A unifying principle relating stress to trabecular bone morphology

Journal o Orthopuedic Research 4304-31l Raven Press, New York 0 1986 Orthopaedic Research Society

A Unifying Principle Relating Stress to Trabecular Bone Morphology

D. P. Fyhrie and D. R. Carter

Design Division, Mechanical Engineering Department, Stanford University, Stanford, California, U.S.A.

~ ~~ ~

Summary: The relationships between cancellous bone apparent density, trabecular orientation, and stress are developed and a mathematical theory de- scribing these relationships is proposed. The bone is assumed to be a self-optimizing material. Using a continuum model, sufficient conditions are developed which ensure that, for a given stress encountered during normal activity, the theory will predict both trabecular orientation and apparent density. Using two special approaches, one based on optimizing strain energy density (stiffness) and the other on optimizing strength, the relationship between apparent density and stress is derived. This is the first time that a single theory has been advanced to predict both the orientation and apparent density of cancellous bone. Key Words: Cancellous bone-Remodeling- Apparent density-Tra- jectory hypothesis- Wolff’s law-Optimal material.

Since the observations of Wolff (28), von Meyer (19), Roux (21), and Culmann (8), it has been generally accepted that the trabecular orientation and apparent density of cancellous bone are functions of the magnitudes and directions of the in vivo loads (20). Two separate theories have been postulated to explain the observed variation of cancellous bone morphology within the whole bone. The first theory is that trabecular orientation is a function of the principal stress directions (the trajectory hypothesis). The second is that apparent density is a function of an effective stress measure. The best quantitative examinations of the trajectory and effective stress hypotheses are due to Hayes and Snyder (13) and Stone et al. (23), who compared the results of finite element analyses of the human patella to measurements of the trabecular orientation and areal density. The former study was two dimensional (2D), supported the trajectory hypothesis, and found that the magnitude of the von

Address correspondence and reprint requests to Dr. D. P. Fyhrie at Palo Alto Veteran’s Administration Hospital 6401153, Rehabilitation Research and Development Center, 3801 Miranda Avenue, Palo Alto, CA 94304, U.S.A.

Mises’ equivalent stress correlated strongly with density. The latter study was three dimensional (3D), supported the trajectory hypothesis, but found that the von Mises’ stress was only weakly correlated to the density.

A dynamic theory of cortical bone remodeling has been proposed by Cowin and Hegedus (6). The constitutive model for the bone remodeling rate was discussed in general terms but no specific form was put forward. In the linear approximation to this theory (7,15) it was assumed that the remodeling rate is a linear function of the strain (or stress). The linear theory has been implemented in a 3D finite element code (11,12) and used to solve cortical bone remodeling problems based on the real geom- etry of an experimental tibia and for an idealized tubular bone. The discussion in Hart et al. (12) describes some of the important aspects of the theory, for example, neglecting the influence of strain his- tory effects on cortical bone remodeling rate. A feature of the linear theory that makes its applica- tion to cancellous bone remodeling difficult is that it does not account for the adaptation of the directions of material anisotropy to the load. This would be a serious consideration in any attempt to use this

304

A UNIFYING PRINCIPLE FOR TRABECULAR BONE 305

theory to predict the response of cancellous bone to a new loading condition.

We will develop a theory that predicts the changes in trabecular orientation and trabecular bone apparent density resulting from a change in applied stress. It will be shown that this new theory is consistent with the trajectory hypothesis (20), which states that cancellous bone trabeculae are oriented with the average or a set of representative principal stresses that the bone experiences due to normal activity. We will show that the trajectory hypothesis and the relationship between effective stress and apparent density can be described by a unifying minimization principle involving a quadratic function in stress similar to strain energy density and a purely quadratic Tsai-Wu failure criterion. First, a theory of cancellous bone optimality will be presented. This will then be specialized to a form similar to well-known failure theories. This specialized form will be used to predict optimal orientation and apparent density using, first, a strain energy interpretation of the theory and, second, a failure stress interpretation of the theory. The results of the two approaches are similar and show that the simultaneous prediction of trabecular orientation and apparent density is possible.

Bone Remodeling Hypothesis

We will assume that cancellous bone is an anisotropic material, which tends to maximize its structural integrity while simultaneously minimizing the amount of bone present. To measure the degree to which the bone meets these conflicting goals we postulate a remodeling objective function, O(p,O,o), which measures the optimality of the bone under a given load, where Q is a positive function of the apparent density p and orientation 8 of the anisotropic material axes with respect to the principal directions of the given static stress (T. That is

Q(p,e,a) 2 o (1)

Q(p,8,a) = o e c = o (2)

where p is the apparent density of bone; 8 is the orientation of material axes; and a is the stress tensor. The orientation of the material axes 8 is the angular information necessary to transform from principal stress axes to material axes and is com- posed of three angles QT = [a p y] for 3D materials and of a single angle 8 = [y] for 2D materials. This is illustrated in Fig. 1 for a 2D material.

The objective function can be used to find the

Pr inc ipa l stress axes

I I

FIG. 1. Material orientation for a two-dimensional material.

optimal orientation 8* and the minimum acceptable apparent density pA, by solving

(3)

By finding the best orientation and the least bone apparent density such that the objective is less than Q,,, the bone is optimized for the given stress. Equations 3 and 4 form the general remodeling principle.

