Effect of Deformation of Red Cell on Nutritional Transport in Capillary-Tissue Exchange System

ABSTRACT

A mathematical analysis of a model for nutritional exchange in a capillary-tissue exchange system is presented in this paper. This model consist of a single file flow of red blood cell in capillary when diameter of red blood cell is greater than tube diameter, In this case, the cell must be deformed. Due to concentration gradients, the dissolved nutrient in substrate diffuses into surrounding tissue. Introducing approximations of the lubrication theory, squeezing flow of plasma in between the gap between cell and capillary wall have been solved with the help of approximate mathematical techniques. The computational results for concentration of dissolved nutrients, diffusive flux and normal component of velocity have been presented and discussed through graphs. We have also shown the effect of deformation parameter and permeability on these results.

A mathematical analysis of a model for nutritional exchange in a capillary-tissue exchange system is presented in this paper. This model consist of a single file flow of red blood cell in capillary when diameter of red blood cell is greater than tube diameter, In this case, the cell must be deformed. Due to concentration gradients, the dissolved nutrient in substrate diffuses into surrounding tissue. Introducing approximations of the lubrication theory, squeezing flow of plasma in between the gap between cell and capillary wall have been solved with the help of approximate mathematical techniques. The computational results for concentration of dissolved nutrients, diffusive flux and normal component of velocity have been presented and discussed through graphs. We have also shown the effect of deformation parameter and permeability on these results.

Cite this paper

nullR. Bali, S. Mishra and M. Mishra, "Effect of Deformation of Red Cell on Nutritional Transport in Capillary-Tissue Exchange System,"*Applied Mathematics*, Vol. 2 No. 11, 2011, pp. 1417-1423. doi: 10.4236/am.2011.211200.

nullR. Bali, S. Mishra and M. Mishra, "Effect of Deformation of Red Cell on Nutritional Transport in Capillary-Tissue Exchange System,"

References

[1] E. M. Renkin, “Transport of Potassium—42 from Blood to Tissue in Isolated Mammalian Skeletal Muscles,” American Journal of Physiology, Vol. 197, 1959, p. 1205.

[2] N. Casson, “A Flow Equation for Pigment-Oil Suspensions of the Printing Ink Type,” In: C. C. Mill, Ed., Rheology of Disperse Systems, Pergamon Press, Oxford, 1959, pp 84-104.

[3] S. E. Charm and G. S. Kurland, “Blood Rheology and Cardiovascular,” In: D. H. Bergel, Ed., Fluid Dynamics, Academic Press, London.

[4] E. C. Eringen, “Theory of Micropolar Fluids,” Journal of Mathematical Fluid Mechanics, Vol. 16, 1966, pp. 1-18.

[5] B. B. Gupta, K. K. Nigam and H. M. Jaffrin, “A Three Layer Semi Empirical Model for Flow of Blood and Other Particulate Suspension through Narrow Tubes,” Journal of Biomechanical Engineering, Vol. 104, No. 2, 1982, pp. 129-135. doi:10.1115/1.3138326

[6] H. R. Haynes, “Physical Basis of the Dependence of Blood Viscosity on Tube Radius,” American Journal of Physiology, Vol. 198, No. 6, 1966, pp. 1193-2000.

[7] P. N. Tandon and R. Agarwal, “A Study of Nutritional Transport in Synovial Joints,” Computers & Mathematics with Applications, Vol. 17, No. 7, 1989, pp. 1131-1141.

[8] P. N. Tandon, M. Mishra and A. Chaurasia, “A Model for Nutritional Transport in Capillary-Tissue Exchange System,” International Journal of Bio-Medical Computing, Vol. 37, No. 1, 1994, pp. 19-28. doi:10.1016/0020-7101(94)90068-X

[9] H. S. Lew and Y. C. Fung, “The Motion of the Plasma between Red Cells in the Bolus Flow,” Biorheology, Vol. 6, 1969, pp. 109-119

[10] M. J. Lighthill, “Pressure Forcing of Tightly Fitting Pellets along Fluid Filled Elastic Tubes,” Journal of Fluid Mechanics, Vol. 34, No. 1, 1968, pp. 113-143. doi:10.1017/S0022112068001795

[11] A. C. Barnard, L. Lopez and J. D. Hellums, “Basic Theory of Blood Flow in Capillaries,” Microvascular Research, Vol. 1, No. 1, 1968, pp. 23-24. doi:10.1016/0026-2862(68)90004-6

[12] P. R. Zarda, S. Chien and R. Skalak, “Interaction of Viscous Incompressible Fluid with an Elastic Body,” T. Belystschko and T. L. Geers, Eds., Computational Methods for Fluid-Solid Interaction Problems, American Society of Mechanical Engineers, New York, 1977, pp. 65-82.

[13] K. L. Lin, L. Lopez and J. D. Hellums, “Blood Flow in Capillaries,” Microvascular Research, Vol. 5, No. 1, 1973, pp. 7-19. doi:10.1016/S0026-2862(73)80003-2

[14] T. W. Secomb and R. Skalak, “A Two Dimensional Model for the Capillary Flow of an Axisymmetric Cell,” Microvascular Research, Vol. 24, No. 2, 1982, pp 194-203. doi:10.1016/0026-2862(82)90056-5

[15] T. W. Secomb, R. Skalak, N. Ozakaya and J. F. Gross, “Flow of Axisymmetric Red Blood Cells in Narrow Capillaries,” Journal of Fluid Mechanics, Vol. 163, 1986, pp. 405-423. doi:10.1017/S0022112086002355

[16] T. W. Secomb, R. Hsu and A. R. Pries, “Motion of Red Blood Cell in Acapillary with an Endothelial Surface Layer: Effect of Flow Velocity,” American Journal of Physiology: Heart and Circulatory Physiology, Vol. 28, No. 2, 2001, pp. H629-H636.

