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1
Department of Electrical Engineering, Rm. 238, National Taiwan University, No.
1, Sec. 4, Roosevelt Rd., Taipei, Taiwan 10764, R. O. C.
2
Department of Nuclear Medicine, National Taiwan University Hospital, Taipei,
Taiwan, R. O. C.
Received July 8, 1999;
accepted after revision September 8, 1999.
H. W. Chung is supported in part by the National Science Council grant
NSC-87-2314-B002-330-M08.
Abstract
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MATERIALS AND METHODS. Fractal dimensions were computed for 108 sets of radionuclide imaging data from 28 patients according to the methods in a previous report, and were then correlated with the ratio of tissue areas segmented at two thresholds (15% and 35% of maximal radioactivity).
RESULTS. Fractal dimension was found to linearly correlate with the ratio natural logarithm of tissue areas segmented at two different threshold levels (n = 108, r = 0.999), with regression slope accurately predicted (error = 0.06%). Bland-Altman analysis showed that fractal dimensions ranging from 0.2 to 1.9 can be explained by this area ratio with disagreement of only 5.13% at two standard deviations; thus, fractal dimension seems to be an over-simplified parameter unrelated to spatial heterogeneity of radioaerosol distribution.
CONCLUSION. The analysis of this study suggested that the fractal dimension defined in a previous report was limited to the indication of the percentage area of low-radioactivity regions with respect to total tissue area in the image. Because the fractal dimension partially reflects, but is not specific to, a certain degree of focal spots of low radioactivity, we suggest using fractal analysis in clinical practice only with careful control and thorough understanding of the physical meanings.
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Recently, Nagao et al. [1] presented fractal analysis, an approach for obtaining a reproducible and sensitive quantification of emphysematous severity, and reported their analysis results of 40 patients. Fractal analysis is potentially suitable for an objective quantification of spatial heterogeneity because it is believed to be effective in helping to characterize complex systems that are hard to describe using conventional Euclidean geometry [7]. In the work of Nagao et al., the heterogeneous distribution of a 99mTc-labeled ultrafine carbon particle radioaerosol, Technegas (Daiichi Radioisotope, Tokyo, Japan) [8], on single-photon emission computed tomography (SPECT) images was assessed using a parameter termed "fractal dimension." Derived from the analysis method developed by Nagao et al., the fractal dimension was found to be significantly larger in emphysematous patients than in healthy volunteers (p < 0.0005) [1]. Furthermore, fractal dimension was reported to be sensitive in detecting mild impairments in the ventilation status in patients with suspected emphysema [1], indicating its clinical value in early diagnosis and in the monitoring of the progression of pulmonary emphysema.
Further investigation of the origin of the efficacy of the fractal analysis
method, considering its promising diagnostic potential and easy
implementation, seems worthwhile. Briefly, the theory is based on an equation
relating a measure (M) to the scale (
) of the ruler that
measures M:
 | (1) |
Nagao et al. [1] used five
cutoff levels, 15%, 20%, 25%, 30%, and 35% of the maximal pixel intensity in
the ventilation SPECT images, to segment the lung tissue; therefore, the total
apparent area occupied by the lung tissue varied as a function of the
intensity thresholds. The five chosen thresholds were used as the ruler scale
in equation 1, and the total numbers of pixels with intensity higher
than the corresponding thresholds were used as M(). Clearly,
M() decreases as increases; hence a linear regression on
the M() versus graph, when plotted on a natural
logarithm versus natural logarithm scale, yields a negative slope with
magnitude equal to the fractal dimension D.
Figure 1A shows a typical lung
perfusion scintigraphy image, and Figure
1B is the corresponding intensity histogram, with dashed lines
marking the five thresholds. The total number of pixels on the right side of
these dashed lines is M(). The graph exemplifying the
calculation of fractal dimension D for this image is shown in
Figure 1C, but notice that the
generation of Figure 1C, and
hence the computation of fractal dimension D, can be accomplished
solely from the intensity histogram (Fig.
1B). Because the histogram simply plots the number of pixels in
the entire field of view versus the gray level possessed by these pixels, the
intensity histogram of a nuclear medicine image reflects only the
radioactivity statistics and is thus totally unrelated to spatial
heterogeneity. Likewise, the fractal dimension D derived from the
histogram is also unrelated to spatial heterogeneity. The application of
fractal dimension to the diagnosis of pulmonary emphysema may therefore face
problems in interpretation because the fractal dimension does not convey any
positional information required to indicate the extent of focal impairments of
alveolar function.
