The History Of Modern Stereology Part Two: The Second And Third Decades

In the 1970s biologists began to favor the newer stereology approaches over more

rough assessments by so-called experts, and subjective (biased) sampling methods. Two peer-review journals were established that focused primarily on stereology Journal of Microscopy and Acta Stereologica (now Image Analysis & Stereology).

Stochastic Geometry And Probability Theory

An important breakthrough occurred in the 1970s when mathematicians joined the ISS, and began to apply their unique expertise and perspective to problems in the field. Mathematicians, also known as theoretical stereologists, recognized the fault in the traditional approaches to quantitative biology based on modeling biological structures as classical shapes (spheres, cubes, straight lines, etc.), for the purpose of applying Euclidean geometry formulas, e.g., area = r2. These formulas, they argued, only applies to objects that fit the classical models, which biological objects did not. They also rejected so-called correction factors intended to force biological objects in the Euclidean models based on false and non-verifiable assumptions.

Instead, they proposed that stochastic geometry and probability theory provided the correct foundation for quantification of arbitrary, non-classically shaped biological objects. Furthermore, they developed efficient, unbiased sampling strategies for analysis of biological tissue at different magnifications (Table 3).

The combination of these unbiased sampling and unbiased geometry probes were then used to quantify the first order stereological parameters (number, length, area, and volume) to anatomically well-defined regions of tissue. These studies showed for the first time that it might be possible to use assumption- and model-free approaches of the new stereology to quantify first-order stereological parameters (number, length, surface area, volume), without further information about the size, shape, or orientation of the underlying objects.

The Third Decade of Modern Stereology (1981-1991)

By the 1980s, biologists had identified the most severe sources of methodological bias that introduced systematic error into the quantitative analysis of biological tissue. Yet before the field could gain greater acceptance by the wider research community, stereologists would have to resolve one of the oldest, well-known, and most perplexing problems: How to make reliable counts of 3-D objects from their appearance on 2-D tissue sections?

The Corpuscle Problem

The work of S.D. Wicksell in the early 20th century (Wicksell, 1925) demonstrated the Corpuscle Problem -- the number of profiles per unit area in 2-D observed on histological sections does not equal the number of objects per unit volume in 3-D; i.e., NA NV. The Corpuscle Problem arises from the fact that not all arbitrary-shaped 3-D objects have the same probability of being sampled by a 2-D sampling probe (knife blade). Larger objects, objects with more complex shapes, and objects with their long axis perpendicular to the plane of sectioning have a higher probability of being sampled (hit) by the knife blade, mounted onto a glass slide, stained and counted.

Correction Factors

A close examination of classical geometry reveals a number of attractive formulas that, if they could be applied to biological objects, would provide highly efficient but assumption- and modelbased approaches for estimation of biological parameters of tissue sections. Since the work of S.D. Wicksell in the 1920s, many workers have proposed a variety of correction factors in an effort to fit biological objects into classical Euclidean formulas. This approach using correction formulas requires assumptions and models that are rarely, if ever, true for biological objects. These formulas simply add further systematic error (bias) to the results. For example, imagine that we decide that a group of cells has, on average, shapes that are about 35% non-spherical. Unless these assumptions fit all cells, then correcting raw data using a formulas based on this assumption would lead to biased results (e.g., Abercrombie 1946). The problems arise immediately when one inspects the underlying models and assumptions required for all correction factors. How does one quantify the nonsphericity of a cell? How does one account for the variability in nonsphericity of a population of cells? Or in the case of a study with two or more groups, should not different effects on cells require different factor to correct for relative differences in nonsphericity between groups? To verify these assumptions is so difficult, impossible, or time- and labor-intensive that it prohibits their use in routine biological research studies. The bottom line is that correction factors fail because the magnitude and direction of the bias cannot be known; if it could, there would be no need for the correction factor in the first place! Note, however, that if the assumptions of a correction factor were correct, the correction factor would work.

Despite numerous attempts using so-called correction factors, this approach failed to overcome the Corpuscle Problem. By the early 1980s, the Corpuscle Problem remained a significant test for the credibility of the newly emerging field of unbiased stereology.

