Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy

The method of considering a single loading condition in the study of stress/morphology relationships in trabecular bone is expanded to include the multiple loading conditions experienced by bone in vivo. The bone daily loading histories are characterized in terms of stress magnitudes or cyclic strain energy density and the number of loading cycles. Relationships between local bone apparent density and loading history are developed which assume that bone mass is adjusted in response to strength or energy considerations. Three different bone maintenance criteria are described which are formulated based upon: (1) continuum model effective stress, (2) continuum model fatigue damage accumulation density, and (3) bone tissue strain energy density. These approaches can be applied to predict variations in apparent density within bone and among bones. We show that all three criteria have similar mathematical forms and may be related to the density (or concentration) of bone strain energy which is transferred (dissipated) in the mineralized tissue. The loading history and energy transfer concepts developed here can be applied to many different situations of growth, functional adaptation, injury, and aging of connective tissues.     

Analysis of microstructural and mechanical alterations of trabecular bone in a simulated three-dimensional remodeling process

Bone remodeling is a complex dynamic process, which modulates both bone mass and bone microstructure. In addition to bone mass, bone microstructure is an important contributor to bone quality in osteoporosis and fragility fractures. However, the quantitative knowledge of evolution of three-dimensional (3D) trabecular microstructure in adaptation to the external forces is currently limited. In this study, a new 3D simulation method of remodeling of human trabecular bone was developed to quantitatively study the dynamic evolution of bone mass and trabecular microstructure in response to different external loading conditions. The morphological features of trabecular plate and rod, such as thickness and number density in different orientations were monitored during the remodeling process using a novel imaging analysis technique, namely Individual Trabecula Segmentation (ITS). We showed that the volume fraction and microstructures of trabecular bone including, trabecular type and orientation, were determined by the applied mechanical load. Particularly, the morphological parameters of trabecular plates were more sensitive to the applied load, indicating that they played the major role in the mechanical properties of the trabecular bone. Reducing the applied load caused severe microstructural deteriorations of trabecular bone, such as trabecular plate perforation, rod breakage, and a conversion from plates to rods.     

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

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

Large-scale microstructural simulation of load-adaptive bone remodeling in whole human vertebrae

Identification of individuals at risk of bone fractures remains challenging despite recent advances in bone strength assessment. In particular, the future degradation of the microstructure and load adaptation has been disregarded. Bone remodeling simulations have so far been restricted to small-volume samples. Here, we present a large-scale framework for predicting microstructural adaptation in whole human vertebrae. The load-adaptive bone remodeling simulations include estimations of appropriate bone loading of three load cases as boundary conditions with microfinite element analysis. Homeostatic adaptation of whole human vertebrae over a simulated period of 10 years is achieved with changes in bone volume fraction (BV/TV) of less than 5 %. Evaluation on subvolumes shows that simplifying boundary conditions reduces the ability of the system to maintain trabecular structures when keeping remodeling parameters unchanged. By rotating the loading direction, adaptation toward new loading conditions could be induced. This framework shows the possibility of using large-scale bone remodeling simulations toward a more accurate prediction of microstructural changes in whole human bones.     

The effects of misalignment during in vivo loading of bone: Techniques to detect the proximity of objects in three-dimensional models

Theories of mechanical adaptation of bone suggest that mechanical loading causes bone formation at discrete locations within bone microstructure experiencing the greatest mechanical stress/strain. Experimental testing of such theories requires in vivo loading experiments and high-resolution finite element models to determine the distribution of mechanical stresses. Finite element models of in vivo loading experiments typically assume idealized boundary conditions with applied load perfectly oriented on the bone, however small misalignments in load orientation during an in vivo experiment are unavoidable, and potentially confound the ability of finite element models to predict locations of bone formation at the scale of micrometers. Here we demonstrate two different three-dimensional spatial correlation methods to determine the effects of misalignment in load orientation on the locations of high mechanical stress/strain in the rodent tail loading model. We find that, in cancellous bone, the locations of tissue with high stress are maintained under reasonable misalignments in load orientation (p<0.01). In cortical bone, however, angular misalignments in the dorsal direction can alter the locations of high mechanical stress, but the locations of tissue with high stress are maintained under other misalignments (p<0.01). We conclude that, when using finite element models of the rodent tail loading model, small misalignments in loading orientation do not affect the predicted locations of high mechanical stress within cancellous bone.     

