NUMERICAL STUDY ON RED BLOOD CELLS AND …math.cts.nthu.edu.tw/Mathematics/khoo.pdf2 NUS...
Transcript of NUMERICAL STUDY ON RED BLOOD CELLS AND …math.cts.nthu.edu.tw/Mathematics/khoo.pdf2 NUS...
NUS Presentation Title 2001NUS Presentation Title 2001
NCTS Workshop on Fluid-Structure Interaction Problems - 2011
NUMERICAL STUDY ON RED BLOOD CELLS AND DRUG-LOADED CAPSULES UNDER STOKES
FLOW CONDITIONFLOW CONDITIONBy
B C KhB.C. KhooCo-Authors: P.G. Jayathilake, G. Liu, & Z. Tan
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O tliOutlineI t d tiIntroduction
-IBM, IIM
Model formulationModel formulation-Governing equations, Boundary conditions
Numerical methodNumerical method-Projection method coupled with IIM, Jump conditions
Code validationCode validation-Simple shear flow, capsule-substrate adhesion
RBC & IRBC (impermeable)RBC & IRBC (impermeable)Drug-loaded capsule (permeable)
-Effect of permeability, initial concentration, flow velocity, p y, , y,initial position, and adhesion
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Introductiony
Permeable- -Applicationsx
Permeable Wall
Biological processes such as inflammatory response
Ω+outer domain
Malaria-infected Red Blood Cell adhesion onto capillary Capsule p ywalls
D d li i d
Ω- inner domain
membrane
Drug delivery using drug loaded capsules
(e.g. for bladder cancer LadPermeable ( gtreatments)
Permeable Wall
33
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Introduction
The Immersed interface method (IIM) is a second order i l h It h t b b t f d f blnumerical scheme. It has proven to be robust for deformable
and impermeable boundary problems.
There is a lack of research work on IIM for permeable and deformable boundary problemsdeformable boundary problems.
-Layton (2006), Jayathilake et al. (2010 a, b),Jayathilake et al (2011)Jayathilake et al. (2011)
It is needed to extend the IIM for a permeable capsule moving along a capillarymoving along a capillary.
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Introduction
Peskin’s Immersed Boundary Method (IBM)Fluid dynamics of blood flow in human heartFluid dynamics of blood flow in human heartBiological flows: platelet aggregation, bacterial organismsorganismsRigid boundaries
I d I t f M th d (IIM b L V d Li)Immersed Interface Method (IIM by LeVeque and Li)Elliptic equations, PDEsStokes flows with elastic boundariesNavier-Stokes equations with flexible boundariesStreamfunction-vorticity equations on irregular domains
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Peskin’s IBMU di t d lt f ti t d th f d it tUse a discrete delta function to spread the force density to
nearby Cartesian grid points.
( )1
( , ) ( ) ( )N
k h kk
t t D t s= − Δ∑F x f x XΩ+1k=
( ) ( ) ( )h h hD x yδ δ=x
⎧ ⎛ ⎞⎛ ⎞
Ω+
X(s,t)s
Ω1 1 cos , 2( ) 4 2h
r r hr h h
πδ
⎧ ⎛ ⎞⎛ ⎞+ ≤⎪ ⎜ ⎟⎜ ⎟= ⎝ ⎠⎨ ⎝ ⎠⎪
Γ(t)
Ω-
0, otherwise⎪⎩ Ω∂
Smearing out sharp interface of O(h).
First-order accurate for problems with non-smooth solutions66
First-order accurate for problems with non-smooth solutions.
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Immersed Interface Method
Incorporate the jumps in the solutions and their derivativesinto the finite difference scheme near the interface
( ) 1 1
2i i
x iv vv x
h+ −−
= x iC v+( )2x i h
x i
)( )],[],[],([ αxxxix vvvCvC =
Avoid smearing out sharp interface
Maintain second-order accuracy
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Model formulation
Assumptions
Fluid density and viscosity, and mass diffusivity are constant throughout the computational domain.g p
Thickness of the membrane is negligible, Neo-Hookean rheology is assumed for the membrane.
A i i b t th l d th llA minimum gap between the capsule and the walls are assumed to maintain the numerical stability of the code.
Uniform inlet flow is assumed for the capillary flows.
