Allosteric control model of bone remodelling containing periodical modes

To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodelling process prevalent in native bone. We have formulated a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo and could set a basis for the more detailed allosteric models. The model is extended towards a bio-cybernetic vision of basic multicellular unit (BMU) action, when some of the regulation loops have been modified to reflect the allosteric control mechanisms, developed by Michaels-Menten, Hill, Koshland-Nemethy-Filmer, Monod-Wyman-Changeux. By implementation of this approach a four-dimensional system was obtained that shows steady cyclic behaviour using a wide range of constants with clear biological meaning. We have observed that a local steady state appears as a limiting cycle in multi-dimensional phase space and this is discussed in this paper. Physiological interpretation of this limiting four-dimension cycle possibly related to a conservative-like value has been proposed. Analysis and simulation of the model has shown an analogy between this conservative value, as a kind of substrate-energy regenerative potential of the bone remodelling system with a molecular nature, and to the classical physical value--energy. This dynamic recovery potential is directed against both mechanical and biomechanical damage to the bone. Furthermore, the current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Hill mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This result indicates the viability of the proposed existence of a conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. Linear stability analysis has been performed in order to determine the robustness of the basic state, however, additional work is required to study a wider range of constants.     

On allosteric control model of bone turnover cycle containing osteocyte regulation loop

One approach to developing a mathematical model that predicts osteoactivity both in bio-scaffolds, as well as the in bone tissue in vivo, is based on a bio-cybernetic vision of basic multicellular unit (BMU) action. In the case of the model presented in this paper, some of the loops of regulation have been modified to reflect the range of allosteric control mechanisms: Michaelis-Menten, Hill, Adair, Koshland-Nemethy-Filmer (KNF), Monod-Wyman-Changeux (MWC). This approach has resulted in a four-dimensional system that shows steady cyclic behaviour using a range of constants with clear biological meaning. The initial findings suggesting that a steady state appears as a cycle in multidimensional phase space and this is discussed in this paper. The existence of this cycle in the osteoclasts-osteoblasts-osteocytes-bone subspace indicates that there is a conservative value along steady trajectories for this dynamic system. Biophysical interpretation of this conservative value has been proposed as a kind of substrate-energy regenerative potential of the bone remodelling system with a similarity to the classical physical value-energy. Such a recovery "potential" is directed against both mechanical and biomechanical damage to the bone. The current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Michaelis-Menten mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This finding demonstrates the viability of the suggestion of the subsistence of conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. The results indicate that the robust behaviour of the model is maintained from the simple cellular level to the molecular biochemical level of regulation.     

A Computational Model for Cortical Endosteal Surface Remodeling Induced by Mechanical Disuse

In mechanical disuse conditions associated with immobilization and microgravity in spaceflight, cortical endosteal surface moved outward with periosteal surface moving slightly or unchanged, resulting in reduction of cortical thickness. Reduced thickness of the shaft cortex of long bone can be considered as an independent predictor of fractures. Accordingly, it is important to study the remodeling process at cortical endosteal surface. This paper presents a computer simulation of cortical endosteal remodeling induced by mechanical disuse at the Basic Multicellular Units level with cortical thickness as controlling variables. The remodeling analysis was performed on a representative rectangular slice of the cross section of cortical bone volume. The pQCT data showing the relationship between the duration of paralysis and bone structure of spinal cord injured patients by Eser et al. (2004) were used as an example of mechanical disuse to validate the model. Cortical thickness, BMU activation frequency, mechanical load and principal compressive strain for tibia and femur cortical models were simulated. The effects of varying the mechanical load and maximum BMU activation frequency were also investigated. The cortical thicknesses of femur and tibia models were both consistent with the clinical data. Varying the decreasing coefficient in mechanical load equation had little effect on the steady state values of cortical thickness and BMU activation frequency. However, it had much effect on the time to reach steady state. The maximum BMU activation frequency had effects on both the steady state value and the time to reach steady state for cortical thickness and BMU activation frequency. The computational model for cortical endosteal surface remodeling developed in this paper can be further used to quantify and predict the effects of mechanical factors and biological factors on cortical thickness and help us to better understand the relationship between bone morphology and mechanical as well as biological environment.     

