Human Tissue Hyperelastic Analysis

Transcription

1 Human Tissue Hyperelastic Analysis João de Sousa Crespo a a Instituto Sperior Técnico, Lisboa Portugal 1. Abstract Various mathematical models have been made to accurately describe the human skin behaviour, using numerous different approaches and hypotheses (behaviour laws as well as mechanical tests).many times the results achieved are complicated and cannot be applied in real life. In order to present a model that can assist plastic surgeons prior to their surgeries, we will describe a model for the human skin, where some simplifications, based on literature, were made but with no loss of accuracy. It is stated in literature that the collagen fibbers in the dermis are largely responsible for the nonlinear mechanical stress-strain response of skin, so we will consider only one layer of the skin, where collagen fibbers are responsible for the nonlinear elasticity of skin whereas the elastin fibbers are responsible for the prestressed phenomenon. Our model considers the skin to be a homogeneous isotropic elastic quasi-incompressible medium, where both the viscous response and anisotropy behaviour are neglected. The deformation processes are restricted to quasi-static processes under axisymmetrical conditions. Thus, no inertial effects are accounted. The model we propose is derived from the Yeoh form. In the end our results will be compared with those present in published papers in order to be validated. After careful analysis we concluded that the proposed hyperelastic model was accurate to predict human skin behaviour. Both of our models presented results accordingly to the ones present on published papers, although model 1 should only be used when small strains are applied. 2. Introduction The skin is the largest organ in the human body in terms of surface area. There are about two square metres of it covering an average adult and it accounts for about an eighth of the total body weight [BBC-h2g-Human skin]. So it is clear how important it is to know its mechanical properties, in order to identify certain diseases, for assessing therapeutic intervention, or for predicting the effect of trauma. There are two distinct parts of the skin, known as epidermis and the dermis and a third layer, the hypodermis or subcutaneous tissue. The part we can see and touch is the epidermis, beneath it is the dermis that consists of connective tissue and cushions the body from stress and strain, and finally the third and deepest layer is known as the hypodermis composed mainly of fat with the purpose to attach the skin to underlying bone and muscle. According to the literature, we can simplify the skin model, and assume the dermis to be the principal structure contributing to the mechanical properties of skin. Some studies propose to analyze each structure of the dermis (collagen fibres, elastin and matrix of proteoglycan), but to achieve a reliable behaviour we must take all of the structures simultaneously into account. As stated on [Delalleau et al. (2007)] the main component of the dermis are the collagen fibbers, (80% of the dermis dry weight), which orientate in the direction of the stress when subjected to small strains. In this case, the resulting low stiffness is mainly due to the elastin fibbers. For large strains, both collagen and elastin fibbers are stressed, and then the high elastic modulus of the collagen fibbers modifies the elastic response of the skin, which becomes stiffer. This behaviour is shown in Fig.1 and is shown by our mathematical model. 1

2 Fig.1- Nonlinear elasticity due to the orientation of the collagen fibber Many mechanical laws, which usually require complex backgrounds, have been analyzed to model soft tissues behaviour, some focused on specific behavioural aspects of the tissue, such as viscoelasticity or nonlinear elasticity, while other studies have shown that the quasistatic viscoelastic effects could only be appreciable at nonphysiologic stress levels. Even if these formulations are based on physical assumptions their related parameters are generally difficult to analyze and identify (time-consuming algorithms, uniqueness of the solution, etc.). To circumvent this problem, many simplifications have been proposed in the literature with the aim of deriving a simple model that can be applicable and still be able to represent the human skin behaviour in a trust worthy way. With these simplifications the mechanical parameters needed to achieve a accurate model, can be found by performing in vivo test such as suction test [Delalleau et al. (2007)], or using an extensometer [Jeffrey E. Bischoff et al (2000)]. We will explain how our model was developed next, present some of the results found, then we will present a discussion where the results are compared to published data, and finally we will draw some conclusions regarding the accuracy of our model. 3. Model Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. In a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modelled assuming the materials exist as a continuum. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.in this work it was used a Lagrangian description, since it is often preferred in elasticity, where it is assumed that there exists a natural state to which the body would return when it is unloaded. To obtain a proper model for the characterization of the human skin behaviour, we must relate the laws of physic in a reference configuration and the current configuration. To do so we started by using the Piola-Kirchhoff stress tensors to express stress and derive the equations of motion relative to the reference configuration as opposed to the Cauchy stress tensors that express stress relative to the actual configuration. The First Piola- Kirchhoff stress tensor,p relates forces in the actual configuration with areas in the reference configuration r r df = PNdA r =σ n o da t as for the second Piola-Kirchhoff stress tensor, S, it relates forces in the reference configuration to areas in the reference configuration SN da o = F 1 df these stress tensors can then be related to the Cauchy stress by and σ = σ = det det 1 T [ F] [ F] PF. 1 T FSF.. 2

