Kenneth

C. Faulkner,

PhD

#{149} Christopher

E. Cann,

PhD

#{149} Bruce

H. Hasegawa,

PhD

Effect of Bone Distribution on Vertebral Strength: Assessment with Patient-Specific Nonlinear Finite Element Analysis’ Three-dimensional

quantitative (QCT) studies of the lumbar spine were extended with finite element analysis (FEA) to include bone distribution in assessment of vertebral body strength. Fifty-nine FEA models were created from data from 43 patients, 28 with no evidence of osteoporosis and 15 with previous vertebral fractures. Simulated loads were applied to the vertebral models to estimate vertebral strength. Yield strength in the models from patients with osteoporosis was 0.221.05 MPa (average, 0.57 MPa ± 0.26 [mean ± standard deviation]), compared with 0.80-2.79 MPa (1.46 ± 0.52, P < .001) in patients with normal bone. Yield strength of vertebrae in patients with osteoporosis uniformly fell below approximately 1.0 MPa, with minimal overlap between patients with osteoporosis and those with normal bone cornpared with the overlap in bone mmeral content and trabecular mineral density. Reproducibility of the FEA technique was 12.1% in a subgroup of patients with normal bone. A constant relationship between corticomputed

cal

and

observed rosis but

tomographic

trabecular

contributions

was

in patients with osteoponot in control patients.

Index

terms: Bones, measurement, 331 .92 #{149} Bones, mineralization #{149} Computed tomography (CT), computer programs #{149} Computed tomography (CT), image processing #{149} Finite element analysis (FEA) #{149} Fractures, stress, 331.415 Model, mathematical #{149} Osteoporosis, 30.56. Spine, CT, 331.1211 #{149} Spine, fractures, 331.411, 331.415 Radiology

1991; 179:669-674

From the Department San Francisco, P0 Received November

nia, bly.

1991; accepted February DK39964 and National requests to C.E.C. C

RSNA,

1991

T

HE determination

of an individfor future onteoporotic fracture is limited by insufficient knowledge of the relationship among the many factors contributing to bone strength. One important factor in this determination is the amount of mineral in the bone. Densitometric studies of the spine, hip, forearm, or heel with any of several established techniquesincluding single-energy photon absorptiometry, dual-energy photon absorptiometry or dual-energy x-ray bone densitometry, and quantitative computed tomography (QCT)-are used to make this measurement. Each technique assumes a relationship between the bone mineral density or content and the strength of the bone at either the measurement site or some other skeletal location. Several researchers have shown that such a relationship exists (1-6); however, an overlap in the noninvasively measured bone mineral density makes the complete separation of patients with normal bone and those with osteoporosis difficult (3,7-9) (Fig 1). Patients with equal bone mass may or may not experience osteoporotic fractures. Whether a fracture occurs likely depends on a difference in cither the loading conditions or the structural characteristics of the bone. Differences in loading conditions depend on the weight of the patient, activity level, and susceptibility to falling. Naturally, if the patient falls often or is frequently required to negotiate many flights of stairs, the risk for fracture is increased. If patients are subject to severe enough loading conditions (such as falling ual patient’s

risk

of Radiology and Bioengineering Graduate Group, University of CaliforBox 0628, San Francisco, CA 94143. From the 1990 RSNA scientific assem15, 1990; revision requested December 17; revision received January 22,

18. Supported in part by National Institutes Aeronautics and Space Administration grant

of Health grant i-P01NAG 9-327. Address reprint

from a cliff), fractures can be expected regardless of their bone mass. Structural differences can be both macroscopic and microscopic. The macroscopic properties refer to the way the bone is distributed through the structure; for the vertebral body, differences in the amount of cortical bone, continuity of the trabecular lattice, and inhomogeneities in trabecular density all influence the macroscopic structure of the bone. The microscopic structure of bone refers to the quality of the bone, which depends on its proportion of mineral to osteoid,

mineralized

microstructure,

and turnover rate. The degree of contribution of each of these factors to skeletal strength remains unknown and is an active area of research. QCT is unique among the current techniques for bone mineral measurement in that it provides true three-dimensional information on density and distribution. Conventional QCT examinations, performed with a single section through the center of the vertebrae, can provide limited data about the distribution of the bone (10,11) but do not represent the complete vertebral structure. Only three-dimensional QCT methods provide a basis for such a representation (12,13). According to the theory of structural mechanics, both the density and distribution factors are important to the overall strength of any structure. The role of structure and distribution in vertebral strength has been previously studied by several researchers through mechanical testing and density information from noninvasive and invasive techniques (3,6,13).

