Skeletal structural adaptations to mechanical usage SATMU2. Redefining Wolff's LawThe remodeling problemкод для вставкиСкачать
THE ANATOMICAL RECORD 226:414-422 (1990) Skeletal Structural Adaptations to Mechanical Usage (SATMU): 2. Redefining Wolff’s Law: The Remodeling Problem H.M. FROST Department of Orthopaedic Surgery, Southern Colorado Clinic, Pueblo, CO 81004 ABSTRACT Basic multicellular unit (BMUI-based remodeling of lamellar bone causes bone turnover, net gains and losses of bone on some bone surfaces or “envelopes,” and a remodeling space comprising bone temporarily absent due to evolving resorption spaces and incomplete refilling of them by new bone. Those features depend a) on how many new BMU arise annually, b) on how much bone each BMU has resorbed and c) formed upon its completion, and d) on how long the typical BMU takes to become completed. Because a, b, and c have limiting or maximal values in life that direct and/or indirect effects of mechanical usage of the skeleton can change, the theory presented here derives mechanical usage functions that express what fractions of those maxima a given mechanical usage history allows to happen. The theory predicts some changes in bone formation, resorption, balance, turnover, and remodeling space that depend on how remodeling responds to the vigor of a subject’s mechanical usage. The theory can predict specific effects of specific mechanical challenges that experiments can test, and it fits abundant published evidence. As the kernel of a new approach to the problem it awaits critique and refinement by others. It plus the 3-way rule can redefine Wolff‘s law conceptually and also in mathematical and quantifiable form. Between 1964-1972 i t became clear that two biologTHE PERTINENT PHYSIOLOGY ically different activities, namely modeling and basic The Remodeling BMU multicellular unit (BMU)-based remodeling, can t u r n A special multicellular mediator mechanism called bone over and modify its architecture, meaning its size, shape, and content and distribution of bone tissue the BMU turns lamellar bone over in small packets.** (Frost, 1964a,b, 1972; Jaworski, 1984;Jee, 1988).Those As in Figure 1, top, a remodeling BMU begins when two activities differ in their anatomical locations, ef- some stimulus suitably Activates cells close to a bone fects, and responses to mechanical usage, disease, and surface, whereupon Resorption by osteoclasts removes aging.** As for mechanical effects, modeling can adapt a local packet of bone. Then those cells disappear and bones to overloading while remodeling can adapt them Formation by osteoblasts refills the local hole or reto underloading and disuse** (Frost, 1985, 1988, 1989; sorption bay. In humans, that ARF sequence normally Uhthoff, 1987). The first article in this series discussed consumes a time period named sigma of -3-4 bone modeling and a theory and algebra that correctly months** (Recker, 1983). When completed it has predict many known modeling responses to mechanical turned over -0.05 mm3 of bone and it happens on all 4 challenges (Frost, 1988). This article reviews a theory bone “envelopes” shown in Figure 1D: the trabecular, and algebra for how BMU-based bone remodeling re- cortical-endosteal, haversian, and periosteal surfaces sponds to mechanical usage. It accounts for some bio- (Courpron, 1981; Eriksen, 1986; Frost, 1964a; Jee, logic and biomechanical realities unknown to Wolff and 1988; Recker, 1983; Sedlin, 1964).** In the whole huhis contemporaries (Frost, 1989a; Wolff, 1892).As in the man adult skeleton some 2 x 106 BMus should act at previous report, a double asterisk (**) identifies those any moment** and some 6 x 106 BMus would become realities here. This text briefly reviews some pertinent completed annually (Frost, 1989b).** Bone turnover physiology, vital biomechanics, and a theory called the provided by the above packet-like mechanism defines “4-way rule,” which applies to how remodeling in intact remodeling or BMU-based remodeling in the new skelbones and bone tissue responds to mechanics in vivo in etal lexicon, to distinguish it from bone modeling. In healthy mammals. It concludes by discussing some of its the older literature, remodeling signified both activinuances. The millimeter and year will provide the basic ties. This relation can encode the remodeling features: dimensions, steady states are assumed (Frost, 1973, 1986; Parfitt, 1980; Recker, 19831, and Table 1lists and defines the symbols used below. Received February 9, 1989; accepted July 18, 1989. 0 1990 WILEY-LISS, INC. 415 THE 4-WAY RULE form nearly equal amounts and form all secondary osteons (Jee, 1988).** The net excess or deficit of bone left Dimensions by a completed BMU defines the AB.BMU,** which rho will signify later on. Normally i t remains negative throughout life on cortical-endosteal and trabecular surfaces in all mammals including man (Frost, 1986, 1989b,c; Jee, 1988; Johnson, 1964; Smith and Walker, mm3/mm2/yr 1964).