ANNALS O:F BIO?CIEDICAL ENGINEERING ~, 4 1 0 - - 4 2 0
(1977)
Stress Distribution in Porous Surfaced Medullary Implants S. C. ANAND
Department of Civil Engineering, Clemson University, Clemson, South Carolina 29631 K . ~ . ST. JOHN AND D . D . MOYLE
Division of Interdisciplinary Studies, Clemson University, Clemson, South Carolina 29631 AND
D. F. WILLIAMS School of Dental Surgery, University of Liverpool, England Received April 13, 1977 Finite element stress analysis has been applied to examine the stress patterns in a prosthesis requiring fixation in the medullary shaft of a l~ng bone. No specific prosthesis is considered but rather a generally applicable geometry has been chosen. This consists of a cylindrical section of cortical bone within which is implanted a prosthesis composed of a solid central rod surrounded by a porous coating. The finite element analysis utilized an axisymmetrie model to determine the dist ribution of stresses throughout the system. T h e effect of changes in length of prosthesis, thickness of porous coating, d e p t h and type of tissue ingrowth, and type of porous coating material were studied under conditions of axisymmetrie loading. The resldts indicate t h a t with complete bone ingrowth, the maximum shear stress and the distance necessary for load transfer are b o t h independent of implant length. However, with ineolnplete ingrowth, increasing implant length reduces shear. Incomplete growth also produces lower shear stresses b u t higher shear strains in areas without ingrowth. In addition, a porous polyethylene coating gives a more even load transfer and lower shear than a porous coating of a high modulus material. INTRODUCTION
It is now several years since the concept of prosthesis fixation to the musculoskeletal system by tissue ingrowth into porous materials was developed (Klawitter and Hulbert, 1971; Hulbert et al., 1973). Experimental work in this area had identified many of the characteristics of the porous material that are conducive to ingrowth (Klawitter, 1969; Predecki et al., 1972) and it has been shown that tissue can be induced to grow into a wide range of materials, including examples of metals (Nilles et al., 1974), ceramics (Hentrich et al., 1971), and polymers (Sauer et al., 1974). 410 Copyright 9 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
ISSN 0090-6964
STRESSES IN MEDULLARY IMPLANTS
411
One difficulty involved in utilizing this concept in clinical practice is that there is no satisfactory evidence to suggest which type of material is the most suitable to use in porous form. Most of the early work was performed with ceramics, partly because of their inertness and partly because of the relative ease with which they may be made with controlled porosity. However, they do have some disadvantages, notably their brittleness, which is not improved by the porosity. The fact that only a small depth of tissue ingrowth is necessary for fixation and that there are some potential disadvantages of uniformly porous prostheses has led to the use of thin porous coatings on solid substrates. In this ease materials giving an optimal compromise between rigidity, strength, and toughness may be used for the substrate, usually implying the use of a metal. The choice of porous coating can then be made on the grounds of biomeehanieal compatibility, tissue compatibility, its own mechanical properties and the ease of bonding to the stem. There is a certain attractiveness in making the porous layer of the same material as the stem. A major disadvantage of an all metal system is that the bioeompatibility of most metals is marginal and the use of a much larger surface area may accentuate any problems. Also, it has been suggested that a superior biomechanieal situation exists if a low modulus material is used at the interface between metal stem and bone (Homsy et al., 1973). For this reason, the use of a polymer coating on a metal stem has been advocated. Since this aspect of biomeehanieal compatibility is of considerable importance, an analysis of the load transfer between an intramedullary prosthesis and bone, as a function of porous layer material and geometry, has been performed. METHOD OF ANALYSIS The finite element stress analysis technique has been applied to study the stress distribution in and around a generalized medullary implant. In the finite element method, a continuum is treated as an assemblage of discrete members connected only at a small number of joints called nodal points. By this procedure, the geometry of the object being modeled can be accurately portrayed and variation in material properties can be accounted for by the assignment of different properties to the various elements. With the advent of the high-speed digital computer, the solution of the many simultaneous equations involved has been faeilit.ated. Plane stress, plane strain, axisymmetrie solid, shell, and three-dimensional problems have been effectively solved through the use of the finite element method (Zienkiewiez, 1971). The technique has also been used in stress analysis studies of the biological systems (Ghista et al., 1975). The finite element model, shown in Fig. 1, is a two-dimensional representation of a three-dimensional axisymmetrie system. The prosthesis consists of a central rod of steel with a diameter of 16 ram, coated with a porous material of 4-ram thickness, implanted in the medullary shaft with the surrounding cortical bone having a uniform thickness of 4 ram. This prosthesis is subjected to a uniform axisymmetrie load of 784 N (80 kg) at its upper end. Although simplified, this model can be related to a number of physiologically identifiable situations, such as a femoral intramedullary implant.