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Model formulation
Governing equations
( ) Fupuuut
rrrrr+∇+−∇=∇+ 2μρ
------------Motion equations0=∇ur (N-S and mass conservation equations)
cDcuc 2∇∇+ Transport equationcDcuct ∇=∇+ ------------------------Transport equation
dstsXxtsftxF D )),((),(),( −= ∫ δ ----------------Interfacial force
Force strength
dstsXxtsftxF D )),((),(),( ∫Γ
δ
( ) nWtsntsTtstsTtsf rr ∂−+
∂= )()()()()( τ ---------Force strength( ) n
ytsntsTtstsT
stsf be ∂
+∂
= ),(),(),(),(),( τ
Elastic Bending Adhesive99
Elastic tension
Bending tension
Adhesive force
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Model formulation
Membrane tension & mass fluxEl ti d l Bending modulus
Membrane tension(Zhang et al., 2008)
Elastic modulus Bending modulus
),( 5.15.1 −−= εεee ET [ ]Rbb Es
T κκ −∂∂
= ((Zhang et al., 2008)
Adhesion potential(Seifert 1991)
s∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
24
2y
dy
dWW mm
ad(Seifert, 1991)
Solvent flux across theAdhesion strength
⎥⎦⎢⎣ ⎠⎝⎠⎝ yy
[ ] [ ]( )Solvent flux across the membrane(Kedem and Katchalsky, 1958)
[ ] [ ]( )cRTpLJ absPv σ−−=
Reflection coefficient ( y )Hydraulic conductivity
][)1( cRTcJJ absavvs ωσ −−=Solute flux across the membrane
1010(Kedem and Katchalsky, 1958)Solute diffusivity
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fModel formulation
Boundary conditionsΩ∂Along boundary-Ω∂ ∂Along boundary
y = b2
Ω+
x
y Periodic boundary condition in the xdirection and the no-slip boundary
Γ
x
=(X,Y)
p ycondition in the y direction are imposed.
0 b b
Along Membrane- ΓΩ-
Γ (X,Y)c = 0 at y = b1, b2
x = a1 x = a2
y = b1
)i ()())(())((
)cos().,()),,(()),,((
θ
θ
JXXV
tsJttsXuttsXU v−=1 x a2 )sin().,()),,(()),,(( θtsJttsXvttsXV v−=
[ ] [ ]( )absPv cRTpLJ −−= σ
1111( ) snv JDccJ =− ±Γ
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fModel formulation
Non-dimensionalization0*∇
r 0. =∇ ur**2***
*
*
Re1. Fupuu
tu
+∇+−∇=∇+∂∂ rrrr
Ret∂*********** )),((),(),( dstsXxtsftxF D −= ∫
Γ
δrr
*
for Motion
*2*** 1c∂ r
( ) nyWtsntsTtstsT
stsf be
rr*
**********
**** ),(),(),(),(),(
∂∂
−+∂∂
= τ
*2***
1. cPe
cutc
∇=∇+∂∂ r
( )][ *** cCCfJ σ= ][)1( **** cCcJJ = σfor Mass
distribution( ),][21 cCCfJ nv σ−−= ][)1( 3 cCcJJ avvs −−= σ
RTCLcRT
CLE
CUrPeUr abspabspe ωρ===== 321 ,,,,Re
1212
UUUdDμ 321 ,,,,
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Numerical methodA pressure-increment projection algorithm is employedA pressure increment projection algorithm is employed
on a MAC staggered grid to solve N-S equations. The transport equations are solved as reported in Jayathilake et al. (2010 a, b).
1k+ 1i jv +
p,i ju
p1,i ju +
k1k+ , 1i j+
,i jp
,i jv
1,i jp +
1k− u -mesh pointk vp
-mesh point
-mesh point,c
1313
control point
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Numerical method
Solving equations of motionFirst the membrane configuration is updated explicitly and then the
mmmmmm VtYYUtXX *,**,1*,*,**,1*, .,. Δ+=Δ+= ++
First the membrane configuration is updated explicitly, and then the Projection method combined with IIM is employed to solve N-S equations.
( ) ( ) **,21*,22/1*,2/1***
*,1*,
Re21. t
mmmmmm
uQuupuut
uu rrrrrrr
−∇+∇+−∇=∇+Δ− +++
+
1* +mr 0. 1*, =∇ +mur
( ) *2/1*,22/1*,2/1***
*,'*,
R1. t
mmmm
uQupuutuu rrrrrr
−∇+∇−∇−=Δ− +++( )
RetΔ [ ] ;..
