Statistical analysis of reduced pair correlation functions of capillaries in the prostate gland


Prof. Dr T. Mattfeldt. Tel: +49 731 5002 3322; fax: +49 731 5002 3384; e-mail:


Blood capillaries are thread-like structures that may be considered as an example of a spatial fibre process in three dimensions. At light microscopy, the capillary profiles appear as a planar point process on sections. It has recently been shown that the observed pair correlation function g(r) of the centres of the fibre profiles on two-dimensional sections may be used to estimate the reduced pair correlation function of stationary and isotropic fibre processes in three dimensions. In the present study, we explored how this approach may be extended to statistical analysis of reduced g-functions of capillaries from multiple specimens of different groups and with replicated observations. The methods were applied to normal prostatic tissue compared with prostate cancer. Confidence intervals for the mean reduced g-functions of groups were estimated for fixed r-values parametrically using the t-distribution, and by bootstrap methods. Each estimated reduced g-function was furthermore characterized in terms of its first maximum and minimum. The mean length of capillaries per unit tissue volume was significantly higher in prostate cancer tissue than in normal prostate tissue. Significant differences between the mean reduced g-functions of malignant and benign lesions could be demonstrated for two domains of r-values. In general, bootstrap-based confidence intervals were slightly wider than parametrically estimated confidence intervals. Falsely negative lower bounds of the intervals, which sometimes arose using the parametric approach, could be avoided by the bootstrap method. Testing of group mean values for significant differences by the bootstrap method yielded more conservative results than multiple t-tests. The functional value of the first maximum of the reduced g-function and a global statistical parameter of short-range ordering was significantly reduced in the carcinoma group. Prostate cancer tissue is more densely supplied with capillaries than normal prostate tissue and the three-dimensional arrangement of the vessels differs with respect to interaction at various distance ranges. In the local approach used here, bootstrap methods can be used as a robust statistical tool for the computation of confidence intervals and group comparisons of mean reduced g-functions at specific ranges of interaction.


Capillaries are defined as blood vessels whose wall consists only of endothelial cells and a basement membrane. The density of capillarization is an important factor for the oxygen supply of a tissue. The growth of new vessels in general is denoted as angiogenesis. Capillary angiogenesis in tumours is a topic of central importance in tumour biology. In the exploration of mechanisms of angiogenesis, the basic structural background remains the capillary network itself, which can be visualized by microscopy. To obtain objective findings in such investigations, it is obligatory to quantify the capillarization. For this purpose methods of quantitative stereology are relevant (Mattfeldt & Mall, 1984; Mattfeldt et al., 2004a,b). These methods are rooted in the mathematical domain of stochastic geometry, where capillaries may be considered as an example of a three-dimensional (3D) fibre process.

Fibre processes are random geometrical models for fibrous structures. They are used in applications in biology, medicine and material science, for example (Mattfeldt et al., 1994; Stoyan et al., 1995; Krasnoperov & Stoyan, 2004). Fibres may be intuitively defined as thread-like structures, i.e. filamentous or thin tubular structures whose length greatly exceeds their width. After cutting, such fibres appear on a microscopic section as dots, e.g. small ellipses when the true fibres are circular or elliptical cylinders. In the case of isotropic and stationary capillary networks, three simple first-order parameters can be estimated by using information from sections of arbitrary location and orientation by using elementary stereological equations

VV = AA(1)
SV = (4/π)BA(2)
LV = 2QA(3)

(see, e.g. Weibel, 1979; Mattfeldt et al., 1990, 2004a,b; Howard & Reed, 2005). Here the stereological shorthand denotes: VV, the volume of capillaries per unit reference volume; AA, the area of capillary profiles per unit reference area on sections; SV, the surface area of capillaries per unit reference volume; BA, the boundary length of capillary profiles per unit reference area on sections; LV, the length of capillaries per unit reference volume, i.e. the intensity of the 3D fibre process and QA, the number of capillary profiles per unit reference area on sections. The three parameters on the left side of Eqs (1)–(3) express the density of capillary supply in 3D space in different terms.

Even the combination of all three model parameters, VV, SV and LV, does not provide a complete geometrical characterization of a capillary network. The first-order parameters tell nothing about the geometrical architecture (pattern) of the blood vessels, i.e. their spatial arrangement relative to each other. To describe arrangements of random sets in space, a well-established approach consists of methods of second-order stereology. Such techniques have hitherto been used mostly for random sets with positive volume fraction (volume processes) (Cruz-Orive, 1989; Mattfeldt et al., 1993, 1996, 2000; Mattfeldt & Stoyan, 2000; Mattfeldt, 2003). In principle, however, they may also be used for the second-order characterization of surface processes and fibre processes in 3D space (Mattfeldt et al., 1994; Stoyan et al., 1995). A recent study showed how second-order stereological inference on isotropic spatial fibre processes may be performed on the basis of observations on two-dimensional sections (Krasnoperov & Stoyan, 2004). The ordinary planar pair correlation function g(r) of the sectional profiles of the fibres can be used to estimate the reduced pair correlation function g3(r) of the 3D fibre process (Krasnoperov & Stoyan, 2004). In this methodological study, the emphasis was put on point estimation of the reduced pair correlation function. However, in an experimental or clinical research project with more data, it is desired to provide confidence intervals of the function for groups of cases and to test for significant differences between groups. Such an attempt at statistical inference was made in the present study.

