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A method to determine the 3-D stiffness of fracture fixation devices and its application to predict inter-fragmentary movement
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 moderate axial movement stimulates the healing process. To be able to control the mechanical situation at a fracture site and to achieve optimal bony healing it is essential to understand the relationship between inter-fragmentary movement, bony loading and fixation stiffness. A 6]6 stiffness matrix is introduced which completely describes the linear relationship between the 6 inter-fragmentary movements and the resulting bony loading (3 forces and 3 moments). Further, it is illustrated that even in relatively stiff external fixateur constructs simple axial loading of the bony fragments leads to complex inter-fragmentary movement. When the 3-D stiffness description is multiplied by the load state in sheep tibiae, movements similar to those measured in vivo are calculated. The relationship between axial compression and medio-lateral or dorso-ventral shear varies depending on the mounting plane of the external fixateur. The authors conclude that a single value is not sufficient to describe the mechanical relationship between inter-fragmentary movement and bony loading. Only a complete description of fixation stiffness allows prediction of inter-fragmentary movement and differentiation 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, bone healing is generally subjected to complex inter-fragmentary movements. It is well accepted that this movement influences the fracture healing process both in its type and rate of healing (Claes et al., 1995; Goodship and Kenwright, 1985; Kenwright and Goodship, 1989). Axial movement using fixateur externe configurations has 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; Kenwright et al., 1991; Kristiansen et al., 1987; Lippert and Hirsch,
* Correspondence address: Forschungslabor der Unfall- und Wiederherstellungschirurgie, Charite´, Humboldt Universita¨t zu Berlin, Augustenburger Platz 1, D-13353 Berlin, Germany. Tel.: #49 30 450 59079; fax: #49 30 450 59969; e-mail: [email protected]. 0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII S0021-9290(97)00115-2
1974). It has been shown that a large stiffness resulting in small axial movements minimizes the risk of pseudarthroses (Schenk et al., 1986; Stu¨rmer, 1988). However, a certain amount of inter-fragmentary movement is necessary to achieve sufficient mechanical stability in the newly formed bone (Kenwright and Goodship, 1989; Molster and Gjerdet, 1984; Molster et al., 1982). Animal experiments have shown that an axial interfragmentary movement within the range of 0.2—1.0 mm seems to be optimal for fracture healing (Claes et al., 1995; Goodship et al., 1988). However, it remains unclear how fracture healing is related to the movement components other than that in the axial direction. From in vivo experiments and clinical experience the impact of axial and shear movements on the healing callus has been qualitatively described (Yamagishi and Yoshimura, 1955). Although a large number of fixation devices in various configurations are clinically used, the 3-D interfragmentary movements actually occurring in vivo are mainly unknown.
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G.N. Duda et al. / Journal of Biomechanics 31 (1998) 247—252
Since axial and shear movements appear to influence the fracture healing processes differently, it would be beneficial to know the 3-D inter-fragmentary movement prior to mounting of the fixation device (Hoffmann et al., 1991). Due to the highly asymmetric nature of fixateur externe devices an axially rigid construct may experience large shear deformations under bending loads. Only the complete description of the 3-D fixation stiffness would allow the prediction of the full set of interfragmentary movements occurring under complex in vivo loading (Gardner et al., 1996). Since the bony load state is primarily independent of the method of fracture fixation, knowledge of the constructs’ 3-D stiffness would allow differentiation between those fixation divices that provide lesser or greater inter-fragmentary movement. Fixation devices could then be configured pre- or intra-operatively to meet the desired specifications, e.g. axial movement between 0.2 and 1 mm, minimized shear. The goal of this study was to develop a method to determine the 3-D stiffness of fracture fixation devices and to predict inter-fragmentary movement as a function of fracture location and fixateur mounting.
2. Materials and methods For determination of the 3-D fixateur stiffness, an ASIF external fixateur (double tube, steel rods, two Schanz screws per fragment) was selected. The specific dimensions are given in Fig. 1 and are identical to those used previously in an in vivo recording of inter-fragmentary movement in sheep (Stu¨rmer, 1988). The external fixateur was mounted to a pertinax rod (diameter 20 mm, length 300 mm) which was then osteotomized at midspan between the inner Schanz screws, creating a 4-mm fracture gap. Prior to testing, a 3-D goniometer system (accuracy 0.1 mm, 0.1°) was attached to the fixateur configuration using the proximal and distal Schanz screws (Fig. 2, Wilke et al., 1994). A tight fit of the Schanz screws in the pertinax rod eliminated pin bone interface motion in this in vitro experiment. Mathematically, the loads between the proximal and distal fracture fragments (three forces and three moments) can be related to the inter-fragmentary movements (three translations and three rotations) by a stiffness matrix. If the loads and gap movements are written as column vectors each, the fixation stiffness is described by a 6]6 stiffness matrix with 36 unknown stiffness values [Eq. (1)]. These values are constant while the load-displacement relationship is linear. To simplify the relationship between the 3-D loading and 3-D interfragmentary movement, the current analysis was performed under the assumption that the entire system behaved linear.
