Mathematical Problems in Engineering
Volume 4 (1998), Issue 5, Pages 437-459

A colsed form solution of convective mass transfer model for intracellular calcium response of endothelial cells

Vineet Kumar,1,3 R. N. Pandey,2 and S. N. Upadhyay1

1Department of Chemical Engineering & Technology, Institute of Technology, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India
3School of Biochemical Engineering, India

Received 7 November 1997; Revised 26 June 1998

Copyright © 1998 Vineet Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Endothelial cells, lining the entire vascular system, respond to change in concentration of specific agonist like adenosine triphosphate (ATP) by increasing cytosolic Ca2+ concentration, producing prostacyclin, endothelium derived relaxing factor and ecto-enzymes. Three different ecto-enzymes metabolize ATP in three steps (ATP ADP AMP adenosine ) . Normally experiments with endothelium are carried out in a rectangular flow chamber provided with a cell surface at one of its walls and feed stream containing ATP. The ATP concentration near the cell surface depends upon two factors, rate of its degradation and the rate at which it reaches from upstream. Closed form solutions for the concentration profile of ATP in such a flow chamber indicates that concentration near the cell surface is lower than the bulk concentration depending on the activity of ecto-enzymes and it increases with increase in tangential flow rate (shear stress). This indicates that shear induced response of endothelial cell (at least for low shear rate) may be due to change in ATP concentration near the cell surface which is sensed by purinoreceptors instead of a mechanoreceptor. Several workers have tried to investigate this problem analytically. Unfortunately, solutions obtained by these workers have limited success. In the present work, exact solution of the problem has been obtained in terms of a confluent hypergeometric function. Solution of the transformed equation gives accurate results even in the entrance region of the flow chamber which eliminates the need of solutions based on approximate methods like perturbation or finite difference techniques.