Repositorio Institucional de Documentos

2014 81 Clara Valero Lázaro Mechanochemical modeling of wound healing: Multiphysics finite element simulations Departamento Director/es Ingeniería Mecánica Gómez Benito, María José Javierre Pérez, Etelvina Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Clara Valero Lázaro MECHANOCHEMICAL MODELING OF WOUND HEALING: MULTIPHYSICS FINITE ELEMENT SIMULATIONS Director/es Ingeniería Mecánica Gómez Benito, María José Javierre Pérez, Etelvina Tesis Doctoral Autor 2014 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA  Programa Oficial de Posgrado en Mec´ anica Computacional Mechanochemical modeling of wound healing: multiphysics finite element simulations. Clara Valero L´ azaro Doctoral degree in Computational Mechanics Faculty Advisors: Dra. Mª Jos´ e G´ omez Benito and Dra. Etelvina Javierre P´ erez Departamento de Ingenier´ ıa Mec´ anica Escuela de Ingenier´ ıa y Arquitectura Universidad de Zaragoza. Abril de 2014 ii Do˜ na Mar´ ıa Jos´ e G´ omez Benito, Profesora Titular de Universidad del Departamento de Ingenier´ ıa Mec´ anica de la Escuela de Ingenier´ ıa y Arquitectura, de la Universidad de Zaragoza, y Do˜ na Etelvina Javierre P´ erez, Profesora del Centro Universitario de la Defensa, centro Adscrito a la Universidad de Zaragoza, CERTIFICAN Que la memoria de Tesis Doctoral presentada por do˜ na Clara Valero L´ azaro con el t´ ıtulo ”Mechanochemical modeling of wound healing: multiphysics finite element simulations” ha sido realizada bajo nuestra direcci´ on en el Departamento de Ingenier´ ıa Mec´ anica, de la Escuela de Ingenier´ ıa y Arquitectura, de la Universidad de Zaragoza, y se corresponde con el Proyecto de Tesis aprobado por la Comisi´ on de Doctorado en Febrero de 2014, por lo que autorizamos su presentaci´ on en la modalidad de compendio de publicaciones y con la Menci´ on de ”Doctor Internacional” cumpliendo por lo tanto las condiciones requeridas para que su autora pueda optar al grado de Doctora de la Universidad de Zaragoza. Firmado, en Zaragoza, a 28 de Abril de 2014. Fdo: Mar´ ıa Jos´ e G´ omez Benito Fdo: Etelvina Javierre P´ erez iv Abstract Wound healing is an important health matter that affects millions of patients and generates high costs to the health system and society. Wounds appear in the skin as a consequence of damage in traumatic accidents but also as a result of surgery incisions, which sometimes are not easy to heal. Wound healing is usually divided into three stages overlapped in time that can lasts for several months or years. These stages are named inflammation, tissue formation and tissue remodeling and they normally follow a predictable progress. Going into more detail, different biological, chemical and mechanical processes can be identified inside every stage. These processes are usually guided by the interaction between cells, chemical factors and mechanical stimulus and the healing process can not be completed in the absence of any of those agents. There exists many different wound types which depending on their severeness heal faster or slower. For instance, superficial cuts or abrasions heal quickly following the usual progression. In other occasions, if the regular time scheme is not followed or the injured person suffers from any kind of physiological disorder, healing can lead into difficulties such as pathological scars (hypertrophic scars and keloids) or chronic wounds, which take longer to heal or even never heal completely. Research in the wound healing field has been focused in experimental and computational works that try to understand the influence of the different variables in the process. Differences between wound types and geometries require a general procedure, customizable, that can be applied to analize different wounds. To this purpose, computational models are the best approach, as they can be easily particularized to the desired wound geometry and characteristics. Computational models of wound healing have been widely studied during the lasts years, from simple geometry models including a unique variable to more complex models with several variables and different geometries. However, these models have been barely extended out of the two-dimensional planar geometries and just a few of them reproduced deep wounds. Moreover, existing models have other limitations that reduce their predictive potential. For instance, most of the existent models consider the skin as a viscoelastic or hyperelastic isotropic material, neglecting the contributions of the collagen fibers and their orientation. Thus, the aim of this thesis is the study of wound healing in the skin through computational simulation and using a multiphysics approach. In order to achieve xi this objective, a computational model that allows to reproduce wound healing under different conditions has been proposed and implemented. More specifically, the proposed model focuses on the wound contraction phase, as it has a strong mechanical component. The model also allows to include more processes that take place simultaneously to wound contraction, for instance angiogenesis. The proposed model includes biological factors (cellular species, growth factors and collagen) and also mechanical factors (characterization of the skin mechanical properties and cellular contraction). A physically based formulation of cell mechanosensing and mechanotransduction has been used to describe both cellexerted stresses and cell differentiation signals. The constitutive model of the skin has been updated from the commonly used viscoelasticy to a more realistic anisotropic hyperelastic description. The resulting governing equations are solved through the Finite Elements Method, and its formulation has been implemented in two and three dimensions in order to be able to study any wound geometry. With all these characteristics, the model here presented allows to analyze wound healing evolution from a much wider perspective, taking into account not only morphological aspects but also specific characteristics of the chemical and mechanical cues on cellular and tissue evolution. Keywords: wound healing, skin, contraction, angiogenesis, ECM collagen matrix, biomechanics, fibroblasts, myofibroblasts, endothelial cells, collagen, growth factor, mechanosensing, mechanotransduction, matrix stiffness, constitutive model, viscoelasticity, anisotropy, hyperelasticity, fibers, cell-traction, cell-induced stress, convection-diffusion-reaction, finite elements, three-dimensional formulation, multiphisics xii Agradecimientos En primer lugar, me gustar´ ıa dar las gracias a mis directoras, MªJos´e yEtel por su constante apoyo durante este tiempo. Por estar en todo momento disponibles para atender mis dudas y ofrecer ayuda y consejo ante todas las incertidumbres que han aparecido a lo largo de estos cuatro a˜ nos. No s´ olo por ofrecerme soluciones ante las dificultades, si no tambi´ en por ayudarme a aprender a encontrar las soluciones por m´ ı misma. Tambi´ en quiero expresar mi agradecimiento a Manu, que siempre ha estado dispuesto a ayudarme en lo que hiciera falta y encontrar un momento para atender a mis preguntas. Gracias tambi´ en a Mª ´ Angeles por estar ah´ ı y preocuparse siempre por saber si todo iba bien. Me gustar´ ıa especialmente agradecer a los profesores Krishna Garikipati de la Universidad de Michigan y Andreas Menzel de la TU de Dortmund, que me acogieron en sus grupos de investigaci´ on durante varios meses. Las estancias en sus grupos me permitieron no s´ olo ampliar mi experiencia investigadora, sino tambi´ en disfrutar de una experiencia dif´ ıcilmente repetible Bel´en, Carmelo, H´ector, Mar´ıa, Marta, Noelia, Rosa, Samuel, Sara,... que durante estos a˜ nos no s´ olo han demostrado ser buenos amigos en los buenos momentos, sino tambi´ en apoyo en los momentos de mayor dificultad. Por todas las tardes de quedadas, viajes para desconectar y conversaciones para desahogarnos. Gracias por hacer que estos a˜ nos hayan sido f´ aciles. No me olvido tampoco de todos los compa˜ neros que a lo largo de estos a˜ nos han formado parte del grupo de becarios del ´ area, hayamos coincidido m´ as o menos tiempo, de todos guardo buen recuerdo. Hemos compartido los momentos duros y estresantes que todos pasamos a lo largo de la tesis pero tambi´ en buenos ratos, dentro y fuera de la universidad. Con estos me quedo. Finalmente, me gustar´ ıa agradecer a mi familia, que me han apoyado en todo momento siendo una motivaci´ on constante. Gracias por entender lo que es importante para m´ ı. xiii xiv ´ Indice general - Table of contents 1 Introducci´ on 3 1.1 Lapiel................................ 3 1.1.1 Estructura de la piel . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Propiedades mec´ anicas de la piel . . . . . . . . . . . . . . 9 1.2 Cicatrizaci´ ondeheridas....................... 13 1.2.1 Etapas de la cicatrizaci´ on de heridas . . . . . . . . . . . . 14 1.2.2 Complicaciones en la cicatrizaci´ on de heridas . . . . . . . 19 1.2.3 Tratamientos para la cicatrizaci´ on ............. 23 1.2.4 Propiedades mec´ anicas del tejido cicatrizado . . . . . . . 24 1.2.5 Mecanorrecepci´ on y mecanotransducci´ on en cicatrizaci´ on deheridas.......................... 25 1.3 Modelos computacionales de cicatrizaci´ on de heridas . . . . . . . 26 1.3.1 Modelos de cicatrizaci´ on de heridas epid´ ermicas . . . . . 27 1.3.2 Modelos de contracci´ on de heridas . . . . . . . . . . . . . 36 1.3.3 Modelos computacionales de angiog´ enesis durante la cicatrizaci´ ondeheridas.................... 39 1.3.4 Limitaciones de los modelos existentes . . . . . . . . . . 42 1.4 Trabajos experimentales . . . . . . . . . . . . . . . . . . . . . . 44 1.5 Objetivos .............................. 45 1.6 Financiaci´ on............................. 47 2 Conclusiones 49 2.1 Conclusiones generales . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Contribuciones originales . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Publicaciones y comunicaciones . . . . . . . . . . . . . . 52 2.3 L´ ıneasfuturas............................ 53 3 Introduction 61 3.1 Theskin............................... 61 3.1.1 Structure of the skin . . . . . . . . . . . . . . . . . . . . 62 3.1.2 Mechanical properties of the skin . . . . . . . . . . . . . 67 xv 3.2 Woundhealing ........................... 70 3.2.1 Stages of wound healing . . . . . . . . . . . . . . . . . . 71 3.2.2 Complications of wound healing . . . . . . . . . . . . . . 74 3.2.3 Healing strategies . . . . . . . . . . . . . . . . . . . . . 78 3.2.4 Scar mechanical properties . . . . . . . . . . . . . . . . . 80 3.2.5 Mechanosensing and mechanotransduction in wound healing ............................. 81 3.3 Computational models of wound healing . . . . . . . . . . . . . . 82 3.3.1 Epidermal wound healing models . . . . . . . . . . . . . 83 3.3.2 Wound contraction models . . . . . . . . . . . . . . . . . 91 3.3.3 Computational models of angiogenesis during wound healing ............................. 94 3.3.4 Limitations of existing models . . . . . . . . . . . . . . . 96 3.4 Experimentalworks......................... 98 3.5 Objectives.............................. 99 3.6 Financialsupport ..........................100 4 Conclusions 103 4.1 Generalconclusions.........................103 4.2 Original contributions . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.1 Publications and communications . . . . . . . . . . . . . 106 4.3 Futurelines .............................111 Work 1: Nonlinear finite element simulations of injuries with free boundaries: Application to surgical wounds 125 Work 2: Numerical modelling of the angiogenesis process in wound contraction 145 Work 3: Stress evaluation during angiogenesis in skin wound healing 159 Work 4: A cell-regulatory mechanism involving feedback between contraction and tissue formation guides wound healing progression 167 Work 5: Modeling anisotropic wound healing: effect of the relative position of wounds with respect to collagen fibers orientation 185 Work 6: A computational study of stress fiber-focal adhesion dynamics governing cell contractility 201 xvi INTRODUCCI ´ ON GENERAL Y CONCLUSIONES Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas: simulaci´ on multif´ ısica con elementos finitos. 2 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas CAP´ ITULO 1 Introducci´ on Las heridas en la piel son lesiones que la mayor´ ıa de las personas sufren a lo largo de la vida. Cuando la persona herida no tiene problemas serios de salud y la herida no es muy profunda, las heridas se curan sin implicaciones m´ as relevantes que una cicatriz en la piel. Sin embargo, cuando el proceso de curaci´ on no sigue su evoluci´ on natural puede dar lugar a diferentes complicaciones, tales como heridas cr´ onicas y otros problemas. Las heridas cr´ onicas afectan a 6,5 millones de pacientes en los Estados Unidos generando un coste anual de 25000 millones de d´ olares (Sen et al., 2009), reducir estos n´ umeros es un objetivo permanente del sistema de salud. Encontrar c´ omo mejorar el proceso de curaci´ on necesita una inversi´ on de tiempo y un elevado gasto econ´ omico. Adem´ as, las implicaciones ´ eticas est´ an siempre presentes cuando los experimentos implican seres vivos. Por ello, los modelos matem´ aticos de curaci´ on de heridas han aumentado su popularidad en las ´ ultimas d´ ecadas. Los modelos matem´ aticos permiten estudiar un amplio n´ umero de casos modificando par´ ametros con un reducido coste econ´ omico y de tiempo, a˜ nadiendo la posibilidad de estudiar casos espec´ ıficos, individualizados para cada paciente. Por ello, el objetivo de esta tesis es el desarrollo de un modelo de cicatrizaci´ on de heridas para predecir la evoluci´ on de la cicatrizaci´ on de heridas bajo diferentes condiciones y que permita el estudio de un amplio rango de heridas y factores que influyen en su curaci´ on. 1.1 La piel La piel es la cubierta exterior de los animales vertebrados y presenta diferente estructura y propiedades en cada especie. En los humanos, la piel es el ´ organo m´ as grande y representa el 8 % de la masa corporal (Gray et al., 1995). La piel cubre la 3 4 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas totalidad del cuerpo humano y su espesor var´ ıa seg´ un la parte del cuerpo (Odland, 1991), entre 1,5 mm y 4 mm considerando la epidermis y la dermis. El tejido inferior tiene una gran variabilidad en cuanto a su profundidad y composici´ on en funci´ on de su localizaci´ on anat´ omica. La piel se clasifica normalmente en dos tipos: fina y gruesa. La piel fina se encuentra en partes tales como los p´ arpados o zonas donde hay fol´ ıculos capilares, mientras que la piel gruesa no presenta normalmente fol´ ıculos capilares y est´ a localizada principalmente en las palmas de las manos y las suelas de los pies. La piel constituye una interfaz entre el cuerpo y el medio ambiente, es una barrera natural que protege los ´ organos internos de agresiones externas. Adem´ as de ser una barrera f´ ısica, la piel tiene varias funciones; evita la deshidrataci´ on regulando la p´ erdida de agua y act´ ua como barrera t´ ermica. Los mecanismos que la piel utiliza para regular la temperatura corporal son principalmente la evaporaci´ on de sudor y la convecci´ on, regulando el flujo sangu´ ıneo en la superficie corporal. Una de las funciones m´ as importantes de la piel es su papel como parte del sistema inmunol´ ogico, evitando la entrada de part´ ıculas extra˜ nas y pat´ ogenos (cualquier agente que pueda causar da˜ no) en el cuerpo (Fore-Pfliger, 2004). Adem´ as, numerosos sistemas fisiol´ ogicos necesarios para el funcionamiento correcto del cuerpo, tales como nervios, capilares o el sistema linf´ atico, est´ an localizados en la piel. Por lo tanto, mantener la estabilidad de la piel y evitar cualquier factor que pueda reducir sus propiedades es altamente importante. Adem´ as, tambi´ en es de gran importancia conseguir una recuperaci´ on r´ apida y funcional de la piel cuando sufre da˜ no. 1.1.1 Estructura de la piel La piel humana est´ a formada por col´ ageno, elastina y un n´ umero variable de especies celulares (Tabla 1.1). Est´ a organizada en tres capas principales, desde la m´ as externa a la m´ as interna: epidermis, dermis e hipodermis (Wilkes et al., 1973). Las heridas en la piel normalmente atraviesan la epidermis y alcanzan la dermis. La epidermis es la capa m´ as externa de la piel y es la que est´ a en contacto con el medio ambiente, y por lo tanto tiene un papel principal en la funci´ on protectora de la piel. La epidermis constituye una barrera frente a la p´ erdida de agua y la entrada de substancias extra˜ nas. A pesar del papel importante de la epidermis, ´ esta es tambi´ en la capa m´ as fina de la piel, con un espesor de entre 75-150µm (Odland, 1991), y se divide en 5 capas: Stratum corneum, Stratum lucidum, Stratum granulosum, Stratum spinosum yStratum germinativum (Figura 1.1). La epidermis est´ a constituida principalmente por keratinocitos, que son tambi´ en parte del sistema inmunitarios y producen mediadores anti-inflamatorios. Puesto CAP´ ITULO 1. Introducci´ on 11 Figura 1.4: Distribuci´ on de las l´ ıneas de tensi´ on en el tronco del cuerpo humano (Langer, 1861). es definido como una variable que tiene una parte independiente y otra parte dependiente de la frecuencia. Silver et al. (2001) estudi´ o las propiedades viscoel´ asticas de los componentes de la piel a partir de datos experimentales (Dunn and Silver, 1983), estimando que la constante el´ astica para el col´ ageno de la piel es de 4,4 GPa y para la elastina es 4,0 MPa. Para realizar los ensayos de tensi´ on-deformaci´ on utilizaron piel del t´ orax y del abdomen. Otros trabajos caracterizan la piel como un material hiperel´ astico. Estos materiales no muestran propiedades el´ asticas lineales en su relaci´ on tensi´ ondeformaci´ on y permiten definir otras propiedades del material tales como anisotrop´ ıa o incompresibilidad. Flynn et al. (2011a) midieron la relaci´ on fuerzadesplazamiento en la piel del antebrazo al aplicar deformaciones. M´ as tarde, encontraron los par´ ametros del material que se ajustan al modelo hiperel´ astico de Ogden y al modelo de Tong y Fung (Flynn et al., 2011b). Gahagnon et al. (2012) estudiaron la anisotrop´ ıa de la piel del antebrazo in-vivo usando pruebas elastogr´ aficas. Para ello estiraron la piel paralela y perpendicularmente a las l´ ıneas de Langer encontrando comportamiento anis´ otropo. 12 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Los ensayos in-vivo para determinar las propiedades de la piel son dif´ ıciles de realizar y la variabilidad entre los sujetos es alta. El amplio rango de valores obtenidos se debe a las diferentes localizaciones anat´ omicas de la piel, la uni´ on de la dermis con el tejido subyacente y las caracter´ ısticas espec´ ıficas del paciente, lo que hace dif´ ıcil encontrar una caracterizaci´ on ´ unica v´ alida para todas las pieles. Mientras que los estudios in-vivo proporcionan informaci´ on acerca de la piel en su entorno natural, sin la eliminaci´ on de los procesos naturales, los estudios in-vitro proporcionan experimentos m´ as controlados, donde diferentes aspectos pueden ser aislados y se pueden realizar ensayos destructivos. Por lo tanto, los ensayos in-vitro dan informaci´ on acerca de diferentes propiedades tales como la resistencia o la elasticidad de la piel, mientras que los ensayos in-vivo muestran la reacci´ on de la piel frente a cargas externas en situaciones reales (Edwards and Marks, 1995). Annaidh et al. (2012b) investigaron la influencia de la ubicaci´ on y la orientaci´ on de la piel en sus propiedades, centr´ andose en la anisotrop´ ıa de la piel. Realizaron ensayos in-vitro de tracci´ on a muestras de piel humana obtenidas de diferentes partes de la espalda (Figura 1.5), encontrando una correlaci´ on entre la orientaci´ on de las l´ ıneas de Langer y las fibras de col´ ageno. M´ as tarde, Annaidh et al. (2012a) relacionaron sus resultados con el modelo hiperel´ astico desarrollado por Gasser et al. (2006). Figura 1.5: Orientaci´ on de las muestras de la espalda estudiadas por Annaidh et al. (2012b) con respecto a las l´ ıneas de Langer (Langer, 1861). (b) Dimensiones de la probeta (mm) utilizada por Annaidh et al. (2012b) Groves et al. (2013) realizaron ensayos de tracci´ on utilizando probetas circulares de piel humana para medir las propiedades de la piel. Para observar el efecto de la anisotrop´ ıa realizaron pruebas a lo largo de tres ejes de carga diferentes. Encontraron que la piel tiene un comportamiento hiperel´ astico CAP´ ITULO 1. Introducci´ on 13 anis´ otropo y utilizaron el modelo de Weiss et al. (1996) para caracterizar este comportamiento. Otros estudios se han centrado en investigar fen´ omenos puntuales que se producen en procesos como la cicatrizaci´ on de heridas. Hinz et al. (2001) estudi´ o el efecto de la tensi´ on en el tejido granular y en la diferenciaci´ on de los fibroblastos a miofibroblastos, mientras que Graham et al. (2004) estudiaron el comportamiento de las fibras de col´ ageno cuando se someten a distintas deformaciones. A pesar de la gran cantidad de datos que se ha generado como resultado de los estudios anteriormente mencionados, la b´ usqueda de un conjunto ´ unico de par´ ametros para definir las propiedades de la piel humana es casi imposible debido a la alta variabilidad de los resultados entre los diferentes sujetos y localizaciones anat´ omicas y la dificultad de preservar la piel sin necesidad de modificar sus propiedades. 