The purpose of this work was to understand how fractal dimension of two-dimensional (2D) trabecular bone projection images could be related to three-dimensional (3D) trabecular bone properties such as porosity or connectivity. Two alteration processes were applied to trabecular bone images obtained by magnetic resonance imaging: a trabeculae dilation process and a trabeculae removal process. The trabeculae dilation process was applied from the 3D skeleton graph to the 3D initial structure with constant connectivity. The trabeculae removal process was applied from the initial structure to an altered structure having 99% of porosity, in which both porosity and connectivity were modified during this second process. Gray-level projection images of each of the altered structures were simply obtained by summation of voxels, and fractal dimension (Df) was calculated. Porosity (φ) and connectivity per unit volume (Cv) were calculated from the 3D structure. Significant relationships were found between Df, φ, and Cv. Df values increased when porosity increased (dilation and removal processes) and when connectivity decreased (only removal process). These variations were in accordance with all previous clinical studies, suggesting that fractal evaluation of trabecular bone projection has real meaning in terms of porosity and connectivity of the 3D architecture. Furthermore, there was a statistically significant linear dependence between Df and Cv when φ remained constant. Porosity is directly related to bone mineral density and fractal dimension can be easily evaluated in clinical routine. These two parameters could be associated to evaluate the connectivity of the structure.
Trabecular bone microarchitecture plays a main role in osteoporosis and prediction of fractures.(1,2) In recent years, important advances have been made concerning three-dimensional (3D) imaging techniques. Automated serial sectioning,(3) quantitative computerized tomography,(4) microcomputed tomographic scanning,(5,6) X-ray tomographic microscopy,(7,8) or magnetic resonance imaging (9–11) are all recent techniques designed to obtain high-resolution 3D reconstruction of trabecular bone micro-architecture. Several image-analysis tools can characterize these 3D images and give some morphological and topological information about trabecular bone. Concerning morphological evaluation, the most current indicators are porosity, trabecular thickness, trabecular spacing,(12) and anisotropy.(13) Concerning topological evaluation, the most current indicator is the connectivity number.(14) Another technique evaluates the fractal dimension of bone-marrow surface,(15,16) giving some information on the degree of complexity or disorder of the bone microarchitecture. Three-dimensional trabecular microarchitecture characteristics are a potent source of information concerning bone strength. Several authors have reported high correlation between 3D microarchitecture characteristics and bone strength.(11,17–19) Finite element analysis was recently used to estimate mechanical properties of bone from its 3D reconstruction.(20–23)
All of these 3D techniques are well accepted. However, they actually remain in a research stage and are not used in clinical routine. Another recent trend characterizes trabecular bone microarchitecture on a two-dimensional (2D) radiographic image. This technique based on plain radiograph analysis can currently be performed in routine evaluation. It is very convenient for the patient and would be suitable for large populations. The 2D radiographic projection image appears as a gray-level texture. Several techniques for analyzing such a texture have been reported and can be classified according to the type of the 2D indicators measured. Structural measurements(24–26) characterize the distribution and shape of the radiographic patterns appearing on the texture after thresholding. Fractal analysis(16,27–29) expresses the roughness of the texture and characterizes the self-similarity of its gray-level variations over different scales.
