The respiratory system is one of the most exposed organ systems of the human body (West, 2012). It has a complex anatomy which can be broadly subdivided into two regions - the upper respiratory tract (URT) and the lower respiratory tract (LRT). The URT comprises the nose, paranasal sinuses, mouth, nasal cavity, pharynx and larynx. The LRT can be further demarcated into the tracheobronchial tree and the alveolar region. The tracheobronchial tree (also called upper or conducting airways) comprises the trachea and the dichotomous bifurcating bronchial structure (see Figures 1a,b), exhibiting self-similar fractal properties. The alveolar region consists of the terminal airways and the alveoli. The bronchial network, along with the alveolar region, is commonly referred to as the lung. The alveolar region is sometimes also termed the deep lung due to its terminal location (West, 2012; Weibel et al., 1963). Pertinent information on lung anatomy is summarized in Table 1.

The large surface area of the lung enables easier respiration through a mechanism which draws in ambient air during inhalation and releases expired air during exhalation. However, it also makes the pulmonary system (and in turn, other internal organs of the body) vulnerable to health hazards from polluting particles and infected droplets, among others, which may enter the respiratory tract during inhalation along with the ambient air. Air, along with suspended particles and droplets, is inhaled into the respiratory tract through the URT, which are transported deeper into the LRT through the conducting airways to reach the deep lung (Pedley, 1977). Gas exchange with the blood stream takes place in the deep lung. The reverse happens during exhalation. While some of the inhaled particles/droplets may get exhaled out, a considerable fraction of these may get deposited in the airway and alveoli as a result of various forces acting on them (Hofmann, 2011). Their deposition takes place in the respiratory mucosa - a thin layer of mucosal fluid that separates the airway lumen from the epithelial tissue (Mauroy et al., 2011; Karamaoun et al., 2018; Chakravarty et al., 2022). The thickness of the mucosal layer is O ( 50 ) μ m in the trachea and reduces as one moves deeper becoming negligible in the terminal airways (Mauroy et al., 2011). The mucosal fluid is also subjected to periodic ciliary beating which results in advective transport of the fluid towards the trachea. The deposited particles/droplets are, as such, transported along with the mucosal fluid, in addition to diffusion, resulting in a self-clearance mechanism (Mauroy et al., 2011) Pulmonary drug delivery systems also rely on this mechanism to deliver aerosolised medicinal drugs to the lung in a non-invasive manner (He et al., 2022).

A large body of literature exists detailing the extensive investigations that have been carried out on gas exchange dynamics as well as particle/droplet transport and deposition within the respiratory tract (Pedley, 1977; Hofmann, 2011; Guha, 2008; Kleinstreuer and Zhang, 2010). Nonetheless, certain aspects - such as the dynamics of viral respiratory infections - still require further investigations (Neelakantan et al., 2022). Infected virus-laden droplets are the main source for transmission of viral respiratory infections such as SARS-CoV-2 (Wiersinga et al., 2020; Harrison et al., 2020; Mittal et al., 2020; Wang et al., 2021; Leung, 2021). These droplets are formed within the respiratory tract of an infected individual (Abkarian et al., 2020; Darquenne et al., 2022a; Morawska et al., 2022) and are subsequently exhaled out during breathing, coughing, sneezing, talking, etc. This leads to formation of turbulent clouds of air containing suspended virus-loaded droplets (Bourouiba, 2021). These droplets may be inhaled by other individuals causing the inhaled droplets to deposit in the respiratory mucosa releasing the entrained viruses and starting an infection in the new subject (Wiersinga et al., 2020; Grant et al., 2021). A complete spectrum of respiratory virus transmission, thus, includes the following components: a) droplet formation from infected respiratory mucosa inside the respiratory tract, b) external transmission of infected droplets through respiratory motions and the effect of preventive measures (such as masks) on such transmission, and c) internal transmission of the inhaled infected droplets within the respiratory tract. A thorough understanding of all of these components is, as such, required in order to fully comprehend the fluid dynamics of respiratory virus transmission.

