Homogenized finite element analysis of distal tibia sections: Achievements and limitations.

High-resolution peripheral quantitative computed tomography (HR-pQCT) based micro-finite element (µFE) analysis allows accurate prediction of stiffness and ultimate load of standardised (∼1 cm) distal radius and tibia sections. An alternative homogenized finite element method (hFE) was recently validated to compute the ultimate load of larger (∼2 cm) distal radius sections that include Colles' fracture sites. Since the mechanical integrity of the weight-bearing distal tibia is gaining clinical interest, it has been shown that the same properties can be used to predict the strength of both distal segments of the radius and the tibia. Despite the capacity of hFE to predict structural properties of distal segments of the radius and the tibia, the limitations of such homogenization scheme remain unclear. Therefore, the objective of this study is to build a complete mechanical data set of the compressive behavior of distal segments of the tibia and to compare quantitatively the structural properties with the hFE predictions. As a further aim, it is intended to verify whether hFE is also able to capture the post-yield strain localisation or fracture zones in such a bone section, despite the absence of strain softening in the constitutive model. Twenty-five fresh-frozen distal parts of tibias of human donors were used in this study. Sections were cut corresponding to an in-house triple-stack protocol HR-pQCT scan, lapped, and scanned using micro computed tomography (µCT). The sections were tested in compression until failure, unloaded and scanned again in µCT. Volumetric bone mineral density (vBMD) and bone mineral content (BMC) were correlated to compression test results. hFE analysis was performed in order to compare computational predictions (stiffness, yield load and plastic deformation field pattern) with the compressive experiment. Namely, strain localization was assessed based on digital volume correlation (DVC) results and qualitatively compared to hFE predictions by comparing mid-slices patterns. Bone mineral content (BMC) showed a good correlation with stiffness (R2 = 0.92) and yield (R2 = 0.88). Structural parameters also showed good agreement between the experiment and hFE for both stiffness (R2 = 0.96, slope = 1.05 with 95 % CI [0.97, 1.14]) and yield (R2 = 0.95, slope = 1.04 [0.94, 1.13]). The qualitative comparison between hFE and DVC strain localization patterns allowed the classification of the samples into 3 categories: bad (15 sections), semi (8), and good agreement (2). The good correlations between BMC or hFE and experiment for structural parameters were similar to those obtained previously for the distal part of the radius. The failure zones determined by hFE corresponded to registration only in 8 % of the cases. We attribute these discrepancies to local elastic/plastic buckling effects that are not captured by the continuum-based FE approach exempt from strain softening. A way to improve strain localization hFE prediction would be to use longer distal segments with intact cortical shells, as done for the radius. To conclude, the used hFE scheme captures the elastic and yield response of the tibia sections reliably but not the subsequent failure process.

A multi-stack registration technique to improve measurement accuracy and precision across longitudinal HR-pQCT scans.

BACKGROUND: Recent applications of high-resolution peripheral quantitative computed tomography (HR-pQCT) have demonstrated that changes in local bone remodelling can be quantified in vivo using longitudinal three-dimensional image registration. However, certain emerging applications, such as fracture healing and joint analysis, require larger multi-stack scan regions that can result in stack shift image artifacts. These artifacts can be detrimental to the accurate alignment of the bone structure across multiple timepoints. The purpose of this study was to establish a multi-stack registration protocol for evaluating longitudinal HR-pQCT images and to assess the accuracy and precision error in comparison with measures obtained using previously established three-dimensional longitudinal registration. METHODS: Three same day multi-stack HR-pQCT scans of the radius (2 stacks in length) and tibia (3 stacks in length) were obtained from 39 healthy individuals who participated in a previous reproducibility study. A fully automated multi-stack registration algorithm was developed to re-align stacks within a scan by leveraging slight offsets between longitudinal scans. Stack shift severity before and after registration was quantified using a newly proposed stack-shift severity score. The false discovery rate for bone remodelling events and precision error of bone morphology and micro-finite element analysis parameters were compared between longitudinally registered scans with and without the addition of multi-stack registration. RESULTS: Most scans (82 %) improved in stack alignment or maintained the lowest stack shift severity score when multi-stack registration was implemented. The false discovery rate of bone remodelling events significantly decreased after multi-stack registration, resulting in median false detection of bone formation and resorption fractions between 3.2 to 7.5 % at the radius and 3.4 to 5.3 % at the tibia. Further, precision error was significantly reduced or remained unchanged in all standard bone morphology and micro-finite element analysis parameters, except for total and trabecular cross-sectional areas. CONCLUSION: Multi-stack registration is an effective strategy for accurately aligning multi-stack HR-pQCT scans without modification of the image acquisition protocol. The algorithm presented here is a viable approach for performing accurate morphological analysis on multi-stack HR-pQCT scans, particularly for advanced application investigating local bone remodelling in vivo.

