Three-dimensional (3D) simulations on blood flow in a complex patient-specific retina vascular network were performed considering deformable red blood cells, white blood cells (WBCs), and obstructed vessels. First, the impact of blockage on flow rate distribution (without cells) was investigated. It showed that the blockage might change the flow rate significantly on distant vessels that were not directly connected with the blocked vessel. The flow rate in some vessels could increase up to 1200% due to an obstruction. However, with cells, it showed a fluctuating flow pattern, and the cells showed complicated transport behavior at bifurcations. Cell accumulation might occur in some bifurcations such as a T-shaped junction that eventually led to a physical blockage. The addition of WBCs impacted the local flow rate when they were squeezed through a capillary vessel, and the flow rate could be decreased up to 32% due to the larger size of WBCs. The simulation of flow under stenosis with cells showed that cells could oscillate and become trapped in a vessel due to the fluctuating flow. Finally, a reduced order model (ROM) with multiple non-Newtonian viscosity models was used to simulate the blood flow in the network. Compared with the 3D model, all ROMs reproduced accurate predictions on hematocrit and flow rate distribution in the vascular network. Among them, the Fåhræus–Lindqvist model was found to be the most accurate one. The work can be used to build a multiscale model for blood flow through integration of ROMs and 3D multiphysics models.

1.
J.
Xu
,
S. L.
Murphy
,
K. D.
Kochanek
, and
E.
Arias
, “
Mortality in the United States, 2018
,” NCHS Data Brief No. 355 (
NCHS
,
2020
).
2.
Ş.
E.
Erdener
,
J.
Tang
,
A.
Sajjadi
,
K.
Kiliç
,
S.
Kura
,
C. B.
Schaffer
, and
D. A.
Boas
, “
Spatio-temporal dynamics of cerebral capillary segments with stalling red blood cells
,”
J. Cereb. Blood Flow Metab.
39
(
5
),
886
900
(
2019
).
3.
L.
Østergaard
,
N. B.
Finnerup
,
A. J.
Terkelsen
,
R. A.
Olesen
,
K. R.
Drasbek
,
L.
Knudsen
,
S. N.
Jespersen
,
J.
Frystyk
,
M.
Charles
,
R. W.
Thomsen
 et al., “
The effects of capillary dysfunction on oxygen and glucose extraction in diabetic neuropathy
,”
Diabetologia
58
(
4
),
666
677
(
2015
).
4.
S. N.
Jespersen
and
L.
Østergaard
, “
The roles of cerebral blood flow, capillary transit time heterogeneity, and oxygen tension in brain oxygenation and metabolism
,”
J. Cereb. Blood Flow Metab.
32
(
2
),
264
277
(
2012
).
5.
M. O.
Bernabeu
,
J.
Köry
,
J. A.
Grogan
,
B.
Markelc
,
A.
Beardo
,
M.
d'Avezac
,
R.
Enjalbert
,
J.
Kaeppler
,
N.
Daly
,
J.
Hetherington
 et al., “
Abnormal morphology biases hematocrit distribution in tumor vasculature and contributes to heterogeneity in tissue oxygenation
,”
Proc. Natl. Acad. Sci.
117
(
45
),
27811
27819
(
2020
).
6.
A.
Guevara-Torres
,
A.
Joseph
, and
J.
Schallek
, “
Label free measurement of retinal blood cell flux, velocity, hematocrit and capillary width in the living mouse eye
,”
Biomed. Opt. Express
7
(
10
),
4228
4249
(
2016
).
7.
F.
Schmid
,
G.
Conti
,
P.
Jenny
, and
B.
Weber
, “
The severity of microstrokes depends on local vascular topology and baseline perfusion
,” bioRxiv (
2020
).
8.
J.
Reichold
,
M.
Stampanoni
,
A. L.
Keller
,
A.
Buck
,
P.
Jenny
, and
B.
