Repositorio Institucional de Documentos

Trabajo Fin de M´aster M´aster en Mec´anica Aplicada Programa Oficial de Posgrado en Mec´anica Computacional Curso 2010-2011 Finite Element study of the angiogenesis process: application to wound healing. Clara Valero L´azaro Septiembre de 2011 Directora: Dra. MªJos´e G´omez Benito Co-Directora: Dra. Etelvina Javierre P´erez Departamento de Ingenier´ıa Mec´anica Escuela de Ingenier´ıa y Arquitectura Universidad de Zaragoza Finite Element study of the angiog´enesis process: application to wound healing. Resumen Este trabajo se ha desarrollado en el departamento de Ingenier´ıa Mec´anica y su principal objetivo ha sido el desarrollo e implementaci´on de un modelo multif´ısico del proceso de angiogenesis. La piel es el ´organo m´as grande del cuerpo humano, encargada del mantenimiento de la integridad de los ´organos en su interior. Es de gran importancia su capacidad de autorregeneraci´on ante lesiones. Dadas las particularidades de cada herida y con el fin de obtener el resultado deseado con la mayor eficiencia, deben plantearse tratamientos espec´ıficos para cada situaci´on. Para ello, ser´a de gran importancia disponer de t´ecnicas num´ericas de simulaci´on que permitan una evaluaci´on r´apida de cada caso. El proceso de cicatrizaci´on de heridas se produce en tres etapas: inflamaci´on, proliferaci´on y remodelaci´on. Dentro de la fase de proliferaci´on tienen lugar numerosos procesos, entre ellos la angiogenesis. El proceso de angiogenesis consiste en la formaci´on de vasos sangu´ıneos a partir de vasos ya existentes. Tras producirse una herida los vasos sangu´ıneos localizados en el tejido da˜nado quedan cortados, de manera que se impide la correcta distribuci´on de ox´ıgeno y nutrientes. Por ello es de gran importancia conocer en qu´e medida afectan los factores presentes en la evoluci´on del proceso de cicatrizaci´on. Durante este proceso adem´as se produce una contracci´on de la herida debida a las tensiones que ejercen las c´elulas presentes, tanto en la herida como en la piel que la rodea. Este proceso es el objeto de estudio de este trabajo y depende tanto de las propiedades mec´anicas como de las biol´ogicas y qu´ımicas de la piel. En este trabajo se ha implementado en primer lugar un modelo que incluye solo los factores bioqu´ımicos que afectan al proceso de angiog´enesis y en segundo lugar un modelo que tiene en cuenta tanto los factores mec´anicos como los biol´ogicos para simular el proceso de angiog´enesis en la cicatrizaci´on de heridas en la piel. Para ello se parti´o de un modelo matem´atico de comportamiento de todos los elementos presentes tanto en la herida como en la piel sana, y se evalu´o utilizando el M´etodo de los Elementos Finitos (MEF). Por ´ultimo, se han comparado los resultados obtenidos con los dos modelos. Para ello se ha prestado atenci´on principalmente a la influencia del ox´ıgeno en el proceso y al efecto de las tensiones mec´anicas creadas por las c´elulas en las concentraciones de las especies presentes. Tambi´en se ha estudiado la diferencia en la geometr´ıa final de la herida de los dos modelos. Table of contents 1. Introduction 1 1.1. Angiogenesis in wound healing . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Angiogenesismodels .............................. 3 1.3. Objectives.................................... 4 2. Biochemical model of angiogenesis 5 2.1. Conservation of species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1. Oxygen (u1)............................... 5 2.1.2. Macrophage-derived growth factor (MDGF) (u2)........... 6 2.1.3. Capillary density (u3) ......................... 6 2.1.4. Fibroblasts(n) ............................. 6 2.2. Numerical implementation of the biochemical model ...................................... 7 2.2.1. System of equations nonlinearly coupled . . . . . . . . . . . . . . . 8 2.2.2. Jacobian matrices of the internal and external forces . . . . . . . . 9 2.3. Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4. Numerical results of the biochemical model . . . . . . . . . . . . . . . . . . 