12

Relationship between the hip-joint loading and the capital epiphyseal growth plate orientation

Y.H. Sniekers

BMTE 02.17

Supervisors: Wouter Wilson Rik Huiskes

Eindhoven, May 2002

/mhj

Relationship between the hip-joint loading and the capital epiphyseal growth plate orientation

Y.H. Sniekers

Abstract During growth, the orientation of the growth plate changes. When this reorientation does not occur correctly, skeletal abnormalities can arise. It is important to understand what causes the reorientation of the growth plate. According to the literature, growth may be influenced by hydrostatic stress and octahedral shear stress. Flow related variables like pore pressure, distortional strain, and fluid velocity may play an important role in tissue differentiation. The aim of this research is to verify whether these variables can explain the orientation of the capital femoral growth plate. For this purpose a three-dimensional finite element model, representing the proximal femur of a 16 year-old child, was used. It was found that the hydrostatic stress, the pore pressure and the fluid velocity are unlikely to stimulate the reorientation of the growth plate. However, the octahedral shear (or distortional) stress and strain may govern the reorientation of the growth plate. This information can be useful for the prevention and treatment of skeletal abnormalities. Introduction The orientation of the capital femoral growth plate changes during growth. [1] Sometimes this reorientation does not occur correctly, which can cause an abnormal orientation of the growth plate and eventually various skeletal abnormalities [2,3]. Some patients undergo an osteotomy to correct this. There are various types of osteotomy for the femoral head. Essentially, all methods are based on the same principal, namely, correction of the deformity to restore anatomy. This means that the femoral head is broken and fixed under a better angle. However, the results of such treatment are not always satisfactory. As time passes, the femoral head turns back to its old orientation.

It is important to understand what causes the reorientation of the growth plate. There are several hypotheses in mechanobiology that might be used to explain the reorientation of the growth plate. Pauwels [4] suggested that the combination of hydrostatic stress and shear stress determines the differentiation pathway of the mesenchymal origin to fibrous, fibro-cartilaginous or cartilaginous tissue. According to Carter et al [5], the maximum principal stresses in the growth plate are perpendicular to the growth plate. This means that a reorientation of the resultant hip force, and thereby a reorientation of the principal stresses in the femoral head, should lead to a reorientation of the growth plate. Carter et al [6] proposed a theory that endochondral growth and ossification are inhibited in areas of intermittent hydrostatic compression and accelerated in areas of intermittent high shear stress. They specified this in their ‘osteogenic index’ [7].

Fluid flow is known to stimulate anabolic cell expressions during in vitro testing [8]. It was hypothesized by Prendergast et al [9] and Huiskes et al [10] that distortional strain and interstitial fluid flow are the mechanical stimuli governing tissue differentiation. This was specified in the mechano-regulation index [10].

The aim of this study is to investigate whether the reorientation of the growth plate could be governed by hydrostatic stress, pore pressure, distortional stress and strain, or fluid velocity in the growth plate, or by a certain combination of these. A second aim is to investigate whether these variables can explain the failure of osteotomy. Materials and Methods

For this purpose a finite element model of the human femoral head and neck of a 16-year-old child was made (figure 1). The geometry of the femoral head was simplified to

1

represent a sphere with a radius of 19 mm covered with a 1 mm layer of articular cartilage. The epiphyseal growth plate was modelled as a part of a sphere with a radius of 38.5 mm and a thickness of 2 mm, based on CT-scans. For the inclination angle of the growth plate in the frontal plane, the angle between the growth plate and the horizontal line, a value of 20° was used [1]. Based on Oguz [2], the angle between the femoral neck and the horizontal line was set to 40°.

Articular cartilage

Epiphyseal growth plate

x

Figure 1: Finite element mesh, front view (left) and side view (right).

