Characterization of the Spatial Arrangement of Secondary Osteons in the Diaphysis of Equine and Canine Long Bones



The blood supply of bone cells in compact bone is provided primarily by blood vessels located within Haversian canals forming the centre of osteons. Mid-diaphysial cross-sections of radii and third metacarpal bones from two horses and radii from two mature dogs were studied using reflective light microscopy to quantify the spatial ordering of canals and compared to a computational model. The distributions of canals were analyzed using: 1) the autocorrelation function (ACF), which describes the probability of finding two canals separated by a given distance and 2) the shortest distance distribution (SDD), which describes the probability that a site within bone is located at a given distance from the nearest canal. The order in the investigated horse radii, as characterized by the oscillations of the ACF, was found to be independent of the anatomical location although, in the metacarpal bone the order was higher in the lateral than in the cranial location. Among the dogs, marked differences were only found in the ACF. An analysis of the SDD demonstrates that ordering of canals minimizes the distance of osteocytes from a blood vessel. This suggests that the efficiency of the blood supply can be adapted through differences in the order of the Haversian canals. In our model, the ordering of canals is achieved via an exclusion zone around each canal, imposed upon newly formed osteons. Simulations demonstrate that differences in the observed order can be explained by either a larger size or a larger variability of this exclusion zone. Anat Rec,, 2011. © 2011 Wiley-Liss, Inc.


In the postnatal period, the cortex of long bones of horses and dogs consists primarily of fibrolamellar and primary osteonal bone (Stover et al., 1992; Currey, 2002; Locke, 2004). Throughout life, the bones of the skeleton continuously undergo modeling and remodeling to respond adaptively to the loads that are imposed on them (Currey, 2002). Considering compact bone, modeling refers to changes in the gross shape of the bone, occurring by removal and accretion of material to the external (periosteal or endosteal) surfaces of the bone. Remodeling, on the other hand, is responsible for changes in the internal architecture of the bone (Robling et al., 2006; Fratzl and Weinkamer, 2007; Chen et al., 2010). The most likely function of this process is to remove parts of the bone which have accumulated excessive damage. In the remodeling process of compact bone, osteoclasts resorb bone, creating a “hole” of a diameter of about 200 μm. In the wake of the resorption process, osteoblasts are recruited to the area and begin to deposit concentric layers of bone lamellae from the periphery inwards (Parfitt, 1994). The filling process is incomplete, leaving a central channel which contains a blood vessel that provides nutrients to the bone cells distributed in the lamellae, and allows removal of their metabolic by-products. The bone cells are interconnected by a network of small canals (canaliculi). The entire structure thus formed is termed secondary osteon. The secondary osteon has a distinct outer sheath, called the cement line (Currey, 2002). Its location marks the position of the border of the erosion caused by the osteoclasts. Secondary osteons are the predominant bone type found in the cortex of bones of mature large mammals like horses, dogs and man.

Secondary osteons affect the mechanical properties of bones, in particular by contributing to their resistance to failure (so-called toughening) as developing microcracks generally do not cross cement lines and are deflected as they reach them (Martin et al., 1998). Although several toughening mechanisms have been proposed for cortical bone (Peterlik et al., 2006), cement sheath deflection seems to be one of the most potent of these (O'Brien et al., 2007; Koester et al., 2008).

The distribution of secondary osteons has been intensively investigated with the aim of linking structural features with the loading history of the bone, on the one hand, and with the turnover rates and the age of the bone, on the other. The later is of particular interest for anthropologists and forensic examinations (Frost, 1987; Burr, 1992; Stout and Paine, 1992; Havill, 2004). Quantities that are studied comprise the secondary osteon population density (N.On/Ar), the fractional area of secondary bone (On.B/Ar), and the mean osteon area (On.Ar). Investigation of the calcaneus of the adult mule deer showed that the part of the cortex under compression or tension has an increased osteon population density, N.On/Ar, compared with the parts of the cortex that experience lower strains. Also, the fractional area of secondary bone was found to be greater in the areas of the cortex that receive more deleterious loading conditions. (Skedros et al., 1994, 2001). In the calcanei, significantly smaller osteons were found in the highly strained part of cortex (Skedros et al., 2001), a result also obtained theoretically using a mechano-biological model of bone remodeling (van Oers et al., 2008). However, this relation was not found in the equine third metacarpal and radius (Mason et al., 1995). Another study found that the size of an osteon in a human rib was found to be positively correlated with the size of its Haversian canal (Qiu et al., 2003).

