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Research Papers

Pelvic Construct Prediction of Trabecular and Cortical Bone Structural Architecture OPEN ACCESS

[+] Author and Article Information
Dan T. Zaharie

The Royal British Legion Centre for
Blast Injury Studies,
Imperial College London,
London SW7 2AZ, UK;
Structural Biomechanics,
Department of Civil and
Environmental Engineering,
Imperial College London,
Skempton Building, South Kensington Campus,
London SW7 2AZ, UK
e-mail: dan.zaharie10@imperial.ac.uk

Andrew T. M. Phillips

The Royal British Legion Centre for
Blast Injury Studies,
Imperial College London,
London SW7 2AZ, UK;
Structural Biomechanics,
Department of Civil and
Environmental Engineering,
Imperial College London,
Skempton Building, South Kensington Campus,
London SW7 2AZ, UK
e-mail: andrew.phillips@imperial.ac.uk

1Corresponding author.

Manuscript received December 5, 2016; final manuscript received November 10, 2017; published online May 24, 2018. Assoc. Editor: Steven D. Abramowitch.

J Biomech Eng 140(9), 091001 (May 24, 2018) (11 pages) Paper No: BIO-16-1497; doi: 10.1115/1.4039894 History: Received December 05, 2016; Revised November 10, 2017

The pelvic construct is an important part of the body as it facilitates the transfer of upper body weight to the lower limbs and protects a number of organs and vessels in the lower abdomen. In addition, the importance of the pelvis is highlighted by the high mortality rates associated with pelvic trauma. This study presents a mesoscale structural model of the pelvic construct and the joints and ligaments associated with it. Shell elements were used to model cortical bone, while truss elements were used to model trabecular bone and the ligaments and joints. The finite element (FE) model was subjected to an iterative optimization process based on a strain-driven bone adaptation algorithm. The bone model was adapted to a number of common daily living activities (walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit) by applying onto it joint and muscle loads derived using a musculoskeletal modeling framework. The cortical thickness distribution and the trabecular architecture of the adapted model were compared qualitatively with computed tomography (CT) scans and models developed in previous studies, showing good agreement. The sensitivity of the model to changes in material properties of the ligaments and joint cartilage and changes in parameters related to the adaptation algorithm was assessed. Changes to the target strain had the largest effect on predicted total bone volumes. The model showed low sensitivity to changes in all other parameters. The minimum and maximum principal strains predicted by the structural model compared to a continuum CT-derived model in response to a common test loading scenario showed good agreement with correlation coefficients of 0.813 and 0.809, respectively. The developed structural model enables a number of applications such as fracture modeling, design, and additive manufacturing of frangible surrogates.

FIGURES IN THIS ARTICLE
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The pelvic construct is the region of transition between the trunk and the lower limbs and plays multiple roles, from protecting the organs and vessels in the lower abdomen to distributing the upper body weight to the lower limbs. Furthermore, it facilitates the transfer of forces between the lower limbs and the upper body during activities such as walking, running, and stair climbing, withstanding loads at the hip joint up to around six times body weight for running [13].

The structure of bone has been studied extensively since the 19th century, with an emphasis on the femur. It was hypothesized that the trabecular architecture of the proximal femur follows a set of trajectories corresponding to the compressive and tensile stresses [46]. The general consensus is that bone continuously updates its structure in response to the mechanical environment it is exposed to [7] until it reaches a state in which it is able to withstand the forces acting on it with a minimum amount of material [8].

As opposed to long bones, which have thick cortical shafts to optimally resist the forces and moments applied on them [911], the pelvis can be viewed as a sandwich structure consisting mainly of trabecular bone encased within a thin layer of cortical bone [1214].

Continuum Modeling Approaches.

Finite element (FE) modeling is a common approach to investigate the behavior of pelvic bone in a wide range of scenarios. Early pelvic models were either two-dimensional [15,16], axisymmetric [17,18], or used simplified geometries [1921]. A common approach over the past three decades has been to develop models based on geometries and material properties extracted from medical imaging data (e.g., computed tomography (CT)) and to build the model using continuum solid elements.

The first computational model developed using subject specific geometry and material properties from a CT scan was presented by Dalstra et al. [22]. The model was validated against experimentally obtained strains from a cadaveric pelvis, although the specimen used for the experiment was different from the one used for generating the FE mesh and material properties. Similarly, Anderson et al. [14] developed an FE model of the pelvis based on CT data with location-dependent cortical thickness and validated it against experimental results obtained from the same specimen. In both studies, specimens were fixed at the superior part of the ilium with the load applied on the acetabulum.

The majority of published models only partially represent the pelvic ring, focusing primarily on either the hip joint [1921,2325] or the hemi-pelvis [14,21,22,2628]. For the purpose of investigating the structural response of the pelvis in a variety of loading conditions, the development of a complete model of the pelvic construct is considered to provide an improved understanding of its behavior in complex loading scenarios, compared to models which represent an isolated part of the construct. An important aspect of the pelvic girdle is the ligaments holding the three components of the pelvis (sacrum and two pelvic innominates) together, the sacroiliac ligaments, and the pubic ligament. A number of other ligaments (sacrospinous, sacrotuberous, iliolumbar, and inguinal) provide additional stability to the pelvic ring under load [29]. Phillips et al. [30] developed a model of the pelvis including both hemipelves, the ligaments mentioned above, a structure representing the sacrum, and the muscles attached to the pelvis in an explicit manner. They observed that the use of ligaments and muscles in the model as opposed to fixed boundary conditions resulted in a more even stress distribution across the bone. A number of complete models have been developed and used to investigate pelvic injury mechanisms [3134].

Structural Modeling Approaches.