Materials and methods

Explorative statistical analysis of planar point patterns

After digitizing the coordinates of the midpoints of the fibre profiles on sections, exploratory methods of data analysis can be applied to characterize the two-dimensional point process of the fibre profile midpoints. The most basic information is an estimate of the intensity λ of the point process, i.e. the mean (= E = expected) number of points per unit reference area. Recall that if X = {Xn} is a motion-invariant point process and W is a sampling window then


where X(W) measures the number of points of X located in W and | W | denotes the area of W. A natural estimator for the intensity λ is given by


Notice that in the case of a Poisson point process, usually λ2 is not estimated as (inline image)2 but as


(Stoyan & Stoyan, 1994, p. 277). Whereas the intensity is a single quantity, second-order functions (summary statistics) provide a series of values as a function of the interpoint distance r. For the estimation of the summary statistics it is advisable to rely on stationarity and isotropy as mathematical model assumptions. One of the most popular functions of explorative spatial point pattern analysis is Ripley's K-function K(r) (reduced second moment function) (Ripley, 1988; Stoyan et al., 1995). Intuitively, K(r) is the mean number of other points of the process lying within a circle of radius r, centred about a typical point (x,y) of the process, divided by the intensity of the process


where the symbol ‘|’ denotes ‘conditional to’. Notice that the conditioning here is not to be regarded in a strict mathematical sense as the probability of a stationary point process X having a point in a particular location (x,y) equals 0. More formally one can define K(r) for a motion-invariant point process X = {Xn} as


where b(Xn,r) is the disc around Xn with radius r. The estimation of K(r) has to be performed using edge-corrected estimators as described previously, e.g. in Diggle (2003) and Stoyan & Stoyan (1994). In particular we used




Note that || Xj − Xi || represents the Euclidean distance between the points Xj and Xi and that inline image is the set W translated by Xj. The usage of the denominator ensures edge correction. Often instead of K(r) the function L(r) given by


is regarded.

As reference model (null hypothesis) for isotropic and stationary point processes, the model of a stationary Poisson point process is used. In this case there is no interaction between the points at all distances and the number of points in a sampling window W is Poisson distributed with parameter equal to the intensity λ of the Poisson point process times the area of W. The points are distributed independently at random, isotropically and homogeneously in the plane, a state which has rightfully been denoted as complete spatial randomness of points (Diggle et al., 1991, 2000; Diggle, 2003; Schladitz et al., 2003). It is easy to see that, under these conditions, a circle with radius r around a typical point of a Poisson point process contains λπr2 points on average, namely the product of the area of the circle and the intensity. After division of this value by λ, the K-function for the planar Poisson point process is obtained

KPoi(r) = πr2(12)

Let us now consider the non-Poisson case. An initial curve segment with K(r) = 0 indicates that the interpoint distance does not attain values below a certain minimum. In the case of biological structures such as cells, cell nuclei or capillaries, this behaviour may simply result from their physical size, if no overlapping is possible. Such curve segments may hence be interpreted as a sign of a hard-core property. The lowest r-value for which the sample K-function reaches a positive value, r0, may be considered as an estimate of the hard-core distance.

In analogy to a probability density function, which is the derivative of a cumulative distribution function, there is a counterpart to the K-function, namely the pair correlation function g(r), which may be obtained after differentiation and normalization of K(r)


In the case of a planar Poisson point process, we obtain


for all r [by insertion of K(r) = πr2 into Eq. (13)]. Values of g(r) below 1 indicate repulsion and values above 1 indicate clustering for point pairs of such a distance r. A hard-core effect leads to an initial segment with zero values of g(r). Similar to a probability density function, hills and valleys above and below the constant value 1 indicate domains of r-values with tendencies of the points for aggregation and repulsion, respectively. The pair correlation function may also be defined as the product density of second order of the point process, divided by the square of the intensity for the purpose of normalization (Stoyan & Stoyan, 1994, p. 249; Stoyan et al., 1995, p. 129). Hence an estimator for g(r) is given by




is an estimator for ρ(2)(r), the product density of second order. For the estimator inline image, similarly to the estimator for K(r), the denominator inline image ensures edge correction. The term kh(x) denotes a kernel function, which is used for smoothing. We used the Epanechnikov kernel


with a bandwidth inline image according to Krasnoperov & Stoyan (2004). Clearly there is a wide choice of kernels available. Most often the Epanechnikov kernel is used, whereas in some cases other functions such as the box kernel may be slightly more favourable (Stoyan & Stoyan, 2000, pp. 649–650).

