Abstract
How human respiratory physiology and the transport phenomena associated with inhaled airflow in the upper airway proceed to impact transmission of SARSCoV2, leading to the initial infection, stays an open question. An answer can help determine the susceptibility of an individual on exposure to a COVID2019 carrier and can also provide a preliminary projection of the stillunknown infectious dose for the disease. Computational fluid mechanics enabled tracking of respiratory transport in medical imagingbased anatomic domains shows that the regional deposition of virusladen inhaled droplets at the initial nasopharyngeal infection site peaks for the droplet size range of approximately 2.5–19 \(\upmu \). Through integrating the numerical findings on inhaled transmission with sputum assessment data from hospitalized COVID19 patients and earlier measurements of ejecta size distribution generated during regular speech, this study further reveals that the number of virions that may go on to establish the SARSCoV2 infection in a subject could merely be in the order of hundreds.
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Introduction
Severe acute respiratory syndrome coronavirus 2 (SARSCoV2) has been identified as the causative agent for coronavirus disease 2019 (COVID19), that has inflicted a global pandemic with nearly 114 million confirmed infections and over 2.5 million deaths worldwide, as of lateFebruary 2021; for details, see^{1}.
As is wellknown by now, transmission of respiratory infections such as COVID19 occurs through carriage of pathogens via droplets of different sizes produced during sneezing, coughing, singing, normal speech, and even, breathing^{2}. Accordingly, the means of persontoperson infection are projected to be threeway^{3}: (a) inhalation of virusladen droplets emitted by an infected individual at closerange; (b) inhalation of vaporized droplet nuclei that can float in air for hours; and (c) contaminating the respiratory mucosa through physical contact to external surfaces (fomites) with droplet deposits sitting on them. While (a) is valid for shortdistance exposures to the COVID19 carrier, transmission through modes (b) and (c) can happen over larger distances and longer time scales. However, clustering trends of infection spread (e.g. in industrial units^{4}, in closed groups^{5}, and inside households^{6}) suggest that closerange exposures, through inhalation of respiratory ejecta from infected individuals, is a critical determinant in worsening of the pandemic. A followup question might be—what entails an exposure? A key component therein are the respiratory droplet sizes one is exposed to. Coughing and sneezing typically generate droplets with lengthscales of \(\mathcal {O}(10^2)\) to \(\mathcal {O}(10^3)\) \(\upmu \), while oral droplets ejected during normal speaking can range over \(\sim \) 0.1–500 \(\upmu \)^{3,7}. Some of the competing environmental effects determining the fate of these droplets are the ambient temperature and humidity (e.g. low relative humidity induces fast evaporation and shrinkage of the droplets), along with the size of the droplet that controls its inertia and the gravitational force acting on it. While smaller droplets would stay airborne for longer, the larger droplets tend to fall fast ballistically; with the critical size for this transition being in the vicinity of 100 \(\upmu \)^{8,9}. Of note here, this study does not insist on any nomenclatural distinction between “aerosols” and “droplets”, owing to ambiguities^{10} in common perception, and simply refers to all expiratory liquid particulates as droplets.
For tracking what range of virusbearing droplet sizes might be more potent for SARSCoV2 transmission and to eventually induce infection, it is crucial that we identify the initial infection sites. At least two 2020 studies^{11,12} reveal a striking pattern of relatively high SARSCoV2 infectivity in ciliated epithelial cells along the nasal passage lining in the upper airway, to less infectivity in cells lining the throat and bronchia, and finally to relatively low infectivity at the lung cells. Such viral tropism is decidedly governed through angiotensinconverting enzyme 2 (ACE2), which is a singlepass type I membrane protein and is the surface receptor that the virus utilizes to intrude into cells. ACE2 is abundant on ciliated epithelial cells, but is relatively scarce on the surface of the lower airway cells. While these findings are for in vitro samples, deposition of virusladen droplets along the anterior nasal airway might not be so effective as to launch an infection despite the presence of ciliated cells, since the mucus layer provides some protection against virus invasion and infection^{3}. This sets up nasopharynx (i.e. the region in the upper airway posterior to the septum and comprising the superior portion of the pharynx; for reference, see Fig. 1) as the main initial infection site; it acts as the seeding zone for subsequent infection of the lower airway, mediated by the aspiration of virusladen boluses of nasopharyngeal fluids^{11,13,14,15}. The ansatz steering the identification of nasopharynx, as the dominant initial site of infection during SARSCoV2 pathogenesis, is supported by the efficacy^{16} of nasopharyngeal swab testing for COVID19 diagnosis, when compared to oropharyngeal swabs. So at this point, a valid question to ask would be: what are the dominant inhaled droplet sizes that are making their way to the nasopharynx?
