Chapter 2
Basic Stereological Procedures
What Stereology Is
As stated in Chapter 1, stereology is a collection of tools that make measurements of geometric properties of real world microstructures practical.
In its typical application, information about the structure under study is
available as a collection of images prepared from that structure. Microscopes and
other imaging devices usually present us with images that are two dimensional. The
visual information they contain is a biased sample of the three dimensional structure. The image may be obtained from a section viewed in reflection, a slice viewed
in transmission, or the projection of some more or less rough external surface of
the structure. The geometry of features in the image may be quantified by measuring one or more geometric properties that may be defined for individual features, or
for a set of features.
Although most uses of stereology seek geometric information about microstructures, its application is not limited to microstructures. Stereology has been used
to study the geometry of rock structures in mine roofs, in astronomy, in geology, in
agronomy, but the analysis of microstructures is its forte. Thus, every field of
endeavor that deals with microstructures has found stereology useful, including
materials science, mineralogy, physiology, botany, anatomy, pathology, histology,
and a variety of other -ologies in the life sciences.
With modern image analysis software it is possible to define and measure
several dozen such geometric properties of features in a two dimensional image.
Indeed, because a picture is worth “a thousand words,” the central problem in image
analysis is the reduction of a few megabytes of information that constitutes a grey
scale or color image to a few meaningful and useful numbers. Stereology provides
one answer to the question, “Of all of these dozens of numbers that can be defined
and measured on an image, which are useful to me?” The specific answer to that
question depends explicitly upon the application under consideration, but stereology limits the numbers that have useful meaning for the three dimensional
microstructure sampled.
If your interest lies in obtaining quantitative information about the three
dimensional structure that the image samples then the answer to that question is
limited. Only a small subset of the geometric properties that can be measured in a
two dimensional image have stereological meaning, that is, are unambiguously
related to geometric properties of the three dimensional structure which the
image samples. Only those relatively few image properties that are identified in the
19
20
Chapter 2
Figure 2.1. Point, line and plane probes are constructed by superimposing a grid of
lines on a plane section through the microstructure. (For color representation see the
attached CD-ROM.)
fundamental equations of stereology have potential for yielding quantitative three
dimensional information.
How Stereology Works
Insight into the geometry of a three dimensional microstructure is acquired
by sampling the structure with probes. Typically the structure is probed by an array
of plane sections, or by a collection of line probes, point probes or small thin volume
elements called disector probes. Other probes have been devised for specific applications and will be treated in later chapters, but these four probe types, points, lines,
planes and volumes, are classical in stereology. Operationally, these probes are generated by sectioning the structure with a plane and preparing the revealed image for
observation, as shown in Figure 2.1. In a given field of view a plane probe is delineated by framing an area for examination in the field of view. Thus, a field on a
microstructural section is viewed as a sample of the population of planes that can
be constructed in the three dimensional space that the specimen occupies. Line and
point probes are obtained in such a field by superimposing a grid consisting of an
arrangement of lines and points, as shown in Figure 2.1. A variety of grids have
been devised for these purposes, and examples will be examined and recommendations made in later chapters.
The disector volume element probe, as shown in Figure 2.2, consists of a
pair of planes spaced closely enough so that unambiguous inferences may be made
about how features appearing on the two planes are connected in the volume
between the planes.
These probes interact with features in the image to produce outcomes or
events of stereological interest. Figure 2.1 shows a microstructure composed of two
feature sets. Call the white areas a features and the shaded areas b features.
Basic Stereological Procedures
21
Figure 2.2. A disector probe is constructed by comparing the microstructural features
observed on two closely spaced plane sections, to characterize the volume contained
between them. In this example two features continue through both planes, two features
appear only on the upper plane, and three features appear only in the lower plane, indicating that their tops must lie in the volume between the planes. (For color representation see the attached CD-ROM.)
1. A point on the grid may exhibit two outcomes in this structure: it may lie in
the a feature set, or in the b set1.
2. Lines in the grid interact with traces of the ab boundary (an interface surface
in three dimensions) that separates the two feature sets; here the event of
interest is the intersection of the line probe with such a trace.
3. The area of the plane delineated by the boundary of the grid interacts with
a and b features in the structure; the event of interest is the intersection of
the plane probe with such three dimensional features.
4. A small volume of the structure called a disector is contained between
matched fields delineated on two plane sections a small known distance apart
interacts with a or b features in the volume; the event of interest is the
appearance of the top of a particle within this volume.
This brief list of probe/event combinations is by no means an exhaustive list
of the interactions that are of interest in stereology; others will be developed in
appropriate chapters. However, these combinations are most frequently applied and
serve to illustrate how stereology works.
