Numerical Simulation for Intranasal Transport Phenomena

More than 10million people in Japan suffer some of nasal diseases every year (Haruna (2003)); the paranasal sinusitis (so-called the empyema), hypertrophic rhinitis and inferior concha inflammation. Nebulizer treatment has been used for the nasal diseases. The effectiveness of the nebulizer treatment has been confirmed from clinical view points until now. However there are a few researches that evaluate the effect of the nebulizer treatment theoretically and quantitatively, i.e., the transport characteristics of medicinal droplets and their deposition on the inflammation areas of nasal wall. The development of medical image processing technique is in progress and now gives us exquisitely detailed anatomic information. Some researchers calculated blood flow inside vital arteries and intranasal airflow characteristics by means of the medical image processing technique as well as Computed Fluid Dynamics (CFD) analysis. As for the CFD analysis of intranasal flow, Weinhold et al. constructed both a transparent resin model and numerical three-dimensional anatomy model with nasal cavities using a patient’s CT data (Weinhold & Mlynski (2004)). They subsequently made clear airflow characteristics in the nose experimentally and numerically, and found that pressure drop was a main factor of nasal airflow. Lindemann et al. focused at a case which underwent radical sinus surgery (Lindemann et al. (2004; 2005)). In their case, both the lateral nasal wall and the turbinates, inhibiting physiological airflow, were removed by the surgery to realize the enlargements of the nasal cavity volumes and to increase the ratio between nasal cavity volume and surface area. However the researches mentioned here dealt with only a few patient-cases even though there are individual differences in the shape of human nasal cavity and in the grade of medical conditions. The past researches considered the individual differences insufficiently. The authors analyzed intranasal transport phenomena for several patient cases and compared each others in the past study (Monya et al. (2009); Yamamoto et al. (2009)). From the results characteristics of airflow and medicinal droplet transportation strongly depend on inflow conditions such as inflow angle, velocity and size of particle even if there are the individual differences for the shape of patient’s nasal cavity. 24


Introduction
More than 10 million people in Japan suffer some of nasal diseases every year (Haruna (2003)); the paranasal sinusitis (so-called the empyema), hypertrophic rhinitis and inferior concha inflammation. Nebulizer treatment has been used for the nasal diseases. The effectiveness of the nebulizer treatment has been confirmed from clinical view points until now. However there are a few researches that evaluate the effect of the nebulizer treatment theoretically and quantitatively, i.e., the transport characteristics of medicinal droplets and their deposition on the inflammation areas of nasal wall. The development of medical image processing technique is in progress and now gives us exquisitely detailed anatomic information. Some researchers calculated blood flow inside vital arteries and intranasal airflow characteristics by means of the medical image processing technique as well as Computed Fluid Dynamics (CFD) analysis. As for the CFD analysis of intranasal flow, Weinhold et al. constructed both a transparent resin model and numerical three-dimensional anatomy model with nasal cavities using a patient's CT data (Weinhold & Mlynski (2004)). They subsequently made clear airflow characteristics in the nose experimentally and numerically, and found that pressure drop was a main factor of nasal airflow. Lindemann et al. focused at a case which underwent radical sinus surgery (Lindemann et al. (2004;2005)). In their case, both the lateral nasal wall and the turbinates, inhibiting physiological airflow, were removed by the surgery to realize the enlargements of the nasal cavity volumes and to increase the ratio between nasal cavity volume and surface area. However the researches mentioned here dealt with only a few patient-cases even though there are individual differences in the shape of human nasal cavity and in the grade of medical conditions. The past researches considered the individual differences insufficiently. The authors analyzed intranasal transport phenomena for several patient cases and compared each others in the past study (Monya et al. (2009);Yamamoto et al. (2009)). From the results characteristics of airflow and medicinal droplet transportation strongly depend on inflow conditions such as inflow angle, velocity and size of particle even if there are the individual differences for the shape of patient's nasal cavity.  This chapter shows you detail features of nasal cavity and some nasal diseases, subsequently how to yield three-dimensional model of nasal cavity and calculation mesh for CFD analysis. Finally the characteristics of intranasal transport phenomena are presented in this chapter.

