Detection of defects in cellular solids using highly nonlinear solitary waves: a numerical study of the proximal femur.

This study investigates the suitability of a relatively new non-destructive evaluation (NDE) technique for the detection of non-visible defects in cellular solids using highly nonlinear solitary waves (HNSWs) in a one-dimensional granular chain. Specifically, the HNSW-based NDE approach is employed to identify the existence of micro-fractures in trabecular bone within the femoral neck (FN) and the intertrochanteric (IT) region of the proximal femur which are fracture-prone sites due to their relatively low bone density, particularly in osteoporosis patients. The availability of a HNSW-based bone quality assessment tool could not only help in early diagnosis of osteoporosis but also affect surgical decisions and improve clinical outcomes in joint replacement surgeries which motivated this study. To obtain a realistic representation of the trabecular microstructure, high-resolution finite-element (FE) models of the FN and the IT region are first constructed using a topology optimization-based bone reconstruction scheme. Then, artificial defects in the form of fractured ligaments are generated in the FN and IT models by selectively disconnecting various struts within the trabecular network. Using the FE models as the inspection medium, hybrid discrete-element/finite-element (DE/FE) simulations are performed to examine the interaction of the HNSWs with the cellular bone samples through two different inspection modes, i.e., inspection via direct contact with the sample and indirect contact through an adequately chosen face sheet inserted between the cellular sample and the granular chain. The delays and amplitudes of the HNSWs are used to estimate the effective elastic moduli of the cellular samples and these estimates were found to be reasonably accurate only in case the face sheet was applied. For the latter case, it was shown that the HNSW-based modulus estimates can be used as indicators for defect detection, allowing to discern between pristine and damaged cellular solids. These results suggest that HNSW-based NDE is a reliable and cost-effective technique for the identification of defects in cellular solids, and is expected to find applications in various fields, such as non-invasive screening of bone diseases and fractures, or damage detection in additively manufactured cellular structures.

Framework of sampling the subject-specific static loads from the gait cycle of interindividual variation.

BACKGROUND AND OBJECTIVE: Numerous techniques for bone remodeling simulation have been developed based on Wolff's law. However, most studies have been conducted with empirically determined static loads, which cannot reflect subject-specific characteristics. We recently proposed a new concept of representative static loads (RSLs) to efficiently consider the effect of cyclically repeated dynamic loads on bone remodeling simulation. Based on this concept, the goal of this study is to sample the subject-specific static loads (SSL) from a general gait cycle of interindividual variation. METHODS: A total of 15 gait cycles (ten normal and five abnormal cycles) obtained from the public database were used in this study. Each gait cycle was applied to a femur FE model constructed from the clinical CT scan data to evaluate the strain energy distribution as a reference. Then, a natural coordinate was introduced to maintain the predefined locations of extreme points (i.e., two peaks and one valley) for both normal and abnormal gait cycles. To determine the RSLs in the natural coordinate, five out of ten normal gait cycles were used. Through an inverse transformation for each gait cycle, the RSLs in the natural coordinate were converted to the SSLs in the original coordinate. Topology optimization results with the proposed SSLs were compared with those with a single full gait cycle (reference). For comparison, topology optimization was also conducted with empirically determined loads (EDLs) which have been widely used in the literature. RESULTS: For normal gait cycles, the proposed SSLs reduced the average computing cost by 95.86% while suppressing the errors of bone mass distribution and apparent stiffness below maximum 4.24% and 1.72%, respectively. Even for abnormal gait cycles, the errors of bone mass distribution and apparent stiffness were suppressed below maximum 9.49% and 2.12%, respectively. Conversely, the conventional EDLs (peak loads selected in this study) showed significantly larger errors of maximum 47.28% and 30.31% in bone mass distribution and apparent stiffness for normal gait cycles. CONCLUSION: By virtue of using the coordinate transformation for each gait cycle, the proposed SSLs achieved a higher accuracy in the bone mass distribution and apparent stiffness than the previous RSLs and EDLs. Furthermore, this approach can be used for abnormal gait cycles which have higher interindividual variation.