The diagonal values of the stiffness matrix refer to the ventral shear stiffness, lateral shear stiffness, compression stiffness bending stiffness around an axis perpendicular, bending stiffness around an axis parallel and torsional stiffness. To determine the 36 unknown values of the stiffness matrix six mechanically independent load cases (consisting of 3 forces and 3 moments each) and correlating inter-fragmentary movements (3 translations and 3 rotations) are necessary. In this analysis the six load cases were selected to be axial compression (F ), torsion (M ), Z Z 4-point-bending around an axis parallel (M ) and an axis Y perpendicular (M ) to the fixateur plane, cantilever benX ding around an axis parallel (F , M ) and an axis perpenX Y dicular (F , M ) to the fixateur plane (Fig. 3). The loads Y X and movements were oriented according to a coordinate system displayed in Fig. 1.
AB A
F s x 11 F s y 21 F s z " 31 M s x 41 M s y 51 M s z i 61 i"1, 2 , 6.
s 12 s 22 s 32 s 42 s 52 s 62
s 13 s 23 s 33 s 43 s 53 s 63
s 14 s 24 s 34 s 44 s 54 s 64
s 15 s 25 s 35 s 45 s 55 s 65
s 16 s 26 s 36 s 46 s 56 s 66
BA B
L x L y L z c x c y c z i (1)
The units of the stiffness values are N mm~1 for s to 11 s , s to s , s to s , N deg~1 for s to s , s to s , 13 21 23 31 33 14 16 24 26 s to s , N m mm~1 for s to s , s to s , s to 34 36 41 43 51 53 61 s and N m deg~1 for s to s , s to s , s to s . 63 44 46 54 56 64 66 All experiments except those in torsion were performed on a materials testing machine (Zwick 1454, Zwick GmbH, Ulm, Germany) with specially designed jigs to allow compression, 4-point-bending and cantilever bending. Torsional testing was performed on a custom built device. The inter-fragmentary movements were recorded during loading and unloading at 2 mm min~1 or 2° min~1. Maximum loads were chosen to be within the linear behavior of the external fixateur constructs (Table 1). Inter-fragmentary movements were recorded with a personal computer and custom software (Wilke et al., 1994). To analyze the influence of the free length of the Schanz screws, the carbon rods were moved incrementally 2.5 mm towards and away from the rod axis creating 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 load state was derived from a 3-D model of the hind limb of a sheep (Duda and Claes, 1996). It was assumed as a simplification that the bony loads acting on an intact tibia are similar to those in one that has been osteotomized and stabilized with an external fixateur. To be able to compute inter-fragmentary movements, the
G.N. Duda et al. / Journal of Biomechanics 31 (1998) 247—252
249
in bending parallel to the fixateur plane. The complete stiffness matrix for a ventrally mounted fixateur was:
A
S"
Fig. 1. ASIF fixateur externe construct as used in in vivo measurements of inter-fragmentary movements in diaphyseal fractures of sheep tibiae (Stu¨rmer, 1988). All dimensions are in mm. The origin of the coordinate system used is at the center of the fracture gap. The x-axis is pointing to ventral, and y-axis to lateral (left hind limb) and the z-axis is pointing to proximal. The free length of the Schanz screws is increased from 54 to 56.5 and 59 mm and decreased to 51.5 and 49 mm total length each.
inverse of the stiffness matrix [Eq. (1)] had to be calculated. Inter-fragmentary movements were calculated from the inverse stiffness matrix multiplied by the diaphyseal load state of the sheep tibia (Duda and Claes, 1996; F : 0 N shear to ventral, F : !50 N shear to lateral, x y F : !1377 N axial compression, M : !8.8 N m bending z x to lateral, M : 2.5 N m bending to ventral, M : 0 N m y z torsional moment). Finally, the influence of the fixateur mounting plane was analyzed. By mathematically rotating the inverse stiffness 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 rotated in steps of 10° to a lateral position.