1.2 Cicatrizaci´ on de heridas La cicatrizaci´ on de heridas es un tema de salud importante que afecta a millones de pacientes y genera un alto coste para el sistema sanitario y la sociedad (Sen et al., 2009). Cada a˜ no se realizan millones de procedimientos quir´ urgicos con el posterior tratamiento de la herida creada y las cicatrices. Tambi´ en se estima que alrededor de 6,5 millones de pacientes se ven afectados por heridas cr´ onicas en EE.UU., lo que genera un coste de 25000 millones de d´ olares al a˜ no. Se calcula tambi´ en que el cuidado de cicatrices en la piel genera un gasto anual de alrededor de 12000 millones de d´ olares. Las heridas pueden aparecer como consecuencia de da˜ nos en accidentes traum´ aticos, pero tambi´ en como resultado de las incisiones necesarias en cirug´ ıa, que muchas veces no son f´ aciles de curar. Por otra parte, las heridas como las ´ulceras por presi´on pueden aparecer a causa de largos per´ ıodos de inmovilidad y pueden conducir a infecciones y complicaciones m´ as graves, incluso la muerte. Las complicaciones en la curaci´ on de heridas tienden a aparecer en personas con enfermedades cr´ onicas como la diabetes y la obesidad, que presentan dificultades para una correcta cicatrizaci´ on. La cicatrizaci´ on de heridas se produce despu´ es de que ocurra el da˜ no en la piel. Las heridas leves se cicatrizan generalmente a trav´ es de una serie de procesos bien 14 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas organizados sin necesidad de tratamiento especial, pero en aquellos casos en los que las heridas causan un da˜ no excesivo es necesario aplicar diferentes t´ ecnicas para lograr su curaci´ on. Adem´ as, los procesos de curaci´ on anormales, como lesiones fibroproliferativas (queloides ocicatrices hipertr´oficas) o las heridas cr´ onicas (´ulceras venosas,´ulceras por presi´on, y ´ulceras de pie diab´etico), no pueden lograr una cicatrizaci´ on adecuada sin ayuda externa debido a su incorrecta actividad fisiol´ ogica. Por lo tanto, es de gran importancia entender los mecanismos de cicatrizaci´ on con el fin de proponer nuevas t´ ecnicas que favorezcan el proceso de cicatrizaci´ on y prevenir la aparici´ on de complicaciones. Estos avances incluyen la aplicaci´ on de diferentes factores de crecimiento o f´ armacos a los pacientes, la elecci´ on de una geometr´ ıa de la incisi´ on que facilite la cicatrizaci´ on en los procedimientos de cirug´ ıa, creando menos cicatrizaci´ on, o el desarrollo de t´ ecnicas de sutura que permitan reducir al m´ ınimo la formaci´ on de la cicatriz (Hall-Findlay, 1999; Pollock and Pollock, 2000). Sin embargo, cuando no se produce la cicatrizaci´ on normal ni siquiera con ayuda, deben aplicarse otras t´ ecnicas m´ as complejas y costosas. 1.2.1 Etapas de la cicatrizaci´ on de heridas La cicatrizaci´ on de heridas se suele dividir en tres etapas superpuestas en el tiempo que pueden durar varios meses o a˜ nos (Figura 3.6). Estas etapas se denominan: inflamaci´ on,epitelizaci´ on yremodelaci´ on (Singer and Clark, 1999). Los procesos bien definidos que tienen lugar durante cada etapa se explican a continuaci´ on: Figura 1.6: Distribuci´ on temporal de las fases de cicatrizaci´ on de heridas, incluyendo los procesos m´ as importantes, a lo largo del tiempo. Fuente: http://commons.wikimedia.org/wiki/File :Wound healing phases.png. 1. Inflamaci´ on: Esta etapa comienza cuando aparece la lesi´ on. Cuando se produce una herida, la matriz extracelular se sustituye por un co´ agulo de sangre proveniente de los vasos rotos, y factores inflamatorios tales como el factor de crecimiento derivado de las plaquetas (PDGF) (Figura 1.7(a)) para estimular la actividad celular. Posteriormente, los vasos se cierran CAP´ ITULO 1. Introducci´ on 15 para detener el sangrado (hemostasia) y un co´ agulo de fibrina reemplaza el co´ agulo de sangre (Gurtner et al., 2008). En este punto, el ox´ ıgeno ha desaparecido del tejido de la herida induciendo hipoxia, y no puede volver a ser suministrado ya que los capilares no se han regenerado todav´ ıa. Las c´ elulas inflamatorias (neutr´ ofilos y macr´ ofagos) limpian la herida y eliminan bacterias, tejido muerto y otras part´ ıculas extra˜ nas (Singer and Clark, 1999). La inflamaci´ on dura aproximadamente 48 horas, hasta que se eliminan todas las part´ ıculas extra˜ nas. Al final de la fase inflamatoria se liberan factores de crecimiento tales como el TGF-βy VEGF, necesarios para guiar la actividad de las c´ elulas en la siguiente etapa (Gray et al., 1995). 2. Epitelizaci´ on: La epitelizaci´ on o formaci´ on de nuevo tejido comienza algunas horas despu´ es de que aparezca la herida y se superpone con la etapa final de la inflamaci´ on, durando habitualmente entre 2 y 10 d´ ıas (Gurtner et al., 2008). Esta etapa se caracteriza por la migraci´ on y proliferaci´ on de varias especies celulares, principalmente fibroblastos y c´ elulas endoteliales, a la zona de la herida, atra´ ıdas por los factores de crecimiento liberados al final de la fase inflamatoria. El co´ agulo de fibrina es eliminado y reemplazado por tejido granular (Figura 1.7(b)). M´ as tarde, el tejido de granulaci´ on es sustituido por una nueva matriz extracelular constituida principalmente por col´ ageno. Los fibroblastos que se han infiltrado en el sitio de la herida se activan y diferencian a miofibroblastos. Ambas especies celulares secretan col´ ageno, inicialmente de tipo III, que se sustituye a lo largo del tiempo por col´ ageno m´ as fuerte, de tipo I, el cual forma inicialmente una red de fibras desorganizada, que da soporte mec´ anico para la formaci´ on del nuevo tejido. En esta etapa, las c´ elulas epiteliales estimuladas por el VEGF forman nuevos vasos sangu´ ıneos a partir de los da˜ nados, en el proceso llamado angiog´ enesis (Risau, 1997). Este proceso permite restablecer el flujo normal de sangre y nutrientes al tejido (Gray et al., 1995) y tambi´ en el suministro de ox´ ıgeno necesario para la actividad celular. Sin embargo, la angiog´ enesis tambi´ en es un fen´ omeno importante en otros procesos fisiol´ ogicos tales como la embriog´ enesis y la formaci´ on de tumores (Carmeliet and Jain, 2000), que puede resultar negativo cuando es excesivo. Por otra parte, en esta etapa se produce la contracci´ on de la nueva matriz extracelular que contiene los nuevos capilares y varios tipos de c´ elulas. La contracci´ on del tejido se debe a las fuerzas que las c´ elulas (fibroblastos y miofibroblastos) ejercen en respuesta al cambio en las propiedades del material en la zona da˜ nada. De hecho, los miofibroblastos son capaces de ejercer fuerzas de tracci´ on m´ as elevadas que los fibroblastos. Por lo tanto, una proporci´ on adecuada entre ambos tipos de c´ elulas es crucial 16 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Figura 1.7: Etapas de la cicatrizaci´ on de heridas a) inflamaci´ on, b) epitelizaci´ on, c) remodelaci´ on. Dibujo de Gurtner et al. (2008). para evitar trastornos en la cicatrizaci´ on. Al final de la reepitelizaci´ on las c´ elulas innecesarias desaparecen del tejido de granulaci´ on, principalmente por apoptosis. Una vez concluido este proceso, se forma la cicatriz inicial. CAP´ ITULO 1. Introducci´ on 17 Nombre Funci´ on Etapa de curaci´ on de heridas /Proceso Otros Factor de crecimiento epid´ ermico (EGF) Estimula la proliferaci´ on y diferenciaci´ on de c´ elulas epiteliales y mesenquimales Formaci´ on de tejido granular Proliferaci´ on Producido por macr´ ofagos y keratinocitos Factor de crecimiento de fibroblastos (FGFs) Quimiotaxis y proliferaci´ on de los fibroblastos Migraci´ on y proliferaci´ on de los keratinocitos Estimula la angiog´ enesis, contracci´ on de heridas y deposici´ on de matriz Angiog´ enesis Producido por macr´ ofagos, c´ elulas endoteliales y fibroblastos entre otros Factor de crecimiento de keratinocitos (KGF) Migraci´ on, proliferaci´ on y diferenciaci´ on de keratinocitos Epitelizaci´ on Producido por keratinocitos Factor de crecimiento derivado de los macr´ ofagos (MDGF) Estimula la proliferaci´ on de fibroblastos, c´ elulas de m´ usculo liso y c´ elulas endoteliales Inflamaci´ on-Epitelizaci´ on Liberado por macr´ ofagos Factor de crecimiento derivado de plaquetas (PDGF) Estimula la proliferaci´ on de fibroblastos Regula el crecimiento y divisi´ on de m´ ultiples c´ elulas Influye en la formaci´ on de vasos sangu´ ıneos (angiog´ enesis) y la remodelaci´ on de tejido Inflamaci´ on/Hemostasis Producido por plaquetas Contin´ua en la p´agina siguiente 18 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Tabla 1.2 – Contin´ua de la p´agina anterior Nombre Funci´ on Etapa de curaci´ on de heridas /Proceso Otros Factor de crecimiento transformante beta (TGFβ) Estimula la diferenciaci´ on de fibroblastos a miofibroblastos Controla la proliferaci´ on, diferenciaci´ on, apoptosis y otras funciones en la mayor´ ıa de las c´ elulas Estimula la producci´ on de ECM Epitelizaci´ on/Diferenciaci´ onProducido por fibroblastos, miofibroblastos y macr´ ofagos entre otros Factor de crecimiento endotelial vascular (VEGF) Estimula la angiog´ enesis y vasculog´ enesis, dirige la formaci´ on de capilares Promueve la migraci´ on de c´ elulas endoteliales y fibroblastos Quimiot´ actico para los macr´ ofagos Hipoxia-Angiog´ enesis Producido por c´ elulas que reciben ox´ ıgeno insuficiente Tabla 1.2: Lista de factores de crecimiento qu´ ımicos que intervienen en la curaci´ on de heridas. CAP´ ITULO 1. Introducci´ on 19 3. Remodelaci´ on: Una vez que el nivel de col´ ageno en la herida recupera el de la piel sana, el col´ ageno comienza a reorganizarse. Al comienzo del proceso, las fibras de col´ ageno est´ an orientadas aleatoriamente. Sin embargo, las fibras de col´ ageno tienden a orientarse con el tiempo a lo largo de direcciones preferenciales, por lo general paralelas a las l´ ıneas de tensi´ on de la piel. Las propiedades del tejido evolucionan aumentando su resistencia a tracci´ on. Sin embargo, la funcionalidad del tejido se recupera despu´ es de varios meses o a˜ nos dependiendo de la herida (Figura 1.7 (c)) aunque una recuperaci´ on completa nunca se logra debido a que las propiedades del tejido reci´ en formado siguen siendo ligeramente inferiores a las propiedades del tejido sano. Durante cada etapa tienen lugar simult´ aneamente varios fen´ omenos que se rigen por la presencia de especies celulares y factores de crecimiento (ver Figura 3.6). La mayor´ ıa de los procesos celulares que ocurren durante la curaci´ on de heridas, tales como la proliferaci´ on, la diferenciaci´ on y el crecimiento, est´ an altamente influenciados por factores de crecimiento. Los factores de crecimiento son sustancias qu´ ımicas secretadas normalmente por las c´ elulas, que tambi´ en estimulan la actividad de estas. Se ha incluido una lista de los principales factores de crecimiento que gu´ ıan los procesos de cicatrizaci´ on de heridas (Tabla 1.2). 1.2.2 Complicaciones en la cicatrizaci´ on de heridas Las etapas de la cicatrizaci´ on de heridas normalmente siguen un esquema predecible. Los cortes o abrasiones superficiales se curan r´ apidamente siguiendo la progresi´ on habitual. Si esta progresi´ on temporal normal no se sigue o si la persona lesionada sufre alg´ un tipo de trastorno fisiol´ ogico pueden aparecer complicaciones en la cicatrizaci´ on como cicatrices patol´ ogicas (cicatrices hipertr´oficas yqueloides) o heridas cr´ onicas (tales como ´ulceras por presi´on,´ulceras de pie diab´etico y´ulceras venosas). La cicatrizaci´ on de heridas termina con la creaci´ on de la cicatriz, que se compone de una masa no-funcional de tejido fibr´ otico (Gurtner et al., 2008). Este tejido est´ a compuesto principalmente por fibroblastos y matriz extracelular que contiene la misma prote´ ına (col´ ageno) que el tejido sano pero con una composici´ on diferente (Gauglitz et al., 2011). Las cicatrices anormales pueden aparecer cuando la producci´ on y la degradaci´ on de ECM no est´ an bien equilibradas. Los trastornos fibroproliferativos generan cicatrizaci´ on excesiva, que puede conducir a diferentes tipos de cicatrices caracterizadas por la cantidad y el tipo de col´ ageno sobreproducido. Mientras que las cicatrices hipertr´oficas 20 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas est´ an formadas por col´ ageno tipo III, las queloides contienen col´ ageno de tipo I y tipo III. Las cicatrices hipertr´oficas est´ an causadas por una producci´ on excesiva de col´ ageno. Se producen cuando hay una lesi´ on grave en la piel, como infecci´ on de la herida o tensi´ on excesiva en la herida (Gauglitz et al., 2011) y el40-70 % de las cicatrices quir´ urgicas puede conducir a cicatrices hipertr´oficas. Las cicatrices hipertr´oficas normalmente s´ olo crecen hacia arriba, mientras que las queloides son un tipo de cicatrices hipertr´oficas tumorales caracterizadas por su crecimiento excesivo, que afecta a los tejidos que en un principio no pertenec´ ıan a la herida. Estas cicatrices suelen aparecer en zonas de elevada tensi´ on y causar contractura, con la consiguiente disminuci´ on de la calidad de vida del paciente (Gauglitz et al., 2011). Igualmente problem´ atico que la cicatrizaci´ on excesiva es la cicatrizaci´ on impedida, que da lugar a las heridas cr´ onicas. Las heridas cr´ onicas tardan m´ as tiempo en curarse que las heridas normales y por lo general no se curan completamente sin ayuda. Hay una serie de factores que contribuyen a la aparici´ on de heridas cr´ onicas tales como enfermedades sist´ emicas (diabetes), insuficiencia arterial, infecciones o la edad avanzada. La primera causa de las heridas cr´ onicas son las ´ulceras venosas (Snyder, 2005), causadas por el mal funcionamiento de las v´ alvulas venosas y un crecimiento vascular no adecuado. Las ´ulceras venosas suelen aparecer en las piernas, son m´ as probables en pacientes diab´ eticos y en casos cr´ ıticos pueden terminar en la amputaci´ on de la extremidad. Los pacientes diab´ eticos tienen alrededor de un 15% de probabilidades de sufrir una ´ulcera de pie diab´etico, una llaga abierta situada en la parte inferior del pie. Los principales factores que dan lugar al pie diab´etico son la diabetes y las neuropat´ ıas vasculares, que producen disminuci´ on de la sensaci´ on de dolor causada por el da˜ no de los nervios despu´ es de mantenerse los niveles de glucosa en sangre excesivamente altos. Por otra parte, la diabetes afecta a la evoluci´ on normal del proceso de curaci´ on, prolongando la fase inflamatoria, lo que retrasa la formaci´ on de tejido de granulaci´ on y reduce la fuerza de contracci´ on en la herida (Ogawa and Hsu, 2013). Debido a su alto impacto, se han desarrollado una serie de soluciones para el tratamiento de heridas cr´ onicas en pacientes diab´ eticos, incluida la aplicaci´ on de factores de crecimiento, el uso de sustitutos de la piel, las terapias de presi´ on negativa y la terapia hiperb´ arica de ox´ ıgeno (HBOT) . Las ´ulceras por presi´on son tambi´ en una de las heridas m´ as complicadas de curar, y se producen generalmente cuando el paciente debe permanecer inm´ ovil durante largos per´ ıodos de tiempo, sobre todo en cama o silla de ruedas. Por este motivo, las ´ulceras por presi´on aparecen por lo general en lugares donde hay una prominencia ´ osea, tales como el sacro, el coxis, las caderas o los talones, CAP´ ITULO 1. Introducci´ on 27 Durante los ´ ultimos a˜ nos la curaci´ on heridas y, concretamente, algunos procesos tales como la contracci´ on de heridas y la angiog´ enesis han sido ampliamente modelados utilizando modelos continuos. Sin embargo, no se ha desarrollado todav´ ıa un modelo que incluya todas las etapas de la cicatrizaci´ on, ya que el n´ umero de procesos y variables que intervienen es demasiado grande, y requerir´ ıa una elevada capacidad computacional y grandes tiempos de c´ alculo. Los aspectos m´ as relevantes de los modelos revisados se resumen en la Tabla 1.4. 1.3.1 Modelos de cicatrizaci´ on de heridas epid´ ermicas El primer trabajo sobre la cicatrizaci´ on de heridas epid´ ermicas fue desarrollado por Sherratt and Murray (1990). El cierre de heridas epid´ ermicas se debe s´ olo a la migraci´ on de c´ elulas epid´ ermicas y no se ve afectado por la contracci´ on de la herida, que s´ olo tiene lugar cuando las capas m´ as profundas de la piel se ven afectadas. Sherratt and Murray (1990) propusieron un modelo continuo de reacci´ on-difusi´ on estableciendo que un factor de crecimiento qu´ ımico producido por las c´ elulas epid´ ermicas difund´ ıa a trav´ es del tejido. Adem´ as, la variaci´ on de la cantidad de c´ elulas proviene de la migraci´ on y la mitosis regulada por el factor producido y tambi´ en por la apoptosis de las c´ elulas (Clark, 1988; Folkman and Moscona, 1978). El modelo se utiliz´ o para predecir la cicatrizaci´ on de una herida epid´ ermica con geometr´ ıa circular considerando que la herida estaba cicatrizada al alcanzar una densidad celular superior al 80% de la densidad de c´ elulas del tejido sano. Sherratt and Murray (1990) compararon sus resultados con uno de los pocos trabajos experimentales existentes sobre la cicatrizaci´ on de heridas epid´ ermicas, realizado por Van den Brenk (1956) en orejas de conejo. Sherratt and Murray (1990) propusieron el primer modelo incorporando el control qu´ ımico, y este ha sido la base para la mayor´ ıa de los modelos de curaci´ on y contracci´ on de heridas. Adem´ as, su modelo ha sido adaptado tambi´ en para simular la cicatrizaci´ on de heridas en otros tejidos. Por ejemplo, Dale et al. (1994) lo adapt´ o para estudiar la cicatrizaci´ on de heridas epiteliales en la cornea en funci´ on del factor de crecimiento epitelial (EGF). Adem´ as, la cicatrizaci´ on de heridas epid´ ermicas se ha simulado junto con otros procesos que tienen lugar al mismo tiempo, tales como la angiog´ enesis (Maggelakis, 2003). 