P. Moreo

Modeling mechanosensing and its effect on the migration and proliferation of adherent cells.

The behavior of normal adherent cells is influenced by the stiffness of the substrate they are anchored to. Cells are able to detect substrate mechanical properties by actively generating contractile forces and use this information to migrate and proliferate. In particular, the speed and direction of cell crawling, as well as the rate of cell proliferation, vary with the substrate compliance and prestrain. In this work, we present an active mechanosensing model based on an extension of the classical Hills model for skeletal muscle behavior. We also propose a thermodynamical approach to model cell migration regulated by mechanical stimuli and a proliferation theory also depending on the mechanical environment. These contributions give rise to a conceptually simple mathematical formulation with a straightforward and inexpensive computational implementation, yielding results consistent with numerous experiments. The model can be a useful tool for practical applications in biology and medicine in situations where cell-substrate interaction as well as substrate mechanical behavior play an important role, such as the design of tissue engineering applications.

Bone ingrowth on the surface of endosseous implants. Part 1: Mathematical model

Osseointegration, understood as an intimate apposition and interdigitation of bone to a biomaterial, is usually regarded as a major condition for the long-term clinical success of bone implants. Clearly, the anchorage of an implant to bone tissue critically relies on the formation of new bone between the implant and the surface of the old peri-implant bone and depends on factors such as the surface microtopography, chemical composition and geometry of the implant, the properties of the surrounding bone and the mechanical loading process. The main contribution of this work is the proposal of a new mathematical framework based on a set of reaction-diffusion equations that try to model the main biological interactions occurring at the surface of implants and is able to reproduce most of the above mentioned biological features of the osseointegration phenomenon. This is a two-part paper. In this first part, a brief biological overview is initially given, followed by the presentation and discussion of the model. In addition, two-dimensional finite element simulations of the bone-ingrowth process around a dental implant with two different surface properties are included to assess the validity of the model. Numerical solutions show the ability of the model to reproduce features such as contact/distance osteogenesis depending upon the specific surface microtopography. In Part 2 [Moreo, P., García-Aznar, J.M., Doblaré, M., 2008. Bone ingrowth on the surface of endosseous implants. Part 2: influence of mechanical stimulation, type of bone and geometry. J. Theor. Biol., submitted for publication], two simplified versions of the whole model are proposed. An analytical study of the stability of fixed points as well as the existence of travelling wave-type solutions has been done with both simplified models, providing a significant insight into the behaviour of the model and giving clues to interpret the effectiveness of recently proposed clinical therapies. Furthermore, we also show that, although the mechanical state of the tissue is not directly taken into account in the model equations, it is possible to analyse in detail the effect that mechanical stimulation would have on the predictions of the model. Finally, numerical simulations are also included in the second part of the paper, with the aim of looking into the influence of implant geometry on the osseointegration process.

On the Modelling of Biological Patterns with Mechanochemical Models: Insights from Analysis and Computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613–621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.

Bone ingrowth on the surface of endosseous implants. Part 2: Theoretical and numerical analysis.

The study of osseointegration of endosseous implants is a matter of great interest, mostly due to the increase in the use of many types of implants in clinical practice. Bone ingrowth results from a complex process, in which mechanics and biology play a major role. A wide variety of diverse factors can affect the development of the process, such as the properties or geometry of the implant surface, the mechanical stimulation or the initial cell conditions. In the first part of this article [Moreo, P., García-Aznar, J.M., Doblaré, M., 2008. Bone ingrowth on the surface of endosseous implants. Part 1: mathematical model. J. Theor. Biol., in press] a model composed of a set of reaction-diffusion equations was proposed to simulate the formation of bone around implants, specially focused on the early stages of bone healing, that was able to contemplate the effects of surface microtopography. The goal of this second part is to use the model to analyse the effect of factors such as cell stimulation, the initial cell concentration in the host bone and the geometry of the implant. For this purpose, two different simplified versions of the model are here analysed theoretically and further insight is gained from the study of the stability of fixed points and existence of travelling waves. Additionally, numerical simulations by means of the finite element method have been performed to examine the osseointegration of a dental implant with grooves at the surface of the threads. Results obtained from the analysis and simulations show that the model can reproduce some features of peri-implant bone ingrowth.