[17] T. W. Secomb, R. Hsu and A. R. Pries, “A Model for Red Blood Cell Motion in Glycocalyx-Lined Capillaries,” American Journal of Physiology: Heart and Circulatory Physiology, Vol. 274, No. 3, 1998, pp. H1016-H1022.

[18] P. N. Tandon, P. Nirmala, M. Tiwari and U. V. Rana, “Analysis of Nutritional Transport through a Capillary—Normal and Stenosed,” Computers & Mathematics with Applications, Vol. 22, No. 12, 1998, pp. 3-13. doi:10.1016/0898-1221(91)90142-Q

[1] E. M. Renkin, “Transport of Potassium—42 from Blood to Tissue in Isolated Mammalian Skeletal Muscles,” American Journal of Physiology, Vol. 197, 1959, p. 1205.

[2] N. Casson, “A Flow Equation for Pigment-Oil Suspensions of the Printing Ink Type,” In: C. C. Mill, Ed., Rheology of Disperse Systems, Pergamon Press, Oxford, 1959, pp 84-104.

[3] S. E. Charm and G. S. Kurland, “Blood Rheology and Cardiovascular,” In: D. H. Bergel, Ed., Fluid Dynamics, Academic Press, London.

[4] E. C. Eringen, “Theory of Micropolar Fluids,” Journal of Mathematical Fluid Mechanics, Vol. 16, 1966, pp. 1-18.

[5] B. B. Gupta, K. K. Nigam and H. M. Jaffrin, “A Three Layer Semi Empirical Model for Flow of Blood and Other Particulate Suspension through Narrow Tubes,” Journal of Biomechanical Engineering, Vol. 104, No. 2, 1982, pp. 129-135. doi:10.1115/1.3138326

[6] H. R. Haynes, “Physical Basis of the Dependence of Blood Viscosity on Tube Radius,” American Journal of Physiology, Vol. 198, No. 6, 1966, pp. 1193-2000.

[7] P. N. Tandon and R. Agarwal, “A Study of Nutritional Transport in Synovial Joints,” Computers & Mathematics with Applications, Vol. 17, No. 7, 1989, pp. 1131-1141.

[8] P. N. Tandon, M. Mishra and A. Chaurasia, “A Model for Nutritional Transport in Capillary-Tissue Exchange System,” International Journal of Bio-Medical Computing, Vol. 37, No. 1, 1994, pp. 19-28. doi:10.1016/0020-7101(94)90068-X

[9] H. S. Lew and Y. C. Fung, “The Motion of the Plasma between Red Cells in the Bolus Flow,” Biorheology, Vol. 6, 1969, pp. 109-119

[10] M. J. Lighthill, “Pressure Forcing of Tightly Fitting Pellets along Fluid Filled Elastic Tubes,” Journal of Fluid Mechanics, Vol. 34, No. 1, 1968, pp. 113-143. doi:10.1017/S0022112068001795

[11] A. C. Barnard, L. Lopez and J. D. Hellums, “Basic Theory of Blood Flow in Capillaries,” Microvascular Research, Vol. 1, No. 1, 1968, pp. 23-24. doi:10.1016/0026-2862(68)90004-6

[12] P. R. Zarda, S. Chien and R. Skalak, “Interaction of Viscous Incompressible Fluid with an Elastic Body,” T. Belystschko and T. L. Geers, Eds., Computational Methods for Fluid-Solid Interaction Problems, American Society of Mechanical Engineers, New York, 1977, pp. 65-82.

[13] K. L. Lin, L. Lopez and J. D. Hellums, “Blood Flow in Capillaries,” Microvascular Research, Vol. 5, No. 1, 1973, pp. 7-19. doi:10.1016/S0026-2862(73)80003-2

[14] T. W. Secomb and R. Skalak, “A Two Dimensional Model for the Capillary Flow of an Axisymmetric Cell,” Microvascular Research, Vol. 24, No. 2, 1982, pp 194-203. doi:10.1016/0026-2862(82)90056-5

[15] T. W. Secomb, R. Skalak, N. Ozakaya and J. F. Gross, “Flow of Axisymmetric Red Blood Cells in Narrow Capillaries,” Journal of Fluid Mechanics, Vol. 163, 1986, pp. 405-423. doi:10.1017/S0022112086002355

[16] T. W. Secomb, R. Hsu and A. R. Pries, “Motion of Red Blood Cell in Acapillary with an Endothelial Surface Layer: Effect of Flow Velocity,” American Journal of Physiology: Heart and Circulatory Physiology, Vol. 28, No. 2, 2001, pp. H629-H636.

[17] T. W. Secomb, R. Hsu and A. R. Pries, “A Model for Red Blood Cell Motion in Glycocalyx-Lined Capillaries,” American Journal of Physiology: Heart and Circulatory Physiology, Vol. 274, No. 3, 1998, pp. H1016-H1022.

[18] P. N. Tandon, P. Nirmala, M. Tiwari and U. V. Rana, “Analysis of Nutritional Transport through a Capillary—Normal and Stenosed,” Computers & Mathematics with Applications, Vol. 22, No. 12, 1998, pp. 3-13. doi:10.1016/0898-1221(91)90142-Q