The minimization of the apparent density (Eq. 3) represents the tendency to use the least amount of bone tissue. The minimization of the objective (Eq. 4) represents the tendency to maximize the bone's structural integrity. The smaller Q is the greater the structural integrity.

The value of the objective in Eq. 4 is bounded by Q,,, to put a lower limit on the apparent density. If we reduce the bone apparent density to decrease the total amount of tissue that the system must sup- port (Eq. 3), the objective should increase to indicate that the remaining bone is more heavily stressed (Eq. 4). The limit on the increase of the objective, Q,, in Eq. 4, corresponds to the physiologically normal apparent density for bone under the given stress. That is, the value of the objective for optimized bone is sQma. We assume that Q,, is independent of the apparent density and orientation of the material, but it is not necessarily a constant. It is possible that the value of Q,, is a function of age, general health, and other global parameters that affect local bone metabolism. It represents the effect that the general metabolism of the body has on the determination of optimal local bone density and trabecular orientation. It sets the level that the bone seeks. It will probably be different for each individual and change throughout life. Since the actual numerical value of this parameter is not needed to predict how apparent density

pA = min p such that

Q(p,8*,a> = min Q(p,8,0) Q m a (4)

.I Orthop Res, Vol . 4, N o . 3 , 1986

306 D. P . FYHRIE AND D. R. CARTER

and trabecular orientation vary with stress, we will not examine the effects of a change in the value of Qmm. In this paper the discussion is limited to the effects that changes in stress have on cancellous bone apparent density and orientation.

If we assume that for any given orientation, 8, and stress, c, that a bone of zero apparent density is infinitely poorly adapted to carry a load, i.e., limp0 Q(p,8,e) = a, we can combine Eqs. 3 and 4 to obtain

(~(PA$*,G) = min Q (min p,e , c ) = e m a x ( 5 ) In Eq. 5 the optimal orientation 8* is determined by the minimization of the objective function. The minimum acceptable apparent density pA is determined by the minimization of the apparent density, controlled by the equality of the objective with the physiological parameter QmaX. This more compact form of the general remodeling principle is used in the remainder of this paper'.

To become more concrete we will assume that the objective function Q(p,O,e) can be expanded as

Q(P,o) = Q(P,W) + Q(P,w%,:c 1 2 + - G:Q(p,e,O),~,:G + . . . (6)

where the comma subscript indicates partial differ- entiation. The operator : is the second order tensor inner product as described in Appendix A. We will eliminate this operator by converting the tensors to a matrix form. The constant first term in Eq. 6 is zero by Eq. 2. Since the constant term is zero, the linear second term is zero by Eq. 1. If the linear term is not zero, it will dominate the expansion for stresses that are small enough. This will cause the objective to be negative for some small uniaxial stress, violating Eq. 1. The approximate objective is

1 Q(p,O,c) = Q = ; c:M:o (7)

L

where

= &(P,8,0),,, (8)

M is a fourth order tensor that governs the bone remodeling characteristics. We will call it the re-

' If the objective monotonically decreases with p, the general remodeling principle can also be rewritten as Q(pa, 0*, a) = min Q(min p, 6, a) = max,{min, Q(p, 6, cr)} = Q,, where the min and max operators represent successive operations over the variables 8, and p (9,lO).

modeling tensor. Based on the historical observations of many investigators, we assume that the remodeling tensor is related to the mechanical properties of the bone tissue (e.g., the compliance). Quadratic functions such as Eq. 7 are common in the analysis of the deformation and failure of materials. For example, the strain energy of a linear material and a purely quadratic Tsai-Wu failure criterion (26,27), can be written in this form. Quadratic failure criteria have also been used to predict the strength of cortical bone (5,14).

The stress tensor c can be written as a column matrix

(9)

where uk = uij using the transformation in Table 1. Assuming the remodeling tensor has major and

minor symmetries*, it can be written as a remodeling matrix using the same transformation. If the orientation 8 only affects the material as a tensor rotation, we can separate the variables p, and 0 and write the material matrix as a product of matrices. Using a rotation matrix (17), R(8), written as a rotation from principal stress coordinates to material coordinates and an as yet unknown matrix, F(p), we have,

[ (T2 1 2D [ u2 u3 0 4 u 5 u(j 13D

c = = {

M = R(e)Tzqp)R(e) (10)

where, in 2D where, OT = [yl

R(e) = R ( ~ ) =

1 cos2 y sin2 y - 2sin y cos y sin2 y cos2 y 2sin y cos y

sin y cos y -sin y cos y cos2 y - sin2 y (1 1)

Using Eq. 10 in Eq. 7, we can write the quadratic

[ with a similar form for 3D where, OT = [a f3 71.

remodeling principle as

1 Q(pA,8*,c) = - 2 min{c@(min p, 8)cp} = Q,,

are the principal stresses.

A major symmetry is Fbld = Fuji' and a minor symmetry is Fij, = Fjw. If the tensor has all the major and minor symmetries, FijH = Fja = FMij = = . . . , then the fourth order tensor in the approximate objective (Eq. 7) can be greatly sim- plified.

J Orthop Res, Vol. 4, N o . 3, 1986


TABLE 1 . Transformation from tensor to matrix

k ‘J

1 1 22 33

23132 31/13 1212 1

With an appropriate choice of F in Eq. 10, Eq. 12 will predict cancellous bone morphology.

