Stereology Info and Terms


Accuracy: the lack of systematic error (bias) in a method. With continuing sampling in a systematic-random manner, results from unbiased stereology approaches progressively converge on the expected (true) value.

Anisotropy: refers to biological objects with a preferred spatial orientation. Examples include blood vessels, fibers, and cell/tissue surfaces. For quantification of length and surface area, anisotropy can be overcome through the use of computer-generated isotropic probes such as Space Balls (sphere probes) for Length (Mouton et al., Journal of Microscopy, 206: 54-64, 2002) and virtual cycloids for surface area (Gokhale et al, Journal of Microscopy, 216: 25-31, 2004).

Arbitrary-shapes: The vast majority of biological objects have arbitrary-shapes, as opposed to the classical shapes of spheres, cylinders, and rectangles of Euclidean objects.

Area (surface): first-order stereological parameter for planar (2-D) structures. Unbiased quantification on tissue sections and images requires probes with minimum 1 dimensions (lines). Quantified in an unbiased manner on isotropic sections using straight line probes and on vertical sections using cycloid line probes.

Area fraction: profile area relative to a reference area.

Area Sampling Fraction (ASF): area sampled divided by the total area.

Ascertainment bias: systematic error cause by biased sampling from a target population.

Assumption-based morphometry: morphometry based on nonverifiable assumptions.



Between-section variation: error caused by between-section variability in an estimated parameter.

Bias: systematic (nonrandom) error.

Biological variation (BV): random error (variation) expressed as true differences in a stereology parameter between individuals in the same treatment group. Due to genotype and environment differences and their interactions, subjects in the same control or treated group show real biological variation, as distinct from variation due to sampling (sampling error).

Biological variance (BV^2): proportion of total variance in a parameter due to true individual differences between subjects in the same treatment group. Calculated as the difference between total variance (CV^2) and sampling error (CE^2).

Buffon, George Louis LeClerc, Compte de (1707 – 1788): French naturalist and mathematician; considered the “father of probability theory.” Buffon first posed his Needle Problem, one of the earliest solved problems in geometric probability, as a question to the Royal Academy of Sciences in Paris (1733):

Suppose a floor consists of parallel strips of wood, each the same width. If we drop a needle onto the floor, what is the probability that the needle will cross the lines on the floor?

Knowing either the distance between the lines or the length of the needle, the unknown quantity can be estimated to any degree of precision. With a Monte Carlo simulation, the solution for Buffon’s Needle Problem allows for approximating the number π. The Needle Problem provides the basis for estimating all first-order and second-order stereology parameters based on random intersections between: a) a probe of known dimensions, e.g., disector; and b) the unknown population of interest.



Cardinality: structural property for Number.

Coefficient of error (CE): second-order stereological parameter for within- sample variation.

Coefficient of variation (CV): second-order stereological parameter for total observed variation.

Correction factor: assumption-based component of a morphometric equation.

Cycloid: a sine-weighted line probe for estimation of surface area and length from vertical sections.



Density: a ratio estimator involving a first-order stereological parameter (numerator) per unit area or volume (denominator), e.g., length density, (Lv).

Design-based stereology: morphometric approach to estimate structural parameters in fixed, defined populations using sampling and probes that conform to probability and stochastic geometric theory.

Dependent (systematic, nonrandom) sampling: scheme in which adjacent structures are sampled in serial order.

Digitization: process that converts a light image to picture elements (pixels).

Disector: a three-dimensional sampling and counting probe. Distribution – the spread of values across a target population.

Do more, less well: optimal strategy for reducing observed variance in structural parameters with high biological variance by sampling less within each individual in favor of sampling a greater number of individuals.



Embedding: immobiliation of tissue in a hard matrix for sectioning.

Error: variation in a parameter estimate.

Error variance: random error in a population; noise.

Error analysis: post-hoc process in which total observed error is partitioned into sources (biological variance and error variance).

Estimate: the approximate determination of a parameter in a small random sample from the population.

Exclusion lines: lower and left lines on an unbiased counting frame.

Expected value: the true value of a parameter at the level of the population.

Euclidean (assumption- and model-based) geometry: the field of geometry that deals with classically shaped structures; model- based.



Fixation: chemical stabilizing of tissue proteins prior to analysis.

First-Order Stereology Parameters: Volume, Surface area, Length, Number.

Fractionator: a three-level sampling procedure in which total parameters are estimated from measurements in a known fraction of the total reference space.



Guard distance: a known distance from the upper and lower section surfaces; used to avoid sectioning artifacts (e.g., lost caps).



Inclusion lines: upper and right lines on an unbiased counting frame.

Individual differences: true variability between individual members of a population.

Intensity: in microscopy the relative balance light and dark.

Interpretation bias: systematic error in an estimate caused by the experimenter favoring a particular outcome.

Isotropic sphere probe (Space balls): sphere probe for unbiased estimation of length (L) and length density (Lv). First introduced by Mouton et al. (2002).

Isotropy: lack of a preferred spatial orientation in 3-D, e.g., sphere.

Isotropic-Uniform-Random (IUR): method for sampling structure in which all orientations in three dimensions have equal weight.



Length: first-order stereological parameter for lineal (1-D) structures; estimated without bias using a 2-D probe.

Line probe: grid of uniformly oriented test lines used for sampling one- or two- dimensional structures.



Mean (average): the central tendency of a normal distribution.

Mean object volume (MOV): average volume for a sample of objects from a target population; for neurons, mean neuron volume. Quantified on tissue sections and images using Nucleator method (Gundersen, 1985) and Rotator (Jensen, 1993) based on the second theorem of Pappas-Guldinus centroid principle.

Methodological bias: systematic variation in observed data caused by method conditions, e.g., assumptions, models, correction factors.

Microtome: device for sectioning tissue.

