Respiratory Fluid Mechanics

J.B. Grotberg, Ph.D., M.D.

Department of Biomedical Engineering

University of Michigan

Airw ay Branching Netw ork

Respiratory Zone

Airw ay and Alveolar Liquid Lining

Cultured human tracheobronchial airw ay epithelia

ML-mucus layer, PCL-periciliary layer, GCgoblet cell, CC-ciliated cell, BC-basal cell,

T-col-semipermeable supports

Alveolar liquid lining and

surfactant system

Airw ay Liquid Plugs: How Do They Occur?

• Intrinsic

– Normal people at full expiration

– Congestive Heart Failure → Pulmonary Edema

– Asthma, Emphysema

• Extrinsic

–

–

–

–

Surfactant Replacement Therapy

Delivery of Drugs, Genetic Material, Stem Cells

Liquid Ventilation

Drowning

Example of Extrinsic Plug:

Surfactant Replacement Therapy

mad

-- Surfactant replacement therapy

in premature infants.

-- Partial or total liquid ventilation.

-- Drug and gene delivery.

Biofluid Mechanics Lab

Experimental Setup – Lung Close-up

• The excised lung suspended and

ventilated from tracheal cannula.

• A small diameter tube attached

to a syringe w as inserted into the

cannula ( upper center of figure) .

• A surfactant bolus w as formed in

the cannula by injecting 0.05 ml of

surfactant through the small

diameter tube.

Cassidy, K.J., J.L. Bull, M.R. Glucksberg, C.A. Dawson,

S.T. Haworth, R.B. Hirschl, N. Gavriely and J.B.

Grotberg. J. Appl. Physiol. 2001.

(Courtesy of Robert Mothen, Medical College

of Wisconsin, Milwaukee, WI)

Anderson, J.C., R.C. Molthen, C.A. Dawson, S.T.

Haworth, J.L. Bull, M.R. Glucksberg and J.B. Grotberg.

J. Appl. Physiol. 2004.

Micro-CT Movie of Surfactant

I nstillation – Single Dose

Experimental Conditions

• Excised Rat Lung

• Lung Suspended Vertically

• Normal Bolus Volume

• Surfactant = Survanta

• Ventilation Rate - 60 br/ min

(Click image to start/stop movie)

Example of I ntrinsic Plug: Airw ay Closure

Liquid film

Air

Air

Cassidy et al., J. Appl. Physiol. 1999

Airway closure

Airway reopening

Instability & plug formation

plug propagation & rupture

Asymptotic Approach to Liquid Plug Propagation

Re=0, Ca1/3<<1, wall elastance & tension

Plug core viscous effects O(Ca), negligible

Transition regions dominate, G1→0 rigid

Generalized Landau-Levich Eq

Howell, Waters, Grotberg JFM 2000

h2 ( input) vs h1 ( output)

Outer, inner, intermediate regions

d (Vol plug )

dt

Plug Rupture Criteria:

Airw ay Reopening h1 > h2

2-D Steady Plug Propagation in Channel

Fujioka, Grotberg PoF 2005

Γ

C

y

H

U

x

P1

h1

•

•

•

•

•

σ(Γ)

Lp

n

∆P=P1-P2 drives plug at constant speed U.

h2=h1, for the steady state.

Soluble surfactant, Newtonian fluid.

Surfactant in far precursor film is prescribed.

Lp/H, h2/H=h1/H prescribed for steady state.

P2

h2

Flow Governing Equations

*

• Scaling with p = p /

µU

H

, u = u * / U , x = x* / H

• Dimensionless Navier-Stokes and the continuity equations.

Re ( u ⋅∇ ) u = −∇p + ∇ 2u

∇⋅u = 0

• Boundary condition along free interface.

− pn + ( ∇u + ∇uT ) ⋅ n = Ca −1 (σκ n + ∇ sσ ) − Pa n

n Normal vector on interface

u ⋅n = 0

κ

Curvature of interface

σ M Surface tension in surfactant-free interface

• Dimensionless Parameters

ρHU

µU

Re =

, Ca =

µ

σM

ρHU

Re ρσ M H

or Re =

,λ =

=

µ2

Ca

µ

Surfactant Transport Equations

• Scaling with

*

C = C * / Ccmc

, Γ = Γ* / Γ*∞

• Bulk surfactant transport equations.