In the next section we will present the general restrictions on the matrix F so that the trajectory criterion is satisfied.

Optimal Orientation

For the theory to be correct it must predict the alignment of the material axes (and hence the trabeculae) with the principal stress directions. We will call this requirement the trajectory criterion. To ensure that the optimal orientation of the material meets the trajectory criterion, certain restrictions on the entries of the matrix F(p) must be satisfied. We will present these restrictions for the 2D and 3D orthotropic cases. The details of the derivations are presented in Appendixes B and C.

General Results for the 2 0 Orthotropic Case

For a plane stress problem we can use the rotation matrix R(y), Eq. 1 1 , and assume that F is an orthotropic matrix

F,l(P) FlLP) 0 F(P) = F12(P) F2AP) 0

0 0 F ~ P )

These form the quadratic remodeling principle

Q(PA ,Y* ~ Q J

1 2 = - min{o,TRT(y)F(min p)R(y)a,} = Q,, (15 )

where Q,’ = [uPl uP2] is the principal stress matrix in 2D.

The condition y* = 0, implies that the principal material directions, and hence the trabeculae, have aligned with the principal stresses. We want the conditions on the entries of F that ensure that this alignment occurs.

Saving the details for Appendix B, a sufficient re-

quirement for the satisfaction of the trajectory criterion for a given stress, Q, is

F I I - F22)(u;z - u;,> -t (-F11 - F22 + 2F1, + F66)

(up2 - UP$ 2= 0 (16) This is the alignment criterion for the plane stress case.

As a demonstration of the alignment criterion, we will use the 2D compliance of cancellous bone as the matrix F. The remodeling objective function is then the strain energy density of the bone, which we can plot as a function of material orientation angle y. We will consider two sets of cancellous bone moduli, both resulting from the finite element calculations of Beaupre and Hayes ( 1 ) . In their work a 3D finite element model of a porous structure was used to calculate the moduli of cancellous bone. It was assumed that the material was cubic, with three independent material constants: the Young’s modulus E, the shear modulus G , and the Poisson’s ratio v. In 2D the compliance matrix for a cubic material is

S = [ 1lE 0 ] (17) symm. 11G

Beaupre and Hayes found that their model gave E = 1.83 GPa, u = 0.08, under all circumstances. Calculation of the shear modulus using two different shear deformation patterns gave two values, G = 3.37 GPa and G = 0.45 GPa. They concluded that both of these shear moduli were too large since their finite element model was displacement based and inherently too stiff.

The first term of the alignment criterion, Eq. 16, is zero for a cubic material since, F , , - F22 = 11E - 1lE = 0. Substituting the compliance terms into the second term we have

1lE -VIE 0

[-2(1 + u)lE + l l G ] ( ~ ~ 2 - 3 0 (18)

Since the stress term is always positive, this equation only depends on the material parameters.

Using their calculated Young’s modulus, Poisson’s ratio, and the larger shear modulus, the alignment criterion for this case, Eq. 22, is not satisfied since (-211.83)(1 + 0.08) + 1/3.37 = -0.9 3k 0. A plot of the normalized objective of this system as a function of orientation with up2 = lOcr,, (Fig. 2) shows that the normalized objective IS

maximum rather than minimum at y = 0, d 2 . Using the smaller shear modulus, the alignment

criterion is satisfied since (-2/1.83)(1 + 0.08) + 110.45 = 1.04 5 0. Plotting the objective for this

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308 D. P . FYHRIE AND D. R . CARTER

OT = [a P rl (21)

and E(0) is derived in Appendix C. With these defi- nitions the quadratic remodeling principle is

Q(PA lo* 9op) 1 2

Y where

G = 0.45 GPO

= - min{o,Ti~(~)~(min p)R(0)op) = Q,,, (22)

7r 0 0,’ = [up, u p 2 Up3I (23) FIG. 2. Polar plot of normalized objective for two shear moduli.

are the principal stresses. Saving the details of the derivation for Appendix

case as a function of orientation (Fig. 2) we see that it is indeed minimum at y = 0, d 2 .

Since the smaller shear modulus is presumably more accurate, the compliance matrix calculated by Beaupre and Hayes satisfies the trajectory criterion. This is interesting since it suggests that the strain energy density may be used as the objective function for cancellous bone. We will develop this idea further in a later section.

Noting that the shear modulus for an isotropic material is (17) Giso = E/2(1 + u), the cubic compliance form of the alignment criterion can be written as Giso 2 Gcubic, where Gcubic is the shear modulus of the cubic material. That the shear modulus of the optimal cubic material is not larger than that of the equivalent isotropic material is reasonable since in the principal stress axes and in the material axes of an aligned material, there is no shear stress to sup- port. Under a static load we could imagine a material with a principal material axis shear modulus equal to zero supporting an applied stress with thin rods of material along the principal stress axes. In vivo, cancellous bone will never reach this extreme form since the actual stresses due to daily activity have varying principal stress directions.

General Results for the 30 Orthotropic Case

For the 3D orthotropic case the remodeling matrix is

where M(P,0) = m e > F ( P > m (19)

F12 F22 F23

F13 F23 F33 0 0 0 F 4 , O 0 0 0 0 O F 5 , O 0 0 0 0 O F 6 6

C, sufficient conditions for the satisfaction of the trajectory criterion for a given stress, op, are in Table 2.