11 2.4.1. Evolution of the species in the wound centre . . . . . . . . . . . . . 11 2.4.2. Oxygeninfluence ............................ 12 2.4.3. Species density in the whole geometry . . . . . . . . . . . . . . . . 12 3. Mechanobiochemical model of angiogenesis 14 3.1. Mechanosensing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2. Material model of the skin . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3. Implementation of the mechanobiochemical model ...................................... 16 3.4. Results of the complete model . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4.1. Comparison of the species densities in the centre of the wound . . . 18 3.4.2. Evolution of the species densities in the whole domain . . . . . . . 19 4. Conclusions and future work 21 4.1. Conclusions ................................... 21 4.1.1. Conclusions from the biochemical model . . . . . . . . . . . . . . . 21 4.1.2. Conclusions from mechanobiochemical model . . . . . . . . . . . . . 21 4.2. Futurework................................... 22 4.3. Applications................................... 22 4.3.1. Medical applications . . . . . . . . . . . . . . . . . . . . . . . . . . 22 I 4.3.2. Technological applications: self-healing materials . . . . . . . . . . . 23 II List of figures 1.1. Scheme of the skin layers. ........................... 1 1.2. Scheme of the wound healing phases in time. ................. 2 2.1. Computational domain of the 2D problem. .................. 11 2.2. Evolution of the species densities in the centre of the wound. ........ 11 2.3. Influence of the oxygen in the fibroblast density. ............... 12 2.4. Distribution of species at day one. ....................... 13 3.1. Scheme of the complete model. ......................... 17 3.2. Effect of the mechanics in the fibroblasts concentration. ........... 18 3.3. Distribution of the four species at day one. .................. 19 3.4. Fibrobalsts distribution at day 30. ....................... 20 3.5. Mesh of the wound at day 30. ......................... 20 III List of tables 2.1. List of normalised model parameters related to the biochemical model. . . 7 3.1. List of normalised model parameters related to the mechanical behaviour. . 15 3.2. Values of the hyperelastic parameters. . . . . . . . . . . . . . . . . . . . . . 16 IV Chapter 1 Introduction 1.1. Angiogenesis in wound healing Skin is the largest organ in human body, it covers the whole external body surface and constitutes about the 8 % of the human body mass [14]. Skin has a critical function in maintaining the integrity of the internal organs under it and in preventing the transmission of infections and dehydration. It is equally important its autoregenerative capacity after an injury, getting a perfect healing in almost every case. Skin is an elastic tissue that can be stretched and compressed under certain limits. Figure 1.1: Scheme of the skin layers. ADAM, http://www.adam.com Skin consist of various tissue layers (Figure 1.1): the epidermis, the dermis and the hypodermis, and can reach a thickness of 1,5 mm to 4 mm depending on the body part [14]. When important injuries occur in the skin, damage can reach the dermis and the hypodermis, and wound healing is more difficult. The dermis is an irregular connective soft tissue, with a collagen fiber matrix. In this matrix are also allocated nerves, blood vessels and several cellular species. From a mechanical point of view, due to the number and location of the collagen fibers, dermis provides to the skin a high resistance to traction and also a high ability to contract elastically. 