(568j) On Damped Wave Diffusion of Oxygen in Iselts of Langerhans: Part-IV Comparison of Parabolic and Hyperbolic Models in a Finite Slab for Convective Boundary Condition | AIChE

(568j) On Damped Wave Diffusion of Oxygen in Iselts of Langerhans: Part-IV Comparison of Parabolic and Hyperbolic Models in a Finite Slab for Convective Boundary Condition


Sharma, K. R. - Presenter, Prairie View A & M University

In some tissue regions within the human anatomy, the oxygen supply becomes limited. The metabolic reactions demad oxygen to a larger extendt compared with the oxygen that has diffused to that region. Oxygenation is a diffusion limited process. Growth of multicellular systems over 100 microns does not come about. Hypoxia condition has been found in Brackmann bodies in fish. The necrosis condition where the cells begin to die for lack of oxygen supply can be measured using microelectrodes. Oxygen is carried in blood by convection. Islets of Langerhans are spheroidal aggregates of cells located in the pancreas. They secrete harmones that are involved in glucose metabolism, particularly insulin. Transplantation of isolated cells is a promising treatment for some forms of Type I diabetes. Islets removed from teh pancreas are devoid of their internal vacularization. The metabolix requirement of the cells requires oxygen to diffuse from the external environment and through the oxygen-consuming islet tissue. The oxygen supply is a critical limiting factor for the functionality and feasibility of islets that are encapsulated, placed in devices for implantation, cultured and used in anaerobic conditions. Theoretical models are needed to describe the oxygen diffusion. The parameters of the model require knowledge of the consumption rate of oxygen, oxygen solubility and the effective diffusion coefficient to oxygen in the tissue. During the time scales associated with the transport of oxygen from the blood stream through the wall, into the tisse region the non-Fick finite speed wave diffusion effects have not been accounted for. The following is a series of studied on modeling wave diffusion of oxygen in the iselts of langerhans. In Part-IV, at asymptotic limits of low and high oxygen concentration, the parabolic Fick diffusion and hyperbolic wave diffusion models are developed for a finite slab for the case of convective boundary condition. The governing equation for oxygen diffusion and reaction in cartesian coordinates in one dimension can be written by combining the Michaelis-Menton kinetic rate expression and Fick's second law of diffusion. The equation is a second order PDE in space and time variables. The oxygen consumption rate is assumed to obey the Michaelis-Menten kinetics. The governing equation describes the interplay of transient diffusion and metabolic consumption of oxygen in the tissue. The concentration of oxygen can be expressed in terms of its partial pressure and a Bunsen solubility coefficient. The product of solubility coefficient and diffusion coefficient gives the permeability of oxygen in tissue. The concentration profile in a finite slab at the asymptotic limit of high oxygen concentration is obtained by the method of separation of variables. An infinite Fourier series solution results. The integration constants are solved for from the initial condition. The Eigen values are calculated from the boundary conditions. Transcedental equations can be reduced to algebraic equations by suitable assumptions. The damped wave diffusion and relaxation effects are included in the governing equation for oxygen concentration. This corrects for the finite speed of propagation of oxygen species. At the asymptotic limit of high oxygen concentration closed form, direct analytical solution is obtained for the hyperbolic PDE. The infinite Fourier series solution in space and time is within the bounds of second law of thermodynamics. Compared with other reports in the literature for similar problems in this study a final condition in time was used to capture the steady state attainment. This may be lead to well bounded solutions. The product of wave concentration and decaying exponential in time term can be shown to tend to zero at infinite time using L'Hospital's rule. Anoxic regions are derived. Expressions for Penetration length and inertial time lag are presented. When the relaxation times of the tissue material becomes larger than a threshold value subcritical damped oscillations in concentration can be expected. Expressions for time taken to steady state are derived. The conditions where the concentration profile changes from convex to concave curvature are derived.