Numerical analysis and simulations in bone remodeling models

Conclusions In the course of this Ph.D. thesis we studied several bone remodeling models, trying to develop a complete study from the mathematical and physical points of view. In Chapter 2, the Cowin and Hegedus model was introduced. In this model, the bone is considered as an elastic material. A variational formulation was provided, obtaining an elliptic variational equation for the displacement ¯eld and an ordinary di®erential equation which describes the evolution of the bone density. Applying the ¯nite ele- ment method and an Euler scheme to approximate the spatial variable and the time derivatives, respectively, we obtained a fully discrete problem and we proved an error estimates result. Moreover, under additional regularity assumptions, we derived the linear convergence of the algorithm. Numerical simulations in one, two and three dimensions were presented to show the accuracy and the behavior of the approximations. In the second part of this chapter, we considered a similar problem assuming now that the bone may come into contact with a rigid or a deformable obstacle. In order to model these two contact conditions, we used the classical Signorini condition and the normal compliance contact law, respectively. The variational formulation was obtained for both problems and the convergence of the solution to the contact problem with a deformable obstacle, when the deformability coeficient tends to zero, to the solution of the Signorini's problem was established. We introduced fully discrete aproximations and we proved an error estimates result for both problems. Finally, under additional regularity assumptions, we obtained the linear convergence of the algorithm and some simulations were also presented. The third chapter dealt with the numerical analysis, including numerical simulations in one and two dimensions, of a bone remodeling model introduced byWeinans, Huiskes and Grootenboer in [66]. A numerical algorithm for the variational problem, based on the ¯nite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, was proposed, an error estimate on its solutions was obtained and its linear convergence was established under suitable regularity assump- tions. The numerical simulations demonstrated the accuracy of the approximations and some properties related to the behavior of the solution. Finally, in the last chapter, we proposed a new bone remodeling model in which we considered the bone as an piezoelectric material. This property of the bone tissue was suggested in 1957. However, it was not normally used to understand bone remodeling and there are not many models that justify bone remodeling based on bone piezoelec- tricity. We continued the work developed in the previous chapter, using this model to characterize the evolution of the bone density and the mechanical properties of the bone. Then, we extended the classical electro-mechanical dependence adding a func- tion ®(½) = ½°, which regulates the coupling between the mechanical and electric ¯elds. This function guarantees that the electric ¯eld increases with the density of the bone. The variational formulation for this model was derived and a numerical algorithm was proposed, coupling the electric and displacement ¯elds. Finally, error estimates were proved and the linear convergence was established under adequate regularity condi- tions. Again, the numerical results shown the accuracy of the approximations as well as the behavior of the solution, giving also a numerical justi¯cation of the electro- mechanical bone remodeling model. All the algorithms proposed in this Ph.D. thesis were implemented using MATLAB code and a good number of examples were computed. First, the one-dimensional exam- ples were chosen in such a way as to show the numerical convergence of the algorithms and also their linear convergence. Then, two-or three-dimensional examples were per- formed in order to show the behavior of the models. The existence and uniqueness of weak solutions for the discrete problems were ob- tained applying classical results on linear variational equations or nonlinear variational inequalities (see [44]). However, we remark that the existence and uniqueness results of weak solutions for the continuous variational formulations are open problems. In the Cowin and Hegedus model, this result was obtained for a similar variational formula- tion in which stronger assumptions were made over the data. Recently, Fern¶andez and Kuttler dealt with the model proposed byWeinans, Huiskes and Grootenboer obtaining an existence and uniqueness result for a regularized problem

Unified Approach to the Biomechanics of Dental Implantology

The human need for safe and effective dental implants is well-recognized. Although many implant designs have been tested and are in use today, a large number have resulted in clinical failure. These failures appear to be due to biomechanical effects, as well as biocompatibility and surgical factors. A unified approach is proposed using multidisciplinary systems technology, for the study of the biomechanical interactions between dental implants and host tissues. The approach progresses from biomechanical modeling and analysis, supported by experimental investigations, through implant design development, clinical verification, and education of the dental practitioner. The result of the biomechanical modeling, analysis, and experimental phases would be the development of scientific design criteria for implants. Implant designs meeting these criteria would be generated, fabricated, and tested in animals. After design acceptance, these implants would be tested in humans, using efficient and safe surgical and restorative procedures. Finally, educational media and instructional courses would be developed for training dental practitioners in the use of the resulting implants

