Supplementary Material
Biomimetics of Fetal Alveolar Flow Phenomena using
Microfluidics
Janna Tenenbaum-Katan1, Rami Fischler1, Barbara Rothen-Rutishauser2, and Josué
Sznitman1
1
Department of Biomedical Engineering, Technion – Israel Institute of Technology,
32000 Haifa, Israel
2
Adolphe Merkle Institute, University of Fribourg, 1700 Fribourg, Switzerland.
Wall shear stress (WSS) in in vitro rectangular microchannels
To calculate the fluid shear stress acting on a wall of our in vitro model, we
consider the well-known problem of pressure-driven (laminar) Poiseuille flow through a
rectangular channel at steady-state1. The uniaxial velocity profile along the x-direction
is given as
y
cosh n
4h P
1
h sin n z
1
U y, z 3
3
w h
L n ,odd n
cosh
n
2h
2
(1)
where is the fluid's dynamic viscosity, ∆P represents the pressure drop across the
channel, L is the channel length and h and w represent the height and width of the
rectangular cross-section, respectively. Here z is the coordinate of the channel width and
h h
y its height ( y , ). The corresponding flow rate Q (i.e., flow rate perfusing the
2 2
channel as generated by the syringe pump) can be calculated as
Q U y, z da U y, z
A
h 3 wP 192h 1
w
1 5 5 tanh n
12 L w n ,odd n
2h
(2)
A useful approximation for our configuration is to consider a flat and wide
channel, where w≫h such as in our microfluidic models. For such conditions, we may
refer to flow in the middle of the microchannel as approximated by a parallel-plate
Poiseuille flow. Thus the axial velocity reduces to
1
Corresponding author: sznitman@bm.technion.ac.il
U y 1.5
Q 4 y2
1 2
wh
h
(3)
The corresponding shear stress within the channel is defined as
dU y
(4)
dy
Substituting for the approximated velocity-profile, we obtain
12
where the wall shear stress (WSS) at y
Qy
w h3
(5)
h
has a magnitude of
2
6Q
h2 w
(6)
In the specific case of our experiment, the main channel is perfused with a flow rate of
̅
Q=1 µL/min, corresponding to Re=0.1 (according to = 𝑈𝐷ℎ⁄𝜈 = 2𝑄⁄(ℎ + 𝑤)𝜈 , Dh being
the hydraulic diameter) and τ=0.17 dyn/cm2.
Wall shear stress (WSS) in prenatal fetal airways
We can approximately estimate the fluid shear stress, and corresponding WSS,
that acts upon fetal airways, under the following assumptions: (i) flow is fully
developed and follows the Hagen-Poiseuille law for ideal cylindrical (liquid-filled)
airways; (ii) the pulmonary tree is described by a symmetric and dichotomous
branching airway network.
Following these assumptions, the steady, uniaxial velocity profile within
a cylindrical airway is given as
U r
2Q
r2
1
,
R2 R2
(7)
where r is the radial position and R represents the airway radius. In this coordinate
frame, fluid shear stress is defined as
dU r
dr
.
Hence, the magnitude of the WSS is calculated at the airway wall (r=R) as
(8)
4Q
.
R3
(9)
For each breathing cycle, the movement of fluid within the fetal lungs is
reported to be approximately 2% of the total lung volume2 (TLV), such that the specific
volume excursion is C=0.02 and corresponds to (Vmax-Vmin)/Vmin, where Vmax and Vmin
are the maximum and minimum lung volumes, respectively. Thus, the total volume flow
rate associated with each fetal breathing movement (FBM) can be estimated as
Qtotal 0.02 f ,
(10)
where is the total lung volume and f represents FBM frequency. For a symmetric
dichotomous branching tree, the volume movement at the nth generation is given as:
Q
Qtotal
.
2n
(11)
Thus, WSS magnitude is obtained as
wall 4
0.02 f
.
2n R 3
(12)
An estimate of the magnitude of τwall can be obtained for morphometric data established
from fetal lungs3–6 at the onset of the respiratory zone with the occurrence of respiratory
bronchioles, where alveolated structures begin to appear (i.e., generation n approx. 1820). In turn, the magnitude of the WSS acting on the epithelium along alveolar fetal
airways yields the range of approximately τ = 0.05 to 0.9 dyn/cm2, as an order of
magnitude estimate. The Reynolds number in such case can be estimated according to
̅
̅ is defined as the average velocity at
𝑅𝑒 = 𝑈𝐷⁄𝜈 , where D is the air-duct diameter and 𝑈
the examined generation:
U
Q
0.02 f
.
2
R
2n R 2
(13)
As a result, linear velocities within the first acinar generations span U = 1×10-4 m/s to
1×10-3 m/s across developmental stages, coinciding with Re=0.01-0.1 for the range of
WSS considered.
Owing to the principal self-similar nature of rhythmic lung distention during
breathing motion7–9, the length-scale expansion factor is defined10,11 as 𝛽 =
(𝐶 + 1)1⁄3 − 1. Under FBMs, these conditions yield 𝛽 ≈ 0.66%, such that linear
displacements of the fetal environment are negligible (compared for example tidal
breathing in an average male adult with 𝛽 ≈ 5.3% and 𝐶 ≈ 16.7%). Following such
considerations, we have not incorporated movement of airway walls in our model and
have only considered static airways since linear strains are less than 1%.
