Leila Malekmotiei Department of Civil Engineering, Sharif University of Technoiogy, P.O. Box 11155-9313, Tefiran, Iran e-maii: [email protected]
Farzam Farahmand Schooi of Mechanicai Engineering, Sfiarif University of Technoiogy, P.O. Box 11155-9567, Tehran, Iran e-maii: [email protected]
Hossein M. Shodja^ Department of Civii Engineering, Sharif University of Technoiogy, P.O. Box 11155-9313, Tehran, Iran; Institute for Nanoscience and Nanotechnoiogy, Sharif University of Technoiogy, P.O. Box 11155-9161, Tehran, Iran e-mail: [email protected]
Aref Samadi-Dooki Department of Civii Engineering, Sharif University of Technoiogy, P.O. Box 11155-9313, Tehran, iran e-mail: [email protected]
An Analytical Approach to Study the Intraoperative Fractures of Femoral Shaft During Total Hip Arthroplasty An analytical approach which is popular in micromechanical studies has been extended to the solution for the interference fit problem of the femoral stem in cementless total hip arthroplasty (THA). The multiple inhomogeneity problem of THA in transverse plane, including an elliptical stem, a cortical wall, and a cancellous layer interface, was formulated using the equivalent inclusion method (ElM) to obtain the induced interference elastic fields. Results indicated a maximum interference fit of about 210 lun before bone fracture, predicted based on the Drucker-Prager criterion for a partially reamed section. The cancellous layer had a significant effect on reducing the hoop stresses in the cortical wall; the maximum press fit increased to as high as 480 ¡imfor a 2 mm thick cancellous. The increase of the thickness and the mechanical quality, i.e., stiffness and strength, of the cortical wall also increased the maximum interference fit before fracture significantly. No considerable effect was found for the implant material on the maximum allowable interference fit. It was concluded that while larger interference fits could be adapted for younger patients, care must be taken when dealing with the elderly and those suffering from osteoporosis. A conservative reaming procedure is beneficial for such patients; however, in order to ensure sufficient primary stability without risking bone fracture, a preoperative analysis might be necessary. [DOI: 10.1115/1.4023699] Keywords: equivalent inclusion method, misfit strain, intraoperative fracture, hip stem, press-fit, interference
Introduction Primary fixation stability is an essential requirement for short and long term success of cementless total hip arthroplasty (THA). Excessive micromotions at the bone-implant interface, caused by a poor primary fixation, often lead to thigh pain and may postpone or even prevent the secondary fixation, achieved by bone ingrowth into the implant surface [1]. While the secondary implant fixation is biologic in nature, the primary fixation depends only upon the mechanical interactions of the implant and the adjacent bone structure. Thus, in a simple mechanical model of two assembling parts, i.e., implant and surrounding bone, a key factor affecting the fixation stability would be the interference fit at the time of assembly, i.e., implantation. In practice, the implant-bone interference fit is introduced intraoperatively by reaming the intramedullary canal to a smaller size than that of the femoral stem. Obviously, a larger interference fit, i.e., higher under-reaming, would provide a tighter press fitting with higher level of stability and smaller micromotion. However, the extent of the interference fit is limited by the fact that large interferences can initiate and propagate cracks in the femoral bone that might end up with intraoperative or early postoperative periprosthetic fractures. A significant incidence of the intraoperative femoral fracture during cementless THA has been well documented in the clinical literature with reported rates from 1 % to as high as 20% [2-4]. In spite of its importance, there is discrepancy with reference to the maximum allowable interference fit that ensures primary stability without risking femoral fracture. Current surgical practice Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 31, 2012; final manuscript received February 3, 2013; accepted niianuscript posted February 19. 2013; published online April 2, 2013. Assoc. Editor: Pasquale Vena.
