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Three-Dimensional Finite Element Simulation of Total Knee Joint in Gait Cycle
Acta Mechanica Solida Sinica, Vol. 22, No. 4, August, 2009 Published by AMSS Press, Wuhan, China.
ISSN 0894-9166
THREE-DIMENSIONAL FINITE ELEMENT SIMULATION OF TOTAL KNEE JOINT IN GAIT CYCLE Yuan Guo
Xushu Zhang
Weiyi Chen
(The Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China)
Received 16 September 2008; revision received 24 November 2008
ABSTRACT Based on CT scanning pictures from a volunteer’s knee joint, a three-dimensional finite element model of the healthy human knee joint is constructed including complete femur, tibia, fibular, patellar and the main cartilage and ligaments. This model was validated using experimental and numerical results obtained from other authors. The pressure distribution of contact surfaces of knee joint are calculated and analyzed under the load action of ‘heel strike’, ‘single limb stance’ and ‘toe-off’. The results of the gait cycle are that the contact areas of medial cartilage are larger than that of lateral cartilage; the contact force and contact areas would grow larger with the load increasing; the pressure of lateral meniscus is steady, relative to the significant variation of peak pressure in medial meniscus; and the peak value of contact pressure on all components are usually found at about 45% of the gait cycle.
KEY WORDS knee joint, finite element simulation, contact pressure, biomechanics
I. INTRODUCTION The knee joint is the most biggest and complicated joint in the whole human body, lying between hip joint and ankle joint. And it is the junction for activities of human lower extremity. Anyone of the main components damaged will result in abnormal movement of the knee joint. As time passes, osteoarthritis will emerge because of wearing and degeneration of cartilage and meniscus, thereby normal activities of sufferers will be severely affected. The osteoarthritis severely affects the old people’s quality of life, even threatens their life; old patients not only suffer the pains, but brings much burden to family and society. At present, with economy development and improvement of living standard day by day, the lower age people begin to suffer the osteoarthritis, and more and more people will be afflicted by the osteoarthritis in the future. In view of the significance of knee joint and hazardness of related diseases, the pressure distribution of articular cartilage of the knee joint in walking is analyzed using a finite element method in this paper. Bendjaballah et al.[1] constructed a nonlinear finite element model of the human knee joint including femur, tibia, cartilage, menisci, and four main ligament bundles modeled as nonlinear springs to investigate the biomechanical properties of the passive tibiofemoral joint. Beynnon et al.[2] presented an analytical sagittal planar model of the knee to study how cruciate ligament bundles control joint kinematics. Heegard et al.[3] developed a 3D model to analyze the human patella biomechanics during passive knee flexion. P´eri´e and Hobatho[4] , and Li et al.[5] considered the
Project supported by the National Natural Science Foundation of China (No. 10702048).
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2009
joint contact pressure and contact areas on the human knee menisci. Pe˜ na et al.[6] presented a relatively complete 3D model of the healthy human knee joint to analyze the pressure of the articulation and ligament, according to the image of CT and MRI. Shirazi-Adl et al.[7] developed a three-dimensional FE model that included muscles and the main structures of the knee joint to analyze the variation of the contact pressure on the articular surfaces and the stress of the cruciate ligament bundles with the flexion angle varied. At present, the finite element method is widely used to analyze the human knee, but the entire knee model including femur, tibia, fibular, patellar and the main cartilage and ligaments is not reported.
