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		1.4.13— Biological Implications of the Fractal Dimension

		1— Quantitative Measure of Self-Similarity

		The numerical value of the fractal dimension d gives us a quantitative measure of self-similarity. The fractal dimension tells us how many small pieces N(r) are revealed when an object is viewed at finer resolution r. The quantitative relationship is that N(r) is proportional to r-d. The larger the fractal dimension, the larger the number of small pieces are revealed as the object is viewed at finer resolution.

		2— Quantitative Measure of Correlations

		The small pieces of a fractal are each related to the large pieces. Thus the small pieces are also related to each other. The fractal dimension measures the correlations between the small and large pieces. Thus it also measures the correlations between the small pieces themselves.

		3— Quantitative Classification

		The numerical value of the fractal dimension can be used to classify different objects. For example, the fractal dimension of the blood vessels in a normal retina is different from the fractal dimension in a retina changed by disease. The fractal dimension may serve as a method to diagnose different diseases and as an index to quantify the severity of these diseases.

		4— Hints about Mechanisms

		Different mechanisms produce fractals with different dimensions. Hence, measuring the fractal dimension of an object may give us clues about figuring out the mechanism that produced it. For example, the process called diffusion limited aggregation produces fractals with a fractal dimension of about 1.7. The blood vessels in the retina have a fractal dimension of about 1.7. Thus it is worthwhile to consider if the growth of these blood vessels was produced by diffusion limited aggregation. This would mean that the growth of the blood vessels was proportional to the gradient of a diffusible substance, such as oxygen or a growth factor.

 

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