REVIEW

Confocal stereology: an efficient tool for measurement of microscopic structures Lucie Kubínová & Jiří Janáček

Received: 22 October 2014 / Accepted: 27 January 2015 / Published online: 7 March 2015 # Springer-Verlag Berlin Heidelberg 2015

Abstract Quantitative measurements of geometric forms or counting of objects in microscopic specimens is an essential tool in studies of microstructure. Confocal stereology represents a contemporary approach to the evaluation of microscopic structures by using a combination of stereological methods and confocal microscopy. 3-D images acquired by confocal microscopy can be used for the estimation of geometrical characteristics of microscopic structures by stereological methods, based on the evaluation of optical sections within a thick slice and using computer-generated virtual test probes. Such methods can be used for estimating volume, number, surface area and length using relevant spatial probes, which are generated by specific software. The interactions of the probes with the structure under study are interactively evaluated. An overview of the methods of confocal stereology developed during the past 30 years is presented. Their advantages and pitfalls in comparison with other methods for measurement of geometrical characteristics of microscopic structures are discussed. Keywords 3-D images . Confocal microscopy . Geometrical characteristics . Spatial probes . Stereology

Introduction Measurements of geometrical parameters of structural components at the microscopic level, such as volume, surface area, Electronic supplementary material The online version of this article (doi:10.1007/s00441-015-2138-3) contains supplementary material, which is available to authorized users. L. Kubínová (*) : J. Janáček Department of Biomathematics, Institute of Physiology, Academy of Sciences of the Czech Republic, Vídeňská 1083, 14220 Prague, Czech Republic e-mail: [email protected]

and length, are the main prerequisites for quantitative analysis in many studies in biological as well as materials research, especially when the relationships between function and structure are analyzed. Confocal stereology represents a contemporary approach to the evaluation of microscopic structures by using a combination of stereological methods and confocal microscopy. Stereology presents a toolbox of methods for the estimation of geometrical properties of real structures by geometrical sampling. The term stereology as a new scientific discipline was coined in 1961, motivated by the need of investigators in materials and life sciences to establish a rigorous theoretical basis for the solution of problems encountered in morphometry. The basis was found in formulas for unbiased estimation of the properties, provided by a mathematical discipline called integral geometry (Santaló 1976). Traditional stereological methods (Weibel 1979) were based mainly on observations made on two-dimensional (2-D) sections of threedimensional (3-D) objects, as well as their generalization working directly with the 3-D data (Sandau 1987), applying test probes of different dimensions, i.e. zero-dimensional (0D, i.e., points), 1-D (i.e., lines) or 2-D (i.e., planes) and counting the interactions of the probes with the structures under study, e.g., the number of test points falling into the given structure or the number of intersection points of test lines with the structure surface. There are a number of reviews and textbooks on stereology describing various stereological methods using a variety of test probes (Howard and Reed 1998; Schmitz and Hof 2005; Mouton 2011; West 2012). Confocal microscopy is a special type of optical microscopy that enables the obtaining of perfectly registered stacks of thin serial optical sections (having a thickness from approx. 350 nm) within thick specimens (into a depth up to several hundreds of micrometres). The principle of a confocal microscope was patented by Marvin Minsky in 1957 but confocal microscopy did not become a useful and efficient tool until

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almost 30 years later, after the confocal microscope with a laser light source was introduced (confocal laser scanning microscope, CLSM; Åslund et al. 1983). The first commercially available system was the Bio-Rad MRC-500 in 1986. What is the motivation for combining stereology and confocal microscopy? Digital images of stacks of optical sections acquired by confocal microscopy provide 3-D image data suitable not only for computer 3-D reconstructions that can be made without having to solve the tedious problem of alignment of images of successive sections (Pawley 1995; Kubínová et al. 1996a) but also for quantitative measurements. Various automatic methods for measuring geometrical characteristics based on image analysis are available (Kubínová et al. 1999). However, in 3-D confocal images, microscopic structures are often difficult to recognize by image analysis due to low contrast or low signal-to-noise ratio, weak or nonspecific staining and various optical artefacts. On the other hand, specific structures can often be more easily identified by a human operator. Human vision uses a parallel hierarchical system of neuronal circuits optimized by evolution for solving difficult tasks necessary for survival, which cannot always be imitated by current image analysis algorithms. Combining this unique ability with sound statistical principles enables the reliable estimation of quantitative characteristics from a relatively small number of sampled points. Therefore, it is reasonable to apply interactive stereological methods for estimating geometrical characteristics from digitized 3-D image data instead of the tedious development of 3D image analysis methods. This can also be advantageous in comparison with using traditional stereological methods that often require randomizing the direction of cutting physical sections used for the measurement; e.g., methods for length estimation need isotropic orientation of these sections. This is often technically demanding, inefficient or outright impossible. Even if randomized sectioning is feasible (Huang et al. 2013), it is usually better to cut parallel serial sections of the object (e.g., an organ) in one, most convenient, direction. The above problems linked to cutting physical sections in randomized directions can be avoided by using 3-D image data provided by confocal microscopy applying spatial estimators evaluating small 3-D samples of the structure under study. By using special software, it is possible to generate different virtual test probes with arbitrary pre-defined (e.g., random) position and orientation within the stack of sections and to apply them directly to this 3-D image data (Fig. 1). The first application of confocal microscopy for stereological measurements was presented by Howard et al. (1985) in their concept of an unbiased sampling brick. They used a special type of confocal microscope, a tandem scanning reflected light microscope (Petráň et al. 1968) for counting osteocyte lacunae. Yet, though mentioned by several authors (Gundersen 1986; Rigaut 1989), the unique features of