TNAP, TrAP, ecto-purinergic signaling, and bone remodeling

Bone remodeling is a process of continuous resorption and formation/mineralization carried out by osteoclasts and osteoblasts, which, along with osteocytes, comprise the bone multicellular unit (BMU). A key component of the BMU is the bone remodeling compartment (BRC), isolated from the marrow by a canopy of osteoblast-like lining cells. Although much progress has been made regarding the cytokine-dependent and hormonal regulation of bone remodeling, less attention has been placed on the role of extracellular pH (pH(e)). Osteoclastic bone resorption occurs at acidic pH(e). Furthermore, osteoclasts can be regarded as epithelial-like cells, due to their polarized structure and ability to form a seal against bone, isolating the lacunar space. The major ecto-phosphatases of osteoclasts and osteoblasts, acid and alkaline phosphatases, both have ATPase activity with pH optima several units different from neutrality. Furthermore, osteoclasts and osteoblasts express plasma membrane purinergic P2 receptors that, upon activation by ATP, accelerate bone osteoclast resorption and impair osteoblast mineralization. We hypothesize that these ecto-phosphatases help regulate [ATP](e) and localized pH(e) at the sites of bone resorption and mineralization by pH-dependent ATP hydrolysis coupled with P2Y-dependent regulation of osteoclast and osteoblast function. Furthermore, osteoclast cellular HCO3(-), formed as a product of lacunar V-ATPase H(+) secretion, is secreted into the BRC, which could elevate BRC pH(e), in turn affecting osteoblast function. We will review the existing data addressing regulation of BRC pH(e), present a hypothesis regarding its regulation, and discuss the hypothesis in the context of the function of proteins that regulate pH(e).     

Bone Balance within a Cortical BMU: Local Controls of Bone Resorption and Formation

Maintaining bone volume during bone turnover by a BMU is known as bone balance. Balance is required to maintain structural integrity of the bone and is often dysregulated in disease. Consequently, understanding how a BMU controls bone balance is of considerable interest. This paper develops a methodology for identifying potential balance controls within a single cortical BMU. The theoretical framework developed offers the possibility of a directed search for biological processes compatible with the constraints of balance control. We first derive general control constraint equations and then introduce constitutive equations to identify potential control processes that link key variables that describe the state of the BMU. The paper describes specific local bone volume balance controls that may be associated with bone resorption and bone formation. Because bone resorption and formation both involve averaging over time, short-term fluctuations in the environment are removed, leaving the control systems to manage deviations in longer-term trends back towards their desired values. The length of time for averaging is much greater for bone formation than for bone resorption, which enables more filtering of variability in the bone formation environment. Remarkably, the duration for averaging of bone formation may also grow to control deviations in long-term trends of bone formation. Providing there is sufficient bone formation capacity by osteoblasts, this leads to an extraordinarily robust control mechanism that is independent of either osteoblast number or the cellular osteoid formation rate. A complex picture begins to emerge for the control of bone volume. Different control relationships may achieve the same objective, and the 'integration of information' occurring within a BMU may be interpreted as different sets of BMU control systems coming to the fore as different information is supplied to the BMU, which in turn leads to different observable BMU behaviors.     

Urinary bladder matrix promotes site appropriate tissue formation following right ventricle outflow tract repair

The current prevalence and severity of heart defects requiring functional replacement of cardiac tissue pose a serious clinical challenge. Biologic scaffolds are an attractive tissue engineering approach to cardiac repair because they avoid sensitization associated with homograft materials and theoretically possess the potential for growth in similar patterns as surrounding native tissue. Both urinary bladder matrix (UBM) and cardiac ECM (C-ECM) have been previously investigated as scaffolds for cardiac repair with modest success, but have not been compared directly. In other tissue locations, bone marrow derived cells have been shown to play a role in the remodeling process, but this has not been investigated for UBM in the cardiac location, and has never been studied for C-ECM. The objectives of the present study were to compare the effectiveness of an organ-specific C-ECM patch with a commonly used ECM scaffold for myocardial tissue repair of the right ventricle outflow tract (RVOT), and to examine the role of bone marrow derived cells in the remodeling response. A chimeric rat model in which all bone marrow cells express green fluorescent protein (GFP) was generated and used to show the ability of ECM scaffolds derived from the heart and bladder to support cardiac function and cellular growth in the RVOT. The results from this study suggest that urinary bladder matrix may provide a more appropriate substrate for myocardial repair than cardiac derived matrices, as shown by differences in the remodeling responses following implantation, as well as the presence of site appropriate cells and the formation of immature, myocardial tissue.     

