## 3D Morphometry

Quantitative bone morphometry is a method to assess structural properties of cortical and trabecular bone. Morphometry has traditionally been assessed in two dimensions by means of histology, where the structural indices are either inspected visually or measured from sections, and the third dimension is added on the basis of stereology [15]. The conventional approach to morphologic measurements typically entails substantial preparation of the specimen, including embedding in methylmethacrylate, followed by sectioning into slices. Although the method offers high spatial resolution and high image contrast, it is a tedious and time consuming technique. Particularly limiting is the destructive nature of the procedure, preventing the specimen from being used for other measurements such as the analysis in different planes or mechanical testing. The latter is highly desirable because of the anisotropic nature of trabecular bone. While indices like bone volume density (BV/TV) and bone surface density (BS/TV) can be directly obtained from two-dimensional images, a range of important measures such as trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), and trabecular number (Tb.N) are to be derived indirectly assuming a fixed structure model. Typically an ideal plate model is used [20, 14]. But any deviation from the assumed model will lead to an unpredictable error of the indirectly derived indices.

For these reasons and in order to take full advantage of volumetric measuring technique such as microCT, new methods, which make direct use of 3D data, are required. To meet this demand, several new 3D image processing methods have been introduced over the last decade, allowing direct quantification of the microstructural bone architecture [11, 5]. Three-dimensional images enable direct assessment of metric indices by actually measuring distances in the 3D space. These techniques do not rely on an assumed model type and are therefore not biased by eventual deviations of the actual structures from this model.

The following section gives an overview of the most often used morphometric indices. All these indices can be assessed by b-cube. Furthermore, b-cube offers custom analysis and image processing to analyze the samples of our customers.

## Primary Indices

Bone surface area (BS) is determined by triangulating the surface of a voxel image and calculating the total area of the triangles. For the surface triangulation the Marching Cubes method [10] is applied. Bone volume (BV) is calculated in a similar way by defining tetrahedrons inside the structure according to the triangulated surface [2]. Total volume (TV) is the volume of the whole examined sample. This volume is typically defined by a contour or mask, which includes the volume of interest (VOI). To be able to compare samples with different VOIs, the normalized indices, BV/TV and BS/TV are used. The specific bone surface or bone surface to volume ratio is given by BS/BV.

### Directly Assessed Indices

Three-dimensional images enable the direct assessment of metric indices by measuring distances in the 3D space [5]. Trabecular thickness (Tb.Th) is determined by filling maximal spheres into the structure using distance transformation [6]. Then the average thickness of all bone structures is calculated to give an estimate of mean Tb.Th.

Trabecular separation (Tb.Sp) is calculated with the same procedure as used for Tb.Th, but this time the voxels representing non-bone parts are filled with maximal spheres. Tb.Sp can thus be expressed as the average thickness of the marrow cavities.

Trabecular number (Tb.N) is taken as the inverse of the mean distance between the mid-axes of the structure to be examined. The mid-axes are assessed from the binary 3D image using the 3D distance transformation and extracting the center points of non-redundant spheres which fill the structure completely. The mean distance between the mid-axes is then determined in analogy to the Tb.Sp calculation, i.e. the average separation between the mid-axes is calculated.

Trabecular Thickness (Tb.Th) |
Trabecular Separation (Tb.Sp) |

### Directly Assessed Non-Metric Indices

The geometrical degree of anisotropy (DA) is usually defined as the ratio between the maximal and the minimal radius of the mean intercept length (MIL) ellipsoid [4]. The MIL distribution is calculated by superimposing parallel test lines in different directions on the 3D image. The directional mean intercept length is then the total length of the test lines in one direction divided by the number of intersections of these lines with the bone-marrow interface [13]. The MIL ellipsoid is calculated by fitting the directional MIL to a directed ellipsoid using a least square fit. A value of 1.0 for DA indicates overall bone material isotropy. However, sampling problems inherent in the MIL method, proven by Simmons and Hipp [18], lead to variations of as much as 52% in the MIL indices, therefore Laib et al [9] developed a new approach, where a direction distribution of the projected triangulated surface is calculated. This distribution is determined by calculating the scalar product of the area weighted normal vector of each surface triangle with the discretised directions and corresponds to the inverse of the usually computed intercept length distribution. The "quasi MIL" ellipsoid is then calculated by fitting the inverse directional projected surface distribution to an ellipsoid using a least squares fit.

The structure model index (SMI) is an estimation of the plate-rod characteristic of the structure [7]. SMI is calculated by a differential analysis of a triangulated surface of a structure and is defined as

where dS/dr is the surface area derivative with respect to a linear measure r, corresponding to the half thickness or the radius assumed constant over the entire structure. This derivative is estimated by a simulated thickening of the structure by translating the triangulated surface by a small extent in its normal direction and dividing the associated change of surface area with the length of the extent. For an ideal plate and rod structure the SMI value is 0 and 3, respectively. For a structure with both plates and rods of equal thickness, the value is between 0 and 3, depending on the volume ratio of rods to plates.

The trabecular bone pattern factor (TBPf) describes quantitatively the ratio of inter-trabecular connectivity [3]. It was developed for 2D histological sections. Bone area (A1) and bone perimeter (P1) are assessed in the slice, then a dilation of the trabeculae is performed and bone area (A2) and perimeter (P2) are calculated again. TBPf is defined to be the quotient of the differences of the first and the second measurement.

