INCLUSION OF PARTICLE DISPERSION IN ARAT PARTICLE DEPOSITION: MODEL IMPROVEMENT AND VALIDATION

Abstract

Deterministic models have been developed for the predictions of the deposited dose to the respiratory tract from inhalation of airborne materials. Complexity of the lung geometry, ventilation mechanics, and transport processes have required model assumptions and simplifications. Model validation is an integral part of the development process before models can be applied to specific scenarios of interest. While several validation efforts have been reported in the literature for regional deposition in the respiratory tract, there is a desire to refine the models to enhance the power of predictions to smaller regions such as per lobe and generation of the respiratory tract. Thisimperative is contingent on the availability of data. Hence, we refined and validated our lung deposition model developed in rodents by adding additional physical mechanisms missing in the original model. Convective mixing of particles impacts both the distribution and deposited dose of inhaled particles. We developed a semi-empirical mixing model for particle exchange in the pulmonary region between the respiratory ducts and alveoli with parameters determined by fitting the model with recent measurements by Lin et al. (2023). The refined model was used to predict lobar, regional, and site -specific deposition of inhaledtrace metals in the puff of regular and mentholated little cigars. Model refinements yielded increased deposition throughout the lung and matched deposition predictions more closely with measurements. The refined deposition model can be used to study the risk from inhalation of tobacco products as well as environment particles.

INTRODUCTION

The advent of electronic cigarettes in the past two decades has presented fresh challenges for health agencies and regulatory authorities such as the Federal Drug and Administration. Entrusted with safeguarding public health, the agency must evaluate the effects of both existing and newly introduced products and establish appropriate regulatory measures. Assessing the health implications of these devices necessitates studies grounded in safety data gathered from both short- and long-term use. Furthermore, animal studies may provide additional valuable insights. As data on the long-term effects continues to be collected, deposition modeling, in conjunction with information on the toxicity of individual constituents, offers a preliminary yet effective means of evaluating product safety in a swift, efficient, and economical manner.

Over the years, various mathematically driven models, grounded in mechanical principles, have been devised to predict the behavior of inhaled substances within the respiratory systems of both humans and several laboratory animals (Anjilvel and Asgharian, 1995; Asgharian et al. 2001; Egan and Nixon, 1985; Koblinger and Hofmann 1985, 1990; Taulbee and Yu 1975; Yu, 1978, among others). Inhalation dosimetry models serve to link exposure concentrations to biological effects, thereby establishing exposure-dose-response relationships.Existing dosimetry models have been validated for insoluble particles by comparing predicted deposition patterns with in vivo measurements in humans (Asgharian et al. 2006) at broader regional levels, such as the head, tracheobronchial (TB) airways, and pulmonary (PUL) airways. While these models allow for comparisons at finer scales, such as site-specific and generation-by-generation levels, their ability to provide detailed assessments is constrained by the simplifications inherent in the formulation of one-dimensional (1D) models (Asgharian et al. 2001). The omission of certain mechanisms and processes in current 1D models may introduce uncertainty in the prediction of site-specific deposition patterns. Therefore, to facilitate more precise comparisons between model predictions and site-specific experimental measurements, it is essential to identify and address these missing mechanisms, refining the model formulation to enhance its accuracy.

The one-dimensional (1D) formalism of deposition models is founded on the principles of convective and diffusive transport of inhaled materials throughout the airways of the respiratory tract. However, several key mechanisms and processes are inherently absent from the 1D model framework, largely due to the complex structure of lung airflow. These include the development of airflow within individual airways, the emergence of secondary airflows as air passes from parent to daughter branches, and additional effects related to the airflow. Such effects alter the distribution of inhaled particles within the lung, a phenomenon commonly referred to as particle mixing by convective flow.Although mixing occurs throughout the respiratory system, its most significant impact is believed to take place in the pulmonary region (Altshuler et al. 1959). When a bolus of particles is inhaled, its concentration profile becomes dispersed upon exhalation (Darquenneet al. 1999; Heyderet al. 1988; Park and Waxler 2007; Rosenthalet al. 1992; Schulz et al. 1994; Verbancket al. 2001A, 2001B; Taulbee et al 1978). This phenomenon, referred to as particle dispersion or mixing within the lung, is chiefly convective in nature. However, it has frequently been described as a diffusive process and expressed by semi-empirical formulations, which are inherently constrained by the range of the data from which they were developed.

