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Nose air-flow-rate measu rements by means of nose and sinus manometry
Article · April 2017
DOI: 10.1088/2057-1976/ aa6513
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Natural Science , 2014, 6, 685- 690
Published Online June 2014 i n SciRes. http://www.scirp.org/journal/ ns
http://dx.doi.org/ 10.4236/ns.2014.610068
How to cite this paper : De Luca , R., et al. (2014) Nose and Sinus Air Flow Model . Natural Science , 6, 685- 690.
http://dx.doi.org/ 10.4236/ns.2014.610068

Nose and Sinus Air Flow Model
R. De Luca1, M. Gamerra2, G. Sorrentino2, E. Cantone3,4
1Department of Physics "E. R. Caianiello", University of Salerno, Fisciano, Italy
2Divisione di Otorinolaringoiatria, Ospedale “S. Leonardo” -A.S.L. NA 3 sud, Castellammare di Stabia, Italy
3Department of Neuroscience, Reproductive and Odontostomatologic Science, ENT Unit, “Federico II”
Universi ty, Naples, Italy
4Department of Molecular Medicine and Medical Biotechnology, “Federico II” University, Naples, Italy
Email: [anonimizat]

Received 10 April 2014; revised 10 May 2014; accepted 17 May 2014

Copyright © 2014 by author s and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract
Air flow in nose and sinuses is studied by means of a simple model based on the steady-state idea l
fluid flow assumption and repeated use of Bernoulli’s equation. In particular, by describing flow of air
drawn in through the vestibulumnasi during inspiration, we investigate how ventilation of the maxil –
lary sinus is affected by surgical removal of part of the lateral walls of the nasal cavity close to the os –
tiummeatal complex. We find that, according to the model proposed, removal of tissues from this
inner part of the nasal cavity may cause a decrease of the flux rate from the maxillary sinus.

Keywords
Air Flow Model , Nose and Sinus, Bernoulli's Equation

1. Introduction
The human ventilation system works by means of gaseous exchanges, which take s place between the nose and
sinus cavities, and between the latter and the blood circle through the mucosa [1]. During respiratory acts, air
flowing in the nasal cavity reaches the paranasal sinuses through the hosts and their ducts [2]. Therefore, a cor –
rect anatomical and physiological equilibrium which is able to generate effective pressure gradients inside the
nasal cavity plays an important role in the ventilation of the sin us cavities. During a single inhalation , air flows
from the vestibulum towards the coana and produces, by “sucking effect”, negative pressures into the ostium –
metal complex [3]. As a result, at the beginning of each nasal inhalation action, a negative nose pressure is gen –
erated, in such a way that air flows out from the sinus cavities, due to the effect of aspiration . Air successively
re-enters these cavities when the inspiration phase ends, and thus one of the continuous life -long respiratory cy –
cles is completed. A fundamental role for a correct ventilation is thus played by the anatomical conformation of

