Numerical modelling of shape regulation and growth stresses in trees

The main objective of this paper is to present the results of a study of the interactions between the growth and design of a tree with regards to biomechanical factors at the plant level. A numerical incremental model dedicated to the calculation of tree mechanical behaviour has been integrated in the plant architecture simulation software AMAPpara. At any stage of tree growth, a new equilibrium was calculated considering the weight increment applied on the structure, i.e. the mass of new wood layers and vegetative elements, as well as the biomechanical reaction caused by cell maturation strains in both normal and reaction wood. The resulting incremental displacements allowed the tree shape to be modified. The field of growth stresses was calculated within the stem, using a cumulative process taking into consideration the past history of each growth ring. The simulation results of trunk and branch shape, as well as internal stresses, were examined after consideration of different growth strategies. A block of trees was also simulated in order to show the influence of spatial competition on stem curvature and the variability in growth stress.     

Multiscale mechanobiology: Coupling models of adhesion kinetics and nonlinear tissue mechanics

The mechanical behavior of tissues at the macroscale is tightly coupled to cellular activity at the microscale. Dermal wound healing is a prominent example of a complex system in which multiscale mechanics regulate restoration of tissue form and function. In cutaneous wound healing, a fibrin matrix is populated by fibroblasts migrating in from a surrounding tissue made mostly out of collagen. Fibroblasts both respond to mechanical cues, such as fiber alignment and stiffness, as well as exert active stresses needed for wound closure.Here, we develop a multiscale model with a two-way coupling between a microscale cell adhesion model and a macroscale tissue mechanics model. Starting from the well-known model of adhesion kinetics proposed by Bell, we extend the formulation to account for nonlinear mechanics of fibrin and collagen and show how this nonlinear response naturally captures stretch-driven mechanosensing. We then embed the new nonlinear adhesion model into a custom finite element implementation of tissue mechanical equilibrium. Strains and stresses at the tissue level are coupled with the solution of the microscale adhesion model at each integration point of the finite element mesh. In addition, solution of the adhesion model is coupled with the active contractile stress of the cell population. The multiscale model successfully captures the mechanical response of biopolymer fibers and gels, contractile stresses generated by fibroblasts, and stress-strain contours observed during wound healing. We anticipate that this framework will not only increase our understanding of how mechanical cues guide cellular behavior in cutaneous wound healing, but will also be helpful in the study of mechanobiology, growth, and remodeling in other tissues.     

A nonlinear finite element model of cartilage growth

The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.     

A study of the effect of non-linearities in the equation of bone remodeling

In this paper, we introduced a high-order non-linear equation of bone remodeling to combine with FEM by introducing two non-linearities, i.e. the remodeling coefficient B(t) and the order of non-linear remodeling equation. The influence of each non-linearity was tested based on its mechanical and physiological implications discussed. We use two finite element models to investigate the influences of non-linearities in this equation: a plate subjected to a ramp load, and a 2D model of the cross-section of a vertebra. By importing the idea of topology optimization in engineering, their external shapes and internal density distributions were simulated from unfixed configurations. To a certain extent, the high-order non-linear equation of bone remodeling we suggested here can control the remodeling processes of bones in different stages of growth or at different anatomic sites more effectively, and make it more consistent with physiological reality, i.e. express the remodeling characteristic that bone's best morphology is adapted to its mechanical environment. Furthermore, it is likely to describe the process of bone growth and evolution.     

