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.

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.).

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

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.,

- (1)

*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.,

- (2)

*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 (*RS*_{s}), we get:

- (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, *MD*_{s}/*MD*_{y} 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:

- (4)

The quotients *A*_{s}/*A*_{y} and *P*_{s}/*P*_{y} 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:

- (5)

12

The best current estimate of the size of the remodeling space in L2–L4 in young adult normals (*RS*_{y}) 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 *RS*_{y} 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 mineralization^{12} 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 *RS*_{y} and 1.411 g for *Tm* in Eq. 5 and rearranging yields

- (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.

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.