Journal of Biomechanical Engineering
suggests an under-reaming of about 0.5 mm interference fit [5,6] which is consistent with the experimental in vitro studies [7,8]. However, recent biomechanical studies [9-11] have reported that, even an interference fit of about 100/im would cause bone damage due to the high levels of interference induced hoop stresses. They recommended a very small interference fit of 50/xm for press fitting of cementless femoral stems inside the intramedullary canal, assuming a safety factor of 2 [10,11]. There are a limited number of modeling studies, concerning the mechanics of the interference fit in cementless THA implants, which have been all based on finite element analysis [9-13]. In spite of the fact that conventional reamers only remove bones with Hounsfield Units (HU) values of less than 600 [14], neither of these studies has accounted for the cancellous bone layer between the stem and the femoral cortical wall. Moreover, these studies have been mostly concerned with the interfacial micromotion and have adopted simple criteria, e.g., von Mises equivalent stress, to address the risk of femoral fracture associated with a given level of interference fit. The von Mises criterion has been traditionally used to predict yielding of ductile materials and it does not account for the brittle behavior of cortical bone and its tension/compression strength mismatch [15]. Some investigators have developed and implemented strain-based fracture criteria for cancellous [16] and cortical bone [17], based on the experimental evidence that the cancellous bone's yield strains vary less than the corresponding yield stresses within an anatomical site [18,19]. Also, Tsai-Wu quadratic criterion, which is a generalized form of von Mises stress-based criterion, has been employed in some studies to predict the failure of cancellous [20] and cortical bone [21]. Recent studies have used Mohr-Coulomb [22] and Drucker-Prager [23-25] criteria which are able to represent the dilatation and tension/compression asymmetrical behavior of bone tissue more effectively. It has been reported that there is a close agreement
Copyright © 2013 by ASME
APRIL2013, Vol. 135 / 041004-1
between the predictions of Drucker-Prager criterion and the experimental data for bone elasto-plastic behavior [26,27]. In spite of the fact that finite element modeling can provide an accurate 3D representation of the mechanics of interference fit, it is not appropriate for parametric studies, due to the fact that a new model should be constructed for each change in the geometrical parameters. Analytical models, on the other hand, are based on more simplifications but can easily implement geometrical changes and investigate their effect on the mechanics of the press fit THA. Considering the transverse plane configuration of the stem inside the femoral intramedullary canal, an analytical solution for the 2D plane strain interference problem might be obtained using equivalent inclusion method (EIM) [28] of micromechanics. In this method, the terminology eigenstrain field is used when referring to nonelastic strains induced by thermal expansion, initial strains, misfit strains, etc. [29]. A subdomain of a material which contains eigenstrains is called an inclusion. Inhomogeneous inclusions are inclusions with elastic moduli different from those of the matrix. After the pioneering work of Eshelby on single ellipsoidal inclusion problem [28], the mechanics of inclusions and inhomogeneities has been extensively studied in thé literature and the resulting elastic fields have been addressed using analytical solutions [30]. In the present study, we attempted to provide an analytical solution for the interference fit problem of the cementless THA implants, based on an extension of EIM. A 2D plane strain model is used to describe the transverse plane configuration of the hip stem inside the femoral intermeduUary canal, considering the fact that the principal strain in the longitudinal direction of femoral shaft-stem system is negligible. The model accounts for the geometries and properties of the constituent phases consisting of stem, cancellous and cortical bones. The cancellous layer is intrinsically a functionally graded material (FGM) and is modeled as an isotropic and sectionally homogenous material with mechanical properties dependent upon its thickness. This assumption is based on the fact that the mechanical stiffness of the cancellous tissue in the shaft of long bones varies radially with an incremental pattern from endost to periost [14]. Thus, with the reamer removing the bone from the endosteal side, the mean density and stiffness would be lower for a thicker cancellous layer, resulted from a more conservative reaming procedure. The cortical wall of femoral shaft is assumed as a homogenous and isotropic material; the latter assumption is based on the fact that the mechanical properties of cortical bone are almost the same in radial and circumferential directions (transverse isotropy), although significantly different from longitudinal properties. To examine the interfacial stresses induced by the interference misfit strain, the model is described as a multiple inhomogeneity system. Subsequently, the equivalent inclusion method (EIM) [31], which is an extension of EIM [28] for a single ellipsoidal inhomogeneity, is employed. The mechanics of the press fit THA, including the risk of the bone fracture, is then studied in detail, and the interrelations between the affecting parameters, i.e., geometrical and mechanical properties of the hip stem and the cortical and cancellous bones, are investigated. The results are used to establish the maximum allowable interference fit that insures primary stability without risking femoral fracture.
