Fractal Analysis of Nuclear Medicine Images

TO THE EDITOR: The Journal of Nuclear Medicine has recently published 2 closely related studies from a Japanese group, reporting the application of a fractal method to the analysis of SPECT images (1,2). Both studies used an intensity-thresholding approach to derive what these investigators termed the fractal dimension, a parameter that was claimed to quantify the spatially heterogeneous distribution of radioactivity. These 2 studies differ only in the application targets: one on pulmonary emphysema using a carbon particle radioaerosol (1) and the other on cerebral blood flow distribution in Alzheimer’s disease with intravenous injection of 99mTc-labeled hexamethylpropyleneamine oxime (HMPAO) (2). Both studies suggested that a large fractal dimension indicates increased heterogeneity in the spatial distribution of radioisotopes and that the fractal dimension showed a statistically significant difference between patients and healthy volunteers. We have evidence (3,4) that the fractal dimension obtained by the analysis method of Nagao et al. (1,2) is unrelated to spatial heterogeneity, nor does their result suggest the SPECT images to exhibit a fractal form as claimed in these articles. The main weakness of the studies performed by Nagao et al. lies in the fact that their so-called fractal dimension was obtained using an approach based on intensity thresholding. Therefore, the fractal dimension can be derived entirely from the image intensity histogram. Because the image intensity histogram is essentially the probability density function of the pixel values throughout the entire image region, the histogram naturally is unrelated to the heterogeneous spatial pattern of radioactivity distribution. Consequently, the fractal dimension as defined by Nagao et al. is unrelated to spatial heterogeneity of radioactivity as well. In our previous publications (3,4) attempting a reinvestigation of the studies of Nagao et al. (1,2), we have demonstrated that the fractal dimension of Nagao et al. reflected the percentage area (or percentage volume, in the case of 3-dimensional images) of low radioactivity. In addition, the closely associated relationship between fractal dimension and area percentage was independent of imaging modality or the anatomy of interest. Thus, in the study on cerebral blood flow distribution in Alzheimer’s disease with intravenous injection of 99mTc-labeled HMPAO (2), the fractal dimension of Nagao et al. is equivalent to the percentage volume of brain tissue showing less than 50% of peak cerebral blood flow. That is to say, an increased fractal dimension at most represents a “significant presence,” rather than a “heterogeneous distribution,” of local perfusion deficits. Fractal dimension is not a term that can be arbitrarily defined. Demonstration of fractal behavior by a specific system entails measurements of a certain physical quantity with variations in ruler size over a wide range of scale, typically of at least one order of magnitude. In the study of Nagao et al. (2) using threshold values from 35% to 50% of maximum intensity, the variation in intensity scale spans less than only one-sixth of an order of magnitude. Furthermore, because the volume measured with intensity segmentation decreases with increasing threshold values for all images in the world, the conclusions of Nagao et al. on the fractal form of SPECT perfusion images would suggest that all kinds of images in the world exhibit a fractal form. Obviously, this claim would be entirely irrational. Therefore, the fact that the 3-dimensional cerebral blood flow maps obtained with SPECT look somewhat heterogeneous for patients with Alzheimer’s disease does not mean that these images are suitable for all arbitrarily defined fractal dimension analyses. Nor does the success of previous reports by Kuikka et al. on the use of fractal approach based on relative dispersion analysis to quantify spatial heterogeneity (5) justify the appropriateness of the fractal analysis method of Nagao et al. We shall be attempting a theoretic comparison of the relative dispersion of Kuikka et al. and the histogram-based fractal analyses of Nagao et al. so that the readers of The Journal of Nuclear Medicine can be clear about the suitability of fractal analysis methods to quantify spatial heterogeneity. Besides all the above pitfalls of the study of Nagao et al. (2), the fact that Nagao et al. had deliberately ignored our warnings on the weakness of their method surprised us. A letter from me to the editor of The Journal of Nuclear Medicine was published in the January 2001 issue, along with Nagao’s reply (4). The letter was accepted on July 24, 2000. As explicitly stated in the most recent article of Nagao et al. (2), their manuscript was submitted to and received by The Journal of Nuclear Medicine on August 22, 2000, with revision accepted on May 29, 2001. As a consequence, Nagao et al. clearly were well aware of the deficiency of their approach before attempting to publish their study on cerebral blood flow distribution in Alzheimer’s disease. In that article (2), however, not one of the potential weaknesses or the controversies was mentioned. In our opinion, any new analysis method claimed to be effective in clinical practice should undergo a solid validation process or should at least be allowed to face criticisms from the scientific society. The deliberate disregard of our warnings by Nagao et al. violates the ethics of honestly reporting their recognition of potential methodologic pitfalls, as mentioned above. This is an issue regarding not only scientific truth but also research honesty. In conclusion, I would like to emphasize that a rigorous validation of any new analysis method is necessary before broadened clinical applications are attempted. In the case of fractal analysis proposed by Nagao et al. (1,2), a second investigation to prove its methodologic validity is strongly recommended.