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Keywords:

  • Box counting;
  • fractal analysis;
  • histological sections;
  • image analyser;
  • methodology;
  • sectional fractals;
  • trabecular bone

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

A standardized methodology for the fractal analysis of histological sections of trabecular bone has been established.

A modified box counting method has been developed for use on a PC-based image analyser. The effect of image analyser settings, magnification, image orientation and threshold levels was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to calculate objectively more than one fractal dimension from the modified Richardson plot.

The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ < 25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0.

It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than one fractal dimension, describing spatial structural entities. Fractal analysis is a model-independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and complements conventional histomorphometric and stereological techniques.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The use of the fractal dimension to describe the structure of trabecular bone has been reported by a number of workers (Majumdar et al., 1993, 1995, 1996, 1997; Benhamou et al., 1994; Chung et al., 1994; Weinstein & Majumdar, 1994; Fazzalari & Parkinson, 1996, 1997, 1998; Shrout et al., 1997, 1998). These studies employ either direct imaging of histological sections or digitization of radiographic images. There has been no uniformity in methodology between these studies, leading to inconsistency as to what is being described by the fractal dimension or what its relationship is to conventional histomorphometric parameters. This study aims to provide an objective and reproducible methodology for the fractal analysis of histological sections of trabecular bone.

Classic fractal analysis involves estimation of the perimeter of an object using rulers of different lengths. As the size of the measuring unit decreases, the estimated perimeter increases. These data, plotted as log of perimeter versus log of measuring unit are linearly codependent. This is known as the Richardson plot and the fractal dimension = D, where 1 − D is the slope of the regression line. This is illustrated by the Coast of Britain effect which, although reported by Lewis Richardson in the 1920s, was not widely known until the publication of Benoit Mandelbrot's seminal work, The Fractal Geometry of Nature (Mandelbrot, 1977) (Fig. 1).

image

Figure 1. 989. Reproduced by permission of B. H. Kaye).

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Mandelbrot (1977), in formulating the principles of fractal geometry illustrated that natural objects have a finite range over which they are approximate fractal curves (Vicsek, 1998, uses the term effective fractals). This means that the measuring units should range from the magnitude of the smallest feature of interest to the largest feature of interest. The range over which an object exhibits apparent self-affinity or self-similarity is determined by the structural and functional properties of the analysed structure. Therefore, it is imperative when estimating the fractal dimension that the size of the lower and upper limits of the structure have been determined.

The adaptation of box counting methods, for fractal analysis, to computer-based image analysers is not uniform. As in conventional histomorphometry, the effect of suboptimal grey level threshold detection of the trabecular bone (Fisher, 1971), the resolution of the imaging system and the angle of presentation of the specimen to the imaging system potentially affect the fractal analysis. It is necessary to quantify the effect of changes to these parameters in order to specify the optimal conditions by which fractal analysis can be reproducibly performed.

Often the Richardson plot from which the fractal dimension is estimated does not form a simple straight line for natural objects (Paumgartner et al., 1981; Rigaut, 1984; Kaye, 1989a). Rigaut (1984) developed the asymptotic fractal model as an empirical approach, which detects fractal behaviour between lower and upper asymptotic limits. This method is well suited to theoretical models, where no assumptions are made regarding the structural or functional properties of the material being examined. Although this method has been used for ‘real’ objects (Pape et al., 1987; Landini & Rigaut, 1997; Rigaut et al., 1998) it is not well suited to box-counting data from interconnected structures such as trabecular bone. Where there is a priori knowledge of biological processes determining the physical structure, the empirical asymptotic method may not be optimal for characterizing the structure complexity.

