Tissue shrinkage and unbiased stereological estimation of particle number and size


  • K.-A. Dorph-Petersen,

    1. Stereological Research Laboratory and Electron Microscopy Laboratory, University Institute of Pathology, Institute of Experimental Clinical Research, University of Aarhus, Denmark
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  • J. R. Nyengaard,

    1. Stereological Research Laboratory and Electron Microscopy Laboratory, University Institute of Pathology, Institute of Experimental Clinical Research, University of Aarhus, Denmark
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  • H. J. G. Gundersen

    1. Stereological Research Laboratory and Electron Microscopy Laboratory, University Institute of Pathology, Institute of Experimental Clinical Research, University of Aarhus, Denmark
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Karl-Anton Dorph-Petersen, Department of Psychiatry, University of Pittsburgh, W1651 BST, 3811 O'Hara Street, Pittsburgh, PA 15213, USA. Tel.: + 1 412 624 9909; fax: + 1 412 624 9910; e-mail: karl-anton@dorph-petersen.dk*Part of paper presented at the 10th International Congress for Stereology, Melbourne, 1–4 November 1999.


This paper is a review of the stereological problems related to the unbiased estimation of particle number and size when tissue deformation is present. The deformation may occur during the histological processing of the tissue. It is especially noted that the widely used optical disector may be biased by dimensional changes in the z-axis, i.e. the direction perpendicular to the section plane. This is often the case when frozen sections or vibratome sections are used for the stereological measurements. The present paper introduces new estimators to be used in optical fractionator and optical disector designs; the first is, as usual, the simplest and most robust. Finally, it is stated that when tissue deformation only occurs in the z-direction, unbiased estimation of particle size with several estimators is possible.

1. Introduction

In modern biomedical research, stereological methods are of great importance for gaining accurate quantitative information about cells, tissues, organs or organisms. In most cases, the stereological methods are applied using histological sections and microscopy. However, when studying the sections it is very important to remember that the image seen in the microscope may be far from the appearance of the living tissue. The tissue may be deformed, often shrunken, during the histological processing preceding microscopy. The main motivation for writing this paper was the realization that the now widely used optical disector and optical fractionator designs may be affected by tissue deformations, see Fig. 1.

Figure 1.

 A deformed histological section is seen in cross-section. The horizontal dashed line at the top indicates the original position of the upper surface of the section. A region containing cells is marked. Five disectors sample zero, two, three, five and zero cells, respectively. From the figure it is clear that the cells are not sampled uniformly − the larger the shrinkage the larger the effective height sampling fraction. Using classical number estimators with this non-uniform sample will result in a biased estimate.

2. Background

An overview of a typical histological process for light microscopic sections is shown in Fig. 2. The field of interest is the tissue or organ in the living subject. However, before the tissue can be studied in the microscope, it passes through several processing steps, almost all of which may result in deformation of the tissue, cf. the discussion in Ladekarl (1994).

Figure 2.

 (A) Organ in living subject; (B) organ in dead subject; (C) organ removed from subject; (D) formaldehyde fixed organ; (E) fixed organ embedded in agar; (F) slabs cut from fixed organ; (G) blocks of fixed tissue; (H) dehydration of tissue block; (I) embedding of dehydrated tissue block; (J) sectioning of embedded tissue; (K) section of tissue floating on liquid; (L) section mounted on glass slide; (M) section fixed on glass slide-typically by drying; (N) staining of section; (O) section covered by cover slip. Notice the difference in scale between (A)–(F) and (G)–(O).

2.1. Types of deformation

It is important to remember the spatial distribution of deformation (cf. Fig. 3).

Figure 3.

 Different sorts of deformation of tissue blocks or thick sections in the direction of the z-axis. Note that all four examples are anisotropic, as there is no deformation in the xy plane. Here and anywhere else in this paper the z-axis is the direction perpendicular to the section plane and the xy plane is the plane of the section.

Deformation may be either homogeneous or differential. Homogeneous deformation is general deformation affecting the whole tissue in the same way. By contrast, differential deformation varies in the different compartments of the tissue, e.g. varying deformation of white and grey matter in brain tissue. It may also be that some components of the tissue behave systematically in a different way from the rest (Deutsch & Hillman, 1977). Consider the ‘frightening’ example of shrinkage of the cells and swelling of the matrix resulting in no global change.

Deformation may also be either isotropic or anisotropic. Isotropic deformation is the same in all directions, whereas anisotropic deformation is not. This happens typically in layered structures or tissues containing bundles. As illustrated in Fig. 4, a common reason for anisotropic shrinkage is the action of exterior, non-isotropic forces on the tissue.

Finally, deformation may be either uniform or non-uniform. The term uniform deformation is used when there is no local variation in the deformation, whereas non-uniform deformation describes the presence of local differences in the deformation. Non-uniform deformation occurs within the same compartment, in contrast to differential deformation, which is variation in deformation between compartments. Non-uniform deformation may correlate to structural variations in the compartment, e.g. cell density.

Mixes of all three types of change may occur in the same tissue at the same time, and the net result, the sum of the local deformation, may be a more or less pronounced global deformation. The global deformation can be detected by global measures, e.g. as a change in the total volume of the tissue blocks.

In a histological section, several types of deformation in the z-axis may theoretically occur, see Fig. 3. In an example by Andersen & Gundersen (1999) it has been shown that the deformations in vibratome sections are mostly homogeneous but extremely anisotropic with regard to the xy and z directions in the section, i.e. of the types shown to the left in Fig. 3.