1 Wounds in dermis can appear as a result from surgery or from a traumatic accident. In both cases, a successful healing is crucial for a perfect functional and aesthetic recovery. However, in some situations an optimal healing without an appropriate medical treatment is not possible. Understanding the wound healing process and all the involved factors that influence it is crucial to cure these wounds, in special the complicated ones. Thus, it is very important to understand the mechanical behaviour of the skin against aggressions such as wounds and burns, and its capacity to recover the properties of the undamaged tissue in the shortest time possible. Wound healing is a natural process in which human body has to regenerate all the skin layers that have suffered damage. During the tissue recovery process, several complex biochemical processes take place to repair the damage. This phenomena overlap in time and can be divided in three stages: inflammatory, proliferative and remodelling, as can be seen in Figure 1.2. Figure 1.2: Scheme of the wound healing phases in time. After the wound formation both sides of the wound contract towards the centre of the wound, creating mechanical stresses in the skin and inducing the inflammatory respond that initiates the healing process. In the inflammatory phase bacteria are removed and factors that are present in the proliferative stage are released. After that, several phenomena take place in the proliferative phase: angiogenesis, granulation tissue formation, epitheliazation and wound contraction. In this stage new blood vessels and extra cellular matrix (ECM) are created while the wound reduces its size. In the final stage, the collagen fibers are remodelled and realigned and cells that are no needed are removed. 2 Fext u1=ZΩ NT u1λ3,1u3dΩ−ZΩ NT u1λ1,1u1dΩ (2.15) Fext u2=ZΩ NT u2λ2,11−u1 uθ2 1dΩ−ZΩ NT u2λ2,2u2dΩ (2.16) Fext u3=ZΩ NT u31−u3 ue 3qu2u3dΩ (2.17) Fext n=ZΩ −NT nλn,nndΩ−ZΩ rnnNT n1−n Ku11−u1 K1dΩ (2.18) where, working with two-dimensional geometries and being N(∗)(x) the shape functions in the nodes and nnod the total number of nodes, the matrices in the problem have the form Nu1=N1 u1N2 u1· · · Nnnod u1 Nu2=N1 u2N2 u2· · · Nnnod u2 Nu3=N1 u3N2 u3· · · Nnnod u3 Nn=N1 nN2 n· · · Nnnod n ∇Nu1=  ∂N1 u1 ∂x ∂Nnnod u1 ∂x ∂N1 u1 ∂y · · · ∂Nnnod u1 ∂y   ∇Nu2=  ∂N1 u2 ∂x ∂Nnnod u2 ∂x ∂N1 u2 ∂y · · · ∂Nnnod u2 ∂y   ∇Nu3=  ∂N1 u3 ∂x ∂Nnnod u3 ∂x ∂N1 u3 ∂y · · · ∂Nnnod u3 ∂y   ∇Nn=  ∂N1 n ∂x ∂Nnnod n ∂x ∂N1 n ∂y · · · ∂Nnnod n ∂y   The solution of the nonlinear system of equations can be obtained applying a standard Newton-Raphson method [16]. 2.2.2. Jacobian matrices of the internal and external forces The Jacobian matrices corresponding to the internal and external forces with non-zero value resulting from the system of equations are now presented. In the used notation, α represents the characteristic temporal parameter of the trapezoidal integration method. The non-zero terms of the Jacobian matrices ∂Fint/∂Zand ∂Fext/∂Zare: 9 ∂Fint u1 ∂u1 =1 α∆tZΩ NT u1Nu1dΩ + ZΩ ∇NT u1D1∇Nu1dΩ (2.19) ∂Fint u2 ∂u2 =1 α∆tZΩ NT u2Nu2dΩ + ZΩ ∇NT u2D2∇Nu2dΩ (2.20) ∂Fint u3 ∂u3 =1 α∆tZΩ NT u3Nu3dΩ + ZΩ ∇NT u3D3∇Nu3dΩ (2.21) ∂Fint n ∂n=1 α∆tZΩ NT nNndΩ + ZΩ ∇NT nDn∇NndΩ (2.22) ∂Fext u1 ∂u1 =−ZΩ NT u1Nu1λ1,1dΩ (2.23) ∂Fext u1 ∂u3 =ZΩ NT u1Nu3λ3,1dΩ (2.24) ∂Fext u2 ∂u1 =ZΩ NT u2Nu1λ2,1−1 uθ2 1dΩ (2.25) ∂Fext u2 ∂u2 =−ZΩ NT u2Nu2λ2,2dΩ (2.26) ∂Fext u3 ∂u2 =ZΩ NT u3Nu2λ3,2u31−u3 ueq 3dΩ (2.27) ∂Fext u3 ∂u3 =−ZΩ NT u3Nu3λ3,2u21−2u3 ueq 3dΩ (2.28) ∂Fext n ∂n=−ZΩ NT nNnλn,ndΩ + ZΩ rnNT nNnu11−2n K1−u1 K1dΩ (2.29) ∂Fext n ∂u1 =ZΩ rnNT nNu1n1−n K1−2u1 K1dΩ (2.30) From these expressions it is possible to obtain the necessary block matrices to assembly the global stiffness matrix. 2.3. Initial and boundary conditions The computational domain Ω consists of two disjoint parts: the wound Ωwand the undamaged surrounding tissue Ωund (see Figure 2.1). As initial conditions we have considered that the initial concentration of every species is null in the wound site. As a result from the injury, every species are removed from the damaged skin. The undamaged tissue is full of all the species except MDGF (u2), which is not present. MDGF will appear in the wound site after the initiation of the model, due to the low oxygen concentration. As boundary conditions we have considered that the domain is sufficiently large in order to prescribe null fluxes through the frontier and no displacements at the boundary. 