Diego Alexander Garzón Alvarado

This study presents a computational model that integrates bone remodeling dynamics with damage accumulation, focusing on both physiological and pathological conditions. Building upon Komarova’s classical model of osteoclast and osteoblast interactions, this work introduces fatigue-induced damage using a stress-life (S-N) approach. By simulating bone responses under sinusoidal and random mechanical loads, the model captures the cyclical nature of bone turnover. The results show that under normal physiological conditions, bone is able to repair microdamage and maintain structural integrity. However, in pathological scenarios such as osteoporosis and tumors, the remodeling cycle is disrupted, leading to an increase in damage accumulation and eventual structural failure. Through numerical simulations, the study also demonstrates the significant impact of fatigue on bone health, showing that repetitive mechanical loads, even below critical stress levels, can result in bone degradation over time. By capturing the accumulation of microdamage and its repair, the model offers potential applications in personalized medicine to assess fracture risks in varying stress and health scenarios. This approach provides a framework for understanding how different stress patterns contribute to bone damage and offers insights into the progression of bone diseases. The model could be extended using metabolic and age-related characteristics and serves as a potential tool for personalized medicine, helping to predict bone failure risks in individuals when they are submitted to repetitive mechanical loads.

The generation mechanisms of Turing patterns have proven to be instrumental in representing biological systems, particularly those involving morphogenesis and pigmentation. While reactiondiffusion models of the molecules responsible for these biological processes have been extensively studied, the impact of interconnected domains on Turing pattern formation remains unexplored. This article delves into the intricate dynamics of Turing pattern formation within interconnected reactiondiffusion systems using the Schnakenberg model. Two coupling mechanisms are examined: (a) diffusive partition walls with variable width (5-100%) and diffusivity (d = 0.1, 10), and (b) direct contact boundaries where domains share a common interface. We examine how partition walls and direct contact boundaries between domains influence the morphology and stability of the resulting patterns. Finite element simulations of modal combinations (1, 1), (2, 2), (3, 3), and (4, 4) demonstrate that interconnected domains exhibit pattern generation mechanisms fundamentally different from the classical ‘local activation and long-range inhibition’ paradigm. Specifically, we observe modal competition, where lower-frequency modes suppress higher-frequency ones, and critical thresholds beyond which pattern formation is completely inhibited, particularly when the partition width exceeds 50% at low diffusivity (d = 0.1). Direct contact boundaries generate greater morphological diversity, giving rise to lobe-shaped and triangular structures not present in isolated domains, indicating that the immediacy of interdomain interaction strongly influences pattern morphology. These findings provide insights into the self-organization mechanisms governing biological pattern formation and offer potential applications in understanding pigmentation patterns and morphogenesis. Furthermore, these results open new perspectives for exploring alternative geometrical configurations, heterogeneous diffusion processes, and the extension of reaction-diffusion models to both two- and three-dimensional interconnected domains.

Bone remodeling is a fundamental physiological process that maintains skeletal integrity through a tightly regulated balance between bone resorption and formation. Mathematical modeling provides a powerful framework for understanding the complex regulatory mechanisms underlying this process, particularly the interplay between biomechanical stimuli and cellular dynamics. This study introduces a new mathematical model of bone remodeling designed to capture the temporal evolution of key bone cell populations and their response to mechanical stimuli within a simplified yet biologically informed architecture. We formulate a system of coupled nonlinear ordinary differential equations to describe the dynamics of osteoclasts, osteoblasts, and osteocytes within a single basic multicellular unit (BMU). The model incorporates feedback regulation via strain energy density, allowing mechanical input to influence cellular activity and bone surface transitions. Initial conditions are assigned to reflect the sequential activation of bone remodeling phases. Physiological parameters are adopted from well-established literature, while non-sourced parameters are tuned to ensure model stability and biological plausibility. The simulations reproduce the canonical sequence of bone remodeling: initiation, resorption, reversal, and formation. The model captures the coupling between mechanical loading and cellular activity, demonstrating how osteocyte signaling can modulate the recruitment of osteoclasts and osteoblasts. The results also highlight the system’s capacity to stabilize around a dynamic equilibrium, sensitive to both internal parameters and external mechanical inputs. This model offers a minimal yet comprehensive representation of bone remodeling dynamics within a single BMU, integrating mechanical and biological controls into a unified mathematical structure. It provides a foundation for future extensions toward spatially distributed models and applications in mechanobiological simulation and computational bone health assessment.

This work presents the integration of mechanobiological models to predict the natural evolution of bone modeling and remodeling processes to obtain the architecture of trabecular bone from the embryonic stage in mammalians. Bone modeling is simulated in two and three dimensions using a reaction–diffusion mechanism with parameters in Turing space. This approach involves the interaction of two molecular factors (VEGF and MMP13) released by hypertrophic chondrocytes that diffuse and interact within a hyaline cartilage matrix. The bone remodeling process follows the model proposed by Komarova et al. employing a set of differential equations to describe autocrine and paracrine interactions between osteoblastic and osteoclastic cells, determining cellular dynamics and changes in bone mass. Bidimensional and tridimensional results for a cartilage portion predict morphological self‐organization parameters between VEGF and MMP13, similar to those present in the architecture of immature trabecular bone. These findings suggest that the dynamic properties of molecular factors play a crucial role in the temporal self‐organization of bone mineralization metabolism, leading to a heterogeneous trabecular architecture characteristic of primary trabecular bone. Through the three‐dimensional bone remodeling model performed on the surface of trabeculae, it is established that equilibrium in population dynamics leads to asynchronous homeostatic remodeling for bone renewal, culminating in the formation of secondary trabecular bone.