Comparison and combination of scaling index method ...

Comparison and combination of scaling index method and Minkowski Functionals in the analysis of high resolution magnetic resonance images of the distal radius in vitro Irina N. Sidorenko*a, Jan Bauerb, Roberto Monetti a, Dirk Muellerb, Ernst J. Rummenyb, Felix Ecksteinc, Christoph W. Raeth a 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 High resolution magnetic resonance (HRMR) imaging can reveal major characteristics of trabecular bone. The quantification of this trabecular micro architecture can be useful for better understanding the progression of osteoporosis and improve its diagnosis. In the present work we applied the scaling index method (SIM) and Minkowski Functionals (MF) for analysing tomographic images of distal radius specimens in vitro. For both methods, the correlation with the maximum compressive strength (MCS) as determined in a biomechanical test and the diagnostic performance with regard to the spine fracture status were calculated. Both local SIM and global MF methods showed significantly better results compared to bone mineral density measured by quantitative computed tomography. The receiver operating characteristic analysis for differentiating fractured and non-fractured subjects revealed area under the curve (AUC) values of 0.716 for BMD, 0.897 for SIM and 0.911 for MF. The correlation coefficients with MCS were 0.6771 for BMD, 0.843 for SIM and 0.772 for MF. We simulated the effect of perturbations, namely noise effects and intensity variations. Overall, MF method was more sensitive to noise than SIM. A combination of SIM and MF methods could, however, increase AUC values from 0.85 to 0.89 and correlation coefficients from 0.71 to 0.82. In conclusion, local SIM and global MF techniques can successfully be applied for analysing HRMR image data. Since these methods are complementary, their combination offers a new possibility of describing MR images of the trabecular bone, especially noisy ones. Keywords: Osteoporosis, 3D high resolution magnetic resonance imaging, distal radius, trabecular bone structure, texture analysis, scaling index method, Minkowski Functionals

1. INTRODUCTION AND PURPOSE High resolution (HR) magnetic resonance (MR) imaging offers a possibility to reveal structural characteristics of trabecular bone, the major load bearing biological tissue of the human skeleton. The quantification of the bone architecture on micro structural level can be useful for better understanding different biological processes, such as effects of aging, drug treatment or mechanical stresses on human bones, as well as the progression and diagnosis of osteoporosis, which is nowadays of great clinical and research interest. Based on the scaling index[1-3] method (SIM) and Minkowski Functionals[4-6] (MF) non-linear numerical methods are recognized as novel techniques for analysing medical images of the trabecular micro architecture of the bone. These methods are not redundant, but provide complementary information about the trabecular net: the SIM quantifies the bone structure on a local level by discriminating between different structural elements (one dimensional rods and two dimensional plates), whereas the MF yields a global characterisation of the topology of the bone. The purposes of the present work were to apply both the SIM and MF methods to 3D HRMR images of human distal radii in vitro and to compare their performance with bone mineral density (BMD) measured by quantitative *

[email protected] Medical Imaging 2008: Image Processing, edited by Joseph M. Reinhardt, Josien P. W. Pluim, Proc. of SPIE Vol. 6914, 69144V, (2008) 1605-7422/08/$18 · doi: 10.1117/12.769395 Proc. of SPIE Vol. 6914 69144V-1

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computed tomography (QCT). Diagnostic performance of the methods was assessed by means of the receiver operating characteristic technique (ROC) with respect to spine fracture status. The structure measurements derived from the discussed methods were also compared to with the maximum compressive strength (MCS) as determined in a biomechanical test using Pearson correlation analysis. The second purpose was to investigate the performance of SIM and MF when applied to HRMR images containing noise and ramp perturbations of the HRMR images, what may appear in images obtained in vivo. Third, we propose and discuss a novel approach for the characterisation of the trabecular bone structure based on a combination of scaling indices and Minkowski Functionals. This technique is applied to artificially generated images with ramp perturbations and decreased signal to noise ratio.