Respiratory droplets, on being expelled, typically lose water and shrink;—the extent of which partially depends on the fraction of nonvolatile constituents present in the droplets, e.g. dehydrated epithelial cell remnants, white blood cells, enzymes, DNA, sugars, electrolytes etc. So, although sputum is composed of 99.5% water; ejected droplets, on dehydration, have a higher density of 1.3 g/mL^{17}, which is what has been used for the initial set of droplet tracking simulations here, with the assumption that the nonvolatile weight fraction is in the 1–5% range. Such dehydration contracts the expelled droplet diameter to 20–34% of the initial size. Considering 100 \(\upmu \) as the critical size prompting ballistic sedimentation, this study tracks inhaled droplet sizes that could still be airborne after dehydration. Hence, the tested size range is 0.1–30 \(\upmu \), assuming a conservative dehydrationinduced shrunk volume of 30% of the initial size^{18}. Choice of the smallest tracked droplet size is dictated by SARSCoV2 dimension, which is in between 0.08 and 0.2 \(\upmu \), with an average physical diameter of 0.1 \(\upmu \)^{19}.
Next piece in this puzzle involves the breathing parameters. Allometric relations^{20} put the minute inhalation at 18.20 L/min for a 75kg male and 15.05 L/min for a 75kg female, for gentle steady breathing while sitting awake. In general, inspiratory rates can stretch over \(\sim \) 15 to 85 L/min, based on whether the individual is inhaling gently or breathing in forcefully. This study uses computational fluid dynamics (CFD) in anatomically realistic upper airway geometries to simulate droplet transmission at four different inhalation rates, viz. 15, 30, 55, and 85 L/min; notably these discrete flow rates are also the ones traditionally used^{21} for checking filtration capacities of protective facecoverings and respirators. The flow physics undergo a transition over this range; e.g. 15 L/min through nasal conduits is in laminar regime, the transport mechanism however devolves into turbulence at higher inhalation rates.
Methods
Anatomic geometry reconstruction
Computed tomography (CT)based in silico model generation was accomplished according to relevant guidelines and regulations, with the anatomic geometries being reconstructed from existing deidentified imaging data from two CTnormal subjects. The use of the archived and anonymized medical records was approved with exempt status by the Institutional Review Board of the University of North Carolina (UNC) at Chapel Hill, with the requirement of informed consent being waived for retrospective use of the deidentified scans in computational research. The test subjects include a 61 yearold female (subject for anatomic reconstruction 1, or AR1) and a 37 yearold female (subject for anatomic reconstruction 2, or AR2). In context to the imaging resolution, the CT slices were collected at coronal depth increments of 0.348 mm in AR1’s scans and 0.391 mm in AR2’s scans. The nasal airspaces were extracted from the medical grade scans (see the bottom row in Fig. 1 for representative CT slices) over a delineation range of − 1024 to − 300 Hounsfield units, and was complemented by careful handediting of the selected pixels to ensure anatomic accuracy. For this step, the DICOM (Digital Imaging and Communications in Medicine) scans for each subject were imported to the image processing software Mimics Research 18.0 (Materialise, Plymouth, Michigan).
The reconstructed geometries were exported as stereolithography files to ICEMCFD 15.0 (ANSYS Inc., Canonsburg, Pennsylvania), and then meshed spatially into minute volume elements. Conforming with established mesh refinementbased protocols^{22,23}, each computational grid contained more than 4 million unstructured, graded tetrahedral elements (e.g. 4.54 million in AR1, 4.89 million in AR2); along with three prism layers of 0.1mm thickness at the airway walls, with a height ratio of 1. The nostril inlet planes comprised 3015 elements in AR1 (1395 elements on left nostril plane, 1620 elements on right nostril plane) and 3000 elements in AR2 (1605 on left nostril plane, 1395 on right nostril plane).
