Estimation of average particle size from vertical projections

Authors


Abstract

A new stereological relationship is derived for the estimation of average size (average width) of a collection of convex particles in a 3D microstructure. The average size is estimated from measurements performed on projected images of the microstructure generated by total vertical projections. The stereological relationship is as follows: D=ĪC/(2N0β). D is the average width, ¯IC is the average absolute number of intersections between the specifically oriented and regularly spaced cycloid shape test lines and particle boundaries observed in the total vertical projections, N0 is the total number of particles observed in the total vertical projection and the parameter β is a characteristic of the measurement grid; it has units of reciprocal of length. The result is applicable to any arbitrary collection of convex particles; the particle orientations need not be isotropic. Only ‘intersection counts’ are required; it is not necessary to measure sizes of the particles in the projected images.

1. Introduction

In microstructures containing dispersed particles (or voids), the ‘average’ particle size is an important microstructural parameter. Different size parameters and averaging procedures lead to different descriptors of average size, such as arithmetic average width ( DeHoff & Rhines 1968), surface area averaged width ( Gokhale, 1985), volume weighted mean volume ( Gundersen & Jensen, 1985), etc. Among these parameters, the arithmetic average width (further ‘arithmetic’ is omitted) is of significant practical importance for ensembles of convex particles. To most scientists, ‘average size’ implies average width. This parameter is built into a number of structure–property correlation models and microstructural evolution models. Therefore, it is of interest to develop stereological techniques for the estimation of the average width of an ensemble of particles dispersed in 3D space.

A tempting strategy to estimate the average width involves the following steps.

1 Estimation of the total number of particles per unit volume NV using the disector technique ( Sterio, 1984), which involves observations on a pair of parallel sectioning planes that are separated by a distance that is less than the smallest particle size.

2 Estimation of the average number of particles intersected by an IUR sectioning plane of unit area, ¯NA.

3 The average width D is then accessible via the well-known stereological relationship ( DeHoff & Rhines, 1968)

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Estimation of ¯NA requires sampling the 3D structure using isotropic planes. The isotropic sampling is often inefficient, and in a number of cases it is not feasible. Therefore, it is of interest to develop stereological procedures that do not involve sampling the 3D microstructure using isotropic planes.

A number of microscopy techniques yield a projected image of a microstructure rather than 2D sections. For example, X-ray and other radiographic images, transmission microscopy images from thick slices, scanning electron microscopy images from nonplanar surfaces etc. represent projected images of a microstructure and not 2D sectioning planes. Furthermore, the microstructural information obtained from a number of 3D microscopy techniques such as confocal microscopy and tomography can be conveniently cast into projected image form. In the conventional microscopy of biological specimens, the observations on a set of thin focal planes at different depths can be utilized to generate projected image information (e.g. see McMillan et al., 1994 ). Alternatively, the test probes for the projected images can be laid directly onto the 3D microstructure for efficient stereological sampling. Therefore, it is of interest to develop stereological techniques to extract 3D microstructural properties from the projected images of microstructure.

Recently, efficient stereological techniques have been developed for quantitative characterization of 1D lineal microstructural features from projected images of vertical slices ( Gokhale, 1990, 1992, 1993) and from total vertical projections ( Cruz-Orive & Howard, 1991). These techniques have been successfully utilized to estimate the length density of cortical microvessels ( McMillan et al., 1994 ), the length density of capillaries in the heart ( Batra et al., 1993 , 1995), the length density of skeletal muscle fibres ( Artacho-Perula & Roldan-Vilalobos, 1995a, b), the total length of blood vessels from magnetic resonance images ( Roberts et al., 1991 ), the length of epidermal nerve fibres, neural dendrites ( Howard et al., 1992 , 1993), etc. Therefore, it is fruitful to develop unbiased and efficient stereological methods to estimate other important microstructural properties (such as the average width) from observations on projected images, which is the aim of the present paper.

2. The method and an example

The above discussion reveals that: an efficient and general stereological technique is not available to estimate the average size (average width) of a collection of convex particles from non-isotropic observations; a number of important microscopy techniques yield projected images of microstructure, and experimental measurements on projected images are practically feasible and convenient, as demonstrated by recent studies on lineal microstructural features.

It is the purpose of this paper to suggest an unbiased stereological estimate of the average width of an arbitrary ensemble of convex particles from their vertical projections. The relationship derived requires only some simple counting measurements; it is not necessary to measure the particle sizes in the projected images. The particles need not be randomly oriented and the projection directions need not be isotropic.

A particle in 3D space is convex if any segment joining two arbitrary points on its surface is completely contained inside the particle. In any direction there exist parallel tangent planes of a convex particle and their distance is called ‘width’, see Fig. 1. The mean width, ¯D, is the average value of the width of a particle over all possible orientations. For an ensemble of convex particles having different shapes and sizes, the average width, D, is defined as follows ( DeHoff & Rhines, 1968):

Figure 1.

