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ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
Jacques Soddell and Robert Seviour
Biotechnology Research Centre
La Trobe University, Bendigo
PO Box 199, Bendigo Vic 3550, Australia
Email: J.Soddell@bendigo.latrobe.edu.au
When filamentous micro-organisms like actinomycetes and fungi grow on solid substrates, they usually grow as a complex branching mycelium producing round colonies with filamentous edges. The shape and size of these colonies is determined by factors such as branching frequency and branching angles, and these may be influenced by growth rate and nutrient availability, as well as being strain specific. Some actinomycetes produce mycelium that fragments readily, resulting in colonies that are less branched, possibly containing smooth entire edges. The potential for using fractal geometry to describe these complex growth and branching patterns has recently been explored [1, 2, 3, 4, 5, 6, 7, 8, 9].
Figure 1: Fourteen colonies of Nocardia otitidiscaviarum
arranged in order of increasing
size and complexity determined subjectively before their fractal
dimensions were analysed.
Fractal dimension has been used to supply information about underlying mechanisms involved in the development of fungal mycelia [2, 5], in particular in trying to understand spatial relationships which develop between fungi and their substrata, with the value of D reflecting the organisms' compromise between exploration and exploitation strategies. D was calculated using the mass of fungal hyphae within different radii of growing colonies [2, 5] and concluded that the processes controlling branching and the distribution of mass of the hyphae were independent, and that fractal analysis offers a potentially powerful tool for understanding the morphological response of fungi to nutrient status [5].
However, measurement of fractal dimension has been more commonly used to provide a quantitative description of changes in the shape of developing colonies of fungi and actinomycetes [1, 4]. Box counting techniques showed that the fungus Sordaria macrospora grew only as a mass fractal, where box mass and box surface dimensions were equal (Obert et al. 1991), but the fungus, Ashbya gossypii showed a transition from mass fractal during early growth to a surface fractal at later stages of colony development where values for box surface and box mass dimensions differ [1] due to overlap of growing hyphae at the centre of the colony (that is, only the edge is fractal). This work suggested that the fractal dimension is an important morphological measurement of microbial growth and capable of detecting different growth behaviour.
This present study follows an earlier one comparing different methods of D determination in actinomycetes [9], where we showed that box counting was superior to other methods for describing changes of colony structure during growth of three actinomycetes with different space-filling properties. However, one major problem was the time-consuming nature of the data collection, which involved obtaining co-ordinates of outlines of colonies. Therefore, to make box counting a more attractive tool for describing complex changes in morphology under different environmental conditions, faster procedures are necessary. This may be achieved by applying box counting directly to images, removing the need for generating co-ordinates of colony outlines. In this paper, we re-analyse colonies used in the earlier study [9] to determine if the box counting technique is still suitable, and to evaluate and compare two computer programs (FDC and Image Fractal) that allow box counting of images.
Outlines of colonies of Nocardia otitidiscaviarum, Nocardia asteroides and Rhodococcus erythropolis were obtained as previously described [9], recaptured using the JAVA image analysis system (Jandel Scientific) on a 386 based microcomputer, and saved in the TIFF format. These images were then manipulated using Image Fractal to produce either black outlines of colonies for box surface dimension or black-filled images for box mass dimension calculation, and saved as PICT files on a Macintosh Quadra 450. These PICT files were then analysed using Image Fractal (available by anonymous ftp from zippy.nimh.nih.gov) and fractal dimension calculator (FDC) (available from Paul Bourke at pdbourke@ccu1.aukuni.ac.nz) on a Macintosh Quadra 450. We examined 14 colonies of Nocardia otitidiscaviarum (Figure 1) which were previously shown to exhibit considerable variation of shape during growth [9], as well as a single large colony of Nocardia asteroides and Rhodococcus erythropolis with quite different shapes (Figure 2).
Figure 2: Comparing developed colonies of three nocardioform bacteria showing different space-filling
properties. All colonies, although not fully developed, are of similar size.