A parametric modeling concept for predicting biomechanical compatibility in total hip arthroplasty

This work attempts to predict the long-term outcome of total hip arthroplasty based on available patient-specific information and possible installation positions of the prosthesis. For this purpose, a holistic modeling approach for the numerical simulation of osseointegration and long-term stability of endoprostheses, including possible prosthesis positions, is developed. In addition, new, efficient, and reliable methods for the numerical description of adaptive bone remodeling and osseointegration are proposed: The adaptive bone remodeling is described as a geometric-linear, material-nonlinear finite element model, following thermodynamically consistent material modeling guidelines. The resulting constitutive equations are expanded to describe osseointegration and transferred into a contact interface between bone and prosthesis. Finally, the results are projected to an imaging format that is easier to interpret for medical professionals, using a newly developed simulation for X-ray images. The inclusion of possible prosthesis positions spans an infinite-dimensional event space. Therefore, the model is reduced to a finite-dimensional surrogate model sampled with an adaptive sparse-grid collocation method. Without clinical validation, reliable statements cannot be made, and therefore the numerical examples given in this thesis can be regarded as proof of correct implementation and feasibility studies. This dissertation thus provides an answer to how much computational effort is required to provide a real digital decision aid in orthopedic surgery

A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse

We propose a multiscale mechanobiological model of bone remodelling to investigate the site-specific evolution of bone volume fraction across the midshaft of a femur. The model includes hormonal regulation and biochemical coupling of bone cell populations, the influence of the microstructure on bone turnover rate, and mechanical adaptation of the tissue. Both microscopic and tissue-scale stress/strain states of the tissue are calculated from macroscopic loads by a combination of beam theory and micromechanical homogenisation. This model is applied to simulate the spatio-temporal evolution of a human midshaft femur scan subjected to two deregulating circumstances: (i) osteoporosis and (ii) mechanical disuse. Both simulated deregulations led to endocortical bone loss, cortical wall thinning and expansion of the medullary cavity, in accordance with experimental findings. Our model suggests that these observations are attributable to a large extent to the influence of the microstructure on bone turnover rate. Mechanical adaptation is found to help preserve intracortical bone matrix near the periosteum. Moreover, it leads to non-uniform cortical wall thickness due to the asymmetry of macroscopic loads introduced by the bending moment. The effect of mechanical adaptation near the endosteum can be greatly affected by whether the mechanical stimulus includes stress concentration effects or not.Comment: 25 pages, 10 figure

Quantitative analysis of bone reactions to relative motions at implant-bone interfaces

Connective soft tissues at the interface between implants and bone, such as in human joint replacements, can endanger the stability of the implant fixation. The potential of an implant to generate interface bone resorption and form soft tissue depends on many variables, including mechanical ones. These mechanical factors can be expressed in terms of relative motions between bone and implant at the interface or deformation of the interfacial material.\ud \ud The purpose of this investigation was to determine if interface debonding and subsequent relative interface motions can be responsible for interface degradation and soft tissue interposition as seen in experiments and clinical results. A finite element computer program was augmented with a mathematical description of interface debonding, dependent on interface stress criteria, and soft tissue interface interposition, dependent on relative interface motions. Three simplified models of orthopaedic implants were constructed: a cortical bone screw for fracture fixation plates, a femoral resurfacing prosthesis and a straight stem model, cemented in a bone. The predicted computer configurations were compared with clinical observations. The computer results showed how interface disruption and fibrous tissue interposition interrelate and possibly enhance each other, whereby a progressive development of the soft tissue layer can occur.\ud \ud Around the cortical bone screw, the predicted resorption patterns were relatively large directly under the screw head and showed a pivot point in the opposite cortex. The resurfacing cup model predicted some fibrous tissue formation under the medial and lateral cup rim, whereby the medial layer developed first because of higher initial interface stresses. The straight stem model predicted initial interface failure at the proximal parts. After proximal resorption and fibrous tissue interposition, the medial interface was completely disrupted and developed an interface layer. The distal and mid lateral side maintained within the strength criterion.\ud \ud Although the applied models were relatively simple, the results showed reasonable qualitative agreement with resorption patterns found in clinical studies concerning bone screws and the resurfacing cup. The hypothesis that interface debonding and subsequent relative (micro)motions could be responsible for bone resorption and fibrous tissue propagation is thereby sustained by the results

