The Study Of Mathematical Model Of Capillary Growth And Networking Based On Reaction-Diffusion Equation

In vitro capillary network construction plays a key role in the survival of organs in vitro.However,experimental methods for the construction of capillary network can not meet the needs of tissue nourishment in terms of structure and function,so it is urgent to study the formation mechanism of capillary network.Vascular model based on mathematical equations is an important way to explore the principle of angiogenesis and guide experiments.It mainly simulates the branching,anastomosis,network and other structural characteristics of capillary network through numerical methods to explore and predict the principle and process of capillary growth.Capillary network is the most important structural feature to ensure the stable perfusion function of capillaries.At present,most vascular mathematical models have been able to simulate the branching characteristics of capillary network,but few studies have successfully simulated all the anastomosis characteristics of capillary network,especially the characteristics of network formation,so the existing vascular mathematical models are not able to achieve the true simulation of capillary network.In this paper,a blood vessel model based on the traditional reaction-diffusion equation is established to simulate the structural characteristics of capillary network.Then,the factors affecting the vascular network,including the velocity of tip cell movement,the distribution of VEGF distribution,the diffusion coefficient of VEGF and the initial distribution of vascular sprouts,were quantified.The quality of vascular network was measured by counting the number of branches,anastomosis,vascular length and vascular nourishing area.The simulation results show that the tip cells are more conducive to angiogenesis at low speed.The central linear VEGF distribution is more conducive to angiogenesis.When the diffusion coefficient of VEGF was 10^-8cm~2/s,the vessel had the best net-forming effect.Appropriately increasing the number of vascular buds is beneficial to the formation of vascular network.Finally,based on the application background of microfluidic chip vascular research,it is concluded that the flow of more than 10^-7m/s in microfluidic channel is not conducive to the formation of vascular network.In order to further simulate a more realistic capillary network,a 3D vascular model based on the phase field reaction-diffusion equation was constructed and explored in this paper.The influence of the relative parameters of the phase field governing equation,including the upper limit of the order parameter,the diffusion coefficient,the interface coefficient and the proliferation rate,on the vessel diameter and the vessel wall was illustrated.Then,the vascular growth and netting of the phase field vascular model in 2d and 3D space are studied.By controlling the initial distribution and concentration gradient of VEGF,the pattern of vascular growth and netting was analyzed.The simulation results show that the large and small distribution range and concentration gradient of VEGF are not conducive to the formation of vascular network.Only when the distribution range and concentration gradient of VEGF are at an appropriate value,it is conducive to the formation of vascular network.Based on two vascular mathematical models,this paper studies the problem of capillary network formation.On the basis of successfully simulating the characteristics of vascular network,it studies the quantitative influence of factors such as VEGF distribution on vascular network formation,and obtains the most favorable conditions for vascular network formation,and actively explores the process of capillary network formation.It is a valuable reference for the experimental design of constructing capillaries in vitro.