Predictions of the elastic modulus of trabecular bone in the femoral head and the intertrochanter: a solitary wave-based approach.

This paper deals with the numerical prediction of the elastic modulus of trabecular bone in the femoral head (FH) and the intertrochanteric (IT) region via site-specific bone quality assessment using solitary waves in a one-dimensional granular chain. For accurate evaluation of bone quality, high-resolution finite element models of bone microstructures in both FH and IT are generated using a topology optimization-based bone microstructure reconstruction scheme. A hybrid discrete element/finite element (DE/FE) model is then developed to study the interaction of highly nonlinear solitary waves in a granular chain with the generated bone microstructures. For more robust and reliable prediction of the bone's mechanical properties, a face sheet is placed at the interface between the last chain particle and the bone microstructure, allowing more bone volume to be engaged in the dynamic deformation during interaction with the solitary wave. The hybrid DE/FE model was used to predict the elastic modulus of the IT and FH by analysing the characteristic features of the two primary reflected solitary waves. It was found that the solitary wave interaction is highly sensitive to the elastic modulus of the bone microstructure and can be used to identify differences in bone density. Moreover, it was found that the use of a relatively stiff face sheet significantly reduces the sensitivity of the wave interaction to local stiffness variations across the test surface of the bone, thereby enhancing the robustness and reliability of the proposed method. We also studied the effect of the face sheet thickness on the characteristics of the reflected solitary waves and found that the optimal thickness that minimizes the error in the modulus predictions is 4 mm for the FH and 2 mm for the IT, if the primary reflected solitary wave is considered in the evaluation process. We envisage that the proposed diagnostic scheme, in conjunction with 3D-printed high-resolution bone models of an actual patient, could provide a viable solution to current limitations in site-specific bone quality assessment.

Determination of the representative static loads for cyclically repeated dynamic loads: a case study of bone remodeling simulation with gait loads.

BACKGROUND AND OBJECTIVE: Bone has the self-optimizing capability to adjust its structure in order to efficiently support external loads. Bone remodeling simulations have been developed to reflect the above characteristics in a more effective way. In most studies, however, only a set of static loads have been empirically determined although both static and dynamic loads affect bone remodeling phenomenon. The goal of this study is to determine the representative static loads (RSLs) to efficiently consider the statically equivalent effect of cyclically repeated dynamic loads on bone remodeling simulation. METHODS: Based on the concept of two-scale approach, the RSLs for the gait cycles are determined from five subjects. First, the gait profiles at the hip joint are selected from the public database and then are preprocessed. The finite element model of the proximal femur is constructed from the clinical CT scan data to determine the strain energy distribution during the gait cycles. An optimization problem is formulated to determine the candidate static loads that minimize the errors of the spatial strain energy distribution for five gait profiles. Then, all candidate static loads from five gait profiles are partitioned into multiple clusters. The RSLs and the corresponding coefficients can be determined at the center of the densest cluster. For verification, topology optimization is separately conducted with the whole gait cycle (reference), empirically determined loads (conventional), and the RSLs (proposed). The strain energy density-based bone remodeling simulation is also conducted for another comparison. RESULTS: For the gait loads, the use of the RSLs enables a 99% reduction of the function calls with negligible errors in the bone spatial distribution (6.75% for two representative static loads and 6.24% for three representative static loads) and apparent stiffness (4.84% for two representative static loads and 4.47% for three representative static loads), compared with the use of a whole gait cycle as reference. CONCLUSION: This study shows the feasibility of the RSLs and provides a theoretical foundation for investigating the relationship between static and dynamic loads in the aspect of bone remodeling simulation.

Numerical predictions of the interaction between highly nonlinear solitary waves and the microstructure of trabecular bone in the femoral head.

