Custom-made Biomechanical Model of the Knee Joints

Transcription

1 Cusom-made Biomechanical Model o he Knee Joins Chin-Yu Wang, Tsung-Lin Tsai 2, Chien-Fen Huang 3 Deparmen o Mechanical Engineering, Cheng Shiu Universiy, Taiwan, R.O.C., 2 Physical Medicine and Rehabiliaion Deparmen, Mackay Memorial Hospial, Taiung Branch, Taiwan, R.O.C. Absrac: This paper obained he daa o cusom-made ibia, emoral joins and meniscus morphology by sudying relaing researches and consruced several sagial secion o he knee join. Based on he 2D bone morphology o emoral condyle and he conac radius relaionship o he ibial plaeau, i creaed he design o he sphere curvaure o emoral condyle o conorm o he demands o he paiens. To simulae he various meshing curved surace beween bones, his paper considered using sphere o i emoral condyle, using emur image daa o i he emur ino spheres and using he Herz s elasic conac heory o consruc a emur-ibia and emur-meniscus biomechanical model. Lasly, using he conac sress and conac deormaion obained by solving he numeric examples under speciic cases can be used as a reerence in he design and producion o uure proshesis. [Chin-Yu Wang, Tsung-Lin Tsai, Chien-Fen Huang. Cusom-made Biomechanical Model o he Knee Joins. Lie Science Journal. 202;9(): ] (ISSN: ).. 70 Keywords Elasic Conac, Biomechanical Model, Femur, ibia. Inroducion Recenly, he qualiy o compuer medical image or ariicial proshesis grealy improved. Due o hese improvemens, he paien who can only use cruches or canes o walk or even he paiens who are ampuaed in he pas can walk like normal people and his improved he qualiy o heir lie grealy. No only ha knee join is he mos complicaed join consruced and he bigges join o a human body, i has high expecaions in is movemen uncion. Every year, a grea number o paiens have o undergo ariicial knee arhroplasy. A presen, he people ha underwen ariicial knee arhroplasy, he limbs o he paiens can load, srech, abduce and roae in a very sable manner. Mos o he ariicial knee proshesis being clinically used should be impored because here are very ew companies, such as Unied Orhopedic Corporaion, who design and develop ariicial knee producs. Imporing proshesis is quie expensive and he producs are designed based on knee join parameers o Americans and Europeans which someimes are incompaible o Asian knees. To recover he complex aciviies o he knees, he ariicial knee proshesis designed should be compaible in anaomy, biomechanical and kinemaics. Sress analysis allows he sress morphology o he proshesis and he human skeleon o be more compaible o improve he long-erm sabiliy o he proshesis and avoid unreasonable sress peak and conac area morphology. Moreover, because he model o he reconsruced bones can be roaed and cu a any angles, i provides a good mehod o he complex surace geomeric morphology researches and simulaion surgeries o he knee. To design a beer knee proshesis compaible o Taiwanese, cusom-made normal knees underwen bone morphology measuremen and kinemaic researches o creae muli-segmened meshing curved surace and Herz heory was used o derive he biomechanical model o he emur and ibia in dieren meshing secions. In addiion, he model is used o derive he average conac sress, conac area in he meshing area. These resuls can serve as a reerence o clinical diagnosis and cusom-made proshesis manuacure. The sudies relaing o he emur and ibia conour morphology are as ollows: David Siu measured he medial emoral condyle and is disal condyle, radius o he paellar surace and arc angle and he laeral emoral condyle and is disal condyle, radius o he paellar surace and arc angle o 5 cadever s knee joins aer CT scanning a 3D reconsrucion []. The resul o he sudy o Elias where 0 samples o cadavers wih normal knees are disseced and compared o 6 normal adul knees showed paiens o dieren races, dieren sizes or due o curve iing resuls o dieren arcs or dieren emoral condyle joins [2]. Zhou s research shows ha he widh o he medial and laeral emoral condyle and he widh o he medial and laeral ibia plaeau are very similar which proves ha he widh o emoral condyle and ibia plaeau is relaed. Greaer emoral condyle widh shows greaer ibia plaeau widh. Thus, he value o emoral condyle and ibia plaeau can be used as he index o represen he size o he knees. His simpliied model shows ha sagial emoral condyle proile can be ied ino hree secions arc curves. Moeinzadeh e al [3]. researched on he biomechanics o he emur ibia o knee joins where hey are he irs o creae an ariicial knee dynamic model human knee join 469