Numerical study of how creep and progressive stiffening affect the growth stress formation in trees

It is not fully understood how much growth stresses affect the final quality of solid timber products in terms of, e.g. shape stability. It is, for example, difficult to predict the internal growth stress field within the tree stem. Growth stresses are progressively generated during the tree growth and they are highly influenced by climate, biologic and material-related factors. To increase the knowledge of the stress formation, a finite element model was created to study how the growth stresses develop during the tree growth. The model is an axisymmetric general plane strain model where material for all new annual rings is progressively added to the tree during the analysis. The material model used is based on the theory of small strains (where strains refer to the undeformed configuration which is good approximation for strains less than 4%) where so-called biological maturation strains (growth-related strains that form in the wood fibres during their maturation) are used as a driver for the stress generation. It is formulated as an incremental material model that takes into account elastic strain, maturation strain, viscoelastic strain and progressive stiffening of the wood material. The results clearly show how the growth stresses are progressively generated during the tree growth. The inner core becomes more and more compressed, whereas the outer sapwood is subjected to slightly increased tension. The parametric study shows that the growth stresses are highly influenced by the creep behaviour and evolution of parameters such as modulus of elasticity, micro-fibril angle and maturation strain.     

An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth

Plant cell morphogenesis depends critically on two processes: the deposition of new wall material at the cell surface and the mechanical deformation of this material by the stresses resulting from the cell's turgor pressure. We developed a model of plant cell morphogenesis that is a first attempt at integrating these two processes. The model is based on the theories of thin shells and anisotropic viscoplasticity. It includes three sets of equations that give the connection between wall stresses, wall strains and cell geometry. We present an algorithm to solve these equations numerically. Application of this simulation approach to the morphogenesis of tip-growing cells illustrates how the viscoplastic properties of the cell wall affect the shape of the cell at steady state. The same simulation approach was also used to reproduce morphogenetic transients such as the initiation of tip growth and other non-steady changes in cell shape. Finally, we show that the mechanical anisotropy built into the model is required to account for observed patterns of wall expansion in plant cells.     

Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation

The aim of this paper is to develop a multiscale hierarchical hybrid model based on finite element analysis and neural network computation to link mesoscopic scale (trabecular network level) and macroscopic (whole bone level) to simulate the process of bone remodelling. As whole bone simulation, including the 3D reconstruction of trabecular level bone, is time consuming, finite element calculation is only performed at the macroscopic level, whilst trained neural networks are employed as numerical substitutes for the finite element code needed for the mesoscale prediction. The bone mechanical properties are updated at the macroscopic scale depending on the morphological and mechanical adaptation at the mesoscopic scale computed by the trained neural network. The digital image-based modelling technique using μ-CT and voxel finite element analysis is used to capture volume elements representativeof 2 mm³ at the mesoscale level of the femoral head. The input data for the artificial neural network are a set of bone material parameters, boundary conditions and the applied stress. The output data are the updated bone properties and some trabecular bone factors. The current approach is the first model, to our knowledge, that incorporates both finite element analysis and neural network computation to rapidly simulate multilevel bone adaptation.     

Incorporating tissue anisotropy and heterogeneity in finite element models of trabecular bone altered predicted local stress distributions

Trabecular bone is composed of organized mineralized collagen fibrils, which results in heterogeneous and anisotropic mechanical properties at the tissue level. Recently, biomechanical models computing stresses and strains in trabecular bone have indicated a significant effect of tissue heterogeneity on predicted stresses and strains. However, the effect of the tissue-level mechanical anisotropy on the trabecular bone biomechanical response is unknown. Here, a computational method was established to automatically impose physiologically relevant orientation inherent in trabecular bone tissue on a trabecular bone microscale finite element model. Spatially varying tissue-level anisotropic elastic properties were then applied according to the bone mineral density and the local tissue orientation. The model was used to test the hypothesis that anisotropy in both homogeneous and heterogeneous models alters the predicted distribution of stress invariants. Linear elastic finite element computations were performed on a 3 mm cube model isolated from a microcomputed tomography scan of human trabecular bone from the distal femur. Hydrostatic stress and von Mises equivalent stress were recorded at every element, and the distributions of these values were analyzed. Anisotropy reduced the range of hydrostatic stress in both tension and compression more strongly than the associated increase in von Mises equivalent stress. The effect of anisotropy was independent of the spatial redistribution high compressive stresses due to tissue elastic heterogeneity. Tissue anisotropy and heterogeneity are likely important mechanisms to protect bone from failure and should be included for stress analyses in trabecular bone.     