The phenomenon of particle dispersion arising from convective mixing has been incorporated into human lung deposition models for both tidal breathing (Asgharian et al. 2023) and bolus inhalation (Asgharian et al. 2025). However, the aerosolized puff produced by an electronic cigarette comprises a complex mixture of constituents present in both particulate and gaseous states. Deposition models originally designed for insoluble particles have been adapted to account for aerosols and vapors, their behavior governed by the physicochemical properties of the individual components within the mixture (Martonen 1992; Muller et al. 1990).Such mixture models have been employed to predict the fate of a puff of smoke, a transient, multi-component system that undergoes continuous droplet-vapor transformation as it traverses the pulmonary airways (Robinson & Yu 2001; Broday and Robinson,2003; Asgharian et al. 2013, 2018A, B). Yet, for these models to yield accurate and site-specific deposition predictions, they must incorporate the effects of particle dispersion, thereby ensuring a more precise representation of the underlying physical processes.

Before deposition models for electronic cigarettes can be employed in risk and safety assessments, they must first undergo rigorous validation. Direct verification of these models in humans is impractical; thus, validation must instead be conducted in laboratory animals. However, the inherent complexity of the puff mixture presents a significant challenge in verifying these models within such experimental settings.To circumvent the difficulties posed by the unstable dynamics of the puff, trace metals found in the emissions of little cigars -representative of the droplet components within the mixture - may serve as a reliable basis for model validation. Lin et al. (2023) conducted a study measuring the lobar deposition of smoke particles from both regular and mentholated little cigars in twelve-week-old Sprague Dawley rats. In response to the two modeling limitations previously discussed - namely, the need for model refinement and validation - we have utilized this dataset to construct the dispersion component of the deposition model in SD rats. By modeling this process through a semi-empirical approach and determining the missing parameters through the calibration of model predictions against observed measurements, we endeavor to enhance the accuracy and applicability of the deposition model.

In the present study, we refined the deposition model of Miller et al. (2014) by incorporating particle dispersion within the pulmonary region, thereby achieving a more precise representation of particle deposition and distribution in this critical area. Adopting the methodological framework established by Asgharian et al. (2025), the dispersion model accounted for the convective exchange of particles between respiratory ducts and alveoli during both inhalation and exhalation.The model’s parameters were determined through a comparative analysis of its predictions against recent deposition measurements of trace metals within the puff emissions of little cigars, as reported by Lin et al. (2023). Upon validation, the enhanced model was subsequently employed to predict the fate of inhaled particles across various lobes and regions of the rat lung, thus offering a more comprehensive and accurate depiction of particulate deposition.

MATERIALS AND METHODS

Lin et al. (2023) examined the lobar deposition of smoke particles from combusted regular and mentholated Swisher Sweets little cigars in Sprague-Dawley rats. The study involved twelve-week-old male and female rats, with mean body weights of $332\pm 11g$ and $221\pm 13g$, respectively. Rats in the exposure group ($n=5$) were exposed for 2 hours to $84\pm 5mg/m^3$ of smoke particles within a whole-body exposure chamber. Aerosols were generated by a custom-designed smoking machine, which delivered a $50mL$, 3-second puff every 60 seconds. The particle size distribution was measured using a Scanning Mobility Particle Sizer (SMPS), revealing a bimodal size distribution with peaks at $359-372nm$ and $122-146nm$. Deposition fractions were determined by the increase in chromium concentrations within lung tissues, analyzed through Inductively Coupled Plasma Mass Spectrometry (ICP-MS).