An alternative approach to continuum FE modeling is to look at bone from a structural perspective and represent its architecture using a combination of idealized elements such as trusses, beams, and shells. An important aspect of this method is the increased computational efficiency [35], while maintaining the ability to capture the overall structural behavior of bone [11]. Furthermore, structural modeling enables the user to visualize and investigate the trajectories formed by trabecular bone.

This study presents the development of a predictive mesoscale structural model of the pelvic girdle in which the pubic and sacroiliac joints, as well as the ligaments forming the pelvic ring, are included. The development of the model consists of adapting its structure to a physiological loading regime. Phillips et al. [11,36] have previously employed this approach to model the femur, showing promising results in terms of bone architecture and structural response to loading of the bone.

The structure of the trabecular and cortical bone was optimized based on the hypothesis that bone adapts its structure to resist the loading environment caused by daily physical activities. The loading applied on the model was obtained from a musculoskeletal model, validated at the hip joint [3739]. In turn, the response of the finite element model to this loading regime was used to optimize the bone structure using a strain-driven algorithm [11]. The optimized model was expected to represent the architecture and structural response to physiological loading (joint contact forces (JCF), muscles forces, and inertial loading) of a healthy pelvis.

Musculoskeletal Modeling.

To obtain a structural representation of the pelvis, the base model was subjected to physiological loads derived from musculoskeletal simulations of the five most frequent daily activities [40]: walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit. The bilateral musculoskeletal model was based on an ipsilateral model developed by Modenese et al. and validated at the hip joint [39]. The model was based on an anatomical dataset published by Horsman et al. [41] and implemented in OpenSim [42]. It includes thirteen body segments: pelvis, two femurs, two patellas, two tibias, two hindfeet, two midfeet, and two sets of phalanges (Fig. 1). The segments are connected by nine joints: pelvis-ground (6 degrees-of-freedom (DOFs)), hip joints (3 DOF each), knee joints (1 DOF each), patellofemoral joints (1 DOF each), and ankle joints (1 DOF each). The local reference systems of the segments were defined in accordance with the recommendations of the International Society of Biomechanics [43]. The muscle attachment points on the right side of the model were the same as in Ref. [37], while those on the left side were obtained by mirroring the right side points with respect to the sagittal plane. Thus, the total number of muscles and actuators in the bilateral model was 76 and 326, respectively.

Gait data for walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit activities were collected from a male volunteer (age: 23 yr, weight: 93 kg, and height: 188 cm). The activities chosen were identified as the most common daily physical activities [40]. The 59 markers positioned on bony landmarks were tracked using a Vicon system (Oxford Metrics, Oxford, UK) equipped with 10 infrared cameras. Three force plates (Type 9286BA, sampling rate 1000 Hz, and Kistler Instruments Ltd., Hook, UK) were used to measure the ground reaction forces (GRFs). The force plates were positioned to form a walkway which was used to record GRFs during walking (speed: 1.34 m/s, stride length: 0.64 m, and cadence: 115.54 steps/min). To measure stair ascent and descent GRFs, the three force plates were positioned on a staircase (step height 15 cm and step depth 25 cm). A stool with a height of 52.5 cm from the floor instrumented with one force plate along with two force plates on the floor was used to record GRFs during sit-to-stand and stand-to-sit. All data were collected in the Human Biodynamics Lab in the Imperial College Research Labs at Charing Cross Hospital and processed using Vicon Nexus and the Biomechanical ToolKit [44].

The segments of the model were scaled to the anatomical dimensions of the volunteer by calculating the ratios of the lengths between experimental and virtual markers, with the inertial properties of the segments being updated according to the regression equations of Dumas et al. [45]. An inverse kinematics approach [46] was used to calculate the joint angles describing the activities recorded. The muscle forces were estimated using static optimization. The optimization problem was solved by minimizing the sum of squared muscle activations for each frame of the kinematics under the constraints of joint moment equilibrium and physiological limits of the muscle forces [3739]. The hip joint contact forces were calculated using the JointReaction tool available in OpenSim [47]. All simulations were performed in OpenSim (Version 3.3) [42]. For each activity, the loads applied on the pelvis throughout the duration of the gait cycle (hip joint reaction forces and muscle forces) were determined to be applied to the FE model. The muscle insertion points on the pelvis along with the direction and magnitude of each muscle force were extracted using the MuscleForceDirection (v1.0) plugin [11,48].

Finite Element Modeling
Base Model.

A CT scan (399 × 3 mm thick slices, 512 × 512 pixels, 0.91 mm/pixel) of a cadaveric pelvis and lower limbs (Male, age: 55, weight: 94.3 kg, and height: 188 cm) provided by the Royal British Legion Centre for Blast Injury Studies of Imperial College London was processed in mimics (Materialise, Leuven, Belgium) to generate a volumetric mesh of the pelvic girdle, composed of 377,362 four-noded tetrahedral elements with an average edge length of 3.76 mm. The external surface of the mesh was used to define three-noded linear triangular shell elements, representing cortical bone. These were arbitrarily assigned an initial thickness of 0.1 mm. The internal nodes were used to define two-noded truss elements connecting each node to its nearest 16 neighboring nodes. The trusses were arbitrarily assigned an initial radius of 0.1 mm, with the network representing trabecular bone. The initial minimum connectivity of each node was 16, with a maximum of 62 and a mean value of 21.32. Previous work in the research group has shown that a nodal connectivity of 16 provides a sufficient range of element directionalities to enable trabecular trajectories to develop during adaptation [49]. In addition, a mesh sensitivity study, described in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection, was performed to assess the sensitivity of the adaptation algorithm to mesh refinement.

