JOURNAL OF BONE AND MINERAL RESEARCH
Volume 5, Supplement 1, 1990 Mary Ann Liebert, Inc., Publishers
Effects of Fluoride Treatment on Bone Strength DENNIS R. CARTER and GARY S. BEAUPRE
ABSTRACT Bone mass and architecture in appendicular and most axial sites is controlled primarily by the tissue-loading history. We introduce a conceptual framework for understanding how fluoride treatment alters this control and can cause systemic increases in bone mass. Due to possible adverse influences of fluoride on the mineralized tissue physical characteristics, however, the increase in bone mass does not necessarily result in an increase in bone strength. Using engineering analyses of bone trabeculae, we calculate the losses in trabecular strength which can be caused by the presence of hypomineralized or hypermineralized fluorotic tissue. Significant increases in bone volume fraction and bone mass may be required to overcome these strength deficits.
BONE STRENGTH, MORPHOGENESIS AND MAINTENANCE LL BONETISSUE(CANCELWUS ANDCORTICAL) can
be considered as
A a two-phase material consisting of a solid, mineralized
tissue which is permeated by a "fluid" phase of marrow and blood. During normal physiologic loading and during most traumatic fracture episodes, the loading rates are low enough so that the fluid phase does not support a significant amount of the load but flows relatively freely through the pores and fractures of the collapsing solid phase.i" The strength of bone tissue, therefore, is determined by the physical characteristics of the solid, mineralized phase. The most important characteristics of this phase are the bone volume fraction (volume of solid/volume of solid and fluid), the chemical composition, and the ultrastructural and microstructural characteristics. The microstructural characteristics in trabecular bone include a consideration of the characterization of the orientation, size, and connectivity of trabecular struts. Although these features can certainly be altered by fluoride treatment, they will not be a focus of this report. Mechanical testing of normal bone tissue has demonstrated that the compressive and tensile strength of bone is approximately proportional to the square of the apparent density, o, which is defined as the mass of solid phase divided by the total solid and fluid tissue volume.i'! If one assumes that the mineralized tissue density, p, (mass of the solid phase/volume of the solid phase) is constant, then the bone volume fraction = p/p, In normal bone, therefore, the strength is approximately
v,
proportional to the volume fraction squared. Note, however, that when one considers bone tissue which contains significant regional variations in mineralized tissue density, the proportionality between apparent density and volume fraction breaks down. For such cases it is important to specify both the mineralized tissue density and the volume fraction to achieve a better characterization of the tissue. Our recent work on skeletal morphogenesis and maintenance suggests that the loading history is the most important single factor controlling the distribution and organization of bone tissue in appendicular and most axial sites.(2-
0.4
0.2
Stress History
[± n. a:
M
i=1
FIG. 2.
I
] (112M)
I
The influence of fluoride treatment on the relationship between stress history and bone volume fraction.
hypermineralized.Of Hypermineralized fluorotic tissue has a greater true tissue density than normal mineralized tissue. The physicochemical abnormalities of this tissue, however, again raise questions regarding a possible decrease in mechanical strength.(15.16) Evans and WOOd(15) conducted mechanical tests to failure of 25 cortical bone specimens extracted from a 45-year-old man with severe endemic fluorosis. This particular individual probably spent most of his life in his native village where the water contained 0.95 mg% or 9.5 ppm of fluorine. A comparable number of cortical bone specimens from normal individuals were also tested. Despite the fact that this person with fluorosis was bedridden for five years prior to his death, the average dry density of the fluorotic specimens was 2.01 g/crrr' compared with 1.84 g/cnr' for the normal control specimens; consistent with the presumed increase in mineralization. The fluorotic specimens had an increase of about 25% in compressive strength compared to the normal bone. In tensile loading, however, the fluorotic specimens were significantly more brittle and had a 25% loss in elastic modulus, and a 40% loss in strength. This loss in tensile mechanical properties is especially worrisome because bending failure of bone is generally initiated in regions of high tensile stress. Bending often is responsible for long bone fractures. (17) Bending of trabeculae is also generally felt to be the initiating failure mode in compression fractures of cancellous boneY) The efficacy of fluoride treatment in increasing the fracture resistance of bones in osteoporotic individuals depends on
whether the increase in bone volume fraction, apparent density and cortical thickness outweighs the possibly inferior mechanical properties of the mineralized tissue. Figure 3 is a schematic stress versus strain diagram of mineralized tissue from normal, hypermineralized fluorotic, and hypomineralized fluorotic regions. The properties of hypermineralized bone relative to normal is based on the data of Evans and Wood. The presumed characteristics of the hypo mineralized tissue curve are based on data from the literature suggesting that immature bone tissue generally demonstrates reduced strength and stiffness but an increase in the strain at failure. (18) One could, of course, envision a whole family of possible stress/strain curves representing fluorotic tissue at various times after formation at different dosages. Although some have speculated that it may be possible at some dosages to form fluorotic tissue with normal stress/strain characteristics, this has not been shown experimentally.
