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Chapter 1
Introduction
Elements of Microstructure
Stereology is the science of the geometrical relationships
between a structurethat exists in three dimensions and the images
of that structure that are fundamen-tally 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
pro-jection images viewed through it. The most intensive use of
stereology has been inconjunction with microscope images, which
includes light microscopes (conven-tional and confocal), electron
microscopes and other types. The basic methods arehowever equally
appropriate for studies at macroscopic and even larger scales
(thestudy of the distribution of stars in the visible universe led
to one of the stereolog-ical rules). Most of the examples discussed
here will use examples from and the ter-minology 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
measurementson 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 thetotal 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 notbe meaningful unless they are
averaged over many images that represent all possi-ble portions of
the sample and perhaps many directions of view. Stereological
rela-tionships provide a set of tools that can relate some of the
measurements on theimages to important parameters of the actual 3D
structure. It can be argued thatonly 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?
Three-dimensional space is occupied by features (Figure 1.1) that
can be:
1. Three-dimensional objects that have a volume, such as
particles, grains (theusual name for space-filling arrays of
polyhedra as occur in metals andceramics), cells, pores or voids,
fibers, and so forth.
2. Two-dimensional surfaces, which include the surfaces of the
3D objects, theinterfaces and boundaries between them, and objects
such as membranesthat are actually of finite thickness but (because
they are much thinner thantheir lateral extent) can often be
considered as being essentially 2D.
3. One-dimensional features, which include curves in space
formed by the inter-section of surfaces, or the edges of polyhedra.
An example of a
1
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one-dimensional (1D) structure in a metal or ceramic grains
structure is thenetwork of triple lines formed by the meeting of
three grains or grainboundaries. This class also includes objects
whose lateral dimensions are sosmall 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 objectsat one
magnification scale may become essentially 1D at a different
scale.
4. Zero-dimensional features, which are basically points in
space. These may beideal points such as the junctions of the 1D
structures (nodes in the networkof triple lines in a grain
structure, for example) or the intersection of 1Dstructures with
surfaces, or simply features whose lateral dimensions aresmall at
the magnification being used so that they are effectively treated
aspoints. An example of this is the presence of small precipitate
particles inmetals.
In the most common type of imaging used in microscopy, the image
repre-sents a section plane through the structure. For an opaque
specimen such as mostmaterials (metals, ceramics, polymers,
composites) viewed in the light microscopethis is a cut and
polished surface that is essentially planar, perhaps with minor
(andignored) relief produced by polishing and etching that reveals
the structure (Figure1.2).
For most biological specimens, the image is actually a projected
imagethrough a thin slice (e.g., cut by a microtome). The same
types of specimens (exceptthat they are thinner) are used in
transmission electron microscopy (Figure 1.3). Aslong as the
thickness of the section is much thinner than any characteristic
dimen-sion of the structure being examined, it is convenient to
treat these projected imagesas 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
requiresmodification, as discussed in Chapter 14.
When a section plane intersects features in the microstructure,
the imageshows traces of those features that are reduced in
dimension by one (Figure 1.4).
2 Chapter 1
Figure 1.1. Diagram of a volume (red), surface (blue) and linear
structure (green) in a3D space. (For color representation see the
attached CD-ROM.)
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Introduction 3
Figure 1.2. Light microscope image of a metal (low carbon steel)
showing the grainboundaries (dark lines produced by chemical
etching of the polished surface).
Figure 1.3. Transmission electron microscope image of rat liver.
Contrast is producedby a combination of natural density variations
and chemical deposition by stains andfixatives.
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That is, volumes (three-dimensional) are revealed by areas,
surfaces (two-dimensional) by lines, curves (one-dimensional) by
points, and points are not seenbecause the section plane does not
hit them. The section plane is an example of astereological probe
that is passed through the structure. There are other probes
thatare used as welllines and points, and even volumes. These are
discussed in detailbelow and in the following chapters. But because
of the way microscopes work wenearly 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 planewith the 3D structure, it is logical to expect that
measurements on the feature tracesthat are seen there (lower in
dimension) can be utilized to obtain information aboutthe 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 thestructure to be obtained from 2D images. It is
helpful to set out the list of struc-tural parameters that might be
of interest and that can be obtained using stereo-logical
methods.