The base mesh consisted of 33,466 cortical shell elements (12,601 for each innominate and 8264 for the sacrum) and 588,028 trabecular truss elements (247,268 for each innominate and 93,492 for the sacrum) (Fig. 2). The initial structural model was generated from the tetrahedral mesh using matlab (The MathWorks, Inc., Natick, MA). The shell and truss elements were assigned linear isotropic material properties with Young's modulus and Poisson's ratio values of E = 18,000 MPa and ν = 0.3, respectively [14,30,50].

Ligaments.

In addition to the pelvic bones, five ligaments were added to the model as bundles of truss elements: sacroiliac, pubic, sacrospinous, sacrotuberous, and inguinal (Fig. 3). The ligaments are required as they facilitate the load transfer between the three bones of the pelvic construct. The anatomical attachment areas of each ligament were visually defined on the model based on descriptions in the literature [30,5153].

The behavior of ligaments is complex and models which aim to capture their biphasic and viscoelastic nature focus on soft tissue behavior or joint mechanics [5456], rather than the behavior of the pelvis as a whole. Ligaments included in the previous pelvic models were either assigned linear elastic material properties [30,31] or modeled as hyperelastic components [33,34,57,58].

The ligaments and cartilage at the joints were assigned linear elastic material properties and were modeled using truss elements by connecting each surface node on an origin area to the closest node on the corresponding insertion area. The material properties of each ligament were derived from previously published data on ligament stiffnesses [30]. Young's moduli were derived from the stiffnesses, with E being Young's modulus, k the stiffness of the ligament, l the length, and A the origination area of the ligament Display Formula

(1)E=klA
Display Formula
(2)r=AπNl

The radii of each truss element, r, were calculated based on the origination area of the ligaments, A, and the number of truss elements, Nl, forming each ligament (Eq. (2)). As ligaments do not have compressive stiffness, they were assigned zero stiffness in compression and were overlaid with identical sets of elements given a low Young's modulus value of 0.1 MPa to ensure numerical stability. Additional sets of trusses were added at the sacroiliac and pubic joints and assigned compressive stiffness to account for the presence of cartilage. The tensile and compressive material properties assigned to the ligaments and cartilage are shown in Tables 1 and 2, respectively.

Loading and Boundary Conditions.

The muscle forces obtained from the musculoskeletal model were applied as point loads on the closest three nodes to each muscle insertion point [37]. The hip joint contact forces and the inertial loads are calculated at the joint centers and the center of mass of the pelvic segment. In addition, for sit-to-stand and stand-to-sit, the reaction load from the seat was applied at the inferior ischium. To spread these loads over the corresponding bone surface, they were applied using “load applicators” and an “inertia applicator,” as they allow for a reduction in computational time in comparison to modeling contact at each joint. The load applicators were built by first projecting each node on the surface of the hip joint along the direction defined by the point and the joint center four times with a distance of 2 mm between them. Subsequently, four layers of six-noded linear continuum wedge elements were created using the newly generated nodes and the initial surface nodes. The overall structure was taken to represent the bone-cartilage-bone interface at the hip joint. Thus, the bottom two layers (closest to the bone surface) were assigned E = 10 MPa and ν = 0.49, representative of cartilage. The outermost two layers were assigned stiffnesses of 10 MPa (ν = 0.3) and 500 MPa (ν = 0.3), respectively [49]. The loads derived from the musculoskeletal model were applied to the outside node closest to the direction of the load vector for each load case (Fig. 4(a)). The role of the “inertia applicator” is to spread the inertial load of the pelvis over the whole construct. It is designed by linking the center of mass of the pelvis to every node of the model using truss elements with a radius of 0.1 mm and low stiffness (E = 0.1 MPa, ν = 0.3) to prevent over constraining it, with the inertial load vector applied at the center of mass.

Similar to the load applicators, four layers of wedge elements were constructed on the top of the sacrum, representing the L5S1 interface between the lumbar spine and the sacrum. The two layers closer to bone were assigned Young's modulus of 10 MPa (ν = 0.49) to represent cartilage, while the outermost two were assigned stiffnesses of 10 MPa (ν = 0.3) and E = 500 MPa (ν = 0.3). The boundary conditions of the model were applied at the top of the sacrum by constraining the nodes on the outermost layer of the construct in the three translational DOF. This was done to avoid fixing nodes on the bone, which would potentially lead to stress concentrations developing over the surface of the pelvis.

The load cases obtained from the musculoskeletal model were subsampled to maintain computational efficiency. The subset of frames was selected such that the difference between the integrated hip JCF for the full set of frames and the selected set of frames was lower than 5% (Fig. 5). Data reported by Bergmann et al. [3] for the same activities are included for comparison. The derived JCFs show a reasonable match for stair ascent and descent. During walking, the second peak was higher compared to the range reported by Bergmann et al. [3]. The selected load cases were applied in consecutive steps to the FE model.

Bone Adaptation Algorithm.

Based on the Mechanostat principle [8], the base structural pelvis model was subjected iteratively to the load cases obtained from the musculoskeletal model for the activities mentioned in Sec. 2.1 (methodology adapted from Ref. [11]). At each iteration, the thickness of each shell element and the cross-sectional area of each truss element were adjusted according to a strain criterion. The iterative process was implemented using matlab (The MathWorks, Inc.) and python (Python Software Foundation, Beaverton, OR) scripts, and the successive FE models were run using the abaqus/standard solver (Dassault Systèmes Simulia, Johnston, RI) until convergence was achieved (Fig. 6).

For each iteration i, the absolute maximum normal strain value for the jth truss element and the jth shell element over all load cases (λ = 1,…, n) was defined using Eqs. (3) and (4), respectively, Display Formula

(3)|εi,j|max=max(|ε11,j,λ|)
with ε11,j,λ being the axial strain in the jth truss element for load case λDisplay Formula
(4)|εi,j|max=max(|εmax,j,λt|,|εmin,j,λt|,|εmax,j,λb|,|εmin,j,λb|)
with εmax,j,λt,εmin,j,λt,εmax,j,λb,εmin,j,λb being the maximum and the minimum principal strains in the top and bottom surfaces of the jth shell element for load case λ.

