Fractal Dimension for Pathology Images, a Repeatable and Quantitative Measurement of Nuclear Rim Irregularity.

ABSTRACT.
1.  Irregularity of the nuclear rim is an important, distinguishing
 feature of malignant cells in cytologic preparations.
2.  Traditionally, pathologists have employed descriptive terms:
 smooth, irregular, convoluted, indented.
3.  Fractal dimension has been proposed as a measure for irregularity
 of as planar object, such as the nuclear rim in a cytologic image.
4.  We measured fractal dimension for 51 cells from Papanicoloau-
 stained cytology smears, each cell measured 10 times apiece.
5.  By one-way analysis of variance, fractal dimension is a repeatable
 measure of cell irregularity for cytologic preparations, p<0.001.


INTRODUCTION TO FRACTALS.
1.  Irregularity of the nuclear rim is considered to be an important,
 distinguishing feature of malignancy in cytologic preparations.
2.  Fractal is a geometric figure in which an identical motif repeats
 itself on an ever-diminishing scale;  popularized by
 Benoit B. Mandelbrot.
3.  Lewis Fry Richardson (1881-1953), an English mapmaker, attempted
 to determine the length of the west coastline of Britain and length
 of the Spanish-Portuguese land frontier, using fractals.
4.  As the scale of the map becomes more refined, the coastline
 and frontier become longer.
5.  In pathology, used to describe the edge of an irregular,
 2-dimensional object, such as the surface of polyps
 in the colonic mucosa.
6.  See:  Lauwerier H.  Fractals, endlessly repeated geometrical
 figures. Princeton, NJ:  Princeton University Press, 1991.


MATERIALS AND METHODS.
1.  Papanicolaou-stained cytology slides from
 normal and dysplastic cervix, lymphoma, melanoma, hepatocellular
 carcinoma, obtained in routine patient care setting at the Baltimore
 Veterans Affairs Medical center (BVAMC).
2.  In each of 51 cells, 10 measurements per cell, for a total
 of 510 separate analyses.
3.  Black-white images obtained as uncompressed 756 x 486 = 367,416
 pixels, with 256 grayvalues per pixel.  40x microscope objective;
 ccd camera; ATVista image grabber board.


FRACTAL VERSUS CLASSICAL GEOMETRY.
1.  For any 2-dimensional object, you can place an overlying
 measurement grid, and estimate the perimeter at each point where
 the object where the object intersects the measurement-grid.
2.  The estimated perimeter of the object is never larger than
 the length of the actual perimeter of the irregular object.
3.  If the measurement-grid is finer, then the estimated perimeter
 becomes longer.
4.  If the object is a classical (Euclidean) geometric figure,
 then with finer-and-finer grids, the estimated perimeter approaches
 a constant.
5.  For an indefinitely irregular (non-Euclidean) object, the
 perimeter becomes indefinitely longer as the grid becomes finer.


FRACTAL DIMENSION.
1.  For an indefinitely irregular (non-Euclidean) object, the
 estimated perimeter becomes indefinitely longer as the grid
 becomes finer.
2.  If the estimated perimeter becomes longer at a constant rate,
 then this rate+1 (i.e., slope+1) is called the `fractal dimension'
 of the object.
3.  For a classical plane-geometric object, fractal dimension, s=1.
 for a `space-filling-edge', fractal dimension, s=2.  For all other
 objects, 1 < s < 2.
4.  s = 1 + lim(d->0) log2((perimeter at 2*d)/(perimeter at d)),
 for gridsize=d, as gridsize approaches 0.


LIMITS OF FRACTALS IN PATHOLOGY.
1.  Grid-interval cannot exceed the size of the object-to-be-measured
 (i.e., nuclear diameter).
2.  Grid-interval cannot be less than pixel-width.
3.  Two grid-intervals determine a slope.  the two grid-intervals
 must be far enough apart to exceed measurement errors.
4.  More-than-two grid-intervals determines a slope by regression
 analysis.  More grid-intervals are better, but they may also encroach
 on the limits of measurement.


FIGURE 1.  In cytology, a malignant cell is expected to have a more
 irregular nuclear rim than  a benign cell.  On the left is
 a schematic of a normal nucleus;  on the right is a schematic
 of a malignant nucleus;


FIGURE 2.  For a 2-dimensional object with an overlying measurement
 grid, and you can estimate the perimeter (thick line) at each point
 where the nuclear rim intersects the grid.  This diagram illustrates
 nuclear perimeter estimated from the coarse grid.


FIGURE 3.  If the measurement-grid is finer, then the estimated
 perimeter becomes larger.  This diagram illustrates nuclear perimeter
 estimated from the fine grid.  Estimated perimeter is expected
 to increase faster for an irregular nucleus.


FIGURE 4.  Image from a Papanicolaou-stained cytology slide of the
 vagina and uterine cervix.  A single, intermediate-cell nucleus has
 been outlined by the user, and an automatic thresholding algorithm
 has traced the nuclear rim.


FIGURE 5.  Distribution of fractal dimensions, ranging from s=1.08
 to s=1.91.


FIGURE 6.  Typical nuclear rim with a small fractal dimension.
 The nuclear rim appears smooth and regular.


FIGURE 7.  Typical nuclear rim with a large fractal dimension.  The nuclear rim appears coarse and irregular.


RESULTS.
1.  Fractal dimensions ranged in value from 1.08 to 1.91,
 average 1.37.
2.  For a single cell, measured repeatedly, the coefficient
 of variation (standard deviation/mean) of fractal dimension
 varied between 0.8% and 15.5%.
3.  In one-way analysis of variance, within-cell variance was
 significantly smaller than between-cell variance,
 F(50;459) = 14.0, p<0.001.


CONCLUSIONS.
1.  Fractal dimension is a repeatable measure of cell irregularity
 for cytologic preparations.
2.  Further studies are needed to determine the value of fractal
 dimension as a predictor of cytologic malignancy.