Bone, 12, 391400 (1991) Printed in the USA. All rights reserved.

Copyright

8756-3282191 $3.00 + .OO 0 1991 Pergamon Press plc

On the Significance of Remodeling Space and Activation Rate Changes in Bone Remodeling R. B. MARTIN Orthopaedic Research Laboratories, University of California at Davis, Davis, California 95616, U.S.A. Address for correspondence and reprints: R. Bruce Martin, Ph.D., of California at D&is, Davis, CA 95616, U.S.A.

Professor

Abstract

bone-Remodeling-Porosity

-Age

Orthopaedic

Research

Laboratory,

TB-150,

University

Bone remodeling is a process in which “teams” of osteoclasts and osteoblasts work locally to remove sequentially a quantity of old bone and then lay down a similar (but usually not identical) amount of new bone. These teams of cells are known as Basic Multicellular Units, or BMUs (Frost 1964). Because they dig and refill tunnels in cortical bone, as well sculpt the surfaces of trabecular bone, it is taken for granted that the resorptive process must occur first and be followed by the formation process. It has also been taken for granted that, whereas coupling the activities of osteoclasts and osteoblasts helps to ensure that resorption and formation stay in balance throughout the skeleton, equilibrium of bone mass depends critically on equal amounts of resorption and formation at each remodeling site, or within each BMU. Much of the thought about the etiology of osteoporosis and other metabolic bone diseases has been focused on this issue, and rightly so. However, an equally important aspect of bone remodeling is the rate at which BMUs are created or “activated.” This is because the activation frequency determines the number of BMUs working in a region of bone, and the remodeling result depends on the product of (a) the net amount of bone added or removed by the average BMU and (b) the number of BMUs that are active. Villanueva and Frost (1970) showed that activation frequency can be more variable than the intra-BMU dynamics (e.g., mineral apposition rate), and thus may dominate the control of bone remodeling by managing the speed with which bone is gained or lost. In this paper, it will be shown that there is a subtle but important consequence of the fact that resorption occurs first in the BMU: Variations in activation frequency alone can add or remove bone from the skeleton, even if resorption and formation are equal in each BMU. This is because the “remodeling space,” which is comprised of resorbing, reversing, and refilling BMU cavities, varies with activation frequency. Quantitative knowledge of the behavior and magnitude of the remodeling space in relation to the balance of the porosity is important in understanding the many ways that bone remodeling contributes to skeletal development, aging, repair, adaptation to mechanical loading, and disease.

This paper quantifies the relative contributions of the remodeling space and the accumulation of Haversian canals to bone porosity at various ages. It also examines the importance of variations in the rate of bone remodeling that occur during growth and aging, and as a result of trauma and disease. The dependence of the remodeling space (cavities due to resorbing, reversing, and refilling BMUs) and the Haversian canal components of porosity on the Basic Multicellular Unit (BMU) activation frequency are mathematically formulated. A graph is developed using data for the cortex of the human rib which shows the extent to which porosity is primarily due to the remodeling space in children, and to accumulated Haversian canals in adults. It is shown that the diminution of activation frequency between birth and age 35 contributes to the concurrent increase in bone volume fraction, and the increase in activation frequency after age 35 contributes to the subsequent decline of bone volume fraction. An equation is derived for determining the time rate of change of activation frequency using two fluorochrome labels. Key Words: Osteonal

& Director,

-

Haversian canals.

Introduction Bone volume fraction and its complementary variable, porosity, are important attributes of both trabecular and cortical bone because they are major determinants of the tissue’s mechanical properties. For this reason, these variables are frequently measured in experiments concerned with the mechanics and/or the biology of bone, as well as in clinical biopsies. It is important to understand that bone porosity consists of two parts: one due to bone remodeling in progress, and the other resulting from all the completed remodeling. This concept was originally discussed by Frost (1964), as a direct result of his recognition of the sequential nature of bone remodeling. He referred to the porosity caused by ongoing remodeling activity as the “phase lag pool.” Today it is more commonly called the “remodeling space,” a term originally used by Jaworski (1976) who, with Parfitt (1976) and Frost, was one of the first to appreciate its significance. The purpose of this paper is to quantitatively develop this concept further so that it can be. better understood.