These conditions are sufficient to ensure that 0 = 0 is a local minimum of the objective for rotations about all three axes when the material matrix is orthotropic. The case where the material matrix is generally anisotropic is briefly discussed in Ap- pendix D.

To understand these criteria in a physical way we will present two special remodeling principles. The first is based on optimization of the linear strain energy density and the second is based on optimization of a quadratic failure criterion. Before we do this, however, we will briefly discuss the general results regarding determination of the minimum acceptable apparent density pA.

Minimum Apparent Density

Assuming the trajectory criterion is satisfied we now have the problem

Q(pA,oP) = Q(min ~ , o p ) (24) 1 2

= - op(min p)op = Q,, (25)

If Q is a monotonically decreasing function of p, we have a unique pA such that,

G ~ ( P A ) C ~ = 2Qma (26) Depending on how pA enters into the left hand

side of Eq. 26, it may or may not be possible to solve explicitly for pA.

Special Results

Depending on the choice of the matrix F, the objective can be interpreted in different ways. These interpretations of the general theory are based on

J Orthop Res, Vol. 4, No. 3 , 1986


TABLE 2. Three-dimensional orthotropic alignment criteria

Angle Alignment criterion

different assumptions as to the goal of cancellous bone remodeling. Specifically, we want to choose some mechanical characteristic of the bone that will best approximate the bone remodeling tensor. We will first discuss the case Q = strain energy density and then the case Q = (effective stress/ strength)2. The strain energy density principle assumes that the bone has the goal of optimizing its stiffness using the least material for a given stress. The second case is the failure stress principle, which assumes that the bone has the goal of optimizing its strength using the least material for a given load.

We will present both of these interpretations and show under what conditions trabecular alignment will occur. The relationship between stress and apparent density will be determined and shown to be similar for the two cases.

Strain Energy Density Principle

A physical matrix that seems to meet the trajectory criterion is the compliance matrix S of cancellous bone. Using the compliance as the material matrix we have

1 2 Q ( p , 0 , ~ ) = strain energy density = - aTS(p,O)cr

(27)

where the compliance matrix in material coordinates for an orthotropic matrix expressed in engineering constants is (17)

l/El - 4 E 2 -~31/E3 0 0 0 -uI2/EI l/Ez -~32/E3 0 0 0 -u13/E1 -uZ3/Ez 11E3 0 0 0

0 0 0 l/G23 0 0

0 0 0 0 0 l/G12 0 0 0 0 l/G31 0

bijl =

In the above equation, Ei , i = 1, 2, 3 are the Young’s moduli and Gij, vij, i, j = 1, 2, 3 are the shear moduli and Poisson’s ratios, respectively.

Remembering the requirements for trabecular alignment (Table 2), we substitute the appropriate

terms from Eq. 28 to obtain the restrictions on the engineering moduli that are listed in Table 3. There is an alignment criterion associated with each angle. All three criteria must be satisfied for the compliance to meet the trajectory criterion.

The three alignment criteria can be satisfied in a myriad of ways. For example, the first and second terms of each of the criteria might be positive and the last term negative but sufficiently small so that the sum of the three terms is positive. In the next four paragraphs we will examine each of the three terms of the trajectory criteria independently. What each term means will be discussed and an estimate of whether it is positive or negative will be made.

The first term of each of the trajectory criteria restricts the relative size of the Young’s moduli. For these terms to be positive, the largest Young’s modulus (stiffness) must be in the maximum (in ab- solute value) principal stress direction, the second largest stiffness in the second principal stress direction, and similarly with the smallest stiffness and principal stress. If the material is aligned in this way, the direct contribution of the normal stresses to the strain energy is minimized. Several studies (3,18,25) have shown that cancellous bone is stiffest in the expected highest stress direction. The conclusion is that the first terms of the three trajectory criteria are almost certainly positive.

The second term of each of the three alignment criteria restricts the relative sizes of all the terms in the compliance matrix. They are a measure of the relative efficiencies of bearing the applied load solely with normal tractions in the principal material directions or permitting some shearing of the material. If we consider the second terms independently, we have

K, = - ( I + ~23)/E2 - (1 + 1”32)/E3

+ 1/G23 2 0 (29)

+ l/G3, 2 0 (30)

+ 1/G12 3 0 (31)

The physical interpretation of Eqs. 29-31 is that, if each orthotropic shear modulus is small, it is most

K P - - - ( I + ~13)lEi - (I + ~31)/E3

K, = -(1 + v J E ~ - (1 + ~21)/E2

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310 D. P. FYHRIE AND D . R. CARTER

TABLE 3 . Three-dimensional orthotropic alignment criteria for compliance

efficient (in terms of strain energy density) to bear the load solely with normal tractions in the principal material directions.

