Dr Phipps is a full-time employee of Procter & Gamble Pharmaceuticals and a member of the Alliance for Better Bone Health (Procter & Gamble Pharmaceuticals and sanofi-aventis). Funding for the study on the BMDD with risedronate treatment was provided by the Alliance for Better Bone Health. All other authors state that they have no conflicts of interest.
Effect of Temporal Changes in Bone Turnover on the Bone Mineralization Density Distribution: A Computer Simulation Study
Article first published online: 28 JUL 2008
Copyright © 2008 ASBMR
Journal of Bone and Mineral Research
Volume 23, Issue 12, pages 1905–1914, December 2008
How to Cite
Ruffoni, D., Fratzl, P., Roschger, P., Phipps, R., Klaushofer, K. and Weinkamer, R. (2008), Effect of Temporal Changes in Bone Turnover on the Bone Mineralization Density Distribution: A Computer Simulation Study. J Bone Miner Res, 23: 1905–1914. doi: 10.1359/jbmr.080711
- Issue published online: 4 DEC 2009
- Article first published online: 28 JUL 2008
- Manuscript Accepted: 25 JUL 2008
- Manuscript Revised: 15 MAY 2008
- Manuscript Received: 8 JAN 2008
- bone mineralization density distribution;
- bone turnover;
- mathematical model;
- risedronate treatment
The heterogeneous distribution of mineral content in trabecular bone reflects the continuous renewal of bone material in bone remodeling and the subsequent increase in mineral content in the newly formed bone packets. The bone mineralization density distribution (BMDD) is typically used to describe this nonuniform mineral content of the bone matrix. Our mathematical model describes changes of the BMDD of trabecular bone as a function of bone resorption and deposition rates and the mineralization kinetics in a newly formed bone packet. Input parameters used in the simulations were taken from experimental studies. The simulations of the time evolution of the BMDD after increase in bone turnover (perimenopausal period) resulted in a shift of the BMDD toward lower values of the mineral content. Transiently, there was a broadening of the BMDD configuration partly showing two peaks, which points to a strongly heterogeneous distribution of the mineral. Conversely, when the remodeling rate was reduced (antiresorptive therapy), the BMDD shifted toward higher values of the mineral content. There was a transient narrowing of the distribution before broadening again to reach the new steady state. Results from this latter simulation are in good agreement with measurements of the BMDD of patients after 3 and 5 yr of treatment with risedronate. Based on available experimental data on bone remodeling, this model gives reliable predictions of changes in BMDD, an important factor of bone material quality. With the availability of medications with a known effect on bone turnover, this knowledge opens the possibility for therapeutic manipulation of the BMDD.
BMD as evaluated by means of X-ray densitometry is important in the diagnosis of osteoporosis. In recent years, however, it has become apparent that “bone quality” needs to be taken into account for a better estimation of fracture risk. The generic concept of “bone quality” includes all the aspects of bone structure beyond BMD that influence bone strength and toughness.[3, 4] Whereas the detailed relationship between bone structure and mechanical properties is only partially elucidated,[5, 6] it is clear that the architecture of trabecular bone and the material properties of the bone matrix play an essential role. Both architecture and material properties are highly dynamic because of the continuous remodeling of bone, with osteoclasts and osteoblasts resorbing and depositing bone, respectively, in a coordinated fashion. The general concept is that a basic multicellular unit (BMU) starts locally with the resorption of a small bone volume and, after a quiescent period, adds new osteoid.[9, 10]
In addition, the newly deposited osteoid is unmineralized, and the mineral content increases over time at first rapidly (primary mineralization) and then more slowly (secondary mineralization).[8, 11] This means that the remodeling process leads to an inhomogeneous mineralization pattern where older bone packets (or bone structural units [BSUs]) have a high mineral content, whereas younger ones are less mineralized.[12, 13] The mineral content is known to strongly influence the mechanical behavior of bone tissue,[14-16] and so the spatial and temporal variations of mineral content caused by the remodeling process will lead to variations in the mechanical properties of bone material.
