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This study presents a comprehensive multiscale analysis of sandwich beams with a polyurethane foam (PUF) core, delivering a numerical comparison between finite element methods (FEMs) and a meshless method: the radial point interpolation method (RPIM). This work aims to combine RPIM with homogenisation techniques for multiscale analysis, being divided in two phases. In the first phase, bulk PUF material was modified by incorporating circular holes to create PUFs with varying volume fractions. Then, using a homogenisation technique coupled with FEM and four versions of RPIM, the homogenised mechanical properties of distinct PUF with different volume fractions were determined. It was observed that RPIM formulations, with higher-order integration schemes, are capable of approximating the solution and field smoothness of high-order FEM formulations. However, seeking a comparable field smoothness represents prohibitive computational costs for RPIM formulations. In a second phase, the obtained homogenised mechanical properties were applied to large-scale sandwich beam problems with homogeneous and approximately functionally graded cores, showing RPIM's capability to closely approximate FEM results. The analysis of stress distributions along the thickness of the beam highlighted RPIM's tendency to yield lower stress values near domain edges, albeit with convergence towards agreement among different formulations. It was found that RPIM formulations with lower nodal connectivity are very efficient, balancing computational cost and accuracy. Overall, this study shows RPIM's viability as an alternative to FEM for addressing practical elasticity applications.

This study employs a meshless computational model to investigate the impacts of compression and traction on angiogenesis, exploring their effects on vascular endothelial growth factor (VEGF) diffusion and subsequent capillary network formation. Three distinct initial domain geometries were defined to simulate variations in endothelial cell sprouting and VEGF release. Compression and traction were applied, and the ensuing effects on VEGF diffusion coefficients were analysed. Compression promoted angiogenesis, increasing capillary network density. The reduction in the VEGF diffusion coefficient under compression altered VEGF concentration, impacting endothelial cell migration patterns. The findings were consistent across diverse simulation scenarios, demonstrating the robust influence of compression on angiogenesis. This computational study enhances our understanding of the intricate interplay between mechanical forces and angiogenesis. Compression emerges as an effective mediator of angiogenesis, influencing VEGF diffusion and vascular pattern. These insights may contribute to innovative therapeutic strategies for angiogenesis-related disorders, fostering tissue regeneration and addressing diseases where angiogenesis is crucial.

Dental caries and dental restorations possess a long history and over the years, many materials and methods have been invented. In recent decades, modern techniques and materials have brought complexity to this issue, which has created the necessity to investigate more and more to achieve durability, consistency, proper mechanical properties, efficiency, beauty, good colour, and reduced costs and time. Combined with the recent advances in the medical field, mechanical engineering plays a significant role in this topic. This work aims at studying the elasto-static response of a human molar tooth as a case study, respecting the integral property of the tooth and different composite materials of the dental restoration. The structural integrity of the case study will be assessed through advanced numerical modelling resorting to meshless methods within the stress analysis on the molar tooth under different loading conditions. In this regard, bruxism is considered as being one of the most important cases that cause damage and fracture in a human tooth. The obtained meshless methods results are compared to the finite element method (FEM) solution. The advantages and disadvantages of the analysed materials are identified, which could be used by the producers of the studied materials to improve their quality. On the other hand, a computational framework, as the one presented here, would assist the clinical practice and treatment decision (in accordance with each patient's characteristics).

Machine learning (ML) and deep learning (DL) approaches can solve the same problems as the finite element method (FEM) with a high degree of accuracy in a fraction of the required time, by learning from previously presented data. In this work, the bone remodelling phenomenon was modelled using feed-forward neural networks (NN), by gathering a dataset of density distribution comprising several geometries and load cases. The model should output for some point in the domain the its apparent density, taking into consideration the geometric and loading parameters of the model . After training. the trabecular distribution obtained with the NN was similar to the FEM while the analysis was performed in a fraction of the time, having shown a reduction from 1020 s to 0.064 s. It is expected that the results can be extended to different structures if a different dataset is acquired. In summary, the ML approach allows for significant savings in computational time while presenting accurate results., Competing Interests: Declaration of competing interest All authors declare no conflict of interest., (Copyright © 2023 Elsevier Ltd. All rights reserved.)

