Reviewed manuscript: Ochs, M. and C. Mühlfeld (2013) Quantitative Microscopy of the Lung – A Problem-Based Approach Part 1: Basic Principles of Lung Stereology. Am. J. Physiol. Lung Cell Mol. Physiology, 305(1):L15-22.This paper by Ochs Mühlfeld ( 2013) is an excellent guide for using unbiased stereology to estimate the first order characteristics of the lung:

The basic parameters that describe the internal lung structure in quantitative terms are characterized by their dimensions: volume (dimension 3), surface (dimension 2), length (dimension 1), number (dimension 0). Regarding the lung, such parameters may be the volume of alveolar septal tissue, the surface of the alveolar epithelium, the length of nerve fibers innervating the conducting airways or the number of alveoli. (Ochs & Mühlfeld, “The challenges of measuring lung structure by microscopy, and how to handle them by stereology” para. 1)

The data to be collected must be well understood and defined: “target parameters should be defined as endpoints, e.g. hyperplasia of airway smooth muscle cells, hypertrophy of alveolar epithelial type II cells, loss of gas-exhange area, angiogenesis of peribronchial blood vessels. In the next step, it should be defined how these changes can be expressed by simple quantitative parameters, such as cell number for hyperplasia, mean volume for cellular hypertophy, surface area of gas-exchange area, length and number of blood vessels for angiogenesis.” (Ochs & Mühlfeld, “Dissecting lung structure by stereology”).

Two problems that can occur during quantitative microscopy are highlighted: sampling and the loss of a dimension. First, only a small fraction of any organ can be sampled under the microscope, and this can lead to bias. For instance, there is a tendency to sample where the lesions are located during the study of disease:

[…] the problem of size reduction warrants that the chosen samples need to represent the whole organ. Hence they have to be distributed randomly over the whole organ to make sure that each part has an equal chance of being selected and analyzed. (Ochs & Mühlfeld, “Problems of quantitative microscopy”)

Second, each structure we observe under the microscope has ‘lost a dimension.=’:

[…] a volume is represented by an area (the larger a structure the larger the area it occupies on a section), the surface area is represented by a line (the larger the surface area of a structure the longer its boundary line on a section), the length is represented by the number of transects (the longer a structure the higher is its chance to be seen as a transect in a section) and the number of a structure is simply not represented within one two-dimensional section. (Ochs & Mühlfeld, “Problems of quantitative microscopy”)

The solutions to both sampling properly and interpreting lower dimensional data are provided by unbiased stereology.

Ochs and Mühlfeld define [unbiased] stereology as “the science of sampling structures with geometric probes.” The purpose of their review is to “provide specific recommendations for the application of stereology to particular animal models of lung disease such as acute lung injury, lung fibrosis, emphysema, pulmonary hypertension, and asthma” (Ochs & Mühlfeld,“Stereology is the gold standard for lung morphometry”).

Practical stereology is the application of unbiased sampling and measurement principles in order to obtain quantitative data about structures in 3D based on nearly 2D (physical or virtual) sections through the structures by using 3D (or geometric) probes. Thus, stereology provides biologically meaningful 3D data. (Ochs & Mühlfeld)

With regards to sampling, the authors stress two aspects. The first aspect is that every step that reduces the sample size ( from the selection of tissue blocks, tissue sections and microscope fields of view, to where events will be counted) must be carried out in such a way that each portion has an equal chance to be sampled. Systematic random uniform sampling is recommended as the easiest and most efficient way to ensure that the selection of sampling sites is not biased. The second aspect is that to to avoid bias estimating surface or length while sampling, the “randomization of orientation (i.e. giving each spatial orientation an equal chance for being selected by the sectioning process)” is necessary. They point out that isotropic or vertical sections must be used to ensure isotropy of interaction between the probe and the characteristic being probed. (Ochs & Mühlfeld,“Stereology is the gold standard for lung morphometry”) But isotropic and vertical sectioning (which is difficult, inconvenient and sometimes prohibitive when compared to using preferential sectioning) would not be needed to ensure isotropy when estimating surface and length if thick sections can be used. Thick sections allow the use of probes that are themselves isotropic (e.g., Spaceballs and Isotropic Fakir) to estimate length and surface. Thin sections however, are usually used when studying the lung (Weibel et. al., 2007, “Pitfalls of lung morphometry, Thin sections”, Hsia, et. al. 2010, “3.5 Preparation Artifacts,” first sentence). Finally, the fractionator method that requires “keeping track of the sampling fraction” is recommended as an efficient way of sampling. No reference volume is needed as it would be with the NvVref method (Nv is numerical density and Vref is reference volume), since the volume fraction is tracked and the reciprocal is taken and multiplied by the number of events counted. An admonition against reporting ratios obtained with the NvVref method is also given since, for example, the ratios are susceptible to differential artificial changes in the denominator ( i.e., the volume):