10 Figure 2.1: Computational domain of the 2D problem. 2.4. Numerical results of the biochemical model In a first approach, a circular wound has been studied. The diameter of the wound is 2 cm and it is surrounded by a 10 cm diameter circumference of undamaged skin. We have simulated a fourth of the whole geometry as we have considered symmetry in the two axes. 2.4.1. Evolution of the species in the wound centre We analyse the results in the centre of the wound, which is the furthest point to the undamaged skin and supposedly the last point to heal. We show the evolution of the species concentration in the wound centre along the simulated time (see Figure 2.2), in this case 30 days where simulated as it was assumed that it is a long enough period for the wound to be completely vascularized. Figure 2.2: Evolution of the oxygen density (top left), MDGF density (top right), capillary density (bottom left) and fibroblasts density (bottom right) in the centre of the wound during the simulated time. We observe how the oxygen concentration regulates the rest of the variables. While there is no oxygen in the centre of the wound the level of MDGF increases rapidly, which 11 cause the appearance of capillaries. When the capillary density grows, more oxygen is supplied to the wound centre and the MDGF gradually disappears. The capillary density grows rapidly to the capillary density of undamaged tissue (ueq 3) when the oxygen density begins to increase and after that, the fibroblasts density grows indicating that the dermis is being restored. 2.4.2. Oxygen influence We have studied the effect of oxygen in the fibroblasts behaviour. It is clear from the results that oxygen affects the fibroblasts concentration (see Figure 2.3). When oxygen is not included in the fibroblasts kinetics, the model predicts fibroblasts invasion into the wound faster than reality, and hence wound healing rate is overestimated. Figure 2.3: Evolution of the fibroblasts density in the wound centre without the influence of oxygen (continuous line) and the oxygen effect (dashed line.) 2.4.3. Species density in the whole geometry To finish this Chapter, we have studied the distribution of every species one day postwounding. The results are shown in Figure 2.4. Results show an elevated density of MDGF in the wound area, and slowing growing density of oxygen, capillaries and fibroblasts. The advance of the so-called healing unit, with the capillaries in first term followed by the fibroblasts, is visible. After a few days the concentrations of all the species are homogeneous in the whole domain. Both, oxygen and capillary density reach a concentration of 1 (the normalised density of undamaged dermis), while MDGF appears in the wound site during the first days but then disappears. Finally, the fibroblasts concentration progresses more slowly, but finally reaches the fibroblast density of the undamaged dermis. 12 Figure 2.4: Distribution of oxygen (top left), MDGF (top right), capillary (bottom left) and fibroblasts (bottom right) in the whole domain one day after wounding. 13 Chapter 3 Mechanobiochemical model of angiogenesis As it has been said in Chapter 1, angiogenesis is a biological process but it is not only influenced by biochemical factors, it is also influenced by mechanical stimuli. When mechanical stresses appear in the skin the geometry of the wound is modified and it affects also the evolution of the biochemical process. In this work, we propose an angigenesis model that includes biological and mechanical factors. For that, we calculate the variation of the biochemical species that are present in the process with the model proposed in Chapter 2 and we have also implemented the mechanical behaviour of the skin and cells. We have implemented this model in a non-coupled way, so we resolve the biochemical equations and the mechanical analysis separately due to the different time scales at which mechanical and biochemical processes occur. 