Bisphosphonate Effects and the Bone Remodeling Transient

Authors


Abstract

Published randomized clinical trial data for alendronate, given at a dose of 10 mg/day, were fitted by a computer algorithm to the currently accepted model of the bone remodeling process. The purpose was to determine how much of the reported improvement in lumbar spine bone density could be attributed to the inevitable remodeling transient and how much might represent positive bone balance. Very good fits to the clinical data were easily obtained, indicating the general validity of current syntheses of bone remodeling biology. The best fit was provided by simulations produced by combinations of 36–38% suppression of remodeling activation and positive remodeling balance ranging from 1.1 to 1.4% per year. Whole body bone biomarker changes would have suggested both a slightly greater degree of suppression and a higher baseline level of remodeling than could be provided by any of the simulations if they were to fit the clinical data. Either regional skeletal heterogeneity or lack of a one-to-one quantitative relationship between remodeling changes and biomarker changes may explain the discrepancies between the two approaches.

INTRODUCTION

In a recent paper, one of us (R.P.H.) presented results of a comprehensive computer simulation of the currently accepted scheme of bone remodeling,1 by which it was possible to mimic the effects of bone active agents on remodeling activation, and thereby to predict clinically observable changes in bone mass (or density), both acutely and in the new steady state resulting from treatment. In that communication it was noted that most, if not all, of the changes in bone mass published to that date, associated with both calcitonin and etidronate, could be attributed to reduction in the remodeling space of bone. It was also noted that, while reclaiming such bone could be predicted to confer a gain in bone strength, reduction in the remodeling space was not biologically the same as reversing negative bone balance.

Subsequently, Liberman et al.2 published results of a 3-year, randomized trial of alendronate, given at a dose of 10 mg/day, in which a greater than 5% gain of bone occurred in the spine in treated individuals at 1 year after starting treatment. As with other remodeling suppressive regimens, much or all of this improvement could plausibly be attributed to reduction of the remodeling space. However, the report by Liberman et al. showed a continuing, if slower, bone gain during the second and third years of treatment. Questions arose as to whether the entire pattern could be attributed to continuing closure of the remodeling space, more rapidly at first, and then more slowly in subsequent years, or if not, how it was possible for positive bone balance to occur in the face of remodeling suppression.

This communication presents the results produced by employing the previously published computer algorithm to mimic the actual subject data underlying the paper by Liberman et al. and to estimate thereby plausible combinations of quantitative changes in the remodeling apparatus of bone that could produce data such as found in the alendronate trial. Further, it addresses the question of whether some degree of positive bone balance might be compatible with partial remodeling suppression.

In addition to suggesting a set of remodeling values compatible with real world data, this simulation also constitutes a test of current syntheses of bone remodeling biology. The algorithm used for the simulation is simply a systematic codification of current quantitative estimates derived from histomorphometry and from total body mineral kinetics. If this model cannot produce a fit to the changes in bone density and at the same time be at least qualitatively consistent with biomarker changes, then some modification may be needed in the current understanding of bone remodeling biology and/or of the interpretation of the biomarkers.

MATERIALS AND METHODS

Subjects

The subjects whose data were used as a target for this simulation were those described by Liberman et al., except in this instance the data were limited to the 161 subjects who were treatment adherers throughout the trial and for whom bone density data were available out to the 36-month terminal point. For this analysis, we will confine our attention to the changes in bone mineral density (BMD) at the lumbar spine. This site was chosen because it produced the largest signal-to-noise ratio for the changes in bone mass over time, i.e., the dispersion of measured values around each of the means, expressed as percent change, was substantially smaller for the spine than for hip or total body. Thus, the spine served as the measured site most sensitive to the types of curvature in the time course of BMD that would be produced by various parameter value sets.

The findings in this group of treatment-adherent subjects, pertinent to this analysis and very similar to those previously reported,2 are a mean decline in total serum alkaline phosphatase of 29%, a mean decline in urine deoxypyridinoline of 44%, and mean baseline bone density values 70% of young adult normal (YAN). (Since roughly half of total serum alkaline phosphatase is derived from bone, the observed decline may be presumed to be equivalent to a reduction in bone alkaline phosphatase of about 50–60%. In addition, the alendronate-treated subjects exhibited the following relative BMD values at the lumbar spine, expressed as percent of baseline values (±2 SEM): 3 months, 102.62 (±0.545); 6 months, 104.14 (±0.616); 12 months, 105.60 (±0.642); 18 months, 106.35 (±0.675); 24 months, 107.31 (±0.638); 30 months, 107.99 (±0.742); 36 months, 108.45 (±0.756). These are the values we attempted to match with the simulation.