Bone remodeling simulations: challenges, problems and applications

La remodelación ósea es el mecanismo que regula la relación entre la morfología del hueso y sus cargas mecánicas externas. Se basa en el hecho de que el hueso se adapta a las condiciones mecánicas a las que está expuesto. Varios factores mecánicos y bioquímicos pueden regular la respuesta final de la remodelación ósea. De hecho, se considera que la remodelación ósea pretende alcanzar varios objetivos mecánicos: reparar el daño para reducir el riesgo de fractura y optimizar la rigidez y resistencia con el mínimo peso. Durante las últimas décadas, se han propuesto un gran número de leyes matemáticas implementadas numéricamente, pero la mayoría de ellas presentan diferentes problemas como la estabilidad, la convergencia o la dependencia de las condiciones iniciales. Por tanto, el objetivo principal de esta tesis es estudiar los modelos de remodelación ósea, mostrando sus retos, su problemática y su aplicación en el ámbito clínico. En primer lugar, se han revisado dos teorías clásicas de la remodelación ósea (conocidas como modelo de Stanford y modelo de Doblaré y García). En ambos casos, se propone un aspecto novedoso planteando que el estímulo homeostático de referencia no es constante, sino que depende localmente de la historia de carga que cada punto local está soportando. Como consecuencia directa de esta hipótesis, se demuestra que las inestabilidades numéricas que normalmente presentan estos algoritmos, pueden quedar resueltas, mejorando claramente los resultados finales. Esta metodología se aplicó a un modelo de elementos finitos 2D/3D mejorando la convergencia de la solución y asegurando su estabilidad numérica a largo plazo. Por otra parte, en un intento de dilucidar las características de adaptación mecánica del hueso en diferentes escalas, se plantea una relación a nivel órgano y a nivel de tejido que depende de un cambio en el estímulo homeostático de referencia acorde con la densidad aparente, mientras que se considera que la densidad de energía de deformación a nivel de tejido permanece invariante. Esta hipótesis mejora la unicidad de la solución y la hace independiente de las condiciones iniciales, ayudando también a su estabilidad numérica. Además, en esta tesis se aborda el modelado de paciente específico que es un tema que está adquiriendo cada vez más importancia. Una de las principales dificultades en la creación de modelos de paciente específico, es la determinación de las cargas que el hueso está realmente soportando. Los datos relativos a pacientes específicos, como la geometría ósea y la distribución de la densidad ósea, puede ser utilizados para determinar estas cargas. Por lo tanto, se ha estudiado la estimación de la cargas con tres diferentes técnicas matemáticas: regresión lineal, redes neuronales artificiales y máquinas de soporte vector. Estas técnicas se han aplicado a un ejemplo teórico para obtener las cargas a través de la densidad aparente que se predice con los modelos de remodelación ósea. Para concluir, la metodología desarrollada que combina modelos de remodelación ósea con redes neuronales se ha aplicado a la predicción de las cargas de cinco tibias de pacientes. Para ello, se han determinado la geometría y la distribución de la densidad a partir de un TAC y se han introducido los valores de densidad en el modelo previamente desarrollado, obteniendo así, las cargas específicas de las tibias de los pacientes. Con el fin de validar la capacidad de esta novedosa técnica, se han comparado las cargas obtenidas de la técnica propuesta con las cargas obtenidas en un análisis de marcha de dichos pacientes. Los errores obtenidos en las predicciones han sido menores de un 6 %. Por lo tanto, se puede concluir que la metodología aquí propuesta, permite determinar de forma aproximada las cargas que un hueso específico soporta.Bone remodeling is the mechanism that regulates the relationship between bone morphology and its external mechanical loads. It is based on the fact that bone adapts itself to the mechanical conditions to which it is exposed. Several mechanical and biochemical factors may regulate the final bone remodeling response. In fact, bone remodeling is hypothesized to achieve several mechanical objectives: repair damage to reduce the risk of fracture and optimize stiffness and strength with minimum weight. During recent decades, a great number of numerically implemented mathematical laws have been proposed, but most of them present different problems as stability, convergence or dependence of the initial conditions. Thus, the main scope of this Thesis is to study bone remodeling models, showing their challenges, their problematic and their applicability in the clinical setting. Firstly, we revisit two classical bone remodeling theories (Stanford model and Doblaré and García model). In both of them, the reference homeostatic stimulus is hypothesized that is not constant, but it is locally dependent on the loading history that each local point is effectively supporting. As a direct consequence of this assumption, we demonstrate that the numerical instabilities that all these algorithms normally present can be solved, clearly improving the final results. For this reason, we applied this methodology to 2D/3D finite element models. This contribution improves the convergence of the solution, leading to its numerical stability in the long-term. In an attempt to elucidate the features of bone adaptation at the di erent scales, we hypothesize that the relationship between the organ level and tissue level depends on the reference homeostatic stimulus changes according to the density and the tissue effective energy remains unchanged. This assumption improves the uniqueness of the solution, independently of the initial conditions selected and clearly helps in its numerical stability. In addition, patient-specific modeling is becoming increasingly important. One of the most challenging diffculties in creating patient-specific models is the determination of the specific load that the bone is really supporting. Real information related to specific patients, such as bone geometry and bone density distribution, can be used to determine patient loads. Therefore, we studied three different mathematical techniques: linear regression, artificial neural networks (ANN) and support vector machines (SVM). These techniques have been applied to a theoretical femur to obtain the load through the density that came from many bone remodeling simulations. Finally, the application of this novel methodology has been applied for the loading prediction of five real tibias. We are able to determine the subject-specific forces from CT data, from which we obtain bone geometry and density distribuviition of the five tibias. Then, the density values at certain bone regions have been introduced in the methodology developed that combines bone remodeling models and artificial neuronal networks (ANN) for obtaining the predicted subject-specific loads. Finally, in order to validate this novel technique for tibia loading predictions, we compare predicted loads with the loads obtained from the patientspecific musculoskeletal model. The errors between both loads were lower tan 6%. Therefore, the methodology proposed has been validate