Sc Re ∇ ⋅ ( uC ) = ∇ C

2

BC −

Scs

( n ⋅∇ ) C = jn

χ Sc

• Interfacial surfactant transport equations.

Scs Re ∇ s ⋅ ( uΓ ) − ∇ s 2 Γ = jn

• Dimensionless

Γ ∞*

µ

µ

Parameters, Sc = , Scs =

,χ=

ρD

ρ Ds

Ccmc* H

Γ*∞ Maximum monolayer packing value of Γ∗

*

Ccmc

Critical Micelle Concentration

Surfactant Transport Equations ( 2)

• Surfactant flux from the bulk to the interface

⎧⎪ K a Cs (1 − Γ ) − K d Γ

jn = ⎨

−Kd Γ

⎪⎩

( Γ < 1)

( Γ ≥ 1)

• Modified Frumkin equation of state

⎧

1 − EΓ

⎪

σ =⎨

⎡ E

⎤

E

1

exp

1

−

−

Γ

(

)

(

)

⎪

⎢⎣1 − E

⎥⎦

⎩

• Dimensionless Parameters,

ka Adsorption

( Γ < 1)

( Γ ≥ 1)

*

ka Ccmc

H2

kd H 2

Γ*∞ ∂σ *

, Kd =

, E=− *

Ka =

σ M ∂Γ*

Ds

Ds

rate coefficient

kd Desorption rate coefficient

Cs Bulk surfactant concentration in subsurface

Flow and Pressure: No Surfactant

1

|u|=1

y

0.5

0

SP

SP

Π1=0.56

Π:

-0.5

SP

0.5

-3

-2

0

-1

-1

-0.5

0

1

B

0.5

-2

x

1

2

Re=50, λ=1000, LP=0.5

•

•

•

•

•

0.5

Π = Ca ⋅ p = p* H / σ M*

SP

A

-1

-2 -1.5 -1 -0.5 0

4 stagnation points in the half-domain, all on the interface.

Two flow regions: recirculation and flow-through

Recirculation region low velocity

Capillary wave at front

Pressure minimized at A (in capillary wave)

3

4

Flow , Pressure, and Surfactant ( Lp= 0.5)

Re=50,λ=1000, LP=0.5, Sc=10, Scs=100,

Ka=104, Kb=102, χ=10-3, E=0.7, C0=10-4

1

10-3

SP

5x10 -3

2x

10 -3

0.5

SP

SP

C: 1.00E-04 1.00E-03 1.00E-02

|u|=1

SP

y

0

A

-0.5

Π1=0.82

0.5

-1

-3

•

•

•

•

0

-2

-1

-1

-0.5

0

x

Π:

-2 -1.5 -1 -0.5 0 0.5 1

B

Π = Ca ⋅ p = p* H / σ M*

0.5

1

2

3

4 stagnation points on midline, 2 on interface, 2 internal

The recirculation zone is no longer in contact with the interfaces.

Thicker transition region at front, thinnest point at B

The maximum concentration attains at the front meniscus

4

Flow , Pressure, and Surfactant ( Lp= 2.0)

Re=50,λ=1000, LP=2, Sc=10, Scs=100,

Ka=104, Kb=102, χ=10-3, E=0.7, C0=10-4

1

-3

10-3

C: 1.0E-04 1.0E-03 1.0E-02

2x10-3

0.5

|u|=1

SP

0

Π1=1.03

-0.5

y

5 x1 0

-0.5

0.5

-1

-3

-2

Π:

A

-2 -1.5 -1 -0.5 0

0.5

1

B

-1

0

0.5

-1

0

x

1

2

3

• Front meniscus stagnation pts. differ from rear.

• Velocity profiles fully developed at the middle cross-section ~ parabolic.

• Concentric iso-pressure lines appear due to centrifugal forces.

