Chapter 14
Finite Section Thickness
Projected Images
The classic applications of stereology and the principal rules which have been
derived and applied are specific to the case of planar sections through threedimensional solids. This is more-or-less the situation in metallographic microscopy,
where polished sections cut through the material are examined. It is not so appropriate for many other fields. Light microscopy in biological and medical applications usually looks through a slice of material, for instance cut with a microtome,
whose thickness is not negligible. Microscopy of particulates also looks through a
thick layer, in which the matrix is transparent air. Confocal light microscopy images
an optical section from within material, whose thickness is finite and depends on
the numerical aperture of the optics, and whose boundaries are not sharp. Even electron microscopy, which uses much thinner sections than light microscopy, deals with
finite sections (the lateral resolution of the images is improved along with the thinner
sections so that features much smaller than the section thickness are detected). And
in scanning electron microscopy although surfaces are imaged to negligible depth
(at least with secondary electrons, if not backscattered electrons or X-rays), the surfaces are rarely flat.
The introduction of finite sections and viewing of images projected through
them raises some significant problems for stereology. In cases where the section
thickness is much less than the dimensions of any features or details of interest it
may be possible to ignore the differences and treat the images as though they were
ideal sections. Indeed, this is the assumption that is commonly made in applying stereological rules to microscope images, although its appropriateness is rarely
evaluated.
Viewing images that are projections through a very thick section makes some
kinds of measurements quite simple. The outer dimensions of features are seen
directly, so that no unfolding is required to ascertain the actual particle size distribution based on the sizes of sections. There are still some important underlying
assumptions, of course. Most basic is that the dispersion is quite sparse so that particles are visible and do not obscure one another. The specific case of particles
deposited onto a substrate meets this test, since the particles are generally present
as a single layer. The original “thickness” of that layer corresponding to a volume
of material from which the particles have settled onto the substrate is not known,
but it is usually possible to control the deposition conditions to prevent the deposit
from becoming so dense that particles touch or overlap one another.
A second assumption is that the particles have uniformly random orientations with respect to the viewing direction. This may or may not be true when
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viewing through an actual matrix, but is always suspect when particles are deposited
on a substrate since physical forces can act to reorient the particles when they
contact the surface. These forces range from gravity (for large particles) to electrostatic or magnetic attraction, surface tension in the fluid as it is removed, and others.
The general result is that particles may align with their long axis parallel to the
surface or to each other, and may tend to clump together. This presents an image
that is not random and makes the determination of three-dimensional particle sizes
suspect. Special preparation techniques can sometimes be employed to produce well
dispersed samples with unmodified particle orientations. One example is the use of
a camphor-napthalene eutectic wax for dispersing cement powders (Thaulow &
White, 1971). The wax supports the particles and is spread onto a slide as a substrate. It is then evaporated away allowing the particles to settle onto the slide. Obviously this is a specialized method, but similar approaches may be practical for a
variety of applications.
When particulates are viewed in projection, the external diameter or
projected area gives a direct measure of size. This of course is only true for convex
features. Any interior details such as surface pits or voids are hidden, and important topological characteristics may be hidden as well. Consider for example a
distribution of toroids—Cheerios® for instance. As shown in Figure 14.1, some
of the projected images will show the presence of the central hole but most will
not. If it is known a priori that all of the features are the same, the multiple
projections make it possible to determine the three-dimensional shape. In fact this
technique has been used to reconstruct the shape of virus particles from transmission electron microscope pictures, by combining multiple projections of presumably
identical particles using tomographic reconstruction. But when a mixture of shapes
is present, the assessment of the actual shapes and relative populations is difficult
or impossible.
Even if the assumption of random orientation can be met, many projected
images do present the viewer with some overlaps. Figure 14.2 illustrates this
problem. Small particles are more likely to hide behind larger ones, and overlapping
projections of particles reduce the number and alter the size of the features in the
projected image. In the example, the spatial distribution is not isotropically random.
In one direction (Figure 14.2b) the particles are nearly all separately visible, while
in another (Figure 14.2c) they are heavily overlapped. In most cases, of course, it is
not possible to select a particular viewing direction or to compare the results in different directions. Figure 14.3 shows a real example, a section viewed in the TEM
containing latex spheres. The apparent clusters of particles are in fact simply superpositions of images along the particular projection direction and the particles are
actually separated in three dimensions.