The unique properties of highly nonlinear solitary waves in granular chains have prompted extensive research in the area of non-destructive testing and led to the development of new diagnostic schemes with potential applications in the healthcare industry. Here, we study numerically the interaction between highly nonlinear solitary waves in a granular chain and the microstructure of trabecular bone in the femoral head. High-resolution finite element models of bone microstructures with varying bone volume fraction are generated using a topology optimization-based bone microstructure reconstruction scheme. The obtained FE models of the trabecular bone were then used to develop a hybrid discrete/finite element model able to simulate the propagation of highly nonlinear solitary waves in a vertical array of steel particles, and their interaction with the adjacent bone microstructure model was studied. Two test modes were considered, one where the granular chain was placed in direct contact with the bone microstructure model, while in the second test mode, a face sheet was included between the chain and the bone model. For both test modes, we found that the characteristic features of the reflected solitary waves are sensitive to the effective compressive modulus of the bone microstructure models and follow similar trends than those obtained for a homogeneous, non-porous solid. It was also found that the use of the face sheet substantially reduces the sensitivity of the predictions to small changes in the bone topology, making it a robust and reliable method for non-destructive evaluation of the effective elastic modulus of cellular materials with small structural dimensions, as it is required for the site-specific evaluation of the mechanical properties of trabecular bone.

Patient-Specific Phantomless Estimation of Bone Mineral Density and Its Effects on Finite Element Analysis Results: A Feasibility Study.

OBJECTIVES: This study proposes a regression model for the phantomless Hounsfield units (HU) to bone mineral density (BMD) conversion including patient physical factors and analyzes the accuracy of the estimated BMD values. METHODS: The HU values, BMDs, circumferences of the body, and cross-sectional areas of bone were measured from 39 quantitative computed tomography images of L2 vertebrae and hips. Then, the phantomless HU-to-BMD conversion was derived using a multiple linear regression model. For the statistical analysis, the correlation between the estimated BMD values and the reference BMD values was evaluated using Pearson's correlation test. Voxelwise BMD and finite element analysis (FEA) results were analyzed in terms of root-mean-square error (RMSE) and strain energy density, respectively. RESULTS: The HU values and circumferences were statistically significant (p < 0.05) for the lumbar spine, whereas only the HU values were statistically significant (p < 0.05) for the proximal femur. The BMD values estimated using the proposed HU-to-BMD conversion were significantly correlated with those measured using the reference phantom: Pearson's correlation coefficients of 0.998 and 0.984 for the lumbar spine and proximal femur, respectively. The RMSEs of the estimated BMD values for the lumbar spine and hip were 4.26 ± 0.60 (mg/cc) and 8.35 ± 0.57 (mg/cc), respectively. The errors of total strain energy were 1.06% and 0.91%, respectively. CONCLUSIONS: The proposed phantomless HU-to-BMD conversion demonstrates the potential of precisely estimating BMD values from CT images without the reference phantom and being utilized as a viable tool for FEA-based quantitative assessment using routine CT images.

Computational study of estimating 3D trabecular bone microstructure for the volume of interest from CT scan data.

Inspired by the self-optimizing capabilities of bone, a new concept of bone microstructure reconstruction has been recently introduced by using 2D synthetic skeletal images. As a preliminary clinical study, this paper proposes a topology optimization-based method that can estimate 3D trabecular bone microstructure for the volume of interest (VOI) from 3D computed tomography (CT) scan data with enhanced computational efficiency and phenomenological accuracy. For this purpose, a localized finite element (FE) model is constructed by segmenting a target bone from CT scan data and determining the physiological local loads for the VOI. Then, topology optimization is conducted with multiresolution bone mineral density (BMD) deviation constraints to preserve the patient-specific spatial bone distribution obtained from the CT scan data. For the first time, to our knowledge, this study has demonstrated that 60-µm resolution trabecular bone images can be reconstructed from 600-µm resolution CT scan data (a 62-year-old woman with no metabolic bone disorder) for the 4 VOIs in the proximal femur. The reconstructed trabecular bone includes the characteristic trabecular patterns and has morphometric indices that are in good agreement with the anatomical data in the literature. As for computational efficiency, the localization for the VOI reduces the number of FEs by 99%, compared with that of the full FE model. Compared with the previous single-resolution BMD deviation constraint, the proposed multiresolution BMD deviation constraints enable at least 65% and 47% reductions in the number of iterations and computing time, respectively. These results demonstrate the clinical feasibility and potential of the proposed method.

