Upper Airway Dynamics in Subjects with Obstructive
Sleep Apnea Jan 16, 2007 – May 9, 2007
Christopher Phaneuf
Brian Tovar
Advisor
Professor David M. Wootton
INTRODUCTION
From young children to adults, a surprisingly large part of the population cannot last a night
without experiencing an apnea. Translated from its Greek origin as “without breath,” this
occurrence is effecting the health of all those suffering from their body’s inability to sustain
steady breathing patterns during sleep.
The last four months have been dedicated to furthering our understanding of airway collapse in
those suffering from obstructive sleep apnea syndrome. Upon extensive reading and discussion
we found this condition to be much more common and multifarious than originally thought. A
myriad of articles ranging from early exploration of this condition to recent studies looking at the
problem from different angles helped to provide a foundation in relevant medical and
engineering knowledge. Additionally, the work of researchers provided a means of realizing the
scope and complexity of this disorder and the many mathematical models and simplifications
used to describe the occurrence of sleep apnea. Notable publications include:
Steady Flow in Collapsible Tubes, 1977. Ascher H. Shapiro
Static mechanics of the velopharynx of patients with obstructive sleep apnea, 1993. Shiroh
Isono, John E. Remmers, et al
Interaction of cross-sectional area, driving pressure, and airflow of passive velopharynx,
1997. Shiroh Isono, John E. Remmers, et al
Interaction between steady flow and individualised compliant segments: application to upper
airways, 1997. R. Fodil, D. Isabey, et al
Computational fluid dynamics modeling of the upper airway of children with obstructive
sleep apnea syndrome in steady flow, 2005. Chun Xu, David M. Wootton, et al
PRESSURE STUDY
Calling on the fundamentals of fluid mechanics, the results and observations of past research was
compared to sets of simple calculations based on Bernoulli’s principle. There was a close
correlation to empirical measurements for some cases and while others showed considerable
discrepancies. This result could be expected, considering the number of assumptions and the
degree of complexity inherent to the human body. Calculations were repeated taking increasingly
more thorough factors, including head loss due to different geometries.
This was beneficial as it reinforced and expanded our grasp on the basics of fluid behavior
through different levels of accuracy. Looking back, this exercise was our first exposure to the
balance achieved with more advanced problem solving in which necessary simplifications must
be reconciled with the need for accurate results.
COMPUTATIONAL FLUID DYNAMICS
Modern medical and scientific investigations are employing more and more powerful numerical
analyses. Computational Fluid Dynamics (CFD) is one type of these powerful tools and it is
becoming extremely useful in the study of circulatory and respiratory flow. We have chosen to
run a series of CFD tutorials in order to 1) understand the context of published research and 2) to
possibly direct our own research to incorporate the use of this tool.
Gambit and Fluent are two programs used in conjunction with one another to produce numerical
simulations of fluid flow interactions with specified geometry. Gambit is used to create meshes
from physical geometries while Fluent computes flow characteristics using finite-difference
methods and convergence criteria. Finite-Difference (or Finite-Volume) transforms continuous
domain problems into discretized cells which define the variables of flow at only the nodes of the
mesh. It effect, continuous partial differential equations are replaced by coupled algebraic ones.
While the geometry of a problem is usually thought of as the physical entity itself, a mesh is the
space around the physical geometry. Gambit is the program that makes this possible. It can be
used to patch inconsistencies in digital geometry (as is the focus of the sedan tutorial) or it can
simplify axially symmetric geometry, like in the laminar pipe flow tutorial. It can even detail a
boundary layer while minimizing the iteration time, like in the turbulent airfoil example.
When we consider each of these tutorials in Fluent, they all had something different to offer; the
results of each simulation emphasized a different feature of flow and how the program allows the
fluid mechanician to view that feature. For instance, the mesh for the airfoil is built around the
boundary layer; it’s finer as you approach the surface of the wing. This allows the boundary
layer to become a pronounced feature of the velocity profile. Also, plotting streamlines can help
the user in the visualization process, although it was not covered in this tutorial.
More important than the fancy visualizations generated by Fluent is the usefulness of the data. In
the sedan tutorial, the user views a pressure field around the body of a car. Instead of only
looking at the plots for information, we can use the numbers generated to accurately analyze the
mechanical or physical implications of the flow and improve a particular design.
