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Continuing periosteal apposition II The significance of peak bone mass strain equilibrium and age-related activity differentials for mechanical compensation in human tubular bones.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 82:473-484 (1990)
Continuing Periosteal Apposition II: The Significance of Peak
Bone Mass, Strain Equilibrium, and Age-Related Activity
Differentials for Mechanical Compensation in Human
Tubular Bones
RICHARD A. LAZENBY
Department of Anthropology, McMaster Uniuersity, Hamilton, Ontario
Canada, L8S 4L9
Mechanical compensation, Periosteal apposition,
KEY WORDS
Strain equilibrium, Aging
ABSTRACT
It is generally presumed that compensation for the reduction
of bone strength by progressive endosteal bone loss in adults is provided by
continuing periosteal apposition (CPA) of new lamellar bone. However, the
appropriate magnitude of compensatory bone growth, and the parameters that
operate t o determine that magnitude, are unknown. This paper examines the
mechanical compensation hypothesis in a series of right-circular tubular bone
analogues. Under this hypothesis, the stated objective of CPA is maintenance
of the cross-sectional geometric properties of the element. These include the
second and polar moments of area, as well as the cortical area of the section (I,
J, and CA, respectively). This study assumes that, as resorption and apposition
proceed, geometric change is isometric (shape preserving).
The analysis suggests that for a given rate of endosteal bone loss (the
stimulus), the magnitude of periosteal growth (the response) required to
maintain geometric strength is determined by the maximum ratio (CT,) of the
radial distances from the section centroid to the endosteal and periosteal
surfaces (i.e.,cortical thickness prior to the onset of progressive endosteal bone
loss, or peak bone mass). The analysis also indicates that, for any given
individual, the amount of compensatory periosteal gain required may be very
small. This is particularly true for individuals having a large CT, and for
whom the magnitude of dynamic loading imparted to the skeleton declines
with advancing age. This finding is illustrated in a model that relates concepts
of bone surface remodeling equilibria and age-related activity differentials.
The reduction in mass that occurs in the
bones of aging individuals is well established
(e.g., Garn, 1970; Mazess, 1982; Kelsey,
1987; Whedon, 1984). In cortical bone, this
takes place primarily through expansion of
the medullary cavity via endosteal resorption along with increasing intracortical porosity. Less appreciated is that the aging
skeleton also experiences bone growth, as a
result of continuing periosteal apposition
(CPA) upon existing bone surfaces. Often,
both bone gain and loss are observed in
single populations within the same skeletal
structure (Garn, 1970; Garn et al., 1972).
This juxtaposition has led to a general acceptance of the hypothesis that the increased
0 1990 WILEY-LISS, INC.
bone mass accrued through periosteal apposition serves as mechanical compensation
for the reduction in mass resulting from
endosteal or intracortical resorption. In tubular bones, CPA data generally consist of
larger values in older cohorts for midshaft
diameters in radiographs of long or short
tubular bones (e.g., Smith and Walker, 1964;
Garn et al., 1972), or for various geometric
measures such as cortical areas and moments of area (eg., Martin and Atkinson,
1977; Ruff and Hayes, 1983b). These measures, however, do not in themselves constiReceived May 9,1989; accepted November 6,1989
474
R.A. LAZENBY
tute direct evidence for CPA as mechanical
compensation. This is because they are nonspecific vis-a-vis the magnitude, location
and timing of depositional events at the periosteal surfaces to resorptive events occurring endosteally or intracortically. Moreover, they often do not consider potential
changes in the biomechanical environment
underlying the proposed stimulus-response
compensatory mechanism, such as would
occur when people become more sedentary
with advancing age. At the same time, such
measures are suggestive of mechanical compensation, and have in fact been often cited
as more or less conclusive evidence that such
a mechanism exists (Martin and Atkinson,
1977).
In accordance with the general tenets of
Wolffs Law (Treharne, 1981; Roesler, 1987)
extended to cortical bone, the objective of
mechanical compensation would be maintenance of the aging skeleton’s structural integrity in a manner consistent with the biomechanical demands placed on it. At the
same time, the cost of that maintenance, in
both metabolic and material terms, is expected to be minimized. In spite of such
hypothesizing, however, cortical bone fractures in the elderly continue to occur e.g., of
the proximal humerus, distal radius, and
subtrochanteric femur. These occurrences
suggest an apparent contradiction. Specifically, why do some individuals fail to benefit
from the compensation provided by periosteal bone growth in adulthood, if the intent
of such compensation is the preservation of
bone strength? Are they breaking their
bones and in so doing breaking Wolff s Law?
This contradiction is made even more apparent by studies (e.g., Martin and Atkinson,
1977; Ruff et al., 1986) which suggest that
mechanical compensation occurs preferentially in males, thereby leaving women, who
are most predelicted to age-related fracture
(Kelsey, 1987), deprived of its benefits.