Counting of osteons can be refined by focusing on specific osteons. Newly formed osteons are defined as osteons, which are still poorly mineralized, as seen in backscattered electron images (Skedros et al., 2001). In the mature cortex of the deer's calcaneus the cortices under tension showed a greater number of newly formed osteons compared with the cortices under compression (Skedros et al., 2001; Skedros et al., 2004). A second class of osteons is that of “atypical” osteons, which include osteons with a noncentrally located canal, with a strongly elongated canal or even including more than one canal. Investigations on horse radii and metacarpals amongst other bones could not demonstrate a consistent relation between the frequency of atypical osteons and the loading mode (compression, tension, shear) of the bone (Skedros et al., 2007). A promising approach to connect loading and osteon morphology is the use of circularly polarized light images, which gives information about the predominant collagen fiber orientation (Skedros et al., 2009).

The aim of this work is to go beyond simple counting of osteons and their classification within an area of cortical bone, by quantifying their spatial arrangement. The osteonal arrangement is obviously not completely random, but contains a significant amount of short-range order (Bell et al., 2001). Considering the location of the osteon to be marked by its Haversian canal, this order is quantified using the autocorrelation function (ACF). The ACF gives the probability of finding two Haversian canals as a function of distance. A related parameter, the shortest distance distribution (SDD), can also be used. In addition, the SDD provides the distribution of distances of locations within the bone from their nearest Haversian canal. The SDD is of interest as it gives an indication of how well the osteocytes are nutritionally supplied. The osteonal arrangement in different anatomical locations in two different types of bones in horses (radii, third metacarpals) and one type of bone in dogs (radii) were analyzed. These bones are often selected for structural analysis as they represent two different loading types: the radius is loaded habitually in bending while the third metacarpal bone of horses is loaded mostly in torsion (Skedros et al., 2006; Skedros et al., 2007). With this analysis, anatomical locations with higher/lower osteonal order within specific bones could be identified. Using a simple computer model the hypothesis was tested whether an “effective repulsion” between Haversian canals can explain the observed order. In the model, each Haversian canal is surrounded by an “exclusion zone,” within which the formation of a new canal is prevented. With only two parameters, describing the size of the exclusion zone and their variability, the different amount of order could be explained. The combination of experimental observations and theoretical simulations of the quantification of the spatial distribution of the Haversian canals in cortical bone presented here, provides insight into the “laws” governing the process of remodeling in cortical bone.


Bone Samples

The left radii were harvested from two horses that died at the Koret Veterinary Teaching Hospital (Hebrew University of Jerusalem) for reasons unrelated to the musculoskeletal system. In addition, the third metacarpal bone was also harvested from the second horse. The horses were an intact male Arabian 6-year old horse and a 9-year old neutered male horse of local mixed breed. At these ages, equine bones are considered mature, as their growth plates are closed and their growth in length has presumably ceased. Both the horses were pleasure-riding horses, kept mostly in stalls and fed barley hay and concentrated feed. The left radii were also obtained from two dogs that had been euthanized at animal shelters due to population control regulations, and tissue samples were released for research purposes. The dogs were mature and of mixed breed, and weighed 27 and 32 kg, respectively.

All bone samples were fresh, harvested shortly after the death of the animals, and were not fixed. The bones were wrapped in saline-soaked gauze pads and stored at −20°C. Before further processing, each bone was slowly thawed to room temperature. All external soft tissues were removed from the selected bones, and a 40-mm long section was cut off from the mid-diaphysis of each bone using a hand saw. The type of bone in all samples was determined by examining thin and polished transverse cross-sections by light microscopy, and was found to be comprised almost entirely of secondary osteons. A thin (500-μm thick) transverse slice was cut off from the mid-diaphyseal region of each bone using a low-speed water-cooled diamond saw (Buehler Isomet low speed saw) with a diamond wafering blade. All sections were ground and polished (Buehler Minimet Polisher). The orientation of the slices was marked, so that their anatomical locations (medial, lateral, cranial and caudal) could always be identified. We did not adhere strictly to anatomical nomenclature in this manuscript to avoid confusion, thus for instance for the metacarpal bone we use the terms “cranial” and “caudal” instead of “dorsal” and “palmar.”


