Duchenne and Becker muscular dystrophy: Cellular mechanisms, image analysis, and computational models: A review.

The muscle is the principal tissue that is capable to transform potential energy into kinetic energy. This process is due to the transformation of chemical energy into mechanical energy to enhance the movements and all the daily activities. However, muscular tissues can be affected by some pathologies associated with genetic alterations that affect the expression of proteins. As the muscle is a highly organized structure in which most of the signaling pathways and proteins are related to one another, pathologies may overlap. Duchenne muscular dystrophy (DMD) is one of the most severe muscle pathologies triggering degeneration and muscle necrosis. Several mathematical models have been developed to predict muscle response to different scenarios and pathologies. The aim of this review is to describe DMD and Becker muscular dystrophy in terms of cellular behavior and molecular disorders and to present an overview of the computational models implemented to understand muscle behavior with the aim of improving regenerative therapy.

Finite element analysis of the influence of interdigitation pattern and collagen fibers on the mechanical behavior of the midpalatal suture.

The midpalatal suture (MPS) corresponds to the tissue that joins the two maxillary bones. Understanding the mechanical behavior of this tissue is of particular interest to those patients who require orthodontic treatments such as Rapid Maxillary Expansion (RME). The objective of this research was to observe the influence of interdigitation and collagen fibers on the mechanical response of MPS. To this end, a finite element analysis in two-dimensional models of the bone-suture-bone interface was performed considering the characteristics of the MPS. The geometry of the suture was modeled with 4 different levels of interdigitation: null, moderate, scalloped and fractal. The influence of collagen fibers, aligned transversely along the suture, was considered by incorporating linked structures of the bone fronts. According to the results, the factor that has the greatest impact on the magnitude and distribution of stresses is the interdigitation degree. A higher level of interdigitation produces an increase in tissue stiffness and a lower influence of collagen fibers on the mechanical response of the tissue. Therefore, this research contributes to the understanding of the MPS biomechanics by providing information that may be useful to health staff when evaluating the feasibility of procedures such as RME.

Computational model of articular cartilage regeneration induced by scaffold implantation in vivo.

Computational models allow to explain phenomena that cannot be observed through an animal model, such as the strain and stress states which can highly influence regeneration of the tissue. For this purpose, we have developed a simulation tool to determine the mechanical conditions provided by the polymeric scaffold. The computational model considered the articular cartilage, the subchondral bone, and the scaffold. All materials were modeled as poroelastic, and the cartilage had linear-elastic oriented collagen fibers. This model was able to explain the remodeling process that subchondral bone goes through, and how the scaffold allowed the conditions for cartilage regeneration. These results suggest that the use of scaffolds might lead the cartilaginous tissue growth in vivo by providing a better mechanical environment. Moreover, the developed computational model demonstrated to be useful as a tool prior experimental in vivo studies, by predicting the possible outcome of newly proposed treatments allowing to discard approaches that might not bring good results.

Simulation of the electrical stimulation of the rat brain using sleep frequencies: A finite element modeling approach.

A realistic rat brain model was used to simulate current density and electric field distributions under frequencies characteristic of sleeping states (0.8, 5, and 12 Hz). Two anode-electrode setups were simulated: plate vs. screws-anode, both with a cephalic cathode. Our simulations showed that these frequencies have limited impact on electric field and current density; however, the highest frequency evidenced higher values for both variables. The type of electrode setup had a greater effect on current distribution and induced fields. In that sense, the screws setup resulted in higher values of the modeled variables. The numeric results obtained are within the range of available data for rodent models using the finite elements method. These modeled effects should be analyzed regarding anatomical consequences (depth of penetration of the currents) and purpose of the experiment (i.e., entrainment of brain oscillations) in the context of sleep research.

Biomechanical behavior of an alveolar graft under maxillary therapies.

Cleft lip and palate is a congenital defect that affects the oral cavity. Depending on its severity, alveolar graft surgery and maxillary orthopedic therapies must be carried out as a part of the treatment. It is widely accepted that the therapies should be performed before grafting. Nevertheless, some authors have suggested that mechanical stimuli such as those from the maxillary therapies could improve the success rate of the graft. The aim of this study is to computationally determine the effect of maxillary therapies loads on the biomechanical response of an alveolar graft with different degrees of ossification. We also explore how the transverse width of the cleft affects the graft behavior and compare results with a non-cleft skull. Results suggest that stresses increase within the graft as it ossifies and are greater if maxillary expansion therapy is applied. This has consequences in the bone remodeling processes that are necessary for the graft osseointegration. Maxillary orthopedic therapies after graft surgery could be considered as a part of the treatment since they seem to act as a positive extra stimulus that can benefit the graft.

