E. Javierre

A cell-regulatory mechanism involving feedback between contraction and tissue formation guides wound healing progression

Wound healing is a process driven by cells. The ability of cells to sense mechanical stimuli from the extracellular matrix that surrounds them is used to regulate the forces that cells exert on the tissue. Stresses exerted by cells play a central role in wound contraction and have been broadly modelled. Traditionally, these stresses are assumed to be dependent on variables such as the extracellular matrix and cell or collagen densities. However, we postulate that cells are able to regulate the healing process through a mechanosensing mechanism regulated by the contraction that they exert. We propose that cells adjust the contraction level to determine the tissue functions regulating all main activities, such as proliferation, differentiation and matrix production. Hence, a closed-regulatory feedback loop is proposed between contraction and tissue formation. The model consists of a system of partial differential equations that simulates the evolution of fibroblasts, myofibroblasts, collagen and a generic growth factor, as well as the deformation of the extracellular matrix. This model is able to predict the wound healing outcome without requiring the addition of phenomenological laws to describe the time-dependent contraction evolution. We have reproduced two in vivo experiments to evaluate the predictive capacity of the model, and we conclude that there is feedback between the level of cell contraction and the tissue regenerated in the wound.

Challenges in the Modeling of Wound Healing Mechanisms in Soft Biological Tissues

Numerical models have become one of the most powerful tools in biomechanics and mechanobiology allowing highly detailed simulations. One of the fields in which they have broadly evolved during the last years is in soft tissue modeling. Particularly, wound healing in the skin is one of the processes that has been approached by computational models due to the difficulty of performing experimental investigations. During the last decades wound healing simulations have evolved from numerical models which considered only a few number of variables and simple geometries to more complex approximations that take into account a higher number of factors and reproduce more realistic geometries. Moreover, thanks to improved experimental observations, a larger number of processes, such as cellular stress generation or vascular growth, that take place during wound healing have been identified and modeled. This work presents a review of the most relevant wound healing approximations, together with an identification of the most relevant criteria that can be used to classify them. In addition, and looking towards the actual state of the art in the field, some future directions, challenges and improvements are analyzed for future developments.

A level set method for three dimensional vector Stefan problems: Dissolution of stoichiometric particles in multi-component alloys

A sharp interface method is proposed for the dissolution of stoichiometric particles in multi-component alloys occurring during the heat treatments of as-cast aluminium alloys prior to hot extrusion. In the mathematical model, a number of non-linearly coupled diffusion equations are given to determine the position of the particle interface and the interfacial concentrations. A level set method is used to determine the interface position at each time step. Once the front position is known, a fixed-point iteration is used to find the interfacial concentrations. The model is applicable to both complete and incomplete dissolution in two and three spatial dimensions, and handles topological changes in a natural fashion. The numerical solution is compared with steady-state and self-similar exact solutions available for simple particle geometries. Subsequently, the model is applied to an AlMgSi-alloy to investigate the influence of the particle morphology in the dissolution kinetics.

On the construction of analytic solutions for a diffusion-reaction equation with a discontinuous switch mechanism

The existence of waiting times, before boundary motion sets in, for a diffusion-diffusion reaction equation with a discontinuous switch mechanism, is demonstrated. Limit cases of the waiting times are discussed in mathematical rigor. Further, analytic solutions for planar and circular wounds are derived. The waiting times, as predicted using these analytic solutions, are perfectly between the derived bounds. Furthermore, it is demonstrated by both physical reasoning and mathematical rigor that the movement of the boundary can be delayed once it starts moving. The proof of this assertion resides on continuity and monotonicity arguments. The theory sustains the construction of analytic solutions. The model is applied to simulation of biological processes with a threshold behavior, such as wound healing or tumor growth.

Nonlinear finite element simulations of injuries with free boundaries: Application to surgical wounds

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.

Mechanobiological Modelling of Angiogenesis: Impact on Tissue Engineering and Bone Regeneration

Angiogenesis is essential for complex biological phenomena such as tissue engineering and bone repair. The ability to heal in these processes strongly depends on the ability of new blood vessels to grow. Capillary growth and its impact on human health has been focus of intense research from an in vivo, in vitro and in silico perspective. In fact, over the last decade many mathematical models have been proposed to understand and simulate the vascular network. This review addresses the role of the vascular network in well defined and controlled processes such as wound healing or distraction osteogenesis and covers the connection between vascularization and bone, starting with the biology of vascular ingrowth, moving through its impact on tissue engineering and bone regeneration, and ending with repair. Furthermore, we also describe the most recent in-silico models proposed to simulate the vascular network within bone constructs. Finally, discrete as well as continuum approaches are analyzed from a computational perspective and applied to three distinct phenomena: wound healing, distraction osteogenesis and individual cell migration in 3D.

A Cut-Cell Finite-Element Method for a Discontinuous Switch Model for Wound Closure

A mathematical model for epidermal wound healing is considered. The model is based on a moving boundary problem for the wound edge in which the edge moves if a generic epidermal growth factor exceeds a given threshold value. We use a Galerkin finite-element method to solve the equations for the growth factor concentration. The moving boundary (wound edge) is tracked using a level-set method with a local adaptive mesh refinement in the interface region. To deal with the reaction-diffusion equation for the growth factor, a cut-cell method has been implemented. This cut-cell method warrants the integration over a continuous reaction term elementwisely. The results improved with respect to the results that were obtained without the use of the cut-cell method.

An Evaluation of Surgical Functional Reconstruction of the Foot Using Kinetic and Kinematic Systems: A Case Report

Most pedobarographic studies of microsurgical foot reconstruction have been retrospective. In the present study, we report the results from a prospective pedobarographic study of a patient after microsurgical reconstruction of her foot with a latissimus dorsi flap and a cutaneous paddle, with a 42-month follow-up period. We describe the foot reconstruction plan and the pedobarographic measurements and analyzed its functional outcome. The goal of the present study was to demonstrate that pedobarography could have a role in the treatment of foot reconstruction from a quantitative perspective. The pedobarographic measurements were recorded after the initial coverage surgery and 2 subsequent foot remodeling procedures. A total of 4 pedobarographic measurements and 2 gait analyses were recorded and compared for both the noninvolved foot and the injured foot. Furthermore, the progress of the reconstructed foot was critically evaluated using this method. Both static and dynamic patterns were compared at subsequent follow-up visits after the foot reconstruction. The values and progression of the foot shape, peak foot pressure (kPa), average foot pressure (kPa), total contact surface (cm2), loading time (%), and step time (ms) were recorded. Initially, the pressure distribution of the reconstructed foot showed higher peak values at nonanatomic locations, revealing a greater ulceration risk. Over time, we found an improvement in the shape and values of these factors in the involved foot. To homogenize the pressure distribution and correct the imbalance between the 2 feet, patient-specific insoles were designed and fabricated. In our patient, pedobarography provided an objective, repeatable, and recordable method for the evaluation of the reconstructed foot. Pedobarography can therefore provide valuable insights into the prevention of pressure ulcers and optimization of rehabilitation.

Computing Interfaces in Diverse Applications

Mathematical models and computing methods for problems involving movinginterfaces are considered. These occur in a great variety of applications, and mathematical models provide a unifying framework, facilitating interdisciplinary cooperation.We discuss and proposesome genericnumericalmethods for problems involving moving interfaces. The level set method is used for interface capturing. A Cartesian and a finite element mesh are used simultaneously. This facilitates effi- cient local mesh refinement and derefinement for accurate computation of physical effects occurring at the interfaces, that move and may undergo topological change. The method has been implemented in three dimensions. We present examples from materials science (homogenization) and medical technology (wound healing).