Some interesting correlations have been reported between fractal analysis on 2D projection texture and microarchitecture parameters measured by histomorphometry(30,31), or between texture analysis and bone strength.(32–34) Furthermore, clinical studies have shown a high level of statistically significant difference between normal subjects and osteoporotic patients with fractures. Caligiuri et al.(29) calculated fractal dimension on lateral lumbar spine radiographs of 43 osteoporotic patients, with and without spine fractures, by a method based on surface area measurement. They used receiver operating characteristic (ROC) analysis to show that fractal dimension was a better discriminator than lumbar spine bone mineral density (BMD) to distinguish spine fracture cases (statistical level P < 0.01). Khosrovi et al.(35) calculated fractal dimension on radius radiographs by a method based on the Fourier transform. They found that fractal dimension was statistically different (P < 0.002) between a group of 10 osteoporotic cases (bone density below normal and/or vertebral fractures) and a group of 10 controls. Benhamou et al.(36) performed a preliminary study in osteoporosis to validate a fractal analysis based on the fractional Brownian motion.(37) They found that fractal dimension calculated on calcaneus radiographs statistically distinguished (P < 0.0001) the osteoporotic group (n = 17) from the control group (n = 12). Pothuaud et al.(38) have shown that fractal dimension calculated on calcaneus radiographs was a better discriminator than BMD (P < 0.0001) to distinguish osteoporotic patients with vertebral crush fractures (n = 39) from an age-matched control group (n = 39). Furthermore, this discrimination remained with high statistical level (P = 0.006) from osteoporotic and control subgroups with overlapping BMD values. These clinical studies have confirmed the growing interest in the 2D fractal analysis of bone radiographic texture. Nevertheless, such an image offers only 2D projection information of complex trabecular bone structure. The correspondence between gray-level variations on 2D projection texture and 3D trabecular bone microarchitecture remains unclear.
The purpose of this study was to understand how fractal dimension on 2D trabecular bone projection image could be related to 3D structural properties such as porosity or connectivity. In this aim, we used numerical algorithms to simulate trabecular bone microarchitecture changes. Such a numerical approach has already been used to find a correlation between 3D architectural parameters and bone strength,(17,19) and in a recent work(39) to characterize anisotropy relationships between plain radiographic patterns and 3D trabecular microarchitecture.
We have recently developed(40) a new technique based on 3D skeletonization and 3D skeleton graph analysis. This technique permits detection and extraction of each trabecula of the bone network, and thereby permits modification of the bone microarchitecture by trabeculae thinning or thickening, as well as by trabeculae removal. These two alteration processes were applied to 3D bone images obtained by magnetic resonance imaging (MRI). Porosity and/or connectivity were progressively modified to obtain several models of bone structure. All of these 3D models were analyzed, and porosity and connectivity were evaluated, after which they were projected onto a 2D gray-level texture, and fractal dimension was calculated.
MATERIALS AND METHODS
3D trabecular bone images
Nineteen trabecular bone samples were taken from human femoral heads after hip arthroplasty surgery, from patients with hip osteoarthritis (OA, n = 10) or patients with hip osteoporotic fracture (OP, n = 9). These samples were cut at the head-neck boundary with a cylindrical diamond core of 9 mm diameter and 10 mm height. They were cleaned with an organic solvent and saturated in water. We used MRI at a resolution of 78 μm and obtained a 3D MRI image of 128 × 128 × 128 voxels for each sample. These images involved 256 gray levels; the gray level is proportional to the quantity of protons belonging to the corresponding elementary volume in the sample. A 3D thresholding technique, based on 3D Deriche recursive filtering,(41) was developed and applied on the MRI images to obtain 3D binary images. Figure 1 shows a slice of a 3D gray-level MRI image (Fig. 1A) and the same slice obtained after thresholding (Fig. 1B). Three-dimensional binary reconstruction is shown in Fig. 2.
Running and training data
Twelve samples (6 OA and 6 OP) were randomly chosen from the total of 19 trabecular bone samples. They were called “running samples” and their analysis yielded “running data.” The other samples (4 OA and 3 OP) were called “training samples” and their analysis yielded “training data.” The study was essentially performed from running data; the training data were used only at the end of the study to validate the stability of a mathematical model established from the running data.
2D projection texture
The projection axis was arbitrarily fixed as the vertical direction (Z axis). A gray level (n) was attributed to each pixel (i, j) of the projection image. This gray level corresponded to the number of solid voxels belonging to the column of the structure stemming from this pixel:
where i, j, and k are the coordinate indices attributed, respectively, to the X, Y, and Z axes; n(i, j) is the gray level attributed to the pixel of coordinates (i, j) of the 2D projection image; and Ω(i, j, k) is the binary value (1 for solid, 0 for pore) corresponding to the voxel of coordinates (i, j, k) in the 3D binary image.