The strain ranges associated with, bone resorption, the lazy zone and bone apposition [8,36,59] are given in the following equation: Display Formula

(5)ϕi,j={1,if0|εi,j|max<1000με(Boneresorption)0,if1000|εi,j|max1500με(Lazyzone)1,if|εi,j|max>1500με(Boneapposition)

For the subsequent iteration (i + 1), the cross-sectional area of the jth truss element (Ai+1,j) was adjusted based on the value of ϕi,j (Eq. (6)). The thickness of the jth shell element (Ti+1,j) was modified using Eq. (7). The target strain, εt, was given a value of 1250 με, corresponding to the middle of the lazy zone Display Formula

(6)Ai+1,j={Ai,j|εi,j|maxεt,ifϕi,j=1Ai,j,ifϕi,j1
Display Formula
(7)Ti+1,j={Ti,j2(1+|εi,j|maxεt),ifϕi,j=1Ti,j,ifϕi,j1

The adaptation equations for the two types of elements were defined such that trabecular bone adapts in preference to cortical bone. This is important in particular during the first few iterations as it prevents the oscillation of the shell elements thicknesses. In addition, truss elements with a strain lower than 250 με were considered to be in the dead zone.

To increase computational efficiency, the thicknesses of the shell elements were linearly discretized into 256 categories. Based on CT scans, cortical thickness throughout the pelvis has been reported to vary between 0.5 and 4 mm [14,22,60]. The thickness range for the shell elements in this model was set between 0.1 mm and 5 mm as for a thickness lower than 0.5 mm the accuracy of clinical CT scanners decreases, meaning that there could be areas with a cortical thickness lower than 0.5 mm which would not be captured due to the image resolution [61].

Similarly, the cross-sectional areas of the truss elements were linearly discretized in 255 categories. The range for the truss elements was set between π(0.1)2 mm2 and π(2)2 mm2, which is considered reasonable to characterize trabecular bone at mesoscale level [11,62]. In addition, a 256th category with a cross-sectional area of π(0.001)2 mm2 was added to allow for effective removal of the elements in the dead zone while maintaining numerical stability. The elements in the dead zone were assigned a radius of 1 μm, making their stiffness contribution to the model negligible. This enabled regeneration of the elements in the dead zone as they could be reassigned in one of the other 255 categories at a later iteration.

Sensitivity Study.

A sensitivity study was performed to assess the effects of varying model parameters on its structure. The parameters which were varied were Young's modulus of each ligament and joint cartilage, the initial assigned thickness of the cortical shell elements, the target strain, and the number of discrete categories used for section definitions. Each case model was iteratively adapted until convergence was achieved. The final structure and trabecular and cortical bone volumes were compared to the baseline model. The stiffness of each pair of ligaments and joints in turn was increased and decreased by 50%. The initial assigned thickness of the cortical shell elements was increased to 1 mm. The target strain was given values of 1000 με and 1500 με, the two ends of the lazy zone. Finally, 128 and 512 discrete categories for section definitions were assessed.

To ensure that the mesh resolution of the pelvis model was appropriate, a mesh sensitivity study was performed to assess the effect of mesh resolution on the outcome of the adaptation process. Given that a mesh of the full pelvis geometry at a smaller scale than the current one would have a very high computational cost, an analogous structural model of a cantilever beam was developed. Three different meshes were tested. The baseline mesh was built using the same specifications as for the pelvis model, while fine and coarse meshes were built by varying the maximum tetrahedral edge length by −50% and +50%, respectively. The three models were then subjected to the adaptation process with for a bending load case. The bone volumes of the converged structures were compared to the baseline model. Further details of the sensitivity study are given in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

Comparison With Computed Tomography Scan Derived Model.

The tetrahedral mesh of the right hemipelvis was assigned varying material properties derived from the CT data. The response of the resulting FE model and the corresponding structural hemipelvis to a vertical load applied at the acetabulum were compared. Details regarding the development of the CT scan derived model and the loading scenario applied to both models are made available in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

Converged Model Structure.

Cortical thickness was highest at the superior sacrum, along the gluteal surface, around the sacroiliac joint, superior pubic ramus, posterior iliac crest, and greater sciatic notch. Cortical bone had a thickness between 0.1 mm and 0.5 mm at the ischium, acetabulum, pubic tubercle, iliac fossa, and posterior superior iliac spine (Fig. 7). The results for the cortical thickness distribution across the hip bone compare well with the study published by Anderson et al. [14].

Clusters of elements with large radii were found at the superior sacrum, supra acetabular region, pubic tubercle, and greater sciatic notch. High trabecular density in these regions was also reported in a previous study [63]. Regions with a small number of active elements were found at the iliac fossa, ischial tuberosity, and inferior sacrum (Fig. 7). Distinct trajectories were formed in the supra acetabular region (Figs. 8(a) and 8(c)), around the sacroiliac joint extending from the ilium to the sacrum (Figs. 8(d)8(f)), and along the direction between the two locations mentioned earlier (Figs. 8(b) and 8(c)). Similar trajectories were found in a study published by Machiarelli et al. [64], where the authors observed a distinct walking-related trabecular architecture formed at the ilium. The trabecular trajectories reported in the paper were identified in the model as the areas where clusters of truss elements developed the most (Fig. 9).