Theoretical Analysis In order to simplify the discussion, the analysis will focus on cortical bone, where it may be assumed that BMUs form osteons by tunneling perpendicular to the cross sections com391

392

R. B. Martin: Porosity and the bone remodeling

space

Table I. Key to variable symbols Variable name

Symbol

H

Cross sect. region Site (point) in C Resorption period Reversal period

none none Rs.P Rv.P none FP NE=N,+N, none none none none none none none none none none none none none none Ca.Rd

s TR T*

TRI

TR + TI Refilling period # resorb. BMUs in H # revers. BMUs in H # refill. BMUs in 2 BMU resorp. rate BMU refill. rate BMU balance Remodeling balance Osteonal porosity Initial value of pO Dynamic porosity Pri. H. canal por. Sec. H. canal por. Time Present time Time of elevated f, Haver. canal radius Cement line radius Area inside cement line BMU activ. rate Initial value off, Haversian porosity Mn. resorp. cavity area Mn. revers. cavity area Mn. refill. cavity area Misc. constants Remodeling space factor Fluorochrome labels L,, L, admin. times # labeled BMUs ex Natural logarithm of n 3.14... aFrom the standardization

Standard symbol”

TF NR N, NF

none none Ac.f none none none none none none none none none

FRS L,, L2 TL, T,, T, NL, N, >N, exp (x) ln (x) 71 suggested

by Parfitt et al. 1987.

monly used for histologic observations. Subsequently, when the principles are clear, the analysis can be altered to describe trabecular bone remodeling using concepts described by Parfitt (1983). Table I shows a key to the symbols used in this paper, and their standard histomorphometric correlates where appropriate (Pa&t et al. 1987). The standard abbreviations are not used here because they are not available for most of the variables, and because the standard abbreviations are not conducive to use in mathematical derivations.

Quanrifying bone balance We will first write equations for the balance between formation and resorption in individual BMUs and within a region of bone. Consider a portion of a typical cross section of cortical bone that is small relative to the total cross-sectional area, but large enough to contain about lo-30 osteons. Assume that this two-dimensional region, called 2, is experiencing osteonal remodeling. As a BMU passes through site S in 2, three phases

of remodeling are experienced (Fig. 1). First, the BMU’s osteoclasts resorb an area A, of existing bone. Next there is an intermediate or “reversal period” during which osteoblasts are recruited to the site, and finally there is a refilling or formation period during which an area A, of new bone is laid down. (For the purposes of this paper, it is unnecessary to consider the mineralization phase of remodeling.) The times required for these three phases to occur at S are TR, TI, and TF, respectively (expressed here in days). The number of sites in C where resorption, reversal, and formation are occurring at any given time is NR, NI, and NF, respectively (BMUs/mm* of section). If Q, is the mean rate of erosion at each resorption site and Q, is the mean rate of formation at each refilling site (both in mm’/day), then the total area of bone resorbed at each BMU site is A, = QRTR and that put back during refilling is A, = QBTF (in mm*). The net bone area exchanged during the remodeling at each BMU site is obviously

6AnMu

=

A, - A, = QJ'F - QcT,.

(1)

R. B. Martin: Porosity and the bone remodeling

393

space

Fig. 2. Sketch depicting two possible ways in which osteons may be located or “packed” in a cross section. In A they are positioned so that no overlapping occurs. In B they are randomly positioned, so that

newer osteons may overlap existing ones, obliterating lamellae, cement lines, and Haversian canals.

Fig. 1. Diagrams showing the development of a BMU on a cross section of cortical bone. In A the resorption phase is partially completed,

an existing Haversian canal is about to be obliterated. In B the resorption phase has been completed, and the BMU is in the reversal phase. In C the BMU is in the refilling phase, and in D it is completed, leaving a new Haversian canal and cement line.

Considering all the BMUs within 2, the bone balance for the region may be written as

Q = Q&G - Q&t.

(2)

This is the net rate at which bone is being added, in mm*/ mm*/day. It is equal to the time rate of change of the bone volume fraction. Osteonal porosity may be defined as the complement of bone volume fraction, which is to say the ratio of the void area to the total area in 2. Here, voids are understood to include Haversian canals and resorbing, reversing, and refilling BMU spaces, but not canaliculi or osteocyte lacunas. (Volkmann’s canals are also not considered in this analysis.) The porosity of X can be considered to have two components: an osteonal part, p,,. due to completed Haversian canals, and a dynamic part, pD, due to the remodeling space (i.e., the cavities of active BMUs). Osteonal porosity is initially due only to primary Haversian canals formed when the bone was laid down endosteally or periosteally. Subsequently, secondary Haversian canals are added as BMUs are completed. The changes in p0 caused by the accumulation of osteons over time have been analyzed by Martin and Burr (1989). The growth of p, with time depends on the way in which new osteons are positioned in Z. If the resorption spaces of new BMUs never overlapped and obliterated existing Haversian canals (Fig. 2A), the relationship between p0 and time would be Porosity

PO = Poi +

nR,*fat

(3)

where poi is the initial value of pO, R, is the mean radius of the completed Haversian canals (mm*), f,is the BMU activation rate (BMUs/mm*/day), and t is time (days). In contrast to eqn (3)‘s implication, p0 cannot increase for-

ever; it would be theoretically limited by the way in which osor located relative to one another. If teens are “packed,” osteons were circular and packed in the highly ordered way shown in Fig. 2A, the maximum porosity would be (PC&l,, = aRH214R,* = 77~~14

(4)

where R, is the radius of the cement line and p,., = R,*/R,* is defined as the “porosity” of a single, completed osteon (i.e., the ratio of the canal area to the area inside the cement line). It is also conceivable that osteons could be packed randomly, as depicted in Fig. 2B. In this scenario, new resorption cavities would begin to form at random points within Z, irrespective of the location of existing Haversian canals. Completed resorption spaces would sometimes encompass a completed Haversian canal, obliterating it (recall Fig. 1). In that case, the BMU would not add to the number of canals within c. In other cases, no existing canal would be erased, and p, would increase when the BMU was completed. In this kind of random remodeling, it has been shown that (Martin 1984) P,, = PH -

@H-pOi)exp(

-

TRc2fat).

