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Determination of quadriceps forces in squat and its application in contact pressure analysis of knee joint
Acta Mechanica Solida Sinica, Vol. 25, No. 1, February, 2012 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
DETERMINATION OF QUADRICEPS FORCES IN SQUAT AND ITS APPLICATION IN CONTACT PRESSURE ANALYSIS OF KNEE JOINT Yuan Guo
Xushu Zhang
Meiwen An
Weiyi Chen
(Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China)
Received 6 November 2010, revision received 26 September 2011
ABSTRACT While the quadriceps muscles of human body are quite important to the daily activities of knee joints, the determination of quadriceps forces poses significant challenges since it cannot be measured in vivo. Here, a novel approach is presented to obtain the forces in squat through the combination of motion photography, force transducers measuring, multi-rigid-body theory and finite element analysis. Firstly, the geometrical and angular data of human for squat process were obtained through the analysis of photographed pictures for human squat with camera. At the same time, force transducers were used to measure the reaction forces from feet and to determine the center of gravity for identical squat process. Next, based on the multi-rigidbody dynamics, a mathematical model for human right leg and foot was established in order to determine the quadriceps torques under different squat angles. Then, so as to determine the quadriceps forces along with varied squat angles, a simplified three-dimensional finite element model was built, including tibia, fibula, patella, patella ligament and quadriceps tendon. Finally, the contact pressure of knee joint was analyzed for the squat with the established model of knee joint involving the obtained quadriceps forces from finite element analysis. And it showed that in the 0-90 degree squat process, the peak value of contact pressure of articular cartilages and menisci is increased with the increased squat angle. This study can be referenced for further understanding of the biomechanical behaviors of knee, contact pressure effects of daily activities on knee, and is significantly instructive for sports rehabilitation.
KEY WORDS quadriceps force, force transducers, motion analysis, finite element analysis, squat
I. INTRODUCTION The knee joint is the junction for activities of human lower extremity, and any damage of the main components will result in abnormal movements of knee. As time passes, osteoarthritis of cartilage or meniscus will emerge due to wear and degeneration, and normal activities of sufferers will be severely affected. At present, young people begin to suffer the osteoarthritis, and more and more people will be cursed by the osteoarthritis in the future. In order to analyze the mechanism of arthritic pathogenesis and predict the articular degeneration, it is important to determine the conditions and pressure distribution of the contact of knee joints in daily activities such as squat. In squat, the quadriceps muscle plays a
Corresponding author. E-mail: meiwen [email protected] Project supported by the National Natural Science Foundation of China (Nos. 10702048 and 11102126) and Natural Science Foundation of Shanxi (No. 2010021004-1).
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main role, so it is necessary to determine quadriceps force firstly, and then study on the contact pressure of knee joint. Because the quadriceps force can’t be measured by experiments in vivo, the theoretical studies were conducted by many scholars[1–6]. And quadriceps torques of the knee joint were also studied in different motions with theoretical methods[7] . On experimental studies of quadriceps torques, the EMG determination method was mostly used, which revealed the proportion relations of quadriceps torques with myoelectric parameters[8, 9]. Some researches focused on the moment arm of components of knee joint, for example, the moment arm of patellar tendon[10, 11] . Owing to the obtained quadriceps forces or torques for the specific motions of knee joint, they cannot be referenced for other movements. In order to analyze the movement and contact conditions, some scholars established many knee models via the finite element (FE) method. A non-linear model consisting of bony structures, their articular cartilage layers, medial and lateral menisci and four primary ligaments[12–15] , was developed and the ligaments in all above-mentioned models were modeled with spring elements. A knee model was built with ACL, PCL, MCL and LCL, all of which were elastic elements[16] . The contact stresses and contact strains were studied on the menisci of knee joint[17–19] . A 3-D nonlinear model was established for a knee joint with nearly complete parts[20, 21] . All above models for knee joints were just the knee parts of human lower extremity, i.e. local regions, and the positions of loads were not clearly given, thus their loads can not be referenced for various movements. So a 3-D model for the knee joint was developed by us including complete femur, tibia, fibular, patellar, the main cartilages and ligaments, which were hyper-elastic material model. In the meanwhile, the contact pressure of cartilages and stress distribution in ligaments were analyzed in the movement of gait[22] . In this paper, based on the previous researches of a knee joint and through the combination of motion photography, force transducers measuring, multi-rigid-body theory and finite element analysis, the quadriceps forces were determined for different squat angles. Then, the obtained quadriceps forces were exerted on the complete established finite element models of the knee joint, the contact pressure distribution of the joint and the stress distribution of ligaments could be acquired for different squat angles.
II. MATERIAL AND METHODS First of all, based on motion photography, force transducers measuring, and multi-rigid-body dynamics equation, the quadriceps torques under different squat angles were obtained by self-designed measuring devices. Next, via an established simplified finite element model of knee joint, the quadriceps forces were determined in different squat angles by FE analysis. Then, through the previously established 3-D FE model, which included all intact bones of knee joint, the quadriceps tendon and main ligaments, we can estimate the contact pressure and the ligament stresses of knee joint[22] . 2.1. The Squat Experiment The height, body weight, and the dimension of their thighs, legs and feet of three healthy male and three healthy female (Age: 25-35) were recorded. Markers were pasted on anatomy position of hip joint, knee joint and ankle joint of right leg, respectively, and the marker of knee joint was located in the center of lateral knee. The squat angle was defined as 0◦ in standing upright and increased with squatting process. Experimental equipments included two 50 kg and two 100 kg calibrated sensors, a computer, a digital camera, a tripod, wood plates, wires, and so on. Dynamic signal acquisition and analysis system DHDAS 5923 (Donghua, China) was adopted to record and analyze the real-time signals from all sensors. At the same time, the motions of subjects were photographed with digital cameras. 2.1.1. Squat photography and data acquisition The fixed distance between front and back sensors d and the distance between right and left sensors l were 150 mm and 130 mm, respectively. Two 50 kg and two 100 kg sensors were linked under the left and right sides of wood platform, respectively. The positions of sensors were recorded, and the experimental apparatus were shown in Fig.1. The positions of tripod, computer screen, sensors were adjusted, so that the squat movement of right side leg and real-time readings of 4 channels displayed on the screen could be captured simultaneously by digital camera, i.e. time synchronization. The sampling frequency was adjusted to 10 kHz.