All polished bone sections were examined with reflective light microscopy (Leica DM 700 digital microscope and Olympus BX 51 microscope). Images were captured by dedicated digital cameras (Leica and Olympus-DP71) (Fig. 1, left). Two different resolutions were chosen, covering a sample area of 5.8 × 4.35 mm at low resolution and 1.415 × 1.06 mm at high resolution. At high resolution, structures of the size and shape of Haversian canals are more precisely defined. The larger area captured in the low resolution images is statistically favorable in the analysis of the osteonal arrangement. High resolution and low resolution images were saved in TIFF format (1,280 × 960 and 1,230 × 890 pixels) and transferred to ImageJ (NIH) for analysis. For each anatomical position of the horse radii and metacarpal bone, three images of low resolution and three images of high resolution were analyzed. For the dogs, six images with the high resolution were included in each analysis.

Figure 1.

Left: Polished cross-sections of the cortical bone of equine radius lateral position (top), and of canine radius caudal position (bottom) using reflective light microscopy. The scale bars denote 250 μm for the horse bone, and 200 μm for the dog bone. On the right, the corresponding cleaned images showing only the Haversian canals in black.

Image Analysis

All light microscopy images were converted into binary images, in which the Haversian canals are marked in black, whereas the rest of the sample provides a white background. In the process of creating the binary image, the Haversian canals and osteocyte lacunae were distinguished automatically based on their size and circularity using the “particle analysis” tool in ImageJ. Remaining artefacts had to be removed by hand to obtain a “cleaned” image (Fig. 1, right). This binary image was then analyzed to obtain basic morphometric parameters such as canal number (N.Ca), and canal area (Ca.Ar). The images were further analyzed to determine more advanced parameters, characterizing the spatial arrangement of the canals (see for details below), using a custom-written code (Matlab, version 7.30).

Quantities to Characterize Osteonal Arrangement


The ACF is a standard function to describe order in a structure with two phases (Torquato, 2002), for example, objects dispersed in a matrix. The ACF has been used to analyze biological structures such as the arrangement of tubules in dentin (Kinney et al., 2001) and capillaries in the prostate gland (Mattfeldt et al., 2006). In this article, cortical bone is considered as a two-phase material consisting of Haversian canals (depicted black) and bone material (depicted white, including lamellae, osteocyte lacunae, cement lines, etc.). Based on the binarized image (Fig. 1, right) the two-point correlation function, S2(r), is calculated, which quantifies the probability to find two black pixels in the image at a distance r. By subtracting a constant term the ACF is obtained,

equation image(1)

where ϕ denotes the area fraction of the canals. This subtraction has the convenient consequence that the ACF approaches zero for large values of r indicating the loss of spatial correlation over large distances.

Figure 2 shows three instructive examples of extremes in terms of order of circular canals depicted in black (Fig. 2, middle row) and their corresponding ACFs (Fig. 2, top). All three arrangements contain the same number (and size) of canals, but they differ clearly in their arrangement, ranging from perfect (quadratic) order (left) to a completely random arrangement of the canals (right). The central image is an actual image of cortical equine bone, where the canals were replaced by circles of uniform size. This replacement was performed in all binarized images, as our analysis concentrates on the spatial organisation of the canals (i.e., the structure factor) and not on the effect of the different sizes and shapes of the canals (i.e., the form factor). For small distances, the ACF is almost identical for all cases, since only information about the average size and shape of the objects is rendered. The location of the first zero of the ACF is related to the average size of the canals. The first minimum gives the distance at which it is least likely to find another neighbouring canal. For a random arrangement, the ACF becomes zero for values larger than the diameter of the canals, corresponding to a lack of correlation between the positions of the canals (Fig. 2, top row, right). Deviations from zero are only due to the finite size of the image, which is comparable to the size of the experimental images. The ACF for the perfectly ordered arrangement of canals displays clear oscillations with well-defined peaks. The position of the first peak corresponds to the smallest distance between the centres of two canals. Likewise, the position of the second peak, third peak and so forth correspond to the distance between the second nearest canals, third nearest canals, and so forth. The partial order in the arrangement of Haversian canals is reflected in the oscillations of the ACF (Fig. 2, top row, centre). The oscillation dampens out rather quickly, showing only two clear peaks, which suggests that the order in the osteonal arrangement is relatively short-ranged, over a distance of ∼500 μm.