Geometric Properties of Features
The features present in a 3D structure have geometric properties
that fall intotwo broad categories: topological and metric. Metric
properties are generally themore familiar; these include volume,
surface area, line length and curvature. In mostcases these are
measured on a sample of the entire specimen and are expressed asper
unit volume of the structure. The notation used in stereology
employs theletters V, S, L, and M for volume, surface area, length,
and curvature, respectively,
4 Chapter 1
Figure 1.4. Sectioning features in a 3D space with a plane,
showing the area inter-section with a volume (red), the line
intersection with a surface (blue) and the pointintersection with a
linear feature (green). (For color representation see the
attachedCD-ROM.)
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and denotes the fact that they are measured with respect to
volume using a sub-script, so that we get
VV the volume fraction (volume per unit volume, a dimensionless
ratio) of aphase (the general stereological term used for any
identifiable regionor class of objects, including voids)
SV the specific surface area (area per unit volume, with units
of m-1) of asurface
LV the specific line length (length per unit volume, with units
of m-2) of acurve or line structure
MV the specific curvature of surfaces (with units of m-2), which
is discussedin 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 illus-trated below. For example,
measurements on an image are reported as per unitarea 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
subscriptL 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 thatfall on a structure of interest relative to the total
number of points, that would bereported as
PP the point fraction (dimensionless)
Volumes, areas and lengths are metric properties whose values
can be deter-mined by a variety of measurement techniques. The
basis for these measurementsis developed in Chapters 2 through 4.
Equally or even more important in some appli-cations are the
topological properties of features. These represent the
underlyingstructure and geometry of the features. The two principle
topological properties arenumber NV and connectivity CV, both of
which have dimensions of m-3 (per unitvolume). Number is a more
familiar property than connectivity. Connectivity is aproperty that
applies primarily to network structures such as blood vessels
orneurons in tissue, dislocations in metals, or the porosity
network in ceramics. Oneway 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 quitesimple and is often desired, but is not trivial to
obtain. The number of objects seen
Introduction 5
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per unit area NA (referring to the area of the image on a
section plane) has units ofm-2 rather than m-3. NA is an example of
a quantity that is easily determined eithermanually or with
computer-based image analysis systems. But this quantity by
itselfhas no useful stereological meaning. The section plane is
more likely to interceptlarge particles than small ones, and the
intersections with particles that are visibledo not give the size
of the features (which are not often cut at their maximum
diam-eter). The relationship between the desired NV parameter and
the measured NA valueis NV = NA/D where D is the mean particle
diameter in 3D. In some instancessuch as measurements on man-made
composites in which the diameter of particlesis known, or of
biological tissue in which the cells or organelles may have a
knownsize, this calculation can be made. In most cases it cannot,
and indeed the idea ofa mean diameter of irregular non-convex
particles with a range of sizes and shapesis not intuitively
obvious.
Ratios of the various structural quantities listed above can be
used to cal-culate mean values for particles or features. For
instance, the mean diameter valueD introduced above (usually called
the particle height) can in fact be obtained asMV/(2pNV). Likewise
the mean surface area S can be calculated as SV/NV and themean
particle volume V is VV/NV. These number averages and some other
metricproperties of structures are listed in Table 1.1. The
reasoning behind these rela-tionships is shown in Chapter 4.
Typical Stereological Procedures
The 3D microstructure is measured by sampling it with probes.
The mostcommon stereological probes are points, lines, surfaces and
volumes. In fact, it isnot generally practical to directly place
probes such as lines or points into the 3Dvolume and so they are
all usually implemented using sectioning planes. There is avolume
probe (called the Disector) which consists of two parallel planes
with a smallseparation, and is discussed in Chapters 5 and 7. Plane
probes are produced in thesectioning operation. Line probes are
typically produced by drawing lines or gridsof lines onto the
section image. Point probes are produced by marking points onthe
section image, usually in arrays such as the intersections of a
grid.
There probes interact with the features in the microstructure
introducedabove to produce events, as illustrated in Figure 1.4.
