31.3 Mechanics and Computational Issues

Ch-31 Page 6 Monday, January 22, 2001 2:05 PM

31-6

Bone Mechanics

31.2.5 Model Classification: Optimization, Phenomenological, and Mechanistic The models that have been developed for simulating the functional adaptation of bone can be classified into three distinct groups that have different goals and modeling approaches. They are optimization, phenomenological, and mechanistic modeling. Each approach has led to new understanding of bone function and adaptation, yet each has limitations. The application of optimization theory to study the adaptation of bones shows a variety of interesting consequences to the global starting assumptions (e.g., bone adapts to minimize mass; or bone maximizes strength; or bone eliminates notch stresses). These studies give insight into bone as a mechanical structure—and may be especially relevant for simulating evolutionary adaptation. However, when they are applied to an individual’s skeletal adaptation, three major deficiencies become apparent. First, optimization assumes that the adaptation not only seeks to achieve a stated goal (e.g., minimize mass, maximize strength) but that there are physiological processes in place that are directed toward achieving the stated goals. This assumption implies a high level of coordination and environmental awareness without a strong foundation of evidence. Second, implementation of optimization strategies focuses upon achievement of adaptive goals, but not upon the physiological process of the adaptation. Therefore, they are ill-suited to providing strategies that can develop and address questions to advance understanding or manipulation of adaptive processes. Third, by subscribing to an objective strategy to study bone adaptation, fundamental questions—including the degree of site and time dependence—cannot be critically investigated. Consequently, the majority of bone adaptation models have been phenomenological—they seek to describe the stimulus and the response quantitatively, and are useful in developing simulations and predictions of bone adaptation. These models are particularly useful for engineering studies motivated by the desire to be able to design bone implants that will achieve interfaces with surrounding bone tissue that are initially stable and remain so. In addition, these models may provide useful clinical guidance for treatment of a variety of metabolic bone diseases such as the loss of trabecular bone tissue in postmenopausal women or to manipulate the role of tissue generation and consolidation in the fracture repair process. Thus, the development and computational implementation of phenomenological models may be sufficient for describing and predicting the adaptive bone consequences based on changed mechanical loading to improve implant design or to treat some patients.30 However, phenomenological models do not directly contribute to understanding of the biological basis and process of functional bone adaptation. There is also a need to elucidate the interplay of genetics, hormones, drug therapy, and the mechanical environment. This level of understanding requires a mechanistic approach that goes beyond the cause-and-effect-based phenomenological models and instead seeks a detailed understanding of the steps and chemical and biological mechanisms involved in changing bone properties and architecture.30 Further progress will depend upon close coupling of model hypotheses with quantitative experiments to test their validity.

31.3 Mechanics and Computational Issues A number of different models and computational techniques have been proposed for simulating net bone remodeling. To organize the models so that the similarities and differences can be objectively examined and compared, it is useful first to develop general descriptions and also to describe the two different mechanical points of view for simulating functional adaptation. A continuum point of view may consider functional adaptation as a coupled “free-boundary-value problem” (a free boundary has the potential to move, as opposed to a fixed boundary). A continuum approach includes time-dependent descriptions for both the external geometry of cortical bone surfaces (the free boundaries) and the time-dependent descriptions for changes in bone material properties. The material property changes could include changes in the cortical bone material in addition to changes in the structural (apparent) properties of cancellous bone.31 With that point of view, differential


Bone Modeling and Remodeling: Theories and Computation

31-7

equations can be developed to describe changes in cortical surface locations, changes in cortical material properties, and changes in apparent trabecular bone material properties. A micro- or tissue-level point of view acknowledges that the cells responsible for bone resorption and deposition are always on the various bone surfaces—periosteal, endocortical, trabecular, or osteonal (see Chapters 1 and 22). Thus, net remodeling can be described as a free-boundary problem in which the various external geometries—or envelopes—of cortical and cancellous bone surfaces can change over time.32 Differential equations can be developed with this microlevel viewpoint to describe the resorption and deposition that occurs at the bone surfaces.4 When monitored at the continuum level, the change in bone surface locations is manifested as modifications of both the shape and material properties of the bone.

31.3.1 General Assumptions and Equations: Linear Elasticity Using an earlier description for the continuum point of view,31 one can formulate the adaptive problem as an extension of linear elasticity theory (see Chapter 6). Although the constitutive relations could be made more complex to capture nonlinearities associated with material damage or very slow or very fast loading, the simpler linear elasticity equations are generally adequate for describing the mechanical response of bone to many loading conditions.

31.3.2 Adaptation Models A fundamental decision for the theoretical development of adaptation models is whether to consider the process of the adaptation over time, or to merely consider the end point or goals of the adaptation. Most of the phenomenological and mechanistic modeling efforts explicitly incorporate time in their formulation so that the process of the adaptive response can be tracked, beginning from chosen starting conditions. In contrast, the optimization modeling efforts have focused upon the adaptation objectives, so they do not generally present the formulation as time dependent, even though the computational implementation may use iterative procedures that modify the starting conditions to generate the final optimized conditions. 31.3.2.1 Material Properties of Bone can Change over Time In addition to the standard equations of linear, small-strain elasticity, the continuum point of view includes the capability for bone tissue to change its material properties over time. Most generally, this can be written as

[C] [C(t)],

(31.5)

where [C] is the matrix of material properties. As will be described subsequently, a number of specific suggestions have been proposed to describe variations in material properties over time. To incorporate functional adaptation, the nature of the time dependence must be related to a mechanical stimulus. Equations of this form are used to describe changes in continuum-level cortical bone properties, sometimes called net “internal remodeling”33 or “remodeling.”32 Additionally, this general form can be specialized to describe the continuum-level changes in trabecular properties that may be a result of either changes in the degree of trabecular porosity and/or changes in the alignment of the trabecular spicules. In the case of a continuum point of view of trabecular bone, the change in the reorientation of the primary alignment of a group of trabeculae could also be incorporated into Eq. 31.5. 31.3.2.2 Boundary Positions of Cortical Bone Surfaces Can Change over Time The standard linear elasticity formulation may also be modified to include the capability to have bone change its shape, sometimes called “modeling,”32 and sometimes called net “surface remodeling.”34 This feature can be written, most generally, as

S S(t),

(31.6)


31-8

Bone Mechanics

where S is the function describing the boundary positions. As described subsequently, several different specific forms for this equation have been proposed, relating the surface description to the mechanical stimulus. In addition to the use of the surface remodeling equations to describe changes in the external cortices, equations of this form may also be incorporated at the tissue level to describe changes in trabecular spicule size and orientation, and for the shape change of individual osteons, or even the cavities occupied by individual osteocytes.

31.3.3 Computational Implementation The models described subsequently all consider bone adaptation to be a function of mechanical usage. Although analytical methods have been used for some idealized examples, numerical approximation methods are usually employed for quantification of the mechanical usage because of the complexity of bone geometry, material property description and distribution, and time-varying boundary conditions and mechanical loading. The most common computational implementations for net remodeling theories are based on finiteelement methods, which use computational models described by the subdivision of complex shapes into a collection of small but finite-sized elements that constitute the “mesh.” Approximate solutions—in terms of displacements, strains, and stresses—can be found at any point of the structure, with the accuracy of the solution controlled by the mesh refinement. Assuming the appropriate steps are also taken for establishing the finite-element model validity, this represents a general and powerful technique for structural analysis.35–37 To simulate the adaptive response, the finite-element models can be incrementally updated (in terms of geometry and/or material properties) according to equations of the form generally expressed as Eq. 31.5 and/or Eq. 31.6. Because the updated geometry or material properties change the structural behavior, the updated structure must be reanalyzed to find the new displacements, strains, and stresses. The analysis-updating process may be continued until the adaptation is complete, or the user elects to terminate the simulation. The choice for controlling the incremental changes can be time dependent if the time course of the adaptation is of interest, or may be independent of time—as in the case of optimization-based simulations. Although finite-element methods are the most powerful and general numerical technique available for solution of the spatial portion of the adaptation solution, the boundary element technique has also been used.10 The boundary-element technique may be more computationally efficient than the finiteelement method for these free-boundary-value problems because they reduce the dimensionality of the problem by one (e.g., three-dimensional geometries are described by their two-dimensional surfaces). Disadvantages of the boundary-element method include the difficulty of implementing boundary elements for analysis of anisotropic materials, and questions about their convergence behavior and the optimum degree of mesh refinement for accuracy. However, as these methods are developed and as more commercial software packages become available, they can be efficiently coupled with time-stepping methods for simulating the spatial and temporal adaptation of bone.

31.4 Theoretical Models and Simulations: Cortical Bone This section presents a review of many of the theoretical models that have been developed and used to simulate cortical bone adaptation, and includes example applications. To motivate the need for these models, a most-striking example of cortical adaptation is first shown in Fig. 31.2. This human specimen was supplied to Wolff by Roux1,2 and shows that a pseudoarthrosis of the tibia causes a dramatic hypertrophy of the fibula. Wolff 2 describes the specimen: “The fibula, which serves as a substitute for the tibia functionally, is enormous. Its cross section is six to eight times thicker than normal.”



31-9

FIGURE 31.2 A most striking example of compensatory shape changes is the hypertrophy of the human fibula (left) following the development of a pseudoarthrosis of the tibia (right) in this lower-limb specimen supplied to Wolff by Professor Roux. (From Wolff, J., The Law of Bone Remodeling, Springer-Verlag, Berlin, 1986. With permission.)

31.4.1 Flexure-Neutralization Theory and the Three-Way Rule Frost’s “flexure-neutralization theory” was developed to formalize the clinical observations of the realignment of bone and could thus be classified as a phenomenological model. The theory is stated as a rule regarding the relocation of the bone surfaces: repeated nontrivial dynamic bending loads cause strains that cause bone formation drifts to build up upon concave-tending surfaces, while resorption drifts tend to erode convex-tending surfaces,32 as shown in Fig. 31.3. More recently, Frost38 has introduced a “3-way rule” that symbolically represents the observations of his flexure-neutralization theory. The rule states38 that if f(M ) specifies the mechanically controlled kind and rate of modeling activity on a unit area of bone surface, then

M M f ( M )  --------   --------  ,  t   t 

(31.7)

where is the drift operator, characteristic of “every small bone surface region,” calculated by the product of

, the end-load operator, which 1 for frequent compression or zero end loads 1 for frequent tension end loads, and


31-10

Bone Mechanics

FIGURE 31.3 This example of the straightening of a long bone subjected to combined compression and bending is described by the “flexure-neutralization theory” and by the “3-way rule.” Bone formation drifts add to concavetending surfaces, while simultaneous resorption drifts erode convex-tending surfaces. (From Frost, H. M., Introduction to a New Skeletal Physiology, Pajaro Group, Pueblo, CO, 1995. With permission.)

, the local strain operator, which 1 for frequent compression or concave tending bending strains that exceed the absolute value of the “minimum effective signal,” |MES| 0 for strains below |MES| 1 for frequent tensile or convex tending bending strains that exceed the |MES| M - is the drift rate function defined as: The quantity M˙  ------

t

M dk  ------- ( P )  ------ ,  t   dt 

(31.8)

where is the mechanical usage coefficient, P is a biologic proportioning coefficient, and dk dt is the drift rate limit. Frost38 suggests ranges for the values of these parameters. The expected bone response is cataloged in the form of a “truth table” that is reproduced in Table 31.1. However, despite the fact that the “3-way rule” states the flexure-neutralization theory symbolically, it is not yet in a form that can be readily incorporated into a computational net bone remodeling program. The major difficulty that must be addressed before the ideas can be used computationally is the dependence of local information, such as the drift operator, , upon global information—including the end-load operator and whether that site is a region of concave or convex bending. Based solely upon the strain at a point, the bone would have no way of “knowing” or sensing the end load or whether or not the strain is a result of concave or convex bending. But, if Frost’s ideas can be reformulated in terms of components of the strain,


31-11


TABLE 31.1 Gamma Truth Table: A Catalog of the Adaptive Response of Bone to Mechanical Usage

1 1 1 1 1 1

R/O/Fa

1 0 1 1 0 1

1 0 1 1 0 1

F O R R O F

aR/O/F: R resorption drift, O no drift, F formation drift. Source: Frost, H. M., Anat. Rec., 226(4), 403–413, 1990. With permission.

strain rate tensors, or strain gradients, rather than rules that include the loading conditions, then his observations may lead to a more fundamental understanding of the functional shapes of long bones.32 31.4.1.1 Example Application: Curvature Reduction Despite the limitations for computational implementation of the three-way rule, it has been used in a series of “thought experiments.” One “experiment,” shown in Fig. 31.3, is described by Frost as one of the “principal structural adaptations of bone”—the diaphysis of a curved long bone that is overloaded with bending and a compression end load.38 The in vivo response is described: “formation drifts arise on concave-tending surfaces, and resorbing drifts on the convex-tending surfaces. Those drifts reduce the bone’s curvature, and also its subsequent bending under the same loads.”38 For this case, the parameters of the three-way rule needed to describe symbolically the adaptation on the concave-tending surfaces are 1, 1, so that 1 and formation drifts should occur; the parameters on the convextending surfaces are: 1, 1, so that 1 and resorption drifts should occur.

31.4.2 Adaptive Elasticity Beginning in 1976, Cowin and co-workers developed and published a series of papers on “Adaptive Elasticity.”17,33,34,39–44 Adaptive elasticity is a comprehensive phenomenological theory that is based on linear elasticity theory supplemented by additional constitutive equations that allow for changes in the density—hence the stiffness—of bone and also changes in the external shapes of bone. Based on customary loadings, material properties, and boundary conditions, the mechanical condition of remodeling equilibrium could be described by the remodeling equilibrium strain state, E0. Then a new strain state, E, produced by altered loading or implantation of a bone prosthesis, generates local strain differences (E 0 E ). The differences are used to formulate first-order remodeling rate equations that dictate the change in bone density and/or shape. For net surface remodeling, Cowin and Van Buskirk34 proposed the following equation (rewritten to be consistent with the prior notation):

S(Q)

0

B { E E },

(31.9)

where S is the speed (change in position over time) of the external bone surface at point Q, in a direction normal to the existing surface; B is the row vector of remodeling rate parameters at the point Q; E 0 is the strain vector at point Q; and E is the remodeling equilibrium strain vector at point Q. Note that the implementation of this idea could incorporate different degrees of site dependence: 0 if both E and B were everywhere the same, then Eq. 31.9 would give results that are completely independent of site; if only B is constant, then the net remodeling rate would depend only upon the 0 “error” between E and E to determine the amount of net remodeling; if B was a different constant on each of the four different bone envelopes, there would be a degree of site dependence, and if B was different at each point, that would represent absolute dependence upon site.


31-12

Bone Mechanics

For describing the material property changes in cortical bone, Cowin and Hegedus39 proposed the following rate equation:

e˙ a ( e ) tr [ A ( e )E ],

(31.10)

where e˙ is the rate of change in solid fraction of bone, a(e) is a function of the current solid fraction, A(e) is a remodeling rate parameter that is dependent upon the current solid volume fraction, and E is the strain tensor. 31.4.2.1 Example Applications: Ulnar Osteotomy Eq. 31.9 has been used extensively for net surface remodeling simulations. First, an analytical example using changed axial loads demonstrated the capability of the theory to describe the eight stable solutions for changes to an idealized diaphysial cross section (involving various combinations of bone addition and removal at the endocortical and periosteal surfaces).40 The theory was also incorporated into a beamtheory-based computer program for net surface remodeling simulations and used to estimate the remodeling rate parameters, B , for five in vivo experiments that had previously been reported in the literature.45 For these examples, the computational descriptions required site-specific constants to have the simulation match the experimental results, but due to limited information, estimates of the average axial strain, E3avg, were considered. Consequently, most of the remodeling rate constants, B1, B2, B4, B5, and B6 were zero, with only B3 nonzero. An example that simulated the results of an ulnar osteotomy in sheep is shown in Fig. 31.4. To simulate the experimentally observed adaptation in this example, a moderate degree of site specificity was required: because there were no net changes observed on the endocortical surface, B3 on the entire endocortical surface was zero; however, because the rate of bone addition on the periosteal surface displayed differences between the tensile and compressive sites, two different values of B3 were necessary for points on the periosteal surface to account for the observed differences. The adaptive elasticity theory was also incorporated into a special-purpose finite-element program, RFEM3D (Remodeling Finite Element Method–3-Dimensional) that has been used to examine a variety of situations including idealized models of tubular bones showing the same eight stable solutions as previously found analytically;31 experimental rabbit tibia subjected to superimposed bending loads;4 idealized tubular models to study the transient effects of gradually stiffening bone added at the periosteal surface;46 idealized models for straightening curved bone;47 and for blind prediction of bone adaptation from in vivo ulnar osteotomy experiments.48 Fig. 31.549 shows the same adaptation problem as illustrated in Fig. 31.4, and uses the same loading and remodeling rate parameters. However, this implementation is based on the finite-element method and is suitable for a wider range of structures than would be applicable with the beam-theory-based program described previously.

31.4.3 Cell Dynamics Model In an attempt to build upon the phenomenological framework provided by adaptive elasticity, a more mechanistic approach, based upon the schematic of the net remodeling process illustrated in Fig. 31.1, was developed.4,50,51 These complementary ideas are based on the concepts of geometric feedback described by Martin52 that cast the bone adaptation process as being dependent upon cellular activity and number and the surface area available for resorption and formation. The ideas are summarized in the following where a general description is first developed and then specialized. In general, the loads imparted to the bone will be a function of time. The resulting strain, E(x, t), at a point x and time, t, is then dependent on the material properties, C (that may also depend on position for nonhomogeneous material models), and the applied loads. The theory assumes that the strain remodeling potential, denoted by p(x, t) is a function of the strain history. The cumulative nature of the influence of the strain upon the strain remodeling potential can


31-13


FIGURE 31.4 The ulnar osteotomy experiment in mature sheep performed by Lanyon et al.62 Each illustration represents the bone cross section in the forelimb. (Top) The contralateral control limb at 9 months showing the radius with the intact ulna. (Middle) The experimental radius at 9 months, showing marked asymmetric periosteal depostion. (Bottom) The calculated net remodeling. Solid line represents surfaces of the control bone, dashed lines represent the computational description at 9 months. The indicated length is 1 mm. (From Cowin, S. C., Hart, R. T., Balser, J. R., and Kohn, D. H., J. Biomech., 18(9), 665, 1985. With permission.)

be expressed as an integral over time in the following form:

p ( x, t )

t

f { E ( x, t ) } dt

or

p ( x, t )

0

f { E ( x, ( t ) ) } d .