1
*2/1*,22/1*,2*
'*,
*
'*,1*,2
tmmm ppCpC
tuC
tu
∇+∇+∇−Δ∇
+Δ∇
=∇ −++ ξφrr
0. 1*, =∇ +mn φr
tt ΔΔ
( )2/1*,2/1*,*1*,*'*,1*, . −+++ ∇−∇Δ−∇Δ−= mmmm pCpCttuu φrr
( ) 11
u, v
1414( ) '*,'*,1*,2/1*,2/1*, .
Re21.
Re21 uCupp mmm rr
∇−∇−+= +−+ φ p
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Numerical method
Some spatial correction terms for
i, j+1k+1
motion equations
S ti l ti t C l i
i, j i+1, ji-1, j
Ω+
yjGrid point
k
k-1
I1
I2
Spatial correction terms are calculated by Taylor series expansions at intersection points I and I as below:
Control pointi, j-1Ω-
y
h
I1 and I2 as below:
1*21
1*1
1**, 111
])[2/(][][21 +++ ++−= m
IxxmIx
mIxji phphp
hpC
xxi
h Membrane
, 1112j h
1*22
1*2
1**, 222
])[2/(][][21 +++ ++−= m
IyymIy
mIyji phphp
hpC
1*22
1*21
1*2
1*1
1*1*2
*2, 212121
])[2/(])[2/(][][][][1 ++++++ +++++−=∇ mIyy
mIxx
mIy
mIx
mI
mIji phphphphpp
hpC
h h1515
,111 Ii xxh −= + 212 Ij yyh −= +
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Numerical method
Pressure and velocity jump conditions needed for correction terms are given below:terms are given below:
[ ] β *'*Nfp =[ ]
[ ] β
β
*'*Tn
N
fs
p
fp
∂∂
= These jump conditions are derived using the equations of
[ ] θβ sin0][][
**
**
Tfuvu
=
==∂ g q
motion as reported in LeVeque and Li (1997).[ ]
[ ] θβ
θβ
cos
sin**
Tn
Tn
fv
fu
−=
VEe μβ /= rVEe
2'
ρβ =
1616
e μβ ρ
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Numerical method
Solving transport equation
Transport equation is solved as reported in Jayathilake et al.(2010a).
1 )(1 ************** yyxxyxt
ccPe
cvcuc +=++
)()()(1 *******21*2*,1*,
mmmmmmji
mji CCCQ
cc − ++
)()()(1 *,
*,,,
*,,
*,
*,,,
*,,
*2,
1*,,
2,*
,,yji
mjiy
mjixji
mjix
mjiji
mjiji
jiji cCcvcCcucCcPe
Qt
+−+−=∇+∇−+Δ
+
12/*,**,***,* ])[(][1 +<<+± mmm ttttt 12/,1
,1
,* ,])[(][ *
1*1
+<<−+Δ
± mmtt
mt
tttcttct
1*,*12/*,**1*,* ])[(][1 +++ <<−+± mmm tttcttc=jiQ , *1* ,])[(][ *'
1*1
<<+Δ
±ttt
tttcttct
1717
c
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Numerical method
Concentration jump conditions
Approximate jump conditions are computed using the boundary condition at the membrane and Kedem Katchalskyat the membrane and Kedem-Katchalsky relations.
N l li( ) m
ksm
km
km PeJhcc *,,
*1*,2,
1*,1,1,*, 24 −− ++
+−
h
Normal line
k,4
( )mkv
mk PeJh
c *,,
*,,,1,,
23−=+
( ) mmm PeJhcc *1*,1*, 24 +− ++
h
h
h
k
k -1
k+1k+2
k,1
k,3 Ω+
Γ
( )mkv
kskkmk PeJh
PeJhccc *,
,*
,4,3,1,*,
2324
++
=++
k -2
Membrane
k,2 Ω-
1,*,1,*,1*,* ][ +−+++ −= mk
mk
mk ccc
( )1,*,1,*,1*,1* ][ +−++++ −= mmmm ccPeJc1818
( ), ][ = kkvkn ccPeJc
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C S fCode Validation: Simple shear flow
Steady state velocity field at Comparison with Breyiannis shear rate, G = 0.0125 and Pozrikidis (2000)
1919
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CCapsule-substrate adhesion
2020
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CCode Validation: Adhesion
The theoretical solution is derived Comparison with Cantat and Misbah (1999)considering the energy balance for the membrane.
