Direct and efficient stereological estimation of total cell quantities using electron microscopy


Jens R. Nyengaard, Stereology and Electron Microscopy Research Laboratory and MIND Center, University of Aarhus, Denmark, Ole Worms Allé 1185, DK-8000 Aarhus C, Denmark. Tel.: + 45 8942 2955; fax: + 45 8942 2952; e-mail:


Stereological estimation of total subcellular quantities in bioscience is presented in this report. Special emphasis is placed on the use of electron microscopy, which under certain circumstances may be combined with light microscopy. Three strategies based on the Cavalieri principle, the disector and local stereological probes through arbitrarily fixed points for estimation of total quantities inside cells are presented. The quantities comprise (total) number, length, surface area, volume or 3D spatial distribution for organelles as well as total amount of gold particles, various compounds or certain cytochemical markers.


To avoid ambiguous interpretation of cell density estimates, the so-called reference trap (Brændgaard & Gundersen, 1986) dictates that the densities should always be transformed to total quantities. Various methods exist in the stereological literature for obtaining total quantities. First, the total volume of any intracellular structure may either be obtained directly by the principle of Cavalieri (Gundersen & Jensen, 1987) or by multiplying the structural density per cell with the volume of the cell, generated by the principle of Cavalieri. Second, an indirect estimate of cell volume, obtained by disector sampling and point counting, multiplied with structural densities per cell provides the corresponding total cell quantities. Third, the family of ‘local’ stereological probes through arbitrarily fixed points, e.g. the rotator (Jensen & Gundersen, 1993), may be used for direct estimation of total quantities, and the 3D spatial distribution of the structure with respect to the cell nucleus or nucleolus may also be obtained. Fourth and finally, if the sample total is obtained in a known fraction of the cell, then the total is obtained by multiplying the sample total by the inverse of the total sampling fraction (fractionator principle, Gundersen, 1986, 2002). The fractionator principle is ordinarily not efficient at the electron microscopy (EM) level and will not be discussed further here. However, recent developments in fractionator sampling with varying but known sampling fractions may change this in the future (Witgen et al., 2006).

The purpose of this paper is to give a presentation of stereological methods available for estimation of subcellular total quantities based on three different strategies used in known or new ways: (1) the Cavalieri principle; (2) the disector, and (3) local stereological estimators.

Cavalieri's estimator of total volume

The Cavalieri estimator is an unbiased principle for estimating the volume of any structure in cells, for instance (Gundersen & Jensen, 1987). By slicing the entire cell into EM sections with a known microtome advance, MA (Dorph-Petersen et al., 2001) and counting the number of points, P(X), associated with a known area, a/p, and hitting the structure on the cut surfaces of the slices, the total volume of the structure per cell may be estimated:


Evidently, X may be the cell itself, as illustrated in Fig. 1. Unless isolated cells are studied, it is necessary to sample cells using the disector (see below) before estimating the total quantities within them. The practical use of the Cavalieri principle requires: (1) that the position of the first slice hitting the cell must be random; (2) that the slices are parallel; and (3) that the thickness of the slices is constant. Although this ancient principle of Cavalieri has had a tremendous positive impact on stereological estimation in recent years, this procedure, using the complete set of EM sections through each cell, is quite cumbersome and must be considered as a last resort (McCullough & Lucocq, 2005).

Figure 1.

Cavalieri's principle. At the top is shown a 3D sketch of a series of sections through the complete mononucleated cell. The cell is sampled in a disector using the nucleus as the sampling unit; t is the distance between sections. At the bottom is indicated one section with an integral point grid superimposed; a/p is the area associated with a ‘small’ point. If the objective is the volume of the whole cell, one may count the encircled points with A/P = 16 ·a/p.

Total cell volume from a single LM disector combined with EM structural density

Light microscopy (LM) sections are defined in this paper as semithin sections (1–3 µm) cut from tissue blocks embedded for EM, e.g. semithin epon sections stained with toluidine blue. The first aim in this procedure is to generate an indirect estimate of mean cell volume by obtaining a ratio of global cell estimators: numerical density of cells and volume density of cells. The numerical cell density is obtained using the disector, which is a three-dimensional probe that samples structures proportional to their number (number-weighted sampling) without regard to size or shape of the structures (Sterio, 1984). The disector comprises an integral test system (Jensen & Gundersen, 1982) and a counting rule: the number of cell nuclei sampled by the counting frame disappearing from one section plane to another, Q(nuc), is counted. The integral test system comprises the two-dimensional counting frame (Gundersen, 1977) with a certain area, a(frame), with some test points, p, the number of which hit the tissue is ΣP(tiss), and two parallel section planes that are a known distance apart. Using two consecutive LM sections sliced with a known microtome advance, MA, the numerical density of cell nuclei, NV(nuc/tiss), is estimated using the physical disector at the LM level:


The reason for the constant 2 in the denominator is that the nuclei are counted both ways in the physical disector (see Fig. 2). Using point counting on one of the LM sections, the volume density of cells per tissue is estimated:

Figure 2.