Computer simulation

The simulation employed the computer algorithm expression of bone remodeling described in detail by Heaney.1 The mathematical basis of its application to actual data, not previously presented, but necessary to explore solution sets for particular values, is included in the Appendix. (Also in the Appendix is a list of the remodeling terms used in this communication, together with their definitions and how they are used in this paper.) With this computer program, values for baseline bone mass, baseline activation/remodeling rate, duration of remodeling period, and bone remodeling balance can be independently adjusted to match any set of clinical conditions. The program then computes the change in bone mass that would be produced by any specified level of remodeling suppression, i.e., it calculates how large the remodeling space is and how much bone would be gained by closing down some (or all) of it. For purposes of the simulations in this report, the originally described program was modified to allow the treatment agent not only to alter activation but to affect osteoblastic work efficiency as well, i.e., to prolong (or shorten) what has been termed “sigma formation” in the histomorphometry literature.3 While this change amounts simply to an alteration of the remodeling period, it nevertheless must be distinguished from the period length existing at the start of therapy. As the development in the Appendix makes clear, the remodeling period prevailing when therapy is started is an important determinant of the size of the remodeling space and hence of the transient produced by altering activation rate. By contrast, a change in period produced by therapy will have no effect on the size of the remodeling transient, but instead on the rate at which the existing remodeling space is refilled, i.e., how rapidly the bone will reach a new steady state following treatment-induced suppression of activation. While formation period has been reported to be affected by agents such as fluoride,4 it was not certain whether bisphosphonates alter osteoblast work efficiency. Nevertheless, provision needed to be made to explore that possibility. A simulation involving remodeling suppression alone would not be adequate to match observed bone changes if those changes had been the result of both suppression of activation and prolongation of formation.

Fitting

The goal of these simulations was to find sets of values for the parameters of the biological model which would coincide with the means of the measured values. Since the parameters are interconnected,1 this required a large number of iterations. The process was quite distinct from the usual statistical modeling because in this case the model existed prior to the data and the fitting involved both quantitative and qualitative criteria. Thus, any predicted value that fell within the confidence interval for the measured value (i.e., the mean ± 2 SEM) at a certain time is, ipso facto, compatible with that measured value. However, such a fit would not itself validate a given parameter set, since at other time points the same parameter set might have predicted values not compatible with the measured data. Thus, for a parameter set to be characterized as producing a good fit, it had to predict values within the measured confidence intervals at all points. Moreover, the closer those predictions were to the respective means, the better the fit was judged to be.

Constraints

With six variables adjustable over a wide range of values, there could be a very large number of combinations of settings that might fit the observed data reasonably well. To deal with a manageable number of iterations, these variables needed initially to be constrained, both for biological plausibility, and to match independently measured data. Initial constraints employed in the simulations described in this paper (and their justification) are as follows:

  • Bone mass (or density) is set to 70% of the YAN value. The reason is that this is the mean value observed by densitometry in these subjects.

  • Basal activation/remodeling rate is initially set above premenopausal normal (i.e., from 1.0 to 1.5×). The reason is that increased remodeling is commonly reported in postmenopausal women generally, as well as in untreated osteoporotics.5–7

  • Remodeling period is set above premenopausal normal (i.e., to 60 weeks). The reason is that osteoporotic patients have been found to have prolonged values for total sigma.7 In our laboratory, the period for cell-based activity in such women averages about 40 weeks, and the model requires a further 20 weeks for completion of mineralization.

  • Baseline remodeling balance is set to zero. While in theory one would expect untreated osteoporotic patients to be losing bone, the placebo group in this study maintained essentially constant BMD values across the 36 months of study (annual loss less than 0.3%/year), presumably because of cotherapy, calcium supplementation, and placebo effects. Since the treated group would have experienced these same adjunctive effects, a choice of zero balance (exclusive of the effects of therapy itself) allows a better base from which to evaluate the additive effect of alendronate.

  • Remodeling rate (activation frequency) is reduced by therapy (initially in the range of 30–60% reduction from basal values). This is because bisphosphonates are well known to suppress activation and because the observed declines in bone biomarker values in these patients suggest a reduction on the order of about 50%.