3. Results With the free length of 54 mm the ASIF external fixateur had a stiffness of 425.5 N mm~1 in axial compression, 1.3 N m deg~1 in torsion, 7.7 N m deg~1 in bending perpendicular to the fixateur plane and 36.4 N m deg~1
41.6
!36.5
11.5
443.4
!49.4
743.8
10.2
!50.6
!62.9
2.1
!735.8
!64.9
425.5
90.6
!225.7
10.5
0.3
2.9
0.5
7.7
!2.2
!0.2
21.7
4.5
!3.8
3.5
36.4
!3.0
!2.2
1.6
0.3
!0.6
!1.7
1.3
B
2298.9 !70.2
.
(2)
The stiffness values changed if the free length of the Schanz screws was increased or decreased, primarily in the diagonal values of the matrix. To simplify the data, only the diagonal values of the stiffness matrix were given as a function of the distance between fixateur body and bone axis (Fig. 4). Besides the non-linear decrease in axial stiffness with increasing distance the shear and bending stiffnesses perpendicular to the fixateur plane decreased. Little effect could be found for the shear and bending stiffnesses parallel to the fixateur plane and the torsional stiffness. Using the most rigid fixateur construct (free length 49 mm) and the load state in the tibia diaphysis the inter-fragmentary movement was 0.46 mm in compression and 0.22 mm laterally (Fig. 5, ventral mounting). If the fixateur was not completely aligned with the bony axis but an additional parallel shift laterally of 1 mm was introduced, the bending moment around the ventral oriented axis and the torsional moments slightly increased. This resulted in an inter-fragmentary movement of 0.45 mm in compression and 0.19 mm laterally. The influence of the orientation of the fixateur plane is given in Fig. 5 with inter-fragmentary movements for all positions between a ventral and lateral mounting of the external fixateur. The inter-fragmentary movement in compression was reduced to nearly zero from a ventral to a lateral mounting. In contrast, shear movements showed their maximum for a pure lateral mounting of the external fixateur.
4. Discussion and conclusions A method was developed to determine the 3-D stiffness of fracture fixation devices. Based on the 3-D stiffness, inter-fragmentary movement was predicted as a function of fracture location and fixateur mounting. In contrast to single stiffness values, the 3-D stiffness description allows differentiation between more rigid and less rigid load planes. From the diagonal stiffness values the selected fixateur construct appears rather rigid in axial compression but less rigid under shear and bending perpendicular to the fixateur plane. If the distance between fixateur body and bone axis is increased, the
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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-D offset 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 and lateral (shear) directions (Fig. 4). In vivo measurements and analytical studies report that bones are mainly axially loaded with only moderate bending (Duda et al., 1997; Schneider et al., 1990). The load magnitudes selected to predict inter-fragmentary movement (Duda and Claes, 1996) are comparable in magnitude with those reported from a 2-D analysis (Hutzschenreuter et al., 1993). Hutzschenreuter et al. report tibial compression of 1090 N (the present study 1377 N) and bending around a laterally oriented axis of 9.4 N m (here 8.8 N m). Although the loading is mainly axial, the computed inter-fragmentary movements are rather complex (Fig. 5). Nonetheless, the inter-fragmentary movements
recorded in an in vivo study on tibial fractures in sheep (Stu¨rmer, 1988) are comparable to those calculated in this study. Stu¨rmer reports for a diaphyseal fracture during stance phase axial movement of 0.46 mm (here 0.46 mm) and lateral movement of 0.19 mm (here 0.22 mm). From these similarities it appeared warranted to extend the presented method to predict inter-fragmentary movements under various fixateur configurations. As an illustration, the influence of fixateur mounting plane on inter-fragmentary movement was selected. A simple method to achieve optimal mechanical conditions at the fracture site (0.2—1.0 mm axial and minimal shear movements) was the modification of the fixateur mounting plane. In the present fixateur configuration,
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251
Fig. 3. The six linear independent load cases used to determine the stiffness matrix. From left to right: Axial compression, torsion, 4-point-bending parallel and perpendicular to the fixateur mounting plane, cantilever bending parallel and perpendicular to the fixateur mounting plane.