2 and ixed he emur. When he model is used o measure he ixed emur, he ibia is relaive o he moion o he emur and can compue or he conac sress o he knee joins. Wongchaisuwa e al [4]. creaed anoher knee dynamic model where he emur is ixed and he ibia is simulaed as a simple pendulum o analyze he conrol sraegy which he emur is relaive o ibia s sliding and rolling. Komisek e al [5]. creaed a model by applying Kane s mehod where he peak o he conac sress beween emur and ibia was obained by analyzing dieren walking speeds [6,7]. Wang released emur and creaed a wo-dimensional biodynamic model o emur and ibia [8~]. Tumer and Engin creaed a emur-ibia-paellar dynamic model o human knee. The model simulaed ha when he emur is ixed, ibia is relaive o he relaive moion o emur and paellar [2]. Wang released emur again and creaed a emur-ibia-paellar hree knee join 2D meshing moion mahemaical model [3]. Jia creaed a lower limb 3D model o compue or he orce and orque o he emur and ibia during gai cycle [4]. Please reer o [5,6,7] or oher relaed researches. The main ask o his paper is o discuss he sagial plane secion near emoral condyle and ibial plaeau under he 3D geomery o he obained emur and ibia hrough geomeric analysis soware o obain he bone conour shape o he knee join. Numerical mehod was applied on he conour o i he arc wih dieren curvaure and he conour is also ied ino several spheres wih dieren diameer by seing he lengh o an appropriae Z axle. Similar cross secion was also applied o proximal ibia o explore he conac sress o he emoral and ibial meshing. The resuls include:. applying he elasic heory o simpliy he biomechanical model o emur, ibia and meniscus can compue he conac sress o he emur, ibia and meniscus. 2. hrough image processing, use he emoral condyle and ibial plaeau meshing o i dieren spheres and calculae he conac sress and conac area according o dieren paiens. Through he sagial secion o he emoral condyle and ibial plaeau saed above, we can creae sphere radius o conac morphology in any secion o use as he knee sress analysis oundaion which can be used as he design and producion o cusom-made ariicial knee joins. 2. Research Mehodology The research mehod o his paper include consrucion and analysis o he conac morphology o proile iing o he emur and ibia under he same secion and solve he problem by using he meshing surace as a muliple sphere radius according o he bone morphology o he dieren paiens. 2. Femur iing calculaion Figure shows he emur/ibia meshing 3D geomery. By he use o he soware, we can obain he bone morphology o he sagial plane and he bone meshing surace as shown in Figure and 2. Figure. Geomeric model operaion Figure 2. Sagial Skeleal Proile 2.2 The sphere iing explanaion using emur is as ollows Consider he Z-direcion lengh on Figure and wih C given, he sphere equaion is esablished. 2 R ( x A) ( y B ) ( z C ) () R x 2Ax A 2By B z 2Cz C (2) where (A,B,C) are he coordinaes o he cener and R is he radius o he sphere. When a=-2a, B=-2B, c=-2c, d=a2+b2+c2-r2, anoher orm o sphere ormula can be derived as shown in equaion (3) x 2 +y 2 +c 2 +ax+by+cz+d=0 (3) 470

