A graph consists of a set of objects called Description vertices and a list of pairs of vertices, called edges . tures, with vertex A represented by a dotThe edge joining A to A is called a loop ,labelled A and each edge AB representedand the graph is called a loop multigraph . by a curve joining A and B . A general graph is one with possible loopsdata or relationshi
Pii: s8756-3282(01)00613-5A Theoretical Analysis of the Contributions of Remodeling
Space, Mineralization, and Bone Balance to Changes in Bone
Mineral Density During Alendronate Treatment
1Rehabilitation Research and Development Center, VA Palo Alto Health Care System, Palo Alto, CA, USA2Biomechanical Engineering Division, Mechanical Engineering Department, and 3School of Medicine, Stanford University, Stanford, CA, USA Introduction
In patients with osteoporosis, alendronate treatment causes
an increase in bone mineral density (BMD) and a decrease in
Osteoporosis patients treated with alendronate experience in- fracture incidence. Alendronate acts by changing the bone
creased dual-energy X-ray absorptiometry (DXA)-derived areal remodeling process. Changes in bone remodeling resulting in
bone mineral density (BMD) and reductions in vertebral fracture decreased remodeling space, increased bone balance per
incidence of up to 50%.22 The increase in BMD caused by remodeling cycle, and increased mineralization (ash mass/
alendronate has been attributed to decreased bone turnover, bone mass) have all been associated with alendronate treat-
increased focal bone balance, and increased degree of mineral- ment. Understanding the relative contributions of these pa-
ization of the bone tissue (expressed here as the ash fraction, ash rameters to BMD increases could help predict the utility of
mass/bone mass). Although decreases in bone turnover have long-term (>10 years) or intermittent treatment strategies,
been described quantitatively, it is not currently known howmuch focal bone balance and ash fraction contribute to BMD as well as treatment strategies in which another pharmaceu-
increases relative to each other. Understanding the relative im- tical is administered concurrently. We have developed a
portance of these factors can be important for understanding the computer simulation of bone remodeling to compare the
changes in bone mechanical properties caused by alendronate. It contributions of focal bone balance and mineralization on
could also be important for predicting which osteoporosis treat- BMD by simulating alendronate treatment using a bone
ments would be most effective when combined with alendronate.
balance method (decreased remodeling space, increased focal
In this work we summarize the findings of alendronate studies bone balance, uniform bone mineralization) and a mineral-
and use them, in combination with a computer simulation of bone ization method (decreased remodeling space, neutral focal
remodeling, to compare the influences of focal bone balance and bone balance, varying bone mineralization). Although both
ash fraction during alendronate treatment.
methods are able to predict BMD increases caused by alen-
Bone remodeling is a focal phenomenon involving groups of dronate over short periods, our findings suggest that the
osteoclasts and osteoblasts known as basic multicellular units mineralization method may be more descriptive of long-term
(BMUs). Each BMU resorbs a small portion of bone and, soon alendronate treatment. This implies that mineralization may
after, forms new bone. Net changes in bone mass can occur when be a larger contributor to BMD changes caused by alendro-
either the focal bone balance (a measure comparing the bone nate than the focal bone balance. Based on this finding we
volume formed to that resorbed by each BMU) is modified or the offer a hypothesis to describe how remodeling space, focal
rate of bone turnover changes (the size or number of BMUs is bone balance, and mineralization each contribute to alendr-
modified). Alendronate treatment causes a significant decrease in onate-induced BMD changes. Future analyses with this
bone turnover and a possible increase in the focal bone balance.4 method could be used to identify improved dosing regimens
An increase in the focal bone balance causes more bone volume and to predict which osteoporosis treatments would best
to be formed at each remodeling site than is resorbed, increasing complement each other. (Bone 29:511–516; 2001)
the total bone mass. A decrease in bone turnover has two by Elsevier Science Inc. All rights reserved.
important consequences with regard to bone mass. First, adecrease in bone turnover causes a reduction in the remodeling Key Words: Alendronate; Osteoporosis; Computer simulation;
space and a corresponding increase in bone volume.15 The Bisphosphonates; Bone mineral density (BMD).
remodeling space represents the voids and osteoid that appeartemporarily due to the fact that resorption and osteoid formationprecede mineralized bone formation.14 When the rate of boneturnover is decreased fewer of these temporary voids are present,decreasing the size of the remodeling space but increasing thebone volume and bone mass.
