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 [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 sinus 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 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 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 Figure 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 Figure 1, during inspiration air enters the nose through the vestibulumnasi and comes out through the coana (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 flow rates in these sections:

By extending Bernoulli’s equation to a Y-shaped tube [6] , in the absence of gravitational effects, we may write:

where p₀ and p₁ are the pressures in the inner and outer portions of the nasal cavity, respectively, and p₂ is the

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 p_a is the atmospheric pressure. Moreover, by assuming that air flowing in the vestibulumnasi is drawn at constant velocity V_L during inspiration, we have:

where k_V is a constant and S_V is the effective section of the vestibulumnasi. In this respect, we need to specify that the assumption on V_V 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 k_V 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 Equation (4) in the following way:

where Φ_I = S₂V₂ is the flux rate inside the ostiummeatal complex. By implicitly differentiating Equation (7) and by noticing that, by Equations (1) and (6b), dΦ_I = d(S₀V₀), we write:

where the differential quantity dΦ_I 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 dΦ_I 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 dΦ_I = d(S₀V₀), following from Equations (1) and (6b), to the expression for dΦ_I 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 p_aV₀S₀. 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 Figure 2 for various values of the constant parameter k. Considering now Equations (10) and (11), we can set:

which directly relates the infinitesimal change dΦ_I to dS₀. By means of Equation (12) it could be possible to find how the flux rate Φ_I 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 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 pressure ρV₀²/2 is, for this type of system, much smaller than p_a. With this in mind, to first order in the term ρV₀²/2p_a, Equations (12) and (13) can be written in the following way:

Solving for V₀ in Equation (14a), we have:

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/p_a³. In Figure 2, we show the velocity V₀ of air in the inner part of the nasal cavity as a function of the effective area S₀, as given by Equation (15), for ρ = 1.29 kg∙m⁻³, p_a = 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:

By now calling A = p_a⁵/ρ²∙k³ and B = 2 ρk²/p_a³, we may easily integrate Equation (16), obtaining

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 ΔΦ_I 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 ΔΦ_I 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 ΔΦ_I = −8.6 mm³∙s⁻¹. The deriva- tive dΦ_I/dS₀, as it can be obtained from Equation (16), is represented, as a function of S₀, in Figure 3 for ρ = 1.29 kg∙m⁻³, p_a = 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 dΦ_I/dS₀ is negative, so that any positive variation of S₀ 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.

Air flow in the nasal cavity is studied by means of a simple model resting on the stationary ideal fluid flow hy-

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.