Trabecular bone architecture can, in principle, be studied by microtomographic methods, and there are also techniques to evaluate the mineral distribution within the bone material. One well-established technique is the measurement of bone mineralization density distribution (BMDD), which is based on scanning electron microscopic investigation of bone biopsies. The BMDD is a histogram (Fig. 1) expressing the probability that a given BSU has a certain mineral content. Typically, the BMDD is a peaked curve, meaning that most BSUs have a characteristic mineral content (as described by the peak position) but that there is a certain variation around this mean. A large width of the BMDD curve shows a large variation in the mineral content for the BSUs within the sample. The BMDD shows how the heterogeneity in mineral content changes in osteoporotic bone compared with healthy bone (Fig. 1), as well as after osteoporosis treatment, but it is not yet fully clear how changes in the BMDD can be interpreted in terms of the dynamics of bone remodeling and mineralization, as discussed above.
The goal of this study was to develop a theoretical framework that relates the kinetics of bone remodeling and matrix mineralization to the shape of the BMDD curve and to its changes with age and with treatment. Such a framework will eventually allow simulating the effect of bone turnover and of osteoporosis treatments on the mineral distribution in the bone material and, therefore, on its mechanical properties. A first step was already taken in previous work, where we analyzed how the balance between bone turnover and mineralization kinetics could lead to a steady state BMDD, as observed in normal individuals. Here we extend the analysis to a full kinetic treatment with the aim of describing the effect of a change in bone turnover caused by a change in the origination frequency of BMUs, as found in postmenopausal osteoporosis,[23, 24] and with antiresorptive treatment with bisphophonates.[25, 26] Transients in the BMDD, occurring before new steady-state configurations are obtained, are analyzed.
MATERIALS AND METHODS
To understand the model, it is helpful to first look at the effects of the processes of remodeling and mineralization on the BMDD separately. In Fig. 2A, a qBEI measurement of a trabecula was schematized to show more clearly the patchwork-like appearance of trabecular bone. In a qBEI measurement, different grayscales correspond to different calcium contents, and these differences in mineral content are the result of different ages of the bone packets. If only mineralization occurs in the absence of remodeling, the mineral content in each of the packets would increase. The mineralization law that describes the temporal increase of the mineral content as a function of its present mineral content is assumed to be the same for each bone packet (Fig. 1B). In the schematic qBEI measurement (Fig. 2B), the increase in the mineral content would be seen as a whitening of each of the bone packets. In the BMDD, the sole action of mineralization would lead to a shift of the peak-shaped distribution curve to the right (i.e., toward higher values of the mineral content; Fig. 2C). Because it is known that mineralization occurs quickly during primary mineralization and then more slowly during secondary mineralization,[8, 11] a change in shape of the BMDD is expected. This slowing down of the increase in the mineral content leads to a narrowing of the peak in the BMDD diagram (Fig. 2C). If now only remodeling occurs without mineralization of bone matrix, the schematic qBEI measurement shows resorption pits formed by osteoclasts that are refilled later on in the remodeling cycle of the BMU[9, 10] by unmineralized new bone matrix (black in Fig. 2E). Whereas the bone resorption lowers the BMDD (Fig. 2F), the unmineralized new bone deposited by osteoblasts appears at the very left of the BMDD (black bar in Fig. 2F).
Considering the two processes together as they appear in the BMDD diagram, it is mineralization that shifts the BMDD peak to higher values of mineral content on the right. Whereas the BMDD peak moves to the right, bone resorption by osteoclasts lowers the peak, reflecting the reduction of bone volume of higher mineral content. New bone is fed back from the very left in the BMDD diagram because of deposition of unmineralized bone by osteoblasts. The partial differential equation describing the temporal changes of the BMDD is given in the Appendix (Eqs. A6 and A7), and more details have previously been published. The experimental observation that healthy humans show an age-independent BMDD of trabecular bone is interpreted as a steady-state configuration of the BMDD.
Input parameters of the model
To simulate the time evolution of the mineralization distribution as a function of different bone turnover, the following input parameters, based on measurement of bone turnover and heterogeneity of the mineral content, are needed.
The mineralization law, which describes the increase in mineral content with time, fully characterizes the mineralization process in the model. Based on the analysis of 52 BMDDs of healthy adults and the definition of a reference BMDD (Fig. 1A), a time-independent analysis of Eqs. A6 and A7 allowed the determination of a corresponding mineralization law for healthy humans (Fig. 1B). The resulting mineralization law is in full agreement with experimental data showing a large and rapid increase in mineral content at early times followed by a significant reduction during secondary mineralization. Whereas routine histomorphometry with fluorescent labeling only allows determination of mineralization between two very adjacent time points[19, 27] (normally 1- to 2-wk interval), the obtained mineralization law describes the increase in mineral content over several years (Fig. 1B). In all the simulations reported here, we used this mineralization law determined for healthy adults. An assumption for the simulations was that changes in turnover (at menopause or with bisphosphonate treatment) had no effect on the rate of mineralization in newly formed bone. There are no data to refute this assumption.
