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.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Three-dimensional topology optimization model to simulate the external shapes of bone

Elucidation of the mechanism by which the shape of bones is formed is essential for understanding vertebrate development. Bones support the body of vertebrates by withstanding external loads, such as those imposed by gravity and muscle tension. Many studies have reported that bone formation varies in response to external loads. An increased external load induces bone synthesis, whereas a decreased external load induces bone resorption. This relationship led to the hypothesis that bone shape adapts to external load. In fact, by simulating this relationship through topology optimization, the internal trabecular structure of bones can be successfully reproduced, thereby facilitating the study of bone diseases. In contrast, there have been few attempts to simulate the external structure of bones, which determines vertebrate morphology. However, the external shape of bones may be reproduced through topology optimization because cells of the same type form both the internal and external structures of bones. Here, we constructed a three-dimensional topology optimization model to attempt the reproduction of the external shape of teleost vertebrae. In teleosts, the internal structure of the vertebral bodies is invariable, exhibiting an hourglass shape, whereas the lateral structure supporting the internal structure differs among species. Based on the anatomical observations, we applied different external loads to the hourglass-shaped part. The simulations produced a variety of three-dimensional structures, some of which exhibited several structural features similar to those of actual teleost vertebrae. In addition, by adjusting the geometric parameters, such as the width of the hourglass shape, we reproduced the variation in the teleost vertebrae shapes. These results suggest that a simulation using topology optimization can successfully reproduce the external shapes of teleost vertebrae. By applying our topology optimization model to various bones of vertebrates, we can understand how the external shape of bones adapts to external loads.     

Three-dimensional design optimisation of patient-specific femoral plates as a means of bone remodelling reduction

Previous investigations into the optimisation of internal plates have mostly focused on the material properties of the implant. In this work, we optimise the shape, size and placement of the plate for successfully minimising bone remodelling around the implant. A design optimisation algorithm based on strain energy density criterion, combined with the finite element analysis, has been used in this study. The main optimisation goal was to reduce this change and keep it close to the conditions of an intact femur. The results suggest that the anterolateral side of the bone would be the optimum location for the plate, as for the geometry, the optimum moves towards having a thick, wide and short plate. These important results could be directly applicable to orthopaedic surgeons treating a femur fracture with internal plates. Since the optimisation algorithm remains the same for any patient, this advancement provides the surgeon with a tool to minimise the post surgery remodelling by trying to maintain the natural structure of the bone.     

Computational study of Wolff's law with trabecular architecture in the human proximal femur using topology optimization

In the field of bone adaptation, it is believed that the morphology of bone is affected by its mechanical loads, and bone has self-optimizing capability; this phenomenon is well known as Wolff's law of the transformation of bone. In this paper, we simulated trabecular bone adaptation in the human proximal femur using topology optimization and quantitatively investigated the validity of Wolff's law. Topology optimization iteratively distributes material in a design domain producing optimal layout or configuration, and it has been widely and successfully used in many engineering fields. We used a two-dimensional micro-FE model with 50 microm pixel resolution to represent the full trabecular architecture in the proximal femur, and performed topology optimization to study the trabecular morphological changes under three loading cases in daily activities. The simulation results were compared to the actual trabecular architecture in previous experimental studies. We discovered that there are strong similarities in trabecular patterns between the computational results and observed data in the literature. The results showed that the strain energy distribution of the trabecular architecture became more uniform during the optimization; from the viewpoint of structural topology optimization, this bone morphology may be considered as an optimal structure. We also showed that the non-orthogonal intersections were constructed to support daily activity loadings in the sense of optimization, as opposed to Wolff's drawing.     

Preliminary studies on the optimum shape of dental bridges

Several pre-existing anterior and posterior dental bridge models using Finite elements and the new ceramic material In-Ceram have been developed. The mechanical behaviour of these models has been compared with optimised profiles obtained from a newly developed evolutionary algorithm known as Evolutionary Structural Optimisation (ESO). The results show that the mechanical behaviour of the bridges was mainly restricted by the properties of the porcelain veneer and the design of the bridges themselves. For the case of the anterior bridge, it was found that there existed a specific thickness of veneer that minimised the maximum principal stress. This was related to peak stresses that occurred at the bridge surface. Peak stresses also occurred in the material interface between the In-Ceram and the veneer. These extreme stresses were attributed to the notch size and shape. For the case of the posterior bridge, it was concluded that the shape of the bottom of the Pontic tooth is crucial in reducing the magnitude of the maximum principal tensile stress. The ESO process produced bridge designs which have uniformly stressed bridge surfaces, and which also have significantly lower maximum principal tensile stresses compared to the pre-existing designs (up to 44%).     