The size of the 3D image was 128 voxels in each direction, and the 2D projection image (128 × 128 pixels) was encoded with 128 gray levels.
Fractal dimension of 2D texture
We used a fractal analysis based on the fractional Brownian motion (FBM) model.(27,37) This technique estimates the Hurst exponent (H) from the maximum likelihood estimator. The estimation is performed from the fractional Gaussian noise, which is the increment of the FBM. Thirty-six directions (i) were taken and for each of them, the Hi exponent was estimated in one-dimensional signal on several gray-level line profiles. Then the 36 mean Hi values were reported on a polar diagram representation.
For each 2D projection texture, we calculated the Hmean parameter corresponding to the mean value of the 36 Hi estimated values. Fractal dimension (Df) was calculated as:
Architectural changes of 3D bone images
In a recent study,(40) we have implemented a 3D skeletonization algorithm on trabecular bone images and developed a real 3D skeleton graph analysis. This analysis permitted detection of each node and branch of the skeleton graph and extraction of each trabecula of the bone network from its corresponding skeleton branch. This 3D image processing technique was used to introduce porosity and/or connectivity changes in bone structures.
First, we used the 3D digitized skeleton graph to modify the initial 3D bone image by dilation of skeleton branches, which was randomly performed from the border points of the reconstructed branches, with respect to topology conservation, and as a consequence, with respect to connectivity conservation. Only porosity was modified during this dilation process, and 20 porosity levels (φ) were fixed between φ = φmax (corresponding to the 3D digitized skeleton graph) to φ = φmin (corresponding to the initial 3D bone structure). The dilation process was applied to the “running” 3D structures, and 240 altered structures or 3D models (including the 12 initial structures) were obtained. These 3D models were analyzed and porosity was calculated. They were then projected onto a 2D gray-level texture, and fractal dimension (Df) was calculated. Relationships between fractal dimension on 2D projection and texture and porosity of the 3D bone structure were examined.
Second, we used 3D skeleton graph analysis to progressively remove some entire trabeculae of the initial 3D bone structure. Porosity and connectivity were modified during this removal process, and 20 porosity levels (φ) were fixed between φ = φmin (corresponding to the initial 3D bone structure) to φ = φmax = 99% (corresponding to an altered structure with 99% of porosity, obtained by trabeculae removal). This removal process was applied both to “running” and “training” 3D structures, and 380 altered structures or 3D models (240 running models and 140 training models) were obtained. These 3D models were analyzed and porosity and connectivity were calculated. Connectivity was evaluated per unit volume (CV) with respect to topological considerations:(40)
V0 was the volume of the region of interest (ROI) considered in the 3D binary structure, β0 was the first Betti number (number of connected solid components), β2 the third Betti number (number of connected pore components), and N3 the Euler–Poincare number. β0 and β2 were calculated, respectively, in 26- and 6-connexity according to the Hoshen–Kopelman algorithm.(42) In order to consider the maximal connexity (26-connexity) of the 3D trabecular network Ω, N3 was calculated on the complementary porous phase Ωc in 6-connexity(43):
The field of calculation of these topological parameters was reduced to a cylindrical ROI (V0) included in the bone structure. The porosity φ was also calculated in this ROI. The volumetric connectivity (connectivity per unit volume) was normalized by the volumetric fraction of solid (1 – φ). This artificial normalization was introduced in order to amplify the sensitivity of the trabeculae removal process.
Finally, the 380 altered structures were projected onto a 2D gray-level texture, and the fractal dimension was calculated. Relationships between fractal dimension on 2D projection texture and porosity or connectivity of the 3D bone structure were examined.