Cell proliferation is vital for the development and homeostasis of the human body. For such to occur, cells go through the cell cycle during which they replicate their genetic material and ultimately complete cellular division, when one cell divides into two new cells with equal genetic material. However, if there are some errors or abnormalities during the cell cycle that disrupt the balance between cell death and proliferation, severe problems can occur, such as tumour development, which is currently one of the leading causes of death in the world. Nowadays, mathematical and computational models are used to understand and study several biological mechanisms and processes, namely cellular proliferation. Over the last forty-five years, several models have attempted to describe cell proliferation and its regulation. Due to the complexity of the process, numerous assumptions and simplifications have been considered. This work presents a review of some of these models, focusing mainly on mammalian or generic eukaryotic models. Previously published continuum, discrete and hybrid approaches are presented and compared, in order to understand and highlight the relevance and capabilities of these models, their shortcomings and future challenges., (© 2023 John Wiley & Sons Ltd.)

Cellular materials have a wide range of applications, including structural optimization and biomedical applications. Due to their porous topology, which promotes cell adhesion and proliferation, cellular materials are particularly suited for tissue engineering and the development of new structural solutions for biomechanical applications. Furthermore, cellular materials can be effective in adjusting mechanical properties, which is especially important in the design of implants where low stiffness and high strength are required to avoid stress shielding and promote bone growth. The mechanical response of such scaffolds can be improved further by employing functional gradients of the scaffold's porosity and other approaches, including traditional structural optimization frameworks; modified algorithms; bio-inspired phenomena; and artificial intelligence via machine learning (or deep learning). Multiscale tools are also useful in the topological design of said materials. This paper provides a state-of-the-art review of the aforementioned techniques, aiming to identify current and future trends in orthopedic biomechanics research, specifically implant and scaffold design.

Multiscale techniques, namely homogenization, result in significant computational time savings in the analysis of complex structures such as lattice structures, as in many cases it is inefficient to model a periodic structure in full detail in its entire domain. The elastic and plastic properties of two TPMS-based cellular structures, the gyroid, and the primitive surface are studied in this work through numerical homogenization. The study enabled the development of material laws for the homogenized Young's modulus and homogenized yield stress, which correlated well with experimental data from the literature. It is possible to use the developed material laws to run optimization analyses and develop optimized functionally graded structures for structural applications or reduced stress shielding in bio-applications. Thus, this work presents a study case of a functionally graded optimized femoral stem where it was shown that the porous femoral stem built with Ti-6Al-4V can minimize stress shielding while maintaining the necessary load-bearing capacity. It was shown that the stiffness of cementless femoral stem implant with a graded gyroid foam presents stiffness that is comparable to that of trabecular bone. Moreover, the maximum stress in the implant is lower than the maximum stress in trabecular bone.

Background: Total knee arthroplasty (TKA) is one of the most frequently performed orthopedic procedures. The correct positioning and alignment of the components significantly affects prosthesis survival. Considering the current controversy regarding the target of postoperative alignment of TKA, this study evaluated the tension at tibial component interface using two numerical methods., Methods: The stress of the prosthesis/bone interface of the proximal tibial component was evaluated using two numerical methods: the finite element method (FEM) and the new meshless method: natural neighbor radial point interpolation method (NNRPIM). The construction of the model was based on Zimmers NexGen LPS-Flex Mobile® prosthesis and simulated the forces by using a free-body diagram., Results: Tibiofemoral mechanical axis (TFMA) for which a higher number of nodes are under optimal mechanical tension is between 1° valgus 2° varus. For values outside the interval, there are regions under the tibial plate at risk of bone absorption. At the extremities of the tibial plate of the prosthesis, both medial and lateral, independent of the alignment, are under a low stress. In all nodes evaluated for all TFMA, the values of the effective stresses were higher in the NNRPIM when compared with the FEM., Conclusion: Through this study, we can corroborate that the optimal postoperative alignment is within the values that are currently considered of 0 ± 3° varus. It was verified that the meshless methods obtain smoother and more conservative results, which may make them safer when transposed to the clinical practice., Competing Interests: The authors declare no conflicts of interest., (Copyright © 2023 The Authors. Published by Wolters Kluwer Health, Inc. on behalf of PBJ-Associação Porto Biomedical/Porto Biomedical Society. All rights reserved.)