Materials and Methods Formulation. The multiple layer interference problem of the THA in transverse plane (Fig. 1) might be described by a multiple inhomogeneity system consisting of stem, cancellous layers, and cortical bone, using an extension of Eshelby's original formulation of EIM [31]. We assume 4^ to represent an infinite solid with elastic moduli C, which contains an n-phase inhomogeneous elliptical inclusion, and ay.
+
Radial Slresses on Shon Radius Hoop Stresses on Shoit Rmlius Radial Stresses on Long Radius
6
- Hoop Stresses on Long Radius
, 4 I
' 0
10
16
-2 -4
0.6
Radial Strains on Short Radius
0.4
Radial Strains on Long Radius
Hoop Strains oo Short Radius
(10)
- Hoop Strains on Long Radius
where the parameters, P and N are defined by:
P= A
1
0.2
•r (11)
0
10
V 3 V »^i + ö'f -0.2
As suggested by Keyak and Rossi [37], the best values for ^ which can best capture the cortical bone fracture is between 0.7 and 1; as such, we chose (o-t/cc) = 0.85 in Eq. (10).
Results Insertion of the elliptical stem into the femoral canal exerts a uniform strain field on the inner wall of the cancellous bone. In the context of the present theory, this misfit strain field within the stem varies as [38]: (12) in which y is a constant that determines the maximum press-fit strain at the interface of the stem and bone, and a^ and 02 are the semiaxes of the stem. The maximum value of y must not exceed the value at which bone fracture occurs. The results of analyzing the proposed model of THA, with the basic geometry (Fig. 5) and mechanical properties (Table 1), are shown in Fig. 6. The distributions of the induced normalized stress, a/y and strain, e/y fields are illustrated as functions of distance from the origin of the stem along the short and long axes (mediolateral and anterior-posterior directions) of the elliptical cortical wall. Results indicate that the hoop stress and strain were always larger than those of the radial components. The highest hoop (tensile) and radial (compressive) stresses and strains within the cortical bone occurred along the longest radius in the mediolateral direction (r = 9 mm) at the cortical-cancellous bones interface. They then decreased gradually with distance from the origin of the stem towards periosteum. Figure 7 illustrates the maximum interference fit, determined based on the Drucker-Prager fracture criterion, for different implant materials and cancellous layer thicknesses. Assuming the basic geometry (Fig. 5) and mechanical properties (Table 1) for the model components, the interference corresponding to the stress-state of fracture was found to be 210 /¡m. The change of the implant material had a negligible effect on the maximum interference fit. However, a significant effect was due to the thickness of the cancellous bone, at the stem and cortical bone interface. The thicker the cancellous bone layer, i.e., less bone removed during reaming, the higher the interference fit to cause fracture. The maximum interference fits corresponding to the 0.15 mm and 2 mm cancellous thickness were 115 /im and 480 /im, respectively.
Journal of Biomechanical Engineering
-0.4
Fig. 6 Distribution of the interference induced stresses (top) and strains (bottom) along the short and long radii of the eilipticai cortical wail, "d" denotes the distance from the origin of the stem aiong the short and iong axes of the eliiptical cortical wail
The effects of the thickness and mechanical properties of the cortical wall on the maximum interference fit, determined based on the Drucker-Prager fracture, are illustrated in Fig. 8. Results indicate that a 3 mm (60%) increase of the thickness of the cortical wall would increase the maximum interference fit by about 32 p.m. (16%). The conical bone mechanical properties, however, were found to affect the maximum interference fit more considerably. For a higher quality bone with 1.1 GPa (6%) increase of the Young's modulus and 20 MPa (15%) increase of the ultimate stress, the maximum interference fit was increased by about 33/xm(17%). Discussion In this study, we provided an analytical solution for the interference fit problem of the hip stem in cementless THA, using the extended EIM for multi-inhomogeneous inclusions. The presented 500 r
400 -
Ë. 300-
200 •
100
0.5
1
1.5