Weber
, “
Vascular graph model to simulate the cerebral blood flow in realistic vascular networks
,”
J. Cereb. Blood Flow Metab.
29
(
8
),
1429
1443
(
2009
).
9.
A. R.
Pries
,
T. W.
Secomb
,
P.
Gaehtgens
, and
J.
Gross
, “
Blood flow in microvascular networks. Experiments and simulation
,”
Circ. Res.
67
(
4
),
826
834
(
1990
).
10.
M.
Peyrounette
,
Y.
Davit
,
M.
Quintard
, and
S.
Lorthois
, “
Multiscale modelling of blood flow in cerebral microcirculation: Details at capillary scale control accuracy at the level of the cortex
,”
PLoS One
13
(
1
),
e0189474
(
2018
).
11.
L.
Lu
,
M. J.
Morse
,
A.
Rahimian
,
G.
Stadler
, and
D.
Zorin
, “
Scalable simulation of realistic volume fraction red blood cell flows through vascular networks
,” in
Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis
(
ACM
,
2019
), pp.
1
30
.
12.
G.
Závodszky
,
B.
van Rooij
,
B.
Czaja
,
V.
Azizi
,
D.
de Kanter
, and
A. G.
Hoekstra
, “
Red blood cell and platelet diffusivity and margination in the presence of cross-stream gradients in blood flows
,”
Phys. Fluids
31
(
3
),
031903
(
2019
).
13.
C.
Kotsalos
,
J.
Latt
, and
B.
Chopard
, “
Palabos-npFEM: Software for the simulation of cellular blood flow (digital blood)
,”
J. Open Res. Software
9
,
1
12
(
2021
).
14.
A.
Peters
,
S.
Melchionna
,
E.
Kaxiras
,
J.
Lätt
,
J.
Sircar
,
M.
Bernaschi
,
M.
Bison
, and
S.
Succi
, “
Multiscale simulation of cardiovascular flows on the IBM Bluegene/P: Full heart-circulation system at red-blood cell resolution
,” in
Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
(
IEEE Computer Society
,
2010
), pp.
1
10
.
15.
D.
Rossinelli
,
Y.-H.
Tang
,
K.
Lykov
,
D.
Alexeev
,
M.
Bernaschi
,
P.
Hadjidoukas
,
M.
Bisson
,
W.
Joubert
,
C.
Conti
,
G.
Karniadakis
 et al., “
The in-silico lab-on-a-chip: Petascale and high-throughput simulations of microfluidics at cell resolution
,” in
Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis
(
ACM
,
2015
), pp.
1
12
.
16.
G.
Li
,
T.
Ye
, and
X.
Li
, “
Parallel modeling of cell suspension flow in complex micro-networks with inflow/outflow boundary conditions
,”
J. Comput. Phys.
401
,
109031
(
2020
).
17.
J. B.
Freund
, “
Numerical simulation of flowing blood cells
,”
Annu. Rev. Fluid Mech.
46
,
67
95
(
2014
).
18.
Q. M.
Qi
and
E. S.
Shaqfeh
, “
Time-dependent particle migration and margination in the pressure-driven channel flow of blood
,”
Phys. Rev. Fluids
3
(
3
),
034302
(
2018
).
19.
D.
Xu
,
E.
Kaliviotis
,
A.
Munjiza
,
E.
Avital
,
C.
Ji
, and
J.
Williams
, “
Large scale simulation of red blood cell aggregation in shear flows
,”
J. Biomech.
46
(
11
),
1810
1817
(
2013
).
20.
P.
Balogh
and
P.
Bagchi
, “
Direct numerical simulation of cellular-scale blood flow in 3D microvascular networks
,”
Biophys. J.
113
(
12
),
2815
2826
(
2017
).
21.
P.
Balogh
and
P.
Bagchi
, “
Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks
,”
Phys. Fluids
30
(
5
),
051902
(
2018
).
22.
C.