All the above-mentioned types of tissue change lead to some sort of dimensional changes of the tissue. Other types of deformation exist that need not change the dimensions of the tissue but change its shape, e.g. torsion and bending. They are not discussed here as they can mostly be prevented by careful processing of the tissue and because they do not affect the more common stereological estimators that are the scope of this paper.

2.2. Block advance and section thickness

In step J, Fig. 2, the section is cut from the block. When the section is cut, the block is reduced in height perpendicularly to the section plane, see Fig. 4. This reduction is called block advance, BA, as it defines the distance the block is advanced towards the knife in the microtome before the section is cut. The BA determines the hitting probability of the particles within the block. BA can be regarded as the section thickness without the deformations of steps J to O (Fig. 2).

Figure 4.

 Top: the cutting of a section from the block. The block advance, BA, may differ from the initial section thickness, t*. t* typically depends on the temperature and buffer concentration of the water bath into which the section is floated (Gerrits et al., 1987). Bottom: the local final section thickness, t, may vary in different positions in the section and the mean final section thickness, t¯, may be different from both BA and t*.

To determine the average BA, the microtome or vibratome used for sectioning has to be calibrated; the factory specifications of the machines may be precise, but it has to be controlled as a part of good laboratory practice. One method is to cut a large number of sections from a tissue block and measure the change in block height. This change divided by the number of sections is a good measure to determine the average BA. In some cases it is possible to measure the movement of the arm to which the tissue block is fixed using a vernier calliper. Again, a large number of sections are cut and the progression of the arm divided by the number of sections is a good measure of average BA.

Naturally, there may be variations in the actual amount of tissue removed from the tissue block both within a section and among sections. The variation within a section is often very small, but may occasionally be quite large. Sometimes when cutting vibratome sections a specific region of alternating sections may have double or zero thickness. However, a trained technician will in most cases be able to produce high quality sections with very low variation in thickness both within the section and among sections. For the average BA to be valid for the section it is important that the section used for the estimation is chosen independently of its quality, as a ‘good section’ is often thicker than the mean.

The final section thickness, t, is typically measured in the microscope by focusing from the top to the bottom of the section and measuring the vertical movement of the table. This is best done by a microcator attached to the table, not by using the marks on the focus knob of the microscope.

Whenever distances in the z-axis are measured by focusing it is important to use immersion optics, and the immersion fluid should have a refractive index matching that of the mounting medium. Without these, the table movement and the travel of the focal plane within the section will not correspond to each other. Typically sections are mounted using a mounting medium with a refractive index near 1.5 matching that of the cover slip glass. If a dry (air) objective is used to examine such a section a pronounced optical shortening of c. 35% in the z-axis will be observed, cf. e.g. Glaser (1982). Using an oil immersion objective prevents this bias. If an aqueous mounting medium is used (e.g. vibratome or frozen sections, when not dehydrated) a water immersion objective should be used for the measurements. If an oil immersion objective is used instead in this case, the mismatch of the refractive index of the immersion oil (∼ 1.518) and that of water (∼ 1.33) will cause a pronounced optical elongation of 15–20% in the z-axis. The elongation is not constant but varies with distance to the section surface, cf. Hell et al. (1993) and Sheppard & Török (1997).

The section thickness should be measured by uniformly random sampling with a constant sampling fraction in the whole set of sections, i.e. in a fixed fraction of the fields of vision observed. If the variation in the thickness observed is very small a central measurement in every 5–10 fields or less may suffice. Both the intrasectional and the intersectional coefficient of variation, CV = SD/mean, may be calculated.

2.3. Embedding and sectioning techniques

This section contains a discussion of the four main principles of embedding and sectioning tissue, i.e. the methods used from steps G to L (Fig. 2).

Paraffin embedding is the classical method of tissue embedding. The major advantage is that many staining methods have been developed for paraffin over the last century, including many immunohistochemical stains. Moreover, it is possible to make thin sections (2–3 µm) and very thick sections (> 100 µm). The main problem is the very pronounced and unpredictable tissue deformation, often in the form of shrinkage by about 50% in volume. Substantial variations occur, depending not only on the tissue (Haug, 1984; Iwadare et al., 1984; Miller & Meyer, 1990).

Plastic techniques using glycol methacrylate are a very versatile embedding method, cf. Gerrits & Horobin (1996). The major advantage is a much lesser degree of tissue deformation compared with paraffin embedding (Hanstede & Gerrits, 1983; Miller & Meyer, 1990). The global deformation may therefore be minimized, presumably without differential deformation, because the use of plastic results in a quite homogeneous matrix. It is possible to generate both quite thin (0.25–2 µm) and thick (> 100 µm) sections and they have a very uniform quality and even thickness if using a rotatory microtome (Helander, 1983). The main disadvantage is that many specific immunohistochemical stains cannot penetrate into the plastic sections, although special techniques allow immunostaining at the surface of the section.

Freezing has become a more frequently used ‘embedding’ method. Thin (2–4 µm) and thick (> 100 µm) sections may be generated and, most importantly, frozen sections are very well suited for designs using specific immunohistochemical stains. The procedure is fast compared to the previous two methods. It is possible to avoid deformation in the section plane (J. O. Larsen, personal communication) but sometimes a marked compression of the section occurs during the sectioning process (Fig. 2, J). Depending on details in the protocol, the final section thickness may be almost equal to the block advance but often the section collapses considerably during the final steps (L–O, Fig. 2).

