Quantification of remodeling parameter sensitivity—assessed by a computer simulation model

Bone Vol. 19, No. 5 November 1996~505-5 I 1

ELSEVIER

Quantification of Remodeling Parameter SensitivityAssessed by a Computer Simulation Model J. S. THOMSEN,t,*

Li. MOSEKILDE,’

and

E. MOSEKILDE*

’Department of Cell Biology, Institute of Anatomy, University of Aarhus, Aarhus, Denmark ‘Department of Physics, The Technical University of Denmark, Lyngby, Denmark

densitometry. 18X44*45 All these studies have shown an accentuated loss of bone mass during and just after menopause. Histomorphometric investigations are generally cross secand only few studies have been “longitudinal” tional, 8~14~22~28~29 with two biopsies taken from the same patient during treatment regimes.9,13.4*,51 Even those studies with repeated biopsies have not been truly longitudinal, as two biopsies cannot be obtained from exactly the same site. Although much information has been acquired concerning changes in cancellous bone structure and remodeling parameters related to the menopause, it is still not possible to identify the most important factors concerning perforations of the trabecular network. It is therefore also difficult to determine the optimal treatment regimen for preventing perforations of the trabecular network. Several computer simulation models have been introduced in order to describe the importance of changes in remodeling parameters with regard to physiological bone changes (aging and menopause) and with regard to treatment regimens.‘9~2’~26~4’~42~52 All these computer models allow “longitudinal” studies at organ level’9.21*42 or bone tissue leve126~4’~52to be conducted. The model we had used was based on the stochastic model by Reeve.41 However, our model had been adapted to the geometry of the vertebral trabecular network.52 The model parameters were based on cross-sectional histomorphometric data obtained from iliac crest bone biopsies from clinical studies. In the present article, the simulation model has been applied to assess the different types of bone loss in connection with estrogen depletion as well as the relative importance of the remodeling parameters with regard to the output variables (sensitivity analysis). During repeated remodeling cycles, bone mass is lost in several ways:

During normal aging and menopause, cancellous bone is lost at all skeletal sites due to remodeling-related factors: negative formation balance; temporarily increased remodeling space; and osteoclastic perforations. The relative importance of the various factors in inducing bone mass loss and perforations is still controversial. We have previously used a computer simulation model to describe the effect of several bone remodeling parameters on vertebral cancellous bone loss. The model focused on two different scenarios for the menopause and three different treatment regimens. The aim of the present study was to extend the previous study by quantifying remodeling parameter sensitivity for changes in the bone mass with the use of the computer model we had previously formulated. The menopause scenario, with increased activation frequency and increased resorption depth, was chosen as the base case scenario, and the following parameters were investigated in the sensitivity analysis: activation frequency; formation balance; resorption depth; and critical trabecular thickness. Simulations were performed for a period of 20 years starting at the age of 48 years. The analyses showed that the number of perforations and the perforation-related mass loss both exhibited a large sensitivity toward variations in the final resorption depth. However, the formation balance was the factor that was responsible for the greater part of the bone mass loss. The computer model allowed us to quantify the sensitivity of different output variables with respect to changes in some of the model parameters. This can give information about the biological mechanisms responsible for bone mass loss around the surgically induced or natural menopause and also provide an indication of the type of treatment that would be most useful in preventing the deterioration of the cancellous network. (Bone 19.505-511; 1996) 0 I996 by Elsevier Science Inc.

1. Osteoclastic perforations of trabecular structures; 2. Negative bone formation balance, which results in a net bone loss for each completed remodeling cycle; 3. Changes in the remodeling space, i.e., the amount of bone that has been removed by osteoclasts and not yet reformed by the osteoblasts during the remodeling sequence.47

Key Words: Computer simulation; Bone mass; Bone structure; Osteoclastic perforations; Remodeling; Sensitivity analysis.