This a priori knowledge and visual examination of the Richardson plot may reveal two or more successive straight-line segments, which by regression analysis fit the data better than a single line, i.e. there is more than one fractal dimension (Fazzalari & Parkinson, 1997, 1998). The points on the Richardson plot where one straight line ends and the next begins have been named ‘pivot points’ by Fazzalari & Parkinson (1997, 1998). The ‘pivot points’ determine the range over which the fractal dimensions are calculated, by regression analysis, and are directly related to histomorphometric parameters. These successive fractal dimensions are termed ‘sectional’ fractal dimensions.

Previously, the ‘sectional’ fractals have been selected by subjective assessment, which is prone to operator variability (Fazzalari & Parkinson, 1997, 1998). Ideally, an objective method should be formulated to determine the straight-line segments on the Richardson plot so that there is uniformity within and between studies. The computational power of modern PCs allows fast analysis of data, enabling application of complex iterative algorithms and statistical models.

The aims of this study are to devise an objective and reproducible methodology for the fractal analysis of histological sections of trabecular bone. In order to achieve this, a highly automated and integrated approach using software developed for a Quantimet® 500MC image analyser (Leica, Cambridge, U.K.), Microsoft® Excel (Microsoft Corporation, Seattle, WA, U.S.A.), IMSL (Visual Numeric Inc, Texas, U.S.A.) and BMDP® statistics package (SPSS Inc, Illinois, U.S.A.) will be presented. The principles of the methodology are applicable to any facility with access to programmable PC-based image analysers.

Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Analytical method

Fractal analysis was performed using a Quantimet 500MC image analyser. The analysis program was written to run automatically, using the proprietary QUIPS language. The program uses an automated box-counting method to estimate the fractal dimension. The box sizes used in these studies range from 1 to 181 pixels in side length. The largest box size (181 pixels) is based on the convention described by Kaye (1989b), whereby the box size used to estimate the perimeter should not exceed 0.3 times the maximum projection of the object. A checkerboard binary image provides half the number of boxes and the reverse of the binary image the other half. A binary representation of the object to be analysed is stored using a separate acquisition program.

Acquisition of the binary image involves digitizing the object to 256 grey levels and using the detect function to create a binary image of the object. The edit function may be used to remove detected artefacts or to correct imperfections in the detected object. In fractal analysis, the outline of the stored binary image is overlaid on the grid of boxes. A logical exclusive or (XOR) operation delineates the boxes containing the outline of the binary image (Fig. 2). For each box size the reverse binary image is also used. The number of boxes containing the outline of the binary image is proportional to the perimeter of the object (Chung et al., 1994; Fazzalari & Parkinson, 1996). This modified Richardson plot is the log of box size versus the log of the number of boxes, whereas the Richardson plot is the log of measuring unit versus the log of perimeter. The fractal dimension = D, where − D is the slope of the line on the modified Richardson plot, for a range of box sizes.

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Figure 2. . Box counting method on Q500MC, where the number of boxes containing the outline of the trabecular bone is proportional to the perimeter.

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Objects and specimens

Thirty histological sections of trabecular bone from two regions of the femoral head and one region of the iliac crest (IC), or subsets of this group, were used for the studies described below (Fig. 3). The samples from the femoral head were taken superior (S) and infero-medial (IM) to the fovea. The sections were impregnated with silver and counterstained with van Gieson's stain to give maximum contrast between the bone matrix and marrow spaces (Fig. 4). The tissue samples were all orientated in the same direction for sectioning and analysis. Only sections taken from the IM region of the femoral head, three with low bone volume (BV/TV) (< 10%), three with medium bone volume (10–20%) and three with high bone volume (> 20%) were selected for use in largest box size, detection, angle of presentation and magnification studies. Images of individual trabeculae were selected from these sections for use in the smallest box size study. For the plateau removal study and sectional fractal studies all 30 sections were used.

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Figure 3. . Composite radiograph of a femoral head and iliac crest, showing the sites of trabecular bone sampling.

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Figure 4. . Histological section of trabecular bone from superior to the fovea on the femoral head (1 pixel = 25 μm when digitized on image analyser).