It lies in the nature of a ratio that it may be changed by variations in the numerator and/or denominator. Ratios are therefore prone to misinterpretation and should always be converted to total values, e.g. the total number of alveoli in the lung instead of the number of alveoli per unit of lung tissue. Horror examples of misinterpretations based on ratios (the so-called reference trap) are frequent and have a high adverse impact on science. (Ochs & Mühlfeld, “The reference space”)

A further warning is given to avoid using the ‘unshrunken reference volume’ if the numerical density was obtained from data that is affected by artifact, i.e. ‘shrunken’ (Ochs & Mühlfeld, “Prerequisites for proper application of stereology to the lung,” sentence 6). The proportionatorTM variant of the fractionator method for sampling is commented upon: “a very promising technique in theory its implementation into practical lung stereology has been limited in application” (Ochs & Mühlfeld, “How does stereology work”). The idea behind the proportionatorTM is to concentrate on sampling where rare events are present, but the problem is that it is difficult in practice to automatically direct the sampling to such high-signal regions.

When it comes to marking events at the final level of sampling, a lower dimensional geometric shape is used to probe a given characteristic. Points are used to probe volumes, points that ‘hit’ the cross sectional area of the volume are counted (Cavalieri/point-counting), lines are used to probe surfaces; the intersections of the probe lines with the cross sectional lines of the surface are counted (Merz or Cycloids for Sv), planes are used to probe length, the intersections of the planes with the cross sectional transects of the lines are counted (image plane as probe), and volumes are used to probe for number; a unique point on the particle is counted in the disector. “Thus, the measurements are reduced to simple counts” (Ochs & Mühlfeld ) All of these estimate types, except for number, can be done on single thin sections that have been systematically and randomly selected. Number estimation requires a 3D test system with either the physical disector that uses pairs of thin sections or the optical disector that requires thick sections (Ochs & Mühlfeld, ”Which test system for which parameter?”). The disector can be used to sample particles not only for counting but also to estimate their volume and surface. Since the cells are selected without bias to size or orientation, the volume estimates are called number weighted. It is possible to sample particles in a volume weighted manner, for instance with Point-sampled-intercepts, and (unlike with the disector) this can be done in a 2D situation with thin sections. Regardless of how the particles are sampled, the estimate of the particle volume is made with a lower dimensional probe, either with a linear probe (NucleatorPoint-sampled-intercepts) or a planar probe (Planar rotatorOptical rotator), as is the estimation of the particle surface (Surfactor) (Ochs & Mühlfeld, ”Particle size”). Isotropic or vertical sections are required for unbiased particle volume and surface estimation.

Next in the manuscript, the authors explain the difference between eliminating bias ( that we have been discussing above) and considering precision:

[…] the absence of bias (systematic error) in the data […] can neither be detected in the data nor can it be decreased by increasing the sampling or measurement effort. It has to be avoided a priori by using unbiased methods throughout the study.

The authors go on to say: “it can never be assumed that the bias of a method influences the results in the same way in different experimental conditions. Biased methods may or may not give the correct results, the difference, however, cannot be seen in the data”. Precision, on the other hand, is affected by how much sampling is done: “precision can be adjusted as needed in the context of a particular study” (Ochs & Mühlfeld, “Accuracy and precision”).

[Steps to eliminate bias, however] have to take place in advance. Once the samples are embedded or sectioned there is no possibility to influence any of these important preceding steps using the existing sections. Thus if bias has already been introduced here, there is no way to rescue the data, which makes any further efforts at the measurement level useless. (Ochs & Mühlfeld,“Prerequisites for proper application of stereology to the lung”).