For instance, the interactionof a plane probe with a volume
produces section areas. Table 1.2 summarizes thetypes of
interactions that are produced. Note that some of these require
measure-
6 Chapter 1
Table 1.1. Ratios of properties give useful averages
Property Symbol RelationVolume V m3 V = VV/NVSurface S m2 S =
SV/NVHeight D m1 D = MV/2p NVMean Lineal Intercept l m1 l =
4VV/SVMean Cross-Section A m2 A = 2p VV/MVMean Surface Curvature H
m-1 H = MV/SV
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ment but some can simply be counted. The counting of events is
very efficient, hasstatistical precision that is easily calculated,
and is generally a preferred method forconducting stereological
experiments. The counting of points, intersections, etc., isdone by
choosing the proper probe to use with particular types of features
so thatthe events that measure the desired parameter can be
counted. Figure 1.5 shows theuse 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 mea-surement values shown in Table 1.2 may also be used such as
the length of lines or
Introduction 7
Table 1.2. Interaction of probes with feature sets to produce
events
3D Feature Probe Events MeasurementVolume Volume Ends
CountVolume Plane Cross-section AreaVolume Line Chord intercept
LengthVolume Point Point intersection CountSurface Plane Line trace
LengthSurface Line Point intersection CountLine Plane Intersection
points Count
Figure 1.5. Sampling the section image from Figure 1.4 using a
grid: a) a grid of linesproduces line segments on the areas that
can be measured, and intersection pointswith the lines that can be
counted; b) a grid of points produces intersection points onthe
areas that can be counted. (For color representation see the
attached CD-ROM.)
a
b
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the area or intersections. In principle, these alternate methods
provide the sameinformation. However, in practice they may create
difficulties because of biased sam-pling by the probes (discussed
in several chapters), and the precision and accuracyof such
measurements are hard to estimate. For example, measuring the true
lengthof an irregular line in an image composed of discrete pixels
is not very accuratebecause the line is aliased by consisting of
discrete pixel steps. As anotherexample, area measurements in
computer based systems are performed simply bycounting pixels. The
pixels along the periphery of features are determined by
bright-ness thresholding and are the source of measurement errors.
Features with the samearea but different shapes have different
amounts of perimeter and so produce dif-ferent measurement
precision, and it is not easy to estimate the overall precision ina
series of measurements. In contrast, the precision of counting
experiments is wellunderstood and is discussed in Chapter 8.
Fundamental Relationships
The classical rules of stereology are a set of relationships
that connect thevarious measures obtained with the different probes
with the structural parameters.The most fundamental (and the
oldest) rule is that the volume fraction of a phasewithin the
structure is measured by the area fraction on the image, or VV =
AA. Ofcourse, this does not imply that every image has exactly the
same area fraction asthe volume fraction of the entire sample. All
of the stereological relationships arebased on the need to sample
the structure to obtain a mean value. And the sam-pling must be
IURisotropic, uniform and randomso that all portions of
thestructure are equally represented (uniform), there is no
conscious or consistentplacement of measurement regions with
respect to the structure itself to select whatis to be measured
(random), and all directions of measurement are equally
repre-sented (isotropic).
It is easy to describe sampling strategies that are not IUR and
have varioustypes of bias, less easy to avoid such problems. For
instance, if a specimen has gra-dients of the amount or size of
particles within it, such as more of a phase of inter-est near the
surface than in the interior, sampling only near the surface might
beconvenient but it would be biased (nonuniform). If the
measurement areas in cellswere 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 anyplace will do, subject only to the statistical
requirement of obtaining enough mea-surements to get an adequate
measurement precision. But few real-world specimensare actually
IUR, so sampling strategies must be devised to obtain
representativedata that do not produce bias in the result. The
basis for unbiased sampling is dis-cussed in detail in Chapter 6,
and some typical implementations in Chapter 7.
The fundamental relationships of stereology are thus expected
value theo-rems that relate the measurements that can be made using
the various probes to the structural parameters present in three
dimensions. The phrase expected value
8 Chapter 1
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(denoted by ) means that the equations apply to the average
value of the popu-lation of probes in the 3D space, and the actual
sample of the possible infinity ofprobes that is actually used must
be an unbiased sample in order for the measure-ment result to give
an unbiased estimate of the expected value. The basic
rela-tionships using the parameters listed above are shown below in
Table 1.3. Theserelationships are disarmingly simple yet very
powerful. They make no simplifyingassumptions about the details of
the geometry of the structure. Examples of the useand
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 amicrostructure. In biological tissue we may be interested in
making measurementsat the level of organs, cells or organelles. In
a metal or ceramic we may have severaldifferent types of grains
(e.g., of different chemical composition), as well as par-ticles
within the grains and perhaps at the interfaces between them
(Figure 1.6).
Introduction 9
Figure 1.6. Example of a polyhedral metal grain (a) with faces,
edges (triple lines wherethree faces from adjacent grains meet) and
vertices (quadruple points where triple linesmeet and four adjacent
grains touch); (b) shows the appearance of a representativesection
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 facesof the grains (e) they
appear in the section along the boundaries of the grains (f).