Numerical simulations
The study considers droplet transport for four different inhaled airflow rates, viz. 15, 30, 55, and 85 L/min. The lower flow rate (i.e. 15 L/min) corresponds to comfortable resting breathing, with the viscouslaminar steadystate flow physics model standing in as a close approximation^{24,25,26,27,28,29,30,31,32,33,34,35}. At higher flow rates (extreme values of which may sometimes lead to nasal valve collapse), the shear layer separation from the tortuous walls of the anatomic geometries results in turbulence^{36,37,38,39,40}. While accounting for the turbulent characteristics of the ambient airflow, the study averages the droplet deposition percentages from implementation of two distinct categories of numerical schemes, viz. (a) shear stress transport (SST) based k\(\omega \) model, which is a subclass under Reynoldsaveraged Navier Stokes (RANS) schemes that parameterize the action of all turbulent fluctuations on to the mean flow; and (b) Large Eddy Simulation (LES). In this work (see results), the two numerical techniques depict high correlation in terms of droplet deposition at the nasopharynx. However, it should be noted that while the SST k\(\omega \) scheme, a 2equation eddyviscosity model, is computationally less expensive; it averages the short timescale flow artifacts, such as the transient vortices (e.g. the lowpressure Dean vortices that are common in tortuous channels and can act as droplet attractors), and hence the prediction of droplet transport affected by the simulated airflow may at times contain errors. LES is computationally more expensive, it separates the turbulent flow into largescale and smallscale motions, and accounts for the small fluctuations through a subgrid scale model (for this study, Kinetic Energy Transport Model was used as the subgrid scale model^{41}). We took the averaged estimates for regional droplet deposition (along the in silico nasal tissue surfaces) from the two schemes, to minimize probable statistical and algorithmic biases.
The computational schemes implemented in the meshed domains employed a segregated solver on ANSYS Fluent 15.0, with SIMPLEC pressurevelocity coupling and secondorder upwind spatial discretization. Solution convergence was monitored by minimizing the mass continuity and velocity component residuals, and through stabilizing the mass flow rate and static pressure at the airflow outlets. For the pressuredriven flow solutions: typical convergence runtime in a laminar simulation with 5000 iterations was approximately 5–6 h for 4processor based parallel computations executed at 4.0 GHz speed. The corresponding runtime for a RANS simulation was \(\sim \) 12 h; for an LES computation, it was 4–5 days. Note that for the LES work, the simulated flow interval was 0.5 s for the 30 L/min case, with 0.0002 s as the timestep^{42} and it was 0.25 s for the 55 and 85 L/min flow rates with the timestep at 0.0001 s. In the computations, assumed air density was 1.204 kg/m\(^3\) and \(1.825\times 10^{5}\) kg/m s was used as dynamic viscosity of air.
Following set of boundary conditions were enforced during the simulations: (1) zero velocity at the airwaytissue interface i.e. at the walls enclosing the digitized nasal airspace (otherwise commonly referred to as the no slip condition), along with “trap” boundary condition for droplets whereby the tracking of a droplet’s transport would cease once it has landed on the walls; (2) zero pressure at nostril planes, which were the pressureinlet zones in the simulations, with “reflect” boundary condition for droplets to mimic the effect of inhalation on the droplet trajectories if they are about to fall out of the anterior nasal domain; and (3) a negative pressure at the airflow outlet plane, which was the pressureoutlet zone, with “escape” boundary condition for droplets, i.e. allowing for the outgoing droplet trajectories to leave the upper respiratory airspace. Mean inlettooutlet pressure gradients were − 9.01 Pa at 15 L/min, − 26.65 Pa at 30 L/min, − 73.73 Pa at 55 L/min, and − 155.93 Pa at 85 L/min. For a reference on the general layout of the anatomic regions, see Fig. 1.
On convergence of the airflow simulations, inhaled droplet dynamics were tracked by Lagrangianbased discrete phase inert particle transport simulations in the ambient airflow; with the localized deposition along the airway walls obtained through numerically integrating transport equations^{43} that consider contribution of the airflow field on the evolution of droplet trajectories, along with the effects for gravity and other body forces such as the Saffman lift force that is exerted by a flowshear field on small particulates moving transverse to the streamwise direction. Also, the droplet size range is considered large enough to discount Brownian motion effects on their spatial dynamics. Note that the study simulated the transport for 3015 droplets of each size in AR1 and 3000 droplets of each size in AR2, the numbers being same as the number of elements on the nostril inlet planes which were seeded with the tobetracked droplets for the droplet transport simulations. For the numerical tracking, the initial mass flow rate of the inert droplets moving normal to the inlet planes into the nasal airspace was required to be nonzero, and was set at \(10^{20}\) kg/s. After the transport simulations, the postprocessing of the droplet transmission data along the airway walls provided the regional deposition trends at the nasopharynx.