The Disector Principle

The solution to the Corpuscle Problem came in a Journal of Microscopy report in 1984 by D.C. Sterio, the one-time pseudonym of a well-known Danish stereologist. The solution, known as the Disector principle, became the first unbiased method for the estimation of the number of objects in a given volume of tissue (Nv), without further assumptions, models or correction factors. A disector is a 3-D probe that consists of two serial sections a known distance apart (disector height), with a disector frame of known area superimposed on one section. In 1986 Gundersen expanded the Disector principle from two sections a known distance apart (physical disector) to optical planes separated by a known distance through a thick section (optical disector). The number of objects in which the tops fall within the disector volume provides an unbiased estimate of the number per unit volume of tissue. The disector makes use of Gundersens unbiased counting rules (Gundersen 1977), which avoids biases (i.e., double counts) arising from objects at the edge of the counting frame (edge effects).

The fractionator method, a further refinement for counting total object number, eliminated the potential effects of tissue shrinkage in the estimation of total object number in an anatomically defined volume of tissue (Gundersen, 1986; West et al., 1991). The disector and fractionator methods provide reliable estimates of objects in a known volume by repeatedly applying the disector counting method at systematic-random locations through an anatomically defined volume of reference space.

The combination of disector-based counting with highly efficient, systematic-random sampling allowed optimal counting efficiency by counting only about 200 cells per individual. Other techniques introduced in the 1980s included methods for unbiased estimation of object sizes, including the nucleator, rotator, and point-sampled intercepts (Gundersen et al., 1988 a, b).

By this point it became clear that making an unbiased estimate of any stereological parameter required choosing the correct probe, the one that does not miss any objects of interest. By ensuring that the dimensions (dim) in the parameter of interest with a probe containing sufficient dimensions so that the total dimensions in the parameter and probe equal at least 3 (parameterdim + probedim > 3).

All Variation Considered

Biologists realized that by avoiding all source of error (variation) arising from assumptions and models, the total observed variation in their results, as measured by the (coefficient of variation (CV = std dev/mean), could be accurately partitioned into its two independent sources: biological variation (inter-individual) and sampling error (intra-individual).

Inter-individual differences arising from biological sources (evolution, genotype, environmental factors, etc.) typically constitute the largest source of variation in any morphological analysis of biological tissue. By sampling more individuals from the population, this source of variation will diminish, and thereby reduce the total observed variation in the data. However, the cost of analyzing more individuals is high in terms of time, effort, and resources. For this reason it can be important to first examine the second contributor to the total observed variation, sampling error, which is variation arising from the intensity of sampling within each individual. Sampling error is expressed in terms of coefficient, CE. In general terms, reducing sampling error, i.e., by sampling more sections and/or more regions within each section, costs less in terms of time and resources than sampling more individuals. Thus, by partitioning the observed variation in stereological results into variation arising from biological sources and sampling error, bio-stereologists learned to design sampling schemes that were optimized for maximal efficiency.

Do More, Less Well

Prior to the modern era of stereological approaches, the amount of work exerted to make an estimate provided the best means of assessing the value of that estimate. In the 1960s, for example, a worker in one influential publication spent two years counting 242,681 cells in a particular area on one side of the brain! Through the multidisciplinary efforts of biologists, mathematicians, and statisticians in the ISS, stereologist learned that an optimal level of sampling within each individual could be defined, regardless of the organism or structure of interest:

Question: What is the optimal number of animals and sections to analyze to make a reliable estimate of a stereological parameter (number, length, surface area, volume)?

Answer: The sampling intensity that most efficiently reduces the total observed variation per unit of time spent analyzing tissue.

In practice, the starting point is to sample the reference space, i.e., the volume of tissue containing the objects of interest, into about 10 systematic-random sections, quantify the parameter of interest, and then repeat this on about 2-3 individuals for each group. From these results the fraction of the total observed variation contributed by biological and sampling error can be estimated. When the sampling error (CE) achieves a point of diminishing returns, i.e., when further sampling of sections and regions within individuals causes only minor reductions in the observed variation, then time and effort are best shifted toward analyzing more individuals from the population of interest. Once a representative number of individuals have been analyzed, usually n = 5 to 10 per group, the results should provide accurate, precise, and efficient data for statistical testing of the hypothesis of biological interest, such as: Is there a statistically significant difference in cell number between the two groups?

The esteemed Swiss stereologist, Prof. Ewald Weibel, named this approach, Do More, Less Well.

by: Peter R. Mouton, Ph.D

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