# American Institute of Mathematical Sciences

September  2012, 17(6): 1903-1937. doi: 10.3934/dcdsb.2012.17.1903

## Modeling high flux hollow fibers dialyzers

 1 Università degli Studi di Firenze, Dipartimento di Matematica "Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze

Received  March 2011 Revised  May 2011 Published  May 2012

In hollow fibres dialyzers blood flows in the fibres channel and plasma filtrates through their permeable wall to feed the flow of the permeate (dialyzate), which takes place among the fibres in the opposite direction. We investigate this fluid dynamical problem exploiting the existence of two separate scales: the one in the fibres direction ($\sim \,20\,cm$), and the one along the fibre radius ($\sim \,0.1\,mm$). We formulate a mathematical model based on a two-scale approach providing a full description of the flows of the blood, of the dialyzate and the of the plasma through the membrane, as well as of the progressive increase of the hematocrit. The problem is characterized by various rather unusual features like the slip condition of blood on the membrane and the feedback loop of boundary data for the hematocrit. Blood rheology is assumed to be of shear-thinning type, with hematocrit dependent coefficients. Under some simplifications explicit solutions are found. We show how the necessity of respecting several constraints and of reaching some specific targets influences the selection of the geometrical and physical parameters of the system. Once the fluid dynamical problem has been solved, the removal from blood of chemicals like urea, etc. has been studied.
Citation: Antonio Fasano, Angiolo Farina. Modeling high flux hollow fibers dialyzers. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1903-1937. doi: 10.3934/dcdsb.2012.17.1903
##### References:
 [1] G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375. [2] I. Borsi, A. Farina and A. Fasano, Incompressible laminar flow through hollow fibers: A general study by means of a two-scale approach, ZAMP, 62 (2011), 681-706. doi: 10.1007/s00033-011-0143-2. [3] I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkōtosho, Tokyo, (2008), 53-72. [4] R. M. Bowen, Theory of mixtures, in "Continuum Physics," Vol. 3, (ed. A. C. Eringen), Academic Press, New York, 1976. [5] W. J. Bruining, A general description of flows and pressures in hollow fiber membrane modules, Chem. Eng. Sci., 44 (1989), 1441-1447. doi: 10.1016/0009-2509(89)85016-X. [6] N. M. Brown and F. C. Lai, Measurement of permeability and slip coefficient of porous tubes, ASME J. Fluids Eng., 128 (2006), 987-992. doi: 10.1115/1.2234783. [7] R. Davis and D. T. Leighton, Shear-induced transport of a particle layer along a porous wall, Chem. Eng. Sci., 42 (1987), 275-282. doi: 10.1016/0009-2509(87)85057-1. [8] F. Carapau and A. Sequeira, 1D models for blood flow in small vessels using the Cosserat theory, WSEAS Trans. on Mathematics, 5 (2006), 54-62. [9] P. C. Carman, Fluid flow through a granular bed, Trans. Instn. Chem. Engrs., 15 (1937), 150-157. [10] J. Cho, I. S. Kim, J. Moon and B. Kwon, Determining Brownian and shear-induced diffusivity of nano- and micro-particles for sustainable membrane filtration, in "Integrated Concenpts in Water Recycling," (eds. S. J. Khan, A. I. Schäfer and M. H. Muston), Elsevier, 2005. [11] J. Coirer, "Mécanique des Milieux Continus," Dunod, Paris, 1997. [12] S. Eloot, D. De Wachter, I. Van Trich and P. Verdonck, Computational flow in hollow-fiber dialyzers, Artificial Organs, 26 (2002), 590-599. doi: 10.1046/j.1525-1594.2002.07081.x. [13] R. Få hraeus and T. Lindqvist, The viscosity of blood in narrow capillary tubes, A. J. Physiol., 96 (1931), 362-368. [14] A. Fasano, R. Santos and A. Sequeira, Blood coagulation: A puzzle for biologists, a maze for mathematicians, in "Modelling Physiological Flows," (eds. D. Ambrosi, A. Quarteroni and G. Rozza), Springer Italia, to appear. [15] B. Goyeau, D. Lhuillier, D. Gobin and M. G. Velarde, Momentum transport at a fluid-porous interface, Int. J. Heat Mass Transfer, 46 (2003), 4071-4081. doi: 10.1016/S0017-9310(03)00241-2. [16] W. Henrich, "Prinicples