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Fig. 1 Spatial probe applied to 3-D data used in confocal stereology. A virtual spatial grid of (green) test lines with randomized direction (a triplet of so-called fakir probes, such as in Fig. 4b) is applied to a 3D object (obtained by 3D visualization of a tobacco cell chain captured by confocal microscopy). The number of intersection points (marked by yellow balls) of the grid with the surface of the object can be used for estimating the object surface area. The length of intersections of the grid with the object interior (red lines) can be used for the estimation of the object volume

confocal microscopy advantageous for stereological measurements of not only number but also other parameters, such as surface area, were not fully recognized earlier than during the 1990s (Rigaut et al. 1992; Howard and Sandau 1992; Kubínová et al. 1995, 1996b). The idea of estimating stereological parameters from optical sections within a thick slice was first used for counting particles by the optical disector principle (Sterio 1984; Gundersen 1986) and in an unbiased sampling brick rule (Howard et al. 1985) and then in many other stereological methods: e.g., a nucleator (Gundersen 1988) and a planar rotator (Jensen and Gundersen 1993) applied to a stack of optical sections and estimating the mean particle volume; the spatial point grid used for volume estimation (Cruz-Orive 1997; Kubínová and Janáček 2001); methods using an optical rotator (Kiêu and Jensen 1993; Tandrup et al. 1997), spatial grid (Sandau 1987), vertical spatial grid (Cruz-Orive and Howard 1995), virtual fakir probes (Kubínová and Janáček 1998) and virtual cycloids (Gokhale et al. 2004) for estimating the surface area; the method of vertical slices (Gokhale 1990); and the methods of total vertical projections (Cruz-Orive and Howard 1991), global spatial sampling (Larsen et al. 1998) and spherical probes (Mouton et al. 2002) used for the estimation of length. In the present review we will briefly describe the methods of confocal stereology that have been introduced till now, discuss advantages and pitfalls of their practical implementation, including tissue preparation, sampling and image acquisition and their comparison with other methods for estimating the geometrical characteristics of microscopic structures.

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Counting particles

estimated by an unbiased estimator:

An unbiased counting or sampling of 3-D particles (e.g., cells) can be achieved by using stereological methods of the disector (Sterio 1984; Gundersen 1986), fractionator (Gundersen 1986) or unbiased sampling brick rule (Howard et al. 1985). In all these methods, 3-D probes are used and their optical versions are well applicable to confocal 3-D image data acquired from a thick section within the physical slice. In the optical disector (Gundersen 1986; Gundersen et al. 1988; West and Gundersen 1990), the particle is sampled if it fulfills both the following conditions: (1) its top profile is lying inside the thick section within the physical slice; and (2) this top profile is sampled by the unbiased sampling frame (Gundersen 1977) placed into the reference plane moving through the thick section. In practice, the thick section is simply focused through and particle profiles that come into sharp focus are observed and can be evaluated whether or not they should be sampled by the sampling frame. In the unbiased sampling brick rule, the particle is sampled if it is lying at least partly in the sampling brick and at the same time is not intersected by the exclusion planes. In practice, one is focusing through the particles lying within the thick slice and checking to see if any of the particle profiles are intersected by the exclusion line of the sampling frame displayed on the optical sections (Fig. 2). In both methods, if the positions of the sampling frames are uniform over all possible positions in the reference space and a grid of test points is placed in the frame, the number of particles, e.g., cells [N(cell)] can be

estN ðcellÞ ¼

Fig. 2 Counting particles. a An unbiased sampling brick for unbiased counting of particles (i.e., to obtain value Q− (cell) in formula 1). Particles lying within this brick or intersecting its planes are counted, except those intersecting the exclusion planes. The exclusion planes in this scheme are represented by gray planes. b Counting cell nuclei by using the Disector

module in the Ellipse software, one frame shown, nuclei with green labels are sampled; the nucleus marked by a red cross is not sampled because it is intersected by the exclusion plane. The measurement procedure is shown in Movie 1 where the Sampling Window plugin run in ImageJ is applied

Q− ðcellÞ p ⋅ ⋅V ðre f Þ Pðre f Þ a⋅h

ð1Þ

where Q− (cell) is the number of cells sampled by all disector boxes (sampling bricks), p is the number of test points of the point grid placed into one sampling frame, P(ref) is the total number of test points of this grid falling in the reference space in all sampling frames, a is the actual area of one sampling frame, h denotes the height of the disector box (sampling brick) and V(ref) is the volume of the reference space. A practical implementation of the method is demonstrated in Fig. 2b. The second approach uses the fractionator method (Gundersen 1986), which is based on direct counting of particles, e.g., cells in a known predetermined fraction of the organ in which we wish to count cells. The organ is divided arbitrarily into blocks. (These can be histological blocks that will be sectioned.) Then, a fraction (bsf) of blocks (e.g., every third or fifth or more) is designated to be totally cut into thick sections. Next, a fraction (ssf) of the thick sections is sampled (say every third, tenth or twentieth) and then a fraction (asf) of the area of the thick section is sampled by unbiased counting frames of the 3D probes (disector or unbiased sampling brick). The height of the probe (i.e., the virtual box height) is some fraction (hsf) of the mean thickness of the thick sections. Finally, the cells are counted by the disector or unbiased

