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

1


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

22


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


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.

44


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

55


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.


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

77


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.

88


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


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


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


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 ,,,,


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


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


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 −= +


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


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


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


C S fCode Validation: Simple shear flow

Steady state velocity field at Comparison with Breyiannis shear rate, G = 0.0125 and Pozrikidis (2000)

1919


CCapsule-substrate adhesion

2020


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

−−


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).


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


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)


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


Results: Drug loaded capsule

Concentration distributionωRTabs = 1x10-3 m/s, γ = 150, Uin = 0, Wad = 0

2626capsule is stationary (Uin = 0)



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



Effect of membrane permeabilityωRTabs = 1x10-5-1x10-3 m/s, γ = 150, Uin = 0, Wad = 0

2828(c) solute mass absorbed by the walls



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



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



Effect of imposed velocityωRTabs = 1x10-3 m/s, γ = 150, Uin = 100-1000 μm/s, Wad = 0

3131



Effect of imposed velocityωRTabs = 1x10-3 m/s, γ = 150, Uin = 100-1000 μm/s, Wad = 0

3232solute mass absorption by walls



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




3434(c) initial adhesive capsule is near a wall, Wad = 20µJ/m2




3535(a) solute mass of the capsule (b) solute mass of the surrounding field




3636(c) solute mass absorption by walls


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.

3737


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.


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.


THREE-DIMENSIONAL NUMERICALTHREE-DIMENSIONAL NUMERICAL SIMULATION OF HUMAN NASAL

CILIA IN THE PERICILIARY LIQUID LAYERCILIA IN THE PERICILIARY LIQUID LAYER

40


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


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


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.


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.


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.


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

4646


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).

4747

MAC scheme is used for discretization.


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

4848


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)

4949

Gheber & Priel (1997)


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


Numerical Modeling: 2D-2LSome results

Mucus velocity depends linearly on Cilia beat frequency.5151

y p y q yMucus velocity decreases with mucus viscosity.


Numerical Modeling: 3D-1LPrescribed cilia beatingg

Sid i T i5252

Side view Top view


Numerical Modeling: 3D-1LPrescribed cilia beatingg

Sid i T i5353

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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

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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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numerical study on red blood cells and …math.cts.nthu.edu.tw/mathematics/khoo.pdf2 nus...

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