Modeling spray and movement of microparticles of chitosan aerogels in the nasal cavity

Introduction

Administering drugs using the nasal route offers a promising approach to treat a variety of diseases. Nasal delivery of drugs can be used as an alternative route for systemic availability of drugs limited by enteral administration [1]. The advantages of nasal delivery include: accelerated absorption of drugs and a rapid onset of therapeutic effect; potentially high bioavailability of small molecules and increased bioavailability of larger molecules; the ability to deliver drugs with poor oral absorption or low oral bioavailability; the absence of first-pass metabolism through the liver; convenience for patients, the possibility of nasal administration in cases of nausea, vomiting, and difficulty swallowing [2–5]. Medicinal substances are rapidly eliminated from the nasal cavity after nasal administration, resulting in rapid systemic absorption of drugs [6].

The outlook for nasal drug delivery systems used in the treatment of various neurological disorders is rather promising [4]. The desired therapeutic effect requires drug delivery to the olfactory region [7]. However, existing approaches prevent a sufficiently high concentration of particles in the desired area. This is due to both the complex structure of the nasal cavity, in which inhaled particles are filtered along before they reach the olfactory region, as well as the difficulty of controlling the trajectory of drug particles. Various approaches to intranasal drug administration have been studied to improve the efficiency of delivery to the olfactory region. Si et al. [8], used computational fluid dynamics (CFD) to show that targeted administration to the specific nasal cavity increased particle deposition in the olfactory region compared to the traditional administration method. Lovskaya et al. [9] described nasal delivery systems based on protein aerogels obtained by dispersing a protein solution in oil and thermal gelation, followed by supercritical drying. The results of in vivo studies showed that clomipramine via nasal administration reaches maximum concentrations in the blood plasma and brain tissue of rats 30 minutes and supporting the potential of biopolymer aerogels as nasal delivery systems. Menshutina et al. [1] reported the potential of chitosan as a material for drug delivery systems.

However, nasal delivery of dry powders, particularly those based on highly porous materials such as biopolymer aerogels, highlighting the novelty and relevance of this study. Aerogel-based drug delivery systems exhibit properties such as biocompatibility, biodegradability, mucoadhesive properties, increased permeability, and increased solubility of active ingredients [1][5]. Dry powder drug production provides increased stability compared to liquid dosage forms and further reduces the costs of special transportation conditions (such as cold chain), thereby simplifying logistics.

Increased computing power in studying drug behavior in the respiratory system makes it possible to conduct in silico studies, which can overcome the shortcomings of in vivo and in vitro studies. In vivo and in vitro studies are characterized by high costs and significant resource expenditures. In particular, mathematical models based on the solution of CFD equations have been successfully used to predict the movement and sedimentation of drug particles or droplets via various routes of administration [10][11], design of dosing devices [12][13] and construct anatomically similar models of the respiratory system [14][15].

Scientists from BRICS countries are actively conducting research on nasal fluid dynamics modeling. Zhu et al. [16] assessed the impact of differences in the morphology of Caucasian, Chinese, and Indian males on nasal airflow patterns. The results show clear differences in both nasal morphology and airflow patterns. Li et al. [17] demonstrated the feasibility of developing a standardized nasal cavity model based on CT scans of 128 adults (64 man and 64 women) of the same ethnicity for evaluating drug efficacy within a single ethnic group.

CFD is widely used in the development and evaluation of new nasal drug delivery systems. The literature provides results on the distribution of micrometer-sized solid particles in the olfactory regions of the nasal cavities [18]. However, the study focused on high-density, low-porosity particles that are not delivery systems. What is more, the capabilities of mathematical modeling for studying the motion and sedimentation zones of drug delivery systems, in particular highly porous dry aerogel particles, using this approach are limited, indicating the novelty of this work. Therefore, modeling particle flight from dosing devices and predicting nasal deposition zones allows for optimization of pharmaceutical development and increased treatment efficacy.

This study aimed to examines the dispersion of chitosan aerogel microparticles when administered into the nasal cavity using mathematical and computer modeling. For this purpose, a virtual geometry of the human nasal airway was constructed based on computed tomography data, and a mathematical model of particle dispersion and motion in the nasal cavity was developed using CFD.