DYNAMIC MODELING: THE COLLAPSIBLE AIRWAY IN SIMULINK
The diagram below represents the anatomical basis for the dynamic model of the upper airway:
Anne M.R. Agur and Ming J. Lee, M.D. Grant’s Atlas of Anatomy: 10th Edition. 1999
By dividing the canals and cavities of the human airway into primary segments, the important
pressures and flow variables can be examined while maintaining a simple model. Before creating
the model, the characteristics of the airway had to be considered and simplified with a set of
assumptions. Most notably, the overlap region is the only segment treated as compliant tissue.
Once having switched frames from the familiar static analysis to a dynamic study, the process of
narrowing down the governing equations and creating a simulation began.
From the start the goal was simple: To create a model that generates expected trends
corresponding to the theory at the core of the system as well as roughly matching experimental
observations. In the process of developing a working Simulink model, the approach changed
through two key stages.
The first stage in producing a functioning model was creating a spring-mass representation of a
deformable, cylindrical section of the pharynx considered to be the overlap region (see diagrams
below). The model begins at t = 0 with an initial airway diameter that allows the calculation of
the surface area over which the force, F is applied. Pressure applied over this area is calculated
using Bernoulli’s principle between the trachea (at constant pressure and zero velocity) and
overlap region. The flow rate, Q is also a constant value and is used to determine air velocity at
the overlap. With the pressure and surface area, the force is calculated and sent through the
feedback loop for determining the displacement of the wall with an effective mass, m. Spring
and damping constants, k and b respectively, give the tissue its deformable characteristics and
can be altered to simulate a transient vibratory / oscillatory response.
This model is highly simplistic and assumes constant values for parameters that, in actuality,
vary with every breath.
xva
force
1
2
3
d
AcsPtm
Pressure on wall
1
s
1
s1/m
k/m
b/m
di
AcsAw
Area of wall
dA
Area of cross-section
Area
Subsystems
_________________________________________________________________________
1
A
u2
pi1
d
1) Diameter to cross-sectional area
1
Awsqrt pi
L 2
1
Acs 2) Cross-sectional area to inner surface area
1
Ptm
u2
0.5 rho
Q
Ptr
1
Acs
3) Cross sectional area to pressure
___________________________________________________________________________
Relevant Equations:
Fkxxbxm =++ &&&
WOL APF =
πCSW ALA 2=
2
2
1VPP TROL ρ−= (Bernoulli)
CSA
QV =
The second stage of dynamic modeling grew from the first. It focused on the refinement of the
original model in an attempt to produce more realistic airway behavior while maintaining the
spring-mass component at the core of our model. Tracheal pressure was made to drive the flow
through the pharynx. In order to derive an expression for the flow, a system of equations is
extracted from a circuit representation of the airway shown below:
This gives rise to a differential equation that allows the determination of flow at any point in
time from a sinusoidal pressure signal, which simulates both inhalation and exhalation. This is an
improvement over the previous model of a continuous breath. The drawback to this model is the
limitation of the spring mass system in governing the area change. Empirical data from past sleep
apnea studies demonstrate a relationship between pressure and area known as the tube law.
While this model displays a similar correlation, the curve is not the same due to simplifications
inherent in the model. Regardless, the model produces many outputs that one would expect from
a collapsible airway segment. See below for the results.
Relevant Equations:
CH N N N
OL CH CH CH OL OL
TR OL TR TR
P P QR QL
P P QR QL QR QL
P P QR QL
= − −
= − − − −
= − −
&
& &
&
( ) ( )TR N N CH OL TR N CH OL TRP P Q R R R R Q L L L L= − + + + − + + +&
Since 0NP = in gage pressure,
( )N CH OL TRTR
N CH OL TR N CH OL TR
R R R RPQ Q
L L L L L L L L
+ + += − −
+ + + + + +
&
N CH OL TR TOT
N CH OL TR TOT
L L L L L
R R R R R
+ + + =
+ + + =
TOTTR
TOT TOT
RPQ Q
L L= − −&
Here is the Simulink representation of the above differential equation:
Q
Tracheal Pressure
Rtot
Ltot
1
s
Equations remaining from the first model:
Fkxxbxm =++ &&&
x d= ∆
WOL APF =
WA dLπ=
( ) ( ) 2 21 1
2 2OL TR TR OL
P t P t V Vρ ρ= + − (Bernoulli)
( )
( )OL
CS
Q tV
A t=
Equation for tracheal pressure:
( ) ( )0 sinTRP t P wt φ= +
Q
force
3
1
2
4
5
d
Q term2
A
Qterm1
Switch
AwallA
Q
Ptr
Q
SubSystem
Q
Ptr
(input)
Pol
F delta d
Damped
Spring-Mass
SubSystem
di
cutoff
dA
Area of cross-sectionA
Subsytems
a v x1
delta d
1
s
1
s21/mol
k/mol
b/mol
1
F
1) Damped spring mass subsystem
1
A
u2
pi0.51
d
2) Diameter to cross-sectional area
2
Ptr
1
Q
Tracheal Pressure
A Rtot
Ltot
1
s
1
A
3) Tracheal pressure to flow rate
1
term1
u2
Vol
rho/2
2
Q
1
A
4) Bernoulli term – overlap region
1
term2
u2
Vtr
rho/2A
1
Q
5) Bernoulli term – trachea
Results
With a seemingly infinite number of combinations of parameters for this model, many of which
seem equally valid from the perspective of a non-expert and newcomer to the field, the task of
generating a representative set of outputs is a challenge. Even with nine different cases studied
here, multitudes of other (either disparate or similar) outputs are still possible. To keep the scope
of this investigation as simple as possible, three different pairs of constants (stiffness k and
damping b) were selected to set the mechanical response of the airway tissue at the overlap
region. A higher stiffness results in lower compliance. The other variables are the three airway
dimensions. The scale of these dimensions for tracheal diameter (dTR), initial diameter of the
overlap (di), and the length of the collapsible region (L), significantly alters the system response.
