A new kinematic model of pro- and supination of the human forearm

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1 Jounal of Biomechanics 33 (2) 487}491 Technical note A new kinematic model of po- supination of the human foeam A.M. Weinbeg, I.T. Pietsch*, M.B. Helm, J. Hesselbach, H. Tschene Tauma Depatment of the Medical Univesity of Hanove (MHH), Gemany Institute of Machine Tools Poduction Engineeing, Technical Univesity of Baunschweig, D-3816 Baunschweig, Gemany Accepted 12 Octobe 1999 Abstact We intoduce a new kinematic model descibing the motion of the human foeam bones, ulna adius, duing foeam otation. Duing this motion between the two foeam extem-positions, efeed to as supination (palm up) ponation (palm down), e!ects occu, that cannot be explained by the the established kinematic model of R. Fick fom 194. Especially, the motion of the ulna is not popely epoduced by Fick's model. Duing foeam otation an evasive motion of the ulna is obseved by vaious authos, using magnetic esonance imaging (MRI) technology. Ou new kinematic model also simulates this evasive motion. Futhemoe, the model is enlaged to include angulations of the foeam bones. Using these esults the in#uence of foeam factues on the ange of foeam motion can be pedicted. This knowledge can be used by sugeons to choose the optimal theapy in e-establishing fee foeam mobility. 2 Elsevie Science Ltd. All ights eseved. 1. Intoduction Pefomance of the po- supination motion is impotant to the human capability of acting in a "nemotoial way, which is necessay in an inceasing numbe of technical jobs eveyday life. Foeam factues sometimes esult in boken bones healing in angulated positions (Nilsson Obant, 1977; Kudena, 198). Today, a physicians evaluation of the neccesity to opeate is based on a small amount of available infomation clinical studies. As a esult of this lack of knowledge many patients with fesh factues ae opeated on, to foce the boken bone in a cetain position wheeas the physician is not sue about the e!ect of the cuent bone-position (Weine, 1981). This situation motivated ou eseach with the goal of pesenting a kinematic model, which pedicts the limitation of motion caused by a known angulation. With this knowledge unnecessay opeations can be pevented the esults of the necessay ones can be impoved. Foeam motion eseach began in the ealy 2th centuy. Fick (194) pesented the "st kinematic model descibing foeam motion, which is still used in many publications (e.g. Kudena, 198; Geen Swiontkowski, 1998). The model basis equies the ulna to emain "xed duing the whole otation, as shown in the left dawing of Fig. 1. Accoding to Fick, the h should not stay paallel duing otation, which is easily efuted by pesonal expeience. The ight dawing in Fig. 1 depicts the evasive motion of ulna adius, which ensues the paallelism of the h to the elbow. 2. Methods The new kinematic model uses a database of MRIscans of the po- supination that wee evaluated in 3 healthy foeams fo kinematic behavio (Weinbeg et al., 1997). The scans wee taken at di!eent sections of the foeam fo seveal po- supination-angles. Spatial motion of the bones involved in po- supination equie a spatial mechanism Kinematic model fo healthy foeam bones The eseach pogam is suppoted by the Geman Reseach Foundation (DFG). * Coesponding autho. addess: i.pietsch@tu-bs.de (I.T. Pietsch) //$- see font matte 2 Elsevie Science Ltd. All ights eseved. PII: S ( 9 9 ) The degee of feedom F (d.o.f.) fo a geneal spatial mechanism is given by F"6(n!g!1)# f, (1)

2 488 A.M. Weinbeg et al. / Jounal of Biomechanics 33 (2) 487}491 Fig. 2. Kinematic model of the foeam consisting of fou elements, depicted as a vecto chain. Fig. 1. Fick's model fom 194 (left) does not allow the h to stay paallel to the elbow duing foeam otation. Evasive movement of ulna adius to ensue paallelism of h elbow as pefomed in eality (ight). whee n is the numbe of elements, g the numbe of joints f the d.o.f. of each joint. We chose a closed kinematic chain with fou joints as ou model. With n"g"4 the positive gea constaint of F"1, we "nd using (1) that, f "1!6(4!4!1)"7 (2) which means distibution of seven d.o.f. among the fou joints. Fig. 2 depicts ou mechanism. It has one spheical joint (d.o.f."3) on the poximal end of the ulna (1) one otational joint (d.o.f."1) on the ulna's distal end (2). The adius has one cadanic joint (d.o.f."2) on the distal end (3) a pismatic joint (d.o.f."1) on the poximal end (4) (Kele Findt, 1997). Fo the mathematical fomulation of this system, a closed vecto-chain is intoduced in Fig. 2. The angle α indicates the actual position of po- o supination. Theefoe, the total d.o.f. of the system is one, it is now possible to "nd exactly one clea position of the mechanism fo evey angle α. The length of the bones (the length of the vectos ) is called l the distance between thei centes at thei ends (the length of the vectos ) is called l. " "l, " "l, "f (α). (3) The whole set of vectos descibes a closed kinematic chain. Fom this condition aises the equation " (4) ". The initial conditions in supination position (α"93 :"α ) ae: (α )" (α)"l 1 (5), (α )"l 1, (6) (α )"l!1, (7) (α )" (α)"l!1, (8) (α )". (9) The condition ( (α) (α))( (α) (α)) (1) yields (α)"l (11) sin(α) cos(α). 3 can be detemined by making use of (α)# (α)"l (12) (α)# (α)". (13) With (12) (11), espectively, (13) (11) we obtain (α)"l (1!sin(α)) (14) (α)"!l cos(α). (15)