Second-order statistics of spatial fibre processes

In a recent study it was shown that, in the context of fibre processes, an additional stereological interpretation of the observed K-function and the observed pair correlation function g(r) of the profile midpoints is possible (Krasnoperov & Stoyan, 2004). Let us denote the intensity of the fibre process as LV (mean length of fibres per unit volume), as usual in stereology. Its K-function K̃3(r) is the mean length of the fibres in a sphere with radius r, centred about a typical point of the fibre process, divided by LV. In the 3D case it holds for the corresponding pair correlation function 3(r)


It was shown that a ‘reduced variant’g3(r) of 3(r) can be estimated from sections if its definition is adapted in the spirit of the definition of the pair correlation function of a planar point process. Remember that in this setting only the number of ‘other’ points of the process are counted in circles around typical points but the typical point itself is not counted. In analogy, the reduced K-function K3(r) may be defined as the expected length of the ‘other’ fibres in a sphere of radius r centred at a typical point of the fibre process, whereby the length of the fibre running through the typical point itself is not counted, divided by LV. Then, having in mind Eq. (18) and by replacing K̃3(r) with K3(r), the reduced pair correlation function g3(r) can be defined as


In Krasnoperov & Stoyan (2004) estimators for g3(r) of the form


are suggested. In practice, this result means that the estimator ĝ3(r) considered in Eq. (15) for the ordinary planar pair correlation function g(r) of the sectional profiles of the fibres is an estimator of the reduced pair correlation function g3(r) of the 3D fibre process. As model requirements, it is necessary to assume isotropy, strict stationarity and independence among the curve elements of the fibre process. This state has previously been denoted as ‘complete directional randomness’ (Mattfeldt et al., 1994). Hence the method shown here is not applicable if the fibres are anisotropic. In this case, knowledge of the distribution of the local angles that the fibres make with the sectioning planes is required (for an example see Mattfeldt et al., 1994). Furthermore, the concept of a reduced g-function rests on the model assumption that the individual fibres have finite length. For our biological material this model assumption may be taken for granted, as blood capillaries inside an organism cannot be infinitely long.

Null models for spatial fibre processes are Poisson line processes and Boolean segment processes in three dimensions (Stoyan et al., 1995, pp. 83 and 250). They represent complete spatial randomness of lines or line segments in space, respectively, just as a Poisson point process represents complete spatial randomness of points. At complete spatial randomness of a 3D fibre process we have inline image. Hence with Eq. (19) it follows that inline image at complete spatial randomness of the 3D fibre process for all r (Krasnoperov & Stoyan, 2004). In its first application (Krasnoperov & Stoyan, 2004), the new estimator was used to investigate blood capillaries in normal rat thyroid glands. The main requirements of isotropy and stationarity should hold in first approximation for many biological fibre processes, e.g. for capillaries in glandular tissues and in many tumour tissues. The hard-core distance of the point process of the fibre profiles on sections may be interpreted in three dimensions as the minimal distance between the longitudinal axes of the capillaries.

Statistical methods

Bootstrap methods.  The bootstrap method was developed by Efron in 1979 and consists basically of an independent random resampling of the sample data with replacement (Efron & Tibshirani, 1993; Ludbrook, 1995). It is a computer-based method largely free of statistical model assumptions. Typically, an arbitrary statistic of interest (the bootstrap statistic) is computed from 100–10 000 bootstrap samples. It is assumed that the distribution of the bootstrap statistics (usually denoted as D*) approximates the distribution of the statistic D in the population. The bootstrap belongs to the computer-intensive resampling methods; other pertinent examples are jackknife techniques, permutation and randomization tests (Efron & Tibshirani, 1993; Ludbrook, 1995).

Up to now, bootstrap methods have only rarely been exploited in stereology. They have been used in the context of point process statistics (Diggle et al., 1991, 2000; Schladitz et al., 2003) and recently for statistical inference from stereological estimates of volume fraction (Mattfeldt & Fleischer, 2005). Using the bootstrap method, it is possible to provide statistical inference from sets of independent data (e.g. a single estimate of VV per case) or from a series of dependent data (e.g. an estimated K-function of a series of r-values per case; see Diggle et al., 1991, 2000; Schladitz et al., 2003).

Confidence intervals.  An estimate ĝ(i)(r) of the g(r)-function at a given r-value in the ith case of n cases was obtained as the arithmetic mean of the values of the two estimated g(r)-functions resulting from two images per case. To estimate a parametric 95% confidence interval for g(r) for a group of n cases, the following statistic standard procedure was used. Let us denote the mean of the ĝ(i)(r)-values from the n cases for fixed r


If the assumption of a Gaussian distribution of the values of ĝ(i)(r) at fixed r is fulfilled, estimates of the bounds of a 95% confidence interval for this mean value are given by (inline image(r) ± t0.975,n−1 SE(inline image(r))), where t0.975,n−1 is the quantile of the t-distribution used for a 95% confidence interval for (n − 1) degrees of freedom and SE is the standard error of the mean. To obtain a bootstrap confidence interval for g(r), we created 1000 bootstrap samples S1, … , S1000 with n items from the original sample Sorig ={ĝ(1)(r), ... , ĝ(n)(r)}, where in this case n = 12. The sampling is independent and with replacement. For all 1000 bootstrap samples their corresponding mean values


are computed, where ĝ(i,j)(r) denotes the jth item in Si. The 1000 values inline image are sorted by size. The lower and upper bounds of a 95% confidence interval of inline image(r) were estimated by the 26th and 975th value of D* in this sequence (Mattfeldt & Fleischer, 2005).

