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Acta orthop. Iugosl., (20) 2-3:35-39, 1990.IZVORNI ZNANSTVENI CLANAK ORIGINAL SCIENTIFIC PAPERMedicinski fakultet LjubljanaUniverzitetski Klinifki centar za ortopediju u L'ubijaniFakultet prirodnih znanosti i tehnologije u ~ ju il ja ni

Medical Faculty of LjubljanaUniv. Clinical Hospital for unhopaedics in L'ubljanaFaculty of Natural Science and Technology of Ljubljana

MATHEMATICAL ANALYSIS OF CHIARI OSTEOTOMY*MATEMATICKA ANALIZA OSTEOTOMIJE PO CHIARIJUA. Iglit, F. Srakar, V. Antolie, V. Kralj IgliE and V. Batagelj

KljuEne rijeEi: biomehanika, lokomotorni sustav, Key words: biomechanics, 2ocomotor system,osteotomija po Chiariju Chiari osteotomy

SAZETAK SUMMARYKonstmiran je jednostavan statiCki trodimemiona-Ian m ~ t e m a t i ~odel kuka kako bi se omogvdilasimulacija flyl ihe osteotomije po mariju. tomse modelu pretpostavlja da je pmsjeEna napetostrn(i5ida u j e h j miSiCno' skupimi jednaka, Slto amo-guduje da se u model uhjufi v& broj miOi& mgoSto ima jednaebi za ssicle i moment mvnoteite bQurjeta optiimizadje. ProraCunolm je utvrdeno da sepomiiicanjem glavice bedrene kosti medijalno za 1cm dko l @ l o smanjuje vdiEilna reaktivne sile kulca.Stoga se zakljuhje

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METHODS

Picture 2.X-ray analysis of the hip after Chiari osteotomy.The cut bone surface ought to be the new enlargedfemoral head roof. The new hip geometry is fixedby the screw.

It has been established by direct in vivo mea-surements with special total hip endoprosthesis(15) that during walking cycle the value of the'CIR strongly varies. For this reason mqny dynamicmathematical models of the hip (2, 3, 4, 5, 6, 7)were constructed in order to simulate the varia-tion of and muscle forces during gait cycle.In these models the number of unknown quan-tities (i.e. joint and muscle forces) exceeds thenumber of model equations. Therefore the sopti-mization technicsu had to be used (16, 17) inorder to solve model equations. Most of the op-timization functions used are physiologicaly bad-ly justified and are chosen merely for practicalreasons (3).

In everyday clinical practice the use of dyna-mic hip models is often too demanding and com-plicated, therefore static mathematical modelsas that of Pauwels is fawoured due to theirsimplicity (8, 12, 13). On the other hand Pauwelship model seems to be oversimplified becauseit is two-dimensional and contains practicallyno direct musculoskeletal anatomical data. The-refore in this work a simple static three-dimen-sional mathematical model of the hip was con-structed in order to simulate Chiari osteotomy,where the hip musculoskeletal anatomical dataof Dostal and Andrews (18) and Johnston (14)are taken into account.

A simple static threedimensional mathematicalmodel was used in order to calculate the hipjoint reaction force = (Rx, ,, Rz) n mono-podal body position (see Fig. 3) before and afterChiari pelvis osteotomy.

The origin of coordinate system was chosenin the femoral head center (denoted + on Fig. 3)so that x and z axis lie in the frontal planeand x and y axis in the sagital plane of the body.I t is assumed that t he origin of the hip jointreaction force also lies in the femoral headcenter. According to the action-reaction forcelaw, the femoral head exerts an opposite force- R on the acetabulum. Other forces acting on

Picture 3.The model d #theone-legged stance. Here W B is thetotal body weight, WL is the weight of the support-ing limb, is the hip j d eaction force, 1 is hdfthe disbance (between the centers of the two ace-tabula, a is zcoordinate of the force (WR L)attachment paint, b is zcoordinate of the 'force WLattachment mint. c is z-coordinate of the crroundreaction force---WB attachment point and x, is thefemoral length. The meaning of the angles andcan be seen firom figure. Origin of the coordinatesystexns (x, y, z) coinaides w th (the center af thehip joint (+).