Extensive studies have been carried out on external transmission of droplets (Abkarian and Stone, 2020; Bourouiba, 2021), especially since the beginning of the SARS-CoV-2 pandemic. These studies have analysed different physical settings with respect to transmission mechanisms, environmental factors, and physical configurations with a focus on identifying the risk of infection spread and possible preventive measures (Pal et al., 2023; Biswas et al., 2022). While there has been evidence of increased number of aerosols exhaled during SARS-CoV-2 infection (Edwards et al., 2021), relatively fewer studies have dealt with droplet formation from respiratory mucosa inside the respiratory tract (Hamed et al., 2020; Pairetti et al., 2021; Anzai et al., 2022; Haslbeck et al., 2010; Kant et al., 2023; Morawska et al., 2022; Saha et al., 2024). These droplets, once formed, may then be exhaled out or internally transmitted within the respiratory tract (Chakravarty et al., 2023; Anzai et al., 2022). Understanding the mechanism of formation of these droplets from respiratory mucosa is critical to identifying the conditions which aid droplet formation and hence, increase the risk of infection transmission - externally as well as internally. Internal transmission of infected droplets within the respiratory tract is also hypothesized to be a plausible mechanism by which a viral infection may spread to the inner regions of the respiratory tract, including the alveolar region (deep lung), where it may cause serious diseases such as pneumonia, acute respiratory distress syndrome etc. (Grant et al., 2021).

As discussed previously, many studies have been carried out on internal droplet transmission and deposition in the respiratory tract. These include analytical, computational as well as limited experimental studies (in vivo, in vitro as well as ex vitro) (Morawska et al., 2022). However, there are not many studies that have taken into account mucosal flow dynamics coupled with pathogen-specific effects, along with internal droplet transmission, while distinguishing between various types of infections and their spread (Quirouette et al., 2020). While discrete studies (experimental as well as computational) on mucus dynamics and infection kinetics are available (Baccam et al., 2006; Mauroy et al., 2011; Banerji et al., 2023; Looney et al., 2011; Ravi et al., 2025), the coupled effects need to be investigated to fully characterize the underlying phenomena. Experimental investigations of such coupled phenomena is prohibitively difficult due to various reasons (Morawska et al., 2022) necessitating reliance on computational studies. This perspective focuses specifically on this aspect–fluid dynamics of internal droplet transmission within the respiratory tract in addition to gas exchange, coupled with mucus flow dynamics and pathogen-specific effects. One dimensional (1D) models are surveyed due to their ease of use and efficacy in providing useful clinically relevant mechanistic insights.

Different computational techniques have been used in the past to model droplet transport and deposition in the respiratory tract (Hofmann, 2011; Guha, 2008; Koullapis et al., 2019; Neelakantan et al., 2022). Geometrical complexity of the respiratory tract, however, makes it difficult (if not impossible) to carry out high-resolution computational analyses considering the complete lung geometry. High-resolution investigations have, therefore, been carried out separately targeting specific truncated regions of the respiratory tract - the upper bronchial region (Koullapis et al, 2016; Basu, 2021), the central conducting airways (Koullapis et al., 2018b; Ghorui et al., 2024) or the terminal alveolar region (Chakravarty et al, 2019; Sznitman, 2013; Fishler et al., 2015) - with appropriate boundary conditions. Some studies have also considered combinations of these regions in their analyses by employing different coupling techniques (Koullapis et al., 2018a; Koullapis et al., 2020; Kuprat et al., 2021; Koullapis et al., 2023). Several studies have also been carried out on droplet transport and deposition in the URT (Basu et al., 2020; Basu, 2021). These studies are useful in modelling geometrical complexities of the respiratory tract (often using high-resolution data obtained from computed-tomography scans (Tawhai et al., 2004)) and provide useful information regarding the local fluid dynamics as well as insights into the mechanism of droplet deposition in the modelled region (Basu et al., 2020; Basu, 2021). Detailed information of droplet transport and deposition for the complete respiratory tract is expensive and difficult using this approach, although some advances has been made in this direction recently through development of computed tomography-based whole lung deposition models (Zhang et al., 2022) and use of self-similar bifurcating units, based on lung fractal geometry Ghorui et al. (2024). Simplified models of the respiratory tract are, therefore, useful when the goal is to capture the key trends of droplet transport and deposition for the complete respiratory tract. In this context, one-path lung models–which considers a single, representative airway path from the trachea to the alveoli–have been utilised. This concept originates from the “Typical Path Lung Model” introduced by Yeh and Schum (1980). Similar one-path models of the lung have been recently utilized for extensively studying flow perturbations through direct numerical simulations (Atzori et al., 2025) as well as for investigation of aerosol flow and deposition through experiments (Möller et al., 2021).