Table 1 Loads used to measure the inter-fragmentary movement for the six independent load cases. Coordinates are according to Fig. 1: X to ventral, Y to lateral, Z to proximal. Origin is the center of the fracture gap F x (N) Axial compression Torsion 4-point-bending, parallel 4-point-bending, perpendicular Cantilever bending, parallel Cantilever bending, perpendicular
F y (N)
F z (N)
M x (Nm)
0 0
0 0
!500 0
0 0
0
0
!200
0
0
0
!200 !10.1
200 0
0 !90
0 0
0 8.8
M y (Nm) 0 0 !10.1
M z (Nm) 0 2.5 0
0
0
19.6 0
0 0
axial movement was well within the range considered appropriate for optimal fracture healing (Claes et al., 1995) but the shear movement was never adequately minimized in any mounting plane (Fig. 5). A laterally mounted fixateur with minimal axial movement and maximum shear would probably provide the least effective mechanical environment for fracture healing. The data support rather a 20° ventro-lateral mounting, if allowed by anatomical constraints. The fracture model used in this study assumes no load transfer across the fracture gap and, therefore, represents a worst case scenario. The model would have to be adapted to other fracture modes by adjusting the fixation stiffness appropriately. Such adaptations as well as predictions of inter-fragmentary movements are only manageable by 3-D descriptions of the fixation stiffnesses.
Fig. 4. The influence of the free length of the screws on the diagonal values of the stiffness matrix. #F /u : ventral shear stiffness, #F /u : x x y y lateral shear stiffness, !F /u : compression stiffness, #M /L : ventral z z x x ending stiffness, #M /L : lateral bending stiffness, #M /L : torsional y y z z stiffness.
The complete description provides a tool to investigate inter-fragmentary movement under any possible load condition in vivo. With knowledge of the optimal axial and shear movements it further allows the appropriate selection of fixateur configurations to control inter-fragmentary movement.
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Fig. 5. Inter-fragmentary movement of a mid-shaft tibia osteotomy stabilized with an ASIF fixateur mounted in a ventral, ventro-lateral or lateral position. #u : shear to ventral, #u : shear to lateral, !u : x y z compression.
Acknowledgements The authors would like to thank Patricia Horny, Department of Unfallchirurgische Forschung und Biomechanik, University of Ulm, for helping with the graphs and figures and Dr K. Wenger, Department of Unfallchirurgische Forschung und Biomechanik, University of Ulm, for editing. References Claes, L., Wilke, H.-J., Augat, P., Ru¨benacker, S., Margevicius, K.J., 1995. Effect of dynamization on gap healing of diaphyseal fractures under external fixation. Clinical Biomechanics 10, 227—234. Cunningham, J.L., Evans, M., Kenwright, J., 1989. Measurement of fracture movement in patients treated with unilateral external skeletal fixation. Journal of Biomedical Engineering 11, 118—122. Duda, G.N., Chao, E.Y.S., Schneider, E., 1997. Internal forces and moments in the femur during walking. Journal of Biomechanics 30, 933—941. Duda G.N., Claes, L., 1996. Prediction and control of 3-D interfragmentary movement in fracture healing. In Proceedings of the International Society of Fracture Repair. Ottawa, p. 24. Gardner, T.N., Hardy, J.R.W., Evans, M., Richardson, J.B., Kenwright, J., 1996. The static and dynamic behaviour of tibial fractures due to the procedure of dynamisation. Clinical Biomechanics 11, 425—430.
Goodship, A.E., Kelly, D.J., Rigby, H.S., Watkins, P.E., Kenwright, J., 1988. The effect of different regimes of axial micromovement on the healing of experimental tibial fractures. In: Bergmann G., Ko¨lbel R., Rohlmann A. (Eds.), Biomechanics: Basic and applied research, Martinus Nijhoff Publishers, Dordrecht, pp. 441—446. Goodship, A.E., Kenwright, J., 1985. The influence of induced micromovement upon the healing of experimental tibial fractures. Journal of Bone and Joint Surgery 67, 650—655. Goodship, A.E., Watkins, P.E., Rigby, H.S., Kenwright, J., 1993. The role of fixator frame stiffness in the control of fracture healing. An experimental study. Journal of Biomechanics 26, 1027—1035. Hoffmann, R., McKellop, H., Sarmiento, A., Lu, B., Ebramzadeh, E., 1991. Three-dimensional measurement of fracture gap motion. Biomechanical study of experimental tibial fractures with anterior clasp fixator and ring fixator. Unfallchirurg 94, 395—400. Hutzschenreuter, P.O., Sekler, E., Faust, G., 1993. Loads on muscles, tendons and bones in the hind extremities of sheep — a theoretical study. Anat. Histol. Embryol. 