J.M. García-Aznar

On scaffold designing for bone regeneration: A computational multiscale approach

Scaffold design for bone tissue engineering applications involves many parameters that directly influence the rate of bone tissue regeneration onto its microstructural surface. To improve scaffold functionality, increasing interest is being focused on in vitro and in vivo research in order to obtain the optimal scaffold design for a specific application. However, the evaluation of the effect of each specific scaffold parameter on tissue regeneration using these techniques requires costly protocols and long-term experiments. In this paper, we elucidate the effect of some scaffold parameters on bone tissue regeneration by means of a mathematically based approach. By virtue of in silico experiments, factors such as scaffold stiffness, porosity, resorption kinetics, pore size and pre-seeding are analyzed in a specific bone tissue application found in the literature. The model predicts the in vivo rate of bone formation within the scaffold, the scaffold degradation and the interaction between the implanted scaffold and the surrounding bone. Results show an increasing rate of bone regeneration with increasing scaffold stiffness, scaffold mean pore size and pre-seeding, whereas the collapse of the scaffold occurs for a faster biomaterial resorption kinetics. Requiring further experimental validation, the model can be useful for the assessment of scaffold design and for the analysis of scaffold parameters in tissue regeneration.

Collective cell durotaxis emerges from long-range intercellular force transmission

The ability of cells to follow gradients of extracellular matrix stiffness—durotaxis—has been implicated in development, fibrosis, and cancer. Here, we found multicellular clusters that exhibited durotaxis even if isolated constituent cells did not. This emergent mode of directed collective cell migration applied to a variety of epithelial cell types, required the action of myosin motors, and originated from supracellular transmission of contractile physical forces. To explain the observed phenomenology, we developed a generalized clutch model in which local stick-slip dynamics of cell-matrix adhesions was integrated to the tissue level through cell-cell junctions. Collective durotaxis is far more efficient than single-cell durotaxis; it thus emerges as a robust mechanism to direct cell migration during development, wound healing, and collective cancer cell invasion.

Permeability evaluation of 45S5 Bioglass®-based scaffolds for bone tissue engineering

Permeability is a key parameter for microstructural design of scaffolds, since it is related to their capability for waste removal and nutrients/oxygen supply. In this framework, Darcys experiments were carried out in order to determine the relationship between the pressure drop gradient and the fluid flow velocity in Bioglass-based scaffolds to obtain the scaffolds permeability. Using deionised water as working fluid, the measured average permeability value on scaffolds of 90-95% porosity was 1.96 x 10(-9) m(2). This value lies in the published range of permeability values for trabecular bone.

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.

Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism

We present here a mathematical model of chondrocytes hypertrophy, regulated by molecular factors, able to predict the onset of secondary ossification centres during long bones development. From the results obtained, we can conclude that the chemical feedback between two reactive molecular factors through a reaction-diffusion mechanism could explain the stable spatial pattern found in the origination of secondary ossification centres, as well as the dependency of such a pattern on the size and shape of the bone head, in line with what has been observed in different animals.

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.

Modeling distraction osteogenesis: analysis of the distraction rate.

Distraction osteogenesis is a useful technique aimed at inducing bone formation in widespread clinical applications. One of the most important factors that conditions the success of bone regeneration is the distraction rate. Since the mechanical environment around the osteotomy site is one of the main factors that affects both quantity and quality of the regenerated bone, we have focused on analyzing how the distraction rate influences on the mechanical conditions and tissue regeneration. Therefore, the aim of the present work is to explore the potential of a mathematical algorithm to simulate clinically observed distraction rate related phenomena that occur during distraction osteogenesis. Improvements have been performed on a previous model (Gómez-Benito et al. in J Theor Biol 235:105–119, 2005) in order to take into account the load history. The results obtained concur with experimental findings: a slow distraction rate results in premature bony union, whereas a fast rate results in a fibrous union. Tension forces in the interfragmentary gap tissue have also been estimated and successfully compared with experimental measurements.

Dynamic Mechanisms of Cell Rigidity Sensing: Insights from a Computational Model of Actomyosin Networks

Cells modulate themselves in response to the surrounding environment like substrate elasticity, exhibiting structural reorganization driven by the contractility of cytoskeleton. The cytoskeleton is the scaffolding structure of eukaryotic cells, playing a central role in many mechanical and biological functions. It is composed of a network of actins, actin cross-linking proteins (ACPs), and molecular motors. The motors generate contractile forces by sliding couples of actin filaments in a polar fashion, and the contractile response of the cytoskeleton network is known to be modulated also by external stimuli, such as substrate stiffness. This implies an important role of actomyosin contractility in the cell mechano-sensing. However, how cells sense matrix stiffness via the contractility remains an open question. Here, we present a 3-D Brownian dynamics computational model of a cross-linked actin network including the dynamics of molecular motors and ACPs. The mechano-sensing properties of this active network are investigated by evaluating contraction and stress in response to different substrate stiffness. Results demonstrate two mechanisms that act to limit internal stress: (i) In stiff substrates, motors walk until they exert their maximum force, leading to a plateau stress that is independent of substrate stiffness, whereas (ii) in soft substrates, motors walk until they become blocked by other motors or ACPs, leading to submaximal stress levels. Therefore, this study provides new insights into the role of molecular motors in the contraction and rigidity sensing of cells.

A reaction–diffusion model for long bones growth

Bone development is characterized by differentiation and growth of chondrocytes from the proliferation zone to the hypertrophying one. These two cellular processes are controlled by a complex signalling regulatory loop between different biochemical signals, whose production depends on the current cell density, constituting a coupled cell-chemical system. In this work, a mathematical model of the process of early bone growth is presented, extending and generalizing other earlier approaches on the same topic. A reaction–diffusion regulatory loop between two chemical factors: parathyroid hormone-related peptide (PTHrP) and Indian hedgehog (Ihh) is hypothesized, where PTHrP is activated by Ihh and inhibits Ihh production. Chondrocytes proliferation and hypertrophy are described by means of population equations being both regulated by the PTHrP and Ihh concentrations. In the initial stage of bone growth, these two cellular proceses are considered to be directionally dependent, modelling the well known column cell formation, characteristic of endochondral ossification. This coupled set of equations is solved within a finite element framework, getting an estimation of the chondrocytes spatial distribution, growth of the diaphysis and formation of the epiphysis of a long bone. The results obtained are qualitatively similar to the actual physiological ones and quantitatively close to some available experimental data. Finally, this extended approach allows finding important relations between the model parameters to get stability of the physiological process and getting additional insight on the spatial and directional distribution of cells and paracrine factors.

Mechano-sensing and cell migration: a 3D model approach

Cell migration is essential for tissue development in different physiological and pathological conditions. It is a complex process orchestrated by chemistry, biological factors, microstructure and surrounding mechanical properties. Focusing on the mechanical interactions, cells do not only exert forces on the matrix that surrounds them, but they also sense and react to mechanical cues in a process called mechano-sensing. Here, we hypothesize the involvement of mechano-sensing in the regulation of directional cell migration through a three-dimensional (3D) matrix. For this purpose, we develop a 3D numerical model of individual cell migration, which incorporates the mechano-sensing process of the cell as the main mechanism regulating its movement. Consistent with this hypothesis, we found that factors, such as substrate stiffness, boundary conditions and external forces, regulate specific and distinct cell movements.