(J thil k t l 2010 b)
Comparison with Cantat and Misbah (1999)π2/1 TSr =
π/2 eqAr =
)/1( 31L
2121
(Jayathilake et al., 2010 b))/1log().3/1(~)/log(
)/1(~
121
31121
rrrLrrrL
ad
ad
−−
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C ff f CAdhesive Coefficient for IRBC
The detachment force for IRBC is 5 to 10 x 10-11 N (Nash et al., ( ,1992).
The adhesive coefficient for IRBC is estimated as 1 x 10-5 N/m2222
The adhesive coefficient for IRBC is estimated as 1 x 10 N/m (approximately).
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ParametersRBC: impermeable IRBC: impermeable
Drug-loaded capsule (fairly similar to IRBC): permeable) p
/1064
5=
=− mNxE
mr μmNxE
mr
e /1064
5−=
= μ
/1064
6=
=− mNxE
mr μ
108.1
/106
23
19= −b
e
NmxE
mNxE
N
NmxEb
e
/104
108.123
19
−
−=
/104
108.1
/106
23
19= −b
e
NmxE
mNxE
/1000/.1043
23
=
= −
mkgmsNx
ρ
μmkg
msNx/1000
/.1043
23−
=
=
ρ
μ
/1000/.1043
23
=
= −
mkgmsNx
ρ
μ
0/10 5
=== −
ad
RTLmNW
ω kgsmxL
mNWad
/101
/1028
5
−
−
=
=0=adW
N dh i
Impermeable
0== absp RTL ω
smRT
kgsmxL
abs
p
/10
/1013−=
=
ω0== abp RTL ωNo adhesion
pmembrane
smxD /105.10
29−=
=σImpermeable membrane
2323The uniform inlet velocity Uin = 100-500μm/s
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C & CResults: RBC & IRBC
RBC deformation in plasma flow IRBC deformation in plasma flowRBC deformation in plasma flow, Uin= 100 μm/s (uniform inlet velocity)
IRBC deformation in plasma flow, Uin = 100 μm/s (uniform inlet velocity)
2424
velocity) velocity)
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Results: Flow resistanceFlow resistance in the presence of RBC and IRBCFlow resistance in the presence of RBC and IRBC
Impermeable pmembrane
Resistance by IRBC > Resistance by RBC
2525Therefore, IRBC may cause microvascular blockage
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Results: Drug loaded capsule
Concentration distributionωRTabs = 1x10-3 m/s, γ = 150, Uin = 0, Wad = 0
2626capsule is stationary (Uin = 0)
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Results: Drug loaded capsule
Effect of membrane permeabilityωRTabs = 1x10-5-1x10-3 m/s, γ = 150, Uin = 0, Wad = 0
2727(a) solute mass of the capsule (b) solute mass of the surrounding field
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Results: Drug loaded capsule
Effect of membrane permeabilityωRTabs = 1x10-5-1x10-3 m/s, γ = 150, Uin = 0, Wad = 0
2828(c) solute mass absorbed by the walls
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Results: Drug loaded capsule
Effect of initial solute concentration of capsuleωRTabs = 1x10-3 m/s, γ = 50-500, Uin = 0, Wad = 0
2929
(a) solute mass of the capsule (b) solute mass of the surrounding field
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Results: Drug loaded capsule
Effect of initial solute concentration of capsuleωRTabs = 1x10-3 m/s, γ = 50-500, Uin = 0, Wad = 0
3030
(c) solute mass absorbed by the walls
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Results: Drug loaded capsule
Effect of imposed velocityωRTabs = 1x10-3 m/s, γ = 150, Uin = 100-1000 μm/s, Wad = 0
3131
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Results: Drug loaded capsule
Effect of imposed velocityωRTabs = 1x10-3 m/s, γ = 150, Uin = 100-1000 μm/s, Wad = 0
3232solute mass absorption by walls
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Results: Drug loaded capsule
Effect of adhesion & initial positionωRTabs = 1x10-3 m/s, γ = 150, Uin = 500 μm/s
3333(a) initial non-adhesive capsule is away from the walls
(b) initial non-adhesive capsule is near a wall
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Results: Drug loaded capsule
Effect of adhesion & initial positionωRTabs = 1x10-3 m/s, γ = 150, Uin = 500 μm/s
3434(c) initial adhesive capsule is near a wall, Wad = 20µJ/m2
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Results: Drug loaded capsule
Effect of adhesion & initial positionωRTabs = 1x10-3 m/s, γ = 150, Uin = 500 μm/s
3535(a) solute mass of the capsule (b) solute mass of the surrounding field
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Results: Drug loaded capsule
Effect of adhesion & initial positionωRTabs = 1x10-3 m/s, γ = 150, Uin = 500 μm/s
3636(c) solute mass absorption by walls
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CConclusions
The numerical approach is deemed reasonable for simulating capsule-substrate adhesion in the presence of a flow field.