As noted above it is more likely for small particles to be hidden by large ones
than the converse. A statistical correction for the systematic undercounting of small
features is possible. The size distribution of the visible features is constructed. Normally such a distribution would report the number per unit volume, where the
volume is the area of the viewed image times the thickness of the section. However,
for small features the actual volume being viewed is less. The projections of large
features block the view of a portion of the area, and this must be subtracted from
Finite Section Thickness
333
a
b
Figure 14.1. Illustration of a projected view of toroids: a) surface image which reveals
the shapes; b) projected image in which most images do not indicate the three dimensional shape. (For color representation see the attached CD-ROM.)
the volume measured when counting the smaller features that could hide within that
shadow as shown in Figure 14.4. This correction can be made for each bin in the
distribution, increasing the number per unit volume of the smaller particles.
There are actually two variants of this correction. When the projected image
is a shadow or binary image, the volume covered by the large feature extends completely through the section thickness so that the sampled volume is reduced by
the projected area times the thickness. When the image shows the object surfaces
(Figure 14.5), small features lying in front of larger ones can be seen, but particles
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a
b
Figure 14.2. Illustration of particles in a thick section, which overlap in a projected
image: a) perspective view; b) projection along the Z axis; c) projection along the Y
axis. (For color representation see the attached CD-ROM.)
Finite Section Thickness
335
c
Figure 14.2. Continued
Figure 14.3. TEM image of latex spheres, which appear to form clusters but are actually separate in three dimensions.
Figure 14.4. Schematic showing the hidden shadow region above and below a large
particle. As discussed in the text, the area of the feature may be reduced if partially
protruding objects can be distinguished. (For color representation see the attached CDROM.)
a
Figure 14.5. For a surface image, small particles above a larger one are visible and
can be measured while those below are completely or partially hidden: a) perspective
view; b) vertical image. (For color representation see the attached CD-ROM.)
Finite Section Thickness
337
b
Figure 14.5. Continued
lying behind larger ones cannot. The hidden volume is reduced by one half since on
the average the large particles lie in the middle of the section.
In either case, it is clear that particles are hidden if they are entirely within
the shadow area. But if their centers lie within a band equal in thickness to the
radius, they will partially protrude and may be detected depending on the criteria
for visibility. Often, particles whose centers lie outside the shadow but intersect the
projection of the large particle can be seen and measured.
Consequently, it is an application-specific decision whether the area of the
large feature is the proper value to use for the correction, or whether a region smaller
or larger by the radius of the smaller feature for which the correction is to be made,
and whether the hidden volume is equal to this area times the thickness or half the
thickness of the section.
Bias in Stereological Measurements
Finite section thickness introduces two opposite sources of bias into conventional stereological measurements. These have been recognized for a long time
(Holmes, 1927), and various correction procedures have been derived (Cruz-Orive,
1983). For the most part, however, these are not used widely. This is because they
require knowing the section thickness, making some assumptions about feature
shape and uniformity (also isotropy of orientation), and/or performing more
measurement work to obtain results. Determining the thickness of a microtomed
section accurately is difficult, and for many materials specimens techniques used to
prepare TEM sections such as electrochemical thinning or ion erosion produce
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Figure 14.6. Diagram of the cross section of a thick section in which the opaque phase
appears to cover more area because of overprojection. (For color representation see
the attached CD-ROM.)
sections whose thickness varies dramatically and imaging is actually performed on
wedge-shaped regions around the edges of holes.
For the case in Figure 14.6, we have a finite section with one transparent and
one opaque phase. Note that this is exactly equivalent to the case of a polished metallographic section with finite polishing relief due to one phase being harder than
the other (Figure 14.7). In each case, the opaque or harder phase appears to cover
a greater fraction of the viewed area than it should, because of overprojection. The
apparent area fraction A¢A is greater than the actual volume fraction VV by an
amount that depends on the thickness of the section t and the amount of surface
area per unit volume SV (Cahn & Nutting, 1959).
AA¢ = VV +
1
SV ◊ t
4
(14.1)
Detemining SV also requires a correction for the section thickness, which
depends on the number of features present and their mean curvature. In terms
of practical measurement quantities A¢A (the apparent area fraction, which is
typically measured by counting a point fraction P¢P ), B¢A (the phase boundary length
per unit area, which is typically measured by counting linear intercept points P¢L )
and the number of features per unit area N¢A, the true stereological structure
parameters are
t ◊H ˆ
Ê1
VV = PP¢ - t ◊ Á PL¢ - N A¢ ◊
˜
Ë2
t+H ¯
(14.2)
Figure 14.7. Diagram of the cross section of a surface with polishing relief, in which
the harder phase projects from the surface and appears to cover more area because
of overprojection. (For color representation see the attached CD-ROM.)