(Forcolor representation see the attached CD-ROM.)
Table 1.3. Basic relationships for expected values
Measurement Relation PropertyPoint count PP = VV Volume
fractionLine intercept count PL = SV/2 Surface area densityArea
point count PA = LV/2 Length densityFeature count NA = MV/2p = NV D
Total curvatureArea tangent count TA = MV/p Total curvatureDisector
count NV = NV Number densityLine fraction LV = VV Volume
fractionArea fraction AA = VV Volume fractionLength per area LA =
(p/4) SV Surface area density
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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 linearfeatures where those
boundaries intersect, and the points where the lines meet atnodes.
In other structures there may be surfaces such as membranes, linear
featuressuch 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 constructa feature list by writing down all of the phases or
features present (and identifyingthe ones of interest), and then
listing all of the additional ones that result from
theirinteractions (contact surfaces between cells, intersections of
fibers with surfaces, andso on). Even for a comparatively simple
structure such as the two-phase metal shownin Figure 1.7 the
feature list is quite extensive and it grows rapidly with the
numberof distinct phases or classes of features present. This is
discussed more fully inChapter 3 as the qualitative microstructural
state.
Consider the common stereological measurements that can be
performed byjust 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, whichcan be multiplied
by 2 to obtain the total length of the corresponding triple
linesper 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 ofpoints placed on the image can be used to count the
fraction of points that fall ona phase (Figure 1.8). The point
fraction PP is given by the number of events whenpoints (the
intersections of lines in the grid) coincide with the phase divided
by thetotal number of points. Averaged over many fields, the result
is a measurement ofthe volume fraction of the phase VV.
Similarly, a line probe (the lines in the same grid) can be used
to count eventswhere the lines cross the boundaries. As shown in
Figure 1.8, the total number ofintersections divided by the total
length of the lines in the grid is PL. The averagevalue of PL
(which has units of m-1) is one half of the specific surface area
(SV, areaper unit volume, which has identical dimensionality of
m2/m3 = m-1).
Chapters 4 and 5 contain numerous specific worked examples
showing howthese and other stereological parameters can be obtained
by counting events pro-duced by superimposing various kinds of
grids on an image. Chapter 9 illustratesthe fact that in many cases
the same grids and counting procedures can be auto-mated using
computer software.
Intercept Length and Grain Size
Most of the parameters introduced above are relatively familiar
ones, suchas volume, area, length and number. Surfaces within real
specimens can have verylarge amounts of area occupying a relatively
small volume. The mean linear inter-cept l of a structure is often
a useful measure of the scale of that structure, and asnoted in the
definitions is related to the surface-to-volume ratio of the
features, sincel = 4 VV/SV. It follows that the mean surface to
volume ratio of particles (cells,grains, etc.) of any shape is S/V
= 4/l.
10 Chapter 1
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Introduction 11
Figure 1.7. An example microstructure corresponding to a
two-phase metal. Color-coding is shown to mark a few of the
features present: blue = b phase, red = ab inter-face; green = aaa
triple points, yellow = bbb triple points. (For color
representation seethe attached CD-ROM.)
a
b
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The mean free distance between particles is related to the
measured interceptlength of the region between particles, with the
relationship L = l (VVb/VVa) whereb is the matrix and a the
particles. This can also be structurally important, forexample, in
metals where the distance between precipitate particles controls
dislo-cation pinning and hence mechanical properties. To illustrate
the fact that stereo-logical rules and geometric relationships are
not specific to microscopy applications,Chandreshakar (1943) showed
that for a random distribution of stars in space themean 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 isL = 0.5 NA-1/2 where NA is the number per
unit area; this will be used in Chapter 10to test features for
tendencies toward clustering or self-avoidance.
A typical grain structure in a metal consists of a space filling
array of more-or-less polyhedral crystals. It has long been known
that a coarse structure con-sisting of a few large grains has very
different properties (lower strength, higher
12 Chapter 1
Figure 1.8. A grid (red) used to measure the image from Figure
1.7. There are a totalof 56 grid intersections, of which 9 lie on
the b phase (blue marks). This provides anestimate of the volume
fraction of 9/56 = 16% using the relationship PP = VV. The
totallength of grid line is 1106mm, and there are 72 intersections
with the ab boundary (greenmarks). This provides an estimate of the
surface area of that boundary of 2 72/1106 =0.13mm2/mm3 using the
relationship SV = 2 PL. There are 8 points representing bbbtriple
points (yellow marks) in the area of the image (5455mm2). This
provides an esti-mate 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 presentin the
structure. (For color representation see the attached CD-ROM.)
