A Method to Determine the 3 D Stiffness of Fracture Fixation Devices and Its Application to Predict Inter Fragmentary Movement 1997 Journal of Biomech

Journal of Biomechanics 31 (1998) 247—252 A method to determine the 3-D stiffness of fracture fixation devices and its application to predict inter-fragmentary movement Georg N. Duda*, Helmut Kirchner, Hans-Joachim Wilke, Lutz Claes Department of Unfallchirurgische Forschung und Biomechanik, University of Ulm, Ulm, Germany Received in final form 1 October 1997 Abstract Inter-fragmentary movement considerably influences the fracture healing process. Large shear movement delays while moder
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  Journal of Biomechanics 31 (1998) 247 —  252 A method to determine the 3-D stiffness of fracture fixation devicesand its application to predict inter-fragmentary movement Georg N. Duda * , Helmut Kirchner, Hans-Joachim Wilke, Lutz Claes  Department of Unfallchirurgische Forschung und Biomechanik, Uni v ersity of Ulm, Ulm, Germany Received in final form 1 October 1997 Abstract Inter-fragmentary movement considerably influences the fracture healing process. Large shear movement delays while moderateaxial movement stimulates the healing process.To be able to control the mechanical situation at a fracture site and to achieve optimalbony healing it is essential to understand the relationship between inter-fragmentary movement, bony loading and fixation stiffness.A 6  6 stiffness matrix is introducedwhich completely describesthe linearrelationship betweenthe 6 inter-fragmentarymovementsand the resulting bony loading (3 forces and 3 moments). Further, it is illustrated that even in relatively stiff external fixateurconstructs simple axial loading of the bony fragments leads to complex inter-fragmentary movement. When the 3-D stiffnessdescription is multiplied by the load state in sheep tibiae, movements similar to those measured  in vivo  are calculated. The relation-ship between axial compression and medio-lateral or dorso-ventral shear varies depending on the mounting plane of the externalfixateur.The authors conclude that a single value is not sufficient to describe the mechanical relationship between inter-fragmentarymovement and bony loading. Only a complete description of fixation stiffness allows prediction of inter-fragmentary movement anddifferentiation between various configurations of fixation devices and their potential for mechanically promoting bony healing.   1998 Elsevier Science Ltd. All rights reserved. Keywords :  Fracture healing; Inter-fragmentary movement; Fixation stiffness; External fixation 1. Introduction Independent of the method of fracture fixation, bonehealing is generally subjected to complex inter-fragmen-tary movements. It is well accepted that this movementinfluences the fracture healing process both in its typeand rate of healing (Claes et al., 1995; Goodship andKenwright, 1985; Kenwright and Goodship, 1989).Axial movement using fixateur externe configurationshas been analyzed in various  in vitro  as well as  in vivo studies (Cunningham et al., 1989; Goodship et al., 1993;Goodship et al., 1988; Hoffmann et al., 1991; Kenwrightet al., 1991; Kristiansen et al., 1987; Lippert and Hirsch, * Correspondence address: Forschungslabor der Unfall- undWiederherstellungschirurgie, Charite´, Humboldt Universita¨t zu Berlin,Augustenburger Platz 1, D-13353 Berlin, Germany. Tel.: # 49 30 45059079; fax: # 49 30 450 59969; e-mail: duda @ 1974). It has been shown that a large stiffness resulting insmall axial movements minimizes the risk of pseudar-throses (Schenk et al., 1986; Stu¨rmer, 1988). However,a certain amount of inter-fragmentary movement is ne-cessary to achieve sufficient mechanical stability in thenewly formed bone (Kenwright and Goodship, 1989;Molster and Gjerdet, 1984; Molster et al., 1982).Animal experiments have shown that an axial inter-fragmentary movement within the range of 0.2 —  1.0 mmseems to be optimal for fracture healing (Claes et al.,1995; Goodship et al., 1988). However, it remains unclearhow fracture healing is related to the movement compo-nents other than that in the axial direction. From  in vivo experiments and clinical experience the impact of axialand shear movements on the healing callus has beenqualitatively described (Yamagishi and Yoshimura,1955). Although a large number of fixation devices invarious configurations are clinically used, the 3-D inter-fragmentary movements actually occurring  in vivo  aremainly unknown. 