Significance tests.  The classical two-sample t-test (Student's t-test) was used as a parametric test for a significant difference between the mean values of g(r) from two groups A and B with nA = 12 and nB = 12 cases. For the non-parametric bootstrap approach (see below), the test statistic D for the two samples was computed as the difference of the sample means inline imageA and inline imageB : D = inline imageA − inline imageB. For the generation of bootstrap samples, the two samples were united to a common sample of size nA + nB = 24. From this sample, we created 999 pairs of bootstrap samples inline image with nA and nB items and mean values inline image and inline image. The sampling is independent and with replacement. For all 999 bootstrap sample pairs, the difference inline image between the group mean values inline image was computed. Finally, the inline image-values of the bootstrap samples and the D-value from the real sample were ordered by size. If the rank of D in this series was less than or equal to 25, or if the rank was greater than or equal to 976, the result was considered to be significant at a level of 5% (Mattfeldt & Fleischer, 2005).

Evaluation of individual reduced g-functions.  In addition to the local computation of confidence intervals and significance tests for fixed r, each estimated reduced g-function per visual field was evaluated with a method presented by Stoyan & Schnabel (1990) (see also Stoyan & Stoyan, 1994, pp. 250–258). This procedure includes identification of the first maximum gmax and the next following minimum gmin with the corresponding r-values rmax and rmin for each reduced g-function, where rmin > rmax. From these data, the statistic


was computed (Stoyan & Stoyan, 1994, p. 251). The statistic M is related to the global degree of disorder in the spatial point pattern. Large values indicate a high degree of order and may be expected, for example, in the case of point patterns with an element of long-range order. The statistic may be used as a tool to summarize the course of the reduced g-function by a single quantity.

These evaluations provided two values of each statistic per case because two reduced g-functions were computed per case. The statistical handling of these data is not elementary. In contrast to the estimates for LV and g(r) themselves, they represent minimum and maximum properties extracted from g(r). For these data a simple arithmetic averaging might not be appropriate. For gmax and gmin, it was decided to perform no averaging within cases before making significance tests between groups but to keep both values unchanged. The significance tests were performed on the basis of 12 × 2 = 24 equally weighted data per group, which leads to t-tests with 47 degrees of freedom.

Practical methods

For the practical investigations, 12 routine cases of prostatic cancer in whom radical prostatectomies had been performed were chosen. These had been examined histopathologically by the Department of Pathology of the University of Ulm. As a control group with normal tissue, the tumour-free regions of 12 radical prostatectomy specimens with prostatic cancer were used (Figs 1 and 2). Paraffin sections were stained using an antibody against CD34, a routine immunohistochemical marker for endothelial cells that is often used for the estimation of the microvessel density in tumours (Fig. 2). The sections were viewed under a Zeiss Axiophot light microscope (Carl Zeiss AG, Oberkochen, Germany), connected to a CCD camera attached to a PC. About 10–15 visual fields per case were acquired and stored using the software diskus 4.50 firewire under Windows 2000. The two technically best images of these series were selected according to quality criteria (best staining quality of the capillaries, well-preserved morphology of tissue, absence of artefacts, etc.). This approach provided two rectangular visual fields per case with 1240 × 1000 pixels, in which 61–341 capillary profiles could be found per field (Fig. 3). At the final magnification, 1 pixel corresponded to 1.5 µm, hence fields with edgelengths of 1860 × 1500 µm were evaluated.

Figure 1.

(a) Tumour-free prostatic tissue from a radical prostatectomy specimen, removed because of prostatic cancer. Immunohistochemical stain with an antibody against the endothelial antigen CD34. Positively stained structures are visualized as brown dots. The capillaries lie within connective tissue (stroma). In the background can be seen non-neoplastic prostatic glands with epithelial cells and lumina (white holes). (b) The same image after detection of the centres of the capillary profiles. Note that many capillaries are located preferentially near the interfaces of epithelium and stroma, whereas another subset lies deeply within the stroma. The rectangular visual field consists of 1240 × 1000 pixels, which corresponds to 1860 × 1500 µm at the level of the specimen.

Figure 2.

(a) Prostatic cancer tissue from a radical prostatectomy specimen with immunohistochemical CD34 stain as in Fig. 1. Here the epithelial component in the background consists of large neoplastic glands that often show a cribriform (sieve-like) pattern. (b) The same image after detection of the centres of the capillary profiles. Note stromal septa that radiate into glands and contain capillaries.

Figure 3.

