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Originally published as Biophys J. BioFAST on September 17, 2004.
doi:10.1529/biophysj.104.044511
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Biophysical Journal 87:3642-3647 (2004)
© 2004 The Biophysical Society

A Study of Age-Related Architectural Changes that Are Most Damaging to Bones

Yan Song *, Michael A. K. Liebschner {dagger} and Gemunu H. Gunaratne *

* Department of Physics, University of Houston, Houston, Texas; and {dagger} Department of Bioengineering, Rice University, Houston, Texas

Correspondence: Address reprint requests to Yan Song, E-mail: ysong4{at}mail.uh.edu.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Osteoporosis-related bone damage causes major socioeconomic problems. For efficient use of therapeutic agents, it is necessary to be able to reliably identify patients with high propensity for nontraumatic fracture. Age-related bone loss imposes several architectural changes in bone; one of the few ways to estimate damage due to individual changes, and hence determine the most serious types of damage, is via the analysis of suitable mathematical models. Anatomical sites such as the vertebral body, proximal femur, and distal radius are locations where most age-related fractures occur. The inner porous (or trabecular) bone from these sites, which resemble disordered cubic networks, play a significant role in load transmission at these sites. Analysis of a mathematical model of porous bone is used to show that perforation of elements of the network is the most damaging architectural change to a bone. We also show that an expression for bone strength, derived on this basis, can capture changes in strength caused by the inclusion of other features like thinning of trabecular bone and the anisotropy of the network. We finally argue that bone density, which is currently the most routinely used diagnostic, cannot be a reliable surrogate for bone strength.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
The growth of the older population, and the corresponding increase of osteoporosis and prosthesis, has added to the significance of research into the analysis of mechanical properties of bone. Although there have been recent efforts devoted to addressing this phenomenon (e.g., Kiberstis et al., 2000Go; Keaveny and Hayes, 1993Go; Weinstein, 2000Go; Blank, 2001Go; Bolotin and Sievänen, 2001Go), many questions remain to be clarified. One of the more important issues is to determine suitable surrogates for bone strength. The measure most often used for the purpose is bone mineral density, which is routinely estimated using dual energy x-ray absorptiometry (Meunier, 1998Go). Although higher densities in general correspond to a larger strength and hence a smaller propensity for fracture, there are other factors that play a key role in fracture (McCreadie and Goldstein, 2000Go; Sandor et al., 1999Go; Allolio, 1999Go). In particular, there can be significant changes in microarchitecture that are not captured by bone mineral density.

Large bones from most anatomical sites contain an outer solid segment (known as the cortex) and an inner porous region (referred to as the trabecular architecture). Aging causes the cortical bone to become brittle (Mosekilde et al., 1987Go; Keaveny and Yeh, 2002Go), and leads to an increasing role for the trabecular bone in transmitting loads (Kiberstis et al., 2000Go; Keaveny and Hayes, 1993Go). Trabecular bone is an extremely heterogeneous cellular solid (Keaveny and Yeh, 2002Go; Gibson and Ashby, 1997Go), and aging causes changes in factors such as connectivity, trabecular thickness, levels of anisotropy of the network, and changes in material properties. Images of synthetic prototypes of healthy and weak bone samples, shown in Fig. 1, exhibit these architectural changes.



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  		FIGURE 1  Images of synthetic prototypes of a healthy bone (right) and a weak bone (left). The explicit vertical and horizontal axes of the struts suggest the use of a disordered cubic network model. Note that aging has caused the perforation of the bone and increased the anisotropy of the sample.

 
The effective diagnosis and treatment require a comprehensive knowledge of changes that affect bone strength; current experimental techniques are far too rudimentary to achieve this goal. The alternative is to use a suitable mathematical model to first identify the most damaging changes, and use this knowledge to predict changes in bone strength. Confidence in the models can be enhanced by experimental confirmation of these predictions.

Experiments on ex vivo bone samples show that the mean breaking stress {sigma}max of large groups of bone samples reduces with their density {rho}, and a power-law relationship (Bell et al., 1967Go; Carter and Hayes, 1977Go; McBroom et al., 1985Go; McElhaney et al., 1970Go; Rice et al., 1988Go), as seen here,

(1)
has been proposed. The index ß varies from ~1.7 in vertebral trabecular bone (Ebbesen et al., 1999Go) to 2.6 in iliac crest trabecular bone (Ebbesen et al., 1997Go). This simple empirical description has a straightforward interpretation from uniform isotropic thinning of rod-like and plate-like trabecular bone (Gibson, 1985Go), but as argued below it is unable to represent variations in breaking stress due to age-related anisotropy or trabecular reduction (for example, the changes of mean thickness of trabecular elements along the axial direction are quite different from those along the off-axial direction). In particular, our results below show that reduction in the number of trabecular elements is most damaging to bone, and hence it is necessary to study a relationship that is based on this mechanism of bone loss.

