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

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.)

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.

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.)

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

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

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

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

(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

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

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

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.)

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

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

(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.)

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

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.)

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