A topology optimization based model of bone adaptation

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.     

A topology optimization based model of bone adaptation

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.     

The application of topology optimization on the quantitative description of the external shape of bone structure

The aim of this paper was to introduce the idea of topology optimization in engineering to the simulation of bone morphology. The external shape of bone structure was predicted with the quantitative bone functional adaptation theory. The high-order nonlinear equation of bone remodeling proposed by Zhu et al. (J. Biomech. 35(7)(2002)951) combining with the finite element method was adopted to a rectangular design domain, which occupies a larger space than the external shape of bone structure. It was at this point we imported the idea of topology optimization in engineering. The proximal femur was used here as an example, whose external shape and internal density distribution were simultaneously simulated quantitatively to validate that the external shape of bone structure could be successfully predicted in this way. Then the growth of vertebral body from young to old was simulated numerically in its coronal section to discuss the significance of the prediction of external shape. The study in this paper provides computational basis for further studies on osteophyte formation, osteoporosis, osteoarthritis, bone growth and even bone evolution, etc.     

Comparison of different hip prosthesis shapes considering micro-level bone remodeling and stress-shielding criteria using three-dimensional design space topology optimization

Since the late 1980s, computational analysis of total hip arthroplasty (THA) prosthesis components has been completed using macro-level bone remodeling algorithms. The utilization of macro-sized elements requires apparent bone densities to predict cancellous bone strength, thereby, preventing visualization and analysis of realistic trabecular architecture. In this study, we utilized a recently developed structural optimization algorithm, design space optimization (DSO), to perform a micro-level three-dimensional finite element bone remodeling simulation on the human proximal femur pre- and post-THA. The computational simulation facilitated direct performance comparison between two commercially available prosthetic implant stems from Zimmer Inc.: the Alloclassic and the Mayo conservative. The novel micro-level approach allowed the unique ability to visualize the trabecular bone adaption post-operation and to quantify the changes in bone mineral content by region. Stress-shielding and strain energy distribution were also quantified for the immediate post-operation and the stably fixated, post-remodeling conditions. Stress-shielding was highest in the proximal region and remained unchanged post-remodeling; conversely, the mid and distal portions show large increases in stress, suggesting a distal shift in the loadpath. The Mayo design conserves bone mass, while simultaneously reducing the incidence of stress-shielding compared to the Alloclassic, revealing a key benefit of the distinctive geometry. Several important factors for stable fixation, determined in clinical evaluations from the literature, were evident in both designs: high levels of proximal bone loss and distal bone densification. The results suggest this novel computational framework can be utilized for comparative hip prosthesis shape, uniquely considering the post-operation bone remodeling as a design criterion.     

Three-dimensional micro-level computational study of Wolff's law via trabecular bone remodeling in the human proximal femur 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 maximize mechanical efficiency creating a naturally optimum structure. The goal of the current study was to utilize an advanced structural optimization algorithm, called design space optimization (DSO), to perform a micro-level three-dimensional finite element 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 result is a "fully stressed" structure with minimized compliance and increased stiffness. The large-scale computational simulation utilized a 175 μm mesh resolution and the routine daily loading activities of walking and stair climbing. The resulting anisotropic trabecular architecture was compared to both Wolff's trajectory hypothesis and natural femur samples from literature using a variety of visualization techniques, including radiography and computed tomography (CT). The results qualitatively revealed several anisotropic trabecular regions, that were comparable to the natural human femurs. Quantitatively, the various regional bone volume fractions from the computational results were consistent with quantitative CT analyses. The global strain energy proceeded to become more uniform during optimization; implying increased mechanical efficiency was achieved. 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 the Wolff hypothesized.     

Three-dimensional reconstruction and characterization of human external shapes from two-dimensional images using volumetric methods

This work presents a volumetric approach to reconstruct and characterise 3D models of external anatomical structures from 2D images. Volumetric methods represent the final volume using a finite set of 3D geometric primitives, usually designed as voxels. Thus, from an image sequence acquired around the object to reconstruct, the images are calibrated and the 3D models of the referred object are built using different approaches of volumetric methods. The final goal is to analyse the accuracy of the obtained models when modifying some of the parameters of the considered volumetric methods, such as the type of voxel projection (rectangular or accurate), the way the consistency of the voxels is tested (only silhouettes or silhouettes and photo-consistency) and the initial size of the reconstructed volume.     