Simulation of pandemics in real cities: enhanced and accurate digital laboratories.

This study develops a modelling framework for simulating the spread of infectious diseases within real cities. Digital copies of Birmingham (UK) and Bogotá (Colombia) are generated, reproducing their urban environment, infrastructure and population. The digital inhabitants have the same statistical features of the real population. Their motion is a combination of predictable trips (commute to work, school, etc.) and random walks (shopping, leisure, etc.). Millions of individuals, their encounters and the spread of the disease are simulated by means of high-performance computing and massively parallel algorithms for several months and a time resolution of 1 minute. Simulations accurately reproduce the COVID-19 data for Birmingham and Bogotá both before and during the lockdown. The model has only one adjustable parameter calculable in the early stages of the pandemic. Policymakers can use our digital cities as virtual laboratories for testing, predicting and comparing the effects of policies aimed at containing epidemics.

Influence of interdigitation and expander type in the mechanical response of the midpalatal suture during maxillary expansion.

BACKGROUND AND OBJECTIVE: The orthopedic Maxillary Expansion (ME) procedure is used for treating the transverse maxillary deficiency. This pathology consists in a smaller transverse dimension in the maxilla and leads to malocclusion. The treatment takes advantage of the existence of the midpalatal suture (MPS), which corresponds to the junction at the palatine bones of its horizontal portions. The technique employs a device, conventionally a palatal expander attached to the posterior teeth, to separate the two maxillary bones in the MPS. The objective of this study was to analyze, using the Finite Element Method, the biomechanical behavior of the MPS when an expansion is applied. METHODS: A Computer Tomography image of the maxilla was reconstructed, the suture geometry was modeled with different interdigitation levels and types of hyrax devices. A total of 12 geometric models (three levels for interdigitation and four types of hyrax devices) were prepared and analyzed taking into account the chewing forces and the expansion displacement. For each case, maximum principal stresses on the maxilla (bone), and equivalent stresses on the expander device (stainless steel) were observed. In the MPS, maximum principal stresses and directional displacement were evaluated. RESULTS: The results showed that the interdigitation does not have an important influence on the deformation behavior of the maxilla but it affects the stress distribution. In addition, the type of expander device and anchorage have a direct relationship with the treatment effectiveness; larger deformation in the expansion direction was obtained with skeletal when compared to dental anchorage. CONCLUSIONS: A study that allows a better understanding of the oral biomechanics during the application of ME was presented. To our knowledge, it is the first study based on computational simulations that takes into account bone structures, like maxilla and part of the skull, to analyze the interdigitation influence on the MPS behavior when exposed to a ME.

Capacitively coupled electrical stimulation of rat chondroepiphysis explants: A histomorphometric analysis.

The growth plate is a cartilaginous layer present from the gestation period until the end of puberty where it ossifies joining diaphysis and epiphysis. During this period several endocrine, autocrine, and paracrine processes within the growth plate are carried out by chondrocytes; therefore, a disruption in cellular functions may lead to pathologies affecting bone development. It is known that electric fields impact the growth plate; however, parameters such as stimulation time and electric field intensity are not well documented. Accordingly, this study presents a histomorphometrical framework to assess the effect of electric fields on chondroepiphysis explants. Bones were stimulated with 3.5 and 7 mV/cm, and for each electric field two exposure times were tested for 30 days (30 min and 1 h). Results evidenced that electric fields increased the hypertrophic zones compared with controls. In addition, a stimulation of 3.5 mV/cm applied for 1 h preserved the columnar cell density and its orientation. Moreover, a pre-hypertrophy differentiation in the center of the chondroepiphysis was observed when explants were stimulated during 1 h with both electric fields. These findings allow the understanding of the effect of electrical stimulation over growth plate organization and how the stimulation modifies chondrocytes morphophysiology.

Mechanobiological modeling of endochondral ossification: an experimental and computational analysis.

