Isotropic continuum damage/repair model for alveolar bone remodeling

Several authors have proposed mechanical models to predict long term tooth movement, considering both the tooth and its surrounding bone tissue as isotropic linear elastic materials coupled to either an adaptative elasticity behavior or an update of the elasticity constants with density evolution. However, tooth movements obtained through orthodontic appliances result from a complex biochemical process of bone structure and density adaptation to its mechanical environment, called bone remodeling. This process is far from linear reversible elasticity. It leads to permanent deformations due to biochemical actions. The proposed biomechanical constitutive law, inspired from Doblaré and García (2002) [30], is based on a elasto-viscoplastic material coupled with Continuum isotropic Damage Mechanics (Doblaré and García (2002) [30] considered only the case of a linear elastic material coupled with damage). The considered damage variable is not actual damage of the tissue but a measure of bone density. The damage evolution law therefore implies a density evolution. It is here formulated as to be used explicitly for alveolar bone, whose remodeling cells are considered to be triggered by the pressure state applied to the bone matrix. A 2D model of a tooth submitted to a tipping movement, is presented. Results show a reliable qualitative prediction of bone density variation around a tooth submitted to orthodontic forces.     

Optimum selection of the dental implant diameter and length in the posterior mandible with poor bone quality – A 3D finite element analysis

This study aimed to evaluate continuous and simultaneous variations of dental implant diameter and length, and to identify their relatively optimal ranges in the posterior mandible under biomechanical consideration. A 3D finite element model of a posterior mandibular segment with dental implant was created. Implant diameter ranged from 3.0 to 5.0 mm, and implant length ranged from 6.0 to 16.0 mm. The results showed that under axial load, the maximum Von Mises stresses in cortical and cancellous bones decreased by 76.53% and 72.93% respectively, with the increasing of implant diameter and length; and under buccolingual load, by 83.97% and 84.93%, respectively. Under both loads, the maximum displacements of implant-abutment complex decreased by 58.09% and 75.53%, respectively. The results indicate that in the posterior mandible, implant diameter plays more significant roles than length in reducing cortical bone stress and enhancing implant stability under both loads. Meanwhile, implant length is more effective than diameter in reducing cancellous bone stress under both loads. Moreover, biomechanically, implant diameter exceeding 4.0 mm and implant length exceeding 12.0 mm is a relatively optimal combination for a screwed implant in the posterior mandible with poor bone quality.     

Studies on Bone Remodeling Theory Based on Microcracks Using Finite Element Computations

The investigation of the microcrack theory supports the research work of understanding the microstructural behaviour of physiological loaded bones. Microcracks in cortical bone are assumed to have a stimulatory effect on osteocytes – the sensor cells for the bone remodeling process. In this contribution an approch to simulate microcrack initiation and propagation inside a 3D anisotropic and inhomogenious FEA model of cortical bone tissue will be shown. The numerical formulations are based on computational continuum damage mechanics. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary elements and β-spline surface modeling for medical applications

The aim of this paper is to present and discuss an approach based on the integration of the boundary element method (BEM) with β-spline geometric modeling of the different surfaces involved in the external bone remodeling phenomena. The purpose of combining these two techniques is to avoid the jagged edges shapes and thus, to increase the convergence speed of the bone remodeling function. In this study, the external bone remodeling model proposed by Fridez et al. [P. Fridez, L. Rakotomanana, A. Terrier, P.F. Leyvraz, Three dimensional model of bone external adaptation, Comput. Methods Biomech. Biomed. Eng. 2 (1998) 189–196] is used. This model shows the change of the external bone surface remodeling at a boundary point, as a function of the stimulus variable Ψ. This variable is related to the stress tensor and the normal vector to that point. The β-spline surfaces were used because they are simple and reliable to smooth the contour by using the less possible number of geometric constraints. Some numerical examples are presented and discussed in order to show the versatility of the proposed approach.     

Computational simulation of piezo-electrically stimulated bone remodeling surrounding teeth implant

Since the natural ligament responsible for the fixation of teeth in jawbone is destroyed when artificial replacements are implanted, the mechanical stimulation of the bone is reversed. Idea of this research project is the development of active implants which provide additional electrical stimulation for bone adaption. A new electromechanically driven bone remodeling theory will be developed and the osseointegration of bone implants has been simulated by means of bio-active interface theory. The thin bone-implant interface is described by the Drucker-Prager plasticity model. Besides the consistent combination of electromechanical bone remodeling simulation, 3D-finite element model of lower mandible has been reconstructed. As the micromotions at the implant-abutment level are reported to be a major determinant of longterm implant success, the osseointegration process is limited by micromotion threshold. The applicability is indicated on a dental implant in order to optimize new developed techniques for activating implants with piezo-electric coatings. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental and practical biomechanics of the human leg

The results of biomechanical studies of the human tibia in torsion, bending, and combined loading are presented. The dependence of the mechanical properties of the bone on age and the linear paramters has been established. The experimental data are presented in graph form.     

Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.     

Computational Simulation of Bone Remodeling using Design Space Topology Optimization

The law of bone remodeling, commonly referred to as Wolff's Law, asserts that the internal trabecular bone adapts to external loadings, reorienting with the principal stress trajectories to optimize mechanical efficiency creating a naturally optimum structure. The current study utilized an advanced structural optimization algorithm, called design space toptimization (DSO), to perform a three-dimensional computational bone remodeling simulation on the human proximal femur and analyse the results to determine the validity of Wolff's hypothesis. DSO optimizes the layout of material by iteratively distributing it into the areas of highest loading, while simultaneously changing the design domain to increase computational efficiency. The large-scale simulation utilized a 175 µm mesh resolution with over 23.3 million elements. The resulting anisotropic trabecular architecture was compared to both Wolff's trajectory hypothesis and natural femur samples from literature using radiography. The results qualitatively showed several anisotropic trabecular regions that were comparable to the natural human femur. The realistic simulated trabecular geometry suggests that the DSO method can accurately predict bone adaptation due to mechanical loading and that the proximal femur is an optimum structure as Wolff hypothesized. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parallel structural optimization applied to bone remodeling on distributed memory machines

In this paper, we investigate parallel structural optimization methods on distributed memory MIMD machines. We have restricted ourselves to the case of minimizing a multivariate non-linear function subject to bounds on the independent variables, when the objective function is expensive to evaluate as compared to the linear algebra portion of the optimization. This is the case in structural applications, when a large three-dimensional finite element mesh is used to model the structure.This paper demonstrates how parallelism can be exploited during the function and gradient computation as well as the optimization iterations. For the finite element analysis, a torus wrap skyline solver is used. The reflective Newton method, which attempts to reduce the number of iterations at the expense of more linear algebra per iteration, is compared with the more conventional active set method. All code is developed for an Intel iPSC/860, but can be ported to other distributed memory machines.The methods developed are applied to problems in bone remodeling. In the area of biomechanics, optimization models can be used to predict changes in the distribution of material properties in bone due to the presence of an artificial implant. The model we have used minimizes a linear combination of the mass and strain energy in the entire domain subject to bounds on the densities in each finite element.Early results show that the reflective Newton method can outperform active set methods when few variables are active at the minimum.     

Multiphasic Modelling of the Vertebral Bone for Cement-Injection Studies

The stability of the human spine is highly dependent on the cancellous bone structure of the vertebra. In the case of osteoporosis and accompanied weaking of the vertebral structure, compression fractures and other lesions of the affected patient may occur. The reinforcement of the porous cancellous bone by the injection of bone-cement is a common procedure in order to overcome this issues. The modelling and computational simulation of vertebroplasty, i.e., bone-cement-injection into the vertebra, is of major interest to obtain valid and reliable predicitions for this surgery. A detailed micromechanical (and locally single-phasic) model exhibits the drawback that all geometrical and physical transition conditions of the individual parts and their complex microstructure have to be known. Therefore, this study considers a macro-scopic (and multi-constituent) continuum-mechanical model based on the Theory of Porous Media, where the homogenisation of the underlying micro-structure results in a model of three constituents. In particular, these are the solid bone skeleton, which is saturated by bone marrow, where the latter may be displaced by the injected liquid bone cement. The micro-architecture is regarded by heterogeneous and anisotropic permeability tensors and the preferred directions of the trabecular bone structure. The presented strongly coupled macroscopic model offers the opportunity to not only simulate the flow of the pore fluids but also predicts the arising stresses and strains of the solid bone skeleton due to the numerical investigation of the injection process. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A convergence result in the study of bone remodeling contact problems

We consider the approximation of a bone remodeling model with the Signorini contact conditions by a contact problem with normal compliant obstacle, when the obstacle's deformability coefficient converges to zero (that is, the obstacle's stiffness tends to infinity). The variational problem is a coupled system composed of a nonlinear variational equation (in the case of normal compliance contact conditions) or a variational inequality (for the case of Signorini's contact conditions), for the mechanical displacement field, and a first-order ordinary differential equation for the bone remodeling function. A theoretical result, which states the convergence of the contact problem with normal compliance contact law to the Signorini problem, is then proved. Finally, some numerical simulations, involving examples in one and two dimensions, are reported to show this convergence behaviour.     