2. MATERIAL AND METHODS 2.1 Data set Our study was based on a sample of 48 three-dimensional high resolution magnetic resonance image data set of distal radius in vitro with an in-plain resolution of 137 x 137 µm and 500 µm in z-direction. Among those, 23 belonged to donors with osteoporotic spine fractures. All images were obtained using a whole body 1.5T MR scanner (Gyroscan NT with Intera upgrade, Philips Medical Systems, DA Rest, Netherlands) and a quadrature wrist coil (Medical Advances, Milwaukee, WI, USA). A spin-echo pulse sequence was used in all specimens. It was a 3D fast large-angle spin-echo (FLASE) sequence with a flip angle of 140° and TE TR of 21 ms and 100 ms, respectively. This resulted in an imaging time of 21 minutes with one excitation. In addition, for all specimens bone mineral density (BMD) for a distal radius was measured by quantitative computed tomography (QCT) and maximum compressive strength (MCS) was determined in a biomechanical test simulating a fall on the outstretched forearm. Before applying SIM and MF for analysing the trabecular structure the HRMR images of distal radius were segmented using a seeded growing algorithm[7]: 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 used a normalisation procedure and applied SIM and MF to images with normalised grey level distribution, i.e. one with zero mean value and unit standard deviation.

2.2 Noise and ramp perturbations of HRMR images One of the most important requirements of HRMR image quality is a significantly high value of signal to noise ratio given by

SNR =

SIbone _ marrow SDbackground

where SIbone_marrow is the mean value of the intensity of the segmented region and SDbackground denotes the standard deviation of the background. It was shown[8] that better signal to noise ratio yields a better texture contrast. SNR strongly depends on scaning time. Shortening of the scanning time in in-vivo studies leads to a lower SNR and a worse performance of the numerical methods. In order to simulate distortions of images as they might occur in in-vivo studies, we apply two kinds of perturbation to the original set of images. To simulate noise we decrease the SNR of each image by the following factor

f noise =

SNRnew SNRold

where SNRold and SNRnew are values of signal to noise ratio of original HRMR image and perturbed one, respectively. To simulate a ramp distortion we add to a grey level a parabolic function multiplied by the factor f ramp .

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The first noticeable degradation of HRMR image starts at value f noise = 0.3 for noise perturbation and f ramp = 200 for ramp perturbation. We apply SIM, MF techniques and their combination to the original images (see Fig.1a), images with ramp perturbation up to factor f ramp = 300 (see Fig. 1b) and images with decreased SNR up to factor f noise = 0.2 from initial value (see Fig. 1c). With larger perturbation values AUC and correlation coefficient

obtained by SIM and MF methods were much smaller than that of BMD method.

a)

b)

c)

Fig. 1. Segmented region of distal radius: a) original image; b) result of ramp perturbation with noise perturbation with

f ramp =300; c) result of

f noise =0.2.

2.3 Scaling indices The scaling index method[1-3] (SIM) allows us to differentiat 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 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 fourdimensional vector pi = (x, y,z,ν (x, y,z)) and the image can now be regarded as a set of N points

r P = {p i }, 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) =

r r where dij = pi − p j

N voxel

∑e(

2 − d ij r)

j=1

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. The SIM gives an alternative representation of the HRMR images: values of α≈1, α≈2 and α≈3 correspond to rode, 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. 2). 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].

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a)

b)

c)

Fig. 2. 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 values of scaling indices can be compiled in the probability density function (pdf)

P (α ) = prob(α ∈ [α,α + ∆α ]). Analysing the P(α) spectrum of HRMR images and its statistical moments one can distinguish between healthy and osteoporotic bones (see Fig. 3a). 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 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 (black curves on Fig. 3). The presence of the perturbations changes the shape of the P(α) spectrum (see Fig. 3b, 3c) and leads to a lower the diagnostic performance of the methods. 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

∆ w1P (α ) ∆ w 2 P (α )

mP (α ) = ∆ w1P (α ) +

where ∆ w1P(α ) and ∆ w 2 P(α ) are the percentage of points in the parts of the spectrum limited by the windows w1 and w2.