CANCELLOUS BONE STRENGTH AFTER FLUORIDE TREATMENT The tensile (and compressive) strength of normal cancellous bone is approximately proportional to the square of the bone volume fraction. If all the mineralized tissue in a region of fluorotic cancellous bone had uniform physicochemical characteristics, a single stress/strain curve could be used to represent all of the tissue. The strength characteristics of a cancellous bone specimen comprised by such homogeneous tissue would then be
8180
CARTER AND BEAUPRE
140
Normal 120
100
Tensile Stress
Fluorotic (Hypermineralized)
80
(MPa) 60
40
Fluorotic (Hypomineralized) 20
o 0.0
0.0025
0.005
0.0075
0.010
0.0125
Strain [rnrn/rnm] FIG. 3.
Stress/strain curves of mineralized tissue from normal and fluorotic bone.
proportionally changed by changes in the mineralized tissue strength (assuming that volume fraction and architecture are unchanged). For instance, if the mineralized fluorotic tissue had a 30% loss in strength relative to normal tissue, the strength of a single trabeculae would decline 30%, and the cancellous bone which the tissue comprises would also have a 30% strength reduction, as illustrated in Figure 4. Significant increases in bone volume fraction would then be required to overcome this deficit. One histological characteristic of fluoride-treated bone, however, tends to be the nonuniformity of the mineralized tissue. There can be regions of relatively normal bone that are adjacent to either hypo- or hypermineralized tissue. This nonuniformity can lead to even greater losses in cancellous bone strength than would be caused by homogeneous changes. To investigate the possible influence of nonuniform tissue characteristics on cancellous bone strength, we considered an idealized theoretical model of a single trabecula (Fig. 5). We assumed that this trabecula was comprised of an inner core of normal bone and an outer layer of fluorotic bone which could be either hypo- or hypermineralized. The stress/strain characteristics of these tissues were as shown in Figure 3 and the assumed material properties used in our calculations are summarized in Table 1.
It should be emphasized that the theoretical calculations to follow are very sensitive to the tissue properties assumed (Fig. 3). Unfortunately, there is a conspicuous lack of data on the material characteristics of fluorotic tissue. The data on hypermineralized fluorotic tissue(15) was from a person who experienced five years of relative disuse immediately prior to his death. The confounding variable of disuse makes these data more uncertain. The assumed characteristics of hypomineralized bone are even more suspect since they are based merely on reasonable conjecture. The compressive strength of cancellous bone is strongly related to the strength of the individual trabeculae. For a first approximation, therefore, one can conduct a theoretical calculation of the loss in bending strength in a single trabecula to represent the strength reduction which may be found in cancellous bone of constant volume fraction and trabecular architecture. Using this simplified approach, we conducted a theoretical, composite-beam stress analysis of our models.l''" The influence of the thickness of the fluorotic layers on single trabecula bending strength was determined. We assumed that the total trabecular width remained constant in all analyses. When the trabecula was assumed to have a hypomineralized layer, there were significant losses in bone strength (Fig. 6). If
FLUORIDE TREATMENT AND BONE STRENGTH
S181
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-
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120
e,
:2 .s:
100
.....
