Decoding the mechanophysiology for inhaled onset of smallpox with model-based implications for mpox spread

Abstract

Orthopoxviruses can transmit via inhalation of virus-laden airborne particulates, with the initial infection triggered along the respiratory pathway. Understanding the flow physics of inhaled aerosols and droplets within the respiratory tract is crucial for improving transmission mitigation strategies and elucidating disease pathology. Here, we introduce an experimentally-validated physiological fluid dynamics model simulating inhaled onset of smallpox caused by the variola virus of Orthopoxvirus genus. Using high-fidelity Large Eddy Simulations, we modeled inhaled airflow and particulate motion within anatomical airway domains reconstructed from medical imaging. By integrating these simulations with viral concentration and individual immune factors, we estimated the critical exposure durations for infection onset to be between 1−19 hours, aligning with existing smallpox literature. To formalize the broader applicability of this framework, we extended our analysis to mpox virus, a circulating pathogen from the same genus. For mpox, the mechanophysiological computations indicate a typical critical exposure duration of 24−40 hours; however, this can vary significantly–from as short as 8 hours to as long as 127 hours–depending on virion concentration fluctuations within inhaled particulates, assuming happenstance of viral evolution. Predictably longer than the critical exposure durations for smallpox, the mpox findings still strongly suggest the possibility for airborne inhaled transmission during prolonged proximity.

Introduction

Before eradication in the late 1970 s through widespread vaccination efforts^1,2,3, smallpox had historically caused substantial global mortality and morbidity. The variola virus (VARV), which is the etiological agent responsible for smallpox, belongs to the Poxviridae family, with sub-family Chordopoxvirinae and the genus Orthopoxvirus, that also includes mpox virus, vaccinia virus, cowpox virus, and several other animal poxviruses which cross-react serologically⁴. The poxviruses consist of a single, linear, double-stranded DNA molecule ranging from 130 to 375 kilobase pairs and replicate within the cytoplasm of host cells. Under electron microscopy, they appear brick-shaped and typically measure approximately 300 by 250 by 200 nm (e.g., see Fig. 1A, with inset). Notably, the extensive body of research and resulting data on smallpox^5,6,7,8,9 positions it as the ideal model Orthopoxvirus for testing new mechanistic frameworks aimed at predicting its detailed transmission dynamics. These models may then be prospectively applied toward studying the disease onset mechanisms for other related (and, circulating) pathogens^10,11. Among such examples, the mpox virus (MPXV)^12,13 has recently prompted global health emergencies; notably declared as a Public Health Emergency of International Concern (PHEIC) by the World Health Organization (WHO)¹⁴ in 2022 and 2024, in view of its rapid international spread^15,16 with potential to cause severe illness, particularly in immunocompromised individuals. Other poxviruses, such as cowpox and vaccinia, typically cause localized human infections; however, they also do remain significant given their zoonotic potential and crucial role in vaccine development.

Fig. 1

Mechanophysiological domain: Panel (A) depicts a cartoon illustration of a smallpox patient with pustules; the inset close-up shows the variola virus structure with outer membrane. Similar skin lesions can also be found in currently circulating viruses from the Poxviridae family, such as the mpox virus, which bears a similar virion morphology. The visual is adopted with a perpetual license agreement from the Getty Images. Panels (B, C, D) respectively show the sagittal, coronal, and axial views of the anatomical test airway domains (respectively labeled anatomical geometry 1 or AG$_1$ and anatomical geometry 2 or AG$_2$, built from high-resolution, medical-grade computed tomography imaging). The red regions mark the initial infection trigger sites along the upper respiratory tract (URT); downwind particulate penetration across the tracheal outlet to the lower respiratory tract (LRT) is also tracked to account for bronchial infection expression. Note: The length scale for the anatomical domains is included between the coronal visuals.

Returning to VARV¹⁷, its primary mode of transmission–beyond direct contact–involves the inhalation of virus-laden aerosols (typically $\lessapprox 5\, \mu \textrm{m}$ in diameter) and droplets ($> 5\,\mu \textrm{m}$) formed from an infected person’s respiratory secretions^5,18,19. Consequently, a systematic understanding of how the inhaled airflow physics^20,21,22,23 would influence the movement of the virus-bearing airborne particulates within the respiratory tract of an exposed subject is crucial toward elucidating the mechanism behind such infections. Be that as it may, although well-developed models exist for agent-based spatial transmission of smallpox in confined environments^6,24,25, efforts on intra-airway exploration beyond understanding virus tropism and host immunity have been limited²⁶. In particular, relatively little attention has been invested in elaborating the physical mechanics of virus propagation and the onset process of infection within the airway, as influenced by respiratory airflow streamlines. The VARV, when inhaled, primarily targets the mucosal epithelial cells at the oropharyngeal tissues–in the lower segment of the pharynx–along with spreading throughout the downwind tracheobronchial region of the respiratory system^6,27,28. In such a perspective, to derive a physics-guided understanding of the infection onset process, this work aims to address the following mechanistic questions that arise from the complex interplay between the intra-airway dynamics of inhaled pathogen-bearing particulates and the viral biology:

• Q1 What sizes of inhaled virus-laden particulates would preferentially deposit at the infective tissue sites along the airway, and how much of infectious viral load could they carry?

• Q2 More poignantly, what could be the critical exposure duration that may lead to the onset of smallpox infection during airborne inhaled transmission?

To answer $\mathrm {Q_1}$ and $\mathrm {Q_2}$, this study develops high-fidelity computational fluid dynamics simulations of inhaled air and particulate transport within two anatomically realistic and representative respiratory domains (see Fig. 1B-D) built from medical-grade airway imaging data. The test geometries are hereafter referred to as anatomical geometry 1 or AG$_1$ and anatomical geometry 2 or AG$_2$. They have been reconstructed from high-resolution, medical-grade computed tomography (CT) imaging data. As a validation exercise, the resulting inhaled transport findings are also benchmarked against representative physical experiments conducted within a 3D-printed anatomical cast (designed from one of the test geometries, namely AG$_1$). Ultimately, the flow physics-based trends assuming typical inhaled particulate size distribution are translationally integrated with viral concentration embedded in the inhaled particulates and individual immunological factors, for a precise quantification of the critical exposure duration that may trigger infection in an exposed individual. Therein, the established infectious dose of smallpox, typically between ${10-100}$ plaque-forming units (pfu)^25,29,30,31 and quantifying the minimum number of virions capable of triggering a new infection³², serves as a key cross-disciplinary parameter in determining the critical airborne exposure duration for infection onset.

The predictions for the critical exposure duration derived from our mechanics-guided inhalation analysis have been compared with the established exposure duration estimates for smallpox transmission⁶. We do see a near-exact alignment between the exposure thresholds from intra-airway inhalation modeling and the known data, demonstrating the fidelity of our approach. Building on this, the study next explores the potential multifocal application of the in silico framework–using virus-specific data (such as the infectious dose and the viral concentration in host ejecta)–to evaluate the inhaled airborne transmission parameters for MPXV, a virus with similar morphology to the VARV. Specifically, MPXV shares 96.3% identity within the central region of the genome encoding essential genes, and 84.5% identity overall, with VARV³³. Preliminary mechanophysiological findings from this work have been presented at the biofluid mechanics sessions of the 2024 and 2025 Annual Meetings of the American Physical Society’s Division of Fluid Dynamics^34,35 and at the 26$^\mathrm{{th}}$ International Congress of Theoretical and Applied Mechanics (Daegu, S Korea, 2024)³⁶.

Results

Intra-airway regional transmission trend as a function of inhaled particulate sizes

The test geometries (AG$_1$ and AG$_2$) represent typical, disease-free airway cavity shapes in exposed individuals—serving as the numerical domains for simulating the inhalation and transmission of virus-laden particulates, which can potentially lead to infection. Figure 2A-B displays the simulation-derived heatmaps for inhaled particulate deposition and penetration fraction, $\eta _k$ (in %); it measures the net percentage of inhaled particulates that: (a) directly deposit at the infective tissue regions along the upper airway (colored red in Fig. 1B-D); and (b) penetrate through the tracheal outlet (see Fig. 1B) to move downwind into the infective bronchial airspace within the lower airway. These deposition and penetration fractions (with $\eta _k$ summing up the two) are obtained as functions of the monodisperse particulate diameters $d \in [0.1,~50.0]~\mu$m tested computationally. Considering the two modeled inhalation rates of 15 and 30 L/min (respectively mimicking relaxed and moderate breathing conditions³⁹) in the two anatomical test geometries AG$_1$ and AG$_2$, the computational data comprises $k \in \{1, 2, 3, 4\}$. In all geometry-flow combinations, $\eta _k \gtrsim 30\%$ for $d \lessapprox 9~\mu$m; the corresponding heatmap regions are highlighted within red boxes in Fig. 2A-B.