28 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Sherratt and Murray (1990) Curaci´ on de heridas epid´ ermicas C´ elulas epid´ ermicas Factor de crecimiento regulador de la mitosis No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en conejo (Van den Brenk, 1956) Tranquillo and Murray (1992) Curaci´ on de heridas en la dermis Contracci´ on de heridas Fibroblastos Matriz de col´ ageno Desplazamientos de la matriz Factor de crecimiento gen´ erico Viscoel´ astico Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en rata (McGrath and Simon, 1983) Dale et al. (1994) Curaci´ on de heridas epiteliales en la c´ ornea C´ elulas EGF No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en conejo Contin´ua en la p´agina siguiente CAP´ ITULO 1. Introducci´ on 29 Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Olsen et al. (1995) Contracci´ on de heridas Fibroblastos Miofibroblastos Factor de crecimiento gen´ erico Matriz de col´ ageno Viscoel´ astico Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en rata (McGrath and Simon, 1983) Pettet et al. (1996a) Angiog´ enesis Fuentes de vasos sangu´ ıneos Factor quimioatrayente Vasos sangu´ ıneos No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en conejo (Van den Brenk, 1956) Pettet et al. (1996b) Angiog´ enesis Fuentes de vasos sangu´ ıneos Fibroblastos MDGF Ox´ ıgeno Matriz de col´ ageno No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en conejo (Van den Brenk, 1956) Contin´ua en la p´agina siguiente 30 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Maggelakis (2003) Angiog´ enesis Curaci´ on de heridas epid´ ermicas Ox´ ıgeno MDGF Capilares No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D No se incluye validaci´ on con resultados experimentales Manoussaki (2003) Angiog´ enesis Vasculog´ enesis C´ elulas endoteliales Col´ ageno Factor angiog´ enico Viscoel´ astico Cuadradas en 2D No se incluye validaci´ on con resultados experimentales Javierre et al. (2008) Cierre de heridas Angiog´ enesis Curaci´ on de heridas epid´ ermicas Ox´ ıgeno MDGF Capilares EGF No incluido Heridas superficiales en 2D No se incluye validaci´ on con resultados experimentales Contin´ua en la p´agina siguiente CAP´ ITULO 1. Introducci´ on 31 Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Schugart et al. (2008) Angiog´ enesis Fuentes de vasos sangu´ ıneos Ox´ ıgeno C´ elulas inflamatorias Quimioatrayente Fibroblastos Miofibroblastos Matriz de col´ ageno Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D No se incluye validaci´ on con resultados experimentales Flegg et al. (2009) Angiog´ enesis en curaci´ on de heridas (aplicacion de terapia de ox´ ıgeno hiperb´ arico) Ox´ ıgeno Capillary tips Vasos sangu´ ıneos No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D No se incluye validaci´ on con resultados experimentales Contin´ua en la p´agina siguiente 32 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Javierre et al. (2009) Contracci´ on de heridas Fibroblasts Miofibroblastos Matriz de col´ ageno Factor de crecimiento gen´ erico Desplazamiento de la matriz Viscoel´ astico Heridas superficiales en 2D No se incluye validaci´ on con resultados experimentales Xue et al. (2009) Cierre de heridas Ox´ ıgeno PDGF VEGF Macr´ ofagos Fibroblastos Matriz de col´ ageno Puntas de capilares Viscoel´ astico Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en cerdo (Roy et al., 2009) Contin´ua en la p´agina siguiente CAP´ ITULO 1. Introducci´ on 33 Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Flegg et al. (2010) Angiog´ enesis en curaci´ on de heridas Ox´ ıgeno Quimioatrayente Puntas de capilares Vasos sangu´ ıneos Fibroblastos ECM No incluido Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D No se incluye validaci´ on con resultados experimentales Vermolen and Javierre (2010) Contracci´ on de heridas Cierre de heridas Angiog´ enesis C´ elulas epid´ ermicas Capilares Matriz de col´ ageno Fibroblastos Ox´ ıgeno VEGF Viscoel´ astico Heridas profundas en 2D No se incluye validaci´ on con resultados experimentales Contin´ua en la p´agina siguiente 34 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Friedman and Xue (2011) Cierre de heridas (en heridas cr´ onicas) Macr´ ofagos Ox´ ıgeno PDGF VEGF Fibroblastos Matriz de col´ ageno Velocidad del contorno Fuentes de vasos sangu´ ıneos Viscoel´ astico Heridas en 2D con simetr´ ıa radial Heridas tridimensionales con simetr´ ıa axial Validado en cerdo (Roy et al., 2009) Murphy et al. (2011) Curaci´ on de heridas d´ ermincas Contracci´ on de heridas Fibroblastos Miofibroblastos Matriz de col´ ageno TGF−β El´ astico Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en rata (McGrath and Simon, 1983) Murphy et al. (2012) Cierre de heridas Curaci´ on de heridas d´ ermicas Fibroblastos Miofibroblastos TGF−β PDGF Matriz de col´ ageno Colagenasa Viscoel´ astico Heridas superficiales circulares simplificadas en un modelo axisim´ etrico 1D Validado en humano (Catty, 1965) y rata McGrath and Simon (1983) Contin´ua en la p´agina siguiente CAP´ ITULO 1. Introducci´ on 35 Tabla 1.4 – Contin´ua de la p´agina anterior Autor Procesos simulados Variables del modelo Modelo mec´ anico Geometr´ ıa Comparaci´ on con trabajos experimentales Vermolen and Javierre (2012) Contracci´ on de heridas Cierre de heridas Angiog´ enesis Regeneraci´ on d´ ermica C´ elulas epid´ ermicas Capilares ECM Fibroblastos Ox´ ıgeno VEGF EGF Viscoel´ astico Heridas profundas bidimensionales No se incluye validaci´ on con resultados experimentales Tabla 1.4: Lista de modelos previos de curaci´ on de heridas y sus caracteristicas. 36 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas 1.3.2 Modelos de contracci´ on de heridas Aunque las heridas epid´ ermicas representan una buena primera aproximaci´ on para el estudio de la cicatrizaci´ on de heridas, las heridas que afectan a las capas m´ as profundas y presentan m´ as dificultades para cicatrizar tambi´ en son m´ as interesantes para estudiar. Se ha observado que las heridas que llegan a la dermis no se curan s´ olo con el efecto de la migraci´ on celular y son necesarios mecanismos m´ as complejos para una curaci´ on exitosa. La contracci´ on de heridas es uno de los procesos m´ as importantes durante la cicatrizaci´ on de heridas d´ ermicas y est´ a fuertemente influenciada por la mec´ anica. La reducci´ on del tama˜ no de la herida es debida en parte a la retracci´ on del contorno de la herida hacia adentro. Este desplazamiento es creado por el equilibrio entre las fuerzas que generan las c´ elulas en el tejido en el que est´ an alojadas y la resistencia que ofrece el tejido a ser deformado. El primer modelo de contracci´ on de heridas fue propuesto por Tranquillo and Murray (1992). Este ha sido la base para la mayor´ ıa de los modelos de contracci´ on de heridas hasta el momento. El modelo est´ a compuesto por un conjunto de ecuaciones diferenciales que describen la conservaci´ on de una especie celular (fibroblastos) y la densidad de col´ ageno de la ECM y el momento lineal de la matriz. Tranquillo and Murray (1992) incluyeron el efecto de la proliferaci´ on celular, la migraci´ on y la difusi´ on junto con la convecci´ on pasiva debida al movimiento de la ECM. En una primera aproximaci´ on se consider´ o que la variaci´ on de ECM se deb´ ıa s´ olo a su convecci´ on pasiva. M´ as tarde, propusieron una ley de evoluci´ on de la ECM m´ as realista que inclu´ ıa la s´ ıntesis de ECM por los fibroblastos. Por otra parte, en el mismo trabajo propusieron un modelo bioqu´ ımico m´ as completo incluyendo el efecto quimiot´ actico de un factor de crecimiento qu´ ımico. La ley de momento lineal incluye el equilibrio de fuerzas entre la matriz (caracterizado como un material viscoel´ astico) y las fuerzas de tracci´ on ejercidas por las c´ elulas. Consideraron un dominio de estudio dividido en dos partes, la herida y la piel sana circundante. El modelo se utiliz´ o para el estudio de heridas unidimensionales y se sigui´ o el desplazamiento del margen de la herida durante el tiempo para medir el tama˜ no de la herida. El modelo de Tranquillo and Murray (1992) fue extendido m´ as tarde por Olsen et al. (1995). Olsen et al. (1995) propusieron un modelo de contracci´ on de heridas que inclu´ ıa el desplazamiento del tejido contra´ ıdo y lo aplicaron a heridas normales y patol´ ogicos. Una de las principales diferencias entre los trabajos de Tranquillo and Murray (1992) y Olsen et al. (1995) es la incorporaci´ on CAP´ ITULO 1. Introducci´ on 43 frontera que est´ a en contacto con el aire est´ a representada por una frontera libre. La implicaci´ on principal es que se puede mover libremente, ya que no hay tejido unido a ´ el. Cuando se formula el modelo matem´ atico, esta frontera libre debe ser tratada de manera diferente al resto del contorno. Algunas de las simplificaciones supuestas para el contorno no libre no son aplicables a esta frontera libre. Desde una perspectiva num´ erica las heridas planas bajo hip´ otesis de tensi´ on plana son m´ as f´ aciles de modelar ya que no es necesario tener en cuenta las condiciones de contorno naturales. Otra diferencia entre el estudio de heridas planas y profundas es la consideraci´ on de las distintas capas de la piel. Cuando se simulan heridas planas solo las propiedades de la capa m´ as superficial de la piel se consideran, como es el caso de los modelos de curaci´ on de heridas epid´ ermicas (Sherratt and Murray, 1990; Maggelakis, 2003; Javierre et al., 2008) y algunos modelo de cicatrizaci´ on de heridas d´ ermicas (Tranquillo and Murray, 1992; Murphy et al., 2011, 2012). Cuando se simulan heridas profundas, dependiendo de la profundidad considerada de la herida, no es posible caracterizar toda la geometr´ ıa con las mismas propiedades. Se sabe que la capa m´ as externa de la piel, la epidermis, tiene un espesor de micr´ ometros. Sin embargo, la dermis por lo general tiene un espesor de 1,5 mm a 4 mm y la hipodermis alcanza diferentes profundidades dependiendo de la parte considerada del cuerpo. Por lo tanto, cuando se estudian heridas profundas las propiedades de las diferentes capas de la piel deben ser consideradas con el fin de obtener resultados m´ as realistas. Por ´ ultimo, uno de los aspectos m´ as limitantes en el modelado de la cicatrizaci´ on de heridas es la caracterizaci´ on mec´ anica de la piel. Tradicionalmente, se ha utilizado un modelo de material viscoel´ astico para modelar el comportamiento mec´ anico de la piel. Este material reproduce con exactitud algunas de las propiedades de la piel que dependen del tiempo, pero tambi´ en presenta algunas limitaciones. Se ha demostrado experimentalmente que el modelo constitutivo hiperel´ astico encaja mejor en el comportamiento de la piel, aunque su caracterizaci´ on es m´ as complicada. La mayor´ ıa de los modelos de cicatrizaci´ on de heridas que incluyen aspectos mec´ anicos (deformaci´ on del tejido) han considerado la piel (o sus capas) como un material viscoel´ astico (Olsen et al., 1995; Javierre et al., 2009; Murphy et al., 2012). Sin embargo, la simulaci´ on de la piel como un material hiperel´ astico permitir´ ıa reproducir el comportamiento de la piel con m´ as precisi´ on, incluyendo la posibilidad de a˜ nadir anisotrop´ ıa tejido. 44 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas 1.4 Trabajos experimentales A pesar de la existencia de un gran n´ umero de modelos computacionales que simulan los diferentes aspectos de la curaci´ on de heridas, el n´ umero de estudios experimentales en el campo para validar los resultados computacionales es muy reducido. Debido a razones ´ eticas, los experimentos de cicatrizaci´ on de heridas en sujetos vivos no son frecuentes, casi inexistentes en seres humanos. En consecuencia, los ensayos con animales son la mejor opci´ on a pesar de que tambi´ en son objeto de consideraciones ´ eticas. Por otra parte, la piel de los mam´ ıferos presenta diferentes propiedades mec´ anicas a las de la piel humana, y los resultados obtenidos en estos experimentos s´ olo se pueden utilizar como un patr´ on cualitativo para los seres humanos. Uno de los primeros estudios experimentales se debe a Van den Brenk (1956). Van den Brenk (1956) estudi´ o heridas epid´ ermicas en orejas de conejo. Sus resultados fueron utilizados para la validaci´ on de los resultados de Sherratt and Murray (1990). McGrath and Simon (1983) estudiaron la influencia de la forma y el tama˜ no de las heridas profundas en ratas. Observaron la evoluci´ on de heridas con tres geometr´ ıas distintas: una herida cuadrada peque˜ na y otra grande y una herida circular con la misma superficie que la herida cuadrada grande. Para medir el tama˜ no de la herida, McGrath and Simon (1983) siguieron el movimiento de algunos puntos que defin´ ıan la frontera de la herida. Diferenciaron la curaci´ on debida a la contracci´ on de la herida y a la epitelizaci´ on. Despu´ es de una distracci´ on r´ apida de la herida y una fase de seis d´ ıas sin variaciones, las heridas se contrajeron durante alrededor de 30 d´ ıas, y posteriormente permanecieron pr´ acticamente sin variaciones. McGrath and Simon (1983) observaron que despu´ es de 70 d´ ıas todas las heridas hab´ ıan experimentado una tasa de contracci´ on similar, reduciendo su tama˜ no al 35% del tama˜ no inicial. La mayor´ ıa de los mam´ ıferos tales como ratas y conejos presentan una piel m´ as flexible y m´ as el´ astica que los seres humanos, y los niveles de contracci´ on que estos experimentan no son comparables con los de los seres humanos. Sin embargo, la piel del cerdo presenta una estructura m´ as similar a la de la piel humana, por lo tanto el modelo porcino es ampliamente aceptado para las comparaciones. Roy et al. (2009) observaron la evoluci´ on de heridas circulares isqu´ emicas y no isqu´ emicas en cerdos. Se realizaron incisiones y mientras algunas CAP´ ITULO 1. Introducci´ on 45 heridas se dejaron curar normalmente, en otras se introdujeron l´ aminas de silicona entre la dermis y el tejido subyacente. De esta manera se pudieron reproducir las heridas isqu´ emicas, en las que la re-epitelizaci´ on est´ a impedida. Roy et al. (2009) fueron capaces de medir la concentraci´ on de ox´ ıgeno y la presi´ on parcial de ox´ ıgeno en la herida, encontrando niveles de saturaci´ on de ox´ ıgeno m´ as elevados en las heridas no isqu´ emicas. Tambi´ en observaron el cierre de la herida a lo largo de 31 d´ ıas, cuando la herida no isqu´ emica casi hab´ ıa cerrado pero la herida isqu´ emica solo hab´ ıa logrado alrededor del 80% del cierre. Al comparar los resultados experimentales con modelos humanos, hay que tener en cuenta que cada especie tiene diferentes par´ ametros, debido a sus diferentes cin´ eticas celulares y tisulares (Reina-Romo et al., 2010). Por lo tanto, si se obtiene una equivalencia adecuada entre especies, es posible la comparaci´ on de los resultados experimentales en animales y los resultados obtenidos computacionalmente en los modelos humanos. 1.5 Objetivos El objetivo principal de esta tesis es el estudio de la cicatrizaci´ on de heridas en la piel a trav´ es de la simulaci´ on computacional y el uso de un enfoque multidisciplinar. Con el fin de lograr este objetivo, se ha propuesto e implementado un modelo computacional que permite reproducir la cicatrizaci´ on de heridas en diferentes condiciones. M´ as concretamente, el modelo propuesto se centra en la fase de contracci´ on de la herida, ya que tiene un fuerte componente mec´ anico. El modelo tambi´ en permite incluir m´ as procesos que tienen lugar de forma simult´ anea a la contracci´ on de la herida, tales como la angiog´ enesis. El modelo propuesto incluye factores biol´ ogicos (especies celulares, factores de crecimiento y col´ ageno) y tambi´ en factores mec´ anicos (caracterizaci´ on de las propiedades mec´ anicas de la piel y contracci´ on celular). Para encontrar la soluci´ on del problema resultante, se aplica el m´ etodo de los elementos finitos (MEF). El modelo est´ a constituido por dos partes, una relacionada con el an´ alisis bioqu´ ımico del proceso y otra en relaci´ on al an´ alisis mec´ anico de la piel. En primer lugar, la evoluci´ on de las especies bioqu´ ımicas se eval´ ua con un sistema de ecuaciones de reacci´ on-difusi´ on. En segundo lugar, el comportamiento mec´ anico de la piel se modela teniendo en cuenta las relaciones mec´ anicas fundamentales para el modelo de material constitutivo elegido para caracterizarla. Las dos partes est´ an relacionadas a trav´ es de un mecanismo mecanosensor y mecanotransductor, que regula el comportamiento de las c´ elulas en funci´ on de las variables mec´ anicas. El modelo permitir´ a estudiar diferentes casos de curaci´ on y diferentes tipos de heridas. Los objetivos principales se resumen a continuaci´ on: 46 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas •Adaptaci´ on del modelo para el estudio de heridas planas y heridas profundas en dos dimensiones. Las heridas planas se caracterizan por su ´ area superficial, utilizando hip´ otesis de tensi´ on plana. Las heridas largas y profundas se estudian a trav´ es de su secci´ on transversal, bajo un enfoque de deformaci´ on plana y teniendo en cuenta las diferentes capas de la piel que se ven afectadas por la lesi´ on. •Utilizaci´ on de varios modelos constitutivos (viscoel´ astico, hiperel´ astico is´ otropo e hiperel´ astico anis´ otropo) para caracterizar el comportamiento mec´ anico de la piel. Los materiales viscoel´ asticos son una buena opci´ on para reproducir el comportamiento de la piel. Por otra parte, los materiales viscoel´ asticos se pueden utilizar cuando se aplica la teor´ ıa de las peque˜ nas deformaciones, pero no cuando el material trabaja con grandes deformaciones. En este caso, se deben utilizar materiales hiperel´ asticos. •Incorporaci´ on de otros fen´ omenos que tienen lugar de forma simult´ anea a la contracci´ on de la herida, como la angiog´ enesis. Como se sabe, todos los fen´ omenos que tiene lugar durante la cicatrizaci´ on est´ an relacionados entre s´ ı, por lo que la inclusi´ on de la mayor cantidad de procesos proporciona un modelo m´ as completo y realista. El principal inconveniente es el aumento de los recursos necesarios para resolver el problema al incrementar la precisi´ on del modelo. •Incorporaci´ on de nuevas leyes de cin´ etica celular en funci´ on de evidencias f´ ısicas, basadas en estudios experimentales en lugar de las leyes fenomenol´ ogicas propuestas hasta ahora. •Resoluci´ on de los problemas bioqu´ ımicos y mec´ anicos utilizando un enfoque totalmente acoplado o un enfoque no acoplado. Si bien un enfoque totalmente acoplado permite resolver todo el problema al mismo tiempo, un enfoque no acoplado permite dividir el problema y resolverlo en diferentes etapas. Esto es especialmente ´ util para separar el an´ alisis bioqu´ ımico y el an´ alisis mec´ anico debido a sus diferentes escalas de tiempo. Por otra parte, permite utilizar un lagrangiano actualizado y la actualizaci´ on de determinadas variables entre las dos etapas. •Aplicaci´ on del modelo a heridas con diferentes tama˜ nos y formas en dos dimensiones, tanto heridas planas como profundas. La capacidad del modelo para reproducir geometr´ ıas complejas, tales como las consideradas en trabajos experimentales, permite una validaci´ on m´ as completa de los resultados. CAP´ ITULO 1. Introducci´ on 47 •Aplicaci´ on del modelo para simular heridas en tres dimensiones, en las que el efecto de la forma superficial y profundidad de la herida se examinan simult´ aneamente. •Incorporaci´ on de las fibras de col´ ageno en la piel sana. Las fibras de col´ ageno permiten simular la anisotrop´ ıa real de la piel en lugar de utilizar simplificaciones is´ otropas, lo cual es importante en la piel sana que rodea a la herida. 1.6 Financiaci´ on Los estudios incluidos en esta tesis han sido financiados por: •El Ministerio Espa˜ nol de Ciencia e Innovaci´ on a trav´ es de la beca BES2010037281 y el proyecto DPI2009-07514. Este proyecto fue parcialmente financiado por la Uni´ on Europea (a trav´ es de los fondos FEDER). •El Ministerio Espa˜ nol de Econom´ ıa y Competitividad a trav´ es del proyecto DPI2012-32888. Este proyecto est´ a parcialmente financiado por la uni´ on europea (a trav´ es de los fondos FEDER). •El Consejo Europeo de Investigaci´ on (ERC) a trav´ es del proyecto ERC2012-StG 306751. 48 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas CAP´ ITULO 2 Conclusiones La motivaci´ on principal de esta tesis es el estudio de la piel despu´ es de sufrir una lesi´ on. Entender el proceso de cicatrizaci´ on de la herida con el fin de reproducirla mediante modelos computacionales es fundamental para avanzar en el desarrollo de soluciones curativas y procedimientos quir´ urgicos. Para lograr este objetivo, se ha propuesto e implementado un modelo de contracci´ on de heridas en piel humana para ayudar a entender el proceso y sus posibles soluciones cuando este no sigue la evoluci´ on normal. El trabajo se ha centrado en el punto de vista te´ orico del problema, incluyendo ecuaciones espacio-temporales de difusi´ on-convecci´ on para describir el problema biol´ ogico combinado con la formulaci´ on del problema mec´ anico. 