Exploring the Fractal Properties of Embryonic Mice Lungs

Abstract

While the fractal nature of the adult lung has been studied somewhat extensively, very little work has been done in identifying and quantifying the fractal proper- ties of the lung during development. This paper looks at the fractal properties of the mouse lung from embryonic days E11 - E14 to identify scaling rules and relationships implicit in the lung. The fractal dimension of the embryonic lungs were calculated, and found to grow with number of terminal buds as a power curve y = a * xb with a = 1.83 (0.021) and b = 0.043 (.002). This formula predicts the adult lung will have a fractal dimension of around 2.73 (0.05), consistent with some of the literature. The exact scaling patterns of the bronchial tree was also explored. It was found that the orthogonal bifurcation regime of the developing lung displayed monopodial, strict self-similarity with a consistent length to diam- eter ratio of 2.99 and bifurcation ratios of 0.858 for the major branch and 0.686 for the minor branch. The domain branching regime, which is responsible for the first few generations of branching, on the other hand showed no consistent relationship between length and width of branches nor between parent and daughter branch diameters. It was however found that, when fit with a re-normalization model accounting for multiple scales, branches that are the product domain branching can be found to have length and width that both grow with the same power law, subject to harmonic modulations with the same shape. However the scaling factor of the models for length and width were not found to have a consistent value.