We read with great interest the recent “Perspectives in Cancer Research” written by Baish and Jain (1) on the potential value of fractal geometry in characterizing the seemingly random tumor vasculature. However, we are concerned about whether tumor vasculature exhibits clearly defined fractal properties. Specifically, a complicated and tortuous tumor vasculature is not necessarily chaotic and subject to successful fractal analysis (2). As a simple test, we applied the box-counting method, as stated in the paper by Baish and Jain, to their three exemplified skeletonized vascular structures (Fig. 2 in Baish and Jain, digitized with 1260 × 1260 matrix at 2400 dots-per-inch resolution). The log-log plot from which the fractal dimensions (D) are derived is shown in Fig. 1. Note that the slope obtained from a simple regression on the linear portion of the data is presumably equal to −D, i.e., the negative of fractal dimension (1). It can be seen, however, that all three sets of data show a convex shape. Consequently, the behaviors of the three vascular structures in Fig. 1 are nonlinear. A clear definition of the fractal dimension, therefore, seems to be impossible in these cases.

The data nonlinearity was further examined by performing linear regression on different portions of these data sets. In other words, the “local fractal dimensions” were obtained as a function of the spatial scale of box widths (Fig. 2). Notice that the measured fractal dimensions change significantly throughout the region of analysis, indicating that there is essentially no linear portion in Fig. 1. The only exceptions are at the “large box extreme” (D = 2) and “small box extreme” (D = 1), where the measured fractal dimension is relatively insensitive to the spatial scale of measurement. It is to be noted, however, that for the “large box extreme” a fractal dimension of D = 2 suggests that the vessel network would be a two-dimensional object filling the entire region of interest, which is obviously unreasonable. On the other hand, for the “small box extreme” a fractal dimension of D = 1 implies that the skeletonized vasculature is essentially a one-dimensional nonfractal line object for which fractal analysis is of no value. In addition, in the “small box extreme” fractal dimensions of the three vascular structures are mutually indistinguishable. Therefore, judging from the behavior shown in Fig. 1 and 2, the use of fractal dimension to quantify the degree of randomness of the vascular distribution seems to be based on weakly-linked relationships between the tumor vascular structure and the mathematical fractional dimension.

We conclude that the tumor vasculature, and perhaps healthy tissue vascular architecture as well, does not exhibit clearly defined fractal properties suitable for box-counting fractal analysis. Although it is true that fractal objects demonstrate a linear log-log behavior with noninteger slopes when the “ruler size” for a measurement is changed, being able to obtain a noninteger slope in a log-log plot, however, does not prove that the object under investigation is fractal. Consequently, we suggest that fractal analysis methods be applied in biomedical studies only with a thorough understanding of its physical meanings, in order for the new methodology to be useful.

Fig. 1.

Plot of box number versus box width, from which the fractal dimensions are derived, of the vascular networks exemplified in the article by Baish and Jain. Note that all three sets of data show a convex shape.

Fig. 1.

Plot of box number versus box width, from which the fractal dimensions are derived, of the vascular networks exemplified in the article by Baish and Jain. Note that all three sets of data show a convex shape.

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Fig. 2.

Fractal dimensions measured locally from the data in Fig. 1, showing that the data nonlinearity results in poorly defined fractal dimensions that vary significantly with slight changes in the spatial scale of measurement.

Fig. 2.

Fractal dimensions measured locally from the data in Fig. 1, showing that the data nonlinearity results in poorly defined fractal dimensions that vary significantly with slight changes in the spatial scale of measurement.

Close modal

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We thank Chung and Chung for their thoughtful reflections on our recent Perspectives(1) article on the use of fractal analysis to characterize tumor vasculature. They raise a number of important technical issues in the analysis of images. We welcome the chance to address these rarely discussed issues, however, we strongly disagree with Chung and Chung’s assertion that tumor vasculature is not amenable to fractal analysis. Our original findings, published in Physical Review Letters(2) and Microcirculation(3), were based on an analysis of scores of tumors by using multiple algorithms that were carefully verified on standard fractal objects. In contrast, Chung and Chung present an analysis of a “redigitized” version of our Fig. 2(1) using a single algorithm. We will show that even the small illustration in our Perspectives article contains far more information than they obtained.