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The subject put her feet slowly on the wood platform, and bent her knees slowly to physiological limits, keeping for 5 s; then extended the knee slowly to upright, and maintained for 5 s again. That process was repeated for three times for every subject and the picture captured was shown in Fig.1.
Fig. 1. The photographed picture of squat.
Fig. 2. The force diagram of right leg and foot.
2.1.2. The analysis of experimental results The pictures of bmp format were exported from the shot video, using the shared frame extracted software Avi2Jpeg (China, 20 fps). After a fixed image coordinate system defined, the coordinates in pictures from the first frame to last were recorded consecutively, including markers and top ends of front and back sensors of the right side. Then, the obtained data were imported into mathematical analysis software Mathematica (Wolfram, USA), and related angles for lower extremity were obtained for each picture, including the squat angle, etc. (see Appendix for specific data) 2.2. Determination of Position for Gravity Center and Quadriceps Torques in Squat According to the sensor readings and the distances between the sensors, the real-time position of gravity center can be computed by Eq.(1) based on moment-equilibrium in squat, N1 x − N2 (l − x) = 0
(1)
where, N1 , N2 are the readings of front and back sensors, respectively, x is the distance between the position of gravity center and the front sensor. According to the table ‘geometry size, mass, and center of gravity statistics table of Chinese youth’ and ‘Link mass distribution chart of China youth body’[23] , the length and gravity center of shank,, the ratio of the mass of shank and foot to the whole human body and mass center can be obtained. Using the angles between the shank and horizontal axis, the moment arms of gravity force for shank and foot to the marker on knee joint could be obtained for each squat angle. Figure 2 is the simplified forces diagram for human right leg and foot, and it is supposed that quadriceps muscle plays the leading role and torques of other muscles are ignored in squat. Here, Fr is the reaction force to the foot from the ground, acting on the point A; mrx g is the weight of right shank, acting on the point B; mrz g is the weight of right foot, acting on the point C; d1 , d2 and d3 are the moment arm of above-mentioned forces to right mid-point O of knee joint, respectively; Fz and Fx are the reaction force of knee joint in z and x axis, respectively; Mq is the quadriceps torques. Based on the moment balance equation for shank, we can write Fr × d1 = mrx g × d2 + mrz g × d3 + Mq
(2)
By Eq.(2), the quadriceps torques can be obtained corresponding to different squat angles. In Fig.3, the quadriceps torques increased with the squat angle from 0 to 90 degree.
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Fig. 3. Curves of the quadriceps torques with squat angle.
2012
Fig. 4. The force diagram of tibia and fibula.
2.3. The Determination of Quadriceps Forces 2.3.1. Moment of inertia for tibia and fibular In order to determine the quadriceps forces, a simplified finite element model was built from the previous established complete model of knee joint[22], which includes tibia, fibula, patella, patella ligament and quadriceps tendon. In that model, the tibia and fibula bone density were adjusted in order to make the total mass equal to the average mass of women shank 2.646 kg[23] . A local coordinate system with origin O was established. And the moment of inertia of tibia and fibular relative to point O was 3.56 × 10−4 kgm2 from the information file of Ls-Dyna. 2.3.2. The determination of quadriceps forces If the effects of large deformation of bones are ignored, the ratio k(θ) of quadriceps torques Mq (θ) to forces Fq (θ) was a variable unrelated with the quadriceps force, i.e. k(θ) = Mq (θ)/Fq (θ). In order to determine the quadriceps force, the coefficient k(θ) must be firstly determined based on the simplified knee model, with tibia and fibula rotating around the femur from 0˚ to 90˚ with femur fixed. A given thrust vertical to the tibia was exerted on the distal end of tibia and fibula, and a pull force was exerted on the top end of quadriceps tendon. Through the finite element calculation, the variation of the angle with time was obtained. After it was imported into Mathematica, curves of the rotation angle θ versus time was plotted (angle unit is degree). Through the second derivative of function θ(t) to time t, angular acceleration could be obtained. And the specific forces diagram was shown in Fig.4 (rotation center O on tibia platform, the action point A of the thrust, the action point B of quadriceps forces, and rotation angle θ). The distance l from point A to the top of tibia plateau was 0.33 m, and based on rigid body dynamics equation with a fixed axis, the equation of tibia around point O from lateral to medial was followed (units: SI): Jβ(t) = N l − Fq × k(t) ⇒ k(t) =
N l − Jβ(t) Fq
(3)
As is shown in Eq.(3), the coefficient k(t) could be obtained. And after the angle θ(t) and the coefficient k(t) were fitted in Mathematica, the k(θ) curve varied with θ was obtained and shown in Fig.5 (unit: m). At last, the obtained Mq (θ) and k(θ) are imported into the expression Fq (θ) = Mq (θ)/k(θ), the quadriceps forces increased with the squat angle (θ), as is shown in Fig.6. 2.4. Finite Element Analysis of Contact Pressure In order to verify the obtained quadriceps forces and analyze the contact pressure of knee joint, a complete 3-D finite element model for knee joint was established (See Fig.7), and detailed description could be found in the previous paper[22] . But there were several changes in the present paper. In the first place, a few elements were established on the site near the anterior inferior iliac spine of the pelvis and were connected with femoral top head; in the second place, the quadriceps force pointed to the pelvis iliac was exerted on the top end of quadriceps tendon, and the negative force was exerted on the site near the anterior inferior iliac spine with the same magnitude in order to simulate the quadriceps force;
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Fig. 5. Curve of k(θ) vs. θ.