2.1 Conclusiones generales Las conclusiones principales que se han obtenido de esta tesis, considerando los estudios num´ ericos realizados, se presentan a continuaci´ on. Teniendo en cuenta que las heridas que se contraen en menor medida generan cicatrices m´ as peque˜ nas, lo cual es muy conveniente, las principales conclusiones desde el punto de vista biol´ ogico, son las siguientes: •Cuando se estudian heridas profundas a trav´ es de su secci´ on transversal, se observa que las heridas con la misma ´ area pero diferente longitud de superficie libre presentan diferentes porcentajes de contracci´ on. Las heridas que son m´ as profundas y m´ as estrechas se contraen menos que las heridas m´ as anchas y menos profundas. •Cuando la contracci´ on de la herida y la angiog´ enesis son simuladas juntas en heridas planas, se ha encontrado que las heridas m´ as alargadas (el´ ıpticas) 49 50 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas muestran una vascularizaci´ on m´ as r´ apida que las circulares. Sin embargo, la contracci´ on experimentada por ambas heridas estudiadas, en comparaci´ on con el tama˜ no inicial, es similar. La mayor rapidez de los procesos bioqu´ ımicos en la herida el´ ıptica pueden ser causados por la difusi´ on m´ as r´ apida de las sustancias, debido a la relaci´ on m´ as elevada de contorno respecto a la superficie de la herida. •Cuando se analiza el efecto del tama˜ no en las heridas planas, se observa que las heridas grandes comienzan a contraerse antes en el tiempo. Sin embargo, al final del proceso de contracci´ on estas heridas han contra´ ıdo menos que las heridas m´ as peque˜ nas, en relaci´ on con el tama˜ no inicial de la herida. •En heridas profundas, es muy importante tener en cuenta las diferentes capas de la piel. La dermis y los tejidos subyacentes tienen diferentes propiedades mec´ anicas y la dermis es m´ as r´ ıgida que el tejido subyacente. En relaci´ on al modelo, su implementaci´ on y sus aspectos num´ ericos, las conclusiones m´ as importantes son: •Cuando las heridas se analizan en dos dimensiones, la decisi´ on m´ as importante es elegir el estudio de la herida bajo un enfoque de deformaci´ on plana o de tensi´ on plana. Esto implica el estudio de la herida a trav´ es de su secci´ on transversal o de su ´ area superficial. La selecci´ on entre los dos enfoques depende de qu´ e tipo de herida se est´ a estudiando. •El modelo mec´ anico constitutivo del material determina altamente la evoluci´ on del proceso de contracci´ on. El uso de modelos m´ as realistas, como el hiperel´ astico ayuda a obtener tambi´ en resultados m´ as realistas. Incluir las fibras en el modelo hiperel´ astico de la piel permite modelar el comportamiento anis´ otropo real de la piel. Se ha encontrado que la orientaci´ on de las fibras en la piel no modifica altamente la tasa de contracci´ on de la herida, pero modifica la distribuci´ on de las deformaciones volum´ etricas alrededor de la herida. •Las variables mec´ anicas influyen en procesos biol´ ogicos como la diferenciaci´ on celular y la generaci´ on de tensiones. Se ha postulado que la deformaci´ on volum´ etrica es el est´ ımulo mec´ anico que regula este proceso, ya que depende de la rigidez del tejido. •La inclusi´ on de leyes basadas en evidencias experimentales en lugar de las leyes fenomenol´ ogicas existentes proporciona resultados igualmente buenos. CAP´ ITULO 2. Conclusiones 51 2.2 Contribuciones originales El desarrollo de esta tesis ha dado como resultado un modelo computacional para simular la curaci´ on de heridas que permite incluir diferentes procesos y estudiar distintas variables y su efecto en el proceso de curaci´ on. Los aspectos de esta tesis que son completamente nuevos y que no han sido incluidos en modelos previos de cicatrizaci´ on de heridas, por lo que sabemos, son: •La tensi´ on generada por las c´ elulas se ha modelado a trav´ es de una nueva ley basada en evidencias experimentales. Los mecanismos considerados anteriormente para representar estas tensiones eran principalmente fenomenol´ ogicos, y aunque los resultados son igualmente buenos, no exist´ ıa justificaci´ on f´ ısica para estas leyes. El modelo propuesto generaliza los modelos anteriores con un mecanismo de regulaci´ on de la actividad celular regulado por la rigidizaci´ on de la matriz extracelular. •La ley fenomenol´ ogica de diferenciaci´ on de fibroblastos tambi´ en se ha actualizado por una ley basada en evidencias f´ ısicas. Una vez m´ as, la ley propuesta sigue observaciones experimentales regul´ andose a trav´ es de la rigidez del tejido, f´ acilmente medible. •Se ha presentado un modelo de cicatrizaci´ on de heridas, que incluye la angiog´ enesis y la contracci´ on de la herida en tres dimensiones. La mayor´ ıa de los modelos existentes reproducen geometr´ ıas bidimensionales, utilizando simplificaciones de tensi´ on plana o de deformaci´ on plana o incluso simplificaciones unidimensionales. El desarrollo de un modelo tridimensional para estudiar la cicatrizaci´ on permite reproducir geometr´ ıas m´ as complejas y realistas; tambi´ en analizar el comportamiento global de la herida. •Se ha utilizado un material hiperel´ astico que incorpora el efecto anis´ otropo de las fibras de col´ ageno para modelar el comportamiento de la piel. Los modelos existentes de cicatrizaci´ on de heridas no tienen en cuenta este comportamiento, que en parte regula el proceso de curaci´ on. Desde el punto de vista num´ erico las contribuciones principales son: •Se ha formulado el problema de manera totalmente acoplada y por otra parte de manera desacoplada. La forma tradicional de resolver este tipo de problema es utilizando la implementaci´ on totalmente acoplada, que permite evaluar los problemas bioqu´ ımico y mec´ anico al mismo tiempo. La contribuci´ on de este trabajo es el desacoplamiento de los dos modelos. 52 Modelado mecanoqu´ ımico de la cicatrizaci´ on de heridas Este enfoque permite hacer frente a las dos escalas de tiempo diferentes de los problemas bioqu´ ımico y mec´ anico. Adem´ as, permite actualizar las variables entre la soluci´ on de los dos problemas. •Se ha implementado la actualizaci´ on de la malla para resolver el problema de manera m´ as precisa, mediante un algoritmo lagrangiano actualizado, necesario para utilizar las hip´ otesis de peque˜ nas deformaciones. Utilizando esta t´ ecnica se evita que las deformaciones sufridas por la malla sean mayores que las permitidas cuando se utilizan las hip´ otesis de peque˜ nas deformaciones. •Se ha implementado una subrutina de elemento de usuario utilizando el programa comercial de elementos finitos Abaqus®(Hibbit et al., 2008). Este elemento incorpora las ecuaciones del modelo que describen el problema y las condiciones de contorno del problema, que se resuelve con el m´ etodo de los elementos finitos. Por otra parte, la aplicaci´ on de las condiciones de contorno naturales para un problema con frontera libre se han incluido en la formulaci´ on. El elemento desarrollado permite elegir entre los enfoques de tensi´ on y deformaci´ on plana en dos dimensiones, permitiendo simular heridas planas y heridas profundas. 2.2.1 Publicaciones y comunicaciones El trabajo desarrollado en esta tesis ha dado lugar a tres art´ ıculos publicados en revistas JCR y otro art´ ıculo que se encuentra en el proceso de revisi´ on. Adem´ as, un quinto art´ ıculo ha sido publicado en una revista JCR, como resultado del trabajo desarrollado durante una estancia de investigaci´ on en los EE.UU. Estos documentos se enumeran en la Tabla 2.1. Adem´ as de estos art´ ıculos, se han presentado un total de 12 contribuciones en congresos nacionales e internacionales. Las contribuciones est´ an detalladas en la Tabla 2.2. Finalmente, se recibi´ o la invitaci´ on para la colaboraci´ on en la escritura de un cap´ ıtulo de libro con los resultados de este trabajo: Autores: E. Reina-Romo, C. Valero, C. Borau, R. Rey, E. Javierre, M.J. G´ omezBenito, J. Dom´ ınguez, J.M. Garc´ ıa-Aznar T´ ıtulo: Mechanobiological modelling of angiogenesis: impact on tissue engineering and bone regeneration Libro: Computational Modeling in Tissue Engineering, pp.379-404 Fecha: 01/2013 GENERAL INTRODUCTION AND CONCLUSIONS Mechanochemical modeling of wound healing: multiphysics finite element simulations. 60 CHAPTER 3 Introduction Wounds in the skin are a major health matter that most people suffer along their live. When the injured person has not serious health problems wounds heal without more relevant implications than leaving a scar in the skin. Nevertheless, when the healing process does not follow its natural evolution several complications arise from it, leading to chronic wounds and other healing disorders. Chronic wounds affect around 6.5 million patients in the United States with an annual cost of US$25 billion (Sen et al., 2009) and reducing these numbers is a permanent objective for the health system. Finding healing improvements is time consuming and has high economic cost. Moreover, ethical implications are always present when experiments involve living subjects. Thus, mathematical models of wound healing have become more popular during the last decades. Mathematical models allow to study a wide number of cases varying parameters with a low time and economic cost, adding the possibility of studying specific cases for individual patients. Therefore, the goal of this thesis is the development of a wound healing model to predict the healing evolution of wounds under different conditions and that allows the study of a wide range of wounds and factors influencing healing. 3.1 The skin Skin is the outer coverage of vertebrate animals and presents different structure and properties in every species. In humans, skin is the largest organ in the body and it represents 8 % of the body mass (Gray et al., 1995). Skin covers the whole human body and its thickness varies in every anatomical location (Odland, 1991), ranging from 1,5 mm to 4 mm considering the epidermis and the dermis. The underlying tissue has a highly variable thickness and composition depending on the 61 62 Mechanochemical modeling of wound healing anatomical location. Skin is usually classified into two types: thin and thick. Thin skin is found in parts such as the eyelids or where hair follicles are present, while thick skin is usually hairless and it is mainly located on the hand palms or the feet soles. Skin constitutes the interface between the body and the environment and it is a natural barrier that protects internal organs from external aggressions. More than just a physical barrier, skin has several functions; it avoids dehydration regulating water loss and acts as a thermal barrier. The mechanisms that skin uses to regulate the body temperature are mainly the evaporation of sweat and convection, regulating the blood flow along the body surface. One of the most important functions of the skin is its role as a part of the immune system, it avoids the entrance of strange particles and pathogens (any agent that can cause disease) in the body (Fore-Pfliger, 2004). In addition, several physiological systems necessary for the proper body functioning, such as nerves, capillary and lymphatic systems, are allocated in the skin. Thus, keeping the skin stability and avoiding any factor that would decrease its properties is highly important. Moreover, it is also crucial to achieve a fast and functional recovery of the skin when it suffers a damage. 3.1.1 Structure of the skin Human skin consists of collagen, elastin and a variable number of cellular species (Table 3.1). It is organized into three main layers, from the more external to the more internal: epidermis, dermis and hypodermis (Wilkes et al., 1973). Wounds in the skin usually go through the epidermis and reach the dermis. The epidermis is the most external layer and it is in contact with the environment, which makes it to have the principal role in the protection function of the skin. Epidermis constitutes a barrier against water loss and against the entrance of external substances. Despite of its important role, epidermis is also the thinnest layer, presenting a thickness of 75-150µm(Odland, 1991), and it is divided into five layers: Stratum corneum, Stratum lucidum, Stratum granulosum, Stratum spinosum and Stratum germinativum (Figure 3.1). Epidermis is mainly constituted by keratinocytes, which are also part of the immune system and produce anti-inflammatory mediators. As it is in direct contact with the environment, the epidermis capacity of self-renovation is crucial; it presents equal cell production and cell death rates in equilibrium. The dermis is the intermediate skin layer and it is connected to the epidermis by the basement membrane. It has a variable thickness of 1-4mm and it is usually divided into two regions, the papillary region and the reticular region (Figure 3.2). The dermis is mainly made of collagen type I and III embedded on a ground CHAPTER 3. Introduction 63 Figure 3.1: Histologic image of the epidermis layers. Source: http://en.wikipedia.org/wiki/File:Epidermal layers.png substance composed of proteoglicans, fibronectins (Gray et al., 1995) and variable cellular species (fibroblasts, myofibroblasts, endothelial cells, macrophages, neutrofils or linfocits among others (Table 3.1)). While some cellular species, such as fibroblasts, are naturally in the skin other species appear only when they are needed. This is the case of myofibroblasts, that only appear when an injury is produced, or macrophages that migrate to the locations where there are pathogens to eliminate them. One of the main components of the dermis is collagen. Collagen is the most abundant protein in the skin and it is usually organized in a fiber network with elastic properties that gives integrity to the skin. In fact, the mechanical tensile strength of the skin is due to this collagen lattice. Deep cuts and wounds penetrate the dermis and reach the underlying layer, the hypodermis, which is located under the reticular region of the skin. 64 Mechanochemical modeling of wound healing Name Function Location Wound healing stage/process Others Adipose cells/Adipocytes Store energy as fat. Hypodermis Synthesize several hormones. Endothelial cells Form new capillaries during angiogenesis. Blood vessels Epithelialization Motile cells that migrate to the wound site during the epithelialization stage. Fibroblasts Synthesize collagen type III. Generate contraction forces. Dermis Hypodermis Epithelialization Contraction Motile cells that migrate to the wound site during the epithelialization stage. Keratinocytes Main constituent of the epidermis. Form a barrier against environmental damage. Modulates the immune system producing anti-inflammatory mediators. Epidermis Epithelialization Motile cells that migrate to the wound site during the epithelialization stage. Macrophages Phagocytosis. Eliminate bacteria and dead cells at injured sites. Dermis Hypodermis Inflammation Derived from blood monocytes. Enter the damaged site through the endothelium of blood vessels. Chemotactically attracted to the wound site by cytokines released from damaged cells. Part of the immune system. Continued on next page CHAPTER 3. Introduction 65 Table 3.1 – Continued from previous page Name Function Location Wound healing stage/process Others Myofibroblasts Synthesize collagen type III. Generate contraction forces. Secrete factors that induce angiogenesis. Dermis Epithelialization Contraction Angiogenesis Differentiated fibroblasts. Non-motile. Neutrophils Eliminate bacteria and dead cells at injured sites. Bloodstream Inflammation Attracted by inflammation factors. Part of the immune system. Platelets Prevent bleeding forming a plug where there is vascular endothelial damage. Blood coagulation. Dermis Hemostasis Release PDGF. Table 3.1: List of cellular species involved in wound healing. 66 Mechanochemical modeling of wound healing Figure 3.2: Histologic image of the epidermis and the dermis layers. Source: http://upload.wikimedia.org/wikipedia/commons/8/84/Epidermis-delimited.JPG The hypodermis, also called subcutaneous tissue or superficial fascia, is the deepest layer and its thickness can reach the order of the centimeters depending on the anatomical location. It usually consists of subcutaneous fat and connective tissue and it also contains many important physiological systems, such as blood vessels (see Figure 3.3) and nerves. Furthermore, as the epidermis and the dermis, the hypodermis contains several cellular species, mainly fibroblasts, adipose cells and macrophages. Bones, muscles and internal organs lay under the hypodermis, making its recovery after damage to be necessary for a global physiological stability. Figure 3.3: Schematic image of the vascular system through the skin layers, (Gray et al., 1995). CHAPTER 3. Introduction 67 The combined function of the three layers is crucial for the body functioning, and maintaining the integrity of the whole structure is necessary for living. Therefore, preserving the skin undamaged and with a good mechanical properties is one of the most important health matters. 3.1.2 Mechanical properties of the skin The mechanical properties of the skin have been studied since centuries ago. One of the first experiments on this field was performed by Langer (1861), illustrating that skin is naturally subjected to anisotropic stress. Skin’s pre-stress is clearly observed when a wound occurs and the skin relaxes and looses, causing that the initial defect size increases. In addition to that, the mechanical properties of the skin vary depending on the skin location, orientation and depth but also on the age; skin loses its elasticity and recovery capacity along time (Escoffier et al., 1989). Most of the mechanical properties of the skin are due to the fibers that compose its extracellular matrix (ECM), which have an elastic modulus of 150-300 kPa (Wilkes et al., 1973). Matrix fibers have a high tensile strength derived from the organization of three primary proteins chain (collagen, elastin and fibrin) into a superhelix (Kerr, 2010). The main component of these fibers is collagen, that gives most of the tensile strength to the ECM. The second main component of the ECM is elastin which gives elastic properties to the skin and allows it to recover its original state after being stretched. Protein fibers are embedded on a ground substance made of proteoglicans and fibronectins (Gray et al., 1995), that help cells to move through the fibers. Collagen fibers align in the skin following the stress lines or Langer lines, which are present in the body surface and were discovered by Langer (1861). In his research, Langer (1861) performed circular cuts in the skin in all the body surface, finding that these cuts turned into ellipses aligned with tension lines when the skin relaxed. The orientation of these cuts defined the natural orientation of collagen fibers, usually parallel to the underlying muscles (Figure 3.4). The importance of these lines has been experimentally proved in some processes like wound healing, where it has been found that wounds heal differently depending on their relative orientation to Langer lines. The relative position of wounds in the skin has shown that wounds parallel to Langer lines heal better and produce less scaring while wounds that are perpendicular to Langer lines present more difficulties to heal (Motegi et al., 1977). 