Tissue engineering utilizes synthetic scaffolds to serve as the extracellular matrix for culturing cells into tissue,...

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Tissue engineering utilizes synthetic scaffolds to serve as the extracellular matrix for culturing cells into tissue, which may be transplanted into a patient. Since the artificial tissue is not vascularized, transport of nutrients and oxygen to cells is limited by diffusion through the matrix. Consider a collagen scaffold, approximated as a slab, as shown in Fig. 1. Both sides of the slab are exposed in a symmetric manner to nutrients (including glucose) and oxygen that maintain a constant concentration at the surface, diffuse through the scaffold, and are depleted by the metabolism of the growing tissue. The rate of depletion is constant, at 104 g/m³ day for glucose. The diffusivity of glucose is 6 x 10¹¹ m²/s. a) Starting from the most general differential equation, simplify it based on the assumptions applicable to the problem. b) Write the two boundary conditions needed to solve this problem. One of the boundary conditions should be a symmetry condition since both sides of the slab are exposed to the same conditions. c) Solve the differential equation in (a) using the boundary conditions in (b) to get an expression for glucose concentration as a function of position (don't plug in numbers, use given variables). d) Use your equation in (c) to determine the minimum concentration for glucose that must be maintained at the surface so that it reaches across the entire thickness of the scaffold, in g/m³. (Hint: At the minimum surface concentration, the concentration at the center of the slab is zero.) +x 2|7 L = 2.5 mm thickness CAS concentration of glucose at surface Scaffold Fig. 1. A scaffold for artificial tissue through which glucose must diffuse Tissue engineering utilizes synthetic scaffolds to serve as the extracellular matrix for culturing cells into tissue, which may be transplanted into a patient. Since the artificial tissue is not vascularized, transport of nutrients and oxygen to cells is limited by diffusion through the matrix. Consider a collagen scaffold, approximated as a slab, as shown in Fig. 1. Both sides of the slab are exposed in a symmetric manner to nutrients (including glucose) and oxygen that maintain a constant concentration at the surface, diffuse through the scaffold, and are depleted by the metabolism of the growing tissue. The rate of depletion is constant, at 104 g/m³ day for glucose. The diffusivity of glucose is 6 x 10¹¹ m²/s. a) Starting from the most general differential equation, simplify it based on the assumptions applicable to the problem. b) Write the two boundary conditions needed to solve this problem. One of the boundary conditions should be a symmetry condition since both sides of the slab are exposed to the same conditions. c) Solve the differential equation in (a) using the boundary conditions in (b) to get an expression for glucose concentration as a function of position (don't plug in numbers, use given variables). d) Use your equation in (c) to determine the minimum concentration for glucose that must be maintained at the surface so that it reaches across the entire thickness of the scaffold, in g/m³. (Hint: At the minimum surface concentration, the concentration at the center of the slab is zero.) +x 2|7 L = 2.5 mm thickness CAS concentration of glucose at surface Scaffold Fig. 1. A scaffold for artificial tissue through which glucose must diffuse