3 It is vital that all physical restrictions, on the form of constitutive relations, obey the second law of thermodynamics in order to be applied. One of the ways to do so is using the Clausius-Duhem inequality elastic-plastic materials. It states the irreversibility of natural processes, especially when energy dissipation is involved and can be represented in the reference configuration by r 1 q S. C& o ρ ( Ψ+ & Ts & o ). rt 0 X 2 T where S is the second Piola-Kirchhoff stress tensor, is the material derivative of the Helmholtz free energy potential, T and T & are the absolute temperature and its material derivative respectively. Having this in mind we then studied the Hyperelastic materials. A hyperelastic or Green elastic material is an ideally elastic material for which the stress-strain relationship derives from a strain energy density function. The stress in each moment depends only on the strain on that particular moment and not on the history of the strain. To obtain our Hyperelastic model that describes the skin behaviour, we started by studying and modifying the Yeoh model, which is a special case of the reduced polynomial with N = 3, given by W (I 1,I 3 2,J) = i=1 C io (I 1 3) i i=1 (J 1) 2i D i in which I 1 = tr[b iso ] =tr[c iso ] and I 2 = 1 2 (I 1 ) 2 2 tr B iso = 1 2 (I 1 ) 2 2 tr C iso. The reason for choosing this model is due to the fact that it considers two coefficients that can accurately describe the human skin behaviour (see Fig 1). The coefficient C 10 will represent the initial shear modulus, which is followed by upturn at large strains due to the positive third coefficient C 30, thus providing an accurate fit over a large strain range. After this, and considering skin as a homogeneous isotropic elastic quasi-incompressible medium, where the viscous response and anisotropy behaviour are neglected and the deformation processes restricted to quasi-static processes under axisymmetrical conditions, we get S = 2J 2 3 C 1o + 3C 3o (I c 3) 2 I which is the case related to K(J 1) or + K(J 1)J 2 3 J 2 3I c C 1o + 3C 3o (I c 3) 2 C 1 S = 2J 2 3 C 1o + 3C 3o (I c 3) 2 I + K(ln[J]) 2 3 J 2 3I c C 1o + 3C 3o (I c 3) 2 C 1 that is the case related to ln[j]. To assure the accuracy of our model we used a finite element model composed of 232 elements with 531 nodes. The elements used were triangular with six nodes, each of them having a height of 0.764mm and a length of 1.69mm. Our axial model had a total height of 3.056mm and a total length of 49.01mm. The finite element model applied simulated a axysimetric analysis as well as a suction test, similar to those present in published papers, from witch we collected some data, and 3

