Finite element assessment of block-augmented total knee arthroplasty

Loosening and migration of tibial prostheses have been identified as causes of early total knee replacement (TKR) failure. The problem is made more complex when defects occur in the proximal tibia compromising fixation and alignment. Clinical studies using metal augments have shown these to be an alternative to other means of defect treatment. Finite element (FE) analysis can be used to identify regions that may be prone to loosening and migration. In the current work, 3D FE models of TKR uncontained type-2 defects treated with block augments have been constructed and analysed. It has been shown that a metal augment is the most suitable. The use of bone cement (PMMA) to fill proximal defects is not considered suitable as stresses carried by the cement block exceed those of the fatigue limit of bone cement. It has been shown that the stresses in the proximal cancellous bone of block-augmented models are significantly below levels likely to cause damage due to overloading. Furthermore, the use of stem extensions has been shown to reduce the cancellous bone stresses in the proximal region thus increasing the likelihood of bone resorption. Given this, it is recommended that stem extensions are not required unless necessary to mitigate some other problem.     

A three-dimensional finite element analysis of the upper tibia

A three-dimensional finite element model of the proximal tibia has been developed to provide a base line for further modeling of prosthetic resurfaced tibiae. The geometry for the model was developed by digitizing coronal and transverse sections made with the milling machine, from one fresh tibia of average size. The load is equally distributed between the medial and lateral compartments over contact areas that were reported in the literature. An indentation test has been used to measure the stiffness and the ultimate strength of cancellous bone in four cadaver tibiae. These values provided the statistical basis for characterising the inhomogeneous distribution of the cancellous bone properties in the proximal tibia. All materials in the model were assumed to be linearly elastic and isotropic. Mechanical properties for the cortical bone and cartilage have been taken from the literature. Results have been compared with strain gage tests and with a two-dimensional axisymmetric finite element model both from the literature. Qualitative comparison between trabecular alignment, and the direction of the principal compressive stresses in the cancellous bone, showed a good relationship. Maximum stresses in the cancellous bone and cortical bone, under a load which occurs near stance phase during normal gait, show safety factors of approximately eight and twelve, respectively. The load sharing between the cancellous bone and the cortical bone has been plotted for the first 40 mm distally from the tibial eminence.     

The effects of tibial component inclination on bone stress after unicompartmental knee arthroplasty

Unlike the case with total knee arthroplasty, the femorotibial angle (FTA) after unicompartmental knee arthroplasty (UKA) does not directly depend on the inclination of the tibial component when the height of the joint line is maintained. This study analyzed the effects of the inclination of the tibial component in the coronal plane on the contact pressure of the implant-bone surface and the stresses on the proximal tibia. A two-dimensional, coronal plane model of the proximal tibia was subjected to finite-element analysis. Sixteen patterns of finite-element models of equal FTA were developed in which the inclination of tibial components ranged from 5 degrees valgus to 10 degrees varus in increments of 1 degrees. Stress concentration at the proximal medial diaphyseal cortex gradually increased as the inclination changed from valgus to varus. Maximum contact pressure on the metal-bone interface similarly changed and shifted from the lateral edge to the medial edge of the implant as the inclination changed to varus. It was found that even without changing FTA, the inclination of the tibial component might affect stress concentration and contact pressure in the proximal tibia after UKA. The results suggested that slight valgus inclination of the tibial component might be preferable to varus and even to 0 degrees (square) inclination so far as the stress distribution is concerned.     

Finite element and experimental cortex strains of the intact and implanted tibia

Finite Element (FE) models for the simulation of intact and implanted bone find their main purpose in accurately reproducing the associated mechanical behavior. FE models can be used for preclinical testing of joint replacement implants, where some biomechanical aspects are difficult, if not possible, to simulate and investigate in vitro. To predict mechanical failure or damage, the models should accurately predict stresses and strains. Commercially available synthetic femur models have been extensively used to validate finite element models, but despite the vast literature available on the characteristics of synthetic tibia, numerical and experimental validation of the intact and implant assemblies of tibia are very limited or lacking. In the current study, four FE models of synthetic tibia, intact and reconstructed, were compared against experimental bone strain data, and an overall agreement within 10% between experimental and FE strains was obtained. Finite element and experimental (strain gauge) models of intact and implanted synthetic tibia were validated based on the comparison of cortex bone strains. The study also includes the analysis carried out on standard tibial components with cemented and noncemented stems of the P.F.C Sigma Modular Knee System. The overall agreement within 10% previously established was achieved, indicating that FE models could be successfully validated. The obtained results include a statistical analysis where the root-mean-square-error values were always <10%. FE models can successfully reproduce bone strains under most relevant acting loads upon the condylar surface of the tibia. Moreover, FE models, once properly validated, can be used for preclinical testing of tibial knee replacement, including misalignment of the implants in the proximal tibia after surgery, simulation of long-term failure according to the damage accumulation failure scenario, and other related biomechanical aspects.     