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~
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-
Fluoride Treated?
-
Fluoride Treated?
III
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....
40 20 0 0
0.2
0.4
0.6
0.8
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Bone Volume Fraction FIG. 4.
Possible influence of fluoride treatment on the relationship between bone volume fraction and tensile strength.
Single Trabecula Model tlaYef"
Width of Trabecula
FIG. 5. Idealized trabecula used in cancellous bone theoretical strength calculations.
the thickness of the layer were less than about 16% of the width of the trabecula, our calculations predicted that the inner core of well-mineralized bone would begin to fracture before the surface layer. If the layer were as little as 10% of the trabecular width, the trabecula (and hence the cancellous bone) would exhibit more than a 40% loss in strength. For layers greater than 16% of the trabecular width, fracture would be initiated at the surface of the layer. If the layer were 20% of the trabecular width, the bone would lose about 70% of its original strength (Fig. 6). Due to the brittle nature of hypermineralized fluorotic tissue, failure will always be initiated first at the surface of this layer. If the layer thickness is less than about 6% of the trabecular width, however, the inner core of normal bone will have residual strength after the surface layer has failed (Fig. 7). For thin layers, therefore, the strength of the trabecula is determined entirely by the core of normal bone. The additional cancellous bone volume fraction and bone density associated with these layers add nothing to the bone strength. In trabeculae with hypermineralized layers greater than 6% of the width, the trabeculae break catastrophically when the surface layer breaks. Bone with a layer 6% of the width manifests a 28% loss in strength. This loss will gradually increase with further increases in layer thickness until a 40% strength loss is reached with a trabecula composed entirely of hypermineralized bone (Fig. 7). The theoretical calculations of possible cancellous bone losses represented by the data of Figures 6 and 7, are based on idealized trabecular geometry (Fig. 5) and specific assumptions on tissue material behavior (Table 1). The purpose of these calculations is not to provide definitive answers to the changes in bone strength
8182
CARTER AND BEAUPRE 1.
TABLE
ASSUMED MINERALIZED TISSUE MATERIAL PROPERTIES
Normal
Fluorotic (hypermineralized)
Fluorotic (hypomineralized)
13.2
9.8
1.4
(MPa)
125.0
74.0
19.0
Ultimate strain (mm/mm)
0.0095
0.0075
0.014
Elastic Modulus (GPa)
Ultimate stress
Trabecula with Hypomineralized Fluorotic Layer \
\ \
\ \
\
-«
Core breaks first at interface
\
\
-.;-
, , ,
\
\
Surface layer breaks first
i I I I I I I
\ \
, ,
, , ,
I
\
10
!:!
20
tID
... ... to
l~
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I
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j
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\
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U
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~
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I
\
0
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~
30
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I
40
c:: c::
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,
I I
I
I
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be (.f)
I
\
I
\
50
I
\ \
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~
I
60
0
70
~
80
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~
Q..
I
/
r/l r/l
....J
•
... ...
/
/
/
90 100 0.0
0.1
0.2
0.3
0.4
0.5
tjayer / trabecular width FIG. 6.
Loss in strength found in a trabecula with a hypomineralized fluorotic layer (based on Figs. 3 and 5).
SI83
FLUORIDE TREATMENT AND BONE STRENGTH
Trabecula with Hypermineralized Fluorotic layer
I~- - - - - - - - breaks first
~I
Surface Layer
Core provides residual strength
--------+
Catastrophic failure
I--I!-~- - - - - I or;
be c
0
Q,)
.::;
(/)
10
20 30
\ \ \
\ Q,)
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40
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c
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0.2
0.3
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0.5
t1al/ er / trabecular width FIG. 7.
Loss in strength found in a trabecula with a hypermineralized fluorotic layer (based on Figs. 3 and 5).
caused by fluoride treatment but rather to demonstrate that apparently small changes in tissue physicochemical characteristic and histology can have profound effects on bone strength. Most importantly, we wish to emphasize the fact that it can be very misleading to rely on simple measures of bone volume fraction and/or bone density as indicators of bone strength in fluoride treated patients. We do not have definitive answers on the effects of fluoride treatment on bone strength. Careful consideration of the tissues raised in this paper, however, should be addressed in future studies.