4 compared our results to. Our calculations were made on a fortran program written by Prof. Marcelo Alves, and then plotted on a postprocessor program. In Fig.2 an axial representation is shown where the total height and length of our model is shown. Fig. 2- Axial suction test 4. Results The first test we made was to see if our two models could represent the nonlinear elasticity due to the orientation of the collagen fibbers. This test was done by subjecting our models to axysimetric displacement, to see how the elastin and collagen fibbers behaved when subjected to high strains, and if we could identify the collagen stiffness, using the third phase slope proposed in our models. In Fig. 3,model 1 we see that there is a very small increase of the stress until 20% of the total displacement is reached. After this point, an almost linear increase is observed until 60% of the displacement, which is in agreement with published papers [Delalleau et al.(2007)], but for larger displacements we observe that the stress tends to reach a maximum, which is not coincident with human skin behaviour. As for model 2, we observe that initially a very small increase of stress occurs but now until 30% of the total displacement, contrary to the 20% observed in model 1. From here on the skin behaves almost linearly, which is coincident to published data concerning human skin behaviour, as seen on [Delalleau et al.(2007)]. With these results, we can conclude that model 1 is accurate describing human skin but for small displacements only. One the other hand model 2 is a more accurate model that can be used even when large displacements are imposed 4

5 Fig. 3- Stress versus increments of time t (model 1) Fig. 4- Stress versus increments of time t (model 2) In Fig. 3 and Fig. show the evolution of the stress with time. The applied displacement (x) was a linear function of time (t) given by x = 0.02*t (the total applied displacement applied was 20mm for t=1).next a suction test was simulated where a circular patch of skin with a radius of approximately 50mm,was submitted to a 100mbar pressure applied on the inner half of the skin path. With this test we wanted to see if our proposed models, were accurate and if our results were similar to those observed in [Delalleau et al.(2007)] and [L.M. Tham et al. (2006)]. 5

6 Fig. 5- Suction test. (a) Initial boundary and loading conditions. (b) Suction dome and deflection Again in this test both models were tested, our intention was to study how skin would respond when subjected to pressure in a suction chamber. In model 1, Fig. 6 and Fig 7, both displacement and Von Mises stress, were in orders quite similar to those found in published papers namely [L.M. Tham et al. (2006)]. The development of the displacement with pressure, Fig. 8, presented a curve similar to those present in [L.M. Tham et al. (2006)] and [Delalleau et al.(2007)]. Fig. 6- Displacement in Y direction when subjected to pressure and no slip is considered (model 1) 6

7 Fig. 7- Von Mises stress when subjected to pressure and no slip is considered (model 1) Fig. 8- Evolution of the Y displacement with pressure when no slip is considered (model 1) The same analysis made for model 1 was also made for model 2. The results reached were suitable due to the similarity to the results present in published data. Concerning the maximum deformation, Fig. 9, we reached a value slightly smaller but very close to the one observed in model 1. As for the Von Mises stress, Fig. 10, the result observed in model 2 was again very close to the one observed in model 1. Finally relatively to the evolution of the Y displacement with pressure,we observed that the curve of our model, represented by Fig. 11, was alike to that observed in the papers [L.M. Tham et al. (2006)] and [Delalleau et al.(2007)]. It is also worth mentioning that in both model cases, the maximum displacement and Von Mises stress were observed in the centre area. 7

8 Fig. 9- Displacement in Y direction when subjected to pressure and no slip is considered (model 2) Fig. 10- Von Mises stress when subjected to pressure and no slip is considered (model 2) 8

9 Fig. 11- Evolution of the Y displacement with pressure when no slip is considered (model 2) Next we present a table, Tab. 1, where the results observed are shown. Model 1 Model 2 Maximum Y. Displacement Von Mises stress Conclusion In our experiment we started by developing a mathematical formulation, represented by constitutive equations, where we tried to accurately represent the human skin behaviour. We then ran our model and plotted it in a postprocessor program, from where our results were presented. Our model was based on the Yeoh model for Hyperelastic materials. The reason for choosing this model is due to the fact that it considers two coefficients that can accurately describe the human skin behaviour (see Fig 1). The coefficient C 10 will represent the initial shear modulus, which is followed by upturn at large strains due to the positive third coefficient 30 C, thus providing an accurate fit over a large strain range. After analyzing our results we can see that both models have similar results when subjected to pressure. These results are according to the ones present in published works like [L.M. Tham et al. (2006)], although model 1 should only be used when small strains are studied. When analyzing simple axysimetric displacement simulating the response of the collagen fibbers, we observed that model 1 was only accurate for small strains. As for model 2 it resembled the nonlinear elasticity due to the orientation of the collagen fibbers present in [Delalleau et al.(2007)] leading us to think that this model can be used in a wider variety of tests. After this test we preformed a suction test, where we tried to understand if our models were accurate when applying pressures of 100 mbar. After performing this test and comparing it to the results present in [L.M. Tham et al. (2006)], we observed that our models offered results similar to the ones on the published paper. The values of displacement as well as the curve representing the development of the displacement with pressure were also alike to those present in [L.M. Tham et al. (2006)]. Provided with these results, we may conclude that our models were accurate and represent human skin behaviour truthfully. We should also state that it is preferable to employ model 2, as it may be 9