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Iglif, A., Srakar, F. , Antolif, V., Kralj IgliP, V., Batagetj, V.: Mathematical analysis of W in osteotomythe pelvis are the body weight minus the weightof the loaded leg (WB-x)nd the musclevector resultant force F=,2 . . (1)iwhich is the vector sum of different muscleforces. Individual muscles have one attachmentpoint on the pelvis and the other attachmentpoint on the femur. The attachment of the force(WB-m)lies at the distance d in the antero--posterior direction from the frontal plane back-ward, which passes through both femoral headcenters. According to Bombelli (12) the valued = 0 was used here. The forces acting on theloaded leg are the hip joint reaction force R , theweight of the loaded leg W; originating abovethe knee, reaction force exerted by the ground-WB opposite to the body weight) and the re--ultant of the muscle forces -F.In the applied mathematical model of the hipthe muscles piriformis, gluteus medius, gluteusminimus, rectus femoris and tensor fasciae lataeare included (see Tab. 1). Since gluteus mediusTable 1.The muscle icluded in the model are divided intothree groups according to their musde attachmentpaint positions relative to the frontal plane d hebody: a (atenior), t (middle) and p (posterior). Herethe rdative cross- secbiond area of the i-th musdleA; are determined from Bhe data of Johnston et al.(14). The symbol F, denates the Cth muscle force andf f @he varage muscle tension in the i-th muscle.Muscle Group i Fi Ai figluteus mediusaterior a 1 -F, 0,266 fagluteus minimus-antekw a 2 0,113 f atensor fasciae latae a 3 F, 0,12 farectus fernonis a 4 F, 0,40 fagluteus medius-middle t 5 F5 0,266 f tgluteus minimus-middle t 6 F6 0,113 ftgluteus mediius+pouterior p 7 -7 0,266 f,glutemmirrimus-pos'terim p 8 6 0,113 f ppir4formrils p 9 F 0,lO f,and gluteus minimus are attached to the pelvisover a rather large area, in the presented modeleach of the two muscles is divided into threeparts (18). These segments of gluteus mediusand gluteus minimus and the remaining threemuscles are clasified in three groups accordingto their positions: anterior (a), middle (t) andposterior (p) as represented in Table 1.

Further it was assumed that any muscle forceKcan be approximately written by the followingvector expression:- -Fi= f i *AigSi i = 1,. . 9

where Ai is the relative cross-sectional area ofthe i-th muscle, f i is the average tension in thei-th muscle and si is the unit vector in the direc-tion of the i-th muscle. Unit vector is deter-mined by the coordinate vector of the muscleorigin point on the pelvis [E= (xi, yi, zi)] andthe' corresponding coordinate vector of themuscle insertion on the femur [G = (xi,' yi,' zi')]:

The values of coordinates xi, yi, zi, xi', yi', and zi'for v = 0 and v = 0 (see Fig. 3) are given in Table2 (18). While using the data f or Ta n d ?, the refe-Table 2.Coo~dinates f the muvdle pelvis (xi, y,, z,) and femo-ral (x,', y,', z,') abtachment p i n t s (cm), where x, = 42,3cm, ~ W P 8,45 cm, )=0 and v =0 (18).5 xi Yi Zi xi' yi' ~ i '

rence coordinates given in Table 2, must be trans-formed by the corresponding rotational matrixfor given values of angles 9 and V.- n order to evaluate the hip joint reaction forceR and muscle resultant % the statistic force andmomentum equilibrium equations must be solved,

where-a = (0, 0,a), and where the distance a (see Fig. 3)is defined by the expression (19)where the meaning of the parameter c is defi-ned in Figure 3. The values of b and c were de-termined according to McLeish and Charnley (19)

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Acta orthop. Iugosl., (20) 2-3:35-39, 1990. ' - .

as a function of the half the distance between thecenters of the two acetabula 1:

where the value of the angle g, = -O.SO. Theangle is determined by the equation (see Fig.3):

sin v = b/q. (8)When determinig the value of WL. experimental-ly obtained approximative equation WL= 0,161 ..WB is used (20).