Precision of bone mechanoregulation assessment in humans using longitudinal high-resolution peripheral quantitative computed tomography in vivo.

Local mechanical stimuli in the bone microenvironment are essential for the homeostasis and adaptation of the skeleton, with evidence suggesting that disruption of the mechanically-driven bone remodelling process may lead to bone loss. Longitudinal clinical studies have shown the combined use of high-resolution peripheral quantitative computed tomography (HR-pQCT) and micro-finite element analysis can be used to measure load-driven bone remodelling in vivo; however, quantitative markers of bone mechanoregulation and the precision of these analyses methods have not been validated in human subjects. Therefore, this study utilised participants from two cohorts. A same-day cohort (n = 33) was used to develop a filtering strategy to minimise false detections of bone remodelling sites caused by noise and motion artefacts present in HR-pQCT scans. A longitudinal cohort (n = 19) was used to develop bone imaging markers of trabecular bone mechanoregulation and characterise the precision for detecting longitudinal changes in subjects. Specifically, we described local load-driven formation and resorption sites independently using patient-specific odds ratios (OR) and 99 % confidence intervals. Conditional probability curves were computed to link the mechanical environment to the remodelling events detected on the bone surface. To quantify overall mechanoregulation, we calculated a correct classification rate measuring the fraction of remodelling events correctly identified by the mechanical signal. Precision was calculated as root-mean-squared averages of the coefficient of variation (RMS-SD) of repeated measurements using scan-rescan pairs at baseline combined with a one-year follow-up scan. We found no significant mean difference (p < 0.01) between scan-rescan conditional probabilities. RMS-SD was 10.5 % for resorption odds, 6.3 % for formation odds, and 1.3 % for correct classification rates. Bone was most likely to be formed in high-strain and resorbed in low-strain regions for all participants, indicating a consistent, regulated response to mechanical stimuli. For each percent increase in strain, the likelihood of bone resorption decreased by 2.0 ± 0.2 %, and the likelihood of bone formation increased by 1.9 ± 0.2 %, totalling 38.3 ± 1.1 % of strain-driven remodelling events across the entire trabecular compartment. This work provides novel robust bone mechanoregulation markers and their precision for designing future clinical studies.

Personalized loading conditions for homogenized finite element analysis of the distal sections of the radius.