Preliminary Studies on the Optimum Shape of Dental Bridges

Several pre-existing anterior and posterior dental bridge models using Finite elements and the new ceramic material In-Ceram have been developed. The mechanical behaviour of these models has been compared with optimised profiles obtained from a newly developed evolutionary algorithm known as Evolutionary Structural Optimisation (ESO).

The results show that the mechanical behaviour of the bridges was mainly restricted by the properties of the porcelain veneer and the design of the bridges themselves. For the case of the anterior bridge, it was found that there existed a specific thickness of veneer that minimised the maximum principal stress. This was related to peak stresses that occurred at the bridge surface. Peak stresses also occurred in the material interface between the In-Ceram and the veneer. These extreme stresses were attributed to the notch size and shape. For the case of the posterior bridge, it was concluded that the shape of the bottom of the Pontic tooth is crucial in reducing the magnitude of the maximum principal tensile stress. The ESO process produced bridge designs which have uniformly stressed bridge surfaces, and which also have significantly lower maximum principal tensile stresses compared to the pre-existing designs (up to 44%).     

Predicting the external formation of a bone fracture callus: an optimisation approach

The formation of a fracture callus in vivo tends to form in a structurally efficient manner distributing tissues where mechanical stimulus persists. Therefore, it is proposed that the formation of a fracture callus can be modelled in silico by way of an optimisation algorithm. This was tested by generating a finite element model of a transversal bone fracture embedded in a large tissue domain which was subjected to axial, bending and torsional loads. It was found that the relative fragment motion induced a compressive strain field in the early callus tissue which could be utilised to simulate the formation of external callus structures through an iterative optimisation process of tissue maintenance and removal. The phenomenological results showed a high level of congruence with in vivo healing patterns found in the literature. Consequently, the proposed strategy shows potential as a means of predicting spatial bone healing phenomena for pre-clinical testing.     

Successful Application of Evolutionary Algorithms in Engineering Design

Evolutionary or bionic strategies have proven to be powerful tools in many optimisation studies. Starting with some parent generations, producing sets of children, selecting the best children to be new parents yields impressive improvements of the objective when used with some experience and sufficient equipment. During some years of research the parameters of evolutionary optimisation have been investigated. Many successful applications showed where and how to use it. Especially in the case of objective functions with some or many local maxima, evolutionary approaches may propose solutions which gradient based optimisation would hardly find.When used with a large number of optimisation parameters, evolutionary methods seem to be superior to other strategies, as the chances to find good proposals within an acceptable number of trials and within affordable time are much higher.Nevertheless, evolutionary approaches like all optimisation methods require large numbers of studies of individual solutions. The computer power necessary to apply these strategies should not be underestimated. Even with today low cost and high availability of computers, the time to solve problems may be surprisingly long. So all ways of parallel processing, using single computers with many processors or clusters of many computers may speed up the time to do the optimisation.The basic terms of the method are outlined, some problems discussed, some examples given and some proposals made, how to use evolutionary methods in engineering optimisation. Finally some warnings are given trying to prevent potential users from non-realistic expectations. Optimisation is a difficult and consuming process. This holds for evolutionary optimisation as well.     

Fit optimisation of a distal medial tibia plate

An iterative method for the fit optimisation of a pre-contoured fracture fixation plate for a given bone data set is presented. Both plate shape optimisation and plate fit quantification are conducted in a virtual environment utilising computer graphical methods and 3D bone and plate models. Two optimised shapes of the undersurface of an existing distal medial tibia plate were generated based on a dataset of 45 3D bone models reconstructed from computed tomography image data of Japanese tibiae. The existing plate shape achieved an anatomical fit on 13% of tibiae from the dataset. Modified plate 1 achieved an anatomical fit for 42% and modified plate 2 a fit for 67% of the bones. If either modified plate 1 or plate 2 is used, then the anatomical fit can be increased to 82% for the same dataset. Issues pertaining to any further improvement in plate fit/shape are discussed.     

Predicting the external formation of a bone fracture callus: an optimisation approach

The formation of a fracture callus in vivo tends to form in a structurally efficient manner distributing tissues where mechanical stimulus persists. Therefore, it is proposed that the formation of a fracture callus can be modelled in silico by way of an optimisation algorithm. This was tested by generating a finite element model of a transversal bone fracture embedded in a large tissue domain which was subjected to axial, bending and torsional loads. It was found that the relative fragment motion induced a compressive strain field in the early callus tissue which could be utilised to simulate the formation of external callus structures through an iterative optimisation process of tissue maintenance and removal. The phenomenological results showed a high level of congruence with in vivo healing patterns found in the literature. Consequently, the proposed strategy shows potential as a means of predicting spatial bone healing phenomena for pre-clinical testing.     