Bächer
,
L.
Schrack
, and
S.
Gekle
, “
Clustering of microscopic particles in constricted blood flow
,”
Phys. Rev. Fluids
2
(
1
),
013102
(
2017
).
23.
T.-W.
Lee
,
K.-S.
Bae
,
H. S.
Choi
, and
M.-J.
Chern
, “
Computational simulations of flow and oxygen/drug delivery in a three-dimensional capillary network
,”
Int. Scholarly Res. Not.
2014
,
1
.
24.
H.
Tamaddon
,
M.
Behnia
,
M.
Behnia
, and
L.
Kritharides
, “
A new approach to blood flow simulation in vascular networks
,”
Comput. Methods Biomech. Biomed. Eng.
19
(
6
),
673
685
(
2016
).
25.
D. A.
Fedosov
,
B.
Caswell
, and
G. E.
Karniadakis
, “
A multiscale red blood cell model with accurate mechanics, rheology, and dynamics
,”
Biophys. J.
98
(
10
),
2215
2225
(
2010
).
26.
I. V.
Pivkin
and
G. E.
Karniadakis
, “
Accurate coarse-grained modeling of red blood cells
,”
Phys. Rev. Lett.
101
(
11
),
118105
(
2008
).
27.
D. A.
Reasor
, Jr.
,
J. R.
Clausen
, and
C. K.
Aidun
, “
Coupling the lattice-Boltzmann and spectrin-link methods for the direct numerical simulation of cellular blood flow
,”
Int. J. Numer. Methods Fluids
68
(
6
),
767
781
(
2012
).
28.
M.
Dao
,
J.
Li
, and
S.
Suresh
, “
Molecularly based analysis of deformation of spectrin network and human erythrocyte
,”
Mater. Sci. Eng.: C
26
(
8
),
1232
1244
(
2006
).
29.
J.
Tan
,
S.
Sohrabi
,
R.
He
, and
Y.
Liu
, “
Numerical simulation of cell squeezing through a micropore by the immersed boundary method
,”
Proc. Inst. Mech. Eng., Part C
232
(
3
),
502
514
(
2018
).
30.
X.
Yu
,
J.
Tan
, and
S.
Diamond
, “
Hemodynamic force triggers rapid NETosis within sterile thrombotic occlusions
,”
J. Thromb. Haemostasis
16
(
2
),
316
329
(
2018
).
31.
J.
Li
,
M.
Dao
,
C.
Lim
, and
S.
Suresh
, “
Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte
,”
Biophys. J.
88
(
5
),
3707
3719
(
2005
).
32.
J.
Tan
,
T. R.
Sinno
, and
S. L.
Diamond
, “
A parallel fluid–solid coupling model using LAMMPS and Palabos based on the immersed boundary method
,”
J. Comput. Sci.
25
,
89
100
(
2018
).
33.
D. H.
Boal
and
M.
Rao
, “
Topology changes in fluid membranes
,”
Phys. Rev. A
46
(
6
),
3037
(
1992
).
34.
S.
Plimpton
, “
Fast parallel algorithms for short-range molecular dynamics
,”
J. Comput. Phys.
117
(
1
),
1
19
(
1995
).
35.
A. P.
Thompson
,
H. M.
Aktulga
,
R.
Berger
,
D. S.
Bolintineanu
,
W. M.
Brown
,
P. S.
Crozier
,
P. J.
in 't Veld
,
A.
Kohlmeyer
,
S. G.
Moore
,
T. D.
Nguyen
,
R.
Shan
,
M. J.
Stevens
,
J.
Tranchida
,
C.
Trott
, and
S. J.
Plimpton
, “
LAMMPS: A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales
,”
Comput. Phys. Commun.
271
,
108171
(
2022
).
36.
D. A.
Fedosov
,
J.
Fornleitner
, and
G.
Gompper
, “
Margination of white blood cells in microcapillary flow
,”
Phys. Rev. Lett.
