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School of Anatomy and Human Biology - The University of Western Australia

     Blue Histology - A Brief Introduction to Stereology

Theory

Exercises

Introduction

The simulations of the methods to estimate volumes and numbers correspond quite closely to the way the methods would would be used and look like using a computer-controlled microscope.
The simulations may require a screen resolution of 800 x 600 to work properly on some laptops.

Sampling

Measuring

Volume Estimation

Cell Density and Cell Number Estimation

Selected References

 

Introduction

Histological sections are used to define the normal appearances of tissue and organs. Thereby, they also define "abnormal" appearances which may be a consequence of a disease affecting an organ. Histological descriptions on this site often use terms like "large", "small", "many", "few", "absent" or "present". These terms are very helpful and often sufficient to describe the basic features of a section. However, they are not good enough to, for example, statistically test for significant changes of appearance, which may result from a disease or experimental treatments. To do so, we need to attach numbers to the more or less subjective terms used in the descriptions - and almost all terms which could be called subjective can be associated with numbers.

Let's say someone says a potato is beautiful - just ask why. The answer may be that it is beautiful because it is large and has a smooth surface with little dirt adhering to it. We now can produce an objective measure of this subjective statement by estimating the volume of the potato (large), its volume to surface ratio (smooth) and the volume of the dirt adhering to it. In the same way we can determine measures for a "beautiful" or "healthy looking" cell.

Where does stereology come into the picture and, by the way, what is it? Think about your stereo set at home or stereo images. They are not called "stereo" because there are two speakers or two pictures. They are called "stereo", which is derived from the Greek word for a "geometric object", because they try to recreate sounds or objects in space. Stereology tries to recreate or estimate the properties of geometrical objects in space (anything from a fish to a nerve cell, but mainly potatoes). Applying stereological methods to tissue or organ section allows us to estimate the geometrical properties of the objects contained in the sections.

What are these properties of objects in space? Space has three dimensions, and objects within it have properties for each possible number of dimensions. Objects have

Each of these properties can be estimated by stereological methods. These methods usually require a two step-procedure:


Sampling

We usually want to make statements about a structure (an organ, a cell or a potato) or a population (of organs, cells or potatoes) on the basis of a small sample of the structure or population. If our statement is supposed to be valid for the entire structure or population, then the sample has to be a representative sample.

We can only create a representative sample if

  1. we have access to the entire structure or population.
     
  2. we can recognize and/or define the entire structure or population.
     
  3. all parts of the structure or population have the same chance to contribute to our sample.
  1. If we do not have full access, our statements can only be valid for the parts of the structure or population that we have access to. Sometimes it will still make sense to investigate this part and sometimes it will not. If it still makes sense to us, we will have to argue for it.
     
  2. "Recognition" and "definition" are not entirely the same as "access". If we are interested in a population of cells, and if we have the entire organ which contains these cells, then we have access to the entire population. We still have to recognize the cells. If we cannot define and/or recognize the structure or population, we either have to limit us to the parts we can define or we have to accept that we include parts of other structures or populations which we are not interested in. Again, sometimes it will still make sense to investigate the structure or population and sometimes it will not. If we think it does, we will have to argue for it.
     
  3. This is the requirement which is the hardest to argue away and the one that most of the rest of this section will deal with. If we cannot make sure that all parts of the structure or population contribute equally to our measurements, our statements will only be valid (1) if we know how well each part was represented in our measurements and (2) if we adjust our final statement for the unequal representation. In practice, this is often impossible to do. If it is possible, it usually requires a lot of work and, of course, we will have to argue for it.

All parts of the structure or population have to have the same chance to contribute to the sample.

We will start with the sampling within one section and apply the same principle to the sampling of sections.

Let us assume that we have access to all sections of the structure which we are interested in and that we can recognize the structure in all sections. Often, even a single section contains more objects (e.g. cells) than we can or want to count or measure. We now have to create a representative sample of locations in the section to perform the actual measurements. To ensure that all locations have the same chance to be represented in the measurements we have to draw a random sample. A random sample can be drawn in two ways.