Intra-articular Contact Stress Distributions at the Ankle Throughout Stance Phase–patient-specific Finite Element Analysis as a Metric of Degeneration Propensity

A contact finite element (FE) formulation is introduced, amenable to patient-specific analysis of cumulative cartilage mechano-stimulus attributable to habitual functional activity. CT scans of individual human ankles are segmented to delineate bony margins. Each bone surface is projected outward to create a second surface, and the intervening volume is then meshed with continuum hexahedral elements. The tibia is positioned relative to the talus into a weight-bearing apposition. The articular members are first engaged under light preload, then plantar-/dorsi-flexion kinematics and resultant loadings are input for serial FE solutions at 13 instants of the stance phase of level walking gait. Cartilage stress histories are post-processed to recover distributions of cumulative stress-time mechano-stimulus, a metric of degeneration propensity. Consistency in computed contact stress exposures presented for seven intact ankles stood in contrast to the higher magnitude and more focal exposures in an incongruously reduced tibial plafond fracture. This analytical procedure provides patient-specific estimates of degeneration propensity due to various mechanical abnormalities, and it provides a platform from which the mechanical efficacy of alternative surgical interventions can be estimated.     

A method to quantify alteration of knee kinematics caused by changes of TKR positioning

Few in-vitro studies have investigated changes in kinematics caused by total knee replacement (TKR) implantation. The advent of surgical navigation systems allows implant position to be measured accurately and the effects of alteration of TKR position and alignment investigated. A test rig and protocol were developed to compare the kinematics of TKR-implanted knees for different femoral component positions. The TKR was implanted and the component positions documented using a navigation system. The quadriceps was tensed and the knees were flexed and extended manually. Torques and drawer forces were applied to the tibia during knee flexion–extension, while recording the kinematics with the navigation system. The implant was removed and replaced on an intramedullary fixation that allowed proximal–distal, and internal–external rotation of the femoral component without conducting a repeated arthrotomy on the knee. The implant was repositioned using the navigation system to reproduce the previously achieved normally navigated position and the kinematics were recorded again. The recorded kinematics of the knee were not significantly different between both normal implantation and intramedullary remounting for tibial internal–external rotation, varus–valgus angulation, or posterior drawer, at any angle of knee flexion examined. Anterior drawer was increased approximately 2.5 mm across the range 20–35° knee flexion (p<0.05), but was otherwise not significantly different. This method of navigating implant components and of moving them within the closed knee (thus avoiding artefactual effects of repeated soft tissue manipulations) can now be used to quantify the effect on kinematics of alteration of the position of the femoral component.     

The effect of valgus/varus malalignment on load distribution in total knee replacements

Valgus or varus malpositioning of the tibial component of a total knee implant may cause increased propensity for loosening or implant wear and eventually may lead to revision surgery. The aim of this study was to determine the effect of valgus/varus malalignment on tibio-femoral mechanics during surgical trial reduction and simulated gait loading. In seven cadaver legs, posterior cruciate sparing total knee replacements were implanted and tibial inserts representing a neutral alignment and 3 degrees and 5 degrees varus and valgus alignments were sequentially inserted. Each knee with each insert was loaded in a manner representative of a trial reduction performed during knee surgery and loaded in a physiological knee simulator. Simulated gait performed on the simulator demonstrated that internal/external and adduction/abduction rotations showed statistical changes with some of the angled inserts at different points in the walking cycle. Neither medial/lateral nor anterior/posterior translations changed statistically during simulated walking. The pressure distribution and total load in the medial and lateral compartments of the tibial component changed significantly with as little as a 3 degrees variation in angulation when loaded in a manner representative of a trial reduction or with a knee simulator. These results support the need for precise surgical reconstruction of the mechanical axis of the knee and proper alignment of the tibial component. These results further demonstrate that tibial contact pressures measured during a trial reduction method may be predictive of contact mechanics at the higher loading seen in the knee simulator.     

