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Centre for Basic Psychiatric Research, Aarhus Psychiatric University Hospital, Risskov, Denmark and Translational Neuroscience Program, Department of Psychiatry, University of Pittsburgh, Pittsburgh, PA, USA

Prof. A. Baddeley, School of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway, Nedlands WA 6009, Australia. Tel: +61 8 6488 3375; fax: +61 8 6488 1028; e-mail: adrian@maths.uwa.edu.au

Summary

Stereological methods for serial sections traditionally assume that the sections are exactly equally spaced. In reality, the spacing and thickness of sections can be quite irregular. This may affect the validity and accuracy of stereological techniques, especially the Cavalieri estimator of volume. We present a new formula for the accuracy of the Cavalieri estimator that includes the effect of random variability in section spacing. A modest amount of variability in section spacing can cause a substantial increase in estimator variance.

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In the standard theory of stereological methods, serial sections are assumed to be exactly equally spaced and to have exactly constant thickness. However, in reality, any device which physically cuts the experimental material produces some variability in section thickness and section spacing. It appears to be unknown whether such variability has a serious impact on the validity and accuracy of stereological methods.

Serial sections cut by a microtome are quite accurately spaced and fairly constant in thickness. However, many experiments require a preparatory step in which the material is embedded in agarose and cut into thick slabs (see Fig. 1). Some devices for cutting slabs may make substantial errors in the position and angle of successive cuts. When sections are subsequently taken from each slab, unevenness in the slab surface obliges the experimenter to trim away partial sections before a full section can be taken. The resulting sections are no longer evenly spaced in the material.

An important technique that might be affected by these errors is the estimation of volume by Cavalieri's principle. Existing theory for the Cavalieri estimator, guaranteeing its basic validity and giving approximate formulae for its standard error, depends on assuming that the section spacing is exactly constant. It appears to be unknown whether variability in section spacing (for example) affects the validity and accuracy of the Cavalieri method. Some indications were obtained by Pache et al. (1993; p. 24).

This question was discussed in a forum at the International Workshop on Variance Estimation in Stereology, Aarhus, Denmark, November 2004. Some results of that discussion are presented here. Further work is in progress.

2. Existing theory

First we recapitulate the existing theory for the Cavalieri estimator; see, for example, Baddeley & Vedel Jensen (2005; Chapters 7 and 13).

Figure 2 sketches a three-dimensional solid object cut by a series of parallel section planes. The sections are assumed to be equally spaced with constant separation t.

The estimate of the volume V by Cavalieri's principle is:

(1)

where A_{1}, A_{2}, … are the areas of the successive sections of the solid. Thompson (1932) described the method as ‘similar to that used in the estimation of the volume of the hull of a ship from cross-sectional plans to scale.’

This can be reformulated mathematically as follows. The position of each section plane is determined by its displacement x along an axis normal to the plane of section. The positions of the successive planes are at coordinates U, U ± t, U ± 2t, … where U is the initial starting position. It is assumed that U is random, with a uniform distribution on the interval [0, t].

If the object were hypothetically to have been cut by a section plane at a position x along the axis, let A(x) denote the area of the section of the object that would have resulted. Then the Cavalieri estimator can be expressed as:

(2)

where the sum is taken over all signed integers k. The function A(x) is termed the ‘measurement function’. An illustrative example is sketched in Fig. 3.

The exact volume V of the solid is equal to the integral

(3)

The Cavalieri estimator V̂ given in Eq. (2) can be regarded as an approximation to this integral, based on a finite set of sample points U+ kt for k = 0, ±1, ±2, … .

The basic validity of the Cavalieri estimator follows from the fact that V̂ is an unbiased estimator of V. That is, the expected value of the estimate, ��[V̂], is equal to the true value, V.

It is also known that the variance var[V̂] of the Cavalieri estimator equals

(4)

where

(5)

is called the geometric covariogram of the function A(x). Intuitively, the covariogram g(z) reflects the correlation between the sectional areas of two sections separated by a distance z. Figure 4 shows the covariogram corresponding to the measurement function plotted in Fig. 3.

Now suppose that, instead of the intended perfectly periodic sample locations U+kt, the sections are actually located at some other positions x_{k} (see Fig. 5). These positions may be displacements of the intended periodic positions, and there may also be some dropouts.

Our approach is to treat the sample locations as a point process X on the one-dimensional axis (Cox & Isham, 1980). The modified Cavalieri estimator is

(6)

where τ is a constant (to be determined) representing the ‘average’ spacing between points.

First we state two general theoretical results, which will be proved in a separate article. For our first result we need the concept of intensity of a point process. Roughly speaking, suppose that a given, infinitesimal interval [x, x+ dx] on the line has a probability m(x)dx of containing a random point of the point process. Then m(x) is called the (first order) intensity function of the point process. If m(x) is constant, then the point process is called ‘first-order stationary’.

Theorem 1 Suppose that X is a first-order stationary point process with intensity m(x) ≡ m where m > 0. Then the modified Cavalieri estimator (6) with τ = 1/m is an unbiased estimator of V.

This is a straightforward consequence of Campbell's Theorem (Stoyan et al., 1995, p. 103).