A three-dimensional topology optimization model for tooth-root morphology

To obtain the root of a lower incisor through structural optimization, we used two methods: optimization with Solid Isotropic Material with Penalization (SIMP) and Soft-Kill Option (SKO). The optimization was carried out in combination with a finite element analysis in Abaqus/Standard. The model geometry was based on cone-beam tomography scans of 10 adult males with healthy bone-tooth interface. Our results demonstrate that the optimization method using SIMP for minimum compliance could not adequately predict the actual root shape. The SKO method, however, provided optimization results that were comparable to the natural root form and is therefore suitable to set up the basic topology of a dental root.     

Monte Carlo model framework to simulate settlement dynamics

We developed an open source Monte Carlo based model to simulate Settlement Dynamics in Drylands (SeDD). The model assigns partial probabilities to each pixel within a grid region, based on several factors that can influence the establishment and subsistence of settlements: groundwater depth, vegetation type, proximity to rivers, paved roads, old river beds, and existing settlements. Partial probabilities are considered to be independent from each other, and therefore multiplied to calculate an overall probability for each pixel. Settlements are assigned by maximum probabilities or randomly, according to pre-established threshold probability values. We also modeled the gradual reduction of vegetation caused by a new settlement in neighboring pixels, decreasing the probabilities related to vegetation type. The final distribution of settlements is given by an average over multiple Monte Carlo simulations. The model is computationally efficient and could be used to rapidly explore different scenarios of settlement dynamics and vegetation degradation in arid environments, and other environmental factors that can be added to the framework without performing changes in the source code.     

Porous metal block based on topology optimization to treat distal femoral bone defect in total knee revision

Metal block augmentations are common solutions in treating bone defects of total knee revision. However, the stress shielding and poor osteointegration resulted from metal block application could not be neglected in bone defects restoration. In this study, a novel porous metal block was designed with topology optimization to improve biomechanical performance. The biomechanical difference of the topologically optimized block, solid Ti6Al4V block, and porous Ti6Al4V block in treating bone defects of total knee revision was compared by finite element analysis. The inhomogeneous femoral model was created according to the computed tomography data. Combined with porous structures, minimum compliance topology optimization subjected to the volume fraction constraint was utilized for the redesign of the metal block. The region of interest was defined as a 10 mm area of the distal femur beneath the contacting surface. The biomechanical performance of daily motions was investigated. The von Mises stress, the strain energy density of the region of interest, and the von Mises stress of metal blocks were recorded. The results were analyzed in SPSS. In terms of the region of interest, the maximum von Mises stress of the topological optimized group increased obviously, and its average stress was significantly higher than that of the other groups (p < 0.05). Moreover, the topologically optimized block group had the highest maximum strain energy density of the three groups, and the lowest maximum stress of block was also found in this group. In this study, the stress shielding reduction and stress transfer capability were found obviously improved through topology optimization. Therefore, the topological optimized porous block is recommended in treating bone defects of total knee revision. Meanwhile, this study also provided a novel approach for mechanical optimization in block designing.

     

A variant of the integrate-and-fire model to simulate the adaptive neural firing pattern

The integrate-and-fire (IF) based modeling technique has been long employed to study the neural firing activities. This paper introduces a variant of the IF model in order to simulate the adaptive neural firing pattern. An inductor branch is integrated into the IF model to track the variance of the external stimulus, and subsequently control the decrease of the instantaneous firing rate in response to the sustained stimulation. Simulation results demonstrated that the proposed model was able to reproduce the adaptive neural firing pattern. Besides, the inductor branch might adjust the decay rate of the adaptive firing curve and the prominent firing rate upon the onset of the adaptation. The proposed model is characterized by its structure simplicity and simulation efficiency, which would be helpful for the analysis and prediction of the adaptive firing responses of the sensory neurons.     

A new model to simulate the elastic properties of mineralized collagen fibril

Bone, because of its hierarchical composite structure, exhibits an excellent combination of stiffness and toughness, which is due substantially to the structural order and deformation at the smaller length scales. Here, we focus on the mineralized collagen fibril, consisting of hydroxyapatite plates with nanometric dimensions aligned within a protein matrix, and emphasize the relationship between the structure and elastic properties of a mineralized collagen fibril. We create two- and three-dimensional representative volume elements to represent the structure of the fibril and evaluate the importance of the parameters defining its structure and properties of the constituent mineral and collagen phase. Elastic stiffnesses are calculated by the finite element method and compared with experimental data obtained by synchrotron X-ray diffraction. The computational results match the experimental data well, and provide insight into the role of the phases and morphology on the elastic deformation characteristics. Also, the effects of water, imperfections in the mineral phase and mineral content outside the mineralized collagen fibril upon its elastic properties are discussed.     