REFERENCES 1. Carter DR, Hayes WC 1977 The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg 59A:954-962.
2. Carter DR 1987 Mechanical loading history and skeletal biology. J Biomech 20:1095-1109. 3. Carter DR, Fyhrie DP, Whalen RT 1987 Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy. J Biomech 20:785-794. 4. Fyhrie DP, Carter DR in press Femoral head apparent density predicted from bone stresses. J Biomech. 5. Whalen RT, Carter DR, Steele CR 1988 Influence of physical activity on the regulation of bone density. J Biomech 21:825--838. 6. Carter DR, Orr TE, Fyhrie DP 1989 Relationship between loading history and femoral bone architecture. J Biomech 22:231-244. 7. Orr TE, Beaupre as, Fyhrie DP, Schurman DJ 1988 Applications of a bone remodeling theory to femoral and tibial prosthetic components. Trans Orthop Res Soc 13:100. 8. Beaupre as, Carter DR 1988 Computer predictions of long-bone cross section morphogenesis. Proc Eur Soc Biomech, University of Bristol, p. AI.
CARTER AND BEAUPRE
8184 9. Vigorita VJ, Suda MK 1982 The microscopic morphology of fluoride-induced bone. Clin Orthop ReI Res 177:274-282. 10. Hendrikson P-A, Lutwak L, Krook L, Skogerboe R, Kallfelz F, Belanger LF. Marier JR, Sheffy BE, Romanus B, Hirsch C 1971 Fluoride and nutritional osteoporosis: Physicochemical data on bones from an experimental study in dogs. J Nutr 100:631-642. 11. Stein 10, Granik G 1980 Human vertebral bone: Relation of strength, porosity, and mineralization to fluoride content. Calc if Tiss Int 32:189-194. 12. Mosekilde L, Kragstrup J, Richards A 1987 Compressive strength, ash weight, and volume of vertebral trabecular bone in experimental fluorosis in pigs. Calcif Tiss Int 40:318-322. 13. Riggins RS, Rucker RC, Chan MM, Zeman F, Beljan JR 1976 The effect of fluoride supplementation on the strength of osteopenic bone. Clin Orthop Rei Res 114:352-357. 14. Grynpas MD, Simmons ED, Pritzker KPH, Hancock RV, Harrison JE 1986 Is fluoridated bone different from non-fluoridated bone? In: Ali Sy (ed) Cell Mediated Calcification and Matrix Vesicles. Elsevier Science Publishers B.Y. Amsterdam, pp. 409--414. 15. Evans FG, Wood JL 1976 Mechanical properties and density of bone
16. 17.
18. 19.
in a case of severe endemic fluorosis. Acta Orthop Scand 47:489495. Franke J, Runge H, Grau P, Fengler F, Wanka C, Rempl H 1976 Physical properties of fluorosis bone. Acta Orthop Scand 47:20-27. Carter DR, Spengler OM 1982 Biomechanics of fracture. In: Sumner-Smith G (ed) Bone in Clinical Orthopaedics. WB Saunders, Philadelphia, pp. 305-334. Carter DR, Spengler OM 1978 Mechanical properties and chemical composition of cortical bone. Clin Orthop Rei Res 135:192-217. Muskhelishvili NI 1963 Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd, Groningen, The Netherlands.
Address reprint requests to: Professor Dennis R. Carter Design Division Department of Mechanical Engineering Stanford University Stanford, CA 94305
Volume 5, Supplement 1, 1990 Mary Ann Liebert, Inc., Publishers
Effects of Fluoride Treatment on Bone Strength DENNIS R. CARTER and GARY S. BEAUPRE
ABSTRACT Bone mass and architecture in appendicular and most axial sites is controlled primarily by the tissue-loading history. We introduce a conceptual framework for understanding how fluoride treatment alters this control and can cause systemic increases in bone mass. Due to possible adverse influences of fluoride on the mineralized tissue physical characteristics, however, the increase in bone mass does not necessarily result in an increase in bone strength. Using engineering analyses of bone trabeculae, we calculate the losses in trabecular strength which can be caused by the presence of hypomineralized or hypermineralized fluorotic tissue. Significant increases in bone volume fraction and bone mass may be required to overcome these strength deficits.