(5)

In this case, p0 asymptotically approaches a maximum value of pH. (Conversely, the same theory predicts that the porosity due to primary Haversian canals would diminish exponentially with time.) Cross sections of osteonal bone generally look much more like Fig. 2B than Fig. 2A. On the other hand, it is sometimes argued that osteons are not randomly located, but appear in response to deterministic mechanical or genetic factors. Alreasonable to though this may be true, it is nevertheless assume that BMUs are quasi-randomly distributed within 2, for two reasons. First, 2 is assumed to be small relative to the whole section, so that a large-scale remodeling pattern apparent in the whole section will not be apparent in s. Also, BMUs clearly do not tunnel along regular, well-defined paths through the bone, but are slightly diverted this way and that as they proceed (Cohen & Harris 1958). Consequently, from section to section they vary their position relative to neighboring osteons, which also followed irregular paths. Thus, since BMUs tunnel 50-100 times their diameter in forming new osteons (Cohen & Harris 1958), it is reasonable to assume that for most of their course they are quasi-randomly located with respect to nearby osteons. Finally, it is clear from observing histologic sections that tunneling BMUs do overlap existing Haversian canals and obliterate them, and the random remod-

R. B. Martin: Porosity and the bone remodeling

394

S,”

Table II. Values used for calculations Parameter

Value

A Rm

A(t) dt

1 =

7

7~Rz.

(10)

5 dt

s

20 3 0.10 0.02

Resorption time, Ta, days: Intermediate time, TI, days: Osteon cement line radius, mm: Osteon Haversian canal radius, mm:

=

The mean area of the growing resorption cavity is seen to be half of its area at completion. If the rate of erosion were constant during refilling, A,, would be only 25% of the area within the cement line. This exaggerates the remodeling space somewhat, but not substantially, since TR is brief and NR is small. Since the cavity may be assumed to be of constant radius during the reversal stage,

theory agrees very well with data for human long bones (Martin & Burr 1989).

eling

Dynamic porosity

A Im = -n R,‘.

The dynamic porosity is composed of three parts, due to BMUs in the three stages of remodeling: resorption, reversal, and refilling. In steady-state remodeling, the number of BMUs in each of these stages will be Na = f, TR (resorption BMUs) Nr = f, Tr (intermediate BMUs) NF = f, TF (refilling BMUs).

(6a) (6b) (6~)

Now let A,,, A,,,,, and A,, be the mean areas of the cavities presented by BMUs during the respective stages of remodeling. Then one has

(11)

In order to calculate A,,, it is also important to consider the variability in bone apposition during TF. Experimental observation of osteonal remodeling has shown that the rate of refilling exponentially decreases during T, (Manson & Waters 1965). That is, the radius of the refilling cavity may be expressed as R = R, exp( -bt)

(12)

where b is a constant. Solving this equation for b when t = T, and R = R, (the Have&an canal radius) shows that b = ln(RJR,)IT,.

PD = A,,

NR + A,, NI + A,,

NF

(7)

v-

t/TR

(8) where R, is the cement line radius. The area of the developing cavity is A(t) = rRC2tlT,

A Fm

S,”

(14)

(15)

n (RC’ - RH2) =

2 In (RJR”)

(9)

frequency

dt

.

Using typical values of R, (100 mcm) and R, (20 mcm), A,, is about 30% of A,, the area within the cement line. If the apposition rate were constant during refilling, the mean cavity area would be 41% ofA,. This is important because the relative slowness of refilling makes NF much larger than Na and Nr, so that most of the remodeling space is due to refilling BMUs. The reduction in A,, caused by nonlinear refilling reduces this part of the remodeling space by about one third. Substituting the expressions obtained for the mean BMU areas and eqns (3) through (5) into eqn (6), and calling

=

7RC2

c

;TR + T, +

f, 0.052 0.0030 0.0060

1 (16)

RH2/R,*) T 2 In(R,_IR,) F

(1 -

and refilling time in human ribs”

BMU/mn?/day

aFrost 1969.

