| Tree-like networks exist widely in nature (for example, lungs and vascular systems in mammals, river basins and plants) and are believed to be the fundamental roots of many important allometric laws in biology. They have found applications in many man-made systems such as the worldwide web, road networks, water and energy transportation systems. In recent hundred years, tree networks have received increasing attention. It has been shown that in natural systems tree networks often give the minimal resistance and optimal vascular diameter for driving the blood in mammals and water in plants. These mechanisms can be applied in design of energy transport systems and cooling systems of electronic chips due to increasing miniaturization of chips in microelectronic equipment and the production of redundant heat under nature inspired. For a special bifurcate-tumor vascular network, the research of tumour angiogenesis is one of the important subjects, such as tumor, medicine, etc.The main works:Firstly, under the constraint of surface conservation, the relationship between thermal conductivity of composites with self-similar fractal tree-like networks and geometrical parameters of the tree-like network are analyzed by using the thermal-electrical analogy technique. From the study, it is shown that the dimensionless effective thermal conductivity of the tree-like network decreases withthe increase of bifurcation number N, branching length ratioγ, branching levels m or fractal dimensions of channel length D when other parameters are kept constant. It is also found that the dimensionless effective thermal conductivity of the tree-like networks reaches maximum when the diameter ratioβ* satisfiesβ* = N-1/△, where△=0.5, N is the bifurcation number N=2, 3, 4, .... In addition, under the constraint of volume conservation, the relationship between permeability of composites with self-similar fractal tree-like networks and geometrical parameters are analyzed. From the study, it is shown that, the dimensionless effective permeability of the tree-like-network decreases with the increase of parameters (N,γ, m, D) when other parameters are kept constant. It is also found that, the dimensionless effective permeability of the tree-like networks reaches maximum when the diameter ratioβ* satisfiesβ* =N-1/△, where△= 3, N=2, 3, 4, .... This optimal diameter ratio for maximumeffective permeability of the fractal tree-like networks obeys Murry's law.Secondly, fluid flow tree-like networks is simulated by Fluent software. Results show that a highest pressure is near the main stem inlet position, whereas the lowest pressure can be noticed at the outlet position; at the braching postitions of the stem highest velovity can be observed and the liquid reaches highest velocity at the final branch as well as out-let positions.At last, the process of angiogenesis is simulated by 2D discrete mathematical model. The fractal dimensions and multifractal spectra are calculated using simulation tumour images. From the results, it can be seen that with increasing time, the pattern structure is strongly affected, and the fractal dimension D increases. The multifractal spectrum indicates that the cluster grows less regularly and less uniformly when the time is longer. The multifractal spectrum curves entirely rising and a shifted of the multifractal spectrum towards higher singularity exponent values with increasing time. The singularity exponent difference is not a constant, which shows some transformation. The best drug delivery stage is discussed at about four days and a half.
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