To obtain a more comprehensive view of the optimized bone architecture, a number of transverse slices of the model were compared to the corresponding CT scans used to define the initial surface geometry (Fig. 10). Across the acetabulum, the model predicted a thickening of the cortex at the ascending pubis ramus and a thin cortex along the acetabular notch, which is in agreement with the CT scan (Figs. 10(a) and 10(b)). Another similarity can be found along the inferior gluteal line, where the model showed a distinct thickening of the cortex, which is also present in the scan (Figs. 10(g) and 10(h)). The region around the sacroiliac joint compares well with the scan for both the sacrum and the hip bones (Figs. 10(e) and 10(f)). The main discrepancies in terms of cortical thickness between the model and scan can be seen between the anterior superior iliac spine and the anterior inferior iliac spine (Figs. 10(c)10(h)), toward the greater sciatic notch (Figs. 10(g) and 10(h)) and ischial body (Figs. 10(a) and 10(b)). On the other hand, the areas mentioned have a higher trabecular density compared to the CT scan, meaning that the stiffness of the region could still be considered similar to the scan.

In terms of trabecular architecture, a number of trajectories were clearly formed. In the region surrounding the sacroiliac joint, thick trabecular elements were formed following trajectories which intersect both hip bones and the sacrum (Figs. 8(e) and 8(f)). As a consequence, the highest trabecular density in the sacrum was found on the load-bearing S1 segment (Figs. 8(d) and 8(e)), which is in agreement with previous findings [65,66]. Another region with high trabecular density was identified above the acetabulum (Figs. 8(a)8(c)). Given that the loads are transmitted from the hip joint towards the lumbar spine through the sacroiliac joint and sacrum, the presence of these trabecular trajectories seems justified.

Influence of Activities.

Walking was primarily responsible for the thickening of the cortex along the ischial spine, the inferior ramus of the pubis and the posterior part of the iliac fossa. The cortex development between the anterior inferior iliac spine and the iliac crest and along the superior pubic ramus was primarily influenced by stair ascent, while stair descent was the main factor in the thickening of the cortex across the sacrum, between the anterior and the posterior gluteal lines, at the ischium, and along the posterior superior iliac spine. The increased trabecular structure present at the upper sacrum was influenced by both stair ascent and descent (Fig. 11). Walking was mainly responsible for the appearance of trabecular structures along the superior pubic ramus, around the acetabulum, and at the anterior inferior iliac spine and anterior gluteal line. The asymmetric distribution of activity influence can be attributed to the subject specific geometry and gait used in the musculoskeletal model. Additional figures illustrating the level of influence each activity had on the regions of the pelvis can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

Sensitivity Study.

To assess the sensitivity of the ligaments and cartilage, the trabecular and cortical volumes obtained adapting each case model were compared to the baseline model. For all cases, the model showed low sensitivity, with the largest difference in trabecular bone volume being 0.86% for the case when the stiffnesses of all ligaments and cartilage were increased by 50%. The largest difference in cortical bone volume was found when the stiffnesses of all ligaments and cartilage were reduced by 50%, with a value of 3.59%. Data for all cases can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection. Similarly, for the sensitivity cases regarding adaptation related parameters, the total trabecular and cortical volumes were compared to the baseline model. The models with a higher initial thickness and different numbers of categories for section definitions predicted a maximum difference in trabecular bone volume of 2.18% and a maximum difference in cortical bone volume of 2.27%. The model was most sensitive to the target strain value. For a target strain of 1000 με, the model predicted a gain of trabecular and cortical bone of 32.70% and 18.51%, respectively. For a target strain of 1500 με, the model predicted a loss of trabecular and cortical bone of 19.28% and 13.88%, respectively. The complete data are made available in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

The sensitivity of the adaptation algorithm to mesh refinement was assessed by comparing the total trabecular and cortical volumes obtained from the adaptation process of the two sensitivity cases with the baseline cantilever beam model. The finer mesh predicted an increase of 0.91% in trabecular bone volume and 2.93% in cortical bone volume, while the coarse mesh predicted a decrease of 6.36% in trabecular bone volume and 1.82% in cortical bone volume. The distribution of the cortical thicknesses and trabecular radii, and overall structure were consistent across all three models and are illustrated in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

Comparison With Computed Tomography Derived Model.

The maximum and minimum principal strains across the surface of the structural model and continuum model derived from the CT data were compared (Fig. 12). The coefficients of correlation for both minimum (0.813) and maximum (0.809) principal strains show a fairly good agreement between the strains predicted by the structural model and CT-derived model. A detailed diagram showing the surface strains across both models and Bland–Altman plots [67] showing the 95% confidence intervals can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.

The structural model of the pelvis composed of 691,852 elements and 235,914 design variables was found to have a run time of 153 s on a workstation with one Intel Xeon E5-2630 v2 2.60 GHz and 64 GB of RAM. The run time for adapting the model to the full loading regime was around 24 h. A direct comparison to another adaptive model of the pelvis cannot be made due to lack of availability in the literature. Continuum models of the pelvis were developed from CT scans and used in simulating loading scenarios with the number of elements varying between 190,000 and 1.2 × 106 [14,34,58], with the run times not being reported. A potentially better comparison can be made with adaptive models of the femur. Boyle and Kim [68] developed a microscale model of the proximal femur (23.3 × 106 elements) reporting a runtime of around 343 h for a single combined load case. Phillips et al. [11] reported a run time of 52 s for their mesoscale structural model of the femur (229,113 elements) and a duration of 10 h for the full adaptation regime. Although the current model has not been implemented at the microscale, it is reasonable to conclude that it has a low computational cost, comparable to a similar computationally efficient structural model of the femur.

Given that a direct validation of the model was not possible, sensitivity studies were performed to assess the effects of material properties and adaptation parameters on the model. The model was not found to be particularly sensitive to changes in material properties of the joints and ligaments or to changes in initial element thickness and number of categories used for section definitions. The most sensitive parameter was found to be the target strain, as changing its value led to considerable changes in trabecular and cortical bone volume, respectively. This result was expected as the main driver of the adaptation algorithm is strain. Overall, the structural model and the adaptation algorithm were found to be robust in terms of predicting cortical and trabecular architecture caused by a loading environment associated with common daily activities.