Address for correspondence and reprints: Christopher J. Hernandez, The second important consequence of reduced bone turnover Ph.D., The Mt. Sinai School of Medicine, One Gustave L. Levy Place, involves the mineralization process in bone. A slower rate of Box 1188, New York, NY 10029. E-mail: [email protected]
bone turnover allows bone to accumulate more mineral before being resorbed in a succeeding remodeling cycle, thereby in- Contributions to BMD changes during alendronate treatment values are determined parametrically based on comparisons tothe results of clinical studies. The two simulation methods arecompared both by predictive ability (correlation with clinicalresults) and how they predict trends in BMD increases.
Materials and Methods
A BMU-based Model of Cellular Activity A computer simulation of BMU activity in cancellous bone isused in this study. The model describes BMU activity in anarbitrary volume of cancellous bone defined by its bone volumefraction (mineralized bone volume/bulk volume). Nine indepen-dent remodeling parameters are used to describe the progressionof BMUs and the resulting resorption and formation of bone (seeAppendix). A feedback diagram illustrates how the model deter-mines changes in bone volume fraction and osteoid volume Figure 1. Conceptual illustration of the changes in BMD caused by
fraction (Figure 2). The key remodeling parameters in the BMU
alendronate treatment. The solid line represents the total BMD change model are those related to bone balance per remodeling cycle, caused by alendronate, whereas the dashed line represented the BMD bone turnover, and bone mineralization.11 For this reason we changes attributed to changes in bone turnover. The difference between concentrate our analysis on remodeling parameters associated the two lines represents the BMD increase attributed to focal bone with focal bone balance (local resorption and formation rates), balance changes. Variation in the degree of mineralization (ash fraction) bone turnover (origination frequency), and the secondary miner- is considered to be part of the bone turnover contribution. Adapted from We express the focal bone balance as the ratio of bone volume formed to that resorbed per remodeling site (the bone creasing the average ash fraction of the bone tissue and the balance ratio, ⌬BMU.Rt). The bone balance ratio is therefore overall bone mass. Recently, it has been demonstrated that dependent on the total volume of bone resorbed and formed at animals subjected to alendronate treatment have an increased ash each site (V , V in cubic millimeters per remodeling site): fraction compared with placebo-treated controls.16 The magni-tude of ash fraction changes occurring in response to decreased bone turnover is dependent on the length of the secondarymineralization period (P), a variable that has yet to be measured If the volume resorbed (V ) does not equal the volume formed (V ) per remodeling site there is a net gain or loss of bone. In the Previously, Heaney and colleagues illustrated how alendro- present study changes in focal bone balance are modeled as nate treatment could be simulated as a decrease in bone turnover decreases in the volume resorbed per remodeling site (corre- and an increase in focal bone balance.10 Using their method of sponding to decreased osteoclast activity). An investigation of simulating alendronate treatment they were able to attribute part how these parameters are derived from histology measurements of the BMD increase to changes caused by bone turnover and part to modification of the focal bone balance (Figure 1).
Bone turnover is evaluated experimentally using a two-di- Although their model could predict the BMD changes in a mensional histologic measurement known as the activation fre- clinical study it accounted for only small variations in ash quency (the rate of appearance of a BMU in a two-dimensional fraction because it used a short (approximately 6 month) second- section, per day). In this model we use a more physiologic, ary mineralization period. It is likely that a greater portion of the three-dimensional descriptive parameter known as the origina- BMD increase that Heaney and colleagues attribute to bone tion frequency (number of new BMUs per square millimeter of turnover and part of the increase attributed to focal bone balance bone surface/day) to represent the birthrate of new BMUs on the may actually be caused by increased ash fraction. To study this cancellous bone surface. Quantitative values for the origination possibility we propose two methods of simulating alendronate frequency are calculated using a relationship between activation treatment: a bone balance method, in which the ash fraction is frequency and origination frequency that has been derived pre- maintained constant and alendronate causes a decrease in bone viously.13 The origination frequency is calculated using param- turnover and an increase in focal bone balance; and a mineral- eter values based on those measured in healthy postmenopausal ization method, in which alendronate treatment is simulated with a change in bone turnover and a resultant change in ash fraction, The rate at which mineral accumulates in newly formed bone leaving the focal bone balance unchanged.