Thickness of the cancellous bone (mm) Fig. 7 Effects of the canceiious iayer thickness and the stem materiai on the maximum interference fit before bone fracture
APRIL2013, Vol. 135 / 041004-5
300 r
-e-E=15.6 GPa, v=0.25,
Farzam Farahmand Schooi of Mechanicai Engineering, Sfiarif University of Technoiogy, P.O. Box 11155-9567, Tehran, Iran e-maii: [email protected]
Hossein M. Shodja^ Department of Civii Engineering, Sharif University of Technoiogy, P.O. Box 11155-9313, Tehran, Iran; Institute for Nanoscience and Nanotechnoiogy, Sharif University of Technoiogy, P.O. Box 11155-9161, Tehran, Iran e-mail: [email protected]
Aref Samadi-Dooki Department of Civii Engineering, Sharif University of Technoiogy, P.O. Box 11155-9313, Tehran, iran e-mail: [email protected]
An Analytical Approach to Study the Intraoperative Fractures of Femoral Shaft During Total Hip Arthroplasty An analytical approach which is popular in micromechanical studies has been extended to the solution for the interference fit problem of the femoral stem in cementless total hip arthroplasty (THA). The multiple inhomogeneity problem of THA in transverse plane, including an elliptical stem, a cortical wall, and a cancellous layer interface, was formulated using the equivalent inclusion method (ElM) to obtain the induced interference elastic fields. Results indicated a maximum interference fit of about 210 lun before bone fracture, predicted based on the Drucker-Prager criterion for a partially reamed section. The cancellous layer had a significant effect on reducing the hoop stresses in the cortical wall; the maximum press fit increased to as high as 480 ¡imfor a 2 mm thick cancellous. The increase of the thickness and the mechanical quality, i.e., stiffness and strength, of the cortical wall also increased the maximum interference fit before fracture significantly. No considerable effect was found for the implant material on the maximum allowable interference fit. It was concluded that while larger interference fits could be adapted for younger patients, care must be taken when dealing with the elderly and those suffering from osteoporosis. A conservative reaming procedure is beneficial for such patients; however, in order to ensure sufficient primary stability without risking bone fracture, a preoperative analysis might be necessary. [DOI: 10.1115/1.4023699] Keywords: equivalent inclusion method, misfit strain, intraoperative fracture, hip stem, press-fit, interference
Introduction Primary fixation stability is an essential requirement for short and long term success of cementless total hip arthroplasty (THA). Excessive micromotions at the bone-implant interface, caused by a poor primary fixation, often lead to thigh pain and may postpone or even prevent the secondary fixation, achieved by bone ingrowth into the implant surface [1]. While the secondary implant fixation is biologic in nature, the primary fixation depends only upon the mechanical interactions of the implant and the adjacent bone structure. Thus, in a simple mechanical model of two assembling parts, i.e., implant and surrounding bone, a key factor affecting the fixation stability would be the interference fit at the time of assembly, i.e., implantation. In practice, the implant-bone interference fit is introduced intraoperatively by reaming the intramedullary canal to a smaller size than that of the femoral stem. Obviously, a larger interference fit, i.e., higher under-reaming, would provide a tighter press fitting with higher level of stability and smaller micromotion. However, the extent of the interference fit is limited by the fact that large interferences can initiate and propagate cracks in the femoral bone that might end up with intraoperative or early postoperative periprosthetic fractures. A significant incidence of the intraoperative femoral fracture during cementless THA has been well documented in the clinical literature with reported rates from 1 % to as high as 20% [2-4]. In spite of its importance, there is discrepancy with reference to the maximum allowable interference fit that ensures primary stability without risking femoral fracture. Current surgical practice Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 31, 2012; final manuscript received February 3, 2013; accepted niianuscript posted February 19. 2013; published online April 2, 2013. Assoc. Editor: Pasquale Vena.