A different class of simplified models are based on morphometry of the respiratory tract (Weibel et al., 1963; Raabe, 1976) and can be broadly classified into semi-empirical regional compartment models and one-dimensional trumpet models (Hofmann, 2011). While these simplified models cannot account for the effects of heterogeneity in the lung, these are extremely useful for understanding the fundamental mechanisms and capturing key trends of aerosol transport and deposition for the whole lung. The compartment models assume that the lung morphology is comprised of four different compartments - extrathoracic, bronchial, bronchiolar, and alveolar regions (Hofmann, 2011; Neelakantan et al., 2022) Separate mathematical models have been developed for each of these compartments, based on analogy with an equivalent electrical circuit, in order to obtain the gas transport and exchange characteristics (Nunn, 1957; Ben-Tal, 2006). Different semi-empirical models are utilised, along with the gas transport models, for determining the deposition of inhaled aerosols within the compartments (Hofmann, 2011) This technique provides an estimate of the compartment-wise deposition of aerosols.

The trumpet model, on the other hand, is a 1D approximation of the anatomical dichotomous branching network of the complete respiratory tract (Paiva, 1973; Darquenne and Paiva, 1994; Choi and Kim, 2007; Devi et al., 2016; Chakravarty et al., 2022; 2023). This technique considers the respiratory tract to be a continuous channel of variable cross-sectional area (see Figure 1c). The length and cross-sectional area of the approximated channel are determined using different empirical correlations based on airway generation number ( N ) and other anatomical parameters. The present work uses a trumpet model where the number of branches ( N l ) , length ( L ) and the total cross-sectional area ( A ) at each generation ( N ) is calculated using a power-law function as (Chakravarty et al., 2022; Chakravarty et al., 2023) N l = 2 N , L N = L 0 α N , A N = A 0 2 β N , (1) where L 0 and A 0 are the length and cross-sectional area at N = 0 (trachea), respectively. α and β are the length-change and area-change factors selected such that the computed length and area at each generation closely matches Weibel’s morphometric data (Weibel et al., 1963) (see Figure 2a). Alveolation of the distal lung airways is considered N = 17 onwards, consistent with human lung (Weibel et al., 1963), by considering additional surface area in the relevant generations (Chakravarty et al., 2022; 2023). The modeled system of airways and alveoli is also assumed to be lined by a thin mucus layer separating the airway lumen from the underlying periciliary layer and the epithelium (see Figure 1d). The periciliary layer, the epithelium and the ciliary motion driving mucus transport are not explicitly modelled. Instead, mucociliary transport is accounted for by assuming a convective motion in the mucus layer from the deeper generations (larger N ) towards N = 0 . The thickness ( δ ) , the total cross-sectional area of the mucus layer ( A m ) , and the convective mucus velocity ( V m ) at different airway generations are also estimated using power law functions as (Chakravarty et al., 2022; Chakravarty et al., 2023) δ N = δ 0 ζ N , A m N = A m , 0 2 β ζ N , V m N = V m , 0 ε N , for N < 18 , = 0 , for N ≥ 18 , (2) where δ 0 , A m , 0 , and V m , 0 are the mucus thickness, area, and velocity at N = 0 , respectively. The magnitudes of the change factors ζ and ε are chosen based on reported data (Karamaoun et al., 2018; Foster et al., 1980; Yeates et al., 1975). V m is assumed to be zero beyond N = 17 (Equation 2) due to the absence of appreciable mucociliary transport in the deep lung (Mauroy et al., 2011). δ and V m are also assumed to be temporally invariant in this analysis (Karamaoun et al., 2018). The reasoning for this assumption is discussed in detail in Section 3.