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Fig. 6. Curve of the quadriceps forces with θ in squat.
finally, based on the motion photography, the angular displacements and initial velocity of kinematics data were exerted on the hip and ankle joints center to simulate the squat process.
Fig. 8. Positions of peak contact pressure with the squat angle for femur cartilage (M: medial; L: lateral). Fig. 7. Complete FE model of knee joint.
III. RESULTS Through the motion photography, the kinematic data of human squat were obtained, which is fundamental for contact pressure analysis for knee joint. By force transducers measuring, the feet reaction forces were obtained and the positions of gravity center in squat were determined to achieve the quadriceps torques. Based on the multi-rigid-body dynamics and above obtained data, the quadriceps forces were estimated in squat. After the quadriceps forces were achieved and they were exerted on the FE model of knee joint, the contact pressure of articular cartilages and menisci for knee joint was studied with different squat angles in the range of 0◦ -90◦ . The changed positions of peak contact pressure on the femur cartilages were shown in Fig.8. The contact pressure on the femur, tibia, and patella cartilages increased with the squat angle. The contact pressure of lateral meniscus was steady, relative to the significant variation of peak pressure on medial meniscus, and the peak value of contact pressure on all components appeared at 90 degree (as shown in Fig.9).
IV. DISCUSSIONS In this paper, based on motion photography analysis for the squat process of human body and force transducers experimental measurement, the kinematic data and forces at the bottom of feet in squat were obtained, and the quadriceps torques with squat angles were determined via multi-rigid-body dynamics. Then, based on a simplified 3-D FE model and rigid-body dynamics, the quadriceps forces were acquired along with squat angles. Finally, using above obtained data and importing them into a complete finite element model of knee joint, we obtain the contact pressure on articular cartilages and menisci of knee joint in squat. And the present model included complete femur, patella, tibia and fibula,
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Fig. 9. Varied curves of peak contact pressure with the squat angle.
medial and lateral meniscus, and the main ligaments MCL, PCL, ACL, LCL, PL (Patella Ligament), QT (Quadriceps Tendon). At the same time, the motion data of hip and ankle joint and the quadriceps force changed with squat angle were considered. Through comparisons with the existing research results of experiments and finite element method[18–20, 24–31] , the present model not only agrees with anatomical structures of knee joint, but also can be relatively accurately agreed with the squat movement of human body and the obtained contact pressure of articular cartilages and menisci. This model can be provided as an operational platform for further exploration of the mechanical behaviors of the knee joint in daily activities. In squat, the rotation axis and contact positions of femur relative to tibia were changed with real-time squat angle and there is no good solution to this problem whether by theoretical analysis or experimental measurement. So the authors adopted a new method combining motion photography, force transducers experimental measuring, multi-rigid-body dynamics theory and finite element analysis, which firstly determined the torques and moment arms of quadriceps in squat, then determined the quadriceps forces, and finally applied the contact pressure analysis for the squat. Through the analysis of previous research results with the FE analysis in this paper, it shows that the peak contact pressure on each cartilage and meniscus becomes greater with the increasing of squat angle within the range of 0◦ -90◦ , and the peak contact pressure on the medial meniscus is greater than that on the lateral meniscus[24, 25] . Based on the obtained quadriceps forces, the contact pressure of articular cartilages and menisci for the knee joint with squat angel was mainly verified and studied, and only the main and largest loads in z direction were considered without taking into account loads and torques in other two directions. In next researches, the muscles related to the knee joint and the body weight exerted on the pelvis could be added into this FE model in order to simulate the real conditions for knee movements; the further researches on quadriceps torques or forces could be estimated in other movements, the more kinematic data could be added into the model to simulate the movements of human lower extremity; and further researches on contact pressure of knee joint could be done in other motions, such as up/down stairs, jogging and riding bicycle etc., so as to get a deeper and better understanding on the biomechanical movement principles of knee joint.
References [1] Anderson,F.C. and Pandy,M.G., Dynamic optimization of human walking. Journal of Biomechanical Engineering, 2001, 123: 381-390. [2] Anderson,F.C. and Pandy,M.G., Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics, 2001, 34: 153-161. [3] Anderson,F.C. and Pandy,M.G., Individual muscle contributions to support in normal walking. Gait Posture, 2003, 17: 159-169. [4] Komistek,R.D., Mathematical Modeling of the Human Arm: an Aid in the Investigation of the Role of Muscle Forces in the Development of Lateral Epicondylitis. PH.D. Dissertation, University of Memphis, 1992. [5] Komistek,R.D., Stiehl,J.B., Dennis,D.A., et al., Mathematical model of the lower extremity joint reaction forces using Kane’s method of dynamics. Journal of Biomechanics, 1998, 31(2): 185-189.