In a typical application the set of probes of a given class is scanned over a
field and the events of interested are noted and simply counted. For example,
1. The points in the grid are scanned and the number of points that lie in the
b phase is counted and recorded for the field. (In Figure 2.1 four points lie
in the b phase.)
1
The third possibility, that the point lies on the boundary between the features, arises because
in real microstructures the boundaries and grid lines have finite widths. This possibility will
be discussed in Chapter 4.
22
Chapter 2
2. The lines in the grid are systematically scanned along their length and the
number of intersections with ab interface is counted and recorded (34 intersections in Figure 2.1);
3. The area of the field is scanned systematically and the number of b features
is counted (24 features in Figure 2.1)2.
4. Comparisons between features on the top (referen ce plane) and bottom
(look-up plane) of the disector shown in Figure 2.2 reveals the appearance
of three new particle tops within the volume sampled.
Each of these simple counting experiments is repeated on a number of fields
that appropriately sample the three dimensional structure. What is meant by an
appropriate sample is a central issue in stereology. Indeed, as will be clearly shown
in this development, the measurements of stereology are trivially simply; the fundamental mathematical relationships that connect these measurements with three
dimensional geometry are also trivially simple. The design and collection of an
appropriate sample of probes is the hard part of stereology.
Counts for a given probe/event measurement are usually reported as normalized counts by dividing the number of counts by the total quantity of probe
scanned in the field. For the examples cited above for Figure 2.1:
1. The count of points that lie in the b phase is normalized by dividing by the
total number of points in the grid (25 points in Figure 2.1, to give a point
fraction PP of 4/25 or 0.16;
2. The count of the number of intersections of lines is normalized by dividing
by the total length of line probes scanned in the field (34 intersection
points on the 40 mm of probe line length in Figure 2.1) to give a line intercept count PL of 0.85 (counts per micrometer) or 0.85 ¥ 104 = 8,500 (counts
per cm);
3. The count of features contained in the delineated area of the field is normalized by dividing by that area (24 features in the 160 mm2 of area probed
in Figure 2.1) to give the feature count NA of 0.15 (counts per mm2) or 0.15
¥ 108 = 1.5 ¥ 107 (counts per cm2).
4. Each new particle appearance noted in the disector volume identifies a point
which is the upper bound of that particle in the direction perpendicular to
the planes of the disector. Since each particle has a single upper bounding
point, dividing this count by the volume contained within the disector provides an estimate of NV, the number of features per unit volume in the structure. If in Figure 2.2 the area of the counting frame outlined on the top
frame is 130 mm2 and the distance between the planes is 6 mm, the volume of
2
Some of the features in the area intersect the boundaries of the counting frame. Adopting
the rule that features that intersect two adjacent edges of the frame boundary are counted
(in the figure, the left and bottom edges) while those that intersect the other two edges are
excluded provides an unbiased feature count.
Basic Stereological Procedures
23
the disector is 780 mm3. The corresponding estimate of number density is
3/780 = 0.0038 (particles/mm3) = 3.8 ¥ 109 particles per cm3.
For a given measurement, counts are recorded on a number of fields under
the microscope, on photomicrographs, or on a digitized image. The mean value and
standard deviation of the counts from this sample of probes selected for measurement are computed using standard statistical formulae (Chapter 8). The standard
deviation is the basis for estimating the precision of the estimate of the mean value
and deciding whether a sufficient number of fields have been be examined. The mean
value of the counts in the sample is used to estimate the expected value for those
counts for the entire population of probes. The central objective of sample design
in stereology (i.e., the selection of fields to be included and how they are to be measured) is to devise a sample that provides an unbiased estimate of this expected value
or population mean for all such probes that can be constructed in three dimensional
space.
The connection between expected values of counts for a given probe/event
combination and a geometric property of the three dimensional structure is
obtained by applying the appropriate fundamental relation of stereology. These
equations, elegant in their simplicity, have the status of expected value theorems. The
simplest of these relations,
·PPÒ = Vn
(2.1)
where VV is the volume fraction occupied by the phase being counted and the brackets around ·PPÒ signify the expected value for this normalized count, in this case the
point fraction. This equation may be read “The expected value of the fraction of
the population of points that exist in the volume of the structure under study that
lie in the phase of interest is equal to the fraction of the volume of the structure
occupied by that phase.” For the b phase sampled in Figure 2.1, the expected value
of PPb is VVb = 0.16. If the set of fields included in the count is appropriately selected,
the mean value of PP for this sample is an unbiased estimator of the expected value
of PP for the population of points in the three dimensional space occupied by the
specimen and thus of the volume fraction of the b phase in the three dimensional
structure.