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Smallest box size study

This study determined the smallest box size above which trabecular bone is effectively fractal. Images of four individual trabeculae from sections with low, medium and high BV/TV, at image analyser resolutions of 5.4 μm/pixel (2 × microscope objective) and 2.7 μm/pixel (4 × microscope objective), were used. At these resolutions on the image analyser, only portions of the trabecular structure were in the field of view. Therefore, only surface detail of individual trabeculae was resolved. The fractal dimension was estimated for box sizes 1–4, 2–5, 3–6, … , 10–13 pixels for the 2 × objective and 1–8, 2–9, 3–10, … , 10–17 pixels for the 4 × objective, where box size of 1 was the image analyser resolution. The box sizes at which the fractal dimension was greater than 1.0 indicated the lowest image analyser resolution at which trabecular bone is an effective fractal.

Largest box size study

This study determined the largest box size to be used in the estimation of the fractal dimension of trabecular bone. The standard errors of the regression coefficient, R, from regression analysis for 1–4, 5, 6, … , 181 pixels were estimated, where 1 pixel = 25 μm. The upper box size was determined to be the average box size closest to 181 pixels at which the standard error of R was at a minimum for the 30 histological sections.

Plateau removal study

This study determined the effect of removing plateaux from the modified Richardson plot. These plateaux are formed when, for a range of box sizes, the number of boxes counted is constant (Fig. 5). They cause a decrease in R for the regression analysis from which the fractal dimension is estimated. Ten sections taken from each of S, IM and IC sampled skeletal sites were used. The fractal dimension was estimated for box sizes of 1–170 pixels, where 1 pixel = 25 μm. The Richardson plot was modified using three methods of replacing the plateaux with a single data point. Firstly, only the first data point in each plateau was retained. Secondly, only the last data point in each plateau was retained. Thirdly, the data point that corresponded to the average box size in each plateau was retained. Comparison of the estimated fractal dimension was made between each of the three methods of plateau removal. This process was performed automatically, using a program written in Visual Basic for Microsoft® Excel.

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Figure 5. . Modified Richardson plot with standard error of multiple R overlaid.

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Detection study

This study determined the effect of grey-level detection (or threshold) level in the formation of the binary image on the fractal dimension. The optimal detection level was set using the ‘flicker method’ (Fisher, 1971) and then was set at ± 1–5 grey levels. The fractal dimension was estimated for the 11 detection level settings. The fractal dimension was estimated for box sizes of 1–170 pixels, where 1 pixel = 25 μm.

Angle of presentation study

This study determined the effect of object orientation, in relation to the box grid, on the fractal dimension. The object was rotated at 10 ° intervals from 0 to 170 °. The fractal dimension was estimated for the 18 angular intervals. The fractal dimension was estimated for box sizes of 1–170 pixels, where 1 pixel = 25 μm.

Magnification study

This study determined the effect of varying magnification on the fractal dimension. The initial resolution of the image analyser was 25 μm/pixel, where the image analyser resolution is defined as the pixel size on the image analyser. The magnification was decreased to 80, 60, 40 and 20% of the initial resolution and the fractal dimension estimated. The actual range over which the fractal dimension was estimated was 77–2652 μm for all magnification settings. Therefore, the range of box sizes for each magnification setting was different. This means that the magnitude of the estimated fractal dimension is not directly comparable with that estimated in the detection and angle of presentation studies.

Sectional fractal study

This study determined the ‘break points’, which define sectional fractal dimensions on the modified Richardson plot. Ten images of trabecular bone from each of two regions of the femoral head (S and IM) and from one region of the iliac crest (IC) were presented to the fractal analysis program on the Q500MC image analyser. A split-line regression analysis was performed on the modified Richardson plot, which relies on the assumption that there are three contiguous straight lines, which accurately describe the data, on the modified Richardson plot. The data are analysed using an automated algorithm, which yields the two data points indicating the transition from one straight line to the next. These are referred to as the ‘break points’ and are the same as the ‘pivot points’ described by Fazzalari & Parkinson (1997, 1998). Adoption of ‘break point’ terminology is consistent with the mathematical method of analysis.