3 A.M. Weinbeg et al. / Jounal of Biomechanics 33 (2) 487} Eqs. (14) (15), the magnitude of (α) (α)" # #, (16) togethe with (7) esults in 1!sin(α) (α)"l! l!2(1!sin(α)). (17) l!cos(α) Fig. 3. Vectos of the angulated ulna. The vecto is eplaced by the vectos. Fig. 2 sets foth that (α)" (α)", because of the pismatic joint that allows only motions in the diection of the y-axis. In ode to meet (α) (α), (18) complies with (α)" (α)# (α). (19) Hence, l (α)"l! l l!2(1!sin(α)). (2) l 2.2. Angulated bones The basic idea of pedicting the in#uence of angulations on the ange of foeam motion is the calculation of the minimal distance between the two bones of the healthy foeam setting that equal to the minimum allowable distance of the angulated foeam. To do so, we measue the ange of ponation of the healthy foeam calculate the minimal distance between the vectos epesenting ulna adius of ou kinematic model. The ange of otation of the human foeam inceases by the natual bending of the bones, especially of the adius. Although ou model uses staight vectos, we conside this e!ect by calculating the distance of the vectos epesenting the healthy foeam at maximal ponation, which natually can only be achieved though bending of the adius. Theefoe, ou calculation method seves as a type of nomalization fo the in#uence of the bending of the bones. With this value the known size of the patient's bones taken fom the X-aypictues we calculate the distances between the foeam bones in the angulated case as a function of α stop, when eaching the minimal distance of the healthy am Angulated ulna To go futhe fom this point, a model fo the system with angulations is needed. To achieve this an extended Fig. 4. Vectos of the angulated adius. The vecto is decomposed into the vectos. vecto loop, that includes one additional-vecto fo evey angulation is intoduced. The paametes k, k k identify the angulation in the local coodinate-system of the factued bone ae chosen to be taken diectly fom the X-ay-pictues. The local x-component k can be measued fom a.p. X-ay. It is de"ned as positive if it points in diection of the thumb. The local y-component k, taken fom both the a.p. side pictues is measued fom the elbow to point A in supination. The z-component k, visible in the side pictue is positive if the angulation is in palma diection. These vectos ae depicted in Fig. 3 along with one angulation of the ulna. We intoduce the new vectos in Fig. 4. meet the following condition fo evey α: (α)# (α)" (α)# (α). (21) We "nd the initial condition of as (α"α )"!k k. (22) k Using Fig. 3, (5), (2) (22) we obtain (α)" k l ( # ) " k l l #l l l! l l!2(1!sin(α)). (23)

4 49 A.M. Weinbeg et al. / Jounal of Biomechanics 33 (2) 487}491 Only the y-values of (α) is dependent of α, hence, (α)"!k (α). (24) k With (5) (2) the position vecto of point B amounts to (α)" (α)# (α) " l (! l!2(1!sin(α))# 2l ). (25) l l (α) is the di!eence between (α) (25) (α) (24): (α)" (α)! (α) " k (1! k )l (! l l!2(1!sin(α))# 2l ). l l!k (26) Angulated adius The case of the angulated adius is moe complicated, because the adius pefoms a spatial motion, wheeas the ulna moves only in y-diection. To hle this case we de"ne the local coodinate-system of the adius (Fig. 4). The index indicates the local coodinate-system of the adius: e ", e ", e "e e. (27) Now, it is easy to descibe the additional vectos in the new coodinates. "l 1 (28), "k kh, (29) k whee kh"l!k (3) "!k k!k. (31) Using (27), (29)}(31) can be tansfomed as " ) e (32) " ) e. (33) The model can easily be exped to deal with a double-faction of ulna adius by combining (6), (8), (24), (26), (32) (33). 3. Results We lean fom Weinbeg et al. (1997) that the amount of evasive angle of the ulna adius (Fig. 1 * ight half) elative to its supination position is on aveage. This evasive motion in ou model is pefomed by the extension of (Fig. 2). This tanslation can easily be ecalculated fo the natual evasive movement. By using a typical adults elation of 8 : 1 fo the length of the foeam-bones compaed to thei distance in full supination position, we "nd a length incease of 3.1% which is equal to 7.23 of evasion otation. Hence, ou kinematic model is in good ageement with the measued values of the MRI-study. Futhemoe, we will discuss two examples to show the eliability of ou pognoses. 1. The "st patient is female "ve-yeas old has a lefth-side foeam factue of the ulna. Fom the a.p. X-ay-pictues we measued l "153 mm (length of the adius), l "14.1 mm (diamete of the capitulum adii, which we use as a measuement fo l ) the dosal angulation of the factue k "!3.5 mm. Fom the side- X-ay-pictue we "nd the distance fom the factue to the elbow k "46.7 mm the angulation of the ulna towads the adius k "!3.5 mm. The ange of motion of the healthy am amounts to 83 in ponation diection. The angulation of the ulna does not limit the patients ange of motion. By using ou model fo the boken ulna we calculated a ange of motion of 83 in ponation diection, which is seen in eality.

