Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

The use of meshfree and particle methods in the field of bioengineering and biomechanics has significantly increased. This may be attributed to their unique abilities to overcome most of the inherent limitations of mesh-based methods in dealing with problems involving large deformation and complex geometry that are common in bioengineering and computational biomechanics in particular. This review article is intended to identify, highlight and summarize research works on topics that are of substantial interest in the field of computational biomechanics in which meshfree or particle methods have been employed for analysis, simulation or/and modeling of biological systems such as soft matters, cells, biological soft and hard tissues and organs. We also anticipate that this review will serve as a useful resource and guide to researchers who intend to extend their work into these research areas. This review article includes 333 references

A structure-preserving upwind DG scheme for a degenerate phase-field tumor model

In this work, we present a modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables. In addition, we develop an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model. Finally, some computational tests in accordance with the theoretical results are introduced. In the first test, we compare our DG scheme with the finite element (FE) scheme related to the same time approximation. The DG scheme shows a well-behavior even for strong cross-diffusion effects in contrast with FE where numerical spurious oscillations appear. Moreover, the second test exhibits the behavior of the tumor-growth model under different choices of parameters and also of mobility and proliferation functions.Comment: 32 pages, 15 figure

Análise Biomecânica de Calo Ósseo usando Método Sem Malha

O osso é um tecido fisiologicamente dinâmico e que quando lesionado tem a capacidade de se reparar com o próprio tecido, não envolvendo um tecido cicatrizante, ao contrário de outros tecidos. Esta característica torna-o particularmente interessante para investigar os processos inerentes de fraturas ósseas. A maior parte das fraturas cicatriza através de uma sequência de processos de diferenciação de tecidos, desde os processos iniciais de hematoma, aos tecidos conjuntivos, e através da cartilagem ao osso. No entanto, qualquer falha neste processo pode resultar em uniões tardias, más uniões ou não uniões. A compreensão na totalidade deste processo ainda constitui um desafio. Os mecanismos que envolvem os processos de estimulação mecânica não se encontram bem compreendidos, em consequência da complexidade dos testes experimentais in vivo, que se tornam dependentes de dados in vitro, tornando difícil validar os pressupostos biológicos. Consequentemente, os modelos computacionais têm demonstrado serem bastante úteis e eficazes na investigação sobre a cicatrização óssea. Desta forma, com o presente trabalho foi possível analisar as condições mecânicas de um calo ósseo resultante de uma fratura, assim como compreender as metodologias de análise numérica aplicadas. O modelo teve por base um estudo in vivo de forma a obter uma variação temporal progressiva da forma do calo e das propriedades mecânicas durante a cicatrização óssea. Com este modelo obtiveram-se os campos de tensão e deformação nas diferentes fases do processo de regeneração, obtendo-se resultados que se encontram em conformidade com a literatura. Adicionalmente, foi aplicado um algoritmo de remodelação óssea em combinação com o Radial Point Interpolation Method (RPIM) que foi capaz de reproduzir as condições apresentadas pela respetiva imagem histológica nesta fase. Por último, espera-se que os trabalhos desenvolvidos neste âmbito possibilitem a conceção de estratégias mais precisas e eficazes tanto para o tratamento como para aceleração da cura. De forma complementar, encontram-se em desenvolvimento modelos específicos dos pacientes e que incorporam variabilidade genética.Bone is a physiologically dynamic tissue that, when injured, has the ability to repair itself, not involving scar tissue, unlike other tissues. This characteristic makes it particularly interesting for investigating the inherent processes of bone fractures. Most fractures heal through a sequence of tissue differentiation processes, from the initial hematoma, to connective tissues and through cartilage to bone. However, any failure in this process can result in a delayed union, mal-union or non-union. A complete understanding of this process is still a challenge. The mechanisms surrounding the mechanical stimulation processes are relatively poorly understood as a result of the complexity of in vivo experimental tests, which become dependent on in vitro data, making it difficult to validate the biological assumptions. Consequently, computational models have proven to be very useful and effective in the investigation of bone healing. Therefore, in the present work, it was possible to analyse the mechanical conditions of a bone callus as a consequence of a fracture and to understand the methodologies of numerical analysis applied. The model was based on an in vivo experimental study in order to obtain a progressive temporal variation of the callus shape and mechanical properties during bone healing. With this model, the stress and strain fields in the different phases of the regeneration process were obtained, where the results are in agreement with the literature. Additionally, a bone remodelling algorithm was applied in combination with the Radial Point Interpolation Method (RPIM), which was able to reproduce the conditions presented by the respective histological image at this stage. Finally, it is expected that the work developed in this area will enable the design of more accurate and effective strategies for both treatment and accelerating healing. Complementarily, patient-specific models and the incorporation of genetic variability are being developed

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

Radiotherapy cancer treatment model with fractional derivative coupled with linear-quadratic model

A mathematical model that simulates a radiotherapy cancer treatment process is presented in this thesis. The model takes two important radiobiological factors into consideration, which are repair and repopulation of cells. The model is used to simulate the fractionated radiotherapy treatment processes of six patients. The results give the population changes in the cells and the final volumes occupied by the normal and cancer cells. The model is formulated by integrating the Caputo fractional derivative with the previous cancer treatment model. Thereafter, the linear quadratic with the repopulation model is coupled into the model to account for the cells’ population decay due to radiation. The treatment processes are then simulated in MATLAB with numerical variables, numerical parameters, and radiation parameters. The numerical parameters which include the proliferation coefficients of cells, competition coefficients of cells, and the perturbation constant of the normal cells are obtained from a previous research. The radiation parameters are obtained from another previous research that reported clinical data of six patients treated with radiotherapy. From the reported clinical data, the patients had tumor volumes of 24.1cm 3, 17.4cm 3, 28.4cm 3 , 18.8cm 3, 3°.6cm3, and 12.6cm 3 and were treated with fractionated doses of 2.0 Gy for the first two patients and 1.8 Gy for the other four. Next, the integrity of the formulated model is established with the proof of the existence of unique solutions, the stability analysis, the sensitivity analysis, the bifurcation analysis, and the comparative analysis. Also, 96 radiation protocols are simulated by using the biologically effective dose formula. All these protocols are then used to obtain regression equations connecting the value of the Caputo fractional derivative with the fractionated radiation dose, and these regression equations are used to simulate various radiotherapy treatments in four different categories. The final tumor volumes, from the results of the simulations, are 3.58cm3 , 8.61cm3 , 5.68cm3 , 4.36cm3 , 5.75cm3 , and 6.12cm3. Meanwhile the volumes occupied by the normal cells are 23.87cm3, 17.29cm3, 28.1lcm3, 18.68cm3, 30.33cm3 , and 12.55cm3. The stability analysis shows that the model is asymptotically and exponentially stable. Also, the solutions of the simulations are unique and stable even there are changes in initial values. The sensitivity analysis shows that the most sensitive controllable model factor is the value of the Caputo fractional derivative and this model factor has bifurcation values. Furthermore, the comparative analysis shows that the fractional derivative model encompasses the memory effect of the radiotherapy process. The predicted simulated final tumor volumes obtained with the regression equations are then compared with the corresponding reported clinical final tumor volumes. The results of these comparisons show that the predictions have minimal errors, hence they are acceptable. Finally, optimal and complete treatment solutions are simulated and predicted

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described