Fig. 2

Simulated inhaled transport trend: Panels (A and B) respectively show the numerically simulated deposition and penetration percentage $\eta _k$ (with $k\in \{1,2,3,4\}$ for the two simulated inhalation rates in the two test geometries), summing in each case the percentage of inhaled particulates that: (i) directly deposit at the infective tissue regions along the upper airway; (ii) penetrate downwind through the tracheal outlet into the infective bronchial airspace. The four rows in the heatmaps correspond to AG$_1$ and AG$_2$ (as marked), for inhalation rates 15 and 30 L/min. Considering all four geometry-flow combinations, $\eta _k \gtrsim 30\%$ for $d \lessapprox 9~\mu$m. The corresponding heatmap regions are highlighted within red boxes. Panel (C) shows 50 representative simulated airflow velocity magnitude streamlines (with 25 streamlines initiating from each nostril plane) during 15 L/min inhalation. Panel (D) depicts the vorticity field (for the same 15 L/min inhaled flow rate) mapped over a representative mid-lying cross-section, on the yz plane (note the coordinate axis orientation on the left of panel (C)). Panel (E) shows the corresponding pressure field on the same cross-section. Panel (F) depicts the simulated trajectories for ten representative 1.5-$\mu$m particulates (the inhaled airflow still being 15 L/min), with 5 starting from each nostril plane. Panel (G) includes the Q-Q plots for the deposition and penetration trends as a function of the particulate sizes. Therein, the horizontal axis shows theoretical quantiles from a fitted normal distribution, the vertical axis shows sorted sample quantiles, and the red dashed line marks perfect normality.

Fig. 3

Mean simulated deposition trend and exposure parameters: Panel (A) demonstrates the mean deposition and penetration rate $\eta$ (in %, as a function of the inhaled particulate diameters d), averaged across the test geometries and breathing rates; i.e., $\eta = \left[ \sum _1^4 \eta _k\right] / 4$. The red square marks the point when $\eta$ first exceeds 30%. Next, based on the reported particulate size distribution in respiratory ejecta³⁷, panel (B) shows the number of environmentally dehydrated³⁸ particulates of each test size assumed to be inhaled per minute by an exposed subject.

Representative inhaled airflow streamlines, vorticity field maps, pressure field contours, and inhaled particulate trajectories are respectively shown in Fig. 2C-F. Therein, quite striking are the mixing patterns at the junction of the oral cavity and the pharyngeal respiratory tract (see recirculating streamlines in Fig. 2C), with the vorticity magnitudes peaking within the laryngotracheal space downwind from the laryngeal constriction near the vocal folds. For a detailed analysis of the relevant vortex dynamics and local instabilities, see our recent preprint⁴⁰. The simulations herein return the following inlet-to-outlet total pressure gradients ($\Delta P$) based on the flow conditions (characterized by the inhaled airflux Q, in L/min): in AG$_1$, $\Delta P = 31.91$ Pa for $Q = 15$ L/min and $\Delta P = 106.11$ Pa for $Q = 30$ L/min. In AG$_2$, $\Delta P = 16.61$ Pa for $Q = 15$ L/min and $\Delta P = 53.10$ Pa for $Q = 30$ L/min. Here, $\Delta P$ is measured as the absolute difference between the simulated pressure at the tracheal outlet and the mean of the inlet pressure values at the two nostrils in each geometry.

Table 1 Geometry-flow coupling: Rank order test data between AG$_1$ and AG$_2$ deposition fractions at the infective regions, taking combinations of the 15 L/min and 30 L/min simulation findings for $\eta _k$ (in %), as function of the inhaled particulate diameters d (in $\mu$m); see Fig. 2A-B.

Rank order testing to assess flow-geometric coupling and homogeneity in transmission trend

Within AG$_1$ and AG$_2$, the simulated deposition and penetration efficiencies ($\eta _k$, in %) as a function of inhaled particulate diameters d, exhibit similar distribution shapes. This can be seen in the Q-Q (quantile-quantile) plot of each distribution against a normal distribution (note the separate panels in Fig. 2G), which also shows a strong deviation from normality. We formally tested the distributions for AG$_1$ and AG$_2$ for similarity, assessing the rank order correlation between them, using both Spearman’s and Kendall’s rank correlation coefficients, appropriate for non-normal distributions⁴¹. These tests involved ranking the inhaled particulate sizes (d) based on $\eta _k$, and comparing the resulting orderings between AG$_1$ and AG$_2$. Table 1 presents the rank orders of $\eta _k$ values as a function of the corresponding inhaled particulate sizes, comparing the simulated data from AG$_1$ and AG$_2$ at identical inhalation rates.

The Spearman’s rank correlation coefficients were $R = 0.9193$ for 15 L/min inhalation and $R = 0.9244$ for 30 L/min inhalation, both with $p<< 0.00001$, indicating a highly significant monotonic relationship. Similarly, with the robust sample size of 63 data points (the count being the total number of particulate diameters tested for each inhalation rate; see the discrete values along horizontal in Fig. 2A-B) in each simulation, the Kendall rank correlation test yielded $R = 0.8212$ for 15 L/min and $R = 0.8605$ for 30 L/min, both with $p<< 0.00001$. These complementary results demonstrate a strongly consistent association between the rankings across the two geometries. Overall, the data reveals a highly reliable and statistically significant correlation in the ordering and monotonic trends of $\eta _k$ values across different geometries and inhalation rates. The extremely small p-values (see Table 1) also reinforce the likelihood that the derived correlation measurements are genuine and not owing to random chances.

Backed by the consistent rank-ordering of $\eta _k$ across the two test geometries for each inhalation rate and considering an equal mix of 15 L/min (for relaxed breathing) and 30 L/min (for moderate breathing) inhaled airflux in the exposed subject, we have subsequently obtained the averaged deposition and penetration fraction $\eta$ ($= \left[ \sum _1^4 \eta _k \right] / 4$) from the tested set of inhalation rates and geometries. This helps facilitate a generic streamlined trend of the inhaled transmission parameters for the remainder of this analysis. Serving as the primary computation-derived input to our study, $\eta$ dictates inhaled transmission levels of virus-laden particulates to the infection-prone airway sites; the corresponding mean transmission rates are plotted in Fig. 3A.

Fig. 4

Exposure duration for infection onset (with Panels A−C for VARV; Panel D for MPXV): Panel (A) reports the critical exposure duration $\tau _c$ (in hours) for infectious dose $I_D \in [10, 100]$ pfu²⁵ and infectious virion potency $p \in [60, 100]$%. The top trend lines are for lower p, implying longer time needed for infection onset (hence $\tau _c$ elevates). Panel B depicts the same data, but with p placed along the horizontal axis. Therein, each of the trend lines accounts for the entirety of the assumed $I_D$ range, with the bottom-most point corresponding to $I_D = 10$ pfu and the top-most point corresponding to $I_D = 100$ pfu. In both A and B, the red curves are for $p = 63\%$. Panel (C) records the sample frequency for $\beta$, where $10^{\beta }$ pfu/mL is the measured viral concentration in throat swabs, adopted from⁸. The data comprises 147 samples, collected from 32 patients, with a mortality rate of 34.38%. Explained on the right of panel (C), the color code points to the day number during illness when the corresponding sample was collected. From this representation, the averaged index $\beta _\mathrm{{mean}} = 2.952$ is used for calculating $V_L = 10^{\beta _\mathrm{{mean}}}$ (in pfu/mL) while extracting the model projections. Panel (D) (in light yellow) highlights the mpox results on critical exposure duration ($\widetilde{\tau _c}$). Curve I comprises findings with previously measured $V_L$⁴²; II shows the state if $\mathrm{{log}}_{10} V_L$ reduces by 0.5; III shows the state if $\mathrm{{log}}_{10} V_L$ increases by 0.5. The underlying blue region depicts the comprehensive $\widetilde{\tau _c}$ domain from viral concentration evolution, with perturbations on $V_L$ following $\Delta ~\mathrm{{log}}_{10} V_L \in [-0.5, 0.5]$.

Temporal assessment of inhaled virus transmission

Figure 3B describes the size distribution of particulates $\mathbb {N}_{in}(d)$, inhaled per minute–assuming they consist of dehydrated respiratory ejecta from an infected host. The data is adopted from established findings³⁷ on the size distribution of aerosols and droplets emitted during normal speech and silent breathing. By coupling the inhaled size distribution $\mathbb {N}_{in}(d)$ with the deposition and penetration efficiency $\eta (d)$, it is straightforward to show that the inhaled liquid volume reaching the infective airway sites per minute would be:

$$\begin{aligned} \Omega ~({\mathrm{in~mL}} /{\textrm{min}}) = \sum \eta \, \mathbb {N}_{in} \, \frac{1}{6} \pi \, d^3 \times 10^{-12}, \end{aligned}$$

(1)

with the summation implying that equation 1 accounts for all particulate sizes that reach the infective airway sites and d representing the particulate diameters in $\mu$m. Subsequently, considering the viral load carried by the liquid volume (as evaluated in equation 1), we can estimate the critical exposure duration for infection onset:

$$\begin{aligned} {\tau _c} = \frac{I_D}{60 \, p \, \Omega \, V_L}, \end{aligned}$$

(2)

where $I_D$ (in pfu) quantifies the infectious dose, $V_L$ (in pfu/mL) is the mean viral loading within inhaled particulates (in other words, it quantifies the mean viral concentration in the liquid volume of the particulates), $\tau _c$ (in hours) is the critical exposure duration for smallpox infection onset, and p, as a measure of the dose-response relationship^43,44, represents the potency (in %) of the plaque-forming units at triggering infection in an exposed subject. This is because while the plaque count (in pfu) quantifies the number of viral invasions within a monolayer of susceptible host cells, not all plaque-forming units are capable of initiating an infection in a real airway, which involves overcoming local and systemic mucosal defenses and immunological barriers. So, if the pfu potency is (say) p%, then out of every 100 infectious virions reaching the susceptible tissue sites, p of them could be presumed successful at invading the cells launching new infection.