(31.11)

A more concise notation can be achieved by replacing the integral with a symbol denoting the history of strain as follows:

p ( x, t ) H p { E ( x, ( t ) ) }

for

0 ,

(31.12)

where Hp is a history functional that is dependent on the strain, the position, and time. As mentioned previously, the appropriate period over which to integrate these equations is not yet clear due to the various timescales that are important in the net remodeling processes.


31-14

Bone Mechanics

FIGURE 31.5 A finite-element simulation (RFEM3D) of the adaptation to the ulnar osteotomy shown in Fig. 31.4. (Top) the initial configuration showing the radius and intact ulna; contours map the axial strain at remodeling equilibrium, E ozz . (Middle) the forelimb at the start of the remodeling simulation immediately following the removal of the ulna. Contours map the axial strain distribution, Ezz. (Bottom) the simulated net remodeling of the ulna after 9 months, using the same remodeling rate constants as found for the simulation shown in Fig. 31.4, with the axial strain contours close to those for remodeling equilibrium, seen in the top. (From Hart, R. T., in Science and Engineering on Supercomputers, E. J. Pitcher, Ed., Computational Mechanics Publications, Boston, 1990. With permission.)

The relationship between remodeling and the remodeling potential comprises the effects of the potential on both the average cellular activity rate and the effect of the potential on the cellular recruitment. For the average osteoblast activity rate, i b, and the average osteoclast activity rate, i c,

a i H ai [ p ( x, t ) ] a iM a iH a iG ,

i b,c,

0 ,

(31.13)

where ai is the average volume rate of bone that is deposited per active osteoblast (or resorbed per active osteoclast), and Hai is the history functional relating cellular activity to the strain remodeling potential.


31-15


Here it is assumed that the metabolic, hormonal, and genetic factors are additive, and specified by aiM, aiH, and aiG, respectively. A similar relationship can be written for the number of active osteoblasts (or osteoclasts) per forming area as

n i H ni [ p ( x, t ) ] n iM n iH n iG,

i b,c,

0 .

(31.14)

Finally, since in general the entire surface area will not be available for both formation and resorption, that fraction of the entire surface area that is available for formation is denoted b and the fraction available for resorption is c. In general, b and c could be functions of time, and could depend on the current magnitude of remodeling. They are related as follows:

b c 1.

(31.15)

Thus, on the external surface, or the surfaces of internal voids, the velocity of the surface is written as the following rate equation:

d˙ n b b a b n c c a c .

(31.16)

where positive values for d˙ indicate that bone is being deposited, and negative values for d˙ indicate that bone is being actively resorbed on the surface. This equation, first proposed by Martin,52 includes regulation of cellular number and activity by the available surface area per unit volume of bone as well as a strain-induced potential. To apply the basic concepts embodied in the above equations to specific circumstances, the functional forms must be specified. Several specific relationships have been developed.3,51 Only the simplest type is described next, but rather than consider the cell population constant as done previously51 the following case is for constant cellular activity. 31.4.3.1

Constant Cellular Activity: A Simple Adaptive Elasticity Model

The simplest relationship for the general equation (Eq. 31.12) relating the proposed strain-induced remodeling potential and axial strain is linear and can be written as follows:

p Cp E3 ,

(31.17)

where Cp is a constant coefficient. If it is assumed that the cellular population is linearly dependent on the strain-induced potential, then the number of osteoblasts (or osteoclasts) can be expressed as

n i C ni p n i0 .

(31.18)

The simplifying assumption of setting the activity of the cells equal to a constant (representing the original activity of the cells) results in

a i a i0 .

(31.19)

d˙ C 1 E 3 C 2 ,

(31.20)

Then Eq 31.16 is reduced to

where

C 1 C p ( a b0 C nb b a c0 C nc c )

and

C 2 n b0 a b0 b n c0 a c0 c .

(31.21)


31-16

Bone Mechanics 0

The axial strains at which no net remodeling occurs are denoted E 3 and are simply

C2 0 E 3 -----. C1

(31.22)

This simple relationship is then equivalent to the linear relationship between surface remodeling and strain proposed by Cowin and Van Buskirk.34 However, the present model uses constants that are not purely phenomenological, but are related to the indicated biological parameters that could be measured and possibly manipulated. 31.4.3.2

Example Application: Ulnar Osteotomy

The RFEM3D program that has been used with the theory of adaptive elasticity was written to also incorporate the adaptive mechanisms shown in Fig. 31.1. Oden 48,53 has developed and implemented the cell dynamics model into the program, and has made a priori predictions of the adaptation of canine radii following an ulnar osteotomy.19,48,53 A variety of simulations were conducted to study a range of possible adaptive responses for the radius 6 months following the ulnar osteotomy. Some simulations used estimates for the remodeling rate constants from earlier studies45 with the theory of adaptive elasticity. Other simulations were performed using the cell dynamics models with the remodeling rate constants estimated from histomorphometric measures taken 1-month postosteotomy and used with the 6-month simulations. A variety of mechanical stimuli were also used including the axial maximum strain magnitude during gait, E3max, and Eˆ , the “mechanical intensity scalar”13 (also calculated at the peak values during gait). Once the blind predictions were made for the response of the radius 6 months following the ulnar osteotomy, they were compared to the experimental results,53,54 shown in Fig. 31.6. Fig. 31.6a shows the subdivision of a radius cross section with 12 stations for measurement of cortical wall thickness.53 Fig. 31.6b shows the comparison of experimental results with those predicted from using adaptive elasticity theory with remodeling coefficients estimated from Cowin et al.45 and based on axial strain differences. In a qualitative sense, the prediction is quite good—the pattern of adaptation is closely predicted around the sectors, except in sector 9, which experimentally experienced resorption that was not predicted by the model. A somewhat better qualitative prediction is shown in Fig. 31.6c that incorporated the ‘mechanical intensity scalar,’ Eˆ , with remodeling coefficients used for the 6-month prediction that were estimated from the histomorphometric quantification of cell number, activity, and distribution from a specimen 1 month after the ulnar osteotomy.

31.4.4 Site-Dependent, Strain Energy Density Model A phenomenological approach similar to Cowin’s adaptive elasticity theory for shape changes has been developed by Huiskes et al.8 with several differences. One difference was implemented to model the observed bone behavior that shows a threshold of mechanical stimulus is required to start the process of net remodeling, sometimes referred to as a “lazy zone”7 or a “dead zone.” In addition, the mechanical stimulus is not taken to be the strain tensor, but a scalar, the strain energy density,11,55 defined in Eq. 31.1. The rate equation is written as

 C x [ U ( 1 s )U n ]; dX  -------  0; dt   C x [ U ( 1 s )U n ];

U ( 1 s )U n   U n U ( 1 s )U n ,  U ( 1 s )U n 

(31.23)

where X is the surface growth, Cx is the remodeling rate coefficient, 2s is the width of the “lazy zone,” and Un is a “site-specific homeostatic equilibrium strain energy density.” A two-dimensional



31-17

FIGURE 31.6 Comparison of experimental and predicted results for a canine ulnar osteotomy experiment. (Top) The subdivision of a canine radius cross section with 12 stations for measurement of cortical wall thickness for comparison of experimental and predicted results following a ulnar osteotomy. (Middle) Percent difference in wall thickness (comparing experimental and contralateral radii) at 6 months following an ulnar osteotomy, showing experimental and predicted results. Computed results used adaptive elasticity theory with remodeling coefficients estimated from Cowin et al.,45 and based on axial strain differences. (Bottom) Same as above, but the predictions ˆ . Further, the remodeling coefficients were made using the cell dynamics model with the mechanical intensity scalar, E used for the 6-month prediction were estimated based on the histomorphometric quantification of cell number, activity, and distribution from a specimen 1 month after the ulnar osteotomy. (From Oden, Z. M., Ph.D. dissertation, Tulane University, New Orleans, 1994. With permission.)

finite-element implementation based on the use of NONSYS and MARC (Marc Analysis Corporation, Palo Alto, CA) was developed for simulating bone adaptation for both net internal (density changes) and surface (shape changes) remodeling, using a forward Euler integration time-stepping algorithm.8 For surface remodeling, surface points are relocated normal to the surface in each time step as


31-18

Bone Mechanics

follows (with s 0):8 j

j

j

j

j

X X ( t t ) X ( t ) tC x { U ( t ) U n },

1 j m,

(31.24)

where m is the number of surface nodal points concerned. In this implementation, the magnitude for the remodeling rate constant, Cx, was chosen arbitrarily but care was taken to ensure stability of the iteration process by choosing sufficiently small time steps, t. Consequently, only the final result of the remodeling process was considered. The temporal evolution was not a focus of the study.8 31.4.4.1 Example Application: “Stress-Shielding” The finite-element procedures that implemented Eq. 31.23 were used by Huiskes et al.8 to study the potential consequences for design changes of an intramedullary fixation stem. Fig. 31.7a shows the simplified two-dimensional configuration showing a cortical shell with the displacement boundary conditions, an intramedullary rod, and the applied bending moment. Side-plate elements simulated the threedimensional nature of the problem. For net remodeling simulations, a number of parameters were varied: stem diameter, remodeling threshold, and implant–bone contact conditions. Computed results simulated resorption of the cortical shell—a phenomenon described as “stress-shielding”—that was most influenced by differences in stem rigidity. Fig. 31.7b shows the loss of bone that was simulated with the use of the most flexible stem design. In contrast, Figure 31.7c shows much more severe bone loss as a consequence of using the net remodeling simulation with the stiffest design.

31.4.5 Homogeneous Stresses at the Surface An optimization approach has been taken by Mattheck and co-workers56,57 in developing the “CAO,” a computer-aided optimization hypothesis. Mattheck makes the observation that “a good mechanical design is characterized by a homogeneous stress distribution at its surface.” As a result, a computerized implementation of CAO was developed and has been used to simulate shape change of bone (and plants). The implementation is possible with commercial finite-element codes that support swelling materials or thermal expansion (e.g., ABAQUS from Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI). The finiteelement model is constructed with a uniformly thick surface layer. The method is based on changing the volume of the surface layer of elements as follows:

˙ v k ( M ref ),

(31.25)

where ˙ v is defined as the volumetric swelling rate, M is the von Mises equivalent stress, ref is a reference value of the von Mises equivalent stress, and k is a rate constant. 31.4.5.1

Example Application: Curvature Reduction

The CAO implementation has been illustrated with a number of examples, including cortical bone, trabecular bone, and—most extensively—tree adaptation.58 The example shown in Fig. 31.8 is for the surface adaptation of a femur that is broken and then heals with a malposition.57,58 It is very similar to the “thought experiment” application of the three-way rule, previously described and shown in Fig. 31.3. Bilinear four-noded plane strain elements were used with homogeneous and isotropic values for Young’s modulus and Poisson’s ratio. The initial finite-element model is intended to represent the structure shortly after fracture, and at the terminal stage of the simulation, the bone has healed and has a homogeneously distributed von Mises stress along the bone surfaces. Although the changes are applied incrementally, this optimization scheme is not intended to describe the physiological process of the adaptation.



31-19

FIGURE 31.7 Example application—“stress shielding.” An analysis of intramedulary fixation using the site-dependent strain energy density equations. (a) The simplified two-dimensional configuration showing a cortical shell with the displacement boundary conditions, an intramedullary rod, and the applied bending moment. (b) Shows the loss of bone that was simulated with the use of the most flexible stem design. (c) Much more severe bone loss is shown as a consequence of using the net remodeling simulation with the stiffest design, shown in (c). (From Huiskes, R. et al., J. Biomech., 20(11–12), 1135, 1987. With permission.)


31-20

Bone Mechanics

FIGURE 31.8 Simulation of the gradual straightening of a femur that is broken and heals with a malposition. The simulation, not explicitly time-dependent, shows the implementation of the “CAO” program that seeks a shape that minimizes notch stress. (From Mattheck, C., Design in Nature, Springer-Verlag, Berlin, 1998. With permission.)

31.4.6

Strength Optimization: Fatigue Damage and Repair

A more mechanistic method—that stops short of directly addressing the biological processes of adaptation—has been developed based on the hypothesis that “bone adapts to attain an optimal strength by regulating the damage generated in its microstructural elements.”59 Prendergast and Taylor59 hypothesized that there is some damage at RE (remodeling equilibrium), and that the rate of repair is associated with the damage rate.



31-21

Mathematically, at RE, ˙ eff 0 and ˙ RE, where eff is the effective damage, ˙ is the current rate of damage production, and RE is the rate of damage production at RE. Then,59,60

dX ------- C eff C dt

t t0

( ˙ ˙ RE ) dt,

(31.26)

where X measures the extent of the remodeling process. This expression can be implemented as a sitespecific theory by variation of either the constant, C, based upon location, or by including an initial variation in the remodeling equilibrium damage. Interestingly, it has been shown that the use of the damage stimulus is equivalent to using the strain energy density,61 although the damage stimulus may be easier to investigate experimentally. 31.4.6.1

Example Application: Ulnar Osteotomy

The same ulnar osteotomy experiment by Lanyon et al.62 that was simulated using the theory of adaptive elasticity by Cowin et al.45 has also been simulated using the strength optimization method.60 A threedimensional finite-element model of the sheep radius and ulna was developed using eight-noded linear isoparametric elements, shown in Fig. 31.9a. A finite-element analysis of the intact forelimb provided the distribution of damage at the remodeling equilibrium, calculated with a load simulating static stance. Then, following the removal of a portion of the ulna, and using a uniform constant, C, they generated a plot of the stress differences and damage distribution near the endocortical and periosteal surfaces, shown in Fig. 31.9b. Although the analysis shows the distribution of the adaptation stimulus rather than incrementally changing the shape, the strength optimization analysis needed only one remodeling rate constant, C. This is in contrast to three different remodeling rate constants needed to produce the simulation results shown in Figs. 31.4 and 31.5. “In particular the low strains on the endosteal surface, though they generate an appreciable strain and corresponding stress difference, do not generate a significant remodelling stimulus with this approach … this suggests that a damage variable may contain more of the nonlinearity of the remodelling process than does a strain variable.”60

31.5 Theoretical Models and Simulations: Trabecular Bone This section provides a review of many of the theoretical models that have been developed and used to simulate trabecular bone adaptation, and closely follows a recent review of finite-element modeling for trabecular bone.30 Although continuum-level characterizations of the mechanical properties for trabecular bone depend primarily upon the solid volume fraction (i.e., the volume of pores is discounted), they also depend on the predominant orientation of the trabeculae. Thus, adaptation models intended to describe changes in continuum material properties have been developed to describe density changes with and without changes in anisotropy. In contrast, tissue-level descriptions are dependent only upon descriptions of trabecular shape—their size and orientation changes are manifested as volume fraction and orientation changes at the continuum level.

31.5.1 Adaptive Elasticity The theory of adaptive elasticity discussed previously for cortical bone adaptation has been supplemented to also simulate trabecular bone adaptation at the continuum level. In addition to a description for changing material properties, Eq. 31.5, by changing the volume fraction of material, Eq. 31.10, the changing anisotropy of material is also now described.


31-22

Bone Mechanics

FIGURE 31.9 Another finite-element study of the adaptation to the ulnar osteotomy shown in Fig. 31.4. (a) Finiteelement model of the intact sheep radius and ulna was developed using eight-noded linear isoparametric elements and analyzed to provide the distribution of damage at the remodeling equilibrium, corresponding to a load to simulate a static stance. (b) Approximate visualization of the stress difference for the intact and osteotomized conditions, and the corresponding damage-rate distribution calculated using the strength optimization equation. The dashed lines represent the stress; the crosshatched area represents the calculated damage distribution. (From McNamara, B. P. et al., J. Biomed. Eng., 14(3), 209, 1992. With permission.)


31-23


Cowin defined a mathematical expression of Wolff ’s trajectorial theory,41 and the mathematics also have an appealing intuitive description.63 The basis of the mathematical expression is the use of a secondrank tensor, called the fabric tensor,64 defined by stereological measures of predominant trabecular orientation.65,66 (Chapter 13 has a discussion of these measures.) Mathematically, Cowin points out that at remodeling equilibrium—defined in this case as the alignment of the trabecular orientations with the principal stresses—the product of any two of the second-rank tensors that define stress, T, strain, E, and the fabric tensor, H, must be commutative. Then, using the notation of X0 to represent the equilibrium value of X (where X can be T, E, or H) Wolff ’s trajectorial theory can be expressed as 0

0

0

0

T E ET ,

0

0

0

0

T H H T,

0

0

0

0

H E EH .

(31.27)

The intuitive description comes from using an ellipsoid to represent a second-rank tensor geometrically (strain, stress, or fabric) with the normalized length of the axes of the ellipsoid proportional to the value of the eigenvalues of the tensor, and the orientation representing the principal directions.41,63 Then, Fig. 31.10a can be used to represent an initial situation in remodeling equilibrium, indicated by the common alignment of the three ellipsoids used to represent the stress, strain, and the fabric. A perturbation in the stress is represented in Fig. 30.10b as a new alignment of the stress ellipsoid. The strain, which responds instantaneously to the changed stress, also has a new alignment, but is not yet aligned with the stress. The trabecular bone fabric, unable to change instantaneously, retains the original alignment. After some time during which the trabecular bone has time to adapt to the new stress imposed in Fig. 31.10b, a new remodeling equilibrium is achieved, as seen in Fig. 31.10c, where the ellipsoids are once again in alignment with each other, but in the new direction dictated by the new stress imposed in Fig. 31.10b.30 If only the relative degree of anisotropy is of interest, then the fabric tensor H can be normalized by making the trace equal to one, i.e., trH 1, and it becomes easier to manipulate mathematically.44

FIGURE 31.10 Intuitive description of Cowin’s ideas for trabecular realignment. (Top) Representation of an initial situation in remodeling equilibrium, indicated by the common alignment of the three ellipsoids used to represent the stress, strain, and the fabric. A perturbation in the stress is shown (middle) with a new alignment of the stress ellipsoid. The strain, which responds instantaneously to the changed stress also has a new alignment, but is not yet aligned with the stress. The fabric, unable to change instantaneously, retains the original alignment. After some time during which the trabecular bone has time to adapt to the new imposed stress, a new remodeling equilibrium is achieved (bottom). The ellipsoids are once again in alignment with each other, but in a new direction. (From Hart, R. T. and Fritton, S. P., Forma, 12, 277, 1997. With permission.)