Growing skin: tissue expansion in pediatric forehead reconstruction

Tissue expansion is a common surgical procedure to grow extra skin through controlled mechanical over-stretch. It creates skin that matches the color, texture, and thickness of the surrounding tissue, while minimizing scars and risk of rejection. Despite intense research in tissue expansion and skin growth, there is a clear knowledge gap between heuristic observation and mechanistic understanding of the key phenomena that drive the growth process. Here, we show that a continuum mechanics approach, embedded in a custom-designed finite element model, informed by medical imaging, provides valuable insight into the biomechanics of skin growth. In particular, we model skin growth using the concept of an incompatible growth configuration. We characterize its evolution in time using a second-order growth tensor parameterized in terms of a scalar-valued internal variable, the in-plane area growth. When stretched beyond the physiological level, new skin is created, and the in-plane area growth increases. For the first time, we simulate tissue expansion on a patient-specific geometric model, and predict stress, strain, and area gain at three expanded locations in a pediatric skull: in the scalp, in the forehead, and in the cheek. Our results may help the surgeon to prevent tissue over-stretch and make informed decisions about expander geometry, size, placement, and inflation. We anticipate our study to open new avenues in reconstructive surgery and enhance treatment for patients with birth defects, burn injuries, or breast tumor removal.     

Model-based inverse estimation for active contraction stresses of tongue muscles using 3D surface shape in speech production

This paper presents a novel inverse estimation approach for the active contraction stresses of tongue muscles during speech. The proposed method is based on variational data assimilation using a mechanical tongue model and 3D tongue surface shapes for speech production. The mechanical tongue model considers nonlinear hyperelasticity, finite deformation, actual geometry from computed tomography (CT) images, and anisotropic active contraction by muscle fibers, the orientations of which are ideally determined using anatomical drawings. The tongue deformation is obtained by solving a stationary force-equilibrium equation using a finite element method. An inverse problem is established to find the combination of muscle contraction stresses that minimizes the Euclidean distance of the tongue surfaces between the mechanical analysis and CT results of speech production, where a signed-distance function represents the tongue surface. Our approach is validated through an ideal numerical example and extended to the real-world case of two Japanese vowels, /ʉ/ and /ɯ/. The results capture the target shape completely and provide an excellent estimation of the active contraction stresses in the ideal case, and exhibit similar tendencies as in previous observations and simulations for the actual vowel cases. The present approach can reveal the relative relationship among the muscle contraction stresses in similar utterances with different tongue shapes, and enables the investigation of the coordination of tongue muscles during speech using only the deformed tongue shape obtained from medical images. This will enhance our understanding of speech motor control.     

The Biomechanics of Tree Fork Design

Two biomechanically different types of tree fork are described: the “compression fork” where the two jointed stems are pressed against each other at the contact face by the action of reaction wood, and the “tension fork” where the two connected stems are bent away from each other by gravity or wind action leading to tensile stresses in the connective zone. It is well known that trees permanently try to improve their own designs by adaptive growth in order to maintain a state of constant mechanical stress at the tree surface. In the case of these two different types of tree fork, adaptive growth also takes different ways in order to avoid high localized stress peaks which could lead to failure of the tree under wind loading. In this paper only the tension fork is assessed with respect to its shape optimization by computer simulation of adaptive growth. It is shown that the tensile fork is shape optimized in a very perfect way in order to avoid any dangerous localized stress peaks (notch stresses) which could lead to failure of the tree.     