Figure 2.

ACFs (upper row) and shortest distance maps (lower row) for three types of images (middle row). All three images have the same area fraction of black circles. The left column describes a perfectly periodic organization of the black circles, while the right column refers to a completely random distribution. The middle column refers to the arrangement of Haversian canals as observed in an equine image, where the canals are replaced by black circles of uniform size. The shortest distance maps are used to calculate the SDDs between the Haversian canals and the surrounding bone.


The second function used to investigate the structural organization of osteons is the SDD. The SDD describes how far away a region of bone material is from its nearest Haversian canal. As the blood vessel within the canal serves as source of nutrients for bone tissue, the SDD has a direct physiological significance; it reflects the efficiency of the blood supply to the bone. The starting point of the calculation of the SDD is a shortest distance map (Fig. 2, lower row), which gives the minimum distance for each bone pixel to the nearest surface of a Haversian canal. The SDD is then the frequency distribution (or histogram) of distances found within such an image (see examples in Fig. 8).

Modeling the Osteonal Arrangement

To evaluate our basic hypothesis, that the observed two-dimensional order of the osteonal arrangement is the result of an “effective repulsion” between Haversian canals, we created a simple model of the process of Haversian canal ordering. The model was governed by the following rule: each added Haversian canal (osteon) restricts the area allowable for the formation of the next Haversian canal by forming a circular exclusion area around each of the existing canals, with exclusion radius, Rexcl [this is referred to as the cherry-pit model in the physics literature (Torquato, 2002)].

The starting point of the simulation is an empty image with dimensions slightly larger than those of the experimental images (to allow placing canals also at the boundary of the image). Only a region of interest identical to that of the experimental images was then used for the analysis. New canals were created sequentially using an algorithm of three steps. In a first step, a potential position of a new Haversian canal was chosen at random. In a second step, the test was performed, whether the new canal would violate the exclusion radius criterion. In a third step, the algorithm offers now two possibilities. When the randomly selected position of the added canal did not lead to an overlap of the exclusion zones of any of the existing canals (Fig. 3, Case B) it was added to the simulation image. In the case of an overlap between the exclusion zones (Case A), the trial was considered unsuccessful and an alternate position for a new canal was again chosen randomly. This random sequential addition (RSA) (Torquato, 2002) process was continued till the appropriate density of canals (determined from an average of the experimental images) was obtained. Note that a choice of a too large an exclusion radius would make it impossible to reach the designated density of osteons. The model was intended to produce an arrangement of osteons with some order, but not to precisely mimic the remodeling process. Therefore, removal of old osteons by the formation of new ones was not included in the model.

Figure 3.

A sketch of the RSA of osteons highlighting the process of adding a new osteon In the model an osteon is characterized by the size of its Haversian canal (marked in black), Rca, and by its exclusion zone (gray) given by the exclusion radius, Rexcl. A: An unsuccessful addition occurs if the exclusion zones of two osteons overlap. B: A successful addition occurs in the simulation if there is no overlap of the exclusion zones.

As model parameters, the density of canals and the total area covered by the canals were chosen to be the same as those in the experimental images to allow a direct comparison with experimental results. With such a model it is possible to investigate how varying the “repulsion” between osteons (by choosing larger or smaller exclusion radii) influences their order. A certain variability of the exclusion radius was introduced by assuming its values to be normally distributed, with the standard deviation of this distribution being a second parameter of the model. Figure 4 shows the influence of the two parameters of the normal distribution, the mean (exclusion radius) μ and its standard deviation σ, on the osteonal order defined by the ACF. An increase in μ (Fig. 4, left) increases also the order, as demonstrated by the deep first minimum and clear first maximum of the ACF. An increase in σ (Fig. 4, right) reduces the order. In addition, the slope between the first minimum and first maximum distinctly increases as σ increases. To improve statistics, simulation results were averaged over multiple runs.