0021-9290/98/$19.00    1998 Elsevier Science Ltd. All rights reserved. PII  S0 0 21 -9 2 9 0 ( 9 7 ) 0 0 1 1 5 - 2  Since axial and shear movements appear to influencethe fracture healing processes differently, it would bebeneficial to know the 3-D inter-fragmentary movementprior to mounting of the fixation device (Hoffmann et al.,1991). Due to the highly asymmetric nature of fixateurexterne devices an axially rigid construct may experiencelarge shear deformations under bending loads. Onlythe complete description of the 3-D fixation stiff-ness would allow the prediction of the full set of inter-fragmentary movements occurring under complex in vivo  loading (Gardner et al., 1996). Since the bonyload state is primarily independent of the method of fracture fixation, knowledge of the constructs’ 3-D stiff-ness would allow differentiation between those fixationdivices that provide lesser or greater inter-fragmentarymovement. Fixation devices could then be configuredpre- or intra-operatively to meet the desired specifica-tions, e.g. axial movement between 0.2 and 1 mm, mini-mized shear.The goal of this study was to develop a method todetermine the 3-D stiffness of fracture fixation devicesand to predict inter-fragmentary movement as a functionof fracture location and fixateur mounting. 2. Materials and methods For determination of the 3-D fixateur stiffness, anASIF external fixateur (double tube, steel rods, twoSchanz screws per fragment) was selected. The specificdimensions are given in Fig. 1 and are identical to thoseused previously in an  in vivo  recording of inter-fragmen-tary movement in sheep (Stu¨rmer, 1988). The externalfixateur was mounted to a pertinax rod (diameter 20mm,length 300 mm) which was then osteotomized at mid-span between the inner Schanz screws, creating a 4-mmfracture gap.Prior to testing, a 3-D goniometer system (accuracy0.1 mm, 0.1 ° ) was attached to the fixateur configurationusing the proximal and distal Schanz screws (Fig. 2,Wilke et al., 1994). A tight fit of the Schanz screws in thepertinax rod eliminated pin bone interface motion in this in vitro  experiment.Mathematically, the loads between the proximal anddistal fracture fragments (three forces and threemoments) can be related to the inter-fragmentarymovements (three translations and three rotations) bya stiffness matrix. If the loads and gap movements arewritten as column vectors each, the fixation stiffness isdescribed by a 6  6 stiffness matrix with 36 unknownstiffness values [Eq. (1)]. These values are constant whilethe load-displacement relationship is linear. To simplifythe relationship between the 3-D loading and 3-D inter-fragmentary movement, the current analysis was per-formed under the assumption that the entire systembehaved linear.The diagonal values of the stiffness matrix refer to theventral shear stiffness, lateral shear stiffness, compressionstiffness bending stiffness around an axis perpendicular,bending stiffness around an axis parallel and torsionalstiffness.To determine the 36 unknown values of the stiffnessmatrix six mechanically independent load cases (consist-ing of 3 forces and 3 moments each) and correlatinginter-fragmentary movements (3 translations and 3 rota-tions) are necessary. In this analysis the six load caseswere selected to be axial compression ( F  ), torsion ( M  ),4-point-bendingaround an axis parallel ( M  ) and an axisperpendicular ( M  ) to the fixateur plane, cantilever ben-dingaroundan axisparallel ( F  , M  ) and anaxis perpen-dicular ( F  , M  ) to the fixateur plane (Fig. 3). The loadsand movements were oriented according to a coordinatesystem displayed in Fig. 1.   