4

Wall Pressure and Shear Stress

λ=1000, LP=2, Sc=10, Scs=100, Ka=104,

Kb=102, χ=10-3, E=0.7, C0=10-4

0.50

1.25

0.40

1.00

0.30

0.50

0.20

0.25

0.10

0.00

τw

Πwall

0.75

-0.25

0.00

-0.10

-0.50

-0.20

Re=10

Re=30

Re=50

Re=70

-0.75

Re=10

Re=30

Re=50

Re=70

-1.00

-1.25

-1.50

-5

-0.30

-0.40

0

5

x

Wall Pressure

Π = Ca ⋅ p = p* H / σ M*

-0.50

-5

0

5

x

Wall Shear Stress

τ w = Ca

∂u

∂u * H

=µ * *

∂x

∂x σ M

• Wall pressure and shear increase with Re: can affect cells.

• In the front meniscus, both stresses oscillate in the capillary wave.

Unsteady Liquid Plug 2-D Channel: Initial Conditions

Fujioka, Grotberg

C

σ(Γ)

P1

h1

P 2=0

1-h 2

Lp

1-h 2

h2

• Hemi-sphere of the radius 1-h2, uniform film thickness of h=h2

• Initial velocity and bulk surfactant concentration are u=0 and C=C0

The initial interfacial surfactant concentration in equilibrium with C0.

• For no surfactant case, =1.

At t=0, a plug of the initial length, LP=1 starts propagating

with a constant pressure drop, P=P1-P2=1.

Effect of the precursor film thickness on Plug Propagation (no surfactant)

Effect of Surfactant on Plug Propagation

Plug Flow Dynamics Through Bifurcations

Zheng, Fujioka, Grotberg

Plug flows in airways:

deposits liquid along airway wall.

splits unevenly at bifurcation.

determines a final liquid distribution

t>0

t=0

B

A

U

Rs=LB / LA

Parent

V

Splitting ratio Rs = B

VA

LB

LA

(Y. Zheng, et al. JBE, 2005)

Pre-bifurcation asymmetry

α

g

B

Flow

A

Experimental setup

Video

Camera

g

Syringe

A

LB

B

HARVARD

Syringe Pump

A

γ

θ

φ

Computer

LA

Bifurcation Plates with Orientations

•

is branching angle of bifurcation, φ is roll angle and is pitch

angle.

L

Determine Rs (Ca,Bo, Vp, φ, γ)

Splitting ratio:

Rs =

B

LA

Simple Theoretical analysis on plug splitting

y

g

αd2

L2

P1

h

π1

a1

Up

π2

a2

θ

π0

x

a3

L1

P2 U2

π3

L3

P3 U3

αd3

• Compute pressure drops and mass balance between rear and front menisci

• Pressure drops

– Capillary jump + trailing thin film correction across rear mensicus (P1 – π1)

– Poiseuille law + gravity in liquid (π1 – π0, π0 – π2, π0 – π3 )

– Moving contact line effect at front menisci. P2 − π2(3) = ( σ / a 2 ) cos (α d )

cos αs − cos α d

• Empirical correlation for αd

= 2Ca1L 2

(Bracke et. al., Prog. Coll. Polymer Sci.,

1989

α)s + 1

cos

Results (low Re-low Ca expts.)

Effect of roll angle φ and pitch angle γ on Rs vs. Ca

0.9

1

0.8

0.9

0.8

0.7

0.7

0.6

0.6

Rs

0.5

Rs 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

0.02

0.04

0.06

Cap

0.08

0.1

0.12

LB-400-X oil, γ=0º. φ=15º:

(experiments),

(theory); φ =30º: (experiments),

(theory);

φ =60º: (experiments),

(theory).

•

•

•

•

0

0.02

0.04

0.06

0.08

0.1

0.12

Cap

LB-400-X oil, φ=30º. γ =-15º: (experiments),

(theory); γ=0º: (experiments),

(theory);

γ=15º:

(experiments),

(theory); γ=30º:

(experiments),

(theory).

A critical capillary number Cac exists below which Rs=0.

Cap increases, Rs increases.

Larger φ and γ cause smaller Rs but larger Cac.

Theory qualitatively agrees with experiments.