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The question that arises from these observations is, "What does
fractal dimension mean if it is shown to be sensitive to the detection of mild
alteration in the ventilation status, while being unrelated to spatial
heterogeneity in a nuclear medicine image?" Taking a closer look at
Figure 1C, note that the
fractal dimension D was computed as the magnitude of the negative
slope, which is simply the ratio of the height (h) to its width
(w), if the five data points fall exactly on the regression line.
Because the five chosen cutoff thresholds were used as the abscissa ,
the width w is fixed at ln |35| - ln |15|
= ln |35/15| = 0.84730 (note that the natural logarithm was used
here and in the work of Nagao et al.
[1]). In other words, the
fractal dimension D, or the absolute slope of the hypotenuse, is
directly proportional to the height h in
Figure 1C with proportionality
constant 1 / 0.84730 = 1.18022. The height h, being approximately
equal to (ln |M( = 15)| -
ln|M( = 35)|), represents the natural logarithm
of the ratio of lung tissue areas when segmenting at two different threshold
levels (i.e., 15% and 35% of the maximal signal intensity). We therefore infer
that the fractal dimension calculated by Nagao et al. is only an indicator of
this area ratio completely unrelated to spatial heterogeneity, with a
proportionality constant of 1.18022. In addition, because the inferences drawn
from our observations were purely methodologic, we predict that such
situations in nuclear medicine images are independent of the image technique
and the anatomy being examined.
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)| versus ln
|| plot. Note that for the three-dimensional data sets,
the highest pixel intensity in the entire volume was used, whereas in the
two-dimensional images, such as the projection scintigrams, D was
calculated on a slice-by-slice basis. The apparent tissue areas obtained at 15% and 35% threshold values were then computed. The ratio (area ratio) of these two values for each set of images was derived and its natural logarithm correlated with the fractal dimension D. Bland-Altman analysis [10] was used for a rigorous assessment of the agreement between these two parameters. This was accomplished by calculating, for all data points, the percentage of difference between D and its predicted value using the area ratio. The difference in D obtained with two methods was then plotted versus the average of D computed using the two methods.
To assess the degree of self-similarity, the ln
|M()| versus ln || plot of
each set of image data was also obtained with a larger range of threshold
values, namely 5-60% of maximal pixel intensity with 5% increments. The
linearity of such curves was visually examined.
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), is not
self-similar over the range of 5-60% threshold values. In contrast, the data
in Figure 2 seem to be highly
linear within the range of 15-35% thresholds, as shown by the fitted
lines.
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This observation may lead one to conclude that the range of 15-35% thresholds is a proper choice for using fractal analysis as done in the work of Nagao et al. [1]. Indeed, the linearity within this limited range was found to be excellent for all images in our study; however, this also implies that the inference we have described seems to be valid; that is, fractal dimension may be only an indicator of the area ratio, with a proportionality constant of 1.18022. Our prediction is corroborated by the experimental data shown in Figure 3, in which the fractal dimension, computed using the five threshold values, was plotted versus area ratio. Notice that regardless of the image technique or the anatomy examined, the combination of data from different imaging techniques shows a significant correlation, with r = 0.999. Furthermore, the regression equation in Figure 3 shows that the fractal dimension and logarithm of the area ratio are linearly related with a proportionality constant of 1.181, only a 0.06% deviation from the predicted value of 1.18022 plus a negligible intercept of -0.0082.
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If one uses the equation in Figure 3 and the area ratio to predict fractal dimension, a Bland-Altman analysis [10] can be performed to assess the agreement between this prediction and that obtained using the method of Nagao et al. [1]. Figure 4 shows the result of Bland-Altman analysis, which indicates that the percentage of disagreement in D at two standard deviations is only 5.13% (D ranging from 0.2 to 1.9). Because the disagreement is minor (<8% in all cases), we concluded that the fractal dimension as defined in the work of Nagao et al. is effectively equivalent to the ratio of apparent tissue areas segmented at two threshold levels, 15% and 35% of maximal pixel intensity.