412
ANAND
ET
AL.
CONSTANT
STRESS
ELEMENTS
I
.k
IMPLANT S g A F T
--
III1!
16~o,A.
II I' ! ~,
4 M*, THICK
LI,..I-,A ITII-[-'! I,,IL I I ' I I LLENSTH:S" EL~
.I
D;,Ji!l EJ~'~T~t~
i!
L JII
ENT
T
FIG. 1. The finite element model. The following assumptions have been made in the finite element representation of the actual system. Rectangular axisymmetric elements are used, each of which is subdivided into four triangular elements. Stresses and strains are assumed to be uniform in each triangular element. Consequently displacements along element boundaries are linear. I t must be emphasized t h a t only elastic deformations are considered in this analysis. I t is further assumed t h a t there is no slip between different components (i.e., between stem and porous layer and between porous layer and adjacent bone). Finally, it is assumed t h a t the b o u n d a r y effects are negligible ; i.e., t h a t there is a sufficient length of bone between the end of the prosthesis and the fixed boundary. For modeling purposes a steel shaft has been chosen. I n practice, it makes little difference in the analysis whether this shaft is made of steel or any metal or ceramic with an approximately equivalent elastic modulus. Porous high density polyethylene ( H D P E ) has been selected as a coating material. I n order to assess TABLE 1 Elastic Properties of Solid Materials Material
Young's modulus psi
Steel HI) polyethylene
Poisson's ratio
References
Parker (1967) Parker (1967) Sandiford and Willbourn (1960) Herrman and Leibowitz (1972) Sweeney et al. (1965) Sweeneyet al. (1965) J~atz (1971)
MN/m ~
28 1
X 106 X 105
1.93 X 105 6.89 X 102
0.29 0.34
CortieM bone
2
X 106
1.38 X 104
0.30
Osteoid tissue
3
X 104
2.09 X 102
0.35
Fibrous tissue
1.5 X 104
1.04 X 10~
0.35
STRESSES IN MEDULLARY IMPLANTS
413
TABLE 2 Calculated Elastic Properties of Porous Materials and Composites Material
Young's modulus psi
Porous HDPE Bone/HDPE Osteoid/HDPE Fibrous/HDPE Parous st~ed Bone/steel
2.5 6.2 5.6 4.2 7 9.4
Poisson's ratio
MN/m 2
X 104 X 105
1.72 X 4.28 X 3.85 X 2.87 X 4.82 X 6.45 X
X 104 X 104 X 10e X 10e
102 103 102 102 104 104
0.34 0.31 0.345 0.343 0.288 0.29
high modulus coatings, porous steel has also been considered. The choice of porous polyethylene is based primarily on current experimental work with this material at Clemson University. Y o u n g ' s modulus and Poisson's ratio for each material considered are listed in Table 1. T h e elastic moduli for the porous materials when ingrowth of bone has been achieved are not available in the literature. The moduli for these materials have been estimated b y assuming 50% porosity and uniform ingrowth of bone and averaging the values obtained by the constant strain and constant stress models (Moyle et al., 1973). These data are presented in Table 2. RESULTS The finite element method yields a considerable a m o u n t of information regarding the stresses and strains in each element. For the sake of clarity and relevance, most of the results are presented as graphs of m a x i m u m shear stress and strain 2.. BONE
/
INGROWTH
LO o3 L~I ~o
BONE INGROWTH
} w. i
i
I
I
,
I BONE t
LOAD
i
1
I
l
SHAFT
i
/I_L 40
i
56
"
1 80,~
FIG. 2. The effect of implant length on the maximum shear stress in the inner region of a 4-mmthick porous polyethylene coating.
414
ANAND ~
ET
AL.
4
b.j ~
ec
(0 O)
co D,
i.___
3
FIG. 3. Maximum shear stresses and strains in a 4-mm-thiek porous polyethylene coating with 2-mm bone ingrowth.
as a function of distance along the length of the prosthesis. Shear stress has been chosen because shear appears to be the most probable mode of failure for this type of prosthesis and loading condition. However, in some eases axial s t r z n or principal stresses have been chosen as more appropriate. Figure 2 shows the effect of varying the prosthesis length within the medullary cavity using a porous polyethylene coating on a steel shaft. Maximum shear stress in the elements nearest the shaft is plotted for both full bone ingrowth and 2-mm bone ingrowth into a 4-mm porous layer. For full ingrowth it can be
4
------
(~
Q3
4 LO/-D )
j
~
. SHAFT
. . . . . . . .