108
(
2
),
028104
(
2012
).
37.
J.
Tan
,
Z.
Ding
,
M.
Hood
, and
W.
Li
, “
Simulation of circulating tumor cell transport and adhesion in cell suspensions in microfluidic devices
,”
Biomicrofluidics
13
(
6
),
064105
(
2019
).
38.
X.
He
and
L.-S.
Luo
, “
Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation
,”
Phys. Rev. E
56
(
6
),
6811
(
1997
).
39.
S.
Chen
and
G. D.
Doolen
, “
Lattice Boltzmann method for fluid flows
,”
Annu. Rev. Fluid Mech.
30
(
1
),
329
364
(
1998
).
40.
C. K.
Aidun
and
J. R.
Clausen
, “
Lattice-Boltzmann method for complex flows
,”
Annu. Rev. Fluid Mech.
42
,
439
472
(
2010
).
41.
R. S.
Voronov
,
T. J.
Stalker
,
L. F.
Brass
, and
S. L.
Diamond
, “
Simulation of intrathrombus fluid and solute transport using in vivo clot structures with single platelet resolution
,”
Ann. Biomed. Eng.
41
(
6
),
1297
1307
(
2013
).
42.
J.
Latt
,
O.
Malaspinas
,
D.
Kontaxakis
,
A.
Parmigiani
,
D.
Lagrava
,
F.
Brogi
,
M. B.
Belgacem
,
Y.
Thorimbert
,
S.
Leclaire
,
S.
Li
 et al., “
Palabos: Parallel lattice Boltzmann solver
,”
Comput. Math. Appl.
81
,
334
(
2021
).
43.
Y.-H.
Qian
,
D.
d'Humières
, and
P.
Lallemand
, “
Lattice BGK models for Navier-Stokes equation
,”
Europhys. Lett.
17
(
6
),
479
(
1992
).
44.
C. S.
Peskin
, “
Flow patterns around heart valves: A numerical method
,”
J. Comput. Phys.
10
(
2
),
252
271
(
1972
).
45.
C. S.
Peskin
, “
The immersed boundary method
,”
Acta Numer.
11
,
479
517
(
2002
).
46.
R.
Mittal
and
G.
Iaccarino
, “
Immersed boundary methods
,”
Annu. Rev. Fluid Mech.
37
,
239
261
(
2005
).
47.
Z.-G.
Feng
and
E. E.
Michaelides
, “
The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems
,”
J. Comput. Phys.
195
(
2
),
602
628
(
2004
).
48.
W.-X.
Huang
and
F.-B.
Tian
, “
Recent trends and progress in the immersed boundary method
,”
Proc. Inst. Mech. Eng., Part C
233
(
23–24
),
7617
7636
(
2019
).
49.
H.
Ye
,
Z.
Shen
,
W.
Xian
,
T.
Zhang
,
S.
Tang
, and
Y.
Li
, “
OpenFSI: A highly efficient and portable fluid–structure simulation package based on immersed-boundary method
,”
Comput. Phys. Commun.
256
,
107463
(
2020
).
50.
J.
Boyd
,
J. M.
Buick
, and
S.
Green
, “
Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method
,”
Phys. Fluids
19
(
9
),
093103
(
2007
).
51.
N.
Kumar
,
A.
Khader
,
R.
Pai
,
P.
Kyriacou
,
S.
Khan
, and
P.
Koteshwara
, “
Computational fluid dynamic study on effect of Carreau-Yasuda and Newtonian blood viscosity models on hemodynamic parameters
,”
J. Comput. Methods Sci. Eng.
19
(
2
),
465
477
(
2019
).
52.
F. J.
Gijsen
,
F. N.
van de Vosse
, and
J.
Janssen
, “
The influence of the non-Newtonian properties of blood on the flow in large arteries: Steady flow in a carotid bifurcation model
,”
J. Biomech.
32
(
6
),
601
608
(
1999
).