Background and Objective: During cell proliferation, cells grow and divide in order to obtain two new genetically identical cells. Understanding this process is crucial to comprehend other biological processes. Computational models and algorithms have emerged to study this process and several examples can be found in the literature. The objective of this work was to develop a new computational model capable of simulating cell proliferation. This model was developed using the Radial Point Interpolation Method, a meshless method that, to the knowledge of the authors, was never used to solve this type of problem. Since the efficiency of the model strongly depends on the efficiency of the meshless method itself, the optimal numbers of integration points per integration cell and of nodes for each influence-domain were investigated. Irregular nodal meshes were also used to study their influence on the algorithm., Methods: For the first time, an iterative discrete model solved by the Radial Point Interpolation Method based on the Galerkin weak form was used to establish the system of equations from the reaction-diffusion integro-differential equations, following a new phenomenological law proposed by the authors that describes the growth of a cell over time while dependant on oxygen and glucose availability. The discretization flexibility of the meshless method allows to explicitly follow the geometric changes of the cell until the division phase., Results: It was found that an integration scheme of 6 × 6 per integration cell and influence-domains with only seven nodes allows to predict the cellular growth and division with the best balance between the relative error and the computing cost. Also, it was observed that using irregular meshes do not influence the solution., Conclusions: Even in a preliminary phase, the obtained results are promising, indicating that the algorithm might be a potential tool to study cell proliferation since it can predict cellular growth and division. Moreover, the Radial Point Interpolation Method seems to be a suitable method to study this type of process, even when irregular meshes are used. However, to optimize the algorithm, the integration scheme and the number of nodes inside the influence-domains must be considered., Competing Interests: Declaration of Competing Interest No conflict of interest declared., (Copyright © 2022. Published by Elsevier B.V.)

Angiogenesis, the formation of new blood vessels from pre-existing ones, begins during embryonic development and continues throughout life. Sprouting angiogenesis is a well-defined process, being mainly influenced by vascular endothelial growth factor (VEGF). In this study, we propose a meshless-based model capable of mimicking the angiogenic response to several VEGF concentrations. In this model, endothelial cells migrate according to a diffusion-reaction equation, following the VEGF gradient concentration. The chick chorioallantoic membrane (CAM) assay was used to model the branching process and to validate the obtained numerical results. To analyse the angiogenic response, the total vessel number and the total vessel length presented in the CAM images and in the simulations for all the VEGF concentrations tested were quantified. In both the CAM assay and simulation, the treatments with VEGF increased the total vessel number and the total vessel length. The obtained quantitative results were very similar between the two methodologies used. The proposed model accurately simulates the capillary network pattern concerning its structure and morphology, for the lowest VEGF concentration tested. For the highest VEGF concentration, the capillary network predicted by the model was less accurate when compared to the one presented in the CAM assay but this may be explained by changes in blood vessel width at higher VEGF concentrations. This remains to be tested., (Copyright © 2021 Elsevier Ltd. All rights reserved.)

This work proposes a novel tissue-scale mechanobiological model of bone remodeling to study bone's adaptation to distinct loading conditions. The devised algorithm describes the mechanosensitivity of bone and its impact on bone cells' functioning through distinct signaling factors. In this study, remodeling is mechanically ruled by variations of the strain energy density (SED) of bone, which is determined by performing a linear elastostatic analysis combined with the finite element method. Depending on the SED levels and on a set of biological signaling factors ([Formula: see text] parameters), osteoclasts and osteoblasts can be mechanically triggered. To reproduce this phenomenon, this work proposes a new set of [Formula: see text] parameters. The combined response of osteoclasts and osteoblasts will then affect bone's apparent density, which is correlated with other mechanical properties of bone, through a phenomenological law. Thus, this novel model proposes a constant interplay between the mechanical and biological components of the process. The spatiotemporal simulation used to validate this new approach is a benchmark example composed by two distinct phases: (1) pre-orientation and (2) load adaptation. On both of them, bone is able to adapt its morphology according to the loading condition, achieving the required trabecular distribution to withstand the applied loads. Moreover, the equilibrium morphology reflects the orientation of the load. These preliminary results support the new approach proposed in this study., (© 2021. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Bone remodeling is a highly complex process, in which bone cells interact and regulate bone's apparent density as a response to several external and internal stimuli. In this work, this process is numerically described using a novel 2D biomechanical model. Some of the new features in this model are (i) the mathematical parameters used to determine bone's apparent density and cellular density; (ii) an automatic boundary recognition step to spatially control bone remodeling and (iii) an approach to mimic the mechanical transduction to osteoclasts and osteoblasts. Moreover, this model is combined with a meshless approach - the Radial Point Interpolation Method (RPIM). The use of RPIM is an asset for this application, especially in the construction of the boundary maps. This work studies bone's adaptation to a certain loading regime through bone resorption. The signaling pathways of bone cells are dependent on the level of strain energy density (SED) in bone. So, when SED changes, bone cells' functioning is affected, causing also changes on bone's apparent density. With this model, bone is able to achieve an equilibrium state, optimizing its structure to withstand the applied loads. Results suggest that this model has the potential to provide high quality solutions while being a simpler alternative to more complex bone remodeling models in the literature., (Copyright © 2020 Elsevier Ltd. All rights reserved.)