Selected visual field with 1240 × 1000 pixels. The crosses denote centres of capillary profiles.

The colour images were stored as TIF-files and analysed interactively using standard imaging software under Windows NT (Adobe photo-shop, Imagetool). The centres of the capillary profiles were detected interactively (Figs 1 and 2) (see also Krasnoperov & Stoyan, 2004). The coordinates of these points were stored as ASCII data sets. The reduced pair correlation function g(r) was estimated for each image as described above in ‘Explorative statistical analysis of planar point patterns’ at steps of 0.5 for r-values up to 500 pixels, i.e. for r = 0, 0.5, 1, … , 499, 499.5, 500. The estimation of g(r) was performed by using geostoch, a Java-based open-library system developed by the Department of Applied Information Processing and the Department of Stochastics of the University of Ulm (Mayer et al., 2004; Parametric and bootstrap-based 95% confidence intervals as well as tests on significant differences between the group mean values were computed for r = 1, 2, … , 99, 100 and for r = 100, 105, … , 495, 500. The hard-core distance was estimated for each visual field as the minimum value of the interpoint distances.


Qualitative observations

Inspection of the sections immunohistochemically stained with an antibody against CD34 disclosed numerous positively marked (brown-stained) small dot-like areas (Figs 1a and 2a). These varied in shape from elliptical to circular dependent on the direction of cutting. These dots were checked by comparison with the usual haematoxylin-eosin stain from the same areas. It was found that all CD34-positive areas in normal prostatic tissue and prostatic cancer tissue were morphologically compatible with blood vessels. A few larger vessels such as venules were sometimes also stained in addition to the largely preponderant capillaries but these were easily discernible from capillaries by their thicker walls. In all cases, the capillaries lay only within the stromal parts of normal and neoplastic prostatic tissue, i.e. they were always surrounded by at least a thin rim of connective tissue and were never directly enclosed by epithelial cells. The capillaries were often found near the borders between glandular lumina and adjacent stroma fields (Figs 1b and 2b). No preferential directions of the points were visible in either class of specimens (Figs 1b and 2b). By visual inspection alone, it was not possible to detect clear differences between the point patterns of capillary profiles in normal and neoplastic tissue.

Individual reduced g-functions

Figure 4 shows the plots of selected individual reduced g-functions from both groups. Invariably, the curves began with a flat curve segment and thereafter attained positive values, usually quickly mounting to a first maximum gmax and then descending to a first minimum gmin (Fig. 4a and b).

Figure 4.

(a) Estimated reduced g-function from a selected visual field of tumour-free prostatic tissue. Note the hard-core effect in the beginning, then weaker repulsion, and thereafter first maximum and first minimum. (b) Estimated reduced g-function from a selected visual field of prostatic cancer tissue. Here the curve ascends less steeply.

Mean values of groups

It was found that the mean intensity of the capillary fibre process LV was increased in the prostatic cancer group by 47% as compared with tumour-free tissue (P < 0.01) (Table 1). For the further statistical evaluations, estimates of the mean reduced g-functions per group were obtained and 95% confidence intervals were computed by classical and bootstrap methods. Selected results are shown in Table 2. Bootstrap-based confidence intervals computed with 1000 bootstrap samples per r-value lay globally in the same order of magnitude as classically computed confidence intervals. However, in general, the bootstrap intervals were slightly wider than the classically estimated confidence intervals. For low r-values, which led to very small g-values near 0 in both groups (see Table 2 for r = 3–6 pixels in the normal group and for r = 3–7 pixels in the carcinoma group), application of the parametric standard formula for computation of interval bounds led to negative lower bounds (Table 2). In contrast, bootstrapping provided non-negative lower bounds also in these domains (Table 2). The mean reduced g-functions of the two groups are plotted with bootstrap confidence intervals in Fig. 5(a and b).

Table 1.  Group comparisons of summary characteristics.
EstimateNormal groupCancer grouptLevel of significance
  1. The summary characteristics are compared between the groups of tumour-free prostatic tissue and prostatic cancer tissue by means of Student's t-test. N, number; , mean value; SD, standard deviation; NS, not significant. For the other abbreviations see text.

N (capillaries field−1)12738188602.98P < 0.01
LV (capillaries tissue−1) 91.4427.36135.3643.202.98P < 0.01
(mm mm−3)
r0 (pixel) 17.33 4.51 15.33 4.021.62NS
r0 (µm) 26.00 6.77 23.00 6.031.62NS
rmax (pixel) 26.77 6.18 30.7512.321.41NS
gmax (pixel)  1.41 0.29  1.10 0.243.83P < 0.001
rmin (pixel) 40.68 8.53 40.1216.220.15NS
gmin (pixel)  1.02 0.20  0.94 0.211.38NS
M  0.0279 0.0154  0.0139 0.01162.31P < 0.05
Table 2.  Results of parametric and bootstrap methods. Comparison of 95%– confidence intervals.
r(r)SDParametric boundsBootstrap bounds
  1. The mean reduced g-functions are given r-wise with 95% confidence intervals for the tumour-free tissue group and for the prostatic cancer tissue group. In most instances, the bootstrap method yielded slightly wider intervals. Note negative lower bounds of parametric confidence intervals for mean values of g-functions at some low r-values in both groups. This effect was precluded by the bootstrap method. SD, standard deviation. For further abbreviations see text.