Random Independent Samples

This is the way of sampling which usually comes to mind first. We could select a location at random, measure whatever we want to measure, and afterwards determine the next location independent of the first location. Once we feel that we have sampled a sufficient number of locations, we calculate the average of our measurement and do, if necessary, some calculations. We now have an estimate for the measure we are interested in that is valid for this section.

If we would repeat this process several times using the same section it is likely that individual estimates would differ from each other, but the average of the individual estimates will, if we measure correctly, converge upon the true value of the property of our structure.

Random independent samples will give us the correct answer, but they are rarely the most efficient way of sampling.

The retina of the eye is a good example. Measurements are often performed in wholemounts of the retina, i.e. retina is mounted as one piece on a slide - one may say there is only one "section". Retinal ganglion cells (grey dots) are not distributed evenly over the retina. Their density is highest close to the fovea and decreases gradually towards the margin of the retina. If we draw random independent samples from the retina to estimate ganglion cell density, it is possible that all our samples will come from an area of high ganglion cell density (red squares). Alternatively, all samples may come from an area of low ganglion cell density (blue squares), or samples may spread evenly over areas of different ganglion cell densities (black squares).

They key thing is that any combination of sampling locations is as likely as another. Even though the average of the three estimates will be close to the true ganglion cell density, they may differ wildly from each other.

To increase the precision of the individual estimates we could increase the number of samples (more work - sigh!). Although we thereby reduce the likelihood that all samples fall into an area of high or low cell density, we can never completely eliminate the chance that they do, unless we use

Uniform Random Systematic Samples

which are a little tricky to understand, but it's not too bad.

In the above example, squares were chosen to sample areas of the retina for ganglion cells.

What about splitting a square into four little squares (A - red sample) and placing this group at random onto the retina? No problem - it's really the same sample as the one big square. What about moving the four little squares slightly away from each other by a fixed distance(A - blue sample)? It shouldn't be a problem - it's still the same area and we still place it a random onto the retina. Why not split the square into sixteen little squares separated from each other by fixed distances (A - black sample)? If the blue sample wasn't a problem, the black sample shouldn't be a problem either as long as we place the lattice of squares at random onto the retina. It's just the "shape" of the sample that varies.

There is actually something very good happening: spreading of the area of our original square makes it quite difficult for all of the area of our sample to fall onto a part of the retina with a high or a low ganglion cell density.

From here, it is only a small step to a uniform random systematic sample (B). After choosing a random starting point, samples are drawn at regular (or uniform systematic) intervals. The random starting point means that all areas of the retina have the same chance to contribute to our sample. The regular spacing means that our samples can no longer cluster in the retina.

Three of the infinitely many possible urs samples are illustrated in the animated image.

By eliminating the possibility of samples to cluster, the estimates using an urs sample will on average be much closer to the true value than estimates using a random independent sample.

The combined area of all the small squares corresponds pretty much to the combined area of any one (red, blue or black) of the four-square samples illustrated in the random independent sampling example, i.e. we actually do not have to count under a larger area.

Biological structures, like the brain and its divisions, rarely have a perfectly regular shape. Cells contained within them are rarely regularly distributed. Whenever something about our structures or cells is irregular we gain by using an urs sample. If the structure is regularly shaped and if the cells are regularly distributed, we will do as well with an urs sample as with a random independent sample.

In theory, it is possible to construct a situation in which an urs sample performs worse than a random independent sample. Think about it and about how likely it is to occur.

In practice, urs sampling should be performed whenever possible.

Urs samples of sections should be collected when the structure is sectioned. If, for example, every sixth section is collected for the series, the only new requirement would be that the first section collected is a random section of the first six sections which contain the structure because an urs sample needs a random starting point. Section series will actually often already be urs samples "by accident".

Much of the initial scepticism directed at the new stereological methods was concerned more with the validity and effectiveness of the urs sampling schemes than with the actual measurement techniques. As a matter of fact, urs sampling is a fairly old hat, which in other fields of the biological sciences had stood the test of time for decades before its use became popularized in the morphological sciences.

Now that we have a representative and efficient sample, let's start measuring.