Table 2 Explicit range for critical exposure duration for smallpox: Representative critical exposure durations ($\tau _c$, in hours) for $I_D \in [10, 100]$ pfu, for discrete p% pfu potency at triggering infection inside an exposed human airway. Note that the listed numbers comprise a subset of the data plotted in Fig. 4A-B.

Critical exposure duration for inhaled onset of VARV – with cross-technique validation

Figure 4A depicts the trend for τ_c for $I_D \in [10, 100]$ pfu²⁵. The upper trend lines correspond to lower values of p (implying fewer of the plaque-forming units ending up at the infective sites are successful at launching infection)–thereby lengthening $\tau _c$. We conservatively^45,46 implement $p~\mathrm {(in~\%)} \in [60, 100]$. Figure 4B re-views the data from Fig. 4A, but from the perspective of p as the independent variable; here, the top-most and the bottom-most levels of the trend lines correspond, respectively, to the maximum and minimum $I_D$. In addition, Table 2 lists a subset of numbers from the data plotted in Fig. 4A-B. Overall, as evidenced in Fig. 4 and also Table 2, $\tau _c$ varies over the range $\approx$ 1.12−18.59 hours, aligning well with current estimates for smallpox⁶. In fact, strikingly enough in context of prediction alignment, for $p = 63\%$ (red curve in Fig. 4A-B), we have $\tau _c \in [1.77, 17.70]$ hours (see fourth column in Table 2), while the critical exposure duration from the well-established Wells-Riley model^47,48,49 (considering exposure to a standard pathogen concentration in confined air) stands at 1.7 to 16.7 hours for a 63% probability of getting infected, when exposed to 1 quantum of infection^6,49.

For the above calculation, we derived the viral loading $V_L$ (in pfu/mL) by processing the virus titre data of throat swabs in 147 samples collected from 32 patients, reported in a seminal 1970s’ study⁸ (additionally, also see^9,50,51 from the same team). Figure 4C presents the sample frequency for $\beta$, where $10^{\beta }$ pfu/mL is the measured viral concentration. The averaged index $\beta _\mathrm{{mean}} = 2.952$ is used in our analysis while deriving the $\tau _c$-trends in Fig. 4A-B.

Model-based transmissibility of a related pathogen of same genus: Extension to mpox virus

With our model projections for critical exposure durations ($\tau _c$) for smallpox in agreement with existing data–we now extend the framework to analyze the airborne transmission characteristics of MPXV, investigating the ability of this pathogen to also spread through inhalation (i.e., without physical contact). As pointed out, MPXV is closely related to the VARV, with both belonging to the genus Orthopoxvirus. The minimum $I_D$ for MPXV has been estimated to be 200 pfu⁵², with an average viral loading measured at $V_L = 10^{2.92}$ pfu/mL⁴², based on Ct/CN counts from saliva samples. Inserting these values into equation 2 yields an estimated critical exposure duration $\widetilde{\tau _c} \in [24.01, 40.02]$ hours for inhaled onset of mpox; see curve I in Fig. 4D. The minimum and maximum bounds on $\widetilde{\tau _c}$ (calculated using the same mathematical idealization as in VARV, i.e., equation 2) correspond to $p = 100$% and 60%, respectively. Such exposure durations could be plausible in a prolonged close-contact setting. The plausibility of MPXV transmission via the airborne inhalation route has implications for infection control (e.g., through effective ventilation and air change in confined spaces), as current guidelines are focused on fomites and larger respiratory droplets. Notably, while respiratory droplets are temporally and spatially limited indoors, the finer aerosolized particulates can accumulate and persist indoors, potentially retaining their infectivity potential at long ranges.

In addition, there are possible ramifications if viral evolutions were to occur in MPXV. As can be seen from equation 2, if the virus evolves such that the viral load in inhaled particulates ($V_L$) changes, this may induce dramatic impacts on the airborne transmissibility of mpox, as it can significantly alter the length of time required for inhaling viral loads equivalent to $I_D$, resulting in infection launch. For example, if we perturb the index on the viral loading value by $\Delta \beta = \pm 0.5$ (i.e., $V_L = 10^{2.92~\pm ~0.5}$ pfu/mL), the $\widetilde{\tau _c}$ duration fluctuates between $\mathcal {O}(0)$ to $\mathcal {O}(2)$ hours, with the precise model-based range being $\widetilde{\tau _c} \in [7.59, 126.54]$ hours; bounded by the extrema on the curves III and II, respectively, in Fig. 4D. Especially at the lower end of the spectrum with reduced critical exposure requirements, the conditions can significantly enhance the airborne transmissibility of MPXV beyond prevalent norms, via inhalation of respiratory secretions expelled by infected hosts.

Fig. 5

Sensitivity analysis on translational outcome predictions: Panel (A) (for smallpox) illustrates the variation in the model-predicted critical exposure duration, $\tau _c$, with the input parameters ($I_D$, p, $V_L$) perturbed as follows: (a) $I_D \in [10, 100]$ pfu; (b) $p\in [60, 100]$%; and (c) $V_L \in 10^{\beta _\mathrm{{mean}} \pm 0.5}$ pfu/mL, with $\beta _\mathrm{{mean}} = 2.952$ (see⁸). As a result, the critical exposure duration spans from 0.35 to 58.78 hours (see the extremes indicated on the adjoining color scale). Panel (B) (for mpox) shows the variation in the model-predicted critical exposure duration, $\widetilde{\tau _c}$, with the input parameters ($I_D$, p, $V_L$) perturbed as follows: (a) $I_D \in 200 \pm 50$ pfu, with 200 pfu being the reported $I_D$ for aerosolized MPXV⁵²; (b) $p\in [60, 100]$%; and (c) $V_L \in 10^{\beta _\mathrm{{mean}} \pm 0.5}$ pfu/mL, with $\beta _\mathrm{{mean}} = 2.92$ (see⁴²). Accordingly, the critical exposure duration varies from 5.69 to 158.18 hours (see the extremes on the color scale).

Uncertainty quantification: Analysis of parametric sensitivity to $\tau _c$ and $\widetilde{\tau _c}$ projections

Panels A and B in Fig. 5 depict the parametric dependence of the critical exposure durations $\tau _c$ and $\widetilde{\tau _c}$ (both in hours), respectively for VARV and MPXV, with the perturbed input parameters being $\in \{I_D, p, V_L\}$. The sensitivity of $\tau _c$ (and $\widetilde{\tau _c}$) to these parameters (qualitatively representing the infectious dose, the virion potency, and the viral concentration in inhaled particulates) is gauged through the following perturbations:

• (1) in VARV: $I_D \in [10, 100]$ pfu²⁵, $p\in [60, 100]$%, $V_L \in 10^{\beta _\mathrm{{mean}} \pm 0.5}$ pfu/mL (with $\beta _\mathrm{{mean}} = 2.952$, see⁸); and

• (2) in MPXV: $I_D \in 200 \pm 50$ pfu (with 200 pfu being the reported $I_D$ for aerosolized MPXV⁵²), $p\in [60, 100]$%, $V_L \in 10^{\beta _\mathrm{{mean}} \pm 0.5}$ pfu/mL (with $\beta _\mathrm{{mean}} = 2.92$, see⁴²).

With such parametric fluctuations, the $\tau _c$ for VARV ranges between ${0.35-58.78}$ hours; the $\widetilde{\tau _c}$ for MPXV ranges between ${5.69-158.18}$ hours. For an uncertainty quantification of the model predictions, these projections are compared with the estimates corresponding to the original unperturbed input parameters (see earlier in the §Results and also Fig. 4), where the $\tau _c$ for VARV was ${1-19}$ hours and the $\widetilde{\tau _c}$ for MPXV was ${24-40}$ hours. Clearly, the perturbed critical exposure duration magnitudes retain the same order. However, during this sensitivity analysis, the extreme $\tau _c$ (and $\widetilde{\tau _c}$) values consistently decline and elevate by approximately 3$\times$ in VARV and approximately 4$\times$ in MPXV.