The data for evaluation of the constants K,, K,, and K, are quite sparse. To estimate the shear modulus we used a simple strength of materials torsion model to calculate the shear modulus of bovine distal femoral cancellous bone from the torque versus rotation data of Bensusan et al. (2). For the apparent density range of their samples, 0.5g/cm3 G p =s 0.77g/cm3, the strength of materials torsion model gives, 74 MPa S G S 411 MPa. To estimate the Young’s modulus we use the relation for bovine and human cancellous bone of Carter and Hayes (4), E = 3,790 p3MPa, for p in g/cm3. For the same range of apparent density this gives, 474 MPa s E S 1,730 MPa. Assuming all the Young’s moduli equal, all the shear moduli equal, and all the Poisson’s ratios equal (i.e., a cubic material) Eqs. 29-31 all reduce to the same relation, -2(1 + u)/E + l/G 2 0. Using the maximum values of the shear and Young’s modulus this gives as a constraint on the Poisson’s ratio, u s 1 . 1 . Using the minimum values of the shear and Young’s modulus gives v s 2.2. Jasty et al. (16) report Poisson’s ratios between 0.08 and 0.45 for human cancellous bone from the distal femur and proximal tibia. These values are below the necessary limits, so the second terms of the trajectory criteria may be positive.

The third term of each of the trajectory criteria restricts the relative sizes of the off-diagonal coupling terms of the compliance matrix. The physical interpretation of these terms of the trajectory criteria is that Poisson expansion (or contraction) of the material should cause the least possible contribution to the total strain energy density of the material. Jasty et al. (16) have determined the off-diagonal stiffness terms for cancellous bone from the distal femur and proximal tibia. However, whether the third terms of the trajectory criteria are positive for cancellous bone cannot be determined without knowledge of the in vivo stresses.

Although these points are far from conclusive they suggest that the three trajectory criteria are satisfied by the compliance of cancellous bone. In addition, it seems plausible that all three terms of each of the three criteria are independently positive. In the absence of in vivo stress data we suggest that Eqs. 29-31 are sufficient to indicate prob- able fulfillment of the trajectory criteria.

As to the determination of the apparent density, there are several studies (2,4,22) that show that the Young’s modulus is related to the apparent density as, Ei = AipB, i = 1,2,3 with B between 2 and 3. If the trajectory criterion is satisfied and the Poisson’s ratio is constant for some range of apparent density p, Eq. 26 can be written as

where

(34)

Notice that no shear terms appear in Eq. 33. This is a consequence of the trajectory criterion being satisfied by the compliance matrix.

Equation 32 can easily be solved to give a func- tional relationship between pA and np

(1lB)

(35

Since the numerical value of Q,, is not known

1 PA = -~jYnnp/Qmax i: we rewrite Eq. 35 as

which means that the minimum acceptable apparent density is proportional to the strain energy density to the power 1/B.

J Orthoa Res. Vol. 4 , No. 3, 1986


Another interpretation is to define an effective

(37) where A,, = (A, + A , + A3)/3. Using this equation we have

stress, uenergy

2 uenergy = Aavgcpncp

This interpretation of the equations shows that the apparent density is proportional to the effective stress uenergy to the power 2/B. Since 2 s B s 3, the exponent is between two-thirds and one.

Next we will develop an objective function that assumes that orientation and apparent density are determined by failure stress rather than by strain energy.

Failure Stress Principle

A different approach is to use an orthotropic matrix of failure influence coefficients to create an objective function. For example, from Stone et al. (22) and Carter and Hayes (4), we have for the shear failure stress and compressive failure stress of human cancellous bone, S, = 21.6 p1.65 MPa and S, = 68.0 p2 MPa, respectively. The apparent density is measured in g/cm3. If we assume a cubic influence matrix, we can use these relations to define an effective stress failure criterion

9.9 0 0 9.9

O O 9.9 O 1 (41)

To obtain A4 we have assumed that both the shear strength and the compressive strength are proportional to pB and that compressive and tensile strengths are equal. The matrix is normalized with respect to the normal compressive failure stress [note that (S,/S,)2 = 9.91.

Equation 39 forms an objective function by defining

Q = (Ufd/SJ2 (42)

The matrix forming (ufs,>2, Eq. 41, is cubic so the first and third terms of the trajectory criteria in Table 2 are identically zero. The second terms of the criteria all reduce to the same equation, -2(1 - F) + 9.9 3 0 or F 3 -3.95. There are no data on the normal coupling term F for failure of cancellous bone. However, since cancellous bone is a porous, highly compressible material we do not ex- pect a large coupling term. The failure based objective function probably meets the trajectory criterion.

To determine the apparent density we can use Eqs. 39, 40, and 42 to obtain

PA = [ ~ ~ n ~ p / ( 6 8 . 0 ) 2 ] ( ” 2 B ) (43)

= (ajai4(68 .0)2)(’/28) (44)

(Wfai,)(l/B)

where

and 1.65 G B G 2. Equations similar; both approaches result

(45)

38 and 45 are very n a power law rela-

tion between apparent density and the effective stress. For the failure stress principle, 1.65 i B s 2, so the exponent in Eq. 45 is roughly between one-half and three-fifths.

DISCUSSION

Determining which of the two special principles more accurately reflects reality is not possible at this time due to lack of experimental results. There are, however, some arguments in favor of each.