Measuring

The volume, surface, length and number of objects are such basic properties that it may surprise that methods to accurately measure these properties only became available in the mid 1980's and that they only became widely accepted in the 1990's. This does not mean that they were never measured before. It just means that the earlier methods did not necessarily give the correct results.

The main problem of many of the earlier methods is that they treat histological sections as two-dimensional images and that they try to measure the properties of three-dimensional objects in these two-dimensional images. But two-dimensional images just do not contain enough information to do so. The problem corresponds pretty much to trying to generate stereo-sound out of a mono-recording. More or less well-founded assumptions about the "missing dimension" have to be made to derive measurements from two-dimensional images. Methods which require such assumptions are referred to as assumption-based methods. How close the measurements can come to the true value will, of course, depend on how good the assumptions are.

An example illustrating the problem rather well are counts of cell nuclei. Section planes (black lines) passing through the 10 blue nuclei would give us 15 "nuclei" in our series of sections. Sections passing through the 13 red nuclei would give us a count of 14 "nuclei" in our series. How many times a nucleus will appear in a section depends on its size, shape and orientation (see red nuclei). If we know that the nuclei, like the blue ones, have a certain size, shape and orientation we can (almost) calculate their number based on the number we count in the series. If we would want to calculate the number of red nuclei based on the count, we would have to know their shapes, sizes, orientations, numbers and all the combinations thereof. If we know all that already - why would we want to count?

Do we want to find out just to do the count? Very few people do. Instead it is assumed that the nuclei are, for example, spheres of a certain size. How big would the error be? Well, again, we can't tell without knowing about sizes, shapes ....... without knowing we can't even tell the direction of the error.

The methods which replaced assumption-based ones are referred to as design-based methods. Common to these methods, and the key to solving the problem of the earlier methods, is that design-based methods take (series of ) sections for what they are: (series of) chunks of tissue which actually have three dimensions. Even the thinnest section has not just an area but also a thickness. We have a real three-dimensional space to work in. We do not need to guess or assume about what is happening in the third dimension. All that is needed are some good probes to measure with.

What are probes?

Antibodies, which are proteins, can be used to detect other specific proteins in cells and tissues. Antibodies could be called probes for the proteins they detect. Synthetic RNA strands can be used to probe for the presence of the complementary RNA in cells. They are actually called probes. Proteins to probe for proteins, RNA to probe for RNA ..... could it be?.... yes.

Geometrical properties are used to probe for geometrical properties, or, a little less abstract,

points (0 dimensions)

can be used to measure

volumes (3 dimensions)

lines (1 dimension)

areas (2 dimensions)

areas (2 dimensions)

lengths (1 dimension)

volumes (3 dimensions)

numbers (0 dimensions)


The Estimation of Volumes using Points

Estimating volumes using points is conceptually the easiest of all stereological methods. The method of doing so is called the Cavalieri Estimator after one of the first scientists who described it.

First, imagine a rectangular grid. Next, we place points at each of the intersections of the grid lines and associate one of the rectangular areas formed by the grid with each point. By associating one area with a point located at each intersection we will cover the entire area of the grid. Each point actually "represents" an area. That means that we can estimate the area occupied by an object in the section by placing a grid of points on the section and counting the points which fall onto the object.

Sometimes a point will just be outside the object, sometimes it will be just inside the object. How good the area estimate will be depends on the number of points which we count. In an urs sample of points the effects of "just insiders" and "just outsiders" will quickly begin to cancel each other. The last step - since our sections are 3-dimensional we can calculate the volume of the object in the section by multiplying the area estimate with the thickness of the section.

This is the way you do the calculations:

 
Vobj = ap × Pobj×d
 

Vobj

is the volume of the object.

ap

is the area associated with each point.

Pobj     

is the sum of all points counted in all sections.

d

is the distance between two consecutive sections that you have counted points in.

 

Note that d only corresponds to section thickness is you count in every section.
If you count in every 6th section the distance between two consecutive sections will be
section thickness × 6.

Done ... almost, we just have to consider if a phenomenon called overprojection is of relevance for our measurement.

