Air flow in nose and sinuses is studied by means of a simple model based on the steady-state ideal 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 maxillary sinus is affected by surgical removal of part of the lateral walls of the nasal cavity close to the ostiummeatal 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.
The human ventilation system works by means of gaseous exchanges, which takes place between the nose and sinus cavities, and between the latter and the blood circle through the mucosa [
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 continuous 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 fluid 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.
We consider the schematic model of the ostium-meatal complex reported in
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 [
By extending Bernoulli’s equation to a Y-shaped tube [
where p0 and p1 are the pressures in the inner and outer portions of the nasal cavity, respectively, and p2 is the
A schematic representation of the ostium-meatal complex. Air, drawn trough the vestibulim nasi, enters the nasal cavity (N) with velocity V1. Air fromthe ostium-meatal complex, having velocity V2, mixes with inhaled air in the inner nasal cavity towards the coana (C) during inspiration. Air from the maxillary sinus (M) is assumed to be at rest at atmosperic pressure pa
pressure in the ostiummeatal complex. Moreover, in Equation (2) ρ is the density of air, Δτ1 is the volume of the inhaled air, Δτ2 is the volume of air drawn from the ostiummeatal complex and Δτ0 is the volume of air flowing in the coana, given by:
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:
By Bernoulli’s equation, considering a point in M and a point in the ostiummeatal complex, we can write
where pa is the atmospheric pressure. Moreover, by assuming that air flowing in the vestibulumnasi is drawn at constant velocity VL during inspiration, we have:
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 VV. 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 Equation (4) in the following way:
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:
where the differential quantity dΦI 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 dΦI 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 dS0 has taken place, so that:
By now substituting the above expression in Equation (8) we have:
We notice that in Equation (10) dV0 can be related to dS0 by equating the expression dΦI = d(S0V0), following from Equations (1) and (6b), to the expression for dΦI obtained from Equation (10). In this way, we have:
By considering Equation (11), it is now not difficult to show that an implicit functional relation of V0 in terms of S0 is given by the following expression:
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 paV0S0. In this way, we notice that, for V0 ≈ 5.0 cm/s and for 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 V0 vs. S0 curves, which we show in
which directly relates the infinitesimal change dΦI to dS0. By means of Equation (12) it could be possible to find how the flux rate ΦI depends explicitly on S0. However, in the following section, we shall adopt a perturbation approach to obtain this dependence.
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 ΦI 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 pressure ρV02/2 is, for this type of system, much smaller than pa. With this in mind, to first order in the term ρV02/2pa, Equations (12) and (13) can be written in the following way:
Solving for V0 in Equation (14a), we have:
The velocity V0 of air in the inner part of the nasal cavity as a function of the effective area S0 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
In this expression we chose the minus sign in front of the square root, since it correctly gives a decreasing behavior of V0 for small values of S0 and take S02 > 2ρk2/pa3. In
Substituting now Equation (15) in Equation (14), we finally have:
By now calling A = pa5/ρ2∙k3 and B = 2 ρk2/pa3, we may easily integrate Equation (16), obtaining
where c is a constant. Since we are only interested in finite variations of ΦI, 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 ΔΦI due to a finite variation of the nasal cavity effective section ΔS0, so that
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, the 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 ΔΦI = −8.6 mm3∙s−1. The deriva- tive dΦI/dS0, as it can be obtained from Equation (16), is represented, as a function of S0, in
Air flow in the nasal cavity is studied by means of a simple model resting on the stationary ideal fluid flow hy-
The derivative dΦI/dS0 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”
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 ostiummeatal 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.