. Definition of width of a convex particle.

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where Ni is the number of particles having the individual mean width equal to ¯Di.

The following steps describe a practical procedure for estimating the average width of a finite collection of convex particles from the measurements performed on the projected images.

1 Select a direction in 3D space as the ‘vertical’ axis. Any arbitrary direction may be specified as the vertical axis.

2 Enclose the specimen containing the collection of particles in a slab of thickness Δ, having parallel faces of area Γ. Orient the slab so that the vertical axis is parallel to the slab faces, see Fig. 2. The slab is our reference space.

Figure 2.

. Reference space containing convex particles is enclosed in a slab of thickness Δ, having faces of area Γ parallel to the vertical axis.

3 Observe the total projection of the reference space along a projection direction that is perpendicular to the vertical axis and uniformly random among projection directions with this property. Identify the vertical axis and the projected images of particles in the projected planar image. It is assumed that the projected images of all particles are observed in the total vertical projection.

4 On the projected image superimpose a grid containing uniformly spaced cycloids ( Cruz-Orive & Howard, 1991, p. 104, give such a grid), so that the cycloid minor axis (of length a) is perpendicular to the vertical axis, see Fig. 3. The grid must cover the projected image of the reference space completely; alternatively, the projected reference space can be sampled by quadrats. The superimposed grid must have a random position w.r.t. translations in the plane. Calculate the grid parameter β by using the following equation

Figure 3.

. A schematic total vertical projection at 500× together with a cycloid grid.

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where M is the magnification of the projected image and n the number of cycloids on the grid.

5 Count the number of intersections between the cycloids and the boundaries of the projected images of the convex particles, IC.

6 Repeat step 5 for a number of uniform random (or systematic uniform random) projection directions; all of these directions must be perpendicular to the vertical axis. The same result can be achieved by spinning the structure around the vertical axis. From these observations, evaluate the average value, ¯IC.

7 Use the total vertical projections in 5 and 6 to estimate the total number of particles in the projected image, N0. This can be done by using the unbiased counting method of Gundersen(1977).

8 Substitute for β, ¯IC and N0 in the new stereological equation

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and calculate D.

As an example consider the projected reference space in Fig. 3 at a magnification of 500×, then β is equal to 13.33 per mm. Counting the number of intersections we obtain ¯IC = 19. Furthermore, the total number of particles in the projected image is N0 = 23. Finally, using Eq. (2) the estimated value of D is 3.1 × 10−2 mm.

3. Heuristic explanation

It has already been pointed out in the Introduction that the estimation of ¯NA in the fundamental formula

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requires sampling the 3D structure using isotropic planes, which present all possible orientations in 3D space. As in many of the other above-mentioned applications ( Baddeley et al., 1986 ; Gokhale, 1990, 1992; Cruz-Orive & Howard, 1991) sampling using probes with anisotropic orientations is more efficient (cf. Krejčíř and Beneš (1997) for a discussion on estimation variances). Suitable probes for our problem are cycloidal surfaces. Consider a cycloidal surface generated by sweeping a cycloid curve with minor axis length a through a distance Δ, perpendicular to the plane of the cycloid curve. Orient this cycloidal surface such that the minor axis of the generated cycloid is perpendicular to the vertical axis, see Fig. 4. The orientation of the cycloidal surface is then described by a single angle φ varying between 0 and π.

Figure 4.

. Vertical projection of a cycloidal surface yields a cycloid curve.

Denote by ¯NC the average (with respect to orientations φ) number of intersections between the convex particles in the reference space and the cycloidal surface, per unit area of the cycloidal surface. It arises that using the cycloidal test surface is the appropriate compensation for seemingly missing probe orientations in the vertical sampling design, i.e. that it holds

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Consider the total projection of the reference space perpendicular to the vertical axis. Such a projection is called the total vertical projection (cf. Cruz-Orive & Howard (1991)). In the total vertical projection, each cycloidal surface appears to be a cycloid curve having a minor axis perpendicular to the vertical axis, see Fig. 4. Furthermore, each time the cycloidal surface intersects a convex particle in the reference space, we observe intersections between the cycloidal curve and the particle boundaries. Denote ¯IC the total average number of intersections between the cycloid curves and particle boundaries, averaged over all the projection directions perpendicular to the vertical axis. One can then write

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where n is the number of cycloidal curves uniformly superimposed on the projected image, L0 = 2a is the length of each cycloidal curve and Δ is the width of the reference slab.

Moreover, the number of particles per unit volume is equal to N0/(ΓΔ). With these interpretations of NC and number density, Eq. (3) can be written as:

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in Eq. (2), where (without magnification)

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Note that the slab thickness Δ does not appear in Eq. (2) and therefore need not be known. In a grid containing regularly spaced oriented cycloids, see Fig. 3, β in Eq. (5) is simply the length of cycloids per unit area of the grid.

