Multi-fractal analysis of the ossification process in developing skull bone

Different bones in vertebrates have different structural and functional roles. Long bones, such as the femur, exist under near continuous load and their development and formation reflect this role. Skullbone, on the other hand, exists in the entirely different context of occasional impact absorption in the service of protecting the brain. In spite of the importance of this function, the development of this tissue is not well-studied, mostly due to the complexity of the pattern by which the bone grows. We employ 2D multi-fractal analyses to study the ossification pattern of skull bone in developing mouse embryos. We first use the Multi-Fractal Detrended Fluctuation Analysis (MF-DFA) to investigate the multi-fractal features of the ossification patterns and their evolution over time. We then use multiple surrogates to determine the origin of the observed multi-fractality. Moreover, using the wavelet transform modulus maxima method, we obtain spatial information about the singularities whose spectrum is given by MF-DFA. We finally simulate some basic processes contributing to ossification and compare their analysis results to the ones from real data to rule out the processes which do not result in multi-fractality.