Fig. 6

Experimental validation: Panel (A) shows the 3D-printable digital model (derived from the AG$_1$ reconstruction), with a scaled-up view of the separable glottic plug included in (B). Therein, $\mathcal {C}_1$ and $\mathcal {C}_2$ are the peripheral curve lengths at the top and bottom openings of the glottic region; $\mathbb {A}_G$ represents the enclosing wall surface area at the glottis. Panel (C) demonstrates the experimental setup for mimicking inhaled particulate transport using a nebulizer. The location where the glottic plug is inserted in the main 3D-printed structure is indicated within the black rectangle. Panels (D–F) depict the process of removing the glottic plug from the 3D-printed airway cast before immersing it in deionized water for measurement of local deposition fractions (for details, see §Methods). Panels (G and H) compare the simulated (in blue) and experimental (in red) measurements for localized deposition fractions at the glottis. The colored area coverages are proportional to the inhaled glottic deposition fraction (with respect to the total number of particulates administered to the numerical space) for 9.5-$\mu$m particulates (considering a fluctuation of ±0.25 mm in the simulations to account for expected heterogeneity in the experimental particulates; see Table 3). See §Methods for details on the experimental benchmarking and validation.

Representative experimental comparison of the model trend, with error estimation

Figure 6A-B depicts the digitized, 3D-printable design of AG$_1$; the corresponding printed geometry is shown in Fig. 6C-F. In the experimental setup (see Fig. 6C), aerosolized droplets (bearing 9.5-$\mu$m diameters) of an aqueous caffeine solution are nebulized into the printed airspace while a constant airflow of 30 L/min is drawn through the tracheal outlet. The white plug (visible in Fig. 6C-F, embodying the printed version of the design in Fig. 6B) replicates the glottic topology. The glottis comprises one of the sub-sites of the infective regions along the respiratory tree. Post-experiment, the plug is removed and the local deposition fraction at the glottis ($G_E$, in %) is quantified with UV-Vis (ultraviolet–visible) spectroscopy (see §Methods for details of the experimental protocol). Similar inhalation and particle entry conditions (see §Methods) are simulated computationally, and the resulting glottic deposition fraction ($G_C$, in %) is assessed for comparison.

$G_E$ is found to be approximately 80% of the computationally projected $G_C$ (i.e., the relative error is 20%); see Table 3 and additionally Fig. 6G-H for a visual comparison. We consider this level of agreement acceptable. The discrepancy likely arises from post-deposition displacement of droplets on the glottic plug (in the experiment) and from airflow-induced shear during the 2-minute duration of the experimental flux. More significantly, loss of deposited liquid particulates is likely in the experiment as the plug is removed from the main 3D print and transferred to deionized water bath for spectroscopic measurements. Note that detachment of deposits from the smooth inner walls of a 3D-printed plastic cast is easier than from native airways, where mucus and ciliary layers promote retention. For a quantitative critique of liquid particulate loss during the experiment, consider the peripheral curve lengths $C_1 \approx 38$ mm and $C_2 \approx 46$ mm marking the top and bottom openings of the glottic plug (see Fig. 6B). The surface area of the glottic airspace is $\mathbb {A}_G \approx 494$ mm$^2$. To explain a 20% loss as the plug is withdrawn (see Fig. 6D-F), let $\alpha$ be the cumulative streamwise edge length (top + bottom) of the plug from where deposits get removed. Therefore:

$$\begin{aligned} \frac{\alpha \left( \mathcal {C}_1 + \mathcal {C}_2\right) }{\mathbb {A}_G} = 20\%,~\textrm{whereby}~\alpha \approx 1.2~\textrm{mm}. \end{aligned}$$

(3)

Now, per the benchmarked simulations which tracked 1892 monodisperse particulates, the mean deposition fraction is $G_C = 1.603$% (see Table 3), implying a glottic deposition of approximately 30 particulates. A loss of 20% of them implies that 6 particulates are lost during plug extraction. This is plausible assuming removal occurs from a narrow edge band of width $\alpha /2 \approx 0.6$ mm at each edge of the glottic plug, particularly because extricating the plug involves a subtle jerk (e.g., see Fig. 6D–F illustrating the process) to overcome the snug fit of the plug in the main print.

Table 3 Validation test: Comparison data for glottic deposition from representative experiments within a 3D-printed AG$_1$ cast and benchmarked computational simulations (refer to §Methods for details of the numerical and experimental protocols). $C_E$ quantifies the concentration of caffeine markers in the solution where the post-experiment glottic plugs are rinsed; $G_E$ is the experimental glottic deposition fraction (see equation 11 later); and $G_C$ represents the computationally predicted glottic deposition fraction. See Fig. 6 for the experimental setup and data visuals.

Discussion

On the significance of historical context while structuring the inhalation model

Our methodology mechanistically benefits from building upon the extensive body of smallpox research (the disease was formally declared eradicated by the WHO back in 1980³), enabling contextualization and cross-validation of historical data with modern computational methods. The general approach (with appropriate pathogen-specific inputs such as viral loading in host ejecta and the sites of infection origin in the exposed subject’s airway) can also contribute to the rapid assessment of transmission parameters for other Orthopoxviruses in the face of emerging outbreaks. The demonstrated extensibility (with results on mpox included in this study) underscores our framework’s potential as a versatile, physics-guided tool for swift risk evaluation based on detailed anatomical and inhalation fluid dynamics modeling.

Another “advantage” our model has benefited from is the lack of usable prior data on virion concentration in freshly expelled respiratory secretions from smallpox patients and, more generally, within their bodily fluids–specifically in units of DNA copies/mL. This scarcity is partly due to smallpox being no longer a circulating virus, and because genomic sequencing emerged only in the late 1970s^53,54, coinciding with the eradication timeline of smallpox. Genomic techniques can be used to estimate viral concentration by analyzing the amount of viral nucleic acids present in a sample. However, for smallpox, such measurements are typically only available for purified historical viral preparations, as seen in⁵⁵. Nevertheless, the inhalation-based model requires viral concentrations in fresh respiratory secretions to estimate the inhaled viral load for an exposed individual. To address this, we have utilized viral loading data expressed in pfu/mL (obtained from older studies involving swabs from contemporary patients⁸) within equations 1 and 2. Fortuitously enough, these 70s’ era papers^8,9 were recently ‘re-discovered’ in a landmark 2025 Lancet paper²⁵. The availability of the viral concentration data in pfu/mL was functional in maintaining dimensional consistency within the mathematical framework, since the units for smallpox infectious dose data are also in pfu. In fact, the conversion between pfu and DNA copies for VARV remains unclear to this day²⁵.

On modeling MPXV transmission: Insights into respiratory viral burden and data gaps

The airborne transmission potential of MPXV clades and sub-lineages remains debated⁵⁶. Data from the 2022/23 Clade IIb pandemic indicated that transmission of sub-lineage b viruses is predominantly mucosal- and contact-based, with the highest viral loads detected in lesion and rectal swabs. Moderate levels of viable virus were recovered from human oral or respiratory samples^42,57,58. Various animal models also demonstrate respiratory tract infection and oropharyngeal shedding (e.g., positivity of swabs for both DNA as well as infectious virus)^59,60,61. However, human studies conducted till date are yet to conclusively establish a direct evidence of infectious virus in the genomic material detected in exhaled droplets or room air^42,62. Nonetheless, experiments with prairie dogs have suggested that airborne transmission can occur but is inefficient for Clade Ia⁶³. Work in Mastomys natalensis rats using Clade IIb also confirmed oral shedding but a clear correlation was not found linking such shedding to contact transmission efficiency⁶⁴. Further confirming this, environmental sampling studies consistently detect MPXV DNA on surfaces, but infectious virus has only been cultured in nosocomial settings, e.g., from air during bed linen changes inside hospital rooms^62,65.

To examine whether the apparent absence of airborne spread could be explained by an insufficient amount of infectious virus generated in the respiratory tract, we extended our VARV based model to MPXV. This required several simplifying assumptions, as key parameters including respiratory tropism, infectious dose (across clades), and within-host kinetics remain poorly defined for MPXV. We therefore maintained comparable parameters for early replication kinetics, varying only the input virus loads (in the form of the viral concentration $V_L$ embedded in the particulates inhaled by the exposed subject; these particulates are assumed to be sourced from host respiratory fluids) based on available data for Clade IIb^42,66,67,68 and the infectious dose ($I_D$) value, based on data for the Clade I Zaire V79-I-005 strain⁵². Our findings suggest that differences in respiratory viral burden could in part explain the observed epidemiology and environmental sampling results for MPXV. Nevertheless, these simplifications introduce uncertainty, as the biological equivalence of VARV and MPXV in these aspects is (still) unverified.

To improve future iterations of the model, we, first, require improved data on tissue tropism, specifically the initial target sites within the respiratory tract, as well as the sites from where the expelled infectious particulates are fragmented from. Experimental aerosol inoculation, which would force the virus into the LRT, in Cynomolgus macaques with Clade Ia MPXV demonstrated primary lower airway infection. In contrast, autopsy findings from severe human cases with advanced HIV infection showed that deep lung involvement can occur under severe immunosuppression but is otherwise atypical for Clade IIb^52,69,70,71, supporting both clade and route of exposure dependency. However, the initial target site in relation to the specific route of exposure in humans is not clearly defined.