Angiogenic factors in bone local environment

Angiogenesis plays an important role in physiological bone growth and remodeling, as well as in pathological bone disorders such as fracture repair, osteonecrosis, and tumor metastasis to bone. Vascularization is required for bone remodeling along the endosteal surface of trabecular bone or Haversian canals within the cortical bone, as well as the homeostasis of the cartilage-subchondral bone interface. Angiogenic factors, produced by cells from a basic multicellular unit (BMU) within the bone remodeling compartment (BRC) regulate local endothelial cells and pericytes. In this review, we discuss the expression and function of angiogenic factors produced by osteoclasts, osteoblasts and osteocytes in the BMU and in the cartilage-subchondral bone interface. These include vascular endothelial growth factor (VEGF), basic fibroblast growth factor (bFGF), BMP7, receptor activator of NF-κB ligand (RANKL) and epidermal growth factor (EGF)-like family members. In addition, the expression of EGFL2, EGFL3, EGFL5, EGFL6, EGFL7, EGFL8 and EGFL9 has been recently identified in the bone local environment, giving important clues to their possible roles in angiogenesis. Understanding the role of angiogenic factors in the bone microenvironment may help to develop novel therapeutic targets and diagnostic biomarkers for bone and joint diseases, such as osteoporosis, osteonecrosis, osteoarthritis, and delayed fracture healing.     

Osteoclast-osteoblast communication

Cells in osteoclast and osteoblast lineages communicate with each other through cell-cell contact, diffusible paracrine factors and cell-bone matrix interaction. Osteoclast-osteoblast communication occurs in a basic multicellular unit (BMU) at the initiation, transition and termination phases of bone remodeling. At the initiation phase, hematopoietic precursors are recruited to the BMU. These precursors express cell surface receptors including c-Fms, RANK and costimulatory molecules, such as osteoclast-associated receptor (OSCAR), and differentiate into osteoclasts following cell-cell contact with osteoblasts, which express ligands. Subsequently, the transition from bone resorption to formation is mediated by osteoclast-derived ‘coupling factors’, which direct the differentiation and activation of osteoblasts in resorbed lacunae to refill it with new bone. Bidirectional signaling generated by interaction between ephrinB2 on osteoclasts and EphB4 on osteoblast precursors facilitates the transition. Such interaction is likely to occur between osteoclasts and lining cells in the bone remodeling compartment (BRC). At the termination phase, bone remodeling is completed by osteoblastic bone formation and mineralization of bone matrix. Here, we describe molecular communication between osteoclasts and osteoblasts at distinct phases of bone remodeling.     

A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload

Bone is a dynamic tissue which, through the process of bone remodeling in the mature skeleton, renews itself during normal function and adapts to mechanical loads. It is, therefore, important to understand the effect of remodeling on the mechanical function of bone, as well as the effect of the inherent time lag in the remodeling process. In this study, we develop a constitutive model for bone remodeling which includes a number of relevant mechanical and biological processes and use this model to address differences in the remodeling behavior as a volume element of bone is placed in disuse or overload. The remodeling parameters exhibited damped oscillatory behavior as the element was placed in disuse, with the amplitude of the oscillations increasing as the severity of disuse increased. In overload situations, the remodeling parameters exhibited critically sensitive behavior for loads beyond a threshold value. These results bear some correspondence to experimental findings, suggesting that the model may be useful when examining the importance of transient responses for bone in disuse, and for investigating the role fatigue damage removal plays in preventing or causing stress fractures. In addition, the constitutive algorithm is currently being employed in finite element simulations of bone adaptation to predict important features of the internal structure of the normal femur, as well as to study bone diseases and their treatment.     