No constraints were placed on treatment-induced change in osteoblast formation time nor on tissue level bone remodeling balance. Except for starting bone mass, which was fixed at 70% of YAN throughout, the other variables were allowed to vary around their initial values as might be needed to fit the data, and, indeed ultimately (as will be seen) to exceed the initial constraints.

RESULTS

Illustrative simulations

Figure 1 presents four typical results of the simulation process. The corresponding values for remodeling variables producing the simulations in all the panels of Fig. 1 are provided in Table 1. In each instance the measured data points (see above) are represented by symbols with error bars (±2 SEM) and the simulation by a solid line. Goodness of fit is evaluated as described above, i.e., by how closely the line approximates the data. It is important, however, to recall that in this case the line is not generated by the data; it is purely a simulation derived from a priori settings of the parameters of the model. The data points are included with each simulation simply as a referent (and hence a visual check on the prediction of the model). When at any time point the line falls outside the range of the error bars, the prediction is not compatible with the reality.

Table Table 1. Remodeling Variables in Four Illustrative Simulations in Which Treatment Does Not Alter Osteoblast Work
original image
Figure FIG. 1.

Four simulations of lumbar spine BMD change with time during treatment with a remodeling suppressor. Data points (±2 SEM) are from the published trial of Liberman et al.2 Starting bone mass for all simulations was set at 70% YAN. Simulations presume no treatment effect on osteoblast work cycle. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

Thus, for Fig. 1A, the line predicts what the bone density curve would look like if (1) basal BMD were 70% YAN, (2) basal remodeling were 136% YAN, (3) basal remodeling period were 40 weeks, (4) osteoblast formation time were not altered by treatment, (5) bone balance under the new treatment steady state were +1.3%/year, and (6) the treatment suppressed remodeling activation by 34%. The sensitivity of the model to choice of parameter values and the evaluation of fit can be illustrated by comparison of Figs. 1A and 1B. Both have the same value for the size of the remodeling transient. As is visually evident, the fit to the actual data is fairly good for both, especially from 12 to 36 months; however, at 3 and 6 months, the fit is less good for Fig. 1A than for Fig. 1B. In fact, the 3-month prediction in Fig. 1A is outside the confidence interval for the mean of the actual data. The simulation of Fig. 1B, by contrast, falls almost exactly on the mean data values, both early and late. The explanation for the different fits, despite identical values for the transient, is found in the differing period lengths. The longer period of Fig. 1B (60 weeks instead of 40) means simply that the remodeling space fills up more slowly. That is precisely what the discrepancy in Fig. 1A shows: slower actual filling of the remodeling space than would be produced with a remodeling period of 40 weeks.

Similarly, Figs. 1C and 1D illustrate attempts to fit the data with larger transients and smaller steady-state rates of gain, but maintaining about the same degree of remodeling suppression. Neither produces as good a fit as shown in Fig. 1B.

Prolongation of the osteoblast work cycle

Fig. 1B shows a near perfect fit of the model to the actual data, without postulating any treatment-induced prolongation of the osteoblast work cycle. Because such a fit requires positive bone balance under treatment, it seemed important to determine to what extent some degree of prolongation of osteoblast work might have been equally compatible with the data, i.e., a scenario in which the remodeling space was larger but filled more slowly over a longer period of time. Accordingly, further simulations were run incorporating such prolongation. Figure 2 shows four such simulations; the corresponding values for the remodeling variables are given in Table 2. Figures 2A and 2B represent attempts to explain essentially all of the BMD increase, even at 36 months, on the basis of a remodeling transient. They show that it is not possible to produce a close fit with a transient alone, especially in view of the apparently linear character of the data after 12 months (see below). However, Fig. 2C plots a simulation incorporating a much more modest treatment-induced prolongation of the osteoblast work cycle, together with a transient of 5.1% of starting BMD (about 20% larger than the transient in Fig. 1B). As can be seen, the fit is quite good. Figure 2D is an example of a less good fit, also incorporating lengthening of the osteoblast work cycle. (In both Figs. 2C and 2D, the basal period was 50 weeks and the basal osteoblast formation time, accordingly, 27 weeks.) Simulations involving greater degrees of period lengthening (e.g., Figs. 2A and 2B) resulted in deterioration of the fit.

Table Table 2. Remodeling Variables in Four Illustrative Simulations in Which Treatment Alters Osteoblast Work
original image
Figure FIG. 2.