This perspective starts with airway fluid dynamics based on 1D assumptions. It is followed by the transport equations for gas exchange, particle/droplet transport, deposition and clearance and finally, pathogen-specific effects. The goal is to consolidate key insights obtainable from 1D models because of which it has remained a useful model to draw mechanistic insights into lung physiology.

In order to develop the airflow transport equations, it is necessary to a priori examine the nature of airflow within the lung and make reasonable assumptions. Scaling based on relevant physiological parameters shows that Reynolds number ( R e ) varies between 1 0 3 − 6 × 1 0 − 3 within the lung. R e reduces from about 1 0 3 at N = 0 (trachea) to less than 10 after N = 13 (airways) and to less than 1 after N = 17 (deep lung), as established through various studies including microfluidic experiments (Pedley, 1977; Kleinstreuer and Zhang, 2010; Sznitman, 2021; Kassinos and Sznitman, 2025). This implies that airflow within most of the lung remains largely laminar. Neglecting turbulence is, thus, a reasonable assumption in most of the lung. Nonetheless, localised air circulation zones can still form near airway bifurcations and in the deep lung which can impact particle/aerosol deposition (Kassinos and Sznitman, 2025).

The pulsatile nature of breathing introduces unsteadiness to airflow in the upper airways. This allows the use of the Womersley number ( W o ) , which originates from a solution of the linearised Navier-Stokes’ equation for oscillatory laminar flow in an infinitely long pipe and subseuquently applied to characterization of pulsatile blood flow in an arterial tree, in quantifying the unsteady effects of airflow (Pradhan and Guha, 2019; Chakravarty et al., 2019; Sznitman, 2021). W o is defined as the ratio of inertial to viscous forces in a pulsatile flow i.e., W o = D ω / ν , where D , ω and ν denote the airway diameter (see Table 1), pulsatile frequency and kinematic viscosity of air, respectively. Thus, if W o > 1 , inertial forces dominate resulting in significant unsteady effects. On the other hand, if W o < 1 , the viscous forces dominate resulting in the flow-field developing fully during each pulsation cycle. The latter allows a quasi-steady approximation. Analysis of airflow in the airways show that W o ranges between 6 − 0.1 , with W o < 2 for N > 2 . This implies that the unsteady pulsatile flow effects exist only in the first couple of generations beyond which the flow remains largely quasi-steady (Sznitman, 2021). A quasi-steady assumption in modelling lung airflow is, thus, a reasonable approximation.

The 1D transport equations for airflow dynamics are summarized next and the corresponding analysis is presented using the above approximations (also detailed in the subsequent sections). It is to be noted that while the balance equations for mass and momentum are fundamental to modeling lung airflow dynamics, the corresponding energy and entropy balances are typically not invoked. This is because temperature variations during normal respiration are small, viscous dissipation is negligible, and thermal nonequilibrium effects are minimal Pedley (1977), West (2012), Karamaoun et al. (2018) In addition, while some degree of phase change exists in the first few generations Karamaoun et al. (2018), it’s effect on airflow and particle deposition is not significant. Consequently, energy and entropy transport do not exert a meaningful influence on airflow or particle deposition.

Along with airflow dynamics, it is essential to study the mechanism of transport and exchange of various constituent gases (of air) in the lung since breathing induces spatio-temporal variations in gas concentration within the lung (Hasler et al., 2019; Tsuda et al., 2008). The 1D transport equation for gases in the idealized lung geometry (see Figure 1) is expressed as ∂ ρ A x c g ∂ t + ∂ ρ Q c g ∂ x = ∂ ∂ x ρ A x D g ∂ c g ∂ x − ρ L e x c g − c ∞ , (13) where c g , Q , and D g are the gas concentration (in air), volume flow rate of air during breathing, and gas diffusivity (in air), respectively. ρ is the gas density and A x is the cross-sectional area of the lung at a particular location x . The last term on the right hand side of Equation 13 takes into account gas exchange with the blood stream (having a gas concentration c g , ∞ ). L e x is defined as the loss coefficient due to gas exchange with the blood stream and is expressed as L e x = 2 π R D e x , (14) where R   ( = R N 2 N ) is the total perimeter in the N ^th generation and D e x is the rate at which the gas is exchanged with the blood stream. Since gas exchange occurs only in the alveolar region of the lung (Tsuda et al., 2008; West, 2012), the loss term (Equation 14) is also considered only in the alveolar region of the idealized lung geometry and neglected in the upper airways. Here alveolar capacitance (or lung elasticity) effects could be modeled but are not explicitly elaborated here (Neelakantan et al., 2022).