In vivo tibial stiffness is maintained by whole bone morphology and cross-sectional geometry in growing female mice

Whole bone morphology, cortical geometry, and tissue material properties modulate skeletal stresses and strains that in turn influence skeletal physiology and remodeling. Understanding how bone stiffness, the relationship between applied load and tissue strain, is regulated by developmental changes in bone structure and tissue material properties is important in implementing biophysical strategies for promoting healthy bone growth and preventing bone loss. The goal of this study was to relate developmental patterns of in vivo whole bone stiffness to whole bone morphology, cross-sectional geometry, and tissue properties using a mouse axial loading model. We measured in vivo tibial stiffness in three age groups (6, 10, 16 wk old) of female C57Bl/6 mice during cyclic tibial compression. Tibial stiffness was then related to cortical geometry, longitudinal bone curvature, and tissue mineral density using microcomputed tomography (microCT). Tibial stiffness and the stresses induced by axial compression were generally maintained from 6 to 16 wks of age. Growth-related increases in cortical cross-sectional geometry and longitudinal bone curvature had counteracting effects on induced bone stresses and, therefore, maintained tibial stiffness similarly with growth. Tissue mineral density increased slightly from 6 to 16 wks of age, and although the effects of this increase on tibial stiffness were not directly measured, its role in the modulation of whole bone stiffness was likely minor over the age range examined. Thus, whole bone morphology, as characterized by longitudinal curvature, along with cortical geometry, plays an important role in modulating bone stiffness during development and should be considered when evaluating and designing in vivo loading studies and biophysical skeletal therapies.     

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.     

Proximal Femoral Density Patterns are Consistent with Bicentric Joint Loads

We developed an alternate method for density-based load estimation and applied it to estimate hip joint load distributions for two femora. Two-dimensional finite element models were constructed from single energy quantitative computed tomography (QCT) data. Load estimation was performed using five loading regions on the femoral head. Within each loading region, individual nodal loads, normal to the local surface, were supplied as input to the load estimation. An optimization procedure independently adjusted individual nodal load magnitudes in each region, and the magnitudes of muscle forces on the greater trochanter, such that the applied tissue stimulus approached the reference stimulus throughout the model. Dominant estimated load resultant directions were generally consistent with published experimental data for loads during gait. The estimated loads also suggested that loads near the extremes of the articulating surface may be important (even required) for development and maintenance of normal bone architecture. Estimated load distributions within nearly all regions predicted bicentric loading patterns, which are consistent with observations of hip joint incongruity. Remodeling simulations with the estimated loads predicted density distributions with features qualitatively similar to the QCT data sets. This study illustrates how applications of density-based bone load estimation can improve understanding of dominant loading patterns in other bones and joints. The prediction of bicentric loading suggests a very fine level of local adaptation to details of joint loading.     

Proximal Femoral Density Patterns are Consistent with Bicentric Joint Loads

We developed an alternate method for density-based load estimation and applied it to estimate hip joint load distributions for two femora. Two-dimensional finite element models were constructed from single energy quantitative computed tomography (QCT) data. Load estimation was performed using five loading regions on the femoral head. Within each loading region, individual nodal loads, normal to the local surface, were supplied as input to the load estimation. An optimization procedure independently adjusted individual nodal load magnitudes in each region, and the magnitudes of muscle forces on the greater trochanter, such that the applied tissue stimulus approached the reference stimulus throughout the model. Dominant estimated load resultant directions were generally consistent with published experimental data for loads during gait. The estimated loads also suggested that loads near the extremes of the articulating surface may be important (even required) for development and maintenance of normal bone architecture. Estimated load distributions within nearly all regions predicted bicentric loading patterns, which are consistent with observations of hip joint incongruity. Remodeling simulations with the estimated loads predicted density distributions with features qualitatively similar to the QCT data sets. This study illustrates how applications of density-based bone load estimation can improve understanding of dominant loading patterns in other bones and joints. The prediction of bicentric loading suggests a very fine level of local adaptation to details of joint loading.     