Four simulations, as for Fig. 1, except all involve varying degrees of slowing of osteoblast work. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

Evaluation of the steady state

As is generally understood, the change in bone mass produced by the remodeling transient at any given time after exhibition of a remodeling suppressor is a function of the relative proportion of remodeling sites activated before and after the start of therapy. Clearly, soon after starting therapy that proportion will be dominated by sites activated pretreatment. But the later the measurement is made after starting treatment, the higher will be the proportion of new sites to still active old sites. A new steady state is reached when there are no more actively mineralizing sites that had been started pretherapy. This changing proportion of new and old sites with time means that the approach to the new steady state for BMD will be curvilinear. This curvature is apparent both in the first year data themselves and in the simulations of Figs. 1 and 2. The simulations of Figs. 2A and 2B, in which virtually the entire change is attributed to a transient, show curvature throughout the period of observation, as theory predicts they must.

It seemed important, therefore, as a collateral check on the prediction of positive bone balance produced by simulations 1B and 2C, to examine the 12− to 36-month data generated in the alendronate trial for curvature, separately from the early, obviously curving data. Figure 3 presents the actual patient BMD values at each time point from 12 to 36 months, with the least squares regression line fitting them. The slope of this line is +1.46%/year (P < 0.001), while confining the fit to the final 12-month period of observation produces a slightly smaller slope (+1.14%/year). Examination of the distribution of the individual data values around the fit (Fig. 3) revealed no hint of deviation from the linear assumption. The means of the measured values for each time point fell almost exactly on the regression line drawn through all of the data.

Figure FIG. 3.

Plot of the actual patient BMD values from the trial of Liberman et al.,2 together with their means and the least squares linear regression line through the data. n for each time point ranges from 155 to 157. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

Since the model, mimicking the exponential process of secondary mineralization, cannot produce a straight line prior to reaching a new steady state (see Fig. 2, especially parts A and B), the actual data thus indicate that the transient had been virtually completed in these subjects by perhaps as early as the 1-year measurement point, and certainly no later than the 2-year point. This conclusion supports the prediction from the simulations that (1) whatever its basis, the new steady-state bone balance was actually positive, with a bone gain of at least +1.1%/year (and more likely closer to +1.4%/year); and (2) prolongation of osteoblast work time by treatment, if it had occurred, would have elongated the total remodeling period by no more than a factor of ∼1.2 (the situation that produced Fig. 2C).

While the fit of Fig. 1B seems marginally better than that of Fig. 2C, there is scant basis for making a distinction between the two fits. Thus, we take it that any of the cognate parameter sets within the envelope defined by these two simulations would be equally plausible representations of the biology underlining the clinical data. The corresponding sets of ranges for the remodeling variables are presented in Table 3. As can be seen, for certain key variables, the range is small (e.g., remodeling suppression, 36% for both; transient, 4.3–5.1%; steady-state bone balance, +1.1 to +1.4%/year). (It should be borne in mind that choice of one value within a range from Table 3 influences the values available from the other ranges. Hence it would not be appropriate to take a set of values all from the high (or low) end of the ranges in Table 3.)

Table Table 3. Range of Remodeling Variables Producing a Good Fit to the Observed Data
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Finally, and to summarize this process graphically, we have taken the simulation in Fig. 1B and decomposed it into its components, plotting them with the actual trial data as Fig. 4. This way of presenting our results shows directly the relative contributions of the transient and the steady-state bone balance at varying times after starting therapy.

Figure FIG. 4.

Decomposition of the best fit simulation to show the portion of the change produced solely by the remodeling transient (the dashed line). The difference between the two lines reflects positive remodeling balance. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

DISCUSSION

To our knowledge, this communication represents the first attempt to wed the quantitative data from a clinical trial of a bone-active agent to the presumably underlying bone biology. The apparent goodness of fit to actual data which can be produced by current syntheses of bone remodeling is gratifying and constitutes, in a general way, a validation of those syntheses. At the same time, there emerge from this attempt some quantitative discrepancies between remodeling theory and the biomarker data that call for exploration and explanation.