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electrical conductivity, etc.) than one consisting of many small
grains. The size ofthe grains varies within a real microstructure,
of course, and is not directly revealedon a section image. The mean
intercept length seems to offer a useful measure ofthe scale of the
structure that can be efficiently measured and correlated with
variousphysical properties or with fabrication procedures.
Before there was any field known as stereology (the name was
coined about40 years ago) and before the implications of the
geometrical relationships were wellunderstood, a particular
parameter called the grain size number was standard-ized 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 twoaccepted procedures for
determining the ASTM grain size (Heyn, 1903; Jeffries etal., 1916),
which are discussed in detail in Chapter 9. One method for
determininggrain size actually measures the amount of grain
boundary surface SV, and theother method measures the total length
of triple line LV between the grains. The SVmethod is based on the
intercept length, which as noted above gives the surface tovolume
ratio of the grains.
Curvature
Curvature of surfaces is a less familiar parameter and requires
some expla-nation. A fuller discussion of the role of surface
curvature and the effect of edgesand corners is deferred to Chapter
5. The curvature of a surface in three dimensionsis described by
two radii, corresponding to the largest and smallest circles that
canbe placed tangent to the surface. When both circles lie inside
the object, the surfaceis locally convex. If they are both outside
the object, the surface is concave. Whenone lies inside and the
other outside, the surface is a saddle. If one circle is
infinitethe surface is cylindrical and if both are infinite (zero
curvature) the surface is locallyflat. 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 4pover any convex surface. This is based on the fact
that there is an element of surfacearea somewhere on the feature
(and only one) whose surface normal points in eachpossible
direction. As discussed in Chapter 5, this also generalizes to
non-convexbut simply connected particles using the convention that
the curvature of saddlesurface is negative.
MV is the integral of the local mean curvature over the surface
of a struc-ture. For any convex particle M = 2pD, where D is the
diameter. MV is then theproduct of 2pD times NV, where D is the
mean particle diameter and NV is thenumber of particles present.
The average surface curvature H = MV/SV, or the totalcurvature of
the surface divided by the surface area. This is a key geometrical
prop-erty 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
Introduction 13
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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 cur-vature 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
fromthe triple line case is due to the fact that the fibers have
surface area around themon 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 istangent to a line or surface. This is
discussed more in Chapter 5 as it applies tovolumes. For a 2D image
the tangent count is obtained simply by marking andcounting points
where a line of any arbitrary orientation is tangent to the
bound-ary. Positive tangent points (T+) are places where the local
curvature is convex andvice versa. The integral mean curvature is
then calculated from the net tangent countas MV = p(T+ - T-)/A.
Note that for purely convex shapes there will be two T+ andno 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 second-order stereology. Consider the case in which a grid
of points is placed onto a fieldof view and the particles which are
hit by the points in the grid are selected for mea-surement. This
is called the method of point-sampled intercept lengths. The
pointsampling method selects features for measurement in proportion
to their volume(points are more likely to hit large than small
particles). For each particle that isthus selected, a line is drawn
through the selection point to measure the radius fromthat 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 (Figure1.9). If the section plane is a vertical
section as discussed in Chapters 6 and 7, thenthe lines should be
drawn with sine-weighted orientations. If the structure is
itselfisotropic, any direction is as good as another.
The volume of the particle vi = (4/3) pr3 where denotes the
expected valueof the average over many measurements. This is
independent of particle shape,except that for irregular particles
the radius measured should include all segmentsof the particle
section which the line intersects. Averaging this measurement over
asmall collection of particles produces a mean value for the volume
vV = (4/3) pr3where the subscript V reminds us that this is the
volume-weighted mean volumebecause of the way that the particles
were selected for measurement.