Materials and methods

Mathematical model of drug particle spray and movement

One of the objectives of this study is to develop a mathematical model of drug particle spray and movement within the human nasal cavity. The model is based on the mechanics of continuous and heterogeneous media. In addition to the model equations, the assumptions applied to the model, boundary and initial conditions, and additional relationships are specified. Numerical methods used to solve the mathematical model are described. The model equations include a system of equations for the continuous medium and an equation for the dispersed phase, which describes the motion of particles and includes the sum of all forces that influence their motion. When developing a mathematical description of the process, the motion of a two-component system is considered, consisting of a continuous medium (air) and a dispersed medium (aerogel particles with a known particle size distribution).

The system of equations for the continuous medium includes the continuity equation, the momentum conservation equation, and the energy conservation equation:

(1)

where α and αp are the fractions of the continuous medium and the dispersed phase; ρ is the density of the continuous medium, kg/m³; v⃗ is the velocity vector of the continuous medium, m/s; v⃗_p is the particle velocity vector, m/s; p is the static pressure, Pa; τ ̿ is the viscous stress tensor; ρg⃗ is the force of gravity, kg/m²·s²; R_s,l is the force characterizing the influence of the dispersed phase on the continuous medium, kg/m²·s²; f_d is the force characterizing the resistance of the particle, kg·m/s²; V_p is the volume of the particle, m³; K_s,l is the momentum exchange between the phases, kg/(m³·s); µ is the dynamic viscosity, Pa·s; C_p is the specific heat capacity of the mixture, J/kg·K; T is the temperature of the mixture, K; λ is the thermal conductivity coefficient of the mixture, W/m·K; I is the unit tensor.

In this case, heat transfer in continuous and dispersed phases is not considered.

The equation for the dispersed phase is written as follows:

(2)

where m_p is the mass of the particle, kg; g⃗ is the acceleration due to gravity, m/s²; f_p,c is the force of interaction between the continuous medium and the particle, N; f_p,p is the force of interaction between particles, N; f_p,w is the force of interaction between the particle and the wall, N.

The force of collision between particles is described by the stiffness coefficient, which is calculated using the following equation:

(3)

where D is the particle diameter, m; ρ is the particle density, kg/m³; v is the relative velocity between two colliding particles, m/s; εD is the maximum overlap.

The following initial and boundary conditions are specified for the continuous medium and the dispersed phase:

Initial conditions for the continuous medium:

(4)

(5)

Initial conditions for the dispersed phase:

(6)

Boundary conditions for the continuous medium:

(7)

(8)

(9)

(10)

Boundary conditions for the dispersed phase:

(11)

(12)

where x, y and z are the spatial coordinates, m; t is time, s; in is inlet; init is the initial condition; w is the wall; where v⃗_1,p and v⃗_2,p are the particle speeds before and after collision with the wall, m/s.

The presented model consists of a system of differential equations. A finite volume method was used to solve the model’s equations. Variables that describe the state of the system under consideration, such as velocity, pressure, concentration, and particle interaction forces, are calculated within a closed domain and satisfy the accepted mathematical expressions. The method is implemented in the ANSYS Fluent 17.0 software package (Ansys, Inc., USA).

Construction of virtual geometry of the nasal airways

A virtual geometry of the human nasal airway was constructed based on computed tomography (CT) scan results. The CT scan results were provided by the Petrovsky National Research Centre of Surgery under a relevant scientific cooperation agreement. The data was transferred in an anonymized data, devoid of any patient demographic characteristics. Written informed consent was obtained from the patient by the staff of the Petrovsky National Research Centre of Surgery for the use of the medical imaging data for scientific purposes.

The resulting data was processed using 3D Slicer 5.10 (open source platform), a free platform for analyzing and visualizing medical data. Built-in tools allow for the generation of a virtual structure with specified characteristics. The resulting 3D geometry was saved in stereolithography (STL) format. The resulting geometry was imported and further processed in ANSYS SpaceClaim, a geometric piece of modeling software. This processing is necessary to identify and correct errors in the resulting 3D model of the nasal cavity and to define the entry and exit regions.

Smoothing and generation of a computational mesh of the virtual geometry of the nasal cavity

Smoothing of the virtual geometry was performed using ANSYS SpaceClaim tools. The determining parameter for smoothing was the minimum radius for smoothing the curved surface of the geometry. To obtain the desired result, smoothing was performed with the following radii: 0.5 mm, 1 mm, and 1.5 mm. Selecting a smaller radius does not significantly alter the original geometry, while choosing a larger value leads to significant changes (merging of different volumes, the emergence of new joints between different parts of the geometry).