The computational algorithm is a two-step process, where initial estimates enable a formal model to determine the final estimates. The initial estimates are obtained using a fitted linear regression written in FORTRAN using the IMSL module. This program is an iterative process, where the least squares estimate for every possible combination of ‘break points’ is calculated and the data points with the lowest standard error are selected. The initial estimates are applied to a maximum-likelihood model for computation of variance components using BMDP software. The model iterates until convergence occurs. The ‘break point’ estimates are then used to determine the three fractal dimensions (1, 2 and 3).

This process is automated using a combination of batch-file processing and Visual Basic for Microsoft® Excel programs to integrate the results from the FORTRAN program and the BMDP model.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The study which determined the smallest box size at which trabecular bone is effectively fractal shows that the range of box sizes where the estimated fractal dimension becomes greater than 1.00 is approximately 25 μm (Table 1). Therefore, 25 μm was selected as the lower limit at which trabecular bone is effectively fractal.

Table 1.  . Fractal dimension (mean ± SD) of individual trabeculae for a range of box sizes. * denotes statistically significant difference to 1.00.Thumbnail image of

The largest box size was determined as the box size at which the variation in the regression analysis of the modified Richardson plot was a minimum. This is illustrated in Fig. 5, which shows a modified Richardson plot for one section with the standard error of R from a progressive regression analysis superimposed. The largest box size was calculated as the mean box size of 30 sections of trabecular bone, where the standard error of R was a minimum, which was 170 pixels or 4250 μm.

The removal of plateaux from the modified Richardson plot minimizes variance of the regression analysis. The frequency and size of the plateaux increase as the box size increases (Fig. 5). Comparison of the three methods used to remove the plateaux shows no statistical differences between them in the estimation of the fractal dimension (Table 2). The method which selects the first point of each plateau was adopted.

Table 2.  . Fractal dimension (mean ± SD) after replacement of each plateau by retention of the first, the average and the last data points. Thumbnail image of

The studies that determined the effect of change to image analyser settings, namely detection and magnification and the angle of presentation, show that for these parameters there was no significant change in the estimate of fractal dimension (Tables 3 and 4, Fig. 6).

Table 3.  . Fractal dimension (mean ± SD) estimated at 1–5 grey levels above and below the optimal detection level of trabecular bone. Thumbnail image of
Table 4.  . Fractal dimension (mean ± SD) estimated with changing initial magnification for trabecular bone with low, medium and high BV/TV. Thumbnail image of
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Figure 6. . Polar plot of fractal dimension estimated at different angles of presentation. (● = low BV/TV, ▴ = medium BV/TV, ▪ = high BV/TV).

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The sectional fractal study provides an objective method for determining the ‘break points’ on the modified Richardson plot (Fig. 7). These ‘break points’ enable the sectional fractals to be calculated. There were three sectional fractals (D1, D2 and D3). The magnitudes of the sectional fractals were between 1.0 and 2.0 with D1 less than D2, which is less than D3 (Table 5).

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Figure 7. . Modified Richardson plot with three sectional fractal dimensions and two ‘break points’.

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Table 5.  . Sectional fractal dimensions (mean ± SD) and break points (mean ± SD) for trabecular bone from two regions of the femoral head (superior (S) and infero-medial (IM) to the fovea) and the iliac crest (IC). Thumbnail image of

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The results define a methodology for the standardized estimation of sectional fractal dimensions of histological sections of trabecular bone. In particular, the studies determine the lower and upper limits within which trabecular bone is effectively fractal. The effect of changes to image analyser settings and specimen orientation on the estimation of fractal dimension was determined. Also, an objective method for determining the ‘break points’, which define sectional fractal dimensions, is established.