=

dt

F RS

l-9 yrs old 30-39 yrs old 70-89 yrs old

s

rFnRC2 exp( -2bt)

A Fm

and the mean area is Table III. Mean values of activation

(13)

Then the mean cavity area during refilling is

for the dynamic porosity. To find A,,, it is necessary to make an assumption regarding the variability of the rate at which bone is resorbed during Tn. It is clear that Q, is not constant during Tn because if it were, the “cutting cone” would indeed be cone-shaped rather than rounded. The curved shape of the resorption surface when observed in a longitudinal section shows that the rate of erosion is greatest at the start of TR and declines thereafter. For the sake of convenience, it will be assumed that the erosion rate is inversely proportional to the radius (R) of the resorption cavity. It follows that R = R,

space

TF days 51 73 109

R. B. Martin: Porosity and the bone remodeling

space

395

-TFig. 4. Diagram to clarify the calculation of NR and NF from f,. An historical time line is shown, with the present marked T. BMUs that

are presently resorbing were activated during the past TR days. Those currently in their intermediate phase were activated in the interval between T-T,-Ta and T-T,. BMUs that are presently refilling were activated in the interval between T-T,T,-T, and T-T,-T,.

0

20

40

60

80

prior to age 35, and then slowly increase during the balance of the lifetime. The total porosity @) would be the sum of pD and p,,; it would likewise diminish and then increase again after age 35. This graph assumes that the bone in 2 was laid down very early in life, when f, is at its peak. Alternatively, if Z were in bone formed later - for example, at age 10 - then f, would be smaller, and the pl, p2, and p, curves would all approach their equilibrium levels more slowly. The pD values at each age would remain unchanged, however. In any case, the graph shows how cortical bone porosity is dominated by the remodeling space in children, and by completed Haversian canals in adults.

100

Age (years) Fig. 3. Graph of the approximate relationships between various components of cortical bone porosity and age, using the human rib as a model. p, = primary Haversian canal porosity, p2 = secondary Haversian canal porosity, p. = p1 + pz = total osteonal porosity, p. = dynamic porosity (the remodeling space), and p = p. + pD = total porosity. Note that (R&)’ is the Haversian porosity, pH, as de-

Effect of a changing activation ffequency If f, is constant, then NR = faTR and Nr = faTF, so that one has for the bone balance, from eqn (2),

scribed in the text. Q

the remodeling space factor, one has PD

=

FRS

fa.

(17)

For typical values of R, and R, (Table II),

=

-

(18)

QJd

However, let us assume that f, is not constant, but is varying with time, and see what the effect is on Q. For the sake of simplicity, let f, be a linear function of time: f,(t)

F RS = .H RC2 [0.5 7’n + Ti + 0.3 Tr] and it is seen that reversing BMUs contribute to FRs (and thus pD) at a greater rate than do resorbing and refilling BMUs. Ordinarily, TR and TI are small, and most of the remodeling space is comprised of refilling BMUs. However, in osteoporoses, which are characterized by long reversal times, the remodeling space may be greatly expanded. The same may be said for conditions in which TF is prolonged. Using the data in Tables II and III, one finds that for a young child, FRS = 0.89 mm*-day and pD = 4.6%; for an adult, FRS = 1 .lO mm2-day and pD = 0.33%. FRS is slightly larger in adults because refilling is prolonged; pD is much larger in children because f, is elevated.

fa(QaT,

=

fo

+

(19)

kr

where k is the time rate of change of f, and f. is its initial value. Now NR and Nr will depend on the magnitude off, at times in the past. For example, (see Fig. 4),

NR=

s;_, f,(t)

(20)

dt

R

where T is the present time. Substituting

sT

NR = and performing

T-T,

the integration

for

f,(r),

one has

C& + kt) dt

(21)

and setting T = 0 yields

Porosiht vs. ape By combining the above considerations, one obtains a picture of the relative magnitudes of the components of porosity as a function of age in the cortex of the human rib (the only bone for which adequate data are available). Figure 3 shows how p0 would increase exponentially with time, using the data in Tables II and III, and assuming that the primary Haversian canals give the bone an initial osteonal porosity of 1%. Osteonal porosity is the sum of contributions by primary and secondary osteons (p, and p2, respectively). At the same time, the dynamic porosity bD) would decline due to the diminution in f,

1 Na = TRC&- 2kT,). Similarly,

(22)

one has NF =

since Nr depends on

f,

T-T

s

T-TR,-TFfa(t)

dt

(23)

during a period TF that ended TRI =

396

R. B. Martin: Porosity and the bone remodeling

TR + TI days ago (Fig. 4). Substituting one obtains

NF = T&

If Q is now calculated

-

1 k(T,

forf,

+

space

and integrating,

2T,,)l.

(24)

for the case of time-varying

f,, one

250 200

has

Q = QBT&

-

; k(TF+2TR,)]

-

Q,T,C,&

-

; kT,).

