Chapter 1
Introduction
Elements of Microstructure
Stereology is the science of the geometrical relationships between a structure
that exists in three dimensions and the images of that structure that are fundamentally two-dimensional (2D). These images may be obtained by a variety of means,
but fall into two basic categories: images of sections through the structure and projection images viewed through it. The most intensive use of stereology has been in
conjunction with microscope images, which includes light microscopes (conventional and confocal), electron microscopes and other types. The basic methods are
however equally appropriate for studies at macroscopic and even larger scales (the
study of the distribution of stars in the visible universe led to one of the stereological rules). Most of the examples discussed here will use examples from and the terminology of microscopy as used primarily in the biological and medical sciences,
and in materials science.
Image analysis in general is the process of performing various measurements
on images. There are many measurements that can be made, including size, shape,
position and brightness (or color) of all features present in the image as well as the
total area covered by each phase, characterization of gradients present, and so on.
Most of these values are not very directly related to the three-dimensional (3D)
structure that is present and represented in the image, and those that are may not
be meaningful unless they are averaged over many images that represent all possible portions of the sample and perhaps many directions of view. Stereological relationships provide a set of tools that can relate some of the measurements on the
images to important parameters of the actual 3D structure. It can be argued that
only those parameters that can be calculated from the stereological relationships
(using properly measured, appropriate data) truly characterize the 3D structure.
What are the basic elements of a 3D structure or microstructure? Threedimensional space is occupied by features (Figure 1.1) that can be:
1. Three-dimensional objects that have a volume, such as particles, grains (the
usual name for space-filling arrays of polyhedra as occur in metals and
ceramics), cells, pores or voids, fibers, and so forth.
2. Two-dimensional surfaces, which include the surfaces of the 3D objects, the
interfaces and boundaries between them, and objects such as membranes
that are actually of finite thickness but (because they are much thinner than
their lateral extent) can often be considered as being essentially 2D.
3. One-dimensional features, which include curves in space formed by the intersection of surfaces, or the edges of polyhedra. An example of a
1
2
Chapter 1
Figure 1.1. Diagram of a volume (red), surface (blue) and linear structure (green) in a
3D space. (For color representation see the attached CD-ROM.)
one-dimensional (1D) structure in a metal or ceramic grains structure is the
network of “triple lines” formed by the meeting of three grains or grain
boundaries. This class also includes objects whose lateral dimensions are so
small compared to their length that they can be effectively treated as 1D.
Examples are dislocations, fibers, blood vessels, and even pore networks,
depending on the magnification. Features that may be treated as 3D objects
at one magnification scale may become essentially 1D at a different scale.
4. Zero-dimensional features, which are basically points in space. These may be
ideal points such as the junctions of the 1D structures (nodes in the network
of triple lines in a grain structure, for example) or the intersection of 1D
structures with surfaces, or simply features whose lateral dimensions are
small at the magnification being used so that they are effectively treated as
points. An example of this is the presence of small precipitate particles in
metals.
In the most common type of imaging used in microscopy, the image represents a section plane through the structure. For an opaque specimen such as most
materials (metals, ceramics, polymers, composites) viewed in the light microscope
this is a cut and polished surface that is essentially planar, perhaps with minor (and
ignored) relief produced by polishing and etching that reveals the structure (Figure
1.2).
For most biological specimens, the image is actually a projected image
through a thin slice (e.g., cut by a microtome). The same types of specimens (except
that they are thinner) are used in transmission electron microscopy (Figure 1.3). As
long as the thickness of the section is much thinner than any characteristic dimension of the structure being examined, it is convenient to treat these projected images
as being ideal sections (i.e., infinitely thin) as well. When the sections become thick
(comparable in dimension to any feature or structure present) the analysis requires
modification, as discussed in Chapter 14.
When a section plane intersects features in the microstructure, the image
shows traces of those features that are reduced in dimension by one (Figure 1.4).