Initial configuration of the BMDD:
For simulating increased turnover after menopause, the initial configuration chosen was the reference BMDD for healthy adults (Fig. 1A). In simulations of antiresorptive therapy, the starting configuration was a BMDD characteristic for women with postmenopausal osteoporosis at the time of initiation of treatment. This configuration was obtained by averaging five BMDDs obtained from iliac crest biopsies from osteoporotic women with at least one vertebral fracture (Fig. 1A).
Remodeling and changes in turnover are described by the resorbed and deposited bone volume per unit time. These remodeling rates are closely connected to changes in the origination frequency of new BMUs (Eqs. A1 and A2 in the Appendix). Studies have shown an increase in the origination frequency of BMUs at the initiation of the perimenopausal period,[24, 28] whereas administration of bisphophonates reduces the BMU origination frequency. In addition, the time course of the resorbed bone volume differs from the deposited one because of the delayed bone formation in an active BMU (Eq. A2) and an imbalance between the resorbed and the deposited bone volume of a single BMU (Eq. A3).
Experimental data about the change in turnover after menopause shows wide variation when measured with biochemical markers[29-31] and standard bone histomorphometry,[24-26, 32] suggesting an increase of roughly 40–100%. Loss in bone volume fraction, BV/TV, in the lumbar spine after menopause can be reasonably described by a linear decrease of 6.4% per decade.
After administration of standard clinical doses of risedronate, turnover is decreased 50–60% with respect to baseline, where changes are seen as early as 3 mo (with N-telopeptide cross-links [NTX]) and are sustained over 3 and 5 yr of treatment (shown by markers and dynamic histomorphometry).[34, 35] Data of the volume fraction of low mineralized bone (CaLow) as obtained by qBEI or ratios of low- to high-mineralized bone volume fractions (BMR-V) as measured by SRμ-CT show a reduction of 38% in CaLow and 70% in BMR-V after 3 yr of risedronate treatment.[21, 36, 37] Measurements of BV/TV showed a nonsignificant increase.
Available experimental data indicate that both at the initiation of the perimenopausal period and after risedronate administration turnover changes occur faster initially but are followed by a subsequent slowing down. The change in bone turnover (described by changing the origination frequency of new BMUs) was therefore modeled by an exponential function (see Eq. A5), which is characterized only by two parameters. The first dimensionless parameter denotes how much the origination frequency is changed with respect to its initial value. The second parameter, which is a characteristic time, describes how fast this change occurs. To gain insight of how these two parameters influence the time evolution of the BMDD, an intensive parameter study has been performed. Results reported here were performed in the case of menopause with increases of 125%, 170%, and 250% of the BMU origination frequency compared with the healthy premenopausal reference values (which corresponds to a turnover time of 5 yr for healthy adults) (Fig. 4A). Time constants for the change of 4 mo, 1 yr, and 4 yr were chosen (Fig. 4E). A constant negative BMU balance was assumed, giving rise to a bone volume loss of 0.64%/yr in the long term. The effect of decreased turnover with risedronate administration in postmenopausal osteoporosis was simulated by reducing the BMU origination frequency by 13%, 23%, and 35% from its baseline value, which was assumed to be 170% higher than the value for healthy adults (Fig. 7A). To mimic the temporal efficacy of the drug, time constants of this change of 1 mo, 4 mo, and 1 yr were used (Fig. 7E).
The following information is gained from the simulations:
Time evolution of the BMDD:
The main output is a complete time evolution of the BMDD, and therefore, also a time development of all parameters characterizing the BMDD curve.CaPeak denotes the position of the peak, which corresponds to the most frequent Ca content, CaMean is the mean Ca content, and CaWidth is the peak width of the BMDD, defined as the full-width at half maximum (FWHM). In cases where the BMDD displays more than one peak, the heterogeneity of the BMDD is described by its SD, CaStd.