Predicting the external formation of callus tissues in oblique bone fractures: idealised and clinical case studies

It is proposed that the external asymmetric formation of callus tissues that forms naturally about an oblique bone fracture can be predicted computationally. We present an analysis of callus formation for two cases of bone fracture healing: idealised and subject-specific oblique bone fractures. Plane strain finite element (FE) models of the oblique fractures were generated to calculate the compressive strain field experienced by the immature callus tissues due to interfragmentary motion. The external formations of the calluses were phenomenologically simulated using an optimisation style algorithm that iteratively removes tissue that experiences low strains from a large domain. The resultant simulated spatial formation of the healing tissues for the two bone fracture cases showed that the calluses tended to form at an angle equivalent to the angle of the oblique fracture line. The computational results qualitatively correlated with the callus formations found in vivo. Consequently, the proposed methods show potential as a means of predicting callus formation in pre-clinical testing.     

A Bionic Approach for Topology Optimization for Tension-only or Compression-only Design

##正## A new bionic approach is presented to find the optimal topologies of a structure with tension-only or compression-onlymaterial based on bone remodelling theory.By traditional methods,the computational cost of topology optimization of thestructure is high due to material nonlinearity.To improve the efficiency of optimization,the reference-interval with material-replacement method is presented.In the method,firstly,the optimization process of a structure is considered as bone remodellingprocess under the same loading conditions.A reference interval of Strain Energy Density (SED),corresponding to thedead zone or lazy zone in bone mechanics,is adopted to control the update of the design variables.Secondly,a material-replacement scheme is used to simplify the Finite Element Analysis (FEA) of structure in optimization.In the operation ofmaterial-replacement,the original tension-only or compression-only material in design domain is replaced with a new isotropicmaterial and the Effective Strain Energy Density (ESED) of each element can be obtained.Finally,the update of design variablesis determined by comparing the local ESED and the current reference interval of SED,e.g.,the increment of a relativedensity is nonzero if the local ESED is out of the current reference interval.Numerical results validate the method.     

Protein sequence design based on the topology of the native state structure

Computational design of sequences for a given structure is generally studied by exhaustively enumerating the sequence space or by searching in such a large space, which is prohibitively expensive. However, we point out that the protein topology has a wealth of information, which can be exploited to design sequences for a chosen structure. In this paper, we present a computationally efficient method for ranking the residue sites in a given native-state structure, which enables us to design sequences for a chosen structure. The premise for the method is that the topology of the graph representing the energetically interacting neighbours in a protein plays an important role in the inverse-folding problem. While our previous work (which was also based on topology) used eigenspectral analysis of the adjacency matrix of interactions for ranking the residue sites in a given chain, here we use a simple but effective way of assigning weights to the nodes on the basis of secondary connections, along with primary connections. This indirectly accounts for the edge weight in the graph and removes degeneracy in the degree. The new scheme needs only a few multiplications and additions to compute the preferred ranking of the residue sites even for structures of real proteins of sizes of a few hundred amino acid residues. We use HP lattice model examples (for which exhaustive enumeration of sequences is practical) to validate our ranking approach in obtaining sequences of lowest energy for any H-P residue composition for a given native-state structure. Some examples of native structures of real proteins are also included. Quantitative comparison of the efficacy of the new scheme with the earlier schemes is made. The new scheme consistently performs better and with much lower computational cost. An optimization procedure is added to work with the new scheme in a few rare cases wherein the new scheme fails to provide the best sequence, an optimization procedure is added to work with the new scheme.     

Inferring functional linkages between proteins from evolutionary scenarios

Identifying potential protein interactions is of great importance in understanding the topologies of cellular networks, which is much needed and valued in current systematic biological studies. The development of our computational methods to predict protein-protein interactions have been spurred on by the massive sequencing efforts of the genomic revolution. Among these methods is phylogenetic profiling, which assumes that proteins under similar evolutionary pressures with similar phylogenetic profiles might be functionally related. Here, we introduce a method for inferring functional linkages between proteins from their evolutionary scenarios. The term evolutionary scenario refers to a series of events that occurred in speciation over time, which can be reconstructed given a phylogenetic profile and a species tree. Common evolutionary pressures on two proteins can then be inferred by comparing their evolutionary scenarios, which is a direct indication of their functional linkage. This scenario method has proven to have better performance compared with the classical phylogenetic profile method, when applied to the same test set. In addition, predicted results of the two methods are found to be fairly different, suggesting the possibility of merging them in order to achieve a better performance. We analyzed the influence of the topology of the phylogenetic tree on the performance of this method, and found it to be robust to perturbations in the topology of the tree. However, if a completely random tree is incorporated, performance will decline significantly. The evolutionary scenario method was used for inferring functional linkages in 67 species, and 40,006 linkages were predicted. We examine our prediction for budding yeast and find that almost all predicted linkages are supported by further evidence.     

