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 & 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.  doi: 10.1017/S0022112067001375.  Google Scholar

[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.  doi: 10.1007/s00033-011-0143-2.  Google Scholar

[3]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting,, in, 29 (2008), 53.   Google Scholar

[4]

R. M. Bowen, Theory of mixtures,, in, (1976).   Google Scholar

[5]

W. J. Bruining, A general description of flows and pressures in hollow fiber membrane modules,, Chem. Eng. Sci., 44 (1989), 1441.  doi: 10.1016/0009-2509(89)85016-X.  Google Scholar

[6]

N. M. Brown and F. C. Lai, Measurement of permeability and slip coefficient of porous tubes,, ASME J. Fluids Eng., 128 (2006), 987.  doi: 10.1115/1.2234783.  Google Scholar

[7]

R. Davis and D. T. Leighton, Shear-induced transport of a particle layer along a porous wall,, Chem. Eng. Sci., 42 (1987), 275.  doi: 10.1016/0009-2509(87)85057-1.  Google Scholar

[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.   Google Scholar

[9]

P. C. Carman, Fluid flow through a granular bed,, Trans. Instn. Chem. Engrs., 15 (1937), 150.   Google Scholar

[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, (2005).   Google Scholar

[11]

J. Coirer, "Mécanique des Milieux Continus,", Dunod, (1997).   Google Scholar

[12]

S. Eloot, D. De Wachter, I. Van Trich and P. Verdonck, Computational flow in hollow-fiber dialyzers,, Artificial Organs, 26 (2002), 590.  doi: 10.1046/j.1525-1594.2002.07081.x.  Google Scholar

[13]

R. Få hraeus and T. Lindqvist, The viscosity of blood in narrow capillary tubes,, A. J. Physiol., 96 (1931), 362.   Google Scholar

[14]

A. Fasano, R. Santos and A. Sequeira, Blood coagulation: A puzzle for biologists, a maze for mathematicians,, in, ().   Google Scholar

[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.  doi: 10.1016/S0017-9310(03)00241-2.  Google Scholar

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W. Henrich, "Prinicples and Practice of Dialysis,", 3rd edition, (2004).   Google Scholar

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J. Himmelfarb and T. A. Ikizler, Hemodialysis,, N. Engl. J. Med., 363 (2010), 1833.  doi: 10.1056/NEJMra0902710.  Google Scholar

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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.   Google Scholar

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W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111.  doi: 10.1137/S003613999833678X.  Google Scholar

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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.  doi: 10.1137/S1064827599360339.  Google Scholar

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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.  doi: 10.1007/s11242-009-9354-9.  Google Scholar

[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.  doi: 10.1016/j.cam.2010.01.032.  Google Scholar

[23]

A. Kargol, A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure,, J. Membr. Sci., 191 (2001), 61.  doi: 10.1016/S0376-7388(01)00450-1.  Google Scholar

[24]

J. Keener and J. Sneyd, "Mathematical Physiology. Vol. II: System Physiology,", Second edition, (2009).   Google Scholar

[25]

J. K. Leypoldt, Solute fluxes in different treatment modalities,, Nephrology, 15 (2000), 3.   Google Scholar

[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 (5791).  doi: 10.1155/2008/579172.  Google Scholar

[27]

G. Pontrelli, Blood flow through a circular pipe with an impulsive pressure gradient,, Math. Models Methods Appl. Sci., 10 (2000), 187.  doi: 10.1142/S0218202500000124.  Google Scholar

[28]

A. Quarteroni, L. Formaggia and A. Veneziani, eds., "Complex Systems in Biomedicine,", Springer-Verlag Italia, (2006).   Google Scholar

[29]

D. Quemada, General features of blood circulation in narrow vessels,, in, (1983).   Google Scholar

[30]

K. R. Rajagopal and L. Tao, "Mechanics of Mixtures,", Series on Advances in Mathematics for Applied Sciences, 35 (1995).   Google Scholar

[31]

N. P. Reddy, Design of artificial kidneys,, in, (2009).   Google Scholar

[32]

P. Saffman, On the boundary condition at a surface of a porous medium,, Stud. Appl. Math., 50 (1971), 93.   Google Scholar

[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.  doi: 10.1016/0017-9310(79)90120-0.  Google Scholar

[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.  doi: 10.1080/00036811.2010.483765.  Google Scholar

[35]

E. M. Starling, On the absorption of fluids from the convective tissue spaces,, J. Physiol., 19 (1896), 312.   Google Scholar

[36]

Y. Suzuki, F. Kohori and K. Sakai, Computer-aided design of hollow fiber dialyzers,, J. Artif. Organs, 4 (2001), 326.   Google Scholar

[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.   Google Scholar

[38]

K. K. Yeleswarapu, "Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood,", Ph.D dissertation, (1996).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.ijheatmasstransfer.2004.04.017.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall,, J. Fluid Mech., 30 (1967), 197.  doi: 10.1017/S0022112067001375.  Google Scholar