The use of individual trabeculae in the study that determined the lower limit at which trabecular bone is effectively fractal allowed a close examination of the surface texture of the trabeculae. This enabled the smallest box size (25 μm) for fractal analysis to be set at a value which resolves the smallest features of interest. The largest box size (4250 μm) used for fractal analysis was determined to be the value above which variance in the data had increased above the minimum value and was the magnitude of the largest structure of interest.

The plateaux on the modified Richardson plot are present in all studies using a box counting method for fractal analysis. The loss of linearity in the data if they are not removed, significantly affects the estimation of the fractal dimension. Buczkowski et al. (1998) have described the mathematical convention whereby the first point in a plateau is equivalent to the average of the values in the plateau, and this method maintains the box size as an integer.

The detection study shows that with high contrast sections, such as silver-impregnated trabecular bone, the ‘flicker’ method of setting the grey-level threshold gives an accurate binary representation of the trabecular bone. The magnification study shows that, for the same range of box sizes (77–2652 μm), initial resolution has no effect on the estimation of the fractal dimension over the magnification range used in this study. The angle of presentation study shows that the architecture of trabecular bone is not sufficiently anisotropic for the angle of presentation to the image analyser to significantly affect the estimation of the fractal dimension. These studies mean that there does not have to be absolute control over how the sample is presented to the image analyser. However, as in all morphometric studies, it is best if a uniform approach is taken to minimize bias and random error in the measurements.

Using three straight-line segments (D1, D2 and D3) to describe the data of the modified Richardson plot is statistically more accurate than using one straight line. This is reflected in the values for the regression coefficient, R, which are greater for the sectional fractal dimensions than for a single fractal dimension. Previous studies by Fazzalari & Parkinson (1997, 1998) showed that two ‘break points’ correlate well with structural entities in the cancellous structure. The interpretation is that the three fractals (D1, D2 and D3) each describe a different compartment of the cancellous structure.

Image analyser detection level, initial resolution and angle of presentation do not significantly affect the estimation of the fractal dimension. However, in histomorphometric analysis of trabecular bone, estimation of bone perimeter differs according to the image resolution. The anisotropic nature of the cancellous structure means that the angle of presentation of the section to the video-scan direction of the image analyser affects the estimation of trabecular perimeter. Also, the setting of the grey level threshold has a significant effect on estimated trabecular perimeter (Fisher, 1971). Using the standard plate or rod models for cancellous structure, the perimeter estimate greatly influences histomorphometric parameters such as bone surface/total volume, trabecular thickness, trabecular separation and trabecular number (Parfitt et al., 1987). Therefore, this method for estimating the fractal dimension of trabecular bone, which is model independent, does not appear sensitive to suboptimal image analyser detection and has a major advantage over conventional techniques. That is, in contrast to histomorphometric analyses, it should be easier to compare results of studies between research laboratories and to reproduce such results.

The studies which investigate the modified Richardson plot enable an objective examination of the data. In many other published studies on fractal analysis, the range over which the modified Richardson plot is constructed seems to be arbitrary and no attempt has been made to set meaningful lower and upper limits. The increased scatter at large box sizes is commonly seen (Cross et al., 1994; Buczkowski et al., 1998). Minimization of the regression variance in the modified Richardson plot leads to more reliable analysis and fractal estimates.

Objective identification of sectional fractal dimensions by determining the ‘break points’ with a split-line regression method is novel and an improvement over the process involving subjective assessments. Previous studies (Fazzalari & Parkinson, 1997, 1998) determine ‘break points’ using subjective assessment which, while internally consistent, are subject to error, particularly if the assessor is inexperienced. Asymptotic fractal models, while applicable to box counting methods, are better suited to fractal analysis using Minkowski erosion/dilation algorithms. This approach does not make use of a priori knowledge of the material of interest, thus restricting interpretation of results to empirical terms. Other studies, reporting a single fractal dimension estimated from the ‘straightest’ segment of the modified Richardson plot, also make no consideration for the type of structure being examined (Keough et al., 1988; Cross et al., 1993).