Introduction
3
Figure 1.2. Light microscope image of a metal (low carbon steel) showing the grain
boundaries (dark lines produced by chemical etching of the polished surface).
Figure 1.3. Transmission electron microscope image of rat liver. Contrast is produced
by a combination of natural density variations and chemical deposition by stains and
fixatives.
4
Chapter 1
Figure 1.4. Sectioning features in a 3D space with a plane, showing the area intersection with a volume (red), the line intersection with a surface (blue) and the point
intersection with a linear feature (green). (For color representation see the attached
CD-ROM.)
That is, volumes (three-dimensional) are revealed by areas, surfaces (twodimensional) by lines, curves (one-dimensional) by points, and points are not seen
because the section plane does not hit them. The section plane is an example of a
stereological probe that is passed through the structure. There are other probes that
are used as well—lines and points, and even volumes. These are discussed in detail
below and in the following chapters. But because of the way microscopes work we
nearly always begin with a section plane and a 2D image to interpret.
Since the features in the 2D image arise from the intersection of the plane
with the 3D structure, it is logical to expect that measurements on the feature traces
that are seen there (lower in dimension) can be utilized to obtain information about
the features that are present in 3D. Indeed, this is the basis of stereology. That is,
stereology represents the set of methods which allow 3D information about the
structure to be obtained from 2D images. It is helpful to set out the list of structural parameters that might be of interest and that can be obtained using stereological methods.
Geometric Properties of Features
The features present in a 3D structure have geometric properties that fall into
two broad categories: topological and metric. Metric properties are generally the
more familiar; these include volume, surface area, line length and curvature. In most
cases these are measured on a sample of the entire specimen and are expressed as
“per unit volume” of the structure. The notation used in stereology employs the
letters V, S, L, and M for volume, surface area, length, and curvature, respectively,
Introduction
5
and denotes the fact that they are measured with respect to volume using a subscript, so that we get
VV the volume fraction (volume per unit volume, a dimensionless ratio) of a
phase (the general stereological term used for any identifiable region
or class of objects, including voids)
SV the specific surface area (area per unit volume, with units of m-1) of a
surface
LV the specific line length (length per unit volume, with units of m-2) of a
curve or line structure
MV the specific curvature of surfaces (with units of m-2), which is discussed
in detail later in Chapter 5.
Other subscripts are used to indicate the measurements that have been made.
Typically the probes used for measurement are areas, lines and points as will be illustrated below. For example, measurements on an image are reported as “per unit
area” and have a subscript A, so that we can have
AA the area fraction (dimensionless)
LA the length of lines per unit area (units of m-1)
PA the number of points per unit area (units of m-2)
Likewise if we measure the occurrence of events along a line the subscript
L is used, giving
LL the length fraction (dimensionless)
PL or NL the number of points per unit length (units of m-1)
And if we place a grid of points on the image and count the number that
fall on a structure of interest relative to the total number of points, that would be
reported as
PP the point fraction (dimensionless)
Volumes, areas and lengths are metric properties whose values can be determined by a variety of measurement techniques. The basis for these measurements
is developed in Chapters 2 through 4. Equally or even more important in some applications are the topological properties of features. These represent the underlying
structure and geometry of the features. The two principle topological properties are
number NV and connectivity CV, both of which have dimensions of m-3 (per unit
volume). Number is a more familiar property than connectivity. Connectivity is a
property that applies primarily to network structures such as blood vessels or
neurons in tissue, dislocations in metals, or the porosity network in ceramics. One
way to describe it is the number of redundant connections between locations
(imagine a road map and the number of possible routes from point A to point B).
It is discussed in more detail in Chapter 3.