2

3

S

a)

2

3

S

b)

Fig. 3. P(α) spectrum of distal radius in vitro with positions of sliding windows for structure measure image; b) ramp perturbation with

f ramp =300; c) noise perturbation with f noise =0.2.

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3

2

c)

mP(α ) for a) original

S

2.4 Minkowski Functionals The Minkowski Functionals[4-6] (MF) provide a global topological description of structural properties of multidimensional data. In this method we consider HRMR image 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

According to integral geometry we can introduce a set of global topological measures for such an object. For 3D images we obtain four functionals, which represent the volume M1, surface area M2, integral mean curvature M3 and connectivity number or Euler characteristic M4

M1 (ν ) = M 2 (ν ) = M 3 (ν ) = M 4 (ν ) =

∫ dV ,

I (ν )

∫ dS ,

∂I (ν )

∫ 1 2(1 R + 1 R )dS , 1

∂I (ν )

2

∫ (1 R R )dS .

∂I (ν )

1

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 binarised 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. 4). 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. The typical plots of the Minkowski Functionals are shown in Fig. 5.

a) νth=0

b) νth=0.5

c) νth=2

Fig. 4. Segmented region of distal radius described by voxels with normalized grey level less than threshold value νth

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Fig. 6. Combination of SIM and MF for noise perturbation with f noise =0.2. Vertical lines show the cuts for the best values of AUC and correlation coefficient. Note the different scale of the axis.

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2.5 Combination of scaling index method and Minkowski Functionals As outlined in 2.4 the MF are calculated for excursion sets, which are defined according to a threshold value of the intensity νth. In the present study we also define – to our knowledge for the first time – the excursion set with respect to the response of a local filter, namely the scaling indices α (see Fig. 6). By doing so we can determine the global morphological properties of an excursion set of voxels, which were selected according to their local geometrical properties (e.g. rod-like, sheet-like). The largest differences in the plots of Minkowski Functionals for dense and rarefied trabecular structure are expected to appear for the MF in the same threshold region as in case of SIM, i.e. for α ∈ [2; 3].

2.6 Data analysis In order to analyze and compare the methods and techniques described in the previous sections (BMD, SIM, MF and their combination) we perform a Pearson correlation analysis with the maximum compressive strength as determined in a biomechanical test and receiver operating characteristics (ROC) analysis with respect to the spine fracture status of bone donors. For a given measurement (BMD, mean value, texture measure m P (α ) or Minkowski Functionals) ROC determines the best threshold value according to the given feature (in our case spine fracture) and calculates the area under the curve (AUC) in the sensitivity versus 1-specificity plane. The parameter AUC represents the percentage of patients correctly characterized by the defined threshold value of the measurement. The value AUC=1 means that the measured parameter divides patient into two groups without any misclassification.

