The key to effective magnetic drug targeting (MDT) is to improve the aggregation of magnetic drug carrier particles (MDCPs) at the target site. Compared to related theoretical models, the novelty of this investigation is mainly reflected in that the microvascular blood is considered as a two-phase fluid composed of a continuous phase (plasma) and a discrete phase (red blood cells (RBCs)). And plasma flow state is quantitatively described based on the Navier–Stokes equation of two-phase flow theory, the effect of momentum exchange between the two-phase interface is considered in the Navier–Stokes equation. Besides, the coupling effect between plasma pressure and tissue fluid pressure is considered. The random motion effects and the collision effects of MDCPs transported in the blood are quantitatively described using the Boltzmann equation. The results show that the capture efficiency (CE) presents a nonlinear increase with the increase of magnetic induction intensity and a nonlinear decrease with the increase of plasma velocity, but an approximately linear increase with the increase of the particle radius. Furthermore, greater permeability of the microvessel wall promotes the aggregation of MDCPs. The CE predicted by the model agrees well with the experimental results.

References

References
1.
Breen
,
S.
,
Kofoed
,
S.
,
Ritchie
,
D.
,
Dryden
,
T.
,
Maguire
,
R.
,
Kearney
,
N.
, and
Aranda
,
S.
,
2016
, “
Remote Real-Time Monitoring for Chemotherapy Side-Effects in Patients With Blood Cancers
,”
Collegian
,
24
(
6
), pp.
541
549
.
2.
Lindley
,
C.
,
Mccune
,
J. S.
,
Thomason
,
T. E.
,
Lauder
,
D.
,
Sauls
,
A.
,
Adkins
,
S.
, and
Sawyer
,
W. T.
,
1999
, “
Perception of Chemotherapy Side Effects Cancer Versus Noncancer Patients
,”
Cancer Pract.
,
7
(
2
), pp.
59
65
.
3.
Senyei
,
A.
,
Widder
,
K.
, and
Czerlinski
,
G.
,
1978
, “
Magnetic Guidance of Drug-Carrying Microspheres
,”
J. Appl. Phys.
,
49
(
6
), pp.
3578
3583
.
4.
Widder
,
K. J.
,
Senyei
,
A. E.
, and
Scarpeui
,
D. G.
,
1978
, “
Magnetic Microsphere: A Model System for Site Specific Drug Delivery In Vivo
,”
Proc. Soc. Exp. Biol. Med.
,
158
(
2
), pp.
141
146
.
5.
Lübbe
,
A. S.
,
Bergemann
,
C.
,
Riess
,
H.
,
Schriever
,
F.
,
Reichardt
,
P.
,
Possinger
,
K.
,
Matthias
,
M.
,
Dörken
,
B.
,
Herrmann
,
F.
,
Gürtler
,
R.
,
Hohenberger
,
P.
,
Haas
,
N.
,
Sohr
,
R.
,
Sander
,
B.
,
Lemke
,
A. J.
,
Ohlendorf
,
D.
,
Huhnt
,
W.
, and
Huhn
,
D.
,
1996
, “
Clinical Experiences With Magnetic Drug Targeting: A Phase I Study With 4'-Epidoxorubicin in 14 Patients With Advanced Solid Tumors
,”
Cancer Res.
,
56
(
20
), pp.
4686
4693
.https://www.ncbi.nlm.nih.gov/pubmed/8840985
6.
Rotariu
,
O.
, and
Strachan
,
N. J. C.
,
2005
, “
Modelling Magnetic Carrier Particle Targeting in the Tumor Microvasculature for Cancer Treatment
,”
J. Magn. Magn. Mater.
,
293
(
1
), pp.
639
646
.
7.
Xu
,
H.
,
Song
,
T.
,
Bao
,
X.
, and
Hu
,
L.
,
2005
, “
Site-Directed Research of Magnetic Nanoparticles in Magnetic Drug Targeting
,”
J. Magn. Magn. Mater.
,
293
(
1
), pp.
514
519
.
8.
Avilés
,
M. O.
,
Ebner
,
A. D.
, and
Ritter
,
J. A.
,
2007
, “
Ferromagnetic Seeding for the Magnetic Targeting of Drugs and Radiation in Capillary Beds
,”
J. Magn. Magn. Mater.
,
310
(
1
), pp.
131
144
.
9.
Avilés
,
M. O.
,
Chen
,
H.
,
Ebner
,
A. D.
,
Rosengart
,
A. J.
,
Kaminski
,
M. D.
, and
Ritter
,
J. A.
,
2007
, “
In Vitro Study of Ferromagnetic Stents for Implant Assisted-Magnetic Drug Targeting
,”
J. Magn. Magn. Mater.
,
311
(
1
), pp.