The number of discrete objects per unit volume is a quantity that seems quite
simple and is often desired, but is not trivial to obtain. The number of objects seen
6
Chapter 1
per unit area NA (referring to the area of the image on a section plane) has units of
m-2 rather than m-3. NA is an example of a quantity that is easily determined either
manually or with computer-based image analysis systems. But this quantity by itself
has no useful stereological meaning. The section plane is more likely to intercept
large particles than small ones, and the intersections with particles that are visible
do not give the size of the features (which are not often cut at their maximum diameter). The relationship between the desired NV parameter and the measured NA value
is NV = NA/·DÒ where ·DÒ is the mean particle diameter in 3D. In some instances
such as measurements on man-made composites in which the diameter of particles
is known, or of biological tissue in which the cells or organelles may have a known
size, this calculation can be made. In most cases it cannot, and indeed the idea of
a mean diameter of irregular non-convex particles with a range of sizes and shapes
is not intuitively obvious.
Ratios of the various structural quantities listed above can be used to calculate mean values for particles or features. For instance, the mean diameter value
·DÒ introduced above (usually called the particle height) can in fact be obtained as
MV/(2pNV). Likewise the mean surface area ·SÒ can be calculated as SV/NV and the
mean particle volume ·VÒ is VV/NV. These number averages and some other metric
properties of structures are listed in Table 1.1. The reasoning behind these relationships is shown in Chapter 4.
Typical Stereological Procedures
The 3D microstructure is measured by sampling it with probes. The most
common stereological probes are points, lines, surfaces and volumes. In fact, it is
not generally practical to directly place probes such as lines or points into the 3D
volume and so they are all usually implemented using sectioning planes. There is a
volume probe (called the Disector) which consists of two parallel planes with a small
separation, and is discussed in Chapters 5 and 7. Plane probes are produced in the
sectioning operation. Line probes are typically produced by drawing lines or grids
of lines onto the section image. Point probes are produced by marking points on
the section image, usually in arrays such as the intersections of a grid.
There probes interact with the features in the microstructure introduced
above to produce “events,” as illustrated in Figure 1.4. For instance, the interaction
of a plane probe with a volume produces section areas. Table 1.2 summarizes the
types of interactions that are produced. Note that some of these require measure-
Table 1.1. Ratios of properties give useful averages
Property
Volume
Surface
Height
Mean Lineal Intercept
Mean Cross-Section
Mean Surface Curvature
Symbol
·V Ò m3
·S Ò m2
·D Ò m1
·lÒ m1
·AÒ m2
·H Ò m-1
Relation
·V Ò = VV/NV
·S Ò = SV/NV
·D Ò = MV/2p · NV
·lÒ = 4·VV/SV
·AÒ = 2p · VV/MV
·H Ò = MV/SV
Introduction
7
Table 1.2. Interaction of probes with feature sets to produce events
3D Feature
Volume
Volume
Volume
Volume
Surface
Surface
Line
Probe
Volume
Plane
Line
Point
Plane
Line
Plane
Events
Ends
Cross-section
Chord intercept
Point intersection
Line trace
Point intersection
Intersection points
Measurement
Count
Area
Length
Count
Length
Count
Count
ment but some can simply be counted. The counting of events is very efficient, has
statistical precision that is easily calculated, and is generally a preferred method for
conducting stereological experiments. The counting of points, intersections, etc., is
done by choosing the proper probe to use with particular types of features so that
the events that measure the desired parameter can be counted. Figure 1.5 shows the
use of a grid to produce line and point probes for the features in Figure 1.4.
With automatic image analysis equipment (see Chapter 10) some of the measurement values shown in Table 1.2 may also be used such as the length of lines or
a
b
Figure 1.5. Sampling the section image from Figure 1.4 using a grid: a) a grid of lines
produces line segments on the areas that can be measured, and intersection points
with the lines that can be counted; b) a grid of points produces intersection points on
the areas that can be counted. (For color representation see the attached CD-ROM.)