The numerical results in the absence of a flow field show that the solute transfer between the capsule and the vessel walls canthe solute transfer between the capsule and the vessel walls can be modulated by changing the membrane permeability and the initial solute concentration of the capsule.
The capsule moving near to one wall would increase the solute absorption by walls as opposed to the case of the capsuleabsorption by walls as opposed to the case of the capsule moving along the centre line of the vessel. The adhesion between the capsule and walls would further increase marginally th t t l l t t f b t th l d l llthe total solute transfer between the capsule and vessel walls.
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Journals
Related Publications
P. G. Jayathilake, G. Liu, Zhijun Tan, and B. C. Khoo, “Numerical study of a permeable capsule under Stokes flows by the immersed interface method”, Chemical Engineering Science, DOI:10.1016/j.ces.2011.02.005.
P.G. Jayathilake, B.C. Khoo, and Zhijun Tan, “Effect of membrane permeability on capsule substrate adhesion: Computation using immersed interface method”, Chemical Engineering Science 65, 3567-3578, 2010.
P G Jayathilake Zhijun Tan B C Khoo and N E Wijeysundera “Deformation and osmotic swelling of an elasticP. G. Jayathilake, Zhijun Tan, B. C. Khoo, and N. E. Wijeysundera, Deformation and osmotic swelling of an elastic membrane capsule in Stokes flows by the immersed interface method”, Chemical Engineering Science 65, 1237-1252, 2010.
ConferencesConferences
P.G. Jayathilake and B.C. Khoo, “Capsule-substrate adhesion, detachment, and mass transport under Stokes flows: Computation using the immersed interface method”, Proceedings of the 13th Asian Congress of Fluid Mechanics, 2010, Dhaka, Bangladesh.Dhaka, Bangladesh.
B.C. Khoo, P.G. Jayathilake, and Z. Tan, “Numerical study of a permeable capsule/cell under Stokes flows by the immersed interface method”, Workshop on Fluid Motion Driven by Immersed Structures, 2010, Fields Institute, Toronto, Ontario, Canada.
P.G. Jayathilake, B.C. Khoo, and Z. Tan, “Numerical Simulation of Interfacial Mass Transfer Using the Immersed Interface Method”, 6th International Symposium on Gas Transfer at Water Surfaces 2010, Kyoto, Japan.
P G Jayathilake B C Khoo and Zhijun Tan “Capsule substrate adhesion in the presence of osmosis by the immersed
3838
P.G. Jayathilake, B.C. Khoo, and Zhijun Tan, “Capsule-substrate adhesion in the presence of osmosis by the immersed interface method”, International Conference on Computational Fluid Dynamics 2009, Bangkok, Thailand.
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Breyiannis, G. and Pozrikidis, C., 2000. Simple Shear Flow of Suspensions of Elastic Capsules. Theoretical and Computational Fluid Dynamics 13, 327–347.
References
p y ,Cantat, I. and Misbah, C., 1999. Dynamics and similarity laws for adhering vesicles in haptotaxis.
