upper airway dynamics in subjects with obstructive sleep...

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


7 (×102)

8 (×10)

9

Low k

Low b Actual


CASE 1

Flow Rate, Q

Pressure at Overlap, POL

Cross-Sectional Area, AOL

CASE 2

Flow Rate, Q



CASE 3

Flow Rate, Q



CASE 4

Flow Rate, Q



CASE 5

Flow Rate, Q



CASE 6

Flow Rate, Q



CASE 7

Flow Rate, Q



CASE 8

Flow Rate, Q



CASE 9

Flow Rate, Q



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)

upper airway dynamics in subjects with obstructive sleep...

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