22, 67—82. Kenwright, J., Goodship, A.E., 1989. Controlled mechanical stimulation in the treatment of tibial fractures. Clinical Orthopedics 241, 36—47. Kenwright, J., Richardson, J.B., Cunningham, J.L., White, S.H., Goodship, A.E., Adams, M.A., Magnussen, P.A., Newman, J.H., 1991. Axial movement and tibial fractures. A controlled randomised trial of treatment. Journal of Bone and Joint Surgery 73, 654—659. Kristiansen, T., Fleming, B., Neale, G., Reinecke, S., Pope, M.H., 1987. Comparative study of fracture gap motion in external fixation. Clinical Biomechanics 2, 191—195. Lippert, F.G., Hirsch, C., 1974. The three-dimensional measurement of tibia fracture motion by photogrammetry. Clinical Orthopedics 105, 130—143. Molster, A.O., Gjerdet, N.R., 1984. Effects of instability on fracture healing in the rat. Acta Orthopedica Scandinavica 55, 342—346. Molster, A.O., Gjerdet, N.R., Raugstad, T.S., Hvidsten, K., Alho, A., Bang, G., 1982. Effect of instability on experimental fracture healing. Acta Orthopedica Scandinavica 53, 521—526. Schenk, R.K., Mu¨ller, J., Willenegger, H., 1986. Experimentell-histologischer Beitrag zur Entstehung und Behandlung von Pseudarthrosen. Hefte zur Unfallheilkunde 94, 15—24. Schneider, E., Michel, M.C., Genge, M., Perren, S.M., 1990. Loads acting on an intramedullary femoral nail. In: Bergmann G., Graichen F., Rohlmann A. (Eds.), Implantable Telemetry in Orthopaedics, Freie Universita¨t Berlin, Berlin, pp. 221—227. Stu¨rmer, K.M., 1988. Histologie und Biomechanik der Frakturheilung unter den Bedingungen des Fixateur externe. Hefte zur Unfallheilkunde 200, 233—242. Wilke, H.-J., Ostertag, G., Cleas, L., 1994. Three-dimensional goniometer linkage system for the analysis of movements with six degrees of freedom. Biomedizinische Technik (Berlin) 39, 149—155. Yamagishi, M., Yoshimura, Y., 1955. The biomechanics of fracture healing. Journal of Bone and Joint Surgery 37, 1035—1068.
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 moderate axial movement stimulates the healing process. To be able to control the mechanical situation at a fracture site and to achieve optimal bony healing it is essential to understand the relationship between inter-fragmentary movement, bony loading and fixation stiffness. A 6]6 stiffness matrix is introduced which completely describes the linear relationship between the 6 inter-fragmentary movements and the resulting bony loading (3 forces and 3 moments). Further, it is illustrated that even in relatively stiff external fixateur constructs simple axial loading of the bony fragments leads to complex inter-fragmentary movement. When the 3-D stiffness description is multiplied by the load state in sheep tibiae, movements similar to those measured in vivo are calculated. The relationship between axial compression and medio-lateral or dorso-ventral shear varies depending on the mounting plane of the external fixateur. The authors conclude that a single value is not sufficient to describe the mechanical relationship between inter-fragmentary movement and bony loading. Only a complete description of fixation stiffness allows prediction of inter-fragmentary movement and differentiation 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, bone healing is generally subjected to complex inter-fragmentary movements. It is well accepted that this movement influences the fracture healing process both in its type and rate of healing (Claes et al., 1995; Goodship and Kenwright, 1985; Kenwright and Goodship, 1989). Axial movement using fixateur externe configurations has 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; Kenwright et al., 1991; Kristiansen et al., 1987; Lippert and Hirsch,
* Correspondence address: Forschungslabor der Unfall- und Wiederherstellungschirurgie, Charite´, Humboldt Universita¨t zu Berlin, Augustenburger Platz 1, D-13353 Berlin, Germany. Tel.: #49 30 450 59079; fax: #49 30 450 59969; e-mail: [email protected]. 0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII S0021-9290(97)00115-2
1974). It has been shown that a large stiffness resulting in small axial movements minimizes the risk of pseudarthroses (Schenk et al., 1986; Stu¨rmer, 1988). However, a certain amount of inter-fragmentary movement is necessary to achieve sufficient mechanical stability in the newly formed bone (Kenwright and Goodship, 1989; Molster and Gjerdet, 1984; Molster et al., 1982). Animal experiments have shown that an axial interfragmentary movement within the range of 0.2—1.0 mm seems to be optimal for fracture healing (Claes et al., 1995; Goodship et al., 1988). However, it remains unclear how fracture healing is related to the movement components other than that in the axial direction. From in vivo experiments and clinical experience the impact of axial and shear movements on the healing callus has been qualitatively described (Yamagishi and Yoshimura, 1955). Although a large number of fixation devices in various configurations are clinically used, the 3-D interfragmentary movements actually occurring in vivo are mainly unknown.