Most obvious is the inability to provide a fit when both basal remodeling and remodeling suppression are set to match the size of the biomarker values, i.e., a basal level of remodeling 150% of YAN and a 50–60% suppression of activation. No simulation could accommodate changes this large and combine them with the basal lengthening of the remodeling period thought to be characteristic of osteoporosis. As noted in the Appendix, the estimated YAN remodeling space in L2–L4 amounts to slightly more than 3 g of bone mineral. If basal remodeling is elevated to 50% above YAN, and if the remodeling period in individuals with osteoporosis is lengthened from 40 to 60 weeks, the remodeling space rises to over 6.5 g; and if L2–L4 BMC starts at 70% YAN, such a remodeling space amounts to about 20% of the starting bone mass. Accordingly, 50% suppression of remodeling must produce a transient amounting to about a 10% gain in bone mineral. But, as the simulations show, this is at least twice what would be a plausible size for the transient with a parameter set that will fit the actual data.

There are several possible explanations for this discordance. Conceivably, histomorphometric and kinetic measurements, which provide the basis for estimates of the size of the remodeling space, may be high. The error would have to be at least a factor of two, which, as the Appendix points out, seems implausible. Alternately, the degree of change in the biomarkers is quantitatively greater than the underlying degree of change in cell-level remodeling, i.e., either the slope of one on the other is not unity, or the relationship is nonlinear. In support of the second alternative is the finding from total body kinetics6 that remodeling rate rises across menopause by about 15–20%, not by the 50% figure suggested from the biomarkers.

As further explanation for some portion of the discrepancy, it may be noted that biomarker levels result from total skeletal remodeling activity. There is abundant reason to conclude that locally the skeleton may be heterogeneous in its response to therapy. This is clearly the case in the disproportionately large responses of the spine to fluoride8 and to PTH.9 Since this paper has focused exclusively on the lumbar spine, this fact alone may explain some quantitative discrepancies between total body biomarker changes and spine BMD changes. Also, the same skeletal heterogeneity would apply to histomorphometric estimates of remodeling changes produced by any therapy. Moreover, histomorphometric estimates of period length and of basal activation rate are based on data from transiliac bone biopsies. Eventov et al.10 have shown large differences in static remodeling parameters between hip and iliac crest measured simultaneously in the same patients.

Since the linearity of biomarker values with respect to remodeling activity has not been established, and since regional skeletal heterogeneity may itself provide an entirely adequate explanation for the discrepancies described in this paper, further speculation would not be useful. However, by the same token, it seems important to stress that biomarker changes, while directionally accurate, may not bear a one-to-one relationship to the actual changes in remodeling that produce them. As this communication shows, this seems to be particularly true for the skeletal site that is often the prime focus of clinical interest in patients with osteoporosis, i.e., the spine.

Finally, this analysis demonstrates that some degree of positive bone balance is compatible with partial remodeling suppression. Histomorphometric and kinetic estimates of skeletal replacement are in the range of 8–12%/year for the total skeleton. The computer algorithm1 assumes a total skeletal value close to 9%/year and a rate for the heavily trabecular bone of the spine three times the total skeletal value (or a replacement rate of ∼27%/year). A degree of remodeling suppression as large as 36%, such as was found in the simulations that closely fit data of the clinical trial, means a reduction in lumbar spine replacement to about 17%/year. A relative imbalance between formation and resorption of less than 10% of that new rate (i.e., ≤1.7%/year) would be more than sufficient to produce a positive (or negative) remodeling balance of the magnitude found in these simulations and observed in the clinical data for years two and three. Positive remodeling balance under therapy might even be anticipated, inasmuch as any factor that blocks osteoclastic resorption evokes homeostatic adjustments that produce a shift toward positive total body calcium balance. (These would be predicted to include a small rise in 24-h integrated parathyroid hormone levels, with corresponding increases in synthesis of 1,25(OH)2D and in renal tubular calcium reabsorption.) Even if the current estimates of the level of remodeling in the spine are much too high—if the remodeling rate at the spine were in fact no different from that in the predominantly cortical total skeleton—there would still be room for positive remodeling balance at the levels of remodeling suppression predicted by these simulations.

Acknowledgements

This work was supported by an educational grant from Merck and Company.