The numerical methods, discussed and used here, are an extension from one of our recent publications^{43} in this journal. The questions explored in the present study are, of course, very different and new, and the findings can be potentially substantial in our evolving field of knowledge on respiratory pathogen transport, with the current focus being on SARSCoV2 transmission mechanisms. The reader should also note that the numeric protocol has been rigorously validated in the earlier publication^{43}, through comparing the regional deposition trends along the inner walls of similar in silico nasal anatomic domains to the in vitro spray tests performed in 3Dprinted solid replicas of the same reconstructions.
Assessing the probability for at least 1 virion being embedded on a respiratory ejecta droplet emanating from an infected individual
A recent landmark paper^{17} reports the probability that a 10\(\upmu \) undehydrated droplet, constituted from the respiratory ejecta of an infected individual, will contain at least one SARSCoV2 virion is 0.37%. The probability drops to only 0.01%, if the undehydrated droplet diameter is 3 \(\upmu \). We attempt in this study to verify their reported data through a simple mathematical model, while at the same time laying down the calculation framework to estimate the number of virions that are directly carried to the nasopharynx by the droplets that could now be inhaled by an exposed individual.
In the oral fluids of hospitalized COVID19 patients, the mean SARSCoV2 RNA load has been measured^{14} at \(7.0 \times 10^6\) copies/mL. Imposing a continuum approach and assuming a homogeneous concentration of the virions in the undehydrated respiratory ejecta, 1 m\(^3\) of oral ejecta therefore has \(7 \times 10^{12}\) virions (given that SARSCoV2 is a singlestranded RNA virus^{44}). Consequently, an expiratory droplet of diameter \(\delta \) (in \(\upmu \)) would have
In other words, to come across 1 virion in undehydrated droplets of diameter \(\delta \) (in \(\upmu \)), one would need to “check”
Thus, the probability of there being at least 1 virion in an undehydrated droplet of diameter \(\delta \) (in \(\upmu \)) is
Plugging in \(\delta = 3\,\upmu \) and \(10\,\upmu \) in Eqs. (1)–(3) respectively generates \(\mathcal {P} = 0.01\%\) and \(0.37\%\), thereby matching prior findings^{17} and validating the above mathematical idealization. We have used Eq. (1) (see next, “Estimating the number of virions transmitted to the nasopharynx”) for further quantification of the number of virions carried by the droplets that undergo nasopharyngeal deposition.
Estimating the number of virions transmitted to the nasopharynx
The logic diagram in Fig. 2 lays out the modus operandi for the estimation of the number of virions that are carried by the inhaled droplets to the nasopharynx. In essence, we use known data on the distribution of dehydrated droplet sizes that are expelled during regular speaking^{7}. The CFD simulations are then used to quantify what percentage of the droplets, purportedly expelled by a COVIDpositive patient in proximity, will land at the nasopharynx of the exposed individual. We obtain both the number and sizes of the droplets that undergo nasopharyngeal deposition, and then use Eq. (1), to quantify the approximate number of virions that should be transmitted by those droplets. Note that at this point, we do account for the environmental dehydration, e.g. an expelled droplet of diameter \(\delta \) \(\upmu \) would undergo dehydration and with mean shrinkage^{17}, transform into a droplet of diameter \(\mathbb {D} = 0.30 \times \delta \) (in \(\upmu \)) before being inhaled by an exposed subject, by losing bulk of its volatile water components, and yet would carry the same viral load that was embedded in its initial \(\delta \)\(\upmu \) size.
Preprint
A preprint version of this manuscript has been screened for content and posted on medRxiv. https://doi.org/10.1101/2020.07.27.20162362.
Results
Droplet size range that targets the nasopharynx
The overall droplet size range of 2.5–19 \(\upmu \) (in AR1: 2.5–19 \(\upmu \), in AR2: 2.5–15 \(\upmu \)) registers the peak, in terms of the percentage of dehydrated droplets of each size that are deposited at the nasopharynx of an exposed individual. The range is determined by a cutoff of at least 5% deposition for around 3000 tracked droplets (viz. 3015 in AR1, 3000 in AR2) of each size. Panel A in Fig. 3 displays the heatmaps for nasopharyngeal deposition (NPD) for different droplet sizes, during inhalation at the four tested airflow rates. The discrete droplet sizes, that were tracked, have been marked along the horizontal axis of the heatmaps. The patch bounded by the grey lines can, in fact, be a definitive graphical technique to delineate the hazardous droplet size range for various airborne transmissions.
What if there were less or no environmental dehydration?
