Advantage of topological texture measures derived from Minkowski Functionals (MF) and Scaling Index Method (SIM) in comparison with biomechanical Finite Elements Method (FEM) for the prediction of osteoporosis
Irina Sidorenko∗a, Jan Bauerb, Roberto Monettia, Dirk Muellerb, Ernst Rummenyb, Felix Ecksteinc, Christoph Raetha a Max-Planck-Institut fuer extraterrestrische Physik, 85748 Garching, Germany b Department of Radiology, Technische Universitaet Muenchen, 81675 Munich, Germany c Institute of Anatomy and Musculoskeletal Research, Paracelsus Medical Private University, Salzburg, Austria ABSTRACT
The assessment of trabecular bone microarchitecture by numerical analysis of high resolution magnetic resonance (HRMR) images provides global and local structural characteristics, which improve the understanding of the progression of osteoporosis and its diagnosis. In the present work we apply the finite elements method (FEM), which models the biomechanical behaviour of the bone, the scaling index method (SIM), which describes the topology of the structure on a local level, and Minkowski Functionals (MF), which are global topological characteristics, for analysing 3D HRMR images of 48 distal radius specimens in vitro. Diagnostic performance of texture measures derived from the numerical methods is compared with regard to the prevalence of vertebral fractures. Both topological methods show significantly better results than those obtained using bone mineral density (BMD) measurement and the failure load estimated by FEM. The receiver operating characteristic analysis for differentiating subjects with and without fractures reveals area under the curve of 0.63 for BMD, 0.66 for maximum compressive strength as determined in a biomechanical test, 0.72 for critical load estimated by FEM, 0.79 for MF4 and 0.86 for SIM, i.e. local topological characteristics derived by SIM suit best for diagnosing osteoporosis. The combination of FEM and SIM on tissue level shows that in both weak and strong bones the plate-like substructure of the trabecular network are the main load bearing part of the inner bone and that the relative amount of plates to rods is the most important characteristic for the prediction of bone strength. Keywords: osteoporosis, 3D high resolution magnetic resonance imaging, trabecular bone structure, finite elements method, scaling index method, Minkowski Functionals, correlation with maximum compressive strength, spine fracture status
1. INTRODUCTION AND PURPOSE High resolution (HR) magnetic resonance (MR) imaging is widely used for the assessment of the structural characteristics of trabecular bone network1-6, which plays the main role in load carrying and stress distribution inside the bone. Detailed description of the topological and biomechanical properties of the cancellous bone microstructure helps to evaluate deterioration of bone tissue caused by osteoporosis and to predict spine and hip fractures, which are the most common complications of osteoporosis. Different morphological, mechanical and topological concepts propose a large variety of texture measures for the quantification of very porous and irregular tissue. In the present work we discuss several texture measures derived from finite elements method (FEM), which describes the biomechanical response of the bone structure to the compressive loading, and two topological methods. The Minkowski Functionals (MF) provide for 3D structures four global topological characteristics, whereas the scaling index method (SIM) quantifies the local topology of an arbitrary structure and indicate rod- and plate-like trabecular substructures in the inner bone network. Texture measures based on the described methods are applied to 3D HRMR images in vitro. ∗
[email protected] Medical Imaging 2010: Biomedical Applications in Molecular, Structural, and Functional Imaging, edited by Robert C. Molthen, John B. Weaver, Proc. of SPIE Vol. 7626, 762629 · © 2010 SPIE CCC code: 1605-7422/10/$18 · doi: 10.1117/12.840373 Proc. of SPIE Vol. 7626 762629-1 Downloaded from SPIE Digital Library on 15 Jun 2010 to 130.183.136.239. Terms of Use: http://spiedl.org/terms
Comparison of the biomechanical and topological characteristics of the trabecular bone on the tissue level gives a possibility to analyse the redistribution of the deformation energy stored in the structure under the load and to make conclusions about the role of different topological substructures (i.e. rod- and plate-like trabecularae) in the global strength of the bone. Diagnostic performance of the derived texture measures is analysed by correlation analysis in respect to the maximum compressive strength (MCS) determined in uniaxial compressive experiment. Additionally, taking information about biomechanical and topological features of the trabecular bone in the distal radius, a receiver operating characteristic (ROC) analysis is performed to differentiate between patients with and without spine fracture.