The concept of TBPf is to get a value for the convexity or concavity of a structure. There is always an increase of bone area when performing the dilation, but an increase in bone perimeter with convex surfaces only. Therefore, TBPf shows positive values for concave structures and negative values for convex structures. In 3D image data, this concept corresponds to the previously described surface area derivative (dS/dr) and may be expressed as:

The connectivity density (Conn.D) is based on the Euler characteristics of the structure [12]. It can be shown that any naturally occurring 3D body X may be decomposed into convex bodies. The Euler characteristics of the decomposition Χ(X) may be expressed in a simplified definition as

where n_{i} is the number of convex bodies of dimension i in the decomposition of X into convex bodies. This equation is particularly suited for 3D images. If each voxel is considered to have 26 neighbors, n_{3} may be seen as the number of voxels, n_{2} as the number of voxel faces, n_{1} as the number of voxel edges, and n_{0} as the number of voxel corners. The Euler-Poincaré formula combines the Euler characteristics with the Betti numbers as

where the zeroth Betti number, β_{0}, reports the number of separate particles of the structure. The first Betti number, β_{1}, reports the number of handles if the structure is deformed into a sphere with handles (i.e., β_{1} reports the connectivity). The second Betti number, β_{2}, reports the number of cavities enclosed within the structure. Given the assumption that no piece of trabecular bone is completely isolated from the main structure, and that no marrow space exists completely encapsulated by bone, then this equation may be reformulated as

or as a normalized connectivity density:

The higher the value, the more connections are present in the object. However, the calculation is based on the above mentioned assumptions. Therefore one should always perform a component labeling and only keep the largest component before applying this algorithm.

### Area Moment of Inertia

For the cortical section in long bones, the tissue area as well as different area moments of inertia [17] can be calculated in 2D sections. For this purpose, 3D cortical image data is sectioned into slices orthogonal to the long axis of the bone. Then the following geometrical indices are calculated for each of the 2D sections and averaged over the number of slices. The tissue area (T.Ar) is defined as the cut bone surface in the slice plane. The axial moments of inertia (I_{xx}) and (I_{yy}) are calculated as

and the biaxial area moment of inertia (I_{xy}) is described as

The geometrical influence on a bending experiment orthogonal to the long axis of the bone is described by the axial moments of inertia. During such a mechanical test, the bone might also twist under the load, which is considered by the biaxial area moment. The polar moment of inertia (J) detects the resistance of a long bone to torsion and is defined as

Furthermore, the principal moments of inertia I_{max} and I_{min}, respectively are assessed.

### Local Morphometry

A recent development at our lab allows spatial decomposition of the three-dimensional trabecular bone structure into its basic elements (rods and plates) [19]. Therefore, the skeleton of the structure is computed and point-classified. Then this classified skeleton is decomposed by removing the intersection points. The final spatially decomposed structure is generated through a multi-color dilation of the decomposed skeleton. This algorithm enables local morphometric analysis of single elements. In order to follow the naming conventions proposed by Parfitt et al [16], the prefix Pl is used for plates, Ro for rods, and El for elements that can be either rods or plates. The volume, surface and thickness of one single element is denoted with V, S and Th and the sum of the volume and surface over all elements with BV, and BS, respectively. Furthermore brackets (<>) are used to denote mean values that are averaged over all elements of the same type.

Basically, all standard three-dimensional morphometric algorithms as described above may be applied to these elements. The mean volume (<Ro.V>, <Pl.V>), the mean surface (<Ro.S>, <Pl.S>) as well as the mean thickness (<Ro.Th>, <Pl.Th>) are averaged for each structure over all rods and plates separately. The plate volume density (Pl.BV/TV) and the rod volume density (Ro.BV/TV) are defined as total plate volume or rod volume respectively, divided by total volume of interest in percent. The specific plate volume and the specific rod volume are given by Pl.BV/BV and Ro.BV/BV, respectively. Another morphometric index which can be calculated on the decomposed structure is the relative size of major elements (El.V_{max}/BV), which is defined as the volume of the largest element divided by the total bone volume.

## Compartmental Morphometry

When analyzing different compartments within an intact bone (i.e. mouse femur) [1, 8], then typically one of the compartments represents the whole bone (full compartment), one represents trabecular bone (trab compartment) and one cortical bone (cort compartment). Since some of the morphometric indices are calculated for more than one compartment, new names have to be introduced. The bone volume density in the full compartment is called apparent volume density (AVD), the bone volume density in the cortical compartment %BV and the cortical thickness is named Ct.Th. Table 1 gives an overview on the reported morphometric indices in a compartmental analysis.

Compartment | Index | Description | Original Index |
---|---|---|---|

full | AVD | apparent volume density | BV/TV |

trab | BV/TV | bone volume density | |

Tb.Th | trabecular thickness | ||

Tb.Sp | trabecular spacing | ||

Tb.N | trabecular number | ||

Conn.D | connectivity density | ||

cort | %BV | cortical volume density | BV/TV |

Ct.Th | cortical thickness | Tb.Th |

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