Furthermore, values of the dimensionless Womersley number which compare
2𝜋𝑓 1/2
) ,
𝜈
unsteady inertial effects to viscous effects11, where 𝑊𝑜 = Dℎ ∙ (
lie in the range
of 0.1-0.2 across week 29-40 of gestation. Since Wo<<1, unsteady effects due to FBMs
are negligible compared to the highly viscous liquid-filled environment characteristic of
the deep alveolar regions of the fetal lungs. Such considerations further rationalize our
use of unixaxial constant flow and static wall conditions as a first approximation.
Computational Simulation of cyclic breathing motion
To investigate the negligibility of unsteady effects and minor wall strains, we have
conducted CFD simulations of a single breathing cycle to investigate fluidic behavior
and patterns. Following previously published computational methods12, we have
performed time-dependent simulations using Comsol 4.3b. Briefly, a self-similar
expansion of the domain was implemented using a sinusoidal expansion function with a
maximal volume increase of 1.8% (corresponding to 𝛽 ≈ 0.66%, as defined above) and
a breathing period of f=1 sec. Inlet flow velocity conditions were also imposed as
sinusoidal with a maximal flow rate identical to the case of rigid walls (see Section 2.6
in the main manuscript). Figure S3 illustrates the resulting flows at peak inhalation of a
breathing cycle; very minor changes can be noted between both regimes (static versus
dynamic), as anticipated for the case of small 𝛽 (see Section 3.4 in the main manuscript
and above Section 'Wall shear stress (WSS) in prenatal fetal airways' in the
supplementary material). Specifically, attached flow is portrayed in the case of α=97º
(corresponding to 29 weeks of gestation) and separated recirculating flows result for the
cases of α=68º and 45º (weeks 34 and 40 of gestation, respectively). Note the minor
change at the vortex origin, in agreement with previous computational results of
unsteady simulations13. Additionally, examining WSS within the simulated cavities,
shows similar trends as well as values establishing in both steady and time-dependant
simulations. For the reasons mentioned above, we may reasonably assume that such low
changes in both lung volume and expansion length scales in utero lead to negligible
unsteady effects and thus steady-state simulations (i.e., rigid walls) represent a good
first approximation of respiratory flows occurring in the alveolar regions in utero.
Model Limitations
Dimensions of the alveoli are in agreement with average morphometric acinar
measurements14 and data reconstructed from micrographs of fetal lungs4. A relatively
low aspect ratio of the channel width to height is used (i.e., w/h ≈ 3.9, with w
representing a duct width), due to difficulties in microfabricating PDMS-based channels
of high-aspect ratio walls.
Due the microfabrication technique, several characteristics of our microfluidic
model are inevitably different from average features and dimensions of the human
pulmonary acinus. These include amongst other the use of cylindrical micro-cavities
rather than polyhedral-like alveoli which better fit physiological data14; namely our
devices feature (i) a rectangular cavity opening rather than a circular- or polygon-shaped
opening, (ii) a rectangular rather than a circular alveolar duct, (iii) large inter-spacing
between alveoli compared to alveoli that share mutual septa (i.e., walls), and (iv) a
relatively low ratio of alveolar to ductal volume. Another feature of our model inherent
to the fabrication process is the planar nature of the acinar geometry, where all ducts
and alveoli share the same height and lie within a single plane.
Despite the aforementioned limitations, it is stressed here that since a wide and
heterogeneous distribution of alveolar shapes and sizes, as well as duct lengths are
inherently present in the human lungs15, our design is a simplified yet useful geometry
for studying alveolar flow dynamics. In addition, it is important to note that the
underlying hydrodynamic characteristics of alveolar flows are known to be captured by
the fundamental coupling between ductal shear flow and alveolar cavity dynamics11,16.
As such, our microfluidic model is conceived for studying the basic mechanisms of
respiratory fetal acinar flows.
References
1
H. Bruus, Theoretical Microfluidics (Oxford University Press, 2007).
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J.C. Schittny and P.H. Burri, in Fishmans Pulm. Dis. Disord. (2008), pp. 91–114.
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J. Sznitman, J. Biomech. 46, 284 (2013).
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Figure S1
Characteristic flows within fetal alveolar cavities through development in utero for channel
depth of R/h=0.6. Flow topologies are similar for the case of R/h=0.4. Note that at R/h=0.6 the
transition from attached to recirculating flow is observed to occur at a lower value of α,
suggesting a slightly later period during the gestational stage.
Figure S2
A549 cells within our microfluidic model fluorescently marked for viability using fluorescent
dyes, calcein acetoxymethyl (AM) in green and propidium iodide in red (see arrow). Viable
cells are present both along the main duct and within the cavities.
Figure S3
Flow patterns (i.e., streamlines and normalized velocity magnitude field) and
corresponding wall shear stress (WSS) for 3 developmental stages of alveoli in utero, as
obtained from CFD. Alveolar flows are compared between steady-state conditions and
peak inhalation (under moving wall conditions) during a breathing cycle (i.e., FBMs).
Generally, minor changes are noted between both cases, where attached (a) and
recirculating patterns (b)-(c) are maintained between static and moving-wall conditions.
In particular, WSS fields and magnitudes preserve reasonably close similarity for each
case simulated.
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