In addition to the sensitivity study, the response of the converged structural model to a loading scenario was compared to an analogous continuum model with varying material properties derived from the CT data. Although the comparison showed that the two models were in fairly good agreement, the correlation was not found to be as strong as reported in the previous studies which have compared continuum models with experimental data [14,58]. However, the sample size used for comparison in the studies mentioned was limited to the number of strain gauges used, meaning that a comparison between strains across the whole bone surface was not possible. Differences between the structural model and the CT-derived model could be attributed to trabecular bone in the CT-derived model having a minimum stiffness of 50 MPa based on the conversion from the Hounsfield units values, while corresponding regions in the structural model had very sparse trabecular architecture.

The limitations of the musculoskeletal and finite element modeling methodologies presented in this study must be acknowledged. The muscle actuators contained in the musculoskeletal model do not take into account contraction dynamics and force–length–velocity relationships [37]. Although it was shown that these assumptions do not influence the predicted outcome for walking [69], they might have an impact on other activities. A limitation of the combined modeling approach is that the loads applied on the FE model are derived from a rigid multibody system, meaning that the equilibrium condition satisfied in the musculoskeletal model is compromised by the deformation and displacement occurring in the FE model. Furthermore, the muscle forces are not spread over attachment areas and the muscle compression forces onto bone are not accounted for. The musculoskeletal model partially overcomes this limitation by having a high number of actuators for muscles. Finally, the model was developed to contain only the pelvic bones and ligaments. Although the main focus was on bone adaptation, the lack of internal organs and soft tissue connected to the pelvis can have an impact on bone adaptation as loads in certain regions are not spread and compressive loads caused by the presence of muscles and organs are not taken into account. Future work will aim to assess the behavior of bone-soft tissue complex in the pelvic region.

To the authors' knowledge, this is the first application of a bone adaptation algorithm to the pelvis and the first structural model of the pelvis developed using an adaptive approach. The resulting structure shows good agreement with previous findings on pelvic bone architecture and the structural model was found to be computationally efficient. A future application of the model will be to examine fractures occurring in scenarios such as falls, vehicular collisions, or solid blast. Furthermore, the structural model enables the use of additive manufacturing to design and produce frangible surrogates. Finally, massive endoprostheses and scaffolds may be designed to allow bone regeneration in specific areas of the pelvis affected by trauma or disease while maintaining mechanical properties of the region.

The authors thank the Human Performance and Musculoskeletal Biomechanics groups at Imperial College London for assistance with the gait analysis, and Royal British Legion Centre for Blast Injury Studies at Imperial College London for providing the CT data. The authors acknowledge and thank Dr. Luca Modenese for providing assistance with the musculoskeletal modeling and Professor Jon Clasper for providing clinical input throughout the modeling process.

The Royal British Legion Centre for Blast Injury Studies at Imperial College London.

Michaeli, D. A. , Murphy, S. B. , and Hipp, J. A. , 1997, “ Comparison of Predicted and Measured Contact Pressures in Normal and Dysplastic Hips,” Med. Eng. Phys., 19(2), pp. 180–186. [CrossRef] [PubMed]
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Bergmann, G. , Deuretzabacher, G. , Heller, M. , Graichen, F. , Rohlmann, A. , Strauss, J. , and Duda, G. N. , 2001, “ Hip Forces and Gait Patterns From Rountine Activities,” J. Biomech., 34(7), pp. 859–871. [CrossRef] [PubMed]
Culmann, K. , 1866, Die Graphische Statik, Mayer and Zeller, Zurich, Switzerland.
Meyer, G. H. , 1867, “ Die Architektur Der Spongiosa, Archief Fur Den Anatomischen Und Physiologischen,” Wiss. Med., 27(4), pp. 1389–1394.
Koch, J. C. , 1917, “ The Laws of Bone Architecture,” Am. J. Anat., 21(2), pp. 177–298. [CrossRef]
Wolff, J. , 1869, “ Über Die Bedeutung Der Architektur Der Spongiösen Substanz,” Zentralbl. Med. Wiss., 6, pp. 223–234.
Frost, H. M. , 2003, “ Bone's Mechanostat: A 2003 Update,” Anat. Rec. Part A, 275(2), pp. 1081–1101. [CrossRef]
Stephenson, P. , and Seedhom, B. B. , 1999, “ Cross-Sectional Geometry of the Human Femur in the Mid-Third Region,” Proc. Inst. Mech. Eng. H, 213(2), pp. 159–166. [CrossRef] [PubMed]
Treece, G. M. , Gee, A. H. , Mayhew, P. M. , and Poole, K. E. S. , 2010, “ High Resolution Cortical Bone Thickness Measurement From Clinical CT Data,” Med. Image Anal., 14(3), pp. 276–290. [CrossRef] [PubMed]
Phillips, A. T. M. , Villette, C. C. , and Modenese, L. , 2015, “ Femoral Bone Mesoscale Structural Architecture Prediction Using Musculoskeletal and Finite Element Modelling,” Int. Biomech., 2(1), pp. 43–61. [CrossRef]
Jacob, H. A. C. , Huggler, A. H. , Dietschi, C. , and Schreiber, A. , 1976, “ Mechanical Function of Subchondral Bone as Experimentally Determined on the Acetabulum of the Human Pelvis,” J. Biomech., 9(10), pp. 625–626. [CrossRef] [PubMed]
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Anderson, A. E. , Peters, C. L. , Tuttle, B. D. , and Weiss, J. A. , 2005, “ Subject-Specific Finite Element Model of the Pelvis: Development, Validation and Sensitivity Studies,” ASME J. Biomech. Eng., 127(3), pp. 364–373. [CrossRef]
Vasu, R. , Carter, D. R. , and Harris, W. H. , 1982, “ Stress Distributions in the Acetabular Region—Part I: Before and After Total Joint Replacement,” J. Biomech., 15(3), pp. 155–157. [CrossRef] [PubMed]
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References