8
Chapter 1
the area or intersections. In principle, these alternate methods provide the same
information. However, in practice they may create difficulties because of biased sampling by the probes (discussed in several chapters), and the precision and accuracy
of such measurements are hard to estimate. For example, measuring the true length
of an irregular line in an image composed of discrete pixels is not very accurate
because the line is “aliased” by consisting of discrete pixel steps. As another
example, area measurements in computer based systems are performed simply by
counting pixels. The pixels along the periphery of features are determined by brightness thresholding and are the source of measurement errors. Features with the same
area but different shapes have different amounts of perimeter and so produce different measurement precision, and it is not easy to estimate the overall precision in
a series of measurements. In contrast, the precision of counting experiments is well
understood and is discussed in Chapter 8.
Fundamental Relationships
The classical rules of stereology are a set of relationships that connect the
various measures obtained with the different probes with the structural parameters.
The most fundamental (and the oldest) rule is that the volume fraction of a phase
within the structure is measured by the area fraction on the image, or VV = AA. Of
course, this does not imply that every image has exactly the same area fraction as
the volume fraction of the entire sample. All of the stereological relationships are
based on the need to sample the structure to obtain a mean value. And the sampling must be IUR—isotropic, uniform and random—so that all portions of the
structure are equally represented (uniform), there is no conscious or consistent
placement of measurement regions with respect to the structure itself to select what
is to be measured (random), and all directions of measurement are equally represented (isotropic).
It is easy to describe sampling strategies that are not IUR and have various
types of bias, less easy to avoid such problems. For instance, if a specimen has gradients of the amount or size of particles within it, such as more of a phase of interest near the surface than in the interior, sampling only near the surface might be
convenient but it would be biased (nonuniform). If the measurement areas in cells
were always taken to include the nucleus, the results would not be representative
(nonrandom). If the sections in a fiber composite were always taken parallel to
the lay (orientation) of the fibers, the results would not measure them properly
(nonisotropic).
If the structure itself is perfectly IUR then any measurement performed any
place will do, subject only to the statistical requirement of obtaining enough measurements to get an adequate measurement precision. But few real-world specimens
are actually IUR, so sampling strategies must be devised to obtain representative
data that do not produce bias in the result. The basis for unbiased sampling is discussed in detail in Chapter 6, and some typical implementations in Chapter 7.
The fundamental relationships of stereology are thus expected value theorems that relate the measurements that can be made using the various probes to
the structural parameters present in three dimensions. The phrase expected value
Introduction
9
Table 1.3. Basic relationships for expected values
Measurement
Point count
Line intercept count
Area point count
Feature count
Area tangent count
Disector count
Line fraction
Area fraction
Length per area
Relation
·PPÒ = VV
·PLÒ = SV/2
·PAÒ = LV/2
·NAÒ = MV/2p = NV · ·DÒ
·TAÒ = MV/p
·NVÒ = NV
·LVÒ = VV
·AAÒ = VV
·LAÒ = (p/4) · SV
Property
Volume fraction
Surface area density
Length density
Total curvature
Total curvature
Number density
Volume fraction
Volume fraction
Surface area density
(denoted by · Ò) means that the equations apply to the average value of the population of probes in the 3D space, and the actual sample of the possible infinity of
probes that is actually used must be an unbiased sample in order for the measurement result to give an unbiased estimate of the expected value. The basic relationships using the parameters listed above are shown below in Table 1.3. These
relationships are disarmingly simple yet very powerful. They make no simplifying
assumptions about the details of the geometry of the structure. Examples of the use
and interpretation of these relationships are shown below and throughout this text.
It should also be noted that there may be many sets of features in a
microstructure. In biological tissue we may be interested in making measurements
at the level of organs, cells or organelles. In a metal or ceramic we may have several
different types of grains (e.g., of different chemical composition), as well as particles within the grains and perhaps at the interfaces between them (Figure 1.6).
Figure 1.6. Example of a polyhedral metal grain (a) with faces, edges (triple lines where
three faces from adjacent grains meet) and vertices (quadruple points where triple lines
meet and four adjacent grains touch); (b) shows the appearance of a representative
section through this structure. If particles form along the triple lines in the structure (c)
they appear in the section at the vertices of the grains (d). If particles form on the faces
of the grains (e) they appear in the section along the boundaries of the grains (f). (For
color representation see the attached CD-ROM.)