If the particles have a distribution of sizes, the conventional
way to describesuch a distribution is fN(V)dV where f is the
fraction of number of the particleswhose size lies between V and V
+ dV. But we also note that the fraction of thevolume of particles
in the structure with a volume in the same range if fV(V)dV.These
are related to each other by
(1.1)
This means that the volume-weighted mean volume that was
measured aboveis defined by
f dV Vf V dVV N= ( )
14 Chapter 1
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(1.2)
and if we substitute equation (1.1) into (1.2) we obtain
(1.3)
The consequence of this is that the variance s 2 of the more
familiar numberweighted distribution can be computed for particles
of arbitrary shape, since for anydistribution
(1.4)
This is a useful result, since in many cases the standard
deviation or varianceof the particle size distribution is a useful
characterization of that distribution,useful for comparing
different populations as discussed in Chapter 8. Determiningthe
volume-weighted mean volume with a point-sampled intercept method
provideshalf of the required information. The other needed value is
the conventional ornumber-weighted mean volume. This can be
determined by dividing the total
s n n2 2 2= -N N
v V f V dV vV NV
N= ( ) = 2
0
2max
v V f V dVV V
V
= ( )0
max
Introduction 15
Figure 1.9. Point sampled linear intercepts. A grid (green) is
used to locate points withinfeatures, from which isotropic lines
are drawn (red) to measure a radial distance to theboundary. (For
color representation see the attached CD-ROM.)
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volume of the phase by the number of particles. We have already
seen how to deter-mine the total volume using a grid count. The
number of particles can be measuredwith the disector, discussed in
Chapter 7. So it is possible to obtain the variance ofthe
distribution without actually measuring individual particles to
construct the dis-tribution function.
There is in fact another way to determine the number-averaged
mean volumeof features vN without using the disector. It applies
only to cases in which eachfeature contains a single identifiable
interior point (which does not, however, haveto be in the center of
the feature), and the common instance in which it is used iswhen
this is the nucleus of a cell. The method (called the Nucleator) is
similar tothe determination of volume-weighted mean volume above,
except that instead ofselecting features using points in a grid,
the appearance in the section of the selectednatural interior
points is used. Of course, many features will not show these
pointssince the section plane may not intersect them (in fact, if
they were ideal points theywould not be seen at all). When the
interior point is present, it is used to draw theradial line. As
above, if the section is cut isotropically or if the structure is
isotropicthan uniform random sampling of directions can be used,
and if the surface is avertical section then sine-weighted sampling
must be employed so that the direc-tions 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 vN = (4/3)
pr3 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-weightedresult).
Stereology of Single Objects
Most of the use of stereological measurements is to obtain
representativemeasures of 3D structures from samples, using a
series of sections taken uniformlythroughout 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
samemethods 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) thearea of an irregular object such as a leaf (Figure
1.10). The expected value of thepoint count in two dimensions is
the area fraction of the object, or PP = AA. Foran (n n) grid of
points this is just PP = P/n2 where P is the number of pointsthat
lie on the feature. The area fraction AA = A/n2l 2 where l is the
spacing of thegrid. Setting the point fraction equal to the area
fraction gives A = l 2P.This means that the number of points that
lie on the feature times the size of onesquare 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 featureon graph paper and counting the squares
within the feature, or of acquiring a
16 Chapter 1
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digitized image consisting of square pixels and counting the
number of pixels withinthe feature.
When extended to three dimensions, the same method becomes one
of count-ing the voxels (volume elements). If the object is
sectioned by a series of N parallelplanes with a spacing of t, and
a grid with spacing l is used on each plane, then thevoxel size is
t l 2. If the area in each section plane is measured as above then
thevolume is the sum of the areas times the spacing t, or V = t l
2PT where PT is thetotal number of hits of grid points on all N
planar sections. This method, elabo-rated in Chapter 4, is
sometimes called Cavalieris principle, but will also be famil-iar
as the basis for the integration of a volume as V = A dz.
Measurements of the total size of an object can be made whenever
the sam-pling grid(s) used for an object completely enclose it,
regardless of the scale. The
Introduction 17
Figure 1.10. A leaf with a superimposed grid. The grid spacing
is 1/2 inch and 39 pointsfall 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 leafarea.
(For color representation see the attached CD-ROM.)
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method can be used for a cell organelle or an entire organ. The
appropriate choiceof a spacing and hence the number of points
determines the precision; it is not nec-essary that the plane
spacing t be the same as the grid spacing l .
Summary
Stereology is the study of geometric relationships between
structures thatexist in three-dimensional space but are seen in
two-dimensional images. The tech-niques summarized here provide
methods for measuring volumes, surfaces and lines.The most
efficient methods are those which count the number of intersections
thatvarious types of probes (such as grids of lines or points) make
with the structure ofinterest. The following chapters will
establish a firm mathematical basis for the basicrelationships,
illustrate the step-by-step procedures for implementing them, and
dealwith how to create the most appropriate sampling probes, how to
automate the mea-suring and counting procedures, and how to
interpret the results.
18 Chapter 1