Physical Review Letters 83, 235–238. Jayathilake, P.G., Liu, G., Zhijun Tan, Khoo, B.C., 2011. Numerical study of a permeable capsule
under Stokes flows by the immersed interface method Chemical Engineering Scienceunder Stokes flows by the immersed interface method. Chemical Engineering Science, DOI:10.1016/j.ces.2011.02.005.Jayathilake, P.G., Zhijun Tan, Khoo, B.C. and Wijeysundera, N.E., 2010. Deformation and osmotic
swelling of an elastic membrane capsule in Stokes flows by the immersed interface method. Chemical Engineering Science 65, 1237-1252.Chemical Engineering Science 65, 1237 1252.Jayathilake, P.G., Khoo, B.C. and Zhijun Tan, 2010. Effect of membrane permeability on capsule
substrate adhesion: Computation using immersed interface method. Chemical Engineering Science 65, 3567-3578. Kedem O and Katchalsky A 1958 Thermodynamics analysis of the permeability of biologicalKedem, O. and Katchalsky, A., 1958. Thermodynamics analysis of the permeability of biological
membranes to non-electrolytes. Biochimica et Biophysica Acta 27, 229–24.Kondo, T., Imai, Y., Ishikawa, T., Tsubota, K. and Yamaguchi, T., 2009. Hemodynamics analysis of
microcirculation in malaria infection. Annals of Biomedical Engineering, 37, 702-709.Layton A T 2006 Modeling water transport across elastic boundaries using an explicit jumpLayton, A.T., 2006. Modeling water transport across elastic boundaries using an explicit jump
method. SIAM Journal on Scientific Computing 28, 2189–2207. LeVeque, R.J. and Li, Z., 1997. Immersed interface methods for Stokes flow with elastic boundaries
or surface tension. SIAM Journal on Scientific Computing 18, 709–735. N h G B C k B M M h K B dt A N b ld C d St t J 1992 Rh l i lNash, G.B., Cooke, B.M., Marsh, K., Berendt, A., Newbold, C. and Stuart, J., 1992. Rheological
analysis of the adhesive interactions of red blood cells parasitized by Plasmodium falciparum. Blood 79, 798-807. Seifert, U., 1991. Adhesion of vesicles in the two dimensions. Physical Review A 43, 6803–6814.
3939
Wan, K.T. and Liu, K.K., 2001. Contact mechanics of a thin-walled capsule adhered onto a rigid planar substrate. Medical & Biological Engineering & Computing 39, 605–608.
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THREE-DIMENSIONAL NUMERICALTHREE-DIMENSIONAL NUMERICAL SIMULATION OF HUMAN NASAL
CILIA IN THE PERICILIARY LIQUID LAYERCILIA IN THE PERICILIARY LIQUID LAYER
40
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OutlineIntroductionIntroduction
Human lungsAirways surface liquidSt t f iliStructure of ciliumCilia beatingFluid dynamics models of cilia motion
Numerical ModelingGoverning equations & Boundary conditionsSome of 2D modeling workg3D modeling
ResultsStandard problemStandard problem Effect of cilia beat frequencyExtension for a 2 layers problem
ConclusionsConclusions
4141
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IntroductionHuman lungs
During normal breathing, the
g
g gairways transport air into the lungs. However, this air is often polluted
ith i t f ti l f iwith a variety of particles, fungi and bacteria that become deposited on the airway surfacedeposited on the airway surface liquid (ASL).
4242
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IntroductionAirway surface liquidy q
One of the main protections of the upper airway and the lungsOne of the main protections of the upper airway and the lungs is a fluid layer covering the interior epithelial surface of the bronchi and bronchioles. This is known as the airway surface liquid.
Th i f li id hibit t l t t ThThe airway surface liquid exhibits a two-layer structure. There are two types of fluids: the watery PCL and the mucus layer. An array of cilia is immersed in the PCL
4343
array of cilia is immersed in the PCL.
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IntroductionStructure of a cilium
Cilia are microscopic, hair-like organelles projecting from a cell’s surface.
The cilium is moved when outer microtubules slide past each
4444other.
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IntroductionCilia beatingg
Each cilium performs a repetitive beat cycle consisting of recovery and effective stroke.
D i th ff ti t k th ili f dDuring the effective stroke the cilium moves forward.
During the recovery stroke the cilium moves backwardsDuring the recovery stroke the cilium moves backwards.
The work done during the effective stroke is several times the 4545
gamount of work done during the recovery stroke.