II. MATERIAL AND METHODS 2.1. Knee Joint Finite Element Model The geometrical data of the model developed were obtained by computerized tomography (CT) for lower limb bones menisci and articular cartilage, with images taken from a normal woman adult volunteer (45 years old, 1.60 m, 65 kg, healthy). The thickness of CT parallel digital images is 1 mm in the sagittal, coronal and axial planes with the knee at 0◦ flexion. The geometrical models of ligaments were developed according to the knowledge of anatomy and the data gained by other authors. Hexahedral block-structures meshes of the bones and soft tissues were constructed in the finite element software ANSYS, and the element type of ligaments was defined with ‘brick’. A total of 17684 nodes and 13885 eight-node brick elements were used in this knee FE model. 2.2. Material Types of Bones and Soft Tissues Bones were assumed to be a linear elastic and isotropic material with an elastic modulus of E = 11000 MPa, and a Poisson ratio of ν = 0.3 in this paper. Cartilage, menisci and ligaments are viscoelastic tissues. However, in our case, considering that the loading time of interest corresponded to that of a single leg stance is far less than the viscoelastic time constant of cartilage approached 1500 s[6, 8] , so articular cartilage was considered to behave as a single-phase linear elastic and isotropic material with an elastic modulus of E = 5 MPa, and a Poisson ratio of ν = 0.46[5, 6]. For the same reason, Menisci were also assumed to be a single-phase linear elastic and isotropic material with the average properties with elastic modulus of E = 59 MPa and Poisson ratio of ν = 0.49[6] . Ligaments were considered to behave as a transversely isotropic hyperelastic model including the effect of one family of fibers[8] , and the initial strain of ligaments was added, in order to simulate the state of ‘in vivo’. 2.3. Contact Definitions Frictionless nonlinear contact was assumed for all the articulations[9] and 19 potential contact zones were defined: two at the medial zone and two at the lateral (femoral cartilage-meniscus and meniscustibia cartilage), five between ligaments (ACL, PCL, MCL, LCL and QT (quadriceps tendon))and femur, five between ligaments and tibia and one between cruciate ligaments, two between ligaments (QT and
Fig. 1. Contact zones figure illustrations.
Fig. 2. The definition of flexion angle.
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PL) and patella, one between femoral cartilage and the retropatellar articular cartilage, and one between tibia articular and fibular cartilage(Fig.1). Contact types in the model were defined with bolted contact (between ligaments and bones) and general surface — surface contact including finite sliding. 2.4. Loads and Boundary Conditions Loads along z negative direction were applied on the mechanical axis of top femur in order to simulate the weight of human upper body in this model. The initial push force along y direction was applied on the middle part of top femur. In all the analyses, tibia and fibula were kept fixed. The flexion angle θ was defined as the angle from standing position to the flexion state at this time in plane oyz (Fig.2). This finite element model was solved and analyzed with the FE software LS-DYNA.
III. RESULTS 3.1. Loads of Some States in Gait Cycle The contact pressures of the surfaces were obtained in some representative states in the gait cycle, in order to study the contact pressure on the surfaces of the healthy knee joint. (a) Heel strike (HS): 5.5◦ flexion at 2.25 multiples of body weight (BW)[10, 11] along z negative direction was applied on the mechanical axis of top femur, i.e. [2.25 × 650 N] = 1462 N (roundoff) (Fig.3); (b) Single limb stance (SLS): 15.5◦ flexion at 1.85 BW[10, 11] along z negative direction was applied on the mechanical axis of top femur, i.e. [1.85 × 650 N] = 1202 N; Fig. 3 Loading pattern. (c) Toe-off (TO): 4.5◦ flexion at 3.5 BW[10, 12] along z negative direction was applied on the mechanical axis of top femur, i.e. [3.5 × 650 N] = 2275 N. The contact pressures and areas of the surfaces were obtained in above states in the gait cycle (Table 1). Table 1. The contact pressure and area of articulation in some different states
Contact area (mm2 ) FC TC TC M L M L 2.23 177.7 126.9 181.1 167.7 1.83 171.4 138.0 193.9 149.9 2.42 181.9 185.9 210.1 180.4 cartilage, TC: tibia cartilage, M: medial, L: lateral
Contact pressure (MPa) FC HS 2.74 SLS 2.30 TO 2.66 FC: femoral
The contact area of the medial cartilage is larger than the lateral, and the contact pressure would be larger with the increased load[11, 12] from the results of Table 1. 3.2. Loads in Gait Cycle The variation curve of z axial force with time in a gait cycle[13] was showed in Fig.4. The variation loads of z axial force with time were applied on the mechanical axis of top femur, and the variation of the peak value of contact pressure with time during 60% of a gait cycle was obtained, because the load of z axial force is constant after 60% of a gait cycle (Fig.5). The averaged contact pressure of retropatellar cartilage is 0.8 MPa; the peak contact pressure of parts appeared at about 45% of a gait cycle, and the peak value in medial meniscus is 21 MPa. The result of Godest et.al.[13] is 22 MPa, which also appeared at 45% of a gait cycle; the same results were obtained by Jia et al.[14] The contact pressure of lateral meniscus is steady, relative to the significant
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2009
Fig. 4. Variation curve of z axial force with time in a gait cycle[13] .