This paper explores the relationship of
periosteal bone gain as a compensatory solution to endosteal bone loss using a series of
right-circular geometric models (sensu
Cowin, 1984) as analogues for tubular bone
diaphyses. The objective of each solution is
the maintenance of the second moment of
area [I],and will assume that the diaphyses
of tubular bones can be modeled as beams
(Ruff, 1987).The second moment of area is a
measure of geometric rigidity under applied
bending (Wainwright et al., 1981).
The intent of this analysis is to show how
the apparent contradiction noted above, and
hence the mechanical compensation hypothesis, is in fact consistent with what are now
recognized to be significant factors in the
prevention of symptomatic osteoporosis.
These factors include (1) the magnitude of
peak bone mass, which is achieved in the
early years of adulthood (Lindsey, 1987);and
(2) the relationship of mechanical loading to
bone modeling and remodeling (Smith and
Raab, 1986).
CONTINUING PERIOSTEAL APPOSITION AS
MECHANICAL COMPENSATION
The mechanical compensation hypothesis
argues that CPA exists to compensate reduced bone strength resulting from age-related endosteal resorption and progressively
increasing intracortical porosity. The hypothesis is derived from engineering beam
theory, and its proponents reasonably assume that bones, and in particular long tubular bones, are deformed primarily by
bending (Bertram and Biewener, 1988). The
effect of such deformation is to place the
largest strains at a point furthest from the
neutral axis in the plane of bending (Wainwright et al., 1981). The neutral axis is that
axis within a beam a t which stress and
strain are zero, and represents the point of
transition between tensile and compressive
strain.
For any given cross-section, the geometric
rigidity under bending is quantified as the
second moment of area (I), the magnitude of
which is determined by two quanta with
reference to a given axis. These are (1)the
unit area of bone perpendicular to the axis in
question; multiplied by (2) the squared distance of that unit area from the axis in
question. These multiples are then summed
over the entire cross-sectional area on either
side of the axis, with I reported in units to the
4th power (see Martin et al., 1980: their Fig.
1). For irregular shapes, software packages
such as SLICE (Nagurka and Hayes, 1980)
are able to rapidly quantify geometric properties from digitized images (e.g., Ruff and
Hayes, 1983a); for regular geometries, such
as hollow cylinders, I may be calculated using equations available in standard texts on
the mechanics of solids (e.g., Popov, 1978).
An important outcome of this area-distance
relationship, which is fundamental to the
mechanical compensation hypothesis, is
that, for any two equal units of bone tissue,
the unit situated farther from the neutral
475
TUBULAR BONE MECHANICAL COMPENSATION
axis will contribute more to geometric
strength than the unit that is closer.
Other ways in which bones (or beams)
might be deformed include torsion and axial
compression. The geometric resistance t o
torsion about a longitudinal neutral axis is
quantified as the polar moment of area (J),
and in fact equals the sum of the second
moments of area (I) for any two orthogonal
axes. The cross-sectional cortical area (CA)
of a section is considered proportional to that
section’s eometric resistance to axially compressive ?and tensile) loads (Ruff, 1987: 10).
Bertram and Biewener (1988: 75) have noted
that axially com ressive loads are seldom
found in vivo, as t ey tend to be transformed
into bending moments due to the curvature
present in most tubular bones. Similarly,
Cowin (1987: 1119) has suggested that significant torsional loads are not normally exerienced by long bone diaphyses, but are
Eorne by the epiphyseal regions instead.
Given the lrkelihood that axial compression
and torsion do not constitute a significant
type of loading for tubular bone diaphyses in
vivo, this paper concentrates on the case for
bending.
Whatever the force or combination of
forces operating, it would serve an animal
well if it could resist these forces with a
minimum of skeletal material, since ac uiring, distributing, and maintaining ske eta1
mass exerts an energetidmetabolic cost.
This line of argument has led Currey (1984)
and Currey and Alexander (1985) to suggest
that the cross-sectional shape of tubular
bones reflects the direction and magnitude
of predominant time-averaged bending loads
applied to them. Bones that are likely to be
loaded equally in all axes over time are best
designed as cylinders, while those that experience loads predominantly in a single preferred axis are optimally designed as Ibeams, and so on. This log^ underlies the
use of cross-sectional long bone geometry as
a basis for drawing behavioral inferences in
both human (Ruff et al., 1984; Bridges, 1989)
and nonhuman (Schaffler et al., 1985; Burr
et al., 1989) primates. Tissue can thus be
economized by placing it where it will serve
the eatest urpose, given a articular loadinggstory. Eince the more istant units of
bone tissue contribute more to geometric
stren h relative to a neutral axis, an animal
can a so economize on the cost of materials
by building hollow bones; i.e., by placing a
given allotment of bone further from the axis
of bending and leaving empty areas closer to
K
?
i
P
that axis. Were these areas to be filled with
bone, their contribution to overall geometric
strength would not warrant the additional
cost of producing and maintaining the tissue
required (Fig. 1).An important constraint on
this option is the propensity for hollow thinwalled cylinders to fail in Euler buckling,
when the ratio of wall thickness to c linder
diameter exceeds a critical value ( urrey,
1984).
