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Keywords:

  • 3D skeleton graph;
  • 3D skeleton graph analysis;
  • trabecula analysis;
  • trabecular bone architecture;
  • Euler–Poincaré characteristic

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

This paper introduces a new three-dimensional analysis of complex disordered porous media. Skeleton graph analysis is described and applied to trabecular bone images obtained by high resolution magnetic resonance imaging. This technique was developed bearing in mind topological considerations. The correspondence between vertices and branches of the skeleton graph and trabeculae is used in order to get local information on trabecular bone microarchitecture. In addition to real topological parameters, local structural information about trabeculae, such as length and volume distributions, are obtained. This method is applied to two sets of samples: six osteoporosis and six osteoarthritis bone samples. We demonstrate that skeleton graph analysis is a powerful technique to describe trabecular bone microarchitecture.


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

In several fields of science and technology, the objects under investigation can be represented as porous media. The solid and void phases of such media determine their physical properties. For example, trabecular bone may be considered as a porous medium with low solid fraction (0.05–0.25). Note that the trabecular microarchitecture of the bone solid phase is very important in terms of bone mechanical strength. Several imaging techniques are available to obtain three-dimensional (3D) reconstruction of porous media (Kinney et al., 1995; Majumdar & Genant, 1995; Ruegsegger et al., 1996; Peyrin et al., 1998). Image analysis techniques are used in order to obtain morphological (Cruz-Orive, 1979; Levitz & Tchoubar, 1992; Guilak, 1994) and topological (DeHoff et al., 1972; MacDonald et al., 1986; Odgaard & Gundersen, 1993) characterizations. As a general rule, these techniques are applied in a global way and give some mean information. These usual global techniques are interesting but do not capture all the complex microarchitecture properties of disordered porous media.

The morphological techniques are generally based on a physical model (Whitehouse, 1974a; Parfitt et al., 1983; Kragstrup, 1985). Consequently, if the structure differs from this assumed model, the morphological parameters will be biased to an unknown extent (Hildebrand & Ruegsegger, 1997). The mean size evaluation (Cruz-Orive, 1979; Levitz & Tchoubar, 1992) takes into account only a size averaging of various shapes of elements. For example the mean trabecular size evaluation (mean intercept length) does not take into account the real complex shape of the trabeculae. The requirement of independent trabecular size estimation free of any assumption was reported by Kinney et al. (1995) and by Hildebrand & Ruegsegger (1997). Concerning topological characterization, a technique based on 2D skeletonization has been developed (Garrahan et al., 1986; Compston et al., 1987, 1989). The results are expressed as a set of ratios between nodes, termini and loops of the 2D skeleton graph. Nevertheless, these 2D topological parameters must be cautiously interpreted because on a theoretical ground, their topological signification is not valid for the complete 3D object (DeHoff, 1983; Gundersen et al., 1993). Some works have been reported concerning the estimation of the connectivity from serial sections (DeHoff et al., 1972; MacDonald et al., 1986; Gundersen et al., 1993). However, these approaches are relatively complex and involve more efforts than the direct 3D evaluation. In fact, the connectivity can be directly evaluated in 3D from the Euler–Poincaré characteristic (Odgaard & Gundersen, 1993; Vogel, 1997). Nevertheless, a global connectivity index does not capture all the topological properties as the numbers of connection nodes and their relative positions.

3D skeleton graph analysis is a powerful technique to describe architecture of porous media. For example, this approach has been used to characterize granular porous media in the case of sedimentary rock (Lin & Cohen, 1982). Each vertex of this graph represented a grain and each branch represented a contact between grains. Several parameters such as size distributions, aspect ratio distributions or coordination number distributions were calculated from the skeleton graph. In the same way, 3D skeleton graph analysis can be applied to trabecular bone microarchitecture. The analysis of the 3D skeleton graph of trabecular bone microarchitecture allows us to obtain real 3D topological parameters and to extract each trabecula of the bone network. This extraction permits a morphological characterization of each trabecula independently of any model.

The purpose of this study was to develop a 3D skeleton graph analysis of trabecular bone microarchitecture, in order to obtain a local characterization of the trabecular bone network. Bone microarchitecture was imaged by magnetic resonance imaging (MRI) (Section 2.1). A 3D skeletonization technique is reported in Section 2.2. Section 2.3 presents a new 3D skeleton graph analysis taking into consideration several artefacts due to the digitization process. We have introduced the notion of pixels clustering and redefined the notion of pixels neighbourhood in order to eliminate topological artefacts. Taking into consideration these two concepts, we have used a graph point classification to count and localize the vertices and branches of the skeleton graph. Our approach was entirely based on the topology preservation between initial structure and its skeleton graph. A topological equation based on the three Betti numbers and Euler–Poincaré characteristic is used to check topology conservation (Section 2.4). The numbers of vertices and branches are expressed by unit volume. Each trabecula is extracted from its associated skeleton branch (Section 2.5). Local morphological parameters such as length and volume of each trabecula are evaluated (Section 2.6), as well as global morphological parameters (Section 2.7). In the last part, we used our new graph analysis technique to analyse 12 trabecular bone samples (six osteoporosis and six osteoarthritis) (Section 3).