3.1. Mechanosensing model As it has been previously said, cells (in this model we assume the most important cell type in angiogenesis are fibroblasts) can exert forces in the surrounding tissue where they are allocated. This mechanical behaviour is due to their actomyosin mechanism [9]. From the species densities values we can obtain the net stress of one fibroblasts cell per unit of extra-cellular matrix (ECM), pcell, which, in this work, is considered to be the mechanical stimulus that regulates the forces exerted by cells. Since endothelial cells (which form the capillaries) are active in ECM production, we assume that the ECM density is proportional to the capillary density. We take pcell following Moreo et al. [9]. In their work Moreo et al. proposed an active mechanosensing model based on the Hill’s model for skeletal muscle behaviour, applicable to cell-substrate interaction situations. In this work pcell is a piecewise linear function depending on the tissue volumetric strain 14 pcell =             Kpasθ θ < θ1 Kactpmax Kactθ1−pmax (θ1−θ) + Kpasθ θ1≤θ≤θ∗ Kactpmax Kactθ2−pmax (θ2−θ) + Kpasθ θ∗< θ ≤θ2 Kpasθ θ > θ2 (3.1) where θ∗can be defined as θ∗=pmax/Kact. pcell is a mechanosensing variable that depends on the volumetric strain θof the tissue where cells are allocated. As the cell actomyosin machinery responds actively only between certain limits, we have considered θ1and θ2, which are the compresion and traction strain limits within which the machinery works. pmax denotes the maximal contractile force that the actomyosin machinery can exert, and Kact and Kpas the stiffness moduli of the active and passive components of the cell. The values of these parameters [5] are included in Table 3.1. Parameter Description Dimensionless value pmax maximal cellular active stress per unit of ECM 1 Kpas volumetric stiffness moduli of the passive components of the cell 2·10−1 Kact volumetric stiffness moduli of the actin filaments of the cell 1 θ1shortening strain of the contractile element -0.6 θ2lengthening strain of the contractile element 0.5 Rτtraction inhibition collagen density 5·10−3 Kecm ECM production rate per unit of capillary density 1 Table 3.1: List of normalised model parameters related to the mechanical behaviour. pcell and the species densities determine stresses that cells are creating in the surrounding tissue. In this work we have considered that the traction stresses, that are created by punctual forces exerted by cells, can be written as σcell =pcell(θ)nρecm R2 τ+ρ2 ecm (3.2) following Javierre et al [5]. Using Equation (3.2), cell traction forces are considered isotropic, generated by fibroblasts and inhibited at high ECM density. In addition, we have considered that the density of the extracellular matrix can be expressed as a function of the capillaries in the skin, thus it is assumed that capillaries can be present only when there is a substrate to stay, in this case the ECM density (ρecm) is expressed as, ρecm =Kecmu3,(3.3) where Kecm is a constant. 15 3.2. Material model of the skin In the numerous studies where skin is simulated, it has been modelled with different material behaviours. In some of these works, as in Javierre et al. [5], skin is assumed to be viscoelastic and it is characterised by its usual stress-strain constitutive relation. However, other works centered just in the mechanical behaviour of skin [1] [6], assume skin as a hyperelastic material, which is more accurate. Thus, in this work skin has been considered as a hyperelastic material. We have used a second order polynomial strain energy potential model with the form [1] ψ= 2 X i+j=1 Cij(I1−3)i(I2−3)j+ 2 X i=1 1 D(Jel −1)2i(3.4) where ψis the strain energy per unit of reference volume and I1and I2are the first and second deviatoric invariants. The values of the material parameters Cij and Dihave been taken from previous skin studies [1], where the parameters were calculated for a normal stiffness tissue. The values of the parameters