Steady and unsteady solutions to Equation 13 are provided in Sections 2.2.1, 2.2.2, respectively. Fluid motion and hence, gas transport during breathing is inherently unsteady with a time-periodic nature (Chakravarty et al., 2019) - it takes more than 10 breathing cycles (normal breathing cycle ∼ 4 s) for gas concentration to reach a time-periodic state. This warrants unsteady analysis. However, the steady-state solution provides useful insights into the transport mechanism and the governing parameters, and it can be used to identify the limiting solution for different flow conditions.

Transport of particles within the lung is fundamentally similar to transport of gases in the lung. Here, the term particle is used to denote solid particles as well as liquid droplets/aerosols. The one-dimensional transport equation for particles at any location in the idealised lung geometry can be expressed in a similar manner as that for gases (Equation 13) as follows (Chakravarty et al., 2022; Chakravarty et al., 2023; Choi and Kim, 2007; Taulbee and Yu, 1975; Mitsakou et al., 2005; Darquenne and Paiva, 1994) - ∂ A c p , a ∂ t + ∂ Q c p , a ∂ x = ∂ ∂ x A D p , a ∂ c p , a ∂ x − L d c p , a , (25) where, c p , a and D p , a represents the particle concentration (in airways) and particle diffusivity in air, respectively. Q represents the volume flow rate of air in breathing.

A major physiological difference exists between the gas and particle transport mechanisms. While the gas molecules are exchanged between the blood stream and the lung in the alveoli, inhaled particles/droplets are deposited in the airway mucus throughout the lung. Therefore, the gas exchange term in Equation 13 is replaced by a deposition term ( L d c p , a ) in Equation 25, where L d is the coefficient of deposition. L d is calculated taking into account different physical deposition mechanisms viz. diffusional, sedimentation, and impact deposition in the lung airways, as well as diffusional and sedimentation deposition in the alveoli (Chakravarty et al., 2022; Yeh and Schum, 1980; Hofmann, 2011; Devi et al., 2016).

Equation 25 can be expressed in terms of lung generation number ( N ) assuming a power law variation (see Section 2.2.1) as A 0 2 β N ∂ c p , a ∂ t = H ∂ ∂ N A 0 2 β N D p , a H ∂ c p , a ∂ N − Q m q t c p , a − L d c p , a , (26) where, q ( t ) represents the temporal sinusoidal function accounting for airflow variation during breathing such that Q = Q m q ( t ) . Equation 26 is reduced to its dimensionless form by multiplying and dividing Equation 26 with L 0 A 0 D p , a and − α  ln ( α ) 1 − α , respectively, and using the following scaling parameters (Chakravarty et al., 2022) τ = t T b , ϕ p , a = c p , a c p , a , 0 , T a = L 0 A 0 | Q max | , S t a = T a T b , P e p , a = | Q max | L 0 A 0 D p , a , D p , a = k B T C S 3 π μ a d p , (27) where, P e p , a and S t a are the Peclet number for particles (in airways) and Strouhal number for the airways, respectively. ϕ p , a and τ denotes the dimensionless particle concentration (in airways) and time, respectively, while the quantities T a and T b represents the convective airflow time-scale and breathing time-scale, respectively. The expression of D p , a is based on the Stokes-Einstein relation (Chakravarty et al., 2019), where C s represents the Cunningham slip correction, T represents the ambient temperature, μ a denotes air viscosity, and d p denotes the particle diameter.