2. Material and methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

2.1. Trabecular bone images

2.1.1. Bone samples

Twelve trabecular bone samples were extracted from human femoral heads after hip arthroplasty. Six of them were obtained from patients with osteoporotic hip fracture (OP) and six from patients with hip osteoarthritis (OA). The samples were taken at the femoral head–neck boundary by drilling a cylindrical core of 0.9 cm diameter in a standardized region of interest. Then, the height of the samples was adjusted to 1.0 cm by a turning saw. Bone marrow was removed with an organic solvent and the samples were saturated with H2O in a cylindrical tube.

2.1.2. Three-dimensional images obtained by magnetic resonance imaging

The samples were imaged by MRI with a DSX 100 Bruker spectrometer (2.4 T). A field of view of 1.0 × 1.0 × 1.0 cm was digitized on a cubic grid of size 128 × 128 × 128 pixels with 256 grey levels and a resolution of 78 µm pixel−1. The grey level of one pixel was proportional to the number of protons belonging to the corresponding elementary cubic volume of the sample. Consequently, the water filling the void space appeared as high grey levels (light), while the solid phase (without protons) appeared as low grey levels (dark) (Fig. 1a). A global 3D segmentation technique based on the grey level gradients was applied and each pixel was allocated either to solid phase X (white) or to void phase Xc (black) (Fig. 1b).

image

Figure 1. A two-dimensional slice of 3D trabecular bone sample obtained by magnetic resonance imaging. (a) Grey level image: high levels (white) correspond to the pore phase of the structure, while low levels (black) were related to the solid phase; (b) binary image obtained after a global 3D segmentation technique.

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The cubic grid fixed by the MRI technique introduces a loss of symmetry between the two complementary phases X and Xc. Each pixel is surrounded by 26 neighbours at different distances: six neighbours six-connected by a face, 12 neighbours 12-connected by an edge and eight neighbours eight-connected by a corner. In order to take into account the highest degree of connection, the solid phase X was analysed with 26-connection (each pixel of X had 26 neighbours having a face, an edge or a corner in common). Consequently, the void phase Xc was analysed with six-connection (each pixel of Xc had six neighbours having a face in common).

2.2. Three-dimensional skeletonization

2.2.1. Skeleton graph

The skeleton graph (G) is defined as the set of the centres of all maximal spheres included in the solid phase X. This skeleton graph can be approached by an iterative thinning of X. This thinning skeletonization must satisfy both topological and geometrical constraints in order to fit as exactly as possible the skeleton graph.

2.2.2. Topological constraints

A pixel p of the solid phase X is called a border pixel if at least one of its six-connected neighbour belongs to the void phase Xc. The thinning skeletonization consists of suppressing border pixels p of X (p belonging to the solid phase X is classified as a point belonging to the complementary phase Xc) with the constraint that these suppressions do not change the topological properties of the thinned solid phase (X′), defined as: X′ = X − {p}. A border pixel p can be suppressed if and only if it satisfies topological constraints. In our study, these constraints were expressed as a local conservation of the Betti numbers (Barrett & Yust, 1970): β0 is the number of connected parts, β1 is the number of loops and β2 is the number of internal surfaces. If Ω(p) is a field of analysis of size 3 × 3 × 3 pixels and centred on the pixel p, we defined the local solid phase XΩ(p) = X∩Ω(p) and the local complementary void phase Xc/Ω(p)= Xc∩Ω(p). If p is a border pixel, it was suppressible if and only if it verified the following equations:

  • image(1)
  • image(2)
  • image(3)

X′Ω(p) was the local solid phase where the border pixel p was suppressed.

β0 and β2 were calculated following the Hoshen–Kopelman clustering algorithm (Hoshen & Kopelman, 1976). This algorithm was applied to the local solid phase XΩ(p) with 26-connection for β0 evaluation, and to the local complementary void phase Xc/Ω(p) with six-connection for β2 evaluation. β1 cannot be directly evaluated. Nonetheless, the three Betti numbers are related to the Euler–Poincaré characteristic (N3) by the relation:

  • image(4)

Hence, if Eqs. (1) and (3) are fulfilled, Eq. (2) can be replaced by:

  • image(5)

N3 was evaluated following a six-connection algorithm (Vogel, 1997) applied to the local complementary solid phase. Then, Eq. (5) was replaced by:

  • image(6)

The evaluation of N3 characteristic in the local phase limited to the field of analysis Ω(p) was performed taking into account the boundary extension of the local phase XΩ(p). Hence, N3 was really evaluated on a 4 × 4 × 4 pixels field of analysis.