Secondly, data on infectious loads within different segments of the respiratory tract and oropharyngeal cavity are not available using standardized, quality-controlled sampling strategies across both VARV and MPXV, let alone across various MPXV lineages. Methodological differences–including variations in PCR (polymerase chain reaction) targets, extraction methods, sampling sites (e.g., saliva versus oral swabs), and timing post-infection–may complicate direct comparisons of viral load data. These discrepancies can lead to inaccurate estimations of respiratory infectiousness when relying solely on Ct values or DNA detection, especially if such data are not stratified according to the initial exposure route of the patient.

Thirdly, our model inferred infection risk primarily for Clade IIb. Epidemiological data indicate that Clade I lineages are generally more pathogenic than Clade II viruses, with historical case fatality rates averaging about 10% and 3.6%, respectively, prior to 2022. Current genomic surveillance shows that Clades Ib and IIb dominate human to human transmission, mainly through close or sexual contact, while Clades Ia and IIa remain largely zoonotic^72,73,74. The co-circulation of sub-lineages in endemic regions underscores the evolving epidemiological complexity^75,76. Atypical clinical presentations and variable disease severity also suggest that early sampling bias may have exaggerated the perceived pathogenic gap between ancestral Clades Ia and IIa. Nevertheless, future model iterations should incorporate clade-specific modifiers to account for possible differences in respiratory tropism, replication kinetics, and transmission potential. Such refinements would require infection route stratified, quantitative, cross-clade comparative data.

On miscellaneous limitations implicit to the current modeling framework and the future directions

The findings presented here are based on inhaled airflow-particulate transport patterns within only two anatomical respiratory domains. Many previous studies have investigated micron-sized particle transport and deposition in adult airways with larger subject pools, and the current test cohort is clearly inadequate so far as capturing anatomical variability is concerned. Be as that may, it is however critically important to emphasize that this study does not aim to establish the statistical frequency of airborne transmissibility of the poxviruses. Instead, it simply seeks to evaluate the theoretical potential for such transmission within representative and realistic respiratory tracts, serving as a basis for understanding possible pathways rather than quantifying likelihood. Exploring how the variability in transmission dynamics might change as the geometric parameters of the airway regions are altered–owing to factors like age, health status, ethnicity or merely anatomical differences across individuals–would be a valuable direction for future research. Such investigations could help in assessing population-level risks and in tailoring more specific intervention strategies.

To expand on the potential limitations in the modeling framework, a significant one stems from the CT-derived geometric systems being static and rigid; thus neglecting the dynamic compliance of airway tissues and tissue deformation that may occur during actual breathing in living subjects. These biomechanical factors can subtly influence airflow patterns, inhaled particulate trajectories, and local deposition sites; these complex fluid-structure interactions that could be present in vivo are not incorporated into our modeling framework. Additionally, the simulated inhalation regimes did not account for mouth breathing, based on the rationale that oral inhalation accounts for less than 10% of breathing time in healthy adults⁷⁷. While this simplification is reasonable for a holistic take on the typical breathing patterns, it may underestimate inhaled particulate deposition at the infective airway regions in scenarios where mouth breathing is more prevalent, such as during exercise or respiratory distress.

Constraints concerning the experimental validation include potential loss and redistribution of deposited droplets during the glottic plug removal (from the 3D-printed AG$_1$; see Fig. 6D-F). $G_E$ measured by UV–Vis was approximately 80% of the simulated $G_C$ (i.e., 20% relative shortfall), primarily owing to partial loss of deposits when the snugly fitting glottic plug was extracted and transferred to the deionized water bath for analysis. To mitigate the handling-related loss (in future investigations), we recommend the following design and procedural modifications: employ a guided or split removable plug to eliminate jerks during removal; apply hydrophilic or mucus-mimetic surface treatments to better replicate in vivo adhesion; implement in situ imaging (e.g., with techniques such as gamma scintigraphy^78,79) to quantify deposits without plug extraction; and refine the computational protocol to incorporate surface wetting and post-deposition transport. Together, these measures can reduce edge-band detachment, improve recovery efficiency, and yield more robust experimental-computational comparisons.

Several additional key limitations on specific biological factors remain open and can have a strong bearing on the disease transmission trends. For example, a precise estimate of virion potency (p) in triggering new infections is still lacking, as current values are based on limited experimental data. More refined, virus-specific potency assessments could enhance the accuracy of predictions regarding critical exposure durations (by providing more specific inputs to equation 2). The described modeling framework will also gain from consideration of local mucosal defenses and mucus clearance effects. Mucociliary clearance represents a primary mode of innate defense against inhaled pathogens. An intact mucus layer traps pathogen-laden inhaled particulates, while coordinated ciliary beating transports them downstream for removal. Impairment of this system (for instance, from dehydration, viral damage to epithelial cells, or genetic disorders) will prolong the residence time of pathogens on airway surfaces, increasing the opportunity for epithelial attachment and invasion. Consequently, compromised mucus clearance accelerates pathogen retention and replication in the airways, raising the likelihood of infection establishment and downstream sequelae.

Next, to model inhalation of pathogen-laden particulates by an exposed subject (after they have been emitted by an infected host) – this study uses dehydrated parameters corresponding to the expelled particulate size distribution from earlier experiments³⁷ (measured in a mouth-connected box) as a conservative, worst-case input condition representing the maximum pool of particles available for inhalation. This assumption intentionally treats emitted-particle concentrations as potentially inhalable to avoid underestimating exposure when structuring mitigation measures. The reader should however note that real-world inhalability is typically < 100% and depends on a myriad of factors like inter-subject distance, spatial orientation, ambient airflow/turbulence, and ventilation. Results should therefore be interpreted conditionally; for specific settings, spatially resolved transport or measured near-field-to-breathing-zone transfer efficiencies should be used to better model inhalability.

Another key implicit assumption has been the temporal continuity of exposure (vide equations 1, 2) to the infected host, when assessing critical exposure durations for infection onset in the exposed individual. The model does account for fluctuations in breathing cycles, but the aforementioned “temporal continuity” represents a simplifying mathematical ansatz. In real settings, the exposures could be intermittent and their cumulative effects should be taken into consideration (along with accounting for potentially varying immune responses for intermittent exposures) while estimating infection transmission probabilities.

Incorporating immunological factors–such as local mucosal defenses and individual innate and adaptive immune responses–into in silico models like the one proposed here also remains an important future direction. Currently equation 2 does a zeroth-order approximation of the individual immune variability through the assumed range on potency p. However, more direct and quantifiable measurements of subject-specific immune responses could significantly improve the predictive power of this mechanophysiological framework and aid in the development of more nuanced public health strategies for controlling airborne pathogen transmission. The described modeling paradigm also assumes that both of the considered viruses (VARV and MPXV) have similar initial infection sites and tropism–a parameter which would benefit from future refinement once more detailed clinical data on MPXV clades becomes available.

The main takeaways: On the airborne inhaled transmissibility of smallpox and mpox

The variola virus, beyond direct contact, enters the body primarily through the respiratory pathway, triggering the initial smallpox infection within the mucosal epithelial cells along the airspace. Through high-fidelity, physiologically realistic computational fluid dynamics simulations of inhaled transport within CT-derived human respiratory tracts, integrated with virological parameters such as viral loading in host ejecta and established infectious dose levels, this study has yielded the following key insights into the airborne transmission of VARV: (a) smaller inhaled, virus-laden particulates are more effective at reaching the infective tissue sites along the airway; for instance, the relevant deposition and penetration efficiencies exceed 30% (of all inhaled particulates of the same size) for all particulate diameters $\lessapprox 9~\mu$m (see Fig. 2A-B and Fig. 3A); and (b) the critical exposure duration $\tau _c$ for transmission of viral load equivalent to the smallpox infectious dose to the infective intra-airway regions, thus resulting in inhalation-induced infection onset, is approximately between 1.1 to 18.6 hours (see Fig. 4A-B and Table 2). These projected durations (derived by integrating intra-airway inhalation physics with virological parameters) align with existing literature^6,49 that cites a critical exposure duration of 1.7 to 16.7 hours (for a 63% probability of infection) based on the probabilistic Wells-Riley model for spatial transmission. Backed by representative experimental validation, the inferences in (a) and (b) above, thus, directly address the questions $\mathrm {Q_1}$ and $\mathrm {Q_2}$ posed in the §Introduction.