** TABLE 1. Definition of terms in the “4-way rule” Term ARF Definition The activation-resorptionformation sequence in remodeling BMU BMU Bone remodeling packet B Annual bone balance A The activation limit E A bone strain The proportioning coefficients; P scalars Bone surface-to-volume ratio SN Bone turned over per typical v b completed BMU Remodeling space V,, Annual bone turnover AB.BMU Net deficit or gain per completed BMU Epsilon, the mechanical usage E coefficient; a scalar Mu, the BMU activation G. frequency Rho, the same as AB.BMU P Amounts resorbed and formed per Pr, Pf completed BMU U Sigma, the bone remodeling period The symbol meaning “amroximatelv to” .I ” eaual . [I-(t + PI1 The mechanical usage €unction; a scalar vt i (A-tR+F) I I no./mm2/yr mmimm -1%P% 1 + over provided by remodeling equals how many BMU arise or reach completion annually (61, multiplied by how much bone the typical BMU turns over (Vb).**In mm3/mm2 life, the activation function usually exerts more influmm3/mm2/yr ence on bone turnover than the Vb function. mm2/mm3 mm3 * mm3 05€51 no./mm2/yr +mm3 +mm3 years - 0 5 xs-1 (new bone packet) !-6-: :-4 Bone Turnover (VJ It follows t h a t the part of the total annual bone turn- mos+; That packet-like, ARF-style, BMU-based bone turnover occurs throughout life in humans, in contrast to bone macromodeling that normally subsides to trivial levels in compacta after skeletal maturity.** The time the typical BMU needs to complete its ARF stages has important uses in clinical medicine and experimental work (Parfitt, 1980; Uhthoff, 1987).** The Activation Function (b) Since each BMU normally functions for only -4 months, making BMU-based remodeling continue throughout life requires continually creating new BMU in some places to replace those completed in other places.** The activation function defines how many new BMU arise per mmz of bone surface per year, e.g., no./mm2/yr. Histomorphometrists often signify i t by Greek lower-case mu, ti (in this text a dot above a term means a rate as the first derivative) (Albright and Brand, 1987; Frost, 1969; Melsen and Mosekilde, 1981; Recker, 1983). The dB.BMU Function (Frost, 1979, 1985, 1986) On periosteal surfaces completed BMus have usually resorbed slightly less bone than they formed, as in Figure 2G.** On trabecular and cortical-endosteal surfaces they resorb slightly more than they form (Fig. 21) leaving a net deficit of --.003 mm3 of bone per completed BMU. On haversian surfaces they resorb and Bone Balance (6) It also follows that the annual net gain or loss of bone due to remodeling (dB or B) equals how many BMU reach completion annually (fi), multiplied by the net gain or loss per typical completed BMU (e.g., AB.BMU) as shown in the lower rows of Figures 1,2.** Mechanical Usage Effects We finally began to understand these about 1984 (Frost, 1985; Fujita and Takahashi, 1989; Kleerekoper and Krane, 1989; Uhthoff, 1987; Wronski and Morey, 1983; Young et al., 1986). They differ from what intuition suggested earlier. In a bone previously subjected to normal mechanical usage, onset of acute disuse then increases BMU activation from 2 x ->5 x on all 4 bone envelopes, as in Figure 1, bottom right, and Figure 3, right.** It also makes the typical AB.BMU more negative, partly by increasing how much the resorption stage removes and partly by decreasing the amount formed, even to the point of blocking formation in some BMus, and partly by delaying, sometimes for months, the onset of formation after completion of the resorption stage.** When combined, those effects can remove over 40% of the spongiosa in a bone in less then 3-4 months (Jaworski and Uhthoff, 1986; Minaire et al., 1974; Uhthoff and Jaworski, 1978). Resuming normal mechanical usage of the above bone then decreases BMU activation toward normal levels** as in Figure 3, left, and makes the AB.BMU less negative by increasing the amount of bone formed per completed BMU toward the amount resorbed.** The MES for Remodeling Circumstantial evidence suggests the above changes in activation begin when the vigor of mechanical usage causes typical peak bone strains in the 50-100 p E (microstrain) range or even less.