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[6] Komistek,R.D., Kanec,T.R. and Mahfouz,M., Knee mechanics: a review of past and present techniques to determine in vivo loads. Journal of Biomechanics, 2005, 38: 215-228. [7] Farahmand,F., Rezaeian,T., Narimani,R., et al., Kinematic and dynamic analysis of the gait cycle of aboveknee amputees. Scientia Iranica, 2006, 13(3): 261-271. [8] Yang,J.Y., Wang,R.Y., Xiong,K.Y., et al., The EMG determination and analysis of quadriceps isokinetic concentric contraction. Journal of Beijing University of Physical Education, 1995, 18(4): 28-34. [9] Jiang,H.Y., The relationships between integrated EMG activity and mechanics parameters of human quadriceps muscle during voluntary contraction. China Sports Science, 1989, 3: 50-54. [10] Janet,L.K., Marcus,G.P. and John,C.P., Moment arm of the patellar tendon in the human knee. Journal of biomechanics, 2004, 37: 785-788. [11] Dimitrios,E.T., Vasilios,B. and Paula,J.R., In vivo changes in the human patellar tendon moment arm length with different modes and intensities of muscle contraction. Journal of biomechanics, 2007, 40: 33253332. [12] 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. [13] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Finite element analysis of human knee joint in varusvalgus. Clinical Biomechanics, 1997, 12(3): 139-148. [14] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Biomechanical response of the passive human knee joint under anterior-posterior forces. Clinical Biomechanics, 1998, 13: 625-633. [15] Jalani,A., Shirazi-Adl,A. and Bendjaballah,M.Z., Biomechanics of human tibio-femoral joint in axial rotation. The Knee, 1997, 4: 203-213. [16] Shelburne,K.B., Pandy,M.G. and Torry,M.R., Comparison of shear forces and ligament loading in the healthy and ACL-deficient knee during gait. Journal of Biomechanics, 2004, 37(3): 313-319. [17] 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. [18] Li,G., Gil,J., Kanamori,A., et al., A validated three-dimensional computational model of a human joint. ASME Journal of Biomechanical Engineering, 1999, 121: 657-662. [19] Li,G., Lopez,O. and Rubash,H., Variability of a three-dimensional finite element model constructed using magnetic resonance images of a knee for joint contact stress analysis. ASME Journal of Biomechanical Engineering, 2001, 123: 341-346. [20] Pe˜ na,E., Calvo,B., Martinez,M.A., et al., 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. [21] 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. [22] Guo,Y., Zhang,X.S. and Chen,W.Y., Three-dimensional finite element simulation of the total knee joint in the gait cycle. Acta Mechanica Solida Sinica, 2009, 22(4): 347-351. [23] Zheng,X.Y., Jia,S.H., Gao,Y.F., et al., Modern Sports Biomechanics. National Defense Industry Press, 2002, 100-163: 366-407. [24] 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. [25] Hao,Z.X., Jin,D.W. and Zhang,J.C., Finite element analysis of in vivo tibio-femoral contact features with menisci. Journal of Tsinghua University, 2008, 48(2): 176-179. [26] Harrington,I.J., A bioengineering analysis of force actions at the knee in normal and pathological gait. Biomedical Engineering, 1976, 11(5): 167-172. [27] Moeinzadeh,M.H., Engin,A.E. and Akkas,N., Two-dimensional dynamic modeling of human knee joint. Journal of Biomechanics, 1983, 16: 253-264. [28] Thambyah,A., Pereira,B.P. and Wyss,U., Estimation of bone-on-bone contact forces in the tibiofemoral joint during walking. The Knee, 2005, 12(5): 383-388. [29] Tumer,S.T. and Engin,A.E., Three-body segment dynamic model of the human knee. ASME, Journal of Biomechanical Engineering, 1993, 115: 350-356. [30] Taylor,S.J.G., Walker,P.S., Perry,J.S., et.al., The forces in the distal femur and the knee during walking and other activities measured by telemetry. Journal of Arthroplasty, 1998, 13: 428-437. [31] Taylor,S.J.G. and Walker,P.S., Forces and moments telemetered from two distal femoral replacements during various activities. Journal of Biomechanics, 2001, 34(7): 839-848.
APPENDIX
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Table 1. The data table of knee squat angle
No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Results from picture analysis (degree) Squat Angle (thigh Angle (shank angle -level) -level) 5.80 88.84 85.36 4.42 89.71 85.87 7.74 87.69 84.57 8.09 87.42 84.49 8.88 87.37 83.75 12.11 84.29 83.60 13.21 82.77 84.02 15.74 79.35 84.91 21.98 74.09 83.93 33.43 66.18 80.39 45.03 57.79 77.18 59.38 48.87 71.75 65.82 45.60 68.58 72.91 39.55 67.54 80.44 34.25 65.31 89.40 28.36 62.24 99.42 20.73 59.85 111.16 12.37 56.47 121.71 6.07 52.22 130.63 0.56 48.81 135.88 -3.68 47.80 138.65 -6.50 47.84
Results measured by transducers (N) Left front Left rear Right front Right rear (1) (2) (3) (4) 89.42 237.27 103.45 206.60 90.64 240.48 105.90 206.60 85.30 242.61 103.45 208.74 78.89 255.58 100.71 208.51 69.43 258.48 94.60 211.18 100.40 246.89 105.59 187.99 39.83 257.57 96.74 254.67 36.62 278.33 97.05 251.46 36.62 285.19 96.13 221.86 20.60 291.19 80.87 242.31 7.17 303.73 52.19 276.79 16.33 313.57 63.48 272.83 3.05 314.18 3.36 312.19 10.22 236.82 37.23 311.89 10.53 274.96 21.97 324.40 29.30 265.20 48.22 290.22 29.14 257.87 40.89 303.96 43.18 256.81 47.60 298.46 42.11 258.72 39.98 307.92 58.75 241.55 41.81 284.42 81.18 233.76 53.10 274.35 81.94 248.72 54.02 250.55
ISSN 0894-9166
DETERMINATION OF QUADRICEPS FORCES IN SQUAT AND ITS APPLICATION IN CONTACT PRESSURE ANALYSIS OF KNEE JOINT Yuan Guo
Xushu Zhang
Meiwen An
Weiyi Chen
(Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China)
Received 6 November 2010, revision received 26 September 2011
ABSTRACT While the quadriceps muscles of human body are quite important to the daily activities of knee joints, the determination of quadriceps forces poses significant challenges since it cannot be measured in vivo. Here, a novel approach is presented to obtain the forces in squat through the combination of motion photography, force transducers measuring, multi-rigid-body theory and finite element analysis. Firstly, the geometrical and angular data of human for squat process were obtained through the analysis of photographed pictures for human squat with camera. At the same time, force transducers were used to measure the reaction forces from feet and to determine the center of gravity for identical squat process. Next, based on the multi-rigidbody dynamics, a mathematical model for human right leg and foot was established in order to determine the quadriceps torques under different squat angles. Then, so as to determine the quadriceps forces along with varied squat angles, a simplified three-dimensional finite element model was built, including tibia, fibula, patella, patella ligament and quadriceps tendon. Finally, the contact pressure of knee joint was analyzed for the squat with the established model of knee joint involving the obtained quadriceps forces from finite element analysis. And it showed that in the 0-90 degree squat process, the peak value of contact pressure of articular cartilages and menisci is increased with the increased squat angle. This study can be referenced for further understanding of the biomechanical behaviors of knee, contact pressure effects of daily activities on knee, and is significantly instructive for sports rehabilitation.