2. MATERIAL AND METHODS 2.1 Data set Our study is based on a set of three dimensional high resolution magnetic resonance (HRMR) images of distal radius in vitro. HRMR images with a resolution 139 x 139 x 500 μm were obtained from forearm specimens of 48 patients. Among those, 23 showed osteoporotic spine fractures. All specimens were scanned with a 1.5T MR scanner (Gyroscan NT with Intera upgrade, Philips Medical Systems, DA Rest, Netherlands) using a gradient echo sequence. In addition, for all samples the bone mineral density (BMD) was measured by quantitative computed tomography (QCT) and maximum compressive strength (MCS) was determined in a biomechanical uniaxial compressive test simulating a fall on the outstretched forearm. Before applying mathematical methods for analysing the trabecular structure the HRMR images of distal radius are segmented using a seeded growing algorithm7: starting from one or more seed voxels inside the region of interest - in our case the trabecular bone - new voxels are included according to predefined threshold conditions. In order to compare the characteristics obtained from images with different grey level distribution we use a normalisation procedure and apply numerical analysis to images with already normalised grey level distribution, i.e. one with zero mean value and unit standard deviation. Different values of the threshold ν describe different tissues of the bone (see Fig. 1).
a) νth=0
b) νth=0.5
c) νth=2
Figure 1. Segmented region of distal radius described by voxels with normalized grey level less than threshold value νth
2.2 Finite Elements Method (FEM) A finite element model8 is generated, by converting voxels of the binarized image into equally sized and oriented brick elements. Two problems that arise during binarization drastically affect accuracy of the FEM when applied to the MR images. By binarizing the grey level images the trabecular network and consequently the mechanical characteristics of the whole specimen can be over- or underestimated. Another problem, that is especially important for rarefied structures, is loss of trabecular connectivity and decrease of the stiffness of the whole specimen. In order to describe the trabecular network more realistically we provide correlation analysis of binariezed structure in respect to the experimental data. Final binarization for generating of FEM is done by the thresholding with intensity value ν = −0.1 at which BV/TV gives the best correlation with MCS (Pearson’s correlation coefficient r=0.78) and at the same time very good correlation with BMD (r=0.79). Material properties are chosen to be isotropic and elastic. In this approach the relationship between components of stress and strain is linear and described by the generalized Hooke’s law (constitutive equations), which states that stress is proportional to strain up to the elastic limit σ ij = Cijklεkl , (1)
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In the three-dimensional space the strain εij and the stress σij are 2nd order tensors with nine components and Cijkl is 4th order elastic tensor with 81 components. Under the condition of a static equilibrium all forces on the elastic body sum to zero (Newton’s second law of motion) and the displacements, strains and stresses are not functions of time, which simplifies the equations to the form
∂σ ij
3
∑ ∂x j=1
+ f i = 0,
i = 1,2,3 .