J.B. Grotberg, Ph.D., M.D.

Department of Biomedical Engineering

University of Michigan

Airw ay Branching Netw ork

Respiratory Zone

Airw ay and Alveolar Liquid Lining

Cultured human tracheobronchial airw ay epithelia

ML-mucus layer, PCL-periciliary layer, GCgoblet cell, CC-ciliated cell, BC-basal cell,

T-col-semipermeable supports

Alveolar liquid lining and

surfactant system

Airw ay Liquid Plugs: How Do They Occur?

• Intrinsic

– Normal people at full expiration

– Congestive Heart Failure → Pulmonary Edema

– Asthma, Emphysema

• Extrinsic

–

–

–

–

Surfactant Replacement Therapy

Delivery of Drugs, Genetic Material, Stem Cells

Liquid Ventilation

Drowning

Example of Extrinsic Plug:

Surfactant Replacement Therapy

mad

-- Surfactant replacement therapy

in premature infants.

-- Partial or total liquid ventilation.

-- Drug and gene delivery.

Biofluid Mechanics Lab

Experimental Setup – Lung Close-up

• The excised lung suspended and

ventilated from tracheal cannula.

• A small diameter tube attached

to a syringe w as inserted into the

cannula ( upper center of figure) .

• A surfactant bolus w as formed in

the cannula by injecting 0.05 ml of

surfactant through the small

diameter tube.

Cassidy, K.J., J.L. Bull, M.R. Glucksberg, C.A. Dawson,

S.T. Haworth, R.B. Hirschl, N. Gavriely and J.B.

Grotberg. J. Appl. Physiol. 2001.

(Courtesy of Robert Mothen, Medical College

of Wisconsin, Milwaukee, WI)

Anderson, J.C., R.C. Molthen, C.A. Dawson, S.T.

Haworth, J.L. Bull, M.R. Glucksberg and J.B. Grotberg.

J. Appl. Physiol. 2004.

Micro-CT Movie of Surfactant

I nstillation – Single Dose

Experimental Conditions

• Excised Rat Lung

• Lung Suspended Vertically

• Normal Bolus Volume

• Surfactant = Survanta

• Ventilation Rate - 60 br/ min

(Click image to start/stop movie)

Example of I ntrinsic Plug: Airw ay Closure

Liquid film

Air

Air

Cassidy et al., J. Appl. Physiol. 1999

Airway closure

Airway reopening

Instability & plug formation

plug propagation & rupture

Asymptotic Approach to Liquid Plug Propagation

Re=0, Ca1/3<<1, wall elastance & tension

Plug core viscous effects O(Ca), negligible

Transition regions dominate, G1→0 rigid

Generalized Landau-Levich Eq

Howell, Waters, Grotberg JFM 2000

h2 ( input) vs h1 ( output)

Outer, inner, intermediate regions

d (Vol plug )

dt

Plug Rupture Criteria:

Airw ay Reopening h1 > h2

2-D Steady Plug Propagation in Channel

Fujioka, Grotberg PoF 2005

Γ

C

y

H

U

x

P1

h1

•

•

•

•

•

σ(Γ)

Lp

n

∆P=P1-P2 drives plug at constant speed U.

h2=h1, for the steady state.

Soluble surfactant, Newtonian fluid.

Surfactant in far precursor film is prescribed.

Lp/H, h2/H=h1/H prescribed for steady state.

P2

h2

Flow Governing Equations

*

• Scaling with p = p /

µU

H

, u = u * / U , x = x* / H

• Dimensionless Navier-Stokes and the continuity equations.

Re ( u ⋅∇ ) u = −∇p + ∇ 2u

∇⋅u = 0

• Boundary condition along free interface.

− pn + ( ∇u + ∇uT ) ⋅ n = Ca −1 (σκ n + ∇ sσ ) − Pa n

n Normal vector on interface

u ⋅n = 0

κ

Curvature of interface

σ M Surface tension in surfactant-free interface

• Dimensionless Parameters

ρHU

µU

Re =

, Ca =

µ

σM

ρHU

Re ρσ M H

or Re =

,λ =

=

µ2

Ca

µ

Surfactant Transport Equations

• Scaling with

*

C = C * / Ccmc

, Γ = Γ* / Γ*∞

• Bulk surfactant transport equations.