10
Chapter 1
In all cases there are several types of volume (3D) features present, as well as
the 2D surfaces that represent their shared boundaries, the space curves or linear
features where those boundaries intersect, and the points where the lines meet at
nodes. In other structures there may be surfaces such as membranes, linear features
such as fibers or points such as crystallographic defects that exist as separate
features.
Faced with the great complexity of structures, it can be helpful to construct
a feature list by writing down all of the phases or features present (and identifying
the ones of interest), and then listing all of the additional ones that result from their
interactions (contact surfaces between cells, intersections of fibers with surfaces, and
so on). Even for a comparatively simple structure such as the two-phase metal shown
in Figure 1.7 the feature list is quite extensive and it grows rapidly with the number
of distinct phases or classes of features present. This is discussed more fully in
Chapter 3 as the “qualitative microstructural state.”
Consider the common stereological measurements that can be performed by
just counting events when an appropriate probe is used to intersect these features.
The triple points can be counted directly to obtain number per unit area NA, which
can be multiplied by 2 to obtain the total length of the corresponding triple lines
per unit volume LV. Note that the dimensionality is the same for NA (m-2) and LV
(m/m3).
Other measurements are facilitated by using a grid. For example, a grid of
points placed on the image can be used to count the fraction of points that fall on
a phase (Figure 1.8). The point fraction PP is given by the number of events when
points (the intersections of lines in the grid) coincide with the phase divided by the
total number of points. Averaged over many fields, the result is a measurement of
the volume fraction of the phase VV.
Similarly, a line probe (the lines in the same grid) can be used to count events
where the lines cross the boundaries. As shown in Figure 1.8, the total number of
intersections divided by the total length of the lines in the grid is PL. The average
value of PL (which has units of m-1) is one half of the specific surface area (SV, area
per unit volume, which has identical dimensionality of m2/m3 = m-1).
Chapters 4 and 5 contain numerous specific worked examples showing how
these and other stereological parameters can be obtained by counting events produced by superimposing various kinds of grids on an image. Chapter 9 illustrates
the fact that in many cases the same grids and counting procedures can be automated using computer software.
Intercept Length and Grain Size
Most of the parameters introduced above are relatively familiar ones, such
as volume, area, length and number. Surfaces within real specimens can have very
large amounts of area occupying a relatively small volume. The mean linear intercept l of a structure is often a useful measure of the scale of that structure, and as
noted in the definitions is related to the surface-to-volume ratio of the features, since
l = 4 · VV/SV. It follows that the mean surface to volume ratio of particles (cells,
grains, etc.) of any shape is ·S/VÒ = 4/l.
Introduction
11
a
b
Figure 1.7. An example microstructure corresponding to a two-phase metal. Colorcoding is shown to mark a few of the features present: blue = b phase, red = ab interface; green = aaa triple points, yellow = bbb triple points. (For color representation see
the attached CD-ROM.)
12
Chapter 1
Figure 1.8. A grid (red) used to measure the image from Figure 1.7. There are a total
of 56 grid intersections, of which 9 lie on the b phase (blue marks). This provides an
estimate of the volume fraction of 9/56 = 16% using the relationship PP = VV. The total
length of grid line is 1106 mm, and there are 72 intersections with the ab boundary (green
marks). This provides an estimate of the surface area of that boundary of 2 · 72/1106 =
0.13 mm2/mm3 using the relationship SV = 2 · PL. There are 8 points representing bbb
triple points (yellow marks) in the area of the image (5455 mm2). This provides an estimate the length of triple line of 2 · 8/5455 = 2.9 · 10-3 mm/mm3 using the relationship LV
= 2 · PA. Similar procedures can be used to measure each of the feature types present
in the structure. (For color representation see the attached CD-ROM.)