are presented in Table 3.2. As far as we know, there is no work in literature which determine mechanical properties of damaged skin. Thus, in a first approach mechanical properties of the damaged skin were assumed to be the same as the undamaged skin properties. Parameter C10 C01 C20 C11 C02 D Value 0.08556 -0.05841 0.03900 -0.02319 0.00851 3.65273 Table 3.2: Values of the hyperelastic parameters. Units are Nmm2for Cij and mm2N−1 for D. 3.3. Implementation of the mechanobiochemical model In this work we have implemented the biochemical and the mechanical behaviour of the skin in the angiogenesis process separately, due to the different time scales at which biochemical and mechanical events occur during wound healing it is assumed more appropriate. However both parts of the model have some variables in common, and these variables are used to merge the biochemical and mechanical parts and close the model. The scheme of the implementation is shown in Figure 3.1 In the biochemical analysis we are evaluating the evolution of the species densities in a fixed geometry, that does not vary in time. The value of these stresses depends on the volumetric strain θas has been explained in Section 3.1, and at the beginning of the calculation θis zero as the tissue has not been yet deformed. To solve the problem in this non-coupled way we use an updated Lagrangian formulation, so we work with the deformed coordinates in both biochemical and mechanical analyses. 16 Figure 3.1: Scheme of the complete model. The value of the stresses σcell (Eq. (3.2)) in every node of the mesh is calculated in the same subroutine used in the biochemical model and introduced in the hyperelastic analysis as punctual loads in the nodes. In this work we have considered that this stresses are created by punctual forces exerted by cells. As we are working, considering symmetries, with just a fourth of the whole geometry we fix the two symmetry faces as boundary conditions. From the mechanical analysis it is possible to obtain the new position of the mesh nodes, that is the new geometry (deformed) of the wound and the undamaged surrounding tissue. This new geometry is introduced as the initial geometry of the next biochemical step. As 17 initial conditions of every biochemical step we introduce the node concentrations obtained from the last biochemical step, as they have not been modified in the mechanical analysis. In this analysis we also calculate the volumetric deformation θthat the tissue has suffered around every node and we introduce these values in the biochemical analysis to calculate the new stresses and species concentrations. To carry out the full analysis we alternate the two analysis several times to complete the desired period. In this work we are studying a whole time period of 30 days, to compare the results against the results obtained with the biochemical model. As the biochemical evolution of the species is slower than the mechanical response we take periods of one day for the first part of the analysis while the time that we use to evaluate the mechanical behaviour is deprecated as the mechanical respond is instantaneous. We are repeating the process 30 times to complete the 30 days analysis, thus each analysis corresponds whith a day of the angiogenesis process. 3.4. Results of the complete model In this section we present the results of the complete model in the same way as we have studied the results from the biochemical model in Section 2.4, and we compare the outcomes. Thus, we simulate a circular wound with a diameter of 2 cm and it is surrounded by a 10 cm diameter circumference of undamaged skin. We have simulated a fourth of the whole geometry as we have considered symmetry in the two axis. This wound has the same geometry as the studied in Chapter 2 so we can compare the results. 