(25)

In order to isolate the effect of a time-varying f, on the bone balance, one may place into the last equation values of Q, and Q, which make fiABMU = 0. (It is recognized that normally, when the BMU is not following an existing Haversian canal, < 0 for osteonal remodeling because a new Haversian &WJ canal must be created. However, for the sake of generalizing the analysis, it is assumed that 6A,,” may be positive, negative, or zero.) If each BMU is “balanced,” will Q still be zero? One has 6A BMU = QJF

-

QcTF = 0

Q, = QcTRJTF. Substituting

; QCTRk(TF + 2T, + T,J.

Finally, as a matter of convenience, note that QcTR = A, (i.e., the area within the cement line) in the balanced BMU. Then

Q =

-

1 kAR(TF + 2T, + TR).

0.04

' 0

I

I

I

I

50

100

150

200

TIME, days

this into eqn (25), and recalling that TR, = TR +

-

r

(27)

T,, one obtains

Q =

0.07

(29)

It is seen that when k is not zero, neither is Q, even though equal amounts of bone are removed and replaced in each BMU. Furthermore, Q and k are of opposite sign. If f, is decreasing with time, the bone balance will be positive, and vice versa. Note also that the magnitude of the effect depends directly and equally on TR and TF. as well as on 2Ti. If any of these periods becomes exaggerated in an individual or by a disease, Q will be larger than when the BMUs are functioning with normal temporal characteristics. To obtain an idea of the magnitude of this effect, let us consider a hypothetical case in which the rate of remodeling is accelerated in an adult human rib over a two-year period. Using the estimates for normal humans from Table II in eqn (29), one obtains Q = - 1.5 1 k [mm’/mm’/day]. For an adult rib, a typical value off, would be 0.005 BMW mm*/day (Table III). Suppose this were increasing by 1.0% each day. Then k = 0.00005 BMU/mm*/day’ and Q =

Fig. 5. The results of a computer simulation of a 20% increase in f, over a loo-day period beginning on day 10 (curve f,, BMUslmm’i day). The temporal changes in the numbers of resorbing (Na) and retilling (Nr) BMUs, Q (mm2imm2/day), and porosity P are also shown. Since Q is negative when f, is increased, its negative value is plotted for convenience. Na and NF are shown as BMUsilOO mm’ of cross section. The parameters shown in Table I were used in the model.

- 0.0275 mm’/mm2/year, so that the porosity would be increasing by 2.76% per year. If the initial porosity were 5%, the 1 .O% daily increase in f, would, in two years, double the porosity while increasing f, to 0.042 BMU/mm’/day. Bear in mind that this is in spite of the assumption that exactly the same amount of bone is being replaced in each BMU as is removed @A,,, = 0). Again, the basic reason for this phenomenon is that resorption occurs first when a new BMU is activated. Thus, when the rate of activation is changing, the remodeling space changes as well, and the number of resorbing BMUs will not be in proportion to the number of refilling BMUs. If f, is diminishing, the current number of resorbing BMUs will be too small relative to the refilling BMUs, which were activated when f, was greater, and Q will be positive. If f, is increasing, the newer, resorbing BMUs will be too numerous in proportion to the older, refilling BMUs, and there will be net bone loss. It is not entirely clear from the analysis presented so far that the bone lost or gained whilef, is changing will remain so when f, becomes constant once more. To study this. and to confirm the above analysis, a computer program was written to simulate the BMU remodeling process. Using the data in Table II, this model gave results like those shown in Figs. 5 and 6. If k > 0 for a period of time T, and then becomes zero once again, Q remains negative during T,, and for a period TF thereafter, and does not “compensate” for the lost bone by going positive when f, becomes constant again at a new value. Bone volume is returned only if f, returns to its original value. A similar situation exists for k < 0, except that Q is posi-

R. B. Martin: Porosity and the bone remodeling

397

space

L, and La are given at times T, and T2, the number of labeled BMUs will be different for each label (N, and N2, respectively). If one continues to assume that f, = kt + fo, the number of labeled BMUs is

300 250 200

NL =

150 100

T-T,-TN

T-TL-TR,-TF

(kt + fO) dt

(30)

or

50

NL = T& -

0 0.06

s

k(:T, + TaI + T-J1 L

(31)

where TL is the time (before T, when the animal was killed) that the label was given. By writing such an equation for each label (N,_ = N,, TL = T, and NL = N2, T,_ = T2), and solving for k, one obtains

Il

k=

NI -N2

(32)

TF(T2-T,)’

0

50

100

150

200

TIME, days

This relationship can be used to determine k histomorphometrically, assuming that TF is determined by using measurements of mineral apposition rate and mean wall thickness. An expression for f.may be similarly obtained:

Fig. 6. Similar to Fig. 3, but a 20% decrease in f. over a lOO-day period is simulated. In this case, Q is positive and plotted as such.

tive while T, + T,.