[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.  doi: 10.1007/s00033-011-0143-2.  Google Scholar

[3]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting,, in, 29 (2008), 53.   Google Scholar

[4]

R. M. Bowen, Theory of mixtures,, in, (1976).   Google Scholar

[5]

W. J. Bruining, A general description of flows and pressures in hollow fiber membrane modules,, Chem. Eng. Sci., 44 (1989), 1441.  doi: 10.1016/0009-2509(89)85016-X.  Google Scholar

[6]

N. M. Brown and F. C. Lai, Measurement of permeability and slip coefficient of porous tubes,, ASME J. Fluids Eng., 128 (2006), 987.  doi: 10.1115/1.2234783.  Google Scholar

[7]

R. Davis and D. T. Leighton, Shear-induced transport of a particle layer along a porous wall,, Chem. Eng. Sci., 42 (1987), 275.  doi: 10.1016/0009-2509(87)85057-1.  Google Scholar

[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.   Google Scholar

[9]

P. C. Carman, Fluid flow through a granular bed,, Trans. Instn. Chem. Engrs., 15 (1937), 150.   Google Scholar

[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, (2005).   Google Scholar

[11]

J. Coirer, "Mécanique des Milieux Continus,", Dunod, (1997).   Google Scholar

[12]

S. Eloot, D. De Wachter, I. Van Trich and P. Verdonck, Computational flow in hollow-fiber dialyzers,, Artificial Organs, 26 (2002), 590.  doi: 10.1046/j.1525-1594.2002.07081.x.  Google Scholar

[13]

R. Få hraeus and T. Lindqvist, The viscosity of blood in narrow capillary tubes,, A. J. Physiol., 96 (1931), 362.   Google Scholar

[14]

A. Fasano, R. Santos and A. Sequeira, Blood coagulation: A puzzle for biologists, a maze for mathematicians,, in, ().   Google Scholar

[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.  doi: 10.1016/S0017-9310(03)00241-2.  Google Scholar

[16]

W. Henrich, "Prinicples and Practice of Dialysis,", 3rd edition, (2004).   Google Scholar

[17]

J. Himmelfarb and T. A. Ikizler, Hemodialysis,, N. Engl. J. Med., 363 (2010), 1833.  doi: 10.1056/NEJMra0902710.  Google Scholar

[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.   Google Scholar

[19]

W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111.  doi: 10.1137/S003613999833678X.  Google Scholar

[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.  doi: 10.1137/S1064827599360339.  Google Scholar

[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.  doi: 10.1007/s11242-009-9354-9.  Google Scholar

[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.  doi: 10.1016/j.cam.2010.01.032.  Google Scholar

[23]

A. Kargol, A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure,, J. Membr. Sci., 191 (2001), 61.  doi: 10.1016/S0376-7388(01)00450-1.  Google Scholar

[24]

J. Keener and J. Sneyd, "Mathematical Physiology. Vol. II: System Physiology,", Second edition, (2009).   Google Scholar

[25]

J. K. Leypoldt, Solute fluxes in different treatment modalities,, Nephrology, 15 (2000), 3.   Google Scholar

[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 (5791).  doi: 10.1155/2008/579172.  Google Scholar

[27]

G. Pontrelli, Blood flow through a circular pipe with an impulsive pressure gradient,, Math. Models Methods Appl. Sci., 10 (2000), 187.  doi: 10.1142/S0218202500000124.  Google Scholar

[28]

A. Quarteroni, L. Formaggia and A. Veneziani, eds., "Complex Systems in Biomedicine,", Springer-Verlag Italia, (2006).   Google Scholar

[29]

D. Quemada, General features of blood circulation in narrow vessels,, in, (1983).   Google Scholar

[30]

K. R. Rajagopal and L. Tao, "Mechanics of Mixtures,", Series on Advances in Mathematics for Applied Sciences, 35 (1995).   Google Scholar

[31]

N. P. Reddy, Design of artificial kidneys,, in, (2009).   Google Scholar

[32]

P. Saffman, On the boundary condition at a surface of a porous medium,, Stud. Appl. Math., 50 (1971), 93.   Google Scholar

[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.  doi: 10.1016/0017-9310(79)90120-0.  Google Scholar

[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.  doi: 10.1080/00036811.2010.483765.  Google Scholar

[35]

E. M. Starling, On the absorption of fluids from the convective tissue spaces,, J. Physiol., 19 (1896), 312.   Google Scholar

[36]

Y. Suzuki, F. Kohori and K. Sakai, Computer-aided design of hollow fiber dialyzers,, J. Artif. Organs, 4 (2001), 326.   Google Scholar

[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.   Google Scholar

[38]

K. K. Yeleswarapu, "Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood,", Ph.D dissertation, (1996).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.ijheatmasstransfer.2004.04.017.  Google Scholar

[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.   Google Scholar

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