A functional vascular network is essential to the correct wound healing. In sprouting angiogenesis, vascular endothelial growth factor (VEGF) regulates the formation of new capillaries from pre-existing vessels. This is a very complex process and mathematical formulation permits to study angiogenesis using less time-consuming, reproducible and cheaper methodologies. This study aimed to mimic the chemoattractant effect of VEGF in stimulating sprouting angiogenesis. We developed a numerical model in which endothelial cells migrate according to a diffusion-reaction equation for VEGF. A chick chorioallantoic membrane (CAM) bioassay was used to obtain some important parameters to implement in the model and also to validate the numerical results. We verified that endothelial cells migrate following the highest VEGF concentration. We compared the parameters-total branching number, total vessel length and branching angle-that were obtained in the in silico and the in vivo methodologies and similar results were achieved (p-value smaller than 0.5; n = 6). For the difference between the total capillary volume fractions assessed using both methodologies values smaller than 15% were obtained. In this study we simulated, for the first time, the capillary network obtained during the CAM assay with a realistic morphology and structure.

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the "influence-domain" approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two-dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects., (© 2020 John Wiley & Sons Ltd.)

Bone is a complex hierarchical material that can be characterized from the microscale to macroscale. This work demonstrates the application of an enhanced homogenization methodology to the multiscale structural analysis of a femoral bone. The use of this homogenization technique allows to remove subjectivity and reduce the computational cost associated with the iterative process of creating a heterogeneous mesh. Thus, it allows to create simpler homogenized meshes with its mechanical properties defined using information directly from the mesh source: the medical images. Therefore, this methodology is capable to accurately predict bone mechanical behavior in a fraction of the time required by classical approaches. The results show that using the homogenization technique, despite the differences between the used homogeneous and heterogeneous meshes, its mechanical behavior is similar. The proposed homogenization technique is useful for a multiscale modelling and it is computationally efficient.

Purpose: Bone is a hierarchical material that can be characterized from the microscale to macroscale. Multiscale models make it possible to study bone remodeling, inducing bone adaptation by using information of bone multiple scales. This work proposes a computationally efficient homogenization methodology useful for multiscale analysis. This technique is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium., Methods: In this work, a morphology- based fabric tensor and a set of anisotropic phenomenological laws for bone tissue was used, in order to define the bone micro-scale mechanical properties. To validate the developed methodology, several examples were performed in order to analyze its numerical behavior. Thus, trabecular bone and fabricated benchmarks patches (representing special cases of trabecular bone morphologies) were analyzed under compression., Results: The results show that the developed technique is robust and capable to provide a consistent material homogenization, indicating that the homogeneous models were capable to accurately reproduce the micro-scale patch mechanical behavior., Conclusions: The developed method has shown to be robust, computationally less demanding and enabling the authors to obtain close results when comparing the heterogeneous models with equivalent homogenized models. Therefore, it is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

Bone remodeling remains a highly researched topic investigated by many strands of science. The main purpose of this work is formulating a new computational framework for biological simulation, extending the version of the bone remodeling model previously proposed by Komarova. Thus, considering only the biological aspect of the remodeling process, the action of osteoclasts and osteoblasts is taken into account as well as its impact on bone mass. It is conducted a spatiotemporal analysis of a remodeling cycle obtaining a dynamic behavior of bone cells very similar to the biological process already described in the literature. The numerical example used is based on bone images obtained with scanning electron microscopy. During simulation, it is possible to observe the variation of bone's architecture through isomaps. These maps are obtained through the combination of biological bone remodeling models with three distinct numerical techniques-finite element method (FEM), radial point interpolation method (RPIM), and natural neighbor radial point interpolation method (NNRPIM). A study combining these numerical techniques allows to compare their performance. Ultimately, this work supports the inclusion of meshless methods due to their smoother results and its easiness to be combined with medical images from CT scans and MRI., (© 2019 John Wiley & Sons, Ltd.)

