Elsevier Editorial System(tm) for Journal of Biomechanics
Manuscript Draft
Manuscript Number: Title: Constitutive characterization of fibre-reinforced biological soft tissues by using a uniaxial multideformation test machine Article Type: Full Length Article (max 3000 words) Section/Category: Keywords: Biomechanics, soft tissue, myocardium, mechanical test, experimental set-up, test machine, foetal Corresponding Author: Dr Ivano Izzo, MD, PhD Corresponding Author's Institution: Scuola Superiore Sant'Anna First Author: Ivano Izzo, Ph.D. Order of Authors: Ivano Izzo, Ph.D.; Selene Tognarelli, MD; Pietro Valdastri, MD, PhD; Paolo Dario, Full Professor; Arturo N Natali, Full Professor; Piero G Pavan, MD, PhD; Emanuele L Carniel, MD, PhD Manuscript Region of Origin: Abstract:
* Cover Letter
Dear Editor, The proposed work deals with the use of uniaxial experimental tests aiming to fully characterize the constitutive behaviour of fibre-reinforced soft tissues. It is known that the biaxial experimental tests were extensively used in the context of soft tissues, but two limitations occur: they are unsuited to identify constitutive parameters of strain energy functions equipped with three or more deformation tensor invariants, since they allow for only two independent mechanical tests; the electromechanical control of the biaxial test machine is complex to be implemented and managed. Obviously, the former limitation does not occur when the biological soft tissue under investigation is assumed isotropic (i.e. brain, liver, etc…), since, assuming valid the isochoric property, the strain energy function depends only on two invariants (i.e. first and second ones). Actually, many biological soft tissues must be considered to have an anisotropic behaviour because of fibres composing their micro-structure. Nevertheless, some examples in a cardiovascular context (Humphrey, Holzapfel, etc…) use strain energy functions equipped with only two invariants: typically the first one (concerning the isotropic part) and the fourth one (concerning the fibrereinforced part); so that the biaxial test machine is still enough to characterize them. The problem arises when it is necessary the use of a strain energy function more effective in describing the shear behaviour of fibre-reinforced biological soft tissue. In this case, indeed, it is necessary to use a functional dependence not only on the first and fourth invariants but also on the fifth one, with a consequence that the biaxial test machine becomes inadequate to characterize it. The latter limitation, together with the former one, leads us to develop a uniaxial test machine able to perform three independent mechanical tests on soft tissues. The validation of this experimental set-up has been accomplished by adopting a simple strain energy function, which is dependent on three invariants (first, fourth and fifth), and by characterizing the myocardium tissue as fibrereinforced biological soft tissue. The submitted manuscript intends to present main results of this investigation, also by showing a notable congruence among elastic module data carried out by our experiments on foetal lamb and those provided by literature experiments concerning other animal models (the canine and rabbit ones). Hence, the authors are confident that the proposed procedure may become a promising and more exhaustive methodology to investigate the biomechanics of fibre-reinforced soft tissues. Finally, the authors declare that all authors were fully involved in the study and preparation of the manuscript and that the material within has not been and will not be submitted for publication elsewhere.
* Conflict of Interest Statement
The authors declare that they have not conflicts of interest.
* Referee Suggestions
1. Prof. Humphrey, Jay D.,
[email protected], Texas A&M University, Department of Biomedical Engineering. 2. Prof. Miller Karol,
[email protected], University Western Australia, School of Mechanical Engineering, Intelligent Systems for Medicine Lab. 3. Prof. Weiss, Jeffrey A.,
[email protected], University of Utah, Department of Bioengineering.