Increase in bone turnover at the initiation of the perimenopausal period
Figure 3 shows the time evolution of the BMDD when the origination frequency of BMUs is increased by 170% with a time constant of 1 yr (see Eq. A5). On the BMDD diagram, the loss in bone volume is reflected by a lowering of the curve with no modification in the overall shape, because changes in bone volume occur on a longer time scale than do shape changes in the BMDD. Like the measured BMDDs, the plotted BMDD also is renormalized such that the total area under the BMDD curve is constant. In agreement with the previous time-independent analysis, the main difference between the initial BMDD and its final configuration after a turnover increase is a shift of the peak toward lower values of the mineral content. The time evolution of the BMDD, however, shows that changes in the BMDD are far from being just a simple and smooth movement of the peak toward lower Ca values. Initially, the peak at ∼23 wt% Ca decreases in height. At the same time, a new, second peak appears in the BMDD diagram, which moves in from the left and can be seen as a small shoulder at the left side of the peak after 1.5 yr. For the parameters used in Fig. 3, after 3 yr, the two peaks have approximately the same height, whereas after 4 yr, the old peak is visible only as a shoulder of the new left-shifted main peak.
The temporal change of bone turnover with the values chosen as input parameters in the simulations (Figs. 4A and 4E) together with the resulting time development of three key parameters characterizing the BMDD, CaPeak, CaStd, and CaMean, are plotted in Fig. 4. The left column of the figure compares the effect of different turnover increases, whereas on the right, the effect of different rates of turnover change are shown. There is a discontinuity of CaPeak (Figs. 4B and 4F) when the new second peak becomes greater than the old peak. With larger changes in turnover, the new peak position is reached faster, also because of the fact that the new equilibrium positions for larger turnover changes are at lower Ca content. The existence of two separate peaks is reflected in transiently large values for CaStd, in particular when increases in turnover are large (Fig. 4C) and occur rapidly (Fig. 4G). With increased turnover, CaMean first decreases and then attains a new reduced constant value, and it is noteworthy that the new steady state is reached faster the larger the change in turnover (Fig. 4D). This effect can be estimated by fitting the behavior of CaMean with an exponential decay function similar to Eq. A5. The corresponding time constant for CaMean is ∼2.3 yr for a 250% increase in bone turnover, whereas a turnover increase of 125% results in a slower time constant of 4 yr. When the change in turnover occurs faster, also the equilibration of the new peak position and of CaMean occurs faster (Fig. 4H).
Prediction of BMD changes during menopause
With the increase of the origination frequency of new BMUs at menopause, the different contributions influencing the BMD[28, 39] can be analyzed separately with our model (see Appendix, Eq. A8 for the mathematical description). In Fig. 5, the time evolution of the BMD is depicted for four different scenarios (see figure caption). Assuming no quiescent period between increased bone resorption and bone deposition (long dashed line), the effect of a decrease of mean bone mineralization on the BMD can be studied. In reality, bone deposition lags behind resorption by ∼2 mo, leading to a loss in bone volume caused by an increase in remodeling space (solid line). From a total BMD reduction of 12.6% 5 yr after initiation of menopause, 5.2% is caused by a decrease in mean mineral content and 4.2% by the an increase in remodeling space. The latter occurs faster with a time constant of ∼2.1 yr compared with a time constant of 3.9 yr caused by changes in mineral content. After ∼5 yr, further reductions in BMD are caused by bone volume loss resulting from the imbalance between resorbed and deposited bone within 1 BMU.
Decrease in bone turnover after administration of risedronate
The effect of a 23% reduction in BMU origination frequency occurring exponentially with a time constant of 4 mo on the BMDD is shown in Fig. 6. In the long term, after the configuration of the BMDD is re-equilibrated, the reduction of turnover results in a new peak shifted to higher Ca content (peak marked as “final” in Fig. 6). As with increasing turnover, the transient configurations of the BMDD display interesting behavior. The transitory peaks at 1 and 3 yr have increased height and reduced width. Furthermore, during the time evolution, the position of the peak overshoots its final position, as can be seen after 5 yr (black triangles in Fig. 6).