Vibratome sectioning does not require ordinary embedding and thereby it bypasses steps H and I, Fig. 2. This method is cheap, fast and easy and may be used with most stains including the immunohistochemical stains. The main drawbacks are that it is not possible to make thin sections (sections are usually thicker than 30 µm depending on fixation and tissue) and the sections may collapse by as much as 70%, especially in step M (Fig. 2), where the vibratome sections are dried on glass slides. The deformation may be very anisotropic and restricted to the z-axis without deformation in the xy-plane, cf. Andersen & Gundersen (1999) and Dorph-Petersen (1999). In some cases it is possible to minimize or even avoid deformation in the z-axis by not drying the sections at all and using an aqueous mounting medium when fixing the cover slip (J. O. Larsen, personal communication).

3. Deformation and number estimation

When estimating particle numbers, deformation may look like only a minor problem: no matter how much the tissue is deformed, the number of particles (the cardinality) is preserved. A closer look shows some treatable problems depending on the stereological design.

In the following discussion the number estimators are divided into physical designs and optical designs. General as well as specific requirements for the different estimators are given in short lists in the text below. To use e.g. the optical fractionator, the general requirements for unbiased number estimation, the specific requirements for optical designs and the specific requirement for fractionator designs all have to be fulfilled.

A histological design most often has several initial steps of splitting tissue into pieces and subsampling among these as shown in E–G (Fig. 2). In the following the simplified setting with only one step will be used as an example. In that context, all the four main estimators are based on subsampling of the particles at the level of blocks, sections, areas, height (in the two optical designs only) and finally particles. The area of the sections is sampled by unbiased counting frames (Gundersen, 1977), and particles are sampled by disectors. Systematic, uniformly random sampling is applied at all levels in order to generate an unbiased sample in the most efficient way.

The reader who is not familiar with stereological terminology should note that in both the fractionator and the disector designs the sampling probe used is called a disector. Two short and general reviews of particle number estimation without the present focus on deformation problems are given in Dorph-Petersen et al. (1998) and West (1999). A longer review is found in Howard & Reed (1998).

General requirements for unbiased particle number estimation:

1 It must be possible to observe and recognize all particles of interest in the containing space. This implies that:

(a) The containing space must be included in toto before subsampling to ensure that all particles have a constant (uniform) probability of being sampled.

(b) It must be possible to unambiguously identify a particle from the set(s) of profiles produced by one or more sections through it.

2 Sampling must have constant intensity in the reference space, i.e. all cuts and sections must be positioned uniformly random in the reference space (systematically for best efficiency).

3.1. Physical designs

The physical designs are based on physical disectors as sampling unit, i.e. two aligned consecutive and thin sections of a transparent containing space, cf. Sterio (1984). ‘Thin’ is ≤ 25% of particle height, i.e. typically 2–5 µm for cell counts.

Specific requirements for physical designs:

1 The histological technique used must be able to generate consecutive thin sections.2 No particles may be completely lost during cutting.

3 The distance between the two sections in a disector pair must be sufficiently small so that no particles are left undetected in the sections.

Note that all sorts of deformation in the direction of the z-axis are allowed after sectioning because the complete thickness is observed.

It should be mentioned that the number of features other than isolated particles can be estimated using physical disectors, cf. e.g. Gundersen et al. (1993).

The major hurdle in the physical designs is the slow aligning of vision fields. Several solutions to the field registration problem have evolved recently and improved the efficiency of the physical designs. One approach is based on a specially constructed double microscope (Howard & Reed, 1998; Howard et al., 1999). An alternative and maybe easier solution is to use a stereological package such as the CAST® (Olympus Denmark), which includes software enabling one microscope to step between two physical sections both mounted on the same table.

3.1.1. The physical fractionator design

The physical fractionator design (Gundersen, 1986) is based on a direct count of particles in a known predetermined fraction of the containing space. First the containing space is divided arbitrarily into blocks. Second, a predetermined fraction of the blocks is sampled, then the blocks sampled are sectioned exhaustively and again a predetermined fraction of the section is sampled (in consecutive pairs). A predetermined fraction of the area of the sections is sampled by unbiased counting frames, and finally particles are sampled by physical disectors.

Specific requirement for fractionator designs:

1 The sampling fraction at each level must be known.

The size and shape of the containing space are not relevant, it does not even need to have well-defined borders.

The total number of particles N is estimated by multiplying the number of particles sampled by the reciprocal of the sampling fractions:


where bsf is the block sampling fraction, ssf is the section sampling fraction, and asf the area sampling fraction. ΣQ is the total count of particles finally sampled by physical disectors. This number estimate is completely independent of deformation problems. In that sense, the physical fractionator is the gold standard for number estimation.

3.1.2. The physical disector design

The physical disector design (Sterio, 1984) is the classical estimator of the disector × Cavalieri type. In this design, the blocks are not sectioned exhaustively, i.e. only a single section pair per block is typically needed.

Specific requirements for disector designs:

1 The reference volume, V(ref), containing the particles must be known (estimated).

2 It must be possible to recognize its borders at all sampling levels.

3 The exact volume of the sample (disectors) must be known (estimated):

(a) The area of the reference space within the sampling frames must be estimated, usually by counting the number of test points hit.

(b) The block advance, BA, must be known.