2.2.3. Geometrical constraints

The final solid phase G stemmed from the thinning skeletonization must be centred in the initial solid phase X. Consequently, each iteration of the thinning algorithm was decomposed into six steps, each of them corresponding to one of the six main directions (d): up, down, north, south, east, west. A pixel p of the solid phase X is called a d-border pixel if its six-connected neighbours in the d-direction belongs to the void phase Xc. Only the d-border pixels were considered during the d-step of each iteration. The thinned solid phase G must respect the elongation of the initial solid phase X. A pixel p of the solid phase X is called an end-line pixel if it has no more than one 26-connected neighbour. In order to satisfy the elongation constraint, the end-line pixels were not considered during the thinning skeletonization.

2.2.4. Thinning algorithm

A pixel p of the solid phase X is a d-suppressible pixel if it is a d-border pixel and if it is not an end-line pixel and if it satisfied Eqs (1), (3) and (6). During the d-step of one iteration, the whole pixels of the solid phase X were examined. A pixel identified as a d-suppressible pixel was immediately suppressed. This sequential implementation of the thinning skeletonization was necessary to preserve the topology (Tsao & Fu, 1981; Ma, 1994; Saha et al., 1996; Palagyi & Kuba, 1998). During each iteration, the six directions (d) were sequentially considered following the sequence {up, north, east, down, south, west}. The skeleton graph G was obtained when no more remaining pixels of the solid phase can be suppressed.

2.3. Three-dimensional skeleton graph analysis

2.3.1. Interpretation of the skeleton graph in terms of vertices and branches

Our objective is to characterize the skeleton graph G from its vertices (V) and branches (B). Nevertheless, in the case of a cubic grid representation, the digital skeleton graph must be considered as an arrangement of cubic pixels. The connection path between neighbouring pixels must be considered in 26-connection, and some pixel arrangements are irreducible in term of thinning operation. These irreducible pixel arrangements disturb the interpretation of the skeleton graph in terms of vertices and branches. The following approach consists of detecting the irreducible pixels arrangements with the help of some rules of pixels clustering. Two concepts were introduced to define these rules: artificial loops, and order relationship between six-, 12- and eight-connection.

2.3.2. Artificial loops

Figure 2(a) shows particular pixel arrangements with the presence of an artificial loop. This artificial loop has no topological meaning but appeared in the connection path between pixels. It was due to the thinning operation, for which one point of the skeleton graph appeared as several pixels. An artificial loop was searched in the 3 × 3 × 3 neighbourhood of one pixel. For each pixel, we have constructed the adjacency graph and detected all loops. When a loop is detected all the pixels belonging to this loop are clustered at one single point. Nevertheless, a loop must be extended outside the 3 × 3 × 3 neighbourhood (Fig. 2b). In the same way, all pixels belonging to loops and interlocking loops were clustered. Each cluster of pixels was considered as one single point of the skeleton graph; the neighbours of this point in particular are searched outside the corresponding cluster of pixels.

image

Figure 2. Particular configurations of pixels of the digitized skeleton graph: (a) artificial connected loops between neighbouring pixels; (b) interlocking artificial loops between several neighbourhoods of pixels. These artificial loops were related to the digitization process leading to some irreducible points of the skeleton graph. These artificial loops introduced topological artefacts.

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2.3.3. Order relationship between six-, 12- and eight-connection

For one point of the skeleton graph (one single pixel or a cluster of several pixels), we have considered its neighbourhood in a 26-connection examination. However, it was necessary to introduce an order relationship between the six-, 12- and eight-connection. The six-connection (by the faces of cubic pixels) is higher than the 12-connection (by the edges of cubic pixels), itself higher than the eight-connection (by the corners of cubic pixels). Two pixels are neighbouring if their direct connection is not disconnected by another connection path with higher level of connection. For example in Fig. 3(a) the pixels x and y were 12-connected. However, they are both six-connected to the pixel z. So, the path of higher connection connecting the pixels x and y is not the x/y path but the x/z/y path. The pixels x and y are not neighbouring. We formulate this concept in more general cases by the following proposals:

image

Figure 3. Relationship between six-, 12- and eight-connectivity. (a, b) Particular skeleton graph point neighbourhoods with a 12-connection annihilated by a path of six-connection; (c) particular graph points neighbourhood with an eight-connection annihilated by a path of six-connection and/or 12-connection. The number/label of each pixel indicates the number of neighbouring pixels taking into account the relationship between six-, 12- and eight-connection.