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G.N. Duda et al. / Journal of Biomechanics 31 (1998) 247—252
Since axial and shear movements appear to influence the fracture healing processes differently, it would be beneficial to know the 3-D inter-fragmentary movement prior to mounting of the fixation device (Hoffmann et al., 1991). Due to the highly asymmetric nature of fixateur externe devices an axially rigid construct may experience large shear deformations under bending loads. Only the complete description of the 3-D fixation stiffness would allow the prediction of the full set of interfragmentary movements occurring under complex in vivo loading (Gardner et al., 1996). Since the bony load state is primarily independent of the method of fracture fixation, knowledge of the constructs’ 3-D stiffness would allow differentiation between those fixation divices that provide lesser or greater inter-fragmentary movement. Fixation devices could then be configured pre- or intra-operatively to meet the desired specifications, e.g. axial movement between 0.2 and 1 mm, minimized shear. The goal of this study was to develop a method to determine the 3-D stiffness of fracture fixation devices and to predict inter-fragmentary movement as a function of fracture location and fixateur mounting.
2. Materials and methods For determination of the 3-D fixateur stiffness, an ASIF external fixateur (double tube, steel rods, two Schanz screws per fragment) was selected. The specific dimensions are given in Fig. 1 and are identical to those used previously in an in vivo recording of inter-fragmentary movement in sheep (Stu¨rmer, 1988). The external fixateur was mounted to a pertinax rod (diameter 20 mm, length 300 mm) which was then osteotomized at midspan between the inner Schanz screws, creating a 4-mm fracture gap. Prior to testing, a 3-D goniometer system (accuracy 0.1 mm, 0.1°) was attached to the fixateur configuration using the proximal and distal Schanz screws (Fig. 2, Wilke et al., 1994). A tight fit of the Schanz screws in the pertinax rod eliminated pin bone interface motion in this in vitro experiment. Mathematically, the loads between the proximal and distal fracture fragments (three forces and three moments) can be related to the inter-fragmentary movements (three translations and three rotations) by a stiffness matrix. If the loads and gap movements are written as column vectors each, the fixation stiffness is described by a 6]6 stiffness matrix with 36 unknown stiffness values [Eq. (1)]. These values are constant while the load-displacement relationship is linear. To simplify the relationship between the 3-D loading and 3-D interfragmentary movement, the current analysis was performed under the assumption that the entire system behaved linear.
The diagonal values of the stiffness matrix refer to the ventral shear stiffness, lateral shear stiffness, compression stiffness bending stiffness around an axis perpendicular, bending stiffness around an axis parallel and torsional stiffness. To determine the 36 unknown values of the stiffness matrix six mechanically independent load cases (consisting of 3 forces and 3 moments each) and correlating inter-fragmentary movements (3 translations and 3 rotations) are necessary. In this analysis the six load cases were selected to be axial compression (F ), torsion (M ), Z Z 4-point-bending around an axis parallel (M ) and an axis Y perpendicular (M ) to the fixateur plane, cantilever benX ding around an axis parallel (F , M ) and an axis perpenX Y dicular (F , M ) to the fixateur plane (Fig. 3). The loads Y X and movements were oriented according to a coordinate system displayed in Fig. 1.
AB A
F s x 11 F s y 21 F s z " 31 M s x 41 M s y 51 M s z i 61 i"1, 2 , 6.
s 12 s 22 s 32 s 42 s 52 s 62
s 13 s 23 s 33 s 43 s 53 s 63
s 14 s 24 s 34 s 44 s 54 s 64
s 15 s 25 s 35 s 45 s 55 s 65
s 16 s 26 s 36 s 46 s 56 s 66
BA B
L x L y L z c x c y c z i (1)
The units of the stiffness values are N mm~1 for s to 11 s , s to s , s to s , N deg~1 for s to s , s to s , 13 21 23 31 33 14 16 24 26 s to s , N m mm~1 for s to s , s to s , s to 34 36 41 43 51 53 61 s and N m deg~1 for s to s , s to s , s to s . 63 44 46 54 56 64 66 All experiments except those in torsion were performed on a materials testing machine (Zwick 1454, Zwick GmbH, Ulm, Germany) with specially designed jigs to allow compression, 4-point-bending and cantilever bending. Torsional testing was performed on a custom built device. The inter-fragmentary movements were recorded during loading and unloading at 2 mm min~1 or 2° min~1. Maximum loads were chosen to be within the linear behavior of the external fixateur constructs (Table 1). Inter-fragmentary movements were recorded with a personal computer and custom software (Wilke et al., 1994). To analyze the influence of the free length of the Schanz screws, the carbon rods were moved incrementally 2.5 mm towards and away from the rod axis creating 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 load state was derived from a 3-D model of the hind limb of a sheep (Duda and Claes, 1996). It was assumed as a simplification that the bony loads acting on an intact tibia are similar to those in one that has been osteotomized and stabilized with an external fixateur. To be able to compute inter-fragmentary movements, the
G.N. Duda et al. / Journal of Biomechanics 31 (1998) 247—252
249
in bending parallel to the fixateur plane. The complete stiffness matrix for a ventrally mounted fixateur was:
A
S"
Fig. 1. ASIF fixateur externe construct as used in in vivo measurements of inter-fragmentary movements in diaphyseal fractures of sheep tibiae (Stu¨rmer, 1988). All dimensions are in mm. The origin of the coordinate system used is at the center of the fracture gap. The x-axis is pointing to ventral, and y-axis to lateral (left hind limb) and the z-axis is pointing to proximal. The free length of the Schanz screws is increased from 54 to 56.5 and 59 mm and decreased to 51.5 and 49 mm total length each.