** When strains stay below that range BMU activation becomes derepressed so it increases, while the AB.BMU becomes more negative. When strains repeatedly exceed t h a t strain range, activation becomes depressed, the AB.BMU becomes less negative and global bone turnover decreases (Frost, 1987a-c). Accordingly, that strain range can define the minimum effective strain (MES) for the mechanical depression of bone remodeling. Remodeling can become even more depressed as the vigor of mechanical usage increases bone strains toward their normally allowed maximum in the region of 2,000 PE** H.M. FROST 416 Fig. 1. A A remodeling BMU constructs a new secondary osteon inside compacta in the ARF sequence described in the text. B The same ARF sequence occurs on a trabecular or cortical-endosteal surface. C: The increased haversian porosity due to increased creation of new BMus is shown on the left, and the opposite situation is shown on the right. D The 4 bone envelopes. The third row of drawings abstract events in the top row to show the amount of bone turned over by a completed BMU. It has a small resulting deficit, the A.BMU or rho. In the bottom row, increased completed BMus on the right increase net bone loss compared to drawing on the left. (Reprinted, by permission, from H.M. Frost, Osteogenesis imperfecta: The setpoint proposal, Clin. Orthop., 216t280, 1987.) (normal bone fractures when strains reach some 25,000 PE, which corresponds to a compression stress of -19,000 psi or -140 MPa). a unit area of a bone surface (Frost, 1989c; Jaworski, 1984).**In man that limit would lie in the region of 10 new BMU per typical mm2 of bone surface annually.** In fact, dozens of dynamic histomorphometric analyses of BMU-based bone remodeling in humans and other animals since 1964 failed to find activation frequencies or indices of them even close to that limit (see Anderson and Danylchuk, 1978; Eriksen, 1986; Melsen and Mosekilde, 1981; Recker, 1983; Recker et al., 1988). That suggests the above limit is a real property of the system. The Mechanical Usage History Three original proposals by the author seem to be correct (Frost, 1964b, 1972, 1983). First, remodeling does not respond detectably to rare large bone strains, provided they do not damage the tissue. Second, it responds to some time-averaged value of typical repeated peak strains equal to or larger than the MES for remodeling. That is often referred to as a “loading history” (Cowin, 1988).Third, lesser strains than the MES have no presently detectable effect on remodeling no matter how frequent, excepting only their influence on microdamage discussed later on. The Activation Limit (A) As Jaworski and Parfitt (in Recker, 1983) have also noted, geometric and other considerations suggest only a limited number of new BMU could arise annually on The BMU/Volume Limit ( R , Rf) How much bone BMU can resorb and form also demonstrates upper limits of about 0.07 mm3 per completed BMU.** The resorption limit, &, appears in acute disuse states. Both limits occuf near the marrow cavity in bones with thick compacta, such as the femur. As mechanical usage and bone strains increase, BMus tend to resorb and form modestly smaller fractions of that limit.** Since in life the largest bone strains occur -* THE 4-WAY RULE 417 Time d AB BMU = (+) AB.BMU= (0) AB.BMU= ( - ) tFig. 2. Drawings A-C show a BMU on a trabecular surface. The second row (D-F)shows the construction of a BMU graph; the third row (G-I) shows the normal A.BMU values on the periosteal, haversian, and cortical-endosteal/trabecularenvelopes. The stair graphs at the bottom show how accumulating numbers of the completed BMus above affect bone balance and mass. (Reprinted, by permission, from H.M. Frost, Intermediary Organization of the Skeleton, C.R.C. Press, Boca Raton, FL, 1986.) on periosteal surfaces, that phenomenon may partly explain why new secondary osteons close to the marrow cavity usually have larger diameters, e.g., they resorbed and formed more bone, than those that formed close to periosteum in large bones such a s the femur, tibia and humerus. THE 4-WAY RULE The Final Common Path BMU-based remodeling responds to other things besides mechanical usage, including, in part, hormones, sex, age, homeostatic challenge, nutrition, microdamage, drugs, the regional acceleratory phenomenon, adjacent tissues, and toxins (Frost, 1986, 1987b, 1989d; Jaworski, 1984; Martin, 1987; Parfitt, 1980).** Also, some “baseline” remodeling goes on in bones in congenitally paralysed limbs, and apparently too in the calvarium, ethmoid, and turbinate bones in the absence of normal mechanical usage (Frost, 1986, 1987b; Johnson, 1964; Kleerekoper and Krane, 1989). It follows that remodeling activity in a particular bone should combine nonmechanical and mechanical usage effects.