FIG. 4. Maximum shear stresses and axial strains in a 4-mm-thick porous polyethylene coating with complete bone ingrowth.
STRESSES IN MEDULLAI~Y IMPLANTS
415
4MM ,~UM G J '% 4 -
1~
s..~,.~ . . . . . . . . . . .
4--
FIG. 9. The effect of a change in the porous coating material on the maximum shear stress in the region of cortical bone adjacent to the prosthesis. situation m a y arise, however, where only limited ingrowth to a given depth occurs in which case the load transfer mechanism is quite different. This case is shown in Figure 6, where a 3-ram limiting depth of ingrowth has been assumed. F o r a prosthesis with 4-ram-thick porous coating, an area of porous material is left unfilled, whereas complete ingrowth occurs in a 3-ram coating. Clearly, higher shear stresses are found for complete ingrowth. In the case of incomplete ingrowth, however, the inner porous polyethylene region suffers much higher shear strains t h a n does the bone-polymer composite of the former ease. F u r t h e r analysis, in which it is assumed t h a t at a given time during tissue ingrowth into a 3-ram layer there are equal distributions of calcified, osteoid, and fibrous tissue (Fig. 7), indicates t h a t maximum shear strains are much higher in the lower modulus regions t h a n in the bone-polymer region or, indeed, in any area for the case of full ingrowth. I t m a y be advisable to use a solid polyethylene inner layer to facilitate bonding to the steel shaft and to produce a stronger interface. Figure 8 shows that a system incorporating a solid inner layer of polyethylene offers a good compromise with respect to shear stresses and shear strains between full ingrowth and partial ingrowth into a porous layer. The large difference in elastic moduli between steel and polyethylene would suggest t h a t a porous steel coating on a steel stem would give quite different results from those obtained with porous polyethylene on the steel stem. This is in fact the case, illustrated in Fig. 9 where the maximum shear stresses in the inner zone of the cortical bone are plotted. For the steel coating, the highest value of maximum shear stress is greater t h a n t h a t for the polyethylene coating, although this difference is not too great. More importantly, the load is transferred to the bone over a much shorter length in the former case. Figure 10 gives the principal stresses in the bone phase of the bone-porous-material composite which have been calculated using a constant strain model. Both the maximum and minim u m principal stresses have been plotted since it is not known which stress system has the greater influence on osteogenesis. I t can clearly be seen that the bone
418
ANAND ET AL.
~
MAXSTRESS POLYETHYLENE POROUSZ 88
c~u~ c~cb ~ ~- ~ -2.
MAXIsTE=L MIN.]
M
/
N
STRESS POt.YE7HFLE/7~
FIG. 10. The principal stresses in the bone phase in the inner region of the bone-porous-material eomposRe for steel and polyethylene coatings. within a steel-bone composite transmits little, if any, of the load, whereas the bone of the bone-polyethylene composite transmits an appreciable amount. DISCUSSION The finite element m e t h o d has been applied to a variety of composite systems for attaching a medullary shaft imphmt. The results, while valid only for axisymmetric geometry, give us information which allows us to assess the performance capabilities of each a t t a c h m e n t system or ingrowth pattern. It has been seen thut, for a porous polyethylene surface, incomplete ingrowth into a thick layer results in relatively lower shear stresses but higher shear strains than are the ease with complete ingrowth into a somewhat thinner layer. If there is a limiting depth of ingrowth of bony tissue we can achieve either of the above situations b y selecting an appropriate thickness of coating material. It is not immediately clear which of these situations is the more desirable and any selection criteria would have to include the strength of the porous material and the effect of deformations on the vascular fibrous tissue within the coating which m a y be the precursor to bone formation. It is possible t h a t large deformations under conditions of incomplete ingrowth m a y be the limiting factor inhibiting mineralization and viable bone formation. We have also seen that the use of a layer of solid polyethylene next to the steel shaft gives values of shear stresses and strains which are between the two extremes cited above. The comparison of a porous steel with the porous polyethylene is of some interest. It is important to note that steel is representative of high modulus materials and, for the purposes of the analysis carried out here, the same qualitative results would be obtained with other metals or even ceramics of approximately equivalent modulus. The steel coating produces a sharper stress concentration t h a t the lower modulus polyethylene and the stresses on the bone within the pores of the steel coating are nearly negligible. The optimum magnitude of the stress in the bone-porous-materiM composite is not clear. Excessively high stresses are not desirable as t h e y m a y lead to mechanicM failure, particularly in the case of the much weaker polyethylene. However,, v e r y low stresses in the:
STRESSES IN MEDULLARY IMPLANTS
419
ingrown bone itself m a y be undesirable, resulting in poor bone ingrowth according to the principles of stress-induced osteogenesis. M o r e specifically, some degree of load transmission through ingrown tissue is p r o b a b l y necessary. CONCLUSIONS T h e results of this s t u d y indicate that, for conditions of axisymmetric loading, the m a x i m u m shear stress attained is independent of the length of the i m p l a n t with full bone growth into the porous coating and the load is transferred over a constant length. Under conditions of incomplete bone ingrowth, however, the m a x i m u m shear stress is reduced b y increasing the i m p l a n t length. I t has been shown t h a t a coating of porous polyethylene produces a more even load transfer and lower m a x i m u m shear stress t h a n a coating of porous steel. Further, the results indicate t h a t incomplete bone ingrowth produces lower m a x i m u m shear stresses b u t higher shear strains in the inner region of the porous coating t h a n under conditions of complete bone ingrowth. This situation is relieved b y using a layer of solid polyethylene next to the shaft, which gives a good compromise, trading somewhat higher stresses for lower shear strains. Finally, the analysis indicates t h a t the bone which grows into a porous steel coating (i.e., the bone within the pores) is under negligible stress. Since it is necessary for skeletal tissue to sustain some loading in order to maintain viability, this result suggests t h a t high modulus materials m a y not be suitable for porous coatings. REFERENCES Ghista, D. N., Kobayashi, A. S., and Davids, N. Analyses of some biomechanical structures and flows by computer-based finite element method. Computers in Biology and Medicine 1975, 5, 119-161. Hentrich, R. L., Graves, G. A., Stein, H. G., and Bajpal, P. K. An evaluation of inert and resorbable ceramics for future clinical orthopedic applications. Journal of Biomedical Materials Research 1971, 5, 25-51. Hernnan, G., and Liebowitz, H. Mechanics of bone fracture. In H. Liebowitz (Ed.), Fracture. New York: Academic Press, 1972. Homsy, C. A., Kent, J. N., and Hinds, E. C. Materials for oral implantation--biological and functional criteria. Journal of the American Dental Association 1973, 86, 817-832. Hulbert, S. F, Cooke, F. W., Klawitter, J. J., Leonard, R. B., Saner, B. W., Moyle, D. D., and Skinner, H. B. Attachment of prostheses to the musculoskeletal system by tissue ingrowth and mechanical interlocking. Journal of Biomedical Materials Research Symposium No. 4 1973, 1-23. Katz, J. L. Hard tissue as a composite material-I. Bounds on the elastic behavior. Journal of Biomechanics 1971, 4, 455-473. Klawitter, J. J. A basic investigation of bone ingrowth into a porous material. Ph.D. Thesis, Clemson Univ., 1969. Klawitter, J. J., and Halbert, S. F. Application of porous ceramics for the attachment of load bearing internal orthopedic appliances. Journal of Biomedical Materials Research Symposium No. 2, 1971~ 161-229. Moyle, ]). D., Klawitter, J. J., and Halbert, S. F. Mechanical properties of the bone-porous biomaterial interface: elastic behavior. Journal of Biomedical Materials Research Symposium No. 4, 1973, 363-382. Nilles, J. L., Karagianes, M. T., and Wheeler, K. R. Porous titanium alloy for fixation of knee prostheses. Journal of Biomedical Materials Research Symposium No. 5, 1974, 319-328.
420
ANAND E T A L .
Parker, E. R. (Ed.) Materials data book for engineers and scientists. New York : McGraw-Hill, 1967. Predeeki, P., Stephan, J. E., Auslaender, B. A., Mooney, u C., and Kirkland, K. Kinetics of bone growth into cylindrical channels in aluminum oxide and titanium. Journal of Biomedical Materials Research 1972, 5, 375-400. Sandiford, D. J. H., and Willbourn, A. H. General mechanical properties. In A. Renfrew and P. Morgan (Eds.), Polythene: the technology and uses of ethylene polymers. New York : Interscience, 1960. Sauer, B. W., Weinstein, A. M., Klawitter, J. J., Hulbert, S. F., Leonard, R. B. and Bagwell, J. G. The role of porous polymeric materials in prosthesis attachment. Journal of Biomedical 3/laterials Research Symposium No. 5, 1974, 145-153. Sweeney, A. W., Kroon, R. P., and Byers, R. K. Mechanical characteristics of bone and its constituents. ASME Paper 65-WA/HUF-7, 1965. Zienkiewicz, O. C. The finite element method in engineering science. London : McGraw-Hill, 1971.