Journal of Biomechanical Engineering
suggests an under-reaming of about 0.5 mm interference fit [5,6] which is consistent with the experimental in vitro studies [7,8]. However, recent biomechanical studies [9-11] have reported that, even an interference fit of about 100/im would cause bone damage due to the high levels of interference induced hoop stresses. They recommended a very small interference fit of 50/xm for press fitting of cementless femoral stems inside the intramedullary canal, assuming a safety factor of 2 [10,11]. There are a limited number of modeling studies, concerning the mechanics of the interference fit in cementless THA implants, which have been all based on finite element analysis [9-13]. In spite of the fact that conventional reamers only remove bones with Hounsfield Units (HU) values of less than 600 [14], neither of these studies has accounted for the cancellous bone layer between the stem and the femoral cortical wall. Moreover, these studies have been mostly concerned with the interfacial micromotion and have adopted simple criteria, e.g., von Mises equivalent stress, to address the risk of femoral fracture associated with a given level of interference fit. The von Mises criterion has been traditionally used to predict yielding of ductile materials and it does not account for the brittle behavior of cortical bone and its tension/compression strength mismatch [15]. Some investigators have developed and implemented strain-based fracture criteria for cancellous [16] and cortical bone [17], based on the experimental evidence that the cancellous bone's yield strains vary less than the corresponding yield stresses within an anatomical site [18,19]. Also, Tsai-Wu quadratic criterion, which is a generalized form of von Mises stress-based criterion, has been employed in some studies to predict the failure of cancellous [20] and cortical bone [21]. Recent studies have used Mohr-Coulomb [22] and Drucker-Prager [23-25] criteria which are able to represent the dilatation and tension/compression asymmetrical behavior of bone tissue more effectively. It has been reported that there is a close agreement
Copyright © 2013 by ASME
APRIL2013, Vol. 135 / 041004-1
between the predictions of Drucker-Prager criterion and the experimental data for bone elasto-plastic behavior [26,27]. In spite of the fact that finite element modeling can provide an accurate 3D representation of the mechanics of interference fit, it is not appropriate for parametric studies, due to the fact that a new model should be constructed for each change in the geometrical parameters. Analytical models, on the other hand, are based on more simplifications but can easily implement geometrical changes and investigate their effect on the mechanics of the press fit THA. Considering the transverse plane configuration of the stem inside the femoral intramedullary canal, an analytical solution for the 2D plane strain interference problem might be obtained using equivalent inclusion method (EIM) [28] of micromechanics. In this method, the terminology eigenstrain field is used when referring to nonelastic strains induced by thermal expansion, initial strains, misfit strains, etc. [29]. A subdomain of a material which contains eigenstrains is called an inclusion. Inhomogeneous inclusions are inclusions with elastic moduli different from those of the matrix. After the pioneering work of Eshelby on single ellipsoidal inclusion problem [28], the mechanics of inclusions and inhomogeneities has been extensively studied in thé literature and the resulting elastic fields have been addressed using analytical solutions [30]. In the present study, we attempted to provide an analytical solution for the interference fit problem of the cementless THA implants, based on an extension of EIM. A 2D plane strain model is used to describe the transverse plane configuration of the hip stem inside the femoral intermeduUary canal, considering the fact that the principal strain in the longitudinal direction of femoral shaft-stem system is negligible. The model accounts for the geometries and properties of the constituent phases consisting of stem, cancellous and cortical bones. The cancellous layer is intrinsically a functionally graded material (FGM) and is modeled as an isotropic and sectionally homogenous material with mechanical properties dependent upon its thickness. This assumption is based on the fact that the mechanical stiffness of the cancellous tissue in the shaft of long bones varies radially with an incremental pattern from endost to periost [14]. Thus, with the reamer removing the bone from the endosteal side, the mean density and stiffness would be lower for a thicker cancellous layer, resulted from a more conservative reaming procedure. The cortical wall of femoral shaft is assumed as a homogenous and isotropic material; the latter assumption is based on the fact that the mechanical properties of cortical bone are almost the same in radial and circumferential directions (transverse isotropy), although significantly different from longitudinal properties. To examine the interfacial stresses induced by the interference misfit strain, the model is described as a multiple inhomogeneity system. Subsequently, the equivalent inclusion method (EIM) [31], which is an extension of EIM [28] for a single ellipsoidal inhomogeneity, is employed. The mechanics of the press fit THA, including the risk of the bone fracture, is then studied in detail, and the interrelations between the affecting parameters, i.e., geometrical and mechanical properties of the hip stem and the cortical and cancellous bones, are investigated. The results are used to establish the maximum allowable interference fit that insures primary stability without risking femoral fracture.