One of the major challenges in the use of fractal methods on images of natural objects is that the possible range of power-law behavior is limited by practical constraints. We agree with Chung and Chung that the analysis of an image is limited on the small extreme by the pixel size and on the large extreme by the size of the image itself. On the small extreme we expect that the line segments representing the axes of the vessels must be one-dimensional. On the larger extreme, we expect that the image will behave as a two-dimensional object. In order for an image to yield a fractal dimension, there must be a range over which the slope of an appropriate power-law plot reaches a stable plateau between 1 and 2. If the slope undergoes a continuous increase from 1 to 2 with no plateau, then the image is not yielding a reliable measure of the fractal dimension but is instead displaying a crossover effect. On all of these points, we agree with Chung and Chung.

We also agree with Chung and Chung that the box dimension algorithm that we presented in the Perspectives article does not yield a particularly robust measure of the fractal dimension for the illustration in the article. We would share Chung and Chung’s concern about the validity of our previously published results (2, 3) if these results were based solely on a box-counting analysis of the single, small illustration in our Perspectives paper. Instead, our results are based on a thorough analysis of scores of images using a combination of box dimension, sandbox dimension, correlation dimension, and minimum path dimension methods. Of these, we chose to explain the box dimension because it is the mathematically simplest to describe to a broad audience. Unfortunately, when the box dimension is applied to some images the range of power-law scaling is quite narrow or nonexistent. Alternatives, such as the sandbox dimension, correlation dimension, and minimum path dimension have proven to be more robust in our experience. (Actually, the minimum path dimension does not use a separate algorithm. Instead, it is based on any of the other three algorithms but applied to the path of minimum length in the image rather than to the image as a whole.) The figures below show the improvement possible simply by using multiple methods of analysis.

Some discussion of the details of the image analysis methods is necessary to explain our point. The illustration in Fig. 2(1) is already a second-generation version of the original data. The resolution of the MS Word figure in Fig. 2(1) is 280 × 280. Chung and Chung redigitized the illustration to 1260 × 1260, more than 4× the resolution that would be expected to yield any meaningful information. Accordingly, we are not surprised that they find that the image appears to be one-dimensional over a wide range of scales. Several other aspects of Chung and Chung’s analysis are unclear from their description.

(a) The image must be prepared by a process of skeletonizing the vasculature so that only a line segment one pixel in width remains for each vessel segment. The illustration had a resolution of 280 × 280. By digitizing the image at 1260 × 1260, Chung and Chung would probably get vessel segments several pixels thick. They did not state whether they skeletonized the redigitized image before analysis.

(b) The process of counting how many boxes are needed to cover the vessels in the image is not as trivial as it might seem. In tiling over the vessels, it is not sufficient to simply partition the image into smaller and smaller fractions and then count how many are occupied by a blood vessel segment. The proper procedure requires that the tiling be shifted to all possible locations until the tiling that requires the minimum number of tiles is found. This process is conceptually simple, but computationally intensive. For example, if square tiles measuring 40 pixels on a side are used, 40 × 40 = 1600 different positions of these tiles must be tried. The minimum number of tiles needed to cover the vessels from these 1600 possibilities is then recorded as the number of boxes needed at this scale. It is unclear whether Chung and Chung completed this crucial step. Omitting this step can narrow the range over which power-law behavior might be observed. For example, the tumor image has an avascular space that is approximately one-seventh of the image size. By not shifting the tiling until the minimum number of boxes is found, Chung and Chung might easily have missed the upper limit of power-law behavior.

We now present our own analysis of the illustration using the full range of methods that we used in our early publications, the box dimension, sandbox dimension, correlation dimension, and minimum path dimension. (The minimum path dimension was illustrated in the Perspectives article itself.) As will be shown below, these methods taken together give a quite robust measure of the fractal dimension. Prior to use on images of unknown dimension, each of our algorithms was verified on a straight line, the 1-D and 2-D Cantor sets, Koch curve, Sierpinski gasket and carpets, and filled and frame circles and squares (3). The box dimension was verified to be accurate within 0.02, the sandbox dimension within 0.02, and the correlation dimension within 0.07. Chung and Chung have not provided verification that their algorithm can accurately determine the dimension of such standards.