An investigation into the feasibility of implementing fractal paradigms to simulate cancellous bone structure

Cancellous bone consists of a framework of solid trabeculae interspersed with bone marrow. The structure of the bone tissue framework is highly convoluted and complex, being fractal and statistically self-similar over a limited range of magnifications. To date, the structure of natural cancellous bone tissue has been defined using 2D and 3D imaging, with no facility to modify and control the structure. The potential of four computer-generated paradigms has been reviewed based upon knowledge of other fractal structures and chaotic systems, namely Diffusion Limited Aggregation (DLA), Percolation and Epidemics, Cellular Automata, and a regular Grid with randomly relocated nodes. The resulting structures were compared for their ability to create realistic structures of cancellous bone rather than reflecting growth and form processes. Although the creation of realistic computer-generated cancellous bone structures is difficult, it should not be impossible. Future work considering the combination of fractal and chaotic paradigms is underway.     

A model of bone adaptation as an optimization process

The internal bone adaptation of the proximal femur is considered. A three-dimensional finite element model of the proximal femur is used. The bone remodeling in this work is numerically described by an evolutionary remodeling scheme with anisotropic material parameters and time-dependent loading. The memory of past loading is included in the model to account for the delay in the bone response from the load changes. The remodeling rate equation is derived from the structural optimization task of maximizing the stiffness in each time step. Additional information can be extracted from the optimization process; the remodeling equilibrium parameter where no apposition or resorption takes place, is defined as the element optimality conditions and the optimal design is used as an initial design for the onset of the remodeling simulation. Two examples of bone adaptation resulting from load changes are given, and the irreversible nature of the model is illustrated.     

Scale-down model to simulate spatial pH variations in large-scale bioreactors

For the first time a laboratory-scale two-compartment system was used to investigate the effects of pH fluctuations consequent to large scales of operation on microorganisms. pH fluctuations can develop in production-scale fermenters as a consequence of the combined effects of poor mixing and adding concentrated reagents at the liquid surface for control of the bulk pH. Bacillus subtilis was used as a model culture since in addition to its sensitivity to dissolved oxygen levels, the production of the metabolites, acetoin and 2,3-butanediol, is sensitive to pH values between 6.5 and 7.2. The scale-down model consisted of a stirred tank reactor (STR) and a recycle loop containing a plug flow reactor (PFR), with the pH in the stirred tank being maintained at 6.5 by addition of alkali in the loop. Different residence times in the loop simulated the exposure time of fluid elements to high values of pH in the vicinity of the addition point in large bioreactors and tracer experiments were performed to characterise the residence time distribution in it. Since the culture was sensitive to dissolved oxygen, for each experiment with pH control by adding base into the PFR, equivalent experiments were conducted with pH control by addition of base into the STR, thus ensuring that any dissolved oxygen effects were common to both types of experiments. The present study indicates that although biomass concentration remained unaffected by pH variations, product formation was influenced by residence times in the PFR of 60 sec or longer. These changes in metabolism are thought to be linked to both the sensitivity of the acetoin and 2,3-butanediol-forming enzymes to pH and to the inducing effects of dissociated acetate on the acetolactate synthase enzyme.     

Three-dimensional structure topology of the calreticulin P-domain based on NMR assignment

Calreticulin (CRT) is an abundant molecular chaperone of the endoplasmic reticulum. Its central, proline-rich P-domain, comprising residues 189-288, contains three copies of each of two repeat sequences (types 1 and 2), which are arranged in a characteristic '111222' pattern. Here we show that the three-dimensional structure of CRT(189-288) contains a single hairpin fold formed by the entire polypeptide chain. The loop at the bottom of the hairpin consists of residues 227-247, and is closed by an anti-parallel beta-sheet of residues 224-226 and 248-250. Two additional beta-sheets contain residues 207-209 and 262-264, and 190-192 and 276-278. The 17-residue spacing of the beta-strands in the N-terminal part of the hairpin and the 14-residue spacing in the C-terminal part reflect the length of the type 1 and type 2 sequence repeats. As a consequence of this topology the peptide segments separating the beta-strands in the N-terminal part of the hairpin are likely to form bulges to accommodate the extra residues. These results are based on nearly complete sequence-specific NMR assignments for CRT(189-288), which were obtained using standard NMR techniques with the (13)C/(15)N-labeled protein, and collection of nuclear Overhauser enhancement upper distance constraints.     

