2.1 Experimental model

Two couple of fourth-generation femur and tibia (model 3403 and 3401), a couple of fourth generation patella (model 3419), a fourth generation pelvis (model 3415-1), feet and a couple of fibula in a solid foam, produced by Pacific Research Labs., Vashon Island, WA, USA, were used to perform the experimental tests. These bone analogues model natural bone using a mixture of short e-glass fibres and epoxy resin for the corticalis (density 1.7g/cm³, tensile modulus 12.4GPa, compressive modulus 7.6GPa, Poisson ratio 0.3, tensile strength 90MPa, compressive strength 120MPa) and a polyurethane foam core for the cancellous bone (density 0.27g/cm³, compressive modulus 0.1GPa, compressive strength 4.8MPa). Left leg was instrumented with six resistive foil strain gauges (Hottinger Baldwin Messtechnik, Darmstadt, Germany), connected in quarter bridge to a digital data acquisition system (System 6100, Vishay Measurement Group, Raleigh, NC, USA). Unidirectional strain gauges (SGs) were positioned as shown in Fig. 1(a). Before bonding the SGsensors, the area for strain measurement was preliminary treated with sandpaper (#100), carefully cleaned and degreased first with ethanol, then with acetone and 2-propanol, and smoothed with fine sandpaper (#400). A bi-component epoxy adhesive was employed to glue the SGs onto the femur surface. Bones have been connected by elastic rods to simulate tendons by screws, while feet were rigidly constrained to a base; in order to avoid sliding between pelvis and femur, femur and tibia, and tibia and foot extremity of bones were rasped. Hence, a series of tests have been carried out on the entire model, having cared to fix the structure. Simulation of the loading conditions occurring in the stance of gait was obtained by applying a single axial load of 700N on the top of the pelvis. Although this may appear as a rough approximation of the physiological loading condition, in which there are also several thigh muscles acting on the bones at different locations, the experimental set-up was greatly simplified, improving measurement repeatability and reproducibility. A material testing machine (Lloyd Instruments Inc., Fareham, UK); equipped with a 1000N load cell (±0.5% full scale accuracy) was used to impose the vertical load. The model was pre-loaded with 100N, to asses equilibrium conditions, and then a quasi-statically axial loads was imposed ranging from 100 to 700N at 10N/s. Test was repeated 3 times, repositioning the model in the testing machine each time, in order to determine repeatability of the experimental set-up and check for the absence of permanent deformations or damages.

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Fig. 1 Experimental model, finite element model with identification of reference nodes, loading and constrain conditions.

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2.2 Finite element model

The geometrical data of the model developed herein were obtained by a computerised tomography (CT) of artificial bones, separated at intervals of 1.5mm in the sagittal, coronal and axial planes with the knee at 0° flexion (accuracy 0.5mm). These lines were transferred into the commercial code Hypermesh by Altair^® where the main surfaces and solid version of the model were reconstructed; in particular 6.842 elements and 2.345 nodes were used for pelvis, 21.457 elements and 1.302 nodes in femur, 2.843 elements and 564 nodes in patella, 2.945 elements and 987 nodes in fibula, 16.987 elements and 2.454 nodes in tibia and 990 elements and 456 nodes for the foot, see Fig. 1. On the upper zone, the ilio-femoral ligament, the ligament of the hip joint, which extends from the ileum to the femur in front of the joint, was modelled with 273 tetrahedral elements and 161 nodes. The knee joint constituted by the medial collateral ligament, which extends from the medial femoral epicondyle to the tibia, the lateral collateral ligament, which extends from the lateral femoral epicondyle to the head of the fibula, the anterior cruciate ligament which extends postero-laterally from the tibia and inserts on the lateral femoral condyle, and the posterior cruciate ligament which extends antero-medially from the tibia posterior to the medial femoral condyles were modelled with 601 tetrahedral elements and 306 nodes. On the lower zone, the foot joint, constituted by the plantar fascia, the medial and lateral ligaments were modelled with 386 tetrahedral elements and 199 nodes.