Normal prostatic tissue
Prostatic cancer tissue
Figure 5.

(a) Mean values and 95% confidence intervals for the reduced g-functions of the capillaries for tumour-free tissue. The confidence intervals were obtained r-wise by bootstrapping. (b) Analogous plot for the prostatic carcinoma group. (c) The mean reduced g-functions of the tumour-free group (continuous line) and of the prostatic cancer group (dotted line) are shown in superposition. The differences between the mean reduced g-functions are significant in the domains between r = 10–32 pixels and r = 64–69 pixels. In both regions the mean reduced g-function is diminished in the carcinoma group as compared with the normal tissue.

To test for statistical differences between the mean reduced g-functions at fixed r-values, multiple t-tests and bootstrap tests were performed. The results are shown in Table 3 and Fig. 5(c). Significant differences between the group mean values of g(r) were found according to the bootstrap method for the ranges of r = 10–32 and r = 64–69 pixels. In both domains, the g-values were significantly reduced in the carcinoma group. The outcomes of the bootstrap tests were slightly more conservative than the outcomes of multiple t-tests, which yielded significant differences in the ranges r = 9–40 and r = 60–77 pixels.

Table 3.  Group comparisons of g-functions. Parametric and bootstrap methods.
rNormal (r)Cancer (r)DtP(t)Rank of D in 1000 BS
  1. The differences between the group mean values of g(r) were tested locally for significance at fixed r-values. D, difference between sample means; BS, bootstrap sample. *P < 0.05, **P < 0.01. Note slightly more conservative results according to bootstrap tests as compared with t-tests. Both approaches yielded evidence for differences with respect to interaction at two separate domains, see also Fig. 5(c). For other abbreviations see text.


Summary characteristics of reduced g-functions

Using the methods described above in ‘Significance tests’, the summarizing characteristics rmax, gmax, rmin and gmin were computed for all individual reduced g-functions. The statistic M was computed from these values (Stoyan & Schnabel, 1990; Stoyan & Stoyan, 1994). The minimum interpoint distance was also recorded for each reduced g-function. The results are shown in Table 1. For the sake of simplicity only the results for Student's t-test are presented; bootstrap tests led to the same conclusions. There was no significant difference between the mean hard-core distances of the two groups. However, there was a highly significant decrease of mean gmax in the cancer group. This finding led to a significant decline of the statistic M. There were no significant differences between group means with respect to rmax, rmin and gmin. Summarizing, these data corroborate the finding that the capillary patterns of normal and neoplastic prostatic tissue are spatially different, which was also concluded on the basis of the pointwise (local) comparison of the mean reduced g-functions.


Capillary changes after neoplastic transformation

The main results of the present study may be summarized as follows. Compared with normal prostatic tissue, the capillary length density is significantly increased in prostatic cancer. Using classical and bootstrap inference methods for r-wise comparisons of mean g-values, a significant decrease of the mean g-values at various distances could be demonstrated in the prostatic cancer group. Hence, the fibre processes of the capillaries in prostatic cancer tissue and normal prostatic tissue differ by first-order as well as by second-order properties. Parallel to these changes, it could be shown that the height of the first maximum of the pair correlation function declined in the carcinoma group. For an interpretation of the changes in the malignant tumours, the following considerations seem appropriate. (i) The increase of the intensity LV of capillaries in the cancer group was not accompanied by a diminished hard-core distance in the cancerous group, which might have been expected. (ii) It cannot be excluded that the second-order changes are partially due to a higher intensity of the capillaries in the carcinoma group. For the simple case of a Poisson process, the pair correlation function is not affected by the density of the points; we have g(r) ≡ 1 for the Poisson process irrespective of the intensity. However, here we are faced with hard-core point processes. Under this condition second-order statistics such as g(r) may well be influenced by the intensity. (iii) Another explanation for the observed alterations of the reduced g-function is a true change of the inner order of the neoplastic tissue as compared with the normal tissue that affects the capillary arrangement, irrespective of the intensity of the process. To study this question in more depth, it will be informative to fit parametric models of point processes to the images of both groups (see below).