Fully automated segmentation of a hip joint using the patient-specific optimal thresholding and watershed algorithm.

BACKGROUND AND OBJECTIVE: Automated segmentation with high accuracy and speed is a prerequisite for FEA-based quantitative assessment with a large population. However, hip joint segmentation has remained challenging due to a narrow articular cartilage and thin cortical bone with a marked interindividual variance. To overcome this challenge, this paper proposes a fully automated segmentation method for a hip joint that uses the complementary characteristics between the thresholding technique and the watershed algorithm. METHODS: Using the golden section method and load path algorithm, the proposed method first determines the patient-specific optimal threshold value that enables reliably separating a femur from a pelvis while removing cortical and trabecular bone in the femur at the minimum. This provides regional information on the femur. The watershed algorithm is then used to obtain boundary information on the femur. The proximal femur can be extracted by merging the complementary information on a target image. RESULTS: For eight CT images, compared with the manual segmentation and other segmentation methods, the proposed method offers a high accuracy in terms of the dice overlap coefficient (97.24 ± 0.44%) and average surface distance (0.36 ± 0.07 mm) within a fast timeframe in terms of processing time per slice (1.25 ± 0.27 s). The proposed method also delivers structural behavior which is close to that of the manual segmentation with a small mean of average relative errors of the risk factor (4.99%). CONCLUSION: The segmentation results show that, without the aid of a prerequisite dataset and users' manual intervention, the proposed method can segment a hip joint as fast as the simplified Kang (SK)-based automated segmentation, while maintaining the segmentation accuracy at a similar level of the snake-based semi-automated segmentation.

Image resolution enhancement for healthy weight-bearing bones based on topology optimization.

Although high-resolution skeletal images are essential for accurate bone strength assessment, the current high-resolution imaging modalities have critical problems that remain to be solved such as high radiation doses, low signal-to-noise ratios, and long scan times. Resolution enhancement techniques, which have recently received much attention, have also been difficult to obtain acceptable image resolutions. Inspired by the self-optimizing capabilities of bone (i.e. reorienting the trabecula for maximum mechanical efficiency with minimum bone mass), this paper proposes a novel resolution enhancement method that can reconstruct a high-resolution skeletal image from a low-resolution image. In order to achieve this, the proposed method conducts mesh refinement for resolution upscaling and then performs topology optimization with a constraint for the bone mineral density deviation in order to preserve the subject-specific bone distribution data. The numerical results show that the proposed method successfully reconstructs the enhanced images of trabecular architecture in terms of structure similarity and apparent elastic modulus, thereby demonstrating the feasibility of the proposed method for skeletal image resolution enhancement.

Estimation of Local Bone Loads for the Volume of Interest.

Computational bone remodeling simulations have recently received significant attention with the aid of state-of-the-art high-resolution imaging modalities. They have been performed using localized finite element (FE) models rather than full FE models due to the excessive computational costs of full FE models. However, these localized bone remodeling simulations remain to be investigated in more depth. In particular, applying simplified loading conditions (e.g., uniform and unidirectional loads) to localized FE models have a severe limitation in a reliable subject-specific assessment. In order to effectively determine the physiological local bone loads for the volume of interest (VOI), this paper proposes a novel method of estimating the local loads when the global musculoskeletal loads are given. The proposed method is verified for the three VOI in a proximal femur in terms of force equilibrium, displacement field, and strain energy density (SED) distribution. The effect of the global load deviation on the local load estimation is also investigated by perturbing a hip joint contact force (HCF) in the femoral head. Deviation in force magnitude exhibits the greatest absolute changes in a SED distribution due to its own greatest deviation, whereas angular deviation perpendicular to a HCF provides the greatest relative change. With further in vivo force measurements and high-resolution clinical imaging modalities, the proposed method will contribute to the development of reliable patient-specific localized FE models, which can provide enhanced computational efficiency for iterative computing processes such as bone remodeling simulations.