To verify whether the stress invariants used in Carter’s osteogenic index [7] can explain the reorientation of the growth plate, the same specifications as Carter et al [7] had to be used. Therefore, a linear elastic model was used. Flow related mechanisms, as Prendergast et al [9] and Huiskes et al [10] suggested, can not be considered in a linear elastic model. Therefore a biphasic model was made. Thus, two different finite element models were made: One with a epiphyseal growth plate of linear elastic, isotropic material and one in which the epiphyseal growth plate was modelled as two-phasic (solid fluid) poroelastic, isotropic material. In both models the bone and articular cartilage were modelled as linear elastic and isotropic. Both finite element models consisted of 5128 8-node, isoparametric, hexahedral elements.

There is a wide variation in bone density in the femoral head and neck. The density of bone was determined using bone remodelling simulations, based on theories from Huiskes et al [11], Weinans et al [12], and Harrigan and Hamilton [13] (figure 2). The Young’s moduli of bone were coupled to the density distribution.

Figure 2a: Density distribution from remodelling simulation

Figure 2b: CT-scan of a 14-year-old child

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According to Lai et al [14] the permeability of cartilage is strain-dependent. In terms of the void ratio this can be written as:

M

eekk

++

=0

0 11

(1)

where k is the permeability, k0 the intrinsic permeability, e is the void ratio, e0 the initial void ratio en M is the strain-dependency coefficient. The material properties that were used are shown in table 1.

It was assumed that the pressure at the edges of the growth plate is zero, which means that at these edges free fluid flow in and out of the growth plate can occur. The distal end of the femoral neck was fully constraint and the frontal plane was constrained in anterior-posterior direction.

A geometrically nonlinear analysis was applied to the linear elastic model. Due to convergence problems, a geometrically nonlinear analysis could not be used for the biphasic model. Therefore, geometrically linearity was assumed for the biphasic model. Finite element solutions were obtained with ABAQUS/CAE version 6.2. For post-processing ABAQUS and Matlab version 5.3.1.29215a (R11.1) (1999) were used. Table 1: Material properties Linear elastic Biphasic Growth plate

Young’s modulus E (MPa) 6 [6,15] 0.87 [16]

Poisson ratio ν 0.47 [6,15] 0.33 [16]

Initial void ratio e0 - 3 [16]

Intrinsic permeability k0 (10-15 m4/Ns) - 25 [16]

Strain-dependency coeff. M - 11 [16]

Articular cartilage Young’s modulus E (MPa) 6 [6,15] 6 [6,15]

Poisson ratio ν 0.47 [6,15] 0.47 [6,15]

Bone Young’s modulus E (MPa) 6-14500 6-14500 Poisson ratio ν 0.3 [6,15] 0.3 [6,15]

The orientation and magnitude of the resultant hip force during gait were derived from

Bergmann [17]. A body weight of 600 N was assumed. The total load cycle during gait was divided in ten different load cases. The resultant hip force was rotated so that it would be correct for a 16-year-old. This was done until, in the load case with the maximal resultant hip force (load case 4), the maximum principal stresses at the centre of the growth plate were perpendicular to the growth plate.

During each load case the force was distributed over the contact area of the femoral head with the acetabular cup using equations of Ipavec et al [18,19]. The lateral border of the contact area is given by the Centre-Edge angle (CE-angle) of Wiberg [20]. The CE angle is formed by a line from the centre of the femoral head to the outer edge of the acetabulum roof, and a vertical line drawn thru the centre of the femoral head (figure 3). An average CE angle of 28.14° was used [21]. During the gait cycle the CE-angle was adjusted to represent the rotation of the femoral head with regard to the acetabulum. This rotation was taken from Bergmann [17].

To determine the influence of the stresses, strains and fluid velocities on the reorientation of the growth plate, two situations were studied: one in which the orientation of the resultant hip force is correct for a 16-year-old and one in which the resultant hip force was rotated 2.5 degrees in medial direction.

To model a valgus or abduction osteotomy, in which the neck-shaft angle is opened [22], the total finite element model was rotated over 15 degrees in lateral direction, while the resultant hip force was not rotated.

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The growth of the growth plate takes place at the bottom of the growth plate [23]. Therefore, hydrostatic stress, pore pressure, distortional stress and strain, and fluid velocity were calculated at the bottom of the growth plate.