Structure of nasal cavity and nasal sinuses
Figure 2 shows front and side views of human head, nasal cavity, nasal conchae and nasal sinuses. The nasal cavity is a large air filled space, and the pathway of respiration flow. The nasal cavity conditions the air to be received by the other areas of the respiratory tract. The nasal conchae, which has large surface area, warms and cools passing air through the nasal cavity. At the same time, the passing air is humidified and cleaned by nasal conchae and nasal hairs; most parts of dusts and particulate matters are removed there. The nasal sinuses are four caviums existing in the ossa faciei as shown in Fig.2; maxillary sinus, frontal sinus, ethmoidal sinus and sphenoidal sinus. These sinuses connect to the nasal cavity via very small ducts, ostia; both diameter and streamwise scales are several millimeters. Figures 3 represents coronal and sagittal cross-sectional CT data for nasal cavity and nasal sinuses. The white areas in the figure represent bones of the patient's head, and the gray areas indicate muscles and fat. The nasal cavity and sinuses are represented as dark area in the CT data. In paranasal sinusitis, both the sinuses and the ostia are infected by bacillus and the virus. The details of such the diseases are stated in following subsections.

Nasal diseases
The nose often suffers some injuries such as fractures because the nose is protruded forward from the human face. Infection, epistaxis, as well as polyps are also found in the nasal diseases. Almost all people have contract rhinitis which is caused by the inflammation of nasal mucosa. Sinusitis will develop when the inflammation of the rhinitis spread to the sinuses via ostia.  cavity is significantly smaller than the other. Then the patient suffers nasal congestion and drying nasal cavity, consequently epistaxis. In the sever cases operative treatment is required.

Rhinitis (nasal inflammation)
The nasal cavity is the part that easily causes infection in the upper respiratory tracts. Rhinitis (nasal inflammation) is mainly classified into two cases; the one is acute rhinitis and the other is chronic rhinitis. The acute rhinitis is generally caused by the infection of virus and allergies of pollen and house dust. The difference point between both rhinitis is the term of these diseases; the former occurs in a short term and the latter in a long term (two weeks and more). In some cases, the chronic rhinitis occasionally accompanies the chronic sinusitis.  Figure 6 shows cross-sectional CT data for both healthy maxillary sinus and sinusitis one. The sinusitis happens anywhere about four kinds of the sinus paranasalis such as the maxillary sinus, sinus ethmoidales, the master wrestlers caves, and the sphenoidal sinuses. The sinusitis is classified into the acute sinusitis (in a short term) and the chronic sinusitis (in a long term). The acute sinusitis is caused by a variety of bacteria and virus, often develops after the blockage of ostia which are opening of sinuses connecting to nasal cavity. The blockage normally occures as a result of virus infection in the upper respiratory caused by a cold. The cold brings on the swelling of the mucous membrane of nostril, and then the ostia are easily obstructed. In the blocked sinus, the air inside the cavity are absorbed to the bloodstream, and subsequently the pressure inside the sinus decreases. This pressure drop produce sever pains and the sinus fills with secretory fluid. The secretory fluid becomes a breeding ground for viruses. In order to attack the viruses, white blood cell and other matters such as the secretory fluid are aggregated in the blocked sinus, then the pressure in the sinus is increased and pain becomes more sever. The case that a symptom of the sinusitis continues more than 8-12 weeks is called the chronic sinusitis. The developing mechanism of the chronic sinusitis has not been clear, however it has been confirmed that the chronic sinusitis develops after the infection of the virus, severe allergies and the influences of the environmental pollution material. A genetic contributor is also regarded as one of the factors concerned with development of the chronic sinusitis.

