Shape-based Assessment of Vertebral Fracture Risk in Postmenopausal Women Using Discriminative Shape Alignment

Rationale and Objectives

Risk assessment of future osteoporotic vertebral fractures is currently based mainly on risk factors, such as bone mineral density, age, prior fragility fractures, and smoking. It can be argued that an osteoporotic vertebral fracture is not exclusively an abrupt event but the result of a decaying process. To evaluate fracture risk, a shape-based classifier, identifying possible small prefracture deformities, may be constructed.

Materials and Methods

During a longitudinal case-control study, a large population of postmenopausal women, fracture free at baseline, were followed. The 22 women who sustained at least one lumbar fracture on follow-up represented the case group. The control group comprised 91 women who maintained skeletal integrity and matched the case group according to the standard osteoporosis risk factors. On radiographs, a radiologist and two technicians independently performed manual annotations of the vertebrae, and fracture prediction using shape features extracted from the baseline annotations was performed. This was implemented using posterior probabilities from a standard linear classifier.

Results

The classifier tested on the study population quantified vertebral fracture risk, giving statistically significant results for the radiologist annotations (area under the curve, 0.71 ± 0.013; odds ratio, 4.9; 95% confidence interval, 2.94–8.05).

Conclusions

The shape-based classifier provided meaningful information for the prediction of vertebral fractures. The approach was tested on case and control groups matched for osteoporosis risk factors. Therefore, the method can be considered an additional biomarker, which combined with traditional risk factors can improve population selection (eg, in clinical trials), identifying patients with high fracture risk.

Osteoporosis is a disease characterized by decreases in the density and quality of bone, potentially leading to fractures. Vertebral fractures are the most common type of osteoporotic fractures , and they may occur in the absence of trauma or after minimal trauma. In this case, they are also called fragility fractures. Moreover, the presence of vertebral fractures increases the risk for hip fractures and additional vertebral fractures . A growing percentage of elderly people develop osteoporosis and experience osteoporotic fractures, and consequently, the related burden sustained by the medical system is heavy and increasing rapidly . The total cost in 2000 in Europe has been estimated to €31.7 billion, and it is foreseen to grow to €76 billion by 2050 , while the estimated expense in the United States in 2005 was $19 billion . Therefore, the early assessment of fracture risk is crucial for reducing the number of fractures through appropriate treatments, improving the quality of life of patients, and reducing costs to the health system.

Several clinical factors can provide information on fracture risk. These include bone mineral density (BMD), age, prior fragility fractures, smoking, the use of corticosteroids, excessive consumption of alcohol, and rheumatoid arthritis . The World Health Organization has formalized the analysis of such risk factors in the FRAX tool , which gives a probability of hip fracture over a 10-year period and a probability of major osteoporotic fracture (located in the spine, forearm, hip, or shoulder) over a 10-year period as well. The FRAX tool is available as a Web-based service and in a paper version with guided charts .

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Figure 1

On the left (a) , overlay of the lumbar region in an x-ray image (case baseline) and its six-point annotation, given by the stars . On the right (b) , the spine shape that can be obtained from the concatenation of the two-dimensional landmarks. The annotations were performed considering coherently the lowest endplate of the vertebrae. The contrast in the reproduction above is most likely too low to clearly visualize all the lower boundaries. For the experiments reported in this study, a spine shape was used as the collection of these landmarks without any interpolation.

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Figure 2

(a) Example of baseline vertebra that had fractured by follow-up and (b) example of baseline vertebra that had not fractured by follow-up. (c,d) Follow-up radiographs of the same vertebrae depicted, respectively, in (a) and (b) .

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Materials and methods

Data

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Methods

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Shape alignment

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dGPA(xi,xj)=∥Wixi−xj∥2, d

GPA

(

)

‖

−

‖

where Wi is a similarity transformation matrix, which can be found either by using a closed-form solution or in an iterative fashion , and x i and x j are specific shapes.

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Point distribution model

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Σ=VΛVT, Σ

where V = [ v 1 |…| vq ] is the column matrix of eigenvectors, and Λ = diag[(λ 1 ,…, λ q )] is the diagonal matrix of corresponding eigenvalues . These eigenvectors and eigenvalues can be used to represent the shapes into the new coordinate system as

Y=Λ−1−−−√VTX. Y

−

This equation represents the linear transformation principal components analysis is performing on the data X to yield the transformed Y .