Abbreviations: FEA

-

finite

five computed

BMC eleient

mineral

- bone analysis,

tomography,

QCT

SD

TMD trabecular mineral Y2.es = yield stress at an accumulated deformation of 2. viation,

content, quantita-

standard density, plastic

de-

100 FtH

4 LH

H Illil

90 80 60 70 40 50 20 30 100

oE

. 0

0.5 0

3

2

4 Totai

a.

b. 4 Three-dimensional generated from

Figure

model mal

deformed mesh plots for a representative vertebral body the results of FEA. (a) Displacement contours show regions of miniat the inferior surface (where the model is constrained not to move) up to

displacement

the large-displacement the same model indicate and

superior

end

regions at the superior regions of local strain

surface of the model. (b) Strain contours concentration at the anterior vertebral

5 Bone

6 Content

7

8

9

(grams)

Figure 5. Yield stress at 2.0% offset as a function of the total BMC of the vertebral model. Results of analysis in patients with normal bone are indicated by triangles (A), and those in patients with osteoporosis are shown as squares (0).

in wall

plate. 1800

Y20% was determined. Reproducibility was defined as the coefficient of variation for the paired evaluations, expressed as a percentage. To assess

the

contribution

significance

to strength

of the in

both

cortical patients

with normal bone and those with osteoporosis, the FEA yield stresses in both the intact and stripped models were compared. Paired, two-tailed Student t tests were used to evaluate the significance of the differences between both groups.

RESULTS Vertebral

Strength

Estimation

Figure 5 shows the FEA-determined yield stress (Y2.o) as a function of the model BMC. For the same BMC, estimated strength varied by more than a factor of two because the bone distribution was included in the analysis. The models in patients with osteoporosis were associated with predictions of yield stresses lower than those predicted in the patients with normal bone who had comparable BMC. In the 23 models of patients with osteoporosis, Y20% ranged from 0.22 to 1 .05 MPa, with an average of 0.57 MPa ± 0.26 (mean plus

or

minus

standard

deviation

[SD]). Y2.o% for the 36 normal vertebra! finite element models ranged from 0.80 to 2.79 MPa (average, 1.46 MPa ± 0.52). FEA yield estimates in the two groups were significantly different based on a Student t test analysis (P < .001 that no difference exists). Yield strength of vertebrae in patients with osteoporosis uniformly fell below approximately 1.0 MPa, with minimal overlap between patients with osteoporosis and those with normal bone compared with the overlap in model BMC and TMD. For the patients in this study, a fracture 672

#{149} Radiology

threshold

of 0.95

ly separated

the

MPa

more

vertebral

1600

accuratemodels

in

patients with normal bone and those with osteoporosis than a QCT-measured TMD threshold of 100 mg/cm3. Previously published compression data from Hansson et a! (4) showed a mange in the fracture stress of 1.034.95 MPa, with an average of 2.08 MPa ± 0.81. Comparison of the FEAdetermined Y2o% and the vertebral fracture stress in vitro from the vertebral studies of Hansson et a! showed slight differences between the two measurements. Note, however, that measurements

of fracture

derived from different healthy subjects and techniques.

stress

80

1400 0 C/, C

1200

8

0

a,

1000

z

.

0

.3 600

A

400 200 0

were

groups of with different

Reproducibility The 10 duplicate studies had an average Y20% of 1.8 MPa, with a coefficient of variation equal to 0.121. For these premenopausal patients with normal bones, this represents a shortterm reproducibility of 12.1% for the patient-specific FEA.

0

800

0

Trabecular

Strength

Contribution

Linear regression showed an inverse relationship between TMD and cortical contribution to yield stress (Y = 0.686 0.0034X, where Y = cortical contribution and X = TMD; r = .88), showing the impomtance of the cortex in strength for patients with diminished TMD. In the patients with normal bones, the average cortical contribution to yield stress was 12.4% (P < .001 that theme was no contribution). In the group with osteoporosis, the cortical contmibution was 56.2% (P < .001 that no -

200

150

Mineral Density (mg/cm3)

Figure

6. Absolute cortical ( 0 ) loads as a function

() and trabecular of TMD of the vertebral body as determined with FEA. Regression lines for the data show that both the cortical shell and trabecular core are important to the absolute vertebral body strength. In the patients with osteoporosis, the ratio of estimated cortical contribution to trabecular contribution is approximately 2:1. Thick line trabecular regression line, thin line = cortical regression line.

contribution

Cortical

100

50

load area)

existed).

at failure (stress was estimated,

patients

with

osteoporosis

When

total

multiplied we found

by that in

a constant

relationship existed between the cortical and tmabecular contributions, with the cortex contributing more than half of the vertebral body strength, whereas in the control group no apparent relationship exist-

ed (Fig

6). DISCUSSION

The incorporation of the macroamchitecture of bone into analysis of bone mineral density is a logical yet

June

1991

complex extension of conventional bone mineral measurements. The use of normalizing factors as simple as cross-sectional area has been shown to be helpful in prediction of the strength of the vertebrae (3). Other investigators have tried to relate bone mineral density in different megions of the bone to its overall strength (1,13), including complex statistical analysis with multiple correlations (21). Analytic models that describe age-related changes in vertebmal microarchitecture and bone distribution have been developed with data obtained from autopsy specimens (22). However, none of these approaches can incorporate the actual three-dimensional distribution of bone into a structural analysis for data obtained in vivo in a given patient. Although FEA has recently been used to estimate the strength of the femur in vivo (23), this technique has not previously been studied as a diagnostic