Construction of three-dimensional nasal cavity model
In order to construct a three-dimensional model for nasal cavity, data conversion algorithm from two-dimensional CT/MRI data to three-dimensional geometric model as well as smoothing algorithm for the three-dimensional model are required. Furthermore, mesh generation models are needed to execute CFD analysis of intranasal transport phenomena. This section describes the fundamental theories of these algorithms.

Data conversion algorithm from 2-D to 3-D: marching cubes algorithm
Marching cubes algorithm is one of the latest models of surface construction used for viewing three-dimensional data (Lorensen & Cline (1987)). This algorithm extracts a polygonal mesh 541 Numerical Simulation for Intranasal Transport Phenomena www.intechopen.com

Smoothing algorithm
In the image processing and image transmission, noises are physically captured and fed into the processes due to various factors. For instance, noises are generated caused by the image resolution and image slice thickness of CT and MRI in the field of medical engineering. When the noise generation mechanism is clear and mathematically modeled, we can remove the noise using the optimized filter. However, it is difficult to modeled the noise mathematically in almost all situations.f The smoothing processes are needed to reduce and minimize the influence of the noise. The noise makes patterned indented surfaces in the three-dimensional model, therefore affects the quality of the constructed nasal cavity model significantly. There are two smoothing algorithms that often uses in the medical image processing; moving-average algorithm and median-average algorithm. This subsection presents both algorithms as follows.  (2002)). The number inside a cube expresses a scalar variable, for example the concentration of a chemical specie. The case of Fig.9(a) presents a noise included in the data and that of Fig.9(b) expresses an asperity in the data. The moving-average algorithm calculates an average of surroundings of the concentration f (i, j) in the input image, and converts to the concentration g(i, j) in the output image.

Moving
The moving-average algorithm feathers the edge and the boundary of the input image, consequently smoothes the object surfaces. Figure 10 indicates the application of the median-average algorithm for 3 × 3 cubes (Boyle & Thomas (1988); Tamura (2002)). The cases of Fig.10(a) and (b) represent a noise and an asperity in the data, respectively. This algorithm sorts both a concentration value f (i, j) at a target position of the input image and its surroundings in ascending order, subsequently converts the output image g(i, j) using their median value. As for Fig.10(a), the concentration values in input image are rearranged like "1, 1, 1, 1, 1, 1, 1, 1, 20" in ascending order, and then the fifth value "1" is picked up as a median value and used as the concentration g(i, j) in the  1, 1, 1, 1, 20, 20, 20, 20" in ascending order as shown in Fig.10(b). Finally, the fifth value "1" is selected as the concentration g(i, j) in the output image. The median-average algorithm can remove the noise effectively compared with the moving-average algorithm. At the same time, this algorithm involves a risk that the magnitude of rearranging the original CT/MRI data is larger than the moving-average algorithm, consequently could yield different three-dimensional model from the real target.

Mesh generation: Delaunay triangulation model
The Delaunay triangulation model is one of the most-used mesh generation technique. This model is applicable for a wide variety of very complicate geometries. This model mathematically ensures that the circumcircle associated with each triangle contains no other point in its interior. This definition extends naturally to three-dimensional problems. Figure 11 shows a schematic drawing of the Delaunay triangulation model (Nakahashi (1995)). The triangles as shown in Fig.11(a) contains the edges and the edge's endpoints. A new point "P" is added inside the triangle "ABC", then the mathematical theory of the Delaunay triangulation is not satisfied in the triangle "ABC" and its surrounding triangle "BCD". The model can eliminate the edge "BC" which is a common edge for both triangles "ABC" and "BCD", subsequently reconstruct triangles using the added point "P". Consequently, as shown in Fig.11(d), the model can yield four triangles "ABP", "ACP", "BDP" and "CDP". The Delaunay triangulation constructs and reconstructs triangles in target domain simultaneously and give us fine meshes for CFD analysis.