R. De Luca et al .

686 the ostiummeatal complex [4].
In sinus physiology, air exchange is also regulated by diffusive molecular mechanisms related to the chemical
and physical characteristics of the inhaled air mixture. The correct and con tinuous sinus ventilation due to the
physical phenomenon described above is the reason why air in the sinus cavities is always in motion. The shape
of the nasal cavity can be assimilated to that of a tube to which, under stationary conditions and ideal flu id flow,
Bernoulli’s equation can be applied. Under the hypothesis of applicability of Bernoulli’s equation, therefore, the single particles of the fluid are taken to describe laminar trajectories with no energy loss in all ducts.
In the present work , the problem of air flow through the ostiummeatal complex coming from the maxillary
sinus is studied by means of repeated use of Bernoulli’s equation. The work is thus organized as follows. In the following section , we give a detailed description of the model adopted. In the third section , we solve the model
equations by means of a first -order perturbation approach, deriving a direct analytic dependence between the
flux rate in the infundibulum and the nose’s effective section. Conclusions are drawn in the last .
2. The Model
We consider the schematic model of the ostium -meatal complex reported in Fig ure 1. In this figure we sche –
matize the maxillary sinus as a spherical cavity (M), in which air at the atmospheric pressure is present. The os –
tiummeatal complex is seen as a short duct linking the nasal cavity (N) to the maxillary sinus. In the schematic
representation of Fig ure 1, during inspiration air enters the nose through the vestibulumnasi and comes out
through the c oana (C -posterior). This air -flow produces a depression in N close to the ostiummeatal complex,
sucking air from the maxillary sinus.
By denoting with S0 and S1 the effective sections of the inner and outer portions of the nasal cavity, respec –
tively, and by S2 the effective section of the ostiummeatal complex, we assume that air behaves as an ideal fluid
[5] through these cavities. Air flows with velocities V0, V1, and V2 through the correspondingly indexed sections,
so that, by continuity equation we can write the following relations for the flo w rates in these sections:
0 0 11 2 2SV SV SV= + . (1)
By extending Bernoulli’s equation to a Y -shaped tube [6], in the absence of gravitational effects, we may write:
2 22
0 00 1 112 221 11
2 22pV pV pVρτ ρτ ρτ   +∆ = +∆ + +∆      . (2)
where p0 and p1 are the pressures in the inner and outer portions of the nasal cavity, respectively, and p2 is the

Figure 1. A schematic representation of the ostium -meatal
complex. Air, drawn trough the vestibulim nasi, enters the
nasal cavity ( N) with velocity V 1. Air from the ostium -meatal
complex, having velocity V 2, mixes with inhaled air in the i n-
ner nasal cavity towards the coana ( C) during inspiration. Air
from the maxillary sinus ( M) is assumed to be at rest at at-
mosperic pressure p a.

R. De Luca et al .

687 pressure in the ostiummeatal complex. Moreover, in Equation (2) ρ is the density of air, Δ τ1 is the volume of the
inhaled air, Δ τ2 is the volume of air drawn from the ostiummeatal complex and Δτ0 is the volume of air flowing
in the coana, given by:
0 12τττ∆ =∆ +∆ . (3)
Notice that Equation (3) is a mere consequence of Equation (1). In this way, Equation (2) can be rewritten in
terms of the volume flow rates, defined in Equation (1), as follows:
2 22
0 00 0 1 11 1 2 22 21 11
2 22p VV S p VV S p VV Sρρρ    + =+ ++         (4)
By Bernoulli’s equation, considering a point in M and a point in the ostiummeatal complex, we can write
2
221
2app V ρ= + (5)
where pa is the atmospheric pressure. Moreover, by assuming that air flowing in the vestibulumnasi is drawn at
constant velocity V L during inspiration, we have:
22
1111
22Va Vkp Vp V ρρ= += + (6a)
11 VV SV S V= . (6b)
where kV is a constant and SV is the effective section of the vestibulumnasi. In this respect, we need to specify
that the assumption on VV is correlated to the patient’s needs of air intake, which can safely be assumed to be
constant. On the other hand, the value of kV is the sum of the atmospheric pressure and of the dynamical pressure
term linked to V V. This term, though varying from individual to individual, remains constant for a single patient.
By now considering Equations (5) and (6a -b), we may rewrite Equat ion (4) in the following way:
2
0 0 001
2VVV a I p V VS kVS pρ+ = +Φ, (7)
where ΦI = S2V2 is the flux rate inside the ostiummeatal complex. By implicitly differentiating Equation (7) and
by noticing that, by Equations (1) and (6b), dΦI = d(S0V0), we write:
22
0 0 00 0 00 01
2aIp p V d S V dp S V dV ρρ− − Φ= + (8)
where the differential quantity d ΦI accounts for an infinitesimal variation of the flux rate in the ostiummeatal
complex solely due to a corresponding infinitesimal variation S0 of the coana effective section. In order to obtain
a set of equations by which we can directly relate d ΦI with dS0, we introduce one further assumption, i.e., that
air can flow at the same temperature inside the nasal cavity before and after the variation dS 0 has taken place, so
that:
() ()00 0 0 00 0 0I d SV p d p SV pd =⇒ = −Φ (9)
By now substituting the above expression in Equation (8) we have:
22
0 00 01
2aIp V d S V dVρρ− Φ=. (10)
We notice that in Equation (10) dV 0 can be related to dS 0 by equating the expression d ΦI = d(S0V0), following
from Equations (1) and (6b), to the expression for d ΦI obtained from Equation (10). In this way, we have :
2
0
00
200
03
2
1
2a
apVdV dS
VSpVρ
ρ−
=− 
−. (11)
By considering Equation (11), it is now not difficult to show that an implicit functional relation of V 0 in terms
of S0 is given by the following expression:

R. De Luca et al .

688 2
00
01
2akp VVSρ−=. (12)
where k is a constant parameter referring to a specific group of patients. In order to obtain a meaningful order of
magnitude for k , we might consider the main term in Equation (12), namely, the product p aV0S0. In this way, we
notice that, for V 0 ≈ 5.0 cm/s and f or S0 ≈ 100 mm2, we have k of about 5.0 N∙ m∙s−1. Equation (12) can now be
inverted either numerically, either analytically, in order to obtain V 0 vs. S0 curves, which we show in Figure 2
for various values of the constant parameter k . Considering now Equations (10) and (11), we ca n set:
3
0
0
2
03
2I
aVd dS
pVρ
ρΦ= −
−, (13)
which dire ctly relates the infinitesimal change d ΦI to dS0. By means of Equation (12) it could be possible to find
how the flux rate Φ I depends explicitly on S 0. However, in the following section, we shall adopt a perturbation
approach to obtain this dependence.
3. Perturbation Solution and Approximated Results
In the previous section we have obtained Equations (12) and (13), which represent the solution to the proposed
problem of finding how the flux rate Φ I inside the ostiummeatal complex varies with respect to the effective area
of the coana S0. Even though an analytic expression for such dependence can in principle be found, it is not
convenient to proceed in this way, since a perturbation approach can be adopted, given that the dynamic pres-
sure ρV02/2 is, for this type of system, much smaller than pa. With th is in mind, to first order in the term ρ V02/2pa,
Equation s (12) and (13) can be written in the following way:
2
0
00 12a
aVVS p kpρ= +
, (14a)
3
0
0 I
aVd dSpρΦ= − . (14b)
Solving for V0 in Equation (14a), we have:
2 2
0
0 32
0211a
aSp kVk pSρ
ρ
= −−, (15)

Figure 2. The velocity V 0 of air in the inner part of the nasal
cavity as a function of the effective area S 0 for ρ = 1.29 kg ∙m−3,
pa = 1.0 atm and for the following values of the parameter k
(from bottom to top): 6, 7, 8, 9 N∙ m∙s−1.