Introduction Under normal circumstances, a trabecula that has been disconnected cannot reconnect,37 but seems likely soon to be removed due to loss of mechanical forces.31 Hence, the bone mass lost through perforations of trabeculae is considered irreversible. In principle, the bone mass loss due to a negative formation balance is reversible, as a subsequent remodeling process with a positive formation balance at the same site could compensate the lost mass. However, in normal adult individuals the formation balance is seldom positive.‘2,48,5’ Thus, bone mass lost due to a

Age-related and menopause-related bone loss have been followed in longitudinal studies at different skeletal sites by use of

Address for correspondence and reprints: Jesper Skovhus Thomsen, M.Sc., Ph.D., Department of Cell Biology, Institute of Anatomy, University of Aarhus, DK-8000 Aarhus C, Denmark. E-mail: jesper@jst. ana.aau.dk

0 1996 by Elsevier Science Inc. All rights reserved.

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negative bone formation balance is normally considered as irreversible. The bone mass lost due to a change in the remodeling space is a reversible bone loss. The size of the remodeling space can increase, for example, as the activation frequency increases during menopause.5,‘7,49 The apparent bone mass, therefore, seems to decline at the onset of menopause.40~42~44~52After menopause, the activation frequency decreases again and the remodeling space becomes smaller. This implies that the bone mass lost due to the increased remodeling space is (partially or fully) regained.& In this study, osteoclastic perforations of trabecular bone (1, above) will be denoted “bone mass loss due to perforations,” and the sum of the two latter types of bone loss (2 and 3, above) will be denoted “bone mass loss due to remodeling.” The purpose of the study is to investigate the bone mass loss due to perforations during simulation of menopause and to compare this loss with the bone mass loss caused by changes in remodeling space and negative bone formation balance. Materials and Methods The Model We have previously described a stochastic simulation model that can simulate bone remodeling in horizontal trabeculae in human vertebrae.“’ The model contained several parameters characterizing the remodeling process and, to some extent, the bone geometry. Parameter values, obtained through different clinical studies,‘2-‘4,48,” were used as input for the model. In this study, the model has been used without any changes, but we have used only one of the scenarios from our previous work as base case scenario (see Table 1 for parameter values). Parameter

frequency is doubled5 (from l/1096 days-’ to l/548 days-‘) and the resorption depth is increased from 50 to 70 pm.14 At the end of menopause, which is assumed to last 5 years, the activation frequency and resorption depth revert to their initial magnitudes. To keep the model simple, these parameter changes occur over just one simulation time unit, which is 1 day. Furthermore, there are currently no histomorphometry data available that quantify how the transition of the remodeling parameters takes place around the onset and cessation of the menopause, In the model, remodeling processes that have been initiated before the onset of the menopause are not affected by the parameter changes; only the remodeling processes that start after the onset of the menopause are affected by the “new” parameters. Hence, the effect of an abrupt parameter change will be smoothed out. Sensitivity Analysis In order to determine how the results vary with changes in the input parameters, we performed a series of simulations in which parameters were changed one at a time. When the input parameters and the simulation results are normalized relative to their base case values, it is possible to determine the sensitivity of the model to each parameter and hence to determine for which remodeling parameter a small change gives the largest relative change in the number of perforations. Each time a parameter was changed, the random number generator was initiated with the same seed. Mass Calculation In the model, it has been assumed that all trabeculae are cylindrical and of the same length 1. The mass M of a cylindrical trabecula with thickness w, length I, and density p is given by:

Variation M=;lpw’

In order to assess the model, and at the same time extend our understanding of the parameters controlling perforations of the cancellous bone network, it was of interest to establish how the model responded to variations in the input parameters. The parameter space is 12-dimensional (this means that 12 different parameters could be adjusted independently at the same time), and even with modem computers, a complete analysis would not be practicable. Hence, we limited investigation to smaller (but crucial) subsets of the parameter space. In the base case scenario (simulations are shown in Thomsen et aL5* as Figure 4B), the simulation starts at the age of 48, and after 5 years menopause begins. In relation to this, the activation Table 1. Simulation

parameters for the base case. Simulation starts at age 48. After 5 years the menopause is initiated and continues for 5 years. After the menopause, a further 10 years of simulation is performed

For a network of N trabeculae, expression for the total mass:

(2) where wi is the thickness of the ith trabecula. A perforated trabecula is assumed to have thickness 0. This implies that whenever a trabecula has been perforated, the whole trabecula is removed. To simplify the analysis, the expressions for bone mass are normalized with respect to the initial bone mass. The total mass loss (relative to the initial mass) as a function of time then becomes:

CL w,(o)2- z:, W,(t)2 =Iz;“i= wi(o)2 1

mr(t) =

’

Resorption period Reversal period Formation period Trabecular thickness Resorption depth Critical thickness Formation balance Activation frequency Number of trabeculae

Normal

Menopause

Unit

42 9 145 + 4.5 135 + 24 50* 15 (130 * 50)/P -2 l/1096 32000

42 9 145 * 45 135+24 70+ 15 (130 * 50)/n -2 l/548 32000

Days Days Days pm pm km pm Days-’

we then obtain the following

CL, w,(t12 g=, w,(o)2. (3)

Equation (3) is the same expression for mass loss as used in our original paper.52 In order to determine the fraction of mass loss attributable to perforations and remodeling separately, an expression for bone mass lost due to perforations was derived. In the simulations, the resorption depth and activation frequency were fixed, except at the onset and end of the menopause. This meant that during most of the simulation, bone mass loss due to remodeling was mainly caused by the negative formation balance.

Bone Vol. 19, No. 5 November 1996505-5 I1 Thereafter, was addressed. mass loss was had, when the mass lost due

J. S. Thomsen et al. Remodeling parameter sensitivity analysis

the issue of defining mass loss due to perforations When a perforation took place, the corresponding considered to be the mass that the trabecula had simulation was initiated. Consequently, the total to perforations rnp at a given time t is

‘E=

m (t) =

1

where Heaviside’s

I1 - /dwi(t)ll ’

wi(")2

P

(4)

XL, Wi(0)Z

unit function p(x) is defined by: &)

=

OforxSO lforx>O,

1

(5)

The function of the term { 1 - p,[wi(t)]) is to remove the perforated trabeculae from the sum. Finally, from equations (3) and (4) and from the trivial relation ml(t) = m,(t) + m,(t), an expression for the remodelingrelated bone mass loss mAt) could be derived: CE,

w,(0)‘p[wi(t)]

m,(t) =

- xi”= 1w,(t)2 (6)

lx;“=, Wi(0)2

The initial number of trabeculae in each simulation was N = 32,000. With this number of trabeculae, a time step of 1 day, and a simulation horizon of 20 years each simulation takes approximately 13 min on a Intel Pentium 90 MHz PC (with 16 Mb RAM and running UNIX). However, the actual simulations were carried out on a HP 9000 series 735/99 UNIX workstation. Results Mass Loss Due to Perforations (Changes With Time)

and Due to Remodeling

Figure 1 shows how the perforation-related mass loss m,,(t) and the remodeling related mass loss m,(t) change during 20 years of simulation. The input parameters that have been used are the base mass loss case values, shown in Table 1. The perforation-related

-

-15

-

3

0

:

10 -

S

-

is calculated as shown in equation (4), and the remodelingrelated mass loss is calculated as shown in equation (6). From Figure 1, it can be seen that the rate at which the perforation-related mass loss increases is much larger during menopause than it is before and after the menopause. During menopause, the activation frequency is doubled, and there are approximately twice as many remodeling processes. Furthermore, the resorption depth is increased, which leads to a higher probability of perforation. Notice that the slope of the curve is almost identical before and after the menopause. When Figure 1 is compared with the results from our previous work,‘* it is apparent that the perforation-related mass loss reflects very closely the number of perforations. However, the size of the mass loss due to perforations after 20 years of simulation is only about 3% of the initial mass. The mass loss due to remodeling m,(t) that can be seen in Figure 1 is similar to the curve depicting the total mass loss that we have previously obtained (based on only 628 trabeculae).52 At the onset of the menopause, the remodeling-related mass loss increases over a period of approximately 6 months from approximately 5% to 15%. This is due to an increase in remodeling space, resulting from the increased activation frequency and resorption depth. Notice that the increase in remodeling space occurs over a period of time, even though the magnitude of resorption depth and the activation frequency are assumed to change abruptly at the onset of the menopause. This is because it takes some time to initiate the increased number of remodeling processes that corresponds to the increased activation frequency and because the remodeling processes that were initiated before the onset of the menopause remain unaffected by the change in resorption depth. After the simulated menopause, the remodeling space is again decreased, which can be seen as a reduction in the remodelingrelated mass loss. Like the increase at the onset of the menopause, this decrease in the remodeling space takes place over a time span of approximately 6 months. When the perforation-related mass loss is compared with the remodeling-related mass loss in Figure 1, it is apparent that the perforations contribute in only a very minor way to the total mass loss. Sensitivity Analysis (Changes as a Function of Remodeling Parameters)

20 -

G?