F  F  F  M  M  M     s   s   s   s   s   s  s   s   s   s   s   s  s   s   s   s   s   s  s   s   s   s   s   s  s   s   s   s   s   s  s   s   s   s   s   s                  i 1, 2 ,6. (1)The units of the stiffness values are Nmm   for  s   to s  ,  s   to  s  ,  s   to  s  , Ndeg   for  s   to  s  ,  s   to  s  , s   to  s  , Nmmm   for  s   to  s  , s   to  s  , s   to s   and Nmdeg   for  s   to  s  ,  s   to  s  ,  s   to  s  .All experiments except those in torsion were per-formed on a materials testing machine (Zwick 1454,Zwick GmbH, Ulm, Germany) with specially designed jigsto allow compression,4-point-bendingand cantileverbending. Torsional testing was performed on a custombuilt device. The inter-fragmentary movements were re-corded during loading and unloading at 2 mmmin   or2 ° min  . Maximum loads were chosen to be within thelinear behavior of the external fixateur constructs (Table1). Inter-fragmentary movements were recorded witha personal computer and custom software (Wilke et al.,1994).To analyze the influence of the free length of theSchanz screws, the carbon rods were moved incremen-tally 2.5 mm towards and away from the rod axis cre-ating distances between the fixateur body and rod axis of 59.0, 56.5, 54.0, 51.5 and 49.0 mm (Fig. 1).To predict inter-fragmentary movement the bony loadstate was derived from a 3-D model of the hind limb of a sheep (Duda and Claes, 1996). It was assumed asa simplification that the bony loads acting on an intacttibia are similar to those in one that has been osteo-tomized and stabilized with an external fixateur. Tobe able to compute inter-fragmentary movements, the 248  G.N. Duda et al.  /   Journal of Biomechanics 31 (1998) 247  —  252  Fig. 1. ASIF fixateur externe construct as used in  in vivo  measurementsof inter-fragmentary movements in diaphyseal fractures of sheep tibiae(Stu¨rmer, 1988). All dimensions are in mm. The srcin of the coordinatesystem used is at the center of the fracture gap. The  x -axis is pointingto ventral, and  y -axis to lateral (left hind limb) and the  z -axis is pointingto proximal.Thefree lengthof the Schanzscrewsis increasedfrom54 to56.5 and 59 mm and decreased to 51.5 and 49 mm total length each. inverse of the stiffness matrix [Eq. (1)] had to be cal-culated. Inter-fragmentary movements were calculatedfrom the inverse stiffness matrix multiplied by the dia-physeal load state of the sheep tibia (Duda and Claes,1996; F  : 0N shearto ventral, F  : ! 50Nshear to lateral, F  : ! 1377 N axial compression, M  : ! 8.8 Nmbendingto lateral,  M  : 2.5 Nm bending to ventral,  M  : 0Nmtorsional moment).Finally, the influence of the fixateur mounting planewas analyzed. By mathematically rotating the inversestiffness matrix around the long axis of the tibia ( z -axis),different fixateur mounting planes could be simulated.Starting with a ventral mounting the fixateur was rotatedin steps of 10 °  to a lateral position. 3. Results With the free length of 54 mm the ASIF external fix-ateur had a stiffness of 425.5 Nmm   in axial compres-sion, 1.3 Nmdeg   in torsion, 7.7 Nmdeg   in bendingperpendicular to the fixateur plane and 36.4 Nmdeg  in bending parallel to the fixateur plane. The completestiffness matrix for a ventrally mounted fixateur was: S   2298.9 41.6  ! 36.5 11.5 443.4  ! 49.4 ! 70.2 743.8 10.2  ! 50.6  ! 62.9 2.1 ! 735.8  ! 64.9 425.5 90.6  ! 225.7 10.50.3 2.9 0.5 7.7  ! 2.2  ! 0.221.7 4.5  ! 3.8 3.5 36.4  ! 3.0 ! 2.2 1.6 0.3  ! 0.6  ! 1.7 1.3  .