** The 4-way rule described next applies only to the mechanical effects in intact subjects, and i t proposes a basic logical framework for modeling the problem. In 1989 a plausible model should account at least for the features reviewed above. Bone strains may directly generate the physical signals that control mechanically controlled bone modeling (Albright and Brand, 1987; Frost, 1986; Johnson, 1984). However, BMU-based remodeling may also respond to other things directly or indirectly associated with mechanical usage, such as bone blood flow and marrow cavity pressure. Nevertheless, and as Cowin notes (19881, a bone’s strains should provide a reliable index of such mechanical usage-dependent features. Accordingly, this theory defines a special mechanical usage coefficient, epsilon (E), to express a bone’s mechanical usage as a normalized history of its “typical p e a k strains. Appendix A suggests a way to derive it. The Activation Function (j~) Let a mechanical usage function, [ l - ( ~+ PF)l, described in Appendix A, specify what fraction of the maximal possible activation frequency, A, a given mechanical usage state allows to happen, and in this way: i. = A [l-(E+Pp)I Eq. (1) In words, defines how many new BMus mechanical factors would allow to arise annually on 1mm2 of a bone surface, and as a function of the strain history of that surface. H.M. FROST 418 Trabecular aurf ace (Spongy bone),. J ,' , I ".*. ..:'J ........... : . /' 3 mos. Local b a n k a f t e r : -4- bite -+- S1 ow turnover Fa s t t u rnovc r Fig. 3. On a trabecular or cortical-endosteal bone surface or envelope, the left column of drawings shows the effect on net bone mass of reduced numbers of BMU completed annually, while the right column shows the increased bone loss associated with increased numbers of BMU completed annually. Decreased MU changes the left-hand situation to that on the right, while increased MU changes the righthand situation to that on the left. The BMU Fractions (pr, p,) BMU as in the bottom rows of Figures 1,2. Or, in units of mm3/mm2/yr: I3 = lip Eq. (4) Let rho, (p,) signify how much bone the typical completed BMU resorbs, let rhof (pf) signify the amount formed and let rho alone (p) signify any difference (so p substitutes for AB.BMU, e.g., p = AB.BMU). Since the bone balance per completed BMU equals the difference between the amounts resorbed and formed, and since p, has only negative values and pf only positive ones, then: P = Pr + Eq. (2) Pf If the maximum amounts of bone a completed BMU can resorb and form equal R, and Rf, respectively, then "rho coefficients" for resorption and formation, or r, and rf, would define how much of those maxima a given mechanical usage state allows to happen. Accordingly: pr = R, r,; pf = Ri. rf Eq. (3) Appendix A suggests a way to find the rho coefficients as functions of a mechanical usage history and biologic proportioning coefficients. Published data suggest these approximate values in healthy humans: p-.003 mm3; pf-- .047 mm3; p,- -.05 mm3; and R, and Rf .07 mm3. This constraint applies to the rho coefficients: Osr,, r p l . Typically, in healthy humans r,, rf seem to equal -0.7 and have smaller values near periosteal sudaces than on cortical-endosteal surfaces. - The Bone Balance (B) In surface referent this equals how many BMU reach completion in a year on a bone surface (fi), multiplied by the rho or AB.BMU value for the typical completed * Multiplying B by the local bone surface-to-volume ratio, S N , would provide the volume-fractional annual bone balance, e.g., c mm3/mm3/yr. Ignoring some nuances, for most human compacta S/V 2-4 mm2/mm3, and for spongiosa, 8-15 mm2/mm3. In healthy adult humans, the volume-fractional annual bone balance equals about - .75% or - .0075 mm3/mm3/yr (Recker, 1983; Recker et al., 1988). - - Bone Turnover (V,) The author and others defined this in several ways in the past (Frost, 1964a, 1969; Meunier, 1977; Recker, 1983). We use here a new definition that separates the turnover and balance functions. We define bone turnover here as only the bone replaced by new bone annually, so if no resorption or formation occurs no turnover would occur either. Therefore surface referent annual bone turnover equals how many BMU reach completion during the year on a unit of bone surface, fi, multiplied by how much bone the typical completed BMU turns over, Vb. Then: 1 Vb = Pf V - ' t - - kvb 2 (P + Id) Eq. (5) Eq. (6) Thus would have dimensions of mm3/mm2/yr. To obtain the volume-fractional turnover rate, multiply 419 THE 4-WAY RULE TABLE 2. Some general physiologic (not pathologic) mechanical usage (MU) effects on bone mass and architecture - MuT (MS1) t Longitudinal growth' (MS2) (MS3) Cortical modeling' - - -- creates new primary spongiosa MU: I I MU: D D - BMU-based remodeling expands outside diameter MU:I - I MU: D-. D (=) r MU: I + I MU:D+D + increases cortical cross section area - MU: I + I MU: D D (K) replaces primary with permanent spongiosa' MU: I D MU: D 1 (m) (?) t- -j turns over cortical and trabecular bone2 MU: I -. D MU: D I -j - removes trabecular and cortical-endosteal bone2 MU: I D (0) MU:D-.I - - + adds length to shaft Code: MU, mechanical usage; I, increased; D, decreased. Examples: MU: I I means that when MU increases, the biologic activity increases too. MU: I D means increased MU retards the biologic activity. =: biologic activity directly proportioned to change in MU. x : biologic activity inversely proportioned to change in MU. MS1,2,3:Putative mechanisms, named mechanostats for convenience, that allow MU to control biologic activities. (?): The only feature of this table for which the direct evidence seems tenuous a t present. While the rest of its features may be unfamiliar to some readers they are not also dubious. 