Michaeli, D. A. , Murphy, S. B. , and Hipp, J. A. , 1997, “ Comparison of Predicted and Measured Contact Pressures in Normal and Dysplastic Hips,” Med. Eng. Phys., 19(2), pp. 180–186. [CrossRef] [PubMed]
van den Bogert, A. J. , Read, L. , and Nigg, B. M. , 1999, “ An Analysis of Hip Joint Loading During Walking, Running, and Skiing,” Med. Sci. Sports Exercise, 31(1), pp. 131–142. [CrossRef]
Bergmann, G. , Deuretzabacher, G. , Heller, M. , Graichen, F. , Rohlmann, A. , Strauss, J. , and Duda, G. N. , 2001, “ Hip Forces and Gait Patterns From Rountine Activities,” J. Biomech., 34(7), pp. 859–871. [CrossRef] [PubMed]
Culmann, K. , 1866, Die Graphische Statik, Mayer and Zeller, Zurich, Switzerland.
Meyer, G. H. , 1867, “ Die Architektur Der Spongiosa, Archief Fur Den Anatomischen Und Physiologischen,” Wiss. Med., 27(4), pp. 1389–1394.
Koch, J. C. , 1917, “ The Laws of Bone Architecture,” Am. J. Anat., 21(2), pp. 177–298. [CrossRef]
Wolff, J. , 1869, “ Über Die Bedeutung Der Architektur Der Spongiösen Substanz,” Zentralbl. Med. Wiss., 6, pp. 223–234.
Frost, H. M. , 2003, “ Bone's Mechanostat: A 2003 Update,” Anat. Rec. Part A, 275(2), pp. 1081–1101. [CrossRef]
Stephenson, P. , and Seedhom, B. B. , 1999, “ Cross-Sectional Geometry of the Human Femur in the Mid-Third Region,” Proc. Inst. Mech. Eng. H, 213(2), pp. 159–166. [CrossRef] [PubMed]
Treece, G. M. , Gee, A. H. , Mayhew, P. M. , and Poole, K. E. S. , 2010, “ High Resolution Cortical Bone Thickness Measurement From Clinical CT Data,” Med. Image Anal., 14(3), pp. 276–290. [CrossRef] [PubMed]
Phillips, A. T. M. , Villette, C. C. , and Modenese, L. , 2015, “ Femoral Bone Mesoscale Structural Architecture Prediction Using Musculoskeletal and Finite Element Modelling,” Int. Biomech., 2(1), pp. 43–61. [CrossRef]
Jacob, H. A. C. , Huggler, A. H. , Dietschi, C. , and Schreiber, A. , 1976, “ Mechanical Function of Subchondral Bone as Experimentally Determined on the Acetabulum of the Human Pelvis,” J. Biomech., 9(10), pp. 625–626. [CrossRef] [PubMed]
Dalstra, M. , Huiskes, R. , Odgaard, A. , and van Erning, L. , 1993, “ Mechanical and Textural Properties of Pelvic Trabecular Bone,” J. Biomech., 26(4–5), pp. 523–535. [CrossRef] [PubMed]
Anderson, A. E. , Peters, C. L. , Tuttle, B. D. , and Weiss, J. A. , 2005, “ Subject-Specific Finite Element Model of the Pelvis: Development, Validation and Sensitivity Studies,” ASME J. Biomech. Eng., 127(3), pp. 364–373. [CrossRef]
Vasu, R. , Carter, D. R. , and Harris, W. H. , 1982, “ Stress Distributions in the Acetabular Region—Part I: Before and After Total Joint Replacement,” J. Biomech., 15(3), pp. 155–157. [CrossRef] [PubMed]
Carter, D. R. , Vasu, R. , and Harris, W. H. , 1982, “ Stress Distributions in the Acetabular Region—Part II: Effects of Cement Thickness and Metal Backing of the Total Hip Acetabular Component,” J. Biomech., 15(3), pp. 165–170. [CrossRef] [PubMed]
Pedersen, D. R. , Crowninshield, R. D. , Brand, R. A. , and Johnston, R. C. , 1982, “ An Axisymmetric Model of Acetabular Components in Total Hip Arthroplasty,” J. Biomech., 15(4), pp. 305–315. [CrossRef] [PubMed]
Huiskes, R. , 1987, “ Finite Element Analysis of Acetabular Reconstruction. Noncemented Threaded Cups,” Acta Orthop. Scand., 58(6), pp. 620–625. [CrossRef] [PubMed]
Goel, V. K. , Valliappan, S. , and Svensson, N. L. , 1978, “ Stresses in the Normal Pelvis,” Comput. Biol. Med., 8(2), pp. 91–104. [CrossRef] [PubMed]
Oonishi, H. , Isha, H. , and Hasegawa, T. , 1983, “ Mechanical Analysis of the Human Pelvis and Its Application to the Artificial Hip Joint—by Means of the Three Dimensional Finite Element Method,” J. Biomech., 16(6), pp. 427–44. [CrossRef] [PubMed]
Schuller, H. , Dalstra, M. , Huiskes, R. , and Marti, R. , 1993, “ Total Hip Reconstruction in Acetabular Dysplasia. A Finite Element Study,” J. Bone Jt. Surg. Br., 75B(3), pp. 468–474. [CrossRef]
Dalstra, M. , Huiskes, R. , and van Erning, L. , 1995, “ Development and Validation of a Three-Dimensional Finite Element Model of the Pelvic Bone,” ASME J. Biomech. Eng., 117(3), pp. 272–278. [CrossRef]
Thompson, M. S. , Northmore-Ball, M. D. , and Tanner, K. E. , 2002, “ Effects of Acetabular Resurfacing Component Material and Fixation on the Strain Distribution in the Pelvis,” Proc. Inst. Mech. Eng. H, 216(4), pp. 237–245. [CrossRef] [PubMed]
Zant, N. P. , Heaton-Adegbile, P. , Hussell, J. G. , and Tong, J. , 2008, “ In Vitro Fatigue Failure of Cemented Acetabular Replacements: A Hip Simulator Study,” ASME J. Biomech. Eng., 130(2), p. 021019. [CrossRef]
Cilingir, A. C. , Ucar, V. , and Kazan, R. , 2007, “ Three-Dimensional Anatomic Finite Element Modelling of Hemi-Arthroplasty of Human Hip Joint,” Trends Biomater. Artif. Organs, 21(1), pp. 63–72. https://www.researchgate.net/publication/267421019_Three-Dimensional_Anatomic_Finite_Element_Modelling_of_Hemi-Arthroplasty_of_Human_Hip_Joint_Three-Dimensional_Anatomic_Finite_Element_Modelling_of_Hemi-Arthroplasty_of_Human_Hip_Joint
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Figures