tissue is the final parameter that influences BMD predictions. A The primary objectives of this study are to compare the bone volume of osteoid becomes mineralized bone when small min- balance method and mineralization method with regard to their eral crystals first appear within spaces between the collagen ability to predict BMD changes caused by alendronate. To meet molecules. Over the first few days of mineralization, crystals this objective we develop a model of the bone remodeling appear throughout the mineralized bone, taking space that was process utilizing quantitative histologic measurements of BMU previously occupied by water. This initial deposition of mineral activity. The model is implemented in a computer simulation occurs very quickly and is referred to as the primary mineraliza- using both the bone balance method and the mineralization tion phase. After the primary mineralization phase mineral con- method. Each of these methods utilizes one influential parameter tinues to accumulate, most likely due to further increases in the value that has not yet been measured definitively in humans (the number or size of crystals. During this secondary mineralization change in focal bone balance caused by alendronate for the bone phase, mineral is added at an exponentially decreasing rate.17 balance method and the length of the secondary mineralization Because mineral accumulates by displacing water present in the period for the mineralization method). The unknown parameter matrix there is a limit to the amount of mineral that can be Contributions to BMD changes during alendronate treatment Figure 2. A diagram of the computational model is presented with remodeling parameters in boxes and model outputs in ovals. Starting at the left, the
number of progressing BMUs at a point in time is modified by the number of new BMUs forming (based on the origination frequency) and the number
of BMUs that are terminating (based on the BMU lifespan). Each progressing BMU begins a remodeling cycle that continues through a resorption
period, reversal period, mineralization lag time, and formation period. The model tracks the BMU population history so that the total number of BMUs
that are actively resorbing or forming bone can be identified at any point in time. The sum of all actively resorbing bone is used along with the local
resorption rate to determine the volume of bone that is resorbed at any point in time. Likewise, the sum of all actively forming bone or osteoid is used,
along with the local formation rate, to determine the volume of bone that is formed and the change in osteoid volume at any point in time. The difference
between the volume of bone formed and that resorbed determines the change in bone volume fraction. The change in osteoid volume is used to determine
the osteoid volume fraction. The bone volume fraction determines the surface area available for remodeling and therefore influences the number of new
BMUs that originate in the next time step.
present in the bone (this would occur if all water was displaced simulation with a neutral focal bone balance, no initial remod- by the mineral and is referred to as the theoretical maximum eling activity (no osteoid), and a bone volume fraction (0.20) mineralization). We define the secondary mineralization period typical of cancellous bone in the lumbar vertebrae. After the (P, years) as the time between the end of the primary mineral- simulation was initiated, new BMUs originated, and resorption ization phase and the point at which the mineral content reaches and formation occurred, removing and replacing bone volume 95% of the theoretical maximum mineralization (where the and bringing the system to an equilibrium state. The equilibrium theoretical maximum ash fraction is 0.70; Figure 3). The length
state was used as the initial state from which model parameters of the secondary mineralization period has been estimated to be were modified to simulate alendronate treatment. Results are Ͼ6 months,18 but could last many years.8 expressed in terms of percent change in BMD from the equilib-rium state.