'Active mainly during general body growth. 'Active throughout life. See Current Concepts ofBone Fragility, Springer: 1987. This table presented a t the Hard Tissue Workshop in August 1987, sponsored by the University of Utah and organized by Prof. W.S.S. Jee. by the SN ratio. Volume-fractional human bone turnover rates equal about 5%/yr for compacta and 20%iyr for spongiosa, but with considerable variations in different bones and parts of a given bone (Anderson and Danylchuk, 1978; Recker, 1983; Recker et al., 1988). function, and the two rho fractions. The following discussion concerns a few nuances described elsewhere (Frost, 1989b). DISCUSSION The Remodeling Space (V,) Global Mechanical Usage Effects This equals how much bone the holes caused by remodeling have temporarily removed. The temporary deficit or hole created by a n active BMU on a bone surface would equal approximately half the bone in a completed BMU, e.g., YZ vb. Then, the remodeling space should equal how many new BMU arise per year, (L, multiplied by how long the typical BMU takes to reach completion, which equals sigma, U, multiplied by half vb. Or: Table 2 summarizes some observed mechanical usage effects on mammalian bone growth, modeling, and BMU-based remodeling as we understand them in 1989. This theory accounts for all remodeling effects in that table. - v,, = ' YZVb ' U Eq. (7) The surface referent result in Eq. (7) means mm3 of bone missing temporarily per typical mm2 of bone surface. Multiplying that by the surface-to-volume ratio of a bone sample would provide the decimal fraction of the sample's "ideal" bone volume temporarily missing due to remodeling space, "V,,. Thus in units of mm3/mm3: "V,, = v,, ' SIV Eq. ( 8 ) Parfitt (1980) and the author estimate the remodeling space in healthy human adults can range from -1% to -10% of the existing bone volume. As Eqs. 7 and 8 indicate, it increases and decreases with BMU activation, with changes in sigma, and with the S/V ratio. This completes the present description of the 4-way rule, which could be said to depend on four basic features: a mechanical usage function, the activation Accounting for the Envelopes Remodeling occurs on 4 anatomically distinct bone surfaces or envelopes and can differ among them in a given bone, both normally and in some diseases (Anderson and Danylchuk, 1978; Sedlin, 1964; Frost, 1973, 1986; Recker, 1983; Jee, 1988; Uhthoff, 1987). Superscripts could specify a particular envelope for the theory's terms and expressions. Let periosteal = p, haversian = h , cortical-endosteal = ce, trabecular = t, a s in Figure 1. Then as examples, ppwould mean rho on the periosteal envelope; hA the activation limit on the haversian envelope; '"R, the BMU resorption limit on the cortical-endosteal envelope; and 'p. the activation frequency on the trabecular envelope. Obtaining Quantitative Data Proven methods can provide all physical and mechanical usage data needed to study, exploit, and refine this theory, including in vivo bone strains, bone materials properties, and bone architecture. The cited references provide sources andlor reviews (Biewener and Taylor, 1986; Burstein and Reilly, 1976; Carter, 1981; H.M. FROST 420 I I I //eMDx / / / / / .- b 4000 mi cros tr8in- Fig. 4. The creation of new BMus on the vertical axis correlates with typical peak bone strains on the horizontal axis, as the curve on the left suggests. However, some authorities, including the author, believe that curve combines two independent effects, as suggested on the right, where the depression of BMU activation (e.g., k) by in- creased MU is shown by the solid line, and the separate and overriding stimulation of activation by increased amounts of mechanical microdamage when peak strains exceed -3,000 pE is shown by the broken line. Cowin, 1988; Currey, 1984; Hayes and Carter, 1986; Lanyon, 1984; Nunamaker et al., 1987; Rubin, 1984). Other proven methods can provide all static and dynamic histologic data needed by the theory’s equations, and the cited references provide sources andlor reviews (Frost, 1969; Martin, 1987; Melsen and Mosekilde, 1981; Meunier, 1977; Minaire et al., 1974; Parfitt et al., 1984; Recker, 1983; Recker et al., 1988). The above methods already reveal provisional values for every term in this theory’s equations, so we now need systematic studies of the terms in several representative bones of the skeleton. The second half of the text’s conclusion suggests reasons for doing such studies. described, MDx for mechanical microdamage, and NM for all nonmechanical influences. This relation can then express the general situation: The High Strain Effect The derepression and depression of BMU activation during disuse and vigorous mechanical usage, respectively, have appeared consistently in experimental and clinical studies (Albright and Brand, 1987; Frost, 1985; Jaworski and Uhthoff, 1986; Parfitt e t al., 1984; Uhthoff, 1987). However, when typical peak bone strains exceed -3,000 PE, BMU-based remodeling can increase again to over 2 x normal.