The six reference points, see Fig. 1(b), where SGS were positioned have been identified also on the numerical model, and verified comparing different measurements executed with a precision gauge. Non-linear finite element analysis of the models was performed with Abaqus version 5.4 (Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI) using the geometric nonlinearity and automatic time stepping options. Material properties were defined as nonlinear elastic materials for the structures and contact interfaces were imposed at the ilio-femoral (femur-pelvis), knee (femur-patella, patella-tibia, and fibula-tibia) and foot (tibia-feet) joints; defined using a penalty-based method with a weight factor, a coefficient of friction of 0.04 was chosen to be consistent. A distributed, on 70 nodes, vertical (y axes) load of 700N was applied on the left side of the pelvis as depicted on Fig. 1(c), while a fixed constrain, of 200 nodes, was imposed at the lower extremity of the foot.⁹

3.1 Validation of results, strains comparison

Two optical transducers (x and z), focused on the left upper side of pelvis, and the LVDT transducer of the testing machine (y) were used to collect experimental data about complete model displacements along the three axes. The obtained data were compared to the corresponding numerical ones, see Table 1. Results revealed a good agreement, the relative errors ranged from −3.65% (Dz) to −16.15% (Dy). The average error was 9.37%. In Fig. 2, are reported the measured longitudinal (i.e., along the y-axis) strains at full load, and the correspondent numerical strains computed at same locations, as well as the corresponding percentage error between estimated and experimental data. The average relative error between computed and measured strains was 9.37% (n=6).

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Fig. 2 Comparison between experimental and numerical data collected on the different six reference nodes.

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In Fig. 2, a histogram of the comparison between experimental and numerical values of strain is shown. The error indicated in figure ranges between 21 and 31% (average error 25.33%) and a good agreement can be deduced between experimental and numerical approaches. By analysing results numerical model seems generally to be subjected to higher strains, only SG#3 and SG#1 exhibit lower ones. In Fig. 3, strain registered on the SG#6 positioned unser the neck of the femur is depicted. Obviously in this case the main solicitations is compressive and for this reason strains are negative, value reached by the experimental test on the gauge is −13 [μm/m], while the omologous numerical point reaches −16 [μm/m]. Curves showing behaviour of SG#5 and numerical point 5, SG#4 and point 4, located at distal and condilar areas of the femur, present the same increasing trend. Values of 353 and 430 [μm/m], respectively for experimental and numerical models, are reached in SG#5 and point 5, while values of 69 and 90 [μm/m] are reached in SG#4 and point 4. Lower values of strain measured in SG#4 and point 4 compared to the ones registered in SG#5 and point 5 can be explained by the spreading of forces which other bony and linking parts play each other. The effect of contact surfaces or elastic ligaments reduces concentrations of stress around the lower part of the femur. The areas around points 4 and 3 are located closely to the knee articulation and present opposite increasing trend of curves. As it is possible to notice, SG#3 reaches a higher value than point 3, thus experimental model seems to perform a more rigid configuration than the numerical one causing higher strains. The SG#2 and point 2 register significant values of strain, respectively 389 and 470 [μm/m], while in SG#1 and point 1 values are of 140 and 105 [μm/m]. Also in this case numerical model reaches strain values lower than the experimental one. Data confirm flexural solicitations added to the compression imposed by the load ageing in the medial area of femur and tibia.

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Fig. 3 Experimental and numerical comparison of strain–load curves calculated for 6, 5, 4, 3, 2, 1 areas.

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3.2 Stress shielding, bony displacements, and equivalent total strain on the model

In order to evaluate trending of displacements and stresses FE model was divided into three parts upper, concerning the pelvis, medium with femur upper ligaments, femur, knee ligaments, and patella, and finally the lower part concerning tibia, fibula, feet, and ligaments. Each part was sectioned by a crossing plane, as depicted in Figs. 4–6 and values of displacements and equivalent von mises stresses were calculated at regular intervals on the frontal surface. Fig. 4 shows a pick of about 10mm on the top of pelvis (quote 0mm) and a decreasing trend for the remaining part (0–140) till a value of about 7mm. Different behaviour can be identified at the corresponding equivalent von mises stress which has its peak on the acetabulum with a value of about 11MPa. Fig. 4, concerning the medium part, depicts the same trend of displacements with a maximum value of about 8mm on the head of femur (quote 140mm) till a value of about 3mm (quote 540mm) localised on the condilar zone. The maximum stress is located on the medial zone (quote 340mm) and has a value of about 13MPa. Finally the lower part reports separately data of displacements and stresses obtained for tibia and fibula which are shown in Fig. 5. As it is possible to notice fibula undergoes to sustain higher displacements on the lower part (quote from 640 to 890mm) also if values are not worrying. Displacements ranging respectively from about 2.5mm in tibia and 2.2mm in fibula, at quote 540–590mm. At quote 890–940mm displacements values are 0.3mm in tibia and 0.5mm in fibula. Different conclusion can be deduced by comparing stresses, as depicted in Fig. 4, fibula appears much more unloaded than tibia with a maximum value of 2.3MPa reached at quote 740mm. On the corresponding zone also tibia reaches its maximum stress of about 13MPa.