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), does not exhibit
self-similarity over a wide range of scales. Because self-similarity (usually
required over several orders of magnitude
[11]) is a prerequisite for a
justification of fractal properties, the absence of self-similarity may lead
to questionable value of the application of fractal analysis on nuclear
medicine images. On the other hand, because the decrease of apparent tissue
area with the increase of threshold values is a natural trend in all digital
images, the decreasing trend of the data shown in
Figure 1C is not related to
fractal properties. In our opinion, the highly linear behavior of this data
set may be because the small amount of the five data points comes only from a
very limited range of values, namely 15-35% of maximal pixel intensity, which
is only a magnitude of 2.33 compared with several orders of magnitude used in
more rigorous fractal analyses
[7,
9,
11]. Furthermore, because a
logarithm was taken for both variables in
Figure 1C, any curved trend
that can be described by the power law would tend to be linearized, giving
what was observed as an apparent fractal property described in the work of
Nagao et al.
Regardless of its origin, the linearity of the area-threshold curve, that
is, the "ln |M()| versus ln
||" plot, showed that this fractal dimension can at
least be computed with high reproducibility in an objective manner. In such a
case, however, our theoretic analysis and the experimental results in Figures
3 and
4 show that this parameter can
also be precisely predicted using the ratio of apparent tissue areas at two
intensity thresholds (15% and 35% of maximal pixel intensity) in a nuclear
medicine scan. If one denotes the apparent tissue area greater than 15%
radioactivity as A15, and that greater than 35% radioactivity as
A35, a simple equation: can further relate the
 | (2) |
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One property of the area ratio or fractal dimension mentioned earlier is
that this parameter is able to sense the portion of tissues with intensity
values falling between 15% (
1 / 6) and 35% (1 / 3) of maximal pixel
intensity; therefore, this parameter indeed seems to be sensitive to mild to
moderate alveolar impairments resulting in a decrease of signal to one third
to one sixth of the normal tissues. Such characteristics may explain the
finding of Nagao et al. [1] in
the sensitivity of fractal dimension in detecting suspected pulmonary
emphysema. On the other hand, in special cases of severe emphysema with
substantial functional impairments of the alveoli leading to signal reduction
to less than one sixth of its expected value, the computational algorithm of
Nagao et al. might treat these areas as being part of the background, possibly
underestimating emphysematous severity.
In addition to these pitfalls of fractal analysis, because the fractal dimension defined by Nagao et al. [1] was derived using five cutoff levels of intensity thresholds, fractal dimension is a parameter that can be determined solely by the image-intensity histogram. As a consequence, any pre- or postprocessing for improvements of image quality, such as median filtering [13], histogram equalization [13], or the reconstruction routinely used in SPECT [14], would alter the measured fractal dimension, unless the intensity histogram is preserved by the processing algorithms. Furthermore, any inherent or artifactual presence of a single hot spot (single pixel with very high intensity) could affect the automatic selection of threshold values used for a calculation of fractal dimension. Such phenomena, however, may not be associated with pulmonary function to any extent. Under such circumstances, it is inferred that the method of fractal analysis proposed by Nagao et al. may be of questionable diagnostic value.
Finally, this study should not be regarded as a criticism to discredit fractal analysis as a potential morphometric tool in nuclear medicine examinations. In the work by Nagao et al. [1], although our study showed that the fractal dimension seemed unrelated to the spatial heterogeneity of radioaerosol distribution, fractal dimension does partially reflect the existence of a certain degree of focal spots of high and low radioactivity by showing increased percentage of area of low radioactivity in patients with pulmonary emphysema. In particular, on thin-slice SPECT images where partial volume effects are less severe than in projection scintigrams, the fractal dimension may be sensitive to disease severity. Therefore, the fractal analysis, when appropriately combined with conventional visual diagnosis, could be of some value by providing an objective quantity to serve as an index for diagnostic reference. Because the fractal dimension as defined herein has been shown in this study to be nonspecific to spatial heterogeneity and actually unrelated to the nature of fractal objects, we cite Lancet, "Since fractal analysis is essentially mathematical, as with all mathematical models, there must be a close link with the biological event, if the model is to be useful," [15] and strongly suggest usage of such an analysis in clinical practice only with rigorous control, careful interpretation, and thorough understanding of its physical meanings.
Acknowledgments
We thank Soo-Chang Pei (Department of Electrical Engineering, National
Taiwan University, Taipei) and Cheng-Yu Chen (Department of Radiology,
Tri-Service General Hospital, Taipei) for helpful discussions.
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