Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis

Abstract

Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed.

Figure 1.

A cauliflower plant as an example for a naturally occurring fractal structure. A self-similar pattern across different magnification scales (A–C) is visible in the ramification pattern as well as in the surface structure. When trying to measure the surface, we will recognize an astonishing phenomenon: with increasing zoom factor (A–C), we will detect more and more structural details and surface infoldings, which will result in an increase of the surface.

Figure 2.

Schematic representation of the fractal globule model after crumpled folding. Open (transcriptionally active) chromatin drawn in white and closed (silent) chromatin in black symbolizing darker heterochromatin and lighter euchromatin in stained nuclei.

Figure 3.

Microscopic image of a H&E-stained cytologic preparation with (segmented) nuclei of a 60 days old Wistar rat.

Figure 4.

Pseudo-3D representation of Figure 3. In this image, the inverse gray level (256 gray values) of each pixel represents the height on the z-axis. In that way, white pixels correspond to the bottom and black pixels to maximum height.

Figure 5.

Decreasing fractal dimension of the cardiomyocyte chromatin with increasing age of normal rats in H&E-stained cytologic preparations. X-axis shows age in days after conception.

Figure 6.

Regression plot of the fractal determination in a blast nucleus of a patient with acute precursor B lymphoblastic leukemia. The y values of the measured values (diamonds) are rather close to those of the calculated (ideal) regression curve. The goodness-of-fit (R² = 0.976) shows that the observed structure approximates an ideal fractal. Scaling window: 200 nm–6.000 nm.

Figure 7.

Regression plot of the fractal determination in a blast nucleus of another patient with acute precursor B lymphoblastic leukemia. In this case the y values of the measured values (diamonds) are more distant to the calculated (ideal) regression curve. The goodness-of-fit (R² = 0.9395) shows that the observed structure is more distant to an ideal fractal than in Figure 6. Scaling window: 200 nm–6.000 nm.

Figure 8.

Estimation of the fractal dimension of a myeloma cell nucleus. (A) Myeloma cell nucleus after segmentation (left) and its pseudo 3D transformation (right). (B) The log-log-plot (also called log-log plot or log-log graph ) for the determination of the fractal dimension (FD) is estimated from the slope of the ideal regression line (black) obtained by curve fitting. X-axis shows the logarithms of the inverse values of the size of the structuring element and y axis the logarithmic values of the fractal areas. The fractal dimension (FD) is derived from the slope of the regression line. Scaling window: 200 nm–6.000 nm.

Figure 9.

Kaplan–Meier survival plot of patients with multiple myeloma. Shorter survival for patients with higher values of the fractal dimension of nuclear chromatin in routinely stained bone marrow smears.