Theory of computational fluid dynamics
In the nasal cavity, particulate matters such as pollens, house dust as well as medicinal droplets are transported via airflow of the respiration and the nebulizer treatment. In order to execute CFD analysis for such the transport phenomena, we have to consider multi-phase flow. This section describes the basic theory of CFD analysis for two-phase flow.

Governing equations for continuous phase
The Reynolds number of intranasal flow is a few hundred. Therefore the flow is assumed as incompressible and laminar flow. The equation for conservation of mass, or continuity equation, can be written as follows: here ρ F and u F are density and velocity of continuous phase. Equation 2 is the general form of the mass conservation equation and is valid for incompressible as well as compressible flows. Conservation of momentum in an inertial reference frame is described by where p is the static pressure, μ F is viscosity of the fluid, and ρg and F are gravitational body force and external body forces, respectively. F contains interaction with the dispersed phase in multi-phase flow and other model-dependent source terms. The thermal energy transport equation is meant to be used for flows which are low speed and close to constant density.
where c p and λ F are specific heat at constant pressure and heat conductivity, respectively. S FP denotes a source term regarding the interaction between continuous phase and disperse phase.

Governing equations for disperse phase
Consider a discrete particle traveling in a continuous fluid medium. The forces acting on the particle which affect the particle acceleration are due to the difference in velocity between the particle and fluid, as well as to the displacement of the fluid by the particle. The equation of motion for such a particle was derived by Basset, Boussinesq and Oseen for a rotating reference frame (Akiyama (2002)): where, superscript i denote the i-th particle, and F Drag , F Pressure , F Basset , F Buoyancy , F Rotation express drag force acting on the particle, pressure gradient force, Basset force, buoyancy force due to gravity and forces due to domain rotation, respectively. The pressure gradient force term applies on the particle due to the pressure gradient in the fluid surrounding the particle caused by fluid acceleration. The Basset force term accounts for the deviation in flow pattern from a steady state. The aerodynamic drag force on a particle is propotional to the slip velocity between the particle and the fluid velocity where u F and u P are the velocities of the fluid flow and the particle. C D denotes the drag coefficient of the particle.
Re i is the Reynolds number based on the slip velocity between the fluid flow and the i-th particle velocities.
The pressure gradient force results from the local fluid pressure gradient around the particle and is defined as This force is only important if large fluid pressure gradients exist and if the particle density is smaller than or similar to the fluid density.
The buoyancy force is the force on a particle immersed in a fluid. The buoyancy force is equal to the weight of the displaced fluid and is given by here g expresses the gravity vector.
In rotating frame of reference, the rotation term is an intrinsic part of the acceleration in and is the sum ob Coriolis and centripetal forces.
where, ω denotes rotation speed of the reference. In the intranasal flow and heat and mass transfer, the influence of both the pressure gradient term and the Basset force term on the transport phenomena in the nasal cavity is extremely small. The momentum conservation equation for the particle is shown as When the Lagrangian method is applied to the particle movement, the location of the particle is given by As for the almost all particulate matter transported in nasal cavity, the mass fraction of the disperse phase are extremely small compared with the continuous phase. Therefore it is only necessary to consider the one-way interaction between both phases, namely only the continuous phase influences the disperse phase. Heat transfer concerning the disperse phase is governed by three physical processes, namely convective heat transfer, latent heat transfer associated with mass transfer, and radiative heat transfer. Since the heat transfer of intranasal flow is not so high temperature condition, we can neglect both the latent heat transfer and the radiative heat transfer. The convective heat transfer Q C is given by where T i P is the temperature of the i-th particle, and Nu is the Nusselt number given by The convective heat transfer Q C is calculated and used in the source term of the thermal energy transport equation, Eq.4.