507

5m&>

/ 5

t

I

10 (years)

,

,

,

,

15

,

,

,

,

20

Figure 1. Mass loss due to perforations m,(r) and mass loss due to remodeling mJr) as function of time during simulation of the menopause. The simulation starts at age 48 and ends at age 68. Onset and cessation of the menopause are indicated by dashed lines. Simulation parameters are shown in Table 1.

Bone formation balance. Figure 2 shows how the terminal values of different variables change as a function of the bone formation balance. On both axes, the values have been normalized with respect to the corresponding base case values. Hence, all curves pass through (1,l). The slope of the curves quantifies the parameter sensitivity at any given point on the curves. For the input parameters, the base case values are listed in Table 1. From Figure 2, it can be seen that if the formation balance has half the value of the base case (b/b,, = 0.5), then the average trabecular thickness is only about 7% larger than in the base case. Notice how the average trabecular thickness in the considered range has a nearly linear relationship with the formation balance. We can also see from Figure 2 that the bone mass loss and the number of perforations are more sensitive to changes in the formation balance than is the average trabecular thickness. This is also reflected in the curves for bone mass loss due to perforation and to remodeling. Resorption depth. Figure 3 shows a similar set of curves as Figure 2, but now with the normalized resorption depth as the active parameter. Figure 3 also shows that the average trabecular

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depth the number of perforations is also increased, and the thin trabeculae are the ones most likely to be perforated. Even a small change in the resorption depth will give rise to a large change in the number of perforations. For example, it can be seen from Figure 3 that an increase of 5% in the base case value of the resorption depth (from 1.OOto 1.05) results in an increase of 25% in the number of perforations over the base case (from 1.00 to 1.25). The total bone mass loss increases with increasing resorption depth, due to the increased number of perforations. The remodeling-related mass loss, however, is seen to decrease with increasing resorption depth; this is related to the higher number of perforations. The mass loss due to remodeling depends on the number of remodeling sites, which again depends on the number of nonperforated trabeculae. When the number of trabeculae that have been perforated increases, there are fewer trabeculae left from which to lose bone mass, and the bone mass loss due to remodeling activity will therefore decrease with increasing resorption depth. The bone mass loss due to remodeling also decreases slightly when the resorption depth is sufficiently reduced, and this output parameter thus displays a maximum as a function of d/do. Notice that almost all the curves in Figure 2 are linear, whereas the curves depicted in Figure 3 are almost all nonlinear. When the curves are linear, the parameter sensitivity (the slope of the curves) is constant, and the parameter sensitivity for the parameter range can be expressed as a single number. However, when the curves are nonlinear, the parameter sensitivity is different throughout the considered interval and depends on the value of the parameter. resorption

1.25

0 q 1.00 .?a

0.75

0.50 1 .oo

I

b/ho Figure 2. b/b, is the bone formation balance b normalized with respect to the base case. y/y, is model output after 20 years simulation normalized with respect to the base case, where: the filled circle is the average trabecular thickness (w). the open triangle is the bone mass loss (ml), the open circle is the number of perforations, the multiplication sign is the bone mass loss due to remodeling (m,), and the open square is the bone mass loss due to perforations (m,,).

thickness increases with increasing resorption depth. Furthermore, the number of perforations and the perforation-related bone mass loss both show a large sensitivity toward variations in the resorption depth. The reason for this is that with increasing

Critical Trabecular Thickness Figure 4 shows how some of the output values vary with respect to normalized critical trabecular thickness. The critical trabecular

1.50 %

1.50

1.25 1.25

0 2

D

1.00

< 1.00 ;3,

0.75 0.75

o.50

1-0.50

0.75

1.00

1.25

1

d/do Figure 3. d/d” is the resorption depth d normalized with respect to the base case. y/y0 is model output after 20 years simulation normalized with respect to the base case, where: the filled circle is the average trabecular thickness (wi, the open triangle is the bone mass loss (m,), the open circle is the number of perforations, the multiplication sign is the bone mass loss due to remodeling (m,), and the open square is the bone mass loss due to perforations (m,,). Notice that the bone mass loss due to remodeling displays a maximum.