R. De Luca et al .

689 In this expression we chose the minus sign in front of the square root, since it correctly gives a decreasing
behavior of V 0 for small va lues of S 0 and take S 02 > 2ρk2/pa3. In Fig ure 2, we show the velocity V 0 of air in the
inner part of the nasal cavity as a function of the effective area S 0, as given by Equation (15), for ρ = 1.29 kg∙m−3,
pa = 1.0 atm, and the following values of the parameter k (from bottom to top): 6, 7, 8, 9 N∙ m∙s−1.
Substituting now Equation (15) in Equation (14), we finally have:
35 2
2
00 0 23 32a
I
ap kd S S dS
kpρ
ρ
 Φ= − − −, (16)
By now calling A = pa5/ρ2∙k3 and B = 2 ρk2/pa3, we may easily integrate Equation (16), obtaining
()23/242 0
0 003
2IBSAS S S B cΦ= − − − − +
, (17)
where c is a constant. Since we are only interested in finite variations of Φ I, we do not need to calculate the con –
stant c. Moreover, we make take Equation (16) as the first -order approximation of the flux rate variation Δ ΦI
due to a finite variation of the nasal cavity effective section ΔS0, so that
35 2
2
00 0 23 32a
I
ap kSS S
kpρ
ρ
 ∆Φ ≈− − − ∆ (18)
The above equation can be considered as an approximation to the solution to the problem we considered in
the present work. For some typical values of the parameters in Equation (18) and for an effective section of
about one tenth of a square centimeter, th e coefficient linking ΔΦI and ΔS0 is found to be numerically equal to
8.6 × 10−3 m∙s−1. In this way, by taking, for example, ΔS0 = 1.0 mm2, we have Δ ΦI = −8.6 mm3∙s−1. The deriva –
tive dΦI/dS0, as it can be obtained from Equation (16), is represented, as a function of S 0, in Fig ure 3 for ρ =
1.29 kg∙m−3, pa = 1.0 atm, and for the following values of the parameter k (from top to bottom): 6, 7, 8, 9
N∙m∙s−1. From Figure 3 and from Equation (18) it can be seen that the derivative d ΦI/dS0 is negative, so that any
positive variation of S 0 causes a decrease of the flux rate in the ostiummeatal complex. In this respect, one might
argue that surgical removal of anatomical structures close to the ostiummeatal complex might worsen the venti-
lation functional efficacy of the maxillary sinus. This particular aspect has been already observed when analyz –
ing the effectiveness of sinus ventilation with the aid of nose and sinus manometric measurements [7]. In these particular studies it was found that better functional results can be achieved by using a conservative surgical technique preserving nose anatomy rather than a non -conservative endoscopic surgery .
4. Conclusion
Air flow in the nasal cavity is studied by means of a simple model resting on the stationary ideal fluid flow hy –

Figure 3. The derivative dΦ I/dS 0 represented as a fiunction
of S0 for ρ = 1.29 kg∙ m−3, pa = 1.0 atm and for the follow –
ing values of the parameter k (from top to bottom): 6, 7, 8,
9 N∙m∙s−1. The units of the derivative are purposely express –
ed as “(mm3/s)/mm2” instead of “mm/s”.

R. De Luca et al .

690 pothesis. Under these assumptions, Bernoulli’s equation can be used. The model might thus be used to give an
elementary description of nose and sinus ventilation. Under these simplifying assumptions, the present analysis predicts that the flow rate of air sucked from the maxillary sinus towards the nasal cavity decreases as the area
close to the ostium meatal complex increases. In this respect, one might argue that surgical removal of anatomi –
cal structures close to the ostiummeatal complex may worsen the ventilation functional efficacy of maxillary si –
nus, in accordance with experimental observations .
References
[1] Stammberger, H. and Hawke, M. (1993) Essential of Functional Endoscopic Sinus Surgery . Mosby, St. Louis.
[2] Messerklinger, W. (1978) Endoscopy of the Nose . Urban & Schwarzenberg, Baltimore.
[3] Passali, D. (2003) Le rinosinusiti. Pacini Ed., Pisa.
[4] Bruno, R., Gamerra, M., Porpora, D., Pagano, G., Napolitano, B. and Bruno, E. (2004) Nose: Aesthetics and Function.
Anales Otorrinolaringológicos Ibero-Americanos , 31, 307 -323.
[5] Halliday, D. , Resnick, R. and Walker, J. (2001) Foundamentals of Physics. 5 th Edition, John Wiley & Sons, New
York.
[6] Gamerra, M. and De Luca, R. (2004) Un modello semplice per descrivere gli effetti sull a ventilazione sinusale del
flusso d’aria immessa nella cavità nasale. Giornale di Fisica , XLV, 225 -228.
[7] Gamerra, M., De Luca, R. , Pagano, G., Merone, M. and Cassano, M. (2013) The Nose and Sinus Manometry: A
Biophysical Model Applied to Functional Endoscopic Sinus Surgery. Journal of Biological Regulators and Homeos –
tatic Agents , 27, 1021 -1027.

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