Finite Section Thickness
SV = 2 PL¢ - 4 N A¢ ◊
339
t ◊H
t+H
(14.3)
The parameter t is the section thickness and H is the average mean
caliper diameter of the particles or phase regions in the section. It is usually assumed
that H is much less than t, corresponding to the case in which particles are
small compared to the section thickness, but this is not essential. Of course, if
H is not small compared to t it is not likely that H will be seen or measurable in
the section images and will have to be determined separately from other experiments.
The model also assumes that the phase regions or particles are convex and do not
overlap each other (the section thickness is very small, or the dispersion is very
sparse and VV is small).
If measurements on two sections of thickness t1 and t2 are performed, a
further simplification is possible and we obtain
VV =
t1 ◊ AA¢1 - t2 ◊ AA¢ 2
t1 - t2
(14.4)
SV =
4 t2 ◊ BA¢1 - t1 ◊ BA¢ 2
◊
t1 - t2
p
(14.5)
In actual practice, determining the thickness values with sufficient precision
to divide by the difference (t1 - t2) is extremely difficult to accomplish.
R. E. Miles (1976) derived a complete set of stereological equations for
volume density, surface density, and curvature based on measurements on the projected image of a section of thickness t which are more general than the preceding
equations. They apply to the case of an aggregate of opaque, isotropic, random
convex particles in a transparent matrix. The required measurements can be
obtained using a point count using a grid, and intersection count using a grid, and
a tangent count using a sweeping test line. Data must be taken on paired sections
and the estimated values depend on the differences between the pairs of measurements (requiring a large number of counts for adequate precision) and the accuracy
of the thickness measurements.
The above discussion deals only with the problem of overprojection, features
appearing too large in the projected image. It is restricted to convex phase regions
and cannot deal with particles hiding behind others. An additional effect may also
be present, usually referred to as the “end cap” problem. Consider the illustration
in Figure 14.8. The spherical particles are of three types: ones that lie entirely within
the section thickness, ones that project out (and are truncated) but whose centers
lie inside the section thickness, and ones whose centers lie outside the section. The
latter group have only their polar caps in the section. In many cases, the contrast
with which the opaque phase is visible in the projected image is such that these thin
end caps may not be seen, or they may literally fall out of the section.
In terms of the number of features seen per unit area of image, the contribution of the first two types of particles is a function of the section thickness t while
that of the third group is a function of the mean particle diameter D , where NV is
the true number density per unit volume.
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Figure 14.8. Diagram of a cross section of a thick section containing spherical particles that lie entirely within the section, ones that protrude out but have their centers
inside, and ones with their centers outside the section but protrude into it producing
end caps. (For color representation see the attached CD-ROM.)
N A¢ = NV · t + NV · D
(14.6)
Making corrections for the presence of end caps or for the fact that some
may be missing requires making many assumptions. Weibel (1980) derives a family
of curves to correct VV and SV measurements for invisible end caps as a function of
the diameter at which they disappear. In most cases, it is more useful to consider
the thickness at which they disappear since the visibility of the opaque phase is
usually related to this dimension. For the assumption of spherical particles, the two
dimensions are of course related.
The same method can be extended to the sphere unfolding method discussed
in Chapter 12. This is typically done by assuming either that the caps disappear at
a constant latitude angle in all size classes, or that they disappear at a fixed thickness due to insufficient image contrast. Of course, the assumption of spherical particle shape is already a basic part of the unfolding method. If the shape of the
distribution can also be assumed (e.g., normal or log-normal), then an iterative calculation of the size distribution of the circles produced by sectioning spheres can
be used to determine the mean and variance of the sphere size distibution (Nauman
& Cavanaugh, 1998).
Measurements within Sections
With all of the difficulties and assumptions involved in applying classical
stereology to thick sections, it is surprising to find that some very straightforward
measurements are possible. One is the length of linear structures, regardless of their
complexity, connectedness or branching. These may be separate discrete lines such
as fibers or dislocations, or branching structures such as roots of plants (Figure 14.9)
or neurons in stained tissue. Linear structures are also formed by the intersection of
planar ones, such as the triple line that forms a network in grain structures where the
surfaces of three grains meet. This network is important as a path for diffusion and
other processes, but is not often visualized as a three-dimensional linear structure.
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341
Figure 14.9. A projected image of the roots of a plant. The branching structure can be
considered to occupy a volume in which the transparent matrix is air.
The projected image of a linear structure is of course a set of lines, and we
will make the assumption that they are all visible and the magnification and section
thickness are appropriate to allow the lines to be resolved and viewed. But of course
since the lines are embedded in a three-dimensional volume and can run in any direction, the length of the lines may be much greater than the length of the projections.
Without making any assumption about the orientation of the lines, it is possible to
measure the actual value with little effort.