A jaw model for the study of the mandibular flexure taking into account the anisotropy of the bone

Knowledge of the complex biomechanical behaviour of the human mandible is of great importance in various clinical situations. The biomechanical and physical behaviour of mandibles have been investigated by different approaches. Some research have been done to evaluate the functional character of mandibles. Methods such as indirect measurement of deformations performed by intraoral appliances and by holographic interferometry have being employed. Other studies evaluated the mechanical properties and material parameters of small cubes of mandibles. One disadvantage of the experiments using strain gauges or holographic interferometry is the inability to determine strains at defined positions within the specimen. Additionally, research in biomechanics by these methods is limited to surface deformations and neither stresses nor dislocations can be measured.In the course of this study, we have investigated the mandibular flexure under mechanical loads using the results of a Finite Element Analysis (FEA). In order to obtain more accurate and realistic results, the bone anisotropy has being taken into account for the mathematical modelling of the jaw.The objective of this study was to establish a non-invasive procedure to predict precisely the complex biomechanical reactions of mandibles under mechanical loading. In order to achieve this aim, a comparison of the numerical data obtained with the experimental values of previous studies was performed. It showed a good correlation between in vitro measurements and mathematical modelling. Then the Finite Elements (FE) model was used to evaluate some mandibular movements (corporal approximation, dorsoventral shear, and corporal rotation in edentulous subjects).It is concluded that the applied procedure of generating the FE model is a valid and accurate non-invasive method to predict different parameters of the complex biomechanical behaviour of human mandibles.     

A three-dimensional quasicontinuum approach for predicting biomechanical properties of malaria-infected red blood cell membrane

This paper presents the first attempt to comprehensively estimate the elastic properties and mechanical responses of malaria-infected red blood cell (iRBC) membrane when subjected to uniaxial, shear and isotropic area-dilation loading conditions. With the three-dimensional (3D) quasicontinuum approach, we predicted the biomechanical properties of the iRBC membrane for all infection stages. Effect of temperature on the membrane elastic properties during the trophozoite stage was also examined. It is found that a multifold increase in the elastic properties of the iRBC membrane occurs as infection progresses. The axial, shear and area stiffnesses of the iRBC membrane increase exponentially, resulting in semi-logarithmic stress–strain relationship curves. In addition, the rigidity of the iRBC membrane in the trophozoite stage increases as temperature rise. It is concluded that Plasmodium falciparum parasites significantly affect the biomechanical properties of the RBC membrane due to the structural remodeling of the iRBC membrane microstructure.     

Study of posterolateral lumbar arthrodesis by means of a finite element model

Posterolateral lumbar arthrodesis consists in the fixation of the lumbar vertebrae pedicles by means of bars and screws. It is indicated in all those cases in which instability exists previous to the surgery, or in those other cases in which the instability has been caused by the need of bone resections that put the articulations structures in danger.The pedicle fixation of the lumbar arthrodesis is a great advance in the lumbar surgery. It contributes to achieve a stable and biologic fusion. The aim of the present research is the analysis of the contact problem that exists between the screw and the bone as one of the key points to control in order to achieve a good future stability of the arthrodesed spine.In order to achieve such aim, a Finite Elements Model (FEM) of the spine was performed. Such a model was obtained using a computer vision technique that creates 3D bodies using computed tomographies of the sacrum and vertebrae L4 and L5. Not only the bone bodies have been modeled, but also the intervertebral discs that act as the joints of the bones. In order to obtain a complete simulation of the lumbar region the titanium screws and bars have been modeled too.The study of the influence of the contact between bone and screw in the biomechanical behavior of the lumbar column has been studied applying several load conditions simulating different kinds of typical movements of the column. Finally, the stresses on the different elements of the lumbar structure and the relative movements between bone and screw as well as the conclusions of this research are also expounded.     

A mathematical model of the response of the semicircular canal and otolith to vestibular system rotation under gravity

A mathematical model of the system composed of two sensors, the semicircular canal and the sacculus, is suggested. The model is described by three lines of blocks, each line of which has the following structure: a biomechanical block, a mechanoelectrical transduction mechanism, and a block describing the hair cell ionic currents and membrane potential dynamics. The response of this system to various stimuli (head rotation under gravity and falling) is investigated. Identification of the model parameters was done with the experimental data obtained for the axolotl (Ambystoma tigrinum) at the Institute of Physiology, Autonomous University of Puebla, Mexico. Comparative analysis of the semicircular canal and sacculus membrane potentials is presented. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 207–220, 2005.     

Approximation of bone remodeling models

We present convergence results and error estimates concerning the numerical approximation of a class of bone remodeling models, that are elastic adaptive rod models. These are characterized by an elliptic variational equation, representing the equilibrium of the rod under the action of applied loads, coupled with an ordinary differential equation with respect to time, describing the physiological process of bone remodeling. We first consider the semi-discrete approximation, where only the space variables are discretized using the standard Galerkin method, and then, applying the forward Euler method for the time discretization, we focus on the fully discrete approximation.