(c) Any global deformation of the reference volume (after estimation of its volume) must be known precisely.

Any kind of deformation is allowed to be anisotropic. Differential deformation of internal structures inside the reference volume is allowed. Note that the sampling fraction at each level need not be known.

In the disector designs, the reference volume, V(ref), is typically estimated by use of the Cavalieri principle (Gundersen & Jensen, 1987) but may in some cases be measured directly. The total number of particles, N, is estimated as the product of V(ref) and the numerical density in the sample, NV :


If no deformation occurs after the estimation of the reference volume, V(ref), the numerical density, NV , is estimated as the total number of particles sampled by physical disectors, ΣQ, divided by the estimated sample volume, BA·(a/p) ·ΣP:


where a is the area of the counting frame used for the area subsampling, p is the number of reference points per counting frame (one or a few points per frame is often sufficient) and ΣP is the total number of counting frame reference points hitting the reference volume.

If deformation occurs after the estimation of V(ref), the NV of the sample is no longer representative for the original numerical density. Most often V(ref) is estimated at step F (Fig. 2) and the two correcting constants mentioned below must be estimated.

The volume deformation, VD, before sectioning, i.e. from steps F to J (Fig. 2), is estimated as:


where ΣV(after) is the sum of block volumes in step J (Fig. 2) and ΣV(before) is the sum of block volumes in step F (Fig. 2).

The area deformation, AD, after sectioning, i.e. from steps J to O (Fig. 2), is estimated as:


where ΣA(section) is the sum of the section areas in step O (Fig. 2) and ΣA(block) is the sum of the areas of the section surface of the blocks in step J (Fig. 2).

Finally, NV is estimated as:


In practice the estimation of AD is straightforward. The areas of the section surface of the blocks in step J (Fig. 2) as well as the final section areas in step O are easily estimated by point counting (e.g. using a superimposed point grid printed on a transparency). The estimation of VD may be somewhat more difficult. The three dimensions of the tissue blocks in step G are easily measured using a calliper and V(before) calculated. However, in step I only two of the three dimensions of the embedded tissue may be observable for direct measurement. The height of the embedded tissue block may be hidden within the embedding medium (as shown in I and J, Fig. 2). If this is the case, the embedding medium may be trimmed from one side of the blocks making the height measurements possible. Alternatively, each of the blocks may be cut having a uniform random orientation around the y-axis (the depth). Each block is then embedded with a random one of the four sides parallel with the y-axis upwards. The shrinkage in the x-axis (the width) of the blocks observed in step J may then be used as an estimate for the shrinkage in the z-axis (the height). See also Fig. 7 in Braendgaard et al. (1990).

3.2. Optical designs

The optical designs are based on optical disectors as the sampling unit, i.e. one thick section (≥ 25 µm) is scanned using thin optical focal planes (< 1 µm). Optical designs have been the standard of efficient, unbiased number estimation for some years, cf. Gundersen (1986) and West et al. (1991), as this method bypasses the slow aligning of vision fields.

However, optical designs are more vulnerable to deformation. The sampling probe, optical disector, subsamples the thick sections in the z-axis. This means that contrary to the physical designs, deformation in the z-axis must be considered.

Specific requirements for optical designs:

1 No particles must be lost from the disector sample, i.e. depending on the particle height, sufficiently large guard regions near the top and the bottom of the section must be respected.

2 The final section thickness, t, must be known.

3 No differential deformation in the direction of the z-axis may occur.

The last requirement is needed because the optical disector is not able to sample the interior of a section uniformly random, it is always central in the section restricted from the guard regions at the upper and lower surface of the section.

3.2.1. The optical fractionator design

The optical fractionator design (West et al., 1991) is similar to that of the physical fractionator except for an extra sampling level in the z-axis by optical sectioning.

If there is either no deformation or homogeneous, uniform deformation in the direction of the z-axis the total number of particles, N, is estimated by multiplying the number of particles sampled by the reciprocal of the sampling fractions:


where hsf is the height sampling fraction and h is the constant height of the optical disectors.

If there is homogeneous, non-uniform deformation of the sections in the direction of the z-axis, the local section thickness, t, must be known in a sufficiently large fraction of the positions of particle sampling. The total number of particles, N, is estimated by Eq. (7), but now the hsf depends on t¯Q which is the Q-weighted mean section thickness:


where ti is the local section thickness centrally in the ith counting frame with a disector count of inline image. This estimate is unbiased even when systematic differences in section thickness occur in and among sections correlated to the local amount of particles sampled. This is easy to understand: see every optical disector as a small fractionator. The total number of particles sampled in a disector times the local section thickness divided by the disector height (qit/h) is an unbiased estimate of the total number of particles in the full section thickness at the position of the disector. Thus, the average number of cells in the full section thickness per cell sampled in a set of optical disectors is the Q-weighted mean section thickness divided by h.

Note that as the sums in Eq. (10) are only summed over the disectors where section thickness is measured, the sum in the denominator is different from ΣQ in Eq. (7).

The section thickness has to be measured in a sufficient number of places. The exact size of the fraction of the counting frames in which t must be measured depends primarily on its variation. It is probably a good idea to keep the coefficient of error of the section thickness, CE(t) = SEM/mean, well below 5%. inline image may be either larger or smaller than CE(t) depending on the sign of the covariance between ti and Qi.