Although the point-counting principle could also be applied to very small objects like cells, it will require considerably more work to cut the very thin sections necessary to reduce the error of a point count procedure and to identify the object in consecutive section. Different methodological approaches have been developed to cope with small objects.

Overprojection
Imagine an opaque sphere in a section which is thick enough to contain the entire sphere. Looking at the section the sphere will look like a circle. If we measure the area of the circle and multiply it be the thickness of the section we end up with the volume of a cylinder instead of the volume of a sphere - a very large overprojection error.

If we divide the sphere over an increasing number of sections, the error will get smaller and smaller (A and B, overestimate in red). If we could cut infinitely thin sections the error would disappear. Although we can not cut infinitely thin sections we can minimize the error by cutting them as thin as possible/necessary. The size of the remaining error depends on the thickness of the section relative to the size of the object, and the appearance of the object in the section. The error can not be eliminated completely, but many biological objects can be cut into hundreds of sections, i.e. sections are often much thinner than the object.

The sampling of sections (urs, of course) can further reduce the error. When we sample, we want the underestimates (green; top samples in C) to balance the overestimates (green; bottom samples in C). The error that remains is an "overestimate of the underestimates" (red; top samples in C).

How big will the error be in a histological application? It depends. The error will only be there if the object is opaque. Usually, we cut sections to be able to "look into" whatever we cut, i.e. so thin that the structure is no longer opaque. How to define the borders of structures which do not have clearly defined outlines far outweighs concerns about the overprojection error.


The Estimation of Numbers using Volumes

Essentially, estimating numbers with volumes is just the reverse of estimating volumes with points. But matters get a little complicated by the fact that we rarely are interested in counting true 0-dimensional points. Instead we want the numbers of biologically interesting objects. Somehow we need to convert objects into points, which means we have to do 2 or 3 things.

But do we want to count all nuclei in a pair of sections? Well, it depends on how many there are, but the typical answer would be "no". We rather want to count in a small sample (urs, of course) of the section area - usually little squares (called frames). Objects will inevitable cross or touch the borders of our frames, which define the first two dimensions of our volume probes. The rules which determine whether to count or not to count along these dimensions are very similar to the rules which determined the counting across sections.

Let's take the x-axis first. We can modify the rule for the counting along the z-axis to something like this:

As we move from the left to the right, an object will only once be visible for the first time in our counting frame.

That means that everything sticking out of the left side of the frame should not be counted (-) because it was visible for the first time outside the frame. Everything that sticks out on the right side should be counted (+) because it was visible for the first time in the frame. A similar rule can be defined for the y-axis - do it yourself comming from below. This leaves us with a counting frame looking like the frame in A - it has two "forbidden" lines. Objects that touch or cross these lines are not counted. Hmmm ... we are still in trouble if we have a urs sample that e.g. says we should count in every second frame because some objects may be countable in two frames (++ in B).

To avoid this from happening the forbidden lines are extended (C). With a frame like this, every object has a chance to be counted once and only once, which is illustrated in the seamless covering of the area containing objects (a "complete tesselation") in D (- don't count in this frame, + do count in this frame) and which works for objects of any shape, size or orientation.

Now, when you count it is terribly "disappointing" to count only one cell when somehow or another 5 are in touch with your frame but "forbidden". And there is this sinking feeling if it happens three times in a row ... but then there is the exciting and relieving moment of actually counting 5 ... and you know that you should definitely take the weekend off.
Remember - it is all in the statistics. We are not counting the objects in the volume defined by the probe - we are estimating this number. Some estimates will be high, some will be low, but the average will be right.

This is the way you do the calculations:

 
Nobj = Q / Vdis×Vref
 

This approach outlined above corresponds to the Vref×Nv method in which

Nobj

is an estimate of the total number of objects in the structure,

Q

is the sum of all objects in all of the disector probes you placed in the structure,

Vdis

is the sum of the volumes of all the disector probes you placed in the structure and

Vref

is the volume of the structure which you estimated with the Cavalieri estimator.

 

Note that Q / Vdis gives you Number per Volume or Nv - hence the
"shorthand" Vref×Nv for the method.