Current Trends in Improving of Artificial Joints Design and Technologies for Their Arthroplasty

There is a global tendency to rejuvenate joint diseases, and serious diseases such as arthrosis and arthritis develop in 90% of people over 55 years of age. They are accompanied by degradation of cartilage, joint deformities and persistent pain, which leads to limited mobility and a significant deterioration in the quality of life of patients. For the treatment of these diseases in the late stages, depending on the indications, various methods are used, the most radical of which are methods of joint arthroplasty and, in particular, total arthroplasty. Currently, total arthroplasty is one of the most effective and high-quality surgical operations at the relevant medical indications. However, complications may also arise after it, leading, inter alia, to the need for repeated surgical intervention. In order to minimize the likelihood of complications, the artificial joints used in total arthroplasty and the technology of their fabrication are constantly being improved, which leads to the emergence of new designs and methods for their integration with living tissues. At the same time, at the moment, the improvement of traditional designs and production technologies has almost reached the top of their art, and their further improvements can be insignificantly or are associated with the use of the most up-to-day technologies, allowing for friction couples with low tribological properties to provide for them high ones, for example, gradient increase hardness in the couple titanium alloy on titanium alloy. This paper presents the current state of traditional technical means and technologies for joint arthroplasty. The main attention is paid to the analysis of the latest technologies in the field of joint arthroplasty, such as osseointegration of artificial joints, the improvement of materials with the property of osteoimmunomodulation, the improvement of joint arthroplasty technologies based on the modeling of dynamic osteosynthesis, as well as the identification of possible unconventional designs of artificial joints that contribute to these technologies, predictive assessment of areas for technologies improvement.DFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berli

Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications

This review paper intends to gather and organize a series of works which discuss the possibility of exploiting the mechanical properties of distributed arrays of piezoelectric transducers. The concept can be described as follows: on every structural member one can uniformly distribute an array of piezoelectric transducers whose electric terminals are to be connected to a suitably optimized electric waveguide. If the aim of such a modification is identified to be the suppression of mechanical vibrations then the optimal electric waveguide is identified to be the 'electric analog' of the considered structural member. The obtained electromechanical systems were called PEM (PiezoElectroMechanical) structures. The authors especially focus on the role played by Lagrange methods in the design of these analog circuits and in the study of PEM structures and we suggest some possible research developments in the conception of new devices, in their study and in their technological application. Other potential uses of PEMs, such as Structural Health Monitoring and Energy Harvesting, are described as well. PEM structures can be regarded as a particular kind of smart materials, i.e. materials especially designed and engineered to show a specific andwell-defined response to external excitations: for this reason, the authors try to find connection between PEM beams and plates and some micromorphic materials whose properties as carriers of waves have been studied recently. Finally, this paper aims to establish some links among some concepts which are used in different cultural groups, as smart structure, metamaterial and functional structural modifications, showing how appropriate would be to avoid the use of different names for similar concepts. © 2015 - IOS Press and the authors