Simulations of Alendronate Treatment Alendronate’s effects on BMU activity were modeled using both the bone balance method (decreased remodeling space, All model simulations were performed on a Silicon Graphics O2 increased focal bone balance, and uniform bone mineralization) workstation (SGI, Mountain View, CA) using functions defined and the mineralization method (decreased bone remodeling for use with MATLAB (Mathworks, Natick, MA, USA). An initial space, neutral focal bone balance, and varying bone mineraliza- equilibrium state for the model was determined by starting the tion). Chavassieux et al. showed that the activation frequencydecreases by 87% after 2 years of daily treatment with 10 mg oralalendronate.4 In our model we assume that a similar change inthe origination frequency occurs in each simulation method(since the activation frequency is directly related to the origina-tion frequency13). Comparison of the two methods was per-formed using values for the unknown parameters (secondarymineralization period and focal bone balance) that predict theBMD changes found in clinical studies administering 10 mg/dayof oral alendronate.3,6,21,22 The sum-of-squares for error (SSE)was used to determine how well the model predicts the results ofclinical studies: Figure 3. Primary and secondary mineralization phases. The secondary
where BMD is the measured change in BMD, and BMD is the mineralization period is defined as the time required for bone to miner-alize from 70% to 95% of the theoretical maximum ash fraction (a value predicted change. The SSE and trends in the model predictions were used to evaluate the two methods.
Contributions to BMD changes during alendronate treatment Figure 4. The results of initial alendronate simulations using both the bone balance method (left) and the mineralization method (right) are presented.
Data from clinical studies of 10 mg/day of oral alendronate treatment in patients with low initial bone mass3,6,21,22 are plotted along with model
predictions. For the bone balance method, the bone balance ratio (⌬BMU.Rt) was initially unknown. A bone balance ratio of 2.0 gave good predictions
of the clinical results (SSE ϭ 24.65). Bone balance ratio values of 1.0, 1.5, 2.0, and 3.0 corresponded to focal increases in bone volume of 0.0, 2.48e-05,
3.73e-05, and 4.97e-05 mm3 per remodeling site. For the mineralization method, the secondary mineralization period (P) was unknown. A 6 year
secondary mineralization period well described the BMD changes found clinically (SSE ϭ 33.01).
parameter values based on histology data. It is therefore limitedby the assumptions upon which those data are based.19 In Alendronate simulations using the bone balance method pre- addition, the focal bone balance was assumed to be neutral (the dicted greater increases in BMD as the bone balance ratio same amount of bone is formed as is resorbed with each remod- became larger (Figure 4, left). Predicted BMD increases were
eling cycle) in all of the pretreatment simulations (equilibrium sensitive to changes in the bone balance ratio, with changes after states) used in this analysis. Individuals treated with alendronate 10 years predicted to be 12.87% for a bone balance ratio of 2.0, may have a negative pretreatment focal bone balance (more bone and 10.12% for a bone balance ratio of 1.5. A bone balance ratio is resorbed than formed with each cycle). A neutral pretreatment of 2.0 gave predictions of BMD increases similar to those found focal bone balance was used in our analysis because a number of in clinical studies (SSE ϭ 24.65). The bone balance method placebo groups in alendronate studies3,21,22 have shown small or predicted that BMD would continue to increase as long as insignificant decreases in bone mass (most likely due to calcium alendronate treatment was continued. The change in bone mass supplementation and placebo effects) and a neutral pretreatment caused by a reduction in the remodeling space alone is presented focal bone balance allows the simulations to reflect changes in in a simulation using the bone balance method with a neutral bone mass caused by alendronate rather than those caused by focal bone balance (⌬BMU.Rt ϭ 1.0; Figure 4, left). The
increase in BMD caused by remodeling space after 10 years was We presented two different methods of simulating alendro- nate’s effects on bone remodeling. Both methods showed similar Simulations using the mineralization method showed that sum-of-squares for error (SSE ϭ 24.65 for the bone balance longer secondary mineralization periods resulted in greater in- method, and SSE ϭ 33.01 for the mineralization method), creases in BMD (Figure 4, right). A secondary mineralization
suggesting that either method could be used to predict BMD period of 6 years resulted in good predictions of the results from changes over the treatment periods that have already been stud- clinical studies of alendronate (SSE ϭ 33.01). The change in ied (up to 7 years). The two methods gave considerably different BMD predicted with a 6 year secondary mineralization period predictions for longer treatment periods. The bone balance was 11.98% after 10 years, much greater than that predicted method predicted a steady increase in BMD with alendronate using a 1 year mineralization period (6.23%). Trends from the treatment because the focal bone balance caused small increases mineralization method simulations suggest that the rate of BMD in bone volume during each remodeling cycle. The mineraliza- increase was reduced after continued alendronate treatment.