Trabeculae dilation process
The trabeculae dilation process was applied to the 12 “running” 3D bone images. The relationship between Df and φ is shown in Fig. 3. Twelve curves are plotted on Fig. 3, corresponding to each 3D image. Each curve is plotted with 20 points, corresponding to the 20 levels of porosity fixed during the dilation process. Linear interpolation between these points is represented with unbroken lines. An increase in porosity was associated with an increase in Df values. The curves were not identical, suggesting that the porosity was not the unique parameter conditioning Df values. We can observe on Fig. 3 that for one porosity level we had wide variations of Df. For instance, if φ = 95%, Df values varied from 1.80 to 1.86; if φ = 90%, Df values varied from 1.68 to 1.80; and if φ = 85%, Df values varied from 1.58 to 1.74. The lower the φ values, the higher the Df variations. These curves must be interpreted according to the usual porosity values observed in our samples: range 69.5-85.5%, mean value 78.4% in osteoporotic samples, and mean value 70.4% in osteoarthritis samples.
Trabeculae removal process
The trabeculae removal process was applied to the 19 3D bone images (both “running” and “training” samples). Figure 4 plots the relationship between Df and CV which is reported in a log scale. Twelve curves (only “running” data) have been plotted with 20 points each, corresponding to the 20 levels of porosity fixed during the removal process. Linear interpolation between these points is represented with unbroken lines. An increase in connectivity was associated with a decrease in Df values. Two different evolutions of Df values appeared: a low decrease for low connectivity values, and a higher decrease for high connectivity values. These curves must be interpreted according to the usual connectivity values observed in our samples: range 0.41-1.93, mean value 0.87 in osteoporotic samples, and mean value 1.19 in osteoarthritis samples.
The relationship between Df and φ during this trabeculae removal process is plotted in Fig. 5, using plots of 12 curves (only “running” data), each of which was plotted with 20 points, corresponding to the 20 levels of porosity fixed during the removal process. Linear interpolation between these points is represented with unbroken lines. An increase in porosity was associated with an increase in Df values. We can observe that for one porosity level we had wide variation of Df. For instance, if φ = 95%, Df values varied from 1.61 to 1.76; if φ = 90%, Df values varied from 1.57 to 1.71; and if φ = 85%, Df values varied from 1.53 to 1.64. These variations of Df values seem more regular than those observed in Fig. 3 for the variations of Df with porosity, with respect to the trabecular dilation process.
Figure 6A reports the relationships between Df and CV values plotted for different fixed porosity levels (φ = 95 to 75% by 5% increments). For each of these porosity levels (φ), only a part of the 240 “running” models was used, representing the 3D models corresponding to the figure of porosity: 12 models at φ = 95%; 11 models at φ = 90%; 10 models at φ = 85%; 10 models at φ = 80%; and 8 models at φ = 75%. Linear regressions were calculated from this 51 model data set (obtained from a part of the “running” data). For φ = 75%, the slope value of the linear regression was 0.087 with a correlation coefficient R2 = 0.82 (P = 0.0008); for φ = 80%, the slope value was 0.118 with R2 = 0.87 (P < 0.0001); for φ = 85%, the slope value was 0.246 with R2 = 0.87 (P < 0.0001); and for φ = 90%, the slope value was 0.447 with R2 = 0.70 (P = 0.0012). The higher the φ value, the higher the slope value of the linear dependence between Dfand connectivity. For φ = 95%, the slope value was 4.541, but with a low statistical accuracy (R2 = 0.36, P = 0.0390). The points (CV, Df) corresponding to the 19 initial structures are plotted on Fig. 6A: circle points for osteoarthritis samples and triangle points for osteoporotic samples.
The regression lines (Φ = 75, 80, 85, and 90%, respectively) were intersected at different points, although these intersection points were very close to one another (Fig. 6B). Hence, we have considered a theoretical intersection point, defined as the gravity center of all real intersection points between two lines. The coordinates of this gravity center were (CV0 = −0.439, Df0 = 1.403). An axis change was performed such that (CV0, Df0) was the origin of the new axis. Regression lines passing through the origin were calculated in the new axis (Fig. 6C). For φ = 75%, the slope value was 0.084 with a correlation coefficient R2 = 0.82 (P = 0.0008); for φ = 80%, the slope value was 0.125 with R2 = 0.86 (P < 0.0001); for φ = 85%, the slope value was 0.228 with R2 = 0.86 (P < 0.0001); and for φ = 90%, the slope value was 0.414 with R2 = 0.70 (P = 0.0006). The slopes of the regression lines were reported in a same plot (Fig. 6D) and an exponential regression model was obtained with a high correlation coefficient (R2 = 0.992).