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x˜=Vdiag[(λ1−−√,…λs−−√,0,…,0)]y+μ, x

diag

[

(

…

)

]

where x˜ x

˜ is the approximation of x using this model, y is the column vector in Y corresponding to the column vector in X , and diag[(λ1−−√,…,λs−−√,0,…,0)] diag

[

(

…

)

] is the diagonal matrix containing a subset of eigenvalues. The optimal number of used components, s , represents a pivotal parameter that needs to be estimated. We describe in the section “Experiments” how this selection was performed in our experiments.

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Linear discriminant classifier

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Pcase(x˜|μcase,Σ). P

case

(

case

)

By normalizing the score as P¯¯¯case=Pcase/(Pcase+Pcontrol) P

case

(

case

control

) , the probability of fracture risk is in the interval between 0 and 1, where values close to 1 represent high risk for vertebra fracture and those close to 0 represent low risk. Figure 3 depicts, from left to right, some spine shapes ordered from low to high fracture risk as defined by the described probability. The picture illustrates how difficult it is to see the slight differences in the vertebra shapes and spine curvature without accurate measurements.

$Figure 3, Examples of baseline spine shape, ordered from left to right accordingly to their probability of belonging to the case group on the basis of our fracture risk score. For the experiments reported in this study, the spine shapes were used as the collection of the landmarks without any interpolation.$

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Discriminative shape alignment

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∑Ni=1(aTxi−ki)2, ∑

(

−

)

where aT is the discriminative linear regression vector.

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z′i=zizHim. z

′

Here, zHi z

H represents the Hermitian of the shape vector. To use this alignment in conjunction with the minimization of equation 6 , the complex vectors must be converted to real values, yielding the expression:

∑Ni=1(aTZiμref−ki)2, ∑

(

ref

−

)

where Zi=(Re(zizHi)Im(zizHi)−Im(zizHi)Re(zizHi)) Z

(

)

−

(

)

(

)

(

)

) . The algorithm is initiated with the mean shape as the μ ref and iteratively updates the reference shape in conjunction with the projection vector aT . Equation 8 cannot be solved in closed form, because it represents an intertwined linear regression problem .

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Here, F is the multidimensional matrix containing all the Z i matrices of all the shapes multiplied by μ ref , K is the vector containing all the labels k i , and r is a regularization parameter that controlled the degree of regularization.

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Experiments

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Pseudocode for BNLO algorithm using the DSA version

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Statistical Analysis

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Results

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Table 1

Biologic Data Used to Match the Case and Control Groups

Variable Cases ( n = 22) Controls ( n = 91)P ∗ Age (y) 64.7 (66.9 ± 5.4) 65.4 (66.6 ± 5.9) .98 Height (cm) 162 (163 ± 4.6) 161 (161 ± 6.0) .21 Weight (kg) 64.5 (68.3 ± 11.7) 65.4 (65.4 ± 8.4) .53 Body mass index (kg/m 2 ) 24.8 (25.5 ± 4.0) 24.6 (25.0 ± 3.2) .72 Spine bone mineral density (g/cm 2 ) 0.79 (0.81 ± 0.14) 0.84 (0.86 ± 0.14) .20

Data are expressed as median (mean ± standard deviation).

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Table 2

AUCs, P Values, and ORs Obtained Using the GPA Alignment

Data Set AUC_P_ OR (95% CI) P.P. 0.72 ± 0.009 .017 5.85 (3.52–9.71) J.P. 0.54 ± 0.089 .352 1.04 (0.62–1.66) A.O. 0.53 ± 0.061 .384 1.03 (0.60–1.45)

AUC, area under the receiver-operating characteristic curve; CI, confidence interval; GPA, generalized Procrustes analysis; OR, odds ratio.

Reported results are mean values with respect to the same test shape for 10 iterations using 1000 samples of the balanced nested leave-out cross-validation protocol. The reported P values were obtained comparing the scores to chance using the DeLong test.

Table 3

AUCs, P Values, and ORs Obtained Using the DSA

Data Set AUC_P_ OR (95% CI) P.P. 0.71 ± 0.013 .019 4.87 (2.94–8.05) J.P. 0.62 ± 0.025 .043 2.04 (1.21–3.36) A.O. 0.60 ± 0.046 .047 1.92 (1.06–2.87)

AUC, area under the receiver-operating characteristic curve; CI, confidence interval; DSA, discriminative shape alignment; OR, odds ratio.

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Discussion

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Acknowledgment

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Shape-based Assessment of Vertebral Fracture Risk in Postmenopausal Women Using Discriminative Shape Alignment

Rationale and Objectives

Materials and Methods

Results

Conclusions

Materials and methods

Data

Methods

Shape alignment

Point distribution model

Linear discriminant classifier

Discriminative shape alignment

Experiments

Pseudocode for BNLO algorithm using the DSA version

Statistical Analysis

Results

Discussion

Acknowledgment

References

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