68 Mechanochemical modeling of wound healing Moreover, the magnitude of the skin pre-stress found in Langer (1861) experiments has been measured in the arm and forearm by Flynn et al. (2011a), where they found values ranging from 28 to 92 kPa. Figure 3.4: Distribution of the stress lines in the human trunk (Langer, 1861). Further knowledge of the skin properties has been an important issue for experimental researches, and several measurement methods have been reported. During the last decades, both in-vivo and in-vitro experiments have been designed to characterize the mechanical behavior of the skin and find accurate parameter values, necessary for modeling and developing skin substitutes. Most of the in-vivo studies to measure skin properties are carried out in the forearm or the upper arm and use different mechanical assays such as extension, indentation, suction or torsion. Diridollou et al. (2000) used suction tests in the forearm and an inverse method to identify the nonlinear material parameters of the skin. They applied negative pressure at the skin surface and measured the skin deflection, finding that the skin becomes stiffer for higher strains. Boyer et al. (2007) studied the mechanical properties of the epidermis and the dermis using a micro indentation device and characterized it as a viscoelastic material, finding that the complex modulus has values of 47.3 to 128.3 N/m. They defined the complex modulus as a variable that has a part which is independent of the frequency and other part dependent of the frequency. CHAPTER 3. Introduction 75 Name Function Wound healing stage / Process Others Epidermal growth factor (EGF) Stimulates proliferation and differentiation of epithelial and mesenchymal cells. Granulation tissue formation. Epithelialization Secreted by macrophages and keratinocytes. Fibroblasts growth factors (FGFs) Fibroblast chemotaxis and proliferation. Keratinocyte migration and proliferation. Stimulates angiogenesis, wound contraction and matrix deposition. Angiogenesis Secreted by macrophages, endothelial cells and fibroblasts among others. Keratinocyte growth factor (KGF) Keratinocyte migration, proliferation and differentiation. Epithelialization Secreted by keratinocytes. Macrophage derived growth factor (MDGF) Stimulates the proliferation of fibroblasts, smooth muscle cells and endothelial cells. InflammationEpithelialization Secreted by macrophages. Platelet derived growth factor (PDGF) Stimulates fibroblasts proliferation. Regulates growth and division of multiple cells. Influences blood vessel formation (angiogenesis) and tissue remodeling. Inflammation/Hemostasis Secreted by platelets. Continued on next page 76 Mechanochemical modeling of wound healing Table 3.2 – Continued from previous page Name Function Wound healing stage / Process Others Transforming growth factor beta (TGF-β) Stimulates the differentiation of fibroblasts into myofibroblasts. Controls proliferation, differentiation, apoptosis and other functions in most cells. Stimulates ECM production. Epithelialization /Differentiation Secreted by fibroblasts, myofibroblasts and macrophage among others. Vascular endothelial growth factor (VEGF) Stimulates angiogenesis and vasculogenesis, directs the sprouting of capillaries. Promotes migration of endothelial cells and fibroblasts. Chemotactic for macrophages. Hipoxia-Angiogenesis Produced by cells receiving insufficient oxygen. Table 3.2: List of growth factors involved in wound healing. CHAPTER 3. Introduction 77 Equally problematic than excessive scarring is impaired wounding, which give place to chronic wounds. Chronic wounds take longer than normal wounds to heal and usually they do not heal completely without help. There are a number of factors that contribute to the occurrence of chronic wounds such as systemic diseases (diabetes), arterial insufficiency, infection or advanced age. The first cause of chronic wounds are venous ulcers (Snyder, 2005), caused by the malfunction of venous valves and a non adequate vascular ingrowth. Venous ulcers usually appear in the legs, are more likely in diabetic patients and in critical cases they end in the amputation of the limb. Diabetic patients have around 15% of probability of suffering a diabetic foot ulcer, an open sore located in the bottom of the foot. The main factors that give place to diabetic foot are vascular disease and diabetes neuropathy, a decrease in sensing pain caused by nerve damage after maintained high blood glucose levels. Moreover, diabetes impairs the normal evolution of the healing process, prolonging the inflammatory stage, which delays the formation of granulation tissue and reduces wound tensile strength (Ogawa and Hsu, 2013). Due to its high impact, a number of solutions have been developed to treat chronic wounds in diabetic patients, including the application of growth factors, the use of skin substitutes, negative pressure therapies and hyperbaric oxygen therapy (HBOT). Pressure ulcers are also one of the most complicated wounds to heal, and are produced usually when the patient must stay immobile for long periods of time, mostly in bed or wheelchair. Thus, pressure ulcers appear usually in places where there is a bony prominence, such as the sacrum, coccyx, hips or heels, as a result of the constant pressure that is applied to the soft tissue (Bluestein and Javaheri, 2008). As a consequence, blood flow is decreased, leading to ischemia and tissue necrosis. As the pressure ulcer evolves (Figure 3.8), tissue thickness is reduced until the dermis is destroyed and the underlying fat is exposed. In the most severe stages (3 and 4) of pressure ulcer, fat also disappears and bones and muscles are uncovered (Topman et al., 2012). The first stages of pressure ulcer appear after a few hours of immobility. Therefore, the best way to prevent them is a constant change of the patient position, usually every two hours (Bluestein and Javaheri, 2008), to allow a change in the pressure distribution. The most extended treatment for pressure ulcers consists on the debridement of the necrotic tissue to avoid bacterial growth. Finally, another complication that can arises from wound healing is tissue contracture. It is caused when dermal scar tissue continues shortening after healing by shortening of the ECM material caused by an excessive myofibroblasts population (Tomasek et al., 2002; Li and Wang, 2011). The forces generated by my- 78 Mechanochemical modeling of wound healing Stage 1 Stage 2 Stage 3 Stage 4 Figure 3.8: Pressure ulcer stages (Gauglitz et al., 2011) ofibroblasts give place to a permanent excessive contracture that decreases tissue functionality reducing the tissue mobility in extreme cases. These healing disorders together with their solutions are summarized in Table 3.3. 3.2.3 Healing strategies Preventing the appearance of healing complications is the first strategy for a proper recovery after injury. One of the most extended therapies to help impaired healing is applying skin grafts, which consists on the transplantation of tissues from the same patient or from another subject of the same or different species (Billingham and Medawar, 1951) to the affected area. It is usually used in diabetic foot disease, venous ulcers or microsurgery wounds (Lineaweaver, 2013). In this procedure, damaged skin is removed from the patient and replaced by the new tissue. Implants can be half-skin thickness or full-skin thickness when wounds are more complicated (Rose et al., 2014). Less frequently, tissues artificially created with a specific purpose are transplanted instead of the natural ones (Tauzin et al., 2014). Some complications that can arise from this technique are infection, nerve damage or tissue rejection. As it is known that wound healing has a high mechanical component, there exist a number of mechanical-based solutions, that modify the mechanical state of the wound. It has been proved that the use of fixable materials such as sutures, tapes or sheetings that limit skin stretching and reduce tension in the wound help to avoid the appearance of hypertrophic scars (Visscher et al., 2001; Widgerow et al., 2000). Nevertheless, when it is not possible to avoid this wounds to appear a number of treatments are available. CHAPTER 3. Introduction 79 Disorder Cause Effect Treatment PATHOLOGICAL SCARS Hypertrophic scars Fibroproliferative disorders Excessive scaring upward Pain Contracture Pressure therapy HBOT Sutures, tapes and sheetings Keloids (tumoral hypertrophic scars) Fibroproliferative disorders Excessive scaring outward of the wound Pain Contracture Sutures, tapes and sheetings CHRONIC WOUNDS Pressure ulcers Patient immobility Ischemia Tissue necrosis Change of patient position Debridement of necrotic tissue Venous ulcers Malfunction of venous ulcers Skin grafts Diabetic foot ulcer Vascular disease Diabetes Open sore in the bottom of the foot Skin grafts Contracture Excessive myofibroblasts population Shortening of the scar tissue. Decrease of tissue functionality Table 3.3: List of healing disorders and their treatment. 80 Mechanochemical modeling of wound healing One of the most applied devices is the vacuum assisted closure (VAC), which consist on a foam cover that is laid over the wound and sealed with a film. Negative pressure is applied using a vacuum pump and a drainage tube to promote faster tissue growth (Argenta and Morykwas, 1997; Scherer et al., 2002). Hydrostatic pressure gradients and shear forces can be regulated with these devices. Another therapy based on the use of different pressures is the hyperbaric oxygen therapy (HBOT), that consists on applying 100% oxygen to the body at a pressure higher to 1 atmosphere. It has been applied to various wound types, for instance acute wounds, chronic wounds and diabetic wounds (Eskes et al., 2011). Eskes et al. (2011) performed a review to find the effect of HBOT on acute wounds. They found that HBOT is beneficial in patients with crush injuries or burns, and it is more effective when applied together with skin grafts. HBOT has multiple positive effects such as reducing infection and cell death, generation of oxygen free radicals, inhibition of bacterial functions and the improvement of the transport of certain antibiotics. Moreover, HBOT amplifies oxygen gradients around ischemic wounds and improve the formation of collagen matrix (Gill and Bell, 2004). The main negative effects of HBOT are elevated pressure and hyperoxia, which can cause problems in the photoreceptor and hearing system. One of the most used methods for treating hypertrophic scars is pressure therapy. the effects of this therapy are mainly acceleration of the remodeling process and reduction of the scar thickness together with a stiffness decreasing. Finally, although it has not been yet applied to humans, the use of specific growth factors to promote wound healing has been seen to be effective. Kwon et al. (2006) studied the effect of applying certain epidermal growth factor (recombinant human epidermal growth factor rhEGF) in full thickness wounds in rats. They applied EGF topically, finding that wound closure was higher in presence of the growth factor. The use of EGF increased the proliferation of myofibroblasts and the rate of collagen production. Moreover, it has been proved that EGF decreases scar formation. 3.2.4 Scar mechanical properties After a wound heals, the newly created tissue has not the same properties as the initial one. It has been reported that wound tissue shows only about 20 % of their normal strength in the first three weeks. Moreover, healed tissue only presents around 70 % of the resistance of the undamaged tissue (Levenson et al., 1965). CHAPTER 3. Introduction 81 Although there are not many experimental works that measure the scar tissue properties, some authors have done it. Among them, Clark et al. (1996) used the elastic modulus and the strain of the skin to determine the severeness of postburning hypertrophic scars. Hypertrophic scars present different properties from normal scars, becoming stiffer and less extensible as long as the scar is more severe. Clark et al. (1996) chose large scars on the arm and forearm and performed extensometer test of the scars and healthy skin of the unaffected arm when possible. 3.2.5 Mechanosensingandmechanotransductionin woundhealing Physiological processes are dependent of multiple biological and chemical factors. For example, Vaughan et al. (2000) studied the effect of transforming growth factor-β1(TGF-β1) on myofibroblasts. They found that the force generated by myofibroblasts is dependent of the TGF−β1dose, as it regulates the expression of α−SM actin. Moreover, TGF−β1also stimulates fibroblasts differentiation into myofibroblasts and increases collagen production (Petrov et al., 2002). Although these biochemical factors, have a crucial role in physiology, they are not the only regulatory factor. Multiple studies have shown evidence about the influence of mechanics on these processes. Mechanical forces such as stretching tension, shear force, scratch and compression, among others, are perceived by cellular mechanoreceptors/mechanosensors (Ogawa, 2011). Hinz et al. (2001) elucidated whether mechanical tension enhanced fibroblasts differentiation in vivo. With this purpose, they studied full-thickness wounds in rats, using a plastic frame in some wounds to splint them and induce a higher mechanical tension. Hinz et al. (2001) found that higher tension induced greater tissue contractility and myofibroblasts markers, which should be a consequence of the formation of stress fibers. Furthermore, α−SMA expression is favored by mechanical tension (Grinnell, 2000). Experimental evidence suggest that cells feel and react to mechanical stimuli from their environment (mechanosensing) to regulate their activity and to turn their environment into a more comfortable one. Analogously, cells can translate biochemical information into mechanical activity (mechanotransduction). Processes like cell migration, differentiation or force generation have been proved to be influenced by mechanical stimuli. 82 Mechanochemical modeling of wound healing One of the most studied mechanical stimuli is tissue stiffness. The attraction of cells to rigid substrates is called durotaxis, and it is though to be one of the mechanisms involved in tumor growth (Harland et al., 2011). Pelham and Wang (1997) investigated the motility of fibroblasts on substrates with different stiffness. They hypothesized that cells evaluate the stiffness by pushing and pulling their integrin receptors. This observation is consistent with the idea that the mechanical environment is evaluated by the cell through their actin-myosin mechanism. Moreover, they observed that focal adhesions became more stable when the substrate stiffness was increased. Cells are anchored to the substrate through focal adhesions (FA) and show different behaviors depending on the mechanical properties of the substrate. Discher et al. (2005) reviewed the influence of the substrate stiffness in cellular activity. Cells generate contractile forces through their actin and myosin filaments. Moreover, cells adhere only to substrates that have a stiffness higher than a threshold value, creating more stable focal adhesions in stiffer substrates and regulating the force generation in response to the resistance of the substrate. Wells (2008) also summarizes the effect on cell functions such as differentiation and proliferation when the substrate (ECM) stiffness is modified, finding in liver cells similar conclusions to Pelham and Wang (1997). Mitrossilis et al. (2009) also demonstrated that cells on elastic substrates modify their activity according to the substrate stiffness. Mitrossilis et al. (2009) wanted to elucidate whether cellular response was determined by stiffness or forces by applying a uniaxial traction to a single living cell. Their in-vitro experiments demonstrated that the forces exerted by cells increase as the substrate becomes stiffer, and that a saturation force level is reached. In addition, Jones and Ehrlich (2011) performed in-vitro tests on collagen surfaces of different stiffness. Their results demonstrate that substrate stiffness conditions the secretion of integrins. 3.3 Computational models of wound healing For the last decades, mathematical models have been developed to study different biological and physiological processes such as tissues damage and recovery. In particular, a number of these models have focused on skin and wound healing. Theoretical models are of great importance, they can help to elucidate why a wound heals properly or leads into a chronic wound. Moreover, theoretical CHAPTER 3. Introduction 83 models offer the possibility of studying the factors that have influence in wound healing. These factors can be analyzed isolated or together with other factors and it is possible to elucidate which factors are more important. During the last few years wound healing and more particularly some processes such as wound contraction and angiogenesis have been broadly modeled using continuous models. Nevertheless, a model that includes all the wound healing stages has not been developed yet, as the number of processes and variables involved is too large and it would require a high computational capacity and would take large calculation times. The most relevant aspects of the reviewed models are summarized in Table 3.4. 3.3.1 Epidermal wound healing models A first work about epidermal wound healing was developed by Sherratt and Murray (1990). Epidermal wound closure is due only to epidermal cells migration and it is not affected by wound contraction, that only takes place when deeper layers of the skin are affected. Sherratt and Murray (1990) proposed a continuous reactiondiffusion model setting that a chemical growth factor produced by epidermal cells diffuses through the tissue. Furthermore, cell variation comes from migration and mitosis regulated by the produced chemical factor and also by apoptosis (Clark, 1988; Folkman and Moscona, 1978). The model was applied to predict epidermal wound healing in a circular wound considering that the wound had healed when it reached a cell density higher than the 80 % of the cell density of the healthy tissue. Sherratt and Murray (1990) compared their results with one of the few experimental works about epidermal wound healing, performed by Van den Brenk (1956) in rabbits ears. Sherratt and Murray (1990) proposed the first model including chemical control, and it has been the base for most of the subsequent wound healing and contraction models. Moreover, their model has been also adapted to simulate wound healing in other tissues. For instance, Dale et al. (1994) adapted it to study corneal epithelial wound healing as a function of the epithelial growth factor (EGF). In addition, epidermal wound healing has been simulated together with other processes that take place simultaneously, such as angiogenesis (Maggelakis, 2003). 84 Mechanochemical modeling of wound healing Author Simulated processes Model variables Mechanical model Geometry Validation with experimental works Sherratt and Murray (1990) Epidermal wound healing Epidermal cells Mitosisregulating GF Not considered Superficial circular wounds simplified by 1D axisymmetric approach Validated with rabbit (Van den Brenk, 1956) Tranquillo and Murray (1992) Dermal wound healing Wound contraction Fibroblasts Collagen ECM ECM displacement Generic chemical growth factor Viscoelastic Superficial circular wounds simplified by 1D axisymmetric approach Validated with rat (McGrath and Simon, 1983) Dale et al. (1994) Corneal epithelial wound healing Cells EGF Not considered Superficial circular wounds simplified by 1D axisymmetric approach None Continued on next page CHAPTER 3. Introduction 91 3.3.2 Wound contraction models Although epidermal wounds represent a good first approach to study wound healing, those wounds that affect deeper layers and present more difficulties to heal are also more interesting to study. It has been observed that wounds that reach the dermis do not heal only with the effect of cell migration and more complex mechanisms are needed for a success healing. Wound contraction is one of the most important processes during dermal wound healing and it is highly influenced by mechanics. The reduction of the wound size is due in part to the retraction of the wound contour inward. This displacement is created by the balance between the forces that cells generate in the tissue where they are embedded and the resistance that the tissue offers to be deformed. The first wound contraction model was proposed by Tranquillo and Murray (1992) and it has been the base for most wound contraction models until now. The model was composed by a set of differential equations describing the conservation of a cellular species (fibroblasts) and the collagen density of the ECM and the linear momentum of the matrix. Tranquillo and Murray (1992) included the effect of cell growth, migration and diffusion together with the passive convection due to the ECM movement. On a first approach they considered that the ECM variation was due only to its passive convection. Later, they proposed a more realistic ECM evolution law that includes ECM synthesis by fibroblasts. Moreover, in the same work they proposed a more complete biochemical model including the chemotactic effect of a chemical growth factor. The linear momentum law included a force balance between the matrix (characterized as a viscoelastic material) and the traction forces exerted by cells. They considered a domain divided into two parts, the wound and the surrounding healthy skin. The model investigated one dimensional wounds and followed the displacement of the wound margin during time to measure the wound size. The model of Tranquillo and Murray (1992) was later extended by Olsen et al. (1995). Olsen et al. (1995) proposed a wound contraction model that included the displacement of the contracted tissue and applied it to normal and pathological wounds. One of the main differences between the works of Tranquillo and Murray (1992) and Olsen et al. (1995) is the incorporation of the effect