Abstract
Abstract Common procedures enabling the identification of constitutive parameters defined for incompressible and fibre reinforced biological soft tissues usually adopt biaxial test machine. Biaxial testing is able to fully characterize a strain energy function dependent on the first (isotropic) and fourth (anisotropic) invariants of right Cauchy-Green strain tensor C, whereas the functional dependence on the fifth C-invariant, which plays an important role in shear test along fibre preponderant direction, is usually neglected. In the present work, a novel procedure employing multi-deformation uniaxial test machine has been proposed in order to characterize fibre reinforced biological soft tissue equipped with three C-invariants strain energy, as introduced by Merodio and Ogden (2005). The experimental equipment has been designed and developed applying a modular approach which allows the use of the same loading axes to perform three different kinds of test: compression, tensile and shear. The whole procedure has been validated on foetal lamb myocardium as biological soft tissue. Constitutive parameters of strain energy function, resulted from a standard identification process, have been compared with those available in literature for other animal models, such as the canine and rabbit ones. The application of the infinitesimal theory has allowed the recognizing of a monotonic trend of classical elastic modules of myocardium in dependence of the animal weights: for example Young’s modules along fibre direction E1 increases with a weight reduction (canine = 11.12 kPa < rabbit = 46.53 kPa < foetal lamb = 151.9 kPa), according to the corresponding increasing of hear rate. Although the effectiveness of no-slip boundary conditions of compression test may still be improved, the goodness of fit and the biological consistency of results lead to consider the proposed procedure as a promising methodology for soft tissue biomechanics.
* Manuscript
1. Introduction Continuum biomechanics is defined as the development, extension, and application of continuum mechanics for the purpose of a better understanding of the human condition, as well as human physiology and pathophysiology. The most important goal of continuum biomechanics is the formulation of constitutive relations describing the mechanical behaviour of a biological material under a loading state (Fung, 1993). An increasing interest of the biomechanics scientific community concerns the behaviour of soft tissues because of difficulties occurring in a non-linear constitutive characterization (Humphrey, 2003). In fact, important applications of soft tissue biomechanics can be found in recent patho-physiological and surgical contexts. For example, the measurement of the mechanical properties of biological tissues allows for the detection of the healthy status of a specific tissue, giving important clues about the condition of the whole organ (Phipps, 2005; Hayashi, 1997). Whereas, novel micro-robotic technologies, which arise in the field of mini or non-invasive surgery, require more predictive models of mechanical interaction between internal soft tissues and surgical micro-instruments (Stefanini et al., 2006). The biomechanical investigation of soft tissues requires a strict correlation between theoretical modelling and experimental measurements. The histological studies of soft tissues reveal the presence of a high microstructural hierarchy, highly spatially oriented components and liquid phases resulting in high complex tissue behaviour (Fung, 1993) that can be described by means of micro-structured and hyperelastic constitutive equations. The definition of such constitutive models is correlated with both the quality and the quantity of independent mechanical tests able to provide experimental data processed in the identification procedure of material parameters. Theoretical basis of correlation between tests and hyperelastic constitutive models can be found, for example, in Ogden (1972) and Criscione (2003). In particular, an isotropic hyperelastic and thin material can be fully characterized by performing a biaxial test (Sacks, 2000). In many cases of biological soft tissues, the hyperelasticity and the incompressibility are associated with the anisotropy due to the presence of fibres reinforcement (Weiss et al., 1996; Holzapfel, 2000; Natali et al., 2007). For this reason further tests, such as in-plane shear test, must be added to the biaxial one in order to fully characterize the non-linear and anisotropic behaviour. Important applications of such theories regard heart soft tissues, such as endocardium, myocardium and epicardium (Humphrey et al., 1990; Kang et al., 1996; Schmid et al., 2006). Due to the complexity in controlling the multiple boundary conditions of the biaxial set-up, the
uniaxial tensile tests have usually been performed on both isotropic (Miller, 2001) and anisotropic biological soft tissues (Holzapfel et al., 2005), where multiaxial and in-plane deformations are continuously monitored and measured with imaging techniques. Main limitation of the current uniaxial experimental apparatus is the impossibility to perform different and mechanically independent tests (only the standard tensile test is possible), which leads to a notable restriction on the completeness of constitutive equations to be characterized. In the present work, a new experimental apparatus is proposed in order to overcome the limitations of the current uniaxial mechanical test machines. The proposed measurement system allows for performing three different kinds of mechanical tests: compression, tensile and shear ones. The validation of procedure has been accomplished by testing a myocardium specimen and by fitting experimental data with a standard hyperelastic model (Merodio and Ogden, 2005), equipped with the three invariants of the right Cauchy-Green strain tensor C. Further, a comparison of obtained results with those reported in literature is also provided in terms of classical elastic modules.
