Nutrient transport through deformable cylindrical scaffold inside a bioreactor: An application to tissue engineering

Tissue engineering deals with the development and growth of cells to obtain tissue structure or organs. In this process, a porous material known as a scaffold is designed to support the cells to grow in the desired form. This designed structure is incubated inside a bioreactor which provides adequate nutrient supply for the cells to grow. The major challenge in tissue engineering is to model a bioreactor and the scaffold which supports high-density cells so that adequate nutrient can be delivered in inner regions to get the significant cell growth. Mathematical modeling of such processes is essential to understand the entire mechanism. Correspondingly, a mathematical model for fluid flow and nutrient transport is developed inside a cylindrical bioreactor containing tubular scaffold which is degradable as well as deformable due to its elastic property. Living biological cells are assumed to adhere to the solid matrix of scaffold firmly. The volume fraction of the solid phase which includes cells and scaffold matrix is assumed to be constant despite the fact that proliferation of cells and degradation of scaffold matrix are undergoing simultaneously. Fluid flow and deformation of the tissue (scaffold) are modeled based on biphasic mixture theory. Navier-Stokes equation accounts for the free fluid in the annular region of the bioreactor. Advection-diffusion-reaction equation governs the concentration of a nutrient in the scaffold region whereas advection-diffusion equation governs the nutrients in the free lumen. Semi-analytical treatment on the mathematical model includes Laplace transformation to deal with the time dependency of the governing fluid flow equation and consequently, Durbin's algorithm is used to retrieve the time-dependent variables. The model allows the uniform nutrient transport inside the scaffold region so that cells get sufficient amount of nutrient to grow and reduce the cell death. Moreover, Sherwood number has been calculated to analyse the mass transport of solute concentration from scaffold region to the free fluid region and vice-versa through the interface.