31-24

Bone Mechanics

Then, the deviatoric part of the normalized fabric tensor is a measure of the relative degree of anisotropy denoted by K, and is defined as44

K H ( 13 )I.

(31.28)

The stress–strain relationship can then be written as

T D ( K, e )E,

(31.29)

where e 0 is a measure of the change in solid volume fraction, , from a reference solid volume fraction, 0. This shows that the mechanical properties that express Hooke’s law are explicitly expected to be functions of both the amount of material present, as measured by the solid volume fraction, , and the orientation of the material, as quantified by a measure of the fabric, K. If it is assumed that the objective of the remodeling process is to reestablish the remodeling equilibrium strain, then a specific bone location is in RE when the eigenvalues of the strain tensor, E1, E2, E3, are equal 0 to (or within an acceptable range of) the eigenvalues of the remodeling equilibrium strain tensor, E 1. 0 0 44 E 2 , E 3. Then two general rate equations can be written to describe the mechanically adaptive process of trabecular bone. One equation describes how the fabric changes with time,

dK dK ( K, E, e ) ------- ----------------------------- , dt dt

(31.30)

and a second equation describes how the solid volume fraction changes with time,

de ( K, E, e ) de ----- ---------------------------- . dt dt

(31.31)

The process of porosity and fabric changes would continue until the material was entirely void, entirely solid, or at remodeling equilibrium. To specialize and implement the ideas in Eqs. 31.30 and 31.31, Cowin et al.44 wrote them in polynomial form. Then, if higher-order terms are neglected, and if it is assumed that (1) the rate of change of the fabric depends only upon the fabric and the deviatoric strain (which measures distortion), and (2) that the rate of change of the bulk density depends only upon the bulk density and upon the dilatational strain (which measures change in volume), then Eqs. 31.29 through 31.31 can be approximated as

T ( g 1 g 2 e ) ( trE )I ( g 3 g 4 e )E g 5 ( KE EK ) g 6 ( I ( tr ( KE ) ( trE )K ) ), 0 0 0 0 3 dK ------- h 1 ( Eˆ Eˆ ) h 2 I ( trK ) ( Eˆ Eˆ ) -- ( K ( Eˆ Eˆ ) ( Eˆ Eˆ )K ) , 2 dt

de ----- ( f 1 f 2 e ) ( trE trE 0 ), dt

(31.32)

(31.33)

(31.34)

where the circumflex indicates the deviatoric part of the indicated tensor, and g1 to g6, h1, h2, f1, and f2 represent constants. Cowin et al.44 calls this set of equations the “approximate noninteracting microstructure theory.” 31.5.1.1

Example Application: Fabric Reorientation

The RFEM3D program, described previously for simulations of cortical bone adaptation, has been enhanced with the implementation of Eqs. 31.32 through 31.34.67 Several changes were made to the program to allow for evolution of trabecular porosity and orientation in the element stiffness matrices for trabecular bone elements.67,68 To test the implementation of the trabecular remodeling theory, an idealized finite-element



31-25

FIGURE 31.11 RFEM3D simulation of idealized trabecular bone reorientation. (Top row) The initial remodeling equilibrium stress, strain, and fabric tensors. (Middle row) The new fabric tensor applied, with altered stress and strain tensors shown. (Bottom row) The new remodeling equilibrium stress, strain, and fabric tensors. (From Hart, R. T. and Fritton, S. P., Forma, 12, 277, 1997. With permission.)

model of a section of trabecular bone was constructed, representing a small column (10 10 20 mm) of trabecular bone (see Fig. 31.11). To ensure accurate results, a convergence test was conducted and an appropriate model (16 elements and 141 nodes) was chosen. Each element in the model measured 5 5 5 mm, the approximate minimum-length scale for valid fabric tensor measurement.69 For equilibrium conditions, the fabric was aligned with the global axes. Prescribed displacements in the axial direction were applied to the model, with the bottom surface constrained in the axial direction, along with the middle bottom node constrained in all three directions to prevent rigid-body motion. The initial equilibrium strains and stresses were calculated, and are shown along with the initial fabric tensors in the top row of Fig. 31.11. The tensors are illustrated with the principal components emanating from an icon placed at the centroid of each element. To perform an idealized example of trabecular bone adaptation, the stress and strain fields were altered by changing the principal fabric directions to an off-axis angle of 30 (Fig. 31.11, middle row). RFEM3D was then run using the trabecular remodeling routines incorporating Eqs. 31.32 through 31.34 to simulate the changes in the fabric tensor and solid volume fraction for the column. The remodeling rate constants were the same as in Cowin et al.,44 and were chosen so the net remodeling process takes approximately


31-26

Bone Mechanics

160 days. After the simulated 160 days of net remodeling, results show that the stress, strain, and fabric tensor principal directions coincided once again (Fig. 31.11, bottom row). These results indicate that this computational implementation of the trabecular remodeling theory is able to simulate trabecular reorientation in response to altered stress and strain fields. The example was chosen as a convenient way to change the stress and strain fields in the bone, and the program was shown to reorient the bone correctly back to its equilibrium orientation similarly to the intuitive description of the alignment of the tensors shown in Fig. 31.10. 31.5.1.2 Trabecular Density and Orientation Models for Skeletal Morphogenesis Using a fundamentally different approach from adaptive elasticity, Fyhrie and Carter11 developed a theory to predict both the apparent density (or volume fraction) of bone and the trabecular orientation for a continuum description of trabecular bone material. First, they assumed that bone is an anisotropic material that tends to optimize its structural integrity while minimizing the amount of bone present. For trabecular adaptation simulations, they introduced the notion of an objective function of the form Q(,, T), which is dependent upon the apparent density (or volume fraction), , the orientation, , and the stress, T. Then, two special cases were examined: the case where the net bone remodeling objective is based upon the strain energy and the case where the objective is based upon the failure stress.11 An extension of this work to a comprehensive theory for relating local tissue stress history to skeletal biology was developed by Carter et al.12 The theory goes beyond the usual ideas for modeling functional adaptation by relating “tissue mechanical stresses to many features of skeletal morphogenesis, growth, regeneration, maintenance, and degeneration.”12 They make a case for the consistency of the theory with many of the features of skeletal biology, and argue that the role of mechanical influence upon the biological processes is more important than has previously been recognized. This assertion brings to focus the lack of verification for the ideas about the relative importance of genetics and environment in the development, growth, and maintenance of structural tissues. In an effort to use mechanical input as a regulator of the trabecular architecture in the proximal end of a femur, Carter et al.70 used finite-element models with multiple loads, noting that a single loading condition cannot reasonably be expected to be the stimulus for the full trabecular architecture. The time rate of the process was not accounted for, but rather it was assumed that there was a relationship between element density and an “effective stress” written as c

  K  ni M  i  i1 

( 12M )

,

(31.35)

where c is the number of discrete loading conditions, K and M are constants, and n is the number of loading cycles at the “energy stress,”11,12 defined as energy 2E avg U. As described subsequently, the objective of the study was to examine the influence of mechanical factors upon the trabecular architecture, and of note is the use of a non-site-specific rule (i.e., the remodeling objective is the same for all points in the bone). Beaupré et al.71 modified and extended the ideas to develop a time-dependent modeling/remodeling theory. The model is based upon using a daily “stress stimulus” with a continuum level measure, , defined as

 n i mi   day 

1m

,

(31.36)

where ni is the number of cycles of load type i, i is the continuum level “stress” as above, and m is an empirical constant. Then, a piecewise linear approximation to an assumed bone deposition rate vs. tissue



31-27

stress stimulus is proposed that could be used for “modeling,” i.e., net changes in external bone shape similar to Eq. 31.23:

 c 1 ( b bAS ) ( c 1 c 2 )w 1 ;   c 2 ( b bAS ); ˙r   c 3 ( b bAS );  c ( ) ( c c )w ; bAS 3 4 2  4 b

( b bAS w 1 ) ( w 1 b bAS 0 ) ( 0 b bAS w 2 )

(31.37)

( b bAS w 2 ),

where bAS is the attractor state stress stimulus (for remodeling equilibrium), b is the actual stress stimulus, c1, c2, c3, and c4 are empirical rate constants, and w1 and w2 define the width of the normal activity region.71 The normal activity region allows for a “window” of values for the stress stimulus in which there would be no signal for net remodeling responses.29 In addition, by using Eq. 31.37 with histomorphometric relations and the data of Martin72 for descriptions of bone apparent density and surface area density, Beaupré et al.71 write a consistent equation for the rate of change of density (net internal remodeling):

˙ r˙S v t,

(31.38)

where ˙ is the time rate of change of the apparent density, r˙ is the linear rate of bone apposition, S v is the bone surface area density, and t is the “true density” of the bone tissue, assumed to be the density of fully mineralized tissue. Thus, an error between the remodeling equilibrium state (called here the attractor state) and the present mechanical state drives the signals for bone formation or resorption. More recently, Jacobs et al.73 have proposed a different method for simulation of net trabecular remodeling that can account for both changes in the volume fraction of material present and in the alignment of the material. Their approach does not explicitly divide the realignment and the volume fraction changes as done by Cowin et al.44 Rather, Jacobs has chosen to operate directly upon the anisotropic stiffness tensor (with up to 21 independent constants, depending on the degree of anisotropy) that relates the stress and the strain. In that fashion, no a priori assumptions are needed for describing the elastic symmetry of the continuum approximation of trabecular bone behavior. This is significant because, although trabecular bone may be well characterized at the continuum level using orthotropic material properties (9 of the 21 constants are independent), there is reason to believe that during active realignment, a more general degree of anisotropy may be needed. Unfortunately, finding the 21 needed constants may not be practical, and the problem is now to assemble the structural stiffness matrix in its entirety. Thus, the advantage of separating out the measurable material properties (the bone matrix material properties) and the measurable structural properties (such as the fabric tensor) from the structural stiffness is lost. With the general approach of adding evolution equations to the general anisotropic stiffness tensor, two special cases have been studied.73 One approach was to assume that the stiffness changes in the stiffness tensor occur to maximize mechanical efficiency. The second case, more consistent with Wolff ’s ideas, was for stiffness changes to occur in the principal stress directions. For this second case, the time rate of change of the stiffness tensor is written as the sum of the changes in the isotropic and anisotropic portions of the tensor. If the stress, T, is given by the product of the stiffness tensor, C, and the strain tensor, E, then the rate equation is written as the following:73

C˙ C˙ iso C˙ aniso .

(31.39)

The isotropic rate equation is the same as has been used previously, Eq. 31.37 by Beaupré et al.71,74 For the anisotropic portion, the magnitude of each principal stress is compared to the average magnitude of the principal stresses. If a principal stress component is larger (smaller) than the average, then the


31-28

Bone Mechanics

magnitude of the stiffness corresponding to the component in question is increased (decreased). Thus, stiffening of the material property tensor occurs in the direction of overloading, whereas softening occurs in the direction of underloading.73 31.5.1.3 Example Applications: Trabecular Bone in the Femoral Head A 1989 paper by Carter et al.70 is one in a series based on the use of a two-dimensional finite-element model of the proximal human femur as the basis for the application of skeletal morphogenesis theories. The basic idea is to begin with a model in which the elements all have a uniform density distribution, to then apply loads and iterate the density of each finite element based on a remodeling rule, and compare the calculated density pattern to in vivo patterns. Using Eq. 31.35 with multiple loads, Carter et al.70 calculated density distributions shown in Fig. 31.12, which are similar to those seen in the proximal femur. However, the best correspondence was observed after the third iteration of the computational implementation and in subsequent iterations, the density distribution progressed to nonphysiological distributions. (Note that because time was not specifically incorporated into the model, the iteration number has significance only within the context of the implementation the

FIGURE 31.12 Examination of the distribution of trabecular bone density in the femoral head. (Top) The twodimensional finite-element mesh. (Bottom) The predicted density distributions from Eq. 31.35 using multiple loads and a linear relationship between the density and the “effective stress.” The density distribution in the third iteration is closer to the in vivo distribution than that found in subsequent iterations (the seventh iteration is also shown). (From Carter, D. R. et al., J. Biomech., 22(3), 231, 1989. With permission.)



31-29

FIGURE 31.13 Another examination of the distribution of trabecular bone in the femoral head. Density distributions were calculated based upon time-dependent density equations (Eqs. 31.36 through 31.38) with a similar twodimensional finite-element model of the proximal human femur, shown in Fig. 31.12. The results are shown for an implementation using a “lazy zone” for 330 days, left, and 412.5 days, right. In contrast to the simulation shown in Fig. 31.12, this solution was convergent with calculated density distributions remarkably similar to physiological distributions. (From Beaupré, G. S. et al., J. Orthrop. Res., 8(5), 662, 1990. With permission.)

authors describe.) The authors believe that ignoring other biological influences prevented the mechanical model from converging to an equilibrium solution.70 Interestingly, as a consequence of the use of multiple loadings to determine the trabecular architecture, the trabecular orientations are no longer always perpendicular. This finding is at odds with Wolff ’s firmly held presumption, and his consequently flawed observation, that the trabeculae can only intersect at right angles.75 In reality, the angles are not always at right angles (see Chapter 29). In 1990, Beaupré et al.74 used the time-dependent bone density equations (Eqs. 31.36 through 31.38) with a similar two-dimensional finite-element model of the proximal human femur, shown in Fig. 31.13. The implementation also used multiple load cases, but in contrast to the results obtained by Carter et al.70 the solution was convergent with calculated density distributions remarkably similar to physiological distributions. The results are “consistent with the hypothesis that similar stress-related phenomena are responsible for both normal morphogenesis and functional adaptation in response to changes in the bone loading.”74 They also note, however, that the response to changed mechanical loading may not be unique, consistent with Weinans et al.76 Jacobs et al.73 have used Eq. 31.39, which operates directly upon the full stiffness matrix with no a priori assumptions about the degree of anisotropy of the trabecular bone, to simulate the density distribution using a two-dimensional finite-element model of the proximal femur. The results were very similar to those obtained using an isotropic rule for density evolution,74 and are thus also remarkably similar to physiological distributions.

31.5.2 Trabecular Density: Strain Energy Dependence Huiskes et al.8 presented a theory for the density distribution of bone, as well as a computational implementation and preliminary applications in prosthesis design as discussed previously. In addition to the Eq. 31.23 for shape changes, Huiskes et al.8 proposed the following description of modulus changes:

 C e ( U ( 1 s )U n );  dE ------  0; dt   C e ( U ( 1 s )U n );

U ( 1 s )U n ( 1 s )U n U ( 1 s )U n U ( 1 s )U n ,

(31.40)


31-30

Bone Mechanics

where E is the elastic modulus for the point considered, U is the current strain energy density, Un is the homeostatic value of strain energy density, Ce is a remodeling rate constant, and 2s is the width of the “lazy zone” representing a range of strain energy density values near the homeostatic value in which there is no net remodeling response. With no lazy zone, i.e., s 0, the equation can be rewritten as

dE ------ C e ( U U n ). dt

(31.41)

In 1992, Weinans et al.77 expanded upon computational work to simulate adaptive bone remodeling that was first presented in 198978 and 1990.79 They used a time-dependent rule for regulation of density as follows:

U d ------ B  ------a k ,   dt

0 cb ,

(31.42)

where the apparent density is represented by (x, y, z), cb is the maximum density of cortical bone, B and k are constants, and Ua represents the sum of the apparent strain energy density for the number of loading configurations being considered. To calculate the isotropic stiffness, the apparent density was used with the power law relation between density and Young’s modulus.80 31.5.2.1 Example Applications: Discrete Structures In 1992, Weinans et al.76 used Eq. 31.42 in a series of example applications using finite-element models. Rather than explicitly incorporate a new set of equations to account for the changing orientation of the trabeculae, they chose to use dense meshes of finite elements and apply Eq. 31.42 to each element. Each element in the mesh was originally given the same density and during the simulations could then reach one of three outcomes: become completely resorbed ( ≈ 0); become cortical ( ≈ cb); or remain cancellous with intermediate densities (0 cb). The simulations were performed for two-dimensional models of a proximal femur and a simple two-dimensional plate model with a linear loading distribution. The simple plate models, shown in Fig. 31.14, allowed for an investigation of the stability and uniqueness of the net remodeling equation (Eq. 31.42) and gave some surprising results. In particular, the plate model could be manipulated (by choice of the remodeling rate constants and the loading) to develop the appearance of a discrete structure, with some elements at “full” or intermediate densities, and other elements essentially “turned-off ” with zero densities. In addition to creating “trabecular-like” structures, the simulations also sometimes produced “checkerboard” patterns of elements with full and zero densities.76 For these simulations, the solution—in terms of the discrete structures produced—was shown to be dependent upon the methods of postprocessing (averaging results gave the appearance of continuum solutions) and upon mesh refinement. The solutions prompted studies 81–83 to examine in more detail the stability and uniqueness of these solutions that give discrete structures, even though the methodology was initially dependent upon a continuum assumption. Interestingly, the checkerboard and mesh-dependent anomalies have been observed while computationally designing microstructures for new materials.84,85 In these cases, the checkerboarding was attributed to “poor numerical modeling of the stiffness of checkerboards by lower-order finite elements” and the mesh-dependency problem—nonconvergence of the solutions—was eliminated using ideas borrowed from image processing.85 The image-processing technique essentially allows for systematic averaging of results by “blurring” the results. The use of higherorder finite elements as an alternate means to eliminate the checkerboarding pattern has been suggested by Jacobs et al.86 and has also been suggested in the material design literature.84 Another computational remedy that prevents the checkerboarding was suggested by Jacobs et al.83 with the use of a “nodal approach” (density calculated at each node) as opposed to an “element approach” (density calculated at the element centroid). Beyond the computational remedies, Mullender et al.87 justify a “biological averaging” technique based on a spatial influence function, described next. This technique eliminates the checkerboarding by



31-31

FIGURE 31.14 Analysis of the stability and uniqueness of adaptation solutions via density changes for a simple plate model. (a) Shows the plate with four-noded linear elements (5 5) and with additional mesh densities, including 10 10 and 20 20. The linear surface traction and boundary conditions are also shown. (b) The resultant density distributions are shown as a result of dependence of density upon strain energy density, as expressed in Eq. 31.42. The continuous structure has developed into a discrete structure with some elements at “full” or intermediate densities, and other elements essentially “turned-off ” with zero densities. (Redrawn from Weinans, H. et al., J. Biomech., 25(12), 1425, 1992. With permission.)

distinguishing the sensor grid (the spatial arrangement of the osteocytes) from the actor cells (the osteoblasts and osteoclasts) and the finite-element mesh.87 Thus, the results of Weinans et al.76 have focused attention on some of the underlying difficulties in converting theories to computer implementations. Now it is widely recognized that the checkerboarding is simply a numerical artifact. However, the evolution of “trabecularization” in continuous structures is


31-32

Bone Mechanics

a topic of continuing interest. Further research into the optimality, uniqueness, and stability of the proposed discrete structures is needed.