tool

for

detection

of verte-

bral

osteoporosis. We have used FEA with three-dimensional QCT to measure the structural strength of individual vertebral bodies in women with normal bone and women with osteoporosis under conditions approximating uniaxial compression to failure. It was shown that patient-specific FEA estimates of vertebral fracture stress caused a slight underestimation of the observed fracture loads on the basis of previously published in vitro results. This is likely a result of a limitation of the FEA. As the stresses in the model approach the failure point of the structure, the mathematical equations relating the applied stress and observed strain are no longer valid. Consequently,

one

would

expect

the

FEA estimate of strength to be somewhat lower than the actual strength of the structure, as was observed in this study. Of greater significance than the absolute accuracy of the patient-specific FEA is the ability of PEA to show differences in vertebral strength estimates between patients with the same BMC or TMD. Significant differences were observed between patients with osteoporosis and those with normal bone, including those with similar BMC and TMD in whom differences were not detected with bone measurements alone. At the 0.95-MPa fracture threshold, FEA shows a greater diagnostic accuracy for separating models of patients with normal bone and patients with osteoporosis than the traditional QCT 7nlisms

1’7Q

#{149} Number

3

fracture threshold of 100 mg/cm3. The FEA threshold chosen, however, was based only on the limited patient sample used for this study and may not be generally applicable for all patients. It should be noted that an cxtended study including age-matched control subjects as well as specimen testing in vitro are necessary to validate these preliminary results. The FEA technique used for the first time (to our knowledge) in this study has several limitations. A material model that depends only on the bone density and applied strain rate is not likely to be highly accurate meflection of the strength of bone in vivo, but it is currently the best available model based on noninvasive measurements ture research

of bone

mineral.

Fu-

may produce more complex and accurate material property relationships based not only on bone density but also on bone quality, the amount of marrow present, and other noninvasive measurements. The reproducibility error of the yield stress determined with FEA in this study was high compared with that of bone mineralization measurements. The measurement of TMD and vertebral body BMC is about 1% (24), indicating that the FEA is mesponsible for the major portion of the precision error. The current method of reducing the mesh resolution from 1 X 1 mm to 3 X 3 mm is required for computational efficiency, but this technique into the

clearly

introduces

errors

mesh. These errors are due to volume averaging of the vertebral cortex with trabecular bone and to further averaging of the vertebral end plates with the intervertebral disks and more central vertebral trabecular bone. The use of rectangular elements also inaccurately represents the complex geometry of the vertebra. The reproducibility error can be reduced by automating sections of the rotation-and-thresholding software to minimize any subjective operator interaction and by reducing the element size. However, the analysis is currently computationally intense, requiring up to 30 hours of computer time to generate and analyze a single model. Computation time can be significantly reduced, up to a factor of 100, by performance of FEA on a mainframe computer. Current research is centered in all of these areas to refine the patient-specific FEA to produce a sensitive diagnostic tool for assessment of the risk of fracture in osteoporosis. Of clinical importance is the con-

tribution of the vertebral cortex to the overall strength of the vertebral body.

Previous

research

into

this

question has had to rely on destructive testing methods to estimate cortical strength. Rockoff et al (6) tested a series of vertebral specimens with and without the cortical shell and estimated a cortical contribution of 45%-75% unrelated to the density of the vertebral trabecular bone. McBmoom et al (3) performed a similam experiment but used hand sanding, rather than the high-speed saw used by Rockoff et al, to remove the vertebral cortex. They concluded that the cortex accounts for only 10% of the vertebral strength and attributed the apparent disparity between their results and those of Rockoff et a! to the different methods of cortex memoval. FEA has an advantage in that the entire cortical shell can be memoved from the vertebral model with software techniques without adversely affecting the underlying trabecular structure. The cortical strength estimates in the FEA fall between those of the previous studies and show a definite dependence of the cortical contribution on the TMD of the vertebral body. PEA results also show that the cortex becomes significantly more important in bearing loads in patients with osteoporosis in whom the TMD of the vertebral body is decreased. It was determined that in patients who already have fractures, the contribution of the vertebral cortex and trabecular bone follows a constant relationship, with the cortex supporting approximately two-thirds of the load. This may explain the observation in many studies that, although a measurement of TMD may help to distinguish patients with osteoporosis and those with normal bone, there seems to be little advantage to measurement of trabecular bone relative to measurement of total vertebral bone for assessing fracture risk in patients with osteoporosis. CONCLUSIONS Results of this preliminary study with patient-specific FEA of the lumbar spine have shown that the distribution of bone material should be considered when one attempts to asseas vertebral strength in a predictive model. In patients with similar vertebra! BMC and TMD, differences have been shown in the estimated strength that were impossible to pmedid with bone measurements alone. In the patients in this study, FEA was Radiology

#{149} 673

shown to be a more accurate diagnostic test for osteoporosis than the QCT-measured trabecular bone density alone. However, the results are preliminary, pertain to a limited number of patients, and remain to be verified

through

future

research.

5.

6.

cal bone vertebrae.

Ad-

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Iun

1Q1

Effect of bone distribution on vertebral strength: assessment with patient-specific nonlinear finite element analysis.

Three-dimensional quantitative computed tomographic (QCT) studies of the lumbar spine were extended with finite element analysis (FEA) to include bone...
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