** The author and others believe that this stems from increased amounts of mechnical bone microdamage, which can initiate its repair by remodeling BMus independently of any depression by mechanical usage (Frost, 1986, 1989c; Schaffler, 1985;Uhthoff, 1987); see Figure 4. Some others do not agree, so we await studies that reveal the correct interpretation. Combined Influences BMU-based remodeling responds to many things, so a n observed remodeling state sums separate influences, a matter of some importance in interpreting experimental and clinical data and responses to challenges. Let f(R) signify remodeling on a bone surface. Let MU stand for the mechanical usage effects already RR) - = f(MU) + f(MDx) + f(NM) We must also note here that, in essence, BMU-based remodeling can change bone in ways that adapt i t to underloading or disuse, whereas modeling can adapt i t to overloading. Some 4- Way Remodeling Rules This theory suggests rules for mechanically controlled BMU-based remodeling, which complement the 3-way modeling rules. Their combination could redefine Wolffs law. To wit: (i) Normally vigorous mechanical usage depresses BMU activation and makes rho less negative to conserve existing bone. (ii) Disuse derepresses BMU activation and makes rho more negative to remove some of the existing bone. (iii) Derepression and depression tend to occur when typical peak bone strains fall below or exceed, respectively, the MES range for bone remodeling, currently thought to equal 50-100 FE. (iv) Where bone lies in touch with marrow, remodeling tends to have a negative net bone balance. (v) Increased remodeling due to mechanical disuse increases net bone loss; decreased remodeling due to vigorous mechanical usage conserves existing bone. (vi) Other nonmechanical influences can also influence remodeling. Some Problems Refining, testing, andlor exploiting this theory face some unresolved problems, including in part: how to express mathematically nature’s definition of “typical peak strains,” also called (aptly) by others the strain history (Alexander, 1984; Carter, 1984; Cowin, 1988); the nature of the corresponding bone loads and effects on bone blood flow and marrow cavity hydrostatic pres- THE 4-WAY RULE sure (Cowin, 1988; Frost, 1986); the precise strain range and stimulus-response curve for the MES that depresses BMU activation and thus remodeling; what physical signals control the remodeling parameters, meaning I;, pr, pf and E (Albright and Brand, 1987; Cowin, 1988; Johnson, 1984); the values of the proportioning coefficients (in Appendix A) over the range of strains they apply to; and how to model pathologic responses of the remodeling mechanism (Frost, 1988). CONCLUSION On Perspective Since four basic terms, E, I;, pr, and pf, can specify how mechanical effects can influence that part of bone turnover and balance that depends on BMU-based remodeling in intact bones and in vivo, the author called this theory the 4-way rule. It accounts for bone biologic, anatomic, clinical-pathologic, and biomechanical features that previous models or analyses did not account for, so in that respect it stands alone. However, it would be naive to believe that it accounts for all realities of the matter. Rather, it proposes a kernel that invites critique, testing, and refinement. It stands on the shoulders of many authorities past and present, and it simply represents another step in understanding our skeleton (Cowin et al, 1985). We do understand our skeleton in steps, each surpassing its predecessors in some way, each bowing to its successors in some way. Some Clinical Relevance of Wolff’s Law as Redefined The author and others (Jee, 1965-1989; Kleerekoper and Krane, 1989) have noted that how mechanics affects a) bone mass and architecture, b) typical peak bone strains and stresses, c) bone microdamage and mechanical fatigue failures (Frost, 1989d1, and d) susceptibility to injury, affect clinical problems of great economic and medical concern for the peoples of the world. As examples, those effects participate in the bone loss and fragility of osteoporoses and the bone pain and pseudofractures in osteomalacia (Uhthoff, 1987; Albright and Brand, 1987; Urist, 1980, SumnerSmith, 1982); they can limit the service life of the current generation of dental, total joint, and bone replacement systems (Carter, 1984; Cowin, 1988; Frost, 1986, 1989a-c); they influence our resistance and reaction to trauma and some kinds of arthroses (Frost, 1989a-c); they play major roles