Fig. 5. Variation of the peak value of contact pressure with time in a gait cycle (PC: retropatellar cartilage, FC: femoral cartilage, TC: tibia cartilage, MM: medial meniscus, LM: lateral meniscus, CP: contact pressure).
variation of peak pressure in medial meniscus, and the peak value of contact pressure on all components is usually found at about 45% of the gait cycle[12] .
IV. DISCUSSIONS A three-dimensional finite element model of the healthy human knee joint is constructed including complete femur, tibia, fibular, patellar and the main cartilage and ligaments. The pressure distribution of articular cartilage of knee joint in walking is studied. It is proved to be accurate and valid by comparing with research data of other scholars. Results of this paper give an operational platform for three-dimensional modeling of personal human knee joint and in vivo biomechanical studying. The contact area of the medial cartilage is larger than the lateral, and the contact pressure would be larger with the increased load[11, 12] ; the pressure of lateral meniscus is steady, relative to the significant variation of peak pressure in medial meniscus, and the peak value of contact pressure on all components is usually found at about 45% of the gait cycle[12] ; the contact pressure of retropatellar cartilage is smaller than other cartilages[3]; the peak contact pressure of parts appears at about 45% of a gait cycle[13, 14] . Because the flexion angle is small in a gait cycle, and the influence of muscle contraction force is not obvious, the quadriceps femoris is not constructed in this model. The contact pressure distribution is obtained in different motions with z axial loads, not considering other loads and torques.
References [1] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Biomechanics of the human knee joint in compression: reconstruction, mesh generation and finite element analysis. Knee, 1995, 2: 69-79. [2] Beynnon B., Yu J., Huston D., etc., A sagittal plane model of the knee and cruciate ligaments with application of a sensitivity analysis. ASME Journal of Biomechanical Engineering, 1996, 118: 227-239
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[3] Heegard,J., Leyvraz,P.F. and Curnier,A., etc., The biomechanics of the human patella during passive knee flexion. Journal of Biomechanics, 1995, 28: 1265-1279. [4] P´eri´e,D. and Hobatho M.C., In vivo determination of contact areas and pressure of the femorotibial joint using nonlinear finite element analysis. Clinical Biomechanics, 1998, 13: 394-402. [5] Li,G., Gill,J. and Kanamori,A., etc., A validated three-dimensional computational model of a human joint. ASME Journal of Biomechanical Engineering, 1999, 121: 657-662. [6] Pe˜ na,E., Calvo,B. and Martinez,M.A., etc., A three-dimensional finite element analysis of the combined behaviour of ligaments and menisci in the healthy human knee joint. Journal of Biomechanics, 2006, 39: 1686-1701. [7] Shirazi-Adl,A. and Mesfar,W., Effect of tibial tubercle elevation on biomechanics of the entire knee joint under muscle loads. Clinical Biomechanics, 2007, 22: 344-351. [8] Armstrong,C., Lai,W. and Mow,V., An analysis of the unconfined compression of articular cartilage. ASME Journal of Biomechanical Engineering, 1984, 106: 165-173. [9] Murakam,T., The lubrication in natural synovial joints and joint prostheses. JSME International Journal, 1990, 33: 465-474. [10] Taylor,S.J.G., Walker,P.S. and Perry J.S., etc., The forces in the distal femur and the knee during walking and other activities measured by telemetry. Journal of Arthroplasty, 1998, 13: 428-437. [11] Hao,Z.X., Jin,D.W. and Zhang,J.C., etc., Finite element analysis of in vivo tibio-femora l con tact features with menisci. Journal of Tsinghua University, 2008, 48(2): 176-179. [12] Ashvin,T., James,C.H.G. and Shamal D.D., Contact stresses in the knee joint in deep flexion. Medical Engineering & Physics, 2005, 27: 329-335. [13] Godest,A.C., Beaugonin,M. and Haug,E., etc., Simulation of a knee joint replacement during a gait cycle using explicit element analysis. Journal of Biomechanics, 2002, 35(2): 267-275. [14] Jia,X.H., Zhang,M., Lee,W.C.C., Load transfer mechanics between trans-tibial prosthetic socket and residual limb-dynamic effects. Journal of Biomechanics, 2004, 37: 1371-1377.