The microstructure of trabecular bone is known to adapt its morphology in response to mechanical loads for achieving a biomechanical homeostasis. Based on this form-function relationship, previous investigators either simulated the remodeling of bone to predict the resulting density and architecture for a specific loading or retraced physiological loading conditions from local density and architecture. The latter inverse approach includes quantifying bone morphology using computed tomography and calculating the relative importance of selected load cases by minimizing the fluctuation of a tissue loading level metric. Along this concept, the present study aims at identifying an optimal, personalized, multiaxial load case at the distal section of the human radius using in vivo HR-pQCT-based isotropic, homogenized finite element (hFE) analysis. The dataset consisted of HR-pQCT reconstructions of the 20 mm most distal section of 21 human fresh-frozen radii. We simulated six different unit canonical load cases (FX palmar-dorsal force, FY ulnar-radial force, FZ distal-proximal force, MX moment about palmar-dorsal, MY moment about ulnar-radial, MZ moment about distal-proximal) using a simplified and efficient hFE method based on a single isotropic bone phase. Once we used a homogeneous mean density (shape model) and once the original heterogeneous density distribution (shape + density model). Using an analytical formulation, we minimized the deviation of the resulting strain tensors ε(x) to a hydrostatic compressive reference strain ε0, once for the 6 degrees of freedom (DOF) optimal (OPT) load case and for all individual 1 DOF load cases (FX, FY, FZ, MX, MY, MZ). All seven load cases were then extended in the nonlinear regime using the scaled displacements of the linear load cases as loading boundary conditions (MAX). We then compared the load cases and models for their objective function (OF) values, the stored energies and their ultimate strength using a specific torsor norm. Both shape and shape + density linear-optimized OPT models were dominated by a positive force in the z-direction (FZ). Transversal force DOFs were close to zero and mean moment DOFs were different depending on the model type. The inclusion of density distribution increased the influence and changed direction of MX and MY, while MZ was small in both models. The OPT load case had 12-15% lower objective function (OF) values than the FZ load case, depending on the model. Stored energies at the optimum were consistently 142-178% higher for the OPT load case than for the FZ load case. Differences in the nonlinear response maximum torsor norm ‖t‖ were heterogeneous, but consistently higher for OPT_MAX than FZ_MAX. We presented the proof of concept of an optimization procedure to estimate patient-specific loading conditions for hFE methods. In contrast to similar models, we included canonical load cases in all six DOFs and used a strain metric that favors hydrostatic compression. Based on a biomechanical analysis of the distal joint surfaces at the radius, the estimated load directions are plausible. For our dataset, the resulting OPT load case is close to the standard axial compression boundary conditions, usually used in HR-pQCT-based FE analysis today. But even using the present simplified hFE model, the optimized linear six DOF load case achieves a more homogeneous tissue loading and can absorb more than twice the energy than the standard uniaxial load case. The ultimate strength calculated with a torsor norm was consistently higher for the 6-DOF nonlinear model (OPT_MAX) than for the 1-DOF nonlinear uniaxial model (FZ_MAX). Defining patient-specific boundary conditions may decrease angulation errors during CT measurements and improve repeatability as well as reproducibility of bone stiffness and strength estimated by HR-pQCT-based hFE analysis. These results encourage the extension of the present method to anisotropic hFE models and their application to repeatability data sets to test the hypothesis of reduced angulation errors during measurement.

Unified validation of a refined second-generation HR-pQCT based homogenized finite element method to predict strength of the distal segments in radius and tibia.

INTRODUCTION: HR-pQCT based micro finite element (µFE) analyses are considered as "gold standard" for virtual biomechanical analyses of peripheral bone sites such as the distal segment of radius and tibia. An attractive alternative for clinical use is a homogenized finite element method (hFE) based on constitutive models, because of its much shorter evaluation times and modest computational resource requirements. Such hFE models have been experimentally validated for the distal segment of the radius, but neither for the distal segments of the tibia nor for both measurement sites together. Accordingly, the aim of the present study was to refine and experimentally validate an hFE processing pipeline for in vivo prediction of bone strength and stiffness at the distal segments of the radius and the tibia, using only one unified set of material properties. MATERIAL AND METHODS: An existing hFE analysis procedure was refined in several aspects: 1) to include a faster evaluation of material orientation based on the mean surface length (MSL) method, 2) to distinguish cortical and trabecular bone compartments with distinct material properties and 3) to directly superimpose material properties in mixed phase elements instead of densities. Based on an existing dataset of the distal segment of fresh-frozen radii (double sections 20.4 mm, n = 21) and a newly established dataset of the distal segment of fresh-frozen tibiae (triple sections, 30.6 mm, n = 25), a single set of material properties was calibrated on the radius dataset and validated on the tibia dataset by comparing hFE stiffness and ultimate load with respective experimental results, obtained by compressing the samples on a servo-hydraulic testing machine at a monotonic and quasi-static displacement rate up to failure. RESULTS: Using the identified set of material properties, the hFE-predicted stiffness and failure load were in excellent agreement with respective experimental results at both measurement sites (radius stiffness R2 = 0.93, slope = 1.00, intercept = 479 N/mm2/radius ultimate load: R2 = 0.97, slope = 1.00, intercept = 679 N; tibia stiffness R2 = 0.96, slope = 1.01, intercept = -1027 N/mm2/tibia ultimate load: R2 = 0.97, slope = 1.04, intercept = 394 N; combined dataset stiffness R2 = 0.95, slope = 1.01, intercept = -230 N/mm2/combined dataset ultimate load: R2 = 0.97, slope = 1.03, intercept = 495 N). DISCUSSION AND CONCLUSION: In conjunction with unified BV/TV calibration, the established hFE pipeline accurately predicts experimental stiffness and ultimate load of distal multi-sections at the radius and tibia. Processing time for non-linear analysis was substantially reduced compared to previous µFE and hFE methods but could be further minimized by estimating bone strength based on a fast and linear analysis like as is currently done with µ FE.