Given that exposure to host-ejected respiratory particulates constitutes the primary (non-contact) transmission mode for Orthopoxviruses, it is important to distinguish our framework from existing transmission models²⁵. While our study conceptualizes physiologically representational inhalation dynamics effects for intra-airway deposition and penetration of virus-laden particulates leading to a precise quantification of disease transmission parameters, most current models generally rely on agent-based, probabilistic methods to estimate viral exposure within confined external spaces, sometimes with room-level computational modeling of spatial pathogen distribution. The potential extensibility of our micro-scale framework is further illustrated by its application to MPXV (a member of the same Orthopoxvirus genus), with the suggestion that its critical exposure duration could be substantially longer (roughly 24 to 40 hours, under the assumptions used; see Fig. 4D’s data trend I), compared to that of smallpox. Prolonged though it may be, the projected time window still suggests that respiratory aerosol transmission of mpox could be plausible in specific close-contact settings. For instance, the mpox critical exposure duration could decline to as short as $\approx {5-8}$ hours (see Figs. 4D and 5B), if the virion concentration in the inhaled pathogenic particulates fluctuates assuming the happenstance of viral evolution where the virus adapts to higher virus loads in the host airway and the infectious agents manage to become airborne by embedding in respiratory ejecta. These model-based insights therefore highlight scenarios meriting targeted risk assessment of mpox for inhaled infection onset; however the in silico findings should not (yet) be considered as a definitive evidence of routine airborne transmission of the disease, especially in view of the lengthened critical exposure durations ($\widetilde{\tau _c}$, for MPXV) reported in this study.

Methods

Airway geometry reconstruction from medical imaging

The test respiratory tract geometries AG$_1$ and AG$_2$ (shown in Fig. 1B-D), with disease-free airway cavity shapes, were reconstructed from de-identified, high-resolution, medical-grade CT imaging of adult human airways. The process entailed radio-density thresholding from $-1024$ to $-300$ Hounsfield units⁸⁰, with selective manual editing of specific pixels for anatomical precision, to accurately capture the airspace from the CT slices in the DICOM (Digital Imaging and Communications in Medicine) format. The outer contour of mouth was assumed circular in each domain, while the topology for the rest of the upper airway till the tracheal base was guided by CT data. The retrospective computational use of the anonymized, existing scans was approved under exempt status by the Institutional Review Board (IRB) at South Dakota State University; the corresponding IRB determination number is: IRB-2206003-EXM⁸¹. Such in silico anatomical modeling marks a rapidly evolving research niche in human health investigations for a range of pathologies and interventions, e.g., see^{82,83,84,85,86,87,88,89,90,91,92}.

Fig. 7

Grid sensitivity analysis: The region marked by the orange rectangle in panel (A) is used to illustrate the progressively refined meshes (shown in the subsequent visuals for a mid-section, its location marked by the vertical white line in panel (B). Panel (C) shows the mesh with 4.0 million tetrahedral elements; (D) shows the mesh with 5.0 million tetrahedral elements; (E) shows the (eventually selected) mesh with 6.0 million tetrahedral elements; (F) shows the mesh with 7.0 million tetrahedral elements; and (G) shows the mesh with 8 million tetrahedral elements. All these meshes are graded and unstructured. Panel (H) provides a zoomed-in view from E, highlighting the four layers of prismatic cells along the airway walls. Panel I tracks the variation in flow resistance, $\mathcal {R}$ (in Pa$\cdot$min/L), across the different mesh resolutions.

Spatial discretization with mesh sensitivity analysis

To prepare the domains for numerical simulations of intra-airway inhaled transport, a representative grid refinement analysis was conducted on AG$_1$, with the cavity being spatially segregated into approximately 4.0, 5.0, 6.0, 7.0, and 8.0 million graded, unstructured, tetrahedral elements, along with four layers of pentahedral cells (with 0.025-mm height for each cell and an aspect ratio of 1.1) extruded at the airway cavity walls to resolve the near-wall particulate dynamics; e.g., see Fig. 7. While the CT images were segmented on the image processing software Mimics Research 18.0 (Materialise, Plymouth, Michigan), the subsequent spatial meshing of the digitized stereolithography domains was carried out on ICEM CFD 2023 R1 (ANSYS Inc., Canonsburg, Pennsylvania). Across the five progressively refined meshes (for AG$_1$), we further assessed the flow resistance $\mathcal {R}$ (in Pa.min/L) to the simulated inhaled airflow field (moving at 30 L/min, measured at the outlet), calculated as $|\Delta P|/Q$, where $|\Delta P|$ in Pa represents the inlet-to-outlet total pressure gradient driving the airflow and Q denotes the volumetric airflux in L/min. The flow outcome sensitivity findings, as a function of the grid refinements, show:

$$\begin{aligned} \begin{aligned} \overline{\mathcal {R}}&= 3.526~\mathrm {Pa.min/L},&\mathrm{{with}}~~\sigma (\mathcal {R})_\mathrm{{All}}&= 0.043~\mathrm {Pa.min/L},&\sigma (\mathcal {R})_\mathrm{{3}}&= 0.031~\mathrm {Pa.min/L}, \end{aligned} \end{aligned}$$

(4)

where the bar implies arithmetic mean, $\sigma (\cdot )_\mathrm{{All}}$ denotes the standard deviation across all five grids, and $\sigma (\cdot )_\mathrm{{3}}$ denotes the standard deviation of the simulated data from the last 3 grids (i.e., cases with 6.0, 7.0, and 8.0 million unstructured tetrahedral cells); see the corresponding resistance plot in Fig. 7I. Based on the asymptotic convergence trend observed in the final three cases (see equation 4), the simplest 6-million-cell grid resolution therein was selected as the mesh density standard for the overall study. Consequently, the mesh used for the comprehensive analysis in AG$_1$ comprised approximately 6 million tetrahedral elements. With the airspace volume in AG$_1$ being 1.55 times larger than that in AG$_2$, the appropriate number of cells for AG$_2$ is estimated to be 6/1.55 $\approx$ 4 million. To ensure adequate resolution, we conservatively adjusted the AG$_2$ mesh to contain convincingly more cells than this minimum warranted estimate, resulting in precisely 4.88 million tetrahedral elements. These spatial refinements are consistent with similar systems involving grid convergence and computational stability when simulating physiologically realistic airflow and particulate deposition within adult healthy human respiratory tracts (e.g., see^93,94,95). The final meshed domains, with described near-wall refinement, were then exported to ANSYS Fluent 2024 R1 for simulating inhaled transport with physiology-guided boundary conditions (detailed in subsequent subsections).

Computational simulation of inhaled transport within the respiratory system

Mathematical framework for inhaled airflow simulation

Inhalation airflow patterns³⁹ for relaxed (15 L/min) and moderate breathing (30 L/min) were replicated in the discretized test geometries using the Large Eddy Simulation (LES) scheme^96,97, which explicitly resolves field eddies larger than the grid scale. Smaller fluctuations, known as subgrid scales, were filtered out, with their effects on larger scales approximated through the dynamic subgrid-scale kinetic energy transport model^98,99,100. With incompressible and isothermal flow conditions for the inhaled air, the filtered continuity and Navier–Stokes momentum equations are respectively as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial x_i}\left( \rho \overline{u}_i\right)&= 0~ \quad \text {and}\quad \ ~\frac{\partial \overline{u}_i}{\partial t} + \frac{\partial }{\partial x_j}\left( \overline{u}_i \overline{u}_j\right)&= - \frac{1}{\rho }\frac{\partial \overline{p}}{\partial x_i} + \frac{\partial }{\partial x_j}\left( \nu \frac{\partial \overline{u}_i}{\partial x_j}\right) - \frac{\partial \tau _{ij}}{\partial x_j}. \end{aligned} \end{aligned}$$

(5)

In the above equations, the filtered quantities are denoted with overbars; specifically the resolved velocity components and pressure are respectively referred to as the filtered velocity ($\overline{u}_i, \overline{u}_j$) and filtered pressure ($\overline{p}$). The fluid properties are the kinematic viscosity ($\nu$) and the density ($\rho$) of the inhaled, warmed air; $\tau _{ij}$ comprises the subgrid-scale (SGS) stress tensor. As noted, to resolve both transitional behavior and secondary flow structures, we employ a dynamic SGS kinetic energy transport model^98,99,100. In this approach the SGS viscosity is obtained from a Kolmogorov–Prandtl type relation¹⁰¹, proportional to the square root of the SGS kinetic energy and to the local filter length scale, the latter taken as the cube root of the grid-cell volume. The SGS kinetic energy is defined as the unresolved portion of the velocity variance and evolves according to a filtered transport equation that includes advection by the resolved flow, diffusion driven by the SGS viscosity, production by resolved strain-rate interactions with SGS stresses, and a dissipation term scaled by the SGS kinetic energy and the filter size. The model constants appearing in the viscosity and dissipation closures are determined dynamically⁹⁸. See Supplementary Information for detailed mathematical formalism.

Airflow boundary conditions and specific numerical inputs: The inhalation simulations applied no-slip boundary condition (i.e., zero velocity) at the airway walls, with the flow conditions being driven by inlet-to-outlet (i.e., nostrils-to-tracheal base) pressure gradients. Therein, 0 Pa gauge pressure was assigned at the two nostril openings and a negative gauge pressure was assigned at the tracheal outlet, when the simulations were launched with targeted outlet mass flux commensurate with 15 and 30 L/min airflow. See the first subsection under §Results for the eventual total pressure gradients on convergence of the flow solutions. The numerical time steps were maintained at 0.0002 s for a total flow time of 0.35 s, consistent with reported findings¹⁰⁰. Performed on a segregated solver with SIMPLEC pressure-velocity coupling and second-order upwind spatial discretization, the transient formulations used a bounded second-order implicit scheme, balancing between accuracy (owing to the second-order formulation), stability (from the implicit approach), and boundedness (to prevent non-physical oscillations). To monitor solution convergence, we minimized the mass continuity residual to $\mathcal {O}(10^{-4})$ and the velocity component residuals to $\mathcal {O}(10^{-5})$. Additionally, assuming that inhaled air warms as it travels through the complex respiratory pathway, the air density $\rho$ was set at 1.204 kg/m$^3$ and its kinematic viscosity $\nu$ was 15.16$\times$10$^{-6}$ m$^2$/s.