2 Material and Methods 2.1 Experimental methods A new kind of uniaxial experimental set-up has been designed and fabricated to perform multiple mechanical tests on biological soft tissue samples, enabling the parameters identification of the hyperelastic, anisotropic and incompressible constitutive equation. Definition of the quality and the quantity of mechanical tests required for a complete constitutive characterization of soft tissues has been accomplished with both theoretical (Criscione, 2003) and numerical approaches (Natali et al., 2006). The latter identifies constitutive parameters of a hyperelastic and anisotropic material by performing a multiple fitting of stretch-stress data resulting from three-dimensional simulations of different mechanical tests, which were performed on thin specimen of hyperelastic material with known constitutive parameters. The best combination of uniaxial mechanical tests, in terms of fitting quality, consists of in-plane tensile test, through thickness compression and shear tests, in according to theoretical predictions. Thus, the proposed experimental equipment has been developed in order to exploit just uniaxial loading force in performing the aforementioned mechanical tests.
The general scheme of the experimental apparatus is shown in Fig. 1. The alternate current (AC) motor (ESCAP 16N28) is mounted on a uniaxial slide which is fixed to the basement by means of a load cell (SENSOTEC Model 11). Two slim carbon wires (0.1 mm in diameter) are rolled up onto a pulley integral with the shaft of motor. The other ends of carbon wires are mechanically fixed on two clamps constrained to move along a same couple of parallel guides (not shown in Fig. 1) collinear with the motor slide. Thanks to four transversal guides, the carbon wires move away the clamps when motor is switched on and roll onto the pulley. In such a way, the tissue sample mounted between clampers is deformed opposing a mechanical resistance which is transferred to the load cell through the whole kinematics chain. The signal measured by the load cell is acquired by an acquisition board (NI6062E, National Intruments) installed on a personal computer (PC), and the AC motor is driven by a proper hardware (MCDC2805, Faulhaber) controlled by PC through a serial cable. To preserve the mechanical tone of the tissue sample, it is maintained into a physiological solution during both clamping and testing steps. The temperature of the physiological liquid is kept constant to 37° C by means of a thermostat system. Finally, an optical microscope (Power Hiscope KH2700 from Hirox, Japan) is used in monitoring the specimen deformation. The video acquisition of that deformation process is performed by a frame grabber device (IMPERX VCE ANCB01) installed on the PC and connected to the microscope. Calibration procedure of the experimental equipment prescribes five tests obtained by mounting a linear spring (characteristic constant Ks=0.072 N*mm-1 ) between the two clamps. Assuming the hypothesis of friction linearity and constant calibration velocity (2.7 mm*s-1), the measured force can be correlated with the actual resistance force (Fr ) by means of a constant friction coefficient g. In the fitting process, Fr has been obtained as the product of the spring constant Ks and displacement d: Fr(v,t)= Ks *v*t= Ks *d. The averaged value and the standard deviation of the g fitted values are 2.087 and 1.701% respectively. The variety of mechanical tests is guaranteed by means of disposable tips properly developed for each kind of test (Fig. 2). All tips are glued on the plane surfaces of the tissue sample and mechanically constrained on two clamps (Lepetit, 2004; Miller, 2001). Due to its complexity, the compression test requires the use of two auxiliary tips able to invert the loading direction (from pulling to pushing the sample) and to transfer the strength itself from the flat disposable tips to the clamps. Disposable tips have been developed also by tacking into account the specimen shapes (Fig. 2). The shape of thin specimen has been designed to
guarantee the occurrence of a homogeneous deformation state during the uniaxial loading. Hence, the shape of strip used in tensile test has a narrow central part where the in-plane deformation will be optically measured. On the other hand, simple circular and rectangular shapes have been designed for specimens used in compression and shear tests, respectively. The validation of the proposed experimental procedure deals with the multiple testing of a thin, anisotropic biological soft tissue. The recent developments in foetal cardiac surgery (Crucean et al., 2005) lead the scientific community to put more attention on biomechanical study on foetal soft tissues. For this reason, the experimental tests have been performed on specimens excised from heart of a foetal lamb with a weight of 2.8 kg at 17÷19 week’s gestation (125 days). A particular attention has been paid to cut biological samples from the whole foetal heart. First, a whole heart has been removed from a foetal lamb and it has been cut along the lengthways of the interventricular septal part. Since the foetal heart has a little size (1÷2 cm in diameter), both the inner (endocardium) and the outer (epicardium) layers have been removed and only the myocardial portion of the cardiac wall have been used to get samples. Due to the compliance and the superficial viscosity of the cardiac tissue, specific “cutting stamps” in Ergal have been designed and fabricated (Fig 3a). The cutting procedure consist of two steps: first the stamp is placed and pressed onto the spread tissue, second the biological sample is cut out by cutting the tissue along the stamp edges with the use of a scalpel (Fig 3b). 