The dimensionless equation is expressed as (Chakravarty et al., 2022; Chakravarty et al., 2023) | P e p , a | S t a 2 α β N ∂ ϕ p , a ∂ τ = ∂ F p , a ∂ N − L d ′ ϕ p , a , (28) where, L d ′ represents the dimensionless form of droplet deposition coefficient ( L d ) and F p , a represents the total particle flux in airways. These are expressed as follows - L d ′ = L d L 0 2 A 0 D p , a α N , (29) F p , a = 2 β α N 1 − α α ln α 2 ∂ ϕ p , a ∂ N + | P e p , a | q t 1 − α α  ln α ϕ p , a . (30)

Different empirical models available in the literature have been used to estimate L d ′ by suitably modifying them to maintain consistency with the present mathematical formulation (see (Chakravarty et al., 2022) for details). The final dimensionless form of these deposition models are expressed as follows (Chakravarty et al., 2022).

Diffusional deposition in the airways: L d , D ′ = L d , D L 0 2 A 0 D p , a α N = L 0 R 0 2 2 α N 3.66 + 22.305 + 57 . (31)

Sedimentation deposition in the airways: L d , s ′ = L d , s L 0 2 A 0 D p , a α N = 1 3 L 0 R 0 2 2 α β N S g cos ψ N , (32) where, S g is defined as the sedimentation parameter and expressed as S g = R 0 ρ p d p 3 g k B T . (33)

Impact deposition in the airways: L d , i ′ = L d , i L 0 2 A 0 D p , a α N = | P e p , a | q t ln f i N θ , S t 1 − α α  ln α . (34)

Diffusional deposition in the alveoli: L d , D , a l v ′ = γ N η D , a l v | P e p , a | q t 1 − α − α ln α , (35) where, γ N denotes the fraction of alveolated area in the corresponding generation (Chakravarty et al., 2022) and η D , a l v denotes the diffusional deposition efficiency in the alveoli. η D , a l v is expressed as (Devi et al., 2016) η D , a l v = 1 − 6 π 2 ∑ k = 1 ∞ 1 k 2 e x p − 4 k 2 t D p d e q 2 . (36)

Sedimentation deposition in the alveoli L d , s , a l v ′ = γ N η s , a l v | P e p , a | q t 1 − α − α  ln α , (37) where, γ N and η s , a l v denotes the fraction of alveolated area in the corresponding generation (Chakravarty et al., 2022) and sedimentation deposition efficiency in the alveoli, respectively. η s , a l v is expressed as (Devi et al., 2016) η s , a l v = 1 + min d s d e q , 1 2 1 − 0.5 min d s d e q , 1 2 − 1 . (38)

The transport equations (Equations 25 and 28, along with Equations 29–38) are formulated based on the assumption that the particle suspension is dilute and that the particles are mono-dispersed, do not undergo coagulation, and are decoupled from airflow in the lung (Kleinstreuer and Zhang, 2010). Based on the inherent physiology, it is also assumed that there is no additional source of particles present within the lung, and the inhaled particles are either deposited in the airway mucus or washed out of the airways. The developed mathematical model has been validated against in vivo experimental data of Heyder et al. (1986) for different inhaled, mono-dispersed particle sizes [see Chakravarty et al. (2022) for details]. This justifies the assumptions made in formulating the model.

Figure 6a illustrates the transport of particles within the lung airways in a particular breathing cycle. Once inhaled, the particles are carried along with the airflow inwards into the lung during the inhalation phase of the breathing cycle. The particle concentration ( ϕ p , a ) reaches its deepest region in the lung at the end of inhalation. This depends on P e p , a which is determined by the airflow velocity and particle diffusivity. During the exhalation phase of the breathing cycle, the particles are transported towards the upper regions of the lung and are ultimately washed out of the lung. However, it can be observed from Figure 6a that all inhaled particles may not get washed out in a single breathing cycle leading to some residual ϕ p , a within the lung at the end of exhalation. These residual particles are again transported inwards into the lung along with any additional inhaled particles (Chakravarty et al., 2019; Chakravarty et al., 2022).