APPENDIX

Remodeling glossary

activation frequency/rate

The number of new remodeling sites started per unit time. This is estimated to be on the order of 10–20,000/day. Since this number is not directly measurable, and since it is not needed in the simulation, activation is expressed as either percent suppression from baseline or percent of YAN, as the context requires.

period (sigma)

The time (in weeks) from the beginning removal of mineral by resorption at a newly activated site to the completion of mineralization of newly deposited bone at that same locus. The term is also used for the duration of the sequential components of the remodeling process, i.e., osteoclastic resorption, osteoblastic formation, and secondary mineralization (i.e., resorption period, formation period, etc.).

bone remodeling balance

The difference between the quantity of old bone removed and the quantity of new bone deposited.

remodeling space

The volume of bone involved in remodeling at any given time, i.e., the sum of resorption cavities plus incompletely mineralized new bone. Because densitometry detects this space as a mineral deficit, in this paper the remodeling space is expressed in equivalent mass units (i.e., grams of bone mineral).

Mathematics underlying the model

The size of the bone remodeling transient, in grams of bone mineral, is given by the product of the activation suppression factor and the size of the remodeling space, i.e.,

equation image(1)

where

Tm is the transient, in grams of bone mineral,

RS is the remodeling space, also in grams of bone mineral, and

S is the activation suppression factor.

Note that, while mass units might seem an odd way to express a space (RS), the space is occupied by a mineral-matrix composite, and mineral mass is what is measured clinically by densitometry.

The remodeling space itself is a direct function of the activation frequency, the remodeling period, and the mineral deficit during remodeling, i.e.,

equation image(2)

where

A is the activation frequency,

P is the period (in weeks), and

MD is the average mineral deficit of each locus, per unit volume, over its remodeling lifetime, multiplied by the total number of sites.

If we write Eq. 2 twice, once for young adult normals, and once for our group of study subjects, and if we divide one by the other, solving for the size of the remodeling space in our study subjects (RSs), we get:

equation image(3)

where the subscripts identify values for the study subjects (s) and the young adult normals (y).

Because the normal filling of the remodeling space is not a linear process, the average mineral deficit per locus may change slightly as the period changes. However, the quotient, MDs/MDy is always very close to 1.0, and for purposes of this analysis it will be convenient to consider the mineral deficit invariant across period changes.11 Equation 3 then reduces to:

equation image(4)

The quotients As/Ay and Ps/Py can be directly evaluated as relative activation rates and remodeling periods, respectively, that is the values for these parameters in our study subjects expressed as fractions of the young adult reference normal, and will be further abbreviated here as Aq and Pq. Combining Eqs. 4 and 1 yields a multivariate definition of the transient, Tm, as follows:

equation image(5)

22

The best current estimate of the size of the remodeling space in L2–L4 in young adult normals (RSy) is 3.15 g (i.e., with a nominal total of 48 g, the space amounts to 6.6% of the measurable bone present in that region). This is admittedly only an estimate. It is based on fairly firm values for the total skeleton, derived from total body calcium kinetics, but requires partition of skeletal remodeling between cortical and trabecular bone and, specifically, partition into the region of interest (L2–L4). Nevertheless, the differences in response to alendronate between total body and spine in the paper by Liberman et al.2 indicate that the partition estimate must be approximately correct for the subjects in this study. Indeed, the value for the remodeling space in L2–L4 (3.15 g) is based on a spine-to-total-body remodeling ratio of 3:1. As noted above, and as reported by Liberman et al.2, the alendronate response data suggest an even higher ratio, i.e., in the range of 4:1–5:1. Thus, the estimate of 3.15 g is conservatively low.

The estimate of RSy is also strongly dependent on the estimated period length in normal adults. Forty weeks is the default, or reference, value used in the computer algorithm (3 weeks resorption; 17 weeks formation; 20 weeks secondary mineralization). The 17 week value for formation itself assumes a modest amount of osteoblast “off-time” (up to 40% of the 17 weeks allowed for formation in cancellous bone). Given reasonably firm estimates for the duration of secondary mineralization12 and for wall thickness and double-label mineral appositional rate,7 it is unlikely that the total period up to completion of mineralization could be less than 30–35 weeks.

Since the remodeling transient, as elucidated in the body of this paper, amounts to ∼4–6% of the starting bone mass, and since we can take starting mass as 70% of young adult normal, the transient (using 4.2% from the simulation in Fig. 1B), expressed in g bone mineral (Tm) is 0.042 * 0.7 * 48 g, or 1.411 g. Substituting 3.15 g for RSy and 1.411 g for Tm in Eq. 5 and rearranging yields

equation image(6)

This formulation, as is apparent, is an equation in three variables. It is readily visualizable graphically and permits unambiguous exploration of the set of all plausible values for each of the three such that, together, they will produce a transient of 1.411 g. Figure A1 plots the values which solve Eq. 6, as a sheet or surface in a three-dimensional space in which the axes are remodeling period, remodeling suppression, and basal remodeling rate. For ease of interpretation, the variable Aq (or basal remodeling) in Fig. 1 has been expressed as percent of young adult normal, and Pq has been converted to absolute durations, in weeks. (Since the YAN reference value for p = 40 weeks, the 40-week line through the sheet would be equivalent to Pq ≡ 1, or 100% YAN).