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Fig. 1

Lower limb musculoskeletal model in (a) standing and (b) seated positions

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Fig. 2

Transverse 2.5 mm slice of the pelvic for the base FE model. Shell elements representing cortical bone are shown in gray in the border areas; truss elements representing trabecular bone are shown in the interior in red (see color figure online).

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Fig. 3

Pelvis model with ligaments included: Sacroiliac ligaments in blue (center top), pubic ligament in red (center bottom), sacrospinous ligaments in yellow (middle), sacrotuberous ligaments in purple (outer edge of sacrospinous ligaments) and inguinal ligaments in green (diagonal front) (see color figure online)

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Fig. 4

(a) Load applicator at the hip and (b) fixator at the top of the sacrum. The elements representing the bone are shown in light gray, with the two layers of soft elements (from left to right in (a) and top to bottom in (b)) in red and the two layers of stiffer elements in blue (dark blue represents the stiffest layer of elements in the hip applicator). In addition, the black arrow illustrates the peak load applied on the hip joint during walking (see color figure online)

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Fig. 5

Hip JCFs derived from musculoskeletal model for cycles of (a) walking, (b) stair ascent, (c) stair descent, and (d) sit-to-stand and stand-to-sit are shown in green lines for the right leg and blue lines for the left leg. The frames selected for each activity to be used in the FE simulations are highlighted using solid circles. Hip JCFs recorded and reported by Bergmann et al. [3] for the right leg are shown in dashed lines for each activity (see color figure online)

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Fig. 6

Modeling framework

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Fig. 7

Contours of position-dependent cortical thickness ranging from 0.1 mm to 5 mm are shown in the top row: (a) frontal view of pelvis, (b) anterior and (c) medial views of the right innominate. Trabecular elements' radii distribution ranging from 0.1 mm to 2 mm is shown in the bottom row: (d) frontal view, (e) anterior, and (f) medial views of the right innominate. Trabecular elements with a radius < 0.1 mm were excluded for clarity.

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Fig. 8

Selected 5 mm slices of the converged pelvic model illustrating the trabecular structure and cortical thickness. Truss elements with a radius >0.1 mm are shown in red (largest clusters), with elements with a radius of 0.1 mm shown in blue. Elements with a radius of 1 μm are not shown for clarity.

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Fig. 9

Trabecular trajectories formed in the ilium adapted from (a) walking and (b) all activities. The trajectories highlighted correspond to regions in the ilium with a higher trabecular density caused by walking observed by Machiarelli et al. [64]: superior bundle (sb), anterior bundle (ab), sacropubic bundle (spb), iliocotyloid bundle (icb), ilioischial bundle (iib), and trabecular chiasma (tc). Truss elements with a radius >0.1 mm are shown in red (clusters with thick elements). Truss elements with a radius of 0.1 mm appear in blue (see color figure online).

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Fig. 10

Figure illustrating transverse slices from the CT scans (a, c, e, g) shown along with corresponding slices of the pelvic model (b, d, f, h) generated using an in-house code developed in matlab. Shell elements and truss elements with a radius > 0.1 mm are shown in light gray. Truss elements with a radius of 0.1 mm appear in dark gray. Elements with a radius of 1 μm are not shown for clarity.

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Fig. 11

Influence of activities on cortical (a, b, c) and trabecular (d, e, f) structure of converged pelvic model. The elements are color mapped based on the activity which has been most influential in determining their thickness. Further information on the influence of activities on the cortical and trabecular architecture can be found in the supplemental Fig. S1 which is available under the “Supplemental Data” tab for this paper on the ASME digital collection.

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Fig. 12

Comparison between the minimum (a) and the maximum (b) principal strains across the surface of the structural model (x axis) and CT-derived model (y axis). Lines of best fit are shown in solid lines for each case and do not differ greatly from the y = x line shown in dash line.

Tables

Table Grahic Jump Location
Table 1 Material properties of the ligaments in tension (taken from Ref. [30]), with k being the ligament stiffness, l the length of the ligament, A the origination area of the ligament, Nl the number of elements forming each ligament, E Young's modulus, and r the radius of each truss element
Table Grahic Jump Location
Table 2 Material properties of the cartilage in compression (taken from Ref. [30]), with k being the ligament stiffness, l the length of the ligament, A the origination area of the ligament, Nl the number of elements forming each ligament, E Young's modulus, and r the radius of each truss element

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