Materials and Methods Formulation. The multiple layer interference problem of the THA in transverse plane (Fig. 1) might be described by a multiple inhomogeneity system consisting of stem, cancellous layers, and cortical bone, using an extension of Eshelby's original formulation of EIM [31]. We assume 4^ to represent an infinite solid with elastic moduli C, which contains an n-phase inhomogeneous elliptical inclusion, and ay.
+
Radial Slresses on Shon Radius Hoop Stresses on Shoit Rmlius Radial Stresses on Long Radius
6
- Hoop Stresses on Long Radius
, 4 I
' 0
10
16
-2 -4
0.6
Radial Strains on Short Radius
0.4
Radial Strains on Long Radius
Hoop Strains oo Short Radius
(10)
- Hoop Strains on Long Radius
where the parameters, P and N are defined by:
P= A
1
0.2
•r (11)
0
10
V 3 V »^i + ö'f -0.2
As suggested by Keyak and Rossi [37], the best values for ^ which can best capture the cortical bone fracture is between 0.7 and 1; as such, we chose (o-t/cc) = 0.85 in Eq. (10).
Results Insertion of the elliptical stem into the femoral canal exerts a uniform strain field on the inner wall of the cancellous bone. In the context of the present theory, this misfit strain field within the stem varies as [38]: (12) in which y is a constant that determines the maximum press-fit strain at the interface of the stem and bone, and a^ and 02 are the semiaxes of the stem. The maximum value of y must not exceed the value at which bone fracture occurs. The results of analyzing the proposed model of THA, with the basic geometry (Fig. 5) and mechanical properties (Table 1), are shown in Fig. 6. The distributions of the induced normalized stress, a/y and strain, e/y fields are illustrated as functions of distance from the origin of the stem along the short and long axes (mediolateral and anterior-posterior directions) of the elliptical cortical wall. Results indicate that the hoop stress and strain were always larger than those of the radial components. The highest hoop (tensile) and radial (compressive) stresses and strains within the cortical bone occurred along the longest radius in the mediolateral direction (r = 9 mm) at the cortical-cancellous bones interface. They then decreased gradually with distance from the origin of the stem towards periosteum. Figure 7 illustrates the maximum interference fit, determined based on the Drucker-Prager fracture criterion, for different implant materials and cancellous layer thicknesses. Assuming the basic geometry (Fig. 5) and mechanical properties (Table 1) for the model components, the interference corresponding to the stress-state of fracture was found to be 210 /¡m. The change of the implant material had a negligible effect on the maximum interference fit. However, a significant effect was due to the thickness of the cancellous bone, at the stem and cortical bone interface. The thicker the cancellous bone layer, i.e., less bone removed during reaming, the higher the interference fit to cause fracture. The maximum interference fits corresponding to the 0.15 mm and 2 mm cancellous thickness were 115 /im and 480 /im, respectively.
Journal of Biomechanical Engineering
-0.4
Fig. 6 Distribution of the interference induced stresses (top) and strains (bottom) along the short and long radii of the eilipticai cortical wail, "d" denotes the distance from the origin of the stem aiong the short and iong axes of the eliiptical cortical wail
The effects of the thickness and mechanical properties of the cortical wall on the maximum interference fit, determined based on the Drucker-Prager fracture, are illustrated in Fig. 8. Results indicate that a 3 mm (60%) increase of the thickness of the cortical wall would increase the maximum interference fit by about 32 p.m. (16%). The conical bone mechanical properties, however, were found to affect the maximum interference fit more considerably. For a higher quality bone with 1.1 GPa (6%) increase of the Young's modulus and 20 MPa (15%) increase of the ultimate stress, the maximum interference fit was increased by about 33/xm(17%). Discussion In this study, we provided an analytical solution for the interference fit problem of the hip stem in cementless THA, using the extended EIM for multi-inhomogeneous inclusions. The presented 500 r
400 -
Ë. 300-
200 •
100
0.5
1
1.5
Thickness of the cancellous bone (mm) Fig. 7 Effects of the canceiious iayer thickness and the stem materiai on the maximum interference fit before bone fracture
APRIL2013, Vol. 135 / 041004-5
300 r
-e-E=15.6 GPa, v=0.25,