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• when two pixels x and y are six-connected, they are neighbouring, the six-connection being the higher level of connection

• when two pixels x and y are 12-connected they can have in their neighbourhood one or more six-connected neighbours (Fig. 3a). In this case, the two pixels x and y are not neighbouring

• when two pixels x and y are eight-connected, they can have in their neighbourhood one or more pixels six-connected to x and 12-connected to y (Fig. 3b,c), or 12-connected to x and six-connected to y (Fig. 3d). In these cases the pixels x and y are not neighbouring.

2.3.4. Graph point classes

The skeleton graph G was organized into single points (one pixel) and multiple points (cluster of several pixels). A graph point is classified as a vertex or a point belonging to a branch from its number of neighbours (single and/or multiple points) and from the neighbourhood configuration of these neighbours. Consequently, we define several classes of points. The following definitions are illustrated in Fig. 4. A node vertex is a point that has more than two neighbours, while an end vertex is a point that has only one neighbour. The branches are defined as a succession of current points (points with two neighbours) linking two node vertices together or one node vertex and one end vertex. A begin point is a current point for which one and only one of its two neighbours is a node vertex. A final point is topologically equivalent to a begin point plus an end vertex. A connect point is a point that has only two neighbours which are two node vertices. A connection between two neighbouring node vertices is defined as a particular branch. This particular branch has a nil length, but it is taken into account for the branches counting. Finally, we consider the isolate points that are topologically equivalent to two end vertices plus a branch. With all of these definitions the numbers of vertices (V) and branches (B) are expressed as:

image

Figure 4. Particular skeleton graph points configurations. The number/label of each point indicates the number of neighbouring pixels taking into account the relationship between six-, 12- and eight-connection. Each point of the skeleton graph was classified following its number of neighbours and its neighbourhood configuration. The node points have more than two neighbours; the end points has only one neighbour; the current points have two neighbours, belonging to a branch; the begin points are current points of which one and only one of the two neighbours is a node point. The final points have only a node point as a neighbour; the connect points have only two node points as neighbours; the particular branches appear between two neighbouring node points; the isolate points has no neighbour.

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  • image(7)

We distinguish the connection vertices (Vc) and the termini vertices (Vt), as well as the connection branches (Bcc) linking two connection vertices and the termini branches (Bct) linking one connection vertex and one termini vertex. The numbers Vc, Vt, Bcc and Bct are expressed as:

  • image(8)
2.3.5. Graph point classification

Let be p a point (simple or multiple) of the skeleton graph, n the number of its neighbours (simple and/or multiple), and neighi the number of neighbours (simple and/or multiple) of the ith neighbour of p. If n = 0, then p is an isolate point. If n = 1, p is a final point if neigh1 ≥ 3, or an end vertex in the other cases. If n = 2, p is a connect point if neigh1 ≥ 3 and neigh2 ≥ 3, p is a begin point if one of its two neighbours has more or exactly three neighbours and if the second neighbour of p has exactly one or two neighbours, p is a current point in the other cases. If n ≥ 3 then p is a node vertex. In this case we examined its neighbours to count the particular branch.

2.4. Topological validation of the interpretation of the skeleton graph in terms of vertices and branches

2.4.1. Euler–Poincaré characteristic of skeleton graph

The Euler–Poincaré characteristic of the skeleton graph, N3(G), is evaluated following a six-connection algorithm (Vogel, 1997) applied to the complementary set Gc (N3(G) = N3(Gc)) In our particular study concerning trabecular bone samples, the initial solid phase (X) as well as its skeleton graph (G) has only one solid component (β0(X) =β0(G) = 1) and no internal surface (β2(X) = β2(G) = 0). From a continuous representation, the theoretical skeleton graph is organized only into single points, and its interpretation in terms of vertices (V) and branches (B) is justified. The Euler–Poincaré characteristic is theoretically expressed as:

  • image(9)

Nevertheless, the analysis of the skeleton graph through a cubic grid induces irreducible arrangements of pixels. Hence, the digital skeleton graph is organized into both single and multiple points. Its interpretation in terms of vertices (V′) and branches (B′) is constrained by some rules of clustering. The estimate value of the Euler–Poincaré characteristic is defined as:

  • image(10)

where, V′ and B′ are calculated from Eq. (7).

2.4.2. Validity criterion of the skeleton graph analysis

The validity of the interpretation of the skeleton graph in terms of vertices (V′) and branches (B′) is characterized by the relative error ε(N3) between the real value N3(G) directly calculated from Vogel's algorithm, and the estimate value N3′(G) evaluated from the number of vertices (V′) and branches (B′):

  • image(11)

2.5. Trabeculae extraction

2.5.1. Numbering of the skeleton graph branches

The vertices (connections and termini) are sequentially considered. Each vertex linked a number of branches equal to the number of its neighbours. Each of these neighbours is considered as the beginning of a new branch, this branch being covered following a neighbouring analysis. All the points (single and/or multiple) belonging to the same branch are numbered with the same index. When all the branches of the skeleton graph are covered, the set of the points (single and/or multiple) having the same index correspond to the same branch.