2.2 Constitutive modelling The passive mechanics of the myocardium tissue was already extensively studied as a soft tissue equipped with hyperelastic, isochoric and anisotropic properties (Humphry et al., 1990a-b). The constitutive equations used in aforementioned works express the strain-energy function W as a sub-class of transversally isotropic and hyperelastic model, i.e. dependent only on the first I1 and the fourth I4 invariants of right-Cauchy deformation tensor C. The capability of the new experimental set-up to perform three independent mechanical tests allows the characterization of a more general constitutive model, where also the functional dependence of W from the fifth invariant I5 is taken into account (Merodio and Ogden, 2005). A simplest form of that W function can be formulated as: W W iso I1 W f I 4 , I 5
1 1 I1 3 I 4 12 I 5 12 2 2
(1)
being Wiso and Wf the isotropic and the fibrous part of the strain-energy function, and , the constitutive
parameters which will be identified by applying a fitting procedure on experimental data. It is assumed that the fibrous part Wf of Eq. (1) vanishes for deformation in which the forth and the fifth invariants are lower than 1. A standard fitting procedure consists in implementing a Marquardt-Levenberg type algorithm in order to find the values of , which are able to minimize the-sum-of-the-square of residual error between the measured nominal stress and the corresponding theoretical one (first Piola-Kirchoff stress tensor) evaluated as: P pFC1 2F
W pFC1 FI 2 I 4 1M M 2 I 5 1M CM MC M C
(2)
being M the unit vector of the fibers direction, and p the kinematics pressure or Lagrangian multiplier enforcing the isochoric material constraint. Beside the unknown constitutive parameters to be identified, other quantities in Eq. (2), such as C and p, have to be measured and/or calculated by boundary and/or isochoric conditions. In the following, quantities to be measured and calculated are detailed for each mechanical test. Assuming a reference Cartesian coordinate system, whose associated vector space has a basis denoted by (i1, i2, i3), soft tissue specimens are considered oriented so that the preferred fiber direction is parallel to i1 {i.e. (Mi) = (1,0,0)T}, and the mean plane of thin sample is contained in the plane (i1, i2) for compression and tensile tests and in the plane (i1, i3) only for through-thickness shear test. Moreover, during all three mechanical tests it is assumed that the specimen (or a part of it) is homogenously deformed and the relevant C components C13, C23, C31 and C32 are null. In Table 1, a summary of other components of C and P, and the principal C-invariants are collected for each mechanical test. Note that the overline denotes kinematics and stress quantities that have been measured; furthermore, the isochoric condition has been used to evaluate the unknown stretches: 2 for compression and 3 for tensile tests. The boundary conditions reported in Table 1 with a “grey background” are applied to the corresponding stress components of Eq. (2) in order to calculate the pressure values for each test as: ~ p C 12 32 ; p T 12 22 ; pS .
(3)
~ ~ ~ Moreover, the three quantities denoted as 1 , P22 and P11 (“overtilded” quantities in Table 1) must be
calculated for compression, tensile and shear mechanical tests, respectively. While in one hand, the unknown ~ ~ stress components P22 and P11 can be calculated at the end of the identification process by using Eq. (2), in
~ the other hand the unknown stretch 1 appears explicitly, together with the unknown constitutive parameters
, to be identified, in the residual error function of the fitting procedure:
NC ~ f P33,i 3,i 1 1,i2 3,i4 i 1
NS
P12,k ,k 1 k 1
2
NT
P11, j 1,i 1 1, 4j 2,2j 2 12, j 1 2 12, j 12, j 1 j 1
2
(4)
2 2 ,2k
where i, j and k are the enumerators of physical data sampled during compression, tensile and shear test, respectively, whilst NC, NT and NS are the corresponding total number of data samples. So that, for each trial ~ values (’, ’ ’), which is processed in the optimization procedure, the stretch values 1,i must be
calculated as the root of the mathematical expression of stress component P11,i evaluated for the compression test (Merodio and Ogden, 2005):
~ ~ ~ ~ ~ P11,i ' 1,i 1 1,i43,i2 2 12,i 1 '2 ' 12,i 12,i 1 0
i 1,2,...N C .