Quantitative examinations of the trajectory and effective stress hypotheses ( 1 3,23) where results of finite element analyses of the human patella are used to predict trabecular orientation and areal density have had conflicting results. The work of Hayes and Snyder was 2D, supported the trajectory hypothesis, and found that von Mises’ stress correlated strongly with density. The work of Stone et al., however, was 3D, supported the trajectory hypothesis, and found that the von Mises’ stress was only weakly correlated to the density. This contradiction is probably a result of the qualitative differences between 2D and 3D von Mises’ failure criteria. The mathematical form of the von Mises’ stress is

J Orthop Res. Val. 4, NO. 3 , 1986

312 D . P . FYHRIE A N D D . R . CARTER

ut, = t$Mv,cp (47)

where

1 - 1/2 - 1/2 M,, = - 1/2 1 - 1/21 (48)

and the surface at, = constant is a cylinder along the hydrostatic pressure axis, as shown in Fig. 3 . In 2D (plane stress or strain) the von Mises’ failure criterion is closed, forming an ellipse in principal stress space. In 3D the criterion is open, forming a cylinder that has its axis along the line of hydrostatic pressure (dilatational stress). The open failure surface reflects the assumption of the von Mises yield criterion that hydrostatic pressure cannot cause yielding-that only shear stress causes yielding. Since the von Mises yield criterion is open, it will not predict bone apparent density in 3D. For example, a 3D hydrostatic stress state (i.e., c,’ = [u u 01) has a von Mises’ effective stress of zero, and hence the predicted bone apparent density is zero. Since the von Mises’ stress will be zero no matter how large the principal stress u is, we are led to the unlikely conclusion that there will be no remodeling response even when the bone is loaded far beyond the static failure stress.

It is important that any effective stress that is chosen to predict cancellous bone apparent density be closed. If it is not closed, the failure stress principle will predict a zero bone density for a nonzero stress just as in the case of the von Mises’ stress. This problem can not occur for a closed effective stress. A closed effective stress is never zero unless the actual stress state is zero. Consequently, a nonzero bone density will be predicted whenever the in vivo stress is nonzero. This behavior agrees with

[ - 1/2 - 1/2 1

f %*

FIG. 3. Von Mises’ yield surface in three-dimensional principal stress space.

the observations of many observers and we believe it to be a necessary ingredient of a bone remodeling theory.

A necessary and sufficient condition for closure of an effective stress defined by a quadratic form in stress, a t = cTMa, is that the matrix M be positive definite. This condition is always met by the energy stress since the compliance matrix of a real material must be positive definite for conservation of energy. Since the compliance matrix is positive definite, the surface u:nergy = constant is a closed football shape as in Fig. 4. The failure stress afai, will be closed when - 1/2 < F < 1 in Eq. 41.

Both of the special principles, the strain energy and failure stress principles, are (or can be restricted to be) closed. Since the closure requirement for prediction of cancellous bone apparent density is met, other considerations will determine which principle is more correct. It should be real- ized, however, that each principle predicts the trajectory hypothesis and that the resulting distribu- tions of apparent density within the bone will be similar. Both theories give the relationship between the minimum apparent density pA, and an effective stress, ueS, as

PA cE u& (49) This is important since for a uniaxial stress state, e.g., 0,’ = [u 0 01, we have

meenergy “Id (50) ufail mlul (51)

Consequently, for a uniaxial stress state both principles predict pA ~ 1 ~ 1 ~ . The strain energy principle predicts that 2/3 6 C s 1 , implying a greater dependence of apparent density on stress than does the failure stress principle, which predicts 1/2 6 C

FIG. 4. Energy stress constant surface in three-dimensional principal stress space.

J Orthop Res, Voi. 4 , No. 3 , 1986


s 315. This difference in dependence may be used to determine which of the principles is more nearly correct. Since the two principles are mathemati- cally indistinquishable the final word must be left to future experimental work.

CONCLUSION

We have developed a theory relating stress to trabecular orientation and apparent density based on the idea that bone is a self-optimizing material. It is important that the objective function that governs the optimization describes a closed surface in 3D principal stress space. The bone remodeling objective can be approximated using specific mechanical characteristics. Two special applications, one based on strain energy and the other on failure stress, are particularly attractive. The first, the strain energy density principle, optimizes the stiffness and the second, the failure stress principle, optimizes the strength, under a given load. Both predict the orientation of trabeculae consistent with the trajectory hypothesis of Roux, von Meyer, Culmann, and Wolff and define stress measures that can be used to predict apparent density. This is the first time that the trajectory hypothesis and the prediction of apparent density have been unified into a single mathematical theory. Experimental verification of this static theory and its extension to time depen- dent loading may lead to reliable and accurate methods of predicting the response of cancellous bone to changes in stress.

APPENDIX A

Nomenclature

p, the apparent density (bone mass per unit

pA, the minimum acceptable apparent density. 0, the rotations from principal stress to material

0*, the optimal orientation. cr, the applied stress. cr,, the principal stress matrix. M(p,0), the remodeling matrix ( M = R T R ) . R(0),R(0), tensor rotation from principal stress

F(p), a matrix that approximates the dependence

volume).

coordinates.

axes to material axes.

of the remodeling matrix on apparent density.

Q, the exact objective function that measures the

Q, the quadratic approximation to the objective

Q,,, the physiologically normal value of the ob-

a:b, inner product for a second order tensor. a:c:b, if a and b are Cartesian second order

tensors, and c is a fourth order Cartesian tensor, a:b = a& and a:c:b = aGcijklbM, where, the summation convention is assumed.

optimality of the bone.

function.

jective function.