Note that the workload is not increased when using Eqs (7), (9) and (10), compared to the classical approach of Eqs (7) and (8) (except a slight increase in book keeping). We therefore recommend readers use the new estimator in any case and not to be concerned whether or not the z-axis deformation is uniform.

3.2.2. The optical disector design

The optical disector design (Gundersen, 1986) is the analogue of the physical disector, the only difference is the added level of sampling in the z-axis.

As usual in a disector design, the total number of particles, N, is estimated by Eq. (2). If no deformation occurs after the estimation of V(ref), NV is estimated as the total count of particles sampled by optical disectors ΣQ divided by the estimated sample volume h·(a/p) ·ΣP:


If there is no deformation after the estimation of V(ref) except a homogeneous, uniform deformation of the sections in the direction of the z-axis, it must be taken into account that t and BA are different. NV is estimated almost as in Eq. (11) but multiplied by the fraction t¯/BA:


If there is homogeneous, non-uniform deformation of the sections in the direction of the z-axis and no other deformation after the estimation of V(ref), the local section thickness must be known in a sufficiently large fraction of the positions of particle sampling. In this case NV depends on the Q-weighted mean section thickness t¯Q (Eq. 10) and is estimated almost as in Eq. (11) but multiplied by the fraction t¯/BA:


This estimator is unbiased even if systematic differences in section thickness occur in and among sections and are related to the particles sampled.

In the case where all three kinds of deformation are present after the estimation of V(ref) − deformation of the blocks, deformation of the sections in the plane of the section and homogeneous, non-uniform deformation of the sections in the direction of the z-axis − the estimator is analogue to the estimator given by Eq. (6). The volume deformation, VD, is estimated by Eq. (4) and the area deformation, AD, is estimated by Eq. (5). Then NV is estimated as:


The main drawback of the optical disector design is that contrary to the optical fractionator, the measurement of BA and the estimation of V(ref), VD and AD are necessary. The uncertainty of these measurements introduces some noise in the estimator making it less precise (but still unbiased). On the other hand, it is not necessary to use a predetermined and constant sampling fraction at each sampling level, which means that there is no need for exhaustive serial sectioning. This makes it possible to use more sophisticated sampling designs, cf. Dorph-Petersen (1999).

3.3. Alternative designs

It may sometimes be convenient to use a slightly modified version of one of the four designs described above. Two useful modifications are mentioned here.

3.3.1. Estimation of V(ref) after embedding

This is a method that allows disector designs when using paraffin sections (Pakkenberg & Gundersen, 1988; Korbo et al., 1990). Small to medium sized organs may be paraffin embedded in toto in one or several blocks. The tissue embedded in (large) paraffin block(s) is used as the reference space. The blocks are sectioned exhaustively and a subsample of the sections is chosen systematically, uniformly randomly.

It is easy to estimate V(ref) using the Cavalieri principle from the distance between the sampled sections and the sum of the areas of the reference space seen in the sections sampled. Using the same sections the numerical density, NV , is estimated using one of the two disector designs mentioned earlier. Both V(ref) and NV are severely influenced by the inevitable deformation of the tissue introduced by paraffin embedding, but the deformation will have no impact at all on the final estimate of total particle number.

A physical as well as an optical disector design may be used. VD equals zero as these steps are eliminated, but AD, BA and (in an optical disector design) t must be measured.

If the estimation of V(ref) and NV is made in one step combining the Cavalieri estimator and the numerical density estimator, the resulting estimator is very close to the fractionator. They are not identical because the disector and the fractionator are differently influenced by missing tissue in the sections (due to artefacts) and sections hitting the artificial edges of V(ref) (when V(ref) is divided into several blocks). See Pakkenberg & Gundersen (1988) for further details.

3.3.2. Fractionator without exhaustive sectioning

It is possible to make a fractionator design without the time-consuming exhaustive sectioning of the blocks. It is possible when the blocks used for sectioning are positioned systematically, uniformly randomly in the containing space and are made with the same known height. This is the case when the tissue is divided into slabs of a known thickness, T, by systematic, uniformly random cuts. One section is cut from the upper surface of every slab. In this design the section sampling fraction, ssf, is replaced by a tissue sampling fraction, tsf = BA/T. This is the only difference from the fractionator designs described earlier. In addition to the obvious advantage of the reduced workload of the sectioning, it is possible to use the remaining tissue in the slabs for other purposes (cf. Dorph-Petersen, 1999). The disadvantage is the need to know both T and BA, which makes the fractionator a bit less precise (but still unbiased).

4. Deformation and size estimation

4.1. The different size estimators

The estimators of particle size are usually divided into global and local estimators:

The global size estimators are estimators of population totals, e.g. total volume or surface of all cells in a tissue. Combined with an unbiased estimate of particle number, the mean particle size can be estimated. For a review of the global estimators of volume, surface area and length see Gundersen et al. (1988).

The local size estimators are estimators of individual particle size, i.e. besides an estimate of mean particle size, the distribution of unbiased particle size estimates may be obtained. How close this distribution maps the real size distribution depends on the precision of the individual particle size estimates. This precision again depends on the 3D-shape of the particles, the choice of estimator and the number of measurements per particle. Jensen (1998) is a review of the family of local estimators.

In the present context it is more relevant to divide size estimators into planar and spatial estimators.

The planar size estimators only use measures within a thin section or a thin optical focal plane (in a thick section). This group comprises the classical global estimators of volume, surface area and length, and also the classical versions of the local estimators: estimation based on point sampled intercepts (Gundersen & Jensen, 1985), the selector (Cruz-Orive, 1987), the planar nucleator (Gundersen, 1988) and the planar rotator (Jensen & Gundersen, 1993).