KEY WORDS quadriceps force, force transducers, motion analysis, finite element analysis, squat
I. INTRODUCTION The knee joint is the junction for activities of human lower extremity, and any damage of the main components will result in abnormal movements of knee. As time passes, osteoarthritis of cartilage or meniscus will emerge due to wear and degeneration, and normal activities of sufferers will be severely affected. At present, young people begin to suffer the osteoarthritis, and more and more people will be cursed by the osteoarthritis in the future. In order to analyze the mechanism of arthritic pathogenesis and predict the articular degeneration, it is important to determine the conditions and pressure distribution of the contact of knee joints in daily activities such as squat. In squat, the quadriceps muscle plays a
Corresponding author. E-mail: meiwen [email protected] Project supported by the National Natural Science Foundation of China (Nos. 10702048 and 11102126) and Natural Science Foundation of Shanxi (No. 2010021004-1).
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main role, so it is necessary to determine quadriceps force firstly, and then study on the contact pressure of knee joint. Because the quadriceps force can’t be measured by experiments in vivo, the theoretical studies were conducted by many scholars[1–6]. And quadriceps torques of the knee joint were also studied in different motions with theoretical methods[7] . On experimental studies of quadriceps torques, the EMG determination method was mostly used, which revealed the proportion relations of quadriceps torques with myoelectric parameters[8, 9]. Some researches focused on the moment arm of components of knee joint, for example, the moment arm of patellar tendon[10, 11] . Owing to the obtained quadriceps forces or torques for the specific motions of knee joint, they cannot be referenced for other movements. In order to analyze the movement and contact conditions, some scholars established many knee models via the finite element (FE) method. A non-linear model consisting of bony structures, their articular cartilage layers, medial and lateral menisci and four primary ligaments[12–15] , was developed and the ligaments in all above-mentioned models were modeled with spring elements. A knee model was built with ACL, PCL, MCL and LCL, all of which were elastic elements[16] . The contact stresses and contact strains were studied on the menisci of knee joint[17–19] . A 3-D nonlinear model was established for a knee joint with nearly complete parts[20, 21] . All above models for knee joints were just the knee parts of human lower extremity, i.e. local regions, and the positions of loads were not clearly given, thus their loads can not be referenced for various movements. So a 3-D model for the knee joint was developed by us including complete femur, tibia, fibular, patellar, the main cartilages and ligaments, which were hyper-elastic material model. In the meanwhile, the contact pressure of cartilages and stress distribution in ligaments were analyzed in the movement of gait[22] . In this paper, based on the previous researches of a knee joint and through the combination of motion photography, force transducers measuring, multi-rigid-body theory and finite element analysis, the quadriceps forces were determined for different squat angles. Then, the obtained quadriceps forces were exerted on the complete established finite element models of the knee joint, the contact pressure distribution of the joint and the stress distribution of ligaments could be acquired for different squat angles.
II. MATERIAL AND METHODS First of all, based on motion photography, force transducers measuring, and multi-rigid-body dynamics equation, the quadriceps torques under different squat angles were obtained by self-designed measuring devices. Next, via an established simplified finite element model of knee joint, the quadriceps forces were determined in different squat angles by FE analysis. Then, through the previously established 3-D FE model, which included all intact bones of knee joint, the quadriceps tendon and main ligaments, we can estimate the contact pressure and the ligament stresses of knee joint[22] . 2.1. The Squat Experiment The height, body weight, and the dimension of their thighs, legs and feet of three healthy male and three healthy female (Age: 25-35) were recorded. Markers were pasted on anatomy position of hip joint, knee joint and ankle joint of right leg, respectively, and the marker of knee joint was located in the center of lateral knee. The squat angle was defined as 0◦ in standing upright and increased with squatting process. Experimental equipments included two 50 kg and two 100 kg calibrated sensors, a computer, a digital camera, a tripod, wood plates, wires, and so on. Dynamic signal acquisition and analysis system DHDAS 5923 (Donghua, China) was adopted to record and analyze the real-time signals from all sensors. At the same time, the motions of subjects were photographed with digital cameras. 2.1.1. Squat photography and data acquisition The fixed distance between front and back sensors d and the distance between right and left sensors l were 150 mm and 130 mm, respectively. Two 50 kg and two 100 kg sensors were linked under the left and right sides of wood platform, respectively. The positions of sensors were recorded, and the experimental apparatus were shown in Fig.1. The positions of tripod, computer screen, sensors were adjusted, so that the squat movement of right side leg and real-time readings of 4 channels displayed on the screen could be captured simultaneously by digital camera, i.e. time synchronization. The sampling frequency was adjusted to 10 kHz.