in skeletal problems arising in vigorous sports or other physical activities and in animals as well as humans (Frost, 1986; Nunamaker et al., 19871, and they influence how much bone we accumulate during growth, how readily we do it, and how quickly we lose it later in life (Uhthoff, 1989; Kleerekoper and Krane, 1989). In brief, they are involved in numerous skeletal diagnostic and management problems, both medical and surgical and in children and adults. Those facts suggest why the matters discussed in these reviews deserve more study. LITERATURE CITED Albright, J.H., and R.A. Brand 1987 The Scientific Basis of Orthopaedics, 2nd ed. Appleton-Century Crofts, New York, pp. 1-526. Alexander, R. 1984 Optimum strengths for bones liable to fatigue and accidental failure. J. Theor. Biol., 109t621-636. Anderson, C., and K.D. Danylchuk 1978 Bone remodeling rates of the 421 beagle: A comparison between different sites on the same rib. Am. J . Vet. Res., 39:1763-1765. Biewener, A.A., and R. Taylor 1986 Bone strain: A determinant of ’gait and speed? J . Exp. Biol., 123t385-400. Burstein, A.H., and D.T. Reilly 1976 Aging of bone tissue: Mechanical properties. J . Bone Joint Surg. [ I, 58At82-86. Carter, D.R. 1984 Mechanical loading histories and cortical bone remodeling. Calcif. Tissue Int. (Suppl.), 36t19-24. Carter, D.R. 1981 The relationship between in vivo strains and cortical bone remodeling. C.R.C. Crit. Rev. Biomech. Eng., 8:l-28. Courpron, P. 1981 Bone tissue mechanisms underlying osteoporoses. Orthop. Clin. North Am., 12t513-546. Cowin, S.C. 1988 Bone Mechanics C.R.C. Press, Boca Raton, FL, pp. 1-313. Cowin, S.C., R.T. Hart, J.R. Baker, and D.H. Kohn 1985 Functional adaptation in long bones: Establishing in uiuo values for surface remodeling rate coefficients. J. Biomech., 18t665-684. Currey, J.D. 1984 The Mechanical Adaptations of Bone. Princeton University Press, Princeton, NJ, pp. 1-294. Eriksen, E.F. 1986 Normal and pathological remodeling of human trabecular bone: Three-dimensional reconstruction of the remodeling sequence in normals and in metabolic bone disease. Endocr. Rev., 7t379-408. Frost, H.M. 1964a Mathematical Elements of Lamellar Bone Remodelling. Charles C. Thomas, Springfield, IL, pp. 1-127. Frost, H.M. 196413 The Laws of Bone Structure. Charles C. Thomas, Springfield, IL, pp. 1-165. Frost, H.M. 1969 Tetracycline-based histological analysis of bone remodeling. Calcif. Tissue Int., 3r211-227. Frost, H.M. 1972 The Physiology of Cartilaginous, Fibrous and Bony Tissues. Charles C. Thomas, Springfield, IL, pp. 1-249. Frost, H.M. 1973 The origin and nature of transients in human bone remodeling dynamics. In: Clinical Aspects of Metabolic Bone Disease. B. Frame, A.M. Parfitt, and H. Duncan, eds. Excerpta Medica, Amsterdam, pp. 124-137. Frost, H.M. 1979 Treatment of osteoporoses by manipulation of coherent bone cell populations. Clin. Orthop., 143t227-244. Frost, H.M. 1983 A determinant of bone architecture: The minimum effective strain. Clin. Orthop., 175t286-292. Frost, H.M. 1985 Pathomechanics of the osteoporoses. Clin. Orthop., 2OOt198-225. Frost, H.M. 1986 The Intermediary Organization of the Skeleton. C.R.C. Press, Boca Raton, FL, Vol I pp. 1-365; Vol I1 pp. 1-334. Frost, H.M. 1987a Bone “mass” and the “mechanostat”: A proposal. Anat. Rec., 219:l-9. Frost, H.M. 1987b Secondary osteon population densities: An algorithm for determining mean bone tissue age. Yearbook Phys. Anthropol .,30:22 1-238. Frost, H.M. 1987c Vital biomechanics: General concepts for structural adaptations to mechanical usage. Calcif. Tissue Int., 42t145-156. Frost, H.M. 1988 Structural adaptations to mechanical usage: A “three-way” rule for lamellar bone modeling. Comp. Vet. Orthop. Trauma, 1t7-17; 2r80-85. Frost, H.M. 1989a Structural adaptations to mechanical usage: A four-way rule for lamellar bone remodeling. A two-part work of 60 pages circulated privately to other authorities in North America, Europe, and Japan. See Appendix A. Frost, H.M. 198913 Bone mechanics, bone mass, bone fragility: A brief overview. In: Clinical disorders in Bone and Mineral Metabolism. M. Kleerekoper and S.M. Krane, eds. Mary Ann Liebert, New York. pp. 15-40. Frost, H.M. 1989c The intermediary organization: Some roles in disease and research. In 5th International Congress on Bone Morphometry. T. Fujita and H. Takahashi, eds. Nishimura, Niigata, Japan. In press. Frost, H.M. 1989d Transient-steady state phenomena in microdamage physiology: A proposed algorithm for lamellar bone. Calcif. Tissue Int. 44:367-381. Fugita, T. and H. Takahashi 1989 Fifth International Congress on Bone Morphometry. Eds. Nishimura, Niigata, Japan. In press. Fyrhie, D.P., and D.R. Carter 1986 A unifying principle relating stress to bone morphology. J . Orthop. Res., 4t304-317. Hayes, W.C., and D.R. Carter 1986 Post yield behavior of subchondral trabecular bone. J . Biomed. Mater. Res. Symposium, 7t537-546. Jaworski, Z.F.G. 1984 Lamellar bone turnover system and its effector organ. Calcif. Tissue Int., (Suppl.), 36t46-55. Jaworski, Z.F.G., and H. Uhthoff 1986 Reversibility of nontraumatic disuse osteoporosis during its active phase. Bone, 7:431-439. Jee, W.S.S. 1988 The skeletal tissues. In: Cell and Tissue Biology: A Textbook of Histology, 6th ed. L. Weiss, ed. Urban and Schwarzenberg, Baltimore, pp. 211-254. 