Introduction to basic concepts and methods of modern stereology Goran Šimić, CIBR, 21th Apr 2006.

Contents I. What is stereology? II. Basic concepts of modern stereology III. Most commonly used stereological methods IV. Advices and examples

I. What is stereology? Stereology is a statistical methodology for estimating the geometrical quantities of an object such as number, length, profile area, surface area, and volume In practice, stereology is a "toolbox" of efficient methods for obtaining unbiased 3D-nal information from measurements made on 2D microscope sections Stereology is now recognized to be of major value in science and medicine

Origins and applications Developed by materials scientists, mathematicians, and biologists since early 1960s Flourished in late 1980s and early 1990s Can be applied to all biological structures

Why is stereology better than the “classical morphometry”? Focuses on total parameters (e.g., total cell number) not densities Avoids all known sources of methodological biases (e.g., assumption that a cell is a sphere, split cells and “lost caps” ) Avoids inappropriate correction formulas However, tissue processing requirements differ from assumption-based methods (probes, i.e. sections should be IUR = isotropic and uniformly random). Such sections are usually unusable for routine qualitative analysis (e.g. cortical sections)

A histological section is a biased sample of particles In a stack of serial sections, relatively taller cells are represented on relatively more sections. If we choose In contrast to classical (old) quantitative methods, new (modern) stereological methods avoid this bias

II. Some basic concepts of modern stereology 1. Estimation / sampling 2. Accuracy / unbiasedness 3. “Significance” 4. Explanation of results obtained

1. Estimation Usually, we cannot “determine” the value of a parameter unless we exhaustively sample the entire material. Instead, we infer the parameter value from a sample, subject to random error – this is statistical estimation. Variance is a random fluctuation between data values or between successive repetitions of the experiment

Basic definitions Population: a well defined set of elements (“sampling units”) about which we want to make inferences Example 1: All Croatian adults (discrete population) Example 2: Brain of rat no. 179 (continuous population) Example 3: Brains of 25-day-old male Wistar rats (superpopulation) Parameter: a well defined numerical quantity relating to the population Example 1: Average height of Croatian adults Example 2: Total brain volume of rat no. 179 Example 3: Average total neuron volume of 25-day-old male Wistar rat Sample: a set of individuals taken from the population Example 1: Ivo Ivić and Pero Perić Example 2: Twenty sections from brain of rat no. 179 Example 3: Brains from 25-day-old male Wistar rats Uniform sampling: a mechanism for choosing samples randomly so that every sampling unit in the population has the same probability of being selected for the sample

Population and sample Always keep in mind the distinction between population and sample “ The sample looks like the population”: - If it is correctly sampled - If it is large enough Issues: - How do I take a “representative” sample? - How do I get unbiased estimates of population parameters? - How precise are these estimates? - What is an optimal sampling design?

Sampling Estimates should refer to a biologically meaningful reference spaces in a defined population Sampling should be representative i.e. uniform random (every member of the population needs to have an equal chance of being selected for the sample) Requires that the structure of interest can be unambiguously defined

The “reducing fraction” problem of sampling The effect of increasing magnification decreases the proportion of the original object being sampled

Hierarchical nature of sampling for microscopy Uniform random sampling should be employed at every level of the sampling hierarchy (in other words: At no stage should anything within the defined reference space be ‘chosen’) Use nomograms to spare time (Gundersen and Jensen, 1987; Gundersen, 1999)

Sampling can be optimized for maximum efficiency ("Do More Less Well"). In a typical biological experiment the overall observed variance (the “spread” of a distribution around its mean) consists of the following relative components: Message: Concentrating on making very precize individual measurements will at best only increase the precision of the overall experiment by about 2%! 70% inter- individual (biological) 20% between blocks 5% between sections 3% between fields 2% between meas.

Sampling can be biased An illustration of how the moving averages of an unbiased estimator of N behave as an experiment is replicated The magnitude of the bias B is unknown = totally invisible at the end of an experiment (you simply have a numerical estimate and there is no way to determine bias from your data!) The presence or absence of bias depends mostly upon the experimental or sampling method used Systematic error = bias Bias is the difference between the expected value of an estimator and the true parameter value Sources of bias: - Wrong calibration - Observer effects - Incorrect assumptions - Wrong sampling, - Any type of selection

Biases do not cancel each other It is often argued that biased measurements may be used when we want to compare two experimental groups. However, this is only justified if the bias in both cases is equal (what we usually just don’t know)

Unbiased sampling regimes Independent random sampling (randomly generates a sample of fixed size) Systematic random sampling (samples a fixed proportion 1/m of total population; starts at random and count every mth item; the sampling probability is 1/m) Cluster sampling (after grouping items into arbitrary “clusters”, use an unbiased sampling method to select some of the clusters) Stratified random sampling (divide the population into subpopulations or strata and sample every stratum by an unbiased sampling rule)

2. Accuracy / unbiasedness Accuracy cannot be ‘bought’ by working harder: Accuracy can only be guaranteed by using ‘tools’ (methods) that are inherently unbiased In stereology this means start with the uniform random sampling that is followed by the application of a set of unbiased ‘geometrical questions’ in 3D (probes) The a priori guarantee of accuracy of these methods, without the need for validation studies, is a major advantage: They can literary be ‘taken off the shelf’ and used in any situation!