31.5.3 Trabecular Density and Orientation: Spatial Influence Function Recently, a mechanistic model has been proposed for the functional adaptation of trabecular bone by Mullender and co-workers.87,88 The basis of the model is the hypothesis that “osteocytes located within the bone sense mechanical signals and that these cells mediate osteoclasts and osteoblasts in their vicinity to adapt bone mass.”88 Each osteocyte i measures its local mechanical signal, Si(t), assumed to be the strain energy density, and influences all other “actor” cells (osteoclasts and osteoblasts) with decreasing influence based on the distance from the osteocyte. Mathematically, this is achieved by the use of a “spatial influence function” written using an exponential form as

fi ( x ) e

( d i ( x )D )

,

(31.43)

where di(x) is the distance between osteocyte and location x, D is the distance from an osteocyte at which 1 its effect is e , so that the influence of an individual osteocyte, i, decreases exponentially with distance from the cell. The use of the influence function to modify the stimulus amplitude, Si(t), summed from each osteocyte results in the stimulus value, F(x,t): N

F ( x,t )

f ( x ) ( S ( t ) k ), i

i

(31.44)

i1

where k is the reference strain energy density. Then the regulation of the relative density, m(x,t), is

dm ( x,t ) -------------------- F ( x,t ) dt

for 0 m ( x,t ) 1,

(31.45)

where is a rate constant, and the elastic modulus at location x, given by

E ( x,t ) Cm ( x,t ) ,

(31.46)

where C and are constants. As described subsequently, this model has been used to investigate the “trabecularization” of bone in response to mechanical loading. 31.5.3.1 Example Applications: Tissue-Level Adaptations The two-dimensional computational examples give striking results. First, the spatial influence function solves the computational checkerboarding artifact that had been observed with some other finiteelement approaches.87,88 Further, by using a tissue-level approach they show that initial idealized geometries with both uniform density distributions and lattice structures can give very similar “trabecularized” geometries during the computational simulations if they are loaded in the same fashion, as seen in Fig. 31.15. This computational test is a first step in showing the conditions needed for uniqueness of solutions. They also show that changing the direction of loading for the “trabecularized” geometries results in realignment of the trabeculae, and that severing a strut in the structure leads to its resorption and complete removal, as seen in Fig. 31.16. These computer simulations do a remarkably good job in mimicking a variety of the physiological behaviors of trabecular bone, and hold the real promise of providing a computational vehicle for simulation and study of the adaptive behavior of trabecular bone at the microstructural level.



31-33

FIGURE 31.15 Trabecular bone density and orientation simulations from use of the spatial influence function. Trabecular-like patterns are formed from initial idealized goemetries with both uniform density distributions and lattice structures. The final equilibrium directions of the trabeculae match the principal stress orientation from the applied loading. (From Mullender, M. G. and Huiskes, R., J. Orthrop. Res., 13(4), 503, 1995. With permission.)

FIGURE 31.16 Simulation of the trabecular bone response due to a severed trabecular spicule. The unloaded trabecular portion is resorbed, and surrounding spicules are thickened and realigned. (From Mullender, M. G. and Huiskes, R., J. Orthrop. Res., 13(4), 503, 1995. With permission.)


31-34

Bone Mechanics

The notion of a spatial influence function has also been used by Smith et al.89 with models in which the trabecular surface bone is added and removed based on the osteocytic signals. These models explore the consequences of assuming that the “set point” is located exclusively in the bone-lining cells to compare with the assumption used by Mullender and co-workers 87,88 in which the “set point” is located in the osteocytes. The simulations show trabecular reorientations similar to those in Fig. 31.15. However, an interesting consequence of having the set point located in the bone-lining cells is that—even in the equilibrium configurations—the strain energy density varies throughout the structure. This is a result of having the strain energy in the bone-lining cell dependent upon the influence function summation, and thus upon the local configuration of the trabecular architecture. The surprising consequence is that the trabecular structure shows “implicit site dependence even though each cell has the same explicit set point.”89

31.5.4 Boundary-Element Implementation Rather than use finite elements, Luo et al.90 and Sadegh et al.10 have used two-dimensional boundary elements to model idealized trabecular adaptation at the tissue level. The advantage of the boundaryelement method is that due to an extra integration step in derivation of the computational implementation of the method, the dimensional order can be reduced by one. Thus, two-dimensional problems can be solved with one-dimensional elements only on the periphery of the area, and three-dimensional problems can be solved with two-dimensional elements only on the surface of the volume. There is an additional advantage for the boundary-element method when used to model free-boundary problems because boundary elements sidestep many difficulties faced when finite-element mesh distortion leads to poor numerical performance or disappearing elements.17 Sadegh et al.10 demonstrate the implementation and feasibility of incorporating the phenomenological surface remodeling rate equation (Eq. 31.9) into a boundaryelement method. 31.5.4.1 Example Applications: Mergers and Separations; Strain and Strain Rate For simulation of trabecular adaptation, the use of boundary-element methods is especially promising with microscale models. Sadegh et al.10 used a series of idealized trabecular models to show that the method can be used to simulate two adjacent trabeculae that join together in a single column or the separation of one into two separate columns, as shown in Fig. 31.17. Luo et al.90 used similar idealized boundary-element models to compare the computed differences in shape change when using dilatational strain and dilatational strain rate as the stimuli for the net remodeling response. First, they analytically demonstrated the equivalence of the stimuli in all cases except when the cyclic strain is always tensile. Consequently, in the cases when the stimuli are the same, the state of remodeling equilibrium is independent of the choice of dilatational strain or strain rate as the net remodeling stimulus. Luo et al.90 then used models of three idealized trabecular bone patterns (cruciform, square diamond, and “brick-wall pattern”) subjected to biaxial loads, and computationally demonstrated the analytical results. The computational examples (Fig. 31.18) showed that although the final shape was the same (for the cases where the two stimuli were shown to be equivalent), the two stimuli gave remarkably different evolutionary paths. The results seem to indicate this model does not suffer from some of the limitations in terms of uniqueness and stability that were observed with computational implementations of continuum-based models.

31.6 Discussion The review provided in this chapter has described the fundamental concepts and assumptions inherent in development of models to describe the mechanical adaptation of bone. While not comprehensive, a number of adaptation models have been introduced, and example applications provided. As described recently by Currey 91 and seen in this review, many adaptation simulations have been developed and produce solutions that are qualitatively reasonable, and may fit with experimentally reported results. However, Currey points out a major problem with the approach of developing theoretical models and observing their qualitative response. Does any one particular simulation produce a better solution than



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FIGURE 31.17 Boundary-element implementation of adaptation used with microscaled trabecular models. Idealized changes show that an original cruciform structure could either separate into two parallel struts (top) or join into a wide strut. (From Sadegh, A. M. et al., J. Biomech., 26(2), 167, 1993. With permission.)

another approach that has already been developed? The dilemma raised by this question is evident with the examples presented in this chapter. The six models describing cortical bone adaptation illustrate examples of optimization, phenomenological, and mechanistic models. The model applications represent the consequences of osteotomy, malalignment, and intramedullary implants. For trabecular adaptation, five models have been presented, again illustrating the three model types. The applications for trabecular adaptation represent a wide range of examples and use continuum-level and tissue-level models. Although there are a variety of implementation differences—time-dependent and time-independent simulations, site-dependent and site-independent remodeling rate constants, and a variety of different strain and stress stimuli—all the applications show qualitative agreement with some experimentally observed cortical and trabecular bone changes. Currey91 writes, “This is a fundamental difficulty if progress is to be made, because the literature becomes full of algorithms that predict real-life outcomes to some extent, but they are not matched against each other, there is no saying which is better than the others, so everybody can happily think that their own algorithm is satisfactory.” Certainly, a conclusion from the numerous theories and applications described in this chapter is that, at least for the simulations shown, bone models certainly seem to adapt to mechanical usage but seem not to be very picky about the specific mechanical stimuli used. It seems that any description of the mechanical situation can be successfully used to drive a satisfactory adaptation response. This apparently promiscuous response of bone to mechanical stimuli may be due to the choices used for parameters to monitor. For example, for axially loaded structures the areas of high axial strains also correlate to areas of high strain energy and to von Mises stresses. A study of 50 different mechanical parameters (six local strain components, six local stress components, three principal strains, three principal stresses, three strain invariants, three stress invariants, maximum shear strain, maximum shear stress, strain energy


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FIGURE 31.18 Boundary-element implementation of idealized periodic elements to approximate trabecular geometries. In this case, models are used to compare the computed differences in shape change when using dilatational strain and dilatational strain rate as the stimuli for net remodeling response. This example shows that although the final shape was the same when using dilatational strain (left) and dilatatational strain rate (right), the two stimuli gave remarkably different evolutionary paths. (From Luo, G. et al., J. Biomech. Eng., 117(3), 329, 1995. With permission.)

density, 21 spatial strain gradients of the six strain gradients and dilatation in the three local directions, the mechanical intensity scalar, Eˆ , and a signed strain energy density) shows that many of these parameters that have been used in adaptation simulations are highly correlated to one another.92 This correlation of mechanical parameters helps mask the true regulators of the adaptation process so that distinct starting



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assumptions about the nature of the mechanical parameters can result in similar simulation solutions. To be more useful, the modeling assumptions should be more tightly coupled to biological parameters that can be histologically measured and manipulated.93 Currey points out that a second major problem is based on the tendency for simulations to be used with a single remodeling situation, with no indication that the method is appropriate for any other one.91 Again, which methods are best? He suggests that an algorithm be chosen as the “null hypothesis algorithm” that can satisfactorily solve a combination of idealized problems, and mirror experimental results as well. New algorithms should only be introduced to the literature if they prove superior to the “null hypothesis algorithm.”91 A response is offered by Huiskes.94 He points out the merit in using phenomenological models for predicting bone response even if the process is not yet well understood. He argues that the process of learning from each other as new ideas are tried and results published may be confusing, but is necessary while ideas and methods are incrementally refined and improved. Most important, he describes a specific example of how computational simulations may help in guiding experiments that can explore the mechanisms for adaptation. The specific testable hypothesis that arises from the spatial influence function87,88 is that osteocyte density may be related to trabecular architecture. This is an exciting insight, and could be quantitatively explored.94 As mentioned at the outset, the merit from adaptation simulations depends upon recognizing that the different approaches have different objectives and outcomes. Optimization studies help to further understanding of bone as a mechanical structure, but do not provide useful information about the physiological process of adaptation. Phenomenological models may be useful in framing experimental questions, testing the consequences of different starting assumptions, and for the predictions of outcomes. They may even be useful for preliminary stages of implant design. However, as the focus shifts toward introducing mechanisms into the phenomenological models, the real benefits of developing and implementing a variety of different theories for bone adaptation studies begin to emerge. The models can help focus in vivo experiments on quantification of assumed parameters. Specifically, further advances in simulating bone adaptation will come from addressing a number of key steps, concepts, and questions: the transducers involved in converting mechanical usage into cellular responses; the degree of and physiological mechanisms for site specificity in adaptive responses; the time dependence of mechanical stimuli and physiological mechanisms responsible; and the question of a remodeling equilibrium state, how it may be initially set, and how it might be reset over time. As inspired by Currey’s91 comments, the use of a “test suite” of example problems would prove useful as theories are developed and refined. A suite of problems could be solved using a variety of algorithms and assumptions—especially including cases that fail to produce reasonable solutions— and would help to improve these methods incrementally. Specific examples for the test suite may include the gradual straightening of a long bone (shown in Figs. 31.3 and 31.8). In fact, the use of strain energy density, which is always positive, did not allow the coordinated addition and removal of bone in a case where axial strains did produce the correct tendencies.47 Similar difficulties would be expected with use of the strength optimization methods that incorporate damage as the stimulus. A test case for skeletal morphogenesis simulations might be first to determine the influence of various assumptions that are required for producing reasonable trabecular density and orientation outcomes. A test would be then to use the same values for a different region, e.g., the distal femur or proximal tibia to see the predicted patterns. Increasingly, the simulations should be cast in terms of physiological parameters that can be measured and manipulated, allowing tests of their validity as important players in functional adaptation processes. A final suggestion is the use of a “fully quantified” osteotomy experiment that includes measures of the temporal and spatial strains on the intact radius and ulna, followed by those same measures immediately following an ulnar osteotomy, and during use for several months thereafter. This level of quantification of the mechanical environment will allow a host of simulations similar to those already presented (Figs. 31.2, 31.4, 31.5, 31.6, and 31.9) but allow many more of the components of the strain history to be included as the primary stimuli.


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Clearly, continued progress in the study of the functional adaptation of bone will depend upon interdisciplinary studies, and require increasingly sophisticated in vivo, mechanical, chemical, and cellular methods. The continuing fascination with bone—a unique and complex living structural material—will eventually yield quantitative models that allow for prediction of bone responses and for testing the consequences of therapeutic interventions.

Acknowledgments This work was supported, in part, with a Whitaker Special Opportunity Award for Computational Tissue Engineering. The author appreciates the comments and suggestions made by M. D. Roberts, J. C. Coleman, R. B. Martin, R. Huiskes, and S. C. Cowin on an earlier draft of this chapter.

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19. Burr, D. B., M. B. Schaffler, K. H. Yang, M. Lukoschek, N. Sivaneri, J. D. Blaha, and E. L. Radin, Skeletal change in response to altered strain environments: is woven bone a response to elevated strain? Bone, 10(3), 223–233, 1989. 20. McLeod, K. J., C. T. Rubin, M. W. Otter, and Y. X. Qin, Skeletal cell stresses and bone adaptation, Am. J. Med. Sci., 316(3), 176–183, 1998. 21. Qin, Y. X., C. T. Rubin, and K. J. McLeod, Nonlinear dependence of loading intensity and cycle number in the maintenance of bone mass and morphology, J. Orthop. Res., 16(4), 482–489, 1998. 22. Rubin, C. T. and L. E. Lanyon, Regulation of bone formation by applied dynamic loads, J. Bone Joint Surg. [Am.], 66(3), 397–402, 1984. 23. Turner, C. H., Three rules for bone adaptation to mechanical stimuli, Bone, 23(5), 399–407, 1998. 24. Treharne, R. W., Review of Wolff ’s law and its proposed means of operation, Orthop. Revi., X(1), 35–47, 1981. 25. Cowin, S. C., L. Moss-Salentijn, and M. L. Moss, Candidates for the mechanosensory system in bone, J. Biomech. Eng., 113(2), 191–197, 1991. 26. Cowin, S. C., S. Weinbaum, and Y. Zeng, A case for bone canaliculi as the anatomical site of strain generated potentials, J. Biomech., 28(11), 1281–1297, 1995. 27. Gjelsvik, A., Bone remodeling and piezoelectricity—I, J. Biomech., 6, 69–77, 1973. 28. Gjelsvik, A., Bone remodeling and piezoelectricity—II, J. Biomech., 6, 187–193, 1973. 29. Frost, H. A., Dynamics of bone remodeling, in Bone Biodynamics, H. A. Frost, Ed., Little, Brown, Boston, 315–333, 1964. 30. Hart, R. T. and S. P. Fritton, Introduction to finite element based simulation of functional adaptation of cancellous bone, Forma, 12, 277–299, 1997. 31. Hart, R. T., D. T. Davy, and K. G. Heiple, A computational method for stress analysis of adaptive elastic materials with a view toward applications in strain-induced bone remodeling, J. Biomech. Eng., 106(4), 342–350, 1984. 32. Frost, H. M., Intermediary Organization of the Skeleton, CRC Press, Boca Raton, FL, 1986. 33. Cowin, S. C. and W. C. Van Buskirk, Internal bone remodeling induced by a medullary pin, J. Biomech., 11(5), 269–275, 1978. 34. Cowin, S. C. and W. C. Van Buskirk, Surface bone remodeling induced by a medullary pin, J. Biomech., 12(4), 269–276, 1979. 35. Huiskes, R. and E. Y. Chao, A survey of finite element analysis in orthopedic biomechanics: the first decade, J. Biomech., 16(6), 385–409, 1983. 36. Cook, R. D., D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, 1989. 37. Hart, R. T., The finite element method, in Bone Mechanics, S. C. Cowin, Ed., CRC Press, Boca Raton, FL, 1989, chap. 4, 53–74. 38. Frost, H. M., Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff ’s law: the bone modeling problem, Anat. Rec., 226(4), 403–413, 1990. 39. Cowin, S. C. and D. H. Hegedus, Bone remodeling I: Theory of adaptive elasticity, J. Elasticity, 6(3), 313–326, 1976. 40. Cowin, S. C. and K. Firoozbakhsh, Bone remodeling of diaphysial surfaces under constant load: theoretical predictions, J. Biomech., 14(7), 471–484, 1981. 41. Cowin, S. C., Wolff ’s law of trabecular architecture at remodeling equilibrium, J. Biomech. Eng., 108(1), 83–88, 1986. 42. Cowin, S. C. and W. C. Van Buskirk, Thermodynamic restrictions on the elastic constants of bone, J. Biomech., 19(1), 85–87, 1986. 43. Cowin, S. C., Bone remodeling of diaphyseal surfaces by torsional loads: theoretical predictions, J. Biomech., 20(11–12), 1111–1120, 1987. 44. Cowin, S. C., A. M. Sadegh, and G. M. Luo, An evolutionary Wolff ’s law for trabecular architecture, J. Biomech. Eng., 114(1), 129–136, 1992.