In the presented model it is assumed that theaverage muscle tensions of individual musclesare equal in the single muscle group (see Table1). This presumption enables us to include in themodel the number of muscles which exceeds thenumber of model force and momentum equili-brium equations without any optimization me-thod.The variation of femoral head center position(xd, dz) was simulated by changing the inter-hip distance 1 --+ 1 f - z (1 f = 8.45 cm)and by changing the coordinate of all muscleattachment points included in the representedmodel. The value of the angle was kept con-stant during the variation of pelvis configura-tion.RESULTS

Figure 4 shows the dependence of magnitudeof the hip joint reaction force on the femoralhead shifts in the medio-lateral and superior-in-ferior directions, where the femoral head shiftsare presumed to be the consequence of the Chi-ari osteotomy. It can be seen from Figure 4 thatthe magnitude of the hip joint reaction force(and therefore also the corresponding pressureon the femoral head) decreases as the femoralhead is shifted medially, while it increases as thefemoral head is shifted laterally in accordancewith the results of Johnston e t al. (14). On theother hand it can be seen from Figure 4 that fe-moral head shifts in superior-inferior directionshave nearly no influence on the magnitude of thehip joint reaction force.

CONCLUSIONSIt is shown that the simulted femoral headshifts in medio-letaral directions have signi-ficant effect on the hip joint reaction force and

therefore also on the corresponding pressure onthe femoral head bearing area. I t was calculated

Caldlated magnitude of the hip jaint reaotion forcewith respect to the body weight (R/WB) as a func-tion of the femoral head shifts in medio-lateraldirection (dz) and 'superior-inferior direction (Ax).Parameters used in these caloulations are cp =--0.5 a(19), x, = 42.3 cm and lRep = 8.45 om (18). Thereference values of the muscle attachment pointcoodnattes for the given of (p and d&ted valueof (see eq. 8) are determined by the rotationaltransformation of rand 7 values g i v q in Table 2.

that the shifting of the femoral head mediallyfor 1 cm reduces the magnitude of the hip jointreaction force by approx 10O10. It seems thatlong lasting overexertion of the hip joint reac-tion force causes development of osteoarthritis.Since medialization of femoral head as a conse-quence of Chiari osteotomy causes a decrease ofthe magnitude of the hip joint reaction force,this appears to be favourable. We suggest thatreduction of the hip joint reaction force aftermedialization at Chiari osteotomy causes s u pplementary protection from cartilage degenera-tion in addition to the primary effect of the in-creased weight bearing surface area after me-dialization. This is especially important whena hip with an incipient coxarthrosis is con-sidered. Therefore it could be concluded thatwhile performing the Chiari pelvic osteotomy, thefemoral head must be shifted medially as far aspossible in order to reduce the hip joint reactionforce and the-corresponding pressure on the fe-moral head.

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I IgliE, A., Srakar, F., Antolie, V., Kralj IgliC, V., Batagelj, V.: Mathematical analysis of Chiari osteotomy

Symbols-R hip joint reaction forceR magnitude of the hip joint r e a d o n force FRj j-th component of the hip jaint reactionforce, j = x, y, z- 1WL weight of the supporting limb-WB total body weight a-Fi (i-thmuscle force, i = 1,. .. 9- bFi magnitude of @he -th musole force

= (xi',yi', zi'), posi ti ~n vector of &e iithmuscle femoral attachmentpointvector sum of the muscle forces aoting onthe pelvishalf the distance between the centew of thetwo acetabulazcoordinate of the force (W g-WL)attachment pointzcoordinate of the force WL ttachmentpoint

Ai relative cross-sxtional area of the itith c zcoo~&~natef the ground reamim forcemuscle force - WB attachment pointfi average m u d e tension of the i-th muscle x, femoral lengthforce- AX, AZ shifts of 'the femoral head in superior-Si unit vedtor in direction of [the iath musole -inferior and mediio4ateral direobiom-G = '(xi, i, zi), p ~ s i t i i ~ ~ ~eotor of t he iithmuscle pelvis attaahmentpoint