Purpose: Bone tissue is a dynamic tissue, possessing different functional requirements at different scales. This layered organization indicates the existence of a hierarchical structure, which can be characterized to distinguish macro-scale from micro-scale levels. Structurally, both scales can be linked by the use of classic multiscale homogenization techniques. Since in bone tissue each micro-scale domain is distinct form its neighbour, applying a classic multiscale homogenization technique to a complete bone structure could represent an inadmissible computational cost. Thus, this work proposes a homogenization methodology that is computationally efficient, presenting a reduced computational cost, and is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium., Methods: The methodology uses the fabric tensor concept in order to define the material principal directions. Then, using an anisotropic phenomenological law for bone tissue correlating the local apparent density with directional elasticity moduli, the anisotropic homogenized material properties of the micro-scale patch are fully defined. To validate the developed methodology, several numerical tests were performed, measuring the sensitivity of the technique to changes in the micro-patch size and preferential orientation., Results: The results show that the developed technique is robust and capable to provide a consistent material homogenization. Additionally, the technique was combined with two discrete numerical techniques: the finite element method and radial point interpolation meshless method., Conclusions: Structural analyses were performed using real trabecular patches, showing that the proposed methodology is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

In this work, an advanced discretization meshless technique is used to study the structural response of a human brain due to an impact load. The 2D and 3D brain geometrical models, and surrounding structures, were obtained through the processing of medical images, allowing to achieve a realistic geometry for the virtual model and to define the distribution of the mechanical properties accordingly with the medical images colour scale. Additionally, a set of essential and natural boundary conditions were assumed in order to reproduce a sudden impact force applied to the cranium. Then, a structural numerical analysis was performed using the Natural Neighbour Radial Point Interpolation Method (NNRPIM). The obtained results were compared with the finite element method (FEM) and a solution available in the literature. This work shows that the NNRPIM is a robust and accurate numerical technique, capable to produce results very close to other numerical approaches. In addition, the variable fields obtained with the meshless method are much smoother than the FEM corresponding solution.

The occurrence of wounds is a main health concern in Western society due to their high frequency and treatment cost. During wound healing, the formation of a functional blood vessel network through angiogenesis is an essential process. Angiogenesis allows the reestablishment of the normal blood flow, the sufficient exchange of oxygen and nutrients and the removal of metabolic waste, necessary for cell proliferation and viability. Mathematical and computational models provide new tools to improve the healing process. In fact, over the last thirty years, in silico models have been continuously formulated to describe the effect of several biological and mechanical factors in angiogenesis during wound healing. Additionally, with different levels of complexity, these models allow coupling the human skin structure, to distinct cell types and growth factors, to study extracellular matrix composition and to understand its deformation. This paper discusses how in silico models, which are more economical and less time-consuming comparatively to laboratory methodologies, can help test new strategies to promote/optimize angiogenesis. The continuum, cell-based and hybrid mathematical models of wound healing angiogenesis are reviewed in the present paper, in order to identify possible improvements. Accordingly, the development of higher dimension models incorporating multiscale analysis at molecular, cellular and tissue level remains a challenge that future models should consider., (Copyright © 2018. Published by Elsevier Ltd.)