From this 51 “running” model data set (altered structures), a mathematical model linking porosity, fractal dimension, and connectivity was established:
where CV and Df are, respectively, the 3D connectivity and the fractal dimension of 2D projection texture; (CV0, Df0) is the origin of the new axis reported below; k is the slope of the regression lines of constant porosity (in the new axis); and α and β are two coefficients defined in the exponential relation between the slope k and the porosity (φ). The first equation was directly deduced from Fig. 6C, whereas the second equation was deduced from Fig. 6D.
Using numerical values, this model can be written as follows:
This numerical relationship between fractal dimension, porosity, and connectivity was obtained from 51 “running” model data set; it was tested on the 240 “running” data set stemming from the 12 “running” samples. Statistical results expressed as mean relative error, standard deviation, correlation coefficient, and statistical significance were reported in Table 1. Mean relative error was 1.02% for the first 51 “running” data set (R2 = 0.89, P < 0.0001) and 1.84% for the other 189 “running” data set (R2 = 0.86, P < 0.0001). These first results showed the continuity of the mathematical relationship on the “running” data stemming from 12 samples. Then, the same relationship was tested on 140 “training” data set stemming from the 7 “training” samples (Table 1). Mean relative error was 1.52% (R2 = 0.87, P < 0.0001). These second results showed the stability of the mathematical relationship with other samples.
In this study, we have used numerical simulation to modify 3D trabecular bone structure and to project each of these modified structures, the projections appearing as 2D gray-level texture images. Fractal dimension of projection textures was calculated and significant relationships were found between 2D fractal dimension and 3D porosity or 3D connectivity. Furthermore, linear correlations were established between 2D fractal dimension and 3D connectivity when 3D porosity was maintained constant. A mathematical model was deduced, expressing fractal dimension as a function of porosity and connectivity.
The changes introduced in our simulation process concerned random trabeculae thickening and random trabeculae removal. These microarchitecture changes were not perfectly representative of the real changes appearing in bone structure in vivo. However, such random simulated changes were often used particularly in the case of simulation of the bone remodeling process.(44–46) The thickening process could be obtained uniformly, although this method would permit only three or four steps of dilation of the trabeculae. On the contrary, a randomly distributed process of dilation permitted obtaining more progressive variations of the structure, with a large range of porosity changes, and consequently a higher accuracy of fractal dimension evolution in 2D projection.
Table Table 1.. Relative Error of Mathematical Model Linking Fractal Dimension of 2D Projection Texture, Porosity, and Connectivity of 3D Structure
Running data (n = 51)
Running data (n = 189)
Training data (n = 140)
The mathematical model was established from 51 “running” data. It was tested by statistical comparison between real and estimate values of fractal dimension. This test was performed with the initial 51 “running” data, with the other 189 “running,” and finally with 140 “training” data. The “running” data (240) were stemmed from 12 samples, whereas the “training” data (140) were stemmed from 7 other samples. Each data (or 3D structure) corresponded to a 3D model obtained following a trabeculae removal process. Mean, minimal, and maximal relative error values were calculated from the relative errors between real and estimate fractal dimension values. R2 and P values were obtained from a Z-test of correlation.
Mean relative error (%)
Standard deviation (%)
Minimal relative error (%)
Maximal relative error (%)
Correlation coefficient (R2)
Statistical significance (P)
We used a projection-simulating process based solely on the quantity of bone appearing in a column of structure. The real radiographic process involves other physical phenomena,(39,47,48) such as exponential attenuation or nonlinear characteristics of the radiographic film (optical density, modulation transfer function, granularity). Nevertheless, the contrast of a real bone radiographic image is determined primarily by the quantity of bone appearing in a column of structure studied in X-ray projection. Our simplified projection process can be considered as the first approach of the real radiographic projection process,(39) and if no correlation could be detected at this first stage, no correlation could be explained at the final stage on the real bone radiographic 2D image.
