Red blood cells tend to aggregate in the presence of plasma proteins, forming structures known as rouleaux. Here, we derive a constitutive rheological model for human blood which accounts for the formation and dissociation of rouleaux using the generalized bracket formulation of nonequilibrium thermodynamics. Similar to the model derived by Owens and co-workers [“A non-homogeneous constitutive model for human blood. Part 1. Model derivation and steady flow,” J. Fluid Mech. 617, 327–354 (2008)] through polymer network theory, each rouleau in our model is represented as a dumbbell; the corresponding structural variable is the conformation tensor of the dumbbell. The kinetics of rouleau formation and dissociation is treated as in the work of Germann et al. [“Nonequilibrium thermodynamic modeling of the structure and rheology of concentrated wormlike micellar solutions,” J. Non-Newton. Fluid Mech. 196, 51–57 (2013)] by assuming a set of reversible reactions, each characterized by a forward and a reverse rate constant. The final set of evolution equations for the microstructure of each rouleau and the expression for the stress tensor turn out to be very similar to those of Owens and co-workers. However, by explicitly considering a mechanism for the formation and breakage of rouleaux, our model further provides expressions for the aggregation and disaggregation rates appearing in the final transport equations, which in the kinetic theory-based network model of Owens were absent and had to be specified separately. Despite this, the two models are found to provide similar descriptions of experimental data on the size distribution of rouleaux.
REFERENCES
1.
Anand
, M.
and Rajagopal
, K. R.
, “A shear-thinning viscoelastic fluid model for describing the flow of blood
,” Int. J. Cardiovasc. Med. Sci.
4
, 59
–68
(2004
).2.
Apostolakis
, M. V.
, Mavrantzas
, V. G.
, and Beris
, A. N.
, “Stress gradient-induced migration effects in the Taylor-Couette flow of a dilute polymer solution
,” J. Non-Newton. Fluid Mech.
102
, 409
–445
(2002
).3.
Apostolidis
, A. J.
and Beris
, A. N.
, “Modeling of the blood rheology in steady-state shear flows
,” J. Rheol.
58
, 607
–633
(2014
).4.
Barabino
, G. A.
, Platt
, M. O.
, and Kaul
, D. K.
, “Sickle cell biomechanics
,” Annu. Rev. Biomed. Eng.
12
, 345
–367
(2010
).5.
Barshtein
, G.
, Wajnblum
, D.
, and Yedgar
, S.
, “Kinetics of linear rouleaux formation studied by visual monitoring of red cell dynamic organization
,” Biophys. J.
78
, 2470
–2474
(2000
).6.
Baskurt
, O. K.
and Meiselman
, H. J.
, “Blood rheology and hemodynamics
,” Semin. Thromb. Hemostasis
29
, 435
–450
(2003
).7.
Beris
, A. N.
, “Simple non-equilibrium thermodynamics applications to polymer rheology
,” Rheol. Rev.
2003
, 37
–75
.8.
Beris
, A. N.
and Edwards
, B. J.
, Thermodynamics of Flowing Systems With Internal Microstructure
(Oxford University Press
, New York
, 1994
).9.
Beris
, A. N.
and Mavrantzas
, V. G.
, “On the compatibility between various macroscopic formalisms for the concentration and flow of dilute polymer solutions
,” J. Rheol.
38
, 1235
(1994
).10.
Bureau
, M.
, Healy
, J. C.
, Bourgoin
, D.
, and Joly
, M.
, “Rheological hysteresis of blood at low shear rate
,” Biorheology
17
, 191
–203
(1980
).11.
Chen
, S.
, Gavish
, B.
, Mahler
, Y.
, and Yedgar
, S.
, “Monitoring of erythrocyte aggregate morphology under flow by computerized image analysis
,” Biorheology
32
, 487
–496
(1995
).12.
Chien
, S.
, “Shear dependence of effective cell volume as a determinant of blood viscosity
,” Science
168
, 977
–979
(1970
).13.
Chien
, S.
, Usami
, S.
, Taylor
, H. M.
, Lundberg
, J. L.
, and Gregersen
, M. I.
, “Effects of hematocrit and plasma proteins on human blood rheology at low shear rates
,” J. Appl. Physiol.
21
, 81
–87
(1966
).14.
Cho
, Y. I.
, Mooney
, M. P.
, and Cho
, D. J.
, “Hemorheological disorders in diabetes mellitus
,” J. Diabetes Sci. Technol.
2
, 1130
–1138
(2008
).15.
Doi
, M.
and Edwards
, S. F.
, The Theory of Polymer Dynamics
(Oxford
, Clarendon, United Kingdom
, 1986
).16.
Eckmann
, D. M.
, Bowers
, S.
, Stecker
, M.
, and Cheung
, A. T.
, “Hematocrit, volume expander, temperature, and shear rate effects on blood viscosity
,” Anesth. Analg.
91
, 539
–545
(2000
).17.
Fang
, J.
and Owens
, R. G.
, “Numerical simulations of pulsatile blood flow using a new constitutive model
,” Biorheology
43
, 637
–660
(2006
).18.
Fedosov
, D. A.