3. RESULTS The comparison of the diagnostic performance of the BMD, SIM and MF methods in vitro was made by means of ROC analysis in differentiating between the patients with or without spine fractures and correlation analysis with the maximum compressive strength (see Tab. 1). Taking the statistical mean value of the P(α) spectrum of scaling indices (see Fig. 3a) as a parameter for ROC analysis we could obtain AUC= 0.857, which was higher than that for BMD (AUC=0.717). Using the texture measures m P (α ) obtained from the P(α) spectrum by two sliding windows technique as described in 2.3 we could achieve an AUC=0.897 and a correlation coefficient r=0.843 (for BMD r=0.677). The highest diagnostic performance in differentiating between the two groups of patients was yielded by Minkowski Functionals M2, M3 and M4 (Fig. 5). We used a value of grey level as a parameter for the ROC analysis and obtained for the four functionals AUC = 0.829, 0.894, 0.911 and 0.903, respectively. All four MF lead to AUC and correlation coefficients higher than that of BMD. The highest correlation coefficient r=0.772 was obtained for the surface area (M2), which was higher than that for BMD, but lower than that for SIM. The effect of ramp and noise perturbations on SIM and MF are summarised in Tab. 2 and Tab. 3. In differentiating between the two groups of patients by means of ROC analysis both SIM and MF methods are rather robust against the applied perturbations. The values of AUC calculated from perturbed images decreased but always stayed over 0.78, what is higher than the AUC calculated from BMD. The correlation analysis is more sensitive to perturbations than ROC technique. Only SIM for all perturbations kept a correlation coefficient higher than that for BMD method. In the case of the noise perturbation the performance was noticeably improved by combining SIM and MF as described in 2.5 (see Fig. 6). Both AUC and correlation coefficient show a strong tendency to increasing (Tab.4). For example, in the case of noise perturbation with f noise = 0.2 for the connectivity number (M4) AUC increased from 0.854 to 0.892 and the correlation coefficient from 0.651 to 0.799. In general, for combination of SIM and MF the AUC was always higher than 0.86 and the correlation coefficient was higher than 0.75 which is better than that obtained using SIM and MF alone (Tab. 3) and than the performance of BMD.

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TABLES Table 1. AUC and correlation coefficient r for BMD, SIM and MF AUC

r

BMD SIM M1

0.717 0.897 0.829

0.677 0.843 0.7422

M2 M3 M4

0.894 0.911 0.903

0.772 0.741 0.755

Table 2. Effect of ramp perturbation on AUC (a) and correlation coefficient r (b)

f ramp =100

f ramp =300

SIM M1

0.882 0.922

0.863 0.897

M2 M3 M4

0.822 0.877 0.923

0.797 0.805 0.788

f ramp =100

f ramp =300

SIM M1

0.744 0.778

0.698 0.571

M2 M3 M4

0.670 0.493 0.726

0.474 0.246 0.559

a) AUC

b) correlation coefficient r

Table 3. Effect of noise perturbation on AUC (a) and correlation coefficient r (b)

f noise =0.3

f noise =0.2

f noise =0.3

f noise =0.2

SIM M1

0.897 0.866

0.843 0.857

SIM M1

0.831 0.722

0.703 0.673

M2 M3 M4

0.878 0.883 0.882

0.850 0.869 0.854

M2 M3 M4

0.759 0.688 0.712

0.694 0.661 0.651

a) AUC

b) correlation coefficient r

Table 4. Combination of SIM and MF for images with noise perturbation

f noise =0.3

f noise =0.2

f noise =0.3

f noise =0.2

M1

0.878

0.868

M1

0.792

0.751

M2 M3 M4

0.903 0.887 0.909

0.877 0.861 0.892

M2 M3 M4

0.809 0.813 0.821

0.785 0.758 0.799

a) AUC

b) correlation coefficient r

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4. DISCUSSION AND CONCLUSIONS In conclusion, both SIM and MF provide a good description of the trabecular bone structure, which was reflected by high values discriminative power between donors with and without spine fractures expressed by the ROC analysis as well as a good correlation with mechanical properties of the bone. Both characteristics showed a tendency to higher values of AUC and Pearson correlation coefficient than the standard clinical parameter BMD not only for original HRMR images but also for images with decreased signal to noise ration and ramp perturbation. Since the SIM and MF methods describe bone micro architecture on different scales, they are non redundant and offer the possibility to combine local and global characteristics of HRMR images. The combination of Scaling Indices and Minkowski Functionals by using SIM as a filtering procedure and calculating MF as functions of scaling indices α is – to our best knowledge – proposed for the first time to analyse MR images of trabecular micro architecture of the bone. For images with decreased signal to noise ratio it was demonstrated that such a technique increased both the AUC in a ROC analysis and Pearson correlation coefficient.

5. ACKNOWLEDGMENTS This study was supported by the Deutsche Forshungsgemeinschaft (DFG) under the grant MU 2288/2-2.

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