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45. Cowin, S. C., R. T. Hart, J. R. Balser, and D. H. Kohn, Functional adaptation in long bones: establishing in vivo values for surface remodeling rate coefficients, J. Biomech., 18(9), 665–684, 1985. 46. Hart, R. T., A theoretical study of the influence of bone maturation rate on surface remodeling predictions: idealized models, J. Biomech., 23(3), 241–257, 1990. 47. Hart, R. T. and A. M. Rust-Dawicki, Computational simulation of idealized long bone re-alignment, in Computer Simulations in Biomedicine, H. Power and R. T. Hart, Eds., Computational Mechanics Publications, Boston, 1995, 341–350. 48. Oden, Z. M., R. T. Hart, M. R. Forwood, and D. B. Burr, A priori prediction of functional adaptation in canine radii using a cell based mechanistic approach, Trans. 41st Orthopaedic Research Society, Orlando, FL, 1995. 49. Hart, R. T., V. V. Hennebel, N. Thongpreda, and D. A. Dulitz, Computer simulation of cortical bone remodeling, in Science and Engineering on Supercomputers, E. J. Pitcher, Ed., Computational Mechanics Publications, Boston, 1990, 57–66, 565–566. 50. Davy, D. T. and R. T. Hart, A Theoretical Model for Mechanically Induced Bone Remodeling, American Society of Biomechanics, Rochester, MN, 1983. 51. Hart, R. T. and D. T. Davy, Theories of bone modeling and remodeling, in Bone Mechanics, S. C. Cowin, Ed., CRC Press, Boca Raton, FL, 1989, chap. 11, 253–277. 52. Martin, R. B., The effects of geometric feedback in the development of osteoporosis, J. Biomech., 5(5), 447–455, 1972. 53. Oden, Z. M., Computational Prediction of Functional Adaptation in Canine Radii, Ph.D. dissertation, Department of Biomedical Engineering, Tulane University, New Orleans, 1994. 54. Burr, D. B., personal communication, 1994. 55. Carter, D. R., Mechanical loading history and skeletal biology, J. Biomech., 20(11–12), 1095–1109, 1987. 56. Baumgartner, A. and C. Mattheck, Computer simulation of the remodeling of trabecular bone, in Computers in Biomedicine, K. D. Held, C. A. Brebbia, and R. D. Ciskowski, Eds., Computational Mechanics Publications, Southampton, 1991, 291–296. 57. Mattheck, C. and H. Huber-Betzer, CAO: Computer simulation of adaptive growth in bones and trees, in Computers in Biomedicine, K. D. Held, C. A. Brebbia, and R. D. Ciskowski, Eds., Computational Mechanics Publications, Southampton, 1991, 243–252. 58. Mattheck, C., Design in Nature, Springer-Verlag, Berlin, 1998. 59. Prendergast, P. J. and D. Taylor, Prediction of bone adaptation using damage accumulation, J. Biomech., 27(8), 1067–1076, 1994. 60. McNamara, B. P., P. J. Prendergast, and D. Taylor, Prediction of bone adaptation in the ulnarosteotomized sheep’s forelimb using an anatomical finite element model, J. Biomed. Eng., 14(3), 209–216, 1992. 61. McNamara, B. P., Ph.D. dissertation, Department of Mechanical and Manufacturing Engineering, University of Dublin, Trinity College, Dublin, 1995. 62. Lanyon, L. E., A. E. Goodship, C. J. Pye, and J. H. MacFie, Mechanically adaptive bone remodelling, J. Biomech., 15(3), 141–154, 1982. 63. Cowin, S. C., personal communication, 1986. 64. Cowin, S. C., The mechanical properties of cortical bone tissue, in Bone Mechanics, S. C. Cowin, Ed., CRC Press, Boca Raton, FL, 1989, 97–128. 65. Harrigan, T. P. and R. W. Mann, Characterization of microstructural anisotropy in orthopaedic materials using a second rank tensor, J. Mater. Sci., 19, 761, 1984. 66. Harrigan, T. P., Bone Compliance and Its Effects in Human Hip Joint Lubrication, Sc.D. dissertation, Department of Mechanical Engineering, Massachusetts Institute of Technology, Boston, 1985. 67. Fritton, S. P., Computational Simulation of Trabecular Bone Adaptation, Ph.D. dissertation, Department of Biomedical Engineering, Tulane University, New Orleans, 1994.



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68. Fritton, S. P. and R. T. Hart, Simulation of in vivo trabecular bone adaptation, 41st Annual Meeting of the Orthopaedic Research Society, Orlando, FL, 1995. 69. Harrigan, T. P., M. Jasty, R. W. Mann, and W. H. Harris, Limitations of the continuum assumption in cancellous bone, J. Biomech., 21(4), 269–275, 1988. 70. Carter, D. R., T. E. Orr, and D. P. Fyhrie, Relationships between loading history and femoral cancellous bone architecture, J. Biomech., 22(3), 231–244, 1989. 71. Beaupré, G. S., T. E. Orr, and D. R. Carter, An approach for time-dependent bone modeling and remodeling—theoretical development, J. Orthop. Res., 8(5), 651–661, 1990. 72. Martin, R. B., Porosity and specific surface of bone, Crit. Rev. Biomed. Eng., 10(3), 179–222, 1984. 73. Jacobs, C. R., J. C. Simo, G. B. Beaupré, and D. R. Carter, Comparing an optimal efficiency assumption to a principal stress-based formulation for the simulation of anisotropic bone adaptation to mechanical loading, in Bone Structure and Remodeling, A. Odgaard and H. Weinans, Eds., World Scientific, Singapore, 1995, 225–237. 74. Beaupré, G. S., T. E. Orr, and D. R. Carter, An approach for time-dependent bone modeling and remodeling—application: a preliminary remodeling simulation, J. Orthop. Res., 8(5), 662–670, 1990. 75. Cowin, S. C., The false premise of Wolff ’s law, Forma, 12, 247–262, 1997. 76. Weinans, H., R. Huiskes, and H. J. Grootenboer, The behavior of adaptive bone-remodeling simulation models, J. Biomech., 25(12), 1425–1441, 1992. 77. Weinans, H., R. Huiskes, and H. J. Grootenboer, Effects of material properties of femoral hip components on bone remodeling, J. Orthop. Res., 10(6), 845–853, 1992. 78. Weinans, H., R. Huiskes, and H. J. Grootenboer, Convergence and uniqueness of adaptive bone remodeling, Transactions of the 35th Annual Meeting of the Orthopaedic Research Society, Las Vegas, 1989. 79. Weinans, H., R. Huiskes, and H. J. Grootenboer, Numerical comparisons of strain-adaptive boneremodeling theories, Abstracts of the 1st World Congress on Biomechanics, La Jolla, 1990. 80. Carter, D. R. and W. C. Hayes, The compressive behavior of bone as a two-phase porous structure, J. Bone Joint Surg. [Am.], 59(7), 954–962, 1977. 81. Harrigan, T. P. and J. J. Hamilton, An analytical and numerical study of the stability of bone remodelling theories: dependence on microstructural stimulus [published erratum appears in J. Biomech., 26(3), 3656, 1993], J. Biomech., 25(5), 477–488, 1992. 82. Cowin, S. C., Y. P. Arramon, G. M. Luo, and A. M. Sadegh, Chaos in the discrete-time algorithm for bone-density remodeling rate equations, J. Biomech., 26(9), 1077–1089, 1993. 83. Jacobs, C. R., M. E. Levenston, G. S. Beaupré, J. C. Simo, and D. R. Carter, Numerical instabilities in bone remodeling simulations: the advantages of a node-based finite element approach, J. Biomech., 28(4), 449–459, 1995. 84. Jog, C. S. and R. B. Haber, Stability of Finite Element Models for Distributed-Parameter Optimization and Topology Design, TAM Report No. 758, UIL-ENG-94-6014, University of Illinois, Urbana-Champaign, 1994. 85. Sigmund, O., Design of Material Structures Using Topology Optimization, Reports 69, Danish Center for Applied Mathematics and Mechanics, Technical University of Denmark, Lyngby, 1994. 86. Jacobs, C. R., M. E. Levenston, G. B. Beaupré, and J. C. Simo, A new implementation of finite element-based remodeling, G. N. P. J. Middleton, and K. R. Williams, Eds., in Recent Advances in Computer Methods in Biomechanics and Biomedical Engineering, Books and Journals International, Swansea, 1992. 87. Mullender, M. G., R. Huiskes, and H. Weinans, A physiological approach to the simulation of bone remodeling as a self-organizational control process, J. Biomech., 27(11), 1389–1394, 1994. 88. Mullender, M. G. and R. Huiskes, Proposal for the regulatory mechanism of Wolff ’s law, J. Orthop. Res., 13(4), 503–512, 1995. 89. Smith, T. S., R. B. Martin, M. Hubbard, and B. K. Bay, Surface remodeling of trabecular bone using a tissue level model, J. Orthop. Res., 15(4), 593–600, 1997.


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90. Luo, G., S. C. Cowin, A. M. Sadegh, and Y. P. Arramon, Implementation of strain rate as a bone remodeling stimulus, J. Biomech. Eng., 117(3), 329–338, 1995. 91. Currey, J. D., The validation of algorithms used to explain adaptive remodelling in bone, in Bone Structure and Remodelling, A. Odgaard and H. Weinans, Eds., World Scientific, Singapore, 1995, 9–13. 92. Oden, Z. M. and R. T. Hart, 1995, Correlation between mechanical parameters that represent bone’s mechanical milieu and control functional adaptation, in Bone Structure and Remodeling: Recent Advances in Human Biology, vol. 2, A. Odgaard and H. Weinans, Eds., World Scientific, Singapore, 1995, ix, 257. 93. Martin, R. B., The usefulness of mathematical models for bone remodeling, Yearbook Phys. Anthropol., 28, 227–236, 1985. 94. Huiskes, R., The law of adaptive bone remodelling: a case for crying Newton? in Bone Structure and Remodelling, A. Odgaard and H. Weinans, Eds., World Scientific, Singapore, 1995, 15–24. 95. Frost, H. M., Introduction to a New Skeletal Physiology, Pajaro Group, Pueblo, CO, 1995.

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32 Mechanics of Bone Regeneration Patrick J. Prendergast Trinity College Dublin

Marjolein C. H. van der Meulen

32.1 32.2 32.3

Introduction................................................................... 32-1 Fracture Healing ............................................................ 32-1 Mechanobiological Models ........................................... 32-4 Pauwels’ Theory • Interfragmentary Strain Theory • Deformation/Pressure Models • Models Including Fluid Flow

Cornell University

32.4

Future Directions......................................................... 32-10

32.1 Introduction Bone has the capability to regenerate, forming new osseous tissue at locations that are damaged or missing. Although this capability extends to regeneration of whole limbs in some animals,1 in humans regeneration is on a more limited scale and occurs during defect healing, e.g., after the removal of bone screws; fracture healing; distraction osteogenesis (during limb lengthening); and integration of orthopedic implants with the host bone. During the regenerative process, bone tissue can form directly or indirectly. During indirect bone formation, also called endochondral ossification, cartilage is formed, calcified, and replaced by bone tissue. During direct bone formation, or intramembranous ossification, bone tissue forms without the intermediate cartilage stage. The importance of the mechanical environment on tissue regeneration is evident in several ways. For example, during defect healing, when the regenerating bone is shielded from load, intramembranous, not endochondral, bone formation occurs.2 Excessive motion during long bone fracture healing inhibits bone formation. Instead of bone, a layer of cartilage forms at the fracture ends, creating a pseudoarthrosis (‘‘fake joint’’).3 Similarly, if an implant moves excessively relative to the surrounding bone, bone formation is inhibited and a persistent fibrous tissue layer forms at the bone–implant interface.4 The complex nature of bone regeneration has mostly been studied during fracture healing of long bones. For this reason, experiments designed to understand long bone fracture healing will be described in Section 32.2. In Section 32.3, different theories to describe the influence of mechanics on bone regeneration will be reviewed.

32.2 Fracture Healing Bone fracture repair occurs either by: 1. Primary (direct, osteonal) fracture healing, where the fracture gap ossifies via intramembranous bone formation without external callus, or

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32-2

Bone Mechanics

2. Secondary (indirect, spontaneous) fracture healing, where a multistage process of tissue regeneration stabilizes the bone with an external callus and repairs the fracture via endochondral ossification. During primary fracture healing, the fractured cortices repair directly by osteonal remodeling across the cortex. This process is less common than secondary fracture healing because the fracture must be extremely stable and well reduced for direct healing to occur. Clinically this process is observed when fractures are stabilized by rigid fixation such as compression plating. Secondary fracture healing occurs in a sequence of three biological phases: the inflammatory, reparative, and remodeling phases.5 These phases involve the coordinated activity of different cell populations that proliferate, differentiate, and synthesize extracellular matrix components. This cellular activity is spatially and temporally coordinated by a variety of growth factors and other regulatory molecules, including insulin-like growth factors (IGFs), fibroblast growth factors (FGFs), plateletderived growth factors (PDGFs), transforming growth factors (TGFs), and bone morphogenetic proteins (BMPs).6–8 During the inflammatory phase, the periosteum of the bone lifts, and a hematoma forms between the bone ends due to the rupture of blood vessels. This hematoma releases a large number of signaling molecules, including inflammatory cytokines such as interleukin-1 (IL-1) and interleukin-6 (IL-6), and growth factors such as PDGF and transforming growth factor- (TGF-).7 These factors appear to regulate the initiation of fracture healing and the associated cellular response. For example, TGF- has been shown to accelerate fracture healing.9 The first cells present are pluripotent progenitor cells called mesenchymal stem cells: they originate from several possible sources (the inner cambial layer of the periosteum, the endosteum, the bone marrow, or the vascular endothelium10–13). These stem cells then differentiate into chondrocytes and osteocytes, which proliferate and generate the reparative callus. This differentiation of the mesenchymal cells appears to be activated by BMPs.14 During the reparative phase an external (periosteal) callus and an internal (medullary) callus are formed (Fig. 32.1). Intramembranous and endochondral bone formation occur simultaneously in different regions of the fracture calluses. Intramembranous ossification occurs only at the callus periphery, beneath the damaged periosteum but somewhat removed from the fracture site, and appears to be regulated by TGF-.8 This rapidly formed woven bone creates a mineralized hard callus. Endochondral ossification occurs adjacent to the fracture site and initiates with cartilage. This soft callus bridges across the fracture. The rapid cartilage formation of the early callus is accomplished primarily by the differentiation of mesenchymal cells into chondrocytes, and less by proliferation of existing chondrocytes.15 TGF- may regulate this cartilage matrix synthesis. Injection of BMP-2 into fractured rabbit tibiae increases the rate of healing, but not the size of the fracture callus.16 Mineralization of the cartilage involves a mechanism similar to long bone growth at the growth plate11 and requires reestablishment of the vascular and nutrient supply. Chondrocytes hypertrophy, the extracellular matrix calcifies and blood vessels penetrate the matrix. Then the calcified cartilage tissue is resorbed and replaced by woven bone formed by osteoblasts. In the final healing phase, osteonal remodeling of the newly formed bone tissue and of the fracture ends restores the original shape and lamellar structure of the bone. White et al.17 defined four biomechanical stages of fracture repair. At this final healing stage, the failure of a healed bone when loaded is not related to the original fracture site, while at the early stages, failure occurs through the weaker fracture. In the reparative phase, the formation of a callus with a large cross-sectional area increases the rigidity of the bone and stabilizes the fracture ends. As ossification proceeds, the fracture becomes stiffer and the motion of the fracture fragments decreases further. Therefore, fracture stiffness may indicate successful, or unsuccessful, healing. Marsh18 measured callus size and bending stiffness during healing of human tibiae and found no correlation between callus index (ratio of the maximum width of the callus to the diameter of the original shaft) and bending stiffness. However, a bending stiffness of less than 7 Nm/° at 20 weeks indicated a delayed union or a non-union. Similarly, Richardson et al.19 found that patients whose tibial fractures had reached a bending stiffness of 15 Nm/° did not refracture after external fixation. This stiffness was proposed to define successful union of these fractures.


Mechanics of Bone Regeneration

32-3

FIGURE 32.1 Histological section of a mouse fracture callus 5 days postfracture. The femur was pinned and then fractured, so no endosteal callus is present. Intramembranous bone formation is evident under the periosteum at the callus periphery; fibrous and cartilaginous callus is present adjacent to the fracture. OB original bone, NB new membranous bone, FC fibrous callus, CC cartilage callus, MC medullary canal, MU muscle. Trichrome staining.

The importance of the mechanical environment on fracture repair has been demonstrated experimentally and can be examined during each stage of the repair process. The type of fixation device* and the nature of the musculoskeletal loading will determine the mechanical environment within the regenerating tissue. This environment, in turn, will influence callus size, ossification rate, and healed bone strength. Based on a series of sheep experiments, Goodship et al.20 concluded, More flexible fixation may lead to excessive interfragmentary motion … whereas more rigid fixation may impair callus formation contributing to … non-union. The rates of healing, tissue ossification, and stiffness are enhanced by cyclic mechanical loading.21–23 In sheep, Goodship and Kenwright21 found that cyclic axial motion compared to rigid fixation induced earlier ossification and more rapid healing, based on increased whole-bone stiffness (Fig. 32.2). Gardner et al.24 investigated the effect of reducing the displacement across a fracture during healing; they found more rapid healing if the displacement was reduced as healing progressed. Comparing static and cyclic compression of healing fractures, Panjabi and co-workers22 observed a greater rate of strength increase at the middle and later healing stages in bones with cyclic loading. Delaying mechanical stimulation until after the initiation of ossification eliminates the beneficial influence of cyclic loads.25 In the study by Goodship et al.,20 the importance of strain rate was also demonstrated. Fractures stimulated at a moderate strain rate (40 mm/s) increased bone mineral content and stiffness faster than those stimulated at slow (2 mm/s) and fast (400 mm/s) rates. Finally, the size of the fracture gap is critical to healing. Augat et al.26 used external fixation of metatarsus of sheep to compare healing with 1, 2, and 6 mm osteotomies, and two amounts of interfragmentary strain (7 or 31%). They showed that increasing the gap size reduced the bone stiffness and tensile strength. Because the lower interfragmentary strain adequately stimulated the callus, gap size was the more dominant of the two factors in the fracture-healing process. *See Chapter 35 for a description of fracture fixation devices.


