 are strain invariants and C 1, C 2 are material constants. Te pysical meaning of constants C 1, C 2 in te limit of infinitesimal deformation is: m/2=c 1 +C 2, were m is te sear modulus. It is known tat real materials fall somewere in between tese two extremes. Tese relationsips, first obtained by Klingbeil and Sield (1966) are given by Eqs. (4) (7). ð3þ

4 156 ARTICLE IN PRESS K. Miller / Journal of Biomecanics 38 (2005) Fig. 3. Deformed sapes of samples made of Neo Hookean (ligter, exterior curves) and Extreme Mooney (darker, interior curves) materials for / H=0.9, 0.8 and 0.7 for practical purposes deformed sapes for te two extreme cases are te same. For Neo Hookean material=> W=C 1 (I 1 3); m/ 2=C 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ 1 þ l 2 z l 2 z arcsecðl zþ ; ð4þ f ðzþ ¼l z cos arcsec½l zš H Z and for Extreme Mooney material =>W=C 2 (I 2 3); m/2=c 2 H ¼ arccosðl zþ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð6þ l z 1 þ l 2 z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðzþ ¼l z 1 ð 1 þ l2 z ÞZ2 H 2 l 2 ; ð7þ z were l z is a stretc in vertical direction (Fig. 2), is alf of te current eigt of te sample (measured), H is alf of te initial eigt of te sample (known), r te current radius of te sample at elevation Z, R is te initial radius of te sample (known) and f ðzþ ¼r=R is te sape of te side of te deformed sample. Eqs. (4) (7) are different to corresponding equations for extension (Miller, 2001) because te implicit solution (valid for compression) used ere, Eq. (3), is different to te one used in te case of extension. To plot te deformed sapes for bot cases for a given displacement of te macine ead /H one as to compute numerically te vertical stretc in te plane of ð5þ symmetry l z from Eqs. (4) (6), and substitute to Eqs. (5) and (7), respectively. Fig. 3 sows te comparison of te deformed sape for different compression levels for tese two extreme cases. It can be seen tat despite apparent differences in te form of equations te actual deformed sape is almost te same. Te maximum difference in radius for / H=0.7 does not exceed 1%. From te perspective of testing biological materials, wic inerently exibit large variability of mecanical properties (see e.g. Estes and McElaney, 1970; Miller and Cinzei, 1997 for brain; Melvin et al., 1973; Farsad et al., 1999, for liver and kidney), tis difference in sape, and te resulting difference in te cross-sectional area are negligible. For practical purposes, I conclude tat te deformed sape of te cylindrical sample of incompressible biological material is insensitive to te form of te constitutive law defining its mecanical properties. Fig. 4 presents, for te two extreme cases, te relationsipbetween te vertical stretc in te plane of symmetry l z and te displacement of te macine ead /H, Eqs. (4) and (6). Even toug Eqs. (4) and (6) look complicated tey really describe, to ig accuracy, te same linear relationsip. Te vertical stretc in te plane of symmetry is proportional to te cange in total eigt, at least for /H between 1 and 0.7. l z ðz ¼ 0Þ 1 ¼ K c H 1 ; K c ¼ 1:411: ð8þ