If urs sampling was used throughout, there is a second way to do the calculations, which has its own name - the fractionator.

Hmmm .... what we actually counted in the disectors corresponds to (on average) 1/6×1/100×1/2 of all that is there. Now we just multiply what we counted with 6×100×2 to obtain an estimate of the total number of objects that are contained in our structure. Or, in slightly less understandable but academically more proper terms, we obtain an estimate of total number by multiplying our counts with the inverse of the sampling fraction. Putting this into a formula with a decent (at least halfway) notation you get the stuff below.

Here are the calculations for the fractionator:

 
Nobj = 1/ssf×1/asf×1/tsf×Q
 

Nobj

is an estimate of the total number of objects in the structure,

ssf

is the section sampling fraction.

asf

is the area sampling fraction.

tsf

is the thickness sampling fraction.

Q

is the sum of all objects in all of the disector probes you placed in the structure.


The Estimation of Areas and Lengths using Lines and Areas

A nice thing about points is that it doesn't matter how you turn them. They look exactly the same from any angle - a real mathematical point would be invisible from any angle. That it doesn't matter which way we turn our points also means that it doesn't matter which way we turn our sections in relation to the probes when we estimate numbers (i.e. when we count "points") or volumes (i.e. when we use points as probes).

The concepts behind estimates of areas and lengths are as easy as those behind estimates of numbers and volumes. However, since both the property (area or length) and the probe (line or area) have dimensions it does matter how we turn our section in relation to the probe when we estimate areas and lengths because the length and area that are visible in the two-dimensional projections which we use to perform the measurement depend on the orientation of the object that we want to investigate. Valid measurements can only be performed in two-dimensional projections if it is guaranteed that the objects we want to investigate are isotropic, i.e. their orientation in space must be completely random. This is rarely the case and next to impossible to prove. Instead, isotropy has to be guaranteed by either the procedures which we use to prepare the tissue or by the procedure which we use to place our probes in the tissue. This critical difference means that estimates of areas and length are technically a bit more difficult to perform than estimates of numbers and volumes.

Concept

Imagine a space - let's say a room in the basement. Now imagine an area in this space - let's say a sheet hanging to dry in the room. Now imagine some weird person shooting arrows in random directions in this room.

How many times will the arrows pierce the sheet? Well, it depends on the size of the sheet and the number of shoots. Let this one sink in slowly ...... If we count the number of holes in the sheet and if we know the area of the sheet we can actually estimate the length of the paths flown by the arrows or, vice verca, if we count the number of holes in the sheet and know the length of the paths flown by the arrows we can estimate the area of the sheet. Or in scientifically more appropriate terms:

The number of intersections (holes in the sheet) of lines (flown by the arrows) with a surface (of a sheet) is directly proportional to the length of the lines and the area of the surface.

Technicalities

This approach to measuring areas and length only works if we can orient the probes or the sections at random, i.e. if we can guarantee the isotropy of the probe or tissue. In practice, this demand was originally satisfied by subdividing the organ which contained the objects of interest into small chunks. These small chunks were then cut at random angles.

Although quite laborious, this approach works fine if we can recognize the objects even if the sections are cut at random angles. It is hopeless for some organs though. The recognition of structures, for example nuclei in the brain, may depend on the structures being sectioned at a specific angle. It may with experience become possible to identify a brain nucleus when cut at an "odd" angle, but it is pretty much hopeless to identify a brain nucleus in small chunks of brain which all are cut at different "odd" angles.

The labour involved in the preparation of randomly oriented sections and the restriction which thereby may be imposed on our ability to recognize/define the objects of interest have limited the ability the routinely measure areas and length. These problems have been overcome by the introduction of computer-interfaced microscopes and the use of rather thick sections. In this case the angle of the section is fixed, i.e. we can use the section planes traditionally employed to identify our object of interest. The computer takes on the job of placing the probes at random angles within the section. A line would be represented by a dot which moves across the computer screen as we focus through the depth of the section. An area would be a line which moves across the computer screen as we focus through the depth of the section. Our estimates of areas and lengths would be based on the number of times the points move across the lines.