Bone Microarchitecture and Strength in Long-Standing Type 1 Diabetes.

Type 1 diabetes (T1DM) is associated with an increased fracture risk, specifically at nonvertebral sites. The influence of glycemic control and microvascular disease on skeletal health in long-standing T1DM remains largely unknown. We aimed to assess areal (aBMD) and volumetric bone mineral density (vBMD), bone microarchitecture, bone turnover, and estimated bone strength in patients with long-standing T1DM, defined as disease duration ≥25 years. We recruited 59 patients with T1DM (disease duration 37.7 ± 9.0 years; age 59.9 ± 9.9 years.; body mass index [BMI] 25.5 ± 3.7 kg/m2 ; 5-year median glycated hemoglobin [HbA1c] 7.1% [IQR 6.82-7.40]) and 77 nondiabetic controls. Dual-energy X-ray absorptiometry (DXA), high-resolution peripheral quantitative computed tomography (HRpQCT) at the ultradistal radius and tibia, and biochemical markers of bone turnover were assessed. Group comparisons were performed after adjustment for age, gender, and BMI. Patients with T1DM had lower aBMD at the hip (p < 0.001), distal radius (p = 0.01), lumbar spine (p = 0.04), and femoral neck (p = 0.05) as compared to controls. Cross-linked C-telopeptide (CTX), a marker of bone resorption, was significantly lower in T1DM (p = 0.005). At the distal radius there were no significant differences in vBMD and bone microarchitecture between both groups. In contrast, patients with T1DM had lower cortical thickness (estimate [95% confidence interval]: -0.14 [-0.24, -0.05], p < 0.01) and lower cortical vBMD (-28.66 [-54.38, -2.93], p = 0.03) at the ultradistal tibia. Bone strength and bone stiffness at the tibia, determined by homogenized finite element modeling, were significantly reduced in T1DM compared to controls. Both the altered cortical microarchitecture and decreased bone strength and stiffness were dependent on the presence of diabetic peripheral neuropathy. In addition to a reduced aBMD and decreased bone resorption, long-standing, well-controlled T1DM is associated with a cortical bone deficit at the ultradistal tibia with reduced bone strength and stiffness. Diabetic neuropathy was found to be a determinant of cortical bone structure and bone strength at the tibia, potentially contributing to the increased nonvertebral fracture risk. © 2022 The Authors. Journal of Bone and Mineral Research published by Wiley Periodicals LLC on behalf of American Society for Bone and Mineral Research (ASBMR).

Fabric-elasticity relationships of tibial trabecular bone are similar in osteogenesis imperfecta and healthy individuals.