(2)
j
As a result of this static equilibrium, there are no body force moments; shear stresses as well as strains across the diagonal are identical: εij = εji and σij =σji. Due to the symmetry of the tensors, the number of unknown components of strain and stress reduces from nine to six and the components of the elastic tensor from 81 to 21. The next assumption is the isotropy of the bone mineral tissue, i.e. its mechanical properties are independent of the direction in space. For isotropic materials the 4th order elastic tensor C reduces to a [6 × 6] matrix
⎛ σ11 ⎞ ⎛ λ + 2μ λ λ 0 0 0 ⎞⎛ ε11 ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ λ + 2μ λ 0 0 0 ⎟⎜ε22 ⎟ ⎜σ 22 ⎟ ⎜ λ ⎜σ ⎟ ⎜ λ λ λ + 2μ 0 0 0 ⎟⎜ε33 ⎟ σ ij = Cijklεkl ⇒ ⎜ 33 ⎟ = ⎜ ⎟⎜ ⎟ . 0 0 0 2 σ μ 0 0 12 ⎜ ⎟ ⎜ ⎟⎜ ε12 ⎟ ⎜σ13 ⎟ ⎜ 0 0 0 0 2μ 0 ⎟⎜ ε13 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ 0 0 0 0 2μ⎠⎝ε23 ⎠ ⎝σ 23 ⎠ ⎝ 0
(3)
The matrix components depend only on two parameters, which are constant for a given material: Young’s modulus (elastic modulus) Y and Poisson’s ratio ν or Lame’s parameters λ and μ
⎧
{Y,ν } ⇒ ⎨ λ = ⎩
Yν , (1+ ν )(1− 2ν )
2μ =
Y ⎫ ⎬. 1+ ν ⎭
(4)
For the bone mineral material in our simulations, we take Y = 10 GPa and ν = 0.3. We complete our system of equations with six strain-displacement (or kinematic) relations
1 ⎛ ∂u ∂u ⎞ εij = ⎜⎜ i + j ⎟⎟, 2 ⎝ ∂x j ∂x i ⎠
i, j = 1,2,3.
(5)
To get the state of stress at any point of a structure system of the 15 first order partial differential, equations (2)+(3)+(5) must be simultaneously solved with respect to 15 unknowns: stresses σij, strains εij, and displacements ui, (i,j = 1,2,3, εij = εji and σij =σji,). By substituting (5) ⇒ (3) ⇒ (2) one can eliminate strains and stresses from the final formulation. The system of 15 equations is equivalent to a second-order elliptic differential equation with respect to the displacement field u, which is called the Navier-Cauchy equation and constitutes the displacement formulation of the linear elastostatic problem (6) (λ + μ)∇ (∇ ⋅ u) + μ∇ 2 u + f = 0 . In a biomechanical compressive test, body forces f are small in comparison with applied loads and are neglected in numerical simulations. Boundary conditions are set to simulate a high friction compressive test in the uniaxial direction with constant strain prescribed on the top surface. In our simulations we take ε0 = 1%.
⎧ ux = 0 ⎪ z = 0 ⇒ ⎨ uy = 0 ⎪u = 0 ⎩ z
z = z top
⎧u = 0 ⎪ x = ⎨ uy = 0 ⎪ t ⎩ uz = ε0 ⋅ ztop = ub
(7)
By solving the linear elastostatic boundary value problem (6) with boundary conditions (7), we obtain the displacement field u, which is replaced in (5) to solve for strain components εij, which are used then in (3) to restore the stress components σij. As a measure of deformation energy stored in bone tissue we use the effective strain
εeff = 2U Y ,
which is calculated from the strain energy density U normalised on Young’s modulus Y of the tissue
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(8)
U = 1 2(σ11ε11 + σ 22ε22 + σ 33ε33 ) + σ12ε12 + σ13ε13 + σ 23ε23 . (9) From discrete nodal displacements obtained from FEM by means of strain-displacement relations (5) and the constitutive equations (3), strains and stresses can be recovered at any point of the structure. However, because of interpolation and discretization errors, the results, provided by finite element model for displacement formulation based on digital image meshing, oscillate around the exact theoretical solution with average absolute errors up to three per cent and maximum errors up to thirty per cent at the boundary9. To improve the accuracy of the solution we calculate strains, stresses and effective strain for each voxel by averaging over eight unsmoothed values evaluated at Gauss points in the interior of each individual element10. To extract some quantitative characteristics of the structure the tissue level effective strains εeff are compiled in the probability density function (see Fig. 2):
(
)
P (εeff )= prob εeff ∈ [εeff ,εeff + dεeff ] .