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sampling brick rule in the way described above. The number of cells can be estimated by: estN ¼

1 1 1 1 X − ⋅ ⋅ ⋅ ⋅ Q bs f ss f asf hsf

ð2Þ

where ∑Q− is the total number of cells counted. The optical disector is especially suitable for counting human or animal cells by their nuclei using a conventional transmission microscope (see, e.g., West and Gundersen 1990) because such nuclei are usually convex, nearly spherical in shape and only their central optical sections come into sharp focus. For counting objects of more complicated, even nonconvex, shapes, or objects with boundaries seen in sharp focus within their entire depth (e.g., plant cells), it can be recommended to use instead the unbiased sampling brick rule and a confocal microscope. If applying the unbiased sampling brick rule in a transmission microscope, it is often problematic to see the caps of cells sharply enough. In confocal microscopy; usually clear and sharp images of the cell profiles can be observed and so the method is well applicable. Counting particles using the above unbiased rules applied to confocal microscopic images belongs among the most frequently used methods of confocal stereology, being applied, e.g., for counting epithelial cells (Postlethwait et al. 2000), skin epidermal cells (Bauer et al. 2001) or endothelial cells (Howell et al. 2002; Kubínová et al. 2003; Mao et al. 2010), satellite cells in muscles (Malmgren et al. 2000; Sajko et al. 2004), glutamate receptor subunits (Mokin and Keifer 2006), neurones and/or synapses (Everall et al. 1999; Rostkowski et al. 2009; Baquet et al. 2009), gap junctions (Romek and Karasinski 2011), glomerular podocytes (Puelles et al. 2014), leaf mesophyll cells (Kubínová et al. 2002; Albrechtová et al. 2007) and chloroplasts in mesophyll cells (Kubínová et al. 2014). Similar principles can also be used for the estimation of topological characteristics, such as the Euler number of capillaries in terminal placental villi (Jirkovská et al. 1998, 2002) or the number of capillary branchings in skeletal muscle (Kubínová et al. 2001).

Length estimation A reliable estimation of the length of curves in a threedimensional space, such as capillaries or other tubular structures, can also be performed by stereological methods. The estimation of the length of curves in 3-D is based on counting intersection points between the curves and randomized planar probes; the number of intersections is directly proportional to the curve length (Weibel 1979). In general, to obtain an

unbiased length estimate, an isotropic random direction of the test probes is required. This means that every orientation must have the same probability of being chosen for the measurement. The orientator method (Mattfeldt et al. 1990) is usually used to generate isotropic uniform random (IUR) sections and the number of fiber profiles in the sections (i.e., number of intersections between the fibers and IUR sections) is counted. Thus, for the length estimate from thin twodimensional histological or ultrathin sections to be reliable, a randomized direction of cutting such physical sections is required. This is often technically demanding, inefficient or outright impossible. Even if randomized sectioning is feasible (Huang et al. 2013), it is usually more convenient to cut parallel serial sections of the object (e.g., an organ) in one direction. Indeed, most anatomical atlases (featuring important structural landmarks, often obtained by a number of staining protocols) are based on such preferred cutting directions (Bjaalije 2002; Zakiewicz et al. 2011). The above problems linked to cutting physical sections in randomized directions can be avoided by using 3-D image data obtained by confocal microscopy. Specialized interactive stereological methods can then be conveniently applied (Kubínová and Janáček 2001; Kubínová et al. 2003, 2004). A number of methods for curve length estimation in 3-D have been introduced, such as those using vertical slices (Gokhale 1990), total vertical projections (Cruz-Orive and Howard 1991), global spatial sampling (Larsen et al. 1998), spatial grid of cylinder surfaces (Gundersen 2002a) and spherical probes (Mouton et al. 2002). Some of the methods still require randomizing the direction of the thick (physical) slice, which is then to be optically (virtually) cut into thin slices by confocal microscopy. This limits their practical applicability. In the present review, we will show in more detail those stereological methods for length estimation that can be applied to thick slices cut in an arbitrary direction. For simplicity, we will limit the considerations outlined below to measurements within a sampling box placed in a thick tissue section of the structure of interest, assuming a digitized 3-D image of the tissue delimited by this sampling box was obtained by confocal microscopy. The length and length density in the entire reference space (e.g., whole organ) can then be easily measured while exploiting standard (2Dimage-based) stereological approaches required to perform the measurement in a sufficiently high number of properly sampled boxes, e.g., when employing the fractionator principle (Gundersen 1986). The length estimators shown below are unbiased if the position of the sampling boxes is systematic uniform random within the reference space volume. Slicer probe method (based on the idea of Bglobal spatial sampling with sets of isotropic virtual planes^ proposed by Larsen et al. 1998). The slicer probe is a systematic probe consisting in parallel test planes (Fig. 3a). When estimating