and Practice of Dialysis," 3rd edition, Lippincott Williams & Wilkins, Philadelphia, 2004. [17] J. Himmelfarb and T. A. Ikizler, Hemodialysis, N. Engl. J. Med., 363 (2010), 1833-1843. doi: 10.1056/NEJMra0902710. [18] W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 23 (1996), 403-465. [19] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math., 60 (2000), 1111-1127. doi: 10.1137/S003613999833678X. [20] W. Jäger, A. Mikelić and N. Neuß, Asymptotic analysis of the laminar viscous flow over a porous bed, SIAM J. on Scientific and Statistical Computing, 22 (2001), 2006-2028. doi: 10.1137/S1064827599360339. [21] W. Jäger and A. Mikelić, Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization, Transp. Porous Med., 78 (2009), 489-508. doi: 10.1007/s11242-009-9354-9. [22] J. Janela, A. Moura and A. Sequeira, A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries, Journal of Computational and Applied Mathematics, 234 (2010), 2783-2791. doi: 10.1016/j.cam.2010.01.032. [23] A. Kargol, A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure, J. Membr. Sci., 191 (2001), 61-69. doi: 10.1016/S0376-7388(01)00450-1. [24] J. Keener and J. Sneyd, "Mathematical Physiology. Vol. II: System Physiology," Second edition, Interdisciplinary Applied Mathematics, Vol. 8/II, Springer, New York, 2009. [25] J. K. Leypoldt, Solute fluxes in different treatment modalities, Nephrology, Dialysis and Transplantation, 15 (2000), 3-9. [26] M. Massoudi and J. F. Antaki, An anisotropic constitutive equation for the stress tensor of blood based on mixture theory, Math. Problems Engineering, 2008, Art. ID 579172, 30 pp. doi: 10.1155/2008/579172. [27] G. Pontrelli, Blood flow through a circular pipe with an impulsive pressure gradient, Math. Models Methods Appl. Sci., 10 (2000), 187-202. doi: 10.1142/S0218202500000124. [28] A. Quarteroni, L. Formaggia and A. Veneziani, eds., "Complex Systems in Biomedicine," Springer-Verlag Italia, Milan, 2006. [29] D. Quemada, General features of blood circulation in narrow vessels, in "Arteries and Arterial Blood" (ed. C. M. Rodkiewicz), Springer-Verlag, New York, 1983. [30] K. R. Rajagopal and L. Tao, "Mechanics of Mixtures," Series on Advances in Mathematics for Applied Sciences, 35, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [31] N. P. Reddy, Design of artificial kidneys, in "Biomedical Engineering and Design Handbook. Volume 2: Applications" (ed. M. Kutz), McGraw Hill, New York, 2009. [32] P. Saffman, On the boundary condition at a surface of a porous medium, Stud. Appl. Math., 50 (1971), 93-101. [33] R. Singh and R. L. Laurence, Influence of slip velocity at a membrane surface on ultrafiltration performance-II (Tube flow system), Int. J. Heat Mass Transfer, 12 (1979), 731-737. doi: 10.1016/0017-9310(79)90120-0. [34] K. Smith and A. Sequeira, Micro-macro simulations of a shear-thinning viscoelastic kinetic model: Applications to blood flow, Applicable Analysis, 90 (2011), 227-252. doi: 10.1080/00036811.2010.483765. [35] E. M. Starling, On the absorption of fluids from the convective tissue spaces, J. Physiol., 19 (1896), 312-319. [36] Y. Suzuki, F. Kohori and K. Sakai, Computer-aided design of hollow fiber dialyzers, J. Artif. Organs, 4 (2001), 326-330. [37] G. J. Tangelder, D. W. Slaaf, T. Arts and R. S. Reneman, Wall shear rates in arterioles in vivo: Least estimates for platelets velocity profiles, Am. J. Physiology, 254 (1988), 1059-1064. [38] K. K. Yeleswarapu, "Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood," Ph.D dissertation, University of Pittsburgh, Pittsburgh, PA, 1996. [39] F. J. Valdés-Parada, J. Alvarez-Ramírez, B. Goyeau and J. A. Ochoa-Tapia, Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation, Transp. Porous Med., 78 (2009), 439-457. [40] D. Weiping, H. Liqun, Z. Gang, Z. Haifeng, S. Zhiquan and G. Dayong, Double porous media model for mass transfer in hemodialyzers, Int. J. Heat Mass Transfer, 47 (2004), 4849-4855. doi: 10.1016/j.ijheatmasstransfer.2004.04.017. [41] A. Wüpper, F. Dellanna, C. A. Baldamus and D. Woermann, Local transport processes in high-flux hollow fiber dialyzers, J. Membr. Sci., 131 (1997), 81-93.