Minimum mass analysis approaches a design problem from the perspective of minimizing the materials required to erform a
given function (Currey, 1984). A !I ogical extension of this approach applied to the problem of bone loss and bone gain would consider what parameter or parameters would
minimize the cost of maintaining bone
strength by apposition of new bone at the
periosteal surface (CPA) in the face of endosteal bone loss. Two uestions become evident: (1)What factors etermine how much
compensation is necessary? and (2) What
factors determine how much CPA is possible? When “possible” is less than “necessary,” broken bones are likely to result.
MATERIALS AND METHODS
Modeling the relative magnitude of periosteal apposition required to maintain bendin rigidity (I) in response to a specified
re ative magnitude of endosteal resorption
can provide an initial response to the first of
these questions. To this end, right-circular
e
8
B
CA = 7 5 . 0
CA = 7 5 . 0
I
I
- 456.67
rE
0.491
r P = 4.91
-
746.04
r E = 2.82
rP
-
5.64
Fig. 1. Schematic depiction to two right-circular sections of equal cortical area, showing the relationship
between the magnitude of the second moment of area (I)
and the two radii rE and rP. In spite of a -475% increase
in the endosteal radius, rE, the figure on the right has a
much larger resistance to bending (I = 746.04 units4 vs.
456.64 units4) as a result of a more modest -15% increase in the periosteal radius, rP.
476
R.A. LAZENBY
tor(s) relevant to maintaining the geometric
bending strength of tubular bones (Table 1;
Fig. 2). The first series of models specifies
that prior to the onset of endosteal bone loss,
rP is equal, making Total Area equal for all
sections. However, rE is allowed to vary, the
consequences of which is that cortical area
(CAI differs for each section. A second series
I = d 4 (rp4 - rE4)
(1) of models permits TA to vary, while CA is
held constant. The final series of three modwhere rP denotes the radius to the periosteal els allows both radii to vary, such that both
surface, and rE the radius to the endosteal total and cortical areas (TA, CAI differ
surface. As endosteal resorption proceeds, among the sections. For the purpose of comrE increases by an amount, drE. Maintaining parison, the values of rE and rP are set
a given value of I requires that rP also in- arbitrarily in order to keep rE/rP ratios constant (at 0.1,0.25, and 0.5) among the three
crease, by arP, which satisfies the equation
series of models. Using Eq. (2), the absolute
change in the response arP, was determined
for a series of successive increments of drE
(e.g.,drE, = 0.50 rE+rE; drE, = 1.0 rE+rE;
drP =
- rP
(2)
7r
drE, = 1.5 rE+rE. . . .drE, = 4.0 rE+rE).
The tubular bone cross-sections modeled in
4
this analysis are assumed equivalent with
The initial radii (rE and rP) represent the regard to the nongeometric properties; and
situation which exists just prior to the onset that geometric change is isometric (shape
of progressive endosteal bone loss-a mo- preserving).
ment at which the section reflects the peak
RESULTS
bone mass of the element. At any given
moment, the ratio rE/rP is a measure of
Tables 2 4 give the values of rE+drE,
cortical thickness, which for peak bone mass rP+drP and drP expressed as a percentage of
is denoted as CT . As will be seen, only the rP for 50% increments in rE, from 0 to 400%
magnitude of enlosteal resorption, and not (0 to 300% for the rE/rP ratio 0.5). The
the rate at which it proceeds (i.e., the magni- analysis indicates that maintaining the sectude of drE per unit time), has an impact on ond moment of area in these right-circular
the value of drP.
tubular bone analogues is independent of the
Three series of three right-circular models initial values for TA and CA; rather, this
were developed in order to identify the fac- maintenance is dependent on the initial ratio of the endosteal and periosteal radii, i.e.,
'While the method employed in constructing these models is
CTo(Fig. 3). The greater the proportion rE is
described for right-circular sections, it may be extended to cases
of elliptical and triangular section analogues by incorporating
of
rP at CTo, the larger drP must be to
additional reference axes. The results obtained for such anamaintain the section's bending rigidity for
logues are qualitatively similar to those derived for right circular
sections.
any given drE. These results are not unexsections can be considered analogous to tubular bone midshaft cross-sections (Cowin,
1984)' Geometric resistance to bending is
given by the value of I. For a hollow circular
section, the magnitude of I relative to a
bisecting neutral axis is determined by the
formula:
TABLE I . Initial rP and rE Values and Their Corresponding Moment o f Area (I), Total Area (TA), and
Cortical Area (CAI
Series,E
rp
Ao i
Ao 2:
Ao 3
Bo I
Bn 'JG
Bo 5
CII I
GI 25
co 5
rP
rE
I
TA
CA
3.9894
3.9894
3.9894
4.9106
5.0463
5.6419
10.0
5.6
3.0
0.3989
0.9974
1.9947
0.491 1
1.2616
2.8209
1.0
1.4
1.5
198.9238
198.1676
186.5097
456.6662
507.3064
746.0388
7853.1962
769.3823
59.6412
50.0
50.0
50.0
75.7576
79.9999
99.9999
314.1593
98.5203
28.2743
49.5
46.875
37.5
75.0
75.0
75.0
311.0177
92.3628
21.2058
'Valuesasslgned to thethreeseries(A = totalarea,TA, isronstant,corticalarea,CA,varies;B=CAisc(,nstant.TAvarics;C- bothTA and
CA vary) of three models for the periosteal radius (rP) a n d t h e endosteal radius (rE) such t h a t the r E /r P ratios arc 0.1, 0.25, and 0.5.