ISSN 0894-9166
THREE-DIMENSIONAL FINITE ELEMENT SIMULATION OF TOTAL KNEE JOINT IN GAIT CYCLE Yuan Guo
Xushu Zhang
Weiyi Chen
(The Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China)
Received 16 September 2008; revision received 24 November 2008
ABSTRACT Based on CT scanning pictures from a volunteer’s knee joint, a three-dimensional finite element model of the healthy human knee joint is constructed including complete femur, tibia, fibular, patellar and the main cartilage and ligaments. This model was validated using experimental and numerical results obtained from other authors. The pressure distribution of contact surfaces of knee joint are calculated and analyzed under the load action of ‘heel strike’, ‘single limb stance’ and ‘toe-off’. The results of the gait cycle are that the contact areas of medial cartilage are larger than that of lateral cartilage; the contact force and contact areas would grow larger with the load increasing; the pressure of lateral meniscus is steady, relative to the significant variation of peak pressure in medial meniscus; and the peak value of contact pressure on all components are usually found at about 45% of the gait cycle.
KEY WORDS knee joint, finite element simulation, contact pressure, biomechanics
I. INTRODUCTION The knee joint is the most biggest and complicated joint in the whole human body, lying between hip joint and ankle joint. And it is the junction for activities of human lower extremity. Anyone of the main components damaged will result in abnormal movement of the knee joint. As time passes, osteoarthritis will emerge because of wearing and degeneration of cartilage and meniscus, thereby normal activities of sufferers will be severely affected. The osteoarthritis severely affects the old people’s quality of life, even threatens their life; old patients not only suffer the pains, but brings much burden to family and society. At present, with economy development and improvement of living standard day by day, the lower age people begin to suffer the osteoarthritis, and more and more people will be afflicted by the osteoarthritis in the future. In view of the significance of knee joint and hazardness of related diseases, the pressure distribution of articular cartilage of the knee joint in walking is analyzed using a finite element method in this paper. Bendjaballah et al.[1] constructed a nonlinear finite element model of the human knee joint including femur, tibia, cartilage, menisci, and four main ligament bundles modeled as nonlinear springs to investigate the biomechanical properties of the passive tibiofemoral joint. Beynnon et al.[2] presented an analytical sagittal planar model of the knee to study how cruciate ligament bundles control joint kinematics. Heegard et al.[3] developed a 3D model to analyze the human patella biomechanics during passive knee flexion. P´eri´e and Hobatho[4] , and Li et al.[5] considered the
Project supported by the National Natural Science Foundation of China (No. 10702048).
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ACTA MECHANICA SOLIDA SINICA
2009
joint contact pressure and contact areas on the human knee menisci. Pe˜ na et al.[6] presented a relatively complete 3D model of the healthy human knee joint to analyze the pressure of the articulation and ligament, according to the image of CT and MRI. Shirazi-Adl et al.[7] developed a three-dimensional FE model that included muscles and the main structures of the knee joint to analyze the variation of the contact pressure on the articular surfaces and the stress of the cruciate ligament bundles with the flexion angle varied. At present, the finite element method is widely used to analyze the human knee, but the entire knee model including femur, tibia, fibular, patellar and the main cartilage and ligaments is not reported.