The mean free distance between particles is related to the measured intercept
length of the region between particles, with the relationship L = l (VVb/VVa) where
b is the matrix and a the particles. This can also be structurally important, for
example, in metals where the distance between precipitate particles controls dislocation pinning and hence mechanical properties. To illustrate the fact that stereological rules and geometric relationships are not specific to microscopy applications,
Chandreshakar (1943) showed that for a random distribution of stars in space the
mean nearest neighbor distance is L = 0.554 · NV-1/3 where NV is the number of points
(stars) per unit volume. For small features on a 2D plane the similar relationship is
L = 0.5 · NA-1/2 where NA is the number per unit area; this will be used in Chapter 10
to test features for tendencies toward clustering or self-avoidance.
A typical grain structure in a metal consists of a space filling array of moreor-less polyhedral crystals. It has long been known that a coarse structure consisting of a few large grains has very different properties (lower strength, higher
Introduction
13
electrical conductivity, etc.) than one consisting of many small grains. The size of
the grains varies within a real microstructure, of course, and is not directly revealed
on a section image. The mean intercept length seems to offer a useful measure of
the scale of the structure that can be efficiently measured and correlated with various
physical properties or with fabrication procedures.
Before there was any field known as stereology (the name was coined about
40 years ago) and before the implications of the geometrical relationships were well
understood, a particular parameter called the “grain size number” was standardized by a committee of the American Society for Testing and Materials (ASTM).
Although it does not really measure a grain “size” as we normally use that word,
the terminology has endured and ASTM grain size is widely used. There are two
accepted procedures for determining the ASTM grain size (Heyn, 1903; Jeffries et
al., 1916), which are discussed in detail in Chapter 9. One method for determining
“grain size” actually measures the amount of grain boundary surface SV, and the
other method measures the total length of triple line LV between the grains. The SV
method is based on the intercept length, which as noted above gives the surface to
volume ratio of the grains.
Curvature
Curvature of surfaces is a less familiar parameter and requires some explanation. A fuller discussion of the role of surface curvature and the effect of edges
and corners is deferred to Chapter 5. The curvature of a surface in three dimensions
is described by two radii, corresponding to the largest and smallest circles that can
be placed tangent to the surface. When both circles lie inside the object, the surface
is locally convex. If they are both outside the object, the surface is concave. When
one lies inside and the other outside, the surface is a saddle. If one circle is infinite
the surface is cylindrical and if both are infinite (zero curvature) the surface is locally
flat. The mean curvature is defined as 1/2 (1/R1 + 1/R2).
The Gaussian curvature of the surface is 1/(R1 · R2) which integrates to 4p
over any convex surface. This is based on the fact that there is an element of surface
area somewhere on the feature (and only one) whose surface normal points in each
possible direction. As discussed in Chapter 5, this also generalizes to non-convex
but simply connected particles using the convention that the curvature of saddle
surface is negative.
MV is the integral of the local mean curvature over the surface of a structure. For any convex particle M = 2pD, where D is the diameter. MV is then the
product of 2p·DÒ times NV, where ·DÒ is the mean particle diameter and NV is the
number of particles present. The average surface curvature H = MV/SV, or the total
curvature of the surface divided by the surface area. This is a key geometrical property in systems that involve surface tension and similar effects.
For convex polyhedra, as encountered in many materials’ grain structures,
the faces are nearly flat and it might seem as though there is no curvature. But
in these cases the entire curvature of the object is contained in the edges, where
the surface normal vector rotates from one face normal to the next. The total
curvature is the same 2pD. If the length of the triple line where grains meet (which
14
Chapter 1
corresponds to the edges between faces) is measured as discussed above, then
MV = (p/2)·LV. Likewise for surfaces (usually called muralia) in space, the total curvature MV = (p/2) ·LV where the length is that of the edge of the surface. For rods,
fibers or other linear features the total curvature is MV = p ·LV; the difference from
the triple line case is due to the fact that the fibers have surface area around them
on all sides.