'rR and r P are used to calculate the magnitude of t h e a r e a moment of inertia (I).
" I = n/4 (rP' rE4k all values are in arbitrary units.
~
TUBULAR BONE MECHANICAL COMPENSATION
63 rE/rP
=
0.1
rE/rP = 0.25
477
rE/rP = 0.5
C.
Fig. 2. Schematic depiction of the nine right-circular
section tubular bone analogues modeled in this study.
rE/rP is the ratio of the two surface radii, i.e., endosteal
radiudperiosteal radius. The three series (A-C) correspond to the constancy of total area (series A), cortical
area (series B), or neither (series C).
478
R.A. LAZENBY
TABLE 2. Ahsolute and Relative Change in the Endosteal and Periosteal Radii for the Three Series o f Models for
Cumulative 50%Increments in rE, When rE/rP = 0.1
'
drE('R1)
0'
50
100
150
200
250
300
350
400
rF,
+ JrE
0.3989
0.5984
0.7979
0.9974
1.1968
1.3963
1.5958
1.7952
1.9947
An.1
rP JrP
+
drP('R1)
3.9894
3.9898
3.9909
3.9932
3.9974
4.0042
4.0146
4.0296
4.0503
0.0
0.0102
0.0375
0.095
0.1994
0.3706
0.6315
1.0073
1.5248
'As calculated from Eq. ( 2 ) (see text).
'When JrE('Xd - 0, rlS JrE = rE t 0 = rE, a n d rP
+
rE
+ JrE
0.4911
0.7366
0.9821
1.2277
1.4732
1.7187
1.9643
2.2098
2.4553
co 1
Hn.1
rP
+ JrP
JrP(%)
4.9106
4.9111
4.9125
4.9153
4.9204
4.9288
4.9417
4.9601
4.9855
rE
0.0
0.0102
0.0375
0.095
0.1994
0.3706
0.6315
1.0073
1.5248
+ drP = rP + 0 = rP, and drp ('XI)
=
+ JrE
rP
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
+ JrP
10.0
10.001
10.0037
10.0095
10.0199
10.0371
10.0632
10.1007
10.1525
JrP(%)
0.0
0.0102
0.0375
0.095
0.1994
0.3706
0.6315
1.0073
1.5248
0.
TABLE 3. Absolute and Relative Change in the Endosteal and Periosteal Radii for the Three Series o f Models for
Cumulative 50%Increments in rE, When rE/rP = 0.25'
drE('Ri)
02
50
100
150
.
..
200
250
300
350
400
rE
+ JrE
0.9974
1.496
1.9947
2.4994
2.9921
3.4907
3.9894
4.4881
4.9868
An 25
rP JrP
+
JrP('%i)
3.9894
4.0052
4.0466
4.1301
4.2701
4.4747
4.7419
5.0648
5.4321
0.0
0.3945
1.4338
3.5262
7.0349
12.1556
18.8627
26.9568
36.1635
rE
+ drE
1.2616
1.8923
2.5231
3.1539
3.7847
4.4155
5.0463
5.677
6.3078
:As calrulated from Eq. ( 2 ) (see text).
When JrE('R8)= 0, rE t JrE = rE 0 = rE, a n d rP t drP = rP
+
r
Bo L',
rP JrP
+
drP(%)
5.0463
5.0662
5.1186
5.2242
5.4013
5.6597
5.9981
6.4066
6.8712
0.0
0.3944
1.4337
3.5261
7.0348
12.1555
18.8626
26.9567
36.1635
rE
+ JrE
1.4
2.1
2.8
3.5
4.2
4.9
56
6.3
7.0
Cn ~i
rP
+ JrP
5.6
5.6221
5.6801
5.7975
5.9939
6.2807
6.6563
7.1096
7.6252
JrP(%)
0.0
0.3944
1.4337
3.5261
7.0348
12.1555
18.8626
26.9567
36.1635
0 = rP, a n d JrP('%,)= 0
TABLE 4. Absolute and Relative Change in the Endosteal and Periosteal Radii for the Three Series o f Models for
Cumulative 50% Increments in rh', When rE/rP = 0.5'
drE('R1)
0'
50
100
150
200
250
300
rE
+ JrE
1.9947
2.9921
3.9894
4.9868
5.9841
6.9815
7.9788
Ao 5
rP JrP
+
JrP(%)
3.9894
4.2216
4.7067
5.4088
6.2438
7.1498
8.0932
0.0
5.8197
17.9806
35.5795
56.5085
79.2182
102.8674
+ JrE
Bn 5
rP JrP
JrP(%)
2.8209
4.2314
5.7642
7.0524
8.4628
9.8733
11.2838
5.6419
5.9702
6.6563
7.6493
8.8301
10.1113
11.4456
0.0
5.8196
17.9806
35.5795
56.5093
79.2182
102.8675
rE
' A s calculated from Eq. (2) (see text).