tion method predicted that there is a maximum possible BMDincrease because there were limits on the degree of mineraliza- Discussion
tion of bone (Figure 3). Although alendronate treatment has notbeen studied for periods of Ͼ7 years, BMD increases caused by In this work we developed a computational model of bone alendronate do not appear to be unlimited. In addition, the bone remodeling and alendronate treatment. The model was designed balance method required a bone balance ratio of 2.0 to predict the to determine reasonable values for unknown parameters in the results of clinical studies. It is unclear whether such high values bone balance and mineralization methods and determine how can occur for prolonged periods in humans. For these reasons it well each model can predict the BMD increases measured clin- is likely that the mineralization method is a better description of ically. We found that the bone balance method, using a bone the effects of alendronate on bone remodeling.
balance ratio of 2.0, and the mineralization method, using a Our results support the findings of recent studies suggesting secondary mineralization period of 6 years, can each predict the that the degree of mineralization contributes more to the BMD BMD increases observed in clinical studies of daily alendronate increases caused by alendronate than the focal bone balance.
Chavassieux et al. showed that patients taking alendronate for The model presented in this work is based on our current 2–3 years do not show significant increases in bone volume understanding of the process of bone remodeling and uses fraction,4 a result that would be expected in response to changes Contributions to BMD changes during alendronate treatment caused by alendronate may be lost if cellular activity returnsto pretreatment levels. If changes in bone turnover account formost of the increases in bone mass, a patient who discontinuesalendronate treatment may eventually lose most of the bene-fits of treatment. Second, the ash fraction is known to influ-ence bone strength differently than the bone volume fraction(mineralized bone volume/bulk volume).12 Areal BMD mea-surements do not differentiate between increases caused byash fraction and those caused by increased bone volumefraction. Because focal bone balance changes modify the bonevolume fraction, it is possible that BMD increases from ashfraction could change bone strength in a different way than thesame BMD increase caused by focal bone balance. This mayexplain why the decreased rate of fracture after alendronatetreatment (nearly 50% decrease after 3 years) appears to be solarge compared to the observed change in BMD (ϳ6%– 8%after 3 years).22 Figure 5. A theoretical description of the relative contributions of
With proper validation the model used in this work could remodeling space, mineralization, and focal bone balance to BMDincreases during alendronate treatment. The solid line represents the total predict changes in bone mass caused by a number of different change in BMD. The dashed line represents the contribution caused by agents. What we have presented here, however, is a useful tool changes in the remodeling space alone. The dotted line represents the for comparing ideas of how alendronate affects bone mass.
BMD changes predicted to be caused by both the remodeling space and Future simulations based on this model could be used to mineralization (ash fraction). As depicted, the focal bone balance con- identify improved dosing regimens and to predict which other tribution is smaller than the contribution caused by mineralization. It is osteoporosis treatments (hormone replacement, parathyroid possible that the BMD contribution caused by focal bone balance isnegligible.
hormone, and exercise) would best complement alendronatetreatment.