f,is changing,

Measurement

W2 - NJ fo’

and bone volume is gained during

of k usinn tluorochrome labels

Conventional histomorphometric analysis of bone remodeling dynamics assumes that f, = NdTF is constant (k = 0); the calculation off, is not accurate if f, is changing with time. This may be a significant limitation in many cases, since diseases or experimental treatments may cause f, to be changing. It is useful to consider how such changes could be detected and estimated using fluorochrome labels. By now it is clear that when f, is varying, the number of refilling BMUs is also changing with time. Therefore, if labels

Table IV. Effects of skeletal diseases on activation

frequency

NormaP Postmenopausal osteoporosisb Senile osteoporosis, femalesb Senile osteoporosis, maleb Osteomalacia’ Rheumatoid arthritis” Cushing’s disease’ Osteogenesis imperfec& “Pirok et al. 1966. bVillanueva et al. 1966a.b. ‘Ramser et al. 1966b. dRamser et al. 1966a. ‘Klein et al. 1965; Villanueva

et al. 1966a.b; Frost 1963a,b.

N,T,) (33)

TF(Tz - T,)

Note that TRI must be known to find fo. In these equations it is assumed that TF and TRI are constant. This may be much less true in some cases than in others. Table III and Fig. 5 show that fa decreases 2040 fold during childhood, while TF merely doubles. On the other hand, Table IV shows that in humans f, varies much less than TF as a consequence of several diseases. Iff, is changing much more rapidly than TF, one may estimate its rate of change by giving two different fluorochrome labels (e.g., tetracycline and xyleno1 orange) and using eqn (32) to find k. Of course, other formulations of f,(f) may be more appropriate in some situations. For example, during growth one might prefer an exponentially decreasing f,(t):

and refilling time in human ribs

f, Disease

(& +TaJ + W, T2 -

TF

BMU/mn?/day

days

0.0055 0.0077 0.0052 0.0060 0.013 0.0016 0.0005 0.025

91 415 621 601 829 708 124 124

398

R. B. Martin: Porosity and the bone remodeling

l

space

MEN

0 WOMEN t 0

I

10

1

20

I

I

I

I

I

I

30

40

50

60

70

80

Fig. 7. Graph of activation frequency versus age for normal rib biopsies (Frost 1964; male and female data are averaged together). Reproduced with permission from Martin and Burr (1989).

f, = .& exp( - k’t)

(34)

which can be integrated to give

2 exp(k’Tat)

[exp(k’Tr) -

I]

(35)

and

k’

=

ln(NP2) T,

which is independent

-

20

40

AGE,

AGE (years)

NF =

0

(36) T2

of both TF and TRI.

Discussion These results are significant in several ways. First, consider the data shown in Fig. 7. The relationship of f, to age is shown for human rib biopsies. Similar data are not available for other bones, but it is thought that they would follow a similar pattern. Notice that f, declines monotonically from birth to about age 35. The analysis presented here shows that this negative rate of change off, with time would be accompanied by a positive bone balance if 6A,,, 2 0. If SA,,, < 0, the diminishment off, would oppose the bone being lost within each BMU. In either case, the increase in bone mass during growth must be aided by the age-related decrease in f,. To estimate the magnitude of this effect, note thatf, decreases from 0.12 to 0.036 BMU/mm’/day in the years from birth to age 10. The time rate of change is k = -2.3 x lop5 BMU/mm*/ day2. Using this k in eqn (29), and assuming that TF = 51 days (Table III) and SA,,, = 0, one finds that Q = 2.8 X 10e5 mm2/mm2/day. In the initial 10 years of life, this value of Q would decrease p,, by about 0.10 mm*/mm*. While the total volume of bone added in this way may be small when compared to the overall increase due to periosteal apposition and epiphyseal growth, it may be very significant in terms of

80

80

years

Fig. 8. Graph of ash weight per unit volume of vertebral cubes as a function of age for normal male and female human subjects who died acutely. Redrawn from the data of Nordin (1973).

changing the mechanical properties of the bone material. Using the empirical relationship between porosity and elastic modulus developed by Schaffler and Burr (1988) for bovine cortical bone, E = 3.66~~‘.~~

[GPa] ,

(37)