Long bone formation starts early during embryonic development through a process known as endochondral ossification. This is a highly regulated mechanism that involves several mechanical and biochemical factors. Because long bone development is an extremely complex process, it is unclear how biochemical regulation is affected when dynamic loads are applied, and also how the combination of mechanical and biochemical factors affect the shape acquired by the bone during early development. In this study, we develop a mechanobiological model combining: (1) a reaction-diffusion system to describe the biochemical process and (2) a poroelastic model to determine the stresses and fluid flow due to loading. We simulate endochondral ossification and the change in long bone shapes during embryonic stages. The mathematical model is based on a multiscale framework, which consisted in computing the evolution of the negative feedback loop between Ihh/PTHrP and the diffusion of VEGF molecule (on the order of days) and dynamic loading (on the order of seconds). We compare our morphological predictions with the femurs of embryonic mice. The results obtained from the model demonstrate that pattern formation of Ihh, PTHrP and VEGF predict the development of the main structures within long bones such as the primary ossification center, the bone collar, the growth fronts and the cartilaginous epiphysis. Additionally, our results suggest high load pressures and frequencies alter biochemical diffusion and cartilage formation. Our model incorporates the biochemical and mechanical stimuli and their interaction that influence endochondral ossification during embryonic growth. The mechanobiochemical framework allows us to probe the effects of molecular events and mechanical loading on development of bone.

Cellular scale model of growth plate: An in silico model of chondrocyte hypertrophy.

The growth plate is the responsible for longitudinal bone growth. It is a cartilaginous structure formed by chondrocytes that are continuously undergoing a differentiation process that starts with a highly proliferative state, followed by cellular hypertrophy, and finally tissue ossification. Within the growth plate chondrocytes display a characteristic columnar organization that potentiates longitudinal growth. Both chondrocyte organization and hypertrophy are highly regulated processes influenced by biochemical and mechanical stimuli. These processes have been studied mainly using in vivo models, although there are few computational approaches focused on the rate of ossification rather than events at cellular level. Here, we developed a model of cellular behavior integrating biochemical and structural factors in a single column of cells in the growth plate. In our model proliferation and hypertrophy were controlled by biochemical regulatory loop formed between Ihh and PTHrP (modeled as a set of reaction-diffusion equations), while cell growth was controlled by mechanical loading. We also examined the effects of static loading. The model reproduced the proliferation and hypertrophy of chondrocytes in organized columns. This model constitutes a first step towards the development of mechanobiological models that can be used to study biochemical interactions during endochondral ossification.

Cellular automata model for human articular chondrocytes migration, proliferation and cell death: An in vitro validation.

Articular cartilage is characterized by low cell density of only one cell type, chondrocytes, and has limited self-healing properties. When articular cartilage is affected by traumatic injuries, a therapeutic strategy such as autologous chondrocyte implantation is usually proposed for its treatment. This approach requires in vitro chondrocyte expansion to yield high cell number for cell transplantation. To improve the efficiency of this procedure, it is necessary to assess cell dynamics such as migration, proliferation and cell death during culture. Computational models such as cellular automata can be used to simulate cell dynamics in order to enhance the result of cell culture procedures. This methodology has been implemented for several cell types; however, an experimental validation is required for each one. For this reason, in this research a cellular automata model, based on random-walk theory, was devised in order to predict articular chondrocyte behavior in monolayer culture during cell expansion. Results demonstrated that the cellular automata model corresponded to cell dynamics and computed-accurate quantitative results. Moreover, it was possible to observe that cell dynamics depend on weighted probabilities derived from experimental data and cell behavior varies according to the cell culture period. Thus, depending on whether cells were just seeded or proliferated exponentially, culture time probabilities differed in percentages in the CA model. Furthermore, in the experimental assessment a decreased chondrocyte proliferation was observed along with increased passage number. This approach is expected to having other uses as in enhancing articular cartilage therapies based on tissue engineering and regenerative medicine.

Flat bones and sutures formation in the human cranial vault during prenatal development and infancy: A computational model.