We also analyzed how changes in the BMDD parameters depend on the specific change in turnover (Fig. 7). Again, in the top row, the different changes of the magnitude of turnover (Fig. 7A) and the velocity of turnover change (Fig. 7E) as used as simulation inputs are plotted with time. Starting from its initial position at 22.27 wt% Ca, the peak moves to the right much further than its final position (which for a decrease of 23% is located at ∼22.67 wt% Ca). The overshoot of the peak is greatest when the turnover reduction is large (Fig. 7B) and rapid (Fig. 7F). Under these conditions (i.e., with large and abrupt changes in the origination frequency of BMUs), CaWidth becomes smallest (Figs. 7C and 7G). After some initial reductions caused by an incompletely equilibrated initial condition, CaMean increases exponentially and can be fitted with an equation similar to Eq. A5 (Figs. 7D and 7H). A turnover reduction of 35% results in a time constant for attaining the new steady-state value of CaMean of 1.5 yr, whereas a turnover reduction of 13% results in a time constant for CaMean of 9 mo.
BMDD and bone turnover are two important contributors to bone quality. These two different aspects, however, are not independent, and with our model, a quantitative connection between a change in bone turnover and the temporal change in BMDD could be obtained. As expected, increased turnover resulted in a shift of the BMDD peak to lower values of mineral content because the faster turnover leaves less time for mineralization. With reduced turnover, the BMDD shifts toward higher mineral content.[19, 43] Much more surprising are the transient configurations of the BMDD shown after a turnover change, and in particular how these configurations differ in case of increased or decreased turnover. An increase in turnover causes a transient double peak of the BMDD, whereas a reduction in turnover leads temporarily to a very pronounced and narrow BMDD peak. In terms of the changing origination frequency of BMUs, these results can be interpreted as follows. With an increase in turnover, the increase in the resorbed bone volume causes a reduction of the main BMDD peak representing the older BSUs, whereas the increase in the deposited bone matrix leads to a formation of a new peak at low Ca values. The mineralization of this newly formed bone subsequently moves this new peak to higher Ca values. The old peak gradually decreases because it is not further replenished with bone packets of lower Ca content (the time to reach high Ca values being insufficient because of increased turnover). Such double peaked BMDDs have been observed in a case of osteomalacia and bone marrow edema syndrome of the femoral head, both characterized by an increase in bone turnover. The narrowing of the peak with reduced turnover can be explained on the basis of a continuation of mineralization in existing BSUs alone, which proceeds slower and slower within a single bone packet. As already noted, mineralization without remodeling would lead to a shift of the peak toward higher Ca values with progressive narrowing. The subsequent broadening of the peak is a result of the ongoing remodeling and the fact that a newly deposited bone packet now needs to travel a longer “distance” in the BMDD diagram (and therefore needs more time) to reach the new Ca peak shifted to higher Ca values.
A comparison of the simulation results with experimentally obtained BMDDs from triple iliac crest biopsies (i.e., from the same subjects) at baseline and after 3 and 5 yr of treatment with risedronate (symbols in Fig. 7) show reasonable agreement for CaPeak and CaMean when a turnover reduction of 23% with a time constant of 4 mo is assumed. For CaWidth, the simulation correctly predicts that the peak will become narrower but overestimates how sharp the peak becomes compared with the experimental results. It is a general feature of our computational results that simulated BMDDs tend to display narrower peaks than ones measured experimentally. One reason could be that the experimental distribution is always affected by a peak broadening because of counting statistics of signal generation and detection. Furthermore, there could be some biological variability in bone turnover reduction over the long observation period.
To assess the implications of our results concerning antiresorptive therapy, it is important to consider the time needed to reach a new equilibration configuration after a change in turnover. The simulations show that as a general rule the BMDD equilibrates faster after an increase than after a reduction in turnover for an otherwise equal turnover change. Another important difference between increase and decrease in turnover is the effect of the magnitude of turnover change on the time to reach a new BMDD equilibrium. With increased turnover, BMDD reaches a new equilibrium faster the larger the increase in turnover. In contrast, with decreased turnover BMDD reaches a new equilibrium slower the larger the decrease in turnover.
Knowing the time variation of the bone volume fraction, BV/TV, our mathematical framework also allows insights on the sigmoidal behavior of BMD after turnover change. Our work is similar to previous simulational work,[28, 40, 47-49] but with an important addition in the use of an experimentally based mineralization law to describe the mineralization kinetics. The initial large reduction in the BMD seen in the perimenopausal years has two roughly equal contributing factors that impact BMD at different rates. The initial decrease in BMD is caused by the increased number of new BMUs, which increase the amount of resorbed bone. This increase in remodeling space leads to a reduction in bone volume. A slower (by ∼45%) decrease in BMD results from the attainment of the new steady state of the mean mineralization of the bone matrix. Only in the long term do contributions caused by the imbalance between resorbed and deposited volume of bone in a BMU become predominant.
