**7. Quantitative micro-CT, histology and stereology**

184 Injury and Skeletal Biomechanics

**Figure 7.** Vessel skeleton with branching node points.

periphery of the vascular tree.

Currently, we are able to segment a vessel tree formed by a corrosive cast. In the example analyzed in this study, continuous regions of the volumetric model were counted to detect the number of individual objects within the volume. The volume of each individual object was estimated. Using skeletonization, the number of branching nodes was counted, and the number of vascular segments between the nodes was estimated, as was the length of each segment. Knowing the vessel length and volume, it is also possible to compute the average diameter of the vessel. However, there are still challenges for future work, such as estimating the vessel diameter in each voxel of the skeleton using a distance transform. With this information, the diameter distribution with respect to the vessel length or the vessel volume could be acquired. With a known diameter, the surface area of the vessels can also be computed. Tracking the skeleton leading its binary graph construction determines the spatial distribution of the branching nodes from the proximal vascular segments to the

The approach presented in this chapter is based on several assumptions that deserve to be discussed, as they also represent the limitations of this study. We assume, for our computation, that the vessels are in the form of generalized cylinders, which means that the cross-section orthogonal to the medial axis is a circle. The voxels in our data set were cubical; however, a data set with unequal voxel edges may be resampled into cubical voxels. An important question is the choice of the threshold value. The segmented vessel diameter is partially linearly dependent on the chosen threshold value. Setting an excessively low threshold causes too many artifacts to appear and objects to merge into each other; contrariwise, an excessively high threshold makes vessels very thin, such that small vessels disappear. This effect implies that the threshold value influences the absolute diameter of the vessels and biases statistical markers. However, the construction of the skeleton and the

topological analysis do not appear to be affected by the selection of the threshold.

Current micro-CT devices are typically bundled with sophisticated software packages that offer a number of automated quantification procedures. However, correlating the micro-CT results with quantitative histology favors the use of unbiased stereological methods, which are highly standardized and widely accepted in biomedical microscopy research (Howard and Reed, 1998; Mouton, 2002). This chapter illustrates the stereological assessment of micro-CT scans of bone scaffolds and microvascular corrosion casts, including the quantification of the volume fraction (*VV*, dimensionless), surface density (*SV*, m-1), length density (*LV*, m-2), orientation and anisotropy of microvessels (Kochová et al., 2011).

In bone tissue samples, the micro-CT resolution is currently capable of providing images that can be used for both analysis of bone vascular canals, and counting individual osteocyte lacunae. Quantification of bone microporosities is used for testing their effect on the viscoelastic properties of bone tissue (Tonar et al., 2011). The microporosity has at least two functional levels, the vascular porosity (related to the vascular canals; the order of magnitude is 10–1000 µm) and the lacunar-canalicular porosity (surrounding the osteocytes; the order of magnitude is 0.1–10 µm). Mechanical experiments clearly demonstrated that the hierarchical organization of bone architecture is crucial and that bone structural heterogeneity varies with the scale of magnification. Whereas Fig. 8 demonstrates twodimensional sections of compact bone produced by histology and micro-CT, Fig. 9 shows the results of 3-D micro-CT reconstruction of cancellous bone.

The vascular corrosion casts described in section 4 and used in section 5 also can be assessed with spatial stereological methods. Image data acquired by micro-CT are demonstrated in Fig. 10.

After manually tracing the microvessel profiles within a series of consecutive twodimensional micro-CT sections (software Ellipse, ViDiTo, Košice, Slovak Republic), a threedimensional system of oriented lines can be acquired (Fig. 11).

Correlating Micro-CT Imaging with Quantitative Histology 187

**Figure 10.** A micro-CT image (A) and a 3-D reconstruction (B) of a vascular corrosion Mercox cast of

**Figure 11.** Tracing the microvessel profiles in serial micro-CT sections (Fig. 10A) results in oriented lines, which can be visualized either as linear structures (A) or as rods (B). The blood microvessels are abstracted as having one dimension only (the length), whereas the spatial orientation is retained. The thickness of the rods (B) has been set for better visualization only and does not represent the real

Next, the orientation of each skeletonized vessel can be described using a spherical coordinate system (Fig. 12). Each blood vessel segment is described as a vector connecting the center of the coordinate system with the surface of the sphere. This vector is described by its length and a combination of azimuth ranging between [0, 2π] and elevation [0, π/2].