Bone remodeling, particle disease and individual susceptibility to periprosthetic osteolysis

Bone remodeling is a tightly coupled process consisting of repetitive cycles of bone resorption and formation. Both processes are governed by mechanical signals, which operate in conjunction with local and systemic factors in a discrete anatomic structure designated a basic multicellular unit (BMU). The microenvironment around total joint arthroplasty is a dynamic and complex milieu influenced by the chemical and physical stimuli associated with servicing the prosthesis. A key factor limiting the longevity of the prosthesis is polyethylene wear, which induces particle disease, and this may lead to increased and prolonged activity of BMUs resulting in periprosthetic osteolysis. Several pathways regulating BMU function have been reported in the past, including RANKL/RANK/OPG/TRAF6, TNF-alpha/TNFR/TRAF1, and IL-6/CD126/JAK/STAT. Moreover, the expression and functional activity of all these molecules can be affected by variations in their genes. These may explain the differences in severity of bone defects or prosthetic failure between patients with similar wear rates and the same prosthesis. Simultaneously, this data strongly support the theory of individual susceptibility to prosthetic failure.     

Numerical test concerning bone mass apposition under electrical and mechanical stimulus

ABSTRACT: This article proposes a model of bone remodeling that encompasses mechanical and electrical stimuli. The remodeling formulation proposed by Weinans and collaborators was used as the basis of this research, with a literature review allowing a constitutive model evaluating the permittivity of bone tissue to be developed. This allowed the mass distribution that depends on mechanical and electrical stimuli to be obtained. The remaining constants were established through numerical experimentation. The results demonstrate that mass distribution is altered under electrical stimulation, generally resulting in a greater deposition of mass. In addition, the frequency of application of an electric field can affect the distribution of mass; at a lower frequency there is more mass in the domain. These numerical experiments open up discussion concerning the importance of the electric field in the remodeling process and propose the quantification of their effects.     

Osteoclast-derived activity in the coupling of bone formation to resorption

The cells of bone and the immune system communicate by means of soluble and membrane-bound cytokines and growth factors. Through local signalling mechanisms, cells of the osteoblast lineage control the formation and activity of osteoclasts and, therefore, the resorption of bone. Both T and B lymphocytes produce activators and inhibitors of osteoclast formation. A local 'coupling factor' linking bone resorption to subsequent formation in the bone multicellular unit (BMU) has long been proposed as the key regulator of the bone remodelling process, but never identified. There is evidence in support of the view that the coupling mechanism is dependent on growth factors released from the bone matrix during resorption, or is generated from maturing osteoblasts. We argue that osteoclasts contribute in important ways to the transiently activated osteoclast, and stimulate osteoblast lineage cells to begin replacing the resorbed bone in each BMU.     

Bone refilling in cortical basic multicellular units: insights into tetracycline double labelling from a computational model

Bone remodelling is carried out by ‘bone multicellular units’ ( $\text{ BMU }$ s) in which active osteoclasts and active osteoblasts are spatially and temporally coupled. The refilling of new bone by osteoblasts towards the back of the $\text{ BMU }$ occurs at a rate that depends both on the number of osteoblasts and on their secretory activity. In cortical bone, a linear phenomenological relationship between matrix apposition rate and $\text{ BMU }$ cavity radius is found experimentally. How this relationship emerges from the combination of complex, nonlinear regulations of osteoblast number and secretory activity is unknown. Here, we extend our previous mathematical model of cell development within a single cortical $\text{ BMU }$ to investigate how osteoblast number and osteoblast secretory activity vary along the $\text{ BMU }$ ’s closing cone. The mathematical model is based on biochemical coupling between osteoclasts and osteoblasts of various maturity and includes the differentiation of osteoblasts into osteocytes and bone lining cells, as well as the influence of $\text{ BMU }$ cavity shrinkage on osteoblast development and activity. Matrix apposition rates predicted by the model are compared with data from tetracycline double labelling experiments. We find that the linear phenomenological relationship observed in these experiments between matrix apposition rate and $\text{ BMU }$ cavity radius holds for most of the refilling phase simulated by our model, but not near the start and end of refilling. This suggests that at a particular bone site undergoing remodelling, bone formation starts and ends rapidly, supporting the hypothesis that osteoblasts behave synchronously. Our model also suggests that part of the observed cross-sectional variability in tetracycline data may be due to different bone sites being refilled by $\text{ BMU }$ s at different stages of their lifetime. The different stages of a $\text{ BMU }$ ’s lifetime (such as initiation stage, progression stage, and termination stage) depend on whether the cell populations within the $\text{ BMU }$ are still developing or have reached a quasi-steady state whilst travelling through bone. We find that due to their longer lifespan, active osteoblasts reach a quasi-steady distribution more slowly than active osteoclasts. We suggest that this fact may locally enlarge the Haversian canal diameter (due to a local lack of osteoblasts compared to osteoclasts) near the $\text{ BMU }$ ’s point of origin.     