a decrease in porosity from 14% to 4% is found to increase the elastic modulus from 11 to 21 GPa. Of course, other factors must also must be considered in this regard. As Fig. 3 shows, p. can be expected to increase while pD is decreasing, pa&ally offsetting the above effect, and concurrent changes in the mineralization of the bone have not been considered. These considerations need to be combined in a more complete analysis. Figure 7 also shows that f, reaches a minimum and begins to increase at age 35, with a positive slope until about age 60. This protracted increase in f,, though smaller than the decrease associated with growth, would exacerbate any senile bone loss produced by a negative SA,,“. Thus, senile osteopenia appears to be increased somewhat by the fact that f, is continuously increasing after age 35. Furthermore, the fact that peak bone mass is typically achieved at about age 35 (Fig. 8) can now be seen to be, at least in part, a reflection of the minimum in f,, which occurs at that age. This is a potentially important point, and has implications with regard to the theory that those who experience senile and/or postmenopausal osteoporosis did not achieve sufficient peak bone mass to withstand age-related losses. With regard to postmenopausal osteoporosis, it is well established that an effect of estrogen diminution is to increase f, (Dannucci et al. 1987; Wronski et al. 1986). The effect of estrogen withdrawal on SA,,, is not clear, with some investigators finding that osteoblast activity within BMUs is decreased (Malluche et al. 1986), and others not (Dannucci et al. 1987). Regardless of that, any increase in f, caused by menopause would exacerbate the bone loss occurring after age 35. Frost (1983) has emphasized the importance of a related effect, the Regional Acceleratory Phenomenon, or RAP. A RAP occurs when the BMU activation frequency is temporarily in-

R. B. Martin: Porosity and the bone remodeling

space

399

creased for some reason - usually trauma. Frost points out that the elevation off. will be accompanied by increased remodeling space, resulting in a transient loss of bone that will be corrected when f, returns to normal and the surplus BMUs are refilled. The present analysis focuses on the situation in which an increase or decrease in fa is prolonged. In this case, the alteration in remodeling space may be either positive or negative, and is not transient, but persists as long as fa does not reverse itself. Although the effect described in this paper may be seen as an extension of the concept of a RAP, it is important to recognize not only the difference between the two phenomena, but the fact that a RAP is frequently a local effect, whereas the effect described here would often be a consequence of systemic changes accompanying growth and development, menopause, or some other global change (recall Fig. 7). In another vein, Frost (1987) has postulated that when mechanical loads require more bone mass, modeling is stimulated and remodeling is inhibited (i.e., f,is decreased). Conversely, he has suggested that when bone mass needs to be reduced, modeling is inhibited and remodeling is stimulated (i.e., f,is increased). This “Mechanostat Hypothesis” has been viewed as iconoclastic, since the conventional view has been that bone remodeling serves to make bones stronger as well as weaker, depending on the circumstances, and it was expected that these goals would be achieved in a proactive sense. Thus, to say that bone strength can be enhanced by reducing remodeling has seemed wrong to many investigators. Frost recognized that reducing f, would reduce the remodeling space and thus the overstrain. He also argued that, in adults, bone is generally being lost with time because SA,,, < 0, so that reducing the rate of remodeling when bone strains increased would reduce the rate of bone loss and thus reduce (but not necessarily correct) the amount of overstrain that would eventually accrue. The analysis presented here supports Frost’s hypothesis by showing quantitatively that reducing f, quickly or over a prolonged period of time will add bone volume through diminishment of the remodeling space. The effect will be permanent so long as f,does not return to its original value. The original “phase lag pool” term of Frost has the advantage of recognizing that factors other than porosity experience transient changes when remodeling is altered - bone mineral content, for example. Parfitt (1980) and Frost (1989) have discussed in detail the effects of both the remodeling space and the time required for new BMUs to mineralize fully, on measurements of bone mineral content by photon absorptiometry and other methods. Jerome (1989) has also investigated this problem using a computer model. Finally, it is very important to appreciate the existence of the effects described here when trying to develop mathematical or computer models for bone remodeling as a means of understanding many other aspects of remodeling (Kimmel 1985; Martin 1985; Polig & Jee 1987). For example, one cannot assume that if &A,,, is zero, Q must be zero as well. If disease processes or Wolff’s Law effects that alter the rate of remodeling are being studied, then the model should simulate the remodeling space, and the investigator should be aware that it can lead to results that are counter-intuitive. Acknowledgmenrs:

Harold Frost, gestions.

author is art anonymous

The

and

grateful to Drs. David Burr and reviewer,

for

their

helpful

sug-

References Cohen, J.; Harris, W. H. The three-dimensional J. Bone Joint Surg. 4OA:419-434; 1958.

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Dannucci. G. A.; Martin, R. B.; Patterson-Buckendahl. P. Ovatiectomy and trabecular bone remodeling in the dog. C&if. Tiss. fnt. 40:194-199; 1987. Frost, H. M. Bone remodeling dynamics. Springfield, IL: Charles C. Thomas; 1963a. Frost, H. M. Dynamics of bone remodeling. Frost, H. M.. ed. Bone biodynamics. Springfield, IL: Charles C. Thomas; 1963b. Frost, H. M. Mafhemarical elements of lamellar bone remodelling. Springfield, IL: Charles C. Thomas; 1964. Frost, H. M. The laws of bone structure. Springfield, IL: Charles C. Thomas; 1964. Frost, H. M. Tetracycline-baaed histological analysis of bone remodeling. Co!-