Table 4 Length scale standards: Comparison between the Kolmogorov length scale ($\kappa$), the Taylor microscale ($\lambda$), and the mean grid scale (s). To evaluate $\kappa$ and $\lambda$, we have used mean values for k (turbulence kinetic energy) and $\varepsilon$ (turbulence dissipation rate) in the simulated datasets from all the regions barring the inlet and outlet surfaces within the corresponding anatomical domain. Mesh element length scale s is calculated as $(6\sqrt{2}\phi )^{1/3}$ (using the volume formula for a tetrahedron), where $\phi$ is the mean element volume estimated by dividing the total airspace volume ($\mathbb {V}$) mapped by the tetrahedral elements by the number of tetrahedral meshed elements ($n_e$). With the discretized airspace domains, $\mathbb {V} \in \{145283, 93603\}$ mm$^3$ and $n_e \in \{6320797, 4882819\}$, for AG$_1$ and AG$_2$, respectively.

Assessment of the computing scales and grid standards

With the LES approach, finer mesh refinement generally yields more accurate results. At the upper limit, LES can approximate Direct Numerical Simulation (DNS) when the grid spacing is smaller than or comparable to the Kolmogorov length scale, $\mathcal {K}$, which represents the smallest, most dissipative eddies in a turbulent flow. Another relevant length scale is the Taylor microscale, $\lambda$, which typically exceeds $\mathcal {K}$. These scales are mathematically defined as⁹⁶:

$$\begin{aligned} \begin{aligned} \mathcal {K}&= \left( \frac{\nu ^3}{\varepsilon }\right) ^{1/4} \quad \text {and}\quad \ \lambda = \left( \frac{10\nu k}{\varepsilon }\right) ^{1/2}, \end{aligned} \end{aligned}$$

(6)

where $\varepsilon$ comprises the turbulence dissipation rate and k is the turbulence kinetic energy. Simulation data for the inhaled airflow in the test geometries indicate that both $\lambda$ and $\mathcal {K}$ collapse to $\rightarrow \mathcal {O}(10^{-4})$ m, while the mean grid scale s is also $\mathcal {O}(10^{-4})$ m in both AG$_1$ and AG$_2$, implying that the test grids have been sufficiently resolved for reliable estimation of the transport physics. See Table 4 for the geometric parameters used while assessing s, along with specifics on the different length scales.

To further evaluate the meshing standards, we have assessed the ICEM mesh quality for the grids selected via sensitivity analysis. For AG$_1$, only 0.003% of elements fell into the lowest quality bin (${0.001-0.0.05}$). For AG$_2$, only 0.005% of elements fell into the lowest quality bin (${7\times 10^{-5}-0.0.05}$). The negligible fractions in the respective lowest bins, together with higher quality scores for the majority of elements and stable convergence behavior in the flow solution confirm the adequacy of the enforced mesh quality.

Finally, as mentioned earlier, the numerical scheme used time-steps of $\delta t_s = 0.0002$ s, for flow solution times of 0.35 s¹⁰⁰, chosen based on the time-step size needed to fully resolve the unsteady turbulent airflow field in a realistic upper airway model. Post-simulation, the solved data was examined to ensure that the implemented $\delta t_s$ was consistently smaller than the Kolmogorov time scale = $\left( \nu /\varepsilon \right) ^{1/2}$, with the mean Courant number $\lesssim$ 1 confirming stability of the numerical solutions.

Fig. 8

Simulation quality: Panel sets (A-B and C-D) respectively show the LES Index of Quality (LESIQ) in the simulated domains of AG$_1$ and AG$_2$. Therein, panels (A and C) additionally illustrate the airway enclosure, for visualization purposes. The maps in (B and D) are for a mid-lying section in each geometry. The section shown in B is same as the one on which field quantities have been plotted in Fig. 2D-E.

Examination of grid resolution using the LES Index of Quality

The predictive fidelity of LES solutions strongly depends on the adequacy of the computational grid in resolving the energy-containing eddies, with the corresponding SGS model accounting for the unresolved smaller-scale motions^102,103. Distribution of turbulent kinetic energy between the resolved and SGS components provides a useful indicator of numerical resolution adequacy. When the grid is sufficiently fine, most of the kinetic energy is captured within the resolved scales, and the SGS contribution remains small. To quantify this balance, we invoke the LES Index of Quality (LESIQ), which measures the fraction of the total turbulent kinetic energy resolved by the grid. According to the guideline proposed by Pope^104,105, a well-resolved LES should explicitly capture more than 80% of the total turbulent kinetic energy. The LESIQ is computed as:

$$\begin{aligned} \textrm{LESIQ}~(x, y, z;~t) = \frac{k_\textrm{resolved}(x, y, z;~t)}{k_\textrm{resolved}(x, y, z;~t) + k_\textrm{sgs}(x, y, z;~t)}, \end{aligned}$$

(7)

where $k_\textrm{resolved}$ and $k_\textrm{sgs}$ denote the resolved and SGS turbulent kinetic energies, respectively, and $\{x, y, z;~t\}$ represent the spatio-temporal coordinates. The computed results (see Fig. 8) confirm that both configurations meet the recommended resolution criterion. The mean LESIQ was 89.05% for AG$_1$ and 98.45% for AG$_2$. Based on the threshold requirements of 80%, demonstrably (with LESIQ $\gtrsim 90$%) the turbulent kinetic energy has been explicitly resolved in both test domains. It could hence be concurred that the numerical resolution is sufficient to capture the dominant turbulent structures with accurate replication of the large-scale complex flow dynamics.

Mathematical framework for inhaled particulate tracking

The spatiotemporal evolution of inhaled particulates entering the airway through the nostril inlets was numerically tracked against the simulated airflow field. Considering dilute conditions for the resulting particulate dispersion, the momentum transfer was one-way coupled; i.e., the airflow continuum was assumed to impact the particulate motion, while the particulates had no effect on the underlying flow field. A Lagrangian-based inert discrete phase model, employing a Runge-Kutta solver, was utilized to numerically integrate the particle transport equation:

$$\begin{aligned} \frac{d u_{pi}}{dt} = \frac{18\mu }{d^2 \rho _p} \frac{C_D Re_p}{24} (u_i - u_{pi}) + g_i \left( 1 - \frac{\rho }{\rho _p}\right) + f_i. \end{aligned}$$

(8)

In this formulation, $u_{pi}$ denoted the particulate velocity, $\rho _p$ was the material density of the particulates, and d (as before) represented the tracked particulate diameter. The term $Re_p$ referred to the particle Reynolds number, while $g_i$ indicated the gravitational acceleration component in the i-direction. The coefficient $C_D$ corresponded to the drag coefficient, and $f_i$ represented additional body forces per unit particulate mass, such as the Saffman lift force exerted by the local flow shear on small particulates transverse to the airflow. Also, the inhaled particulates were assumed to be sufficiently large for Brownian motion effects to be negligible.

The drag contribution in equation 8 was evaluated using the following forms for $Re_p$ and $C_D$:

$$\begin{aligned} Re_p = \frac{\rho _p d |u_i - u_{pi}|}{\mu } \quad \text {and} \quad C_D = a_1 + \frac{a_2}{Re_p} + \frac{a_3}{Re_p^2}, \end{aligned}$$

(9)

where $\mu$ denoted the molecular viscosity of the ambient air, and $a_1$, $a_2$, and $a_3$ were functions of $Re_p$, determined based on the spherical drag law¹⁰⁶. Subsequently, the particulate trajectories were derived from their spatiotemporal locations, $x_i(t)$, obtained through the numerical integration of the following velocity vector equation:

$$\begin{aligned} u_{pi} = \frac{d x_i}{dt}. \end{aligned}$$

(10)

The tracked particulates were designed to mimic environmentally dehydrated respiratory ejecta from an infected individual, now being inhaled by an exposed subject. When expelled, liquid-based respiratory particulates generally lose water and decrease in size^38,107, with the degree of shrinkage partly influenced by the proportion of non-volatile components within the particulates, such as dehydrated epithelial cell remnants, white blood cells, enzymes, DNA, sugars, and electrolytes. Therefore, while sputum is made up of up to 99.5% water, the post-dehydration particulates that are about to be inhaled have been shown to attain a density as high as 1.3 g/mL³⁸, which is what has been enforced in the intra-airway particulate tracking simulations. The dehydration resulting in such density variation causes the diameter of an emitted particulate to shrink to 30% of its original size³⁸, before being inhaled by the exposed subject. In other words, for example, an inhaled 1-$\mu$m particulate would carry the same viral load embedded in its pre-dehydration 3.33-$\mu$m size. The computationally tracked (inhaled) particulates, which mimicked the dehydrated and shrunken ejecta as described, were monodispersed and had the following diameters d: ${0.1 - 0.9~\mu m}$ (with increments of 0.1 ${\mu \textrm{m}}$); ${1.0 - 4.5~\mu \textrm{m}}$ (with increments of 0.5 ${\mu \textrm{m}}$); and ${5.0 - 50.0~\mu \textrm{m}}$ (with increments of 1.0 ${\mu \textrm{m}}$)–beyond this size, prompt gravitational sedimentation is expected³⁸, thus barring such particulates from being inhaled into the airway by the exposed subject. For each test diameter within the stated range, the total number of particulates tracked were $N_i$: 1372 in AG$_1$ and 2506 in AG$_2$, with $N_i$ being the total number of mesh facets (faces) mapping the nostril inlets. Therein, the spatial coordinates of the initial particulate positions (in the numerical domain) coincided with the centroids of the inlet mesh facets.