'When JrlS('%i)= 0, rE Jrb: = rE t 0 = rE, a n d rP t JrP = rP
+
rE
+ JrE
1.5
2.25
3.0
3.75
4.5
5.25
6.0
co 5
rP
+ JrP
3.0
3.1746
3.5394
4.0674
4.6953
5.3765
6.086
drP('K)
0.0
5.8196
17.9806
35.5795
56.5085
79.2182
102.8674
+ 0 = rP, and JrP('%ij= 0.
pected, since I is not an intrinsic pro erty of
the cross-section but is simply a mat ematical description of the amount and distribution of material relative to some axis of bending. What is important is the relationship
between the relative magnitude of CT, and
the amount of periosteal apposition required
to compensate for a given percentage increase in rE, since CPA is a cellularly based
R
+
phenomenon having intrinsic physical a n d
or bioloDca1 limitations that determine how
much bone can be deposited within a specific
time period. These limitations speak to the
second question posed earlier, how much
CPA is possible?
In the rib, the appositional rate for bone
modeling ranges from 2 to 20 tm-dday (Frost,
1980). This would certainly be sufficient to
TUBULAR BONE MECHANICAL COMPENSATION
479
1
0
I00
200
400
300
drE
500
[%I
Fig. 3. Curves illustrating the relative magnitude of
change ( 8 ) required in the radial distance, rP, in response
to a specified changed in the radial distance, rE, in order
to preserve the second moment of area associated with
the section's peak bone mass. The three curves corre-
spond to the initial rE/rP ratios: .1,0.25 and 0.5. drP is
the postulated product of continuing periosteal apposition, while arE represents the progressive endosteal
resorption associated with normal skeletal aging.
achieve any required increase in rP. For
example, at the lower end of this range
(where the monolayer of osteoblasts would
presumably be operating with least efficiency),it would take just under 14 years to
add 10 mm of bone at any point on the
periosteal surface. Accepting even a quarter
of this amount (in the event that CPA is an
intermittent rather than a continuous process, as suggested by Epker and Frost (1966)
for the rib) makes it unlikely that intrinsic
cellular dynamics would act as a limiting
factor for mechanical compensation. However, a variety of factors "extrinsic" to the
bone cell system, such as raw material availabilit for bone matrix production, or reducediefficacy of the signal transducers activating the bone formation sequence,' may
effectively serve to limit CPA as mechanical
compensation for bone loss. An important
consequence of the presumed existence of
such factors limiting a CPA response is that
skeletons having a lower peak bone mass are
more likely to be threatened with structural
failure than are skeletons having a high
eak bone mass, if maintenance of a speciled level of geometric strength is required,
since they require a greater drP for any given
drE.
Figure 4 depicts drP in bending versus
axial compression, for an initial rP = 4
units, and an rE/rP ratio of .25. These two
curves indicate that the magnitude of drP
required to maintain compressive strength
is much larger than that required to main-
F
6o
lo:
'The nature of these signal transducers is presently unknown;
though several candidates have been nominated, notably piezoelectric effects and streaming potentials (Currey,1984: 249). An
intriguing possibility has recently been proposed by Lanyon and
associates (Lanyon, 19871, involving reorientation of proteoglycan molecules, known to occupy or even invest the bone cell
membrane, in response to osteogenic loadin The intriguing
nature of this mechanism lies in the possibfe role molecular
reorientation might play as a "strain memory," providing "the
means for both 'capturing' strain transients and for presenting
the bone cells with a stimulus related to the 'averaged' strain
patterns accumulated over a 24 h period"(Lanyon, 1987: 1092).
'-
1 I+
COMPRESSION
1
BEND"G
I /
7
/
/
arE [%I
Fig. 4. Curves illustrating the relative magnitude of
arP required in response to a specified change in rE, for a
tubular section having an initial rE/rP ratio of 0.25. The
top curve indicates the response necessary to preserve
the section's initial compressive strength; the bottom
curve indicates the response required to preserve bending strength.
480
R.A. LAZENBY
tain bending stren h. It may also be inferred from this re ationship that massive
increase in I would result as a consequence
of maintaining CA, if axial compressive
strength was the factor determining the
magnitude of drP. Were this the case, tubular bones would become increasingly resistant to bending loads in consequence of preserving a given cortical area. Since a large
body of literature documents a reduction in
cortical area in older groups (e.g., Ruff et al.,
1986), maintaining geometric resistance to
compression likely does not determine drP.