Cyclic mechanical compression increases mineralization of cell-seeded polymer scaffolds in vivo

Despite considerable documentation of the ability of normal bone to adapt to its mechanical environment, very little is known about the response of bone grafts or their substitutes to mechanical loading even though many bone defects are located in load-bearing sites. The goal of this research was to quantify the effects of controlled in vivo mechanical stimulation on the mineralization of a tissue-engineered bone replacement and identify the tissue level stresses and strains associated with the applied loading. A novel subcutaneous implant system was designed capable of intermittent cyclic compression of tissue-engineered constructs in vivo. Mesenchymal stem cell-seeded polymeric scaffolds with 8 weeks of in vitro preculture were placed within the loading system and implanted subcutaneously in male Fisher rats. Constructs were subjected to 2 weeks of loading (3 treatments per week for 30 min each, 13.3 N at 1 Hz) and harvested after 6 weeks of in vivo growth for histological examination and quantification of mineral content. Mineralization significantly increased by approximately threefold in the loaded constructs. The finite element method was used to predict tissue level stresses and strains within the construct resulting from the applied in vivo load. The largest principal strains in the polymer were distributed about a modal value of -0.24% with strains in the interstitial space being about five times greater. Von Mises stresses in the polymer were distributed about a modal value of 1.6 MPa, while stresses in the interstitial tissue were about three orders of magnitude smaller. This research demonstrates the ability of controlled in vivo mechanical stimulation to enhance mineralized matrix production on a polymeric scaffold seeded with osteogenic cells and suggests that interactions with the local mechanical environment should be considered in the design of constructs for functional bone repair.     

Experimental and computational biomechanical characterisation of the tracheo-bronchial tree of the bottlenose dolphin (Tursiops truncatus) during diving

Dolphins have adapted their anatomic structures to survive in the water environment and so far, the behaviour of their respiratory system during diving has not been fully understood, since they being protected species cannot be subjected to invasive analysis. Aim of this work is to model the tracheo-bronchial tree of the bottlenose dolphin Tursiops truncatus to study its behaviour during diving by coupling experimental in vitro mechanical characterisation of airways tissues to finite element computational analyses. Furthermore, a comparison was performed between the mechanical behaviour of tracheo-bronchial trees of dolphins and that of the goat, a terrestrial mammal, whose conformation of the upper airways is similar to that of the human, to determine how different structures respond to pressure in a controlled experimental set-up. The comparison between the goat and dolphin airways' mechanical behaviour highlights a lower collapsibility of the dolphin structure due to higher stiffness, lack of musculature and irregular shape of cartilaginous rings. Our data showed that the air entrapped into the airways plays a key role in avoiding the collapse. This effect is enhanced when accounting for the air flow escaping the alveoli that start to collapse during descent, even at depth as shallow as 10m of sea water. The comparison between airways behaviour of marine and terrestrial mammals may help in shedding a light on the biomechanical behaviour of human airways during breath-holding diving.     

Biomechanical analysis of the shear behaviour adjacent to an axially loaded implant

Good mechanical fixation of an implant to the surrounding bone is important for its longevity, and is influenced by both biological and mechanical factors. This study parametrically evaluates the mechanics of the interface with a computationally efficient analytic structural model of the shear stress field and global shear stiffness of an axially loaded implant. The utility of the analytic model was first established by validating its assumptions with a case-specific finite element model. We then used the analytic model for a sensitivity analysis of the relationship between the pattern of tissue growth and shear properties of the interface for our previously reported loaded in vivo experimental micromotion device. The bone located directly at the implant surface was found to be the most effective site for increasing interface stiffness. This suggests that the implant surface is the most desirable site for bone growth, yet is also the most mechanically challenging environment due to its maximal shear stresses. Thus, these findings support the further investigation of osteo-conductive coatings and other biological stimuli to overcome the challenging mechanics, and to promote bone growth directly at the implant surface. The model also demonstrated that the mechanical contribution to the global implant shear stiffness of a commonly observed isolated sclerotic bone rim is very limited. The results of this sensitivity analysis agree with experimental studies with the micromotion device, and with clinical studies reporting good results with osteo-conductive coatings.     