Do not mix accuracy (unbiasedness) with efficiency (precision) It is possible to have a biased estimator which is ‘efficient’ (converges on to a stable value quickly and has small SD), as well as inefficient unbiased estimator Only if you know that you have an unbiased estimator, you can use variance or SD to measure precision. Low variability High variability

Factors contributing to variance Instrument noise Sampling variation Biological variation Dependence on uncontrolled factors Oberver effects (counting errors), etc. Variance can be estimated empirically, but is unrelated to bias. Random error can be decreased by taking more data; bias will not decrease. So, it is important to minimize bias of estimators.

3. “Significance” “My two experimental groups gave different results. Does this prove they are different, or is it just the result of random variation?” Use: - regressions - formal significance tests for H 0 - other statistical methods

4. Explanation Try to attribute variability to different factors (sources of variability) such as: - Age - Gender - Different experimental conditions, etc.... Analyze response variables vs. explanatory variables Analyze systematic effects vs. random effects

Example Explain brain weight of rat as combination of several influences: Weight = population mean + litter effect + indiv. variation Rats from the same litter differ less than rats from diff. litters Var(weights) = Var (litter effects) + Var (indiv. deviations) Estimate this from all rats Estimate this by comparing litter means Estimate this by comparing rats within a litter

III. Some of the most commonly used stereological methods 1. Cavalieri principle for volume estimation 2. Physical disector for number estimation 3. Optical disector for number estimation 4. Optical fractionator for number estimation

1. Estimation of reference volume using the Cavalieri method V= t. A(p).  P t=average slab thickness =distance between section planes A(p)=area per test point corrected for magnification  P=total number of test-points hitting the structure

Tissue shrinkage Due to dehydration and embedding Differential shrinkage must be calculated (for the 3rd dimension, use the sqare root of the calculated areal shrinkage) Final volume can be only 26% of original (fetal brain has higher % of water) (Pakkenberg, 1966)

Stereological probes Feature = geom. feature of a 3D object to which the probe is sensitive d = n – k d = feature dimension n = dimension in which objects are embedded k = dimension of a probe Only if d=n or k=n, the distribution of estimated objects is irrelevant

The unbiased brick counting rule for number estimation (Howard et al., 1985) Inclusion planes (surfaces) A particle is counted if it is totally inside the brick or if it intersects any of the acceptance (inclusion) planes and does not intersect any of the forbidden surfaces anywhere. The unbiased brick counting rule is a general 3D counting rule that is applicable for particles of any shape and size. If the particles of interest are convex, such as nerve cell nuclei, then the optical disector can be used (Gundersen, 1986).

Optical disector (about 5x faster than physical) a)Not counted (topmost plane is exclusion plane) b)Is counted (nucleus touches the inclusion line) c)Not counted (cuts the forbidden line) d)Profile is sampled e)and f) are counted (bottom plane is inclusive) Dark nuclei are in maximal focus

Fractionator principle (2D example)

Optical fractionator (3D) Outline a region of interest at a low magnification. The region may encompass several fields of view. Once a region of interest has been defined, count cells at high power. The image on the right shows a counting frame for an optical fractionator probe superimposed on a live video image.

Optical fractionator (3D) The overall fraction of the object sampled = ssf. asf. hsf Total number of particles in the object = Q. 1/hsf. 1/asf. 1/ssf

Scheme of the multi-stage fractionator

Example of a local probe Local probes may be integrated with global probes to measure the size of objects as they are counted. Here a nucleator is being used with an optical fractionator to measure cell area and volume.

IV. Advices and examples Šimić G et al. (1997) Volume and number of neurons of the human hippocampal formation in normal aging and Alzheimer's disease. J. Comp. Neurol. 379: OPTICAL DISECTOR Šimić G et al. (2000) Ultrastructural analysis and TUNEL demonstrate motor neuron apoptosis in Werdnig-Hoffmann disease. J. Neuropathol. Exp. Neurol. 59: PHYSICAL DISECTOR Šimić G et al. (2005) Hemispheric asymmetry, modular variability and age-related changes in the human entorhinal cortex. Neuroscience 130: OPTICAL FRACTIONATOR

Properties of modern stereological methods Estimates first-order parameters of biological structures (e.g. volume, surface area, length, number) and their variability from a small sample of the population of interest. Uses highly efficient systematic sampling. Efficiency based on true variability of objects and features of biological interest.

Advances Avoids tissue processing artifacts, e.g., shrinkage/expansion, lost caps. Strong mathematical foundation in stochastic geometry and probability theory, but: Advanced mathematical background not required for users.

Computorized or not? Doesn't require computerized hardware- software systems, but: Computerized stereology systems are very efficient. Statistical power for studies of the same parameter is cumulative across populations.

Last but not least: Considered state-of-the-art by journal editors and grant review study groups. Worldwide cooperation possible, in theory, through Web-accessible databases.

Some of the options on the market C.A.S.T. Grid (Olympus / Zeiss) StereoInvestigator and Neurolucida (MicroBrightField) Stereologer (Systems Planning and Analysis)

1. Some new, simple and efficient stereological methods and their use in pathological research and diagnosis. Gundersen, H.J.G., T.F. Bendsen, L. Korbo, N. Marcusen, A. Moller, K. Nielsen, J.R. Nyengaard, B. Pacakkenberg, F.B. Sorensen, A. Vesterby, and M.J. West. APMIS 96: The new stereological tools: Disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis. Gundersen, H.J.G., P. Bagger, T.F. Bendsen, et. al. APMIS 96: Two most important references in the field:

Reference Sites

Conclusion The key to doing science efficiently is to: Always use valid techniques (strive for techniques that are guaranteed to be unbiased) Balance the accuracy of the estimate against the cost of performing the experiment

Feel free to contact me later on: Possible applications in your own research?