3D sketch of cell counting using a disector using two semithin, toluidine blue-stained sections. The leftmost and the rightmost cells are counted because their nuclei are only seen in one section of the disector, not in the other (two-way sampling). The microtome advance must be known (the microtome must be calibrated) in order to calculate the number density.


When Eqs (2) and (3) are combined, it is possible to estimate the number-weighted mean cell volume provided that the cells have one nucleus:


From here on the estimation of various cell structural densities is carried out at the EM level. The total structural quantity may be estimated by multiplying the density obtained at the EM level with the average cell volume obtained at the LM level. That is why only the densities are provided in the last part of this paragraph. The volume density of the structure X per cell may be estimated by point counting using systematic, uniformly random sampling:


The surface density of the cell structure may be estimated by the use of isotropic test lines. This is usually achieved most efficiently using vertical sections (Baddeley et al., 1986) and cycloid test systems. Isotropic, uniform random sections may also be used. These may easily be made using the isector (Nyengaard & Gundersen, 1992) or the orientator (Mattfeldt et al., 1990). The various techniques for correct sampling for orientation-dependent estimators are descried in detail in Gundersen et al. (1988a,b). The estimator of surface density is in both cases:


where the length of test line per point is denoted ℓ/p, and the number of intersections between test lines and the structural boundary is Σ(X).

The length density of intracellular structures may be estimated by the use of isotropic test planes, which can be generated by the orientator or the isector. The length density is estimated as:


The number of structural profiles is ΣQ(X) and the area of test frame per test point is denoted a/p.

It is also possible to estimate the numerical density of the cell structure, NV(X/cell), at the EM level:


The height of the disector equals microtome advance, MA, the number of structures counted both ways is ΣQ(X), and the test frame with P test points has area a(frame). For all the above structural densities per cell, the corresponding total structural quantity is obtained by multiplying with the average cell volume, whether it is total volume, total surface, total length or total number.

A variant of this method has recently been published (Ochs et al., 2004). The physical disector is still used for sampling the cells on semithin LM sections and then the local stereological estimators, nucleator (Gundersen et al., 1988) or planar rotator (Jensen & Gundersen, 1993), estimate the volume of the cells of interest. The number or size density of the subcellular structures can then be estimated at the EM level as described above and the totals calculated.

The rotator and total intracellular structural volume per cell

The rotator principle (Jensen & Gundersen, 1993) is a two-step, number-weighted, local size estimator. It exists in both isotropic and vertical versions; for simplicity all examples in this text are vertical (Baddeley et al., 1986). First, the cells are sampled uniformly in proportion to their number using the disector as illustrated in Fig. 3. Second, the total volume of the intracellular structure is estimated by measuring distances from the vertical axis through a unique point, e.g. the nucleolus in a cell.

Figure 3.

Top: 3D sketch of a block with two semithin toluidin-blue sections indicated. One cell is sampled because its nucleolus is only seen in the (second) sampling section of the disector, not in the first look-up section. The dashed line indicates the first ultrathin section cut thereafter from the block. Bottom: a low magnification picture of the sampling section from the physical disector (shown to the left) with the sampled cell indicated. The picture is used as a map to locate the sampled cell profile in the EM section, cf. also Fig. 4, below.

The practical procedure for sampling of cells may be carried out by using the nucleolus as a sampling unit and two consecutive EM-sections as a physical disector at low magnification or by using semithin LM sections as already described. The nucleoli present in one section and not in the other and vice versa are sampled as illustrated in Figs 2 and 3. In some instances this procedure may be substituted with a less time-consuming sampling procedure in which the centrioles or centrosomes (each centrosome contains a pair of centrioles) are used as a unique reference ‘double-point’ (Mironov & Mironov, 1998; McCullough & Lucocq, 2005). In this case, cells are sampled number-weighted on a single EM section if the centrosomes/centrioles are present in this section. This is a much more efficient sampling methodology but requires that the centrosomes/centrioles have a constant size and number per cell under study.

When the total volume of intracellular structures is to be estimated, isotropic test lines (on either isotropic, uniform random sections or vertical sections) are required.