BONE STRENGTH, MORPHOGENESIS AND MAINTENANCE LL BONETISSUE(CANCELWUS ANDCORTICAL) can
be considered as
A a two-phase material consisting of a solid, mineralized
tissue which is permeated by a "fluid" phase of marrow and blood. During normal physiologic loading and during most traumatic fracture episodes, the loading rates are low enough so that the fluid phase does not support a significant amount of the load but flows relatively freely through the pores and fractures of the collapsing solid phase.i" The strength of bone tissue, therefore, is determined by the physical characteristics of the solid, mineralized phase. The most important characteristics of this phase are the bone volume fraction (volume of solid/volume of solid and fluid), the chemical composition, and the ultrastructural and microstructural characteristics. The microstructural characteristics in trabecular bone include a consideration of the characterization of the orientation, size, and connectivity of trabecular struts. Although these features can certainly be altered by fluoride treatment, they will not be a focus of this report. Mechanical testing of normal bone tissue has demonstrated that the compressive and tensile strength of bone is approximately proportional to the square of the apparent density, o, which is defined as the mass of solid phase divided by the total solid and fluid tissue volume.i'! If one assumes that the mineralized tissue density, p, (mass of the solid phase/volume of the solid phase) is constant, then the bone volume fraction = p/p, In normal bone, therefore, the strength is approximately
v,
proportional to the volume fraction squared. Note, however, that when one considers bone tissue which contains significant regional variations in mineralized tissue density, the proportionality between apparent density and volume fraction breaks down. For such cases it is important to specify both the mineralized tissue density and the volume fraction to achieve a better characterization of the tissue. Our recent work on skeletal morphogenesis and maintenance suggests that the loading history is the most important single factor controlling the distribution and organization of bone tissue in appendicular and most axial sites.(2-
0.4
0.2
Stress History
[± n. a:
M
i=1
FIG. 2.
I
] (112M)
I
The influence of fluoride treatment on the relationship between stress history and bone volume fraction.
hypermineralized.Of Hypermineralized fluorotic tissue has a greater true tissue density than normal mineralized tissue. The physicochemical abnormalities of this tissue, however, again raise questions regarding a possible decrease in mechanical strength.(15.16) Evans and WOOd(15) conducted mechanical tests to failure of 25 cortical bone specimens extracted from a 45-year-old man with severe endemic fluorosis. This particular individual probably spent most of his life in his native village where the water contained 0.95 mg% or 9.5 ppm of fluorine. A comparable number of cortical bone specimens from normal individuals were also tested. Despite the fact that this person with fluorosis was bedridden for five years prior to his death, the average dry density of the fluorotic specimens was 2.01 g/crrr' compared with 1.84 g/cnr' for the normal control specimens; consistent with the presumed increase in mineralization. The fluorotic specimens had an increase of about 25% in compressive strength compared to the normal bone. In tensile loading, however, the fluorotic specimens were significantly more brittle and had a 25% loss in elastic modulus, and a 40% loss in strength. This loss in tensile mechanical properties is especially worrisome because bending failure of bone is generally initiated in regions of high tensile stress. Bending often is responsible for long bone fractures. (17) Bending of trabeculae is also generally felt to be the initiating failure mode in compression fractures of cancellous boneY) The efficacy of fluoride treatment in increasing the fracture resistance of bones in osteoporotic individuals depends on
whether the increase in bone volume fraction, apparent density and cortical thickness outweighs the possibly inferior mechanical properties of the mineralized tissue. Figure 3 is a schematic stress versus strain diagram of mineralized tissue from normal, hypermineralized fluorotic, and hypomineralized fluorotic regions. The properties of hypermineralized bone relative to normal is based on the data of Evans and Wood. The presumed characteristics of the hypo mineralized tissue curve are based on data from the literature suggesting that immature bone tissue generally demonstrates reduced strength and stiffness but an increase in the strain at failure. (18) One could, of course, envision a whole family of possible stress/strain curves representing fluorotic tissue at various times after formation at different dosages. Although some have speculated that it may be possible at some dosages to form fluorotic tissue with normal stress/strain characteristics, this has not been shown experimentally.
