Box-Counting Fractal Dimensions of Cranial Suture: Effects of Measurement Conditions and Model-Based Reproduction of Fractal-Like Patterns

Cranial sutures are important structures associated with skull growth, and it is widely believed that cranial sutures exhibit fractal structure. However, measurement conditions and analytical procedures have varied among studies, making direct comparison and interpretation difficult. In this study, we reviewed previous measurements and established a standardized box-counting protocol for the fractal dimension (FD) of cranial sutures by quantifying the effects of skeletonization and scale-range selection. Using this protocol, we quantified FD in 45 digitized images of human lambdoid sutures and in eight structure-formation model variants: step, Gaussian, and Mexican-hat kernels; absolute-value and sine $F_{\mathrm{base}}$ fields; low-frequency noise; a time-dependent kernel; and a given-pattern initialization. We tested whether each model can generate structures with FD values at least as high as those of real sutures using one-sided Welch $t$-tests with Bonferroni correction for eight comparisons ($\alpha_{\mathrm{adj}} = 0.00625$), and found that six of eight model variants satisfied this criterion. Analysis of scale-dependent FD further revealed that FD is close to 1 at fine scales and approaches 2 at coarse scales in both real and model-generated patterns, indicating that cranial sutures are better described as finite-range fractal-like structures than as strict fractals, and that the box-counting FD is a scale-conditioned descriptor sensitive to preprocessing choices.