(5)
3 Results The values of constitutive parameters are collected in Table 2 together with the R-square evaluated for all 2 tests ( Rtot ) and for the single ones (compressive RC2 , tensile RT2 and shear RS2 ). High values of R-square
(the minimum one is evaluated for the compression test and it is equal to 0.9146) are also confirmed by analysing both experimental and predicted data sets, plotted in Fig.4. Measurements of hyperelastic parameters of foetal myocardium are still missing in literature, but comparable data can be found for canine (Novak et al., 1994) and rabbit (Kang and Yin, 1996) animal models. Both of them use hyperelastic, isochoric constitutive model introduced by Humphry et al. (1990a), which involved only the first and fourth invariants of C. Direct comparison between the constitutive parameter values obtained by the aforementioned model and the model described by Eq. (1) can be accomplished by applying the infinitesimal theory, which is able to provide a classical elastic modules, such as shear Gij, Young’s modules Ei and Poisson’s ratios ij, in terms of hyperelastic parameters of whatever energy function W (Merodio and Ogden, 2005). Both formal and experimental values of classical elastic modules are reported in Table 3 for canine and rabbit myocardium, together with those calculated for foetal lamb myocardium by
using constitutive hyperelastic parameters , here evaluated. Those formal expressions are obtained by following method described by Merodio and Ogden (2005) and considering i1 as predominant fibre direction. The calculated kinematics and stress quantities of the three mechanical tests are plotted in Fig. 5. Decreasing ~ trend of 1 against 3 (Fig. 5a) corresponds to an analogous behaviour theoretically predicted in the Fig. 3
of Merodio and Ogden (2005). Further, this trend confirms, in a linear approximation, the value of Poisson’s ratio 31=0.1223 since the orthogonal stretch '3
0.1223
1.027 , corresponding to ' 3 0.8 , is very similar
~ to the measured one '1 1.03 . Violation of the theoretical Poisson’s ratio, which should be 0.5 according to
the isochoric deformation hypothesis, can be noted by analysing Fig. 5b: lateral contraction in the cross-fibre direction '2 '1 1.2 0.9698 is lower than what expected by the theory of transversally isotropic material
'1 1.20.5 0.9129 . This violation is due to the experimental inability of clamping the two ends of the thin specimen strip without constraining deformations in cross-fibre direction, with the consequent ~ occurrence of a non-null tensile stress P22 . The Poynting relation of the isotropic theory, expressed in terms ~ of first Piola-Kirchoff stress tensor components, expects the zero-value of P11 , so that the analysis of Fig. 5c, ~ where a non-zero-value of P11 is occurred, leads to a violation of that relation, as well as theoretically
predicted by Merodio and Ogden (2005).