Figure 3: Graphical representation of CE-angle and inclination angle (θR). R is resultant hip force.

Several theories of the mechanobiology might be used to explain the growth in the

growth plate. The osteogenic index (OI) of Carter [7] is given as

)(1

ii

c

ii kDSnOI += ∑

=

, (2)

where ni is the number of the loading cycles for the ith load case, S is the octahedral shear stress, D is the hydrostatic stress. The value of k, the empirical constant, is determined by parametric variation. High values of the index were hypothesized to stimulate endochondral growth and ossification. The octahedral shear stress (S) is defined as

( ) ( ) ( )3

213

232

221 σσσσσσ −+−+−

=S (3)

and hydrostatic stress (D) is defined as

3321 σσσ ++

=D , (4)

where σ1, σ2, and σ3 are the principal stresses. The mechano-regulation index of Huiskes et al [10], which was formulated for tissue

differentiation, is given as

bv

aM +=

γ, (5)

where γ (dimensionless) is the maximal distortional (or octahedral shear) strain, v (µm/s) the interstitial fluid velocity, a=0.0375 and b=3 µm/s. It was proposed that where M>3, fibrous tissue would prevail, where 1≤M≤3, cartilage would prevail and where M<1, bone would form. This theory has two emperical thresholds which determine when transition in the tissue type occurs, and two emperical constants a and b which are interdependent. Figure 4 shows the mechano-regulation diagram.

0 0.02 0.04 0.06 0.08 0.1 0.120

1

2

3

4

5

6

7

8

9

Strain [-]

Flu

id v

eloc

ity [m

icro

met

er/s

ec]

F IBROUS

FIBRO-CARTILAGE

BONE

Figure 4: Mechano-regulation diagram

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Results All results shown in this section were calculated at the bottom of the growth plate in the

frontal plane, for the load case with the maximal resultant hip force after 1 gait cycle. An exception was the osteogenic index which was summed over all load cases of the first gait cycle. ‘0 deg’ is used to indicate that the resultant hip force is oriented such that the maximum principal stresses are perpendicular to the growth plate at the centre of growth. ‘2.5 deg’ means that the resultant hip force is rotated 2.5 degrees in medial direction with regard to ‘0 deg’. ’15 deg’ is used for the finite element model that is rotated 15 degrees in lateral direction with respect to ‘0 deg’.

For the linear elastic model, the hydrostatic stress and octahedral shear stress were calculated. It was found that the orientation of the resultant hip force has little or no effect on the hydrostatic stress (figure 5). The distribution was symmetric over the growth plate. As shown in figure 6, the orientation of the resultant hip force had a large effect on the octahedral shear stress. For the case where the maximum principal stresses were perpendicular to the growth plate, the octahedral shear stress, as a function of the position on the growth plate, formed a oblique curve with some deviation at the edges. When the resultant hip force was rotated over 2.5 degrees, the octahedral shear stress increased at the lateral side and decreased at the medial side of the growth plate.

-20 -15 -10 -5 0 5 10 15 20-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

x [mm]

Hyd

rost

atic

stre

ss [M

Pa]

0 deg 2.5 deg

-20 -15 -10 -5 0 5 10 15 200.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

x [mm]

Oct

ahed

ral s

hear

stre

ss [M

Pa]

0 deg 2.5 deg

Figure 5: Distribution of hydrostatic stress for two orientations of resultant hip force.

Figure 6: Distribution of octahedral shear stress for two orientations of resultant hip force.

The osteogenic index of Carter [6] was calculated for the linear elastic model, for

different values of k (figure 7). When the maximum principal stresses were perpendicular to the centre of the growth plate, the osteogenic index was symmetric with regard to the centre of the growth plate. It can be seen that for small values for k, the osteogenic index, as function of the position on the growth plate, is almost a straight line. The greater the value for k, the more the line curves. When the resultant hip force was rotated over 2.5 degrees, the osteogenic index increased at the lateral side and decreased at the medial side with respect to the 0 degrees case.