Curvature is measured using a moving tangent line or plane, which is
swept across the image or through the volume while counting events when it is
tangent to a line or surface. This is discussed more in Chapter 5 as it applies to
volumes. For a 2D image the tangent count is obtained simply by marking and
counting points where a line of any arbitrary orientation is tangent to the boundary. Positive tangent points (T+) are places where the local curvature is convex and
vice versa. The integral mean curvature is then calculated from the net tangent count
as MV = p(T+ - T-)/A. Note that for purely convex shapes there will be two T+ and
no T- counts for each particle and the total mean curvature HV is 2pNA.
Second Order Stereology
Combinations of probes can also be used in structures, often called secondorder stereology. Consider the case in which a grid of points is placed onto a field
of view and the particles which are hit by the points in the grid are selected for measurement. This is called the method of point-sampled intercept lengths. The point
sampling method selects features for measurement in proportion to their volume
(points are more likely to hit large than small particles). For each particle that is
thus selected, a line is drawn through the selection point to measure the radius from
that point to the boundary of the particle. If the section plane is isotropic in space,
these radial lines are drawn with uniformly sampled random orientations (Figure
1.9). If the section plane is a vertical section as discussed in Chapters 6 and 7, then
the lines should be drawn with sine-weighted orientations. If the structure is itself
isotropic, any direction is as good as another.
The volume of the particle vi = (4/3) ·p·r3Ò where ·Ò denotes the expected value
of the average over many measurements. This is independent of particle shape,
except that for irregular particles the radius measured should include all segments
of the particle section which the line intersects. Averaging this measurement over a
small collection of particles produces a mean value for the volume ·vÒV = (4/3) ·p··r3ÒÒ
where the subscript V reminds us that this is the volume-weighted mean volume
because of the way that the particles were selected for measurement.
If the particles have a distribution of sizes, the conventional way to describe
such a distribution is fN(V)dV where f is the fraction of number of the particles
whose size lies between V and V + dV. But we also note that the fraction of the
volume of particles in the structure with a volume in the same range if fV(V)dV.
These are related to each other by
fV dV = VfN (V )dV
(1.1)
This means that the volume-weighted mean volume that was measured above
is defined by
Introduction
15
Figure 1.9. Point sampled linear intercepts. A grid (green) is used to locate points within
features, from which isotropic lines are drawn (red) to measure a radial distance to the
boundary. (For color representation see the attached CD-ROM.)
Vmax
vV =
Ú V◊f
V
(V )dV
(1.2)
0
and if we substitute equation (1.1) into (1.2) we obtain
Vmax
vV =
ÚV
2
◊ fN (V )dV = v 2
N
(1.3)
0
The consequence of this is that the variance s 2 of the more familiar number
weighted distribution can be computed for particles of arbitrary shape, since for any
distribution
s 2 = n2
N
- n
2
N
(1.4)
This is a useful result, since in many cases the standard deviation or variance
of the particle size distribution is a useful characterization of that distribution,
useful for comparing different populations as discussed in Chapter 8. Determining
the volume-weighted mean volume with a point-sampled intercept method provides
half of the required information. The other needed value is the conventional or
number-weighted mean volume. This can be determined by dividing the total
16
Chapter 1
volume of the phase by the number of particles. We have already seen how to determine the total volume using a grid count. The number of particles can be measured
with the disector, discussed in Chapter 7. So it is possible to obtain the variance of
the distribution without actually measuring individual particles to construct the distribution function.