Adapting a patch model to simulate the sensitivity of Central-Canadian boreal ecosystems to climate variability

Aim To investigate effects of within-season and interannual climate variability on the behaviour of boreal forest ecosystems as simulated by the FORSKA2 patch model. Location Eleven climate station locations distributed along a transect across the boreal zone of central Canada. Methods FORSKA2′s water balance submodel was modified to enable it to behave more realistically under a varying climate. Long-term actual monthly time-series of temperature and precipitation data were detrended and used to drive the modified model. Long-term monthly averages of the same detrended data were used to drive the unmodified model. Results Modifications created significant improvements when simulating species composition at sites in boreal Canada. Simulated forest biomass values were slightly higher than those obtained from the unmodified model using averaged climate records, but resembled the observed distribution of vegetation more closely. Main conclusions Modified FORSKA2 suggests that boreal forest composition and distribution may be more sensitive to changes in monthly rainfall data than to changes in temperature. Climate variability affects seasonal water balances and should be considered when using patch models to forecast vegetation dynamics during and following a period of climate transition. The modified model provided improved representation of the latitudinal trend in spatially averaged biomass density in this region.     

Automatic search for model to simulate the differentiation of T lymphocytes within the thymus

The differentiation of T Lymphocytes within the thymus is an important biological phenomenon during wich these cell acquire their functions to further control the immune system. Numerous experiments under various conditions have been devised to understand the different mechanisms involved in this complex process. Nevertheless, interpretation of these experiments lead to still contradictory debatable hypotheses. Modelisation of this process through classical simulation methods cannot be envisaged because they are not adapted to modifications of the model structure, which is the point of interest. For these reasons, we proposed a new approach of automatic search for model. The program consists of four independent connected modules : The generator produces model, based on the rationale of formal grammars. Protocol and experimental data are stored in a set of experiments. The simulator using a protocol and a model provides simulated results. Finally, the supervisor by comparing simulated results and experimental data, adapts the model parameters to increase their fit and either chooses a new experiment to explore, or modifies the model structure. Change of the model structure is performed among still unexplored models according to their promise level, which is iteratively evaluated relatively to previously explored models through a proposed model distance. The generator is written in Prolog and the other modules in C++. The architecture of the program allows us to modify or complete a module without changing anything in the other modules. As a consequence, the proposed modeling approach conceived to study T lymphocyte differentiation within the thymus remains independent of this biological phenomenon and can be applied to other biological problems.     

A semi-empirical cell dynamics model for bone turnover under external stimulus

The normal periodic turnover of bone is referred to as remodeling. In remodeling, old or damaged bone is removed during a 'resorption' phase and new bone is formed in its place during a 'formation' phase in a sequence of events known as coupling. Resorption is preceded by an 'activation' phase in which the signal to remodel is initiated and transmitted. Remodeling is known to involve the interaction of external stimuli, bone cells, calcium and phosphate ions, and several proteins, hormones, molecules, and factors. In this study, a semi-empirical cell dynamics model for bone remodeling under external stimulus that accounts for the interaction between bone mass, bone fluid calcium, bone calcium, and all three major bone cell types, is presented. The model is formulated to mimic biological coupling by solving separately and sequentially systems of ODEs for the activation, resorption, and formation phases of bone remodeling. The charateristic time for resorption (20 days) and the amount of resorption (~0.5%) are fixed for all simulations, but the formation time at turnover is an output of the model. The model was used to investigate the effects of different types of strain stimuli on bone turnover under bone fluid calcium balance and imbalance conditions. For bone fluid calcium balance, the model predicts complete turnover after 130 days of formation under constant 1000 microstrain stimulus; after 47 days of formation under constant 2000 microstrain stimulus; after 173 days of formation under strain-free conditions, and after 80 days of formation under monotonic increasing strain stimulus from 1000 to 2000 microstrain. For bone fluid calcium imbalance, the model predicts that complete turnover occurs after 261 days of formation under constant 1000 microstrain stimulus and that turnover never occurs under strain-free conditions. These predictions were not impacted by mean dynamic input strain stimuli of 1000 and 2000 microstrain at 1 Hz and 1000 microstrain amplitude. The formation phase of remodeling lasts longer than the resorption phase, increased strain stimulus accelerates bone turnover, while absence of strain significantly delays or prevents it, and formation time for turnover under monotonic increasing strain conditions is intermediate to those for constant strain stimuli at the minimum and maximum monotonic strain levels. These results are consistent with the biology, and with Frost's mechanostat theory.