5 ARTICLE IN PRESS K. Miller / Journal of Biomecanics 38 (2005) Fig. 4. Linear (for practical purposes) relationsip between te measured macine ead movement /H and te vertical stretc in te plane of symmetry l z ðz ¼ 0Þ for samples made of Neo Hookean and Extreme Mooney materials. 4. Discussion and conclusions Two teoretical results presented in tis paper ave important implications for testing in biomecanics of very soft tissues. As sown above, in te uniaxial compression experiment wit no-slip boundary conditions, in te plane of symmetry Z=z=0 (see Fig. 3) te ortogonal state of deformation can be assumed. Tis state of deformation can be described, as in te case of te unconfined compression experiment wit no-friction boundary condition (Miller and Cinzei, 1997), by a diagonal deformation gradient, Eq. (2). Terefore, te results of te uniaxial compression wit no-slip boundary conditions of cylindrical biological specimens can be analysed in analogous way to tat used in te unconfined compression no-friction boundary conditions: Unconfined compression wit no-friction boundary conditions (see Fig. 1) => l z ðz ¼ 0Þ ¼ H : Uniaxial compression wit no-slip boundary condition (see Fig. 2) => l z ðz ¼ 0Þ 1 ¼ K c H 1 ; K c ¼ 1:411: ð10þ To test ow te properties of tissue cange wit te speed of loading (strain rate) one would like to conduct series of experiments for various, but constant, nominal strain rates. Since l z is linearly related to /H, constant velocities of te macine ead =H=constant translate to constant stretc rates in te plane of symmetry l z ðz ¼ 0Þ: Tis is an important feature, wic allows equation for stress to be resolved analytically even for complicated forms of energy function W, used e.g. in quasi-linear, yper-viscoelastic constitutive laws first proposed for biological tissues by Fung (1981). ð9þ Te most common of tese energy functions are polynomials in strain invariants derived basing on Mooney s teory (Mooney, 1940; Mendis et al., 1995; Miller, 1999, 2000). Anoter important class is represented by Ogden-type energy functions in te form of powers of principal stretces (Ogden, 1972; Miller and Cinzei, 2002). For bot cases if ortogonal state of deformation can be assumed one can compute te only non-zero Lagrange stress component from te simple formula: T zz ¼ qw : ð11þ ql z Tis can be done analytically. Te explicit formula for stress in case of polynomial energy function is given in (Miller, 1999); and in te case of Ogden-type energy function in (Miller and Cinzei, 2002). Tese explicit, analytical formulas for stress togeter wit proven ortogonality of te state of deformation in te plane of symmetry of te sample allow analysing measurement results and identifying constitutive models in a straigtforward way. Te limitation of te proposed metod is tat it cannot be used for experiments at very ig strains. In suc case, at te circumference of te sample face attaced to te macine platens ig sear stresses would develop. Tis would result in violation of te assumption (3) tat te planes perpendicular to te direction of te applied force remain plane. Tis limitation does not affect te utility of te metod in very soft tissue biomecanics were te most interesting is te material response for strains of a few to about 20%. Acknowledgements I would like to tank Drs. Sopie Nigen and Guillaume Caidron from National University of Singapore for very elpful discussions. Te financial support of te Australian Researc Council is gratefully acknowledged. References ABAQUS Teory Manual, Version 5.2, Hibbit, Karlsson & Sorensen, Inc. Bilston, L., Liu, Z., Nan Pan-Tien, Large strain beaviour of brain tissue in sear: some experimental data and differential constitutive model. Bioreology 38, Estes, M.S., McElaney, J.H., Response of Brain Tissue of Compressive Loading, ASME Paper No. 70-BHF-13. Farsad, M., Barbezat, M., Flueler, P., Scmidlin, F., Graber, P., Niederer, P., Material caracterization of te pig kidney in relation wit te biomecanical analysis of renal trauma. Journal of Biomecanics 32 (4),

6 158 ARTICLE IN PRESS K. Miller / Journal of Biomecanics 38 (2005) Fung, Y.C., Biomecanics: Mecanical Properties of Living Tissues. Springer, New York, USA. Klingbeil, W.W., Sield, R.T., Large-deformation analyses of bonded elastic mounts. Z.A.M.P. 17 (20), Melvin, J.W., Stalnaker, R.L., Roberts, V.L., Impact injury mecanisms in abdominal organs. SAE Transactions , Mendis, K.K., Stalnaker, R.L., Advani, S.H., A constitutive relationsipfor large deformation finite element modeling of brain tissue. Trans. ASME. Journal of Biomecanical Engineering 117, Miller, K., Constitutive model of brain tissue suitable for finite element analysis of surgical procedures. Journal of Biomecanics 32, Miller, K., Constitutive modelling of abdominal organs. Journal of Biomecanics 33, Miller, K., How to test very soft tissues in extension. Journal of Biomecanics 34/5, Miller, K., Cinzei, K., Constitutive modelling of brain tissue; experiment and teory. Journal of Biomecanics 30 (11/12), Miller, K., Cinzei, K., Mecanical properties of brain tissue in tension. Journal of Biomecanics 35/4, Mooney, M., A teory of large elastic deformation. Journal of Applied Pysics 11, Nasseri, S., Bilston, L., Tanner, R., Lubricating squeezing flow: a useful metod for measuring te viscoelastic properties of soft tissues. Bioreology 40, Ogden, R.W., Large deformation isotropic elasticity on te correlation of teory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A. 326, Pamidi, M.R., Advani, S.H., Nonlinear constitutive relations for uman brain tissue. Transactions of ASME, Journal of Biomecanical Engineering 100, Prange, M.T., Margulies, S.S., Regional, directional, and agedependent properties of te brain undergoing large deformation. ASME Journal of Biomecanical Engineering 124, Saay, K.B., Merotra, R., Sacdeva, U., Banerji, A.K., Elastomecanical Caracterization of Brain Tissues. Journal of Biomecanics 25, Wals, E.K., Scettini, A., Calculation of brain elastic parameters in vivo. American Journal of Pysiology 247, R637 R700.