(1977)
Stress Distribution in Porous Surfaced Medullary Implants S. C. ANAND
Department of Civil Engineering, Clemson University, Clemson, South Carolina 29631 K . ~ . ST. JOHN AND D . D . MOYLE
Division of Interdisciplinary Studies, Clemson University, Clemson, South Carolina 29631 AND
D. F. WILLIAMS School of Dental Surgery, University of Liverpool, England Received April 13, 1977 Finite element stress analysis has been applied to examine the stress patterns in a prosthesis requiring fixation in the medullary shaft of a l~ng bone. No specific prosthesis is considered but rather a generally applicable geometry has been chosen. This consists of a cylindrical section of cortical bone within which is implanted a prosthesis composed of a solid central rod surrounded by a porous coating. The finite element analysis utilized an axisymmetrie model to determine the dist ribution of stresses throughout the system. T h e effect of changes in length of prosthesis, thickness of porous coating, d e p t h and type of tissue ingrowth, and type of porous coating material were studied under conditions of axisymmetrie loading. The resldts indicate t h a t with complete bone ingrowth, the maximum shear stress and the distance necessary for load transfer are b o t h independent of implant length. However, with ineolnplete ingrowth, increasing implant length reduces shear. Incomplete growth also produces lower shear stresses b u t higher shear strains in areas without ingrowth. In addition, a porous polyethylene coating gives a more even load transfer and lower shear than a porous coating of a high modulus material. INTRODUCTION
It is now several years since the concept of prosthesis fixation to the musculoskeletal system by tissue ingrowth into porous materials was developed (Klawitter and Hulbert, 1971; Hulbert et al., 1973). Experimental work in this area had identified many of the characteristics of the porous material that are conducive to ingrowth (Klawitter, 1969; Predecki et al., 1972) and it has been shown that tissue can be induced to grow into a wide range of materials, including examples of metals (Nilles et al., 1974), ceramics (Hentrich et al., 1971), and polymers (Sauer et al., 1974). 410 Copyright 9 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
ISSN 0090-6964
STRESSES IN MEDULLARY IMPLANTS
411
One difficulty involved in utilizing this concept in clinical practice is that there is no satisfactory evidence to suggest which type of material is the most suitable to use in porous form. Most of the early work was performed with ceramics, partly because of their inertness and partly because of the relative ease with which they may be made with controlled porosity. However, they do have some disadvantages, notably their brittleness, which is not improved by the porosity. The fact that only a small depth of tissue ingrowth is necessary for fixation and that there are some potential disadvantages of uniformly porous prostheses has led to the use of thin porous coatings on solid substrates. In this ease materials giving an optimal compromise between rigidity, strength, and toughness may be used for the substrate, usually implying the use of a metal. The choice of porous coating can then be made on the grounds of biomeehanieal compatibility, tissue compatibility, its own mechanical properties and the ease of bonding to the stem. There is a certain attractiveness in making the porous layer of the same material as the stem. A major disadvantage of an all metal system is that the bioeompatibility of most metals is marginal and the use of a much larger surface area may accentuate any problems. Also, it has been suggested that a superior biomechanieal situation exists if a low modulus material is used at the interface between metal stem and bone (Homsy et al., 1973). For this reason, the use of a polymer coating on a metal stem has been advocated. Since this aspect of biomeehanieal compatibility is of considerable importance, an analysis of the load transfer between an intramedullary prosthesis and bone, as a function of porous layer material and geometry, has been performed. METHOD OF ANALYSIS The finite element stress analysis technique has been applied to study the stress distribution in and around a generalized medullary implant. In the finite element method, a continuum is treated as an assemblage of discrete members connected only at a small number of joints called nodal points. By this procedure, the geometry of the object being modeled can be accurately portrayed and variation in material properties can be accounted for by the assignment of different properties to the various elements. With the advent of the high-speed digital computer, the solution of the many simultaneous equations involved has been faeilit.ated. Plane stress, plane strain, axisymmetrie solid, shell, and three-dimensional problems have been effectively solved through the use of the finite element method (Zienkiewiez, 1971). The technique has also been used in stress analysis studies of the biological systems (Ghista et al., 1975). The finite element model, shown in Fig. 1, is a two-dimensional representation of a three-dimensional axisymmetrie system. The prosthesis consists of a central rod of steel with a diameter of 16 ram, coated with a porous material of 4-ram thickness, implanted in the medullary shaft with the surrounding cortical bone having a uniform thickness of 4 ram. This prosthesis is subjected to a uniform axisymmetrie load of 784 N (80 kg) at its upper end. Although simplified, this model can be related to a number of physiologically identifiable situations, such as a femoral intramedullary implant.