Stereology of capillaries: general aspects

Up to now we have discussed the nature of the change in capillarization after transition from normal to neoplastic prostatic tissue. A more elementary question is to ask for the nature of the two point processes, each viewed separately from the viewpoint of exploratory statistics. A first step could be to check whether the observed point patterns are compatible with a planar Poisson point process by a Monte-Carlo test on complete spatial randomness of points (Schladitz et al., 2003; Baddeley & Turner, 2005). Here such tests were not performed because it was obvious at first sight that the observed point processes were not compatible with planar Poisson point processes but were clearly hard-core point processes. An informative approach would consist of a stochastic modelling of the observed point processes. An obvious approach to modelling could consist of parameter fitting on the basis of a known stochastic point process model, which must take into account the hard-core property in the first instance (Baddeley & Turner, 2005). Similar to our previous investigation on point processes of intramembranous particles, a repulsive pattern was found at low r-values and clustering of the points at longer distances (see Schladitz et al., 2003, Figs 4–6 therein). It is natural to consider the flexible class of Gibbs processes for modelling of these processes (see Stoyan & Stoyan, 1994; Stoyan et al., 1995; Schladitz et al., 2003). First results with such models are available (see Mattfeldt & Fleischer, 2006) but a presentation of these findings is beyond the scope of this study. An attractive alternative idea to simulation on the basis of a parametric model is statistical reconstruction of the point process, where one proceeds from extracted data of the empirical images themselves. For example, the intensity and second-order functions of the observed point process are estimated from the images, and simulations of point processes are then performed conditional to these data. Reconstruction methods were extensively applied in the domain of random sets [see the work of Torquato and co-workers (Yeong & Torquato, 1998; Manwart et al., 2000; Torquato, 2001)]. However, they have scarcely been used for point processes (Tscheschel & Stoyan, 2006).

Choice of the reference space

The observation that capillaries lie only in the stroma and never inside the epithelium has the following consequences: it means that the capillaries cannot originate from all parts of the total reference space, i.e. the total tissue, but only from a subset of it, i.e. the stroma. Hence in a study on capillarization, the investigator is free to define either the total tissue or the stroma as reference space. The former decision is the most natural and was chosen in the study of Krasnoperov & Stoyan (2004) and here. The latter choice makes the stereological investigation of capillaries somewhat more complicated. The estimation of second-order statistics such as g(r) is then no longer as straightforward as in this study. In our investigation, the whole rectangular visual fields were used as the reference area. Concerning estimation of the first-order property LV, we would have a ratio estimator with variable denominator, as the stromal fraction varies from field to field within each case. This means that instead of taking the mean of ratios as done here, it would be necessary to sum up the number of capillaries over all fields and divide by the sum of the areas of the stroma of all fields, and to use a ratio-of-sums estimator. In this case similar statistical problems arise as in the case of volume fraction estimation from multiple fields with varying content of the reference phase per case using the Delesse principle (Mattfeldt & Fleischer, 2005). In the latter situation, using the ratio-of-sums estimator with variable reference area may, however, sometimes increase the precision. Whether the same holds for LV of capillaries per unit stromal volume as compared with LV of capillaries per unit tissue volume remains to be determined.

Considerations of sampling design

The main aim of the present study was a comparison of capillary networks in normal prostatic tissue with capillary networks in prostatic cancer tissue in terms of the intensity and the reduced g-function. In the first methodological study (Krasnoperov & Stoyan, 2004), five laboratory animals were investigated with one field per specimen at electron microscopy. Here we were confronted with routine human material, in which larger variability has to be expected. To increase the precision and to have an idea about the intraindividual variance ‘within cases between fields’, it was decided to evaluate not just one visual field but two fields per case. Using the same workload, it would also have been possible to evaluate more small visual fields dispersed over the specimen. This would probably have been the most efficient way towards estimation of LV. However, here the estimation of the reduced g-function was the main concern and thus, the emphasis was put on few, comparatively large windows of observation. When it comes to the estimation of second-order statistics of point processes, the well-known motto ‘Do more less well’ (Gundersen & Østerby, 1981) is of only limited validity. Windows of smaller size may quickly attain a critical level of the numbers of points, in particular if the sizes of the windows are further diminished by minus sampling. With too few points inside a window of observation, second-order statistics finally loses its sense (see, e.g. Baddeley et al., 1993).

Local vs. global bootstrapping

In the present study, classical inference methods and bootstrap techniques were applied to compare mean reduced g-functions of two groups of cases. For this purpose, a pointwise, local approach (consecutive significance tests at all r-values) or a global approach [taking the whole series of g(r)-values into account simultaneously] is possible. In previous bootstrap studies on point process data, the global approach was used (Diggle et al., 1991, 2000; Schladitz et al., 2003). Thus, in the studies of Diggle and co-workers on the brains of schizophrenic people, the global question was addressed: do the brains of schizophrenic people and the brains of normal people differ spatially with respect to various summary statistics of the point process of the neurones as seen in microscopic sections (Diggle et al., 1991, 2000)? In the present study, it was desired not only to find out whether the mean reduced g-functions for normal and neoplastic tissue differ globally but at which specific ranges of interaction the significant differences emerge. Only the local approach gives an answer here. The global approach necessitates complex weighting procedures (Diggle et al., 1991, 2000; Schladitz et al., 2003). Using the pointwise approach, it is possible to compute confidence intervals of reduced g(r) for individual r-values, whereas the global approach would have led to a confidence band. We also wanted to compare bootstrapping with classical tests; this is feasible with multiple t-tests as compared with multiple bootstrap tests but a classical test alternative to the global test on significant difference between two functions is not obvious.