The spatial estimators use three-dimensional measures obtained by focusing through a thick section. Examples of estimators in this group are a global length estimator (Larsen et al., 1998), local estimators of particle surface (Kubínová & Janáček, 1998) and particle volume (Tandrup et al., 1997).

The main advantage of the spatial estimators is the use of arbitrarily orientated sections (Kubínová & Janáček, 1998; Larsen et al., 1998) or an increase in precision (Tandrup et al., 1997). In contrast, the planar estimators of length, surface area and local volume are bound by the requirement of IUR (isotropic uniformly random) sections (Mattfeldt et al., 1989; Nyengaard & Gundersen, 1992) or (except for length) vertical sections (Baddeley et al., 1986).

4.2. Strategies when dealing with deformation

In all sorts of size estimation, the problem related to tissue deformation is obvious. The initial deformation related to isolation and fixation of the tissue (steps A to D, Fig. 2) are unavoidable (with exception of in vivo microscopy and possibly in special freezing techniques, cf. Bald (1985) and Sartori et al. (1993)). When estimating size the only solution is in most cases to use the fixed tissue in step D (Fig. 2), as the reference state. It is important to remember that different fixation deformations between groups may occur, see Haug (1984) and Mendis-Handagama & Ewing (1990).

Thereafter, it seems that the only good method is to avoid tissue deformation, or at least minimize it as much as possible. The global size of the tissue must be monitored carefully throughout all the histological steps using extensive measurements and the procedures must then be repeatedly modified until global deformation is avoided or minimized. Note that lack of global deformation does not eliminate the possibility of local deformation. This may limit the choice of histological methods, and plastic embedding is the first choice for size estimation.

4.2.1. Deformation in the z-axis

A special case is optical sampling and size estimation of particles in thick sections deformed only along the z-axis, i.e. when there is no deformation in the plane of the section. This is often the case in frozen sections and vibratome sections. In this case we recommend that the use of spatial size estimators be avoided. In our opinion, it would be questionable to try to correct the 3D measurements of the deformed tissue structures. It is virtually impossible to determine to what extent fibre length or surface area is changed proportional to the z-axis change or just curled up with minor change in size. However, if the only deformation of the objects measured is in the line of projection, the projected or focal images seen in the microscope are unchanged and planar size estimators may be used. This is because these stereological estimators only use measurements in the non-deformed section plane (see Fig. 5).

Figure 5.

 At top left is shown the original thick section (containing ‘penguin particles’) and top right is its projection on the section plane. If the section is only deformed in the direction of the z-axis (middle left) leaving the projected image unchanged (middle right), it is possible to use planar size estimators, as these by definition only use information from sections in the unchanged xy plane. If shrinkage occurs in all three dimensions (bottom) it is not possible to obtain an unbiased size estimate without exact knowledge about the deformation in all directions.

It is important to remember that in cases of non-uniform z-axis deformation, the sampling may be non-uniform if a standard sampling scheme with a disector of fixed height is used (see Fig. 1). Contrary to the number estimators there is no simple correction factor to estimate in order to make the ordinary sampling unbiased. This is because the non-uniform deformation may be correlated not only to the number of cells but also to differences in type and size, i.e. that different regions of the same tissue shrink differently in relation to variations in the particle size.

Two approaches can be suggested. (1) Uniform sampling using an optical disector of variable height sampling a constant fraction of the local section thickness. In this case the ordinary calculations of mean size and the size distributions are unbiased. (2) Unbiased estimates using a non-uniform sampling scheme. This is achieved by recording the local section thickness, t, in every counting frame and using t/h as weight in the calculations. For a general discussion of non-uniform systematic sampling, see Dorph-Petersen et al. (2000).

For non-uniform sampling, the arithmetic mean and the geometric mean of particle volume, inline image and inline image, and the frequency, F, of particles in a given class of the size distribution are calculated as:


where the sums are over all particles (or particles in the size-class). tj is the local section thickness and hj the disector height corresponding to the jth particle, which has an estimated volume of vj(par). When several particles are sampled in the same disector (e.g. particles j and j + 1) they are assigned identical values of t and h (i.e. tj = tj+1 and hj = hj+1). n is the number of cells in the size-class of interest.

Both solutions suggested result in an increase in the workload, as the section thickness has to be measured in every single field of vision sampled where particles are present for sampling. It should be stated that the bias in the estimate of mean particle size and the size distribution could be substantial when using an ordinary systematic, uniformly random sampling scheme. The bias will be significant if there is a large variation in the z-axis-deformation and the amount of deformation is correlated to the particle size. In any case of varying z-axis-deformation it is recommended that one of the approaches suggested is used to verify that the bias is negligible, before moving on to use an ordinary systematic, uniformly random optical sampling scheme.

5. Practical example

This section is included in order to provide a practical example of a design based on the new optical fractionator estimator, which is robust to shrinkage in the z-axis. In the example we estimate the number of toluidine stained neurones in the dorsal raphe nucleus, which is a group of neurones in the brainstem, cf. Baker et al. (1990) and Jacobs & Azmitia (1992).