4. Discussion and conclusion 2 The overall quality of the fitting procedure, expressed by Rtot , is sufficiently high (0.9946) to conclude that
the calculated constitutive parameters are the effective minimization point of the cost function (4). The analysis of R-square calculated for each mechanical test, together with the analysis of experimental observations, allows for some considerations related to the quality of tests. The compression test accounts for an R-square sensibly lower (0.9146) than those of tensile (0.9937) and shear (0.9809) tests. From the experimental viewpoint, the vertical position of the sample mean plane requires the use of surgical glue to fix the same sample onto disposable tips as shown in Fig.2, by introducing a no-slip (constrained) condition transversally to the uniaxial loading direction (Miller, 2001 and 2005). On the other hand, the impossibility to measure the transversal constrained force (the disposable tips are not sensorized) leads to superimpose a
null stress state in that transversal directions in order to calculate the Lagrangian pressure pC and the ~ transversal deformation 1 (see Eq. 3-5). Hence, these two experimental and theoretical requirements are
opposite and incompatible. A theoretical solution could have been to use a no-slip condition applied onto both the sample circular surfaces and to evaluate the deformation tensor F, accounting for the transversal shear components, by adapting the isotropic solution presented in Miller (2001 and 2005) to the transversally isotropic case, but this would have been out of the scope of present work. Hence, an approximated solution has been adopted by assuming a zero transversal stress state, but relaxing the no-slip condition only on one of the two sample surfaces. This approximation can be considered the main reason for the lower fitting quality of the compression test. Future development of the proposed uniaxial test machine will implement a compression test condition with early zero friction, such as that proposed by Nasseri et al. (2003). The goodness of fit reported in Table 2 is not the only factor considered in discussing the results of present work, but the analysis of data collected in Table 3 and their comparison with corresponding data here calculated are also important in order to validate whole procedure from a biological viewpoint. A first remark rises from the comparison between canine and foetal lamb modules. Both shear and Young’s modules roughly increase of one order of magnitude going from canine to the foetal lamb model, whereas the Poisson’s modules are roughly the same. This can motivated from a biological viewpoint by considering that dogs used in Novak et al. (1994) weight roughly 20 Kg against 2.8 Kg of foetal lamb. In fact, a notable reduction of the animal weight corresponds to an increase of heart rate and, consequently, to an increase of stiffness of heart tissues needed to be adequately resistant against the increased loading conditions. This consideration seems to be confirmed by data from rabbit model: assuming a rabbit weight in between that of dog and foetal lamb (the paper from Kang and Yin doesn’t specify this kind of data), the Young’s module E1 has, instead, the same bounds (dog=11.12 < rabbit=46.53 < foetal lamb=151.9). But, the rabbit data suffer of a singular trend: all classical modules whose theoretical expression containing parameter c3 as multiplicative factor are anomaly lower (i.e. Gij=0.2325 vs. 1.175) than those of canine model calculated with the same formula, also considering that an opposite relation occurs for E1, whose formula doesn’t contain c3 as factor. This deviation is confirmed also by comparing the ratio c1/c3 between canine and rabbit models: for the former is roughly 6.5, whereas the latter is about 197. This could be explained considering that the only c3
data reported in Table 1 of Kang and Yin was expressed in kPa instead of in g/cm2 as it was did for the other ones. In that case, the new values of classical modules are: Gij=4.563 kPa; E1=59.52 kPa; E2= E3=17.26 kPa; 21=31=1.424*10-1; 23=32=8.576 *10-1 which are fully congruent with canine data, thus confirming the biological validity of monotonic trend of elastic modules with animal weight. The present investigation has defined a systematic procedure aiming at fully characterize an anisotropic, hyperelastic, biological soft tissue by exploiting three different uniaxial mechanical tests: tensile, compression and shear. Uniaxial mechanical tests, in fact, provide significant advantages, in terms of usability and ease of control, compared with the biaxial one. Furthermore, the proper modular design of the proposed experimental apparatus allows overcoming main disadvantages of uniaxial test machine, relating to the ability to perform the only tensile test. The strategy adopted in the present work has allowed characterizing myocardial tissue of foetal lamb by identifying a hyperelastic constitutive equation more general than those commonly used in literature. Actually, three different mechanical tests have properly supported the calculation of constitutive parameters of a transversally isotropic strain energy function, depending not only on the first and fourth C-invariants but also on the fifth one (Merodio and Ogden, 2005). Also, a direct comparison between constitutive parameters here calculated with those of literature concerning myocardial tissues from other animal models, has provided a biological validation of our results. The experimental approximation adopted for the compression test, can be highly reduced or totally eliminated in further developments of the set-up. However, despite this approximation, the mentioned features of the proposed procedure lead to consider it as a new and promising methodology in the context of soft tissue biomechanics. Acknowledgments The present work was supported by the Fondazione Cassa di Risparmio di Pisa, in the framework of the “microSURF” project for basic research on foetal heart tissue biomechanics applied to develop innovative technologies in foetal surgery. The authors wish to thank Dr. S. Burchielli and Dr. N. Funaro for their valuable assistance in providing foetal lamb tissues and in supporting the sample preparation.