As discussed, the inhaled particles may also get deposited in the airway mucus while being transported within the lung. As a result of deposition, the net ϕ p , a decreases which is apparent from the comparative plot in Figure 6a. The total amount of particle deposited in the airway mucus is shown in Figure 6b for a representative case. It can be observed that the deposition is non-uniform and it increases towards the deep lung (acinar region) reaching a maximum at N = 22 . This trend of deposition is, however, observed to be dependent on P e p , a , S t a and the exposure duration (Chakravarty et al., 2022). Exposure duration ( τ exp ) is observed to only quantitatively influence particle deposition (other parameters remaining same) with the total deposition increasing linearly as τ exp becomes longer (see Supplementary Figure S6). However, P e p , a as well as S t a are observed to qualitatively influence particle deposition as discussed below.

Particles are observed to deposit in the deep lung only when P e p , a remains in the range of 2.37 × 1 0 6 − 3.07 × 1 0 11 (see Supplementary Figure S7a). The reason being, at small P e p , a , the advection is not strong enough to carry the particles into the deep lung, whereas at large P e p , a the particles deposit primarily in the upper airways due to the impaction mechanism (Chakravarty et al., 2022; Hofmann, 2011; Choi and Kim, 2007; Heyder et al., 1986). This range translates to particle diameters of 0.006 μ m to 20 μ m for normal breathing in a healthy individual (tidal volume of 1000  ml and T b = 4  s ). Within this range, deposition in the deep lung increase up to P e p , a ∼ 1 0 8 and then decreases to reach a minimum when 4.29 × 1 0 9 < P e p , a < 1.6 × 1 0 10 (particle diameters ∼ 0.4 − 1.2   μ m). Beyond this, another increase in deposition in the deep lung take place with a peak at P e p , a = 9.03 × 1 0 10 (particle diameter of 3   μ m) (Chakravarty et al., 2022).

A longer breathing time period (lower S t a ) results in deeper breaths and lead to greater volume of air (and hence, particles) being inhaled, keeping all other parameters same and vice-versa. This increases the probability of the inhaled particle reaching the deep lung. Consequently, the deposition pattern also shifts towards the deep lung with decrease in S t a (see Supplementary Figure S7c, d). The fraction of particles deposited in the deep lung are, as such, observed to increase as S t a decreases (see Supplementary Figure S7b) (Chakravarty et al., 2022).

The above discussed results are also consistent with the observations of various experimental studies on the effects of inhalation flow rate, particle size and inhalation duration on particle deposition in the lung (Heyder et al., 1986; Kim and Jaques, 2004). Further, the applicability of specific model hypotheses and observations from this study can be experimentally verified in the future (with suitable model augmentations, if required) through focused experiments that have been made possible through experimental advances that can mimic the lung structure and function (Sznitman, 2021; Fishler et al., 2015; Möller et al., 2021; Chen et al., 2025; Kant et al., 2023).

The mass and momentum balance equations of the mucus layer within the lung can be written, based on existing literature (Mauroy et al., 2004; Mauroy et al., 2011; Smith et al., 2008; Karamaoun et al., 2018), as

Mass balance: ∂ A m , N ∂ t + ∂ Q m ∂ x = S m , N . (39)

Momentum balance: ρ m ∂ ∂ t Q m A m , N = − ∂ p a ∂ x + f c ′ − f visc ′ , (40) where f c ′ and f visc ′ are the mucociliary propulsive force and the mucosal viscous resistive force per unit mucosal volume, respectively. The convective mucus transport term is neglected due to low Reynolds number flow in the thin viscous mucus layer. Note that the airway pressure gradient ∂ p a ∂ x is assumed to be imposed in the mucus layer. This is similar to how pressure outside the boundary layer gets imposed within it. A m , N and S m , N are the area of the mucus layer and mass source of the mucus at the N t h lung generation. Q m is the mucus volume flow rate and ρ m is the density of the mucus. These quantities are defined as A m , N = 2 π R N δ m , N 2 N ; Q m = A m , N v m , N ; S m , N = 2 π R N G m , N 2 N ; G m , N = G m , 0 ψ g N , where δ m , N is the thickness of the mucus layer at the N t h lung generation and v m is the mucus velocity. G m represents the volumetric mucus source per unit area per unit time. Using these definitions and H N = ∂ N ∂ x , the mass balance equation (Equation 39) is re-written in terms of N as ∂ ∂ t 2 π R N δ m , N 2 N + H N ∂ ∂ N 2 π R N δ m , N 2 N v m = 2 π R N G m , N 2 N . (41) Equation 41 can be simplified into ∂ ∂ t δ m , N − 1 − α L 0 α ln α R N 2 α N ∂ ∂ N R N δ m , N 2 N v m = G m , N . (42)

