Continuing periosteal apposition II The significance of peak bone mass strain equilibrium and age-related activity differentials for mechanical compensation in human tubular bones.код для вставкиСкачать
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. LITERATURE CITED Bertram JEA, and AA Biewener (1988) Bone curvature: Sacrificing strength for load predictability? J . Theor. Biol. 131:75-92. Bouvier M (1985) Application of in vivo bone strain measurement techni ues to problems of skeletal adaptation. Yearb. Phys. hthropol28t237-248. Bridges P (1989) Changes in activities with the shift to agriculture in the southeastern United States. Curr. Anthropol. 30t385-394. Burr DB, Ruff CB, and Johnson C (1989) Structural adaptations of the femur and humerus to arboreal and terrestrial environments in three species of macaque. Am J Phys Anthropol79:357-367. Cowin SC (1984) The mechanical and stress adaptive properties of bone. Ann. Biomed. Eng. 11t263-295. Cowin SC (1987)Bone remodeling of diaphyseal surfaces by torsional loads: theoretical predictions. J. Biomech. 2ot1111-1120. Currey JD (1984)The Mechanical Adaptations of Bones. Princeton, NJ: Princeton University Press. Currey JD, and RMcN Alexander (1985)The thickness of the walls of tubular bones. J. Zool. 206(A):453468. Epker BN, and HM Frost (1966) Periosteal appositional bone growth from age two to age seventy in man. Anat. Rec. 154:573-578. Frost HM (1980) Skeletal physiology and bone remodeling. In MR Urist (ed.):Fundamental and Clinical Bone Physiology. Philadelphia: J B Lippencott, pp. 208-241. Frost HM (1985) The “new bone”: some anthropological potentials. Yearb. Phys. Anthropol. 28t211-226. Garn SM (1970) The Earlier Gain and Later Loss of Cortical Bone. Springfield, IL: Charles C Thomas. Garn SM, Frisancho, AR, Sandusky ST, and McCann MB (1972)Confirmation of the sex difference in continuing subperiosteal apposition. Am. J. Phys. Anthropol. 36: 37 7-380. Kelsey J L (1987)Epidemiology of osteoporosis and associated factors. In W Peck (ed.): Bone and Mineral Research. Vol. 5. Amsterdam: Elsevier, pp. 409444. 484 R.A. LAZENBY Kriska AM, Sandler RB, Cauley JA, LaPorte RE, Hom DL, and Pambianco G (1988)The assessment of historical physical activity and its relation to adult bone parameters. Am. J . Epidemiol. 127:1053-1063. Lanyon LE (1987) Functional strain in bone tissue as a n objective, and controlling stimulus for adaptive bone remodeling. J . Biomech. 20t1083-1093. Lindsay RA (1987) Prevention of osteoporosis. Clin. Orthop. 222:44-59. Martin RB, and Atkinson PJ (1977) Age and sex-related changes in the structure and strength of the human femoral shaft. J. Biomech. 10:223-231. Martin RB, Pickett JC, and Zinaich S (1980) Studies of skeletal remodeling in aging men. Clin. Orthop. 149:26%282. Mazess RB (1982) On aging bone loss. Clin. Orthop. 165:239-252. Nagurka ML, and Hayes WC (1980) An interactive graphics package for calculating cross-sectional properties of complex shapes. J. Biomech. 17:203-213. Popov EP (1978) Mechanics of Materials. Englewood Cliffs, NJ: Prentice-Hall. Roesler H (1987) The history of some fundamental concepts in bone biomechanics. J. Biomech. 20:10251034. Rubin C, and Lanyon L (1984)Regulation of bone formation by applied dynamic loads. J. Bone Joint Surg. 66(Aj :397-402. Rubin C, and Lanyon L (1987) Osteoregulatory nature of mechanical stimuli: Function as a determinant for adaptive remodeling in bone. J. Orthop. Res. 5:300310. Ruff CB (1987) Structural allometry of the femur and tibia in Hominoidea and Mucaca. Folia Primatol. 48:9-49. Ruff CB, and Hayes WC (1983a) Cross-sectional geometry of Pecos Pueblo femora and tibiae-A biomechanical investigation. I. Method and general patterns of variation. Am. J. Phys. Anthropol. 60:359-381. Ruff CB, and Hayes WC (1983b3 Cross-sectional geometry of Pecos Pueblo femora and tibiae-A biomechanical investigation. 11. Sex, age and side differences.Am. J. Phys. Anthropol. 60:383-400. Ruff CB, Larsen CS, and Hayes WC (19841 Structural changes in the femur with the transition to agriculture on the Georgia coast. Am. J. Phys. Anthropol. 64:125136. Ruff CB, Hayes WC, and Lotz JC (1986) Sex differences in age-related remodeling of the femur and tibia. Orthop. Trans. 10:319-320. Sandler RB, Cauley JA, Ham DL, Sashin D, and Kriska AM (1987)The effects of walking on the cross-sectional dimensions of the radius in postmenopausal women. Calcif. Tissue Int. 41:65-69. Schaffler MB, Burr DB, Jungers WL, and Ruff CB (1985) Structural and mechanical indicators of limb specialization in primates. Folia Primatol. 45:61-75. Smith EL Jr, and Raab DM (1986) Osteoporosis and physical activity. Acta Med. Scand. 711(suppl.):149156. Smith RW Jr and Walker RR (1964) Femoral expansion in aging women: Implications for osteoporosis and fractures. Science 145t156-157. SorockGS, Bush TL, Golden&, Fried LP, Breuer B, and Hale WE (1988)Physical activity and fracture risk in a free-living elderly cohort. J . Gerontol. 43:M134-139. Tideiksaar R, and Kay AD (1987) Explaining falls: a logical procedure. Geriatr Med August: 25-30. Treharne RW (1981) Review of Wolffs Law and its proposed means of operation. Ortho Rev. 10:3547. Vincentelli R, and Grigorov M (1985) The effect of Haversian remodeling on the tensile properties of human cortical bone. J. Biomech. 18t201-207. Wainwright SA, Biggs WD, Currey JD, and Gosline J M (1981)Mechanical Design in Organisms. 2nd Ed. New York: John Wiley & Sons. Whedon GD (1984) Disuse osteoporosis: Physiological aspects. Calcif. Tissue Int. 36(supplj:S146-150.