Findings in “Droplet size range that targets the nasopharynx” assume that the postdehydration density of the respiratory droplets (expelled by the carrier and now being inhaled by the exposed individual) is at 1.3 g/mL. If, on the contrary, there is little or no dehydration, the ejected droplet density would remain approximately at \(\sim \) 1 g/mL, since pure saliva contains 99.5% water while exiting the salivary glands. With such material density, the inhaled droplet size range for peak NPD upscales to 3–20 \(\upmu \), as depicted in Fig. 4. Note that for these lighter droplets, the tested droplet size range was expanded to 0.1–35 \(\upmu \) (instead of 0.1–30 \(\upmu \), as used for the dehydrated heavier droplets). The somewhat lighter droplets can now penetrate further into the intranasal airspace, the transport process being aided by the inspiratory streamlines. The underlying physical principle can be assessed in terms of the nondimensional Stokes number and the inertial motion of the droplets.
The Stokes number (Stk) can be mathematized as^{45}
Here U is flow rate divided by flux area, \(\xi _{\mathbb {D}}\) is the material density of the inhaled droplets, \(C_c\) is the Cunningham slip correction factor, \(\upmu \) is the dynamic viscosity of the ambient medium i.e. air, and d is the characteristic diameter of the flux crosssection. All other flow and morphological parameters staying the same, it is straightforward to show from Eq. (4) that
where \((\xi _{\mathbb {D}_i},\mathbb {D}_i)\) for \(i = 1, 2\) are two different droplet density and droplet size pairings. In this study, we have seen that for say, \(\xi _{\mathbb {D}_1} = 1.3\) g/mL, the droplet size range where NPD peaks is 2.5–19 \(\upmu \). Let us now use the scaling argument in Eq. (5), to predict the corresponding size range if the droplets do not undergo dehydration, and as such \(\xi _{\mathbb {D}_2} = 1.0\) g/mL. If the tobepredicted size range that would generate peak NPD for the lighter droplets is represented by \(\mathbb {D}'_{\text {min}}\) to \(\mathbb {D}'_{\text {max}}\) (in \(\upmu \)), then
This results in \(\mathbb {D}'_{\text {min}} = 2.9\) \(\upmu \) and \(\mathbb {D}'_{\text {max}} = 21.7\) \(\upmu \). Despite the simplicity of the scaling analysis, the theoretically predicted range 2.9–21.7 \(\upmu \) is reassuringly close to the CFDbased projection of 3–20 \(\upmu \).
Statistical analysis and data interpretation
Consider representatively the simulated intranasal transmission for regular dehydrated droplets. Panel B in Fig. 3 plots the NPD values from RANS (along horizontal axis) and LES (along vertical axis) schemes, implemented for the higher inhalation rates (i.e. 30, 55, and 85 L/min). The simulation outputs on NPD (%) for different droplet sizes are linearly correlated with an average Pearson’s correlation coefficient of 0.98 for 30 L/min, 0.93 for 55 L/min, and 0.95 for 85 L/min. Subsequent check of the slope m for the linear bestfit trendline, through the scatter plots of RANS and LESbased NPD data, indicates how similar the estimates are quantitatively; the mean measures therein being \(m = 1.113\) for 30 L/min, \(m = 1.052\) for 55 L/min, and \(m = 1.177\) for 85 L/min; with the value 1 signifying exact equivalence. The statistical operations were carried out on Mathematica 12.0 (Wolfram Research, Champaign, Illinois).
Also at this point, to think of a realistic exposure to a COVID19 carrier: the vulnerable individual can be considered to inhale at different airflow rates over the duration of exposure. In such context, Panel A in Fig. 5 extracts the averaged nasopharyngeal deposition for the different tested inhalation rates in the two test subjects. Such inhalationaveraged transmission presents an approximate dehydrated droplet size range of 2.5–15.0 \(\upmu \), for a minimum 2% NPD for each droplet size.
