Respiratory Fluid Mechanics
J.B. Grotberg, Ph.D., M.D.
Department of Biomedical Engineering
University of Michigan
Airway Branching Network
Respiratory Zone
Airway and Alveolar Liquid Lining
Cultured human tracheobronchial airway 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
Airway 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 was inserted into the
cannula (upper center of figure).
• A surfactant bolus was 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
Instillation – 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 Intrinsic Plug: Airway 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:
Airway Reopening h1 > h2
2-D Steady Plug Propagation in Channel
Fujioka, Grotberg PoF 2005
Γ
h1
•
•
•
•
•
P1
C
y
H
U
x
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 =
=
Ca
µ2
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
y
0.5
|u|=1
0
Π1=0.56
Π:
-0.5
-1
SP
SP
SP
0.5
-3
-2
0
-1
-1
0
1
B
A
0.5
-2
x
1
2
Re=50, λ=1000, LP=0.5
•
•
•
•
•
0.5
Π = Ca ⋅ p = p* H / σ M*
SP
-0.5
-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
-1
•
•
•
•
0
0.5
-3
-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
0
|u|=1
SP
Π1=1.03
-0.5
y
5 x1 0
-0.5
-1
0.5
-3
-2
-1
0
-1
Π:
A
0
x
1
-2 -1.5 -1 -0.5 0
0.5
1
B
0.5
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
-0.75
Re=10
Re=30
Re=50
Re=70
-1.00
-1.25
-1.50
-5
Re=10
Re=30
Re=50
Re=70
-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
P1
h1
1-h 2
Lp
σ(Γ)
P 2=0
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, σ=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
B
HARVARD
Syringe Pump
A
γ
LB
θ
φ
Computer
LA
Bifurcation Plates with Orientations
• θ is branching angle of bifurcation, φ is roll angle and γ 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
a1
π1
Up
π2
a2
θ
π0
x
a3
L1
P2 U2
L3
π3
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
cos
α)s + 1
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.
Pre-Bifurcation Asymmetry: Gravity Effects
h4*
h3*
y*
p1*
Γ
h1*
H
x*
C
U
p2*
LP*
n
h2*
α
g
Same governing equations…add gravity → Bo (Bond number)
For steady state require h2+h4=h1+h3
Bo=0, No Gravity
Ca=0.05, Bo=0,LP=0.5,Re=0
1
P: -40
Y
0.5
-17
-6
5
0
Vortex strength (VS)=0.176
-0.5
-1
4
-1
0
1
X
2
τ
-29
0
-2
-4
-1
0
X
1
• Two symmetric recirculation regions.
• Precursor film fluid flows into trailing film
Bo=0.1, 0.6
Ca=0.05, Bo=0.1,LP=0.5,Re=0
α=90°
1
1
P: -40
-17
-6
5
0
0
-0.5
-1
-1
0
1
4
X
-1
0
1
X
2
2
0
τ
τ
-29
VS=0.074
VS=0.098
VS=0.149
-1
4
P: -40
0.5
Y
Y
-29
VS=0.149
0.5
-0.5
Ca=0.05, Bo=0.6,LP=0.5,Re=0
Lower wall
Upper wall
-2
0
Lower wall
Upper wall
-2
-4
-4
-1
0
X
1
-1
0
X
• Bypass flow from upper precursor film to lower trailing film.
• Recirculation decreases with increasing Bo.
• Lower (upper) wall shear stress decreases (increases).
1
-17
-6
5
Effects of Ca for Re=0
Ca=0.03, Bo=0.6,LP=1,Re=0
α=90°
Ca=0.3, Bo=0.6,LP=1,Re=0
1
1
0.5
P: -40
0
-15
0.5
-3
Y
Y
VS=0.244
-28
P: -40
-28
-15
-3
No vortex
0
VS=0.238
-0.5
-0.5
-1
-1
-2
-1
1
2
X
5
0
τ
τ
0
L o w e r w a ll
U p p e r w a ll
-5
-2
-1
0
X
-2
-1
5
2
1
2
1
2
0
L o w e r w a ll
U p p e r w a ll
-5
1
0
X
-2
-1
• Recirculation decreases with increasing Ca.
• No recirculations for high Ca, high Bo.
• Wall shear decreases with increasing Ca.
0
X
Effect of Re
Ca=0.05, Bo=0.6,LP=0.5,Re=50
α=90°Ca=0.05, Bo=0.6,LP=2.0, Re=50
1
1
0.5
P: -40
-14
Y
-28
-15
VS=0.268
0
0
VS=0.120
VS=0.206
-0.5
-0.5
-1
-1
5
P: -40
0.5
0
Y
VS=0.094
-27
-1
0
5
1
-2
-1
0
1
2
1
2
X
X
0
τ
τ
0
-5
-10
-1
0
X
Lower wall
Upper wall
-5
Lower wall
Upper wall
1
-10
-2
-1
0
• Recirculations increase and shift to left with increasing Re.