inverse of the stiffness matrix [Eq. (1)] had to be calculated. Inter-fragmentary movements were calculated from the inverse stiffness matrix multiplied by the diaphyseal load state of the sheep tibia (Duda and Claes, 1996; F : 0 N shear to ventral, F : !50 N shear to lateral, x y F : !1377 N axial compression, M : !8.8 N m bending z x to lateral, M : 2.5 N m bending to ventral, M : 0 N m y z torsional moment). Finally, the influence of the fixateur mounting plane was analyzed. By mathematically rotating the inverse stiffness 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 rotated in steps of 10° to a lateral position.
3. Results With the free length of 54 mm the ASIF external fixateur had a stiffness of 425.5 N mm~1 in axial compression, 1.3 N m deg~1 in torsion, 7.7 N m deg~1 in bending perpendicular to the fixateur plane and 36.4 N m deg~1
41.6
!36.5
11.5
443.4
!49.4
743.8
10.2
!50.6
!62.9
2.1
!735.8
!64.9
425.5
90.6
!225.7
10.5
0.3
2.9
0.5
7.7
!2.2
!0.2
21.7
4.5
!3.8
3.5
36.4
!3.0
!2.2
1.6
0.3
!0.6
!1.7
1.3
B
2298.9 !70.2
.
(2)
The stiffness values changed if the free length of the Schanz screws was increased or decreased, primarily in the diagonal values of the matrix. To simplify the data, only the diagonal values of the stiffness matrix were given as a function of the distance between fixateur body and bone axis (Fig. 4). Besides the non-linear decrease in axial stiffness with increasing distance the shear and bending stiffnesses perpendicular to the fixateur plane decreased. Little effect could be found for the shear and bending stiffnesses parallel to the fixateur plane and the torsional stiffness. Using the most rigid fixateur construct (free length 49 mm) and the load state in the tibia diaphysis the inter-fragmentary movement was 0.46 mm in compression and 0.22 mm laterally (Fig. 5, ventral mounting). If the fixateur was not completely aligned with the bony axis but an additional parallel shift laterally of 1 mm was introduced, the bending moment around the ventral oriented axis and the torsional moments slightly increased. This resulted in an inter-fragmentary movement of 0.45 mm in compression and 0.19 mm laterally. The influence of the orientation of the fixateur plane is given in Fig. 5 with inter-fragmentary movements for all positions between a ventral and lateral mounting of the external fixateur. The inter-fragmentary movement in compression was reduced to nearly zero from a ventral to a lateral mounting. In contrast, shear movements showed their maximum for a pure lateral mounting of the external fixateur.