5.1. Biological material and methods

The brain from a 64-year-old normal male was fixed in 10% phosphate-buffered formalin (4% formaldehyde in 0.1 m phosphate buffer, pH = 7.0) for approximately 2 months. The brainstem was separated from the cerebrum and the cerebellum, embedded in 7% agar and cut into slabs perpendicularly to its longest axis. The mean slab thickness was T¯= 2.93 mm. Each slab was subsequently re-embedded flatly in 7% agar. Using a Bio-Rad Polaron H1200 vibratome, 100 µm sections were cut parallel to the slab surface. The vibratome was calibrated, i.e. the block advance was estimated by measuring the movement of the tissue holder using a vernier calliper. The arm moved 10.00 mm (to the nearest 0.01 mm) when advanced 100 × 100 µm steps, i.e. no difference was detected from the scale on the vibratome and BA = 100 µm.

One set of sections was mounted on chrome-alum coated glass slides and air-dried at room temperature overnight, and was then stained using toluidine blue. For a more detailed description of the design, including the staining protocol, see Dorph-Petersen (1999).

A modified Olympus BH2 microscope equipped with a Heidenhain ND281 microcator and a motorized object table and an object rotator (Olympus, Denmark) was used for the microscopy. A CCD camcorder (JAI-2040, Protec, Japan) connected to a monitor was mounted on the top of the microscope. A computer fitted with a framegrabber (Screen Machine II, Germany) was connected to the monitor and had the CAST-Grid stereology software (Olympus, Denmark) installed. The sections were studied using a 100 × oil-immersion objective (Olympus, S-Plan Apo, NA 1.40) at a final magnification of 2460 ×. Systematic, uniformly random sampling of the fields of vision was performed with X-step= 500 µm and Y-step= 350 µm. In each field of vision, an unbiased counting frame area of a = 3500 µm2 was superimposed and used for sampling of the neurones. The nucleoli of the neurones were used as the sampling unit. No cells with more than one nucleolus were observed.

As the variation in section thickness observed in this particular set of sections was very large (15.5–39 µm), we used a stratified fractionator design with two disectors of different height in order to make the sampling efficient.

The section thickness was evaluated roughly in every field of vision and measured in every third field sampled (close to its centre) using the microcator. If the local section thickness, t, in a field of vision was evaluated to be larger than 20 µm (as in most cases), a disector height of h1 = 10 µm was used. If t was judged to be around 20 µm or less, the section thickness was measured carefully. If it was then confirmed that t < 20 µm, t was recorded (and marked as thin) and a disector height of h2 = 5 µm was used.

The t-values measured were recorded together with the local disector count. In any case, the section thickness was only measured if any cells were sampled in the counting frame.

5.2. Results

The total set consisted of nine transverse sections through the brainstem. Seven of these sections hit the dorsal raphe nucleus. In those seven sections, 245 fields of vision were sampled. ΣQ1 = 205 cells were sampled in 126 counting frames using the disector height h1 = 10 µm and ΣQ2 = 16 cells were sampled in 12 counting frames using the disector height h2 = 5 µm. From the measurements of t in the two sets of disectors (38 and 12 measurements, respectively), t¯Q1 = 25.3 µm and t¯Q2 = 17.7 µm were calculated using Eq. (10), see Table 1.

Table 1.   Calculation of t¯Q −2, i.e. for ti < 20 µm. Note that the difference between the ordinary mean and the number weighted mean is small in this example due to the low variance (CV is small)


  1. t¯Qinline image = inline image = 17.7 µm

Sum16 283.0
Mean 17.5 
CV 0.075 

The total number of neurones, N, was estimated using Eq. (7) modified for this stratified version of the design, see Table 2. The coefficient of error of the number estimate was estimated using the method stated in Gundersen et al. (1999) modified to take the stratified design into account. Var(SURS) in Table 2 refers to the variance in the estimator caused by systematic, uniformly random sampling.

Table 2.   Estimation of the total number of neurones in the dorsal raphe nucleus and the corresponding CE
 SectionQi ,1Qi ,1·Qi ,1Qi ,1·Qi+1, 1Qi ,1·Qi+2, 1
t ≥ 20 µm4193611330
SectionQi ,2Qi ,2·Qi ,2Qi ,2·Qi+1, 2Qi ,2·Qi+2, 2 
  1. tsf: = BA/T¯ = 100 µm/2.93 mm = 10/293
    asf:= a/(X-step × Y-step) = 3500 µm2/(500 µm × 350 µm) = 1/50
    1: = h1/t¯Q1 = 10 µm/25.30 µm = 0.395
    hsf2: = h2/t¯Q2 = 5 µm/17.69 µm = 0.283
    inline image
    Noise1:= ΣQ1 = 205

  2. Var(SURS1):= (3(A1 − Noise1) − 4B1 + C1)/240 = (48744 − 38216 + 2513)/240 = 54.34
    CE1:= SEM/mean = √{Noise1 + Var(SURS1)}/ΣQ1 = √(205 + 54.34)/205 = 0.079
    2:= Q2 = 16
    Var(SURS2):= (3(A2 − Noise2) − 4B2 + C2)/240 = (432 − 192 + 0)/240 = 1.00
    CE2:= SEM/mean = √{Noise2 + Var(SURS2)}/ΣQ2 = √(16 + 1.00)/16 = 0.26
    inline image
    inline image
    inline image

t < 20 µm40000

The present estimate of the total number of neurones in the dorsal raphe nucleus is approximately twice the number reported by Dorph-Petersen (1999). The two studies are not directly comparable, as the previous study used only toluidine stained sections, whereas in the present ongoing study, a parallel set of PH8 immunostained sections is used for a more precise delineation of the dorsal raphe nucleus (DR), cf. Törk et al. (1992). In this way, a somewhat larger region of tissue is included in the reference volume (V(DR) = 129 mm3 in the present case compared with a mean volume of V(DR)= 79.0 mm3 in the earlier study).