Homeostasis-based aging model for trabecular changes and its correlation with age-matched bone mineral densities and radiographs.

PURPOSE: This paper aims (1) to propose a novel bone adaptation model for age-related trabecular changes by adopting two implicit parameters in optimization, (2) to compare the simulated bone volume fraction (BV/TV) with the reported bone mineral density (BMD), and (3) to review the simulated trabecular architectures with the age-matched radiographs. MATERIALS AND METHODS: The proposed model simulated the trabecular changes for an age span of 32-80 years. Quantitative comparison was conducted in terms of BMD and other morphometric indices. Then, two radiologists scored the simulated trabecular architectures using the age-matched radiographs. This protocol was approved by the hospital institutional review board. RESULTS: The simulated BV/TV was well correlated with BMD reported in the literature (R(2)=0.855; p<0.05). In comparison with age-matched radiographs, the consensus scores of agreement of the trabeculae were higher in age groups over the 50s, and the means of the Ward's triangle areas were strongly correlated with those in the age-matched radiographs (R(2)=0.982; p<0.05). CONCLUSION: The proposed model could reflect the targeted trabecular changes in proximal femur with age. With further follow-up measurements, this research would contribute to the development of patient-specific models that assist radiologists in predicting skeletal integrity with aging.

Design improvement of the three-beam detector towards a precise long-range 6-degree of freedom motion sensor system.

This paper presents performance enhancement of the three beam detector through design improvement based on the error analysis. The three-beam detector is a novel sensor system that measures six motions (i.e., 3 translations and 3 rotations) of a remote object with high accuracy by implementing three laser distance sensors and a vision sensor. In this paper, finite element analysis and parameter analysis are applied to the three-beam detector to analyze its structural frames and sensor configuration, respectively, in terms of sensing error. By virtue of such a systematic, quantitative error analysis, this research has successfully improved the performance of the three-beam detector and thoroughly validated through field tests that it can detect the 6-degree of freedom motions of a remote target at 30 m with an accuracy of 1.51 mm and 0.18° and precision of 0.38 mm and 0.13°.

Computational simulation of simultaneous cortical and trabecular bone change in human proximal femur during bone remodeling.

In this study, we developed a numerical framework that computationally determines simultaneous and interactive structural changes of cortical and trabecular bone types during bone remodeling, and we investigated the structural correlation between the two bone types in human proximal femur. We implemented a surface remodeling technique that performs bone remodeling in the exterior layer of the cortical bone while keeping its interior area unchanged. A micro-finite element (microFE) model was constructed that represents the entire cortical bone and full trabecular architecture in human proximal femur. This study simulated and compared the bone adaptation processes of two different structures: (1) femoral bone that has normal cortical bone shape and (2) perturbed femoral bone that has an artificial bone lump in the inferomedial cortex. Using the proposed numerical method in conjunction with design space optimization, we successfully obtained numerical results that resemble actual human proximal femur. The results revealed that actual cortical bone, as well as the trabecular bone, in human proximal femur has structurally optimal shapes, and it was also shown that a bone abnormality that has little contribution to bone structural integrity tends to disappear. This study also quantitatively determined the structural contribution of each bone: when the trabecular adaptation was complete, the trabecular bone supported 54% of the total load in the human proximal femur while the cortical bone carried 46%.

Computational study on the effect of loading alteration caused by disc degeneration on the trabecular architecture in human lumbar spine.