3.4.1. Comparison of the species densities in the centre of the wound The main difference between the two cases, with and without the mechanical influence, is that in the complete model the geometry varies with time according to the deformation that the tissue suffers from the traction that the cells exert on it. First we have plotted the fibroblasts concentration in the centre of the wound (Figure 3.2). This point is the only point of the wound that does not experience any displacement due to the mechanical behaviour. In Figure 3.2 we observe the differences between the concentration of fibroblasts in the centre of the wound when considering just biochemical factors and both biochemical and mechanical ones. At the beginning the evolution is similar in both cases, but when the mechanical factors are included, the fibroblasts density grows more rapidly due to the contraction of the wound. A possible reason is that when the mechanics is considered the boundary between the dermis and the wound is moving towards the centre of the wound, and the cellular densities are dragged in the same direction. 3.4.2. Evolution of the species densities in the whole domain We present here the distribution of the four species densities (oxygen, macrophage-derived growth factor, capillary and fibroblasts) at day one (Figure 3.3) under the effect of cell 18 Acknowledgements I gratefully acknowledge research support from the Spanish Ministry of Science and Technology through the research project, Grant DPI2009-07514. Furthermore, I also thank the Spanish Ministry of Science and Technology for the support to Clara Valero through the grant BES2010-037281. 25 References [1] Cheung, J. T.-M., Zhang, M., Leung, A. K.-L., F., Y.-B., 2005. Three-dimensional finite element analysis of the foot during standing–a material sensitivity study. Journal of Biomechanics 38 1045–1054. [2] Glazier, J.A., Graner, F., 1993. Simulation of the differential adhesion driven arrangement of biological cells. Phys Rev E 47 2128–2154. [3] Hughes, T. J. R., 1987. The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs NJ. [4] Javierre, E., Vermolen, F.J., Vuik, C., Zwaag, S., 2008. Numerical Modelling of Epidermal Wound Healing. In: Numerical Mathematics and Advanced Applications 83– 90. Springer Berlin Heidelberg [5] Javierre, E., Moreo, P., Doblar´e, M., Garc´ıa-Aznar, J.M., 2009. Numerical modeling of a mechano-chemical theory for wound contraction analysis. International Journal of Solids and Structures 46 3597–3606. [6] Lapeer, R.J., Gasson, P.D., Karri, V.,2011. A Hyperelastic Finite-Element Model of Human Skin for Interactive Real-Time Surgical Simulation. IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING 58 1013–1022. [7] Maggelakis, S., 2003. A mathematical model of tissue replacement during epidermal wound healing. Applied Mathematical Modelling 27 189–196. [8] Midwood, K.S., Williams, L.V., Schwarzbauer, J.E., 2004. Tissue repair and the dynamics of the extracellular matrix. International journal of biochemistry & cell biology 36 1031–1037. [9] Moreo, P., Garc´ıa-Aznar, J.M., Doblar´e, M., 2008. Modeling mechanosensing and its effect on the migration and proliferation of adherent cells. Acta Biomateriaalia 4 613–621. [10] Pettet, G.J., Byrne, H.M., Mcelwain, D.L.S., Norbury, J., 1996a. A model of woundhealing angiogenesis in soft tissue. Mathematical Biosciences 136 35–63. [11] Pettet, G., Chaplain, M.A.J., Mcelwain, D.L.S., Byrne, H.M., 1996b. On the role of angiogenesis in wound healing. Proc R Soc London Ser B 263 1487–1493. [12] Schreml, S., Szeimies, R.M., Prantl, L., Karrer, S., Landthaler, M., Babilas, P., 2010. Oxygen in acute and chronic wound healing. British Journal of Dermatology 163 257–268. 26 [13] Schugart, R.C., Friedman, A., Zhao, R., Sen, C.K., 2008. Wound angiogenesis as a function of tissue oxygen tension: A mathematical model. Proc Natl Acad Sci USA 105 2628–2633. [14] Williams, P.L., 1998. Anatom´ıa de Gray. Volumen I. Ed. Elsevier. [15] De, R., Zemel, A., Safran,S. A., 2007. Dynamics of cell orientation 3(9). Nature Physics 58(4) 655–659. [16] Zienkiewicz, O.C., Taylor, R.L., 2000b. The Finite Element Method. Volume 2: Solid Mechanics. Butterworth-Heinemann, Oxford. 27