(10)
Figure 2. Probability density function of effective strain εeff (blue dashed curves correspond to the healthy patient, red solid curves correspond to the patient with spine fractures). Inset: scatter plot for correlation between experimental MCS and failure load estimated in FEM.
To demonstrate the predictive power of the finite element approach we carry out correlation analyses of the results of numerical modelling with the maximum compressive strength (MCS) measured in compressive experiments. As a main quantitative characteristic we use estimated failure load LFEM, which is computed by multiplying the apparent top reaction force Fr with linear scaling factor kcv. The total reaction force is recalculated from normal tissue level stress along load direction at the top face σtzz and the exact configuration of the top surface At according to the formula
Fr =
∫σ
t zz
dA t .
(11) 11
The linear scaling factor kcv is calculated according to the critical value method , which assumes that failure would be initiated as soon as deformation energy εeff exceeds a critical value Ccv in significant part of the bone tissue.
LFEM = Fr ⋅ kcv ,
kcv = 1 P (ε ≥ Ccv )
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(12)
In order to include information about both large and small deformations we do not restrict the critical value Ccv to a particular number, but let it rather slide along the whole effective strain spectrum. The Pearson’s correlation coefficient between failure load estimated by FEM and MCS is r=0.74. 2.3 Minkowski Functionals (MF) The Minkowski Functionals4-6 (MF) provide a global topological description of structural properties of multidimensional data. In this method HRMR images are considered as a union of 3D convex bodies (voxels)
I(x, y,z,ν ) =
N (ν ≤ν th )
U p (x, y,z) . i
i= 1
Figure 3. Minkowski Functionals. Blue (red) curves correspond to the healthy (osteoporotic) patient. Vertical dashed lines correspond to the threshold value νth, which gave the best AUC.
According to integral geometry one can introduce a set of global topological measures for such an object and obtain for 3D images four functionals, which represent the volume (M1), surface area (M2), integral mean curvature (M3) and connectivity number or Euler characteristic (M4):
M1 (ν ) =
∫ dV ,
I (ν )
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M 2 (ν ) =
M 3 (ν ) = M 4 (ν ) =
∫ dS ,
∂I (ν )
∫ 1 2(1 R + 1 R )dS , 1
∂I (ν )
2
∫ (1 R R )dS . 1
∂I (ν )
2
The functionals are functions of an excursion set I(ν) which includes all voxels with grey levels less than a given threshold value νth. Thus the Minkowski Functionals become a function of the threshold intensity value ν and give a global characteristic for the whole image. It can be shown that the calculation of the MF for images binarized according to the threshold value νth is reduced to the calculation of open vertices (nv), edges (ne), faces (nf) and interior of the voxels (np) which belong to the excursion set I(ν). Thus one obtains
M1 (ν ) = n p , M 2 (ν ) = −6n p + 2n f , M 3 (ν ) = 3n p − 2n f + n e ,
M 4 (ν ) = −n p + n f − n e + n v . Different values of the threshold ν describe different tissues of the bone (see Fig. 1). For the discrimination between healthy and osteoporotic bone the most interesting region is ν ∈ [-1; 1], which includes both the trabecular structure and the transition region. 2.4 Scaling Index Method (SIM) The scaling index method1-3 (SIM) provides differentiation between point-, rod-, sheet-like and unstructured (background) image components and is an effective tool for studying the microstructure of the trabecular bone. The SIM uses local scaling properties to characterise structural patterns in a multi-dimensional point distributions. In SIM a 3D magnetic resonance image is described as a set of points in virtual 4D space with three spatial coordinates x, y, z and the value of the grey level of each voxel ν(x,ry, z) as a fourth dimension. Thus both space and intensity information can be combined in a four-dimensional vector pi = (x, y,z,ν (x, y,z)) and the image can now be regarded as a set of N points
r P = {pi }, i = 1,..., N voxels . For each voxel the logarithmic gradients r ∂ log ρ ( pi , r ) , αi = ∂ log r
which are called scaling indices, are calculated by means of a Gaussian shaped weighted cumulative point distribution
r
ρ( pi ,r) =
N voxel
∑e(
2 − d ij r)
.