412
ANAND
ET
AL.
CONSTANT
STRESS
ELEMENTS
I
.k
IMPLANT S g A F T
--
III1!
16~o,A.
II I' ! ~,
4 M*, THICK
LI,..I-,A ITII-[-'! I,,IL I I ' I I LLENSTH:S" EL~
.I
D;,Ji!l EJ~'~T~t~
i!
L JII
ENT
T
FIG. 1. The finite element model. The following assumptions have been made in the finite element representation of the actual system. Rectangular axisymmetric elements are used, each of which is subdivided into four triangular elements. Stresses and strains are assumed to be uniform in each triangular element. Consequently displacements along element boundaries are linear. I t must be emphasized t h a t only elastic deformations are considered in this analysis. I t is further assumed t h a t there is no slip between different components (i.e., between stem and porous layer and between porous layer and adjacent bone). Finally, it is assumed t h a t the b o u n d a r y effects are negligible ; i.e., t h a t there is a sufficient length of bone between the end of the prosthesis and the fixed boundary. For modeling purposes a steel shaft has been chosen. I n practice, it makes little difference in the analysis whether this shaft is made of steel or any metal or ceramic with an approximately equivalent elastic modulus. Porous high density polyethylene ( H D P E ) has been selected as a coating material. I n order to assess TABLE 1 Elastic Properties of Solid Materials Material
Young's modulus psi
Steel HI) polyethylene
Poisson's ratio
References
Parker (1967) Parker (1967) Sandiford and Willbourn (1960) Herrman and Leibowitz (1972) Sweeney et al. (1965) Sweeneyet al. (1965) J~atz (1971)
MN/m ~
28 1
X 106 X 105
1.93 X 105 6.89 X 102
0.29 0.34
CortieM bone
2
X 106
1.38 X 104
0.30
Osteoid tissue
3
X 104
2.09 X 102
0.35
Fibrous tissue
1.5 X 104
1.04 X 10~
0.35
STRESSES IN MEDULLARY IMPLANTS
413
TABLE 2 Calculated Elastic Properties of Porous Materials and Composites Material
Young's modulus psi
Porous HDPE Bone/HDPE Osteoid/HDPE Fibrous/HDPE Parous st~ed Bone/steel
2.5 6.2 5.6 4.2 7 9.4
Poisson's ratio
MN/m 2
X 104 X 105
1.72 X 4.28 X 3.85 X 2.87 X 4.82 X 6.45 X
X 104 X 104 X 10e X 10e
102 103 102 102 104 104
0.34 0.31 0.345 0.343 0.288 0.29
high modulus coatings, porous steel has also been considered. The choice of porous polyethylene is based primarily on current experimental work with this material at Clemson University. Y o u n g ' s modulus and Poisson's ratio for each material considered are listed in Table 1. T h e elastic moduli for the porous materials when ingrowth of bone has been achieved are not available in the literature. The moduli for these materials have been estimated b y assuming 50% porosity and uniform ingrowth of bone and averaging the values obtained by the constant strain and constant stress models (Moyle et al., 1973). These data are presented in Table 2. RESULTS The finite element method yields a considerable a m o u n t of information regarding the stresses and strains in each element. For the sake of clarity and relevance, most of the results are presented as graphs of m a x i m u m shear stress and strain 2.. BONE
/
INGROWTH
LO o3 L~I ~o
BONE INGROWTH
} w. i
i
I
I
,
I BONE t
LOAD
i
1
I
l
SHAFT
i
/I_L 40
i
56
"
1 80,~
FIG. 2. The effect of implant length on the maximum shear stress in the inner region of a 4-mmthick porous polyethylene coating.
414
ANAND ~
ET
AL.
4
b.j ~
ec
(0 O)
co D,
i.___
3
FIG. 3. Maximum shear stresses and strains in a 4-mm-thiek porous polyethylene coating with 2-mm bone ingrowth.
as a function of distance along the length of the prosthesis. Shear stress has been chosen because shear appears to be the most probable mode of failure for this type of prosthesis and loading condition. However, in some eases axial s t r z n or principal stresses have been chosen as more appropriate. Figure 2 shows the effect of varying the prosthesis length within the medullary cavity using a porous polyethylene coating on a steel shaft. Maximum shear stress in the elements nearest the shaft is plotted for both full bone ingrowth and 2-mm bone ingrowth into a 4-mm porous layer. For full ingrowth it can be
4
------
(~
Q3
4 LO/-D )
j
~
. SHAFT
. . . . . . . .