2.5.2. Deletion of the nil length skeleton graph branches

The final points and the particular branches are defined during the topological analysis of the skeleton graph. However, these definitions introduce some nil length branches that have no morphological meaning. Consequently, these nil length branches are deleted before the trabeculae extraction.

The particular branches are fictitious and their deletion introduced no artefact concerning trabecula extraction. On the contrary the deletion of a final point prescribes us to reconsider the node vertex to which the final point is linked (Fig. 4). If this node vertex has more than three neighbours, it remains a node vertex (more than two neighbours) after deletion of the final point. If it has exactly three neighbours, it becomes a current point (only two neighbours) after deletion of the final point. In this last case, the two previous branches linked to the new current point become the same branch (with the same index).

2.5.3. Deletion of the shortest termini branches

The termini branches mainly appear at the boundary of the sample due to the experimental extraction of the sample from the bone. However, some termini branches are localized inside the sample due to surface irregularities of the trabeculae. These internal termini branches are generally shorter than the connection branches and have no interest for trabecula extraction.

A length criterion is defined from the mean intercept length of the solid phase (lS) (Section 2.7). All the termini branches which are shorter than two times lS are deleted. In the same way as in Section 2.5.2, the deletion of a termini branch is related to a reanalysis of the status of the node vertex to which the termini branch is linked. The termini branches remaining after this deletion are considered as real termini branches.

2.5.4. Conditional thickening of the skeleton graph branches

Each pixel of the thinned solid phase G has an index which is the index of the branch at which the pixel is linked. The trabeculae extraction consists of giving to all pixels of the initial solid phase X an index equal to the index of the nearest pixel belonging to G. This operation can be understood as a thickening of the skeleton graph under the condition that all added pixels belong to the initial phase X.

2.6. Morphological analysis of trabeculae

2.6.1. Trabecular length

The length (L) of a trabecula is defined as the length of the branch from which the trabecula is extracted. For a single vertex (one pixel) the spatial position is directly evaluated from the coordinates of the pixel, whereas for a multiple vertex (cluster of several pixels) the spatial position is evaluated from the coordinates of the centre of gravity of all clustered pixels. If V1 and V2 are the two vertices linked to this branch, L is the distance between the spatial positions of V1 and V2:

  • image(12)

δ(V1, V2) is the distance between V1 and V2, δ the spatial resolution (δ = 0.078 mm). Lcc is the length of a connection trabecula (extracted from a connection branch), Lct is the length of a termini trabecula (extracted from a termini branch).

2.6.2. Trabecular volume

The volume (V) of a trabecula is directly evaluated from pixel counting. If np is the number of pixels belonging to the trabecula, the volume V is expressed as:

  • image(13)

Vcc is the volume of a connection trabecula, Vct is the volume of a termini trabecula.

2.7. Global morphological analysis

The solid fraction (fS) is evaluated from the maximal cylindrical volume (V0) that can be located in the entire solid phase X where fS remained stable, fS being the number of pixels belonging to the volume V0 and filling over the solid phase divided by the total number of pixels belonging to the volume V0. Knowing the total number of solid pixels (nS) of the entire solid phase X and the solid fraction (fS) estimated on the reduced cylindrical volume V0, we can deduce the total volume of analysis (V) corresponding to the entire solid phase X:

  • image(14)

Mean intercept lengths of each phase (ls for the solid phase, lp for the void phase) are calculated following a 3D version of the directed secant algorithm (Goulet et al., 1994; Hipp et al., 1996).

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

3.1. Three-dimensional skeleton graph

3D thinning skeletonization was applied to 12 trabecular bone images. Figure 5 shows a 3D visualization of a 3D MRI image (X) (Fig. 5a) and its associated 3D skeleton graph (G) (Fig. 5b). The topological equivalence between X and G was checked for the 12 images, with β0(X) = β0(G) = 1 (only one connected part of solid), β2(X) = β2(G) = 0 (no internal surface) and N3(X) = N3(G) (the values of N3(G) are reported in Table 1. In this particular case (β0 = 1 and β2 = 0), the number of loops (β1) of the solid phase was directly related to N3: β1 = 1 − N3 (Eq. 4).

image

Figure 5. Three-dimensional visualization of a trabecular bone binary image obtained by magnetic resonance imaging (a), and its corresponding 3D skeleton graph (b).