3 As long as a, b,c and d are derived, we can obain he cener and radius parameers: a A 2 b B 2 c C 2 2 D a b c 4 d In ball poin se (X i,y i,z i ),i(,2,3 N), D i is he disance beween midpoin and he cener. D ( X A ) ( Y B ) ( Z C ) (5) 2 i i i i Thus, we can obain he disance beween he cener o poins (X i,y i,z i ) and square dierence o he lengh o radius (δ): i R (4) D (6) Φ(a,b,c) can be obained as he square leas o δ i : ( a, b, c, d) i ( Xi Yi Zi axi byi czi d) (7) Use parial derivaives o obain a,b,c,d values and he value o Φ(a,b,c,d) is he smalles, hus ( a, b, c, d) 2( Xi Yi Zi axi byi czi d) Xi 0 a ( a, b, c, d) 2( Xi Yi Zi axi byi czi d) Yi 0 b (8) ( a, b, c, d) 2( Xi Yi Zi axi byi czi d) Zi 0 c ( a, b, c, d) 2( Xi Yi Zi axi byi czi d) 0 d From equaion (8), he values o a,b,c,d can be derived. Equaion (4) derived he cener and radius which compleed he eaure exracion o he spaial sphere. 2.3 Axis line alignmen The design mehod o he ibial plaeau is similar as he one above. The dierence is ha he diameer o he ibial plaeau is bigger and i can be la, concave or convex aer iing according o he dieren elasic model. In obaining ibia and he ibia cross-secion, several imporan axis need o be i ino one sraigh line, hus he verical iing algorihm is needed. The axis line equaion is Ax+by+=0 (9) In using he leas square i, he deviaion ( ) o equaion (0) should be he smalles. 2 ( axi byi ) (0) Equaion (0) separaely solves or he parial derivaives o a and b seing he parial derivaives o 0. 2 ( axi byi ) xi 0 a () 2 ( axi byi ) yi 0 b Equaion () can obained he smalles a and b values o he deviaion ( ) o i he required sraigh line. 2.4 Bone proile morphology relaed auhenicaion Several researchers conduced sudies regarding he curvaure design o disal emur emoral condyle among which Zhang measured hree iing image as shown in Figure 3 and 4. Similar sudy [8], Chou has he curvaure design o he 3 blocks o he emoral condyle. Table describes he measuremen resuls and errors o he sample paiens in he saisics [3]. Figure 3. The X-ray shows he 3 arcs o he disal emur aricular surace (re. 6) Figure 4. The measured parameers o disal emur Table. Parameers o disal emur (re. 4) Geomeric dimension Values r (RPMFC) ± 3.59 φ (APMFC) 4.72 ± 6.7 r 2 (RDMFC) ± 6.79 φ 2 (ADMFC) ± 7.68 r 3 (RPaMFC) 25.7 ± 2.45 φ 3 (APaMFC) ±

4 2.5 Join elasic conac model The organizaional orm o he join o emur and ibia is shown in Figure 5. The disal emur and ibial plaeau don have a pure rolling meshing relaionship which aecs he lexing posure and sreching ligamens and because he meniscus roll buer is usually a sliding movemen. The moion range o every bone has is limi. Thus, he curvaure geomeric design and he consrucion o elasic mechanical model helps simulae conac sress and conac area morphology hrough dieren lexing needs. The E, is se as he elasic modulus and Poisson s raio o knee emoral elasic meshing, respecively. E, is he occlusion o ibial elasic modulus and Poisson s raio, respecively. R, R ' is he wo main curvaure radius o he elasic emur conac surace, respecively. R, R ' is he wo main curvaure radius o he elasic ibia conac surace, respecively. a and b are he long and shor axis he ibial-emoral join conac spnere surace. is he relaive displacemen o he emur and ibia elasic cener. q 0 is he average sress o he ibial-emoral join conac elasic surace. P is he axial saic pressure. Based rom Herz heory, he maximum sress, elasic occlusal conac area and he deormaion o he soluion beween he meshing conac surace o he knee, shown below, among 3P which he average sress is q0 and a and 2 ab b separaely represens he wo circle axis. a b 3 P 3 4 A E E 3 P 3 4 A E E For sphere,he coeicien a=b. 3P q0 2 ab x y q q a b A P 3 E E A 2 R R ' R R ' B 2 cos 2 2 R R ' R R ' R R ' R R ' B c o s A (9) In he equaion, is he wo curvaure angle o (2) (3) (4) (5) (6) (7) (8) R and R in he beween planes. Moreover,,, is used as he parameers where he values can be based on he value derived rom equaion (9) where he resuls is shown in Table 2. I he wo equaions o he meshing surace o he knee join is known, he maximum conac sress, conac area and meshing deormaion o he ibial-emoral surace o he knee can be solved according o equaions (2)~(9). Figure 5. The inra-aricular organizaional proile Table 2. The coeicien o equaions 3. Resuls and Discussion Combining he resuls o lieraures based on emoral condylar morphology, he hypohesize ibia and emoral condyle soware conac o his sudy separaely considered ibia plaeau radius as -400mm and plaeau as 400mm where he conac radius (R) and average pressure (q 0 ) are shown in Figure 6 and he parameers are shown in Table 3. I he conac beween meniscus and emur is considered, he meniscus coeicien is E=59MPa and ν=0.49. The meniscus and emur conac radius considers he meniscus as -400mm, plane as +400mm and he emur condyle changes 2 each ime rom 20 o 40. The conac radius and average pressure is shown in Figure