Apart from what we assume to be the actual scale, with dTR = 2 cm, di = 1 cm, and L = 2 cm, two
other scales based on factors of ten showed the response of larger airway dimensions. The same
driving pressure at the trachea was used throughout the simulations:
( ) ( )0.5sin 0.75TRP t t π= +
The response is plotted for a period of fifteen seconds. Generally, as theory dictates, pressure
drops at the shrinking overlap and cross-sectional area decreases. From case to case, this trend is
displayed differently.
Case 1 – 3: With a positive flow rate presumably representing inhalation (and negative flow for
exhalation), faster flow leads to a collapse. With a more extreme pressure drop upon inhalation
(which is reasonable due to the sources of resistance between the nostrils and the overlap), area
decreases. This is illustrated clearly with the plots at the largest dimensions. Smaller dimensions
reveal possible airway closure or near closer, where the area curve nears or reaches zero.
Unfortunately, the steadiness observed at the larger dimension breaks down on the more realistic
scale. Pressure fluctuates rather violently and seems to show periodic discontinuities in which the
pressure approaches infinity and this is where closure seems to occur. This can be attributed to
several limitations and flaws in the model. Geometrical simplification is an important factor and
likely the primary source of inaccuracy.
Case 4 – 6: With an increased damping constant, the larger dimension response in not
significantly different. The smaller dimension response characteristics are affected, showing a
noticeable stabilization. While the chaos remains, it is more periodic. Also, complete collapse
does not seem to occur with the large damping.
Case 7 – 9: For both a low stiffness and low damping, the system responses are possibly the least
coherent. The large dimension shows an overlap pressure varying similar to the pressure-driven
flow, with the exception of seemingly random points of drastic change for short periods. For the
most realistic dimensions, drops in pressure predictably coincide with decreasing area but
closure to zero area does not occur.
The following table summarizes the different cases:
Description Case
Compliance Scale
Natural Frequency
(rad / s)
Damped Frequency
(rad / s)
1 (×102)
2 (×10)
3
High k
Low b Actual
141.4 50, overdamped
4 (×102)
5 (×10)
6
High k
High b Actual
141.4 7499, overdamped
7 (×102)
8 (×10)
9
Low k
Low b Actual
44.72 246, overdamped
CASE 1
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 2
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 3
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 4
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 5
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 6
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 7
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 8
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
CASE 9
Flow Rate, Q
Pressure at Overlap, POL
Cross-Sectional Area, AOL
FUTURE
We are in the process of acquiring access to the Visible Human Dataset, a “complete,
anatomically detailed, three-dimensional representations of the normal male and female human
bodies” based on “transverse CT, MR and cryosection images of representative male and female
cadavers…”1 Some sample images are shown below.
These images could be integral in the process of creating a three-dimensional mesh of the upper
airway. Segmentation modeling software such as Mimics and Amira will be an important tool for
future work in the development of this project. Also in the near future is the possibility of a
physical, deformable model as the next step beyond the previously constructed rigid model. The
deformable version of the airway could be correlated with the results of Fluent-based simulations
using the more complex airway models.
Sagittal section of neck and head
Axial section at lower jaw
1 The Visible Human Project. <http://www.nlm.nih.gov/research/visible/visible_human.html> (Accessed May 9,
2007)