The deposition of inhaled particles in the five-lobe lung geometry is modeled using a one-dimensional (1D) formulation, based on the mass balance within each airway of the lung. This balance accounts for particles that enter and exit the airway, as well as those initially present and those remaining after the particles have traveled past the airway. Miller et al. (2014) extended the 1D deposition model originally developed for humans (Asgharian et al. 2001) to Sprague Dawley (SD) rats by replacing the human respiratory tract geometry with that of the SD rat lung. The sink terms in the model are species-independent and are formulated to represent particle deposition in the lung airways, considering various deposition mechanisms (e.g., sedimentation, inertial impaction, and Brownian diffusion) based on fully developed flows. The base 1D model does not account for particle dispersion arising from physical mechanisms that cannot be captured within the 1D framework. Asgharian et al. (2023, 2025) describe modifications to this base model, incorporating dispersion as a physics-based phenomenon arising from various mechanisms, particularly particle exchange between the respiratory ducts and alveoli. This approach contrasts with earlier treatments in the literature, which attributed dispersion solely to diffusion - a concept that is unrealistic. In the proposed model, dispersion is linked to a parameter, α (equations 3 and 5 below), which cannot be determined analytically due to the complexities of flow dynamics between the ducts and alveoli, along with other contributing physical mechanisms. The current study uses data collected by Lin et al. (2004) to estimate α and refine model predictions by fitting the data to the dispersion model.

Dispersion in the TB region has been accounted for by apparent diffusivity, which combines with the molecular diffusion in the mass balance (advection-diffusion) transport equation (Scherer et al. 1975; Lee et al. 2000; Lee and Lee 2001). It is noted that all variables used in the following equations are in CGS (centimeter, gram, second) units. Particle transport model based on mass

$$\left\{\begin{array}{c}\frac{\partial C_p\left(z,t\right)}{\partial t}+\frac{q_d}{A}\frac{\partial C_p\left(z,t\right)}{\partial Z}=\left(D_m+D_a\right)\frac{{\partial }^2{C}_p\left(z,t\right)}{\partial z^2}-{\lambda }_p{C}_p\left(z,t\right)\\ t=0C_p=C_p\left(z,0\right)\\ z=0C_p=C_p\left(0,t\right)\\ z=L_{TB}{C}_p={\left.C_p\left(L_{TB},t\right)\right|}_{TB}\end{array}\right.$$ (1)

where $C_p$ is the average particle concentration in the duct, ${\left.C_p\left(L_{TB},t\right)\right|}_{TB}$ is the concentration of particles at the distal end of terminal bronchioles, $q_d$ is the flow rate through the duct, $L_{TB}$ is the length of TB airways from the trachea to the distal end of terminal bronchioles, $D_m$ and $D_a$ are the molecular and apparent diffusion coefficients, respectively, $A$ is the cross-sectional area of an airway, $z$ is the axial position, $t$ is the time, and ${\lambda }_p{C}_p$ is the number of particles depositing on the walls of the duct per unit volume of the duct per unit time. Expressions for ${\lambda }_p$ were previously developed and reported (Anjilvel &Asgharian, 1995).

Particles that enter the pulmonary region mix irreversibly by convection with the initial air in the alveoli. As a result, particles’ return path during exhalation will differ from that during inhalation. Convective mixing of the particles has been recognized as the dominant mechanism (Taulbee et al. 1978), and thus, is included in the PUL region. The transport equation for particles accounting for convection, diffusion, and dispersion during inhalation is given by the following equation:

$$\left\{\begin{array}{c}\frac{\partial C_p\left(z,t\right)}{\partial t}+\frac{q_d}{A}\frac{\partial C_p\left(z,t\right)}{\partial Z}=D_m\frac{{\partial }^2{C}_p\left(z,t\right)}{\partial z^2}-\left(\frac{q_a}{L·A}+{\lambda }_p\right)C_p\left(z,t\right)\\ t=0C_p=C_p\left(z,0\right)\\ z=L_{TB}{C}_p={\left.C_p\left(L_{TB},t\right)\right|}_{tb}\\ z=L_{tot}\frac{\partial C_p}{\partial z}=0\end{array}\right.$$ (2)