j=1
Here
r r dij = pi − p j
2
is the Euclidean distance between two points in the virtual 4D space. It is the exponential shape
of the weighted function ρ that causes SIM to be a local method: the value of the scaling index depends on the number of the neighbours only in a small vicinity of the point for which α is calculated.
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a)
b)
c)
Figure 4. Segmented region of distal radius: a) original image; b) trabecular structure described by the voxels with values of scaling indices α ∈ [0;2.5]; c) bone marrow described by voxels with α ∈ [2.5;4].
The SIM gives an alternative representation of the HRMR images: values of α≈1, α≈2 and α≈3 correspond to rod, plate and bone marrow structures. In terms of scaling indices trabecular structure can be described by values α∈[0;2.5] and bone marrow by α∈[2.5;4] (see Fig. 4). One can expect the largest differences between healthy and osteoporotic bones in the transition region from trabecular to marrow bone tissues, which corresponds to values α∈[2;3]. The values of scaling indices can be compiled in the probability density function (pdf)
P(α ) = prob(α ∈ [α,α + Δα ]). Analysing the P(α) spectrum of HRMR images and its statistical moments one can distinguish between healthy and osteoporotic bones (see Fig. 5a). Due to the loss of bone mineral content and the structural deterioration of bone tissue, less trabecular structure and more bone marrow are typical for osteoporotic patients. For the weak trabecular structure it is typical (red solid curves on Fig. 5a) that the position of the maximum of the P(α) distribution is shifted to the right and values of P(α) in transition region around α = 2.5 are considerably decreased in comparison with those of a healthy bone (blue dashed curves on Fig. 5a).
a) b) Figure 5. Typical tissue level characteristics of the two groups of patients (blue dashed curves correspond to the healthy patient, red solid curves correspond to the patient with spine fractures): a) scaling indices α with positions of two sliding windows for the best AUC=0.86; b) effective strain eff=, averaged over voxels with the same value of the scaling index α.
Both FEM and SIM methods propose alternative representation of the bone structure on tissue level. Each voxel can be characterised by two new properties: effective strain εeff obtained by FEM and scaling index α obtained by SIM. By averaging values of εeff over voxels with the same topological characteristics α we derive the dependence of biomechanical stresses in the bone from the topology of the underlying structure (see Fig. 5b). Both in healthy and
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osteoporotic bones (blue dashed and red solid lines on Fig. 5b) the largest stresses were observed in substructure with α∈[1.5; 2.], which corresponds to the plate-like trabeculae. Comparison of the two plots (See Fig. 5) lead to the conclusion that in all bones plates are the main load bearing trabecular substructure of the cancellous bone and strength of the whole bone depends on amount of plates in trabecular network. In order to give a quantitative description of the local differences in the P(α) spectra of different images we use a two sliding windows technique and define the following scalar texture measure1
mP(α ) = Δ w1P(α ) +
Δ w1P(α ) . Δ w 2 P(α )
Here Δ w1P(α ) and Δ w 2 P(α ) are the percentage of points in the parts of the spectrum limited by the windows w1 and w2.