For each of the resulting virtual geometries, computational meshes were generated using ANSYS Meshing. The same mesh generator settings were selected for each case. An irregular computational mesh was constructed using the tetrahedral method.

Experimental studies of the spray torch of chitosan aerogel microparticles

Chitosan aerogels were obtained using the previously described method [1][5]. Samples with an apparent density of 0.03 g/cm³ were selected for computational experiments. These samples had varying particle size distributions. The average particle size of the chitosan aerogel was selected as 50, 100, and 150 µm. The spray angle varied from 5 to 15° in 1° increments. The initial particle velocity varied from 5 to 9 m/s in 1 m/s increments. The particle injection duration through the atomizer was set to 120 ms. The atomizer outlet diameter was 2 mm.

As part of this study, an experimental study of the atomization of chitosan aerogel particles was conducted using a special dry powder dosing device from Nasaval^® (Zambon, United Kingdom). For this study, a sample of particles with known characteristics was loaded into the dosing device, which was positioned against a dark background in front of a video camera. The spraying process was then videotaped. The experiments were conducted three times. The spray pattern was then evaluated. The experimental value of the spray cone based on photographic processing was 9 ± 2°. The spray velocity was measured using an anemometer and was 5 ± 1 m/s.

Computational experiment to evaluate the spray pattern of solid aerogel particles

It is assumed that the particle velocity at the injection point decreases linearly with time from the initial value. The particle size distribution was approximated using the Rosin–Rammler equations. The coefficient n of the equation was 3.404, the number of fractions was 10, and the coefficient of determination (R²) of the equation was 0.9762.

These data are necessary for successfully conducting a computational experiment using a mathematical model and CFD methods. For this purpose, a geometry was also constructed within which the spray occurs. The geometry is parallelepiped with a square at the lower base. The square measures 10 × 10 cm, and the parallelepiped is 30 cm high. These dimensions are sufficient for the free flight of the sprayed particles. The base of the parallelepiped serves as a wall. The spray point is located at its center.

Computational experiment on the dispersion of aerogel particles in the nasal cavity

Preliminary computational experiments were conducted to validate the models, select computational methods, and choose output methods. The computational experiment was conducted within the nasal cavity geometry. Spraying into the nasal cavity is accomplished through a dosing device, which is shaped like a cone and located within the entrance to the right nasal passage. The dosing device is positioned at a 45-degree angle to the horizontal plane. The characteristics of the sprayed particles and the spray device correspond to those described previously. It is assumed that the particles are captured upon contact with the nasal cavity walls. The spray angle was varied from 5 to 15° in 1° increments. The initial particle velocity was varied from 5 to 9 m/s in 1 m/s increments.

The primary criterion (output parameter) was the proportion of particles entering the olfactory region of the nasal cavity relative to all particles sprayed into the nasal cavity. Thus, the computational experiment calculated particle accretion (A(x, y, z, t)) – an accumulative parameter that characterizes the mass of particles captured by the geometry wall, divided by the surface area of that wall. To determine particle accretion in the target areas, the geometry was divided into two parts: upper and lower. This division allows for the necessary computational information to be derived. For each computational experiment, the total mass of particles deposited on the surfaces of the upper and lower nasal cavities after 3 seconds is calculated.

To determine the mass of these particles, an integral is taken over the target surface:

(13)

where S_k is the surface area, m²; M_k is the total mass of particles deposited on the surface S, kg; the index k is necessary to designate the surface area of the upper (u) and lower (l) parts of the nasal cavity.

Based on the two obtained values, the proportion of particles entering the upper part of the nasal cavity is calculated as a ratio:

(14)

where R_A is the proportion of particles deposited in the upper part of the nasal cavity; M_u is the total mass of particles deposited on the surface of the upper (u) part of the nasal cavity, kg; M_l is the total mass of particles deposited on the surface of the lower (l) part of the nasal cavity, kg.

Computational experiments aimed to find the most suitable particle size. A series of computational experiments with varying input parameters consisted of 30 calculations. Based on the obtained results, response surfaces were constructed, which were used to determine the optimal values of the input parameters (spray angle, initial particle velocity, and average particle diameter) to achieve the maximum deposition area. During the construction and smoothing of the response surfaces, the data was interpolated based on the particle velocity values.