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##### References:
 [1] G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375. [2] I. Borsi, A. Farina and A. Fasano, Incompressible laminar flow through hollow fibers: A general study by means of a two-scale approach, ZAMP, 62 (2011), 681-706. doi: 10.1007/s00033-011-0143-2. [3] I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkōtosho, Tokyo, (2008), 53-72. [4] R. M. Bowen, Theory of mixtures, in "Continuum Physics," Vol. 3, (ed. A. C. Eringen), Academic Press, New York, 1976. [5] W. J. Bruining, A general description of flows and pressures in hollow fiber membrane modules, Chem. Eng. Sci., 44 (1989), 1441-1447. doi: 10.1016/0009-2509(89)85016-X. [6] N. M. Brown and F. C. Lai, Measurement of permeability and slip coefficient of porous tubes, ASME J. Fluids Eng., 128 (2006), 987-992. doi: 10.1115/1.2234783. [7] R. Davis and D. T. Leighton, Shear-induced transport of a particle layer along a porous wall, Chem. Eng. Sci., 42 (1987), 275-282. doi: 10.1016/0009-2509(87)85057-1. [8] F. Carapau and A. Sequeira, 1D models for blood flow in small vessels using the Cosserat theory, WSEAS Trans. on Mathematics, 5 (2006), 54-62. [9] P. C. Carman, Fluid flow through a granular bed, Trans. Instn. Chem. Engrs., 15 (1937), 150-157. [10] J. Cho, I. S. Kim, J. Moon and B. Kwon, Determining Brownian and shear-induced diffusivity of nano- and micro-particles for sustainable membrane filtration, in "Integrated Concenpts in Water Recycling," (eds. S. J. Khan, A. I. Schäfer and M. H. Muston), Elsevier, 2005. [11] J. Coirer, "Mécanique des Milieux Continus," Dunod, Paris, 1997. [12] S. Eloot, D. De Wachter, I. Van Trich and P. Verdonck, Computational flow in hollow-fiber dialyzers, Artificial Organs, 26 (2002), 590-599. doi: 10.1046/j.1525-1594.2002.07081.x. [13] R. Få hraeus and T. Lindqvist, The viscosity of blood in narrow capillary tubes, A. J. Physiol., 96 (1931), 362-368. [14] A. Fasano, R. Santos and A. Sequeira, Blood coagulation: A puzzle for biologists, a maze for mathematicians, in "Modelling Physiological Flows," (eds. D. Ambrosi, A. Quarteroni and G. Rozza), Springer Italia, to appear. [15] B. Goyeau, D. Lhuillier, D. Gobin and M. G. Velarde, Momentum transport at a fluid-porous interface, Int. J. Heat Mass Transfer, 46 (2003), 4071-4081. doi: 10.1016/S0017-9310(03)00241-2. [16] W. Henrich, "Prinicples and Practice of Dialysis," 3rd edition, Lippincott Williams & Wilkins, Philadelphia, 2004. [17] J. Himmelfarb and T. A. Ikizler, Hemodialysis, N. Engl. J. Med., 363 (2010), 1833-1843. doi: 10.1056/NEJMra0902710. [18] W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 23 (1996), 403-465. [19] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math., 60 (2000), 1111-1127. doi: 10.1137/S003613999833678X. [20] W. Jäger, A. Mikelić and N. Neuß, Asymptotic analysis of the laminar viscous flow over a porous bed, SIAM J. on Scientific and Statistical Computing, 22 (2001), 2006-2028. doi: 10.1137/S1064827599360339. [21] W. Jäger and A. Mikelić, Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization, Transp. Porous Med., 78 (2009), 489-508. doi: 10.1007/s11242-009-9354-9. [22] J. Janela, A. Moura and A. Sequeira, A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries, Journal of Computational