J. Biomechanics Vol. 25, No. 6. pp. 645-650, Printed in Great Britain
1992.
0021-9290/92 Pcrgamon
TECHNICAL
PERIOSTEAL
SS.OO+.tXl Press Ltd
NOTE
AND ENDOSTEAL CONTROL OF BONE REMODELING UNDER TORSIONAL LOADING XIANG-DONG Guo
and STEPHEN C. COWIN
Department of Mechanical Engineering, The City College of The City University of New York, New York, NY 10031, U.S.A. Abstract-The
shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled rightcircular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected t’oan axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.
INTRODUCTION
Two animal studies have shown that chronic raising of the daily level of torsional loading and reducing the level of axial loading in living long bones will cause the long bone to remodel to a shape that has an increased mean radius and a decreased wall thickness. Pead and Lanyon (1988) describe an experiment in which hyperphysiological torsional loading is applied to the functionally isolated avian ulna between the epiphyseal plates for the purpose of investigating the adaptive remodeling response. In some of these experiments it was found that the remodeled effect of increasing the torsional loading history and reducing the axial loading history on the avian ulna to near zero was to increase the mean radius of the ulna and decrease the ulna wall thickness. In similar experiments in which both the axial and torsional loading were reduced to near zero, the cortical bone tissue of the ulna wall resorbed and there was no increase in the mean radius of the ulna. A similar effect has recently been reported by Guichet et al. (1991) in conjunction with an experimental study of a twist-activated intramedullary nail for gradual limb lengthening in the sheep femur. The lengthening mechanism in this intramedullary nail requires that the limb containing the nail be twisted periodically. The nail reduced the axial loading on the sheep femur because of its stress-shielding effect. The remodeled femur showed that the effect of increasing the torsional loading history and reducing the axial loading history on the sheep femur was similar to the effect of these same loadings on the avian ulna. In both cases the mean radius of the long bone increased and the wall thickness decreased. An idealized model that is applicable to such bone remodeling experiments is developed in this note. The object of
Receioed
infinal form 15 August 1991.
the model is to predict the qualitative, rather than quantitative, nature of the results, directions and trends in bone shape change. In the next section a formulation of the problem is presented, and the ideas of endosteal or periosteal control of remodeling are developed. These are different hypothecated strategies for the determination of bone shape due to torsional loading. Sections describing the calculations for periosteal and endosteal control of remodeling are followed by a discussion of the results.
FORMULATION OF THE PROBLEM
The geometric model for the diaphyseal region of a long bone is the hollow right-circular cylindrical shaft illustrated in Fig. 1. The mechanical loading of this cylinder and the stresses induced are familiar from the mechanics of materials. The shaft is subjected to an axial compressive load of magnitude P and a twisting torque of magnitude T. The radii of the periosteal surface and endosteal surface are R and r, respectively. The axial stress (r produced by the axial load P is given by -P (r=-
n(R*-r')'
(1)
and the shearing stress T caused by the torque T has the form T=--,
27’~ n(R4-r4)
(2)
where p is the variable radial coordinate, I < p < R. The formulation of the periosteal (endosteal) control of remodeling is developed easily in terms of the bounds on these stresses. We assume that it is the strain at a specific surface site in the bone that controls the bone remodeling process in the neighborhood of the surface-point. The strain
645
646
Technical Note
Fig. 1. Schematic illustration of the diaphyseal region of a long bone subjected to combined torsional and axial loads.
controls the bone remodeling process the way the temperature controls the thermostat of a heating/cooling system for a house. It is generally accepted that there is a range of strain values associated with remodeling equilibrium RE, corresponding to the range of temperatures that will not activate a heating/cooling system; but if the upper limiting temperature is exceeded, the cooling system is turned on, and if the lower limiting temperature is surpassed, the heating system is activated. Bone RE is a range of conditions under which there is no net deposition or resorption of the bone tissue; see e.g. Carter, 1984; Cowin, 1984; Huiskes et al., 1987. Since the bone is assumed to be linearly elastic, there is a range of RE stress values corresponding to the range of RE strain values. The ranges of the axial normal stress TV on the periosteal and endosteal surfaces at remodeling equilibrium are denoted by a;
1992.