306
311
.
10.
Avilés
,
M. O.
,
Ebner
,
A. D.
, and
Ritter
,
J. A.
,
2008
, “
In Vitro Study of Magnetic Particle Seeding for Implant Assisted-Magnetic Drug Targeting
,”
J. Magn. Magn. Mater.
,
320
(
21
), pp.
2640
2646
.
11.
Avilés
,
M. O.
,
Ebner
,
A. D.
, and
Ritter
,
J. A.
,
2009
, “
In Vitro Study of Magnetic Particle Seeding for Implant-Assisted-Magnetic Drug Targeting: Seed and Magnetic Drug Carrier Particle Capture
,”
J. Magn. Magn. Mater.
,
321
(
10
), pp.
1586
1590
.
12.
Furlani
,
E. P.
, and
Ng
,
K. C.
,
2006
, “
Analytical Model of Magnetic Nanoparticle Transport and Capture in the Microvasculature
,”
Phys. Rev. E
,
73
(
6
), p.
061919
.
13.
Furlani
,
E. J.
, and
Furlani
,
E. P.
,
2007
, “
A Model for Predicting Magnetic Targeting of Multifunctional Particles in the Microvasculature
,”
J. Magn. Magn. Mater.
,
312
(
1
), pp.
187
193
.
14.
Shaw
,
S.
,
Murthy
,
P. V. S. N.
, and
Pradhan
,
S. C.
,
2010
, “
Effect of non-Newtonian Characteristics of Blood on Magnetic Targeting in the Impermeable Micro-Vessel
,”
J. Magn. Magn. Mater.
,
322
(
8
), pp.
1037
1043
.
15.
Shaw
,
S.
, and
Murthy
,
P. V. S. N.
,
2010
, “
Magnetic Drug Targeting in the Permeable Blood Vessel—The Effect of Blood Rheology
,”
ASME J. Nanotechnol. Eng. Med.
,
1
(
2
), p.
021001
.
16.
Shaw
,
S.
, and
Murthy
,
P. V. S. N.
,
2010
, “
Magnetic Targeting in the Impermeable Microvessel With Two-Phase Fluid Model—Non-Newtonian Characteristics of Blood
,”
Microvascular Res.
,
80
(
2
), pp.
209
220
.
17.
Shaw
,
S.
,
Murthy
,
P. V. S. N.
, and
Sibanda
,
P.
,
2013
, “
Magnetic Drug Targeting in a Permeable Microvessel
,”
Microvascular Res.
,
85
, pp.
77
85
.
18.
Avilés
,
M. O.
,
Ebner
,
A. D.
, and
Ritter
,
J. A.
,
2008
, “
Implant Assisted-Magnetic Drug Targeting: Comparison of In Vitro Experiments With Theory
,”
J. Magn. Magn. Mater.
,
320
(
21
), pp.
2704
2713
.
19.
Cregg
,
P. J.
,
Murphy
,
K.
,
Mardinoglu
,
A.
, and
Prina-Mello
,
A.
,
2010
, “
Many Particle Magnetic Dipole-Dipole and Hydrodynamic Interactions in Magnetizable Stent Assisted Magnetic Drug Targeting
,”
J. Magn. Magn. Mater.
,
322
(
15
), pp.
2087
2094
.
20.
Cregg
,
P. J.
,
Murphy
,
K.
, and
Mardinoglu
,
A.
,
2012
, “
Inclusion of Interactions in Mathematical Modelling of Implant Assisted Magnetic Drug Targeting
,”
Appl. Math. Modell.
,
36
(
1
), pp.
1
34
.
21.
Rukshin
,
I.
,
Mohrenweiser
,
J.
,
Yue
,
P.
, and
Afkhami
,
S.
,
2017
, “
Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting
,”
Fluids
,
2
(
2
), pp.
29
31
.
22.
EI-shahed
,
M.
,
2004
, “
Blood Flow in a Capillary With Permeable Wall
,”
Phys. A
,
338
, pp.
544
558
.
23.
Shaw
,
S.
,
Sutradhar
,
A.
, and
Murthy
,
P.
,
2017
, “
Permeability and Stress-Jump Effects on Magnetic Drug Targeting in a Permeable Microvessel Using Darcy Model
,”
J. Magn. Magn. Mater.
,
429
, pp.
227
235
.
24.
Sutradhar
,
A.
,
Mondal
,
J. K.
,
Murthy
,
P.