Strain shielding in proximal tibia of stemmed knee prosthesis: experimental study

Theoretical concerns about the use of cemented or press-fit stems in revision total knee arthroplasty (TKA) include stress shielding with adverse effects on prosthesis fixation. Revision TKA components are commonly stemmed to protect the limited autogenous bone stock remaining. Revision procedures with the use of stems can place abnormal stresses through even normal bone by their constrained design, type of materials and fixation method and may contribute for bone loss. Experimental quantification of strain shielding in the proximal synthetic tibia following TKA is the main purpose of the present study. In this study, cortical bone strains were measured experimentally with tri-axial strain gauges in synthetic tibias before and after in vitro knee surgery. Three tibias were implanted with cemented and press-fit stem augments and solely with a tibial tray (short monobloc stem) of the P.F.C. Sigma Modular Knee System. The difference between principal strains of the implanted and the intact tibia was calculated for each strain gauge position. The results demonstrated a pronounced strain-shielding effect in the proximal level, close to tibial tray with the cemented stem augment. The press-fit stem presented a minor effect of strain shielding but was more extensively throughout the stem. An increase of strains closely to the distal tip of the cemented and the press-fit stem augment was observed. This suggests for a physiological condition, a potential effect of bone resorption at the proximal region for the cemented stem augment. The localized increase of strains in stems tip can be related with the clinical finding of the pain, at the end of stem after revision TKA.     

Finite element analysis of a hemi-pelvis: the effect of inclusion of cartilage layer on acetabular stresses and strain

An appropriate method of application of the hip-joint force and stress analysis of the pelvic bone, in particular the acetabulum, is necessary to investigate the changes in load transfer due to implantation and to calculate the reference stimulus for bone remodelling simulations. The purpose of the study is to develop a realistic 3D finite element (FE) model of the hemi-pelvis and to assess stress and strain distribution during a gait cycle. The FE modelling approach of the pelvic bone was based on CT scan data and image segmentation of cortical and cancellous bone boundaries. Application of hip-joint force through an anatomical femoral head having a cartilage layer was found to be more appropriate than a perfectly spherical head, thereby leading to more accurate stress–strain distribution in the acetabulum. Within the acetabulum, equivalent strains varied between 0.1% and 0.7% strain in the cancellous bone. High compressive (15–30 MPa) and low tensile (0–5 MPa) stresses were generated within the acetabulum. The hip-joint force is predominantly transferred from the acetabulum through the lateral cortex to the sacroiliac joint and the pubic symphysis. The study is useful to understand the load transfer within the acetabulum and for further investigations on acetabular prosthesis.     

Adaptive changes in micromechanical environments of cancellous and cortical bone in response to in vivo loading and disuse

The skeleton accommodates changes in mechanical environments by increasing bone mass under increased loads and decreasing bone mass under disuse. However, little is known about the adaptive changes in micromechanical behavior of cancellous and cortical tissues resulting from loading or disuse. To address this issue, in vivo tibial loading and hindlimb unloading experiments were conducted on 16-week-old female C57BL/6J mice. Changes in bone mass and tissue-level strains in the metaphyseal cancellous and midshaft cortical bone of the tibiae, resulting from loading or unloading, were determined using microCT and finite element (FE) analysis, respectively. We found that loading- and unloading-induced changes in bone mass were more pronounced in the cancellous than cortical bone. Simulated FE-loading showed that a greater proportion of elements experienced relatively lower longitudinal strains following load-induced bone adaptation, while the opposite was true in the disuse model. While the magnitudes of maximum or minimum principal strains in the metaphyseal cancellous and midshaft cortical bone were not affected by loading, strains oriented with the long axis were reduced in the load-adapted tibia suggesting that loading-induced micromechanical benefits were aligned primarily in the loading direction. Regression analyses demonstrated that bone mass was a good predictor of bone tissue strains for the cortical bone but not for the cancellous bone, which has complex microarchitecture and spatially-variant strain environments. In summary, loading-induced micromechanical benefits for cancellous and cortical tissues are received primarily in the direction of force application and cancellous bone mass may not be related to the micromechanics of cancellous bone.     