Network models of trabecular bone have been analyzed in many studies (Reeve, 1986Go; Tayyar et al., 1999Go; Keaveny and Hayes, 1993Go). A recently proposed Voronoi model (Silva et al., 1995Go; Silva and Gibson, 1997aGo,bGo; Vajjhala et al., 2000Go; Guo and Kim, 2002Go; Gibson and Ashby, 1997Go) has been introduced to study variations in bone strength. In this model, the randomly distributed nodes are connected (if their distance is larger than a predetermined value) to form a network, whose struts are constructed using the Voronoi algorithm (Silva et al., 1995Go). Results from computational studies indicate that trabecular removal has a more significant effect on strength than trabecular thinning or anisotropy (Guo and Kim, 2002Go; Vajjhala et al. 2000Go). According to percolation theory (Stauffer, 1985Go), if cell elements are randomly removed, the Voronoi network will become more porous and fragile until it is completely separated into many isolated segments when half of the cells are removed; i.e., a Voronoi network has a percolation threshold {nu}0 = 0.5. Thus, the fracture load {sigma}max will vanish for nonzero bone density; hence, Eq. 1 cannot represent the relationship between density and strength.


   DESCRIPTION OF THE MODEL SYSTEM
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
In this article, we report conclusions based on a model of trabecular bone that is composed of a disordered cubic network. Locations of nodes of the network are obtained by displacing vertices of a rectangular network of size Nx, Ny, Nz, and sides of length Lx, Ly, and Lz by up to {Delta} randomly. Struts joining the nearest-neighbor nodes are assumed to be elastic, and their elastic moduli are chosen randomly from a predetermined range [k1, k2]. Their random values are meant to represent spatial variations in thickness of trabecular elements seen in Fig. 1. Finally, to prevent an entire layer passing through another under compression of the network (glide instability), a bond-bending energy is introduced to change the angles {theta} between adjacent struts. The corresponding linear coefficients {kappa} are chosen randomly within a predetermined range [{kappa}1, {kappa}2] (Gunaratne et al., 2002Go). Under an external compression, the system is made to adiabatically approach an equilibrium state that is determined by minimizing the potential energy as

(2)
where {delta}r denotes the compression or stretching of a strut and {delta}{theta} represents the change in a bond angle from equilibrium.

Results reported in this article are obtained for networks with identical properties in the x and y directions. In particular Nx = Ny and Lx = Ly. We have also chosen the same ranges [k1, k2] for the spring constants in these off-axial directions. We assume that the nodes on the surface of the network cannot move away from it to model the fact that nodes at the cortex do not move inward or outward. The external strain displaces nodes on the top surface by {epsilon} in the z direction but do not move nodes on the bottom surface in the z direction. All the other nodes on the network are allowed to move freely.

Experiments on the fracture load of trabecular bone from specific anatomical sites have shown that they fail at nearly a fixed level of strain, independent of the age of the subject (Ciarelli et al., 2000Go; Hogan et al., 2000Go; Keaveny et al., 1994Go). It is worth noting that the corresponding fractures stresses can differ by nearly two orders of magnitude (Ebbesen et al., 1997Go). These conclusions motivate us to use a strain-based criteria for fracture of struts on the network. Previous studies of failure of thin wires (Harlow and Phoenix, 1981Go) suggest the use of a Weibull distribution, with a cumulative density of to assign the fracture strain {gamma} of struts. Here, the integer {eta} determines the width of distribution; larger values of {eta} correspond to sharper distributions. The mean value of the fracture strains is proportional to {gamma}e. A second Weibull distribution P({gamma};{eta},{gamma}b) is used to set the failure of bonds. We study the behavior of the network under adiabatically imposed uniform external strains. Whenever a strut fails, it will be removed along with all associated bond-bending terms in potential energy and it will never be reinstated. The consequence of bond failure is to remove the side that has the weaker coupling constant. We use different random seeds to generate multiple, nominally identical networks.

Fig. 2 shows an example of stress versus strain under compression of such a network. The stress increases nearly linearly with strain until the yield point where the first strut experiences fracture. The stress typically continues to increase until a critical number of struts have failed; the entire network collapses soon after. The maximum value {sigma}max of the stress is considered to be the fracture load of the network. We study how {sigma}max changes when the network is degraded by different mechanisms.