, and
Gorla
,
R. S. R.
,
2016
, “
Influence of Starling's Hypothesis and Joule Heating on Peristaltic Flow of an Electrically Conducting Casson Fluid in a Permeable Microvessel
,”
ASME. J. Fluids Eng.
,
138
(
11
), p.
111106
.
25.
Gerber
,
R.
,
Takayasu
,
M.
, and
Friedlander
,
F. J.
,
1983
, “
Generalization of HGMS Theory—The Capture of Ultra Fine Particles
,”
IEEE Trans. Magn.
,
19
(
5
), pp.
2115
2117
.
26.
Takayasu
,
M.
,
Gerber
,
R.
, and
Friedlander
,
F. J.
,
1983
, “
Magnetic Separation of Sub-Micron Particles
,”
IEEE Trans. Magn.
,
19
(
5
), pp.
2112
2114
.
27.
Fletcher
,
D.
,
1991
, “
Fine Particle High Gradient Magnetic Entrapment
,”
IEEE Trans. Magn.
,
27
(
4
), pp.
3655
3677
.
28.
Fernández-Pacheco
,
R.
,
Marquina
,
C.
,
Gabriel Valdivia
,
J.
,
Gutiérrez
,
M.
,
Soledad Romero
,
M.
,
Cornudella
,
R.
,
Laborda
,
A.
,
Viloria
,
A.
,
Higuera
,
T.
,
García
,
A.
,
De Jalón
,
J. A. G.
, and
Ricardo Ibarra
,
M.
,
2007
, “
Magnetic Nanoparticles for Local Drug Delivery Using Magnetic Implants
,”
J. Magn. Magn. Mater.
,
311
(
1
), pp.
318
322
.
29.
Wang
,
H.
, and
Skalak
,
R.
,
1969
, “
Viscous Flow in a Cylindrical Tube Containing a Line of Spherical Particles
,”
J. Fluid Mech.
,
38
(
1
), pp.
75
96
.
30.
Chen
,
T. C.
, and
Skalak
,
R.
,
1970
, “
Stokes Flow in a Cylindrical Tube Containing a Line of Spheroidal Particles
,”
Appl. Sci. Res.
,
22
(
1
), pp.
403
441
.
31.
Taylor
,
M.
,
1955
, “
The Flow of Blood in Narrow Tubes II: The Axial Stream and Its Formation, as Determined by Changes in Optical Density
,”
Austral. J. Exp. Biol.
,
33
(
1
), pp.
1
16
.
32.
Sankar
,
D. S.
, and
Lee
,
U.
,
2010
, “
Two-Fluid Casson Model for Pulsatile Blood Flow Through Stenosed Arteries: A Theoretical Model
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
8
), pp.
2086
2097
.
33.
Sankar
,
D. S.
, and
Lee
,
U.
,
2011
, “
Nonlinear Mathematical Analysis for Blood Flow in a Constricted Artery Under Periodic Body Acceleration
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
11
), pp.
4390
4402
.
34.
Sankar
,
D. S.
, and
Ismail
,
A. I.
,
2009
, “
Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study
,”
Boundary Value Probl.
,
2009
(
1
), pp.
1
15
.
35.
Sankar
,
D. S.
,
2011
, “
Two-Phase Non-Linear Model for Blood Flow in Asymmetric and Axisymmetric Stenosed Arteries
,”
Int. J. Non-Linear Mech.
,
46
(
1
), pp.
296
305
.
36.
Sankar
,
D. S.
,
Goh
,
J.
, and
Ismail
,
A. I. M.
,
2010
, “
FDM Analysis for Blood Flow Through Stenosed Tapered Arteries
,”
Boundary Value Probl.
,
1
, pp.
1
16
.
37.
Sankar
,
D. S.
,
2010
, “
Pulsatile Flow of a Two-Fluid Model for Blood Flow Through Arterial Stenosis
,”
Math. Probl. Eng.
,
4
, pp.
242
256
.
38.
Sankar
,
D. S.
, and
Lee
,
U.
,
2010
, “
Pulsatile Flow of Two-Fluid Nonlinear Models for Blood Flow Through Catheterized Arteries: A Comparative Study
,”
Math. Probl. Eng.
,
2010
, p.
21
.
39.
Bugliarello
,
G.
, and
Sevilla
,
J.
,
1970
, “
Velocity Distribution and Other Characteristics of Steady and Pulsatile Blood Flow in Fine Glass Tubes
,”
Biorheology
,
7
(
2
), pp.
85
107
.
40.
Sharan
,
M.