Droplets that are better at carrying the virions
The next pertinent question is: how effective are these droplets at carrying virions? SARSCoV2 belongs to a diverse family of singlestranded RNA viruses^{44,46}, and as noted before, virological assessments^{14} done on the sputum of hospitalized COVID19 patients show an averaged viral load of \(7.0 \times 10^6\) RNA copies/mL of oral fluid, with the peak load being \(2.35 \times 10^9\) copies/mL. For the average load, simple calculations (see “Assessing the probability for at least 1 virion being embedded on a respiratory ejecta droplet emanating from an infected individual” in the methods) show that the probability that a dehydrated 10\(\upmu \) droplet (contracted from its original size of \(\sim \,33.33\) \(\upmu \)) will carry at least 1 virion is 13.6%. The same number is 45.8% for a postshrinkage 15\(\upmu \) droplet (contracted from its original size of \(\sim 50\) \(\upmu \)). The probability drops exponentially to 0.2% for a 2.5\(\upmu \) dehydrated droplet (contracted from its original size of \(\sim 8.33\) \(\upmu \)). Now, with existing data on the size distribution of expelled droplets during normal speaking (see Panel B, Fig. 5), the proportion of virion deposits at the nasopharynx by different droplet sizes can be computed (see Panel C, Fig. 5) by using the transmission data presented in Fig. 3, coupled with the mathematical framework laid out in “Estimating the number of virions transmitted to the nasopharynx”. The virion deposition trends are again from droplets that are being inhaled postdehydration.
Significantly enough: in the absence of environmental dehydration, the probability of 1 virion being embedded in, for instance, a 10\(\upmu \) droplet plummets to 0.37% (see Fig. 6). This rationalizes why in geographic regions with high humidity (and hence relatively less dehydration and shrinkage of respiratory ejecta), the pandemic’s spread has been somewhat measured^{47,48}.
On the SARSCoV2 infectious dose
The infectious dose is a fundamental virological measure quantifying the number of virions that can go on to start an infection; the value of which is still not conclusively known for SARSCoV2^{49}. Theoretically, according to the independent action hypothesis^{50}, even a single virion can potentially establish an infection in highly susceptible systems. Whether the hypothesis is true for humans and specifically for SARSCoV2 transmission is as yet undetermined. The rapid spread of COVID19 though a priori suggests a small infective dose for the disease, that is triggering interhuman transmission.
Since it would be unethical at this point to expose subjects to SARSCoV2 (especially in the absence of wellevidenced remediating therapeutics—as of February 2021), this study proposes a novel strategy synergizing computational tracking and virological data, to estimate the infectious dose. Based on the nasopharyngeal transmission trends (Fig. 3) and the virion transmission data (Panels B,C of Fig. 5), for a 5min exposure: the number of virions depositing at the susceptible individual’s nasopharynx is 11, considering average RNA load in the carrier’s sputum. On the contrary, if the infecting individual is in the disease phase with peak RNA load, as many as 3835 virions will be deposited on the nasopharynx of the exposed individual over 5 min (see Panel D in Fig. 5).
To put the computed prediction on virion transmission into context, consider now the March 2020 Skagit Valley Chorale superspreading incident^{5}, where a COVID19 carrier infected 52 other individuals in a 61member choir group. Exposure time there was reported to be 2.5 h. The subjects were positioned close to each other; which justifies discounting the effect of spatial ventilation for a conservative estimate of the number of virions that a susceptible individual would have been exposed to. Therefore, for an average RNA load (assuming that the carrier had mildtomoderate symptoms), coupled with derived exposure estimates from previous paragraph, the number of virions depositing at a closelysituated individual’s nasopharynx over the 2.5h duration approximates to: (11/5) virions per minute \(\times \) 60 min \(\times \) 2.5 h \(= 330\). So, an order of \(\mathcal {O}(10^2)\) could be reckoned as a conservative estimate for SARSCoV2’s infectious dose; the scale agreeing with preliminary estimates based on the viral replication rates^{51}.
A review of projections for the SARSCoV2 infectious dose, as per other reports
A recent dose titration study^{52} of SARSCoV2 in a ferret model has shown 500 as the lower tested limit for the number of virions needed to launch an infection; the study extracted the dose in terms of the plaqueforming units (PFU)—which is a measure used in virology to describe the number of virus particles capable of forming plaques per unit volume. The quantitative estimate of the infectious dose, being of the order \(\mathcal {O}(10^2)\), agrees with the findings presented here. Moreover, though not yet shown through experimental models in humans, multiple preliminary critiques^{53,54} expect the infectious dose of SARSCoV2 to be similar to that of other coronaviruses, such as SARSCoV. According to a key publication^{55} from the 2010s, the SARSCoV dose that correlated to 10% and 50% adverse responses (i.e., illness) was estimated respectively at 43 and 280 PFU. This confirms the feasibility of the infectious dose (\(\approx 300\), or more conservatively \(\mathcal {O}(10^2)\)) derived here for SARSCoV2.