Figure 4.

ACFs of simulated images of the arrangement of Haversian canals. The exclusion radii of the osteons placed in the image following a RSA process are assumed normally distributed, with mean μ und standard deviation σ. The influence on the ACF function is different in the case of constant σ and different values of μ (left), or constant μ and different values of σ (right).


Table 1 reports basic morphometric parameters characterizing the Haversian canals in the samples used in this study. The parameters used are the mean canal area (Ca.Ar), the canal number density (N.Ca/Ar) and the canal area fraction. Ca.Ar was determined by evaluating all the canals of the high-resolution images, while N.Ca/Ar was determined from all the images used. The cross-sectional area of the Haversian canals in the horse radius is larger than in the horse metacarpal by about 30%, and substantially larger (by a factor of 6×) than canals in the radius of dog A. The large standard deviations show that the Ca.Ar is not a tightly controlled quantity. For the canal number density, the largest values are found in dogs, and are about three times larger than those in horses. This kind of compensation for smaller canals by increasing their numbers has the effect that the area fraction covered by canals becomes similar in dogs and horses (e.g., at the lateral position of the horse metacarpals and the radius of dog B).

Table 1. Basic morphometric quantities for the canal areas and number in the investigated equine radii and metacarpal bone, and dog radii
 Canal area Ca.Ar [μm2]Canal number density N.Ca/Ar [mm−2] 
 MedianMeanStd. devStd. errMedianMeanStd. devArea of canal (%)
  1. Medians and means are given to show that Ca.Ar is not normally distributed.

Horse radius medial1995.82279.11412.471.122.6022.611.184.51
Horse radius cranial2207.42621.01831.972.122.8922.861.965.05
Horse radius lateral2173.82486.61624.281.522.1622.381.674.82
Horse metacarpal cranial1692.82057.61404.666.225.7825.722.184.36
Horse metacarpal lateral1529.31873.71237.760.023.8323.890.693.64
Dog A radius319.7420.4320.911.768.0768.346.842.18
Dog B radius577.0699.6587.222.265.0564.043.863.75

Figure 5 presents the ACF calculated for a region located in the lateral location of the metacarpus of the horse and the radius of dog B. It demonstrates that the ACF provides information about the arrangement of Haversian canals that is independent of their number density. Although the canal density in dogs and horses is quite different (Table 1), their ACF is similar. The same statement holds when looking at different anatomical positions in the radius of the horse (Fig. 6, top). For all three investigated locations—medial, cranial, and lateral—the obtained ACFs are statistically indistinguishable considering the error bars of the measurement. However, the analysis of the metacarpal of the horse showed clear differences (Fig. 6, bottom). For the cranial and the lateral location the first minimum of the ACF is almost identical. The slope of the ACF from the first minimum to the following maximum is steeper for the cranial cortex than for the lateral cortex, resulting in a lower first maximum occurring at a smaller distance.

Figure 5.

Comparison of the ACFs of the lateral region of the equine metacarpal and the radius of dog A. To make the ACFs of horses and dogs comparable, the real Haversian canals were replaced by circular canals of equal size in the images before the ACF was calculated. The error bars denote the standard deviation.

Figure 6.

Top: Comparison of the ACFs of the equine radius at different anatomical positions. Bottom: Comparison of the ACFs at cranial and lateral positions of the metacarpal of the horse. As reference the ACF of the equine radius (data averaged over all three anatomical locations plotted above) is shown (blue).

Evaluation of the ACF for the two investigated dogs showed differences, despite the rather large error bars (due to the higher resolution of the images of dogs, which resulted in a smaller number of canals) The most prominent differences occur in the slope between the first minimum and first maximum (Fig. 7). Dog A demonstrates a less ordered osteonal arrangement than dog B.

Figure 7.

Comparison of the ACFs of two dogs averaged over all anatomical directions. The larger error bars are due to the higher resolution of the images used for the calculation of the ACF.