, and
Popel
,
A. S.
,
2001
, “
A Two-Phase Model for Flow of Blood in Narrow Tubes With Increased Effective Viscosity Near the Wall
,”
Biorheology
,
38
(
5–6
), pp.
415
428
.
41.
Chakravarty
,
S.
,
Sarifuddin
., and
Mandal
,
P. K.
,
2004
, “
Unsteady Flow of a Two-Layer Blood Stream Past a Tapered Flexible Artery Under Stenotic Conditions
,”
Comput. Methods Appl. Math.
,
4
(
4
), pp.
391
409
.
42.
Pozrikidis
,
C.
,
2005
, “
Axisymmetric Motion of a File of Red Blood Cells Through Intravascular
,”
Phys. Fluids
,
17
(
3
), p.
031503
.
43.
Ni
,
J.
,
Wang
,
G.
, and
Zhang
,
H.
,
1991
,
The Basic Theory of Solid Liquid Two-Phase Flow and Its Latest Applications
,
Science Press
,
Beijing, China
, pp.
35
52
.
44.
Vaidheeswaran
,
A.
, and
Bertodano
,
M. L. D.
,
2016
, “
Interfacial Pressure Coefficient for Ellipsoids and Its Effect on the Two-Fluid Model Eigenvalues
,”
ASME J. Fluids Eng.
,
138
(
8
), p.
081302
.
45.
Ni
,
J.
,
Wang
,
G.
, and
Zhang
,
H.
,
1991
,
The Basic Theory of Solid-Liquid Two Phase Flow and Its Latest Applications
,
Science Press
,
Beijing, China
, pp.
28
73
.
46.
Baxter
,
L. T.
, and
Jain
,
R. K.
,
1989
, “
Transport of Fluid and Macromolecules in Tumors—I: Role of Interstitial Pressure and Convection
,”
Microvasc. Res.
,
37
(
1
), pp.
77
104
.
47.
Baxter
,
L. T.
, and
Jain
,
R. K.
,
1990
, “
Transport of Fluid and Macromolecules in Tumors—II: Role of Heterogeneous Perfusion and Lymphatics
,”
Microvasc. Res.
,
40
(
2
), pp.
246
263
.
48.
Baxter
,
L. T.
, and
Jain
,
R. K.
,
1991
, “
Transport of Fluid and Macromolecules in Tumors—III: Role of Binding and Metabolism
,”
Microvasc. Res.
,
41
(
1
), pp.
5
23
.
49.
Darcy
,
H.
,
1856
,
Les Fontaines Publiques de la Ville de Dijon: Exposition et Application
,
Victor Dalmont
, Paris, France.
50.
Zarandi
,
M. A. F.
,
Pillai
,
K. M.
, and
Kimmel
,
A. S.
,
2017
, “
Spontaneous Imbibition of Liquids in Glass‐Fiber Wicks—Part I: Usefulness of a Sharp‐Front Approach
,”
Aiche J.
,
64
(
1
), pp.
294
305
.
51.
Masoodi
,
R.
,
Tan
,
H.
, and
Pillai
,
K. M.
, “
Darcy's Law–Based Numerical Simulation for Modeling 3D Liquid Absorption Into Porous Wicks
,”
Aiche J.
,
57
(
5
), pp.
1132
1143
.
52.
Salathe
,
E. P.
, and
An
,
K. N.
,
1976
, “
A Mathematical Analysis of Fluid Movement Across Capillary Walls
,”
Microvascular Res.
,
11
(
1
), pp.
1
23
.
53.
Selvaggi
,
J. P.
,
Salon
,
S. J.
, and
Chari
,
M. V. K.
,
2010
, “
Computing the Magnetic Induction Field Due to a Radially Magnetized Finite Cylindrical Permanent Magnet by Employing Toroidal Harmonics
,”
PIERS Proceedings
, Cambridge, MA, July 5–8, pp.
244
251
.
54.
Huang
,
Z.
, and
Ding
,
E.
,
2006
,
Transport Theory
,
Science Press
,
Beijing, China
, pp.
154
248
.
55.
Sugii
,
Y.
,
Nishio
,
S.
, and
Okamoto
,
K.
,
2002
, “
In Vivo PIV Measurement of Red Blood Cell Velocity Field in Microvessels Considering Mesentery Motion
,”
Physiol. Meas.
,
23
(
2
), pp.
403
416
.
56.
Polak
,
E.
,
1974
, “
A Globally Converging Secant Method With Application to Boundary Value Problem
,”
SIAM J. Numer. Anal.
,
11
(
3
), pp.
529
537
.
You do not currently have access to this content.