Osteogenesis Imperfecta (OI) is an inherited form of bone fragility characterised by impaired synthesis of type I collagen, altered trabecular bone architecture and reduced bone mass. High resolution peripheral computed tomography (HR-pQCT) is a powerful method to investigate bone morphology at peripheral sites including the weight-bearing distal tibia. The resulting 3D reconstructions can be used as a basis of micro-finite element (FE) or homogenized finite element (hFE) models for bone strength estimation. The hFE scheme uses homogenized local bone volume fraction (BV/TV) and anisotropy information (fabric) to compute healthy bone strength within a reasonable computation time using fabric-elasticity relationships. However, it is unclear if these relationships quantified previously for healthy controls are valid for trabecular bone from OI patients. Thus, the aim of this study is to investigate fabric-elasticity relationships in OI trabecular bone compared to healthy controls. In the present study, the morphology of distal tibiae from 50 adults with OI were compared to 120 healthy controls using second generation HR-pQCT. Six cubic regions of interest (ROIs) were selected per individual in a common anatomical region. A first matching between OI and healthy control group was performed by selecting similar individuals to obtain identical mean and median age and sex distribution. It allowed us to perform a first morphometric analysis and compare the outcome with literature. Then, stiffness tensors of the ROIs were computed using µFE and multiple linear regressions were performed with the Zysset-Curnier orthotropic fabric-elasticity model. An initial fit was performed on both the OI group and the healthy control group using all extracted ROIs. Then, data was filtered according to a fixed threshold for a defined coefficient of variation (CV) assessing ROI heterogeneity and additional linear regressions were performed on these filtered data sets. These full and filtered data were in turn compared with previous results from µCT reconstructions obtained in other anatomical locations. Finally, the ROIs of both groups were matched according to their BV/TV and degree of anisotropy (DA). Linear regressions were performed using these matched data to detect statistical differences between the two groups. Compared to healthy controls, we found the OI samples to have significantly lower BV/TV and trabecular number (Tb.N.), significantly higher CV, trabecular separation (Tb.Sp.) and trabecular separation standard deviation (Tb.Sp.SD), but no differences in trabecular thickness (Tb.Th.). These results are in agreement with previous studies. The stiffnesses of highly heterogeneous ROIs were randomly lower with respect to the fabric-elasticity relationships, which reflects the limit of validity of the computational homogenisation methodology. This limitation does not challenge the fabric-elasticity relationship, which extrapolation to heterogeneous ROIs is probably reasonable but can simply not be evaluated with the employed homogenisation methodology. Moreover, due to their low BV/TV, the potential (unknown) errors on these heterogeneous ROIs would have negligible influence on whole bone stiffness in comparison to homogeneous ROIs which are orders of magnitude stiffer. The filtering of highly heterogeneous ROIs removed these low stiffness ROIs and led to similar correlation coefficients for both OI and healthy groups. Finally, the BV/TV and DA matched data revealed no significant differences in fabric-elasticity parameters between OI and healthy individuals. Moreover, the filtering step did not exclude a particular OI type. Compared to previous studies, the stiffness constants from the 61 µm resolution HR-pQCT ROIs were lower than for the 36 µm resolution µCT ROIs. In conclusion, OI trabecular bone of the distal tibia was shown to be significantly more heterogeneous and have a lower BV/TV than healthy controls. Despite the reduced linear regression parameters found for HR-pQCT images, the fabric-elasticity relationships between OI and healthy individuals are similar when the trabecular bone ROIs are sufficiently homogeneous to perform the computational stiffness analysis. Accordingly, the elastic properties used for FEA of healthy bones are also valid for OI bones.

Conventional finite element models estimate the strength of metastatic human vertebrae despite alterations of the bone's tissue and structure.