Supercritical fluid-assisted controllable fabrication of open and highly interconnected porous scaffolds for bone tissue engineering

Recently tremendous progress has been evidenced by the advancements in developing innovative three-dimensional(3 D)scaffolds using various techniques for addressing the autogenous grafting of bone. In this work, we demonstrated the fabrication of porous polycaprolactone(PCL) scaffolds for osteogenic differentiation based on supercritical fluid-assisted hybrid processes of phase inversion and foaming. This eco-friendly process resulted in the highly porous biomimetic scaffolds with open and interconnected architectures. Initially, a 2~3 factorial experiment was designed for investigating the relative significance of various processing parameters and achieving better control over the porosity as well as the compressive mechanical properties of the scaffold. Then, single factor experiment was carried out to understand the effects of various processing parameters on the morphology of scaffolds. On the other hand, we encapsulated a growth factor, i.e., bone morphogenic protein-2(BMP-2), as a model protein in these porous scaffolds for evaluating their osteogenic differentiation. In vitro investigations of growth factor loaded PCL scaffolds using bone marrow stromal cells(BMSCs) have shown that these growth factor-encumbered scaffolds were capable of differentiating the cells over the control experiments. Furthermore, the osteogenic differentiation was confirmed by measuring the cell proliferation, and alkaline phosphatase(ALP) activity, which were significantly higher demonstrating the active bone growth. Together, these results have suggested that the fabrication of growth factor-loaded porous scaffolds prepared by the eco-friendly hybrid processing efficiently promoted the osteogenic differentiation and may have a significant potential in bone tissue engineering.     

The behavior of adaptive bone-remodeling simulation models

The process of adaptive bone remodeling can be described mathematically and simulated in a computer model, integrated with the finite element method. In the model discussed here, cortical and trabecular bone are described as continuous materials with variable density. The remodeling rule applied to simulate the remodeling process in each element individually is, in fact, an objective function for an optimization process, relative to the external load. Its purpose is to obtain a constant, preset value for the strain energy per unit bone mass, by adapting the density. If an element in the structure cannot achieve that, it either turns to its maximal density (cortical bone) or resorbs completely.

It is found that the solution obtained in generally a discontinuous patchwork. For a two-dimensional proximal femur model this patchwork shows a good resemblance with the density distribution of a real proximal femur.

It is shown that the discontinuous end configuration is dictated by the nature of the differential equations describing the remodeling process. This process can be considered as a nonlinear dynamical system with many degrees of freedom, which behaves divergent relative to the objective, leading to many possible solutions. The precise solution is dependent on the parameters in the remodeling rule, the load and the initial conditions. The feedback mechanism in the process is self-enhancing; denser bone attracts more strain energy, whereby the bone becomes even more dense. It is suggested that this positive feedback of the attractor state (the strain energy field) creates order in the end configuration. In addition, the process ensures that the discontinuous end configuration is a structure with a relatively low mass, perhaps a minimal-mass structure, although this is no explicit objective in the optimization process.

It is hypothesized that trabecular bone is a chaotically ordered structure which can be considered as a fractal with characteristics of optimal mechanical resistance and minimal mass, of which the actual morphology depends on the local (internal) loading characteristics, the sensor-cell density and the degree of mineralization.     

Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds--a modeling approach

Bone tissue engineering is a promising strategy to repair local defects by implanting biodegradable scaffolds which undergo remodeling and are replaced completely by autologous bone tissue. Here, we consider a Keller-Segel model to describe the chemotaxis of bone marrow-derived mesenchymal stem cells (BMSCs) into a mineralized collagen scaffold. Following recent experimental results in bone healing, demonstrating that a sub-population of BMSCs can be guided into 3D scaffolds by gradients of signaling molecules such as SDF-1α, we consider a population of BMSCs on the surface of the pore structure of the scaffold and the chemoattractant SDF-1α within the pores. The resulting model is a coupled bulk/surface model which we reformulate following a diffuse-interface approach in which the geometry is implicitly described using a phase-field function. We explain how to obtain such an implicit representation and present numerical results on μCT-data for real scaffolds, assuming a diffusion of SDF-1α being coupled to diffusion and chemotaxis of the cells towards SDF-1α. We observe a slowing-down of BMSC ingrowth after the scaffold becomes saturated with SDF-1α, suggesting that a slow release of SDF-1α avoiding an early saturation is required to enable a complete colonization of the scaffold. The validation of our results is possible via SDF-1α release from injectable carrier materials, and an adaptation of our model to similar coupled bulk/surface problems such as remodeling processes seems attractive.     

On stiffness of scaffolds for bone tissue engineering—a numerical study

Tissue scaffolds are typically designed and fabricated to match native bone properties. However, it is unclear if this would lead to the best tissue ingrowth outcome within the scaffold as neo-tissue keeps changing the stiffness of entire construct. This paper presents a numerical method to address this issue for design optimization and assessment of tissue scaffolds. The elasticity tensors of two different types of bones are weighted by different multipliers before being used as the targets in scaffold design. A cost function regarding the difference between the effective elasticity tensor, calculated by the homogenization technique, and the target tensor, is minimized by using topology optimization procedure. It is found that different stiffnesses can lead to different remodeling results. The comparison confirms that bone remodeling is at its best when the scaffold elastic tensor matches or is slightly higher than the elastic properties of the host bone.     

Polyvinyl alcohol modified polyvinylidene fluoride-graphene oxide scaffold promotes osteogenic differentiation potential of human induced pluripotent stem cells

Tissue engineering is fast becoming a key approach in bone medicine studies. Designing the ideally desirable combination of stem cells and scaffolds are at the hurt of efforts for producing implantable bone substitutes. Clinical application of stem cells could be associated with serious limitations, and engineering scaffolds that are able to imitate the important features of extracellular matrix is a major area of challenges within the field. In this study, electrospun scaffolds of polyvinylidene fluoride (PVDF), PVDF-graphene oxide (GO), PVDF-polyvinyl alcohol (PVA) and PVDF-PVA-GO were fabricated to study the osteogenic differentiation potential of human induced pluripotent stem cells (iPSCs) while cultured on fabricated scaffolds. Scanning electron microscopy study, viability assay, relative gene expression analysis, immunocytochemistry, alkaline phosphates activity, and calcium content assays confirmed that the osteogenesis rate of hiPSCs cultured on PVDF-PVA-Go is significantly higher than other scaffolds. Here, we showed that the biocompatible, nontoxic, flexible, piezoelectric, highly porous and interconnected three-dimensional structure of electrospun PVDF-PVA-Go scaffold in combination with hiPSCs (as the stem cells with significant advantageous in comparison to other types) makes them a highly promising scaffold-stem cell system for bone remodeling medicine. There was no evidence for the superiority of PVDF-GO or PVDF-PVA scaffold for osteogenesis, compared to each other; however both of them showed better potentials as to PVDF scaffold.     

Stability analysis and finite element simulation of bone remodeling model

Bone remodeling is widely viewed as a dynamic process--maintaining bone structure through a balance between the opposed activities of osteoblast and osteoclast cells--in which the stability problem is often pointed out. By an analytical approach, we present a bone remodeling model applied to n unit-elements in order to analyze the stationary states and the condition of their stability. In addition, this theory has been simulated in a computer model using the Finite Element Method (FEM) to show a relationship between the bone remodeling process and the stability analysis.     

A mathematical model for bone tissue regeneration inside a specific type of scaffold

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.