Discrete phase boundary conditions: The particulate tracking simulations imposed a discrete phase boundary condition of ‘trap’ along the enclosing surfaces of each airspace, mimicking mucosal adhesion. Accordingly, the intra-airway spatial tracking of a particulate pathline was terminated once it entered the mesh element layer adjacent to the enclosing cavity walls and the regional deposition fractions were recorded. In addition, to ensure that these particulates do not move out of the numerical domain at the upwind sites, the discrete phase boundary condition was assigned as ‘reflect’ at the nostrils and the mouth. The particulates leaving the numerical domains through the tracheal outlets (e.g., see Fig. 1B) were also recorded (with the local boundary condition being ‘escape’); the resulting finding represented the fraction of inhaled particulates that were able to navigate downwind to the infective bronchial recesses in the LRT⁹⁷.

Validation of the computational scheme by benchmarking with physical experiments

3D printing of an anatomically realistic respiratory cast

To assess the numerically predicted particulate deposition trend vis-á-vis physical experiments, we generated a 3D-printed solid twin of AG$_1$ (see Fig. 6A-C), made from the stereolithography material Watershed (DSM Somos, Elgin, Illinois); the printing of the main cast being done at ProtoLabs (Maple Plain, Minnesota)⁷⁸. In the glottic region of the airspace (see marked region in Fig. 1B), we incorporated a hollow groove (see Fig. 6A) designed to snugly accommodate a plug (see Fig. 6B) that captures the CT-based internal anatomical topology at the glottis. The primary objective was to collect wall deposits within the plug, measure the localized deposition fraction, and compare these experimental measurements with our computational projections for localized deposition at the same region. The glottic plug was 3D-printed using an Original Prusa i3 Mk3 printer, produced by Prusa Research, with flexible TPU (thermoplastic polyurethane) filament supplied by SainSmart.

Experimental logistics and measurements

Figure 6C shows the 3D-printed AG$_1$ fitted to a negative pressure pump that drew a constant air flux of 30 L/min through the tracheal outlet for a period of 2 minutes before and during nebulization. To obtain intra-airway particulate transport trends, aerosolized droplets with a diameter of 9.5 $\mu$m¹⁰⁸ were delivered via a modified Pari LL jet nebulizer inserted at the mouth opening of the anatomical reconstruction. Each experimental run utilized 2 mL of a nebulized aqueous caffeine solution at a concentration of 10 mg/mL, with a total of three runs conducted. Following nebulization, we measured localized deposition after the droplets settled on the inner walls of the glottic plug (Fig. 6B and D-F) using ultraviolet (UV) absorbance of the caffeine solutions, analyzed with a Thermo Scientific Biomate 5 spectrophotometer at the University of North Carolina’s Biomarker Mass Spectrometry Core Facility. For each of the three samples, the glottic plug was rinsed in 60 mL of deionized water for 10 minutes using an ultrasonic bath. The calibration range was adjusted to bracket the samples, and we recorded the signal for each sample in triplicate. A blank test insert was rinsed in the same manner to ensure there was no background signal at 273 nm, using a 1 mL aliquot for the UV absorbance measurement. The resulting measurements are presented in Table 3 and Fig. 6H, with $C_E$ quantifying the caffeine concentration in the solutions resulting from rinsing the glottic deposits with deionized water and $G_E$ representing the percentage of the nebulized solution that landed in the glottis while inhalation was simulated by the pressure pump drawing in air.

Given the described experimental parameters, the primary concentration measurements $C_E$ (in $\mu$g/mL) from rinsing the glottic plug are utilized to derive the local deposition fractions $G_E$ using the following conversion:

$$\begin{aligned} G_E = \left[ \frac{60\, C_E \times 10^{-3}}{10 - C_E\times 10^{-3}}\right] (\mathrm {in~mL}) \times \frac{1}{2 \,(\mathrm {in~mL})} \times 100~\%. \end{aligned}$$

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Benchmarked computational simulation for comparison with experimental measurements

To establish the reliability of the broader computational results, we conducted a benchmarking exercise by replicating the experimental conditions in silico, utilizing the same numerical flow scheme as implemented in the larger study. For a simulated airflow rate of 30 L/min in AG$_1$, we computationally tracked particulates with diameters $d_C \in 9.50 \pm 0.25~{\mu \textrm{m}}$ (the perturbation on $d_C$ accounting for fluctuations in the monodisperse nature of the nebulized particulates in the experiments). The material density of the simulated inert particulates was set at 1 g/mL, to mimic the aqueous solution used in the experiments. Similar to the nebulized droplets in the experiments, the simulated particulates were administered into the airspace, specifically for this benchmarking, through the mouth with an initial speed of 10 m/s (based on reported data on jet nebulizers)–the velocity vector lying on the yz plane and oriented at 45$^{\circ }$ to both y and z-axes (see Fig. 1B)–ensuring a realistic entry angle through the sectional plane at the mouth. The last two columns in Table 3 report the computationally simulated deposition fractions at the glottis (the precise region marked by the experimental glottic plug); with the total number of particulates administered and tracked for each $d_C$ being 1892, corresponding to the number of mesh facets across the planar cross-section at the mouth opening. These localized deposition fractions ($G_C$) are equivalent to the ratio of the number of particulates depositing at the glottis over the total number of particulates administered, converted to percentage.

The mean glottic deposition fraction from the experiments ($G_E$) is found to be approximately 0.8 times that of the computational projection ($G_C$); see Table 3, and additionally Fig. 6G-H for a visual comparison. We have argued that this comparability between the experiments and simulations could be deemed satisfactory, with the difference primarily attributable to post-deposition surface displacement of the deposited droplets; see the last subsection under §Results.

Virological inputs to the flow physics model

VARV, the causative agent of smallpox, is a classic example of an Orthopoxvirus, with an established infectious dose ($I_D$) ranging between 10−100 pfu²⁵; this range is used in equation 2. Critically though, for variola, the conversion of pfu to DNA copies is still an unresolved question. Earlier findings⁸ reported plaque counts from throat samples of smallpox patients (with variola major), enabling an estimation of the viral load distribution in particulates expelled by a host; accordingly, we have calculated the mean viral load $V_L = 10^{2.952}$ pfu/mL (as an input to equation 2). The raw data comprised 147 throat swab specimen taken from 32 hospitalized patients at progressive stages of the disease over a 2-week period; see Fig. 4C. In the historical study⁸, the cotton swabs were soaked in Hanks’ basal salt solution (BSS) containing 0.5% bovine albumin and antibiotics (penicillin and streptomycin); subsequently, the virus titre was obtained from the fluid obtained by squeezing the swabs after they had been dipped in 1-mL of BSS. Although swabs can possibly carry higher pathogen concentration (than in exhaled particulates by a host) owing to the invasive nature of sample collection, the BSS dilution explains the standard use²⁵ of the eventual titre information as the viral loading parameter in the airborne particulates (composed of dehydrated respiratory ejecta from an infective host) inhaled by an exposed subject.

For MPXV, the input value for $I_D$ in equation 2 was set to 200 pfu, based on measurements for the Zaire V79-I-005 strain⁵². Additionally, the mean viral load ($V_L = 10^{2.92}$ pfu/mL) was estimated from Ct/CN counts in saliva samples⁴² for MPXV belonging to Clade II (subclade IIb)^66,67,68. Herein, we note that the findings for MPXV could be further improved with matched data sets for each clade, which could additionally address clade differences in disease onset mechanisms. In general, for both Orthopoxviruses considered here, the reported ranges for the critical exposure durations ($\tau _c, \widetilde{\tau_c}$) span several hours: approximately 1−19 hours for VARV and 24−40 hours for MPXV; the latter based on current phylogenetic data. This assumes continuous exposure to a live pathogen source–and as such–while pox viruses like MPXV can indeed stay stable for hours¹⁰⁹, the described mechanophysiological model remains agnostic to virus stability-related long-term inactivation of infectious particulates.