The analysis of the SDD of the different cortical bones supports the results for the ACF. For the radius of the horses, the SDDs were found to be similar for all anatomical locations and therefore only the averaged result over all locations is shown (Fig. 8, top). Also, the differences between horse radius and dog radius based on the SDD are smaller than the corresponding error bars. The larger order in the osteonal arrangement of the lateral part of the horse metacarpal compared with the cranial part is reflected by a slightly narrower SDD (Fig. 8, bottom). The SDDs cross each other at about 120 μm. Compared with the cranial cortical bone, the bone material in the lateral location is on average closer to a blood vessel than this distance. All the experimentally obtained SDDs of the osteonal arrangement lie between the extreme cases of a fully ordered quadratic arrangement and a random arrangement of the canals (Fig. 8, top).

Figure 8.

Comparison of the SDDs of the arrangement of Haversian canals in the investigated long bones of dogs and horses. Top: SDDs of dogs and horses in comparison with the SDD of a fully ordered (dashed grey line) and a random arrangement (continuous grey line) of Haversian canals (see Fig. 2). Bottom: SDDs at different anatomical locations of the horse metacarpal bone in comparison with the averaged SDD of the horse radius.

To interpret the differences in the ordered arrangement of Haversian canals as reflected in the ACFs (e.g., Fig. 6) and SDDs, we performed numerical simulations with the theoretical model described in the Methods section. Experimental results were compared with those obtained by analyzing simulated images, where the average exclusion radius and the standard deviation of its distribution were varied. A good agreement of the simulated ACF and the ACF of the horse, averaged over all three locations (medial, lateral, and cranial), was obtained when an exclusion radius of roughly three times the canal radius (R.Ca) was chosen, with a standard deviation of the exclusion radii of 1.4 × R.Ca. (Fig. 9, top). The experimental ACF of the lateral cortex of the metacarpal bone (Fig. 9, bottom), with its steep slope of the ACF between the first minimum and maximum, could be simulated when the exclusion radius was left unchanged (Rexcl = 2.9 R.Ca) in comparison to the radius, but by reducing the standard deviation to σ = 0.58 R.Ca. The simulation, however, predicts a slightly deeper first minimum than was measured experimentally. Finally, the best agreement with the experimental ACF for the cranial cortex of the metacarpal (Fig. 9, middle) is obtained by choosing the same small standard deviation as for the lateral cortex (σ = 0.58), but with an exclusion radius of only 2.18 R.Ca. The simulated images were also evaluated with respect to the SDDs. With the same values of μ and σ as in Fig. 9 good agreement of the experimental (Fig. 8) and simulated SDDs was achieved (result not shown).

Figure 9.

Comparison of experimental and simulated ACFs. In each plot the experimental ACF is in black and error bars denote the standard deviation. The simulated ACF that fits best the experimental ACF is plotted as a continuous line, whereas the two dashed lines correspond to the two other simulated ACF given as reference.


Our analysis of microscopic images of compact bone of horses and dogs detected in a quantitative way the order in the arrangement of Haversian canals. Introducing order into canal arrangement has the physiological advantage that the probability of an osteocyte to be far away from a blood vessel is clearly reduced, compared with a random arrangement of canals (Fig. 8). The different amounts of order that are observed in the equine metacarpal at different anatomical locations suggest an adaptive principle to be active. To interpret the order in the arrangement of Haversian canals, only in terms of an improved nutritional supply by reducing the distance nutrients must cross to be delivered to the bone tissue, is a clear oversimplification. It ignores the role of oscillatory loading of the bone on the interstitial fluid flow through the osteon. This mechanism is strongly dependent on the external “habitual” loading and the anatomical position within a given bone. The resulting time-dependent strain gradients in the cortex strongly influence the interstitial fluid flow.

From the investigated bones, the radius is exposed to rather simple loading under bending (Mason et al., 1995). Major variations in circumferential strain gradients have been measured in radii of adult roosters (Judex et al., 1997). This would imply that a great variation in osteon arrangement would be expected to be seen in different anatomical locations of the radius. However, our finding that the osteonal order in the investigated horse radius is independent of anatomic location does not support this hypothesis. The second investigated bone, the metacarpal of horses, has been shown to be loaded primarily in torsion (Skedros et al., 2006; Skedros et al., 2007). In our analysis, the metacarpal bone displays higher order at the lateral compared with the cranial location. However, this observed anisotropy of the ACFs in the metacarpal bones cannot be explained based on purely torsional loading. It has been reported that some third metacarpals have a loading history of combined torsion with bending (Skedros et al., 2006), which would account for loss of rotational asymmetry.