0.50 0.50

0.75

1.00

1.25

1.50

WC/%,0 Figure 4. W/W,:, is the average critical thickness w,. normalized with respect to the base case. y/v, is model output after 20 years simulation normalized with respect to the base case, where: the filled circle is the average trabecular thickness (w), the open triangle is the bone mass loss (m,), the open circle is the number of perforations, the multiplication sign is the bone mass loss due to remodeling (m,), and the open square is the bone mass loss due to perforations (m,,).

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Remodeling

thickness is the thickness at which a remodeling site will completely encircle a trabecula. ‘* The average trabecular thickness, the total mass loss, and the remodeling-related mass loss show almost no dependence on the critical trabecular thickness. However, the number of perforations and the perforation-related mass loss do increase with increasing critical trabecular thickness. When the trabecular thickness is below the critical thickness, the effective speed of bone erosion is twice the size it would have been above criticality. The probability for perforations will therefore increase with increasing critical thickness. The perforations that will emerge if the critical thickness is increased will all take place on thin trabeculae, and the increase in bone mass loss will therefore be very low. Activation

Period

Figure 5 shows the activation period l/l.t normalized with respect to its base case value l/p,. If the time between two subsequent remodeling events on the same trabecula is high (high value of &CL), then the average trabecular thickness will be higher than in the base case. If the activation frequency is high (low value of F&L), the average trabecular thickness will be lower than in the base case. From Figure 5, it can be seen that the total bone mass loss, the remodeling-related bone mass loss, the number of perforations, and the perforation-related mass loss exhibit an almost identical dependence on the activation frequency. When the activation frequency is increased, the mass loss will also increase, because of the negative formation balance that is assumed here. The number of perforations will increase because of the decreasing trabecular thickness. Discussion The simulation model has shown that the number of perforations and the perforation-related bone mass loss both exhibit a large 1.50

1.25

0.50

11 0.50

0.75

1.00

1.25

1.50

PO/P

Figure 5. JL,J~.L is the activation period l/p normalized

with respect to the base case y/y0 is model output after 20 years simulation normalized with respect to the base case, where: the filled circle is the average trabecular thickness (r-v). the open triangle is the bone mass loss (ml), the open circle is the number of perforations, the multiplication sign is the bone mass loss due to remodeling (m,), and the open square is the bone mass loss due to perforations (m,).

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sensitivity toward variations in the final resorption depth. Even a small change in this parameter will give rise to a large change in the number of perforations. These findings are in accord with the theory of Parfitt37-39 suggesting that increased osteoclastic resorption depth around the menopause causes an increase in trabecular perforations and thereby deterioration of the network. That the resorption depth might be increased in women around the menopause was indicated by the cross-sectional study of Eriksen et al. I4 This was also indicated in the longitudinal study of Compston et aZ.,9 in which connectivity measurements (nodestrut analysis) on iliac crest bone biopsies were performed before and after treatment with Gonadotrophin-releasing hormone (GnRH) agonists. However, their data did not provide any direct evidence to support a role for increased depth of erosion in bone loss associated with the induced estrogen deficiency. Therefore, an increased resorption depth as a result of estrogen depletion has still not been proven in human studies. From studies with minipigs, on the other hand, an increase in final resorption depth leading to an increase in marrow space star volume has been demonstrated in relation to ovariectomy and to mild calcium restriction.3.34 The increase in final resorption depth in the minipig study was of the order of 20% (number of counted lamellae 15.8-19.0). The computer model indicates that it is not the perforationrelated bone loss that accounts for the largest contribution to the total loss of bone mass. As the trabeculae that are disconnected by perforations are the thinner trabeculae in the network, the relatively small impact of the perforation on bone mass is explainable. However, the perforations might have a relatively larger impact regarding biomechanical competence, due to the changes in the trabecular structure.‘0~25~32~33Furthermore, the change in the bone perforations represent an irreversible structure,3’ which gives the perforations additional significance. It has been demonstrated that bone strength declined much more than bone mass during normal aging,33 and that this decline in strength was more pronounced in women than in men after the age of 50 years.32 In order to prove the significance of the perforations, direct structural measurements (e.g., 3D connectivity or finite element methods from 3D data) of trabecular bone are needed. However, these methods are still under development. 2-4,6,11,16.20,23,24,35,36.46,53,54 Notice, that although the perforations show a large sensitivity toward a change in the final resorption depth this does not mean that the perforations are the largest contributing factor to the bone mass loss, as the sensitivity analysis gives information on the dynamics of the output with a parameter change and not on the actual size of the mass loss. Concerning the bone mass loss as such, the computer model has shown that the factor that accounts for the largest part of the bone loss is the (negative) formation balance that results in bone mass being lost with each completed remodeling cycle. This is in accordance with the study of Cohen-Solal et al.,’ which clearly demonstrated that women with osteoporosis had a more negative formation balance (-6.4 km) than normal women (-1.34 km). This difference was found to be statistically significant. There are two ways of preventing this type of bone loss caused by a defective recruitment and/or function of osteoblasts: 1. Reduce the magnitude of the negative formation balance--or, if possible, make the formation balance positive (anabolic agents). 2. Reduce the number of times the negative formation balance is involved-i.e., decrease the activation frequency (antiresorp tive agents).