in the focal bone balance.16 In addition, the mean wall thicknessof bone formed does not appear to increase in response to Appendix
alendronate treatment.4,5 If the erosion depth reached during The BMU-based simulation examined here is adapted from a resorption also remains unchanged the focal bone balance wouldbe neutral during alendronate treatment. Accurate measurement model presented previously.11 The model uses nine indepen- of the erosion depth and the focal bone balance is difficult, dent parameters to describe BMU geometry and the resorption however. Trends in the focal bone balance have been observed, and formation process (Table A1). The volume resorbed and
but it is still unclear whether modification of the focal bone formed per remodeling site (V , V ), shown in equation (1), balance truly occurs in response to alendronate.4 A more recent are related to the local resorption and formation rates (Rs.R, study of patients taking alendronate2 found the changes in degree of mineralization to be similar to the changes in BMD. Theinvestigators in this study concluded that changes in the degree of mineralization may account for a majority of the BMDincrease caused by alendronate. These findings do not rule out the possibility that focal bone balance is changed in response toalendronate. They do imply that, with regard to alendronate The variables Rs.P and FP represent the length of time in which treatment, the influence of focal bone balance on BMD increases resorption and formation occur during a remodeling cycle (re- is small compared to the influence of mineralization or the sorption and formation periods in days) and Rs.R and FR represent the rate at which bone volume is resorbed or formed Heaney and colleagues attributed part of the increase in (local resorption and formation rates in cubic millimeters per day BMD caused by alendronate to the remodeling space and per remodeling site). Initial values for the volume resorbed or another part to the focal bone balance (Figure 1).10 Our formed per remodeling site (V , V ) are calculated by defining findings imply that changes in mineralization (ash fraction) the shape of a BMU in cancellous bone (semiellipsoidal in cross not only contribute to BMD changes after alendronate treat- section with major radius equal to half the BMU width and minor ment, but may explain most of the increase in BMD that was radius equal to the erosion depth).11 The BMU width (equivalent attributed to focal bone balance by Heaney et al. We therefore to the diameter of a cortical BMU, 0.152 mm)1 and erosion depth suggest that changes in remodeling space, changes in miner- (for healthy postmenopausal women, 0.049 mm)7 are based on alization (ash fraction), and increased focal bone balance all contribute to alendronate-induced BMD increases, with the
focal bone balance being the smallest contributor (Figure 5).
The initial values for the local formation and resorption rates Changes in bone turnover are responsible for the changes in are calculated from equations (A1) and (A2). Changes in the remodeling space and ash fraction presented here, making local formation and resorption rates by the same factor represent bone turnover the single most important aspect of BMU changes in the BMU shape (width or depth). Changes in local activity that is modified by alendronate treatment. The impor- formation and resorption rates by different factors result in tance of bone turnover has two consequences with regard to changes in the focal bone balance. This method of representing alendronate treatment. First, changes in bone mass caused by the focal bone balance allow us to differentiate between modi- bone turnover are the result of a reduction in remodeling fications in focal bone balance caused by changes in resorption activity,9 implying that most of the benefits in bone mass and those caused by changes in formation.
Contributions to BMD changes during alendronate treatment Table A1. Independent parameters of basic multicellular unit (BMU) bone remodeling
Volume of bone resorbed per remodeling site per unit time Volume of bone formed per remodeling site per unit time Time during which resorption occurs at a remodeling site Time between osteoclast and osteoblast activity Time between osteoid formation and the start of mineralization Time during which formation occurs at a remodeling site Time required for bone to mineralize from 70% to 95% aInitial value calculated using histology data.
bValue based on histologic data for healthy postmenopausal women.7cValue based on estimates by Parfitt et al.20dEstimated values ranging from 6 months18 to many years.8 logic and metabolic influences on bone adaptation. J Rehabil Res Devel Acknowledgments: This work was supported by the Department of Veterans Affairs (A2424P) and a Dissertation Fellowship from the Ford 12. Hernandez, C. J., Beaupre´, G. S., Keller, T. S., and Carter, D. R. The influence of bone volume fraction and ash fraction on bone strength and modulus. TransOrthopaed Res Soc 29:74 –78; 2001.
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WHAT REGULATE THE GROWTH REGULATORS ? Logos Publisher, Kiev, 1998. ISBN 966-581-101-0Edited by B. A. Kurchii Institute if Plant Physiology and Genetics, 31/17 Vasylkivska Str. 03022 Kiev, UkraineDecember, 1998202 pages, illustratedPrice: $50.00Monograph is published in Russian (30%) and in English (70%). The papers presented in this book deal with the relationships between the structure of a