, Pan
, W.
, Caswell
, B.
, Gompper
, G.
, and Karniadakis
, G. E.
, “Predicting human blood viscosity in silico
,” Proc. Natl. Acad. Sci. U. S. A.
108
, 11772
–11777
(2011
).19.
Germann
, N.
, Cook
, L. P.
, and Beris
, A. N.
, “Nonequilibrium thermodynamic modeling of the structure and rheology of concentrated wormlike micellar solutions
,” J. Non-Newton. Fluid Mech.
196
, 51
–57
(2013
).20.
Grmela
, M.
and Öttinger
, H. C.
, “Dynamics and thermodynamics of complex fluids. I. Development of a general formalism
,” Phys. Rev. E
56
(6
), 6620
(1997
).21.
Herschel
, W. H.
and Bulkley
, R.
, “Konsistenzmessungen von gummi benzollösungen
,” Kolloid-Z.
39
, 291
–300
(1926
).22.
Ju
, M.
, Ye
, S. S.
, Namgung
, B.
, Cho
, S.
, Low
, H. T.
, Leo
, H. L.
, and Kim
, S.
, “A review of numerical methods for red blood cell flow simulation
,” Comput. Methods Biomech. Biomed. Eng.
18
, 130
(2013
).23.
Kaestner
, L.
, Steffen
, P.
, Nguyen
, D.
, Wang
, J.
, Wagner-Britz
, L.
, Jung
, A.
, Wagner
, C.
, and Bernhardt
, I.
, “Lysophosphatidic acid induced red blood cell aggregation in vitro
,” Bioelectrochemistry
87
, 89
–95
(2012
).24.
Kaliviotis
, E.
, “Mechanics of the red blood cell network
,” J. Cell. Biotechnol.
1
, 37
–43
(2015
).25.
Kaliviotis
, E.
and Yianneskis
, M.
, “Fast response characteristics of red blood cell aggregation
,” Biorheology
45
, 639
–649
(2008
).26.
Kaliviotis
, E.
and Yianneskis
, M.
, “Blood viscosity modelling: Influence of aggregate network dynamics under transient conditions
,” Biorheology
48
, 127
–147
(2011
).27.
Kaliviotis
, E.
, Dusting
, J.
, and Balabani
, S.
, “Spatial variation of blood viscosity: Modelling using shear fields measured by a μPIV based technique
,” Med. Eng. Phys.
33
, 824
–831
(2011
).28.
Kaliviotis
, E.
, Sherwood
, J. M.
, and Balabani
, S.
, “Partitioning of red blood cell aggregates in bifurcating microscale flows
,” Sci. Rep.
7
, 44563
(2017
).29.
Mavrantzas
, V. G.
and Beris
, A. N.
, “A hierarchical model for surface effects on chain conformation and rheology of polymer solutions. I. General formulation
,” J. Chem. Phys.
110
, 616
–627
(1999a
).30.
Mavrantzas
, V. G.
and Beris
, A. N.
, “A hierarchical model for surface effects on chain conformation and rheology of polymer solutions. II. Application to a neutral surface
,” J. Chem. Phys.
110
, 628
–638
(1999b
).31.
Mehri
, R.
, Laplante
, J.
, Mavriplis
, C.
, and Fenech
, M.
, “Investigation of blood flow analysis and red blood cell aggregation
,” J. Med. Biol. Eng.
34
, 469
–474
(2013
).32.
Merrill
, E. W.
and Pelletier
, G. A.
, “Viscosity of human blood: Transition from Newtonian to non-Newtonian
,” J. Appl. Physiol.
23
, 178
–182
(1967
).33.
Merrill
, E. W.
, Gilliland
, E. R.
, Cokelet
, G.
, Shin
, H.
, Britten
, A.
, and Wells
, R. E.
, Jr., “Rheology of human blood, near and at zero flow: Effects of temperature and hematocrit level
,” Biophys. J.
3
, 199
–213
(1963a
).34.
Merrill
, E. W.
, Cokelet
, G. C.
, Britten
, A.
, and Wells
, R. E.
, Jr., “Non-Newtonian rheology of human-blood-effect of fibrinogen deduced by ‘subtraction’
,” Circ. Res.
13
, 48
–55
(1963b
).35.
Merrill
, E. W.
, Benis
, A. M.
, Gilliland
, E. R.
, Sherwood
, T. K.
, and Salzman
, E. W.
, “Pressure-flow relations of human blood in hollow fibers at low flow rates
,” J. Appl. Physiol.
20
, 954
–967
(1965
).36.
Moyers-Gonzalez
, M.
, Owens
, R. G.
, and Fang
, J.
, “A non-homogeneous constitutive model for human blood. Part 1. Model derivation and steady flow
,” J. Fluid Mech.
617
, 327
–354
(2008
).37.
Murata
, T.
and Secomb
, T. W.
, “Effects of shear rate on rouleau formation in simple shear flow
,” Biorheology
25
, 113
–122
(1988
).38.
Oldroyd
, J. G.
, “On the formulation of rheological equations of state
,” Proc. R. Soc. A
200
, 523
–541
(1950
).39.
