Goodness-of-Fit of the Fractal Dimension as a Prognostic Factor

One of the most important tasks in quantitative pathology is to elaborate methods which help to In a previous investigation we found that the frac-tal dimension (based on the Minkowski–Bouligand method extended to three dimensions) of nuclear chro-matin was not a prognostic factor in patients with acute precursor B lymphoblastic leukemia [1]. When looking at the log–log plots (area versus ε) containing 30 points used to calculate the regression line, we noticed that in most of the cases the real values were not located very closely to the calculated linear regression. This prompted us to investigate, whether the goodness-of-fit of the regression line could be of prognostic value. Therefore we calculated the coefficient of determination , i.e., the R 2 value of the regression between the real and the estimated values on the y-axis. R 2 characterizes the goodness-of-fit of the regression line and is therefore an estimate of the " quality of fractality ". It could be demonstrated that this goodness-of-fit is an independent prognostic factor for overall survival when applied to the chromatin texture of MGG-stained routine bone marrow smears of patients with acute precursor B lymphoblastic leukemia [1]. Furthermore, it could be shown that this variable is relatively robust against variations of the staining process [13]. We agree with the arguments of Prof. Albregtsen. Therefore we reanalyzed the data [1] in the following way: The fractal dimension (FD) was determined as described earlier: FD was estimated by the volume/2ε (being ε the radius, varying between 1 and 30 pixels) of the non-planar structuring element in form of a ball. The linear regression was calculated in a log–log plot (area versus ε) containing 30 points. The difference between the " real " y value and its corresponding " ˆ Y " on the regression line was called " res ". The square value of the correlation coefficient between " y " and " ˆ Y " was called R 2 (the coefficient of determination). Now we calculated the sum of the square values of the 30 residuals per case " Sum res 2 ". In a second step we determined the perpendicular residuals " res p " , calculating in analogy the variable " Sum res p 2 ". In a third step the above mentioned point distribution was rotated so that the slope was at 45 degrees and then we calculated the coefficient of determination between " …