of myofibroblasts in the contraction process by Olsen et al. (1995). Myofibroblasts are non-motile activated fibroblasts that due to their similarities to muscle cells are able to exert higher traction forces than fibroblasts. It has been reported that a proper contraction level is not reached without myofibroblasts forces (Tomasek 92 Mechanochemical modeling of wound healing et al., 2002). Olsen et al. (1995) model describes the temporal evolution of fibroblasts, myofibroblasts, a chemical growth factor and the extracellular matrix during wound contraction. To model the cellular evolution they included cell mitosis, differentiation, apoptosis, passive convection for both species and fibroblasts migration due to random dispersal and chemotaxis. They predicted the evolution of the wound until reaching a steady state. Following Tranquillo and Murray (1992), they applied the model to unidimensional geometries, which reduces its applicability to circular wounds assuming axisymmetry. Cook (1995) proposed a mechanochemical model for dermal repair accounting tissue contraction. Cook (1995) uses the zero stress state approach, that assumes an unstressed state for all the elements, to which a relative strain is measured. Despite, these models included a great number of the factors that guide wound healing, wound contraction is highly influenced not only by cellular and chemical stimuli, but also mechanics. Cells are able to feel the mechanical environment where they are and regulate their activity, for instance force generation, in function of certain mechanical properties or stimuli. Processes like cell migration, differentiation or force generation have been proved to be influenced by mechanical stimuli (Discher et al., 2005; Mitrossilis et al., 2009; Harland et al., 2011). Therefore, wound healing and wound contraction models have evolved including the regulatory effect of mechanical variables in different biochemical processes (Javierre et al., 2009; Murphy et al., 2011, 2012). Nevertheless, there is not a common opinion about which is the involved mechanical stimulus that regulates the cellular mechanosensing mechanism. Following the work of Olsen et al. (1995), Javierre et al. (2009) proposed a mathematical model of wound contraction including fibroblasts, myofibroblasts, collagen, a generic growth factor and the tissue displacements. They included the effect of mechanical stress to regulate cellular processes, defining the traction stresses generated by cells through the concept of net stress of one fibroblasts cell per unit of ECM matrix introduced by Moreo et al. (2008). Moreo et al. (2008) proposed a mechanosensing model applicable to cellular processes such as migration or proliferation, based on the Hill’s model for skeletal muscle behavior. On their work, they proposed a model to evaluate the octahedral stresses exerted by cells as a function of the tissue stiffness. They evaluated this stress trough two components. The first measures the contractile stresses generated by the myosin machinery transmitter thorough actin bundles and a term related to the contractile stress supported by the passive resistance of the cell (absorbed by the micro- CHAPTER 3. Introduction 93 tubules). These two contributions give place to the stress that the cell effectively transmit to the ECM. This approach considers that the strain that the substrate and the cell suffer is the same. They also included this factor in the expression of fibroblasts differentiation into myofibroblasts, as it had been experimentally observed that differentiation is guided by mechanical tension (Hinz and Gabbiani, 2003; Tomasek et al., 2002). Javierre et al. (2009) also focused their model on the study of different wound geometries, creating one of the first 2D wound contraction mechanochemical models. Thus, they studied how the elongation of elliptical wounds affects the wound contraction. Following Cook (1995), Murphy et al. (2011) developed a 1D model reproducing the interaction between the cellular, chemical and mechanical phenomena adding new factors. As it has been proved that TGF-βis critical in dermal repair (Shah et al., 1992), they included its kinetics on their model. They proposed a differentiation mechanism, from fibroblasts to myofibroblasts, activated by TGFβand tissue stress. They applied the model to investigate how the wound heals producing certain disorders. When there is excessive TGF-βit gives place to contracture caused by excessive myofibroblasts forces. On the other hand, the wound contracts insufficiently when TGF-βdisappears too fast and there are not enough myofibroblasts to generate forces. They found the same effect when the myofibroblasts kinetics were modified by changing its differentiation from fibroblasts and death rate. As a main difference with previous models, Murphy et al. (2011) used linear elasticity to model the skin behaviour instead of the viscoelastic model chosen by other authors (Tranquillo and Murray, 1992; Olsen et al., 1995). Following the work by Tranquillo and Murray (1992), Murphy et al. (2012) proposed a more complex model that included the effect of more factors. In addition to the TGF-βkinetics, they added the effect of collagenase (an enzyme that contributes to new tissue formation) on the collagen concentration. One of the main differences of Murphy et al. (2012) model with previous models was the differentiation mechanism from fibroblasts to myofibroblasts. They set that the mechanical stimuli which guides differentiation was the positive elastic stress. This idea was first introduced by Hall (2008), while previous approaches proposed that this stimulus was cell traction stress (Javierre et al., 2009). They followed the approach proposed by Tranquillo and Murray (1992) to evaluate the traction forces exerted by cells, taking them proportional to the collagen density and cell concentration and neglecting the saturation terms dependent on cell densities (Tranquillo and Murray, 1992) or ECM densities (Olsen et al., 1995; Javierre et al., 2009). As a main difference with previous models (Olsen et al., 1995; Javierre et al., 2009), Murphy et al. (2012) assumed that myofibroblasts generate traction forces even 94 Mechanochemical modeling of wound healing in the absence of fibroblasts, as it is experimentally evidenced (Tomasek et al., 2002). They also assumed that there is not differentiation back from myofibroblasts to fibroblasts. Following Olsen et al. (1995) they considered the skin to be a viscoelastic material, although Murphy et al. (2012) uses a evolutive law to evaluate the elastic modulus proportional to the collagen concentration. As in previous models, they studied superficial circular wounds approximated by an unidimensional axisymmetric model. 3.3.3 Computationalmodelsofangiogenesis during wound healing Vascular growth has been also broadly modeled and simulated, mainly angiogenesis and vasculogenesis. While vasculogenesis is the formation of new blood vessels when there is no pre-existing vasculature, angiogenesis consists on the formation of new blood vessels from pre-existing ones, being this the case during wound healing. Angiogenesis is also involved in other biological processes, for instance tumor growth or embryogenesis, and most of the computational works about angiogenesis are focused on simulating angiogenesis in cancer (Anderson and Chaplain, 1998; Chaplain, 2000; Mantzaris et al., 2004) due to its social impact nowadays. Nevertheless, angiogenesis in wound healing has been also broadly modeled. Pettet et al. (1996a) presented the first angiogenesis model applied to wound healing, based on the fungal growth model of Edelstein (1982). It comprises a set of differential equations to model the evolution of capillary-tip endothelial cells migrating chemotactically to a macrophage-derived chemoattractant regulated by the vasculature density. Oxygen supply was not included in the model and oxygen levels were assumed proportional to the vessel density and later included as a primary variable in a model extension (Pettet et al., 1996b), in which they also included the effect of fibroblasts and the extracellular matrix. Both models were used to reproduce two-dimensional wound geometries approximated by a one-dimensional model, and allowed to reproduce the evolution of normal wounds and also non-healing wounds. Maggelakis (2003) developed an angiogenesis model that included the effect of macrophage-derived growth factors (MDGF), the capillary density and the tissue oxygen concentration in one dimension. She proposed that oxygen is provided to the wound by capillaries and consumed by macrophages, that appear when there is a low oxygen concentration and secrete MDGF, at the same time that diffuses through the tissue. Finally, high levels of MDGF enhanced the appearance of capillaries regulated also by a negative feedback loop until reaching the maximum capacity. The model was applied to study the dependence of the healing of a circu- CHAPTER 3. Introduction 95 lar wound with the oxygen supply. A subsequent model was proposed by Javierre et al. (2008), adding a new variable: the epidermal growth factor (EGF) concentration. They proposed a coupling between angiogenesis and a wound interface due to cell migration to study the effect of oxygen availability on cell function during epidermal wound healing in two dimensions. One of the factors that impairs wound healing is hypoxia (the lack of oxygen) (Schreml et al., 2010), which is necessary for cellular activity. Oxygen is mainly supplied by capillaries and thus angiogenesis is crucial for restoring the regular oxygen flux. Therefore, many authors have proposed angiogenesis models focused on the role of oxygen during angiogenesis. Schugart et al. (2008), proposed a seven-variables model, following Pettet et al. (1996b), adding the effect of macrophages. They studied for the first time the role of the tissue oxygen tension in the wound healing process and set that there is an optimal level of hypoxia for vascular growth and wound healing. In some cases, natural oxygen supply is not enough for an adequate healing, causing hipoxia and ending subsequently in chronic wounds. Flegg et al. (2009) simulated one of the techniques used to treat this pathology, the Hyperbaric Oxygen Therapy (HBOT) which consist on the deliberated elevation of oxygen levels. They studied the effect of the HBOT in angiogenesis in acute and chronic wounds. On a first approach Flegg et al. (2009) studied the evolution of oxygen density, capillary tips and blood vessels. They predicted that intermittent HBOT helps chronic wounds to heal but normobaric oxygen has not positive effects. Later, Flegg et al. (2010) expanded the model including the effect of a chemoattractant, fibroblasts and the ECM, and applied it to chronic diabetic wounds. They studied the effect of different pressures, oxygen percentage, exposition time and frequency. From all the studied cases they found that the therapy is beneficial only under certain conditions. In a similar way that wound contraction depends on mechanical factors, angiogenesis is also influenced by mechanics. Although there exist multiple mechanochemical wound healing models (Tranquillo and Murray, 1992; Olsen et al., 1995; Javierre et al., 2009; Murphy et al., 2011, 2012) there is still a lack of mechanochemical models of angiogenesis in wound contraction. Vermolen and Javierre (2010) developed a model including angiogenesis, wound contraction and wound closure. They combined the model by Tranquillo and Murray (1992) to model the contraction process and Maggelakis (2003) model for the angiogenesis process. However, they treated these processes isolated, without considering their interaction. They studied the processes separately in the dermis and the epidermis, which allows the study of deep wounds. Later, Vermolen and Javierre (2012) modified the model adding the dermal regeneration process coupling it with the angiogenesis process. 96 Mechanochemical modeling of wound healing Manoussaki (2003) developed a mechanochemical model of angiogenesis taking into account the traction forces exerted by cells on the ECM, which was modeled as a viscoelastic material. Moreover, cells movement was caused by chemotaxis guided by a chemical factor regulated by themselves. Although Manoussaki (2003) did not focused on wound healing, her model could be adapted to this application. Xue et al. (2009) extended the model by Schugart et al. (2008) incorporating the mechanical behaviour of the skin. 3.3.4 Limitations of existing models Mathematical models of biological processes usually come from a same initial model. Although authors extend these models adding new factors and variables, sometimes they do not focus their effort on solving model limitations, that are transmitted from one model to the subsequent ones. One of the main limitations of wound contraction models is that they are limited to one dimensional geometries (Murray et al., 1998; Sherratt and Murray, 1990; Schugart et al., 2008; Murphy et al., 2011; Olsen et al., 1995, 1996; Murphy et al., 2012). This is useful for studying simple axisymmetric geometries because they allow the spatial problem to be reduced to a one-dimensional model, which is less time and resources consuming. However, this oversimplification of the real wound morphology limits their true predictive capacity. In this direction, some recent works allow to study two dimensional geometries (Javierre et al., 2008, 2009; Vermolen and Javierre, 2010, 2012), bringing models closer to realistic situations, despite three dimensional models are required to obtain accurate representation of the involved phenomena. Another limitation of prior wound healing works is that they are focused on tracking the evolution of superficial wounds, studying the evolution of the superficial area and not taking into account the influence of wound depth on the contraction kinetics. Thus, Olsen et al. (1995); Javierre et al. (2009); Murphy et al. (2012) use a plane stress approach to simplify the geometry of real wounds. However, the phenomena that take place along the wound depth are equally important, although have not been modeled due to its complexity. To our knowledge, only Vermolen and Javierre (2012) study deep wounds, though the plane strain hypotheses are adopted to reduce the wound geometry to two dimensions. As a matter of fact, while using three dimensional geometries is the most realistic approach, using a plane strain approach to study wounds along its depth can reveal useful information without the complexity and computational cost of three-dimensional models. CHAPTER 3. Introduction 97 There are some main differences when plane stress and plane strain approaches are assumed. In first place, when simulating deep wounds there is part of the boundary that is not surrounded by skin, while plane wounds are completely surrounded by non-wounded tissue. This part of the boundary is in contact with the air and it is represented by a free boundary. The major implication to this is that it can move freely, as there is no tissue attached to it. When the mathematical model is formulated, this free boundary must be treated differently to the constrained boundaries. Some of the simplifications that are assumed for the attached boundaries are not applicable to this free boundary. From a numerical perspective planar wounds under plane stress hypotheses are easier to model as natural boundary conditions do not need to be taken into account. Another difference between the study of planar and deep wounds is the consideration of the different skin layers involved in the model. When plane wounds are simulated only the properties of the most superficial layer of the skin are taken into account, as it has been the case in models of epidermal wound healing (Sherratt and Murray, 1990; Maggelakis, 2003; Javierre et al., 2008) and some models of dermal wounds healing (Tranquillo and Murray, 1992; Murphy et al., 2011, 2012). When deep wounds are represented, depending on the depth of the considered injury it is not possible to characterize the whole geometry with the same properties. It is known that the most external layer of the skin, the epidermis, has a thickness of micrometres. However, the dermis usually has a thickness from 1,5 mm to 4 mm and the hypodermis reaches different depths depending on the considered part of the body. Thus, when deep wounds are studied and the properties of the different skin layers need to be considered in order to obtain more realistic results. Finally, one of the most limiting aspects in wound healing modeling is the mechanical characterization of the skin. Traditionally, a viscoelastic material model has been used for modeling the mechanical behavior of the skin. This approach captures accurately some time-dependent properties of the skin but also presents some limitations. It has been experimentally proved that a hyperelastic constitutive model fits better the skin behavior, although its characterization is more complicated. Most of the wound healing models that include mechanics (tissue deformation) have treated the skin (or its layers) as a viscoelastic material (Olsen et al., 1995; Javierre et al., 2009; Murphy et al., 2012). However, implementing skin as a hyperelastic material would allow to reproduce the skin behavior more accurately, including the possibility of adding tissue anisotropy. 98 Mechanochemical modeling of wound healing 3.4 Experimental works Despite there is a high number of computational models that simulate different aspects of wound healing, there is a lack of experimental works in the field that can help to corroborate the computational outcomes. Because of ethical reasons, wound healing experiments in living subjects are not frequent, almost inexistent in humans. Consequently, animal assays are the best approach although they are also subjected to strong ethical considerations. Moreover, mammals skin presents different mechanical properties to human skin, and results obtained in these experiments can be used only as a qualitative pattern for humans. One of the firsts experimental studies is due to Van den Brenk (1956). Van den Brenk (1956) studied epidermal wounds in rabbits ears. Their results were used to validate Sherratt and Murray (1990) results. McGrath and Simon (1983) studied the influence of shape and size in deep wounds on rats. They observed the evolution of three wound geometries: a small square, a large square and a circular wound with the same area as the large square. To measure the wound size, McGrath and Simon (1983) tracked the movement of some points defining the wound border. They differentiated healing due to wound contraction and epithelialization. After a rapid wound distraction and a six-days plateau phase, the wounds contracted during around 30 days, followed by a long almost steady plateau. McGrath and Simon (1983) observed that after 70 days all the wounds had experienced a similar contraction rate, reducing its size to the 35% of the initial size. Additionally, most mammals such as rats and rabbits present a looser and more elastic skin than humans, and the contraction levels are not comparable with those from humans. However, pigs present a skin structure more similar to human, thus the porcine model is widely accepted for comparisons. Roy et al. (2009) observed the evolution of ischemic and non-ischemic circular wounds in pigs. They performed the incisions and while some wounds were left to heal normally, others were disrupted introducing silicon sheets between the dermis and the underlaying tissue. In this way, ischemic wounds -in which re-epithelialization is impairedwere reproduced. Roy et al. (2009) were able to measure the oxygen concentration and the partial oxygen pressure in the wound, finding higher oxygen saturation levels in the non-ischemic wounds. They observed also wound closure along CHAPTER 3. Introduction 99 31 days, when the non-ischemic wound had almost closed but the ischemic wound had achieved only around 80% of closure. When comparing experimental results with human models, it must be taken into consideration that each species has different time parameters due to their different cellular and tissue kinetics (Reina-Romo et al., 2010). Thus, if a proper equivalence between species is obtained, it is possible the comparison of experimental results on animals to computational results on human models. 