Two important aspects of fractal dimension calculation are the choice of box size and the positioning of the grid over an object [10]. Image Fractal offers only one starting position for the grid, and one default set of box sizes (2, 4, 8, 16, 32, 64 and 128 pixels wide), with no ability to change these. FDC also offers a default set of box sizes, determined automatically according to an algorithm based on the size of the image, and a default set of randomly chosen offsets for positioning the grid. However, there is an option to substitute any box size, although this is limited to a maximum of 12 sizes in a single run. FDC also allows the user to select the number of "offsets" for each box size to allow for placement of the grid at different starting positions, and to also carry out an exhaustive count with every possible starting position. We compared box surface and box mass dimensions obtained with Image Fractal to those obtained with FDC using various options. (1) The same box sizes as used by Image Fractal with only one starting position (offset). (2) This was repeated with 10 offsets, and (3) the exhaustive option where all possible offsets were used. (4) FDC's default step sizes and offsets, which are chosen by the software to reflect the size of the image. Although box sizes below 4 were possible, counts obtained with these were not used because these points often deviated from the straight line required in the log-log curve. Box sizes above 25% of the size of the colonies were also not used because they give poor results [10]. Control images of known fractals were also tested to validate these methods.
Two different fractal dimensions were calculated - box mass dimension, in which all boxes covering the colony were counted; and box surface dimension, in which only boxes covering the perimeter of the colony were counted. Both programs provide output files containing box sizes and the corresponding counts. These were imported into Sigma Plot (Jandel Scientific), and graphed as log (box count) vs log (1/box size), and the fractal dimension calculated from the slope, m, of the line of best fit, according to the relationship D=m. Analysis of image outlines produced the box surface dimension, whereas analysis of filled images resulted in a box mass dimension.
Table 1: Box mass and box surface dimension of fourteen
colonies of Nocardia otitidiscaviarum of various sizes and
a large colony of Nocardia asteroides and Rhodococcus
erythropolis, using different computer software. (a,g) Image
Fractal (IF) with preset box sizes, (b,h) Fractal dimension
calculator (FDC) without offsets using same box sizes as Image
Fractal, (c,i) FDC with ten offsets using same box sizes as
Image Fractal (d,j) FDC with exhaustive (exh) analysis of all
possible offset positions (e,k) FDC with own default box sizes
and offsets. (f,l) Colony, previously published data.
Fourteen colonies of Nocardia otitidiscaviarum which ranged from very early stages of filamentous growth to small developed colonies (Figure 1) were first analysed using Image Fractal and FDC at the same box sizes used by Image Fractal, with no provision for extra offsets for placing the grid. Values obtained for box mass (Table 1(a), (b)) and box surface (Table 1(g), (h)) dimensions showed considerable variation, with differences between the two programs reaching 0.12 in one instance. Plotting these values (Figure 3(a), (b)) showed that Image Fractal produced a smoother correlation between D values and complexity (that is, the subjectively determined ranking of colonies according to complexity resulted in higher D for higher complexity). However, Image Fractal does not allow any further choice with respect to box sizes and offsets.
Using the same box sizes as Image Fractal, FDC was also tried with the option allowing an exhaustive search for every possible offset for the grid (Table 1(d), (k)), as well as 10 randomly chosen offsets (Table 1(c), (i)). Although many values were similar to those obtained with only one offset, differences up to 0.08 were seen in some instances. When D was plotted against colony complexity (Figure 3(c), (d)) and compared to the original calculations using only one offset (Figure 3(b)), a smaller scatter of points was seen when multiple offsets were used. The default option in FDC, where the software determines a reasonable spread of box sizes and offsets based on the size of the image being examined, was also tested (Table 1(e), (j)) and produced results which in most cases differed from the exhaustive analysis by only 0.01 or 0.02 for N. otitidiscaviarum, suggesting that the extra time required for the exhaustive analysis (hours per image) was not significantly increasing accuracy.
Data from the earlier study of the same colonies [9], in which the computer program "Colony" analysed co-ordinates of the outlines of these colonies, is also included (Table 1(f), (l)). The results agree with those in the current work, with the exception of the box mass dimensions obtained for the smaller colonies where the results obtained by "Colony" were a little lower, and also showed a smoother correlation between fractal dimension and complexity (Figure 3(f)).