MMM1 encodes a mitochondrial outer membrane protein essential for establishing and maintaining the structure of yeast mitochondria

In the yeast Saccharomyces cerevisiae, mitochondria are elongated organelles which form a reticulum around the cell periphery. To determine the mechanism by which mitochondrial shape is established and maintained, we screened yeast mutants for those defective in mitochondrial morphology. One of these mutants, mmm1, is temperature- sensitive for the external shape of its mitochondria. At the restrictive temperature, elongated mitochondria appear to quickly collapse into large, spherical organelles. Upon return to the permissive temperature, wild-type mitochondrial structure is restored. The morphology of other cellular organelles is not affected in mmm1 mutants, and mmm1 does not disrupt normal actin or tubulin organization. Cells disrupted in the MMM1 gene are inviable when grown on nonfermentable carbon sources and show abnormal mitochondrial morphology at all temperatures. The lethality of mmm1 mutants appears to result from the inability to segregate the aberrant-shaped mitochondria into daughter cells. Mitochondrial structure is therefore important for normal cell function. Mmm1p is located in the mitochondrial outer membrane, with a large carboxyl-terminal domain facing the cytosol. We propose that Mmm1p maintains mitochondria in an elongated shape by attaching the mitochondrion to an external framework, such as the cytoskeleton.     

Investigation of cancellous bone architecture using structural optimisation

The optimality of bone's internal architecture and its remodelling is investigated by applying topological optimisation. The os-calcis (heel bone) which is subject to a relatively simple loading environment was used as the test problem. The solution is compared with X-ray and CT images of the os-calcis and is demonstrated to agree favourably with the bone's typical trabecular orientation in the coronal plane. The optimality of bone remodelling is further investigated by applying a perturbation during optimisation which leads to a different optimum topology. This is compared with in-vivo experimental results from a model in which the same perturbation was applied. The findings indicate that there may exist more than one optimum state for the same mechanical condition and that structural optimisation is able to find these multiple states. It is therefore suggested that structural optimisation can be used to investigate the mechanisms of bone remodelling.     

Quartet-mapping, a generalization of the likelihood-mapping procedure.

Likelihood-mapping (LM) was suggested as a method of displaying the phylogenetic content of an alignment. However, statistical properties of the method have not been studied. Here we analyze the special case of a four-species tree generated under a range of evolution models and compare the results with those of a natural extension of the likelihood-mapping approach, geometry-mapping (GM), which is based on the method of statistical geometry in sequence space. The methods are compared in their abilities to indicate the correct topology. The performance of both methods in detecting the star topology is especially explored. Our results show that LM tends to reject a star tree more often than GM. When assumptions about the evolutionary model of the maximum-likelihood reconstruction are not matched by the true process of evolution, then LM shows a tendency to favor one tree, whereas GM correctly detects the star tree except for very short outer branch lengths with a statistical significance of >0.95 for all models. LM, on the other hand, reconstructs the correct bifurcating tree with a probability of >0.95 for most branch length combinations even under models with varying substitution rates. The parameter domain for which GM recovers the true tree is much smaller. When the exterior branch lengths are larger than a (analytically derived) threshold value depending on the tree shape (rather than the evolutionary model), GM reconstructs a star tree rather than the true tree. We suggest a combined approach of LM and GM for the evaluation of starlike trees. This approach offers the possibility of testing for significant positive interior branch lengths without extensive statistical and computational efforts.     

The ups and downs of protein topology; rapid comparison of protein structure

Protein topology can be described at different levels. At the most fundamental level, it is a sequence of secondary structure elements (a "primary topology string"). Searching predicted primary topology strings against a library of strings from known protein structures is the basis of some protein fold recognition methods. Here a method known as TOPSCAN is presented for rapid comparison of protein structures. Rather than a simple two-letter alphabet (encoding strand and helix), more complex alphabets are used encoding direction, proximity, accessibility and length of secondary elements and loops in addition to secondary structure. Comparisons are made between the structural information content of primary topology strings and encodings which contain additional information ("secondary topology strings"). The algorithm is extremely fast, with a scan of a large domain against a library of more than 2000 secondary structure strings completing in approximately 30 s. Analysis of protein fold similarity using TOPSCAN at primary and secondary topology levels is presented.