Next, the combinations of vessel lengths and their 3-D orientation can be assessed using various 2-D plots (Fig. 13). These plots are very useful when assessing the directionality and anisotropy of the vascular segments. Should the anisotropy be quantified, several methods are available, such as the ellipsoidal anisotropy, fractional anisotropy, or a chi-square method comparing the observed length densities of lines with the discrete uniform

distribution of an isotropic line system (Kochová et al., 2011).

human intestinal mucosa. The scale bars indicate 300 µm (A) and 100 µm (B).

thickness of the original microvessels.

**Figure 8.** Comparing a histological ground bone section stained with basic fuchsin (A, human femur) with a micro-CT image of compact bone (B, human tibia). In the compact bone, two types of microporosities can be quantified – the osteocyte lacunae and the vascular canals. Both levels of microporosities are clearly visible in either method. The volume fraction of the vascular canals can be quantified stereologically with a point counting method, whereas the numerical density of the osteocyte lacunae can be assessed by the 3-D counting method called disector, which is not biased by the variation in size and orientation of the lacunae (Sterio, 1984; Tonar et al., 2011). The scale bars indicate 60 µm (A) and 200 µm (B).

**Figure 9.** Micro-CT reconstruction of cancellous bone (human tibia) – an overall view (A) and a detail of the surface of bone trabeculae (B). The density and 3-D arrangement of bone trabeculae can be easily assessed with micro-CT. In contrast to scanning electron microscopy of bone surfaces, micro-CT is not biased by perspective or the depth of the 3-D sample. A dry bone sample does not require any laboratory processing prior to micro-CT scanning. The scale bars indicate 1 mm (A) and 200 µm (B).

186 Injury and Skeletal Biomechanics

and 200 µm (B).

**Figure 8.** Comparing a histological ground bone section stained with basic fuchsin (A, human femur)

**Figure 9.** Micro-CT reconstruction of cancellous bone (human tibia) – an overall view (A) and a detail of the surface of bone trabeculae (B). The density and 3-D arrangement of bone trabeculae can be easily assessed with micro-CT. In contrast to scanning electron microscopy of bone surfaces, micro-CT is not biased by perspective or the depth of the 3-D sample. A dry bone sample does not require any laboratory processing prior to micro-CT scanning. The scale bars indicate 1 mm (A) and 200 µm (B).

with a micro-CT image of compact bone (B, human tibia). In the compact bone, two types of microporosities can be quantified – the osteocyte lacunae and the vascular canals. Both levels of microporosities are clearly visible in either method. The volume fraction of the vascular canals can be quantified stereologically with a point counting method, whereas the numerical density of the osteocyte lacunae can be assessed by the 3-D counting method called disector, which is not biased by the variation in size and orientation of the lacunae (Sterio, 1984; Tonar et al., 2011). The scale bars indicate 60 µm (A)

**Figure 10.** A micro-CT image (A) and a 3-D reconstruction (B) of a vascular corrosion Mercox cast of human intestinal mucosa. The scale bars indicate 300 µm (A) and 100 µm (B).

**Figure 11.** Tracing the microvessel profiles in serial micro-CT sections (Fig. 10A) results in oriented lines, which can be visualized either as linear structures (A) or as rods (B). The blood microvessels are abstracted as having one dimension only (the length), whereas the spatial orientation is retained. The thickness of the rods (B) has been set for better visualization only and does not represent the real thickness of the original microvessels.

Next, the orientation of each skeletonized vessel can be described using a spherical coordinate system (Fig. 12). Each blood vessel segment is described as a vector connecting the center of the coordinate system with the surface of the sphere. This vector is described by its length and a combination of azimuth ranging between [0, 2π] and elevation [0, π/2].

Next, the combinations of vessel lengths and their 3-D orientation can be assessed using various 2-D plots (Fig. 13). These plots are very useful when assessing the directionality and anisotropy of the vascular segments. Should the anisotropy be quantified, several methods are available, such as the ellipsoidal anisotropy, fractional anisotropy, or a chi-square method comparing the observed length densities of lines with the discrete uniform distribution of an isotropic line system (Kochová et al., 2011).

**Figure 12.** The angular description of the directions of vascular line systems using a spherical coordinate system. Each blood vessel segment is represented by a vector with a known length and a combination of azimuth (longitude) and elevation (latitude). This figure was redrawn and modified according to Kochová et al. (2011).