• The streamline pattern is similar.
• Shear stress increases with Re.
X
-3
Effects of surfactant
Ca=0.05, Bo=0.6,LP=0.5,Re=0, E=0.7, C0 = 10-4
1
α=90°
1
S1
0.75
VS=0.122
0.7
0.5
0.65
Y
0.5
0.6
P: -31
0
-19
-7
6
0.5
Y
Y
0.55
0
0.45
0
VS=0.0014
-0.5
-1
0.3
0.4
C: 1.2x10-049.8x10-041.8x10-032.7x10-03
-1
-1
0
1
-1
X
0
1
X
2
τ
0.2
X
-0.5
4
0.1
S2
Color: surfactant concentration contour
0
Lower wall
Upper wall
-2
-4
-1
0
X
1
• Recirculations are weaker with soluble surfactant ( E=0.7).
• Saddle points exist inside the plug core.
• Surfactant accumulates at the lower front transition region.
Cellular Damage Induced by Liquid Plug
Propagation and Rupture in a Microfluidic
Airway System
Dongeun Huh, Hideki Fujioka, James B. Grotberg & Shuichi Takayama
Department of Biomedical Engineering
Department of Macromolecular Science and Engineering
University of Michigan, Ann Arbor
Micro-engineered In Vitro Small Airway System
100 µm
300 µm
PDMS upper
chamber (apical)
Cell differentiation
& shearing
Porous
polyester
membrane
Culture to confluence
PDMS lower
chamber (basal)
Culture with air-liquid
interface and basal feeding
Media (upper)
Media (lower)
Liquid
perfusion
culture
Media (lower)
Plug
generator
Air-liquid
interface
(ALI)
culture
Culture
chamber
Micro-engineered In Vitro Small Airway System
100 µm
300 µm
PDMS upper
chamber (apical)
Cell differentiation
& shearing
Porous
polyester
membrane
(400 nm)
Culture to confluence
PDMS lower
chamber (basal)
Culture with air-liquid
interface and basal feeding
Air ON
Liquid
flow
Plug
generator
Culture
chamber
Cell shearing with plug
propagation & rupture
Air OFF
Air ON
Air flow
Detrimental Effect of Plug Rupture
~ 42 %
Average viability
from B, C, D
Viability
from A
120
Rupture region
100
Viability (%)
A (rupture region)
B CD
~ 17 %
Up- & mid-stream
A
Plug
propagation
80
60
40
20
0
Liquid culture
ALI
5 PR/min
10 PR/min
8 % (5 PR/min) or 9 % (10 PR/min)
lower in rupture region
5 PR/min
~ 50 %
B (mid-stream)
C (upstream)
D (far upstream)
10 PR/min
~ 26 %
Surfactant Spreading on a
Thin Viscous Film
• Drops of surface active substances land on the lungs
air-liquid interface:
– Oil Fog, Surfactant Aerosol, Drugs
• Fronts of surface active substances are introduced
into airways (last step to alveoli)
– Surfactant Instillation, Other Ingestions
• How do they spread?
• Answering this question led to identifying a new fluiddynamical shock wave
Steady Surfactant “Spreading” Model
Borgas & Grotberg, JFM 1988
Steady state “spreading” model
Clean Surface
Surfactant Monolayer~ Γ(x),σ(x)
Barrier
H-=2
H+=1
− 13 p x H 3 + 12 σ x H 2 + H = 1
(− 12 p x H 2 + σ x H + 1)Γ =
1
Γx
Pes
p x = −ε 2 (σH xx ) x , ε =
Lubrication theory (ε→0), px~0, no diffusion (Pes→∞)
σ x = − 12 , u s ( x) = 0, H ( x) = 2
Shock
H
L
Unsteady, Axisymmetric Surfactant
Monolayer Spreading on a Thin Film
Evolution Equations
Same assumptions as before, shock wave develops
Shock speed given in terms of jump in volume flow
rate divided by jump in cross-sectional area of the film
drf
dt
=
but ,
2πrf ( 12 vs− H − − 12 vs+ H + )
2πrf ( H − − H + )
dr f
dt
= vs−
vs− H −
=
(H − − H + )
1
2
→ H − = 2 when
H + =1
Gaver & Grotberg, JFM 1990
Surface Speed and Critical Thinning
Behind Shock
Gaver & Grotberg, JFM 1992
Cycle-Induced Flow and
Surfactant Transport
in an Alveolus
Wei, Fujioka, Grotberg PoF 2005
η =0
R(t ) = R0 (1 + ∆f (t ))
•
•
(i)
(ii)
“High surface tension” ~ interface circular
The bc’s of Γ at the alveolar opening :
No flux ∂Γ ∂s = 0
Fixed Γ=1
ξ
α
a(t )
η
η =∞
x
Velocity Field: Stokes Flow
Stream Function Formulation
bipolar coordinates
Assume (following Haber et al)
ψ1 represents the flow contribution without the alveolar wall while ψ2
is the disturbance flow field caused by the alveolar boundary.