3.5 Objectives The main objective of this thesis is the study of wound healing in the skin through computational simulation and using a multiphysics approach. In order to achieve this objective, a computational model that allows to reproduce wound healing under different conditions has been proposed and implemented. More specifically, the proposed model focuses on the wound contraction phase, as it has a strong mechanical component. The model also allows to include more processes that take place simultaneously to wound contraction, for instance angiogenesis. The proposed model includes biological factors (cellular species, growth factors and collagen) and also mechanical factors (characterization of the skin mechanical properties and cellular contraction). To find the solution of the resulting governing equations, the finite element method (FEM) is applied. The model is constituted by two parts, one related to the biochemical analysis of the process and the other regarding the mechanical analysis of the skin. First, the evolution of the biochemical species is evaluated with a reaction-diffusion system of equations. Second, the mechanical behavior of the skin is modeled assuming the fundamental mechanical relationships for the constitutive material model chosen to characterize it. The two parts are related through a mechanosensing and mechanotransductor mechanism, that regulates the behavior of the cells as a function of mechanical variables. The model will allow to study different healing cases and different wound types. The main objectives are summarized next: •Adaptation of the model for the study of planar and deep wounds, in two dimensions. Planar wounds are characterized by their superficial area, using plane stress approach. Long and deep wounds can be studied through their transversal section, under a plane strain approach and taking into account the different skin layers that are affected by the injury. •Use of several constitutive models (viscoelastic, hyperelastic isotropic and hyperelastic anisotropic) to characterize the mechanical behavior of the skin. 100 Mechanochemical modeling of wound healing Viscoelastic materials are a good approximation to reproduce skin behavior. Moreover, viscoelastic materials can be used when small deformations theory is applied but not when the material works with large deformations. In this case, hyperelastic materials should be used. •Inclusion of other phenomena that take place simultaneously to wound contraction, such as angiogenesis. As it is known that every phenomena which takes place during healing are interrelated, the inclusion of as much processes as possible gives a more complete and realistic model. As a disadvantage, the resources needed to solve the problem increase together with the model precision. •Incorporation of new cellular kinetics laws depending on physical evidences, based on experimental studies instead of the phenomenological laws proposed until now. •Solving the biochemical and mechanical problems with a fully coupled approach or a non-coupled approach. While a fully-coupled approach allows to solve the whole problem at the same time, a non-coupled approach allows to divide the problem and solve it in different stages. This is specially useful to separate the biochemical and the mechanical analysis due to their different time scales. Moreover, it allows us to use an updated Lagrangian and updating certain variables between the two stages. •Application of the model to different size and shape wounds in two dimensions, both planar and deep wounds. The model capacity of reproducing complex geometries, such as those considered in experimental works, allows for a more thorough validation of the results. •Application of the model to simulate three-dimensional wounds, in which the effect of the superficial shape and wound depth are considered simultaneously. •Incorporation of the collagen fibers in the healthy skin. Collagen fibers allow to simulate the real anisotropy of the skin instead of isotropic simplifications, which is important in the healthy skin surrounding the wound. 3.6 Financial support The research included in this thesis was supported by: CHAPTER 4. Conclusions 107 Work Title Authors Journal IF∗Publication date 1 Nonlinear finite element simulations of injuries with free boundaries: Application to surgical wounds C.Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito Int. J. Numer. Meth. Biomed. Engng. (2014). DOI: 10.1002/cnm.2621 1.310 Online 2 Numerical modelling of the angiogenesis process in wound contraction C.Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito Biomech Model Mechanobiol. DOI: 10.1007/s10237-0120403-x 3.331 3 A cell-regulatory mechanism involving feedback between contraction and tissue formation guides wound healing progression C.Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito PLOS ONE. DOI:10.1371/journal.pone.0092774 3.730 Online 4 Modeling anisotropic wound healing: effect of the relative position of wounds with respect to collagen fibers orientation C.Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito, A. Menzel Journal of the Mechanics and Physics of Solids 3.406 Submitted 5 A computational study of stress fiber-focal adhesion dynamics governing cell contractility M.Maraldi, C.Valero, K.Garikipati Biophysical Journal 3.668 Accepted for publication ∗Journal Impact Factor. Table 4.1: Articles included in this thesis. 108 Mechanochemical modeling of wound healing Title Authors Conference Date and Place Participation 1 Finite element modelling of wound contraction: a nonlinear mechano-chemical approach C. Valero, J.M. Garc´ ıa Aznar, M.J. G´ omezBenito, E. Javierre Congress on Numerical Methods in Engineering CMNE 2011 June 2011, Coimbra (Portugal) Speaker 2 Three dimensional modelling of wound contraction. C. Valero, J.M. Garc´ ıa Aznar, M.J. G´ omezBenito, E. Javierre XXIII congress International Society of Biomechanics (ISB 2011) July 2011, Brussels (Belgium) Speaker 3 Finite Element Analysis of the angiogenesis process in wound healing C. Valero, E. Javierre, M.J. G´ omez-Benito XVIII congress of the European Society of Biomechanics (ESB 2012) July 2012, Lisbon (Portugal) Speaker 4 Modelling skin healing: Effect of cell traction C. Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito XIVrd congress of the European Society of Biomechanics (ESB 2013) August 2013, Patras (Greece) Speaker 5 A cell-regulatory mechanism between wound contraction and tissue formation C. Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito V International Conference on Computational Bioengineering (ICCB 2013) September 2013, Leuven (Belgium) Coauthor 6 Modelado multif´ ısico de la contracci´ on de heridas (Multi-physical modeling of wound contraction) C. Valero, E. Javierre. XV Reuni´ on de Usuarios del Programa October 2010 Speaker Continued on next page CHAPTER 4. Conclusions 109 Table 4.2 – Continued from previous page Title Authors Conference Date and Place Participation 7 Simulaci´ on por elementos finitos del proceso de angiog´ enesis aplicado a la cicatrizaci´ on de heridas C. Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito Reuni´ on del Cap´ ıtulo de la ESB November 2011, Zaragoza (Spain) Speaker 8 In-silico Models of Wound and Bone Healing: Examples from Nature M.J. G´ omez-Benito , C. Valero, E. Reina-Romo, J.M. Garc´ ıa Aznar, E. Javierre Euro Bio-inspired Materials - International School and Conference on Biological Materials Science. March 2012, Potsdam (Germany) Coauthor 9 Stress Evaluation During Angiogenesis in Skin Wound Healing C. Valero, E. Javierre, J.M. Garc´ ıa Aznar, M.J. G´ omez-Benito 10th International Symposium Computer Methods in Biomechanics and Biomedical Engineering (CMBBE 2012) April 2012, Berlin (Germany) Coauthor 10 A Mechano-Chemical Model of Wound Contraction for Superficial and Acute Wounds E. Javierre, C. Valero, M.J. G´ omez-Benito, J.M. Garc´ ıa-Aznar XVIIIrd congress of the European Society of Biomechanics (ESB 2012) July 2012, Lisbon (Portugal) Coauthor 11 Mathematical analysis of physiological and pathological wound healing. Application to diabetic foot ulcers E. Javierre, C. Valero, M.J. G´ omez-Benito, F.J. Vermolen Mathematical Modelling in Engineering & Human Behavior September 2012, Valencia (Spain) Coauthor Continued on next page 110 Mechanochemical modeling of wound healing Table 4.2 – Continued from previous page Title Authors Conference Date and Place Participation 12 Finite element analysis of the mechanoc-chemical regulation of the wound contraction in surgical wounds E. Javierre, C. Valero, M.J. G´ omez-Benito, J.M. Garc´ ıa-Aznar Conference on the Mathematics of Finite Elements and Applications MAFELAP 2013 June 2013, Uxbridge (United Kindom) Coauthor Table 4.2: Contributions in national and international conferences. CHAPTER 4. Conclusions 111 4.3 Future lines Considering the results and conclusions obtained from the work developed in this thesis and its limitations, some lines of future work could be defined: •Extension of the contraction two-dimensional model to three dimensional geometries. The use of plane strain and plane stress simplifications allow to study a large number of wound geometries and cases, but they also imply loss of information in the simulation process. The actual three-dimensional model includes angiogenesis and wound contraction guided only by the fibroblasts. The model is ready to be extended with the myofibroblasts and collagen kinetics. •The volumetric cell behavior could be improved by a more complete approach, considering the anisotropic cell behavior. The model proposed in the thesis considers that cells exert the same traction in every direction independently of the direction in which they are been deformed. A more evolved model will distinguish the magnitude in which the cell is been deformed along each direction and thus will regulate the forces exerted as a result of these different strains. •Incorporation of the remodeling phenomena in the wound tissue after healing. The actual model includes the fibers effect in the dermis while the wound heals, but it does not reproduce the production and remodeling of fibers in the healed wound. A more realistic approach will include the fiber production and orientation in the tissue as long as it presents an adequate healing level. Moreover, the fiber orientation will be regulated by the direction of higher stress levels. •Include all the skin layers in the three dimensional geometry. In a same way than the dermis and the underlying tissue are distinguish in the plane strain model, these layers can be separated in the three dimensional model, presenting different properties. This modification will provide a more realistic behavior. •Study of wound pathologies, in which healing is impaired, excessive or presents other difficulties such as pressure ulcers or keloids. The study of these cases could be simulated by properly adjusting the biochemical kinetics or the skin properties. •Study of different healing solutions, such as sutures or other devices that modify the mechanical conditions in which wounds are healed. The simulation of these phenomena will be of great help for clinicians to choose favorable solutions in specific cases. 112 Mechanochemical modeling of wound healing Bibliography Anderson, A., Chaplain, M., 1998. Continuous and discrete mathematical models of tumor-induced angiogenesis rid a-5355-2010. Bulletin of Mathematical Biology 60 (5), 857–899. Annaidh, A. N., Bruyere, K., Destrade, M., Gilchrist, M. D., Maurini, C., Ottenio, M., Saccomandi, G., 2012a. Automated estimation of collagen fibre dispersion in the dermis and its contribution to the anisotropic behaviour of skin. Annals of Biomedical Engineering 40 (8), 1666–1678. Annaidh, A. N., Bruyere, K., Destrade, M., Gilchrist, M. D., Ottenio, M., 2012b. Characterization of the anisotropic mechanical properties of excised human skin. Journal of the Mechanical Behavior of Biomedical Materials 5 (1), 139– 148. Argenta, L., Morykwas, M., 1997. Vacuum-assisted closure: A new method for wound control and treatment: Clinical experience. Annals of Plastic Surgery 38 (6), 563–576. Billingham, R., Medawar, P., 1951. The technique of free skin grafting in mammals. Journal of Experimental Biology 28 (3), 385–402. Bluestein, D., Javaheri, A., 2008. Pressure ulcers: Prevention, evaluation, and management. American Family Physician 78 (10), 1186–1194. Boyer, G., Zahouani, H., Le, B. A., Laquieze, L., 2007. In vivo characterization of viscoelastic properties of human skin using dynamic micro-indentation. 2007 Annual International Conference of the Ieee Engineering in Medicine and Biology Society, Vols 1-16, 4584–4587. Carmeliet, P., Jain, R., 2000. Angiogenesis in cancer and other diseases. Nature 407 (6801), 249–257. Catty, R. H. C., 1965. Healing and contraction of experimental full-thickness wounds in human. British Journal of Surgery 52 (7), 542. 113 114 Mechanochemical modeling of wound healing Chaplain, M., 2000. Mathematical modelling of angiogenesis rid a-5355-2010. Journal of Neuro-Oncology 50 (1-2), 37–51. Clark, J. A., Cheng, J. C. Y., Leung, K. S., 1996. Mechanical properties of normal skin and hypertrophic scars. Burns 22 (6), 443–446. Clark, R., 1988. Overview and general considerations of wound repair. In: Clark, R., Henson, P. (Eds.), The Molecular and Cellular Biology of Wound Repair. Springer US, pp. 3–33. Cook, J., 1995. A mathematical model for dermal wound healing: wound contraction and scar formation. Ph.D. thesis. Dale, P., Maini, P., Sherratt, J., 1994. Mathematical-modeling of corneal epithelial wound-healing. Mathematical Biosciences 124 (2), 127–147. Diridollou, S., Patat, F., Gens, F., Vaillant, L., Black, D., Lagarde, J., Gall, Y., Berson, M., 2000. In vivo model of the mechanical properties of the human skin under suction. Skin Research and Technology 6 (4), 214–221. Discher, D., Janmey, P., Wang, Y., 2005. Tissue cells feel and respond to the stiffness of their substrate. Science 310 (5751), 1139–1143. Dunn, M., Silver, F., 1983. Viscoelastic behavior of human connective tissues - relative contribution of viscous and elastic components. Connective tissue research 12 (1), 59–70. Edelstein, L., 1982. The propagation of fungal colonies - a model for tissuegrowth. Journal of theoretical biology 98 (4), 679–701. Edwards, C., Marks, R., 1995. Evaluation of biomechanical properties of human skin. Clinics in dermatology 13 (4), 375–380. Escoffier, C., Derigal, J., Rochefort, A., Vasselet, R., Leveque, J., Agache, P., 1989. Age-related mechanical-properties of human-skin - an invivo study. Journal of Investigative Dermatology 93 (3), 353–357. Eskes, A. M., Ubbink, D. T., Lubbers, M. J., Lucas, C., Vermeulen, H., 2011. Hyperbaric oxygen therapy: Solution for difficult to heal acute wounds? systematic review. World journal of surgery 35 (3), 535–542. Flegg, J. A., Byrne, H. M., McElwain, L. S., 2010. Mathematical model of hyperbaric oxygen therapy applied to chronic diabetic wounds. Bulletin of mathematical biology 72 (7), 1867–1891. Bibliography 115 Flegg, J. A., McElwain, D. L. S., Byrne, H. M., Turner, I. W., 2009. A three species model to simulate application of hyperbaric oxygen therapy to chronic wounds. Plos Computational Biology 5 (7), e1000451. Flynn, C., Taberner, A., Nielsen, P., 2011a. Measurement of the forcedisplacement response of in vivo human skin under a rich set of deformations. Medical engineering& physics 33 (5), 610–619. Flynn, C., Taberner, A., Nielsen, P., 2011b. Modeling the mechanical response of in vivo human skin under a rich set of deformations. Annals of Biomedical Engineering 39 (7), 1935–1946. Folkman, J., Moscona, A., 1978. Role of cell-shape in growth-control. Nature 273 (5661), 345–349. Fore-Pfliger, J., 2004. The epidermal skin barrier: implications for the wound care practitioner, part i. Advances in Skin & Wound Care 17 (8), 417–425. Friedman, A., Xue, C., 2011. A mathematical model for chronic wounds. Mathematical Biosciences and Engineering 8 (2), 253–261. Gahagnon, S., Mofid, Y., Josse, G., Ossant, F., 2012. Skin anisotropy in vivo and initial natural stress effect: A quantitative study using high-frequency static elastography. Journal of Biomechanics 45 (16), 2860–2865. Gasser, T., Ogden, R., Holzapfel, G., 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the Royal Society Interface 3 (6), 15–35. Gauglitz, G. G., Korting, H. C., Pavicic, T., Ruzicka, T., Jeschke, M. G., 2011. Hypertrophic scarring and keloids: Pathomechanisms and current and emerging treatment strategies. Molecular Medicine 17 (1-2), 113–125. Gill, A., Bell, C., 2004. Hyperbaric oxygen: its uses, mechanisms of action and outcomes. Qjm-an International Journal of Medicine 97 (7), 385–395. Graham, J., Vomund, A., Phillips, C., Grandbois, M., 2004. Structural changes in human type i collagen fibrils investigated by force spectroscopy. Experimental cell research 299 (2), 335–342. Gray, H., Williams, P., Bannister, L., 1995. Gray’s Anatomy: The Anatomical Basis of Medicine and Surgery. Gray’s Anatomy. Churchill Livingstone. Grinnell, F., 2000. Fibroblast-collagen-matrix contraction: growth-factor signalling and mechanical loading. Trends in cell biology 10 (9), 362–365. 116 Mechanochemical modeling of wound healing Groves, R. B., Coulman, S. A., Birchall, J. C., Evans, S. L., 2013. An anisotropic, hyperelastic model for skin: Experimental measurements, finite element modelling and identification of parameters for human and murine skin. Journal of the Mechanical Behavior of Biomedical Materials 18, 167–180. Gurtner, G. C., Werner, S., Barrandon, Y., Longaker, M. T., 2008. Wound repair and regeneration. Nature 453 (7193), 314–321. Hall, C. L., 2008. Modelling of some biological materials using continuum mechanics. Ph.D. thesis, Queensland University of Technology. Hall-Findlay, E., 1999. A simplified vertical reduction mammaplasty: Shortening the learning curve. Plastic and Reconstructive Surgery 104 (3), 748–759. Harland, B., Walcott, S., Sun, S. X., 2011. Adhesion dynamics and durotaxis in migrating cells. Physical Biology 8 (1), 015011. Hibbit, Karlsonn, Sorensen, 2008. Abaqus user´s guide, v.6.9. Pawtucket, RI, HKS Inc. Hinz, B., Gabbiani, G., 2003. Mechanisms of force generation and transmission by myofibroblasts. Current opinion in biotechnology 14 (5), 538–546. Hinz, B., Mastrangelo, D., Iselin, C., Chaponnier, C., Gabbiani, G., 2001. Mechanical tension controls granulation tissue contractile activity and myofibroblast differentiation. American Journal of Pathology 159 (3), 1009–1020. Javierre, E., Moreo, P., Doblare, M., Garcia-Aznar, J. M., 2009. Numerical modeling of a mechano-chemical theory for wound contraction analysis. International Journal of Solids and Structures 46 (20), 3597–3606. Javierre, E., Vermolen, F. J., Vuik, C., van der Zwaag, S., 2008. Numerical Modelling of Epidermal Wound Healing. Springer-Verlag Berlin, Berlin; Heidelberg Platz 3, D-14197 Berlin, Germany. Jones, C., Ehrlich, H. P., 2011. Fibroblast expression of alpha-smooth muscle actin, alpha 2 beta 1 integrin and alpha v beta 3 integrin: Influence of surface rigidity. Experimental and molecular pathology 91 (1), 394–399. Kerr, J., 2010. Functional Histology. Elsevier Mosby Australia. Kwon, Y., Kim, H., Roh, D., Yoon, S., Baek, R., Kim, J., Kweon, H., Lee, K., Park, Y., Lee, J., 2006. Topical application of epidermal growth factor accelerates wound healing by myofibroblast proliferation and collagen synthesis in rat. Journal of Veterinary Science 7 (2), 105–109. COMPENDIUM OF PUBLICATIONS Mechanochemical modeling of wound healing: multiphysics finite element simulations. 123 Work 1: Nonlinear finite element simulations of injuries with free boundaries: Application to surgical wounds Journal: International Journal for Numerical Methods in Biomedical Engineering(2014). Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2621 Journal Impact factor: 1.310 Contribution of the author of the thesis: the author was in charge of rewriting the formulation for the plane strain approach, making a review of the existing literature, choosing the mechanical variables, performing all the computational simulations, analyzing the results and determining their implications. Everything was done under the supervision of the other authors. 