**Figure 13.** Polar plots can be constructed (A) using the Lambert azimuthal equal area projection. Radial lines are azimuths and concentric lines are elevations, whereas the arrows indicate individual directions. Color histograms can also be used (B) with a color scale corresponding to the microvessel lengths in the given combination of elevation and azimuth (a dark color represents a high value of the microvessel length). In this example, the prevailing directions demonstrate the anisotropy of the microvessels, as both plots exhibit preferential combinations of azimuth and elevation.

Using the skeletons of the microvessels, their lengths *L* within a reference volume *V(ref)* can be expressed as the length density *LV*; see equation 3:

$$L\_V = \frac{L}{V(ref)}.\tag{3}$$

Correlating Micro-CT Imaging with Quantitative Histology 189

*<sup>Q</sup> <sup>Q</sup> <sup>A</sup>* . (5)

*V(ref )* . (6)

sections. When estimating the area *A*, the points hitting the profiles are counted, and their sum is multiplied by the area corresponding to each point. At least 200 points must be counted to obtain a reliable volume estimate (Gundersen & Jensen, 1987); see Fig. 14A.

To simulate the histological measurement of microvessel density *QA* (Fraser et al., 2012), i.e., the number of microvessel profiles *Q* per area unit *A*, an unbiased counting frame can be

*A*

**Figure 14.** Estimating the volume fraction and the microvessel density. A - When estimating the area and volume of the microvascular cast, the points hitting the profiles (marked red) are counted, and their sum is multiplied by the area corresponding to each point. B – Counting the microvessel profiles per section area simulates the histological assessment of microvessel density. This procedure can be performed using the projection of an unbiased counting frame consisting of two admittance (green) and two forbidden (red) borders. Marked profiles of microvessels (red outlines) situated inside the frame or those touching admittance borders and not touching the forbidden lines are counted in the software

The surfaces of the microvascular casts can also be estimated using stereological methods. However, several of these methods require isotropic uniform random sections or vertical uniform random sections. Randomized orientation of the sections cannot always be guaranteed in micro-CT, as the sample is typically oriented with its long axis perpendicular to the X-ray beam (Fig. 1). The section plane is often arbitrary and cannot be regarded as random. A suitable solution without randomizing the cutting plane is using an isotropic virtual test probe named fakir (Larsen et al., 1998; Kubínová & Janáček, 1998; Kubínová & Janáček, 2001); see Fig. 15. The ratio between the surface area *S* and the reference volume

> *V <sup>S</sup> <sup>S</sup>*

*V(ref)* is called the surface density *SV*; see equation 6:

Ellipse.

applied (Fig. 14B), and the microvessel density can be expressed using equation 5,

The volume fraction occupied by the microvascular corrosion cast easily can be estimated using the Cavalieri principle (Howard & Reed, 2005), as shown in equation 4,

$$testV = T \cdot (A\_1 + A\_2 + \dots + A\_m) \, , \tag{4}$$

where *estV* is the estimated volume of the microvessels, *T* is the distance between the sections sampled for the estimation, and *A* is the area of the cast profiles in *m* individual sections. When estimating the area *A*, the points hitting the profiles are counted, and their sum is multiplied by the area corresponding to each point. At least 200 points must be counted to obtain a reliable volume estimate (Gundersen & Jensen, 1987); see Fig. 14A.

188 Injury and Skeletal Biomechanics

according to Kochová et al. (2011).

**Figure 12.** The angular description of the directions of vascular line systems using a spherical coordinate system. Each blood vessel segment is represented by a vector with a known length and a combination of azimuth (longitude) and elevation (latitude). This figure was redrawn and modified

**Figure 13.** Polar plots can be constructed (A) using the Lambert azimuthal equal area projection. Radial

Using the skeletons of the microvessels, their lengths *L* within a reference volume *V(ref)* can

The volume fraction occupied by the microvascular corrosion cast easily can be estimated

where *estV* is the estimated volume of the microvessels, *T* is the distance between the sections sampled for the estimation, and *A* is the area of the cast profiles in *m* individual

. (3)

*estV T ( A A ... A )* 1 2 *<sup>m</sup>* , (4)

lines are azimuths and concentric lines are elevations, whereas the arrows indicate individual directions. Color histograms can also be used (B) with a color scale corresponding to the microvessel lengths in the given combination of elevation and azimuth (a dark color represents a high value of the microvessel length). In this example, the prevailing directions demonstrate the anisotropy of the

> *V <sup>L</sup> <sup>L</sup> V(ref )*

using the Cavalieri principle (Howard & Reed, 2005), as shown in equation 4,

microvessels, as both plots exhibit preferential combinations of azimuth and elevation.