Finite element simulations of a focal knee resurfacing implant applied to localized cartilage defects in a sheep model

Articular resurfacing metal implants have recently been tested in animal models to treat full thickness localized articular cartilage defects, showing promising results. However, the mechanical behavior of cartilage surrounding the metal implant has not been studied yet as it is technically challenging to measure in vivo contact areas, pressures, stresses and deformations from the metal implant. Therefore, we implemented a detailed numerical finite element model by approximating one of the condyles of the sheep tibiofemoral joint and created a defect of specific size to accommodate the implant. Using this model, the mechanical behavior of the surrounding of metal implant was studied. The model showed that the metal implant plays a significant role in the force transmission. Two types of profiles were investigated for metal implant. An implant with a double-curved profile, i.e., a profile fully congruent with the articular surfaces in the knee, gives lower contact pressures and stresses at the rim of the defect than the implant with unicurved spherical profile. The implant should be placed at a certain distance into the cartilage to avoid damage to opposing biological surface. Too deep positions, however, lead to high shear stresses in the cartilage edges around the implant. Mechanical sealing was achieved with a wedge shape of the implant, also useful for biochemical sealing of cartilage edges at the defect.     

Finite elements modeling of olive tree mechanical harvesting using different shakers

The aim of this study is the optimization of stem shaking conditions in the mechanical harvesting of olive fruits. In order to study the influence of different shakers on the dynamic response of “Chemlali”, the main olive variety in Tunisia, a finite element numerical modeling was undertaken. The olive tree is modeled by three-dimension beams. Each beam has two nodes and 6 freedom degrees for each node. The structure is built by 560 elements and 561 nodes. For each part of the tree, the wood’s mechanical characteristics are determined experimentally. Orbital and multidirectional shakers were the mechanical harvesting tools tested. For each type, we developed the excitatory force equation as a function of the shaker’s characteristics: the eccentric value, the unbalanced weights and their rotation frequencies. In this paper, we present the structure response in relation to the shaker’s type, its placement and its excitation frequencies. This response was controlled, firstly by the acceleration and displacement values at the ends of secondary branches, and secondly by the reaction force at the peduncles level. These results are useful for the design of new shakers adapted to the studied olive variety.     

Mechanical property determination of bone through nano- and micro-indentation testing and finite element simulation

Measurement of the mechanical properties of bone is important for estimating the stresses and strains exerted at the cellular level due to loading experienced on a macro-scale. Nano- and micro-mechanical properties of bone are also of interest to the pharmaceutical industry when drug therapies have intentional or non-intentional effects on bone mineral content and strength. The interactions that can occur between nano- and micro-indentation creep test condition parameters were considered in this study, and average hardness and elastic modulus were obtained as a function of indentation testing conditions (maximum load, load/unload rate, load-holding time, and indenter shape). The results suggest that bone reveals different mechanical properties when loading increases from the nano- to the micro-scale range (microN to N), which were measured using low- and high-load indentation testing systems. A four-parameter visco-elastic/plastic constitutive model was then applied to simulate the indentation load vs. depth response over both load ranges. Good agreement between the experimental data and finite element model was obtained when simulating the visco-elastic/plastic response of bone. The results highlight the complexity of bone as a biological tissue and the need to understand the impact of testing conditions on the measured results.     

Computational simulation of dental implant osseointegration through resonance frequency analysis

The aim of this study is to predict the evolution of the resonance frequency of the bone-implant interface in a dental implant by means of finite element simulation. A phenomenological interface model able to simulate the mechanical effects of the osseointegration process at the bone-implant interface is applied and compared with some experimental results in rabbits. An early stage of slow bone ingrowth, followed by a faster osseointegration phase until final stability is predicted by the simulations. The evolution of the resonance frequency of the implant and surrounding tissues along the simulation period was also obtained, observing a 3-fold increase in the first principal frequency. These findings are in quantitative agreement with the experimental measurements and suggest that the model can be useful to evaluate the influence of mechanical factors such as implant geometry or implant loading on the indirect evaluation of the process of implant osseintegration.