Discussion

On the practicality of viral load used in the calculations—Through tissue culture examinations for respiratory infections, it is fairly well recognized^{56} that only a small fraction of virions are actually able to infect a human cell, and that this fraction decreases rapidly with increasing duration from the time of initial infection of the carrier. So, the SARSCoV2 infectivity is being conjectured to peak well before the viral load reaches a maximum. This substantiates the use of averaged viral load (i.e. \(7.0 \times 10^6\) RNA copies/mL) in the carrier’s sputum for the virological calculations, while deducing the conservative upper estimate for the SARSCoV2 infectious dose.

On the significance of the hazardous droplet sizes—Whereas the computed data is postprocessed to specifically extract the droplet sizes that tend to target the nasopharynx, a vastly larger remainder (comprising predominantly the droplets that are smaller than 5 \(\upmu \)) actually go further down the respiratory tract (considering that the air passageways narrow down to just a few microns in the lower airway). However, the significantly pronounced surface area of the lower airspaces, together with the relative scarcity of ACE2 receptors there, validates the robustness of the modeling approach i.e. focusing on the droplets that deposit on the ACE2rich epithelial cells at the nasopharynx. Also, the probability of droplets smaller than 5 \(\upmu \) to carry a virion is often trivial; e.g. the probability of containing a virion is only 1.7% for a 5\(\upmu \) dehydrated droplet; for the related analytical framework, see “Assessing the probability for at least 1 virion being embedded on a respiratory ejecta droplet emanating from an infected individual”.

On the number of simulated droplets—The reader should note that the modeling framework to track approximately 3000 monodispersed droplets of discrete sizes is a mathematical idealization employed to obtain statistically robust data on the intranasal droplet transmission trends for each size and identify the specific size range for which a higher fraction of the inhaled droplets would land directly at the infectionprone nasopharynx in the upper respiratory pathway. Hence, the focus was not on replicating a realistic quantity and size distribution of droplets, which would be functions of a wide range of environmental and locational factors^{57}, such as wind and interindividual distances.

On the “trap” boundary condition in the simulations—The numerical scheme used in this study tracks the inhaled droplets as long as they are airborne inside the intranasal space and stops the tracking when the droplets land on the airway tissue surfaces, which have a noslip boundary condition. In other words, the droplet trajectories would cease once they touch a wall and the effects of any possible accretion and erosion of droplets at the walls were not considered. The feasibility of such entrapment is backed by the physicochemical properties^{58} of mucus which washes the upper airway walls. Mucus in the nose exhibits^{59} a pH range of 5.5–6.5. Mucins, which are the main component of the mucus secretions from goblet and epithelial cells, are known for inducing drastically different rheological and mechanical properties to mucus based on several ambient factors, of which the dominant one is the medium pH. The pH range in the nasal mucus is associated^{60} with a distinct hydrophilic behavior, which justifies the assumption of the airway walls to be perfectly absorbing via surface tension forces. The absorbed droplets will spread on impact^{61,62,63,64}, and mechanistically, the virions embedded in the droplets could be conjectured to disperse over and percolate through the top gellike layer of mucus.
Additionally, note that this study considers a droplet to “land” on the airway walls, only when the droplet center enters the layer of mesh cells that adhere to the walls. To enhance accuracy of tracking, the mesh at the airwaytissue interfaces comprises three layers of prism cells, with the thickness of each layer being approximately 30 \(\upmu \). With similar mesh refinement, the use of the entrapment boundary condition at the walls during nasal transport is also supported by a long history of publications on particle and droplet tracking by multiple research groups^{65,66,67,68}, along with several such studies^{43,69} presenting a strong agreement between CFD simulations and in vitro experiments.

On the limitations of the virion exposure estimate—Spatiotemporal parameters, such as ambient airflow and ventilation rates; and subjectspecific biological variables, such as innate immune response and anthropometric differences, are only a subset of the many critical factors that can have a strong correlation on quantifying viral exposure and infectious dose, and as such, the estimates for virion transmission presented here has not been substantiated by an epidemiological model. To that end, obtaining an exact and conclusive measure of the SARSCoV2 infectious dose will require a wide in vivo study. The crossdisciplinary strategy presented here (i.e. integration of numerical simulations of transport in complex anatomic pathways with virological assessments and respiratory ejecta data) could however be potentially extended as a subcomponent of a fullscale epidemiological model, for an exact quantification of virologic parameters, such as the infectious dose.