be expressed as the length density *LV*; see equation 3:

To simulate the histological measurement of microvessel density *QA* (Fraser et al., 2012), i.e., the number of microvessel profiles *Q* per area unit *A*, an unbiased counting frame can be applied (Fig. 14B), and the microvessel density can be expressed using equation 5,

$$Q\_A = \frac{Q}{A} \,. \tag{5}$$

**Figure 14.** Estimating the volume fraction and the microvessel density. A - When estimating the area and volume of the microvascular cast, the points hitting the profiles (marked red) are counted, and their sum is multiplied by the area corresponding to each point. B – Counting the microvessel profiles per section area simulates the histological assessment of microvessel density. This procedure can be performed using the projection of an unbiased counting frame consisting of two admittance (green) and two forbidden (red) borders. Marked profiles of microvessels (red outlines) situated inside the frame or those touching admittance borders and not touching the forbidden lines are counted in the software Ellipse.

The surfaces of the microvascular casts can also be estimated using stereological methods. However, several of these methods require isotropic uniform random sections or vertical uniform random sections. Randomized orientation of the sections cannot always be guaranteed in micro-CT, as the sample is typically oriented with its long axis perpendicular to the X-ray beam (Fig. 1). The section plane is often arbitrary and cannot be regarded as random. A suitable solution without randomizing the cutting plane is using an isotropic virtual test probe named fakir (Larsen et al., 1998; Kubínová & Janáček, 1998; Kubínová & Janáček, 2001); see Fig. 15. The ratio between the surface area *S* and the reference volume *V(ref)* is called the surface density *SV*; see equation 6:

$$S\_V = \frac{S}{V(ref)}.\tag{6}$$

Correlating Micro-CT Imaging with Quantitative Histology 191

*Laboratory of Tissue Engineering, Institute of Experimental Medicine, Academy of Sciences of* 

*Department of Surgery, Charles University Prague, University Hospital in Pilsen, Pilsen,* 

*Department of Imaging Methods, Faculty of Medicine in Pilsen, Charles University in Prague,* 

*Department of Anatomy, Third Faculty of Medicine, Charles University in Prague, Prague,* 

*Department of Cybernetics, Faculty of Applied Sciences, University of West Bohemia, Pilsen,* 

*Sciences, University of West Bohemia in Pilsen, Pilsen, Czech Republic* 

the Charles University, Project GAUK No. 96610.

heart. *Mol. Imaging*, Vol. 4, pp. 110-116., ISSN 1535-3508.

*European Centre of Excellence NTIS - New Technologies for Information Society, Faculty of Applied* 

This work was supported by the European Regional Development Fund (ERDF) project "NTIS - New Technologies for Information Society", European Centre of Excellence, CZ.1.05/1.1.00/02.0090. The micro-CT technique was developed within the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, which is cofunded from the ERDF within the OP RDI program of the Ministry of Education, Youth and Sports. The corrosion casting was funded by the Charles University in Prague, Project No. SVV 264808, and by the Internal Grant Agency of the Ministry of Health of the Czech Republic under Project No. IGA MZ ČR 13326. The quantification of vascular trees was funded by the Grant Agency of the Czech Republic, Project No. 106/09/0740. The bone scaffold research was funded by the Grant Agency of the Czech Republic, Project No. P304/10/1307, and by the The Grant Agency of

Badea, C. T.; Fubara, B.; Hedlund, L.W.; Johnson G. A. (2005). 4-D micro-CT of the mouse

*Institute of Biophysics, Charles University in Prague, Prague, Czech Republic* 

Eva Prosecká

Václav Liška

*Czech Republic*  Hynek Mírka

David Kachlík

*Czech Republic* 

*Czech Republic*  Zbyněk Tonar

**Acknowledgement** 

**9. References** 

*the Czech Republic, v.v.i., Prague, Czech Republic* 

*Faculty Hospital in Pilsen, Pilsen, Czech Republic* 

Ivan Pirner and Petr Zimmermann

**Figure 15.** Estimating the surface in a series of micro-CT sections with arbitrary orientation using an isotropic triple spatial grid of orthogonal lines with a random initial orientation (fakir probe). The test lines of the fakir probe are green. The violet points denote intersections between the test lines and the current section. The left window shows a 3-D view, the right window shows the current section. Only one third of the triple line system is shown (software Ellipse).