where $q_a$ is the radial air flow rate from the duct to the alveolar space, and $L_{tot}$ is the total length of the lung, extending from the trachea to the end of the alveolar sacs. The term $\frac{q_a}{L·A}{C}_p$ accounts for the mass of particles in the duct that enters the alveoli. It is the average mass flow per unit duct volume. Particle transport in the alveolar space of an airway during inhalation is described by

$$\left\{\begin{array}{c}\frac{d{C}_a}{dt}+\left(\frac{q_a}{V_{al}}+{\lambda }_a\right)C_a=\frac{q_a}{V_{al}}{C}_p\\ C_a=C_a\left(z,0\right)\end{array}\right.$$ (3)

in which $C_a$ is the average particle concentration in the alveoli, $V_{al}$ is the volume of the alveoli in each respiratory duct, and ${\lambda }_a{C}_a$ is the number of particles lost to the walls in the alveoli per unit volume of the alveoli per unit time.

There is no particle dispersion during pause due to the absence of flow convection. The transport equation in the respiratory ducts during exhalation is given by

$$\left\{\begin{array}{c}\frac{\partial C_p\left(z,t\right)}{\partial t}+\frac{q_d}{A_d}\frac{\partial C_p\left(z,t\right)}{\partial z}=D_m\frac{{\partial }^2{C}_p\left(z,t\right)}{\partial z^2}+\frac{q_a}{L·A}{C}_a\left(z,t\right)-{\lambda }_p{C}_p\left(z,t\right)\\ t=t_{inh}+t_{pau}{C}_p=C_p\left(z,t_{inh}+t_{pau}\right)\\ z=0\frac{\partial C_p}{\partial z}=0\\ z=L_{tot}\frac{\partial C_p}{\partial z}=0\end{array}\right.$$ (4)

where $t_{inh}$ and $t_{pau}$ are the inhalation and pause times, respectively. The sign difference between equations (2) and (4) reflects that particles flow out of and into the duct during inhalation (minus sign) and exhalation (plus sign), respectively. Particle dispersion in the alveolar region occurs due to the irreversibility of particle travel path during inhalation and exhalation. Particles in the alveoli at the end of inhalation may be slow to return or be partially trapped in the alveoli during exhalation. We let $\alpha$ denote the fraction of particles leaving the alveolar region and entering the duct per unit time. It indicates the degree of mixing. All particles return when $\alpha =1$ and none return when $\alpha =0$. Particle transport in the contracting alveolar space during exhalation is described by

$$\left\{\begin{array}{c}\frac{d{C}_a}{dt}+\left(\alpha \frac{q_a}{V_{al}}+{\lambda }_a\right)C_a=\frac{q_a}{V_{al}}{C}_a\\ C_a=C_a\left(z,t_{inh}\right)\end{array}\right.$$ (5)

The parameter $\alpha$ introduces a correction factor to enhance particle exchange in each lung lobe, potentially accounting for some asynchronous filling and emptying of the lobes. It is determined by varying it in the model until predictions and measurements are matched.It is noted that the model defined by equations (3) – (5) in this work is identical to that presented in Asgharian et al. (2022). However, in Asgharian et al. (2022), all particles within the alveoli are returned to the respiratory tract (i.e., $\alpha =1$ in equation 5), whereas in the present manuscript, only a fraction of the particles is returned, as defined by $\alpha$ in equation 5.

RESULTS AND DISCUSSION

The same lung geometry as in the base 1D model (Miller et al. 2014) was used to predict the deposition of trace metal particles from both regular and mentholated cigars in male and female Sprague Dawley (SD) rats. Lung and breathing parameters were determined using allometric expressions derived by Miller et al. (2014), based on the animals’ weight as reported in Table I of Lin et al. (2023). While the lung structure was identical in both sexes, airway dimensions differed due to scaling with body weight (and consequently lung size), which varied between male and female rats, as noted in Lin et al. (2023).