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Table 1.   Estimation of the Euler–Poincaré characteristic (N3′) from 3D skeleton graph analysis (vertices V′ and branch B′ counting) for 12 trabecular bone samples. Samples 1–6 were taken from osteoarthritic subjects (OA), whereas samples 7–12 were taken from osteoporotic subjects (OP). N3 was the Euler–Poincaré characteristic evaluated from Vogel's algorithm. ε(N3) was the relative error between N3 and N3′, expressed in percent (ε(N3) was considered as a validity criterion of the 3D skeleton graph analysis). ε1(N3) was the relative error calculated without clustering pixels and without taking into consideration the order relationship between six-, 12- and eight-connectivity (connexity priority). ε2(N3) was the relative error calculated without clustering pixels and taking into consideration the order relationship between six-, 12- and eight-connectivity.
 3D skeleton graph analysis with connection priority and pixel clustering  Relative error (%)
Bone   Initial 3DWith priority/Without priority/Only with
samplesV′B′N3image N3cluster ε(N3)cluster ε1N3)priority ε2(N3)
1 OA1034012273−1933−19360.15 606 358
2/OA1124912705−1456−14630.481026 677
3/OA1554118470−2929−29611.08 609 366
4/OA 957310532 −959−9630.421085 664
5/OA1003710900 −863−8670.461237 772
6/OA1378015833−2053−20680.73 897 592
7/OP1071312452−1739−17430.23 813 524
8/OP 59086522 −614−6140.0017771383
9/OP1164613389−1743−17430.00 737 427
10/OP 62087924−1716−17200.23 447 273
11/OP1618919149−2960−29860.87 559 302
12/OP 8727 9903−1176−11800.34 818 488
  (12504.3)Mean relative error0.42884 569

3.2. Three-dimensional skeleton graph analysis

The values of V′, B′ and N3′(G) are reported in Table 1 for the 12 skeleton graphs. N3′(G) was estimated from the number of vertices (V′) and branches (B′) (Eq. 10), while N3(G) was directly evaluated from Vogel's algorithm. The relative error ε(N3) between N3(G) and N3′(G) was expressed in percent. ε1(N3) was the relative error calculated without clustering of pixels and without taking into consideration the order relationship between six-, 12- and eight-connection. ε2(N3) was the relative error calculated without clustering of pixels and taking into consideration the order relationship between six-, 12- and eight-connection. The use of this order relationship allowed us to improve the relative error: ε2(N3) = 569% vs. ε1(N3) = 884%. ε2(N3) remained high due to an important number of artificial loops appearing between neighbouring pixels. In fact, when clustering of pixels was used the relative error strongly decreased and reached a value as small as ε(N3) = 0.42%.

3.3. Three-dimensional global morphological analysis

The 12 trabecular bone samples were classified into two groups: one osteoarthritis group (OA, n = 6) and one osteoporosis group (OP, n = 6). The global morphological parameters (lS, lP and fS) are reported in Table 2 for each of the 12 samples as well as for each group of samples (mean value ± standard deviation). The fraction of solid (fS) was lower in OP samples than in OA samples (fS = 0.22 ± 0.04 vs. 0.30 ± 0.05), this loss of solid being characterized by a decrease of the mean intercept length of the solid phase (lS = 0.27 ± 0.05 mm vs. 0.35 ± 0.06 mm) and an increase of the mean intercept length of the void phase (lP = 0.92 ± 0.25 mm vs. 0.83 ± 0.17 mm).

Table 2.   Global morphological parameters consisting of mean intercept lengths of the solid phase (lS) and of the complementary void phase (lP), solid fraction (fS), as well as topological parameters consisting of Euler–Poincaré characteristic (N3), numbers of branches (Bcc, Bct) and numbers of vertices (Vc, Vt). Bcc and Bct were, respectively, the numbers of connection and termini branches, while Vc and Vt were, respectively, the numbers of connection and termini vertices. These parameters were evaluated for 12 trabecular bone samples. Samples 1–6 were taken from osteoarthritic subjects (OA), whereas samples 7–12 were taken from osteoporotic subjects (OP). V was the volume of analysis, evaluated as the volume of solid divided by the solid fraction. The topological parameters were expressed per unit of volume (N3/V, Bcc/V, Bct/V, Vc/V, Vt/V). The mean values of both morphological and topological parameters were reported in each group (OA, OP) of samples.
Bone sampleslS (mm)lP (mm)fSV (mm3)N3/V (mm−3)Bcc/V (mm−3)Bct/V (mm−3)Vc/V (mm−3)Vt/V (mm−3)
1/OA0.310.910.25385.6−5.0229.5210.4016.4110.40
2/OA0.440.660.40349.9−4.1830.4314.8917.2614.89
3/OA0.290.680.29436.6−6.7839.5114.6620.9414.66
4/OA0.431.010.29422.4−2.2819.8910.8211.8410.82
5/OA0.341.070.25440.9−1.9719.2010.9811.7810.98
6/OA0.320.670.31431.5−4.7931.9014.4217.5114.42
Mean value0.350.830.30(411.1)−4.1728.4112.7015.9612.70
 ± 0.06± 0.17± 0.05 ± 1.65± 7.05± 1.97± 3.25± 1.97
7/OP0.260.480.24414.1−4.2126.5111.1214.7511.12
8/OP0.351.070.25334.0−1.8415.248.629.078.62
9/OP0.280.850.25439.4−3.9726.6711.3315.1811.33
10/OP0.191.100.14278.2−6.1829.556.7215.596.72
11/OP0.260.810.24484.2−6.1737.1913.5519.8913.55
12/OP Mean value0.27    0.27 ± 0.051.24    0.92 ± 0.250.17    0.22 ± 0.04432.2 (397.0)−2.73  −4.18 ± 1.6119.76  25.82 ± 6.998.85 10.03 ± 2.2211.35 14.30 ± 3.428.85 10.03 ± 2.22