The box dimension is illustrated in Fig. 1,a, which shows the minimum number of boxes needed to tile over the occupied pixels for various box sizes. The local slope of this curve is shown in Fig. 1,b. The curve fits in Fig. 1 a represent the best fit to seven consecutive box sizes. For the tumor illustration, the largest scale for which fractal behavior might be observed would be one-seventh of the image (L = 40 pixels of a possible 280 and log(L) = 1.6). Beyond this scale, the results show only edge effects. Even with our uncertainty about the details of Chung and Chung’s methods, we agree with them that the box-counting method appears to shows only a narrow plateau, especially for the arteries.

Fig. 2, a and b show plots for the sandbox dimension. The sandbox algorithm works by first selecting an occupied pixel (a point on a blood vessel) as the center of a box with side L. The number of occupied pixels within the box is then counted. The number of occupied pixels within boxes of progressively larger size is counted until the box of interest extends to the edge of the image. The process is then repeated for many other center pixels. The mean number of occupied pixels N(L) within regions of side L is then calculated. The fractal dimension is then obtained from the slope of the log(N(L)) versus log(L) plot, that is

\[N(L)\ {\propto}\ L^{d_{\mathit{sandbox}}}\]

Only that region of the plot that is linear is used to estimate the slope. Because L must be limited by the size of the image, at no time does the actual edge of the image affect the results. Fig. 2 b shows that the slope of the tumor illustration has a clear plateau at 1.87. The capillaries show a crossover to a dimension of 2, as expected for a compact object, and the arteries again show a weak plateau at a slope of 1.60.

The correlation dimension is applied to the Perspectives illustration in Fig. 3, a and b. The correlation dimension is estimated from an image by counting the number of occupied pixels Mi(r) within a circular ring of radius r and width 0.1r. The use of a ring of width 0.1r is arbitrary; it need only be narrow. As with the sandbox method, the count is repeated over n circular rings each centered on an occupied pixel. The correlation function c(r) is then estimated from

\[c(r)\ {\propto}\ \frac{N(r)}{r^{2}}\ {=}\ \frac{1}{r^{2}{\cdot}n{\cdot}N_{pixels}(r\ {\rightarrow}\ 1.1r)}\ {{\sum}_{i{=}1}^{n}}\ M_{i}(r)\]

where the result has been normalized by n and the total number of occupied and unoccupied pixels within the ring Npixels (r → 1.1r). The correlation dimension dcorr is then obtained from the slope of the log(N(r)) versus log(r) plot, that is

\[N(r)\ {\propto}\ r^{d_{corr}}\]

Again, only the linear region of the plot is used. Fig. 3 b shows that all three vessel types have a clear plateau. The plateau for the arteries remains relatively narrow.

Although the methods differ somewhat in their ability to show a plateau, taken together they yield clear evidence of power-law behavior and consistent values for the fractal dimension. In our experience, the sandbox method has been the most robust method in terms of yielding accurate numbers for standard fractal objects and giving a clear plateau for the slope of the power-law plot. Tumor vasculature has generally shown more robust fractal behavior than normal arteries and veins, but when numerous images are considered, the estimates of fractal dimension are consistently different for the two vessel types. We have found that normal s.c. capillaries consistently yield compact, two-dimensional patterns. Most of the actual images used in our original study showed power-law behavior over about 1-and-a-half orders of magnitude. Do we wish that observed range of power-law behavior was greater? Of course we do, but Chung and Chung’s suggestion that tumor vasculature does not display power-law behavior is not well supported when the data are analyzed more thoroughly. In fact, a recent publication by Sabo et al.(4) confirms that fractal analysis of tumor vasculature can provide prognostic information that was not readily available by traditional methods of image analysis.

We regret that our attempt to present the use of fractal analysis in Cancer Research was oversimplified to the extent that it raised doubts about our earlier results. The difficulty of obtaining reliable measures of the fractal dimension from images is considerable and little discussed in the literature. We fear that many published results claiming fractal behavior of natural objects may be flawed by the lack of attention to detail and multiple checks that we have found necessary. We would welcome the opportunity to carry on a more detailed dialogue with Chung and Chung on our original methods and findings in a suitable technical journal.

1

To whom requests for reprints should be addressed, at Department of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837. Phone: (570) 577-1163; Fax: (570) 577-1822; E-mail: baish@bucknell.edu

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