II. MATERIAL AND METHODS 2.1. Knee Joint Finite Element Model The geometrical data of the model developed were obtained by computerized tomography (CT) for lower limb bones menisci and articular cartilage, with images taken from a normal woman adult volunteer (45 years old, 1.60 m, 65 kg, healthy). The thickness of CT parallel digital images is 1 mm in the sagittal, coronal and axial planes with the knee at 0◦ flexion. The geometrical models of ligaments were developed according to the knowledge of anatomy and the data gained by other authors. Hexahedral block-structures meshes of the bones and soft tissues were constructed in the finite element software ANSYS, and the element type of ligaments was defined with ‘brick’. A total of 17684 nodes and 13885 eight-node brick elements were used in this knee FE model. 2.2. Material Types of Bones and Soft Tissues Bones were assumed to be a linear elastic and isotropic material with an elastic modulus of E = 11000 MPa, and a Poisson ratio of ν = 0.3 in this paper. Cartilage, menisci and ligaments are viscoelastic tissues. However, in our case, considering that the loading time of interest corresponded to that of a single leg stance is far less than the viscoelastic time constant of cartilage approached 1500 s[6, 8] , so articular cartilage was considered to behave as a single-phase linear elastic and isotropic material with an elastic modulus of E = 5 MPa, and a Poisson ratio of ν = 0.46[5, 6]. For the same reason, Menisci were also assumed to be a single-phase linear elastic and isotropic material with the average properties with elastic modulus of E = 59 MPa and Poisson ratio of ν = 0.49[6] . Ligaments were considered to behave as a transversely isotropic hyperelastic model including the effect of one family of fibers[8] , and the initial strain of ligaments was added, in order to simulate the state of ‘in vivo’. 2.3. Contact Definitions Frictionless nonlinear contact was assumed for all the articulations[9] and 19 potential contact zones were defined: two at the medial zone and two at the lateral (femoral cartilage-meniscus and meniscustibia cartilage), five between ligaments (ACL, PCL, MCL, LCL and QT (quadriceps tendon))and femur, five between ligaments and tibia and one between cruciate ligaments, two between ligaments (QT and
Fig. 1. Contact zones figure illustrations.
Fig. 2. The definition of flexion angle.
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PL) and patella, one between femoral cartilage and the retropatellar articular cartilage, and one between tibia articular and fibular cartilage(Fig.1). Contact types in the model were defined with bolted contact (between ligaments and bones) and general surface — surface contact including finite sliding. 2.4. Loads and Boundary Conditions Loads along z negative direction were applied on the mechanical axis of top femur in order to simulate the weight of human upper body in this model. The initial push force along y direction was applied on the middle part of top femur. In all the analyses, tibia and fibula were kept fixed. The flexion angle θ was defined as the angle from standing position to the flexion state at this time in plane oyz (Fig.2). This finite element model was solved and analyzed with the FE software LS-DYNA.
III. RESULTS 3.1. Loads of Some States in Gait Cycle The contact pressures of the surfaces were obtained in some representative states in the gait cycle, in order to study the contact pressure on the surfaces of the healthy knee joint. (a) Heel strike (HS): 5.5◦ flexion at 2.25 multiples of body weight (BW)[10, 11] along z negative direction was applied on the mechanical axis of top femur, i.e. [2.25 × 650 N] = 1462 N (roundoff) (Fig.3); (b) Single limb stance (SLS): 15.5◦ flexion at 1.85 BW[10, 11] along z negative direction was applied on the mechanical axis of top femur, i.e. [1.85 × 650 N] = 1202 N; Fig. 3 Loading pattern. (c) Toe-off (TO): 4.5◦ flexion at 3.5 BW[10, 12] along z negative direction was applied on the mechanical axis of top femur, i.e. [3.5 × 650 N] = 2275 N. The contact pressures and areas of the surfaces were obtained in above states in the gait cycle (Table 1). Table 1. The contact pressure and area of articulation in some different states
Contact area (mm2 ) FC TC TC M L M L 2.23 177.7 126.9 181.1 167.7 1.83 171.4 138.0 193.9 149.9 2.42 181.9 185.9 210.1 180.4 cartilage, TC: tibia cartilage, M: medial, L: lateral
Contact pressure (MPa) FC HS 2.74 SLS 2.30 TO 2.66 FC: femoral
The contact area of the medial cartilage is larger than the lateral, and the contact pressure would be larger with the increased load[11, 12] from the results of Table 1. 3.2. Loads in Gait Cycle The variation curve of z axial force with time in a gait cycle[13] was showed in Fig.4. The variation loads of z axial force with time were applied on the mechanical axis of top femur, and the variation of the peak value of contact pressure with time during 60% of a gait cycle was obtained, because the load of z axial force is constant after 60% of a gait cycle (Fig.5). The averaged contact pressure of retropatellar cartilage is 0.8 MPa; the peak contact pressure of parts appeared at about 45% of a gait cycle, and the peak value in medial meniscus is 21 MPa. The result of Godest et.al.[13] is 22 MPa, which also appeared at 45% of a gait cycle; the same results were obtained by Jia et al.[14] The contact pressure of lateral meniscus is steady, relative to the significant
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2009
Fig. 4. Variation curve of z axial force with time in a gait cycle[13] .