Similarly, the momentum balance equation (Equation 40) can be converted in terms of lung generation number ( N ) using the above parameters and the converted equation is expressed as ρ m ∂ ∂ t Q m A m , N = 8 π μ a Q a 2 N A 0 2 β 2 N + k c μ m h c v c N c 0 N c ′ δ m , N − k v μ m v m δ m , N 2 , (43) where ∂ p a ∂ x = − 8 π μ a Q a 2 N A 0 2 β 2 N , f c ′ = k c μ m h c v c N c 0 N c ′ 1 δ m , N , f visc ′ = k v μ m v m δ m , N 1 δ m , N .

In the expressions above, μ m is the mucosal viscosity, h c is the height of each cilia, v c is the cilia driving velocity scale, N c 0 is the number of cilia per unit surface area in generation 0, N c is the number of cilia in generation N , and N c ′ = N c / N c 0 .

In the expression for f c ′ , the term in the first parenthesis is the scale of the total mucociliary driving force, the term in the second parenthesis arises due to conversion to force per unit volume, and k c is the proportionality constant. Similarly, in the expression for f visc ′ , the term in the first parenthesis is the scale of the viscous shear resistance to mucosal flow, the term in the second parenthesis arises due to conversion to force per unit volume, and k v is the proportionality constant.

In order to obtain the dimensionless form of Equations 42 and 43, we define the following τ = t T b , δ m , N ′ = δ m , N Δ m , s , v m , N ′ = v m , N v m , s , (44) where T b is the breathing time scale. Δ m , s and v m , s are the mucus layer thickness scale and the velocity scale, respectively. These scales are estimated based on steady-state scalings implied by mass and momentum balance equations. The dominant steady-state scaling is between the last two terms in each of the Equations 42 and 43, which gives Δ m , s = k v G m , 0 L 0 k c h c v c N c , 0 , v m , s = G m , 0 L 0 Δ m , s . (45)

The dimensionless equations, thus, obtained using the scales defined in Equations 42, 43 can be written as S t m ∂ ∂ t δ m , N ′ − 1 − α α ln α 2 α 2 N ∂ ∂ N δ m , N ′ v m , N ′ 2 α N = ψ g N , (46) R e m S t m ∂ v m , N ′ ∂ t = P Q − N c ′ δ m , N ′ − 1 δ m , N ′ v m , N ′ δ m , N ′ , (47) where S t m = T m T b = L 0 v m , s T b and R e m = ρ m G m , 0 Δ m , s k v μ m are the Strouhal number and Reynolds number for the mucus layer, respectively. The parameter P Q is defined as 8 π μ a Q a Δ m , s 2 ( 2 β 2 ) N A o 2 k v μ m v m , s . Equations 46 and 47 are solved in a coupled manner to obtain the spatio-temporal change in δ m ′ and v m ′ within the lung. It is initially assumed that δ m ′ and v m ′ follow a power-law spatial change, similar to the assumptions made for airway length and area (Equation 1), in accordance with the following equations δ m , N ′ = δ m , 0 ′ ζ N , (48) v m , N ′ = v m , 0 ′ ε N , (49) where δ m , 0 ′ and v m , 0 ′ are the dimensionless mucus thickness and mucus velocity, respectively, at N = 0 . The magnitudes of ζ and ε in Equations 48, 49 are considered to be 0.9 and 0.87, respectively, based on reported data (Karamaoun et al., 2018). As discussed previously in Section 1 (Equation 2), mucus velocity ( v m , N ′ ) is assumed to be zero beyond N = 17 due to the absence of appreciable mucociliary transport in the deep lung Mauroy et al. (2011). The spatial change due to the above assumptions is represented in Figure 7a. Note that the mucus velocity has a negative magnitude since mucus flow is in a direction opposite to increasing depth of the lung and becomes zero in the deep lung. N c ′ is assumed to vary as ψ n N where ψ n = ε ζ . ψ g is defined as ε ζ α .