32-4

Bone Mechanics

FIGURE 32.2 Mean in vivo increase in fracture stiffness in two groups of sheep with experimental tibial fractures. Axial movement at the fracture site different in the two groups. The difference between the two groups was significant at the 8 to 10 week period. (From Goodship, A.E. and Kenwright, J., J. Bone Joint Surg., 67B, 650, 1985. With permission.)

Particularly interesting questions arise from the interaction of growth factors involved in the normal healing cascade and mechanical loading during fracture healing.27 Growth factor expression may be regulated by mechanical stimuli, but is difficult to study in fracture healing. Distraction osteogenesis induces a bone formation response similar to fracture healing and has a better-defined spatial gradient of bone formation. Increased mechanical stimulus during distraction osteogenesis increases expression of BMP-2 and BMP-4, which may be responsible for enhanced ossification.28 During distraction osteogenesis, BMP-4 is found at sites of proliferation and differentiation of mesenchymal cells suggesting a regulatory role in the early fracture callus.29 The development of rodent fracture models will lead to a better understanding of these interactions.30,31 In particular, knockout mice and other genetic manipulations offer exciting possibilities32 and will greatly improve knowledge of fracture healing.

32.3 Mechanobiological Models According to Carter et al.,33 mechanobiology is the study of how mechanical or physical conditions regulate biological processes. During bone regeneration, physical forces contribute to the differentiation of the cell populations that arise over the healing period. For musculoskeletal tissues, Caplan and Boyan34 have hypothesized that mesenchymal stem cells differentiate into one of several musculoskeletal tissues (Fig. 32.3). The mechanics of bone regeneration involves understanding how the osteogenic pathway shown in Fig. 32.3 is regulated by mechanical forces within the tissue.

32.3.1 Pauwels’ Theory Much of present-day understanding of the regulative effect of mechanical forces on tissue differentiation comes from Friedrich Pauwels. Pauwels recognized that physical factors cause stress and deformation of the mesenchymal cells, and that these stimuli could determine the cell differentiation pathway. Using simple experimental models of a fracture callus,35 he demonstrated how shearing results in a change of shape of a


32-5

MSC Proliferation

Proliferation

Osteogenesis

Chondrogenesis

Transitory Osteoblast

Transitory Chondrocyte

Tendogenesis/ Ligamentagenesis

Myogenesis

Marrow Stroma

Myoblast

Transitory Stromal Cell

Other

Commitment

Lineage Progression

Transitory Fibroblast

Chondrocyte Myoblast Fusion

Osteoblast

Differentiation

Unique Micro-niche

Maturation Osteocyte

Hypertroph Chondrocyte

T/L Fibroblast

BONE

CARTILAGE

TENDON/ LIGAMENT

Myotube

Stromal Cells

Adipocytes, Dermal and Other Cells

MUSCLE

MARROW

CONNECTIVE TISSUE

Mesenchymal Tissue

Mesenchymal Stem Cell (MSC)

Bone Marrow / Periosteum


FIGURE 32.3 Mesenchymal stem cells have the potential to differentiate into a variety of tissues, such as bone, cartilage, tendon, muscle, fat, and dermis. Commitment to a particular pathway, as well as progression through the lineage, is dependent not just on specific growth factors and/or cytokines but also on biophysical stimuli acting on the cells. Continuous differentiation from the mesenchymal cell pool ensures that changing biophysical stimuli will elicit changing tissues. (From Caplan, A.I. and Boyan, B.D., in Mechanism of Bone Development and Growth, B. K. Hall, Ed., CRC Press, Boca Raton, FL, 1994. With permission.)

FIGURE 32.4 A direct compressive force (denoted D) “deforms the elementary particles of a cube of elastic material” into ellipsoids, as does a tensile force (denoted Z) and a shear force (denoted S). (Adapted from Pauwels, F., in Biomechanics of the Locomotor Apparatus, translated by Maquet, P. and Furlong, R., Springer, Berlin, 1980, 375–407.)

piece of tissue (Fig. 32.4) whereas hydrostatic pressure results in a change in volume. Using this concept, he proposed a quantitative hypothesis for the influence of mechanical factors on tissue differentiation as follows: 1. Shear, which causes a change in cell shape, stimulates mesenchymal cell differentiation into fibroblasts, and 2. Hydrostatic compression, which causes a change in volume without a change in shape, stimulates mesenchymal cell differentiation into chondrocytes. By decomposing mechanical stimuli into shear and hydrostatic compression, Pauwels also predicted that a combination of shear and hydrostatic pressure would stimulate differentiation of fibrocartilage, such as occurs in the menisci of the knee or in the intervertebral disk.36


32-6

Bone Mechanics

FIGURE 32.5 Pauwels’ concept of tissue differentiation. Bone regeneration occurs only after stabilization of the mechanical environment by the formation of soft tissue. (From Weinans, H. and Prendergast, P.J., Bone, 19, 143, 1996. With permission.)

Pauwels’ theory, as illustrated in Fig. 32.5, also predicted the following: 1. Primary bone formation requires a stable mechanical environment so endochondral bone formation will proceed only if the soft tissues create this low strain environment. 2. Ligamentous tissue ossifies by the process of desmoid ossification. 3. Both ossified tissues remodel and are replaced by secondary lamellar bone (indicated by arrows). Pauwels’ concept that tissue differentiation is evoked by mechanical factors drew on the ideas of earlier German biologists such as Roux and Krompecher; see Prichard.37 Roux proposed that cells compete for a functional stimulus within the tissue.38 Interpreting this in the context of the Pauwels theory, the functional stimulus favoring fibroblast differentiation from the mesenchymal cell pool is high shear. Endochondral bone formation, on the other hand, occurs in low strain/high compression environments.

32.3.2 Interfragmentary Strain Theory When a fractured bone is loaded, the fracture fragments displace relative to each other. The strain produced in the fracture gap was named the ‘‘interfragmentary strain (IFS)’’ by Perren.39 If the width of the fracture gap is given by L and the change in the fracture gap after loading is given by L, then the strain in the longitudinal direction (IFS) is

L IFS ------- . L

(32.1)

Perren’s interfragmentary strain theory proposes that the fracture gap can be filled only with a tissue capable of sustaining the IFS without rupture.40 The strain tolerance is greatest for granulation tissue, intermediate for cartilage, and least for bone (Fig. 32.6). If the IFS is high, only granulation tissue can form according to the interfragmentary strain theory. If granulation tissue does form, the callus will



32-7

FIGURE 32.6 Strain tolerance of repair tissues. A tissue cannot exist in an environment where the interfragmentary strain exceeds the strain tolerance of the extracellular matrix of the tissue. (From Perren, S.M. and Cordey, J., in Current Concepts of Internal Fixation of Fractures, Springer, Berlin, 1980, 65. With permission.)

reduce the IFS. This reduction in the motion of the fracture fragments generates the environment where chondrocytes can sustain the IFS and proliferate. In this way, endochondral bone formation proceeds as the IFS is ‘‘gently and progressively limited.’’40 Whether or not the presence of soft tissue will actually reduce the IFS depends on many factors, including the nature of the musculoskeletal loading on the bone, and the structural behavior of the fracture fixation device.41 Although the interfragmentary strain theory is useful for understanding the relationship between motion and bone healing, the theory is not a general theory of bone regeneration because 1. The strain field in the fracture gap is multiaxial,42 2. Mechanical stimuli vary spatially in the fracture callus.43,44 Furthermore, if the displacement is not very small relative to the gap size, the longitudinal strain in a fracture gap cannot be accurately calculated using Eq. 32.1 because of geometric and material nonlinearity.45 It is noteworthy that Claes et al.,46 who were able to determine the IFS with a telemeterized external fixation system, concluded that the IFS did not significantly influence the success of fracture healing, although inferior tissue properties were observed histologically for larger IFS.

32.3.3 Deformation/Pressure Models Investigating the mechanics of endochondral ossification, Carter and colleagues47,48 proposed that intermittent or cyclic mechanical loading occurring over a period of time (load history) stimulates tissue differentiation. The loading history was decomposed into discrete loading conditions, denoted c. Using the concepts proposed by Pauwels,35 the stress acting on the regenerating tissue was described as a combination of two scalar quantities: 1. Hydrostatic stress (related to the dilatational strain) denoted Di and 2. Octahedral shear stress also called the deviatoric stress (related to the distortional strain) denoted Si , where the subscript denotes the ith load case, and i 1, 2, 3, …. c.


32-8

Bone Mechanics

There are six independent stresses in a material* written as x , y , z , xy , yz, and xz, where x, y, z is an orthogonal axis. The hydrostatic stress is given by

1 D -- ( x y z ) 3

(32.2)

and the octahedral shear stress is given by

1 2 2 2 2 2 2 S -- ( x y ) ( x z ) ( y z ) 6 ( xy yz xz ) . 3

(32.3)

The hydrostatic and octahedral shear stresses are invariants of the stress—i.e., they will have the same numeric value no matter what coordinate system is used. Carter and colleagues proposed that cyclic octahedral shear stress encourages cartilage ossification whereas the action of a cyclic hydrostatic stress inhibits ossification. The driving force for bone formation may be described by a linear combination of the stress invariants. This is termed the osteogenic index (OI), given as c

OI

n ( S kD ), i

i

i

(32.4)

i1

where ni is the number of loading cycles for the ith load case. The value of k, the empirical constant, is determined by parametric variation in computer models of fracture healing,48,49 joint formation,50 and endochondral ossification of the sternum.51 High values of OI are caused by high shear stress or by tensile hydrostatic stresses—therefore, according to the osteogenic index theory, these stimuli favor endochondral bone formation. Bone formation is inhibited in the presence of large compressive hydrostatic stresses. In a finite-element analysis of the mechanical stimuli on ossifying surfaces during fracture healing, Claes and colleagues52,53 hypothesized that magnitudes of hydrostatic pressure and strain regulate the selection of either intramembranous or endochondral bone formation processes. By quantitative analysis of a real callus geometry, they found that if the compressive hydrostatic pressure (negative) exceeded 0.15 MPa at the regenerating bone surface then endochondral bone formation (i.e., prior formation of cartilage) occurred, whereas if the hydrostatic pressure was below this threshold then intramembranous bone formation proceeded (Fig. 32.7a). While the osteogenic index theory refers specifically to ossification within cartilage, the concept of a stimulus combined from hydrostatic stress and shearing due to tension may be applied more generally to tissue differentiation. Tissue formation in tendons,54 at implant interfaces,55 and during distraction osteogenesis56 has been analyzed using a relationship between mechanical stimuli and tissue differentiation shown in Fig. 32.7b. Stress and strain are combined on these diagrams even though these are not, in general, decoupled.57

32.3.4 Models Including Fluid Flow Although models of tissues as solid elastic materials may give adequate macroscopic load/deformation responses, the cells respond to cell level deformations and fluid flows (see Chapter 27). Since tissues are composed of a solid phase (collagen, with hydroxyapatite in the case of bone) and a significant amount of fluid (mostly water), they may be analyzed as mixtures of solid and fluid constituents. When bone58 or cartilage59 is loaded, the fluid component may flow and cause complex mechanical stimuli within tissues at the cellular level. Several in vitro cell studies have demonstrated the potential regulatory role of mechanical stimulation of bone cells,60 in the form of both fluid flow (osteocytes61 and osteoblasts62) and *See Chapter 6 for an introduction to mechanics of materials.



32-9

FIGURE 32.7 (a) Relationship between mechanical stimuli and bone formation. (Adapted from Claes, L.E. and Heigele, C.A., J. Biomech., 32, 255, 1999.) (b) Relationship between loading history and tissue phenotype. (Adapted from Carter, D.R. et al., Trans. Orthop. Res. Soc., 234, 1998.)

FIGURE 32.8 (a) A mechanoregulation diagram for bone formation and resorption based on mechanical strain and fluid flow; (b) application to simulation of fracture healing. (Courtesy of Damien Lacroix.)

substrate strain (osteocytes63 and osteoblasts64). Owan et al.65 have found that the responsiveness to fluid flow may dominate that of mechanical strain in bone cells. The biomechanical stress acting on cells will be high if the fluid flow is high: therefore, deformation and fluid flow must be taken together to define the mechanical milieu on the cell.66,67 Following Pauwels and the interfragmentary strain theory, high strains may be hypothesized to create the mechanical environment for fibroblast differentiation from the mesenchymal cell pool and the subsequent emergence of a fibrous connective tissue phenotype. Similarly, intermediate strains will allow cartilage formation and low strains will allow bone formation. The influence of fluid flow is to increase the deformation of the cells further. Therefore, the presence of high fluid flow will increase the biomechanical stress and thus decrease the potential for tissue differentiation in the case of each tissue phenotype. If the strain or fluid flow becomes low, then the lack of mechanical stimulation to the cells may initiate a resorptive process, as indicated by the resorption field on Fig. 32.8a.


32-10

Bone Mechanics

This regulatory scheme can simulate the time course of bone regeneration around implants using a poroelastic biphasic theory to calculate the stress in the fluid and solid phases of a tissue.68,69 For a biphasic material, the solid stress, denoted s, and fluid stresses, denoted f , are given by

s pI e I 2 ,

(32.5)

f pI,

(32.6)

s

s

s

f

where e and denote the dilatational strain and the total strain in the solid phase, p is the apparent pressure in the fluid, with denoting the volume fraction, and and being the Lamé constants. This model has been used to simulate the time-course of fracture healing70 where the influence of loading on bone regeneration in the callus can be predicted, see Fig. 32.8b.

32.4 Future Directions This chapter has introduced the biology of bone healing and reviewed the theories that describe the regulation of bone regeneration by mechanical forces. Fracture repair is increasingly well understood from the point of view of the molecular biologist; the influence of the mechanical factors and the interaction between biophysical and biochemical regulatory pathways have yet to be fully elucidated. Further development of models is needed to relate continuum stress to subcontinuum stimuli at the cellular level. With such models, mechanical stimuli determined from in vitro cell culture experiments could be used to compute how cells react within regenerating bone. These models have exciting potential not only for understanding fracture healing and tissue formation but also for designing bioreactors used to grow tissue-engineered constructs.71

Acknowledgments The authors acknowledge financial support from the Wellcome Trust for a Biomedical Research Collaboration Grant between Trinity College and Cornell University. We thank Alicia Bailon-Plaza and Damien Lacroix for their contributions to this chapter.

References 1. Wolpert, L., Regeneration, in The Triumph of the Embryo, Oxford University Press, Oxford, 1991, chap. 13. 2. Johner, R., Zur Knochenheilung in abhängigkeit von der defektgrösse, Helv. Chir. Acta, 39, 409, 1972. 3. Pauwels, F., Grundriss einer biomechanik der frakturheilung, in 34th Kongress der Deutschen Orthopädischen Gesellschaft, Ferdinand Enke, Stuttgart, 1940, 62–108. (Translated by Maquet, P, Furlong, R. as Biomechanics of fracture healing, in Biomechanics of the Locomotor Apparatus, Springer, Berlin, 1980, 375–407.) 4. Søballe, K., Hydroxyapatite ceramic coating for bone implant fixation. Mechanical and histological studies in dogs, Acta Orthop. Scand., 64, Suppl. 255, 1993. 5. Brand, R.A. and Rubin, C.T., Fracture healing, in Surgery of the Musculo-skeletal System; Vol. 1, McCollister, E.C., Ed., Churchill-Livingstone, London, 1990, 93–114. 6. Bolander, M.E., Regulation of fracture repair and synthesis of matrix macromolecules, in Bone Formation and Repair, Brighton, C.T., Friedlander, G., and Lane, J.M., Eds., American Academy of Orthopaedic Surgeons, Rosemont, IL, 1994, chap. 11. 7. Bolander, M.E., Regulation of fracture repair by growth factors, Proc. Soc. Exp. Biol. Med., 200, 165, 1992.