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The subject put her feet slowly on the wood platform, and bent her knees slowly to physiological limits, keeping for 5 s; then extended the knee slowly to upright, and maintained for 5 s again. That process was repeated for three times for every subject and the picture captured was shown in Fig.1.
Fig. 1. The photographed picture of squat.
Fig. 2. The force diagram of right leg and foot.
2.1.2. The analysis of experimental results The pictures of bmp format were exported from the shot video, using the shared frame extracted software Avi2Jpeg (China, 20 fps). After a fixed image coordinate system defined, the coordinates in pictures from the first frame to last were recorded consecutively, including markers and top ends of front and back sensors of the right side. Then, the obtained data were imported into mathematical analysis software Mathematica (Wolfram, USA), and related angles for lower extremity were obtained for each picture, including the squat angle, etc. (see Appendix for specific data) 2.2. Determination of Position for Gravity Center and Quadriceps Torques in Squat According to the sensor readings and the distances between the sensors, the real-time position of gravity center can be computed by Eq.(1) based on moment-equilibrium in squat, N1 x − N2 (l − x) = 0
(1)
where, N1 , N2 are the readings of front and back sensors, respectively, x is the distance between the position of gravity center and the front sensor. According to the table ‘geometry size, mass, and center of gravity statistics table of Chinese youth’ and ‘Link mass distribution chart of China youth body’[23] , the length and gravity center of shank,, the ratio of the mass of shank and foot to the whole human body and mass center can be obtained. Using the angles between the shank and horizontal axis, the moment arms of gravity force for shank and foot to the marker on knee joint could be obtained for each squat angle. Figure 2 is the simplified forces diagram for human right leg and foot, and it is supposed that quadriceps muscle plays the leading role and torques of other muscles are ignored in squat. Here, Fr is the reaction force to the foot from the ground, acting on the point A; mrx g is the weight of right shank, acting on the point B; mrz g is the weight of right foot, acting on the point C; d1 , d2 and d3 are the moment arm of above-mentioned forces to right mid-point O of knee joint, respectively; Fz and Fx are the reaction force of knee joint in z and x axis, respectively; Mq is the quadriceps torques. Based on the moment balance equation for shank, we can write Fr × d1 = mrx g × d2 + mrz g × d3 + Mq
(2)
By Eq.(2), the quadriceps torques can be obtained corresponding to different squat angles. In Fig.3, the quadriceps torques increased with the squat angle from 0 to 90 degree.
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Fig. 3. Curves of the quadriceps torques with squat angle.
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Fig. 4. The force diagram of tibia and fibula.
2.3. The Determination of Quadriceps Forces 2.3.1. Moment of inertia for tibia and fibular In order to determine the quadriceps forces, a simplified finite element model was built from the previous established complete model of knee joint[22], which includes tibia, fibula, patella, patella ligament and quadriceps tendon. In that model, the tibia and fibula bone density were adjusted in order to make the total mass equal to the average mass of women shank 2.646 kg[23] . A local coordinate system with origin O was established. And the moment of inertia of tibia and fibular relative to point O was 3.56 × 10−4 kgm2 from the information file of Ls-Dyna. 2.3.2. The determination of quadriceps forces If the effects of large deformation of bones are ignored, the ratio k(θ) of quadriceps torques Mq (θ) to forces Fq (θ) was a variable unrelated with the quadriceps force, i.e. k(θ) = Mq (θ)/Fq (θ). In order to determine the quadriceps force, the coefficient k(θ) must be firstly determined based on the simplified knee model, with tibia and fibula rotating around the femur from 0˚ to 90˚ with femur fixed. A given thrust vertical to the tibia was exerted on the distal end of tibia and fibula, and a pull force was exerted on the top end of quadriceps tendon. Through the finite element calculation, the variation of the angle with time was obtained. After it was imported into Mathematica, curves of the rotation angle θ versus time was plotted (angle unit is degree). Through the second derivative of function θ(t) to time t, angular acceleration could be obtained. And the specific forces diagram was shown in Fig.4 (rotation center O on tibia platform, the action point A of the thrust, the action point B of quadriceps forces, and rotation angle θ). The distance l from point A to the top of tibia plateau was 0.33 m, and based on rigid body dynamics equation with a fixed axis, the equation of tibia around point O from lateral to medial was followed (units: SI): Jβ(t) = N l − Fq × k(t) ⇒ k(t) =
N l − Jβ(t) Fq
(3)
As is shown in Eq.(3), the coefficient k(t) could be obtained. And after the angle θ(t) and the coefficient k(t) were fitted in Mathematica, the k(θ) curve varied with θ was obtained and shown in Fig.5 (unit: m). At last, the obtained Mq (θ) and k(θ) are imported into the expression Fq (θ) = Mq (θ)/k(θ), the quadriceps forces increased with the squat angle (θ), as is shown in Fig.6. 2.4. Finite Element Analysis of Contact Pressure In order to verify the obtained quadriceps forces and analyze the contact pressure of knee joint, a complete 3-D finite element model for knee joint was established (See Fig.7), and detailed description could be found in the previous paper[22] . But there were several changes in the present paper. In the first place, a few elements were established on the site near the anterior inferior iliac spine of the pelvis and were connected with femoral top head; in the second place, the quadriceps force pointed to the pelvis iliac was exerted on the top end of quadriceps tendon, and the negative force was exerted on the site near the anterior inferior iliac spine with the same magnitude in order to simulate the quadriceps force;
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Fig. 5. Curve of k(θ) vs. θ.