Morphometry: the methodology used to measure structure.

Multi-level: random within-sample variation; estimated by CE.



Nugget Effect: variability in an estimate caused by the smallest part of a structure (nugget) that contributes to the observed variation in a parameter; within-section variation.

Number: first-order stereological parameter for cardinality (0-D); estimated without bias using a 3-D probe.

Number-weighted: sampling approach in which all values in a defined number distribution have an equal probability of being sampled.

Number-weighted mean volumes (arithmetic mean volume): Mean object volume quantified on objects sampled from their number distribution. In biological stereology, mean cell volume quantified cells counted using the director and fractionator methods.



Object: the level of structure identified at high magnification.

Orientation bias: systematic error caused by a preferred three- dimensional orientation.

Optical disector: an unbiased method for estimating density and, with Vref, total Number; uses a stack of optical sections to sample the top-most points of three- dimensional objects in a known volume of a defined reference space.

Optical fractionator: an unbiased method for estimating total Number; uses a stack of optical sections to determine the number of events in a known fraction of a defined reference space.

Optimization: post hoc strategy to direct sampling effort at the primary sources of variability, for the purpose of reducing observed variation in an estimate.

Overprojection: information in an image caused by excess thickness of transparent embedding material relative to an opaque object of interest. Also known as the Holmes effect.



Pappas-Guldinus theorem: mathematical tenet which holds that the mean object volume for a defined population can be estimated without bias from average lengths of isotropic lines radiating from a fixed point to the border on random profiles through the objects.

Pattern (feature) recognition: technique of computerized image analysis for automatic identification of structure based on defined characteristics.

Paraffin-embedding: a processing technique using paraffin to physically stabilize tissue for sectioning; allows histochemistry and immunocytochemistry staining of objects.

Physical disector: method to estimate number of objects in a defined volume using physically separate look-up and reference sampling planes.

Pilot study: preliminary analysis in a small number of samples.

Plastic-embedding: a processing technique using water-soluble plastic to harden tissue block for sectioning.

Point counting:  probability-based technique for unbiased estimation of area and, with the Cavalieri principle, volume.

Population: a collection of objects that have at least one attribute in common.

Postmortem artifact: tissue changes that begin at the death of the organism or separation of tissue from its blood supply.

Precision: observed variation (error) in an estimate.

Probe: a zero-to-three-dimensional test system of points, lines, or frames used for sampling and estimating structural parameters.

Profile: cross-section through an object sampled by a two- dimension plane (e.g., knife blade).

Projection bias: error caused by projection of 3-D structures using light microscopy.



Quadratic approximation formula: equation used to estimate within-sample variation (method error) for dependent sampling (Gundersen and Jensen, 1987; Gundersen et al., 1999).



Random sampling: sampling process in which each sample is independent.

Recognition bias: systematic error caused by the inability to identify structures of interest.

Reference space: biological region defined by natural borders.

Reference trap: systematic error arising from use of density estimators to measure absolute quantities.

Region: the level of structure identified at low magnification.

Region of interest: in image analysis, the boundaries of structure to be analyzed.

Rotator: an application of the Pappas-Guldinus theorem for efficient estimation of mean object volume on random profiles.



Sample: a subset of the population (noun); to obtain a subset of the population (verb).

Sampling error (within-sample error): random variation related to the intensity of sampling within an individual.

Sampling interval (k): the period between successive samples through a reference space.

Section sampling fraction (SSF): the number of sections analyzed divided by the total number of sections through the reference space.

Shape bias: systematic error caused by variability in shape of objects.

Size bias: systematic error caused by variability in size of objects.

Snap freezing: a procedure that uses liquid nitrogen to rapidly freeze tissue; retains water and ions within cells and prevents shrinkage/expansion of reference space.

Standard error of the mean (SEM): the standard deviation divided by the square root of the number of individuals in the sample.

Stereology: the analysis of 3-D objects (from Greek stereos”).

Stereology Resource Center (SRC)global provider of resources for quantifying changes in biological structure.

STEREOLOGER – an integrated hardware-software system for unbiased stereological analysis of biological structure.

Stochastic geometry: the analysis of variable-shaped objects using probability- based formulas.

Surface area: first-order stereological parameter with two dimensions.

Systematic-Uniform-Random (SUR): sampling approach that allows a reference space to be sampled in an efficient manner and without methodological bias.

Systematic error: non-random variation (bias); causes sample data to deviate from expected values.



Thickness sampling fraction (TSF): section thickness sampled divided by the total section thickness.

Three-level sampling design: fraction-based (multi-level) approach to estimate total parameters in known fractions of a reference space.

Two-stage sampling: volume-based approach for estimation of absolute parameters as the product of a density estimator and the reference volume.



Unbiased method: an approach that avoids systematic error; with increased sampling, estimates approach their expected values.

Unbiased counting frame: an areal frame to ensure that the number of objects is sampled with equal probability.

Underprojection: lack of information caused by embedding media opaque relative to the object of interest.



Variation (variability, variance): error; may be systematic (bias) or nonsystematic (random).

Vertical axis: the axis of rotation in a defined direction.

Vertical section: a random plane parallel to the axis of rotation.

Vertical-Uniform-Random (VUR): sampling method to avoid systematic error when estimating surface area and length of anisotropic structures in conjunction with cycloid line probes.

Virtual sectioning: optical sectioning technique that uses a stack of parallel focal planes to probe objects.

Volume: a first-order stereological parameter for 3-D structures; estimated without bias using a zero dimensional  (0-D) point grid.

Volume-weighted sample: a subset of objects or regions sampled on the basis of volume.



Within-sample error: random variation within an individual; measured by CE.

Within-section variation: random variation within a section caused by the smallest part (nugget) of a region or object; the Nugget Effect.