APPENDIX B

Optimal Orientation for the 2D Orthotropic Case

It will be shown that, if F is an orthotropic matrix with its minimum axial compliance values in the principal material directions, the best orientation of the material is 0* = 0. That is, its lines of anisotropy align with the principal stresses-just as the trabeculae line up with the principal stress directions in cancellous bone. We will say that any matrix that behaves in this way meets the trajectory criterion.

Two-Dimensional Orthotropic Case

For purposes of illustration we will assume a 2D stress field and an orthotropic matrix F. In material coordinates this matrix is

Using this along with the 2D rotation matrix, Eq. 1 1 , and substituting into Eq. 7 the objective function becomes

1

2 Q(P,Y,C,) = - o,TRT(r)F(p)R(r)crp (53)

where cr; = [a,, a,,] are the principal stresses.

rearrangement Using Eqs. 1 1 and 52 in Eq. 53, we have after

Q(P,Y,~~) = Q~(P,c,) + AQ(P,Y,~J (54) where

J Orthop Res , Vol. 4 , No. 3 , 1986

314 D. P . FYHRIE AND D . R. CARTER

sin2 y - D + A C O S ~ y - A C O S ~ y - A cos2 y D + A cos2 ,] AQ(p,y,oP) = 7 tup1up21 [

and satisfy Eq. 67 are those for which both F,, aniF2, are local minima when y = 0. Defining F = RT(y)FR(y), we see from Eqs. 55 and 56 that the normal stress diagonal terms of F are

Fii(3') = Fi, - (Fii - F22)sin2Y + A sin2? cos2Y (68)

= F,, cos2 y + F22 sin2 y + A sin2 y cos2 y (69)

F22(y) = F22 + (F,, - F2,)sin2 y + A sin2 y cos2y (70)

A = -F11 - F 2 2 + 2F12 f F66 (57) (58) (59)

D = F11 - F 2 2 upi: i = 1,2 (principal stresses)

- Equation 54 will be extreme when Q,? = 0.

Taking the derivative and rearranging we have

- Q,y = sin y cos y[u u ] [ B1l '12] (60) p 1 p 2 symm. B,, up2

where

B2, = 2F,, sin2 y + (2F12 + FM>(cos2 y - sin2 y) - 2F22 C O S ~ y (63)

Due to the common term sin y cos y in Eq. 60 there will always be extrema at y = 0, d 2 and perhaps at other angles.

The trajectory hypothesis is the observation that the trabeculae align with the principal stresses. If the minimization principle is correct, Q(p,y,a,) will be minimal when y = 0. Examining Eq. 54 we see that this will be the case when AQ(p,y,op) 3 0 for all y. Contracting and rearranging Eq. 56 gives

= F22 cos2 y + F,, sin2 y + A sin2 y cos2 y (71)

Examination of Eqs. 69 and 71 shows that F,,(O) and F22(0) will be minima if F,, 3 0, F2, 3 0, and A > 0. That is, the diagonal values are positive and Eq. 67 is satisfied by A being positive. Therefore, if the normal stress diagonal values of the matrix are both minima when y = 0, the trajectory criterion can be satisfied.

APPENDIX C

Optimal Orientation for the 3D Orthotropic Case

It will be shown using a second order approximation to the rotation matrix that, if F is an orthotropic matrix meeting certain conditions, the best

which is satisfied for all y when

(F1i - F 2 2 ) ( ~ ; 2 - u;,) + A(upp2 - 3 0 (65)

This is called the alignment criterion for the matrix F.

The alignment criterion is satisfied when both terms of Eq. 65 are greater than zero:

(66) (67)

(F11 - F22)(~;2 - u;,) a 0 A = -F11 - F 2 2 + 2F12 + F M 3 0

This is an interesting case since these requirements have interesting physical interpretations. Equation 66 requires that the smaller diagonal material constant correspond to the larger principal stress and vice versa. Equation 67 restricts the possible values that the matrix entries may have.

A group of 2D orthotropic material matrices that

orientation of the material is 8" = 0. That is, its principal material directions align with the principal stresses-just as the trabeculae line up with the principal stress directions in cancellous bone. We will say that any matrix that behaves in this way meets the trajectory criterion.

Three-Dimensional Orthotropic Case

In summation notation the objective function (Eq. 7) can be written as

1 (72) Q = - u..F.. (J- 2 IJ 1Jkl kl

1 - - - 2 umnumiu~jFijklukpulqu~q (73)

J Orthop Res, Vol. 4, No. 3 , 1986


where

a = [aij] a11 = cos(P) cos(y) u12 = sin(a) sin@) cosfy) - cos(cr) sin(-y) uI3 = -cos(a) sin@) cos(y) - sin(a) sin(y) u21 = cos(P) sin(y) u22 = sin(a) sin@) sin(?) + cos(a) cos(y) a 2 3 = -cos(a) sin@) sin(?) + sin(a) cos(y) a 3 1 = sin@) a 3 2 = -sin(a) cos(p) a 3 3 = cos(a) COS(P)

is the rotation matrix for sequential rotations (a , P, y) about the (1, 2 , 3 ) axes. We define OT = [a P y ] such that Eq. 74 represents the rotation from principal stress coordinates to material coordinates.

To determine under what conditions the minimum of the objective occurs when 8 = 0 we will assume that it does and derive the conditions on uij and Fijkl that ensure the assumption to be true.