Case data and three-dimensional model
Three-dimensional models of nasal cavity were constructed by means of five actual patient's CT data. These cases suffer several grades of deviated nasal septum symptom, hypertrophic rhinitis, chronic sinusitis and inferior nasal concha swelling. The case data are summarized in Table 1. Since head CT data involves not only nasal cavity but also skin, fat, bone and purulent matter, it is required to eliminate them. This study used three-dimensional medical image processing software "Mimics (Materialize Inc.)" and the marching cubes algorithm to determine the surfaces of nasal wall. The moving-average algorithm is also applied to smoothing the surface. Figure 12 shows the three-dimensional nasal model of case-I constructed in this study. The anterior and the posterior of the figure show anterior nares and upper pharynx, respectively. The three-dimension model has curved nasal septum which corresponds to the curvature as shown in Fig.12. The cross-sectional areas of nasal cavities 547 Numerical Simulation for Intranasal Transport Phenomena

Computational fluid dynamics model
This study used CFD analysis code "CFX-11 (ANSYS, Inc.)" to analyze the intranasal transport phenomena of the nebulizer treatment. This code is able to solve partial differential equations for mass, momentum and energy transportations with appropriate boundary conditions. Since the Reynolds number of intranasal flow was several hundred, normal laminar flow analysis was used in this study. The Lagrange approach is adopted to solve transportation of the droplets of medicinal mist. Then the model employs assumptions that a droplet is sensitive with fluid flow, accelerating by velocity difference between fluid flow and the droplet. Air inlet velocity is set at 0.5m/s. This velocity condition corresponds to the condition of actual nebulizer treatment. As shown in Fig.16, four air inlet angles are considered in this study; 90, 60, 30 and 25 degree. In the nebulizer treatment, medicinal mist is atomized and transported by dry air. This study considers dry air (298K, 0.1MPa) as a fluid medium. No-slip condition, where fluid velocity on the wall surface is set at zero, is adopted to the nasal wall. Droplet diameter of medicinal mist is uniformly 50 micro meters. This value also corresponds to the condition of actual nebulizer treatment. nebulizer angle conditions flow velocity increases near the nares. Especially, the increasing flow velocity is most obvious in the inflammation regions. Circulation flow is found in the lower inlet angle conditions. This circulation flow will be able to extend residence times of airflow and medicinal droplet in nasal cavity, consequently enhance the effect of the nebulizer treatment. Figure 20-   tendencies even though there are the individual differences such as the shape of nasal cavity and the state of nasal disease. Therefore these parameters are useful to organize the results mentioned above. The maximum velocity has direct proportional relationship with pressure drop. Meanwhile pressure drop and maximum velocity have strong correlation with deposition ratio. These results lead us to conclude that the nebulizer conditions have same-level influence on the intranasal transport phenomena for all cases.

Conclusion
This chapter presents not only the procedure of CFD analysis for intranasal transport phenomena but also medicinal droplets transport characteristic in patient's nasal cavity using actual CT/MRI data. Furthermore, the result of correlation analysis for the nebulizer treatment condition is also presented here. As for the results of CFD analysis, nebulizer angle, inlet velocity and size of droplet significantly affect on the intranasal transport phenomena. The condition of low nebulizer angle achieves that the medicinal droplets are transported all over the nasal cavity, that is this condition will increases therapeutic response to the nasal diseases. The correlation among velocity of intranasal flow, pressure drop and deposition ratio, which are main parameters for intranasal transportation of medicinal droplets in the nebulizer treatment, indicates that the basis of the intranasal transportation show similar characteristics in all patient's cases. These results mean that there is an optimum condition of nebulizer treatment for all patients who suffer nasal diseases. Further research of the intranasal transport phenomena for many patient's cases is required to survey the best treatment condition.
Since the development of the medical image processing technique has been in progress, the CFD analysis for human body, such as blood flow from the heart, respiration flow in the pharynx and the lung, will be widely spread and achieve tangible results in clinical practices within the next several years.

Acknowledgement
This work was supported by the grant from JSPS KAKENHI 19760114.