A robotic cadaveric flatfoot analysis of stance phase

The symptomatic flatfoot deformity (pes planus with peri-talar subluxation) can be a debilitating condition. Cadaveric flatfoot models have been employed to study the etiology of the deformity, as well as invasive and noninvasive surgical treatment strategies, by evaluating bone positions. Prior cadaveric flatfoot simulators, however, have not leveraged industrial robotic technologies, which provide several advantages as compared with the previously developed custom fabricated devices. Utilizing a robotic device allows the researcher to experimentally evaluate the flatfoot model at many static instants in the gait cycle, compared with most studies, which model only one to a maximum of three instances. Furthermore, the cadaveric tibia can be statically positioned with more degrees of freedom and with a greater accuracy, and then a custom device typically allows. We created a six degree of freedom robotic cadaveric simulator and used it with a flatfoot model to quantify static bone positions at ten discrete instants over the stance phase of gait. In vivo tibial gait kinematics and ground reaction forces were averaged from ten flatfoot subjects. A fresh frozen cadaveric lower limb was dissected and mounted in the robotic gait simulator (RGS). Biomechanically realistic extrinsic tendon forces, tibial kinematics, and vertical ground reaction forces were applied to the limb. In vitro bone angular position of the tibia, calcaneus, talus, navicular, medial cuneiform, and first metatarsal were recorded between 0% and 90% of stance phase at discrete 10% increments using a retroreflective six-camera motion analysis system. The foot was conditioned flat through ligament attenuation and axial cyclic loading. Post-flat testing was repeated to study the pes planus deformity. Comparison was then made between the pre-flat and post-flat conditions. The RGS was able to recreate ten gait positions of the in vivo pes planus subjects in static increments. The in vitro vertical ground reaction force was within ± 1 standard deviation (SD) of the in vivo data. The in vitro sagittal, coronal, and transverse plane tibial kinematics were almost entirely within ± 1 SD of the in vivo data. The model showed changes consistent with the flexible flatfoot pathology including the collapse of the medial arch and abduction of the forefoot, despite unexpected hindfoot inversion. Unlike previous static flatfoot models that use simplified tibial degrees of freedom to characterize only the midpoint of the stance phase or at most three gait positions, our simulator represented the stance phase of gait with ten discrete positions and with six tibial degrees of freedom. This system has the potential to replicate foot function to permit both noninvasive and surgical treatment evaluations throughout the stance phase of gait, perhaps eliciting unknown advantages or disadvantages of these treatments at other points in the gait cycle.     

A combined RSA and FE study of the implanted proximal tibia: correlation of the post-operative mechanical environment with implant migration

There is strong evidence to suggest that inducible displacements, migration and implant loosening are closely related to the initial mechanical environment of the implanted tibia. If this is true, then it should be possible to predict the likelihood of implant migration using patient-specific finite element models. Finite element models of the proximal implanted tibiae were analysed based on pre-operative quantitative computed tomography data of four patients entered into a radiographic migration study. These four patients were also part of an radiostereometric analysis (RSA) study. A variety of load cases were analysed and the risk of bone failure determined for a 2 mm layer of bone immediately beneath the tibial tray. The results were compared with the RSA data measured 1 year post-operatively for each patient. For each patient, an appropriate load case was selected based on patient weight and on the varus-valgus migrations observed in the migration study. The two patients with press-fit implants were predicted to have the highest risk of failure and were found to migrate the most. The two patients with bonded implants (one HA coated and one cemented) were found to have a low risk of failure and these implants migrated the least. This study suggests that the degree of implant migration is dependent on the initial mechanical environment and can be determined using patient-specific finite element analysis.     