Simulation of the behaviour of the L1 vertebra for different material properties and loading conditions

Three-dimensional finite element models of the thoracolumbar junction (T12–L2) and isolated L1 vertebra were developed to investigate the role of material properties and loading conditions on vertebral stresses and strains to predict fracture risk. The geometry of the vertebrae was obtained from computed tomography images. The isolated vertebra model included an L1 vertebra loaded through polymethylmethacrylate plates located at the top and bottom of the vertebra, and the segment model included T12 to L2 vertebrae and seven ligaments, fibrous intervertebral discs and facet joints. Each model was examined with both homogeneous and spatially varying bone tissue properties. Stresses and strains were compared for uniform compression and flexion. Including material heterogeneity remarkably reduced the stiffness of the isolated L1 vertebra and increased the magnitudes of the minimum principal strains and stresses in the mid-transverse section. The stress and strain distributions further changed when physiological loading was applied to the L1 vertebra. In the segment models, including heterogeneous material properties increased the magnitude of the minimum principal strain by 158% in the centre of the mid-transverse section. Overall, the inclusion of heterogeneity and physiological loading increased the magnitude of the strains up to 346% in flexion and 273% in compression.     

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses

Micro-finite element (\(\upmu \)FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone (“embedded configuration”). Thirteen cubic volumes (10.6 mm side length) were collected from \(\upmu \)CT reconstructions of human vertebrae and femora and converted into \(\upmu \)FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear \(\upmu \)FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3 mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (?10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.     

A contact algorithm for density-based load estimation

An algorithm, which includes contact interactions within a joint, has been developed to estimate the dominant loading patterns in joints based on the density distribution of bone. The algorithm is applied to the proximal femur of a chimpanzee, gorilla and grizzly bear and is compared to the results obtained in a companion paper that uses a non-contact (linear) version of the density-based load estimation method. Results from the contact algorithm are consistent with those from the linear method. While the contact algorithm is substantially more complex than the linear method, it has some added benefits. First, since contact between the two interacting surfaces is incorporated into the load estimation method, the pressure distributions selected by the method are more likely indicative of those found in vivo. Thus, the pressure distributions predicted by the algorithm are more consistent with the in vivo loads that were responsible for producing the given distribution of bone density. Additionally, the relative positions of the interacting bones are known for each pressure distribution selected by the algorithm. This should allow the pressure distributions to be related to specific types of activities. The ultimate goal is to develop a technique that can predict dominant joint loading patterns and relate these loading patterns to specific types of locomotion and/or activities.     

Improved prediction of the collagen fiber architecture in the aortic heart valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.     

Trabecular surface remodeling simulation for cancellous bone using microstructural voxel finite element models

A computational simulation method for three-dimensional trabecular surface remodeling was proposed, using voxel finite element models of cancellous bone, and was applied to the experimental data. In the simulation, the trabecular microstructure was modeled based on digital images, and its morphological changes due to surface movement at the trabecular level were directly expressed by removing/adding the voxel elements from/to the trabecular surface. A remodeling simulation at the single trabecular level under uniaxial compressive loading demonstrated smooth morphological changes even though the trabeculae were modeled with discrete voxel elements. Moreover, the trabecular axis rotated toward the loading direction with increasing stiffness, simulating functional adaptation to the applied load. In the remodeling simulation at the trabecular structural level, a cancellous bone cube was modeled using a digital image obtained by microcomputed tomography (microCT), and was uniaxially compressed. As a result, the apparent stiffness against the applied load increased by remodeling, in which the trabeculae reoriented to the loading direction. In addition, changes in the structural indices of the trabecular architecture coincided qualitatively with previously published experimental observations. Through these studies, it was demonstrated that the newly proposed voxel simulation technique enables us to simulate the trabecular surface remodeling and to compare the results obtained using this technique with the in vivo experimental data in the investigation of the adaptive bone remodeling phenomenon.     

High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone

The ability to predict trabecular failure using microstructure-based computational models would greatly facilitate study of trabecular structure–function relations, multiaxial strength, and tissue remodeling. We hypothesized that high-resolution finite element models of trabecular bone that include cortical-like strength asymmetry at the tissue level, could predict apparent level failure of trabecular bone for multiple loading modes. A bilinear constitutive model with asymmetric tissue yield strains in tension and compression was applied to simulate failure in high-resolution finite element models of seven bovine tibial specimens. Tissue modulus was reduced by 95% when tissue principal strains exceeded the tissue yield strains. Linear models were first calibrated for effective tissue modulus against specimen-specific experimental measures of apparent modulus, producing effective tissue moduli of (mean±S.D.) 18.7±3.4 GPa. Next, a parameter study was performed on a single specimen to estimate the tissue level tensile and compressive yield strains. These values, 0.60% strain in tension and 1.01% strain in compression, were then used in non-linear analyses of all seven specimens to predict failure for apparent tensile, compressive, and shear loading. When compared to apparent yield properties previously measured for the same type of bone, the model predictions of both the stresses and strains at failure were not statistically different for any loading case (p>0.15). Use of symmetric tissue strengths could not match the experimental data. These findings establish that, once effective tissue modulus is calibrated and uniform but asymmetric tissue failure strains are used, the resulting models can capture the apparent strength behavior to an outstanding level of accuracy. As such, these computational models have reached a level of fidelity that qualifies them as surrogates for destructive mechanical testing of real specimens.     

