The brain cancer is a very serious disease and difficult to treat. There are over 120 different types of brain tumors, which are making treatment complicated. Therefore, numerical approaches based on mathematical models of the drug penetration into the brain tumor can give results much faster than real experiments, which is very important if a tumor is progressive. However, there are still several problems, for example, how deep given drugs can penetrate into the brain tumor and how to overcome physiological barriers which hinder the motion of drugs inside of the tumor.

The mathematical model of drug delivery to brain tumors is not new, there are various configurations of different complexity, for example, in [1] authors consider the full momentum equation instead of Darcy’s law, but they neglect a few terms in the drug concentration equation. Authors in the article [3] take up Darcy’s law instead of full momentum equation. 

In our work, the governing system of equations was taken from [1] and consists of:
-	the mass conservation equation
 

-	the momentum equation 

 

-	the drug concentration into the brain tumor

  ,

where   and   are the net gain of fluid from blood vessels and the net fluid loss to the lymphatic per unit volume of tissues, correspondingly. The fluid sources  were described by Starling’s law:

 ,

where   and   are the interstitial and the vascular pressures,   and   are the osmotic pressures of plasma and interstitial fluid,    is the hydraulic conductivity of the microvascular wall multiplied by the exchange area of blood vessels per unit volume of tissues,   is the osmotic reflection coefficient for the plasma proteins,  and   are the fluid density and viscosity,   and   are the stress tensor and gravity acceleration,   and  are prescribed matrices for the inertial and the viscous loss term, respectively. 
The domains include such areas as:
-	wafers, where  ,  ;
-	cavity after surgery, where  ,  ;
-	remnant tumor tissues and surrounding normal tissues, where  ,    

This system describes the penetration of drugs into the brain tumor (there is a cavity after surgical removal of a cancer tumor), which fill up the cavity after a surgery. We consider full mass conservation and momentum equations which are coupled with the concentration equation depending on the velocity and prescribe at the interior boundaries continuity of pressure, velocity, concentration and drug flux. We do not neglect any terms in drug concentration equation. We want to describe the blood brain barrier (BBB) in the given mathematical model of drug delivery to brain tumor and to see the BBB influences on the penetration of the drug to brain tumor and surrounding brain tissues.

We use techniques of computational fluid dynamics (CFD) to get a solution of the derived partial differential equations (Navier-Stokes equation with the added equations and terms). First of all, we are non-dimensionalized the governing equations. After discretization of the governing system of equations with finite elements (discretizing in space and in time (using Crank-Nicolson method or Backward Euler)) we obtain following system of equations:

 ,

where . We can easily see that the given problem is a saddle point problem. It is well know problem for CFD, which request special technique [5]. We use modern CFD tools like FEATFLOW to get numerical solutions of this problem. Before using Featflow we make theoretical analysis the system of equations. The analysis shows us that we should add additional equations into the code of Featflow that is a trace equation. All rearrangement of the code will be in the post-processing part (pure post-processing problem), because the additional equation does not couple with Navier-Stokes equation, i.e. concentration of drug does not have any influence on the pressure and velocity.

We have obtained a closed system of partial differential equations (PDE’s) for pressure, vector of velocity and drug concentration with defined boundary and initial conditions. We are non-dimensionalized the given equations, which helped to simplify the given equations. We found out that the saddle-point problem requires techniques for incompressible flow problems and we can solve these CFD problem using numerics methods for CFD. We got the simple analytical solutions of the given problem, which will help to estimate the accuracy of the numeric solution.