On the continuum assumption related to the ejecta generation from oral liquids—The mathematical approach for the estimation of virion contamination in the respiratory ejecta has, by the very nature of it, presumed a simplistic estimate of viral load in the ejected droplets, based on a continuumbased argument that the spatial distribution of virions could be considered uniform in the sputum. In reality, how the complex rheology of oral fluids might affect the ejecta generation and subsequent breakdown^{70}, and the resultant volumetric concentration of virions embedded in the expiratory remnants—are also critical open questions.

On the size of the test cohort—This study is somewhat limited by the small sample size, primarily owing to the lack of CT scans in subjects with otherwise diseasefree airways. To get a realistic insight on the intranasal transport phenomena at the onset of any respiratory infection, it is preferred that we base the in silico cavity reconstructions on CTnormal images. Nonetheless, the preliminary findings presented here could be considered an important step in the mechanistic characterization of the transmission dynamics for inhaled pathogens, such as SARSCoV2.
The main takeaways
To highlight the main finding from this study: the detection of the inhaled droplet sizes (\(\sim \) 2.5 to 19 \(\upmu \), see “Droplet size range that targets the nasopharynx”) that specifically target the infectionprone nasopharynx, can provide a pivotal resource in mitigating the pandemic. For instance, the information on the droplet sizes that tend to launch the initial infection at the nasopharynx could be utilized to inform public policy on social distancing and in the design^{71,72,73,74} of novel masks and facecoverings that can execute targeted screening of only the hazardous droplet sizes and in the process, be more breathable and userfriendly than the mask respirators available now. The findings can additionally extend salient inputs to the mechanistic design of topical antiviral therapeutics^{75,76,77,78} and targeted intranasal vaccines^{79,80,81,82,83}, that could be tailored to undergo regional deposition directly over the infected nasopharynx, thereby significantly pronouncing their therapeutic indices, especially when compared to systemically administered vaccines and other prophylactics.
Obtained as a corollary, the \(\mathcal {O}(10^2)\) estimate for the SARSCoV2 infectious dose (see “On the SARSCoV2 infectious dose”) is also noteworthy. The low order underlines the high communicability of this disease, especially if discerned in the perspective of other airborne transmissions, e.g. the infective dose for influenza A virus, when administered through aerosols to human subjects lacking serum neutralizing antibodies, is at least an order greater and ranges between 1950 and 3000 virions^{84}.
Data availability
This project has generated simulated, quantitative, deidentified data on regional deposition over nasal tissues. The digitized anatomic geometries, the simulation datasets (including Fluent .cas and .dat files), and the numeric protocols; along with MATLAB codes, Wolfram Mathematica notebooks, and Microsoft Excel spreadsheets used for data postprocessing—are available onrequest from the author, through a sharedaccess Google Drive folder^{85}.
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Acknowledgements
Computing facilities at both South Dakota State University and UNC Chapel Hill were used for the simulations. Illustrator 2020 (Adobe Inc., Mountain View, California; link to software homepage) was availed to polish and label the generated visuals and graphics. The author acknowledges Dr. Brent Senior, MD, FACS, FARS (Professor and Chief, Division of Rhinology, Allergy, and Endoscopic Skull Base Surgery at the Department of Otolaryngology / Head and Neck Surgery, UNC Chapel Hill) for the clinical insights and Dr. Brato Chakrabarti (Research Fellow at the Center for Computational Biology, Flatiron Institute, New York) for reading/critiquing the original first draft of the manuscript. Dr. Sunny Jung (Associate Professor at the Department of Biological and Environmental Engineering, Cornell University) and Dr. Leonardo Chamorro (Associate Professor at the Department of Mechanical Science and Engineering, University of Illinois at UrbanaChampaign) were collaborators on the National Science Foundation (NSF) grant supporting the study. A preliminary report on the findings discussed here has been accepted through peer review—for presentation at the International Congress of Theoretical and Applied Mechanics (ICTAM) 2020+1 (to be held in August 2021; Milan, Italy), organized by the International Union of Theoretical and Applied Mechanics (IUTAM).
Funding
The material is based on work supported by the NSF RAPID Grant 2028069 for COVID19 research. Any opinions, findings, and conclusions or recommendations expressed here are, however, those of the author and do not necessarily reflect NSF’s views. Supplemental assistance for the project came from S.B.’s faculty startup package at South Dakota State University.
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Basu, S. Computational characterization of inhaled droplet transport to the nasopharynx. Sci Rep 11, 6652 (2021). https://doi.org/10.1038/s41598021857657
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DOI: https://doi.org/10.1038/s41598021857657
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