Periosteal and endosteal control of bone remodeling under torsional loading.

The shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled right-circular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected to an axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.     

Characterization of in vivo strain in the rat tibia during external application of a four-point bending load.

This paper describes a technique for characterizing strains and stresses induced in vivo in the rat tibia during application of an external four-point bending load. An external load was applied through the muscle and soft tissue with a four-point bending device, to induce strain in a 11 mm section of the right tibiae of ten adult female Sprague-Dawley rats. Induced strains were measured in vivo on the lateral surface of the tibia. Inter-rat difference, leg positioning and strain gage placement were evaluated as sources of variability of applied strains. Beam bending theory was used to predict externally induced in vivo strains. Finite element analysis was used to quantify the magnitude of shear stresses induced by this type of loading. There was a linear relationship between applied load and induced in vivo strains. In vivo strains induced by external loading were linearly correlated (R2 = 0.87) with the strains calculated using beam bending theory. The finite element analysis predicted shear stresses at less than 10% of the longitudinal stresses resulting from four-point bending. Strains predicted along the tibia by finite element analysis and beam bending theory were well-correlated. Inter-rat variability due to tibia size and shape difference was the most important source of variation in induced strain (CV = 21.6%). Leg positioning was less important (CV = 9.5%).     

The fate of mechanically induced cartilage in an unloaded environment.

According to mechanobiologic theories, persistent intermittent mechanical stimulation is required to maintain differentiated cartilage. In a rat model for bone repair, we studied the fate of mechanically induced cartilage after unloading. In three groups of rats, regenerating mesenchymal tissue was submitted to different loading conditions in bone chambers. Two groups were immediately killed after loading periods of 3 or 6 weeks (the 3-group and the 6-group). The third group was loaded for 3 weeks and then kept unloaded for another 3 weeks (the (3 + 3)-group). Cartilage was found in all loaded groups. Without loading, cartilage does not appear in this model. In the 3-group there was no clear ongoing endochondral ossification, the 6-group showed ossification in 2 out of 5 cartilage containing specimens, and in the (3 + 3)-group all cartilage was undergoing ossification. These results suggest a tendency of the cartilage to be maintained also under unloaded conditions until it is reached by bone that can replace it through endochondral ossification.Additional measurements showed less amount of new bone in the loaded specimens. In most of the loaded specimens in the 3-group, necrotic bone fragments were seen embedded in the fibrous tissue layer close to the loading piston, indicating that bone tissue had been resorbed due to the hydrostatic compressive load. In some specimens, a continuous cartilage layer covered the end of the specimen and seemed to protect the underlying bone from pressure-induced resorption. We suggest that one of the functions of the cartilage forming in the compressive loaded parts of a bone callus is to protect the surrounding bone callus from pressure-induced fluid flow leading to resorption.     

Mechanical implications of pneumatic neck vertebrae in sauropod dinosaurs

The pre-sacral vertebrae of most sauropod dinosaurs were surrounded by interconnected, air-filled diverticula, penetrating into the bones and creating an intricate internal cavity system within the vertebrae. Computational finite-element models of two sauropod cervical vertebrae now demonstrate the mechanical reason for vertebral pneumaticity. The analyses show that the structure of the cervical vertebrae leads to an even distribution of all occurring stress fields along the vertebrae, concentrated mainly on their external surface and the vertebral laminae. The regions between vertebral laminae and the interior part of the vertebral body including thin bony struts and septa are mostly unloaded and pneumatic structures are positioned in these regions of minimal stress. The morphology of sauropod cervical vertebrae was influenced by strongly segmented axial neck muscles, which require only small attachment areas on each vertebra, and pneumatic epithelia that are able to resorb bone that is not mechanically loaded. The interaction of these soft tissues with the bony tissue of the vertebrae produced lightweight, air-filled vertebrae in which most stresses were borne by the external cortical bone. Cervical pneumaticity was therefore an important prerequisite for neck enlargement in sauropods. Thus, we expect that vertebral pneumaticity in other parts of the body to have a similar role in enabling gigantism.     

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.