The processes of flat bones growth, sutures formation and interdigitation in the human calvaria are controlled by a complex interaction between genetic, biochemical and environmental factors that regulate bone formation and resorption during prenatal development and infancy. Despite previous experimental evidence accounting for the role of the main biochemical factors acting on these processes, the underlying mechanisms controlling them are still unknown. Therefore, we propose a mathematical model of the processes of flat bone and suture formation, taking into account several biological events. First, we model the growth of the flat bones and the formation of sutures and fontanels as a reaction diffusion system between two proteins: TGF-ß2 and TGF-ß3. The former is expressed by osteoblasts and allows adjacent mesenchymal cells differentiation on the bone fronts of each flat bone. The latter is expressed by mesenchymal cells at the sutures and inhibits their differentiation into osteoblasts at the bone fronts. Suture interdigitation is modelled using a system of reaction diffusion equations that develops spatio-temporal patterns of bone formation and resorption by means of two molecules (Wnt and Sclerostin) which control mesenchymal cells differentiation into osteoblasts at these sites. The results of the computer simulations predict flat bone growth from ossification centers, sutures and fontanels formation as well as bone formation and resorption events along the sutures, giving rise to interdigitated patterns. These stages were modelled and solved by the finite elements method. The simulation results agree with the morphological characteristics of calvarial bones and sutures throughout human prenatal development and infancy.

Growth plate stress distribution implications during bone development: a simple framework computational approach.

Mechanical stimuli play a significant role in the process of long bone development as evidenced by clinical observations and in vivo studies. Up to now approaches to understand stimuli characteristics have been limited to the first stages of epiphyseal development. Furthermore, growth plate mechanical behavior has not been widely studied. In order to better understand mechanical influences on bone growth, we used Carter and Wong biomechanical approximation to analyze growth plate mechanical behavior, and explore stress patterns for different morphological stages of the growth plate. To the best of our knowledge this work is the first attempt to study stress distribution on growth plate during different possible stages of bone development, from gestation to adolescence. Stress distribution analysis on the epiphysis and growth plate was performed using axisymmetric (3D) finite element analysis in a simplified generic epiphyseal geometry using a linear elastic model as the first approximation. We took into account different growth plate locations, morphologies and widths, as well as different epiphyseal developmental stages. We found stress distribution during bone development established osteogenic index patterns that seem to influence locally epiphyseal structures growth and coincide with growth plate histological arrangement.

Numerical simulation of electrically stimulated osteogenesis in dental implants.

Cell behavior and tissue formation are influenced by a static electric field (EF). Several protocols for EF exposure are aimed at increasing the rate of tissue recovery and reducing the healing times in wounds. However, the underlying mechanisms of the EF action on cells and tissues are still a matter of research. In this work we introduce a mathematical model for electrically stimulated osteogenesis at the bone-dental implant interface. The model describes the influence of the EF in the most critical biological processes leading to bone formation at the bone-dental implant interface. The numerical solution is able to reproduce the distribution of spatial-temporal patterns describing the influence of EF during blood clotting, osteogenic cell migration, granulation tissue formation, displacements of the fibrillar matrix, and formation of new bone. In addition, the model describes the EF-mediated cell behavior and tissue formation which lead to an increased osteogenesis in both smooth and rough implant surfaces. Since numerical results compare favorably with experimental evidence, the model can be used to predict the outcome of using electrostimulation in other types of wounds and tissues.

Numerical investigation into blood clotting at the bone-dental implant interface in the presence of an electrical stimulus.

The insertion of a dental implant activates a sequence of wound healing events ending with bone formation and implant osseointegration. This sequence starts with the blood coagulation process and the formation of a fibrin network that detains spilt blood. Fibrin formation can be simplified as the kinetic reaction between thrombin and fibrinogen preceding the conversion of fibrinogen into fibrin. Based on experimental observations of the electrical properties of these molecules, we present a hypothesis for the mechanism of a static electrical stimulus in controlling the formation of the blood clot. Specifically, the electrical stimulus increases the fibrin network formation in such a way that a preferential region of higher fibrin density is obtained. This hypothesis is validated by means of a numerical model for the blood clot formation at the bone-dental implant interface. Numerical results compare favorably to experimental observations for blood clotting with and without the static electrical stimulus. It is concluded that the density of the fibrin network depends on the strength of the static electrical stimulus, and that the blood clot formation has a preferential direction of formation in the presence of the electrical signal.

Mathematical model of electrotaxis in osteoblastic cells.