Lin et al. (2023) characterized their experimental aerosols as bimodal, with a primary peak ranging from 359 to 372 nm and a secondary peak between 122 and 146 nm. The exposure concentrations remained nearly constant between 100 and 400 nm for both aerosol types (regular and mentholated) and for both sexes. Given that the mass for the first mode was significantly less than that of the second due to its smaller diameter, we used a mean particle diameter of 359 nm, a geometric standard deviation of 1.79, and a particle mass density of 1.18 g/cm³ (as reported in Table I of Lin et al., 2023) for all scenarios in our calculations.

Breathing parameters and particle information, as presented in Table I of Lin et al. (2023), are mirrored here. This article utilizes the experimental conditions and results from Lin et al. (2023), which may give the impression that the two studiesareintertwined. However, the inclusion of Table I in this study, along with the lobar deposition measurements, allows readers to examine each article independently.

Table I.

SD rat weight, lung and breathing parameters, and particle properties used in deposition predictions provided by Lin et al. (2023).

Parameter	Male	Female
Body weight (g)	332	221
Functional residual capacity (mL)	3.69	3.02
URT volume (mL)	0.44	0.34
Particle density (g/cm3)	1.18	1.18
Particle diameter (nm)	359	359
Breathing frequency (1/min)	113	125
Tidal volume (mL)	2.32	1.62
Inspiratory fraction	0.5	0.5
Pause fraction	0	0

Model refinement.

The dispersion model embedded in the transport equations for the TB and PUL regions as described by equations (1)–(5) was used to improve deposition predictions in lobar-specific rat lung geometry. Parameter $\alpha$ in equation(5), which denoted the degree of mixing,had to be different in each lobe due to differences inlung geometry and flow distribution. The particle deposition fractions were calculated independently for each lobe, for various values of α ranging from zero to one. The predicted deposition fraction for each α value was then compared with the corresponding measurements reported by Lin et al. (2023). The value of α that resulted in the best agreement between the predictions and the measurements was chosen as the appropriate α value for that lobe. Given that mixing in one lobe occurred independent of that in other lobes as evident by equation (5), we conducted the parametric study for $\alpha$ one lobe at a time to significantly reduce the number of runs. The value of $\alpha$ in each lobe that made a match between prediction and measurement of deposition fraction denoted the degree of mixing in that lobe and thus was selected for that lobe.

The development of the dispersion model involved certain simplifications, assumptions, and approximations, which introduced uncertainties in both its mechanics and predictions. As noted by Lin et al. (2023), one source of uncertainty may stem from variations in breathing parameters. Furthermore, breathing parameters were determined based on the average age of the animal using an allometric equation, adding another layer of uncertainty. These factors, along with other approximations, may impact the accuracy of the model’s predictions. Thus, our findings for the parameter $\alpha$ should be considered an initial estimate, with future studies aimed at refining these values through the integration of additional mechanisms and processes informed by new experimental data.

Figure 1 shows the predicted and measured deposition fraction of particles in different lobes of male and female Sprague-Dawley (SD)rat lungs for regular and mentholated little cigars. The panels are as follows:1A - left lung (LL), 1B - right cranial (RCr), 1C - right middle (RM); 1D - right caudal (RCa), and1E - right accessory (RA). The positions of the lobes were also described in relation to the lung entrance. Specifically, the cranial lobe was considered the proximal lobe, while the caudal and accessory lobes were categorized as distal lobes. In theory, values for measurements and predictions should match. In reality, differences in particle size distribution, distribution of trace metals over the particles size distribution,and lung structure between males and females as well as inter-subject variability regarding lung geometry caused some variation even among the measurements. In addition, we used a mean particle diameter (i.e., monodisperse) in our predictions when particles were polydisperse. These uncertaintiescontributed to differences in deposition fractions between lab measurements and model predictions. As a first order approximation, we determined an $\alpha$ value in each lobe in male and female rats for regular and mentholated little cigars. The negative measurements in Figures 1B and 1C were excluded from the analysis and are not displayed in the figures. Additionally, the measurements and predictions in the right middle lobe for female rats intersected at α > 1, which is not physically possible, and that data point was also excluded from the analysis. The values of α for male and female rats exposed to regular and mentholated little cigars were averaged and presented in Table II to provide an overall value for each lobe. Table II lists $\alpha$value and volume for each lobe. The degree of mixing in each lobe could be assessed by the value of $\alpha .$ The lower the value, the higher mixing would be. Greater mixing was observed in the more distal lobes (RCa and RA) in comparison to the proximal lobe (RCr) and the middle lobe, which exhibited no mixing. The $\alpha$ values were 0.75 and 0.69 for the distal lobes, versus 0.83 and 1 for the proximal and middle lobes, respectively.We also aimed to explore the relationship between lobar volume and the degree of mixing, but no such correlation was found. Therefore, it is likely that mixing is influenced by a range of interconnected factors, including lobar volume and flow rate, airway shape and resistance, alveolar movement due to compliance, and particle size characteristics. Consequently, a thorough three-dimensional analysis is essential to identify the underlying mechanisms of mixing and to develop a physics-based analytical model.