Comparison with previous results

The present findings were compared with those of the first methodological study (Krasnoperov & Stoyan, 2004). In general, both studies disclosed a hard-core effect for very low r-values, whereas at somewhat higher r-values weaker repulsion was found. The mean reduced g-function of the thyroid capillaries did not cut the reference line g ≡ 1 in the studied range of r-values (Krasnoperov & Stoyan, 2004, Fig. 6 therein). This happened, however, in the case of the prostate capillaries (Fig. 5). The reason may lie in the fact that in the electron microscopic study, the r-values ranged only from 0 to 40 µm, whereas here we used light microscopy and evaluated higher values of r from 0 to 300 µm. The estimated length density of capillaries was higher in the electron microscopic study and a lower hard-core distance was found there (Krasnoperov & Stoyan, 2004). The two studies differ in the type of tissue as well as in the microscopical methods (light microscopic vs. electron microscopic resolution, thin vs. ultrathin sections). The estimate of r0 amounted to ≈ 7.5 µm in Krasnoperov & Stoyan (2004, see Fig. 5 therein), whereas we found group mean values of 23–26 µm (Table 1). At first sight it might appear contradictory that the estimation of g(r) using Eqs (16) and (17) provided positive values for r-values distinctly below r0 (see Table 2). No interpoint distances can occur below the hard-core distance. However, when estimating g(r) for stationary planar point processes, kernel methods should be used (Stoyan & Stoyan, 1994; Stoyan et al., 1995; Diggle, 2003; Baddeley & Turner, 2005). The application of any kernel leads to a smoothing of the g-values within the chosen bandwidth. This effect is also operative for the lowest values of r and thus it fully explains the small positive values of ĝ(r) in the region where (r < r0). A viable means of ensuring that functional values of g(r) vanish for r < r0 consists of the so-called reflection method (Stoyan & Stoyan, 1994, pp. 288–290, 368). In comparison to a previous investigation using immunohistochemistry and light microscopy (Mattfeldt et al., 2004a), the estimated length densities of capillaries per unit tissue volume in the prostate were very similar.

Bootstrapping vs. parametric and non-parametric alternatives

When comparing bootstrap confidence intervals with classical confidence intervals, we found slightly wider intervals according to the bootstrap method (Table 2). In our previous study on volume fractions (Mattfeldt & Fleischer, 2005), the bootstrap method provided narrower confidence intervals as compared with parametric bounds. This finding was explained as a result of a non-Gaussian distribution of the data in the previous study (Mattfeldt & Fleischer, 2005). For the present data set, the distribution of the g(r) data between cases within groups appeared roughly consistent with a Gaussian distribution. Under this condition, it may well happen that the bootstrap method leads to less sharp confidence bounds, because a non-parametric technique is applied to data that essentially follow a Gaussian distribution. In general, bootstrapping is nevertheless recommended because of its robustness, as the compatibility of biological data with a Gaussian distribution is never granted a priori. Moreover, a useful side-effect of the bootstrap method was found for low r-values, which led to very small g-values near 0 in both groups (see Table 2 for r = 3–6 pixels in the normal group and for r = 3–7 pixels in the carcinoma group). In these r-ranges, application of the parametric standard formula for computation of interval bounds led to negative lower bounds (Table 2). However, these bounds make no sense for our g-functions, which assume only values ≥ 0. This finding may also be relevant for other applications, where confidence intervals for mean values of data sets with very small non-negative values near 0 are desired. Clearly the method shown here is not the only approach for robust statistical inference from empirically estimated reduced g-functions. A different computer-intensive approach consists of randomization tests, where resampling is performed without replacement (see, e.g. Manly, 1997; Pitt & Kreutzweiser, 1998). Regrettably, randomization tests can only be used for significance testing and not for the computation of confidence intervals. However, the latter option is important in the context of second-order statistics because the computation of confidence intervals allows testing of the null hypothesis of absence of spatial correlation for a given distance. For this purpose, it may be checked whether the confidence interval of the reduced g-function at r includes the reference line g ≡ 1 (see Table 2, and Fig. 5a and b). This purpose can be achieved by parametric methods or bootstrapping but not by randomization tests.


Reduced pair correlation functions of capillaries in glandular tissue may be estimated from ordinary histological sections using light microscopy and immunohistochemistry. In contrast to the classical approach based on fundamental stereological equations, it enables the characterization not only of the density but also of the spatial arrangement of isotropic and stationary fibre processes, thus providing valuable extra information. In the local approach used here, bootstrap methods can be used as a robust statistical tool for the computation of confidence intervals and group comparisons of mean reduced g-functions. Thus, the data analysis allows judgement not only of whether two fibre processes differ at all spatially but also provides details of at which specific ranges of interaction the differences occur.


Thanks are due to Gabriele Ehmke and Rolf Kunft for skilful technical assistance. S.E. is supported by a grant of the graduate college 1100 at the University of Ulm.