The total structural volume per average cell can be estimated in two ways when the correct number-weighted sample of cells is obtained:

(1) The distances from the axis through the unique point to the structure profiles are measured, + and see Fig. 4, and the total structural volume is estimated:

Figure 4.

The direct estimate of total volume per cell using the rotator. Left: uniformly positioned test lines with distance h intersect subcellular structural profiles. To each side of the axis the distances to the near and the far end and of the intercept are measured, ℓ+ and -, respectively. Right: a set of systematic points is superposed in a random position but with point columns parallel to the vertical axis. The column numbers c are indicated as is the area per point, a/p, and the side S of that square. For each column of points the distances d to the axis is computed as d = abs(Δ + c · s), where Δ is the random distance from the nucleolus to the column marked 0.


The height between test lines is denoted h and the number of cells studied is denoted n.

(2) As shown in Fig. 4 right, the structural profiles are superposed with a set of systematic points. Where a test point hits a profile the distance, d, from the test point to the axis through the unique point in the cell is measured. The direct estimator of total structural volume per cell is the surprisingly simple equation:


The area per test point in the counting grid is denoted a/p and n is the number of cells investigated.

The rotator and the total number of subcellular structures per cell

This setting requires number-weighted sampling of cells, which can be performed by a physical disector at the LM or EM levels or by using the centrosomes/centrioles as a unique reference ‘double-point’ at a single EM section, as already described. When the cells are sampled, the rest of the procedure is performed at the EM level and requires isotropic, uniform random sections or vertical sections. The subcellular structures are sampled in a physical EM disector. The distances, d, from the centre of the sampled intracellular structural profiles to the vertical axis through the nucleolus are measured (see Fig. 5). The number of structures per sampled cell, Ni(X, cell), and the average total number of structures per cell, NN(X, cell), are estimated as:

Figure 5.

Total number of subcellular structures per cell using the rotator. The cells are sampled either in a physical disector at the LM level (see Fig. 3) or at the EM level. The picture may be used as a map to locate the sampled cell profile in the EM section In the second step, the subcellular structures under study are sampled in a disector at the EM level if they are present in one section but not in the other look-up section (not shown), as indicated. The distance, d, from the centre to the vertical axis is measured for each of the sampled subcellular structures.


The height of the physical disector for sampling of the cell structures is denoted h(dis) and n is the number of cells studied. In practice, two consecutive sections are used and h(dis) is the thickness of the ultrathin sections.

The rotator and total amount per cell

This methodology requires uniformly random sampling of EM sections of a known thickness t and systematic, uniformly random sampling of test squares. The section thickness may be measured by Small's smallest fold (Small, 1968). The test areas have size a, and the sampling area for each test area is denoted A (see Fig. 6). In each test area, the jth, the amount, mj, of the chemical compound or physical structure under study is measured. Measurements can be performed as counting of gold particles per test area, elementary analysis of various compounds per test area or point counting of certain cytochemical markers. The distance from the centre of each small test area, dj, to the axis is measured. The total amount per sampled cell, Mi(X, cell), and the average total amount per cell, MN(X, cell), are estimated:

Figure 6.

Total mass of substance per cell using the rotator. The real mass, mj, is estimated in small, uniformly positioned test areas of area a, each sampled in a larger area A. The distance d from the vertical axis to the centre of each (row of) test area is measured.


The number of cells studied is n.

The greatest advantage of immuno-EM is the use of colloidal gold markers, which with their excellent contrast are perfectly suited for stereology. Some of the latest developments in quantification of colloidal gold markers are discussed in Mayhew et al. (2002).


It will be exciting to introduce in electron microscopy some of the ‘local’ stereological probes through arbitrarily fixed points for direct estimation of total quantities, and the 3D spatial distribution of the structure with respect to the cell nucleus or nucleolus. Nowadays, most electron microscopes have been equipped with digital cameras, which will greatly improve the implementation and efficiency of these methods. New developments in stereology software combined with image analysis (Gardi et al., 2006) will further improve the efficiency of these methods. In this regard, it will be important to analyse the error variance of the methods according to the rules of nested sampling, see e.g. Gundersen et al. (1999) or Nyengaard (1999). Some practical guidelines for estimating the error variances of the various stereological methods are provided in the latter reference.


The study was supported by Novo Nordisk Foundation, Eva & Henry Frænkels Mindefond, Mindefonden for Alice Brenaa and Foundation of 17.12.1981. MIND is supported by the Lundbeck Foundation. The skilful assistance of A. Berg, H. Andersen, L. Lysgaard, M. Lundorff, H. Krunderup, K.Ø. Kristensen and A. Larsen is greatly appreciated.