The 3D-based scaling index algorithm: a new structure measure to analyze trabecular bone architecture in high-resolution MR images in vivo

Abstract

Introduction

The purpose of this study was to obtain different structure measures as the three-dimensional (3D)-based scaling index method (SIM) and standard two-dimensional (2D) bone histomorphometric parameters from high-resolution (HR) magnetic resonance (MR) images of the distal radius and to compare these parameters with bone mineral density (BMD) in their diagnostic performance to differentiate postmenopausal patients with and without vertebral fractures.

Methods

Axial HR-MR images of the distal radius were obtained at 1.5 T in 40 postmenopausal women (17 with osteoporotic spine fractures and 23 controls). Trabecular microarchitecture analysis was performed using the new structure measure $m_{{P(\alpha )}}$, derived from the SIM, as well as standard morphological 2D parameters. BMD of the spine was obtained using quantitative computed tomography (QCT). Receiver operating characteristic (ROC) analyses were used to determine diagnostic performance in differentiating both groups. Results were validated by bootstrapping techniques.

Results

Significant differences between both patient groups were obtained using $m_{{P(\alpha )}}$, 2D parameters, and spine BMD (p<0.05). In comparison with the 2D texture parameters [area under the curve (AUC) up to 0.67], diagnostic performance was significantly higher for $m_{{P(\alpha )}}$(AUC=0.85; p<0.05). There was a trend for a higher AUC value for $m_{{P(\alpha )}}$compared with BMD of the spine (AUC=0.71; p=0.81).

Conclusion

$m_{{P(\alpha )}}$ yielded a robust measure of trabecular bone microarchitecture for HR-MR images of the radius, which significantly improved the diagnostic performance in differentiating postmenopausal women with and without osteoporotic spine fractures compared with standard 2D bone histomorphometric parameters. This 3D characterization of trabecular microarchitecture may provide a new approach to better assess the strength of human cancellous bone using HR-MR image data.

Appendix

Calculation of weighted scaling indices

The 3D HR-MR image data are represented as a virtual 4D point distribution, where the fourth dimension is depicted by the grey level of each point. Thus, spatial and intensity information of each voxel i is integrated in a 4D vector ${\overrightarrow{p}} _{i} = (x_{i} ,y_{i} ,z_{i} ,g_{i} )$, where x, y, and z denote the spatial variables and g the intensity of the considered voxel. Given this representation of the 3D image data, we calculated for each point with coordinate vector ${\overrightarrow{p}} _{i}$ the weighted cumulative point distribution $\rho (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle{\rightharpoonup}}$}} {p} _{i} ,r)$ using a Gaussian shaping function:

$$\rho (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _{i} ,r) = {\sum\nolimits_{j = 1}^N {e^{{ - (\frac{{d_{{ij}} }} {r})^{2} }} } },d_{{ij}} = {\left\| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _{i} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _{j} } \right\|}_{2} .$$

d _ij denotes the (Euclidean) distance between two points ${\overrightarrow{p}} _{i}$ and ${\overrightarrow{p}} _{j} .$

This formula can be interpreted in the following way: For each point ${\overrightarrow{p}} _{i}$, one counts the number of adjacent points located in the vicinity of this point. Shaping, or equivalently, the kernel function introduces a weight, with which the point ${\overrightarrow{p}} _{j}$ contributes to the total sum. Thereby, it is ensured that points with a smaller distance to ${\overrightarrow{p}} _{i}$ have a higher weight than those with higher distances.

Local scaling properties, which can be regarded as a measure for the dimensionality of the point set, are, as usual, assessed by analyzing the change in the point distribution $\rho {\left( {{\overrightarrow{p}} _{i} ,r} \right)}$ with varying scale parameter r. Therefore, the weighted scaling indices $\alpha {\left( {{\overrightarrow{p}} _{i} ,r} \right)}$ for each point are given by the logarithmic derivative of $\rho (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _{i} ,r)$ with respect to$\alpha {\left( {{\overrightarrow{p}} _{i} ,r} \right)} = \frac{{\partial \ln \rho {\left( {{\overrightarrow{p}} _{i} ,r} \right)}}}{{\partial \ln r}}$.

Using the differentiable Gaussian kernel function, we obtained an analytic expression for arbitrary values of r:

$$\alpha {\left( {{\overrightarrow{p}} _{i} ,r} \right)} = \frac{{{\sum\nolimits_{j = 1}^N {2{\left( {\frac{{d_{{ij}} }} {r}} \right)}} }^{2} e^{{ - {\left( {\frac{{d_{{ij}} }} {r}} \right)}^{2} }} }} {{{\sum\nolimits_{j = 1}^N {e^{{ - {\left( {\frac{{d_{{ij}} }} {r}} \right)}^{2} }} } }}}$$

Since the quality of the measure strongly depends on the choice in relationship to the objects of the analysis—in our case: trabeculae—it is essential to have a certain amount of initial information about the nature of the investigated point distribution with respect to the scale of the structuring elements.

Using the 4D representation of image data and exploiting local scaling properties in this embedding space, the SIM offers the possibility of taking advantage of a fully 3D analysis of bone structure. Scaling indices for the whole point set under study form the probability distribution

$$P(\alpha ) = \Pr ob(\alpha \in {\left[ {\alpha ,\alpha + d\alpha } \right]})$$

.

This representation of point distribution can be regarded as structural decomposition of the point set where points are differentiated according to local morphological features of the structure elements to which they belong to. Thus, the spectrum reveals the structural content of a point set and can consequently characterize the image data under study.