3.4. Three-dimensional topological analysis

The topological parameters (N3, Bcc, Bct, Vc and Vt) are reported in Table 2 for each of the 12 samples as well as for each group of samples (mean value ± standard deviation). These parameters (without units) are expressed per unit of volume (divided by the volume of analysis V, Eq. 14). This normalization allows us to compare several samples together. The topological parameters are evaluated from finite samples, and consequently they are biased by boundary effects. Nevertheless, these boundary effects were identical for the 12 samples.

The two groups of samples have approximately the same mean value of N3/V (N3/V = − 4.18 ± 1.61 mm−1 for OP samples, N3/V = − 4.17 ± 1.65 mm−1 for OA samples), indicating a same number of loops. The mean numbers of vertices (Vc/V and Vt/V) and branches (Bcc/V and Bct/V) are slightly higher in OA samples. Nevertheless, the mean number of coordination (Nc) defined as two times the ratio of Bcc over Vc is identical in the two groups: Nc = 3.58 ± 0.15 for OP samples and Nc = 3.53 ± 0.17 for OA samples.

3.5. Three-dimensional local morphological analysis

The length (Lcc, Lct) and volume (Vc, Vt) of the trabeculae are reported in Table 3 for each of the 12 samples as well as for each group of samples (mean value ± standard deviation). The mean lengths are identical in the two groups (Lcc = 0.46 ± 0.04 mm, Lct = 0.41 ± 0.06 mm for OP samples; Lcc = 0.46 ± 0.04 mm, Lct = 0.46 ± 0.06 mm for OA samples), this result being confirmed by the mean length distributions (Fig. 6). By contrast, the mean volumes are lower in OP samples than in OA samples (Table 3, Fig. 7).

Table 3.   Local morphological parameters consisting of individual length (Lcc, Lct) and volume (Vcc, Vct) of trabeculae, for 12 trabecular bone samples. Samples 1–6 were taken from osteoarthritic subjects (OA), whereas samples 7–12 were taken from osteoporotic subjects (OP). Lcc and Lct were, respectively, the mean length of connection and termini trabeculae calculated from one sample, while Vcc and Vct were the mean volume of connection and termini trabeculae. The mean values of both length and volume of trabeculae are reported in each group of samples (OA, OP). This local characterization was performed from only a part of the skeleton graph branches (shorter branches being deleted before the extraction of the trabeculae): Bcc analysed connection branches for the characterization of the connection trabeculae, and Bct analysed termini branches for the characterization of the termini trabeculae.
 Length (mm)Volume (mm3)Analysed branches
Bone samples Lcc Lct Vcc Vct Bcc Bct
1/OA0.460.410.0320.0272819255
2/OA0.480.540.0540.0582330172
3/OA0.410.380.0290.0223891618
4/OA0.460.520.0590.0551873197
5/OA0.450.480.0430.0431986543
6/OA0.440.440.0350.0343059696
Mean value0.460.460.0420.040(2659.7)(413.5)
 ± 0.04± 0.06± 0.011± 0.013  
7/OP0.450.400.0300.0282617729
8/OP0.550.530.0530.0511221378
9/OP0.450.400.0340.0312714616
10/OP0.450.340.0160.0132245402
11/OP0.400.370.0250.0214019774
12/OP0.440–400.0290.0261971609
Mean value0.460.410.0310.028(2464.5)(584.7)
 ± 0.04± 0.06± 0.011± 0.012  
image

Figure 6. Mean length (connection (a) and termini (b)) distributions of trabeculae in each group of samples (six osteoarthritic samples (OA) and six osteoporosis fracture samples (OP)).

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image

Figure 7. Mean volume (connection (a) and termini (b)) distributions of trabeculae in each group of samples (six osteoarthritic samples (OA) and six osteoporosis fracture samples (OP)).