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the length of a linear structure in a digitized 3-D image, the intersections between the linear structure of interest (e.g., blood capillary) and the slicer probe are counted. A combination of slicer probes with different orientations can be used to increase the counting precision by an antithetic effect (Mattfeldt et al. 1985). If the spatial grid consists in n slicer probes (n=3 in Fig. 3b), the length, L, in the sampling box can be estimated by the formula: n X

estL ¼ 2⋅

Qi

i¼1 n

X

ð3Þ sV ;i

i¼1

where sV,i is the surface area of the i-th slicer probe per unit volume (1/sV,i =di, i.e., the distance between neighboring parallel slicer planes in the i-th slicer probe) and Qi (i=1,2,…,n) is the number of intersections between the linear structure and the i-th slicer probe. The above length estimator is unbiased if the orientation of the spatial grid is isotropic random and its position uniform random. If three slicer probes are used (n=3) and the distance d=1/sv between neighboring parallel planes in all three slicer probes is the same, e.g., if we apply an orthogonal (mutually perpendicular) triplet of otherwise identical slicer probes (Fig. 3b), Eq. (3) becomes simplified: estL ¼ 2⋅d⋅

Q1 þ Q2 þ Q3 3

ð4Þ

A practical implementation of the method is demonstrated in Fig. 3c. Spherical probe method The spherical probe (Mouton et al. 2002) is a probe consisting in a sphere surface. It is possible to consider a set of spherical probes with centers forming a point lattice (see fig. 2c in Kubínová et al. 2013), placed within or intersecting a sampling box. This multi-sphere approach is more flexible than the one proposed by Mouton et al. (2002), making it possible to modify the layout of spheres and the sphere diameter. When estimating the length of a linear structure in a digitized 3-D image of the sampling box, the intersections between the linear structure and the spherical probes are counted. Eq. (3) is then applied, with sV denoting the surface area of the sphere probe per unit volume and Q denoting the number of intersections between the linear structure and the spherical probe. The above length estimator is unbiased if the position of the spherical spatial grid is uniform random. A practical implementation of the method is demonstrated in Fig. 3d.

The above measurement requires to use a specific software generating an isotropic set of virtual slicer or spherical probes applied to stacks of transmission, or preferably, confocal images. The necessity of special software can be the reason why the length estimation using confocal stereology has not been yet widely used (Kubínová and Janáček 2001; Kubínová et al. 2002, 2005). The slicer probe method using isotropic virtual planes was used for estimation of capillary length in skeletal muscle (Kubínová et al. 2001; Čebašek et al. 2004, 2005, 2006, 2007, 2010; Eržen et al. 2011) and brain capillary length (Kubínová et al. 2003, 2013; Mao et al. 2010). A comparison of different methods for length estimation including those of confocal stereology was presented by Čebašek et al. (2010) and Kubínová et al. (2013).

Surface area estimation The stereological estimation of the surface area of a 3-D object (e.g., the cell surface area) is based on counting intersection points between the object surface and linear probes or on the measurement of the length of the intersection between the surface and test planes going through the object. The orientation of test lines, planes or projections can be arbitrary only if the structure has the same geometrical properties in all directions of space, i.e., if it is isotropic. Otherwise, isotropic random orientation of test probes is required for unbiased estimation. This means that every orientation must have the same probability to be chosen for the measurement. The most frequently used stereological methods for the unbiased estimation of the surface area suitable for practical application on 2-D sections are the method of vertical sections (Baddeley et al. 1986), ensuring the isotropic orientation of test lines and the orientator method (Mattfeldt et al. 1990), ensuring the isotropic orientation of test planes. Several efficient stereological methods based on using spatial grids of test lines that can be conveniently applied to 3-D image data acquired by confocal microscopy have been introduced: spatial grid (Sandau 1987), optical rotator (Kiêu and Jensen 1993), vertical spatial grid (Cruz-Orive and Howard 1995), isotropic fakir (Kubínová and Janáček 1998) and the virtual cycloids method (Gokhale et al. 2004). Some of the above methods still require randomizing the direction of the thick slice, which limits their practical applicability. We will show in more detail methods for surface area that can be applied to slices cut in an arbitrary direction, i.e., the fakir method (Kubínová and Janáček 1998) and the virtual cycloids method (Gokhale et al. 2004). For simplicity, we will restrict ourselves to measurements within a sampling box placed in a thick tissue section of the given structure, assuming a digitized 3-D image of the tissue

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Fig. 3

Length estimation. a Slicer probe. b Triple slicer probe. c Measurement of the capillary length in a skeletal muscle by slicer probe using our stand-alone Slicer program. During the measurement, a magenta mark is placed by a mouse into the intersection of the capillary with the probe. The number of intersections is Qi in formula 3. The measurement procedure is shown in Movie 2. d Measurement of the capillary length by the spherical probe method by using Ellipse software, see also Movie 3

delimited by this sampling box was acquired by confocal microscopy. The estimation of the surface area or surface density can then be easily extended to obtain the global values, exploiting standard stereological approaches, such as the fractionator principle (Gundersen 1986), requiring the performing of the measurement in a sufficiently high number of properly sampled boxes. The surface area estimators shown below are unbiased if the position of the sampling boxes is systematic uniform random within the reference volume (e.g., the organ under study). Fakir method The fakir probe (named by Cruz-Orive 1993) is a systematic probe consisting in parallel test lines (Fig. 4a). When estimating the surface area, as suggested by Kubínová and Janáček (1998), the intersections between the surface and the fakir probe are counted. Different combinations of fakir probes can be used to increase the efficiency. If the spatial grid consists in n fakir probes, the surface area, S, in the sampling box can be estimated by the formula: n X Ii i¼1 estS ¼ 2⋅ X n l V ;i

concept, we can consider a spatial grid consisting in n cycloid arcs (for n=6 ,see Fig. 5b). For simplicity, we will call this spatial grid Bspider^ and cycloid arcs Blegs^. The surface density can be estimated by the formula: n 2 X estS V ¼ ⋅ Ii n⋅l cycl i¼1