Our results therefore suggest that the ordering of Haversian canals is relatively independent of the nonuniformity of the strain gradients of the radii. This observation coincides with that of the Skedros et al. (Skedros et al., 2005) who showed no correlation between the spatial distributions of osteocytes with loading history. This of course does not rule out the possibility that time-dependent strain gradients influence the architectural arrangement of the bone in ways not measured in this study, for example in the relationship between diameters of the Haversian canal and the complete osteon.

The theoretical model illustrates how the observed order in the canal arrangement can be explained through the introduction of an exclusion zone. Interestingly, only poor agreement could be obtained between experiment and simulation when assuming a single value of the exclusion radius. It was necessary to introduce some degree of variability into the model by assuming a normal distribution of the exclusion radii. The physiological interpretation of the exclusion radius would be that a new, secondary osteon could be created in an area as long as this area is not already well supplied by an existing osteon (i.e., within the so-called exclusion radius). Such a mechanism could produce a partly ordered arrangement of canals in a self-organized way. A simple biological mechanism for creating such an exclusion zone would be an inhibitory signalling factor, which migrates into the bone from the canal and would therefore exhibit a decreasing concentration gradient as distance from the canal increases. Such factors have been proposed already in bone, for example the effect of sclerostin in controlling the final size of the Haversian canal (Power et al., 2010). A larger exclusion radius would correspond to a stronger source of the inhibitory factor at the canal. The variability of the exclusion zone could be partly the result of the large variability in the size of the canals. Other factors that are not included in our simple model could also modify the size of the exclusion zone, for instance, differences in osteon size. The area of osteon pull-out was shown to be different between cranial and lateral cortices of equine third metacarpals (Hiller et al., 2003). Although the size of the osteon is determined by the action of osteoclasts, the diameter of the canal results from the infilling of the osteoblasts. The model does not account for any interplay between the different bone cells. Also not considered in the model is the arrangement of the osteocyte canaliculi, which affects the efficiency of interstitial flow through the tissue (Mishra and Moshe-Tate, 2003). A further complication is introduced by the effect of age. Older bone could contain a reduced number of viable osteocytes, which have to be supplied with nutrients.

Based on this model, the drawback of introducing too much order in the canal arrangement by increasing the exclusion radius is that it becomes more and more difficult to find an allowed region for creating a new osteon. Similar self-organising processes on larger scales leading to patterning of a collection of organisms are observed in many other systems in nature. The observed order can be explained by a combination of attractive and inhibitory factors. Examples include the patterning seen in vegetation in arid regions (von Hardenberg et al., 2001), of tree density in boreal forest landscapes (Pastor et al., 1999), the self-organisation of nests of the African fish tilapia (Barlow, 1974) and in the pattern formation within mussel beds (van de Koppel et al., 2008).

The main limitation of our analysis of the spatial arrangement of osteons is the rather small number of samples. Therefore, we have to abstain from general conclusions about the order of Haversian canals in different bones and different animals. To draw specific firm conclusions about intraindividual, interindividual and interspecies differences, more samples have to be analyzed. Only then can a conclusive connection between the canal arrangement and the local loading of this region of compact bone be made. In the analysis, the limiting factor is the binarization and subsequent cleaning of the microscopic images (Fig. 1), which has to be done partly by hand. Automating this process is a challenging computational problem.

The limitations of our computational model are, first, that it is two-dimensional. Second, once Haversian canals are placed in the model, they are fixed and cannot be removed by further remodeling events. This could readily be tested by running time dependent simulations, which introduce more realistic dynamics of bone remodeling, to see how the ordering of Haversian canals varies with time.

In this article, we have investigated the spatial ordering of Haversian canals within cortical bones from dogs and horses using a combination of image analysis and computer simulation. It was found that the Haversian canals are partially ordered in both species. The order seen could be reproduced by our computer simulations by using an exclusion radius in which the new osteons are unlikely to be formed in the close vicinity of another one. The proposed analysis method shows an approach of how static images of bone and its osteonal arrangement can be used to obtain information about how the remodeling process is controlled.