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Of these two different treatment schedules, treatment with and PTH43) has shown an increase anabolic agents (fluoride”” of 15%-20% in spinal bone density. The effect of antiresotptive therapy has, on the other hand, been of only 3%-7% when measured as spinal bone density.27S48,50 However, the studies with antiresorptive therapy27,48,50 have additionally shown a reduced fracture rate despite little effect on bone mass, again suggesting the importance of preventing perforations of the cancellous network. From the present study, it can be deduced that an antiresorp tive agent capable of decreasing both resorption depth and activation frequency would be more effective in preventing perforations than an agent that changes only the activation frequency. As a “spin off’ of our study of parameter sensitivity, an explanation was obtained for the findings of many crosssectional histomorphometric studies: that trabeculae are often thicker in women after the menopause than in younger women.iS8 The model clearly showed that the reason for this is that with increasing final resorption depth, the number of perforations is also increased, and it is the thin trabeculae that are most susceptible to perforation. When perforations have caused the thin trabeculae to vanish, a larger proportion of the thicker trabeculae remains; and therefore the average trabecular thickness will also be larger. The intention of the present study was to investigate the already formulated model in detail before making changes to the model. The simulations in the present article and the sensitivity analysis were performed on our previous model: During menopause only changes in activation frequency and resorption depth were considered, and a simple shift in parameters was used in onset and cessation of the menopause. The model must still be considered as preliminary in some aspects, and we recognize that the simple shift in parameters during menopause might not be entirely biologically accurate, as some longitudinal studies based on bone densitometry and biochemical markers have indicated a slower change in bone remodeling at organ level at the cessation of the menopause.44,49 However, as previously mentioned, there are currently no histomorphometric data available that describe how the transition in the parameters at tissue level takes place around the menopause. Furthermore, the impact of the exact temporal development of the change in parameters will be minor regarding the sensitivity analysis. Performed as a computer simulation at histomorphometric level, this “longitudinal” study has been able to illustrate the biological background for findings from cross-sectional histomorphometric studies, e.g., increasing trabecular thickness after the menopause. In conclusion, the computer simulation has illustrated that the bone formation balance seems to have the largest impact on loss of bone mass. Furthermore, the sensitivity analysis has disclosed that even a small change in resorption depth will give rise to a relatively large change in number of perforations, leading to a larger deterioration of the trabecular network.

Acknowledgment: Michael Hewitt is gratefully revising this manuscript.

acknowledged

for help in

References 1. Aaron, I. E., Makins, N. B., and Sagreiya, K. The microanatomy of trabecular bone loss in normal aging men and women. Clin Orthop Rel Res 215:26C-271; 1987.

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Date Received: January 26, 1996 Date Revised: May 15, 1996 Date Accepted: July 5, 1996