ð6Þ

where lcycl is the length of one cycloid arc (spider leg) and Ii is the number of intersections between the surface and the i-th spider leg (i=1,…,n). A practical implementation of the method is demonstrated in Fig. 5c. The fakir method has been used for the estimation of the surface area of tobacco cell walls (Kubínová et al. 1999; Kubínová and Janáček 2001), rat skeletal muscle fibers (Kubínová et al. 2001; Čebašek et al. 2004, 2005, 2006, 2007; Kubínová and Janáček 2009; Eržen et al. 2011), cortical microvessel walls (Kubínová et al. 2003), the internal surface area in Norway spruce leaves (Albrechtová et al. 2007; Lhotáková et al. 2008, 2012) and the mesophyll surface area in spinach leaves (Bandaru et al. 2010). The virtual cycloids method was applied to estimate the surface area of rat skeletal muscle fibers (Kubínová and Janáček 2009) acquired by confocal microscopy. Another stereological method for surface area estimation, the method of spatial grid, proposed by Sandau (1987), was used to measure the surface area of osteocyte lacuna (Howard and Sandau 1992), lymphocytes (Delorme et al. 1998) and capillary walls (Tomori et al. 2000).

ð5Þ

i¼1

Volume estimation where lV,i is the length of the i-th fakir probe per unit volume and Ii is the number of intersections between the surface and the i-th probe (i=1,…,n). The above surface area estimator is unbiased if the orientation of the spatial grid is isotropic random and its position is uniform random. Clearly, the surface density, SV, in the sampling box can be estimated by est S/v(box), where v(box) is the volume of the sampling box. A practical implementation of the method is demonstrated in Fig. 4c. Virtual cycloids (spider) method The virtual cycloid probe was introduced by Gokhale et al. (2004), who showed the application of a spatial grid consisting in three cycloid arcs placed within a sampling box lying in vertical planes forming an angle of 120° one with another (Fig. 5a). The vertical planes are parallel to the vertical axis, which is chosen to be perpendicular to the tissue block surface. In a more general

The volume of a 3-D object can be expressed by the integral: Z

h

V ¼

AðxÞdx

ð7Þ

0

where A(x) is the area of the section of the object passing through the point x ∈ [0,h] and h is the caliper diameter of the object perpendicular to section planes. If systematic uniform random points are selected from the interval [0,h], it is possible to estimate the object volume by the Cavalieri estimator (Gundersen and Jensen 1987): estV ¼ T ⋅

n X j¼1

Aj

ð8Þ

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Fig. 4 Surface area estimation by the fakir method. a Fakir probe. b Orthogonal triplet of half-way shifted fakir probes. c Measurement of the skeletal muscle fiber surface area by the fakir probe using our stand-alone Fakir program. During the measurement, a magenta mark is placed by a mouse into the intersection of the surface with the probe. The number of intersections is Ii in formula 5. The measurement procedure is shown in Movie 4

where T denotes the distance between sections, Aj is the area of the j-th section (j=1,…,n) and n is the number of sections. In practice, the area Aj is usually estimated by the pointcounting method when a point test system is superimposed

on the j-th section and the number of test points (Pj) falling into the section is counted. The following relationship holds: estA j ¼ a⋅P j

ð9Þ

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Fig. 5 Surface area estimation by the virtual cycloids method. a “Spider” probe with 3 cycloidal arcs. b “Spider” probe with 6 cycloidal arcs. c Measurement of the skeletal muscle fiber surface area by the probe with 3 cycloidal arcs using the Spider module in Ellipse software. During the measurement, a magenta mark is placed by a mouse into the intersection of the surface with the probe. The number of intersections is Ii in formula 6. The measurement procedure is shown in Movie 5

where a is the area corresponding to a single test point. Thus, it is possible to estimate the object volume by the formula: estV ¼ T ⋅a⋅

n X

Pj

ð10Þ

j¼1

The Cavalieri principle is usually used for volume estimation from series of physical sections. If digitized images of perfectly registered serial optical sections encompassing the object are available, such as 3-D image data acquired by

confocal microscopy, efficient stereological methods using spatial grids can also be used for volume estimations (CruzOrive 1997). Such methods can be applied to estimate, e.g., volume of cells from optical sections, provided the cell profiles are visible and sharp as it is under a confocal microscope. Spatial grid of points The method of volume estimation using spatial grid of points (Cruz-Orive 1997) is a modification of the Cavalieri principle. If a cubic spatial grid of points is applied, the object volume can be estimated by the formula: estV ¼ u3 ⋅P

ð11Þ

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where u is the grid constant (distance between two neighboring points of the grid) and P is the number of grid points falling into the object. The position of the point grid should be uniform random. Fakir method The fakir probe (introduced above) can also be used for estimating the object volume, when the total length of intercepts between the object and the probe is measured (Cruz-Orive 1993, 1997). Different combinations of fakir probes (Fig. 4b) can be used to increase the efficiency, i.e., to lower the variance of the method. If the spatial grid consists in n fakir probes, the object volume can be estimated by the formula: estV ¼