Reference
Criscione, J., 2003. Rivlin's representation formula is ill-conceived for the determination of response function via biaxial testing, Journal of Elasticity 70, 129-147. Crucean, A., Murzi, B., Giorgi, A., Burchielli, S., Trivella, M., Coceani, F., 2005. Cardiopulmonary bypass in ewe's fetus: Advances and setbacks in our learning curve, Asaio Journal 51 (5), 649-653. Fung, Y.C., 1993. Biomechanics: Mechanical properties of living tissues (2nd ed.), New York: SpringerVerlag. Hayashi, K., Imai, Y., 1997. Tensile property of atheromatous plaque and an analysis of stress in atherosclerotic wall, Journal of Biomechanics 30 (6), 573-579. Holzapfel, G.A., Sommer, G., Gasser, C.T., Regitnig, P., 2005. Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling, American Journal of Physiology-Heart and Circulatory Physiology 289 (5), H2048-H2058. Holzapfel, G.A., 2000. Nonlinear Solid Mechanics: A Continuum Approach for Engineering, John Wiley & Sons, LTD. Humphrey, J.D. Strumpf, R.K., Yin, F.C.P., 1990a. Determination of a constitutive relation for passive myocardium.I. A new functional form, Journal of Biomechanical Engineering-Transactions of the ASME 112 (3), 333-339. Humphrey, J.D. Strumpf, R.K., Yin, F.C.P., 1990b. Determination of a constitutive relation for passive myocardium.II. Parameter estimation., a new functional form, Journal of Biomechanical EngineeringTransactions of the ASME 112 (3), 340-346. Humphrey, J.D., 2003. Continuum biomechanics of soft biological tissues, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 459 (2029), 3-46. Kang, T., Humphrey, J.D., Yin, F.C.P., 1996. Comparison of biaxial mechanical properties of excised endocardium and epicardium, American Journal of Physiology-Heart and Circulatory Physiology 39 (6), H2169-H2176. Kang, T., Yin, F.C.P., 1996. The need to account for residual strains and composite nature of heart wall in mechanical analyses, American Journal of Physiology-Heart and Circulatory Physiology 40 (3), H947-H961. Lepetit, J., Favier, R., Grajales, A., Skjervold, P.O., 2004. A simple cryogenic holder for tensile testing of soft biological tissues, Journal of Biomechanics 37 (4), 557-562.
Merodio, J., Ogden, R., 2005. Mechanical response of fiber-reinforced incompressible non-linearly elastic solids, International Journal of Non-Linear Mechanics 40, 213-227. Miller, K., 2001. How to test very soft biological tissues in extension?, Journal of Biomechanics 34 (5) 651657. Miller, K., 2005. Method of testing very soft biological tissues in compression, Journal of Biomechanics 38 (1) 153-158. Nasseri, S., Bilston, L., Tanner, R., 2003. Lubrificated squeezing flow: a useful method for measuring the viscoelastic properties of soft tissues, Biorheology 40 (5), 545-551 Natali, A., Carniel, E., Pavan, P., Dario, P., Izzo, I., Menciassi, A., 2006. Hyperelastic models for the analysis of soft tissue mechanics: definition of constitutive parameters., IEEE/RAS- EMBS International Conference on Biomedical Robotics and Biomechatronics.. Natali, A.,N., Carniel E.,L., Pavan, P.G., Bourauel, C., Ziegler, A., Keilig, L., 2007. Experimental-numerical analysis of minipig’s multi-rooted teeth, Journal of Biomechanics 40, 1701-1708. Novak, V.P., Yin, . F.C.P., Humphrey, J.D. 1994. Regional mechanical properties of passive myocardium, Journal of Biomechanics 27 (4), 403-412. Ogden, R., 1972. Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids., Proc. R. Soc. Lond. A 326, 565-584. Phipps, S., Yang, T.H.J., Habib, F.K., Reuben, R.L., McNeill, S.A., 2005. Measurement of tissue mechanical characteristics to distinguish between benign and malignant prostatic disease, Urology 66 (2), 447-450. Sacks, M.S., 2000. Biaxial mechanical evaluation of planar biological materials, Journal of Elasticity 61 (13), 199-246. Schmid, H., Nash, M.P., Young, A.A., Hunter, P.J., 2006. Myocardial material parameter estimation - a comparative study for simple shear, Journal of Biomechanical Engineering-Transactions of the ASME 128 (5), 742-750. Stefanini, C., Menciassi, A., Dario, P., 2006. Modeling and experiments on a legged microrobot locomoting in a tubular, compliant and slippery environment, International Journal of Robotics Research 25 (5-6) 551560.