53.
P.
Ballyk
,
D.
Steinman
, and
C.
Ethier
, “
Simulation of non-Newtonian blood flow in an end-to-side anastomosis
,”
Biorheology
31
(
5
),
565
586
(
1994
).
54.
B. M.
Johnston
,
P. R.
Johnston
,
S.
Corney
, and
D.
Kilpatrick
, “
Non-Newtonian blood flow in human right coronary arteries: Steady state simulations
,”
J. Biomech.
37
(
5
),
709
720
(
2004
).
55.
B. M.
Johnston
,
P. R.
Johnston
,
S.
Corney
, and
D.
Kilpatrick
, “
Non-Newtonian blood flow in human right coronary arteries: Transient simulations
,”
J. Biomech.
39
(
6
),
1116
1128
(
2006
).
56.
J.
Venkatesan
,
D.
Sankar
,
K.
Hemalatha
, and
Y.
Yatim
, “
Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries
,”
J. Appl. Math.
2013
,
583809
.
57.
V.
Srivastava
and
M.
Saxena
, “
Two-layered model of Casson fluid flow through stenotic blood vessels: Applications to the cardiovascular system
,”
J. Biomech.
27
(
7
),
921
928
(
1994
).
58.
F.
Ali
,
N. A.
Sheikh
,
I.
Khan
, and
M.
Saqib
, “
Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: A fractional model
,”
J. Magn. Magn. Mater.
423
,
327
336
(
2017
).
59.
P.
Ghassemi
,
J.
Wang
,
A. J.
Melchiorri
,
J. C.
Ramella-Roman
,
S. A.
Mathews
,
J. C.
Coburn
,
B. S.
Sorg
,
Y.
Chen
, and
T. J.
Pfefer
, “
Rapid prototyping of biomimetic vascular phantoms for hyperspectral reflectance imaging
,”
J. Biomed. Opt.
20
(
12
),
121312
(
2015
).
60.
P.
Cignoni
,
M.
Callieri
,
M.
Corsini
,
M.
Dellepiane
,
F.
Ganovelli
,
G.
Ranzuglia
 et al., “
Meshlab: An open-source mesh processing tool
,” in
Eurographics Italian Chapter Conference
, Salerno, Italy (
Eurographics Digital Library
,
2008
), Vol.
2008
, pp.
129
136
.
61.
J.
Schindelin
,
I.
Arganda-Carreras
,
E.
Frise
,
V.
Kaynig
,
M.
Longair
,
T.
Pietzsch
,
S.
Preibisch
,
C.
Rueden
,
S.
Saalfeld
,
B.
Schmid
 et al., “
Fiji: An open-source platform for biological-image analysis
,”
Nat. Methods
9
(
7
),
676
682
(
2012
).
62.
A. G.
Koutsiaris
,
S. V.
Tachmitzi
, and
N.
Batis
, “
Wall shear stress quantification in the human conjunctival pre-capillary arterioles in vivo
,”
Microvascular Research
85
,
34
39
(
2013
).
63.
T. V.
Colace
,
R. W.
Muthard
, and
S. L.
Diamond
, “
Thrombus growth and embolism on tissue factor-bearing collagen surfaces under flow: Role of thrombin with and without fibrin
,”
Arterioscler., Thromb., Vasc. Biol.
32
(
6
),
1466
1476
(
2012
).
64.
M.
Cates
,
J.-C.
Desplat
,
P.
Stansell
,
A.
Wagner
,
K.
Stratford
,
R.
Adhikari
, and
I.
Pagonabarraga
, “
Physical and computational scaling issues in lattice Boltzmann simulations of binary fluid mixtures
,”
Philos. Trans. R. Soc. A
363
(
1833
),
1917
1935
(
2005
).
65.
M.
Hecht
,
J.
Harting
,
T.