This is not unexpected given Bertram and
Biewener's (1988)suggestion that axial compressive loads seldom occur in vivo, and
given that bone tissue is materially stronger
(i.e.,independent of shape) under axial compression than in either bending or torsion
(Currey, 1984).
P
ACTIVITY DIFFERENTIALS AND STRAIN
EQUILIBRIUM WINDOWS: EXPLAINING
MECHANICAL COMPENSATION
The relation for stress (a)in a beam can be
expressed as (T = My/I (Wainwright et al.,
1981: 247), where I is the second moment of
area, M is the applied bending moment and y
is the distance from the neutral axis of bending at which (T is measured, Larger magnitudes of stress (hence of strain) will result
from increasing M or decreasing I. When M
and I are constant, larger strains will occur
in proportion to the distance from the neutral axis, y. In the present study, M was
assumed to be constant for the purpose of
determining the magnitude of compensatory
apposition at the periosteal surface to a
given magnitude of endosteal resorption. A
further assumption that underlies the
present analysis is that the objective of CPA
is to maintain the geometric bending rigidity, I, of a given cross-section (and thus that
of the diaphysis). As stipulated by contemporary interpretations of Wolff s Law (Roesler,
1987) and suggested by numerous studies
involving experimentally modified loading
regimens (see Bouvier, 1985; Lanyon, 1987,
for reviews), tubular bones are seen to be
minimum mass solutions adapted to particular levels of functional stresdstrain derived
from their time-averaged loading histories
(Frost, 1985; Rubin and Lanyon, 1987). As
such, the above objective can be restated in
terms of stress (a)rather than strength: CPA
is assumed to act to limit increases in a gven
a specified loading regimen and a reduction
in the value of I engendered by endosteal
bone loss. This assumption is justified only
so long as the time-averaged dynamic forces
(the bending moments, M) applied to the
structure do not change. Casual observation
clearly does not support the premise that
bending moments remain constant, as the
fre uency and the magnitude of dynamic
loa8ing that peo le im art to their skeletons
decline noticeab y wit age. It could also be
argued that the neuromuscular control over
this loading becomes less precise with increasing age, and is one factor redisposing
older individuals to falls (Tiifeiksaer and
Kay, 1987). As far as our bones are concerned, the rest of the body becomes less
predictable. Thus, in addition to bones experiencing applied loads which are both less
frequent and less strenuous, there may also
be altered strain distributions within the
bone tissue. Such a scenario may act to raise
or lower the level of strain at a given point in
the bone tissue which was previously recognized as normal by the bone cell system.
Stephen Cowin and associates developed a
series of theoretical models for surface remodeling as a function of strain history (see
Cowin, 1987, for discussion and references).
A basic tenet of their approach is that a bone
surface exists either in equilibrium or disequilibrium, depending upon the level of normal strain. Cowin suggests that there is a
window of normal strain recognized by the
bone cell system within which equilibrium
obtains, and within which no net modeling/
remodeling occurs. The upper and lower
bounds of this range are denoted as E' and
E-. Strain that exceeds E' prompts a net
de osition of bone tissue, that which falls
be ow E- results in net resorption. The individual magnitude, and degree of separation,
is unknown for these limits; as is whether
they are set phylogenetically or ontogenetically.
In bending, the magnitude of stresdstrain
increases along a gradient proportional to
the distance y from the centroidal neutral
axis. For a cross-section at equilibrium experiencing no net modeling/remodeling (e.g.,
when CToobtains),the endosteal surface can
be visualized as E-, and the periosteal surface as E'. The absence of bone matrix between the section centroid and the endosteal
surface reflects the existence of strain levels
below E-, leading to net resorption, which
ensures the presence of a medullary cavity.
At the same time, the absence of periosteal
apposition in the equilibrium state reflects
the equivalence of E and the normal strains
P R
f
TUBULAR BONE MECHANICAL COMPENSATION
that occur at the periosteal surface. This
range of normal strain can be visualized as
the product of a range of activity differentials, A+ to A-, i.e., a range of dynamic
loading events averaged over a given time
period to give a mean value of A. The length
of time over which bone cells ‘average’ the
cyclic loading inputs received is unknown,
however, it is clear that it must be sufficiently short as to enable differentiation between excessive use (leading to net deposition), use (remodeling equilibrium) and
disuse (net resorption).
Experimental work on the in vivo response
to altered loading environments (Rubin and
Lanyon, 1984) indicates that a remodeling
‘equilibrium’,in which the morphology of the
experimental limb = that of the control limb,
can be maintained in the functionally isolated turkey ulna by a pl ing a relatively
small number (four) o p ysiological load
reversals (cycles) per day. Thirty-six such
cycles produced a significant positive remodeling response at both the periosteal and
endosteal surfaces, while zero cycles led to
an eventual negative remodeling response a t
the endosteal and intracortical surfaces in
moving toward a “genetic” baseline level of
bone mineral content which was -88% below control levels. From these results and
related studies (Rubin and Lanyon, 19871,
they suggest that the significant parameters
recognized by the bone remodeling system
appear to be both the magnitude and the
distribution of dynamic loading events imparted to the skeleton. Increasing the former
andlor altering the latter from that which
has been perceived as “normal” elicits an
adaptive remodeling response with the presumed aim of equilibrating the bone’s mass
and geometry to accommodate the “new”
criteria. For example, Rubin and Lanyon
(1984) consider their results to have derived
from alterations in the strain distribution,
since the loads applied to the isolated element were within the physiolo ical range for
the turkey ulna as registered uring normal
wing-flapping. Presumably, this altered
strain distribution effectively established a
state of disequilibrium at those surface locations which subsequently underwent net
deposition or net resorption.