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Frost, H. M. The mechanostat: A proposed pathogenic mechanism of osteoporoses and the bone mass effects of mechanical and nonmechanical agents. Bone Min. 2:73-85; 1987. Frost, H. M. Some effects of basic multicellular unit-based remodeling on photon absorptionretry of trabecular bone. Bone Min. 7:47-65; 1989. Jaworski. Z. F. G. Parameters and indices of bone resorption. Meunier, P. J., ed. Bone Histomorphomewy. Second International Workshop. Lyon: Armour Montague; 1976. Jerome, C. P. Estimation of the bone mineral density variation associated with changes in turnover rate. C&if. Tin. Inr. 44:406-410; 1989. Kimmel, D. B. A computer simulation of the mature skeleton. Bone 6:369-372: 1985. Klein, M.; Villanueva, A. R.; Frost, H. M. A quantitative histologic study of rib from 18 patients treated with adrenal cortical steroids. Acta Orthop. Sand. 35:171-184; 1965. Malluche. H. H.: Faugere, M. C.; Rush, M.; Friedler, R. Osteoblastic insufficiency is responsible for maintenance of osteopenia after loss of ovarian function in experimental Beagle dogs. Endocrinol. 119:2643-2654: 1986. Manson, J. D.; Waters, N. E. Observations on the rate of maturation of the cat osteon. J. Anat. (London) 99:539-549; 1965. Martin, R. B. Porosity and the specific surface of bone. CRC Crit. Rev. Biomed. Eng. 10:179-222; 1984. Martin, R. B. The usefulness of mathematical models for bone remodeling. Yearbook

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Martin, R. B.; Burr, D. B. The fun&m, structure, and adaptation of compact bone. New York: Raven Press; 1989. Nordin, B. E. C. Metabolic bone and ~fone disease. Baltimore: Williams and Wilkins; 1973. Parfitt, A. M. The actions of parathyroid hormone on bone. Relation to bone remodeling and turnover, calcium homeostasis and metabolic bone disease. 111. PTH and osteoblasts, the relationship between bone turnover and bone loss, and the state of bones in primary hyperparathyroidism. Merabolism 25: 1033-1069;

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Parfitt, A. M. The morphologic basis of bone mineral measurements. Transient and steady state effects in the treatment of osteoporosis (editorial). Min. Elecrro. Metab. 4:273-287; 1980. Parfitt, A. M. The physiologic and clinical significance of bone histomorphometric data. Reeker, R. R., ed. Bone hisfomorphometry: Techniques and interpreration. Baca Raton, FL: CRC Press; 1983. Partitt, A. M.; Drezner. M. K.; Glorieux, F. H.; Kanis. J. A.; Malluche. H.; Meunier, P. J.; Ott, S. M.; Reeker, R. R. Bone histomorphometry: Standardization of nomenclature, symbols, and units. J. Bone Min. Res. 2:595610; 1987. Pirok, D. J.; Ramser, J. R.; Takahashi, H.; Villanueva, A. R.; Frost, H. M. Normal histological tetracycline and dynamic parameters in human mineralized bone sections. Henry Ford Hosp. Med. BULL l&195-218; 1966. Polig, E.; Jee. W. S. S. Bone age and remodeling: A mathematical treatise. C&if. Tiss. Int. 41:130-136; 1987. Ramser, J. R.; Duncan. H.; Landeros, 0.; Epker. B.; Frost, H. M. Measurements of bone dynamics in seven patients with salicylate treated rheumatoid arthritis. Arth. Rheum. 9424-429; 1966a. Ramser, J. R.: Villanueva, A. R.; Frost, H. M. Cortical bone dynamics in osteomalacia, measured by tetracycline bone labeling. C/in. Orth. Rel. Res. 49:89-102: 1966b. Schaffler, M. B.; Burr, D. B. Stiffness of compact bone: Effects of porosity and density. J. Biomech. 21:13-16; 1988. Villanueva, A. R.: Frost. H. M. Evaluation of factors determining the tissuelevel haversian bone formation rate in man. .I. Dem. Res. 49t836-846; 1970.

R. B. Martin: Porosity and the bone remodeling space Villanueva. A. R.; Frost. H. M.; Ilnicki, L.; Frame, B.; Smith, R.; Amstein, R. Cortical bone dynamics measured by means of tetracycline labeling in 21 cases of osteoporosis. J. Lab. Clin. Med. 68599616; 1966a. Villanueva, A. R.; Ilnicki, L.; Duncan, H.; Frost, H. M. Bone and cell dynamics in the osteoporoses: A review of measurements by tetracycline bone labeling. Clin. Orrhop. Rel. Res. 49~135-150; 1966b. Wronski, T. J.; Walsh, C. C.; Ignaszewski, L. A. Histologic evidence for os-

teopenia and increased bone turnover in owiectomized 123; 1986.

rats. Bone 7: 119-

Date Received: December 12, 1990 Date Revised: March 27, 1991 Dare Accepted: May 6. 199 1