Figure 3: Changes in box mass (solid line) and box surface
(broken line) dimension with increasing size/complexity of
colonies of Nocardia otitidiscaviarum using six different
combinations of computer programs and conditions (a) Image
Fractal with preset box sizes, (b) Fractal dimension calculator
(FDC) with same box sizes as Image Fractal, with no allowance
for extra offsetting boxes, (c) FDC with same box sizes, but
allowing 10 randomly chosen offsets, (d) FDC with same box sizes
but carrying out exhaustive calculations using all possible
offsets, (e)FDC with default box sizes and offsets determined by
the software on the basis of image size, (f) previously published
data for Colony, based on 10 randomly chosen starting points for
the grid.
In addition to these natural fractals, a number of geometric objects with known fractal dimension were also analysed. These were Koch island (D=1.5), Koch coastline (D=1.26), Koch boxes (D=1.77), a straight line (D=1), and a filled rectangle (D=2). Both Image Fractal and FDC gave results that approached the expected values of D for these objects, but this was only achieved by careful selection of box sizes ( Table 2), making the validity of answers obtained with natural fractals difficult to assess.
It is for this reason that a program like FDC, which allows a choice of box sixes and offsets, can be more useful than Image Fractal. However, before attempting to use the results of fractal analysis, a wide range of box sizes needs to be assessed to determine if any points deviate from the straight line necessary in the double log plot used to calculate the dimension. We were forced to omit boxes below 4 pixels for this reason, and this may be related to the diameter of the filaments in the colony which are slightly less than four pixels in the images examined and, therefore, a major influence on the irregularities in the shape of a colony. Others [10] have demonstrated two linear elements in a full analysis of carbon particles and suggest these represent two different dimensions: one describing a structural fractal, which is determined at the larger sizes, and the other a textural fractal, which describes the smaller irregularities on the particle surface and is measured by the smaller box sizes.
Table 2: Fractal dimension analysis of known fractal objects using Image Fractal (IF) and
Fractal
dimension calculator (FDC) at different box sizes (n = not determined).
Therefore, it is important to emphasise that fractal dimension is a generic term without strict definition [11] and covers a number of different and related measures [12]. As a result, fractal analysis must always be used cautiously and related to the object being measured. What aspect of a colony's complexity does D actually measure? Smith and co workers [13, 14] suggest that in neuron cells, which bear some resemblance to growing colonies of branching filamentous micro-organisms, the ruggedness of the border (that is, rugged, jagged or uneven border), the amount of branching, and space filling properties (which may reflect extent of branching) contribute to the complexity, with an increase in each contributing to an increase in D.
Results comparing three large colonies of different nocardioform bacteria (Figure 2, Table 1) agree with our previous study [9] using different box counting programs. As a result of increased branching frequency, organisms like N. otitidiscaviarum quickly move from being a mass fractal where the two dimensions measured are similar, to exhibiting only surface fractal properties as the colony centre fills out. N. asteroides branches less frequently, and remains a mass fractal at the same colony size as the largest colony size examined here. As the colony becomes larger, one would expect the centre to fill out and a change from mass fractal to occur, albeit at a much later stage than with N. otitidiscaviarum. On the other hand, R. erythropolis, shows only rudimentary branching and rapidly fills out the colony centre (Figure 2) so that at no stage can it be considered a mass fractal, and because its shape approaches that of a circle, its box surface and box mass dimensions are close to 1 and 2, respectively.
Differences in branching can also occur in the same organism when exposed to different environmental conditions, particularly those affecting growth rate, and fractal analysis should be invaluable for describing such differences. The attraction of fractal analysis is in its ability to quantify aspects of the development of a growing colony, and in doing so providing us with a new tool to objectively quantify the effect of changes in environment and nutrients on growth of micro-organisms.
Fractal dimension analysis shows considerable potential as a tool for describing the development of filamentous, branching micro-organisms, particularly in measuring the effect of changes in branching behaviour on the complexity of the colony shape. However, extreme care must be taken in choice of box sizes for the analyses. In addition, different programs may produce different results in fractal analysis, even though they use the same algorithmic approach. Programs like FDC and "Colony", which allow choice of box sizes and random starting points for laying grids, allow more flexibility than Image Fractal, but those like "Colony", which rely on co-ordinates of outlines, are less useful than FDC because of extra manipulations involved in obtaining the co-ordinates. Image Fractal, despite having less flexibility in choosing operating parameters, includes excellent image-manipulating facilities (it is based on Image produced by the National Institute of Health), and does allow other fractal analyses.
Using Box Counting Techniques for Measuring Shape of Colonies of Filamentous Micro-organisms
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