Electrotaxis is the cell migration in the presence of an electric field (EF). This migration is parallel to the EF vector and overrides chemical migration cues. In this paper we introduce a mathematical model for the electrotaxis in osteoblastic cells. The model is evaluated using different EF strengths and different configurations of both electrical and chemical stimuli. Accordingly, we found that the cell migration speed is described as the combination of an electrical and a chemical term. Cell migration is faster when both stimuli orient cell migration towards the same direction. In contrast, a reduced speed is obtained when the EF vector is opposed to the direction of the chemical stimulus. Numerical relations were obtained to quantify the cell migration speed at each configuration. Additional calculations for the cell colonization of a substrate also show mediation of the EF strength. Therefore, the term electro-osteoconduction is introduced to account the electrically induced cell colonization. Since numerical results compare favorably with experimental evidence, the model is suitable to be extended to other types of cells, and to numerically explore the influence of EF during wound healing.

Comparative analysis of numerical integration schemes of density equation for a computational model of bone remodelling.

In this study, a computational model of bone remodelling problem as proposed by Weinans et al. (1992) is described and solved by other temporal integration techniques different from the Euler scheme. This model considers three types of numerical integration schemes of the evolution of the material density during the remodelling: Euler, Heun and Runge-Kutta methods. Also the strain and the density field are obtained inside each element, at Gauss points or at the nodes of the mesh. A square plate with 1.00 m of side subjected to non-uniform pressure is simulated with two meshes of quadrilateral element with size [Formula: see text] and [Formula: see text] m. Two increments time size: [Formula: see text] and [Formula: see text] days are used. The results show that Euler, Heun and Runge-Kutta's methods correctly approached the problem of bone remodelling and that there were no appreciable differences in the patterns obtained by the mesh and time step used. In contrast, using an element-based approach and node-based approach, substantial differences were produced in bone remodelling density pattern. 'Chess board' type discontinuities were found in the element approach near the applied pressure area, as were well-defined columns away from this. The node-based approach showed continuity in density distribution. These patterns were well represented by the methods for resolving the density equation. This study concluded that any method of time integration could be used for these meshes and time steps size.

A mathematical model of medial collateral ligament repair: migration, fibroblast proliferation and collagen formation.

The partial rupture of ligament fibres leads to an injury known as grade 2 sprain. Wound healing after injury consists of four general stages: swelling, release of platelet-derived growth factor (PDGF), fibroblast migration and proliferation and collagen production. The aim of this paper is to present a mathematical model based on reaction-diffusion equations for describing the repair of the medial collateral ligament when it has suffered a grade 2 sprain. We have used the finite element method to solve the equations of this. The results have simulated the tissue swelling at the time of injury, predicted PDGF influence, the concentration of fibroblasts migrating towards the place of injury and reproduced the random orientation of immature collagen fibres. These results agree with experimental data reported by other authors. The model describes wound healing during the 9 days following such injury.

A mechanobiological model of epiphysis structures formation.

Developing bone consists of epiphysis, metaphysis and diaphysis. The secondary ossification centre (SOC) appears and grows within the epiphysis, involving two histological stages. Firstly, cartilage canals appear; they carry hypertrophy factors towards the central area of the epiphysis. Canal growth and expansion is modulated by stress on the epiphysis. Secondly, the diffusion of hypertrophy factors causes SOC growth. Hypertrophy is regulated by biological and mechanical factors present within the epiphysis. The finite element method has been used for solving a coupled system of differential equations for modelling these histological stages of epiphyseal development. Cartilage canal spatial-temporal growth patterns were obtained as well as the SOC formation pattern. This model qualitatively agreed with experimental results reported by other authors.

A phenomenological mathematical model of the articular cartilage damage.

Articular cartilage (AC) is a biological tissue that allows the distribution of mechanical loads and movement of joints. The presence of these mechanical loads influences the behavior and physiological condition of AC. The loads may cause damaged by fatigue through injuries due to repeated accumulated stresses. The aim of this work is to introduce a phenomenological mathematical model of damage caused by mechanical action. It is considered that tissue failure is a consequence of chondrocyte death and matrix loss, taking into account factors modifying fatigue resistance such as age, body mass index (BMI) and metabolic activity. The model was numerically implemented using the finite elements method and the results obtained allowed us to predict tissue failure at different loading frequencies, different damage sites and variations in damage magnitude. Qualitative concordance between numerical results and experimental data led us to conclude that the model may be useful for physicians and therapists as a prediction tool for prescribing physical exercise and prognosis of joint failure.