Internal mechanical conditions in the soft tissues of a residual limb of a trans-tibial amputee

Most trans-tibial amputation (TTA) patients use a prosthesis to retain upright mobility capabilities. Unfortunately, interaction between the residual limb and the prosthetic socket causes elevated internal strains and stresses in the muscle and fat tissues in the residual limb, which may lead to deep tissue injury (DTI) and other complications. Presently, there is paucity of information on the mechanical conditions in the TTA residual limb during load-bearing. Accordingly, our aim was to characterize the mechanical conditions in the muscle flap of the residual limb of a TTA patient after donning the prosthetic socket and during load-bearing. Knowledge of internal mechanical conditions in the muscle flap can be used to identify the risk for DTI and improve the fitting of the prosthesis. We used a patient-specific modelling approach which involved an MRI scan, interface pressure measurements between the residual limb and the socket of the prosthesis and three-dimensional non-linear large-deformation finite-element (FE) modelling to quantify internal soft tissue strains and stresses in a female TTA patient during static load-bearing. Movement of the truncated tibia and fibula during load-bearing was measured by means of MRI and used as displacement boundary conditions for the FE model. Subsequently, we calculated the internal strains, strain energy density (SED) and stresses in the muscle flap under the truncated bones. Internal strains under the tibia peaked at 85%, 129% and 106% for compression, tension and shear strains, respectively. Internal strains under the fibula peaked at substantially lower values, that is, 19%, 22% and 19% for compression, tension and shear strains, respectively. Strain energy density peaked at the tibial end (104kJ/m(3)). The von Mises stresses peaked at 215kPa around the distal end of the tibia. Stresses under the fibula were at least one order of magnitude lower than the stresses under the tibia. We surmise that our present patient-specific modelling method is an important tool in understanding the etiology of DTI in the residual limbs of TTA patients.     

Multilevel finite element modeling for the prediction of local cellular deformation in bone

The underlying mechanisms by which bone cells respond to mechanical stimuli or how mechanical loads act on osteocytes housed in lacunae in bone are not well understood. In this study, a multilevel finite element (FE) approach is applied to predict local cell deformations in bone tissue. The local structure of the matrix dictates the local mechanical environment of an osteocyte. Cell deformations are predicted from detailed linear FE analysis of the microstructure, consisting of an arrangement of cells embedded in bone matrix material. This work has related the loads applied to a whole femur during the stance phase of the gait cycle to the strain of a single lacuna and of canaliculi. The predicted bone matrix strains around osteocyte lacunae and canaliculi were nonuniform and differed significantly from the macroscopically measured strains. Peak stresses and strains in the walls of the lacuna were up to six times those in the bulk extracellular matrix. Significant strain concentrations were observed at sites where the process meets the cell body.     

A three-dimensional finite element stress analysis for tunnel placement and buttons in anterior cruciate ligament reconstructions

This communication reports the results of a three-dimensional finite element (FE) model of stresses in a surgically altered femur and tibia. The model incorporated a novel approach in implementing orthotropic and inhomogeneous bone properties and non-uniform distributed loading. Cortical, cancellous, and subchondral bone of the femur and the tibia were modeled. Mechanical properties for the cortical and cancellous bone were mapped from published data characterizing the anisotropy and inhomogeneity of the bone properties. Mesh adequacy was determined using stress convergence and strain energy error convergence. Qualitatively, the results of the study compare well with experimental principal compressive strains from the literature. With respect to tunnel placement in anterior cruciate ligament reconstruction, the model predicted stress-shielding at the postero-lateral region of the tunnel wall, and increased stress at the postero-medial region of the tunnel wall. The stresses in the cancellous bone beneath the tunnel were, in general, lower than those above the tunnel. Prolonged stress shielding leads to bone resorption of the posterior tunnel wall leading to tunnel enlargement, and possible compromise of the ACL reconstruction. The stresses on the femoral cortex produced from a button-type fixation were noticeable for low levels of loading; the stress levels were very similar in models incorporating bone properties of patients aged 45 and 65. Repeated compression of the femoral cortex at these stress levels may cause microdamage to the cortex eventually resulting in fatigue failure.     

Three dimensional patient-specific collagen architecture modulates cartilage responses in the knee joint during gait

Site-specific variation of collagen fibril orientations can affect cartilage stresses in knee joints. However, this has not been confirmed by 3-D analyses. Therefore, we present a novel method for evaluation of the effect of patient-specific collagen architecture on time-dependent mechanical responses of knee joint cartilage during gait. 3-D finite element (FE) models of a human knee joint were created with the collagen architectures obtained from T₂ mapped MRI (patient-specific model) and from literature (literature model). The effect of accuracy of the implementation of collagen fibril architecture into the model was examined by using a submodel with denser FE mesh. Compared to the literature model, fibril strains and maximum principal stresses were reduced especially in the superficial/middle regions of medial tibial cartilage in the patient-specific model after the loading response of gait (up to ?413 and ?26%, respectively). Compared to the more coarsely meshed joint model, the patient-specific submodel demonstrated similar strain and stress distributions but increased values particularly in the superficial cartilage regions (especially stresses increased >60%). The results demonstrate that implementation of subject-specific collagen architecture of cartilage in 3-D modulates location- and time-dependent mechanical responses of human knee joint cartilage. Submodeling with more accurate implementation of collagen fibril architecture alters cartilage stresses particularly in the superficial/middle tissue.     