Results

Results of constructing the virtual geometry of the nasal cavity

The most interesting aspect of the study’s objective is the hydrodynamics of particle flight through the main nasal passages, including the ethmoidal labyrinth and sphenoid sinus. This is because particle deposition on the superior wall of the ethmoidal labyrinth, in close proximity to the fascia nerve, can accelerate targeted drug delivery. Furthermore, particle penetration into the maxillary and frontal sinuses during spraying is hampered by their small size and the specific location of their connections to the nasal cavity. Therefore, the virtual geometry of the nasal cavity was analyzed without considering these sinuses.

The resulting virtual geometry is characterized by increased curvature of the outer surface, the presence of numerous small details, and high complexity. The geometry contains numerous regions with small angles and small distances between adjacent faces. Some faces are excessively beveled and close to degeneracy. To achieve smoother geometry and reduce the number of small, insignificant details, it was decided to smooth it at various radii. The resulting mesh characteristics after smoothing are presented below (Table).

The obtained data demonstrates that using a smoothing radius of 0.5 mm results in a dense computational mesh, which will incur increased computational costs. This is particularly important when solving the final problem – calculating the hydrodynamics of particle spraying within the nasal cavity. It is also recognized that 76,000 cells are insufficient for an accurate calculation. Moreover, with a smoothing radius of 1.5 mm, a closer inspection reveals minor articulations between different geometric volumes, which is unacceptable for obtaining an adequate solution. Therefore, for further adaptation, a computational mesh generated using geometry with a smoothing radius of 1.0 mm was selected for computational experiments. The appearance of the final geometry and the computational mesh are shown in Figure 1.

Results of calculation of hydrodynamics of air movement

Below are some results from calculating the hydrodynamics of air movement in the developed geometry. These results include velocity vector distributions across longitudinal (Figure 2) sections of the nasal cavity. The figure shows the air velocity at each local point in the nasal cavity, from blue to red. The resulting velocity vector distributions can be used to identify areas with more active hydrodynamics and stagnant zones.

Within the ethmoid labyrinth and sphenoid sinus, air velocity is significantly lower than in other parts of the nasal cavity. This can hinder the effective penetration of particles into these spaces.

Flowlines within the nasal cavity are shown below (Figure 3). The color in the figure also reflects the velocity values. The flowlines confirm the previously obtained result: the velocity of the medium is greatest along the shortest airflow path from the entrances to the exits of the nasal cavity. These velocities are significantly lower in the surrounding spaces.

The calculation also yields pressure fields within the nasal cavity. The data obtained shows that the maximum vacuum occurs near the nasal cavity exit, and the minimum occurs at the entrance.

Results of a computational experiment assessing the spray of solid aerogel particles and its spray plume

Calculating the model equation yielded velocity and pressure fields, as well as particle trajectories at each instant. Figure 4 shows particle locations at specific instants of time, obtained from the calculated and experimental data. Experimental studies and simulation results showed that the spray patterns are similar, so Figure 4 shows one example of a spray pattern for the selected spray angle of 9° and initial velocity of 9 m/s. This figure allows for visual comparison. The color in the calculated data reflects particle size.

Based on the data obtained, it can be concluded that the particle sputtering observed in the computational and field experiments is similar. Uneven particle size distribution along the spray plume height is noted, which is likely due to the varying influence of the environment and gravity on particles of different sizes and, consequently, different masses. The data obtained confirm that conducting a computational experiment on particle sputtering will allow for a more detailed study of this process and the influence of various factors, particle characteristics, and their size distribution.

Сomputational experiment on the dispersion of aerogel particles in the nasal cavity

The results obtained during the computational experiments visually differ little from each other. Therefore, below, as an example, are the results of one of the calculations using a spray angle of 10°, a particle diameter of 100 µm, and an initial particle velocity of 5 m/s (Figure 5). These figures show the particle trajectories in the human nasal cavity at various points in time. The color in the figures indicates the particle diameter.

