Quantitation of the renal arterial tree by fractal analysis

To determine whether the renal arterial system has a fractal structure, the fractal dimension of renal angiograms from 52 necropsy cases was measured using an implementation of the box-counting method on an image analysis system. The method was validated using objects with known fractal dimensions. The method was accurate with errors of less than 1.5 per cent and reproducible with initial values within 1.2 per cent of the mean of ten sets of measurements (reliability coefficient 0.968, 95 per cent confidence limits 0.911-0.984). In the 36 satisfactory angiograms the mean fractal dimension was 1.61 (SD 0.06), which was significantly greater than the topological dimension of 1 (P < 0.0001), indicating that the renal arterial tree has a fractal structure. There was no significant relationship between age (P = 0.494), sex (P = 0.136), or systolic (P = 0.069) or diastolic (P = 0.990) blood pressure, but two congenitally abnormal kidneys (hypoplastic dysplasia and renal artery stenosis) had fractal dimensions at the lower end of the normal range (third percentile). Since the renal arterial tree has a fractal structure, Euclidean geometric measurements, such as area and boundary length, are invalid outside precisely defined conditions of magnification and resolution.