To conclude, this model approach has universal applicability. Whereas here we studied the effect of risedronate on the BMDD, all drugs that have effects on turnover but no significant direct effect on bone mineralization can be studied. Similarly, our studies focused on human trabecular bone, but the model can also be applied to cortical bone and to animal bone to study effects of turnover on mineralization. The demanding task to perform clinical studies of bone medications over longer and longer time periods emphasizes the importance of modeling. To have a tool to rapidly assess drug effects on bone mineralization would be valuable for future development of therapeutic strategies. Once the effect of a medication on bone turnover is well characterized, the model allows the manipulation of the BMDD toward a designated target BMDD. Moreover, to have new techniques to interpret experimentally measured BMDDs will be helpful in diagnosis of bone diseases.
- 1WHO Study Group 1994 Assessment of Fracture Risk and Its Application to Screening for Postmenopausal Osteoporosis. World Health Organization, Geneva, Switzerland.
- 451999 Response to the letter to the editor by E. G. Vajda and J. G. Skedros. Bone 24: 620–621., , ,
- 542002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK.
The relation between the origination frequency of new BMUs, Or.f, and the resorbed and deposited bone volume per time, jRs and jF, taking tissue volume as referent is given by[40, 48, 49, 51, 52]:
where RP, QP, and FP denote the resorption period, quiescent period, and formation period, respectively. Values for the different periods were 42 days for RP, 24 days for QP, and 130 days for FP. Note that the different integration limits in time for resorption and deposition take into account the quiescent period between resorption and deposition. This leads to a change in remodeling space once the origination frequency of new BMUs changes. The bone volume resorbed and formed by a BMU, BV.Rs.BMU and BV.F.BMU, respectively, may be different, resulting in a BMU imbalance,[40, 48, 49, 51, 52]
The change of the total bone volume is simply the difference between deposited and resorbed bone volume,
After the initiation of the perimenopausal period or administration of bisphosphonates, the origination frequency of new BMUs is chosen to increase or decrease in an exponential manner:
where Or.f−i denotes the initial and Or.f−f denotes the final origination frequency of new BMUs. A potential coupling between the origination frequency and the available bone surface could be considered by modifying Eq. A5. The temporal change in bone turnover is characterized by two parameters, Or.f−f/Or.f−i, which describe the amount of turnover change, and the time constant, tEFF, which characterizes how fast this change occurs. The corresponding half-life is equal to ln 2 tEFF.
The mineralization velocity, v(c), is obtained by differentiating the mineralization law,c = m(t), i.e., v(c) = (dm)/(dt).
The output of the calculations is the time evolution of the BMDD, ρ(c,t), which is a function of time, t, and Ca content, c. In the plots of Figs. 3 and 6, the BMDD as a function of c, ρ(c), is plotted for different times.
The changes of the BMDD can be described by a partial differential equation of the following form (derivation of the equation and more details are shown elsewhere:
along with the boundary condition at c = 0:
Slowing down of mineralization from primary to secondary mineralization can render the first term of Eq. A6 positive. The last term of Eq. A6 ensures that bone is resorbed independently of the Ca content. The time evolution of the BMDD is obtained by solving Eqs. A6 and A7 using finite volume methods to avoid numerical instabilities.
The time evolution of the volumetric bone mineral density (vBMD) is given by
where vBMD is defined as mineral mass, MHA (i.e., mass of hydroxyapatite [HA] per tissue volume TV; first equal sign (i) in Eq. A8), and is measured in grams per cubic centimeter. Rewriting the mineral mass as product of mineral volume, VHA, and mineral density, ρHA, establishes the equality (ii). Expanding the equation by the bone volume, BV, equality (iii) is obtained. The last equality (iv) must be read as a definition of HA, which denotes the mean mineral volume per bone volume and which is the parameter describing BMC entering in the equation for the BMD. This parameter can be obtained from the BMDD, ρ(c,t), by first expressing c in terms of the mineral instead of the Ca content. Second, c has to be expressed in terms of volume and not of weight percent.HA is the mean value of this distribution function rewritten in terms of mineral volume and time.