For the main result, we need the concept of the pair correlation function. Consider two nonoverlapping intervals [x, x + dx] and [y, y+ dy] on the line. Suppose there is a probability k(x, y)dx dy that the two intervals each contain a random point. Then ρ(x, y) = k(x, y)/(m(x)m(y)) is called the pair correlation function of the point process. If the process is first-order stationary and also ρ(x, y) depends only on x–y, the process is called ‘second-order stationary’.

Theorem 2 Suppose that X is a second-order stationary point process with intensity m(x) ≡ m and pair correlation function ρ(x, y) =ρ(x − y). Then the variance of the modified Cavalieri estimator (6) is

(7)

This is a fairly straightforward consequence of the Campbell-Mecke formula (Stoyan et al., 1995, p. 119).

Equation (7) is our main theoretical result; it gives the exact variance of the Cavalieri estimator, including the effect of irregularity in the section spacing.

To apply this general theory to obtain numerical values for the variance, we would need to know the intensity m and pair correlation function ρ, which will depend on information or assumptions about the nature of the errors in placement of the serial sections. The next section discusses one example application of the theory.

The classical theory for exactly constant section spacing (Section 2) can also be obtained as a special case of Eq. (7). Suppose X consists of the points U+ kt for k = 0, ± 1, ± 2, … , as in the classical theory. Then X is a second-order stationary point process with intensity m(x) = 1/t. The estimator (6) is simply the usual systematic sampling estimator (2). The pair correlation function ρ(x) of this point process is a sum of delta functions with mass t at the values kt for each integer k ≠ 0. The variance formula (7) reduces to the usual identity (4) for the variance under systematic sampling.

4. Perturbed systematic sampling

For some cutting devices it is plausible to assume that the actual position of each cut deviates from its intended location by a random displacement. We called this ‘perturbed systematic sampling’. We will now calculate the variance of the Cavalieri estimator for this case.

Formally, assume that the intended, equally spaced locations x_{k} = U + kt are perturbed by random errors D_{k} so that the actual section locations are y_{k} = x_{k} + D_{k}. Assume the deviations D_{k} are statistically independent (from one another and from x_{k}), and all deviations have the same probability density h. The resulting point process X is first- and second-order stationary, with intensity m = 1/t. Therefore, by Theorem 1, the Cavalieri estimator is still unbiased. By Theorem 2, the variance of the Cavalieri estimator is, using (7),

(8)

where the sums are over all nonzero integers k = ±1, ±2, … , and

is the so-called convolution of g with h^{(2)}, the probability density of D_{k}–D_{m} for any k ≠ m. For example, if the deviations are Normally distributed with variance σ^{2}, then h^{(2)} is the probability density of the Normal distribution with mean 0 and variance 2σ^{2}, and g * h^{(2)} is a smoothed version of g. Intuitively, (g · h^{(2)})(z) reflects the correlation between the sectional areas of two sections separated by a nominal distance z, and allows for error in the actual placement of the two sections.

5. Example

For a numerical example of Eq. (8), we assumed the solid object was a sphere of unit radius. For this object the measurement function is

if |x| 1, and zero otherwise. The covariogram g is

for |z| 2, and zero otherwise.

We computed the variances of the Cavalieri estimators under ‘exact’ and ‘perturbed’ systematic sampling. For exact systematic sampling we assumed that the sections were evenly spaced at an exact distance t apart. For perturbed systematic sampling we assumed the random displacements D_{k} were uniformly distributed (constant probability density) over [−s/2, s/2] where s = 0.1t. That is, the maximum displacement error is plus or minus 5% of the nominal section spacing; the coefficient of error of the section spacing is CE = 0.1/ = 2.9%, because a uniform random variable on an interval of length L has standard deviation L/.

The variances of the exact and perturbed Cavalieri estimators are plotted in Fig. 6. The lower graph is the variance of the classical Cavalieri estimator based on exact systematic sampling with section spacing t, plotted as a function of t. It was computed using the classical formula (4). The upper graph is the variance of the Cavalieri estimator based on the perturbed periodic sample, computed using the modified formula (8).

The two plots in Fig. 6 have different slopes, indicating that the variance of the Cavalieri estimator goes to zero at a slower rate in the perturbed case than in the exact case. The ratio of the two variances is plotted in Fig. 7. In this example, when the section spacing is about 0.02 units (i.e. when the sphere is cut into about 100 serial sections) the variance in the perturbed case may be as much as 20 times higher than the variance in the exact case. This suggests that the standard asymptotic variance formulae for systematic sampling may yield substantial underestimates of the true variance.

Curiously, the variance of the perturbed estimator is actually smaller than that of the exact estimator in some cases (with large section spacing). This is presumably due to suppression of the Zitterbewegung effect.

6. Discussion

The main result of this paper is a general mathematical technique for dealing with the effects of variability in section spacing. Subsequent papers will explore applications of the technique.

The preliminary calculations in Sections 4 and 5 suggest that variability in section spacing or slab spacing may have a substantial impact on the variance of the Cavalieri estimator. The Cavalieri method is still an unbiased and very accurate estimator of volume, but the classical theory for calculating its variance may underestimate the true variance by an order of magnitude.

The numerical example shown in Figs 6 and 7 is conservative, in the sense that it assumes that the random displacement error is a fixed fraction of the nominal section spacing. In practice, the magnitude of displacement error is probably constant for a given cutting device. This would imply an even greater inflation of the variance when the section spacing is small.

Further results will be reported in a subsequent paper.