One of the rules of classical stereology is that a plane probe can be used to
intersect lines, and that the expected value of the length of the lines per unit volume
LV is equal to 2 · NA where NA is the number of intersections that the lines make with
the plane. These intersections appear as points, of course, and sometimes they can
be used directly for classical measurement. For instance, etching a piece of semiconductor to produce pits where dislocations intersect a polished surface makes it
possible to count them. The factor two arises because an isotropic distribution of
lines would intersect the surface at many different angles. So the classical measurement of LV from the value of NA on a section surface is dependent on the assumption of isotropy.
This highlights the usual difficulty with plane probes—it is very difficult
to make them isotropic in three-dimensional space, if the specimen itself is not
isotropic. Fortunately with thick sections there is a way to accomplish this. As shown
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in Figure 14.10, a line drawn on the projected image represents a surface extending
through the thick section. If the line is straight the plane is flat; if the line is curved
the surface is a cylindrical one. The area of the surface is the product of the length
of the line times the section thickness. Counting points where the linear structure
of interest crosses the surface can be accomplished by counting the intersections of
the projection of the linear structure with the projection of the surface, which is the
drawn line.
Drawing a uniform set of straight lines on the projected image does not
produce a set of random planes in three dimensions, so the problem of anisotropic
sampling is still present. All such surfaces share a common direction through the
thickness of the section, and so are far from isotropic. However, the vertical section
method discussed in other chapters can be used here as well. Select one direction in
the plane of the section as the nominal “vertical” direction, cut several sections that
include that direction (experiments show that as few as three sections cut 60 degrees
apart provide a reasonably sample set), and then draw cycloids on projected images
of each section as shown in Figure 14.11.
Note that the vertical direction is shown on the image and the orientation
of the cycloids with respect to that direction is not the same as when cycloids are
drawn on a planar vertical section. The cycloidal surfaces have more of their area
parallel to the vertical direction than perpendicular to it, since every thick section
cut to include the same vertical section will have the same planes perpendicular to
that direction, which would be oversampled by planes corresponding to isotropic
lines drawn on the projection.
Counting the intersections of the linear structure with the cycloids allows an
easy and straightforward measurement of the total length of the structure. Sometimes this is performed manually and sometimes by using image processing to
combine the image of the grid with the linear structure (a Boolean AND) to mark
Figure 14.10. The relationship between a line drawn on a projected image and the
surface that line represents in the original thick section. (For color representation see
the attached CD-ROM.)
Finite Section Thickness
343
Figure 14.11. Transmission electron microscope image of nerve axons with a superimposed grid of cycloids. (For color representation see the attached CD-ROM.)
Figure 14.12. Projected image of a wire tangle superimposed on a cycloid grid, with
intersections marked. (For color representation see the attached CD-ROM.)
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the crossings for automatic counting. Both methods have been discussed in other
chapters.
In some cases it is not even necessary to cut multiple thick sections. In the
case of the root in Figure 14.9, the entire volume of the root can be treated as a
(very) thick section with a transparent (air) matrix. Rotating the entire structure to
three orientations about 60 degrees apart provides the isotropic sampling around
some selected vertical direction; in this case the plant stem would provide a logical
reference. Superimposing a cycloid grid on each of the three projected images and
counting the intersections can be used to measure the total length of the structure.
As a practical demonstration of this method, it is convenient to take a wire
or other structure that can later be measured and bend it into a complex threedimensional shape, place it on a grid (using a transparency and overhead projector
is a good classroom demonstration method), and count the intersections in each of
three rotational orientations. As shown in the example of Figure 14.12, this produces a number of counts that can be combined with the known length of the lines
and the area of the grid.
If the volume occupied by the wire is assumed to be a cube whose face is the
area of the grid, then the key relationships can be written in terms of the length per
unit volume of the linear structure LV, the length of the grid line G, the number of
counts N (averaged over the three orientations of the specimen), and t the size of
the grid (and the assumed section thickness). Equation (14.7) is simply the classical
relationship between LV and the number of intersections, and equation (14.8) gives
the actual length L in terms of the number of counts and the length of grid line per
unit area (a known constant for any test grid).
LV =
L=
2◊N
G ◊t
2◊N
GA
(14.7)
(14.8)
In the example of Figure 14.12, the number of intersections is 23. The total
length of grid line is 68.054 cm in an area of 7.5 ¥ 7.5 cm, so GA = 1.209 cm-1. That
produces an estimate of the wire length of 19 cm. The uncertainty in this estimate
comes from the number of counts ( 23 23 is about 20%, or 19 ± 4 cm (the actual
length of the wire was 20 cm). Repeating the measurement with other placements
of the grid and other orientations around the same vertical direction would produce
more counts and thus improve the statistical precision of the measurement.