6. Discussion

Figure 6 is an overview of the problems of the different histological and stereological techniques and the approach suggested.

Figure 6.

 Schematic overview of the problems of the different histological and stereological techniques and the recommended approach. *See Section 3.3.1 for an exception.

Concerning the new estimators of particle number it is obvious to consider sampling a constant fraction of the section thickness. Sampling the particles with a disector of variable height, say the middle third of the thickness of the sections, would suffice. However, it is crucial that the fraction is strictly constant. If not, the estimator will be biased. Also, it is required that the section thickness is measured in every single counting frame and the observer (or the computer) has to calculate the local disector height and guard regions before the particles are sampled. It is obvious that this simpler approach is less robust and will result in an unacceptable increase in workload compared to the methods described in this paper, as the Q-weighted mean section thickness may be estimated from observations in a suitable fraction of the vision fields.

A more serious problem is the assumption of homogeneous deformation of the z-axis, which must be verified, or at least discussed, before using the new methods for number estimation. It is easy to imagine a situation where the section is deformed asymmetrically as the free surface could behave differently from the surface against the object glass. Surprisingly little attention has been paid to this problem, and further investigations are needed to quantify the magnitude of the problem in relation to various embedding techniques. A recent paper by Hatton & von Bartheld (1999) addresses this particular problem. The authors report finding a U-shaped distribution of neuronal nucleoli and nuclei along the z-axis in plastic and paraffin sections but not in frozen sections. Hatton & von Bartheld interpret this finding as a differential deformation of the sections due to the sectioning process. That interpretation raises the question why the effect is not observed in frozen sections, which are easier to deform than plastic sections. In our previous studies using the optical disector we have sometimes observed U-shaped distributions along the z-axis when the sections were not stained well due to imperfect penetration, a frequent problem when dealing with thick sections. In fully stained sections we have, in contrast, observed a decrease in density close to the surfaces of the sections due to lost caps. This is not observed by Hatton & von Bartheld. In their paper ‘the upper and the lower surfaces were defined by focusing on any stained particles.’ This may imply that they may have had a tendency to neglect tiny regions at the top and bottom of the sections depleted of stained particles. Also, one should not overlook that when a U-shaped density is observed through a thick section, poor stain penetration is the most likely explanation. Anyway, using the approach of measuring the distribution of, e.g. nucleoli as a function of relative depth within the well-stained section, as used in Andersen & Gundersen (1999), it is straightforward and almost always possible to implement a robust method to determine the size of the guard zones needed to avoid problems due to lost caps.

Another problem can be found in the local section thickness measurement. The thickness is measured by focusing up and down through the section observing where the section starts and ends. The problem is that different observers may have differing conceptions of the top and bottom of the section, and that the process is asymmetrical. This implicates that different observers may get different results when measuring the thickness of the same section, see Fig. 7. In addition, the accommodation of the imaging device (eye, photographic plate, CCD) is important for the precision. The problem of observer perception of depth in optical sectioning has been known since at least 1955 (Haug, 1955). As discussed by West et al. (1991), the bias is small when using oil immersion optics with high numerical apertures.

Figure 7.

 The section thickness is usually measured by focusing up and down through the section. The true section thickness is t. In the case of too early a recognition of top and bottom, the thickness will erroneously be too large, tE. Too late a recognition will result in a thickness measurement which is too small, tL.

Unbiased sampling of particles for size estimation has to be made as suggested in section 4.2.1: use one of the two methods suggested. Alternatively, use one of the unbiased methods to evaluate the size of the bias in an ordinary sampling scheme. It may show that the ordinary scheme has only a negligible bias and can be used with some care.

Anyway, it should be stated that it is very important to make the necessary observations for detection of deformation. If not, and deformation is present, the study may very well end up biased. The new estimators are well suited for use with frozen sections and vibratome sections. It is possible to meet the requirements in both cases, and both techniques result in more or less shrinkage in the z-axis. We therefore recommend the use of the new estimators suggested here if an optical disector/fractionator design is used, and the section thickness varies or deformation is detected.


The skilful technical assistance of Maj-Britt Lundorf, Helene Andersen and Charlotte Løchte is greatly appreciated. Eva B.V. Jensen is thanked for statistical guidance. The Institute of Forensic Medicine at the University of Aarhus is thanked for collecting human brains. This study was supported by grants from ‘Århus Universitetshospitals Forskningsinitiativ’, ‘Overlæge, dr.med. Einer Geert-Jørgensen og hustru Ellen Geert-Jørgensens Forskningslegat’, ‘Psykiatrisk Forskningsfond’, ‘P. Carl Petersens Fond’, ‘Pulje til Styrkelse af Psykiatrisk Forskning’, ‘Eli Lilly Danmarks Psykiatriske Forskningsfond’, ‘Lægeforeningens Forskningsfond’, ‘Fonden til Lægevidenskabens Fremme’ and ‘The Lundbeck Foundation’.


  1. *Temp. Part of paper presented at the 10th International Congress for Stereology, Melbourne, 1–4 November 1999.