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Fig. 6. Curve of the quadriceps forces with θ in squat.
finally, based on the motion photography, the angular displacements and initial velocity of kinematics data were exerted on the hip and ankle joints center to simulate the squat process.
Fig. 8. Positions of peak contact pressure with the squat angle for femur cartilage (M: medial; L: lateral). Fig. 7. Complete FE model of knee joint.
III. RESULTS Through the motion photography, the kinematic data of human squat were obtained, which is fundamental for contact pressure analysis for knee joint. By force transducers measuring, the feet reaction forces were obtained and the positions of gravity center in squat were determined to achieve the quadriceps torques. Based on the multi-rigid-body dynamics and above obtained data, the quadriceps forces were estimated in squat. After the quadriceps forces were achieved and they were exerted on the FE model of knee joint, the contact pressure of articular cartilages and menisci for knee joint was studied with different squat angles in the range of 0◦ -90◦ . The changed positions of peak contact pressure on the femur cartilages were shown in Fig.8. The contact pressure on the femur, tibia, and patella cartilages increased with the squat angle. The contact pressure of lateral meniscus was steady, relative to the significant variation of peak pressure on medial meniscus, and the peak value of contact pressure on all components appeared at 90 degree (as shown in Fig.9).
IV. DISCUSSIONS In this paper, based on motion photography analysis for the squat process of human body and force transducers experimental measurement, the kinematic data and forces at the bottom of feet in squat were obtained, and the quadriceps torques with squat angles were determined via multi-rigid-body dynamics. Then, based on a simplified 3-D FE model and rigid-body dynamics, the quadriceps forces were acquired along with squat angles. Finally, using above obtained data and importing them into a complete finite element model of knee joint, we obtain the contact pressure on articular cartilages and menisci of knee joint in squat. And the present model included complete femur, patella, tibia and fibula,
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Fig. 9. Varied curves of peak contact pressure with the squat angle.
medial and lateral meniscus, and the main ligaments MCL, PCL, ACL, LCL, PL (Patella Ligament), QT (Quadriceps Tendon). At the same time, the motion data of hip and ankle joint and the quadriceps force changed with squat angle were considered. Through comparisons with the existing research results of experiments and finite element method[18–20, 24–31] , the present model not only agrees with anatomical structures of knee joint, but also can be relatively accurately agreed with the squat movement of human body and the obtained contact pressure of articular cartilages and menisci. This model can be provided as an operational platform for further exploration of the mechanical behaviors of the knee joint in daily activities. In squat, the rotation axis and contact positions of femur relative to tibia were changed with real-time squat angle and there is no good solution to this problem whether by theoretical analysis or experimental measurement. So the authors adopted a new method combining motion photography, force transducers experimental measuring, multi-rigid-body dynamics theory and finite element analysis, which firstly determined the torques and moment arms of quadriceps in squat, then determined the quadriceps forces, and finally applied the contact pressure analysis for the squat. Through the analysis of previous research results with the FE analysis in this paper, it shows that the peak contact pressure on each cartilage and meniscus becomes greater with the increasing of squat angle within the range of 0◦ -90◦ , and the peak contact pressure on the medial meniscus is greater than that on the lateral meniscus[24, 25] . Based on the obtained quadriceps forces, the contact pressure of articular cartilages and menisci for the knee joint with squat angel was mainly verified and studied, and only the main and largest loads in z direction were considered without taking into account loads and torques in other two directions. In next researches, the muscles related to the knee joint and the body weight exerted on the pelvis could be added into this FE model in order to simulate the real conditions for knee movements; the further researches on quadriceps torques or forces could be estimated in other movements, the more kinematic data could be added into the model to simulate the movements of human lower extremity; and further researches on contact pressure of knee joint could be done in other motions, such as up/down stairs, jogging and riding bicycle etc., so as to get a deeper and better understanding on the biomechanical movement principles of knee joint.
References [1] Anderson,F.C. and Pandy,M.G., Dynamic optimization of human walking. Journal of Biomechanical Engineering, 2001, 123: 381-390. [2] Anderson,F.C. and Pandy,M.G., Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics, 2001, 34: 153-161. [3] Anderson,F.C. and Pandy,M.G., Individual muscle contributions to support in normal walking. Gait Posture, 2003, 17: 159-169. [4] Komistek,R.D., Mathematical Modeling of the Human Arm: an Aid in the Investigation of the Role of Muscle Forces in the Development of Lateral Epicondylitis. PH.D. Dissertation, University of Memphis, 1992. [5] Komistek,R.D., Stiehl,J.B., Dennis,D.A., et al., Mathematical model of the lower extremity joint reaction forces using Kane’s method of dynamics. Journal of Biomechanics, 1998, 31(2): 185-189.
