Nutrient diffusion and simple nth-order consumption in regenerative tissue and biocatalytic sensors

This contribution addresses intra-tissue molar density profiles for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. One-dimensional transient and steady state models of the reaction-diffusion equation are solved to correct a few deficiencies in the first illustrative example of diffusion and zeroth-order rates of consumption in tissues with rectangular geometry, as discussed in Ref. [(Griffith and Swartz, 2006) 1]. The functional form of the molar density profile for each species depends on geometry and the magnitude of the species-specific intra-tissue Damköhler number. The tissue's central core is reactant starved at high consumption rates and low rates of intra-tissue diffusion when the Damköhler number exceeds its geometry-sensitive critical value. Ideal tissue engineering designs avoid the diffusion-limited regime such that attached cells are exposed to all of the ingredients required for proliferation everywhere within a regenerative matrix. Analytical and numerical molar density profiles that satisfy the unsteady state modified diffusion equation with pseudo-homogeneous nth-order rates of intra-tissue consumption (i.e., n = 0,1,2) allow one to (i) predict von Kármán-Pohlhausen mass transfer boundary layer thicknesses, measured inward from the external biomaterial surface toward its central core, and, most importantly, (ii) estimate the time required to achieve steady state conditions for regenerative tissue growth and biocatalytic sensing.

Graphical AbstractAnalytical and numerical molar density profiles that satisfy the unsteady state modified diffusion equation with pseudo-homogeneous nth-order rates of intra-tissue consumption (i.e., n = 0,1,2) allow one to (i) predict von Kármán-Pohlhausen mass transfer boundary layer thicknesses, measured inward from the external biomaterial surface toward its central core, and, most importantly, (ii) estimate the time required to achieve steady state conditions for regenerative tissue growth and biocatalytic sensing.Numerical analysis of dimensionless nutrient molar density vs. spatial coordinate η for one-dimensional diffusion and simple first-order consumption at various dimensionless diffusion times τ via Eq. (22). The intra-tissue Damköhler number ΛA is greater than its critical value for regenerative tissue and biocatalytic sensors with rectangular geometry. The thinnest spatial dimension (i.e., x) is divided equally into 101 mesh points for finite-difference calculations. The analytical molar density profile at steady state is given by [4,5,9]: ΨA(η,ΛA) = cosh(ΛAη) / cosh(ΛA), uppermost curve, which yields ΨA(η = 0,ΛA) = 1 / cosh(ΛA) = 0.266 in the tissue's central core when ΛA = 2.Research Highlights► Reaction-diffusion equation: A modification of Fick's second law is analyzed in porous biomaterials that contain anchorage-dependent cells. ► Critical intra-tissue Damköhler number: The rate of nutrient consumption relative to the rate of intra-tissue diffusion is quantified for 3 biomaterial geometries. ► von Kármán-Pohlhausen mass transfer boundary layer thicknesses are calculated to estimate the time required to achieve stable conditions for regenerative tissue growth. ► Effect of reaction kinetics and biomaterial geometry on time-dependent mass transfer boundary layer thickness.