The higher levels of the sampling cascade (number of animals per group, number of tissue blocks per animal) contribute significantly more to the overall variation of the data than the lower levers of the sampling cascade (number of sections per tissue blocks, number of fields of view per section, number of counts per field of view) – so this is where the effort should go into in order to be efficient” (Ochs & Mühlfeld,“Accuracy and precision” and also see: Gundersen and Osterby, 1981).

At the end of this paper, the non-trivial and ubiquitous question regarding precision is asked: “How much counting is enough counting. There is no general answer that works in all cases. This is actually determined by the specific conditions of each individual study“ (Ochs & Mühlfeld, “How much counting is enough counting?”). Good practical advice is given on how much to sample: “Do a pilot study in 2 animals to determine an appropriate sampling design (number of samples at each level).” A possible place to start the pilot study is suggested: “100-200 well-distributed counting events per individual for each parameter of interest.” This formula is considered:

CVobs2 = CVbiol2 + CEmeth2

CV is the coefficient of variation

CE is the coefficient of error

“[…] the interindividual variation between the subjects of an experimental group can be observed in the data from the standard variation or the coefficient of variation” (Ochs & Mühlfeld, ” How much counting is enough counting?”). The observed coefficient of variation (CVobs) is the standard deviation among the data in the group divided by the mean:

CVobs = SD/mean

SD is the standard deviation

The observed coefficient of variation can be thought of as consisting of two parts, the biological (CVbiol) and the sampling or methodological (CEmeth) contributions. The biological contribution is unknown at first, but the methodological contribution can be estimated using various formulas for the coefficient of error. Once the CEmeth is estimated, and the CVobs calculated, one can solve for the CVbiol. Now the relative contribution of the sampling and the natural biological contribution to the variance within a group can be considered by looking at:

CEmeth/ CVbiol2

The authors suggest keeping this ratio between one-fifth and one-half (Ochs & Mühlfeld, ” How much counting is enough counting?”; and also see: West, chapter 10, Optimizing the Sampling Scheme: Amount of Individuals, Sections, and Probes).

This manuscript gives a great explanation of the major pitfalls that can cause bias while carrying out unbiased stereology, and also gives useful advice about how much to sample, i.e. how to think about the precision of the sampling. Once this theory is studied please see the companion paper for practical, worked-out examples showing how to use unbiased stereology on thin sections to estimate the number, length, surface and volume of various pulmonary structures in normal and diseased lungs. (Mühlfeld and Ochs, 2013)


Hsia, C.C.W., Hyde, D.M., Ochs, M. and E.R. Weibel, on behalf of the ATS/ERS joint Task Force on the Quantitative Assessment of Lung Structure (2010) An Official Research Policy Statement of the American Thoracic Society/European Respiratory Society: Standards for Quantitative Assessment of Lung Structure. Am. J. Respiratory Critical Care Med., 181, 394 – 418.

Gundersen, H.J. and R. Osterby (1981) Optimizing Sampling Efficiency of Stereological Studies in Biology: or ‘do more less well!’. J. Microscopy, 121, 65 – 73.

Ochs, M. and C. Mühlfeld (2013) Quantitative Microscopy of the Lung – A Problem-Based Approach Part 1: Basic Principles of Lung Stereology. Am. J. Physiol. Lung Cell Mol. Physiology, 305(1):L15-22.

Mühlfeld, C. and M. Ochs (2013) Quantitative Microscopy of the Lung — a Problem-Based Approach Part 2: Stereological Parameters and Study Designs in Various Diseases of the Respiratory Tract. Am. J. Physiol. Lunc Cell Mol. Physiology, 305(3):L205-21.

Weibel, E.R., Hsia, C.C.W., and M. Ochs (2007) How much is there really? Why stereology is essential in lung morphometry. J. Applied Physiology, 102, 459 -467.

West, M.J. (2012) Basic Stereology for Biologists and Neuroscientists, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, N.Y.



Note: to estimate the arithmetic mean thickness or harmonic mean thickness of membranes or barriers in the lung, use the orthogonal intercepts probe (Ferrando, etal. 2003).

Ferrando, R.E., Nyengaard, J.R., Hays, S.R., Fahy, J.V. and P.G. Woodruff, 2003, Applying Stereology to Measure Thickness of the Basement Membrane Zone in Bronchial Biopsy Specimens. J. of Allergy and Clinical Immunology, Vol. 112, pp. 1243-1245.