Numerical simulation of blood flow through permeable vascular network embedded in tumour porous interstitium

Tumour blood flow plays a critical role in tumour growth and cancer therapies. Computational fluid dynamics is an efficient method to study blood behaviour by modelling fluid flow through numerical simulations. A mathematical model is developed to study the blood flow through a three-dimensional permeable vascular network embedded in a solid tumour, and its transvascular movement and spread within tumour interior in context with cancer therapies. The vasculature is described by the parametric equations in terms of vessel centre lines. The flow through each tumour vessel is approximated with the leading component in the longitudinal direction of the vessel, and its governing equation becomes an ordinary differential equation based on the parameter of the parametric equation for the vessel centre line. The pressure continuity and mass conservation conditions are imposed at every junction within tumour vascular network. The interstitial flow is described by the Darcy’s law which is converted into the Laplace equation. The coupling effect between the flows through tumour vasculature and within tumour interstitial due to the vascular permeability is described by the Starling’s law. A coupling mathematical model is then developed. Based on mass conservation, a differential equation for pressures on both sides of vascular surface is obtained. Transforming the Laplace equation into the boundary-integral form by using the Green’s function offers another equation linking the pressures inside and outside vessels. The numerical procedure is developed, and the discretised differential and integral equations are solved by finite difference method and boundary element method respectively. The model is applied to investigate how different types of physical parameters and special characters of tumour vasculature affect tumour blood flow. Finally, an approximation model by ignoring the term with small value of the fully coupling model is developed, and its validity and simulation efficiency are examined