Experimentally tested computer modeling of stress fractures in rats

The objective of this study was to develop a finite-element (FE) modeling methodology for studying the etiology of a stress fracture (SF). Several variants of three-dimensional FE models of a rat hindlimb, which differed in length or stiffness of tissues, enabling the analyses of mechanical strains and stress in the tibia, were created. We compared the occurrence of SFs in an animal model to validate locations of peak strains/stresses in the FE models. Four Sprague-Dawley male rats, age ~7 wk, were subjected to mechanical cyclic loads of 1.2 Hz and ~6 N, which were delivered to their hindlimb for 30 min, 3 times/wk, up to 12 wk, by using a specially designed apparatus. The results showed that 1) FE modeling predicted the maximal strains/stresses (~220,0 με and ~29 MPa, respectively) between the mid- and proximal thirds of the tibia; 2) in a longer shin, greater and more inhomogeneous tensile strains/stresses were evident, at the same location; 3) anatomical variants in shin length influenced the strain/stress distributions to a greater extent with respect to changes in mechanical properties of tissues; and 4) bone stiffness was more dominant than muscle stiffness in affecting the strain/stress distributions. In the animal study, 35,000 loading cycles were associated with the formation of a SF. The location of the identified SF in the rat limb verified the FE model. We find the suggested model a valuable tool in studying various aspects of SFs.     

The use of a force-controlled dynamic knee simulator to quantify the mechanical performance of total knee replacement designs during functional activity

The experimental evaluation of any total knee replacement (TKR) design should include the pre-implantation quantification of its mechanical performance during tests that simulate the common activities of daily living. To date, few dynamic TKR simulation studies have been conducted before implantation. Once in vivo, the accurate and reproducible assessment of TKR design mechanics is exceedingly difficult, with the secondary variables of the patient and the surgical technique hindering research. The current study utilizes a 6-degree-of-freedom force-controlled knee simulator to quantify the effect of TKR design alone on TKR mechanics during a simulated walking cycle. Results show that all eight TKR designs tested elicited statistically different measures of tibial/femoral kinematics, simulated soft tissue loading, and implant geometric restraint loading during an identical simulated gait cycle, and that these differences were a direct result of TKR design alone. Maximum ranges of tibial kinematics over the eight designs tested were from 0.8mm anterior to 6.4mm posterior tibial displacement, and 14.1 degrees internal to 6.0 degrees external tibial rotation during the walking cycle. Soft tissue and implant reaction forces ranged from 106 and 222N anteriorly to 19 and 127N posteriorly, and from 1.6 and 1.8Nm internally to 3.5 and 5.9Nm externally, respectively. These measures provide valuable experimental insight into the effect of TKR design alone on simulated in vivo TKR kinematics, bone interface loading and soft tissue loading. Future studies utilizing this methodology should investigate the effect of experimentally controlled variations in surgical and patient factors on TKR performance during simulated dynamic activity.     

Automated three-dimensional finite element modelling of bone: a new method

Three-dimensional finite element stress analysis of bone is a key to understanding bone remodelling, assessing fracture risk, and designing prostheses; however, the cost and complexity of predicting the stress field in bone with accuracy has precluded the routine use of this method. A new, automated method of generating patient-specific three-dimensional finite element models of bone is presented — it uses digital computed tomographic (CT) scan data to derive the geometry of the bone and to estimate its inhomogeneous material properties. Cubic elements of a user-specified size are automatically defined and then individually assigned the CT scan-derived material properties. The method is demonstrated by predicting the stress, strain, and strain energy in a human proximal femur in vivo. Three-dimensional loading conditions corresponding to the stance phase of gait were taken from the literature and applied to the model. Maximum principal compressive stresses of 8–23 MPa were computed for the medial femoral neck. Automated generation of additional finite element models with larger numbers of elements was used to verify convergence in strain energy.