As is immediately apparent from inspection of Fig. A1, there is only a limited set of values for P and S which would produce a transient of 1.411 g from a skeletal region with a starting remodeling rate above YAN. All the available value sets are in the rear corner of Fig. 1. The higher the basal remodeling rate, the lower must be the degree of activation suppression and the shorter the period. In fact, at the reference YAN period of 40 weeks, all available basal remodeling values fall below 140% YAN.

Figure FIG. A1.

Solution sheet for Eq. 6 using a value of 3.15 g for the reference young adult normal remodeling space. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

Figure FIG. A2.

Solution sheet for Eq. 6 using a value of 1.575 g for the reference young adult normal remodeling space. (Copyright Robert P. Heaney, 1996. Reproduced with permission.)

As Eqs. 4–6 show, the transient is a linear function of its component factors. Thus, if the YAN value for remodeling space is not correct (for example, if the period is shorter than has been estimated to date, or the basal level of remodeling is lower than had been thought), the solution sheet to Eq. 6 will be displaced up (or down) in a linear fashion by the size of the error factor. Thus, if the basal remodeling space were 1.575 g (only half as large as the estimate used for Fig. 4), Eq. 6 would be set equal to 0.9174 rather than 0.4587. The resulting solution set is presented as Fig. A2. Now, a much larger region of the solution sheet is compatible with basal remodeling above YAN. For example, 50–60% suppression of remodeling, and a period in the range of 60 weeks, can be seen to provide an appropriate solution to Eq. 6.

However, given the body of data available from both total body calcium kinetics and histomorphometry, it seems very unlikely that the current estimates of the YAN remodeling space can be erroneously high by a factor of 2×, as required for Fig. A2. As already noted, the 40-week period value cannot be high by more than a factor of 1.2–1.3, and the daily rate of mineral removal and replacement from the total skeleton is fairly firmly established from whole body calcium kinetic studies. Also, as already noted, the estimate of 3.15 g is based on a spine:TBBM remodeling ratio of 3:1, while the clinical trial data suggest a value which, if anything, might be even higher. Thus, 3.15 g, instead of being spuriously high, could even be low. Furthermore, histomorphometric measurements of the remodeling period in postmenopausal women often yield estimates that yield total remodeling periods substantially greater than even the 60–70 week range that was available for solution in Fig. A2. These histomorphometric estimates, however, are derived from the iliac crest and are, additionally very tentative, since they involve large increases in osteoblast off-time which, while a firm enough concept, is extremely difficult to measure accurately.

Footnotes

  1. 1

    For uniform slowing of remineralization, the average MD does not change at all. For the more likely possibility that slowing is predominantly in the osteoblast work cycle, average MD rises. However, a treatment-induced halving of appositional rate increases the quotient by only 20% and a 25% decrease in appositional rate increases the quotient by just 12%. Thus, its value, rather than actually being invariant at 1.0, is likely to lie between 1.0 and 1.2. This small error has no effect on the simulations in the body of the paper, since no assumptions are made about the MD in the computer program. It is in the exhaustive search for solution sets, which follows, where it is useful to simplify Eq. 3. The consequence of this simplification is that the transient used in exploring the solution sets in what follows could be from 0 to 20% larger than would be predicted under an assumption of an invariant remodeling mineral deficit. This error is small enough so as not to affect the conclusions that follow.

  2. 2

    It is interesting to note that this equation expresses the transient in these osteoporotic subjects—or, indeed, in any study group—in terms of the young adult normal remodeling space as modified, not by differences in bone mass or density, but by changes in activation frequency and remodeling period. This clearly reflects the independence of the remodeling transient of both bone mass and bone remodeling balance. Moreover, the expression of the remodeling parameters for the study group in terms of the young adult normal values is consistent with the consensus practice in the field, as reflected, for example, in the WHO definitional standards for osteoporosis.11

Ancillary