The obtained results demonstrate how particles emitted at a certain velocity from the dosing device are distributed throughout the nasal cavity. They also enter the upper nasal cavity. Over time, the particles come into contact with the walls of the geometry, and some of the particles, under the influence of gravity, fall back to the lower nasal cavity. According to the model equations, particles that contact the wall are considered to be captured by the mucosal surface of the nasal cavity. The obtained response surfaces are presented in the Supplementary Figures 1–3 to the article (supplementary materials on the journal website https://doi.org/10.47093/3034-4700.2026.3.1.50-63-annex). As a result, the values of the input parameters were obtained (the spray angle of the dosing device was 5°; the initial velocity of the particles was 6.95 m/s; the average diameter of the microparticles of chitosan aerogels was 100 µm), which made it possible to achieve the angle of the maximum proportion of microparticles deposited in the upper part of the nasal cavity, which was 14.67%.

Discussion

The present study demonstrates the feasibility of using the developed approach based on mathematical modeling to evaluate the deposition zones in the nasal cavity of highly porous chitosan aerogel particles, which act as drug delivery systems. The obtained data demonstrate how the proportion of particles deposited in the upper nasal cavity changes with varying spray parameters. It can be noted that in all cases, a decrease in spray angle leads to an increase in the proportion of particles. This is obvious, since the narrower the spray cone, the fewer particles will contact and be absorbed by the lateral walls of the nasal cavity before reaching the upper part. The effect of the initial particle velocity during spraying on the output parameter is not as obvious – a certain extreme can be observed, which is especially pronounced at small spray angles. Changes in average particle size are also not clearly reflected in the change in the proportion of particles deposited in the upper nasal cavity. The literature contains numerous studies evaluating the distribution of solid particles or droplets in the lungs or droplets in the nasal cavity [11][14]. However, modeling the flight and deposition of solid, highly porous particles is remains underrepresented in the literature. This study has expanded scientific knowledge in this area and identified avenues for further development of this research.

The obtained spray pattern calculation data clearly allows for the evaluation of spray patterns at any given time, depending on the physicochemical properties of solid particles, their size distribution, and the type of dosing device. The developed approach can be used to study the influence of the composition and characteristics of the dosing device on the resulting spray pattern and particle size distribution. The results support the characterization of drug prototypes during pharmaceutical development. The target delivery region depends on the active pharmaceutical ingredient used and the drug being developed [6][19]. For active pharmaceutical ingredients aimed at combating brain diseases, the target region could be the olfactory region, for example [8][18]. When using other active pharmaceutical ingredients, the goal may be to achieve an optimal deposition zone, for example, to ensure rapid onset of therapeutic effect with topical application [6]. Mathematical modeling can also be applied in the design of devices for delivering dry powders into the nasal cavity [14][20]. While research and patents in this area are currently available, few commercially available devices are available on the market. This further expands the possibilities for future research in this area.

The optimal parameters found are applicable to the developed nasal cavity geometry, which may be a limitation of the developed approach. Future experiments could explore similar cavity geometries with different geometric dimensions and anatomical features, or develop a standardized nasal cavity geometry. This would allow the influence of geometry on particle distribution in the nasal cavity and identify regions of maximum deposition for further evaluation of the developed approach.

Additional limitations include the complexity of implementation and the required computational power, which, despite its drawbacks, may still be cheaper and faster than conducting a series of experimental studies. During the experimental studies, the comparison of experimental and calculated data was performed visually because limitations in the available video-recording equipment. We believe that these limitations do not diminish the validity of the findings but rather define directions for future studies aimed at improving model accuracy. This research may include analytical equipment, such as SprayView^® (Proveris Scientific, USA) and SprayTec^® (Malvern Panalytical, UK) to obtain spray patterns and particle size distributions [14].

Conclusion

The obtained results allow us to evaluate the distribution of particles (precise coordinates and the proportion of deposited microparticles) within the nasal cavity. A computational experiment using a model for predicting spray hydrodynamics with various initial parameters of the sprayed material and spray device will allow us to determine the distribution of particles across the inner surface of the nasal cavity. These data will be used as initial conditions for calculating the release kinetics or pharmacokinetics. Combining these approaches will enable reverse engineering for efficient nasal drug delivery.

The proposed approach can be used to select spray parameters for solid drugs in the nasal cavity. It can reduce experimental work, facilitate the early identification of suitable delivery devices, and accelerate the selection of their characteristics. Future research in this area may include developing a standardized nasal cavity model based on ethnicity, gender, and age; expanding the range of study subjects depending on the powdered drugs being developed; and design devices for targeted dry-powder delivery. All of these studies will improve pharmaceutical development in the BRICS countries and globally.