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8. Barnes, G.L, Kostneuik, P.J., Gerstenfeld, L.C., and Einhorn, T.A., Perspective: growth factor regulation of fracture repair, J. Bone Miner. Res., 14, 1805, 1999. 9. Joyce, M.E., Terek, R.M., Jingushi, S., and Bolander, M.E., Role of transforming growth factor—in fracture repair, Ann. N.Y. Acad. Sci., 593, 107, 1990. 10. McKibbin, B., The biology of fracture healing in long bones, J. Bone Joint Surg., 60B, 150, 1978. 11. Einhorn, T.A., The cell and molecular biology of fracture healing, Clin. Orthop., 355S, 7, 1998. 12. Yoo, J.U. and Johnstone, B., The role of osteochondral progenitor cells in fracture repair, Clin. Orthop., 355, 73, 1998. 13. Brighton, C.T. and Hunt, R.M., Early histological and ultrastructural changes in the medullary fracture callus, J. Bone Joint Surg., 73A, 832, 1991. 14. Bostrom, M.P.G., Expression of bone morphogenetic proteins in fracture healing, Clin. Orthop., 355S, S116, 1998. 15. Sandberg, M., Aro, H., Multimäka, P., Aho, H., and Vuorio, E., In situ localization of collagen production by chondrocytes and osteoblasts in fracture callus, J. Bone Joint Surg., 71A, 69, 1989. 16. Bax, B.E., Wozney, J.M., and Ashurst, D.E., Bone morphogenetic protein-2 increases the rate of callus formation after fracture of rabbit tibiae, Calcif. Tissue Int., 65, 83, 1999. 17. White, A.A., Panjabi, M.M., and Southwick, W.O., The four biomechanical stages of fracture repair, J. Bone Joint Surg., 59A, 188, 1977. 18. Marsh, D., Concepts of fracture union, delayed union, and nonunion, Clin. Orthop., 355S, 22, 1998. 19. Richardson, J.B., Cunningham, J.L., Goodship, A. E., O’Connor, B.T., and Kenwright, J., Measuring stiffness can define healing of tibial fractures, J. Bone Joint Surg., 76B, 389, 1994. 20. Goodship, A.E., Watkins, P.E., Rigby, H.S., and Kenwright, J., The role of fixator frame stiffness in the control of fracture healing. An experimental study, J. Biomech., 26, 1027, 1993. 21. Goodship, A.E. and Kenwright, J., The influence of induced micromovement upon the healing of tibial fractures, J. Bone Joint Surg., 67B, 650, 1985. 22. Wolfe, S.W., Lorenze, M.D., Austin, G., Swigart, C.R., and Panjabi, M.M., Load-displacement behaviour in a distal radial fracture model—the effect of simulated healing on motion, J. Bone Joint Surg., 81A, 53, 1999. 23. Probst, A., Janson, H., Ladas, A., and Spiegel, H.U., Callus formation and fixation rigidity: a fracture model in rats, J. Orthop. Res., 17, 256, 1999. 24. Gardner, T.N., Evans, M., and Simpson, H., Temporal variation of applied inter fragmentary displacement at a bone fracture in harmony with maturation of the fracture callus, Med. Eng. Phys., 20, 480, 1998. 25. Goodship, A.E., Cunningham, J.L., and Kenwright, J., Strain rate and timing of stimulation and mechanical modulation of fracture healing, Clin. Orthop., 355S, S105, 1998. 26. Augat, P., Margevicius, K., Simon, J., Wolf, S., Suger, G., and Claes, L., Local tissue properties in bone healing: Influence of size and stability of the osteotomy gap, J. Orthop. Res., 16, 475, 1998. 27. Cornell, C.N. and Lane, J.M., Newest factors in fracture healing, Clin. Orthop., 277, 297, 1992. 28. Sato, M., Ochi, T., Nakase, T., Hirota, S., Kitamura, Y., Nomura, S., and Jasui, N., Mechanical tension-stress induces expression of bone morphogenetic protein (BMP)-2 and BMP-4, but not BMP-6, BMP-7, and GDF-5 mRNA, during distraction osteogenesis, J. Bone Miner. Res., 14, 1084, 1999. 29. Li, G., Berven, S., Simpson, H., and Triffitt, J.T., Expression of BMP-4 mRNA during distraction osteogenesis in rabbits, Acta Orthop. Scand., 69, 420, 1998. 30. Bonnarens, F. and Einhorn, T.A., Production of a standard closed fracture in laboratory animal bone, J. Orthop. Res., 2, 97, 1984. 31. Aro, H., Eerola, E., and Aho, A.J., Determination of callus quantity in 4-week-old fractures of the rat tibia, J. Orthop. Res., 3, 101, 1985. 32. van der Meulen, M.C., Bailón-Plaza, A., and Hunter, W.L., Long bone fracture healing in bone morphogenetic protein-5-deficient mice, J. Bone Miner. Res., 13, S352, 1998.


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33. Carter, D.R., Beaupré, G.S., Giori, N.J., and Helms, J.A., Mechanobiology of skeletal regeneration, Clin. Orthop. Relat. Res., 355S, 41, 1998. 34. Caplan, A.I. and Boyan, B.D., Endochondral bone formation: the lineage cascade, in Mechanism of Bone Development and Growth. B. K. Hall, Ed., CRC Press, Boca Raton, FL, 1994. 35. Pauwels, F., Eine neue Theorie über den einfluss mechanischer Reize auf die Differenzierung der Stützgewebe, Z. Anat. Entwickl., 121, 478, 1960. (Translated by Maquet, P., Furlong, R. as A new theory concerning the influence of mechanical stimuli in the differentiation of supporting tissues, in Biomechanics of the Locomotor Apparatus, Springer, Berlin, 1980, 375–407. 36. Pauwels, F., Atlas zur Biomechanik der Gesunden und Kranken Hüfte, Springer, Berlin, 1973. 37. Prichard, J.J., Bone, in Tissue Repair, McMinn, R.M.H., Ed., Academic Press, London, 1969, chap. 4. 38. Weinans, H. and Prendergast, P.J., Tissue adaptation as a dynamical process far from equilibrium, Bone, 19, 143, 1996. 39. Perren, S.M., Physical and biological aspects of fracture healing with special reference to internal fixation, Clin. Orthop., 138, 175, 1979. 40. Perren, S.M. and Cordey, J., The concept of interfragmentary strain, in Current Concepts of Internal Fixation of Fractures, H. K. Uhthoff, Ed., Springer, Berlin, 1980, 63–77. 41. Tencer, A.F. and Johnson, D., Biomechanics in Orthopaedic Trauma, Martin Dunitz, London, 1994. 42. DiGioia III, A.M., Cheal, E.J., and Hayes, W.C., Three-dimensional strain fields in a uniform osteotomy gap, J. Biomech. Eng., 108, 273, 1986. 43. Biegler, F.R. and Hart, R.T., Finite element modelling of long bone fracture healing, in Computer Methods in Biomechanics and Biomedical Engineering, Books and Journals International, Swansea, J. Middleton, G.N. Pande, K.R. Williams, Eds., 1992, 30–39. 44. Gardner, T.N., Stoll, T., Marks, L., and Knote-Tate, M., Mathematical modelling of stress and strain in bone fracture repair tissue, in Computer Methods in Biomechanics and Biomedical Engineering II, Gordon and Breach, Amsterdam, 1998, 247. 45. Meroi, E.A. and Natali, A.N., A numerical approach to the biomechanical analysis of fracture healing, J. Biomed. Eng., 11, 390, 1989. 46. Claes, L., Augat, P., Suger, G., and Wilke, H.-J., Influence of size and stability of the osteotomy gap on the success of fracture healing, J. Orthop. Res., 15, 577, 1997. 47. Carter, D.R., Mechanical loading history and skeletal biology, J. Biomech., 20, 1095, 1987. 48. Carter, D.R., Blenman, P.R., and Beaupré, G.S., Correlations between mechanical stress history and tissue differentiation in initial fracture healing, J. Orthop. Res., 6, 736, 1988. 49. Blenman, P.R., Carter, D.R., and Beaupré, G.S., Role of mechanical loading in the progressive ossification of a fracture callus, J. Orthop. Res., 7, 398, 1989. 50. Carter, D.R. and Wong, M., The role of mechanical loading histories in the development of diarthrodial joints, J. Orthop. Res., 6, 804, 1988. 51. Wong, M. and Carter, D.R., Mechanical stress and morphogenetic endochondral ossification of the sternum, J. Bone Joint Surg., 70A, 992, 1988. 52. Claes, L.E., Heigele, C.A., Neidlinger-Wilke, C., Kaspar, D., Seidl, W., Margevicius, K.J., and Augat, P., Effects of mechanical factors on the fracture healing process, Clin. Orthop., 355S, S132, 1998. 53. Claes, L.E. and Heigele, C.A., Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing, J. Biomech., 32, 255, 1999. 54. Giori, N.J., Beaupré, G. S., and Carter, D.R., Cellular shape and pressure may medicate mechanical control of tissue composition in tendons, J. Orthop. Res., 11, 581, 1993. 55. Giori, N.J. Ryd, L., and Carter, D.R., Mechanical influences of tissue differentiation at bone–cement interfaces, J. Arthroplasty, 10, 514, 1995. 56. Carter, D.R., Helms, J.A., Tay, B.K., and Braupré, G. S., Stress and strain distributions predict tissue differentiation patterns in distraction osteogenesis, Trans. Orthop. Res. Soc., 44, 234, 1998. 57. Cowin, S.C., Deviatoric and hydrostatic mode interaction in hard and soft tissue, J. Biomech., 23, 11, 1990. 58. Cowin, S.C., Bone poroelasticity, J. Biomech., 32, 217, 1999.



32-13

59. Mow, V.C., Ratcliffe, A., and Poole, A.R., Cartilage and diarthrodial joints as paradigms for hierarchical materials and structures, Biomaterials, 13, 67, 1992. 60. Brown, T.D., Techniques for mechanical stimulation of cells in vitro: a review, J. Biomech., 33,3, 2000. 61. Klein-Nulend, J., van der Plas, A., Seimens, C.M., Ajubi, N.E., Frangos, J.A., Nejweide, P.J., and Burger, E.H., Sensititivity of osteocytes to biomechanical stress in vitro, FASEB J., 9, 441, 1995. 62. Jacobs, C.R., Yellowley, C.E., Davis, B.R., Zhou, Z., Cimbala, J.M., and Donahue, H.J., Differential effect of steady versus oscillating flow on bone cells, J. Biomech., 31, 969, 1998. 63. Pitsillides, A.A., Mechanical strain-induced NO production by bone cells: a possible role in adaptive bone (re)modelling, FASEB J., 9, 1614, 1995. 64. El Haj, A.J., Mitner, S.L., Rawlinson, S.C.F., Suswillo, R., and Lanyon, L.E., Cellular responses to mechanical loading in vitro, J. Bone Miner. Res., 5, 923, 1990. 65. Owan, I., Burr, D.B., Turner, C.H., Qui, J., Tu, Y., Onya, J.E., and Duncan, R.L., Mechanotransduction in bone: osteoblasts are more responsive to fluid flow than mechanical strain, Am. J. Physiol., 273, C810, 1997. 66. Prendergast, P.J. and Huiskes, R., Finite element analysis of fibrous tissue morphogenesis—a study of the osteogenic index using a biphasic approach, Mech. Composite Mater. (Riga), 32, 209, 1996. 67. Prendergast, P.J., Huiskes, R., and Søballe, K., Biophysical stimuli on cells during tissue differentiation at implant interfaces, J. Biomech., 30, 539, 1997. 68. van Driel, W.D., Huiskes, R., and Prendergast, P.J., A regulatory model for tissue differentiation using poroelastic theory, in Poromechanics: A Tribute to Maurice A. Biot, J.-F. Thimus, Y. Abousleiman, A.H.-D. Cheng, O. Coussy, and E. Detournay, Eds., A.A. Balkema, Rotterdam, 1998, 409–413. 69. Huiskes, R., van Driel, W.D., Prendergast, P.J., and Søballe, K., A biomechanical regulatory model for peri-prosthetic tissue formation, J. Mater. Sci. Mater. Med., 8, 785, 1997. 70. Lacroix, D. and Prendergast, P.J., A model to simulate the regenerative and resporption phases of long bone fracture healing, Trans. Orthop. Res. Soc., 25, 869, 2000. 71. Patrick, C.W., Mikos, A.G., and McIntire, L.V., Frontiers in Tissue Engineering, Pergamon, Oxford, 1998.

ch-32 Page 14 Friday, February 2, 2001 2:38 PM

Section-VI Page 1 Monday, January 22, 2001 2:24 PM

VI Clinically Related Issues 33 Applications of Bone Mechanics Marta L. Villarraga and Catherine M. Ford ....................................................................................................... 33-1 Introduction • Parameters Related to Applied Loads • Ex Vivo Behavior and Imaging Parameters • In Vivo Fracture Risk and Imaging Parameters • Development of Models of Skeletal Structures

34 Noninvasive Measurement of Bone Integrity Jonathan J. Kaufman and Robert S. Siffert ........................................................................ 34-1 Introduction • X-Ray Densitometry • Ultrasonic Techniques • Alternative Techniques • Summary

35 Bone Prostheses and Implants Patrick J. Prendergast .................................................... 35-1 Introduction • Biomaterials • Design of Bone Prostheses • Analysis and Assessment of Implants • Future Directions

36 Design and Manufacture of Bone Replacement Scaffolds Scott J. Hollister, Tien-Min Chu, John W. Halloran, and Stephen E. Feinberg ..................................................................................................... 36-1 Introduction • Designing Bone Scaffolds • Fabricating Bone Scaffolds • Bone Scaffolds: An Example from Design to Testing • Conclusion

VI-1

Section-VI Page 2 Monday, January 22, 2001 2:24 PM

CH-33 Page 1 Monday, January 22, 2001 2:13 PM

33 Applications of Bone Mechanics Marta L. Villarraga Exponent Failure Analysis Associates

Catherine M. Ford Exponent Failure Analysis Associates

33.1 33.2

Introduction ................................................................... 33-1 Parameters Related to Applied Loads........................... 33-2 Proximal Femur • Spine

33.3

Ex Vivo Behavior and Imaging Parameters.................. 33-9 Proximal Femur • Spine

33.4

In Vivo Fracture Risk and Imaging Parameters......... 33-16 Proximal Femur • Spine

33.5

Development of Models of Skeletal Structures ......... 33-18 Proximal Femur • Spine • Future Trends in Modeling Skeletal Structures

33.1

Introduction

From youth to old age, the human skeleton is routinely exposed to mechanical environments that challenge its structural integrity. Among younger individuals, bone fractures typically occur only when the applied loads substantially exceed those associated with routine daily activities. Common events that produce fractures among younger individuals include motor vehicle accidents, falls from heights, sports-related injuries, and occupational trauma.1 Efforts to reduce fractures associated with these types of activities often focus on designing environments and protective equipment that limit the applied loads to a level below the tolerance of the bone. In contrast, fractures among older individuals often result from events whose associated energies are substantially lower, including falls from standing height or lifting activities. In these cases, questions arise about the contribution of age-related reductions in bone strength in determining whether fracture will occur. Efforts to reduce these fractures focus on affecting the intrinsic strength of skeletal structures, for example, through pharmacological or dietary interventions, as well as on protective systems such as hip padding and energy-absorbing floors. This is in contrast to the strictly environment-oriented intervention efforts associated with higher-energy fractures in younger people. Any intervention effort to prevent bone fracture, whether aimed at the mechanical environment, bone strength, or both, must be based on a sound understanding of how the bone responds to in vivo mechanical loads. Failure of skeletal structures can be described in terms of the factor of risk,2 which is the ratio of the force applied to a bone during a specific activity to the force at which the bone will fracture. A factor of risk greater than 1 indicates that the bone will be overloaded and will most likely fail, whereas a factor of risk lower than 1 indicates that fracture is less likely to occur. This is the inverse of the factor of safety, traditionally used in engineering to characterize the safety of a structure. From an engineering mechanics perspective, material properties, geometry, and loading and boundary conditions are the key pieces of information needed for the design of a safe structure. All of these factors contribute to the stress distribution within the bone, and therefore influence the likelihood of bone failure (Fig. 33.1). 0-8493-9117-2/01/$0.00+$.50 © 2001 by CRC Press LLC

33-1


33-2

Bone Mechanics

FIGURE 33.1 Characteristics of the spine that determine its capacity to carry load: material properties, structural design, and loading conditions. (From Myers, E.R. and Wilson, S.E., Spine, 22(24 Suppl.), 25S–31S, 1997. With permission.)

Investigators have used multiple approaches to examine the relative contributions of material properties, geometry, and loading conditions to the mechanical behavior of skeletal structures. For example, substantial knowledge has been derived from examining relationships between empirically determined structural properties of whole bones ex vivo and densitometric, geometric, and architectural measurements. This has been facilitated by the evolution of bone imaging technologies that allow investigators to examine the intricate details of the microstructure and material of bone. Clinical studies of factors related to in vivo fracture risk have complemented the understanding of whole-bone mechanics gained by examining ex vivo structural behavior. Issues that have emerged as important from both ex vivo and in vivo work include the relative contributions of cortical and trabecular bone, loading conditions, geometry, and age-related density variations. A parallel approach to understanding the mechanical behavior of skeletal structures utilizes engineering models based on fundamental principles of mechanics. These models utilize the bone material properties and geometry to predict stresses under a variety of loading conditions, which is advantageous since stress cannot be directly measured in bone. Models have evolved, with ever-increasing complexity, from two-dimensional idealizations with closed-form solutions into three-dimensional, patient-specific finite-element models. While shortcomings related to these models currently limit their use in accurate prediction of whole-bone failure, such models have contributed substantially to the understanding of the factors that govern whole-bone behavior. The aim of this chapter is to review and synthesize the research that has led to the current state of the art in the understanding of whole-bone mechanics, in particular as it relates to whole-bone fracture. Specific approaches used to study whole-bone mechanics (including experimental, clinical, and modeling work) are emphasized, using research involving the proximal femur and spine to illustrate these approaches (Fig. 33.2). The chapter begins with a review of the in vivo loads associated with fracture, followed by a discussion of how parameters related to applied loads affect mechanical behavior of the femur and spine. Work that has been performed to relate ex vivo structural properties and in vivo fracture risk to imaging parameters is then reviewed, followed by a discussion of the development of mathematical models of the proximal femur and spine.