INTRODUCTION: Pathologic vertebral fractures are a major clinical concern in the management of cancer patients with metastatic spine disease. These fractures are a direct consequence of the effect of bone metastases on the anatomy and structure of the vertebral bone. The goals of this study were twofold. First, we evaluated the effect of lytic, blastic and mixed (both lytic and blastic) metastases on the bone structure, on its material properties, and on the overall vertebral strength. Second, we tested the ability of bone mineral content (BMC) measurements and standard FE methodologies to predict the strength of real metastatic vertebral bodies. METHODS: Fifty-seven vertebral bodies from eleven cadaver spines containing lytic, blastic, and mixed metastatic lesions from donors with breast, esophageal, kidney, lung, or prostate cancer were scanned using micro-computed tomography (µCT). Based on radiographic review, twelve vertebrae were selected for nanoindentation testing, while the remaining forty-five vertebrae were used for assessing their compressive strength. The µCT reconstruction was exploited to measure the vertebral BMC and to establish two finite element models. 1) a micro finite element (µFE) model derived at an image resolution of 24.5 µm and 2) homogenized FE (hFE) model derived at a resolution of 0.98 mm. Statistical analyses were conducted to measure the effect of the bone metastases on BV/TV, indentation modulus (Eit), ratio of plastic/total work (WPl/Wtot), and in vitro vertebral strength (Fexp). The predictive value of BMC, µFE stiffness, and hFE strength were evaluated against the in vitro measurements. RESULTS: Blastic vertebral bodies exhibit significantly higher BV/TV compared to the mixed (p = 0.0205) and lytic (p = 0.0216) vertebral bodies. No significant differences were found between lytic and mixed vertebrae (p = 0.7584). Blastic bone tissue exhibited a 5.8% lower median Eit (p< 0.001) and a 3.3% lower median Wpl/Wtot (p<0.001) compared to non-involved bone tissue. No significant differences were measured between lytic and non-involved bone tissues. Fexp ranged from 1.9 to 13.8 kN, was strongly associated with hFE strength (R2=0.78, p< 0.001) and moderately associated with BMC (R2=0.66, p< 0.001) and µFE stiffness (R2=0.66, p< 0.001), independently of the lesion type. DISCUSSION: Our findings show that tumour-induced osteoblastic metastases lead to slightly, but significantly lower bone tissue properties compared to controls, while osteolytic lesions appear to have a negligible impact. These effects may be attributed to the lower mineralization and woven nature of bone forming in blastic lesions whilst the material properties of bone in osteolytic vertebrae appeared little changed. The moderate association between BMC- and FE-based predictions to fracture strength suggest that vertebral strength is affected by the changes of bone mass induced by the metastatic lesions, rather than altered tissue properties. In a broader context, standard hFE approaches generated from CTs at clinical resolution are robust to the lesion type when predicting vertebral strength. These findings open the door for the development of FE-based prediction tools that overcomes the limitations of BMC in accounting for shape and size of the metastatic lesions. Such tools may help clinicians to decide whether a patient needs the prophylactic fixation of an impending fracture.

In vivo repeatability of homogenized finite element analysis based on multiple HR-pQCT sections for assessment of distal radius and tibia strength.

INTRODUCTION: Micro finite element analysis (µFE) is a widely applied tool in biomedical research for assessing in vivo mechanical properties of bone at measurement sites, including the ultra-distal radius and tibia. A finite element approach (hFE) based on homogenized constitutive models for trabecular bone offers an attractive alternative for clinical use, as it is computationally less expensive than traditional µFE. The respective patient-specific models for in vivo bone strength estimation are usually based on standard clinical high-resolution peripheral quantitative CT (HR-pQCT) measurements. They include a scan region of roughly 10 mm in height and are referred to as single-sections. It has been shown, that these small peripheral bone sections don't reliably cover the fracture line in Colles' fractures and therefore the weakest region at the radius. Recently introduced multiple section (multiple adjacent single-sections) measurements might improve the evaluation of bone strength, but little is known about the repeatability of hFE estimations in general, and especially for multiple section measurement protocols. Accordingly, the aim of the present work is to quantify repeatability of clinical in vivo bone strength measurement by hFE on multiple section HR-pQCT reconstructions at the distal radius and tibia. METHODS: Nineteen healthy Swiss women (43.6y ± 17.8y) and twenty men (48.2y ± 19.4y) were examined with HR-pQCT at 61 µm isotropic voxel resolution. Each subject was first scanned three times using a double-section (336 slices) at the distal radius and then three times using a triple-section (504 slices) at the distal tibia. The multiple section HR-pQCT reconstructions were graded for motion artefacts and non-linear hFE models (radius and tibia) and linear µFE models (only radius) were generated for estimation of stiffness and ultimate load. Then in vivo repeatability errors were computed in terms of root mean square coefficients of variation (CV). RESULTS: In vivo repeatability errors of non-linear hFE stiffness (S) and ultimate load (F) were significantly higher at the radius (S: 2.71% and F: 2.97%) compared to the tibia (S: 1.21%, F: 1.45%). Multiple section linear µFE at the radius resulted in substantially higher repeatability errors (S: 5.38% and F: 10.80%) compared to hFE. DISCUSSION/CONCLUSION: Repeatability errors of hFE outcomes based on multiple section measurements at the distal radius and tibia were generally lower compared to respective reported single-section µFE repeatability errors. Therefore, hFE is an attractive alternative to today's gold standard of µFE models and should especially be encouraged when analyzing multiple section measurements.