10 used for a wider variety of tests, due to the inaccurate behaviour of model 1 when subjected to large displacements. 6. Acknowledgements João Crespo would like to thanks Professor Marcelo Alves (UFSC) and Professor Paulo Fernandes (I.S.T) for all the help provided. 7. References 1. Delalleau, G. Josse, J.M. Lagarde, H. Zahouani and J.M. Bergheau: A nonlinear elastic behaviour to identify the mechanical parameters of human skin in vivo, Skin Research and Technology 2008,vol. 14, pp accessed on April skin - accessed on April accessed on April Jeffrey E. Bischoff, Ellen M. Arruda, Karl Grosh: Finite element modelling of human skin using an isotropic, nonlinear elastic constitutive model, Journal of Biomechanics 2000, vol 33, pp Cormac Flynn and Brendan A. O. McCormack: Finite element modelling of forearm skin wrinkling, Skin Research and Technology 2008,vol. 14, pp Fouad Khatyr, Claude Imberdis, Paul Vescovo, Daniel Varchon, and Jean-Michel Lagarde: Model of the viscoelastic behaviour of skin in vivo and study of anisotropy, Skin Research and Technology 2004,vol. 10, pp Wilkes GL, Brown IA, Wildnauer RH: The biomechanical properties of skin. Crit Rev Bioeng Fung Y. C., Biomechanics: Mechanical Properties of living tissues, second edition, Springer- Verlag, New York, Inc, USA 10. Energy and finite element methods in structural mechanics, Hemisphere publishing corporation, Washington, New York, USA 11. Reddy, J. N.: Energy principles and variational methods in applied mechanics, John Wiley & Sons, Inc, Chichester, England. 12. Reddy, J. N.: An introduction to the finite element method, second edition, McGraw-Hill, Inc, New York, USA 13. Coimbra, Alberto Luiz, Novas Lições de mecânica do continuo: Editora edgard blucher ltda, São Paulo, Brasil. 14. L.M. Tham, H.P. Lee, C. Lua, Cupping: From a biomechanical perspective, Journal of Biomechanics 2006, vol 39, pp Pascon João Paulo. Modelos constitutivos para materiais hiperelasticos: estudo e implementação computacional, Universidade de São Paulo, Escola de Engenharia de São Carlos, Departamento de Engenharia de Estruturas, Sharipov R A, Quick Introduction to Tensor analysis, Pinho, Silvestre T,Dinis, Dra Ana Lucia: Mestrado em Engenharia Mecânica apontamentos de hiperelasticidade,faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecânica e Gestão Industrial, Kolecki, Joseph C: An introduction to tensor for students of Physics and Engeneering,Nasa Team, Glen research centre Cleveland Ohio, September Alves, Marcelo Krajnc: Apontamentos das aulas de Introdução à mecânica do Continuo, Meijer, Riske: Characterization of anisotropic an non-linear behaviour of human skin, Personal Care Institute of Philips Research Laboratories in Eindhoven, Joseph J O'Hagan and Abbas Samani: Measurement of the hyperelastic properties of tissue slices with tumour inclusion, Physics in Medicine and Biology,2008, Vol 53, pp