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  		FIGURE 2  Example of stress and strain relation under uniform compression. To compare damage from distinct architectural changes, we relate the reduction in fracture load to the corresponding loss of bone mass.

 

   TYPES OF DAMAGE
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
The aim of our study is to determine the relationship between the density and fracture load of a network when individual modes of bone removal are considered separately. In all our analyses, we assume that material properties of bone tissue remain unchanged.

Isotropic thinning of trabecular elements
Let us first consider consequences of isotropic trabecular thinning. Suppose the strength (fracture stress) of the original, complete network is {sigma}max(0) and its mass is m(0). Now consider a second network constructed by reducing the cross-sectional area of all struts by a fraction t; thus, the mass of the network is reduced by a fraction t (the length of the rods are constant during thinning) and we assume that the bond-bending coefficient is reduced by t2. The thinning is implemented in a series of steps; typically t (initially set to 1) is reduced by 0.03 for 25 steps. The breaking stress and mass of each of these 25 networks are calculated as described. Because both the elastic modulus and the mass of each strut depends linearly on its cross-sectional area, {sigma}max and the mass should maintain a linear relationship. Hence, assuming that the contribution to fracture load from the bond-bending terms is not significant, the reduction in fracture load {sigma}max/{sigma}max(0) will be proportional to the reduction m/m(0) of the mass of the network.

Anisotropic thinning of trabecular elements
There are several distinct ways for a network to be made anisotropic. The mean length of axial trabecular elements can be different from that of off-axial elements. Alternatively, struts along axial and off-axial directions can degrade at different rates. We introduce the following notations to study effects of such changes:

  1. Horizontal and vertical struts are thinned by amounts tH and tV at each step.
  2. The mean length of struts in vertical and horizontal directions are LV and LH.
  3. Elastic moduli in vertical and horizontal struts, kV and kH, span different ranges.

The mass of the system is reduced by the combination of these different factors; for example, in considering anisotropic thinning, reduction in mass will be estimated by using (SVLV/3 + 2SHLH/3). Here SV is the cross-sectional area in vertical elements and SH is that in horizontal elements. After n levels of thinning, cross-sectional areas of vertical and horizontal struts are SV = SV0(1–tVn) and SH = SH0(1–tHn), where SH0 and SV0 are the original values of the variables. The corresponding change in mass of the network is

(3)
In this derivation we have noted that one-third of the struts lie along the axial direction, and the remainder in the off-axial directions.

The equivalent spring constant in the anisotropic media is ~K = kV + kH({delta}z/LH) (the approximation to first term of horizontal contribution as ({delta}x/LH)2), where {delta}z is the deviation of the springs in vertical direction. The breaking stress is thus

(4)
where the model parameter {Delta} = <{delta}z2>. Solving Eq. 3 for n and inserting it into Eq. 4 gives

(5)
Thus, the slope of the {tau} versus m curve will depend on the levels of anisotropy. Nonlinear corrections to the expression can also be relevant.

Random removal of struts
We next consider consequences of random removal of struts from a network. In this scenario, all remaining elements are assumed to retain their strength. Suppose the fraction of elements removed is represented by {nu}, then the reduction of mass is

(6)

Previous analysis (Espinoza et al., 2002Go) on model networks have shown that the reduction in fracture load due to trabecular removal is given by

(7)
where {sigma}max({nu}) denotes the maximum stress of a network with a fraction {nu} of struts removed, and {sigma}max(0) is that of the original unchanged network. The values a1, a2, and {psi} are parameters that depend on the model. Here

(8)
where N is the total number of nodes in the network, and {nu}0 = 0.75 is the bond percolation threshold (Stauffer, 1985Go) for cubic networks. When {nu} approaches {nu}0, u becomes large, and Eq. 7 approaches the power-law {sigma}max ~ ({nu}0–{nu}){psi}.

Damage due to combination of architectural changes
We have studied reductions in fracture load due to a combination of architectural changes. Previous analyses on electrical networks (Espinoza et al., 2002Go) show that the expression in Eq. 7 can accurately capture the reductions in strength. We will test the validity of this conclusion in our networks.

We use combinations of types of degradation described earlier, and reductions in mass are estimated accordingly. For example, if we are dealing with anisotropic removal of trabecular elements, the mass will be proportional to (SVLV{nu}V/3 + 2SHLH{nu}H/3). Here {nu}H and {nu}V are the fractions of horizontal and vertical trabecular removed. We also assume that thinning and trabecular removal are independent processes; thus their combined effect will reduce bone mass according to (SVLV{nu}VtV/3 + 2SHLH{nu}HtH/3).