By the definition of 8 we have

[aij] = [(JmnUmiUnj] = [a-]T IJ (5' (T:, ] [aij] (84) 0 0 u p 3

Using Table 1 we define

bT = [(TI (T2 (T3 (T4 (T5 (Ts] (85 )

6,' = [upl up2 up3] (principal stresses) (86) Using Eqs. 84 and 85 we have

where

4 2 4 2

R(8) = 4 3 4 3 4 3

a12a11 u22421 u32'31

L ' 1 3 u l l 423u21 u33'31]

u13412 u23'22 u33'32

and the aij are from Eq. 74. Using the orthotropic symmetry property of F and the renumbering scheme of Table 1 we can write the objective function as

1 - -

2 Q = - C ; R ~ F R C , (89)

where plI F12 F13 0 0 0 1 F12 F22 F23

F13 F23 F33 0 0 O F - 0 0 F = 1

OI 0 0 0 0 F 5 5 L 0 0 0 0 O F ,

To expand Eq. 89 to the basic level of sines and cosines is tedious and not particularly useful due to its overwhelming complexity. More informative is the second order expansion of Eq. 89. If we assume rotations, 8, which are small enough, then to the second order in 8

sina = a (91)

(92)

with equivalent approximations for the other rotation angles as well. Let, F = R(0)TFR(8) with R(8) and F as in Eqs. 88 and 90. Using the second order approximations to the sine and cosine terms in Eq. 88 we have to the second order in 0

I 2

cosol = 1 - - a2

F = F + A F (93)

where F is as in Eq. 90 and

where the undefined terms are symmetric and

Defining

J Orthop Res, Vol. 4, No. 3, 1986

316 D . P . FYHRIE AND D. R . CARTER

we have for small rotations FG = F + FA (1 12)

Q(p,8,0p) = €?(p,O,o,) + AQ(p,8,0p) (105)

If Q(p,O,ap) is the minimum value, AQ(p,8,ap) must be greater than zero for all 8 # 0. Now, AF is not

where F is identical to the 3D orthotropic matrix, Eq. 90, and

positive definite. This is shown by the submatrix test (24). Since the matrix is not positive definite, the values of up enter into the conditions for AQ to be positive.

For a particular stress state, op, AQ will be greater than or equal to zero if the following three inequalities hold

FA =

- F14 F15 F16

F24 F25 F26

0 F34 F35 F36 F45 F46

symm. F56 - 0 -

where Ha, H,, H y are as in Eqs. 101-103. These inequalities come directly from the contraction of Eq. 104 in a manner identical to that used in the 2D orthotropic case, Appendix B , Eq. 64. These equations are the alignment criteria for the matrix F.

Equations 106-108 can be satisfied in a myriad of ways. For example, in Eq. 106 the first and second terms might be positive and the last negative but sufficiently small so that the sum of the three terms is positive. We will only examine the case where each of the nine terms is independently positive. This will result in a set of sufficient requirements on F that have elegant physical interpretations.

The equations are simple permutations of one another, so we will examine Eq. 106 only. Re- quiring all of its terms to be greater than or equal to zero gives

(F33 - F22)(u;2 - u;3) 2 0

(F13 - F12)(up2 - up3)up,, 2 0

( 109) Ha = - FZ2 - F33 + 2FZ3 + F44 3 0 (1 10)

(1 11)

The first of the three, Eq. 109, requires that the smaller diagonal material value be associated with the larger principal stress. The second, Eq. 110, restricts the relative size of the matrix entries. The last, Eq. 11 1, requires that the normal stress coupling terms aid in reducing the objective.

APPENDIX D

Three-Dimensional Generally Anisotropic Case

A generally anisotropic material matrix is full. Unlike the orthotropic matrix, Eq. 90, there is no orientation that leads to a large number of zero ele- ments. Define

are the terms that make F , generally anisotropic rather than orthotropic.

Since there are no principal material directions for an anisotropic material, the concept of alignment is not as clear for this case. It is, however, still reasonable to seek the material orientation that is optimal for a given stress. If we choose arbitrary material axes and assume that the optimal orientation of the material is alignment of these material axes with the principal stress directions, then pro- ceeding in a manner identical to that used for the 3D orthotropic case we obtain conditions on the entries of the anisotropic matrix that ensure that this is the optimal orientation. That the material axes are not principal material axes is no bar to the existence of an optimum when the sets of axes are aligned.

Using Eq. 113, the objective function is

= - 1 - o,TR(0)TFiT(O)~p + - 1 - o,’R(~)~F,E(O)O~ 2 2

It is clear that by using the same second order rotation approach as for the 3D orthotropic case that sufficient conditions on the matrix entries can be determined which ensure that the optimal orientation is with the material axes aligned with the principal stress directions. In fact, the first term of Eq. 115 will lead to conditions identical to those of the orthotropic case. The second term of Eq. 115 will lead to additional conditions on the entries of

J Orthop Res, Vol. 4 , N O . 3 , 1986


FA which ensure that the aligned orientation is optimal. Since it is not likely that the necessary material constants will ever be determined, we have not expricitly derived these conditions. This appendix was included to show that the theory we have developed is not restricted to orthotropic material symmetry.

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A unifying principle relating stress to trabecular bone morphology