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IntroductionFluid dynamic models of cilia motion
No of No of Cilia
y
No. of layers-L
No. of dimensions-D
Cilia beating
1L 2L 3L1L 2L 3L
PCL PCL PCL
2D 3DVolume force Discrete cilia
+
Mucus
+
Mucus Prescribed b ti
Couple-internal mechanics/fluid
+
Air
beating mechanics/fluid-structure interaction
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Numerical Modeling PCL & mucus motion is governed by NS equations
Slip BCz
g y q
Mucus
PBCPBCPBC
z
y
Mucus
PCL
⎞⎛ No slop BC
PBCx
;. 2 Fupuutu rrrrr
+∇+−∇=⎟⎠⎞
⎜⎝⎛ ∇+∂∂ μρ 0. =∇ur
No-slop BC
In this numerical study, the time-dependent incompressible fluid-flow problem is solved by the projection method.p y p j
The interaction between cilia and PCL is imposed using the Immersed Boundary Method (IBM).
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MAC scheme is used for discretization.
NUS Presentation Title 2001NUS Presentation Title 2001
Numerical Modeling MAC Scheme in 2D
i ju 1i juk1k+ , 1i jv +
,i jp,i ju
1,i jp +
1,i ju +
,i jv1k− u
v-mesh points
-mesh points
p -mesh points
control points
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Numerical ModelingInteraction force
nnU 1' rr⎞⎛ +
( ) ( ) ( )n kjin
kjin
kji
nkji
nkjin
kji upuutuU
F ,,2
,,,,,,
1,,,1
,, . rrrr∇−∇+⎟
⎟⎠
⎞⎜⎜⎝
⎛∇−
Δ
−=
++ μρ
⎠⎝
Distributed cilium velocity at a yCartesian grid point
)().,,,(1,',, XUXkjifU nkji
rrrr=+
Cilium velocity
Sanderson & Sleigh
hXx
rrXkjif kji
rrr −
=−−
= ,,;2
)5.0/)1tanh((1),,,(
g(1981)
Or
Gheber & Priel (1997)
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Gheber & Priel (1997)
NUS Presentation Title 2001NUS Presentation Title 2001
Numerical Modeling: 2D-2LCilia motion
Mucus velocity = 4.4.x10-3 cm/s5050
yMucus velocity in Smith et al. (2007) = 3.8.x10-3 cm/s
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Numerical Modeling: 2D-2LSome results
Mucus velocity depends linearly on Cilia beat frequency.5151
y p y q yMucus velocity decreases with mucus viscosity.
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Numerical Modeling: 3D-1LPrescribed cilia beatingg
Sid i T i5252
Side view Top view
NUS Presentation Title 2001NUS Presentation Title 2001
Numerical Modeling: 3D-1LPrescribed cilia beatingg
Sid i T i5353
Side view Top view
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1.STANDARD
Results: 3D-1LSTANDARD Problem
C t ti l d i [ 24 24] [ 6 6] [0 8] 3Computational domain = [-24, 24] x [-6, 6] x [0, 8] μm3
The length of the cilium = 6.0 μm
PCL density = 1 g/cm3
PCL i it 0 01 PPCL viscosity = 0.01 P
Cilia beating frequency = 10 Hz
No. of cilia = 24 (in two rows, i.e. 12x2)
Mesh = 120x30x20Mesh = 120x30x20
No. of control points on each cilium = 20
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Results: 3D-1LComputational domainp
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Results: 3D-1LCilia & Velocity fieldy
5656Side view Top view
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Results: 3D-1LCilia & Pressure field
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Results: 3D-1LThe total PCL flow QQ
∫∫∫xQ )(
∫∫∫= dydzdtzyxu ),,(
Average PCL velocity= 2 3x10-5 cm/s 2.3x10 cm/s
Average PCL velocity in Smith et al (2007)Smith et al. (2007) = 3.1x10-5 cm/s
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Results: 3D-1LEffect of Cilia Beat Frequency (CBF)
CBF = 5, 10, 15, 20, 25 Hz
q y ( )
The average PCL velocity is linearlyvelocity is linearly dependent on CBF.
It has been shown experimentally that ciliary forces depend linearly on CBF (Teff et al 2008)et al., 2008).
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Results: 3D-2LA more accurate model
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Conclusions
In the present work, multiple human nasal cilia immersed in p , pthe PCL are simulated by using 2D and 3D Projection Method combined with the Immersed Boundary Method.
The present results are found to be in reasonable agreement with previously reported results for the mucus andagreement with previously reported results for the mucus and PCL velocities of human respiratory system.
The present model is a basic one. A more realistic model including non-Newtonian mucus layer is being developed.
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CConclusion
OTHANK YOU
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