125 126 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. (2014) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2621 Nonlinear finite element simulations of injuries with free boundaries: Application to surgical wounds C. Valero1,*,†, E. Javierre2, J. M. García-Aznar 1and M. J. Gómez-Benito1 1Multiscale in Mechanical and Biological Engineering, Aragón Institute of Engineering Research, University of Zaragoza, Zaragoza, Spain 2Centro Universitario de la Defensa, Academia General Militar, Zaragoza, Spain SUMMARY Wound healing is a process driven by biochemical and mechanical variables in which a new tissue is synthesised to recover original tissue functionality. Wound morphology plays a crucial role in this process, as the skin behaviour is not uniform along different directions. In this work, we simulate the contraction of surgical wounds, which can be characterised as elongated and deep wounds. Because of the regularity of this morphology, we approximate the evolution of the wound through its cross section, adopting a plane strain hypothesis. This simplification reduces the complexity of the computational problem; while allows for a thorough analysis of the role of wound depth in the healing process, an aspect of medical and computational relevance that has not yet been addressed. To reproduce wound contraction, we consider the role of fibroblasts, myofibroblasts, collagen and a generic growth factor. The contraction phenomenon is driven by cell-generated forces. We postulate that these forces are adjusted to the mechanical environment of the tissue where cells are embedded through a mechanosensing and mechanotransduction mechanism. To solve the nonlinear problem, we use the finite element method (FEM) and an updated Lagrangian approach to represent the change in the geometry. To elucidate the role of wound depth and width on the contraction pattern and evolution of the involved species, we analyse different wound geometries with the same wound area. We find that deeper wounds contract less and reach a maximum contraction rate earlier than superficial wounds. Copyright © 2014 John Wiley & Sons, Ltd. Received 13 May 2013; Revised 12 November 2013; Accepted 20 November 2013 KEY WORDS: finite elements; free boundary problem; wound healing; nonlinear convection-diffusionreaction; mechanosensing and mechanotransduction 1. INTRODUCTION Skin is the protective barrier between internal organs and external aggressions. Occasionally, this barrier is damaged and a number of complicated processes are needed to recover the initial functionality of the skin. Different injuries such as burns, cuts, ulcers and surgery scars cause a reduction in skin quality, making the wound healing process to recover the appropriate properties [1]. Wound healing is mainly driven by different cellular species (fibroblasts, myofibroblasts, epithelial cells and macrophages) and growth factors (macrophage-derived growth factor, transforming growth factor (TGF)˛,TGFˇ, platelet-derived growth factor and vascular endothelial growth factor). These species undergo several processes (proliferation, differentiation, migration and apoptosis) that modify their concentration, regulating the evolution of the wound. Wound healing is usually divided into three stages: inflammation, tissue formation and scar remodelling [2]. During the inflammation stage, a fibrin clot is formed at the wound site and a number of growth factors are released. During the second stage, different cellular species are attracted to the clot by the growth *Correspondence to: C. Valero, Multiscale in Mechanical and Biological Engineering, Aragón Institute of Engineering Research, University of Zaragoza, Maria de Luna s/n, 50018 Zaragoza, Spain. †E-mail: [email protected] Copyright © 2014 John Wiley & Sons, Ltd. 127 C. VALERO ET AL. factors released during inflammation. Epidermal cells proliferate into the wound area, granulation tissue appears and new blood vessels begin to grow to supply oxygen and nutrients to the new tissue [1]. Fibroblasts secrete collagen to create a new extracellular matrix (ECM) that will replace the temporary fibrin clot. In this stage, the wound reduces its size and acquires a tensional state that will slowly relax. Finally, in the remodelling stage, previously synthesised collagen fibres align with tension lines in such a way that the damaged tissue gradually recovers most of its initial functionality. In the last stage, previously initiated processes end, and cell types that are no longer needed die and are removed. Wound contraction is one of the most important processes during wound healing. This process is strongly influenced by not only cellular and chemical species but also mechanics. Cells feel the mechanical changes on the substrate in which they are embedded and regulate the forces they exert corresponding to this mechanical environment [3–5]. Therefore, most wound healing studies include both biological and mechanical factors in their models [6–10]. In addition, wound geometry is one of the most important characteristics in determining the evolution of the healing process. Wounds in the skin can be classified according to their dimensions. Wounds with a large superficial area and a shallow depth should be treated differently than deep wounds with a relevant depth. For a number of years, mathematical models have been developed to study different biological and physiological processes [11]. Prior wound healing works have focused on tracking the wound superficial area over time, neglecting any influence of wound depth on the contraction kinetics [6, 9, 12]. Previous works on wound healing [8,13–15] and wound contraction [9, 12,16] have mostly studied wounds from a one-dimensional (1-D) perspective or under the assumption of plane stress. These models consider simple axisymmetric geometries and allow the spatial problem to be reduced to a 1-D model. Recent works have considered more realistic wound geometries [6, 10], solving the two-dimensional (2-D) spatial problem but neglecting the wound depth. 2-D models allow the study of complex wounds with more realistic geometries. From a numerical perspective, planar wounds are easier to model as the boundary conditions are the same for the whole boundary and the natural boundary conditions do not need to be taken into account. Therefore, in this work, we focus on the numerical solution of the governing system without boundary simplification to obtain a model applicable to both wound types. Hence, we present a mathematical model that can reproduce the evolution of both (superficial and deep) wounds. We focus on the simulation of deep and elongated wounds to consider the different behaviours of wounds along different directions. The healing of deep and elongated wounds varies from that of planar wounds because all involved phenomena mainly occur along the wound depth. From a mechanical perspective, the hypotheses of plane strain and a free boundary on the top surface of the wound are adopted. Note, moreover, that this approach to wound healing is the closest approximation to a three-dimensional spatial model of wound healing. Three-dimensional models are desirable to capture more realistic and complex wound morphologies. However, they are much more complex to develop and more expensive computationally. There are still many hypotheses on cell and tissue behaviour that need to be properly addressed using the predictive power of three-dimensional wound healing models. Therefore, from a modelling and simulation point of view, 2-D models present the most affordable option with less simplification of hypotheses than 1-D models. To the best of our knowledge, a model with these characteristics has not been developed previously. The FEM is used to conveniently manage the complex wound geometries and free boundary. A nonlinear finite element is implemented to solve the mechanical equilibrium of the tissue and evolution of the chemical and cellular species simultaneously. The finite element approximation of the highly nonlinear and coupled convection-diffusion-reaction governing equations allows us to propose a strict linearisation of the governing equations, which yields a linear system of equations that are easy to solve on each time increment. 2. MATHEMATICAL MODEL The aim of this work is to address the difference between planar and long deep wounds, specifically the effects of wound morphology and mechanical behaviour. Superficial wounds are studied Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2014) DOI: 10.1002/cnm 128 NONLINEAR FINITE ELEMENT SIMULATIONS OF INJURIES WITH FREE BOUNDARIES assuming a plane stress approach because their depths are much smaller than their other dimensions. In contrast, long deep wounds are studied under a plane strain approach because the wound behaviour is assumed to be the same along their lengths, and thus, only the transversal section should be studied. Furthermore, in long deep wounds, the upper part of the wound edge is in contact with the external environment with no constrains on its movement, which allows a different contraction pattern. In this work, we present the formulation and numerical solution of a model that reproduces the contraction of elongated and deep wounds. The model formulation used here, based on the mechanochemical coupling of different cellular species, growth factors and the ECM, is similar to those in earlier works [6,9,12], but fewer simplifications are assumed in its implementation than in previous models. Thus, to reproduce the wound status over time, we take into account biological and mechanical factors that affect both the cellular kinetics and ECM mechanical evolution. This mechanochemical coupling is supported by experimental results proving that cells not only respond to biochemical stimulus but also modify their behaviour depending on the mechanical evolution of the skin [3, 4]. Moreover, the mechanical contribution of the skin determines the way the wound contracts, a crucial element of the healing process. 2.1. Governing equations The model reproduces the temporal and spatial evolution of four different species within the wound space and surrounding tissue. Following previous works [6,9, 12], we considered two different cellular species: fibroblasts and myofibroblasts. Fibroblasts (n) are motile cells inside the skin that secrete ECM and exert traction forces on the tissue in which they are embedded. Myofibroblasts (m) are nonmotile cells that appear in the skin because of the combined action of inflammatory growth factors and mechanical stimulus and amplify the forces exerted by fibroblasts [17, 18]. The secreted ECM is mainly composed of collagen (), which gives structural support to the skin and determines its mechanical properties. Collagen forms fibres that are synthesised and degraded by fibroblasts and myofibroblasts [19]. Thus, we assume that collagen is the main component in the skin that causes the stiffening effect of the newly synthesised ECM. The elastic modulus (E) is therefore dependent on the collagen density through the equation EDE0=0,whereE0and 0 are the elastic modulus and collagen density of the undamaged skin, respectively, considering that the skin becomes stiffer as the collagen density increases [20, 21]. Finally, we consider a generic growth factor (c) accumulated at the wound site during the inflammatory phase that regulates cell migration and cell function during the contraction process. Each of the aforementioned cellular and chemical species follows a conservation law that incorporates the biological cues previously described. In general terms, this conservation law can be expressed as @Q @t CrJQDfQ,(1) where JQdenotes the net flux of the species Qand its net production, fQ. If we single out this equation for each of the four species, we find that the corresponding laws can be written for fibroblasts (Eqn (2)), myofibroblasts (Eqn (3)), collagen (Eqn (4)) and growth factor (Eqn (5)) as follows, @n @t Cr0 B B B @ Dnrn „ƒ‚… random migration Can .bnCc/2nrc „ƒ‚ … chemotaxis Cn@u @t „ƒ‚… passive convection 1 C C C A DrnCrn,maxc C1=2 Ccn1n K „ƒ‚ … proliferation k1,maxc CkCc pcell./ dCpcell./n „ƒ‚ … differentiation Ck2m „ƒ‚… dedifferentiation dnn „ƒ‚… death (2) Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2014) DOI: 10.1002/cnm 129 C. VALERO ET AL. @m @t Cr0 B B @m@u @t „ƒ‚… passive convection 1 C C A DrrnCrn,maxc C1=2 Ccm1m K „ƒ‚ … proliferation Ck1,maxc CkCc pcell./ dCpcell./n „ƒ‚ … differentiation k2m „ƒ‚… dedifferentiation dmm „ƒ‚… death (3) @ @t Cr0 B B @@u @t „ƒ‚… passive convection 1 C C A DrCr,maxc CCcnCbm R2 C2 „ƒ‚ … production d.nCdm/ „ƒ‚ … degradation (4) @c @t Cr0 B B @ Dcrc „ƒ‚… diffusion Cc@u @t „ƒ‚… passive convection 1 C C A Dkc.n Cm/c Cc „ƒ‚ … production dcc „ƒ‚… degradation (5) where urepresents the tissue displacements. The parameter values and descriptions can be found in Tables I and II. As the model reproduces wound contraction, we consider that wound healing is driven by not only biochemical laws but also mechanical stimuli. Regarding the mechanical behaviour of the system, we express the balance between the internal stresses and the external forces as r.ecm Ccell/Dfext,(6) where ecm denotes the ECM stress contribution. Following most works in wound contraction [6, 9, 12], we assume that the skin behaves as a viscoelastic material. Therefore, ecm can be written as ecm D1 @" @t C2 @ @t ICE 1C"C 12 I,(7) Table I. List of model parameters related to fibroblasts and myofibroblasts kinetics. Parameter Description Value Observations n0Fibroblasts density in undamaged dermis 104cells/cm3[12] DnFibroblasts diffusion rate 2102cm2/day [22] anTogether with bndetermines the maximal chemotaxis rate per unit of GF concentration 41010 g/cm day [6] bnGF concentration that produces 25% of the maximal chemotactic response 2109g/cm3[6] rnFibroblasts proliferation rate 0.832 day1[22] rn,max Maximal rate of GF-induced fibroblasts proliferation 0.3 day1[6] C1=2 Half-maximal GF enhancement of fibroblasts proliferation 108g/cm3[12] KFibroblasts maximal capacity in dermis 107cells/cm3[12] k1,max Maximal rate of fibroblasts differentiation 0.8 day1[6] CkHalf-maximal GF enhancement of fibroblasts differentiation 108g/cm3[6] k2Myofibroblasts dedifferentiation rate 0.693 day1[6] dnFibroblasts death rate 0.831 day1dnDrn1n0 K rProportionality factor 0.5 [12] dmMyofibroblasts death rate 2.1102day1[6] GF, growth factor. Adjusted to fit reported migration rate with a traveling wave model. Determined fibroblasts proliferation kinetics to remain in equilibrium away from the wound. Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2014) DOI: 10.1002/cnm 130 NONLINEAR FINITE ELEMENT SIMULATIONS OF INJURIES WITH FREE BOUNDARIES Table II. List of model parameters related to collagen and growth factor kinetics. Parameter Description Value Observations 0Collagen concentration in undamaged dermis 0.1 g/cm3[12] ini Initial collagen concentration in the wound 103g/cm3[12] c0GF concentration in the wound 108g/cm3[12] rCollagen production rate 7.591010 g3ıcm6cell day rDd0R2 C2 0 r,max Maximal rate of GF-induced collagen production 7.59109g3ıcm6cell day [12] CHalf-maximal GF enhancement of collagen synthesis 109g/cm3[12] Proportionality factor 2 [12] RHalf-maximal collagen enhancement of ECM deposition 0.3 g/cm3[12] dCollagen degradation rate per unit of cell density 7.59108cm3/cell day [12] DcGF diffusion rate 5102cm2/day [12] kcGF production rate per unit of cell density 7.5106cm3/cell day [6] Proportionality factor 1 [12] Half-maximal enhancement of net GF production 108g/cm3[12] dcGF decay rate 0.693 day1[6] GF, growth factor; ECM, extracellular matrix. Determined collagen degradation kinetics to remain in equilibrium away from the wound. Downestimated to prevent fibro-proliferative disorders [16] with the used GF decay rates. where the elastic modulus of the ECM varies with the collagen density, denotes the volumetric strain of the ECM, "denotes the ECM deformation and Irefers to the second-order identity tensor. In contrast, cell represents the stress exerted by the cells in the ECM cell Dpcell./ .1Cm/n R2 C2I.(8) Cell densities play a crucial role in determining the contractile stress [18, 19], which is limited by the collagen density and modulated by a mechanical stimulus, pcell. This stimulus represents the net stress of one cell per unit of ECM [23] and is a function of the matrix volumetric strain (), pcell./ DKactpmax Kact1pmax .1/Œ1,./ CKactpmax Kact2pmax .2/.,2./ CKpas. (9) In this expression, the contributions of two different components of the cell to the generation of stresses are considered, following the assumptions proposed by Moreo et al. (2008) [23]. The active contribution is a linear stiffness-dependent actuator that corresponds to the contractile mechanism, simulating the force provided by the actin and myosin cross-bridges at the sarcomere level during shortening. In fact, pmax is the maximum force provided by the actomyosin system. Following this approach, the series element Kact corresponds to the stiffness of the actin components that are aligned with the actomyosin motors. This model also considers a parallel component, Kpas,which corresponds to the stiffness of different mechanical components of cells, such as the membrane, microtubules and cytoplasm. Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2014) DOI: 10.1002/cnm 131 C. VALERO ET AL. Table III. List of model parameters related to the mechanical behaviour of cells and extracellular matrix. Parameter Description Value Observations pmax Maximal cellular active stress per unit of ECM 105Ng ıcm2cell [6] Kpas Volumetric stiffness moduli of the passive components of the cell 2105Ng ıcm2cell [23] Kact Volumetric stiffness moduli of the actin filaments of the cell 104Ng ıcm2cell [23] 1Shortening strain of the contractile element 0.6[6] 2Lengthening strain of the contractile element 0.5 [23] dHalf-maximal mechanical enhancement of fibroblast differentiation 105Ng ıcm2cell [6] 1Undamaged skin shear viscosity 200 N day/cm2[6] 2Undamaged skin bulk viscosity 200 N day/cm2[6] EdDermis Young’s modulus 33.4 N/cm2[24] dDermis skin Poisson’s ratio 0.3 [24] Eut Underlying tissue Young’s modulus 0.82 N/cm2[25] ut Underlying tissue Poisson’s ratio 0.459 [26] Myofibroblasts enhancement of traction per unit of fibroblasts density 103cm3/g [12] RTraction inhibition collagen density 5104g/cm3[12] sDermis tethering factor 101N/cm g Estimated ECM, extracellular matrix. Finally, fext denotes the tethering forces created by the attachments to the underlying tissue, which are proportional to the collagen density and the tissue displacements, fext Dsu. (10) The parameter values and descriptions of the mechanical equilibrium equations can be found in Table III. 2.2. Boundary conditions When modelling deep wounds, special attention should be paid to the evolution of species and mechanical tension along the wound depth. Thus, detailed descriptions of the boundary conditions for the different species are given in this section. Given the fixed morphology of the wound (deep and elongated), we assume that the mechanical evolution of the wound follows the plane strain hypotheses, neglecting deformations along its longitudinal direction. Furthermore, we simulate only the cross section of the wound. The wound is assumed to behave in the same way in all its cross sections, except near the two ends, considering the wound length is much larger than its other two dimensions (Figure 1). Let us denote the computational domain as (Figure 1). The computational domain consists of two parts, the wound and the surrounding undamaged tissue. The undamaged tissue consists of two layers, the dermis and the underlying tissue. The boundary of the domain @ consists of two distinct and nonintersecting parts: the free boundary (t), which is in contact with the environment and can move freely, and the fixed boundary (u), which is attached to the underlying tissue. Mathematically, it can be written as @ Du[twith u\tD;. The computational domain is large enough so that boundary effects on the evolution of the species can be disregarded. Hence, on the outer boundary u, diffusive fluxes and displacements are not allowed: rQn?D0and uD0on u, (11) for all species Qthat migrate or diffuse through the tissue (that is, fibroblasts nand the generic growth factor c). Here, n?denotes the normal vector pointing out from . On the upper boundary Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2014) DOI: 10.1002/cnm 132