CANCELLOUS BONE STRENGTH AFTER FLUORIDE TREATMENT The tensile (and compressive) strength of normal cancellous bone is approximately proportional to the square of the bone volume fraction. If all the mineralized tissue in a region of fluorotic cancellous bone had uniform physicochemical characteristics, a single stress/strain curve could be used to represent all of the tissue. The strength characteristics of a cancellous bone specimen comprised by such homogeneous tissue would then be
8180
CARTER AND BEAUPRE
140
Normal 120
100
Tensile Stress
Fluorotic (Hypermineralized)
80
(MPa) 60
40
Fluorotic (Hypomineralized) 20
o 0.0
0.0025
0.005
0.0075
0.010
0.0125
Strain [rnrn/rnm] FIG. 3.
Stress/strain curves of mineralized tissue from normal and fluorotic bone.
proportionally changed by changes in the mineralized tissue strength (assuming that volume fraction and architecture are unchanged). For instance, if the mineralized fluorotic tissue had a 30% loss in strength relative to normal tissue, the strength of a single trabeculae would decline 30%, and the cancellous bone which the tissue comprises would also have a 30% strength reduction, as illustrated in Figure 4. Significant increases in bone volume fraction would then be required to overcome this deficit. One histological characteristic of fluoride-treated bone, however, tends to be the nonuniformity of the mineralized tissue. There can be regions of relatively normal bone that are adjacent to either hypo- or hypermineralized tissue. This nonuniformity can lead to even greater losses in cancellous bone strength than would be caused by homogeneous changes. To investigate the possible influence of nonuniform tissue characteristics on cancellous bone strength, we considered an idealized theoretical model of a single trabecula (Fig. 5). We assumed that this trabecula was comprised of an inner core of normal bone and an outer layer of fluorotic bone which could be either hypo- or hypermineralized. The stress/strain characteristics of these tissues were as shown in Figure 3 and the assumed material properties used in our calculations are summarized in Table 1.
It should be emphasized that the theoretical calculations to follow are very sensitive to the tissue properties assumed (Fig. 3). Unfortunately, there is a conspicuous lack of data on the material characteristics of fluorotic tissue. The data on hypermineralized fluorotic tissue(15) was from a person who experienced five years of relative disuse immediately prior to his death. The confounding variable of disuse makes these data more uncertain. The assumed characteristics of hypomineralized bone are even more suspect since they are based merely on reasonable conjecture. The compressive strength of cancellous bone is strongly related to the strength of the individual trabeculae. For a first approximation, therefore, one can conduct a theoretical calculation of the loss in bending strength in a single trabecula to represent the strength reduction which may be found in cancellous bone of constant volume fraction and trabecular architecture. Using this simplified approach, we conducted a theoretical, composite-beam stress analysis of our models.l''" The influence of the thickness of the fluorotic layers on single trabecula bending strength was determined. We assumed that the total trabecular width remained constant in all analyses. When the trabecula was assumed to have a hypomineralized layer, there were significant losses in bone strength (Fig. 6). If
FLUORIDE TREATMENT AND BONE STRENGTH
S181
140
-
l'I:l
120
e,
:2 .s:
100
.....
b.O
s::::: CIJ
... .....
80
~
60
lf)
-
Fluoride Treated?
-
Fluoride Treated?
III
s::::: CIJ
....
40 20 0 0
0.2
0.4
0.6
0.8
1.0
Bone Volume Fraction FIG. 4.
Possible influence of fluoride treatment on the relationship between bone volume fraction and tensile strength.