After imposing BCs
The effect of soluble surfactant
• Conservation of bulk surfactant
• Conservation of interfacial surfactant
• SBCs normal (capillarity), tangential (Marangoni)
• Bulk diffusion “fast”, C=1
• Sorption controlled
a0kd
K=
U
Sorption
Parameter
Competing Effects:
• Wall motion
• Pressure/capillarity
• Marangoni (interfacial gradients of Γ and σ)
Instantaneous Streamlines
Inhalation
Exhalation
Flow is mostly “reversible” during inhalation vs. exhalation
Not “reversible” at turn-around
Opposite recirculation directions
Thinner vs thicker layer
Cycle-Averaged Streamlines – K=0 (Insoluble Surfactant)
1
alveolar
cells
y
0.75
0.5
Mixing & Cell-Cell
Communication
0.25
One Recirculation
(clockwise) Region
• fixed end conc.
0
0
0.25
0.5
x
0.75
1
• I:E =1:1)
V=0.9, ∆=0.2, α=π/2, Ma=4.0, Ps=10.0, K=0 (insoluble)
Cycle-Averaged Streamlines - K=0.5 (Soluble)
1
Large molecules (drugs, genetic
material) Dm ~ 10-8 cm2/s)
Steady Péclet number O(1) to O(102)
y
0.75
0.5
0.25
Second Recirculation Region
0
0
0.25
0.5
x
0.75
1
V=0.9, ∆=0.2, α=π/2, Ma=4.0, Ps=10.0, K=0.5
Cycle-Averaged Streamlines – K=0.7
1
y
0.75
0.5
0.25
Three Recirculation Regions
0
0
0.25
0.5
x
0.75
1
V=0.9, ∆=0.2, α=π/2, Ma=4.0, Ps=10.0,K=0.7
Cycle-Averaged Streamlines – K=1.0
1
Counterclockwise
y
0.75
0.5
0.25
Saddle Point
0
0
0.25
0.5
x
0.75
1
V=0.9, ∆=0.2, α=π/2, Ma=4.0, Ps=10.0, K=1.0
Models of Airway Closure: Effects of
Tube Elasticity, Surfactants, and
Oscillatory Core Flow
James B. Grotberg1 & David Halpern2
1Department
of Biomedical Engineering
University of Michigan
Ann Arbor, MI
2Department
of Mathematics
University of Alabama
Tuscaloosa, AL
Effects of Flexibility and Surfactant
on Airway Closure
Halpern & Grotberg, JBE 1993
Fluid:
Elastic Tube:
Insoluble Surfactant:
σ
Eh0
rigid
Airway Closure in Normal Gravity
•
•
•
•
Occurs in lower lung only
Local atelectesis
Regional ventilation inhomogeneity
Mismatch of ventilation and perfusion
Airway Closure in Microgravity
• Can occur everywhere
Effects of Oscillatory Core Flow
in a Liquid-Lined Rigid Tube
Parameters:
λ = VT / πb 3
ε=
Ω = ω a /(ε 3σ / µ f )
a −b
, α 2 = ωb 2 / υ c , Ca = µ cωLT / ε 2σ
a
Model Assumptions
1. Film consists of incompressible Newtonian fluid
2. Oscillatory core flow provides shear-stress at interface
3. Flow in thin film governed by lubrication theory
4. Low-viscosity core fluid perceives the thin, high-viscosity
film as a rigid boundary, uncoupled problem
5. Interface deformation is quasi-steady with respect to the
core fluid motion
6. Long wave approximation
7. Fixed tidal volume
Unperturbed Closure
Oscillatory Core Flow
Minimum core radius versus time for various
frequencies, with β = 43.05 , ε = 0.13, γ = 0.0018
1
0.9
0.8
rmin0.7
Ω
Increasing frequency ω
0.6
8.49
6.79
5.09
4.25
3.40
0.5
0
0.4
0
250
500
t
750
1000
Ω
3.40
4.25
5.09
6.79
8.49
0.6
0.55
rmin
0.5
0.45
980
981
982
t
983
984
985
Mechanism “Reversing Butter Knife”
During stroke: trailing film thicker than leading film → bulge diminishes
During turn around: capillary instability → bulge grows
Needs tuning with high enough frequency
Students, Grand-Students,
Great-Grand-Students….
NSF, NIH, NASA, Whitaker Foundation