While the quantitative results obtained
from experimental studies carried out on the
turkey ulnae should not be translated directly to the human skeleton, the qualitative
aspects are intriguing in the present context
of age-related bone loss and gain. Consider-
PK
f
48 1
ation should be gwen to the possibility that
reductions in activity differentials with advancing age predispose anindividual to a
lower mean activity state, A, and in consequence thereof to a lower strain level (freuency and magnitude). Also, altered strain
%istributions may result from age-related
loss of neuromuscular control over body
movement. The possibility that these occurences may at least in part by responsible
for both endosteal bone loss and continuing
periosteal apposition warrants investigation.
Figure 5 applies Cowin’s theoretical construct to the question of CPA as mechanical
com ensation. Given the increasing strain
gra(Pient that exists from the centroid outward, the reduction in time-averaged strain
that results from becoming increasingly sedentary will first be perceived at the endosteal surface. Conversely, an increase in
strain would first be perceived a t the periosteal surface. A widening or narrowing of
the strain e uilibrium window width is registered at on y one surface because the strain
parameters at the o posite surface remain
within the recognize range of E+ to E-, and
4
E+
E-
a
€+
E-
E+
E-
Fig. 5 . Schematic representation of age-related
change in activity differentials to strain equilibrium,
resulting in endosteal bone loss and subsequent periosteal ap osition. A Peak bone mass (CT,). In young
adulthoot the magnitude of the mean activit differen
tial, A, establishes the strain equilibrium winJow width,
E’ to E-. The time-averaged strains associated with A
always fall within this equilibrium window. B: Endosteal
bone loss. WiJh increasing age and decreasing activity,
the value of A decreases, effectively shifting the lower
limit of the strain e uilibrium window, E-, toward the
periosteal surface. T%is leads t o a net resorption of bone
tissue from the endosteal surface, and a net reduction in
the section’s geometric resistance to bendng, I. C: Continuing periosteal apposition (CPA). The reduction in I
results in an increase in strain levels wit_hinthe remaining bone tissue, and the existing level of A leads to levels
of strain at the periosteal surface which exceeds E’. This
excess promotes a net apposition of bone, and the repetition of this and the above sequence with aging leads to
the phenomenon recognized a s CPA.
482
R.A. LAZENBY
periosteal surface, and not directly by the
magnitude of endosteal bone loss.
As far as the models described previously
are concerned, the ratio rE/rP (initially a
reflection of peak bone mass, CTo), can be
viewed as a measure of our time-averaged
activity levels established in young adulthood; hence of the initial strain equilibrium
window width, E- to E+. When the ratio
rE/rP is small (i.e., cortical thickness is
large) the endosteal resorption which follows
from lowering activity differentials towards
the minus side will at first register only
slight decrements in the magnitude of the
area moment of inertia, I. However, when
the initial rE/rP ratio is large, say 0.5, a
situation exists in which the strain equilibrium window is narrow at the outset. Any
further reduction in the width of E- to E'
will necessitate a much larger CPA res onse
if and when E' is, on occasion, exceede for a
sufficient period to elicit a response. The
roblem, however, is that individuals with
k g h initial rE/rP ratios are likely to have
been habitually less active as young adults.
In all probability, their mean activity differential was low at the outset. Should these
individuals become even more sedentary
with increasing age, they face the potential
SUMMARY AND CONCLUSIONS
result of maximally narrowing the strain
This interpretation of age-related change equilibrium window without exceeding E
in tubular bone cross-sectional geometr has and thus fail to elicit a biologically signifiimplications for the hypothesis of mec ani- cant amount of CPA as compensation. While
cal compensation, as well as for the public these individuals will be most susceptible to
health sphere where agin bone loss is a breaking their bones, it cannot be argued
major concern, and where p ysical activity is that they have broken Wolff's Law!
often cited as a useful measure against roIn a comparison of age-related geometric
gressive osteoporotic bone loss. First, PA
!?