-20 -15 -10 -5 0 5 10 15 20-10

-8

-6

-4

-2

0

2

x [mm]

Ost

eoge

nic

inde

x [M

Pa]

0 deg 2.5 deg

k=0.1

k=0.5

k=1.0

Figure 7: Osteogenic index for two orientations of resultant hip force and different values for k.

5

For the biphasic model, the pore pressure, distortional stress, distortional strain, and fluid

velocity were calculated. It was found that the orientation of the resultant hip force has little or no effect on the pore pressure (figure 8). The distribution of the pore pressure was symmetric with respect to the centre of the growth plate. As shown in figures 9 and 10, there was a large effect on the distortional stress and the distortional strain. In the case of the rotated resultant hip force over 2.5 degrees, both the distortional stress and strain increased at the lateral side of the growth plate and decreased at the medial side. The orientation of the resultant hip force seems to have little effect on the fluid velocity (figure 11). When the resultant hip force was rotated over 2.5 degrees in medial direction, the distribution changed especially at the edges.

-20 -15 -10 -5 0 5 10 15 200

0.5

1

1.5

2

2.5

x [mm]

Por

e pr

essu

re [M

Pa]

0 deg 2.5 deg

-20 -15 -10 -5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

x [mm]

Dis

torti

onal

stre

ss [M

Pa]

0 deg 2.5 deg

Figure 8: Distribution of pore pressure for two orientations of the resultant hip force.

Figure 9: Distribution of distortional stress for two orientations of the resultant hip force

-20 -15 -10 -5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

x [mm]

Dis

torti

onal

stra

in [-

]

0 deg 2.5 deg

-20 -15 -10 -5 0 5 10 15 200

1

2

3

4

5

6x 10

-3

x [mm]

Flu

id v

eloc

ity [m

m/s

]

0 deg 2.5 deg

Figure 10: Distribution of distortional strain for two orientations of resultant hip force.

Figure 11: Distribution of fluid velocity for two orientations of resultant hip force.

To investigate if the tissue differentiation theory of Huiskes et al [9] can also be used to

model the growth of the growth plate, their mechano-regulation index was calculated. When the maximum principal stresses at the centre of the growth plate were perpendicular to the growth plate, the distribution of the mechano-regulation index was symmetrical with respect to the centre of the growth plate (figure 12). The index was smaller than 1 in the middle of the growth plate. At the edges, the index was greater than 3. In the phase diagram for the 0 degrees case, the measure points were approximately on one line (figure 13), which corresponds to the symmetric distribution in figure 12. A few points were situated in the ‘bone region’. The points in the ‘fibrous region’ correspond with points at the edges of the growth plate. When the resultant hip force was rotated over 2.5 degrees, the mechano-regulation index increased at the lateral side and decreased at the medial side (figure 12). In the phase diagram, more points of the lateral side and less points of the medial side were in

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the ‘fibrous region’ (figure 14). The mechano-regulation index profile over the growth plate (figure 12) corresponds much to the distribution of the distortional strain (figure 10)

-20 -15 -10 -5 0 5 10 15 200

1

2

3

4

5

6

7

x [mm]

Mec

hano

-reg

ulat

ion

inde

x

0 deg 2.5 deg

BONE

FIBROCARTILAGE

FIBROUS

Figure 12: Mechano-regulation index for two orientations of resultant hip force.

0 0.02 0.04 0.06 0.08 0.1 0.120

1

2

3

4

5

6

7

8

9

Strain [-]

Flu

id v

eloc

ity [m

icro

met

er/s

ec]

0 deg

FIBROCARTILAGE

FIBROUS

BONE 0 0.05 0.1 0.15 0.2

0

1

2

3

4

5

6

7

8

9

Strain [-]

Flu

id v

eloc

ity [m

icro

met

er/s

ec]

2.5 deg

FIBROUS

FIBROCARTILAGE

BONE

Figure 13: Phase diagram of mechano-regulation index for ‘0 degrees’.

Figure 14: Phase diagram of mechano-regulation index for ‘2.5 degrees’.