Having the number of intersections of lines (probes) or objects (the length of which we are interested in) with surfaces (of objects) or areas (probes) we can calculate the length of the structures or their surface area using the formulas below.

This formula is used to calculate the length of the structures:

 
Lobj = 2×QA×Vref
 

 
 
 
 

Selected References

Two broad early reviews

Gundersen HJG, Bendtsen TF, Korbo L, Marcussen N, Moller A, Nielsen K, Nyengaard JR, Pakkenberg B, Sorensen FB, Vesterby A, West MJ (1988) Some new simple and efficient stereological methods and their use in pathological research and diagnosis. APMIS 96:379-394

This reference is an introduction to the basic concepts of quantitation, probes and sampling. APMIS is the abbreviation for Acta Pathologica, Microbiologica et Immunologica Scandinavica - some libraries will use this name for the catalogue.

Gundersen HJG, Bagger P, Bendtsen TF, Evans SM, Korbo L, Marcussen N, Moller A, Nielsen K, Nyengaard JR, Pakkenberg B, Sorensen FB, Vesterby A, West MJ (1988) The new stereological tools: disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis. APMIS 96:379-394

This reference is a bit more application oriented.

Estimating the number of objects

West MJ (1999) Stereological methods for estimating the total number of neurons and synapses: issues of precision and bias. Trends Neurosci 22:51-61

A discussion and explanation of the principles based on the experience gained in many discussions and explanations of the principles. The best review I've seen so far, it's relatively easy reading which gives everyone a chance to understand.

West MJ, Gundersen HJG (1990) Unbiased sterological estimation of the number of neurons in the human hippocampus. J Comp Neurol 296:1-22

By combining new methodology with human material and an interesting topic like aging this paper did a lot to introduce assumption-free stereological techniques into neuroscience. Is is also a good description of the combination of volume estimates with estimates of neuron density to arrive at total neuron number - the Nv × Vref method to count objects.

West MJ, Slomianka L, Gundersen HJG (1991) Unbiased stereological estimation of the total number of neurons in the subdivisions of the rat hippocampus using the optical fractionator. Anat Rec 231:482-497

This is the paper which introduced the optical fractionator into neuroscience. Hmm ... I had my "hands" in it too. Although true literally speaking, it probably wasn't even the tip of my little finger intellectually speaking. But as they say ... each fly on the pizza counts.

Estimating the length of objects

Calhoun ME, Mouton PR (2000) Length measurement: new developments in neurostereology and 3D imagery. J Chem Neuroanat 20:61-69

A nice review of the history and problems of length estimations. It does cover the new approaches to length estimations by isotropic probes and the ways in which they can be implemented in practice.

Tang Y, Nyengaard J R (1997) A stereological method for estimating to total length and size of mylin fibers in human brain white matter. J Neurosci Meth 73:193-200

In this study isotropy is guaranteed at the levels of the objects that we want to measure, i.e. small chunks of tissue are embedded at random angles - a now comparatively rare and labour intensive process, but less demanding in terms of the equipment needed to perform the actual estimation.

Larsen JO, Gundersen HJG, Nielsen J (1998) Global spatial sampling with isotropic virtual planes: estimators of length density and total length in thick, arbitrarily oriented sections. J Microsc 191:238-248

This is the first description of how length measurements can be performed in "normal" sections with the help of a computer-interfaced microscope and the appropriate software. It is a well illustrated paper with emphasis on the "how to" and plenty of schematic drawings which make it easier to understand the principles behind the method.

Mouton PR, Gokhale AM, Ward NL, West MJ. (2002) Stereological length estimation using spherical probes. J Microsc 206:54-64

This paper describes how the length of objects in a volume can be estimated by a using a probe which has an isotropic surface, i.e. a sphere - the "space balls" concept. Like the above implementation of length estimations, the method heavily relies on a computer-interfaced microscope and software to generate virtual spheres as the focal plane of the microscope lens is moved through thick tissue sections.


page content and construction: Lutz Slomianka
last updated: 5/08/09