Other fundamental relations are equally simple. The expected value of the
line intercept count for the population of line probes in space is proportional to the
surface area density of the interfaces being counted:
PL =
1
SV
2
(2.2)
where SV is the surface area of per unit volume of the set of interfacial features in
the three dimensional structure whose intersections were counted in the experiment.
In Figure 2.1, the line intercept count estimates the surface area per unit volume of
the ab boundaries separating the a and b feature sets. Sv = 2PL = 2 · 8,500 = 17,000
(cm2/cm3).
The feature count is related to the number density NV of features in the structure and their average caliper diameter ·DÒ:
·NAÒ = NV ·DÒ
(2.3)
24
Chapter 2
This latter relation is limited in its application to structures with features that
are convex bodies, although it is otherwise independent of their shape. For features
of more general shape ·NAÒ measures a property called the integral mean curvature,
discussed in Chapter 5.
Similarly, three dimensional features or particles that are convex have a single
“upper bound point” in any direction. The expected value of the number of points
per unit volume of disector probes is a direct estimate of the number of such points
per unit volume in the structure:
·NVÒ = NV
(2.4)
The expected value relationships of stereology are powerful because the
require no assumptions about the geometry of the features being characterized. That
point bears repeating. The expected value of counts made on any of the probe/event
combinations in an appropriately designed sample of probes provide an unbiased
estimate of the corresponding geometric property of the three dimensional feature
set without geometric assumptions.
The typical stereological experiment begins by devising a procedure for
selecting a set of fields to be examined that provides an appropriate sample of the
population of probes to be used in the measurement. Each field is examined by overlaying an appropriate grid containing those probes. Interactions of the probes with
features of interest in the structure are noted and counted. The mean of these counts
for the sample of probes is computed and normalized. This sample mean is then
used to estimate the population mean for the same count for the full set of probes
in three dimensions. The result is inserted in the appropriate fundamental relationship to yield an estimate of the corresponding geometric property for the three
dimensional structure. The standard deviation of the sample measurements is used
to compute the precision of this estimate.
Why Stereology Works
Why do simple counts of the number of probe/event occurrences on an
appropriate sample of probes report geometric properties of the three dimensional
structure they sample? This question goes to the core of stereology, which is contained in its collection of fundamental relationships. In this overview brief arguments are presented with the intention of making these relationships plausible; more
rigorous derivations of these relationships are presented in later sections.
Keep in mind that the stereological relationships are expected value theorems. This means that the result applies when the whole population of probes is
included in the sample. The task is to design a sample of that population which will
yield an unbiased estimate of that expected value.
Consider the population of points in three dimensional space. In a two phase
a + b structure this population of points fills the volumes occupied by the a and b
phases. If points are uniformly distributed in space the number of points in the a
part of the structure is proportional to the volume of the a part. The number of
points in the total specimen is proportional to the total volume of the specimen.
The volume fraction of the a phase may be thought of as the ratio of the number
Basic Stereological Procedures
25
of points in the a phase to the number of points in the structure. Equation (2.1)
follows. If a sample of points is drawn uniformly and randomly from this population, i.e., without bias, then the fraction of points in a in this sample will estimate
the fraction of points in the population, which in equation (2.1) is shown to be the
volume fraction of the a phase in the structure.
To make equation (2.2) plausible consider the population of vertical lines in
three dimensional space, as shown in Figure 2.3. An individual member of this set
of lines may be located by its point of intersection P with a horizontal plane. Now
focus on an element (a small piece) of the ab surface dS somewhere in the three
dimensional structure. In the application of equation (1.2) the event of interest is
the intersection of a line probe with such a surface in space. In Figure 2.3, dA is the
area of the projection of dS on the horizontal plane. The event of interest, i.e., an
intersection with dS, occurs for the subset of vertical line probes that pass through
the area dA on the horizontal plane. The number of lines that produces an
intersection with dS is evidently reported by the number of points in the horizontal plane that lie within its projected area on that plane, which is evidently determined by the area dA. The projected area dA is simply related to the area of the
surface element dS:
dA = dS cos F
(2.5)
where F is the angle between the direction that is perpendicular to the surface
element in the three dimensional structure and the direction of the line probes. Averaging this result over the full population of lines in three dimensional space averages the cosine of F over the sphere of orientation. In Chapter 4 this average is
shown to be 1/2. Thus parallel line probes in space sense the projected area of surfaces (on a plane that is perpendicular to the direction of the set of lines) they
encounter in the structure. Including all orientations of lines in the probe set reports
(one half of) the true area of the surfaces in three dimensions.
Equation (2.3) derives from interactions of plane probes with features in the
structure. To make this result plausible focus on the subset of the population of
plane probes that all share the same normal direction (i.e., are all parallel to each
other). For example, the set of horizontal planes in space is illustrated in Figure 2.4.