n 1 Xv ⋅ ⋅Li n i¼1 l i

ð12Þ

where li / v is the length of the i-th fakir probe per unit volume and Li is the length of the intercepts between the object and the i-th probe (i=1,…,n). The position of the spatial grid should be uniform random. Although the method of spatial grid as well as the fakir method can yield volume estimation in an efficient and convenient way from confocal microscopic images, they have been rarely applied, e.g., the spatial grid method for estimating tobacco cell volume (Kubínová et al. 1999; Kubínová and Janáček 2001) and rat skeletal muscle fiber volume (Kubínová et al. 2001; Kubínová and Janáček 2009; Eržen et al. 2011). The fakir method has been used for the measurement of tobacco cell volume (Kubínová et al. 1999; Kubínová and Janáček 2001), erythrocyte volume (Difato et al. 2004) and the biovolume of testate amoebae (Burdíková et al. 2010). In addition, the Cavalieri principle has been applied to estimate the volume of lymphocytes from their confocal images (Delorme et al. 1998). Nucleator method The volume of convex particles can be estimated from the endpoints of the intersection of the isotropic randomly oriented line segment through the internal pivotal point with the particle: 2π X ⋅ d ð yi ; O Þ 3 3 i¼1

pivotal volume estimation (Hansen et al. 2011) and surface estimation (Dvořák and Jensen 2013) for isotropic particles have been applied to confocal images of hippocampal neurons.

Software for confocal stereology All methods of confocal stereology require the use of a specific software generating relevant spatial probes (Peterson 1999; Tomori et al. 2001). Such dedicated software is included in several commercially available systems for stereology, e.g., in Stereo Investigator (MBF Bioscience, Williston, VT, USA, http://www.mbfbioscience.com/stereo-investigator), Ellipse (ViDiTo, Košice, Slovakia, http://www.ellipse.sk), newCAST (Visiopharm, Hoersholm, Denmark, http://www. visiopharm.com), Mercator (Explora Nova, La Rochelle, France, http://www.exploranova.com) and Stereologer (SRC, Tampa, FL, USA, http://www.stereologyresourcecenter. com/). All these systems support counting by using the optical disector or unbiased sampling brick rule and offer an isotropic set of virtual slicer and/or spherical probes applied to stacks of images for the length estimation. The isotropic fakir method is available in Stereo Investigator software and in Ellipse: we implemented the fakir method in the Ellipse Fakir plug-in module, enabling us to apply an orthogonal triplet of half-way shifted fakir probes (Fig. 4b and Kubínová and Janáček 1998) or a quadruple spatial grid of fakir probes, which both were shown to be highly efficient (Kubínová and Janáček 2001). Further, we implemented the virtual cycloids method in the Ellipse Spider plug-in module, enabling us to set the size and position of the spider, as well as the angle of rotation and number of spider legs. Our standalone free MS Windows programs for volume, surface area and length estimation by fakir and slicer probes can be downloaded from Jiří Janáček’s web page: http://www2. biomed.cas.cz/~janacek/fakir/3dtools.htm. Plugins running in ImageJ (http://imagej.nih.gov/ij/), free software, well known to the microscopic community, can also be used for some of the methods of confocal stereology, e.g., the Sampling Window plugin for optical disector and the Grid plugin for volume measurement.

2

estV ¼

where d is distance, O is the pivotal point and yi are endpoints of the line segment. When the particles are isotropically randomly oriented, lines in horizontal sections through the pivotal points can be used. The borders of particles are best viewed in such sections, because the lateral resolution of the confocal microscope is better than axial resolution. Semiautomatic methods for

Practical aspects of confocal stereology When using confocal stereology, a number of practical aspects connected with the specific features of confocal imaging have to be taken into account as explained below (for more details in connection with cell counting, see Peterson 2014). Tissue preparation Proper tissue preparation is crucial in confocal stereology. Cells and tissues have to be prepared so that

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they keep their original dimensions as much as possible; especially, tissue shrinkage has to be coped with (Geuna and Herrera-Rincon 2015). In experimental studies (e.g., when comparing tissue in health and disease), the deformations should be carefully rectified (Dorph-Petersen et al. 2001). A specific problem encountered in confocal stereology is caused by the compression of the thick tissue sections necessary for the acquisition of 3-D image data. Axial shrinkage is usually encountered in thick 3-D slices, unless the sample is fresh, perfectly fixed and mounted. If tissue shrinkage takes place and native-state capillary length is to be reliably measured, some corrections prior to the measurements are necessary. If the shrinkage rate is homogeneous throughout the sample, a linear rescaling (in one direction only) to the original slice thickness known from the microtome setting is sufficient (Janáček et al. 2012). Further, cells and tissues should be stained suitably, so that the structures under study can be clearly detected in confocal images. Confocal microscopy is usually based on the detection of fluorescence in the specimen. In special cases, autofluorescence can be exploited, e.g., in plant specimens (Albrechtová et al. 2007; Kubínová et al. 2014), also a non-specific staining like eosin can be used (Jirkovská et al. 1998); however, in many cases, the structures under study (e.g., the cell surface) can be visualized by immunohistochemistry or by the transgenic expression of fluorescent proteins in tissue such as the green fluorescent protein (GFP). For example, when measuring the cell surface area, antibodies can be bound to integral membrane proteins of the cellular plasma membranes and labeled by a common fluorescent dye. Ideally, the dye should be distributed evenly throughout the entire section thickness. Above all, insufficient penetration of antibodies in the entire depth of the specimen should be avoided. This can be achieved by prolongation of the incubation time. For instance, in studies of capillary supply of skeletal muscle fibers, we found that triple staining with CD31, Griffonia (Bandeira) simplicifolia lectin (GSL I) and laminin efficiently distinguishes vascular endothelium from the basal lamina of skeletal muscle fibers, while the incubation time for each antibody has to be adjusted to the section thickness (from 30 to 60 μm) from 3–6 days for CD31 and laminin and from 24 h to 3 days for lectin (Čebašek et al. 2004). There are other possibilities on how to achieve staining in the entire depth of specimens; e.g., for stereological estimation of capillary length in rat brain, the capillaries can be stained by using an intravenous injection of biotin-labeled Lycopersicon esculentum lectin (Kubínová et al. 2013). Image acquisition There are practical differences when using confocal microscopy for stereological measurements instead of the traditional approach based on image acquisition by transmission light microscopy when online measurements are usually performed. On the other hand, 3-D confocal image data are acquired first and the evaluation goes on offline, using