To study the effect of osteotomy, the hydrostatic stress and octahedral shear stress were

calculated for the rotated linear elastic model, representing the femoral head and neck after osteotomy. The rotation of the model had little effect on the distribution of the hydrostatic stress over the growth plate (figure 15). For the rotated model, the hydrostatic stress was less symmetric with respect to the centre. As shown in figure 16, the octahedral shear stress increased much at the lateral side and increased a little at the medial side, with regard to the original, unrotated model.

-20 -15 -10 -5 0 5 10 15 20-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

x [mm]

Hyd

rost

atic

stre

ss [M

Pa]

0 deg 15 deg

-20 -15 -10 -5 0 5 10 15 200.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x [mm]

Oct

ahed

ral s

hear

stre

ss [M

Pa]

0 deg 15 deg

Figure 15: Distribution of hydrostatic stress for two orientations of the femoral neck and head.

Figure 16: Distribution of octahedral shear stress for two orientations of the femoral neck and head.

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The biphasic model was also rotated, and pore pressure, distortional stress and strain, and

fluid velocity were calculated. The rotation of the biphasic model had little effect on the distribution of the pore pressure (figure 17), but had a large effect on the distortional stress and strain (figure 18 and 19). The latter two increased much at the lateral side and a little at the medial side. The fluid velocity was also influenced by the rotation of the model (figure 20). At the lateral side the fluid velocity increased, at the medial it decreased, except at the edges.

-20 -15 -10 -5 0 5 10 15 20-2.5

-2

-1.5

-1

-0.5

0

0.5

x [mm]

Por

e pr

essu

re [M

Pa]

0 deg 15 deg

-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x [mm]

Dis

torti

onal

stre

ss [M

Pa]

0 deg 15 deg

Figure 17: Distribution of pore pressure for two orientations of the femoral neck and head

Figure 18: Distribution of distortional stress for two orientations of the femoral neck and head

-20 -15 -10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

x [mm]

Dis

torti

onal

stra

in [-

]

0 deg 15 deg

-20 -15 -10 -5 0 5 10 15 200

0.005

0.01

0.015

x [mm]

Flu

id v

eloc

ity [m

m/s

]

0 deg 15 deg

Figure 19: Distribution of distortional strain for two orientations of the femoral neck and head

Figure 20: Distribution of fluid velocity for two orientations of the femoral neck and head

Discussion

The objective of this research was to find whether the hydrostatic stress, pore pressure, octahedral shear (or distortional) stress or strain, or fluid velocity, alone or combined in the osteogenic or mechano-regulation index, can explain the reorientation of the growth plate. To do this, two finite element models were made: one with a linear elastic growth plate, and one with a biphasic growth plate.

For the linear elastic model, it was found that the hydrostatic stress was not influenced by the orientation of the resultant hip force. It is therefore unlikely that the hydrostatic stress influences the reorientation of the growth plate. However, the orientation of the resultant hip force had a large effect on the octahedral shear stress. If the octahedral shear stress determines the rate of growth in the growth plate, a reorientation of the resultant hip force will lead to an increase in growth at the lateral side and a decrease in growth at the medial side, and therefore will lead to a reorientation of the growth plate. Thus, the octahedral shear stress could be the stimulus that determines the reorientation of the growth plate.

The osteogenic index was little influenced by the orientation of the resultant hip force. After rotation of the resultant hip force, the osteogenic index increased at the lateral side and

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decreased at the medial side, which means that ossification was more stimulated at the lateral side. The osteogenic index would explain the reorientation of the growth plate best when the value for k is zero, because then the influence of the hydrostatic stress, which is unlikely to influence reorientation, is zero. For high values of k, the difference in osteogenic index between the centre and the edges was greater than for small values of k. This means, that for high values of k, much more ossification occurred at the edges than in the centre of the growth plate. This would lead to straightening of the growth plate. For small values of k, the osteogenic index curved less, so the difference in osteogenic index between edges and the centre was smaller. This would not result in straightening of the growth plate. This is another reason why an osteogenic index with small values of k would better explain the reorientation of the growth plate than with high values of k.