0021-9290/92 Pcrgamon
TECHNICAL
PERIOSTEAL
SS.OO+.tXl Press Ltd
NOTE
AND ENDOSTEAL CONTROL OF BONE REMODELING UNDER TORSIONAL LOADING XIANG-DONG Guo
and STEPHEN C. COWIN
Department of Mechanical Engineering, The City College of The City University of New York, New York, NY 10031, U.S.A. Abstract-The
shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled rightcircular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected t’oan axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.
INTRODUCTION
Two animal studies have shown that chronic raising of the daily level of torsional loading and reducing the level of axial loading in living long bones will cause the long bone to remodel to a shape that has an increased mean radius and a decreased wall thickness. Pead and Lanyon (1988) describe an experiment in which hyperphysiological torsional loading is applied to the functionally isolated avian ulna between the epiphyseal plates for the purpose of investigating the adaptive remodeling response. In some of these experiments it was found that the remodeled effect of increasing the torsional loading history and reducing the axial loading history on the avian ulna to near zero was to increase the mean radius of the ulna and decrease the ulna wall thickness. In similar experiments in which both the axial and torsional loading were reduced to near zero, the cortical bone tissue of the ulna wall resorbed and there was no increase in the mean radius of the ulna. A similar effect has recently been reported by Guichet et al. (1991) in conjunction with an experimental study of a twist-activated intramedullary nail for gradual limb lengthening in the sheep femur. The lengthening mechanism in this intramedullary nail requires that the limb containing the nail be twisted periodically. The nail reduced the axial loading on the sheep femur because of its stress-shielding effect. The remodeled femur showed that the effect of increasing the torsional loading history and reducing the axial loading history on the sheep femur was similar to the effect of these same loadings on the avian ulna. In both cases the mean radius of the long bone increased and the wall thickness decreased. An idealized model that is applicable to such bone remodeling experiments is developed in this note. The object of
Receioed
infinal form 15 August 1991.
the model is to predict the qualitative, rather than quantitative, nature of the results, directions and trends in bone shape change. In the next section a formulation of the problem is presented, and the ideas of endosteal or periosteal control of remodeling are developed. These are different hypothecated strategies for the determination of bone shape due to torsional loading. Sections describing the calculations for periosteal and endosteal control of remodeling are followed by a discussion of the results.
FORMULATION OF THE PROBLEM
The geometric model for the diaphyseal region of a long bone is the hollow right-circular cylindrical shaft illustrated in Fig. 1. The mechanical loading of this cylinder and the stresses induced are familiar from the mechanics of materials. The shaft is subjected to an axial compressive load of magnitude P and a twisting torque of magnitude T. The radii of the periosteal surface and endosteal surface are R and r, respectively. The axial stress (r produced by the axial load P is given by -P (r=-
n(R*-r')'
(1)
and the shearing stress T caused by the torque T has the form T=--,
27’~ n(R4-r4)
(2)
where p is the variable radial coordinate, I < p < R. The formulation of the periosteal (endosteal) control of remodeling is developed easily in terms of the bounds on these stresses. We assume that it is the strain at a specific surface site in the bone that controls the bone remodeling process in the neighborhood of the surface-point. The strain
645
646
Technical Note
Fig. 1. Schematic illustration of the diaphyseal region of a long bone subjected to combined torsional and axial loads.
controls the bone remodeling process the way the temperature controls the thermostat of a heating/cooling system for a house. It is generally accepted that there is a range of strain values associated with remodeling equilibrium RE, corresponding to the range of temperatures that will not activate a heating/cooling system; but if the upper limiting temperature is exceeded, the cooling system is turned on, and if the lower limiting temperature is surpassed, the heating system is activated. Bone RE is a range of conditions under which there is no net deposition or resorption of the bone tissue; see e.g. Carter, 1984; Cowin, 1984; Huiskes et al., 1987. Since the bone is assumed to be linearly elastic, there is a range of RE stress values corresponding to the range of RE strain values. The ranges of the axial normal stress TV on the periosteal and endosteal surfaces at remodeling equilibrium are denoted by a;