33.2 Parameters Related to Applied Loads 33.2.1

Proximal Femur

The ability to predict the stress distribution within a skeletal structure is of little practical value unless the relevant in vivo loading conditions are known. In the same way, studies of ex vivo behavior are most valuable when the loading conditions employed reflect approximate in vivo loading conditions. Early efforts to define loads acting on the proximal femur involved the use of mathematical models and reaction


Applications of Bone Mechanics

FIGURE 33.2

33-3

Femur and vertebra whole bones. (Illustration © Paul Chang. With permission.)

force measurements to estimate forces across the hip joint during stance and gait. A thorough review of the earliest work in this area is contained in Rydell.3 Williams and Lissner4 calculated a resultant force on the femoral head of 2.4 times body weight during single-leg stance. McLeish and Charnley5 predicted values of 1.8 to 2.7 times body weight with the pelvis in a normal walking attitude, and found that the pelvis angle and the position of the trunk and limbs had a considerable effect on femoral forces. These predicted forces are in general agreement with those reported by Rydell,3 whose first attempt at direct measurement of hip joint forces using an instrumented hip prosthesis yielded force magnitudes of 2.3 to 2.9 times body weight for single-leg stance and 1.6 to 3.3 times body weight for level walking. Later measurements for single-leg stance and gait using telemetered hip prostheses were generally consistent with those of Rydell for loads associated with stance and walking.6–9 Davy et al.7 noted that measured values for gait were often lower than previous predictions using optimization routines,10–13 which could be explained in part by gait differences between normal subjects and those with prostheses. The results from mathematical models and early instrumented hips have provided a fundamental understanding of the magnitude and direction of loads applied to the femur during routine (and typically noninjurious) activities of daily living. Although these data are applicable to determining stress distributions under physiological loads, the loads under which many femur fractures occur are substantially different from simple stance and gait configurations in magnitude and direction. Evidence from both clinical and experimental studies suggest that loads associated with muscular contraction can be sufficient to result in femur fracture among elderly people.14–18 Studies using optimization methods as well as instrumented hip prostheses have demonstrated the importance of muscle forces as determinants of load transmission across the joint and therefore the stress distribution within the bone.9,19,20 Recent investigations using instrumented hip prostheses have shown that although patients in the early postoperative


33-4

Bone Mechanics

period can execute planned activities of daily living with relatively low joint contact forces, unexpected events such as stumbling or periods of instability during single-leg stance can generate resultant forces in excess of eight times body weight as a result of muscular contraction.8,21 Forces that are within the capacity of the muscles crossing the hip joint have been shown to produce clinically realistic fracture patterns in ex vivo experiments,18 consistent with retrospective clinical studies in which femur fractures were attributed to turning, stumbling, rising from a seated position, or climbing.16,17,22 These data suggest that femur fracture can occur as a result of muscular contraction in individuals with reduced bone strength. However, attempts to rule out impact from a fall as a cause of proximal femur fracture based on either primitive stress analyses23 or ex vivo fracture studies18 should be interpreted with caution. Despite compelling evidence that the majority of femur fractures are associated with falls,24–27 surprisingly little is known about the magnitude and direction of loads applied to the femur during a fall. Robinovitch and colleagues28 combined pelvis release experiments with a lumped parameter model of the body to predict peak impact forces up to approximately 8000 N or more, depending on fall height and trochanteric soft tissue thickness. By comparing their predictions to measurements of the forces required to fracture the femur under fall loading conditions, these investigators concluded that any fall from standing height with direct impact to the greater trochanter has the potential to fracture the elderly femur. van den Kroonenberg and co-workers29 used dynamic link models of the body to predict hip impact velocities, effective mass, and peak femur impact forces during falls from standing height. Using their most-sophisticated model, which reflected experimentally determined values for impact configuration and impact velocity,30 they predicted peak forces on the greater trochanter of 2.9 and 4.3 kN for females of 5th and 95th percentile height and weight, respectively. The stress distribution within the femur, and therefore the likelihood of fracture, depends on the direction of the applied loads as well as their magnitude. Despite its potential importance, however, few studies have directly addressed the configuration of loads applied to the femur during a fall, and how changes in this configuration influence failure load. Courtney and co-workers31 derived a loading configuration (Fig. 33.3) based on body positions at impact during a fall,30 which produced clinically relevant fracture patterns. Pinilla and colleagues32 extended this work by using cadaveric femurs to examine the

FIGURE 33.3 Diagram of the mechanical testing setup for failure tests of the proximal femur. The femoral shaft is aligned 10° from horizontal and femoral neck is internally rotated 15° relative to vertical. The femoral shaft is fixed distally but remains free to slide and rotate. (From Bouxsein, M.L. et al., Calcif. Tissue Int., 56(2), 99–103, 1995. With permission.)



33-5

effect of subtle variations in impact direction from a fall. They found a significant reduction in failure load with a 30° change in the angle between the line of action of the applied force and the angle of the femoral neck, as viewed from the superior aspect of the femoral head. This reduction in failure load was noted to be comparable in magnitude to approximately 25 years of age-related bone loss after age 65. These experimental results have been confirmed by parametric finite-element analyses.33 Loading rate is an additional variable related to applied load that must be considered in determining the stress distribution in the femur and the likelihood of fracture. Weber et al.34 found that, under loading conditions approximating a fall, the force required to fracture the femur increased with increasing loading rate. This general information regarding the influence of loading rate was refined by Courtney and co-workers,31 who used a predicted loading rate based on time to peak force during a fall.28 These authors reported that fracture forces for both younger adult and older adult femurs increase by about 20% with a 50-fold increase in applied displacement rate. Differences in both loading configuration and loading rate may partly explain why investigators have reported mean structural capacities of the proximal femur under both stance and fall loading conditions that vary by as much as a factor of four to five (Table 33.1). The wide range in failure loads reported in these ex vivo studies shows that failure load can be dramatically influenced by several variables; however, substantial future work is necessary to characterize precisely the contributions of these variables.

33.2.2 Spine Every routine activity, including sitting, standing, walking, or lifting a light or heavy object, creates loads on the spine. Knowledge of relevant in vivo loading conditions on the spine is necessary to predict adequately the stress distribution in this structure, in both in vivo and ex vivo studies. Schultz et al.64 predicted in vivo lumbar spine compressive loads ranging from 440 N (relaxed standing position) to 2350 N (forward bending with the trunk at 30° and holding an 8-kg weight at arm’s length) using a combination of electromyography (EMG) and intradiscal pressure measurements on volunteers and mathematical modeling. In a later study using volunteers, Hansson et al.65 estimated that the compressive loads on L3 during squat lifting and lifting while bent at the waist range from 5000 to 11,000 N. Using a biomechanical analysis, validated against EMG and intradiscal pressure measurements, they estimated that the compressive load on L3 during these types of exercises was about eight times greater than the shear load. Actual magnitudes of loads on an individual’s lumbar vertebrae depend on the anatomy and weight of the individual, as well as on muscle forces. Characterizing activities as specific events leading to spine fractures is the subject of current research. Complications arise in part from the fact that osteoporotic compression fractures of the spine are usually associated with minimal trauma and with loads no greater than those encountered during normal activities of daily living.66 Another contributing factor is fatigue, which may in part explain vertebral fractures that are reported as spontaneous or are detected incidentally.67 Loading of the spine during falls is currently under investigation, with particular emphasis on the transmission of impact forces along the spine.67 To understand how loads are distributed in the spine, it is important to consider the structures that form the spine and how each of these components contributes to load transmission. The majority of the compressive forces in the spine are transmitted from the intervertebral disks to the vertebral end plates, and are then distributed between the trabecular bone and cortical shell that make up the vertebral bodies. The exact percentage of contribution of the cortical shell to load sharing is still not well established and is currently under investigation. A portion of the compressive loads can also be transmitted along a parallel path, through the posterior elements and facet joints. The overall contribution of the posterior elements to vertebral body strength is currently under investigation with the help of mathematical models.68 In evaluations of the effects of compressive loads on vertebrae, stress levels associated with end plate failure are of special concern, since the end plate has been shown to fail first under high compressive loads.65,69 Also of concern are stress levels that exceed the strength of trabecular bone, because most of the axial compressive load on a vertebral body is carried by the trabecular bone.67,70

d

— 33 14 10d 39 61 32 19 10 20j 10 4 6

Hirsch (1960)36 Mizrahi (1984)37 Yang (1996)18

Vose (1963)38

Phillips (1975)39 Dalen (1976)40 Leichter (1982)41

Sartoris (1985)42 Werner (1988)43 Delaere (1989)44

Weber (1992)34

22 17

22 26d 21d 13d 20d 13d 6 28

n

0.7 14 4000

— 0.33 0.03

0.20 0.25 0.08

37 N/s

— 0.08 50

— —

Gradual Impact Gradual Impact Gradual Impact Gradual —

Load Rate (mm/s)

2930 3510 6020

— — 11,000

— — 6100

6200

1770–4940 2190–5950 3110–9720

— — 5100–19,500

3070–13,300 — 1600–12,600

3830–10,000

2940–7850 1600–12,800 —

14–59 980–7850

29 Nm 3830

4900 — 3040

4310–18,700 170–475e 2670–15,000 233–624 2270–10,700 24–102 — 1280–11,900

9470 371 J 8230 415 J 4000 49 J 3410 3860

Range (N)

Failure Load Mean (N)

DPA

DESPR DPA DPA

X-ray XRS CS

X-ray

— — —

— —

— — — — — — — —

Imaging Methoda

BMD, intertrochanteric

Density, Ward’s triangle Average area porosity BMC Trabecular density, neck Neck density index BMC BMD, proximal diaphysis BMD, neck

— — —

— —

— — — — — — — —

Correlated Parameterb

0.60

0.89

0.29 NS 0.55

— 0.79 0.66

0.64

— — —

— —

— — — — — — — —

Max r 2(c)

(continued)

0.071q

0.0001q

0.05 — —

— — 0.001

0.005g

— — —

— —

— — — — — — — —

p Value

33-6

Fall

Stance Fallh Stance

Stance Fallh Stance

Bending, shear Bending, shear, compression Torsion Bending, shear, compression, torsion Stance Stance Muscle contractionf Stance

Anterior bending

Fall

Stance

Loading Configuration

Reported Failure Properties of the Proximal Femur and Correlations with Imaging Parameters

Backman (1957)35

Smith (1953)

14

Author (year)

TABLE 33.1


Bone Mechanics

Fall

13

17 16 33

64

Courntey (1994)31

Courtney (1995)56

Bouxsein (1995)57

Pinilla (1996)32

Cheng (1997), Nicholson (1997)58,59

Fall

18

Fall, 0° n Fall, 15° Fall, 30° Fall

Fall

Fall

Fall

Stance

Stance

18

25 26 19

Cody (1999)55

Keyak (1998)54

Stance

13

Lang (1997)53

Compressionm

22

Smith (1992)52

Stance Rotationl Rotationl Stance Stance Fall

36 — 50 26 8 12

Alho (1988)45–47 Alho (1988)47 Husby (1989)48 Alho (1989)49 Esses (1989)50 Lotz (1990)51

14

100

2

2

100

0.21

—

—

—

0.5

0.21

0.08 0.88 0.70

— 5°/min 15° /min

4050 3820 3060 3980

3680

5430

9920 9200 —

—

—

—

—

—

5490k — 74.6(Nm) 6100 — 2110

— — — —

1730–6730

—

— — —

—

—

—

—

4940–16,100

2750–9610 — — — — 778–4040

DXA QCT US

DXA US DXA

DXA

DXA

QCT

QCT

QCT

QCT

QCT

QCT QCT QCT QCT QCT QCT

BMD, trochanter CA, trochanter BUA, calcaneus

CSA, neck BMD, neck BMD, neck CSA, neck BMD, neck BUA, calcaneus BMD, total

HU CSA, neck HU CSA, midshaft HU CSA, midshaft HU CSA, head HU, subcapital HU CSA, intertrochanteric BMD, superomedial head Neck BMD (trabecular) min CSA FNAL Trochanter BMD (trabecular) min CSA BMD, subcapital FE Model BMD CSA, intertrochanteric FE model FE model

0.0001 0.0001 0.0001 0.0001 — — — — 0.001 — 0.001 0.002 — — — 0.001 0.001 0.01 0.90 0.82 0.84 0.77 0.72 0.92 0.79 0.79 0.51 0.68 0.68 0.71 0.88 0.83 0.47

—

0.89

0.61 0.75 0.82

—

—

0.001 0.001 0.001 0.001 0.02 0.001

0.93

0.66

0.62 0.50 0.50 0.71 0.64 0.93

CH-33 Page 7 Friday, February 2, 2001 2:42 PM

Applications of Bone Mechanics 33-7

22 58d 25 45d

Beck (1998)60 Lochmuller (1998)61

Bouxsein (1999)62

Lochmuller (1999)63 Stance

Fall

Fall Stance

Loading Configuration

60

100

0.5 1.0

Load Rate (mm/s)

3200

2640

— 3150

933–7000

933–7000

Range (N)

Failure Load Mean (N)

DXA US US

DXA DXA

Imaging Methoda Failure load BMD, trochanter and neck BMD, trochanter BUA, calcaneus SOS/BUA, calcaneusq

p

Correlated Parameterb

0.92 0.70 0.52

0.83 0.45

Max r2(c)

0.0001 0.0001 0.001

— 0.001

p value

a

XRS X-ray spectophotometry; CS Compton scattering; DESPR dual-energy scanned projection radiography; DPA dual-photon absorptiometry; QCT quantitative computed tomography; DXA dual-energy X-ray absorptiometry; US sultrasound. b BMC bone mineral content; BMD bone mineral density; HU Hounsfield units; CSA cross-sectional area; FNAL femoral neck axis length; FE finite element; CA cortical area; BUA broadband ultrasound attenuation; SOS speed of sound. c NS not significant. d Embalmed specimens. e Seven specimens dislocated from pelvis without fracture. f Simulated gluteus medius and iliopsoas contraction. g p value calculated from raw data. h Femur embedded up to neck; force applied perpendicular to shaft. j Dried and defatted specimens. k Median value. l Femoral head cemented in polyurethane blocks and rotated about femoral neck axis. m Compression applied approximately along femoral neck axis. n Angle between femoral neck axis and applied load, as viewed from superior aspect of femoral head. p Failure load calculated using a DXA-based curved beam model. q Composite value of SOS and BUA.

n

Reported Failure Properties of the Proximal Femur and Correlations with Imaging Parameters (Continued)

Author (year)

TABLE 33.1


33-8 Bone Mechanics



33-9

Although much of the loading imposed on the spine during daily activities involves compression, many lifting activities also involve bending, which introduces a flexion component as well. Flexion has been shown to produce greater peak compressive stresses on vertebral bodies than pure compresssion.64 To determine the effects of flexion, Granhed et al.69 examined whether the compressive strength of thoracolumbar vertebrae varied with the direction of the applied load by comparing compressive loads applied axially to those applied in different degrees of flexion. In their study, the first components to fail were the end plates and the underlying trabecular bone. These results were similar to those of previous studies in which vertebral bodies with 3 mm of adjacent disks covering each end plate were loaded strictly axially.71 The intervertebral disks play an important role in the transfer of loads to the vertebral body. To examine the influence of the presence of disk material (annulus and nucleus pulposus) on load transfer and the consequent effects on the strength and stiffness of lumbar vertebral bodies, Keller et al.72 examined the mechanical properties of trabecular bone samples from different locations with respect to the annulus or nucleus pulposus. They found regional variations in compressive properties of trabecular bone samples as a function of location with respect to the disk, with posterocentral samples having greater stiffness and strength than peripheral samples. This heterogeneous distribution of compressive mechanical properties in segments with normal disks was different from the more homogeneous distribution in segments with degenerated disks. In addition to load magnitude and direction, loading rates and the resulting differences in strain rates must be considered when evaluating the effects of prescribed loading conditions on the spine (Table 33.2). This is of particular significance since it has been hypothesized that elderly individuals could be subjected to loads at relatively low strain rates.73 Furthermore, strain rate has been found to increase as a result of muscular fatigue at other skeletal sites, suggesting a causal relationship with stress fractures.74,75 Occasional overloads may increase the risk of vertebral fracture by substantially degrading the mechanical properties of trabecular bone.73 Examples include non-fracture-producing falls that could potentially have a deleterious effect on the mechanical competence of vertebral bodies. Keaveny et al.73 quantified the percentage reductions in modulus and strength and the development of residual strains due to overloading. Reductions in modulus of up to 85% and reductions in strength of up to 50% occurred with overloads of as much as 3% of the total applied strain, and residual strains developed depended mainly on the magnitude of the applied strains. Work that has been performed to characterize the magnitude and direction of loads applied to the proximal femur and spine during both daily activities and falls provides an important foundation for ex vivo experimental work. However, further research is necessary to characterize the loads leading to femur and spine fractures. Further consideration of boundary conditions is also warranted, including, for example, the effects of irregular end plates and compliant disks in the spine, as well as the effects of constraints at the knee joint and acetabulum on the behavior of the femur.

33.3 Ex Vivo Behavior and Imaging Parameters 33.3.1 Proximal Femur Investigations of the mechanical behavior of the proximal femur ex vivo and the relationship to imaging parameters have provided insight into factors that are important in determining the stress distribution in the bone and resultant failure properties. Early ex vivo investigations of the proximal femur were primarily oriented toward determining the tolerance of the femur under different loading conditions, as well as examining the associated fracture patterns.14,35,36,91 The results of these early investigations suggested the importance of the complex interplay between loading configuration, variations in local strength, and variations in geometry in determining the tolerance of the femur as well as the resulting fracture pattern, despite a lack of sophisticated imaging techniques.

L2

L2

L2

L1–L4

T12–L5 L2–L4

Mosekilde (1988)77

Vesterby (1991)78

Korstjens (1996)79

Hansson (1980)71

Granhed (1989)69 Ortoft (1993)80

Mosekilde (1990)76

Mosekilde (1986)

T5 T7 L2 L2

Spine Level

52 46

FSU-NE VB-NE

VB-D

VB-NE

VB-NE

VB-NE

VB-NE

VB-NE

12h 4.5

5

4.5

4.5

4.5

4.5

4.5

Loading Rate (mm/min)

Fmax max Fmax Fmax max Fmax max

max

3850 2.27 5430

1520–11,000 1.02– 4.95 810–10090

2400–8600 3940–9860

6290

Fmax Fmax

1.5–7.8

Range Loads(N), Stress (MPa)

2.00–7.04

8260 (50 yrs) 4590 (50–75yrs) 3110 (75 yrs) 5.83 (50 yrs.) 2.96 (50–75 yrs) 1.95 (75 yrs) 6940 (50 yrs) 3260 (50–75 yrs) 2800 (75 yrs) 5.70 (50 yrs) 2.62 (50–75 yrs) 2.29 (5 yrs)

Mean Loads (N), Stress (MPa)

max

max

Fmax

max

Fmax

max

Mechanical Parameterb

DPA DPA

DPA

Imaging Method c

BMC BMCT BMCT BMCB BMCB

BV/TV (%) (L1) CTh (L1) Vm (L1) TbTh (L1) Vtr (L1) Radiographic trabecular pattern (L3) BMC

Iliac crest TBV (%)

Age (females)

Age (females)

Age (females)

Age (males)

Age

Correlated Parameter d f

0.74 0.48 positive 0.48 0.22 0.55 0.24

0.01 0.01 0.0001 0.001 0.001 0.001 0.001

0.001 0.007 0.001 0.002 0.05

0.56 f, g 0.38 g 0.52 g 0.50 f 0.59

g

0.01 0.002

0.001 0.35 g 0.49

0.48

0.001

0.001 0.69f

0.71f

0.001

0.001

31.3 Mechanics and Computational Issues

31.3 Mechanics and Computational Issues

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