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[6] Komistek,R.D., Kanec,T.R. and Mahfouz,M., Knee mechanics: a review of past and present techniques to determine in vivo loads. Journal of Biomechanics, 2005, 38: 215-228. [7] Farahmand,F., Rezaeian,T., Narimani,R., et al., Kinematic and dynamic analysis of the gait cycle of aboveknee amputees. Scientia Iranica, 2006, 13(3): 261-271. [8] Yang,J.Y., Wang,R.Y., Xiong,K.Y., et al., The EMG determination and analysis of quadriceps isokinetic concentric contraction. Journal of Beijing University of Physical Education, 1995, 18(4): 28-34. [9] Jiang,H.Y., The relationships between integrated EMG activity and mechanics parameters of human quadriceps muscle during voluntary contraction. China Sports Science, 1989, 3: 50-54. [10] Janet,L.K., Marcus,G.P. and John,C.P., Moment arm of the patellar tendon in the human knee. Journal of biomechanics, 2004, 37: 785-788. [11] Dimitrios,E.T., Vasilios,B. and Paula,J.R., In vivo changes in the human patellar tendon moment arm length with different modes and intensities of muscle contraction. Journal of biomechanics, 2007, 40: 33253332. [12] 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. [13] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Finite element analysis of human knee joint in varusvalgus. Clinical Biomechanics, 1997, 12(3): 139-148. [14] Bendjaballah,M.Z., Shirazi-Adl,A. and Zukor,D.J., Biomechanical response of the passive human knee joint under anterior-posterior forces. Clinical Biomechanics, 1998, 13: 625-633. [15] Jalani,A., Shirazi-Adl,A. and Bendjaballah,M.Z., Biomechanics of human tibio-femoral joint in axial rotation. The Knee, 1997, 4: 203-213. [16] Shelburne,K.B., Pandy,M.G. and Torry,M.R., Comparison of shear forces and ligament loading in the healthy and ACL-deficient knee during gait. Journal of Biomechanics, 2004, 37(3): 313-319. [17] 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. [18] Li,G., Gil,J., Kanamori,A., et al., A validated three-dimensional computational model of a human joint. ASME Journal of Biomechanical Engineering, 1999, 121: 657-662. [19] Li,G., Lopez,O. and Rubash,H., Variability of a three-dimensional finite element model constructed using magnetic resonance images of a knee for joint contact stress analysis. ASME Journal of Biomechanical Engineering, 2001, 123: 341-346. [20] Pe˜ na,E., Calvo,B., Martinez,M.A., et al., 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. [21] 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. [22] Guo,Y., Zhang,X.S. and Chen,W.Y., Three-dimensional finite element simulation of the total knee joint in the gait cycle. Acta Mechanica Solida Sinica, 2009, 22(4): 347-351. [23] Zheng,X.Y., Jia,S.H., Gao,Y.F., et al., Modern Sports Biomechanics. National Defense Industry Press, 2002, 100-163: 366-407. [24] 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. [25] Hao,Z.X., Jin,D.W. and Zhang,J.C., Finite element analysis of in vivo tibio-femoral contact features with menisci. Journal of Tsinghua University, 2008, 48(2): 176-179. [26] Harrington,I.J., A bioengineering analysis of force actions at the knee in normal and pathological gait. Biomedical Engineering, 1976, 11(5): 167-172. [27] Moeinzadeh,M.H., Engin,A.E. and Akkas,N., Two-dimensional dynamic modeling of human knee joint. Journal of Biomechanics, 1983, 16: 253-264. [28] Thambyah,A., Pereira,B.P. and Wyss,U., Estimation of bone-on-bone contact forces in the tibiofemoral joint during walking. The Knee, 2005, 12(5): 383-388. [29] Tumer,S.T. and Engin,A.E., Three-body segment dynamic model of the human knee. ASME, Journal of Biomechanical Engineering, 1993, 115: 350-356. [30] Taylor,S.J.G., Walker,P.S., Perry,J.S., et.al., The forces in the distal femur and the knee during walking and other activities measured by telemetry. Journal of Arthroplasty, 1998, 13: 428-437. [31] Taylor,S.J.G. and Walker,P.S., Forces and moments telemetered from two distal femoral replacements during various activities. Journal of Biomechanics, 2001, 34(7): 839-848.
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Table 1. The data table of knee squat angle
No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Results from picture analysis (degree) Squat Angle (thigh Angle (shank angle -level) -level) 5.80 88.84 85.36 4.42 89.71 85.87 7.74 87.69 84.57 8.09 87.42 84.49 8.88 87.37 83.75 12.11 84.29 83.60 13.21 82.77 84.02 15.74 79.35 84.91 21.98 74.09 83.93 33.43 66.18 80.39 45.03 57.79 77.18 59.38 48.87 71.75 65.82 45.60 68.58 72.91 39.55 67.54 80.44 34.25 65.31 89.40 28.36 62.24 99.42 20.73 59.85 111.16 12.37 56.47 121.71 6.07 52.22 130.63 0.56 48.81 135.88 -3.68 47.80 138.65 -6.50 47.84
Results measured by transducers (N) Left front Left rear Right front Right rear (1) (2) (3) (4) 89.42 237.27 103.45 206.60 90.64 240.48 105.90 206.60 85.30 242.61 103.45 208.74 78.89 255.58 100.71 208.51 69.43 258.48 94.60 211.18 100.40 246.89 105.59 187.99 39.83 257.57 96.74 254.67 36.62 278.33 97.05 251.46 36.62 285.19 96.13 221.86 20.60 291.19 80.87 242.31 7.17 303.73 52.19 276.79 16.33 313.57 63.48 272.83 3.05 314.18 3.36 312.19 10.22 236.82 37.23 311.89 10.53 274.96 21.97 324.40 29.30 265.20 48.22 290.22 29.14 257.87 40.89 303.96 43.18 256.81 47.60 298.46 42.11 258.72 39.98 307.92 58.75 241.55 41.81 284.42 81.18 233.76 53.10 274.35 81.94 248.72 54.02 250.55