A thorough understanding of age-related phenomena on the trabecular architecture in the human lumbar spine can help the diagnosis and prognosis of age-related architectural changes, and provide an insight into the corresponding clinical assessments. In this paper we considered the different loading conditions of the young and old lumbar spines mainly caused by disc degeneration and studied the effect of loading alteration on trabecular architecture in lumbar spines. A two-dimensional muFE models with a 40mum pixel resolution were built to represent the full trabecular architecture in the human lumbar spine, and a topology optimization with the aid of finite element method was conducted to numerically investigate the trabecular morphological changes. Topology optimization iteratively distributes material in a design domain producing optimal layout or configuration, and it has been widely and successfully used for the study of bone remodeling. As a result of adaptive response of bone remodeling due to different loading conditions, we obtained two distinctively different trabecular architectures for the young and old lumbar spines, and we observed a strong correlation between our numerical results and the actual trabecular architecture in the literature. The proposed numerical framework and results demonstrated the potential use of the topology optimization-based numerical tool for putative treatments in advance of actual clinical procedures for the patients.

Computational simulation of trabecular adaptation progress in human proximal femur during growth.

There are a large number of clinical and experimental studies that analyzed trabecular architecture as a result of bone adaptation. However, only a limited amount of quantitative data is currently available on the progress of trabecular adaptation during growth. In this paper, we proposed a two-step numerical simulation method that predicts trabecular adaptation progress during growth using a recently developed topology optimization algorithm, design space optimization (DSO), under the hypothesis that the mechanisms of DSO are functionally equivalent to those of bone adaptation. We applied the proposed scheme to trabecular adaptation simulation in human proximal femur. For the simulation, the full trabecular architecture in human proximal femur was represented by a two-dimensional microFE model with 50 microm resolution. In Step 1, we determined a reference value that regulates trabecular adaptation in human proximal femur. In Step 2, we simulated trabecular adaptation in human proximal femur during growth with the reference value derived in Step 1. We analyzed the architectural and mechanical properties of trabecular patterns through iterations. From the comparison with experimental data in the literature, we showed that in the early growth stage trabecular adaptation was achieved mainly by increasing bone volume fraction (or trabecular thickness), while in the later stage of the development the trabecular architecture gained higher structural efficiency by increasing structural anisotropy with a relatively low level of bone volume fraction (or trabecular thickness). We demonstrated that the proposed numerical framework predicted the growing progress of trabecular bone that has a close correlation with experimental data.

Analogy of strain energy density based bone-remodeling algorithm and structural topology optimization.

In bone-remodeling studies, it is believed that the morphology of bone is affected by its internal mechanical loads. From the 1970s, high computing power enabled quantitative studies in the simulation of bone remodeling or bone adaptation. Among them, Huiskes et al. (1987, "Adaptive Bone Remodeling Theory Applied to Prosthetic Design Analysis," J. Biomech. Eng., 20, pp. 1135-1150) proposed a strain energy density based approach to bone remodeling and used the apparent density for the characterization of internal bone morphology. The fundamental idea was that bone density would increase when strain (or strain energy density) is higher than a certain value and bone resorption would occur when the strain (or strain energy density) quantities are lower than the threshold. Several advanced algorithms were developed based on these studies in an attempt to more accurately simulate physiological bone-remodeling processes. As another approach, topology optimization originally devised in structural optimization has been also used in the computational simulation of the bone-remodeling process. The topology optimization method systematically and iteratively distributes material in a design domain, determining an optimal structure that minimizes an objective function. In this paper, we compared two seemingly different approaches in different fields-the strain energy density based bone-remodeling algorithm (biomechanical approach) and the compliance based structural topology optimization method (mechanical approach)-in terms of mathematical formulations, numerical difficulties, and behavior of their numerical solutions. Two numerical case studies were conducted to demonstrate their similarity and difference, and then the solution convergences were discussed quantitatively.

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.