Sc Re ∇ ⋅ ( uC ) = ∇ C

2

BC −

Scs

( n ⋅∇ ) C = jn

χ Sc

• Interfacial surfactant transport equations.

Scs Re ∇ s ⋅ ( uΓ ) − ∇ s 2 Γ = jn

• Dimensionless

Γ ∞*

µ

µ

Parameters, Sc = , Scs =

,χ=

ρD

ρ Ds

Ccmc* H

Γ*∞ Maximum monolayer packing value of Γ∗

*

Ccmc

Critical Micelle Concentration

Surfactant Transport Equations ( 2)

• Surfactant flux from the bulk to the interface

⎧⎪ K a Cs (1 − Γ ) − K d Γ

jn = ⎨

−Kd Γ

⎪⎩

( Γ < 1)

( Γ ≥ 1)

• Modified Frumkin equation of state

⎧

1 − EΓ

⎪

σ =⎨

⎡ E

⎤

E

1

exp

1

−

−

Γ

(

)

(

)

⎪

⎢⎣1 − E

⎥⎦

⎩

• Dimensionless Parameters,

ka Adsorption

( Γ < 1)

( Γ ≥ 1)

*

ka Ccmc

H2

kd H 2

Γ*∞ ∂σ *

, Kd =

, E=− *

Ka =

σ M ∂Γ*

Ds

Ds

rate coefficient

kd Desorption rate coefficient

Cs Bulk surfactant concentration in subsurface

Flow and Pressure: No Surfactant

1

|u|=1

y

0.5

0

SP

SP

Π1=0.56

Π:

-0.5

SP

0.5

-3

-2

0

-1

-1

-0.5

0

1

B

0.5

-2

x

1

2

Re=50, λ=1000, LP=0.5

•

•

•

•

•

0.5

Π = Ca ⋅ p = p* H / σ M*

SP

A

-1

-2 -1.5 -1 -0.5 0

4 stagnation points in the half-domain, all on the interface.

Two flow regions: recirculation and flow-through

Recirculation region low velocity

Capillary wave at front

Pressure minimized at A (in capillary wave)

3

4

Flow , Pressure, and Surfactant ( Lp= 0.5)

Re=50,λ=1000, LP=0.5, Sc=10, Scs=100,

Ka=104, Kb=102, χ=10-3, E=0.7, C0=10-4

1

10-3

SP

5x10 -3

2x

10 -3

0.5

SP

SP

C: 1.00E-04 1.00E-03 1.00E-02

|u|=1

SP

y

0

A

-0.5

Π1=0.82

0.5

-1

-3

•

•

•

•

0

-2

-1

-1

-0.5

0

x

Π:

-2 -1.5 -1 -0.5 0 0.5 1

B

Π = Ca ⋅ p = p* H / σ M*

0.5

1

2

3

4 stagnation points on midline, 2 on interface, 2 internal

The recirculation zone is no longer in contact with the interfaces.

Thicker transition region at front, thinnest point at B

The maximum concentration attains at the front meniscus

4

Flow , Pressure, and Surfactant ( Lp= 2.0)

Re=50,λ=1000, LP=2, Sc=10, Scs=100,

Ka=104, Kb=102, χ=10-3, E=0.7, C0=10-4

1

-3

10-3

C: 1.0E-04 1.0E-03 1.0E-02

2x10-3

0.5

|u|=1

SP

0

Π1=1.03

-0.5

y

5 x1 0

-0.5

0.5

-1

-3

-2

Π:

A

-2 -1.5 -1 -0.5 0

0.5

1

B

-1

0

0.5

-1

0

x

1

2

3

• Front meniscus stagnation pts. differ from rear.

• Velocity profiles fully developed at the middle cross-section ~ parabolic.

• Concentric iso-pressure lines appear due to centrifugal forces.