Vertiginous symptoms are one of the most common symptoms in the world, therefore investing in new ways and therapies to avoid the sense of insecurity during the vertigo episodes is of great interest. The classical maneuvers used during vestibular rehabilitation consist in moving the head in specific ways, but it is not fully understood why those steps solve the problem. To better understand this mechanism, a three-dimensional computational model of the semicircular ducts of the inner ear was built using the finite element method, with the simulation of the fluid flow being obtained using particle methods. To simulate the exact movements performed during rehabilitation, data from an accelerometer were used as input for the boundary conditions in the model. It is shown that the developed model responds to the input data as expected, and the results successfully show the fluid flow of the endolymph behaving coherently as a function of accelerometer data. Numerical results at specific time steps are compared with the corresponding head movement, and both particle velocity and position follow the pattern that would be expected, confirming that the model is working as expected. The vestibular model built is an important starting point to simulate the classical maneuvers of the vestibular rehabilitation allowing to understand what happens in the endolymph during the rehabilitation process, which ultimately may be used to improve the maneuvers and the quality of life of patients suffering from vertigo.

This work has the objective to compare the mechanical behaviour of a brain impact using an alternative numerical meshless technique. Thus, a discrete geometrical model of a brain was constructed using medical images. This technique allows to achieve a discretization with realistic geometry, allowing to define locally the mechanical properties according to the medical images colour scale. After defining the discrete geometrical model of the brain, the essential and natural boundary conditions were imposed to reproduce a sudden impact force. The analysis was performed using the finite element analysis and the radial point interpolation method, an advanced discretization technique. The results of both techniques are compared. When compared with the finite element analysis, it was verified that meshless methods possess a higher convergence rate and that they are capable of producing smoother variable fields.

Purpose: The vestibular system is the part of the inner ear responsible for balance. Vertigo and dizziness are generally caused by vestibular disorders and are very common symptoms in people over 60 years old. One of the most efficient treatments at the moment is vestibular rehabilitation, permitting to improve the symptoms. However, this rehabilitation therapy is a highly empirical process, which needs to be enhanced and better understood., Methods: This work studies the vestibular system using an alternative computational approach. Thus, part of the vestibular system is simulated with a three dimensional numerical model. Then, for the first time using a combination of two discretization techniques (the finite element method and the smoothed particle hydrodynamics method), it is possible to simulate the transient behavior of the fluid inside one of the canals of the vestibular system., Results: The obtained numerical results are presented and compared with the available literature. The fluid/solid interaction in the model occurs as expected with the methods applied. The results obtained with the semicircular canal model, with the same boundary conditions, are similar to the solutions obtained by other authors., Conclusions: The numerical technique presented here represents a step forward in the biomechanical study of the vestibular system, which in the future will allow the existing rehabilitation techniques to be improved.

In this work the maxillary central incisor is numerically analysed with an advance discretization technique--Natural Neighbour Radial Point Interpolation Method (NNRPIM). The NNRPIM permits to organically determine the nodal connectivity, which is essential to construct the interpolation functions. The NNRPIM procedure, based uniquely in the computational nodal mesh discretizing the problem domain, allows to obtain autonomously the required integration mesh, permitting to numerically integrate the differential equations ruling the studied physical phenomenon. A numerical analysis of a tooth structure using a meshless method is presented for the first time. A two-dimensional model of the maxillary central incisor, based on the clinical literature, is established and two distinct analyses are performed. First, a complete elasto-static analysis of the incisor/maxillary structure using the NNRPIM is evaluated and then a non-linear iterative bone tissue remodelling analysis of the maxillary bone, surrounding the central incisive, is performed. The obtained NNRPIM solutions are compared with other numerical methods solutions available in the literature and with clinical cases. The results show that the NNRPIM is a suitable numerical method to analyse numerically dental biomechanics problems.

In this work, a novel anisotropic material law for the mechanical behaviour of the bone tissue is proposed. This new law, based on experimental data, permits to correlate the bone apparent density with the obtained level of stress. Combined with the proposed material law, a biomechanical model for predicting bone density distribution was developed, based on the assumption that the bone structure is a gradually self-optimising anisotropic biological material that maximises its own structural stiffness. The strain and the stress field required in the iterative remodelling process are obtained by means of an accurate meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). Comparing with other numerical approaches, the inclusion of the NNRPIM presents numerous advantages such as the high accuracy and the smoother stress and strain field distribution. The natural neighbour concept permits to impose organically the nodal connectivity and facilitates the analysis of convex boundaries and extremely irregular meshes. The viability and efficiency of the model were tested on several trabecular benchmark patch examples. The results show that the pattern of the local bone apparent density distribution and the anisotropic bone behaviour predicted by the model for the microscale analysis are in good agreement with the expected structural architecture and bone apparent density distribution.