Fig. 5. Variation of the peak value of contact pressure with time in a gait cycle (PC: retropatellar cartilage, FC: femoral cartilage, TC: tibia cartilage, MM: medial meniscus, LM: lateral meniscus, CP: contact pressure).
variation of peak pressure in medial meniscus, and the peak value of contact pressure on all components is usually found at about 45% of the gait cycle[12] .
IV. DISCUSSIONS A three-dimensional finite element model of the healthy human knee joint is constructed including complete femur, tibia, fibular, patellar and the main cartilage and ligaments. The pressure distribution of articular cartilage of knee joint in walking is studied. It is proved to be accurate and valid by comparing with research data of other scholars. Results of this paper give an operational platform for three-dimensional modeling of personal human knee joint and in vivo biomechanical studying. The contact area of the medial cartilage is larger than the lateral, and the contact pressure would be larger with the increased load[11, 12] ; the pressure of lateral meniscus is steady, relative to the significant variation of peak pressure in medial meniscus, and the peak value of contact pressure on all components is usually found at about 45% of the gait cycle[12] ; the contact pressure of retropatellar cartilage is smaller than other cartilages[3]; the peak contact pressure of parts appears at about 45% of a gait cycle[13, 14] . Because the flexion angle is small in a gait cycle, and the influence of muscle contraction force is not obvious, the quadriceps femoris is not constructed in this model. The contact pressure distribution is obtained in different motions with z axial loads, not considering other loads and torques.
References [1] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Biomechanics of the human knee joint in compression: reconstruction, mesh generation and finite element analysis. Knee, 1995, 2: 69-79. [2] Beynnon B., Yu J., Huston D., etc., A sagittal plane model of the knee and cruciate ligaments with application of a sensitivity analysis. ASME Journal of Biomechanical Engineering, 1996, 118: 227-239
Vol. 22, No. 4
Yuan Guo et al.: Three-Dimensional Finite Element Simulation
· 351 ·
[3] Heegard,J., Leyvraz,P.F. and Curnier,A., etc., The biomechanics of the human patella during passive knee flexion. Journal of Biomechanics, 1995, 28: 1265-1279. [4] P´eri´e,D. and Hobatho M.C., In vivo determination of contact areas and pressure of the femorotibial joint using nonlinear finite element analysis. Clinical Biomechanics, 1998, 13: 394-402. [5] Li,G., Gill,J. and Kanamori,A., etc., A validated three-dimensional computational model of a human joint. ASME Journal of Biomechanical Engineering, 1999, 121: 657-662. [6] Pe˜ na,E., Calvo,B. and Martinez,M.A., etc., A three-dimensional finite element analysis of the combined behaviour of ligaments and menisci in the healthy human knee joint. Journal of Biomechanics, 2006, 39: 1686-1701. [7] Shirazi-Adl,A. and Mesfar,W., Effect of tibial tubercle elevation on biomechanics of the entire knee joint under muscle loads. Clinical Biomechanics, 2007, 22: 344-351. [8] Armstrong,C., Lai,W. and Mow,V., An analysis of the unconfined compression of articular cartilage. ASME Journal of Biomechanical Engineering, 1984, 106: 165-173. [9] Murakam,T., The lubrication in natural synovial joints and joint prostheses. JSME International Journal, 1990, 33: 465-474. [10] Taylor,S.J.G., Walker,P.S. and Perry J.S., etc., The forces in the distal femur and the knee during walking and other activities measured by telemetry. Journal of Arthroplasty, 1998, 13: 428-437. [11] Hao,Z.X., Jin,D.W. and Zhang,J.C., etc., Finite element analysis of in vivo tibio-femora l con tact features with menisci. Journal of Tsinghua University, 2008, 48(2): 176-179. [12] Ashvin,T., James,C.H.G. and Shamal D.D., Contact stresses in the knee joint in deep flexion. Medical Engineering & Physics, 2005, 27: 329-335. [13] Godest,A.C., Beaugonin,M. and Haug,E., etc., Simulation of a knee joint replacement during a gait cycle using explicit element analysis. Journal of Biomechanics, 2002, 35(2): 267-275. [14] Jia,X.H., Zhang,M., Lee,W.C.C., Load transfer mechanics between trans-tibial prosthetic socket and residual limb-dynamic effects. Journal of Biomechanics, 2004, 37: 1371-1377.