Sprout Branching of Tumour Capillary Network Growth: Fractal Dimension and Multifractal Structure

A tumour vascular network, characterized as an irregularly stochastic growth, is different from the normal vascular network. We systematieally analyse the dependence of the branching. It is found that anastomosis of tumour on time is according to a number of tumour images, and both the fractal dimensions and multifractal spectra of the tumours are obtained. In the eases studied, the fractal dimensions of the tumour vascular network increase with time and the multifractal spectrum not only rises entirely but also shifts right. In addition, the best drug delivery stage is discussed according to the difference of the singularity exponent δα（δα = αmax - αmin）, which shows some change in the growth process of the tumour vascular network. A common underlying principle is obtained from our analysis along with previous results.     

Analysis of thermal conductivity in tree-like branched networks

Asymmetric tree-like branched networks are explored by geometric algorithms. Based on the network, an analysis of the thermal conductivity is presented. The relationship between effective thermal conductivity and geometric structures is obtained by using the thermal-electrical analogy technique. In all studied cases, a clear behaviour is observed, where angle (δ ,θ ) among parent branching extended lines, branches and parameter of the geometric structures have stronger effects on the effective thermal conductivity. When the angle δ is fixed, the optical diameter ratio β^* is dependent on angle θ . Moreover, γ and m are not related to β ^* . The longer the branch is, the smaller the effective thermal conductivity will be. It is also found that when the angle θ < δ / 2, the higher the iteration m is, the lower the thermal conductivity will be and it tends to zero, otherwise, it is bigger than zero. When the diameter ratio β ₁< 0.707 and angle δ is bigger, the optimal k of the perfect ratio increases with the increase of the angle δ ; when β ₁> 0.707, the optimal k decreases. In addition, the effective thermal conductivity is always less than that of single channel material. The present results also show that the effective thermal conductivity of the asymmetric tree-like branched networks does not obey Murray's law.     

Tortuosity for streamlines in porous media

An analysis of tortuosity for streamlines in porous media is presented by coupling the circle and square models. It is assumed that some particles in porous media do not overlap and that fluid in porous media is incompressible. The relationship between tortuosity and porosity is attained with different configurations by using a statistical method. In addition, the tortuosity fractal dimension is expressed as a function of porosity. Those correlations do not include any empirical constant. The percolation threshold and tortuosity fractal dimension threshold of porous media are also presented as: c = 0.32, D T c = 1.07. The predicted correlations of the tortuosity and the porosity agree well with the existing experimental and simulated results.