Öttinger
, H. C.
, Beyond Equilibrium Thermodynamics
(Wiley-Interscience
, New York
, 2004
).40.
Öttinger
, H. C.
, “Thermodynamic formulation of wall slip
,” J. Non-Newtonian Fluid Mech.
152
, 66
–75
(2008
).41.
Öttinger
, H. C.
and Grmela
, M.
, “Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism
,” Phys. Rev. E
56
(6
), 6633
(1997
).42.
Owens
, R. G.
, “A new microstructure-based constitutive model for human blood
,” J. Non-Newtonian Fluid Mech.
140
, 57
–70
(2006
).43.
Pan
, W.
, Caswell
, B.
, and Karniadakis
, G. E.
, “A low-dimensional model for the red blood cell
,” Soft Matter
6
, 4366
–4376
(2010
).44.
Perkkiö
, J.
and Keskinen
, R.
, “Hematocrit reduction in bifurcations due to plasma skimming
,” Bull. Math. Biol.
45
(1
), 41
–50
(1983
).45.
Rajagopal
, K. R.
and Srinivasa
, A. R.
, “A thermodynamic frame work for rate type fluid models
,” J. Non-Newtonian Fluid Mech.
88
, 207
–227
(2000
).46.
Samsel
, R. W.
and Perelson
, A. S.
, “Kinetics of rouleau formation. I. A mass action approach with geometric features
,” Biophys. J.
37
, 493
–514
(1982
).47.
Samsel
, R. W.
and Perelson
, A. S.
, “Kinetics of rouleau formation. II. Reversible reactions
,” Biophys. J.
45
, 805
–824
(1984
).48.
Schmid Schönbein
, H.
, Gaehtgens
, P.
, and Hirsch
, H.
, “On the shear rate dependence of red cell aggregation in vitro
,” J. Clin. Invest.
47
, 1447
–1454
(1968
).49.
Shalak
, R.
, Keller
, S. R.
, and Secomb
, T. W.
, “Mechanics of blood flow
,” J. Biomech. Eng.
103
, 102
–115
(1981
).50.
Shelby
, J. P.
, White
, J.
, Ganesan
, K.
, Rathod
, P. K.
, and Chiu
, D. T.
, “A microfluidic model for single-cell capillary obstruction by Plasmodium falciparum-infected erythrocytes
,” Proc. Natl. Acad. Sci. U. S. A.
100
, 14618
–14622
(2003
).51.
Shiga
, T.
, Imaizumi
, K.
, Harada
, N.
, and Sekiya
, M.
, “Kinetics of rouleaux formation using TV image analyzer. I. Human erythrocytes
,” Am. J. Physiol.
245
, H252
–H258
(1983a
).52.
Shiga
, T.
, Imaizumi
, K.
, Maeda
, N.
, and Kon
, K.
, “Kinetics of rouleaux formation using TV image analyzer. II. Rat erythrocytes
,” Am. J. Physiol.
245
, H259
–H264
(1983b
).53.
Steffen
, P.
, Verdier
, C.
, and Wagner
, C.
, “Quantification of depletion-induced adhesion of red blood cells
,” Phys. Rev. Lett.
110
, 018102
(2013
).54.
Stephanou
, P. S.
, “How the flow affects the phase behaviour and microstructure of polymer nanocomposites
,” J. Chem. Phys.
142
, 064901
(2015
).55.
Stephanou
, P. S.
, “The rheology of drilling fluids from a non-equilibrium thermodynamics perspective
,” J. Pet. Sci. Eng.
(published online 2017
).56.
Stephanou
, P. S.
, Baig
, C.
, and Mavrantzas
, V. G.
, “A generalized differential constitutive equation for polymer melts based on principles of nonequilibrium thermodynamics
,” J. Rheol.
53
, 309
−337
(2009
).57.
Stephanou
, P. S.
, Mavrantzas
, V. G.
, and Georgiou
, G. C.
, “Continuum model for the phase behavior, microstructure, and rheology of unentangled polymer nanocomposite melts
,” Macromolecules
47
, 4493
−4513
(2014
).58.
Stephanou
, P. S.
, Tsimouri
, I. Ch.
, and Mavrantzas
, V. G.
, “Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermodynamics
,” Macromolecules
49
, 3161
−3173
(2016
).59.
Szołna-Chodór
, A.
, Bosek
, M.
, and Grzegorzewski
, B.
, “Kinetics of red blood cell rouleaux formation studied by light scattering
,” J. Biomed. Opt.
20
(2
), 025001
(2015
).60.
Usmani
, R.
, “Inversion of a tridiagonal jacobi matrix
,” Linear Algebra Appl.
212-213
, 413
–414
(1994
).61.
Usami
, S.
, Chien
, S.
, and Bertles
, J. F.
, “Deformabolity of sickle cells as studied by micro-sieving
,” J. Lab. Clin. Med.
86
, 274
–279
(1975
).62.
Yilmaz
, F.
and Gundogdu
, M. Y.
, “A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions
,” Korea-Australia Rheol. J.
20
, 197
–211
(2008
).© 2018 Author(s).
2018
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