4. Discussion

The main result of this contribution, Eq. (2), permits an efficient and unbiased stereological estimation of the average size of an ensemble of convex particles from the measurements performed on the total vertical projections. The stereological procedure does not involve IUR sampling; it is based on the vertical uniform random (VUR, see Baddeley et al., 1986) sampling of the 3 D microstructure. The result is applicable to any arbitrary collection of convex particles in the reference space of any arbitrary geometry. The orientations of the particles may be anisotropic. The average width D can be estimated from simple counting measurements performed on the projected images; it is not necessary to measure the size of any particle in the projected image (or in the 3D structure) to estimate the average size.

Although the present method is applicable to the convex particles of any arbitrary anisotropy, the efficiency of the sampling design can be further increased if some information about the anisotropy is available a priori. If the anisotropy of particle orientations has an axis of symmetry, then it is most efficient to select the symmetry axis as the vertical axis. In such a case, all the total vertical projections are statistically similar, and therefore the average particle size can be estimated from a single total vertical projection. If the anisotropy of the particle orientations does not have an axis of symmetry, then using systematic random vertical projections should yield an efficient estimate of the average particle size.

In the present analysis, it is assumed that the projected images of all particles are observed in the total vertical projections. However, if there are overlaps in the particle projections, the images of some particles may not be observed in the projection; some particles may be hiden behind others. If such overlaps are significant, then the present method should not be used to estimate the average size from the total vertical projections. By contrast if 3D microscopy techniques such as confocal microscopy or computed tomography are utilized, then Eq. (2) can be applied directly to the 3D images to estimate the average size, and in such a case, there is no need to resort to the projected images.

Acknowledgements

This research is supported by the U.S. National Science Foudation through the research grant DMR-9301986. Dr B. MacDonald is the project monitor. The financial support is gratefully acknowledged. The second author was supported by the Grant Agency of the Czech Republic, project no. 201/96/0226.

Appendix

We shall prove Eq. (2) rigorously. While in the heuristic explanation it was convenient to consider Eq. (3) based on cycloidal surface probe, in the mathematical proof of Eq. (2) we may skip this step and use the total projections directly. This makes the derivation simpler.

Let a finite ensemble of bounded particles be given. Since it is assumed that for each vertical plane the projections of all particles are completely accessible, the total number of particles is known and it is sufficient first to consider the problem for one particle K. The mean width ¯D(K) is the average of widths over all orientations

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where D(K, v) is the width of K in direction v ∈ S2 and ω is the uniform measure on the unit sphere S2 in the Euclidean space R3. Using the total vertical projections we need to decompose ω into measures μL on the sphere S1(L), where L runs over the planes containing a given line l (vertical axis). Using the spherical coordinates (u, L) in R3 (u is the colatitude and L the longitude) we obtain this decomposition from Eq. (A1) in a form (we write D(K, u) = D(K, (u, L)))

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where the family of vertical planes {L: l ⊂ L} can be represented by the sphere in the plane l orthogonal to l and ω1 is the uniform measure on that sphere. From Eq. (A2) it holds that

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where ωL is the uniform measure on S1(L). Equation (A2) belongs more generally in a family of decompositions based on formulae of Blaschke–Petkantschin type ( Jensen & Kiêu, 1991). The inner integral in Eq. (A2)

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is the ‘weighted’ mean width of particle projection on a given L; this projection is denoted by KL.

Furthermore observe that

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where N(KL ∩ ux) is the number of intersections between KL and a line ux ⊂ L perpendicular to u (system of such parallel lines is indexed by the shift x). It is well known ( Gokhale, 1990; Cruz-Orive & Howard, 1991; Beneš, 1993) that the system of test lines ux when u is distributed according to μL has the orientation distribution equal to the tangent orientation distribution of a cycloid curve with minor axis perpendicular to the vertical axis.

The inner integral in Eq. (A5) is a mean value of β−1iN(KL ∩ uxi), where uxi are equidistant of distance β−1 and x1 is uniform random on 〈0, β−1〉 . In the rectangular projection of a reference slab with sides s, t and area Γ = st we have the length of such a test system equal to tsβ; therefore β has the interpretation of test system length per unit area (without magnification).

Hilliard (1962) has shown that the expected number of intersections per unit area when two lineal arrays are superimposed randomly w.r.t. translations is determined by the lengths and orientation distributions of the arrays. This argument enables one to substitute the mean number of intersections in Eq. (A5) by the mean number of intersections IC on test cycloids, despite the fact that for an appropriate convex particle K the intersection number between the boundary of KL and a cycloidal curve can be arbitrarily large with positive probability.

Summing over a set of N0 particles we thus obtain from Eq. (A5) IC/β = 2∑i¯D1(K) and averaging over vertical planes L finally D = ¯IC/2βN0 in Eq. (2).

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