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special software (see BSoftware for confocal stereology^). The image acquisition has to be performed in a number of fields of view, suitably sampled (see below) and therefore it is time consuming. This can be speeded up by applying fast scanning of confocal image stacks, e.g., using lower but still sufficient image quality during image acquisition by confocal laser scanning microscopes; sometimes, it can be useful to apply fast scanning microscopes, such as the spinning disk confocal microscope. It is also desirable to use a programmable motorized xy stage during the image acquisition, enabling automatic capturing of image data (using, e.g., the Stage XY plug-in module of Ellipse; see Mao et al. 2010). For more details concerning these issues, see Peterson (2014). Further, in confocal image acquisition, we have to take care of artifacts in microscopic images, caused by aberrations of microscope optics and/or a refractive index mismatch between the specimen and immersion oil (Sheppard and Török 1997; Wan et al. 2000); they are more pronounced at greater depths of the specimen than at sites closer to the objective and can lead to serious misinterpretations. Such problems with thick sections, chiefly due to spherical aberration, can be tackled by using suitable objectives permitting the use of immersion fluids (e.g., water or glycerin) closely matching the refractive index of the specimen. Moreover, images often need to be rectified by compensating for light attenuation at greater depths in the specimen (Čapek et al. 2006). Sampling Whether interactive stereological or automatic image analysis techniques are applied, it is always necessary to follow proper sampling, i.e., the fields of view chosen for evaluation must be selected in a representative, unbiased manner. Well-designed and carefully accomplished multilevel sampling schemes have been developed in stereology giving sufficiently precise results with reasonable amounts of work. Usually, systematic (random) sampling is a good and efficient way to select the sections and sampling frames for analysis (Gundersen and Jensen 1987). For a detailed practical application of systematic sampling in confocal stereology, see Peterson (2014). In some cases, more sophisticated ways of sampling are efficient, e.g., using the smooth fractionator (Gundersen 2002b) or proportionator (Gardi et al. 2008) principle. It is also necessary to define precisely the conditions and aims of the study. The above methods lead to reliable results only if the reference space and structures under study can be unambiguously identified.

Discussion It has been demonstrated how useful confocal microscopy can be in estimating different geometrical parameters of microscopic structures. In comparison with conventional optical

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microscopy, confocal microscopy offers not only higher resolution and examination of thicker specimens but also a better possibility to analyze living, fresh or more easily prepared specimens. An advantage of the methods of confocal stereology is the possibility to extract accurate data from thick sections cut in an arbitrary direction, i.e., it is possible to choose the one that is practically convenient. The fact that quantitative measurements of structures can be easily combined with 3-D reconstructions from a series of physical sections represents an additional bonus. It is the case that 3-D microscopic image data per se enable a generation of properly randomized virtual sections, subsequent superposition of relevant test systems and evaluation by standard stereological methods (i.e., those utilizing 2D images only). However, we may have to choose between a convenient direction dictated by the morphology/ anatomy of the tissue/organ under study and image quality, as axial resolution is ca 3-fold worse than the lateral resolution (Pawley 2006). The practical implementation of the presented methods is demanding and prone to errors, as discussed in BPractical aspects of confocal stereology^, as far as tissue preparation and image acquisition are concerned. As for the quality of confocal microscopic images, it should be noted that a drawback exists because the axial resolution (though higher than in a conventional optical microscope) is lower than the lateral resolution. The shape of the point spread function of a confocal microscope is elongated in the direction of the z-axis (Shaw 1994), which causes defocusing that can possibly result in an overestimation of the surface area and volume of the examined objects. This can be eliminated by applying special deconvolution algorithms to perform 3-D deblurring of the images before the measurements. In two-photon excitation fluorescence microscopy, the axial resolution is higher than in confocal microscopy (Denk et al. 1990; Nakamura 1999; Diaspro 2002) and it is also possible to penetrate more deeply into the specimen with decreased bleaching of fluorescence dyes. However, possible deblurring of 3-D images should be considered even here. It should be mentioned that the recent advancement of diverse super-resolution optical microscopic techniques makes it possible for considerable improvement of resolution in both the lateral and axial directions (for an overview, see Nature Methods, Vol. 6, No. 1, 2009). It should also be noted that there is a variety of further 3-D imaging modalities, such as electron tomography, computed tomography (CT) and micro-computed tomography (μCT) and magnetic resonance imaging (MRI) that can provide 3-D image data suitable for evaluation by the presented stereological methods exploiting confocal microscopic data. Besides interactive methods of confocal stereology, a number of automatic digital methods for estimating the geometrical characteristics of the microscopic structure have been introduced and compared from the point of view of their