All of our in vitro and in vivo studies have shown an increase of Df values measured on radiographic images in the case of osteoporosis,(36,38) bone strength loss,(34) and, more recently, aging and menopause. However, this increase of Df remained unexplained. Our hypothesis was that an increase of fractal dimension could be related to a disorganization of the trabecular bone microarchitecture and, as a consequence, to a loss of connectivity.(36,38) In this study, we have shown that an increase of Df values was really related to a loss of connectivity. Furthermore, with the porosity and the connectivity evolving together, an increase of Df could be considered to reflect porosity decrease only and, as a consequence, Df would give no more information than bone density; yet, this was not the case. When porosity was maintained constant, the Df values remained related to connectivity (Fig. 6a). There was a linear dependence between fractal dimension and connectivity, and the higher the fixed value of porosity, the higher the slope value of this linear dependence.
There was a differential evolution of Df values with both porosity (φ) and connectivity (CV). First, there was a global increase of Df values when connectivity decreased (Fig. 4) and porosity increased (Fig. 5). In other words, the fractal dimension measured on projection texture increased when the 3D structure was altered. Second, when φ was maintained constant, a decrease of connectivity was linked to a decrease of fractal dimension (Fig. 6A). This second evolution of Df values was in opposition with the usual changes observed on projection images, and was difficult to explain in terms of 3D/2D correlation.
Df values were linked to porosity and connectivity. From our data, we can express a mathematical relationship between connectivity, fractal dimension, and porosity [Eq. (5)]. This mathematical model obtained from 51 “running” model data set (R2 = 0.86, P < 0.0001) remained with high statistical accuracy, both with the other 189 “running” data set (R2 = 0.86, P < 0.0001) and with 140 “training” data set (R2 = 0.87, P < 0.0001). The first statistical accuracy obtained from “training” data show both the validity of the mathematical model (with 51 “training” data set) and its continuity in the initial data (240 “training” data set stemming from 12 samples). Furthermore, the second statistical accuracy (140 “training” data set stemming from 7 samples) show the stability of the mathematical model with other samples. Such a mathematical model seems very interesting in terms of clinical application. Porosity is directly linked to bone mineral density, which is evaluated by double-energy X-ray absorptiometry (DEXA). Fractal dimension could be easily evaluated by radiographic image analysis. These two parameters, simply evaluated in clinical routine, could be associated to obtain a third characteristic, the connectivity. The connectivity measures the number of connection loops appearing in the 3D structure and, as a consequence, gives indications on the trabecular network. Hence, the connectivity being linked to bone strength,(19,49) its in vivo evaluation could be very efficacious in terms of diagnosis of bone diseases.
The three main components of trabecular microarchitecture characterization are porosity, connectivity, and anisotropy. Our study did not involve anisotropy evaluation, although a recent work has specifically addressed this consideration,(39) using the same approach: characterization of the 3D structure by micro-CT on one calcaneus sample, morphological operations of erosion dilation leading to 15 models, then simulation of radiographic projection. The 3D and 2D images have been studied by the mean intercept length (MIL) method. The MIL anisotropy values in 2D and 3D were strongly correlated with a linear correlation parameter (R2 = 0.98). In parallel to our study, this work brings complementary information. Both studies suggest that trabecular bone microarchitecture can be adequately assessed by plain radiographs either for porosity, connectivity, or anisotropy. These three parameters conditioning bone strength,(19,49,50) analysis of texture on radiographic images could prove very useful estimator of bone status in humans. This method could improve the detection of patients at risk for osteoporotic fractures.
We thank the Orthopeadic Surgery Department of the Hospital of Orleans (France) for the femoral samples. The numerical calculations were carried out on a workstation financed by the Regional Council of Region Centre (France), to which we are thankful. We acknowledge Hologic Europe Company for its highly appreciated help.