Weiss, J.A., Maker, B.N., Govindjee, S., 1996. Finite element implementation of incompressible, transversely isotropic hyperelasticity, Computer Methods in Applied Mechanics and Engineering 135 (1-2), 107-128.
Figure(s)
Fig. 1. General scheme of experimental apparatus. The basement is fabricated by machining a thick Teflon sheet. Two clamps are free to move onto one couple of stainless steel guides, by means of a couple of Teflon bearings for each clamp. The steel guides and transversal guides (made by Teflon material) are fixed onto four lateral walls of physiological bath. Both clamps and disposable tips are fabricated by rapid prototyping with the use of an acrylic resin.
Fig. 2. Tissue-clamping mechanisms. Each mechanism prescribes specific shape and dimensions of the relevant tissue sample: the traction sample has a rectangular shape (overall dimension 40x20 mm2) with a central taper (10x10 mm2) (a); the shear sample has a rectangular shape (20x10 mm2) (b); the compression sample has a circular shape (diameter = 10 mm) (c). In the latter case, the disposable tips have to be mounted onto the respective auxiliary tips (made by rapid prototyping), by matching the pairing sites as shown in (c).
(a)
(b)
Fig. 3. Cutting protocol of tissue samples. The three cutting stamps shown from left to right in (a), correspond to shear, mono-axial tensile and compression tests, respectively. (b) The heart tissue is spread on a Teflon plate, the specific cutting stamp is pushed against tissue and the sample is finally cut out.
(a)
(b)
(c) Fig. 4. Nominal stress vs. stretch curves of experimental and fitted data, corresponding to compression (a), tensile (b) and shear (c) tests.
(a)
(b)
(c) Fig. 5. Unknown kinematics and stress quantities calculated with the use of equations (2) and (5), and constitutive parameters resulted by curve fitting procedure, and corresponding to compression (a), tensile (b) and shear (c) tests.
Table(s)
Table 1: Kinematics and dynamics parameters of different mechanical tests
Tests Compression Tensile Shear
C11 ~ 12
C12 0
12
0
1
C22
~ 12 32
22
1+
2
C33
~ ~ 12 12 32 32
I4
I5
32
12
14
12 22
12 22 12 22
12
14
1
I1
3
2
1
1+
P11 0 2
P11 ~ P11
P12 0
P22 0
0
~ P22
P12
0
P33 P33
0 0
Table 2: Constitutive parameter values and R-square fitting errors resulted from the identification procedure.
(kPa) 9.895
1.587
0.3752
2 Rtot
0.9946
RC2
RT2
RS2
0.9146
0.9937
0.9809
Table 3: Comparison among classical elastic modules calculated by fitting experimental stress-strain curves with the modelling predicted ones. The ci parameters of Humphry’s constitutive model have been calculated as: the mean value of the best-fit ones reported in table 2 of Novak et al. (1994) with concern the canine animal model; the mean value of the group C ones reported in table 1 of Kang and Yin (1996) with concern the rabbit animal model. Poisson’s modules 12 and 13 are omitted because of both equal to 0.5 for every isochoric constitutive models, whereas shear modules Gij are independent from indices I and j ranging from 1 to 3.
Gij (kPa) E1 (kPa) E2=E3 (kPa) 21=31 23=32
Humphry’s model Theoretical Canine 2c3= 1.175 2(c1+3c3)= 11.12 8c3(c1+3c3)/(c1+4c3)= 4.251 2c3/(c1+4c3)= 1.911*10-1 (c1+2c3)/(c1+4c3)= 8.088*10-1
Rabbit 2.325*10-1 46.53 9.256*10-1 9.945*10-3 9.900*10-1
Merodio-Ogden’s model Theoretical Fetal lamb 9.895 151.9 37.16 -1 0.5 1.223*10 -1 8.777*10