4

Wall Pressure and Shear Stress

λ=1000, LP=2, Sc=10, Scs=100, Ka=104,

Kb=102, χ=10-3, E=0.7, C0=10-4

0.50

1.25

0.40

1.00

0.30

0.50

0.20

0.25

0.10

0.00

τw

Πwall

0.75

-0.25

0.00

-0.10

-0.50

-0.20

Re=10

Re=30

Re=50

Re=70

-0.75

Re=10

Re=30

Re=50

Re=70

-1.00

-1.25

-1.50

-5

-0.30

-0.40

0

5

x

Wall Pressure

Π = Ca ⋅ p = p* H / σ M*

-0.50

-5

0

5

x

Wall Shear Stress

τ w = Ca

∂u

∂u * H

=µ * *

∂x

∂x σ M

• Wall pressure and shear increase with Re: can affect cells.

• In the front meniscus, both stresses oscillate in the capillary wave.

Unsteady Liquid Plug 2-D Channel: Initial Conditions

Fujioka, Grotberg

C

σ(Γ)

P1

h1

P 2=0

1-h 2

Lp

1-h 2

h2

• Hemi-sphere of the radius 1-h2, uniform film thickness of h=h2

• Initial velocity and bulk surfactant concentration are u=0 and C=C0

The initial interfacial surfactant concentration in equilibrium with C0.

• For no surfactant case, =1.

At t=0, a plug of the initial length, LP=1 starts propagating

with a constant pressure drop, P=P1-P2=1.

Effect of the precursor film thickness on Plug Propagation (no surfactant)

Effect of Surfactant on Plug Propagation

Plug Flow Dynamics Through Bifurcations

Zheng, Fujioka, Grotberg

Plug flows in airways:

deposits liquid along airway wall.

splits unevenly at bifurcation.

determines a final liquid distribution

t>0

t=0

B

A

U

Rs=LB / LA

Parent

V

Splitting ratio Rs = B

VA

LB

LA

(Y. Zheng, et al. JBE, 2005)

Pre-bifurcation asymmetry

α

g

B

Flow

A

Experimental setup

Video

Camera

g

Syringe

A

LB

B

HARVARD

Syringe Pump

A

γ

θ

φ

Computer

LA

Bifurcation Plates with Orientations

•

is branching angle of bifurcation, φ is roll angle and is pitch

angle.

L

Determine Rs (Ca,Bo, Vp, φ, γ)

Splitting ratio:

Rs =

B

LA

Simple Theoretical analysis on plug splitting

y

g

αd2

L2

P1

h

π1

a1

Up

π2

a2

θ

π0

x

a3

L1

P2 U2

π3

L3

P3 U3

αd3

• Compute pressure drops and mass balance between rear and front menisci

• Pressure drops

– Capillary jump + trailing thin film correction across rear mensicus (P1 – π1)

– Poiseuille law + gravity in liquid (π1 – π0, π0 – π2, π0 – π3 )

– Moving contact line effect at front menisci. P2 − π2(3) = ( σ / a 2 ) cos (α d )

cos αs − cos α d

• Empirical correlation for αd

= 2Ca1L 2

(Bracke et. al., Prog. Coll. Polymer Sci.,

1989

α)s + 1

cos

Results (low Re-low Ca expts.)

Effect of roll angle φ and pitch angle γ on Rs vs. Ca

0.9

1

0.8

0.9

0.8

0.7

0.7

0.6

0.6

Rs

0.5

Rs 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

0.02

0.04

0.06

Cap

0.08

0.1

0.12

LB-400-X oil, γ=0º. φ=15º:

(experiments),

(theory); φ =30º: (experiments),

(theory);

φ =60º: (experiments),

(theory).

•

•

•

•

0

0.02

0.04

0.06

0.08

0.1

0.12

Cap

LB-400-X oil, φ=30º. γ =-15º: (experiments),

(theory); γ=0º: (experiments),

(theory);

γ=15º:

(experiments),

(theory); γ=30º:

(experiments),

(theory).

A critical capillary number Cac exists below which Rs=0.

Cap increases, Rs increases.

Larger φ and γ cause smaller Rs but larger Cac.

Theory qualitatively agrees with experiments.

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