There is in fact another way to determine the number-averaged mean volume
of features ·vÒN without using the disector. It applies only to cases in which each
feature contains a single identifiable interior point (which does not, however, have
to be in the center of the feature), and the common instance in which it is used is
when this is the nucleus of a cell. The method (called the “Nucleator”) is similar to
the determination of volume-weighted mean volume above, except that instead of
selecting features using points in a grid, the appearance in the section of the selected
natural interior points is used. Of course, many features will not show these points
since the section plane may not intersect them (in fact, if they were ideal points they
would not be seen at all). When the interior point is present, it is used to draw the
radial line. As above, if the section is cut isotropically or if the structure is isotropic
than uniform random sampling of directions can be used, and if the surface is a
vertical section then sine-weighted sampling must be employed so that the directions are isotropic in 3D space as discussed in Chapters 6 and 7.
The radial line distances from the selected points to the boundary are
used as before to calculate a mean volume ·vÒN = (4/3) ·p··r3ÒÒ which is now
the number-weighted mean. The technique is unbiased for feature shape. The
key to this technique is that the particles have been selected by the identifying
points, of which there is one per particle, rather than using the points in a grid
(which are more likely to strike large features, and hence produce a volume-weighted
result).
Stereology of Single Objects
Most of the use of stereological measurements is to obtain representative
measures of 3D structures from samples, using a series of sections taken uniformly
throughout a specimen, and the quantities are expressed on a per-unit-volume basis.
The geometric properties of entire objects can also be estimated using the same
methods provided that the grid (either a 2D array of lines and points or a full
3D array as used for the potato in Figure 7.4 of Chapter 7) entirely covers the
object.
In two dimensions this method can be used to measure (for example) the
area of an irregular object such as a leaf (Figure 1.10). The expected value of the
point count in two dimensions is the area fraction of the object, or ·PPÒ = AA. For
an (n ¥ n) grid of points this is just ·PPÒ = ·PÒ/n2 where ·PÒ is the number of points
that lie on the feature. The area fraction AA = A/n2l 2 where l is the spacing of the
grid. Setting the point fraction equal to the area fraction gives A = l 2·PÒ.
This means that the number of points that lie on the feature times the size of one
square of the grid estimates the area of the feature. Of course, as the grid size
shrinks this is just the principle of integration. It is equivalent to tracing the feature
on graph paper and counting the squares within the feature, or of acquiring a
Introduction
17
Figure 1.10. A leaf with a superimposed grid. The grid spacing is 1/2 inch and 39 points
fall on the leaf, so the estimated area is 39 · (0.5)2 = 9.75 in2. This compares to a
measured area of 9.42 in2 using a program that counts all of the pixels within the leaf
area. (For color representation see the attached CD-ROM.)
digitized image consisting of square pixels and counting the number of pixels within
the feature.
When extended to three dimensions, the same method becomes one of counting the voxels (volume elements). If the object is sectioned by a series of N parallel
planes with a spacing of t, and a grid with spacing l is used on each plane, then the
voxel size is t · l 2. If the area in each section plane is measured as above then the
volume is the sum of the areas times the spacing t, or V = t · l 2·PTÒ where ·PTÒ is the
total number of hits of grid points on all N planar sections. This method, elaborated in Chapter 4, is sometimes called Cavalieri’s principle, but will also be familiar as the basis for the integration of a volume as V = ÚA · dz.
Measurements of the total size of an object can be made whenever the sampling grid(s) used for an object completely enclose it, regardless of the scale. The
18
Chapter 1
method can be used for a cell organelle or an entire organ. The appropriate choice
of a spacing and hence the number of points determines the precision; it is not necessary that the plane spacing t be the same as the grid spacing l .
Summary
Stereology is the study of geometric relationships between structures that
exist in three-dimensional space but are seen in two-dimensional images. The techniques summarized here provide methods for measuring volumes, surfaces and lines.
The most efficient methods are those which count the number of intersections that
various types of probes (such as grids of lines or points) make with the structure of
interest. The following chapters will establish a firm mathematical basis for the basic
relationships, illustrate the step-by-step procedures for implementing them, and deal
with how to create the most appropriate sampling probes, how to automate the measuring and counting procedures, and how to interpret the results.