FIG. 4. Maximum shear stresses and axial strains in a 4-mm-thick porous polyethylene coating with complete bone ingrowth.
STRESSES IN MEDULLAI~Y IMPLANTS
415
4MM ,~UM G J '% 4 -
1~
s..~,.~ . . . . . . . . . . .
4--
FIG. 9. The effect of a change in the porous coating material on the maximum shear stress in the region of cortical bone adjacent to the prosthesis. situation m a y arise, however, where only limited ingrowth to a given depth occurs in which case the load transfer mechanism is quite different. This case is shown in Figure 6, where a 3-ram limiting depth of ingrowth has been assumed. F o r a prosthesis with 4-ram-thick porous coating, an area of porous material is left unfilled, whereas complete ingrowth occurs in a 3-ram coating. Clearly, higher shear stresses are found for complete ingrowth. In the case of incomplete ingrowth, however, the inner porous polyethylene region suffers much higher shear strains t h a n does the bone-polymer composite of the former ease. F u r t h e r analysis, in which it is assumed t h a t at a given time during tissue ingrowth into a 3-ram layer there are equal distributions of calcified, osteoid, and fibrous tissue (Fig. 7), indicates t h a t maximum shear strains are much higher in the lower modulus regions t h a n in the bone-polymer region or, indeed, in any area for the case of full ingrowth. I t m a y be advisable to use a solid polyethylene inner layer to facilitate bonding to the steel shaft and to produce a stronger interface. Figure 8 shows that a system incorporating a solid inner layer of polyethylene offers a good compromise with respect to shear stresses and shear strains between full ingrowth and partial ingrowth into a porous layer. The large difference in elastic moduli between steel and polyethylene would suggest t h a t a porous steel coating on a steel stem would give quite different results from those obtained with porous polyethylene on the steel stem. This is in fact the case, illustrated in Fig. 9 where the maximum shear stresses in the inner zone of the cortical bone are plotted. For the steel coating, the highest value of maximum shear stress is greater t h a n t h a t for the polyethylene coating, although this difference is not too great. More importantly, the load is transferred to the bone over a much shorter length in the former case. Figure 10 gives the principal stresses in the bone phase of the bone-porous-material composite which have been calculated using a constant strain model. Both the maximum and minim u m principal stresses have been plotted since it is not known which stress system has the greater influence on osteogenesis. I t can clearly be seen that the bone
418
ANAND ET AL.
~
MAXSTRESS POLYETHYLENE POROUSZ 88
c~u~ c~cb ~ ~- ~ -2.
MAXIsTE=L MIN.]
M
/
N
STRESS POt.YE7HFLE/7~
FIG. 10. The principal stresses in the bone phase in the inner region of the bone-porous-material eomposRe for steel and polyethylene coatings. within a steel-bone composite transmits little, if any, of the load, whereas the bone of the bone-polyethylene composite transmits an appreciable amount. DISCUSSION The finite element m e t h o d has been applied to a variety of composite systems for attaching a medullary shaft imphmt. The results, while valid only for axisymmetric geometry, give us information which allows us to assess the performance capabilities of each a t t a c h m e n t system or ingrowth pattern. It has been seen thut, for a porous polyethylene surface, incomplete ingrowth into a thick layer results in relatively lower shear stresses but higher shear strains than are the ease with complete ingrowth into a somewhat thinner layer. If there is a limiting depth of ingrowth of bony tissue we can achieve either of the above situations b y selecting an appropriate thickness of coating material. It is not immediately clear which of these situations is the more desirable and any selection criteria would have to include the strength of the porous material and the effect of deformations on the vascular fibrous tissue within the coating which m a y be the precursor to bone formation. It is possible t h a t large deformations under conditions of incomplete ingrowth m a y be the limiting factor inhibiting mineralization and viable bone formation. We have also seen that the use of a layer of solid polyethylene next to the steel shaft gives values of shear stresses and strains which are between the two extremes cited above. The comparison of a porous steel with the porous polyethylene is of some interest. It is important to note that steel is representative of high modulus materials and, for the purposes of the analysis carried out here, the same qualitative results would be obtained with other metals or even ceramics of approximately equivalent modulus. The steel coating produces a sharper stress concentration t h a t the lower modulus polyethylene and the stresses on the bone within the pores of the steel coating are nearly negligible. The optimum magnitude of the stress in the bone-porous-materiM composite is not clear. Excessively high stresses are not desirable as t h e y m a y lead to mechanicM failure, particularly in the case of the much weaker polyethylene. However,, v e r y low stresses in the:
STRESSES IN MEDULLARY IMPLANTS
419
ingrown bone itself m a y be undesirable, resulting in poor bone ingrowth according to the principles of stress-induced osteogenesis. M o r e specifically, some degree of load transmission through ingrown tissue is p r o b a b l y necessary. CONCLUSIONS T h e results of this s t u d y indicate that, for conditions of axisymmetric loading, the m a x i m u m shear stress attained is independent of the length of the i m p l a n t with full bone growth into the porous coating and the load is transferred over a constant length. Under conditions of incomplete bone ingrowth, however, the m a x i m u m shear stress is reduced b y increasing the i m p l a n t length. I t has been shown t h a t a coating of porous polyethylene produces a more even load transfer and lower m a x i m u m shear stress t h a n a coating of porous steel. Further, the results indicate t h a t incomplete bone ingrowth produces lower m a x i m u m shear stresses b u t higher shear strains in the inner region of the porous coating t h a n under conditions of complete bone ingrowth. This situation is relieved b y using a layer of solid polyethylene next to the shaft, which gives a good compromise, trading somewhat higher stresses for lower shear strains. Finally, the analysis indicates t h a t the bone which grows into a porous steel coating (i.e., the bone within the pores) is under negligible stress. Since it is necessary for skeletal tissue to sustain some loading in order to maintain viability, this result suggests t h a t high modulus materials m a y not be suitable for porous coatings. REFERENCES Ghista, D. N., Kobayashi, A. S., and Davids, N. Analyses of some biomechanical structures and flows by computer-based finite element method. Computers in Biology and Medicine 1975, 5, 119-161. Hentrich, R. L., Graves, G. A., Stein, H. G., and Bajpal, P. K. An evaluation of inert and resorbable ceramics for future clinical orthopedic applications. Journal of Biomedical Materials Research 1971, 5, 25-51. Hernnan, G., and Liebowitz, H. Mechanics of bone fracture. In H. Liebowitz (Ed.), Fracture. New York: Academic Press, 1972. Homsy, C. A., Kent, J. N., and Hinds, E. C. Materials for oral implantation--biological and functional criteria. Journal of the American Dental Association 1973, 86, 817-832. Hulbert, S. F, Cooke, F. W., Klawitter, J. J., Leonard, R. B., Saner, B. W., Moyle, D. D., and Skinner, H. B. Attachment of prostheses to the musculoskeletal system by tissue ingrowth and mechanical interlocking. Journal of Biomedical Materials Research Symposium No. 4 1973, 1-23. Katz, J. L. Hard tissue as a composite material-I. Bounds on the elastic behavior. Journal of Biomechanics 1971, 4, 455-473. Klawitter, J. J. A basic investigation of bone ingrowth into a porous material. Ph.D. Thesis, Clemson Univ., 1969. Klawitter, J. J., and Halbert, S. F. Application of porous ceramics for the attachment of load bearing internal orthopedic appliances. Journal of Biomedical Materials Research Symposium No. 2, 1971~ 161-229. Moyle, ]). D., Klawitter, J. J., and Halbert, S. F. Mechanical properties of the bone-porous biomaterial interface: elastic behavior. Journal of Biomedical Materials Research Symposium No. 4, 1973, 363-382. Nilles, J. L., Karagianes, M. T., and Wheeler, K. R. Porous titanium alloy for fixation of knee prostheses. Journal of Biomedical Materials Research Symposium No. 5, 1974, 319-328.
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ANAND E T A L .
Parker, E. R. (Ed.) Materials data book for engineers and scientists. New York : McGraw-Hill, 1967. Predeeki, P., Stephan, J. E., Auslaender, B. A., Mooney, u C., and Kirkland, K. Kinetics of bone growth into cylindrical channels in aluminum oxide and titanium. Journal of Biomedical Materials Research 1972, 5, 375-400. Sandiford, D. J. H., and Willbourn, A. H. General mechanical properties. In A. Renfrew and P. Morgan (Eds.), Polythene: the technology and uses of ethylene polymers. New York : Interscience, 1960. Sauer, B. W., Weinstein, A. M., Klawitter, J. J., Hulbert, S. F., Leonard, R. B. and Bagwell, J. G. The role of porous polymeric materials in prosthesis attachment. Journal of Biomedical 3/laterials Research Symposium No. 5, 1974, 145-153. Sweeney, A. W., Kroon, R. P., and Byers, R. K. Mechanical characteristics of bone and its constituents. ASME Paper 65-WA/HUF-7, 1965. Zienkiewicz, O. C. The finite element method in engineering science. London : McGraw-Hill, 1971.