(2)The stiffness values changed if the free length of theSchanz screws was increased or decreased, primarily inthe diagonal values of the matrix. To simplify the data,onlythe diagonalvalues of the stiffness matrixwere givenas a function of the distance between fixateur body andbone axis (Fig. 4). Besides the non-linear decrease in axialstiffness with increasing distance the shear and bendingstiffnesses perpendicular to the fixateur plane decreased.Little effect could be found for the shear and bendingstiffnesses parallel to the fixateur plane and the torsionalstiffness.Using the most rigid fixateur construct (free length49 mm) and the load state in the tibia diaphysis theinter-fragmentary movement was 0.46 mm in compres-sion and 0.22 mm laterally (Fig. 5, ventral mounting). If the fixateur was not completely aligned with the bonyaxis but an additional parallel shift laterally of 1 mm wasintroduced, the bending moment around the ventraloriented axis and the torsional moments slightly in-creased. This resulted in an inter-fragmentary movementof 0.45 mm in compression and 0.19 mm laterally.The influence of the orientation of the fixateur plane isgiven in Fig. 5 with inter-fragmentary movements for allpositions between a ventral and lateral mounting of theexternal fixateur. The inter-fragmentary movement incompressionwas reducedto nearly zero from a ventral toa lateral mounting.In contrast,shear movementsshowedtheir maximum for a pure lateral mounting of theexternal fixateur. 4. Discussion and conclusions A method was developedto determinethe 3-Dstiffnessof fracture fixation devices. Based on the 3-D stiffness,inter-fragmentary movement was predicted as a functionof fracture location and fixateur mounting.In contrast to single stiffness values, the 3-D stiffnessdescriptionallows differentiationbetween more rigid andless rigid load planes. From the diagonal stiffness valuesthe selected fixateur construct appears rather rigid inaxial compression but less rigid under shear and bendingperpendicular to the fixateur plane. If the distancebetween fixateur body and bone axis is increased, the G.N. Duda et al.  /   Journal of Biomechanics 31 (1998 )  247  —  252  249  Fig. 2. Test setup to measure the axial stiffness with the goniometer system attached to the distal and proximal Schanz screws of the fixateur. The 3-Doffset of the measurement points of the goniometer system were used to transform the data into relative displacements at the centre of the fracture gap(Fig. 1). construct stiffness rapidly decreases both in the axial andlateral (shear) directions (Fig. 4). In vivo  measurements and analytical studies reportthat bones are mainly axially loaded with only moderatebending (Duda et al., 1997; Schneider et al., 1990). Theload magnitudes selected to predict inter-fragmentarymovement (Duda and Claes, 1996) are comparable inmagnitude with those reported from a 2-D analysis (Hut-zschenreuter et al., 1993). Hutzschenreuter et al. reporttibial compression of 1090 N (the present study 1377 N)and bending around a laterally oriented axis of 9.4 Nm(here 8.8 Nm).Although the loading is mainly axial, the computedinter-fragmentary movements are rather complex(Fig. 5). Nonetheless, the inter-fragmentary movementsrecorded in an  in vivo  study on tibial fractures in sheep(Stu¨rmer, 1988) are comparable to those calculated inthis study. Stu¨rmer reports for a diaphyseal fractureduring stance phase axial movement of 0.46 mm (here0.46 mm) and lateral movement of 0.19 mm (here0.22 mm).From these similarities it appeared warranted to ex-tend the presented method to predict inter-fragmentarymovements under various fixateur configurations. As anillustration, the influence of fixateur mounting plane oninter-fragmentary movement was selected. A simplemethod to achieve optimal mechanical conditions at thefracture site (0.2 —  1.0 mm axial and minimal shearmovements) was the modification of the fixateurmounting plane. In the present fixateur configuration, 250  G.N. Duda et al.  /   Journal of Biomechanics 31 (1998) 247  —  252

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