Complexity, fractals, disease time, and cancer

Despite many years of research, a method to precisely and quantitatively determine cancer disease state remains elusive. Current practice for characterizing solid tumors involves the use of varying systems of tumor grading and staging and thus leaves diagnosis and clinical staging dependent on the experience and skill of the physicians involved. Although numerous disease markers have been identified, no combination of them has yet been found that produces a quantifiable and reliable measure of disease state. Newly developed genomic markers and other measures based on the developing sciences of complexity offer promise that this situation may soon be changed for the better. In this paper, we examine the potential of two measures of complexity, fractal dimension and percolation, for use as components of a yet to be determined “disease time” vector that more accurately quantifies disease state. The measures are applied to a set of micrographs of progressive rat hepatoma and analyzed in terms of their correlation with cell differentiation, ratio of tumor weight to rat body weight and tumor growth time. The results provide some support for the idea that measures of complexity could be important elements of any future cancer “disease time” vector.

Figure 1

Disease progression vs chronological time.

Figure 2

Disease progression vs “disease time.”

Figure 3

(Color online) Image 1.

Figure 4

(Color online) Image 2.

Figure 5

(Color online) Image 3.

Figure 6

(Color online) Image 4.

Figure 7

(Color online) Image 5.

Figure 8

(Color online) Image 6.

Figure 9

(Color online) Image 7.

Figure 10

(Color online) Image 8.

Figure 11

(Color online) Image 9.

Figure 12

(Color online) Image preprocessing.

Figure 13

Fractal dimension of the black pixel sets vs image order for the 4 pathologists.

Figure 14

Deviation of all black pixel BCA curves from their average.

Figure 15

Deviation of all white pixel BCA curves from their average.

Figure 16

Fractal dimension vs cell differentiation for black and white pixel sets with Linear fits.

Figure 17

Fractal dimension vs cell differentiation for black and white pixel sets with quadratic fits.

Figure 18

Rate of change of $I_{\min }$ vs cell differentiation.