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The trabeculae are extracted from skeleton graph branches, but only a part of these branches are considered, the shorter of them being deleted before the trabeculae extraction. When we compare the number of skeleton graph branches (Table 1) and the number of analysed branches (Table 3), only 24.5% of the initial branches are considered for the extraction and local analysis of trabeculae. Hence, skeleton graph analysis induces a lot of branches that have topological meaning but no morphological interest for trabecula extraction (more than 75% of all the branches).

The volume of solid (VS) can be expressed as the number of trabeculae multiplied by the volume of these trabeculae, or the number of connection trabeculae (Bcc) multiplied by the volume of connection trabeculae (Vcc) plus the number of termini trabeculae (Bct) multiplied by the volume of termini trabeculae (Vct). We have performed this calculation for each of the 12 samples and calculated the relative error concerning the volume of solid. The mean relative errors in the two groups of samples are, respectively, 0.25 ± 0.18% for OP samples and 0.62 ± 0.43% for OA samples.

4. Discussion and conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

In this study, we have developed a new technique to analyse 3D digitized skeleton graph of porous media. This technique is validated from a topological criterion with a relative error of 0.42%. In the particular case of trabecular bone microarchitecture, the skeleton graph gives a template of the trabecular bone network. Our 3D skeleton graph analysis, which localizes each vertex and branch of the skeleton graph, allows us to extract each trabecula of the bone network as related to a branch.

A 3D skeletonization technique is used, based on topological invariant, and implemented with a sequential thinning algorithm. The topology is tested from β0, β2 and N3 numbers, and is rigorously preserved between the initial 3D image and its 3D digitized skeleton graph. This validation is performed from a topological analysis of the initial image and the skeleton graph.

We then developed a 3D skeleton graph analysis to count and to isolate all the vertices and branches of the skeleton graph. This critical step is disturbed by topological artefacts (Thovert et al., 1993) due to both the digitization process and a loss of symmetry between solid and void phases in the case of a cubic grid. Hence, we introduce the concept of pixels clustering involving detection of artificial loops between neighbouring pixels. As a result, the digitized skeleton graph is analysed as a thin network, where each point of the network was either a single point (one pixel) or a multiple point (cluster of several pixels). Furthermore, we have introduced an order relationship between six-, 12- and eight-connectivity in order to define the degree of proximity between two skeleton points. The graph points are classified according to the number of their neighbours and the neighbourhood configuration of these neighbours. The numbers of vertices and branches are directly deduced from this classification. A topological criterion is used in order to validate the counting of vertices and branches.

A global analysis applied to our 12 femoral head samples permits us to distinguish the OP samples from the OA samples by global morphological parameters (Table 2). This distinction, indicating a loss of solid in OP samples, was confirmed by local analysis of the volume of trabeculae (Fig. 7). On the contrary, the length of the trabeculae (Fig. 6) indicates no difference about the relative position of the vertices, interpreted as connecting site of the trabeculae. The topology of the two groups evaluated from the number of loops and the number of coordination (mean number of branches linked to a same vertex) is the same. These results show that the two groups of samples have equivalent trabecular network, and only the thickness of the trabeculae is lower in OP samples than in OA samples. This interpretation must be cautious: firstly due to the limited size of the two groups of samples, and secondly because the osteoarthritis alterations at the head–neck boundary have not been previously studied (Ciarelli et al., 2000), the preferential site of osteoarthritis changes being the subchondral area of the femoral head.

Our new technique consisting of the use of the 3D skeleton graph allows local information on each trabecula independently of a size or shape model (Kinney et al., 1995; Hildebrand & Ruegsegger, 1997). The localization of each trabecula of the bone network is interesting firstly to get a local morphological analysis but allows several new interesting analyses. Such an approach could be used either to perform a 3D simulation of the bone remodeling process (Reeve et al., 1993; Thomsen et al., 1994, 1996; Smith et al., 1997), or a 3D simulation of architectural changes due to bone diseases or bone treatments (Steiniche et al., 1989; Boyce et al., 1995b), or in order to provide an evaluation of bone mechanical strength (Silva & Gibson, 1997; Jiang et al., 1998; Ladd et al., 1998; Mullender et al., 1998).

Finally, such a 3D skeleton graph analysis is not limited to trabecular bone study. It can be extended to the case of natural disordered porous media (Levitz, 1998).

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References

We wish to acknowledge the contribution of J. P. Jernot for valuable remarks concerning topology. We thank the Orthopaedic Surgery Department of the Hospital of Orléans (France) for the head femoral samples. We acknowledge Hologic Company for financial help during this work.

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  3. 1. Introduction
  4. 2. Material and methods
  5. 3. Results
  6. 4. Discussion and conclusion
  7. Acknowledgements
  8. References
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