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Fig. 4 Contour maps and quantitative representation of global displacements and Eq. Von Mises Stress versus Position (0–140) calculated for the upper part of the FE model (pelvis).

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Fig. 5 Contour maps and quantitative representation of global displacements and Eq. Von Mises Stress versus. Position (140–540) calculated for the medium part of the FE model (femur, patella, ligaments).

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Fig. 6 Contour maps and quantitative representation of global displacements and Eq. Von Mises Stress versus. Position (540–940) calculated for the lower part of the FE model (tibia, fibula, foot).

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4.1 Experimental and numerical strain field

Results confirm a bending effect, opposite to the compression load applied, present under the neck of the femur which determines a negative value of strain. Medial portions of femur and tibia manifested higher values of deformations, while lower and upper extremity result almost unloaded, because of tensions are probably concentred on the contact surfaces. Although the validation was restricted to only few sparse points on the femur surface and to a single load case, the obtained results provide evidence that it is possible to successfully predict the structural response.

4.2 Numerical stress and displacements

Comparison among results permits to evaluate if the multi body investigation approach, followed in this paper, obtains results too different from the traditional ones largely used in literature. The problem consists in comparing the different setup configurations adopted by authors. The obtained results reveal interesting consequences deriving by taking into account the complete bony chain. The stress shielding effects, displacements, and equivalent total strain induced in each part of the model are summarised in Table 2. As it is possible to notice the maximum total displacement is localised on the upper zone of the leg, concerning the pelvis and the femur, with values ranging from 10 to 8mm. The lower part undergoes a minor effect producing displacements of the order of 2–3mm localised in knee, tibia and fibula. Different consideration must be done for the Equivalent Von Mises Stress which does not follow a continuum trend like displacements. Maximum values of about 19MPa are reached on the knee ligaments, while values of about 16MPa were found in pelvis and fibula. Femur, femoral ligaments, and tibia register values of about 13MPa, while patella, feet ligaments and feet are solicited with less than 8MPa.

4.3 Comparison among different results

A comparison among the literature data and our model, carried on the works studying femur, knee, and tibia show generally less stressing conditions ageing on the structure, and moreover can furnish a complete vision of the global spreading of stresses along the various bony components. In order to carry on a useful comparison same simplification should be due. The elastic work expressed by Eq. (1), can be used in order to compare displacements by assuming a perfectly elastic behaviour of simulations, thus a linear dependence is defined.(1)W(J)=F(N)*d [mm]

Similarly by assuming a similar area where forces can spread within an analogue geometry, stresses can be compared by Eq. (2):(2)F(N)=σ [MPa]A [mm]Results, in terms of displacements and equivalent von mises stress, obtained by different authors at different loading conditions, and calculated by using Eqs. (1) and (2) are reported in Table 2.

Rapperport et al.¹ by finite element analysis for the human upright stance position evaluated pressure distributions across the articular surface for a resultant femoral load magnitude of 1.000N and 40° angled on the medial of vertical. Results obtained by applying Eq. (2) compared with those ones presented in this paper show an error of −28% concerning the peak of Eq. Von Mises Stress. Montanini et al.² considered the effect of muscle action proposed by Bergmann et al.³ This load profile, which basically consists of four distinct muscle groups in addition to the hip joint force, is based on a validated musculoskeletal analysis and provides consistent torsion loading of the femur in addition to bending, loaded on the head with a force of 980N. Results obtained by applying Eq. (1) revealed an error of −7% in displacements and 43% for Eq. Von Mises Stress.

Filardi,⁴ in a precedent paper found a value of 9.29mm for displacement in the femur and a Von Mises stress of about 14MPa, by using different material properties.

Other results, Filardi,⁵ confirm a good agreement in displacements calculated in meniscus, error range around 2% for displacements and 5% for stresses.

Bendjaballah et al.⁶ considered a compression load of 1.300N over the femur to measure compression stress on the meniscus in the external and internal periphery, respectively, for the healthy joint. Results, in this case, evidenced an error of −12% referred to the maximum Eq. Von Mises Stress.

Li et al.⁷ obtained a peak stress at the tibial plafond producing an error of −27%, calculated by imposing a load estimated in about 3% of the body weight. The peak stress, measured by Vrahas et al.,⁸ in cadaveric tibia statically loaded with 1360N, produced an error of 18%.

Filardi⁴ found stresses ageing on the foot of 4.12MPa with a good agreement with results herein presented. Cheung et al.^9,10 developed a geometrical accurate 3D FE model of the human foot and ankle, and for a subject with body mass of 70kg, a vertical force of approximately 350N is applied on each foot during balanced standing, producing in this case an error 18% in the peak of stress.