5 Table 3. The coeicien o wo cases Case Case2 E = E = 5MPa E = 59Mpa ; E = 5Mpa ν = ν = 0.46 ν = 0.46 ; ν = 0.49 h = h (carilage hickness) = mm h = mm F: bearing capaciy F: bearing capaciy Figure 6. Conac o emur and ibia Figure 7. Conac o emur and meniscus 4. Conclusion The demand or ariicial knee in he marke and he qualiy o is demand rapidly works oward he direcion o cusom-made. According o he human skeleon conour modeling echniques and hen according o he paien s weigh, dieren liesyle and habis and movemen paerns o he knee, designing and consrucing knee proshesis is quie complex and challenging. Wih a main direcion o cusom-made and wih he help o he resuls medical images, his paper designed a emur condyle sphere and sphere or concave sphere o ibia plaeau biomechanical model hrough change curvaure value o he bone processing echnique and wih he elasic heory o Herz beween balls o calculae he conac sress, conac area and deormaion in he meshing o he joins o serve as a reerence in he design, producion and analysis o cusom-made proshesis. 2. Elias SG, Freeman MAR, Gokcay EI: A correlaive sudy o he geomery and anaomy o he disal emur. Clinical Orhopaedic 990; 260: Zhou, Feihu, Measuremen and relaed research o he hree-dimension morphology o he healhy Chinese knee, The Orhopedic Journal o China, 2004, 2(8). 4. M.H Moeinzadeh, A.E Engin, N. Akkas, Two-dimensional dynamic modeling o human knee join, Journal o Biomechanics, 983 vol.6, C. Wongchaisuwa, H. Hemami and H.J. Buchner, Conrol o sliding and rolling a naural joins, ASME, Journal o Biomechanical Engineering, 984, vol.06, R.D. Komisek, Mahemaical modeling o he human arm: an aid in he invesigaion o he role o muscle orces in he developmen o laeral epicondyliis, [PH.D. Disseraion], Universiy o Memphis, R.D. Komisek, J.B.Siehl, D.A. Dennis, Mahemaical model o he lower exremiy join reacion orces using Kane s mehod o dynamics, Journal o Biomechanics, 992, vol. 3, X.S. Wang, N. Akkas, Two-dimensional biodynamic model o he human emur-knee join-ibia sysem, French: The 4h Inernaional Conerence on Biomechanics, X.S Wang, Two-dimensional biodynamic model o human lower exremiy [Ph D] Disseraion, Ankara, Turkey: Middle Eas Technical Universiy, S.T. Tumer, X.S. Wang, N. Akkas, A planar dynamic anaomical model o he human lower limb, Biomedical Engineering, 995, vol. 7, Wang, Xishi, Bai, Ruipu, S.T. Tumer, The mahemaical model o a human knee join and is biomechanical response, Medical Biomechanics, 998, vol. 3(), S.T. Tumer, A.E. Engin, Three-body segmen dynamic model human knee, ASME, Journal o Biomechanical Engineering, 993, vol. 5, Wang, Xishi, Bai, Ruipu, S. Turgu Tumer, e.al., Femur-ibia-paellar hree knee join meshing moion mahemaical model, Biomedical Engineering, 998, vol. 8(4), X.H. Jia, M. Zhang, W.C.C. Lee, Load ranser mechanics beween rans-ibial prosheic socke and residual limb-dynamic eecs, Journal o Biomechanics, 2004, vol. 37, Bo Wun Huang, Huang Kuang Kung, Kuan-Yu Chang, Po Kai Hsu, Jung-Ge Tseng, Human Cranium Dynamic Analysis, Lie Science Journal, 2009; 6(4): Chang Kuan Yu, Chang Hong Chang, Yu Pu Ping, Yen Ke Tien and Huang Bo Wun, Naural Properies in a Micro Drill Cuing ino Bones, Lie Science Journal, 2009; 6(4): Jeng-Nan Lee, Hung-Shyong Chen, Chih-Wen Luo and Kuan-Yu Chang, Rapid Prooyping and Muli-axis NC Machining or The Femoral Componen o Knee Proshesis, Lie Science Journal, 200; 7(): Zhang, Xiuli, Liu, Yong, The correlaion beween he geomery and anaomy in he disance o he disal end o emur, Hebei Medical Journal, Jan 2008, vol. 30, No.. Received December 30, 20 Reerences. David Siu,e al. Femoral aricular shape and geomery-a hree-dimensional compuerized analysis o he knee. Journal o Arhroplasy 996; (2):