Figure 1.

Predicted and measured deposition fraction of particles in lung lobes of the Sprague Dawley rat ((1A: left lung; 1B: right cranial; 1C: right middle; 1S: right caudal; 1E: right accessory).

Table II.

Values of parameter of the dispersion model for different lung lobes of Sprague Dawley rats.

Lung lobe	α value	Lobar volume (cm3)
LL	0.93	1.57
Right cranial	0.83	0.71
Right middle	1	0.73
Right caudal	0.75	1.46
Right accessory	0.69	0.38

The trend in the right lobes was partly correlated to the lobar volume and airflow into the lobe as shown in Table II. Distal lobeshad the lowest volumes and thus had more mixing. In contrast, right middle lobe, which was the second largest lobe, experienced the least or no mixing.

Model predictions.

Values of $\alpha$ ranging from 0 to 1 were inserted in equation (5) foreach rat lung lobe to predict deposition fractions with and without inclusion of dispersion. Predictions were then compared with measurements of Lin et al. (2023) in male and female SD rats.Comparisons for males and females are givenin Figures 2A and 2B, respectively.Inclusion of the dispersion model improved predictions and gave a better agreement with measurements. Because of differences in reported measurements for regular and mentholated little cigars, predicted deposition fraction using averaged $\alpha$ value should lie between these two measurements, which was the case in all lobes in male SD rats except for the non-mixing ($\alpha =1$), and right accessory lobe (Figure 2A). Similar findings were observed when predictions were compared with measurements in female SD rats (Figure 2B). Predicted deposition fraction increased when dispersion was included to reach closer to measurements. There was uncertainty regarding the $\alpha$ value in the right caudal lobe. Predicted value of deposition fraction fell short of reported measurements.

Figure 2.

Comparison of predicted lobar deposition fractions of inhaled regular and mentholated cigars with measurements of Lin et al. (2023) in male SD rats. A. Male rats and B. female rats.

Addition of the dispersion mechanisms changed deposition predictions significantly. Figure 3 gives regional deposition of $1\mu m$ particles in the lung of a 300g SD rat. Figure 3A is the predicted deposition fraction fordifferent lobes within the rat TB regionwith and without inclusion of particle dispersion in the deposition model. Particle deposition is directly linked to the residence time of particles in the lung airways. Ingham (1975) and Pich (1972) developed models for particle deposition in the airways by Brownian diffusion and gravitational settling, respectively. Model predictions indicated that deposition was dependent on the residence time of particles in the airways. The trapping of particles in the alveoli, caused by dispersion and mixing, delays their exit, thereby increasing their residence time. Consequently, increased deposition is anticipated as a result of particle mixing, as illustrated in Figure 3, where predictions are compared for both scenarios - with and without mixing. Significant increase in deposition was observed in LL and RCa lobes and a modest increase in RCa and RA when dispersion was included. Little or no mixing occurred in the RM lobe; thus, no change occurred in the particle deposition. The largest increase in depositionfor the lobes within the PUL regionoccurred in RCa lobe while there was a noticeable increase in other lobes as shown in Figure 3B. LRT (lung)deposition increased significantly in LL and RCa (Figure 4C) with slight increase in other lobes.