3. RESULTS The diagnostic performance of the described methods is assessed by means of a linear correlation analysis with maximum compressive strength (MCS) measured in biomechanical experiment and by means of a receiver operating characteristic (ROC) analysis with respect to the spine fracture status. The best correlation with MCS is showed by the global topological characteristics, obtained with the MF: Pearson’s correlation coefficient gives r=0.77 for M1 and M2, what is significantly better than r=0.68 for the BMD. All other texture measures showed a lower, but also good correlation with MCS (see Table 1). The ROC analysis for the separation of the patients with and without spine fractures shows that all numerical methods applied to the HRMR images had higher predictive power, than experimental MCS and BMD. The maximum value of the area under the curve (AUC) calculated with the FEM is 0.72, which is better than those calculated for MCS (AUC=0.66) and for BMD (AUC=0.63). The moderate value of AUC could be explained by the observation that the distribution curves of effective strain for strong and weak bones are very close to each other (solid red and dashed blue curves on Fig. 2). However, a different situation is obtained when considering topological characteristics of the trabecular bone structure. P(α) spectra of osteoporotic bones are shifted in the region of higher values of scaling indices (see Fig. 5a). This shift reflects the fact that osteoporotic bones have less trabecular structures than healthy ones. Taking the mean value of the P(α) distribution calculated from the distal radius as a texture measure for discrimination between vertebrae with and without fractures, AUC=0.79 is obtained. Further improvement up to AUC=0.86 is achieved by using texture measures calculated by means of a two sliding windows technique. When considering Minkowski Functionals as discriminative texture measures, AUC are 0.78, 0.77, 0.78 and 0.79 respectively (see Fig. 3). In order to understand the redistribution of the deformation energy accumulated inside the trabecular structure during compressive loading, we calculate the average effective strain for the voxels with the same topological values of scaling indices (see Fig. 5b). Both in healthy and osteoporotic bones maximum average effective strain is accumulated in substructures with α≈2, which correspond to the plate-like trabeculae. The amount of plates in osteoporotic bones is, however, decreased in comparison with healthy ones, which is reflected in shift of the P(α)-spectrum (see Fig. 5a) and causes decreasing of the global bone srength. Table 1. Diagnostic performance of different numerical methods.
MCS BMD FEM SIMmean SIM2win
M1
M2
M3
M4
rMCS
1
0.68
0.74
0.72
0.75
0.78
0.74
0.77
0.75
AUC
0.66
0.63
0.72
0.78
0.86
0.78
0.77
0.78
0.79
4. CONCLUSIONS Comparison on the tissue level of the effective strain calculated by the FEM and scaling indices provided by SIM, shows that the plates are the main load bearing substructure of the trabecular network both in healthy and osteoporotic bones, but the amount of plates in osteoporotic bones is decreased in comparison with healthy ones, which leads to the
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decrease of the bone strength and stability on the global level. Among the texture measures discussed in the present work the best diagnostic performance (AUC=0.86) in differentiating between patients with and without spine fractures is shown by the scaling index method (SIM), which describes the local topology of the structure. This preference of SIM in comparison with FEM and MF when implemented to the MR images follows from the two advantages of the SIM. Calculation of local indices α does not need thresholding of the image (which usually leads to over- or underestimation of trabecular structure) and is rather based on the real trabecular structure. Another problem, which arises with decreasing image resolution, is loss of trabecular connections, which leads to the decrease of stiffness and subsequent underestimation of elastic modulus and failure load in FEM. These effects are stronger for rarefied structures, which are essential the case of osteoporosis. SIM calculates local topological characteristics not from the exact number of neighbours, but from the neighbourhood with some radius and consequently does not suffer from the loss of connectivity of the object. These conclusions suggest that local topological characteristics of the distal radius described by scaling indices α reflect the general bone quality. Whereby it turns out that SIM is more robust for coarser image resolution and can be proposed for diagnosis of osteoporosis based on the distal radius HRMR images.
5. ACKNOWLEDGMENTS This study was supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant MU 2288/2-2. The authors are grateful to Andrew Burghardt (University of California, San Francisco, USA) for collaboration in benchmarking of the FEM numerical code with official software “Scanco FE Software v1.12” developed by Dr. B. van Rietbergen and provided by Scanco Medical AG, Bruettisellen, Switzerland.
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