An individual member of this set of planes may be uniquely identified by its point
of intersection with the vertical axis. The population of planes in this set is thus the
population of points on the vertical axis. A feature in the b phase will be observed
on a plane probe if the probe intersects the feature. A plane intersects the feature
in Figure 2.4 if it lies within the vertical interval bounded by the two planes that
are tangent to the top and bottom of the b feature. The number of planes that satisfy
this condition is the same as the number of points that lie within the interval on the
vertical axis between the two tangent planes, since each plane is uniquely represented
by its point on this axis. Assuming the population of points along this line is uniformly distributed, the number of points in this interval is proportional to the length
of this interval. The length of the interval is equal to the tangent diameter D (i.e.,
the distance between the two tangent planes) in the vertical direction. Thus the
feature count on horizontal planes senses tangent diameters in the vertical direction
a
b
Figure 2.3. The set of vertical line probes (a) intersect an element of surface area dS
if their positions lie within the projected area dA on the horizontal plane (b). The relationship between dA and dS depends on the angle F between the line probe and the
surface normal. (For color representation see the attached CD-ROM.)
Basic Stereological Procedures
27
Figure 2.4. The set of horizontal planes intersect a particle if their positions on the vertical axis lie between the top and bottom tangent planes, i.e., within the caliper diameter D in the vertical direction. (For color representation see the attached CD-ROM.)
of the three dimensional features. If the sample of planes includes the full set of
orientations of plane probes then the tangent diameters will be averaged over the
set of orientations in space See equation (2.3) on page 23.
The disector probe is a direct sample of the three dimensional volume
of the structure. Equation (2.4) simply recognizes that the expected value of
the NV for the population of disectors is the same as the value for the
structure.
Point probes sample features according to their volume; line probes sense the
area of surfaces and interfaces; plane probes select features according to their
tangent diameter; disector probes select features according to their number. The
demonstrations in this section show that these counting measurements sense directly
their corresponding geometric properties. That is the basis for the fundamental relationships in stereology.
Ground Rules for Applying Stereology
The fundamental relationships of stereology make no geometric assumptions. The interactions of probes with features are free of such assumptions precisely because the probe/event combinations that are counted sense specific
geometric properties directly, regardless of how the features that display those properties are arrayed in the structure. However the connecting relationships are expected
value theorems relating the average value of counts for the entire population of
probes to a corresponding geometric property. In a given experiment some subset
of this population of probes is selected for inclusion in a sample. The mean of the
sample is used to estimate the mean of the probe population. In order to insure that
this sample mean is an unbiased estimate of the expected value it is essential to obey
the ground rules for stereological sample design.
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Chapter 2
The ground rules for sample design center around the word uniform. The
fundamental relationships assume that the elements of the population of each class
of probes,—points, lines, planes, disectors, etc.,—are uniformly distributed in space.
Positions of probes are uniformly distributed in space in the derivation of these
equations. Probes that have directions, like lines and planes, have in addition orientations that are uniformly distributed over the sphere of orientations in three dimensional space, i.e., are isotropically distributed.
If the structures being probed are uniformly distributed in space and
isotropic in orientation, then any set of sample probes can be used. However, it is
difficult to know a priori whether this condition is met, and for many real structures
of interest it is not. A discussion of anisotropy and gradients is presented in Chapters 3, 6 and 13.
For statistical purposes it is also convenient that sample probes be selected
randomly from these uniform distributions because this simplifies the statistical
interpretation of results. In some cases it has been shown that relaxation of this
requirement samples the structure more efficiently.
The typical design of the selection of fields, grids and probes for examination in a stereological experiment strives to attain an Isotropic, Uniform, Random
sample of the corresponding population of probes. In seeking an unbiased estimate
of the expected value of an event count the primary requirements are that the probe
sample be “IUR”.
Summary
Stereology is the methodology that is required to obtain quantitative estimates of geometric properties of features in three dimensional structures from measurements that are made on essentially two dimensional images.
The structure is sampled with point, line, plane, disector or other probes.
Events that result from interactions of these probes with features in the structure
are counted. Normalized averages of these counts are used to estimate the corresponding average count for the full population of probes. Fundamental relations of
stereology connect these expected values for the probe population to a geometric
property of the structure being sampled.
Point probes sense the volume of three dimensional features. Line probes
sense the area of interfaces in the structure. Plane probes sense the average tangent
diameter of features. Disector probes sense the number of features in the structure.
The relationships of stereology are geometrically general because each kind
of probe senses its corresponding geometrical property directly. The result is thus
independent of how the features that exhibit the property are distributed in the
structure.
Unbiased estimation of the stereological counting measurements requires
that the selection of probes employed in any experiment be chosen Isotropically,
Uniformly and Randomly from the corresponding population of those probes.