For the biphasic model, the orientation of the resultant hip force had little or no effect on the pore pressure and the fluid velocity. Therefore, the pore pressure and the fluid velocity are unlikely to influence the orientation of the growth plate. The distortional stress and strain, on the other hand, were highly influenced by the orientation of the resultant hip force. Just as the octahedral shear stress in the linear elastic model, the distortional stress and strain could be the stimuli for the reorientation of the growth plate.

The mechano-regulation index of Huiskes et al [9] was influenced by the orientation of the resultant hip force. Rotation of the resultant hip force caused more fibrous tissue at the lateral side of the growth plate and fibrocartilage instead of fibrous tissue at the medial side. It is unlikely that during growth the cartilage of the growth plate turns into fibrous tissue. Therefore it is unlikely that the mechano-regulation index, which was meant to explain tissue differentiation, could explain the growth and the reorientation of the growth plate.

When the resultant hip force was correct for 16-year-old, the fluid velocity, distortional stress and strain, and thereby also the mechanoregulation index, were not constant over the growth plate. If these variables govern the growth in the growth plate, the edges of the growth plate would grow stronger than the centre, and this would flatten the growth plate. An explanation may be that the presented results were all taken in the first gait cycle. In the biphasic model, the variables may change after several gait cycles because the interstitial fluid needs time to flow away due to the low permeability. Only a little change in the fluid velocity was seen after 25 gait cycles (Appendix I, figure 21). In a geometrically nonlinear analysis with a permeability that was 1000 times to high, the fluid velocity profile flattened during 4 gait cycles (Appendix I, figure 22). It is unclear if this flattening does not occur in the geometrically linear analysis with the right permeability only due to the permeability or that the kind of analysis also has influence on this. It is possible that the fluid velocity and the other variables in the correct biphasic model flatten after more gait cycles, for example 1000 gait cycles.

For the biphasic model, a geometrically linear analysis was performed, due to convergence problems with a geometrically nonlinear analysis. This means that the nonlinearity is ignored in the element calculations - the kinematic relationships are linearized. Linear analyses are used in cases where the strains and rotations are small. However, the strains in this analysis vary from 0 to 0.5. This might have an effect on the results presented here.

For the model representing a femoral head and neck after osteotomy, almost the same results were found as for the rotation of the resultant hip force. All studied variables are influenced by the rotation of the model, some more than others. The octahedral shear stress in the linear elastic model, the distortional stress and the distortional strain in the biphasic model all increased more at the lateral side than at the medial side. If these variables determine the growth, the growth plate will reorient and therefore the femoral head will turn back to its old position.

The finite element model is a simplification of the reality. In the finite element analyses, the material properties of the elements in the femoral head and neck were modelled isotropic. As a high degree of anisotropy exists in the femoral head, this simplification might have influence on the results presented in this article. However, in earlier studies where bone was

9

modelled as an anisotropic material, similar results were obtained as when using isotropic material properties [24].

Further, deviations were found at the edges of the growth plate, as can be seen in figures 6, 9, and 20. These deviations occur probably because the finite element model is not enough refined at the edges, where materials with different properties are adjacent to each other.

In this study it was found that the hydrostatic stress, the pore pressure and the fluid velocity are unlikely to influence the reorientation of the growth plate. The distortional (or octahedral shear) stress and strain could be the stimuli that determine the reorientation of the growth plate. Appendix I: Results of fluid velocity in two different analyses

-20 -15 -10 -5 0 5 10 15 200.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x [mm]

Flu

id v

eloc

ity [m

m/s

]

1st cycle 25th cycle

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0.5

1

1.5

2

2.5

x [mm]

Flu

id v

eloc

ity [m

m/s

]

1st cycle2nd cycle3rd cycle4th cycle

Figure 21: Fluid velocity for 1st and 25th gait cycle using geometrically linear analysis and correct permeability.

Figure 22: Fluid velocity for 1st to 4th gait cycle using geometrically nonlinear analysis and wrong permeability.

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