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precision and efficiency (Kubínová et al. 1999, 2002, 2005, 2013). It should be stressed that no absolutely universal method optimal for any type of structure exists. In general, the automatic methods are faster than interactive stereological methods but require automatic segmentation (i.e., recognition) of analyzed objects from the images or at least a high contrast between the object and the background. Often, there is no easy way to recognize objects and tedious software developments and parameter tuning are necessary to obtain good segmentation of the objects under study, which is inefficient, especially if a relatively small number of samples are to be analyzed. For surface area estimation, the fakir method appears to be the most universal of all the presented methods and so it might be recommended for testing the applicability of other, less time-consuming, methods. The method using a spatial grid of points is very good for interactive volume estimation. The triangulation method applied to grayscale images (Kubínová et al. 1999) appears to be suitable for measuring the volume and surface area of isotropic as well as anisotropic objects, provided a high contrast between the object and the background is achieved. If the object segmentation is feasible, the voxel-counting method is suitable for the volume and the digital Crofton method (Kubínová et al. 1999) for the surface area measurements. Unlike volume and surface area measurements, it appears that the length measurement is very sensitive to segmentation, resolution and degree of smoothing. As to the number estimation, e.g., automatic counting of cells, this requires automatic segmentation of individual cells, which in some cases is not easy; e.g., Schmitz et al. (2014) discovered that the automated 3-D cell detection methods they applied were not a suitable replacement for manual cell counting by stereological methods. It should be stressed that, before applying any automatic method, it should be very carefully tested to see if it gives reasonably precise results for the given type of image data. In the practical application of the presented methods, the question of efficiency is important, as it is always desirable to obtain sufficiently precise results with the least effort. Therefore, more and more attention is being paid to the study of the variances of stereological estimators. The variances of the estimators based on the Cavalieri principle, the spatial grid of points (so-called systematic sampling) and the disector principle have been studied in detail using Matheron’s (1965) theory of regionalized variables (Gundersen and Jensen 1987; Cruz-Orive 1989, 1993, 1999; Kiêu et al. 1998; Gundersen et al. 1999) and it turns out that the variance of the volume estimate of randomly oriented objects is roughly proportional to their surface area and a power function of the grid density. For the estimators based on measuring intersections of the object with isotropic uniform random grids (i.e., the fakir and slicer methods), the situation is more complicated; their variance can be split into the component due to the grid orientation and to the residual component due to the

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grid position (Hahn and Sandau 1989). The first component depends on the mutual orientation of the fakir probes applied and on the shape of the object under study, namely the anisotropy of the surface, which can be expressed by the rose of directions of normals to the surface. The second, residual component of variance is dependent mainly on the grid density and arrangement, e.g., on the mutual shift of the fakir probes. The first component of variance is clearly the highest for a totally anisotropic object, the rose of directions of which is represented by a single vector, i.e., for the flat surface in 3-D. It can be proven that in this special, ‘worst’ case the coefficient of variance of the surface area estimate by applying a single fakir probe is 57.74 %, while in the case of three orthogonal fakir probes (Fig. 4b), it is only 10.16 % (Mattfeldt et al. 1985; Janáček 1999). For less anisotropic surfaces, the first component of variance is decreasing, e.g., for the triple grid estimator of the surface area of an ellipsoid with diameter ratios of 1:4:4, the coefficient of variance was calculated to be 4.90 %, while for an ellipsoid with diameter ratios of 1:1.6:1.6, it is already very close to zero (Hahn and Sandau 1989). The second, residual component of variance reflects the arrangement of the spatial grid applied; an optimally designed grid may decrease the variance of estimates of regular objects, e.g., it can be shown that the shifted orthogonal triplet of fakir probes (Fig. 4b) is much more efficient than a nonshifted orthogonal triplet of fakir probes (Janáček 1999). However the estimates of objects, which are strongly irregular (changing substantially within the period of the grid), have residual variance comparable to the simple random sampling, i.e., the variance of the number of intersections of the object and the grid is equal to the mean of the number of intersections. The variance of the estimators calculated from this simple relationship can often serve as the upper bound for the residual variance of the grid estimators. The precise formulas for the variance of estimation of characteristics other than volume by systematic sampling (Janáček and Kubínová 2010; Kubínová et al. 2013) are useful for the design of efficient estimator grids rather than for the practical estimation of variance in microscopic studies, because they require further characteristics of the object under study, e.g., those describing shape, curvatures, anizotropy or orientations. The variance in specific studies can be estimated using a small number of pilot measurements with different grid types and values of parameters (such as the grid density). Experiments with a hierachical design can then be planned using general stereological procedures (Gundersen and Østerby 1981). In summary, if properly applied, confocal stereology can yield important parameters of different types of structures. Thanks to the spread of confocal microscopy and other techniques providing 3-D image data, it is believed that its usefulness will be identified by the broad scientific community and the methods will be applied more frequently when reliable evaluation of geometrical characteristics is needed. The spread

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of these methods, dependent on special software, should be reinforced by suitable open source software modules. Acknowledgements The present study was supported by the Czech Republic’s public funds provided by the Czech Academy of Sciences (RVO:67985823) and the Ministry of Education, Youth and Sports (KONTAKT LH13028). The authors wish to thank Dr. Ida Eržen (Institute of Anatomy, Faculty of Medicine, University of Ljubljana) for providing the skeletal muscle specimen shown in Figs. 3, 4 and 5.

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