and Applied Mathematics, 234 (2010), 2783-2791. doi: 10.1016/j.cam.2010.01.032. [23] A. Kargol, A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure, J. Membr. Sci., 191 (2001), 61-69. doi: 10.1016/S0376-7388(01)00450-1. [24] J. Keener and J. Sneyd, "Mathematical Physiology. Vol. II: System Physiology," Second edition, Interdisciplinary Applied Mathematics, Vol. 8/II, Springer, New York, 2009. [25] J. K. Leypoldt, Solute fluxes in different treatment modalities, Nephrology, Dialysis and Transplantation, 15 (2000), 3-9. [26] M. Massoudi and J. F. Antaki, An anisotropic constitutive equation for the stress tensor of blood based on mixture theory, Math. Problems Engineering, 2008, Art. ID 579172, 30 pp. doi: 10.1155/2008/579172. [27] G. Pontrelli, Blood flow through a circular pipe with an impulsive pressure gradient, Math. Models Methods Appl. Sci., 10 (2000), 187-202. doi: 10.1142/S0218202500000124. [28] A. Quarteroni, L. Formaggia and A. Veneziani, eds., "Complex Systems in Biomedicine," Springer-Verlag Italia, Milan, 2006. [29] D. Quemada, General features of blood circulation in narrow vessels, in "Arteries and Arterial Blood" (ed. C. M. Rodkiewicz), Springer-Verlag, New York, 1983. [30] K. R. Rajagopal and L. Tao, "Mechanics of Mixtures," Series on Advances in Mathematics for Applied Sciences, 35, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [31] N. P. Reddy, Design of artificial kidneys, in "Biomedical Engineering and Design Handbook. Volume 2: Applications" (ed. M. Kutz), McGraw Hill, New York, 2009. [32] P. Saffman, On the boundary condition at a surface of a porous medium, Stud. Appl. Math., 50 (1971), 93-101. [33] R. Singh and R. L. Laurence, Influence of slip velocity at a membrane surface on ultrafiltration performance-II (Tube flow system), Int. J. Heat Mass Transfer, 12 (1979), 731-737. doi: 10.1016/0017-9310(79)90120-0. [34] K. Smith and A. Sequeira, Micro-macro simulations of a shear-thinning viscoelastic kinetic model: Applications to blood flow, Applicable Analysis, 90 (2011), 227-252. doi: 10.1080/00036811.2010.483765. [35] E. M. Starling, On the absorption of fluids from the convective tissue spaces, J. Physiol., 19 (1896), 312-319. [36] Y. Suzuki, F. Kohori and K. Sakai, Computer-aided design of hollow fiber dialyzers, J. Artif. Organs, 4 (2001), 326-330. [37] G. J. Tangelder, D. W. Slaaf, T. Arts and R. S. Reneman, Wall shear rates in arterioles in vivo: Least estimates for platelets velocity profiles, Am. J. Physiology, 254 (1988), 1059-1064. [38] K. K. Yeleswarapu, "Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood," Ph.D dissertation, University of Pittsburgh, Pittsburgh, PA, 1996. [39] F. J. Valdés-Parada, J. Alvarez-Ramírez, B. Goyeau and J. A. Ochoa-Tapia, Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation, Transp. Porous Med., 78 (2009), 439-457. [40] D. Weiping, H. Liqun, Z. Gang, Z. Haifeng, S. Zhiquan and G. Dayong, Double porous media model for mass transfer in hemodialyzers, Int. J. Heat Mass Transfer, 47 (2004), 4849-4855. doi: 10.1016/j.ijheatmasstransfer.2004.04.017. [41] A. Wüpper, F. Dellanna, C. A. Baldamus and D. Woermann, Local transport processes in high-flux hollow fiber dialyzers, J. Membr. Sci., 131 (1997), 81-93.
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