LITERATURE1. Chiari, K.: Medial displacement osteotoony of th epelvis, CKn. Onthap., 1974; 98:55-71.2. Cmwninslhidld, R. D., Johnston, R. C., Andrews,J. S. in Brand, R. A.: A ibiameohanical investiga-'tion d he human hip, J. Biomechanics, 1978; 11:75-45.3. Crcrwninlshidd, R. D. in Brand, R. A.: A physic+lo$ical bawd cniterion af muscle force prdationin loaxnotion, J. Biomechd'os, 1981; 14:793--801.4. Brand, W. A. in Pedersen, D. R.: Computer model-ing of surgery and a consideration of the meoha-nical effects of proximal femoral osteotonnies, v:The Hip Socidty Award Papers tUSA), 1984; ~ p .193-210.5. Seireg, A. in AAmrilkar, R. J.: The pred idw ofmuscular load shaning lanand joint forces in thelower extremities during wdlking, J. Bimnecha-nics, 1975; 8:89-102.6. Gilksta, D. N., Tonidis, T. G. in Stinivasan, T. M.:Human Gait Analysis: Determination d nstanta-

n m 6 joint reactive forces, musale forces and $$hestress distribution in bone segments, Part II.,Biwned. Techn., 1976; 2156-74.7. Pedmti, A., Krishnan, V. V. tin Stark, L.: Opti-mization af Musole-Force Sequendng in HrltmanLocomobion, Math. Biosci., 1978; 3857-76.8. Pauwds, F.: Bimechanics of the Normal andDiseased Hip, Spninger-Verlag, Berlin, Hddelberg,N w onk 1976; pp. 1-276.9. ~Paul, . P.: Eoengineening s tudies of the farcestransminted by Dhe jlaimts, 11. Engineering analysis,

Y: Kenedi, R. M. (eds.): Bimeclhmics and relatedbimgineening topics, Pergamon Press, Oxford,1%5; pp. 369-380.

10. W~illiamls,J. F. in Svensson, N. L.: A Force Ana-lysis of )the Hip Joint, Homed. Engng., 1968; 3:-170-- -. .11. goal, V. K. h SYensson, N. L.: Forces on thepdvis, J. Biomechanics, 1977; 10:1%-200.12. BiambeE, R.: Osteoafithritiis of the Hip, Spf ige rVeflag, Berlin, Haidelberg, New Yovk 1983; pp.1-386.13. Maquet, P. G. J.: B~iomechanics of the Hip,Springer-Verlag, Be lf h, Heidelberg, New Y d ,T&YO 1985; p ~ .-309.14. Johnston, R. C., Brand, R. A. in Cm~ nins hiel d,R. D.: Reconstruatim of th e Hip, J. Bone andJ&t Surg., 1979; 61A:639652.15. Rydeill, N. W.: Forces atin%on \the femoral head-pwsthesis, Aota Orthap. !!&ind., 37 (Suppl. 88):1%; 1-132.16. Seireg, A. in Arvjlkar, R. J.: A mathematical mo-del for evaluation of forces in the lower extre-&ties of h e m u ~ ~ s k e l a W 1ystem, J. 'Bitme-chan~ios,1973; 6:313-326. 017. Penmd, D. D., Davy, D. T.h Singh, D. P.: An wti-mizartion approach to itendon force anwlysis, J. Eo-meuhamics. 1974: 73123-129.18. Dostal, w.' F. in Andrews, J. G.: A three.dimaiio-nd ulomechadic~lmodel of hip musculature, J.~fiomeahamics, 981; 14:803--812.19. MdLaish, R. D. in Gharnley, J.: Abduation ~KHUXin the oneilegged stance, J. Birrmeohanios, 1970; 3:191-209.20. lalauser, 13. E., MoCormille, J. T. in Young, J. W.:Weight, Valume an d cent re of mass af segmentsof the human bady, Aerospace Medical ResearchLaboratory, Aerospace medical Divis~im, WPM'Base, OH @JSA), 1969.