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Trabecular thinning
The results reported below are obtained by computations on five nominally identical model networks. They each have identical control parameters, but are generated using distinct random seeds. The mean and standard deviations of the variables are provided in Table 1.


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  		TABLE 1  Model parameters on thinning-related processes

 
The star symbols in Fig. 3 represent reduction in fracture load due to isotropic thinning of struts. As expected, the fractional reduction of strength {sigma}max/{sigma}max(0) bears a linear relationship to the reduction m/m(0) of mass.



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  		FIGURE 3  Relationships between fracture load and trabecular mass during isotropic (star symbols) and anisotropic (plus signs and solid circles) thinning as well as trabecular removal (open circles). The best fit of Eq. 7 for the last case is shown by the dashed line.

 
The solid circles represent damage due to anisotropic thinning that is predominantly in the axial direction. At each successive stage the thickness tH of off-axial struts are reduced by 0.017 and that of the axial struts tV are decreased by 0.030. The plus-sign symbols denote results for networks whose axial and off-axial struts are thinned by 0.017 and 0.030 at each stage. Reductions in mass in these networks are calculated using Eq. 3. Nonlinear corrections to Eq. 5 are apparent in these curves.

Trabecular removal
The open circles in Fig. 3 represents reductions in strength caused by random removal at trabecular elements. It is clear that damage caused by removal is far more significant than those due to other factors. As an example, a 30% reduction in mass corresponds to 1), a 30% reduction in fracture load with isotropic thinning; 2), 25% and 40% reductions in fraction loads for the two cases of anisotropic thinning described; and 3) a 70% reduction in fracture load with trabecular removal.

The results from the analysis are displayed in Fig. 3, and the parameters used are summarized in Table 1.

Combination of architectural changes
We have studied the damage caused by the implementation of multiple types of architectural changes. Fig. 4 shows results for four such sets described in Table 2. The star and plus in Fig. 4 correspond to isotropic strut removal only, whereas the others include additional types of damage (Table 2). The variations between runs confirm the architecture-dependent behavior of the system, which reinforce previous observations that architecture of the trabecular bones are important in determining the strength (Dempster, 2000Go; Mosekilde, 2000Go, 1989Go). The dashed lines in Fig. 4 represent the best fits of the data to Eq. 7. The parameters a1, a2, and {psi} are given in Table 2 for each case.



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  		FIGURE 4  Relationship between fracture stress and trabecular mass when isotropic trabecular removal is combined with other architectural changes. The dashed lines in each case represent the best fit to Eq. 7.

 

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  		TABLE 2  Simulation and fitting parameters at removal-related processes

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
ACKNOWLEDGEMENTS
 REFERENCES
 
With aging, the solid cortical segments of bone become brittle, and the porous trabecular bone takes a significant role in load transmission. It is thus necessary to be able to reliably estimate the strength of trabecular bone to determine the need for therapeutic intervention. It is difficult to identify such diagnostic tools because the damage to bone arises out of several distinct types of architectural changes. It is thus necessary to first determine the types of age-related architectural changes that are most damaging, and how important the remaining factors are. Use of mathematical models is perhaps the only way to reach this goal.

The analysis of the model system shows that among known age-related architectural changes in bone, trabecular perforation is most damaging. As an example our results show that the loss of strength due to a 30% reduction in mass caused by trabecular removal is more than double the loss due to other architectural changes. This observation suggests that an expression relating the strength and mass of bone based on an expression of trabecular perforation will be able to describe damage due to multiple types of architectural changes. This expectation was validated in the analysis of model networks. However, it is also clear from Fig. 4 that the relationship between fracture load and density of trabecular bone depends on changes in factors such as the level of trabecular thinning and anisotropy. Consequently, bone density (mass per unit volume) cannot be expected to be a reliable surrogate for the fracture load of a bone.


   ACKNOWLEDGEMENTS
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
REFERENCES
 
The authors thank J. Espinoza-Ortiz and C. Rajapaske for discussion.

This research was partially funded by a grant from the National Science Foundation (Y.S., G.H.G.).

Submitted on May 27, 2004; accepted for publication August 9, 2004.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 DESCRIPTION OF THE MODEL...
 TYPES OF DAMAGE
 RESULTS
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
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M. A. K. Liebschner, R. Muller, S. J. Wimalawansa, C. S. Rajapakse, and G. H. Gunaratne
Testing Two Predictions for Fracture Load Using Computer Models of Trabecular Bone
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