change in modern autopsy lower limb bone
can be viewed as mechanical compensation, samples with archaeological (Pecos Pueblo)
more specifically as a proximate res onse to samples, Ruff et al., (1986) suggested that
intermittent higher strain leve s peri- the absence of significant incremental
osteally resulting from reducing th_e time- chan e at the periosteal surface in modern
averaged mean activity differential (A).This fema es, in spite of their experiencing signifacts to narrow the strain equilibrium win- icant endosteal bone loss, possibly reflects a
dow (E' to E-) within which no net more sedentary lifestyle for this population;
modelin /remodeling occurs, by moving E- i.e., CPA was not required. The analysis
toward t e periosteal surface. The endosteal presented here identifies an additional quesresorption which follows reduces the geo- tion which, if possible, should be asked of
metric strength of the bone predisposing, but these data; is the insignificant finding for
not predicting, higher levels of stress and modern females a product of their relatively
strain within the remaining tissue. That is, greater inactivity, or is it due to their having
these elevated strain levels will be propor- a characteristically high average peak bone
tional to the redudion in I within the context mass that would lessen the magnitude of
of the value of A, and not simp1 to the CPA required as activity levels decreased
reduction in I, i.e., to the amount of one lost with age, possibly to an amount below that
endosteally. This would see the magnitude which would register as statistically signifiand frequency of CPA set by the frequency cant? The point is that the presence (or
and magnitue of excursions beyond E at the absence) of measurable geometric change is
thus no net remodeling occurs at the nonaffected surface. With increasing age, the
change to a lower mean activity state (A)
effectively shifts the value of E- our bones
ise as the lower equilibrium limit towar the periosteal surface. The bone tissue
which exists between the centroidal neutral
axis and this new location of E- now exists in
remodeling disequilibrium, and resorption
follows. This resorption reduces bone mass,
and as a result the second moment of area, I
is also reduced. Given this loss of geometric
bending strength, subsequent bending moments will produce a larger strain gradient
within the remaining tissue, with the result
that the strain ma itude at the periosteal
surface now excee s E+, and net periosteal
apposition ensues. The scenario envisioned
here is an age-progressive and continual
reduction in A, promoting endosteal resorption, which in turn reduces bending strength
(I)thus engendering strain at the periosteal
surface which exceeds E+. These successive
excursions beyond E+ a t the periosteal surface elict successive net depositional responses, and the accumulation of these intermittent events are measured post facto as
CPA.
recofi"
CP
i
+
i
a
P
f
fl
B
+
TUBULAR BONE MECHANICAL COMPENSATION
a necessary, but not a sufficient criterion for
drawing inferences regarding past or
present patterns of behavior or changes in
behavior. It is also necessary to consider the
magnitude of average peak bone mass for the
population(s) in question.
Moderate physical exercise has been suggested as an inexpensive and easily delivered prophylaxis for osteoporosis (Smith and
Raab, 1986; Sandler et al., 1987; Sorock
et al., 1988).It should be clear from the above
arguments that it is highly desirable to start
out with a high mean activity differential
(i.e., more active than inactive), and a wide
strain equilibrium window (i.e., low rE/rP
ratio). Kriske et al. (19881,for example, were
able to positively correlate differences in historical activity patterns (assessed through
participant recall) to parameters of bone
quantity in the radius, especially with regard to cross-sectional area. However, the
above analysis clearly supports a common
sense approach attendent upon designing
activity programs for individuals with lower
mean activity differentials, and thus narrow
strain equilibrium windows. Specifically,
care must be taken to generate excursions
beyond E + which will elicit a CPA response,
and hence strengthen a bone, without exceeding its yield strength.
While serving as a useful first approximation to placing recognized age-related geometric skeletal changes into a behavioral
context, the analysis presented here is admittedly simplistic. It assumes that CPA is
a uniformly circumferential phenomenon,
which is contradicted by studies indicatin
that CPA is une ually distributed aroun
the periosteal su ace (e.g.,Epker and Frost,
1966, for ribs), or along the diaphysis (e.g.,
Sumner, 1984, for femora). Such intraelement variability possibly reflects both the
nature of the applied load as well as that of
the mechanism(s) which monitors and/or
transduces the mechanical signal into biological action. It also does not consider the
fact that compressive yield strength of compact bone tissue exceeds that of tensile yield
strength (Wainwright et al., 19811, and thus
ignores the effects of potentially different
strain equilibrium windows for tension and
compression created in pure bending of hollow circular cylinders. Nor does this analysis, consider the fact that intrinsic differences exist in strength for bone of different
histological character; i.e., lamellar bone has
been shown to be stronger in tension than
Haversian bone (Vincentelli and Grigorov,
4
fi
483
1985). Research in progress will address
these issues, as well as how the models developed above are altered with respect to
irregular rather than right circular geometries, and especially their consistency with
empirical histological data that relate endosteal, periosteal, and intracortical remodeling changes to reconstructed mechanical
loading histories.
ACKNOWLEDGMENTS
The ideas and opinions expressed in this
paper, while the sole responsibility of the
author, owe much to discussions along the
way with Shelley Saunders, Emoke Szathmary, Susan Pfeiffer, and Mark Skinner,
among others. The final version has benefitted greatly from the comments of two anonymous reviewers, as well as those of the
Editor. This paper was prepared while I was
supported in part by the Ontario Graduate
Scholarship Program, whose assistance is
gratefully acknowledged.
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