Figure 3.

Comparison of predicted regional deposition fractions of inhaled regular and mentholated cigars with measurements of Lin et al. (2023) in female SD rats. A. TB region, B. PUL region, and C. LRT region.

Figure 4.

Predicted deposition fraction of inhaled particles in the SD rat lung lobes as function of lung airway generation number (lung depth) with and without presence of particle dispersion. A. Left lung, B. Right lung, and C. All lobes.

Particle dispersion impacted both deposition and distribution of deposited particles in the lung. To inspect the distribution in the lung, deposition fraction was calculated as a function of airway generation number in the left lung (Figure 4A), right lung (Figure 4B), and total lobar deposition (Figure 4C) in different airway generations. Particle deposition was increased in all airway generations when dispersion mechanism was included in the overall deposition model. The increase was significant in the first airway generations of the left lung, increased moderately in the remaining generations in the TB region, and showed a large increase in the PUL region (Figure 4A). The trend was similar in the right lung, but the degree of increase was large for all airway generations except for the first few large airways of the lung (Figure 4B). The depth of particle deposition increased for the right lobe compared with that for the left lobe (generation 21 versus generation 18), which was mainly due to differences in lobar geometry and airflow transport. The peak deposition occurred in the same airway generation with or without dispersion in both the left and right lobes. Total lobar deposition increased significantly in all airway generations when dispersion mechanism was accounted for (Figure 4C).

Figures 2 – 4 showed the deposition trend for $1\mu m$ particle diameters. To extend to the entire respirable size range, we ran simulations from $0.01\mu m$ to $5\mu m$. Results for all 5 lobes of the SD rats are given in Figure 5 for both cases of with and without particle dispersion. The deposition pattern in each lobe showed two peaks: one at diameters smaller than $0.1\mu m$ and the other for particle diameters larger than $1\mu m$. This pattern–which is consistent among all species -was related to transition in deposition mechanism from Brownian diffusion to gravitational settling and inertial impaction when particle size increased from nano-sized to micro-sized. Left lung and right caudal gave the highest deposition followed by rightcranial. Least deposition was observed in the right accessoryand right middle lobes. Deposition ranking was related to the lung size, state of the lung distension, and amount of inhaled air flowing into the lobe. In addition, deposition was increased for all particles diameters when particle dispersion (mixing) was included. The increase depended on the particle size: small for particles approaching the limits of $0.01\mu m$ and $5\mu m$ particle diameters. The greatest increase was seen between $0.1\mu m$ and $1\mu m$ size particles where deposition fraction was least. Hence the revised model with inclusion of particle dispersion improved the existing model and thus is more reliable for deposition predictions at lobar, regional and airway generation levels. While not attempted here, the deposition model can also be extended to polydisperse particles for which measurements were based on. However, model run times will increase significantly. If impractical due to limited computational resource, calculations for the particle monodisperse scenario based on mean particle diameter can replace that for polydisperse particles.

Figure 5.

Deposition fraction indifferent lung lobes of SD rats for the respirable particle sizes.

CONCLUDING REMARKS

Commonly developed 1D deposition models to predict the deposited dose of inhaled materials, while they have been shown to give good predictions for regional deposition, have limitations in predicting deposition at finer levels such as in the lobes or sites of the respiratory tract. Limitations are due to exclusion of other mechanisms and processes that do not directly appear in the transport model in 1D modeling approaches. To improve and refine, we included in the deposition model, particle dispersion occurring in the TB and PUL regions of the lung. As a result, deposition predictions increased and matched better with available measurements in the literature (Lin et al. 2023). Model simulations showed that dispersion by convective mixing was different among the 5 lobes of SD rats. Most mixing was observed in the right caudal lobe where more airflow into the lobe was observed. There was little or no mixing in the right middle lobe.The improved deposition model can aid in various applications such as drug delivery and risk assessment from exposure to hazardous materials.

DATA AVAILABILITY STATEMENT

Data generated from the study is available on request.