422 H.M. FROST Jee, W.S.S. 1965-1989 Organizer of a series of Bone Workshops sponsored by the University of Utah. Seminal, unpublished, informal, multidisciplinary, and influential. Johnson, L.C. 1964 Morphologic analysis in pathology. In: Bone Biodynamics. H.M. Frost, ed. Little, Brown, Boston, pp. 543-654. Johnson, M.W. 1984 Behavior of fluid in stressed bone and cellular stimulation. Calcif. Tissue Int. (Suppl.), 36:72-76. Kleerekoper, M., and S.M. Krane 1989 Clinical Disorders of Bone and Mineral Metabolism. Mary Ann Liebert, New York. pp 1-647. Lanyon, L.E. 1984 Functional strain as a determinant for bone remodeling. Calcif. Tissue Int. (Suppl.), 36:56-61. Martin, R.B. 1987 Osteonal remodeling in response to screw implantation in canine femora. J . Orthop. Res., 5t445-454. Melsen, F., and L. Mosekilde 1981 The role of bone biopsy in the diagnosis of metabolic bone disease. Orthop. Clin. North Am., 12t571-602. Meunier, P.J. 1977 Bone Histomorphometry/l976. Armour-Montagu, Paris, pp. 1-476. 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APPENDIX A The Mechanical Usage Coefficient (E) Let Greek lower-case epsilon, E, specify a bone’s mechanical usage history as a normalized value ranging from zero to unity, where zero means that typical peak bone strains stay below the lower limit of the MES for remodeling, and unity means that strains rise to the point above which pathologic tissue reactions to mechanics begin. Epsilon would equal some still undefined function of a bone’s strain magnitude, range, rate, frequency, and time, a matter discussed by Carter (1984) and Cowin (1988)among others. Let I Eobslmean the absolute value of a bone’s observed typical peak strains, let I Esatl mean the absolute value of the peak strains above which pathologic bone remodeling responses begin, and let a gamma operator (y) equal unity for strains above Esat or below the MES but otherwise zero. Then this general expression can provide epsilon: E = (1 - Y) [(I Eobsl - I MESI) (1 E,,tI - I MES/ )-‘I Eq. (9) The Proportioning Coefficients (P,,, Pr, P,) When mechanical usage changes by, say, 10%, the biologic parameters might change by, say, 6%or 15%. That applies to BMU activation and to the BMU resorption and formation amounts pr and pf. Then suitable proportioning coefficients would convert a value of epsilon provided by Eq. (9) into another value between zero and unity that defined what fractions of the biologic maxima correspond to epsilon. That value could be written thus: (E + P). Whereas mechanical usage can stimulate otherwise dormant bone modeling, it can depress otherwise very active BMU-based remodeling. If for the moment “x” stands for a biologic activity or feature that responds to a mechanical usage history expressed a s epsilon, then for the modeling responses a n expression of the form E . x would apply, but for the remodeling responses expressions of the form (1-E)x, or x ( E ) - ~would apply. The former expression is suggested here. Then the mechanical usage functions for BMU-based remodeling in the text would be written thus: [ l - (E+P)]. Then these expressions for the effects of mechanics on the remodeling terms or features would result: = A [l - (E + P,)]. For the rho For activation, coefficients in Eq. (3) in the main text, r, = [l - (E + P,)], and rf = [l - (E + P,)]. In a first approximation one could set the P coefficients equal to zero. To repeat, the terms [l - (E + P)] would be the normalized and dimensionless mechanical usage functions for remodeling. The features of space-time-probability and of limits and thresholds are introduced into the theory by the limits defined in the terms A, R, ESat,and the MES. Nota bene: Refinements of this theory might substitute nonlinear probability andlor other compound functions for some of the terms in its equations, including for epsilon and the MES. To repeat, this theory only suggests a new logical framework for modeling the problem, on which one could build further. That framework accounts for the biologic realities described in the second part of the text. A 60-page, two-part manuscript circulated privately in 1988 to a n international cohort of scientists discusses other nuances of this matter and suggests solutions for most of them. C. Anderson, D.B. Burr, H. Duncan, B.N. Epker, W.S.S. Jee, Z.F.G. J a worski, A.M. Parfitt, R.B. Martin, F. Melsen, G. Sumner-Smith, H. Takahashi, E.L. Radin, and D. Vansickle received copies.