5 A.M. Weinbeg et al. / Jounal of Biomechanics 33 (2) 487} The second example is a 31-yea old male with a adius factue on the ight foeam. The length of the adius amounts l "259 mm, the diamete of the capitulum adii is l "24 mm, measued fom the a.p. X-ay. The paamete of the factue ae k "!5 mm (a.p. X-ay), k "11 mm k "17 mm (k k fom side X-ay). The ange of motion of the left foeam is 753 in ponation diection. The ange of otation of the ight am is esticted to 3 in ponation diection. Calculating the ange of motion using the equations fo the boken adius yields 33 of ponation. 4. Discussion In 1872, Duchenne (1949) descibed the motion of the ulna duing ponation}supination as an ac of a cicle, which involved "st an extension, then a lateal motion lastly a #exion of the ulna. This desciption of the ulna motion was substantiated in 1884 by Heibeg (1884) Dwight (1884). Although this fact was known long ago, many authos "xed the ulna when examining the foeam otation. With this simpli"ed assumption they found a "xed otation axis fo the ponation}supination motion. But fo a coect desciption the motions of both foeam bones, adius ulna ae impotant. New expeiments like the MRI-studies pefomed by Nakamua et al. (1994) Weinbeg et al. (1997) with a moveable ulna descibe the ulna motion fom supination to ponation as the combined abduction extension/#exion motion, which is a cooboation of the ealy esults of Duchenne Hiebeg. Futhemoe, instead of a "xed otational axis fo the foeam otation, a otation-angle dependent scew-axis fo the foeam otation is depicted. We exped ou kinematic model to the case of a boken ulna o boken adius. This novelty can be used to pedict the e!ects of foeam factues on the ange of foeam motion. A compaison with a still small patient database showed that the calculating method fo ponation comes vey close to natue. We cannot make a statement fo supination yet. The kinematic model is mainly based on MRI-studies at this stage. We ae going to make expeiments on dead pobs fo a futhe e"nement of ou model, especially fo the angulated case. A compute tool will be developed that pedicts motion estictions due to foeam factues fom knowledge in human foeam otational kinematics befoe the best theapy is chosen, in espect to helping in the pevention of unnecessay opeations theeby cutting health sevice costs patient isk. Futhemoe, the kinematic model could be used to fo futhe investigations of the ole of the humeo-ulna joint duing ponation}supination. This could help to impove existing elbow-joint postheses. Refeences Duchenne, G. B. A. (1949). Physiologiy of Motion, Demonstated by Means of Electical Stimulation Clinical Obsevation Applied to the Study of Paalysis Defomities. Lippicott, Philadelphia (tanslated by Kaplan, E. B.). Dwight, T., The movements of the ulna in otation of the foeam. Jounal of Anatomy Physiology 19, 186}189. Geen, N. E., & Swiontkowski, M. F. (1998). Skeletal Tauma in Childen. Vol. 3, Saundes, London. Heibeg, J., The movement of the ulna in otation of the foeam. Jounal of Anatomy Physiology 19, 237}24. Kele, H., & Findt, M. (1997). Zu Kinematik eines biomechanischen Modells fuk den menschlichen Unteam. Poceedings of Getiebetechnik: WanemuK nde, Rostock, Gemany. pp. 115}123 (in Geman). Kudena, H. (198). Zusammenhang zwischen Achsenfehlen und FunktionseinschaK nkungen nach Vodeamfaktuen. Unfallchiugie 6(1), 7}13 (in Geman). Nakamua, T., Yabe, Y., & Hoiuchi, Y. (1994). A biomechanical analysis of ponation}supination of the foeam using magnetic esonance imaging: dynamic changes of the inteosseous membane of the foeam duing ponation}supination. Nippon Seikeigekagakuai Zasshi, Japan (in Japanese). Nilsson, B.E., Obant, K., The ange of motion following factue of the shaft of the foeam in childen. Acta Othopaedica Scanavia 48, 6}62. Weinbeg, A.-M., Helm, M.B., Rzesacz, E., Reilmann, L., Kele, H., & Reilmann, H. (1997). Can a limited po- supination caused by axis-defomation of the foeam-bones be pedicted using compute-simulation? Compute Assisted Radiology Sugey. Elsevie, Amstedam. p Weine, B. (1981). SchaftbuK che am Unteam im Kindesalte. Thesis, Klinikum echts de Isa, MuK nchen, Gemany (in Geman).