4. Discussion and conclusions A method was developed to determine the 3-D stiffness of fracture fixation devices. Based on the 3-D stiffness, inter-fragmentary movement was predicted as a function of fracture location and fixateur mounting. In contrast to single stiffness values, the 3-D stiffness description allows differentiation between more rigid and less rigid load planes. From the diagonal stiffness values the selected fixateur construct appears rather rigid in axial compression but less rigid under shear and bending perpendicular to the fixateur plane. If the distance between fixateur body and bone axis is increased, the
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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-D offset 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 and lateral (shear) directions (Fig. 4). In vivo measurements and analytical studies report that bones are mainly axially loaded with only moderate bending (Duda et al., 1997; Schneider et al., 1990). The load magnitudes selected to predict inter-fragmentary movement (Duda and Claes, 1996) are comparable in magnitude with those reported from a 2-D analysis (Hutzschenreuter et al., 1993). Hutzschenreuter et al. report tibial compression of 1090 N (the present study 1377 N) and bending around a laterally oriented axis of 9.4 N m (here 8.8 N m). Although the loading is mainly axial, the computed inter-fragmentary movements are rather complex (Fig. 5). Nonetheless, the inter-fragmentary movements
recorded in an in vivo study on tibial fractures in sheep (Stu¨rmer, 1988) are comparable to those calculated in this study. Stu¨rmer reports for a diaphyseal fracture during stance phase axial movement of 0.46 mm (here 0.46 mm) and lateral movement of 0.19 mm (here 0.22 mm). From these similarities it appeared warranted to extend the presented method to predict inter-fragmentary movements under various fixateur configurations. As an illustration, the influence of fixateur mounting plane on inter-fragmentary movement was selected. A simple method to achieve optimal mechanical conditions at the fracture site (0.2—1.0 mm axial and minimal shear movements) was the modification of the fixateur mounting plane. In the present fixateur configuration,
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251
Fig. 3. The six linear independent load cases used to determine the stiffness matrix. From left to right: Axial compression, torsion, 4-point-bending parallel and perpendicular to the fixateur mounting plane, cantilever bending parallel and perpendicular to the fixateur mounting plane.
Table 1 Loads used to measure the inter-fragmentary movement for the six independent load cases. Coordinates are according to Fig. 1: X to ventral, Y to lateral, Z to proximal. Origin is the center of the fracture gap F x (N) Axial compression Torsion 4-point-bending, parallel 4-point-bending, perpendicular Cantilever bending, parallel Cantilever bending, perpendicular
F y (N)
F z (N)
M x (Nm)
0 0
0 0
!500 0
0 0
0
0
!200
0
0
0
!200 !10.1
200 0
0 !90
0 0
0 8.8
M y (Nm) 0 0 !10.1
M z (Nm) 0 2.5 0
0
0
19.6 0
0 0
axial movement was well within the range considered appropriate for optimal fracture healing (Claes et al., 1995) but the shear movement was never adequately minimized in any mounting plane (Fig. 5). A laterally mounted fixateur with minimal axial movement and maximum shear would probably provide the least effective mechanical environment for fracture healing. The data support rather a 20° ventro-lateral mounting, if allowed by anatomical constraints. The fracture model used in this study assumes no load transfer across the fracture gap and, therefore, represents a worst case scenario. The model would have to be adapted to other fracture modes by adjusting the fixation stiffness appropriately. Such adaptations as well as predictions of inter-fragmentary movements are only manageable by 3-D descriptions of the fixation stiffnesses.
Fig. 4. The influence of the free length of the screws on the diagonal values of the stiffness matrix. #F /u : ventral shear stiffness, #F /u : x x y y lateral shear stiffness, !F /u : compression stiffness, #M /L : ventral z z x x ending stiffness, #M /L : lateral bending stiffness, #M /L : torsional y y z z stiffness.
The complete description provides a tool to investigate inter-fragmentary movement under any possible load condition in vivo. With knowledge of the optimal axial and shear movements it further allows the appropriate selection of fixateur configurations to control inter-fragmentary movement.
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Fig. 5. Inter-fragmentary movement of a mid-shaft tibia osteotomy stabilized with an ASIF fixateur mounted in a ventral, ventro-lateral or lateral position. #u : shear to ventral, #u : shear to lateral, !u : x y z compression.
Acknowledgements The authors would like to thank Patricia Horny, Department of Unfallchirurgische Forschung und Biomechanik, University of Ulm, for helping with the graphs and figures and Dr K. Wenger, Department of Unfallchirurgische Forschung und Biomechanik, University of Ulm, for editing. References Claes, L., Wilke, H.-J., Augat, P., Ru¨benacker, S., Margevicius, K.J., 1995. Effect of dynamization on gap healing of diaphyseal fractures under external fixation. Clinical Biomechanics 10, 227—234. Cunningham, J.L., Evans, M., Kenwright, J., 1989. Measurement of fracture movement in patients treated with unilateral external skeletal fixation. Journal of Biomedical Engineering 11, 118—122. Duda, G.N., Chao, E.Y.S., Schneider, E., 1997. Internal forces and moments in the femur during walking. Journal of Biomechanics 30, 933—941. Duda G.N., Claes, L., 1996. Prediction and control of 3-D interfragmentary movement in fracture healing. In Proceedings of the International Society of Fracture Repair. Ottawa, p. 24. Gardner, T.N., Hardy, J.R.W., Evans, M., Richardson, J.B., Kenwright, J., 1996. The static and dynamic behaviour of tibial fractures due to the procedure of dynamisation. Clinical Biomechanics 11, 425—430.
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