Ihle
, and
H. J.
Herrmann
, “
Simulation of claylike colloids
,”
Phys. Rev. E
72
(
1
),
011408
(
2005
).
66.
T.
Secomb
,
R.
Hsu
, and
A.
Pries
, “
Motion of red blood cells in a capillary with an endothelial surface layer: Effect of flow velocity
,”
Am. J. Physiol.
281
(
2
),
H629
H636
(
2001
).
67.
D. A.
Fedosov
,
M.
Peltomäki
, and
G.
Gompper
, “
Deformation and dynamics of red blood cells in flow through cylindrical microchannels
,”
Soft Matter
10
(
24
),
4258
4267
(
2014
).
68.
K.-I.
Tsubota
, “
Elongation deformation of a red blood cell under shear flow as stretch testing
,”
J. Mech. Phys. Solids
152
,
104345
(
2021
).
69.
R.
Skalak
and
P.
Brånemark
, “
Deformation of red blood cells in capillaries
,”
Science
164
(
3880
),
717
719
(
1969
).
70.
S. M.
Recktenwald
,
K.
Graessel
,
F. M.
Maurer
,
T.
John
,
S.
Gekle
, and
C.
Wagner
, “
Red blood cell shape transitions and dynamics in time-dependent capillary flows
,”
Biophys. J.
121
(
1
),
23
36
(
2022
).
71.
F.
Reichel
,
J.
Mauer
,
A. A.
Nawaz
,
G.
Gompper
,
J.
Guck
, and
D. A.
Fedosov
, “
High-throughput microfluidic characterization of erythrocyte shapes and mechanical variability
,”
Biophys. J.
117
(
1
),
14
24
(
2019
).
72.
T.
Krüger
,
M.
Gross
,
D.
Raabe
, and
F.
Varnik
, “
Crossover from tumbling to tank-treading-like motion in dense simulated suspensions of red blood cells
,”
Soft Matter
9
(
37
),
9008
9015
(
2013
).
73.
Y.
Kodama
,
H.
Aoki
,
Y.
Yamagata
, and
K.
Tsubota
, “
In vitro analysis of blood flow in a microvascular network with realistic geometry
,”
J. Biomech.
88
,
88
94
(
2019
).
74.
E.
Kaliviotis
,
J. M.
Sherwood
, and
S.
Balabani
, “
Local viscosity distribution in bifurcating microfluidic blood flows
,”
Phys. Fluids
30
(
3
),
030706
(
2018
).
75.
J. M.
Sherwood
,
D.
Holmes
,
E.
Kaliviotis
, and
S.
Balabani
, “
Spatial distributions of red blood cells significantly alter local haemodynamics
,”
PLoS One
9
(
6
),
e100473
(
2014
).
76.
L.
Rolfes
,
M.
Riek-Burchardt
,
M.
Pawlitzki
,
J.
Minnerup
,
S.
Bock
,
M.
Schmidt
,
S. G.
Meuth
,
M.
Gunzer
, and
J.
Neumann
, “
Neutrophil granulocytes promote flow stagnation due to dynamic capillary stalls following experimental stroke
,”
Brain, Behav., Immun.
93
,
322
330
(
2021
).
77.
P.
Bagher
and
S. S.
Segal
, “
Regulation of blood flow in the microcirculation: Role of conducted vasodilation
,”
Acta Physiol.
202
(
3
),
271
284
(
2011
).
78.
A.
Bucaro
,
C.
Murphy
,
N.
Ferrier
,
J.
Insley
,
V.
Mateevitsi
,
M. E.
Papka
,
S.
Rizzi
, and
J.
Tan
, “
Instrumenting multiphysics blood flow simulation codes for in situ visualization and analysis
,” in
2021 IEEE 11th Symposium on Large Data Analysis and Visualization (LDAV)
(
IEEE
,
2021
), pp.
88
89
.

Supplementary Material

You do not currently have access to this content.