Single Trabecula Model tlaYef"
Width of Trabecula
FIG. 5. Idealized trabecula used in cancellous bone theoretical strength calculations.
the thickness of the layer were less than about 16% of the width of the trabecula, our calculations predicted that the inner core of well-mineralized bone would begin to fracture before the surface layer. If the layer were as little as 10% of the trabecular width, the trabecula (and hence the cancellous bone) would exhibit more than a 40% loss in strength. For layers greater than 16% of the trabecular width, fracture would be initiated at the surface of the layer. If the layer were 20% of the trabecular width, the bone would lose about 70% of its original strength (Fig. 6). Due to the brittle nature of hypermineralized fluorotic tissue, failure will always be initiated first at the surface of this layer. If the layer thickness is less than about 6% of the trabecular width, however, the inner core of normal bone will have residual strength after the surface layer has failed (Fig. 7). For thin layers, therefore, the strength of the trabecula is determined entirely by the core of normal bone. The additional cancellous bone volume fraction and bone density associated with these layers add nothing to the bone strength. In trabeculae with hypermineralized layers greater than 6% of the width, the trabeculae break catastrophically when the surface layer breaks. Bone with a layer 6% of the width manifests a 28% loss in strength. This loss will gradually increase with further increases in layer thickness until a 40% strength loss is reached with a trabecula composed entirely of hypermineralized bone (Fig. 7). The theoretical calculations of possible cancellous bone losses represented by the data of Figures 6 and 7, are based on idealized trabecular geometry (Fig. 5) and specific assumptions on tissue material behavior (Table 1). The purpose of these calculations is not to provide definitive answers to the changes in bone strength
8182
CARTER AND BEAUPRE 1.
TABLE
ASSUMED MINERALIZED TISSUE MATERIAL PROPERTIES
Normal
Fluorotic (hypermineralized)
Fluorotic (hypomineralized)
13.2
9.8
1.4
(MPa)
125.0
74.0
19.0
Ultimate strain (mm/mm)
0.0095
0.0075
0.014
Elastic Modulus (GPa)
Ultimate stress
Trabecula with Hypomineralized Fluorotic Layer \
\ \
\ \
\
-«
Core breaks first at interface
\
\
-.;-
, , ,
\
\
Surface layer breaks first
i I I I I I I
\ \
, ,
, , ,
I
\
10
!:!
20
tID
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l~
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I
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60
0
70
~
80
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I
/
r/l r/l
....J
•
... ...
/
/
/
90 100 0.0
0.1
0.2
0.3
0.4
0.5
tjayer / trabecular width FIG. 6.
Loss in strength found in a trabecula with a hypomineralized fluorotic layer (based on Figs. 3 and 5).
SI83
FLUORIDE TREATMENT AND BONE STRENGTH
Trabecula with Hypermineralized Fluorotic layer
I~- - - - - - - - breaks first
~I
Surface Layer
Core provides residual strength
--------+
Catastrophic failure
I--I!-~- - - - - I or;
be c
0
Q,)
.::;
(/)
10
20 30
\ \ \
\ Q,)
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40
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90
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Q,)
, '"
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100 0.0
0.1
0.2
0.3
---0.4
0.5
t1al/ er / trabecular width FIG. 7.
Loss in strength found in a trabecula with a hypermineralized fluorotic layer (based on Figs. 3 and 5).
caused by fluoride treatment but rather to demonstrate that apparently small changes in tissue physicochemical characteristic and histology can have profound effects on bone strength. Most importantly, we wish to emphasize the fact that it can be very misleading to rely on simple measures of bone volume fraction and/or bone density as indicators of bone strength in fluoride treated patients. We do not have definitive answers on the effects of fluoride treatment on bone strength. Careful consideration of the tissues raised in this paper, however, should be addressed in future studies.
REFERENCES 1. Carter DR, Hayes WC 1977 The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg 59A:954-962.
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Address reprint requests to: Professor Dennis R. Carter Design Division Department of Mechanical Engineering Stanford University Stanford, CA 94305
