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Skeletal structural adaptations to mechanical usage SATMU2. Redefining Wolff's LawThe remodeling problem

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THE ANATOMICAL RECORD 226:414-422 (1990)
Skeletal Structural Adaptations to Mechanical
Usage (SATMU): 2. Redefining Wolff’s Law:
The Remodeling Problem
Department of Orthopaedic Surgery, Southern Colorado Clinic, Pueblo, CO 81004
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
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.
form nearly equal amounts and form all secondary osteons (Jee, 1988).** The net excess or deficit of bone left
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”
The activation-resorptionformation sequence in
remodeling BMU
Bone remodeling packet
Annual bone balance
The activation limit
A bone strain
The proportioning coefficients;
Bone surface-to-volume ratio
Bone turned over per typical
v b
completed BMU
Remodeling space
Annual bone turnover
AB.BMU Net deficit or gain per completed
Epsilon, the mechanical usage
coefficient; a scalar
Mu, the BMU activation
Rho, the same as AB.BMU
Amounts resorbed and formed per
Pr, Pf
completed BMU
Sigma, the bone remodeling
The symbol meaning
” eaual
[I-(t + PI1 The mechanical usage €unction; a
i (A-tR+F)
-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.
* mm3
0 5 xs-1
(new bone packet)
Bone Turnover (VJ
It follows t h a t the part of the total annual bone turn-
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**
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
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
Time d
= (+)
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 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
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:
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.
Trabecular aurf ace
(Spongy bone),.
3 mos.
Local b a n k a f t e r :
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,
P =
Eq. (2)
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:
Vb = Pf
V - '
t -
+ Id)
Eq. (5)
Eq. (6)
Thus would have dimensions of mm3/mm2/yr. To
obtain the volume-fractional turnover rate, multiply
TABLE 2. Some general physiologic (not pathologic) mechanical usage (MU) effects on
bone mass and architecture
Longitudinal growth'
Cortical modeling'
creates new primary spongiosa
BMU-based remodeling
expands outside diameter
MU:I - I
MU: D-. D (=)
MU: I + I
+ increases cortical cross section area
MU: I + I
replaces primary with permanent spongiosa'
MU: D 1 (m) (?)
turns over cortical and trabecular bone2
MU: I -. D
removes trabecular and cortical-endosteal bone2
+ 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).
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.
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:
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
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;
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
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-
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).
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
Those facts suggest why the matters discussed in
these reviews deserve more study.
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
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
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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
(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.
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