The split-line regression analysis used for determination of ‘break points’ is very computer intensive as it calculates every combination of ‘break points’ to find the best fit. The time for this process is machine dependent: when using a Digital® Alpha 1000 the computation time was approximately 10 s per specimen. This objective method for ‘break point’ determination enables automation of the entire process of fractal analysis, with subsequent minimization of human error.

The magnitude of the ‘break points’ relates to the compartment of the cancellous structure which is described by each of the fractal dimensions. Fractal 1 (D1) is estimated using box sizes ranging from 25 μm to 150–350 μm, which encompasses the size range of resorption pits, as estimated in other studies (Ericksen et al., 1985; Palle et al., 1989; Yamaguchi et al., 1993). This fractal dimension describes the surface texture of individual trabeculae. Fractal 2 (D2) is estimated using box sizes ranging from 150–350 μm to 500–1000 μm. The lower and upper box sizes vary considerably; however, the range encompasses the size range of individual trabecular elements, as estimated in other studies (Weinstein & Hutson, 1987; Mosekilde, 1988; Moore et al., 1992). This fractal describes the shape or form of trabeculae. Fractal 3 (D3) is estimated using box sizes from 500–1000 μm to 4250 μm, which encompasses several structural units of the cancellous architecture. This fractal describes the overall spatial arrangement of cancellous bone.

The results of this study compare closely with other reported studies describing sectional fractal dimensions for cancellous bone. However, the methodology of Fazzalari & Parkinson (1997, 1998) used a much lower sampling density (15 box sizes over a range of 15 μm to 4500 μm) than this study. Also, the ‘break points’ on the modified Richardson plot were subjectively assessed. Histomorphometric parameters, as estimated by other workers, can exhibit large variability, which may be due to differences in the site of sampling or whether the subjects are diseased or normal. For example, resorption pit magnitude has been estimated at 15 μm to 100 μm in depth and 50 μm to 200 μm in linear extent (Ericksen et al., 1985; Yamaguchi et al., 1993). Trabecular separation ranges from approximately 300 μm to 1000 μm (Moore et al., 1992). These examples illustrate that while ‘break points’ are indicative of the transition between ‘compartments’ of the structure, they are influenced by the dynamics of cancellous bone architecture.

The box counting technique from which the modified Richardson plot is constructed is easily performed by any commercial PC-based image analyser. However, it is the manipulation and analysis of the regression line to produce three fractals which is novel in this study. The level of automation and reproducibility through an objective process provides a tool for comparative studies of subjects afflicted with diseases affecting trabecular bone structure at different sites throughout the skeleton.

The identification of sectional fractal dimensions, which describe distinct morphological entities within the cancellous structure, enables the effect of the cellular processes which drive changes in the structure to be studied (Fazzalari & Parkinson, 1997, 1998). Sectional fractals provide an index of complexity of known entities of bone structure, namely surface texture, trabecular shape or form and overall structure. A single fractal dimension calculated over the entire range does not relate as easily to the biology or structure of trabecular bone.

The studies described in this paper to determine a stable and reproducible methodology for fractal analysis of trabecular bone show that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than one fractal dimension describing spatial structural entities.

Unlike conventional histomorphometry, fractal analysis is a model-independent method for describing a complex multifaceted structure using an objective methodology within clearly defined operational parameters. Although this study describes a method specifically for trabecular bone, it can be adapted to the fractal analysis of other biological systems. This may be at the cell, tissue or organ level, giving insight to complex processes. Conventional histomorphometric and stereological techniques would be complemented by fractal analysis.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The authors would like to thank Phil Leppard, Department of Statistics, University of Adelaide, for assistance with automation of split-line regression.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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