Statistical Uncertainty of Inhalation Dose Coefficients: Impact of Particle Deposition in ICRP 66 Human Respiratory Tract Model

Abstract

Inhaled radioactive materials can pose a long-term health concern, as the material can be incorporated into the body’s metabolic pathways and remain in organs and tissues for extended durations. During the retention period, the radioactive material may localize in a source organ and irradiate adjacent target organs and tissues. Distribution of these materials changes over time, requiring biokinetic modeling to evaluate their movement through various tissues and organs. The evolving distribution depends on multiple inputs characterizing the inhaled material, such as particle size and size distribution, particle density, aspect ratio, specific radionuclide, the chemical form, and solubility. In addition, biological parameters such as breathing rate, breathing type (nasal or nasal/oral), respiratory system morphometry, tidal volume, functional residual capacity, and anatomical dead space all influence material transport. These aerosol properties and physiological characteristics of the respiratory tract jointly define a range of initial conditions that influence the time-dependent distribution of radioactive material. To evaluate both uncertainty in the initial conditions of inhalation exposure and the final output (committed effective dose) from biokinetic models, a Python-based software tool, Radiological Exposure Dose Calculator (REDCAL), was developed to propagate uncertainty within the human respiratory tract model. Focusing on deposition fraction uncertainty, the primary objective was to characterize the initial activity distribution across respiratory regions as a function of anticipated particle sizes and distributions. The impact of the deposition fraction uncertainty was propagated to committed effective dose coefficients for selected radionuclides in a companion publication. For each particle size, a lognormal distribution, characterized by its geometric mean as defined within ICRP Publication 66, serves as the basis for introducing uncertainty into the physical processes governing deposition in various lung regions. This study addresses the deposition process and examines how uncertainty in deposition mechanisms affects activity distribution in the airways, ultimately presenting the expected range and standard deviation of deposited activity as a function of particle size.

INTRODUCTION

Inhaled radioactive material poses a distinct radiological hazard, where once inhaled, the radioactive material is transported throughout the body, integrating into metabolic and immune processes. During metabolism, the radioactive material emits radiation, exposing nearby tissues. Because the distribution of radioactive material changes dynamically over time, biokinetic modeling techniques are used to track its movement within internal organs and tissues. For inhalation exposure, the region of particle or aerosol deposition directly affects the estimated radiation dose from internal contamination.

Internal radiation dose assessment is governed by two primary modeling components. The first is a biokinetic model that simulates the movement of radioactive material in the body. The second is a dosimetric model that calculates how radioactive decay within organs delivers dose to their tissues and nearby organs. The dosimetric model relies on organ- and tissue-specific source-target activities and the biokinetic model is constructed by solving systems of ordinary differential equations (ODEs) based on compartmental models. Since the exposure pathway of primary interest in this study is inhalation, constructing the initial conditions for the biokinetic model requires data on the deposition of radioactive particles in the airways. Lung deposition is influenced by factors such as lung region volumes, breathing rate, and breathing mode (mouth vs. nasal inhalation and exhalation) (Eckerman 1994; ICRP 1994).

Although individual anatomical and physiological variability exists, reference biokinetic and dosimetric models have been created to estimate organ and effective doses in occupational and public exposure scenarios. The International Commission on Radiological Protection (ICRP) uses a combination of biological data and experimental evidence to predict the deposition and behavior of radionuclides, even when limited human data are available. The ICRP’s Human Respiratory Tract Model (HRTM) (ICRP 1994) provides methodologies and data for estimating:

1. The fraction of inhaled activity deposited along the linings of the extrathoracic and respiratory airways;

2. The removal of deposited material via mechanical mucociliary transport to the esophagus and direct absorption into the bloodstream through airway tissues (Fig. 1); and

3. Dosimetric calculations for energy deposition in stem cells following radionuclide decay within airway tissues.

Fig. 1.

The Human Respiratory Tract Model (HRTM) illustration of material transport between respiratory compartments, clearance of material to the gastrointestinal tract, and material transfer to regional lymph nodes (ICRP 1994). The arrows depict the transfer of materials between compartments with an assigned transfer rate. AI₁: relatively short-term retention (half-time about 35 d); AI₂: long-term retention (half-time about 700 d); AI₃: very long-term retention (half-time about 6,000 d); BB₁: short-term retention (half-time about 100 min) of particles in the BB region; bb₁: short-term retention (half-time about 8 h) of particles in the bb region; BB₂: intermediate retention (half-time about 20 d) of particles in the BB region; bb₂: intermediate retention (half-time about 20 d) of particles in the bb region; BB_SEQ: long-term retention (half-time about 70 d) in airway walls of a small fraction of the particles deposited in the BB region; bb_SEQ: long-term retention (half-time about 70 d) in airway walls of a small fraction of the particles deposited in the bb region; ET₁: retention of material deposited in the anterior nose (region ET1); ET’₂: short-term retention (half-time about 10 min) of the material deposited in the posterior nasal passage, larynx, pharynx and mouth (region ET2); ET_SEQ: long-term retention (half-time about 700 d) in airway tissue of a small fraction of particles deposited in the nasal passages; LN_ET: lymphatics and lymph nodes that drain the extrathoracic region; LN_TH: lymphatics and lymph nodes that drain the thoracic regions.

Specifically, ICRP Publication 66’s HRTM divides the respiratory tract into compartments, modeling the movement of inhaled radionuclides through these regions. The model accounts for deposition, mechanical clearance, dissolution/absorption, and radioactive decay within the lungs (ICRP 1994).

The ICRP deposition model is based on reference morphometry and physiology, which limits representation of individual variability. Despite this constraint, the ICRP model remains a widely adopted tool for estimating deposition fractions in internal radiation dose estimations following radionuclide inhalation. In ICRP Publication 66, Annex D and Annex E, James et al. (1994) and Bailey & Roy (1994), respectively, investigated the impact of variability and uncertainty on respiratory tract model parameters (ICRP 1994).

Drawing on both human and animal experimental data, the annex demonstrates how model parameters can be expressed using measurable quantities. Annex D of the ICRP Publication 66 (1994) provides a detailed discussion of the experimental data on regional deposition of aerosol particles in the human respiratory tract, which forms the basis of the deposition model. Rather than relying solely on the tabulated deposition fractions in Table F of the ICRP Publication 66, the annex introduces an algorithm to define deposition in various regions of the respiratory tract using filter efficiency models, enabling parametric variation for uncertainty and sensitivity analysis. Furthermore, expressing particle dissolution rate in terms of measurable variables, such as particle surface area, mass, diameter, density, and the dissolution rate constant, facilitates its direct integration into sensitivity analyses of the respiratory tract model. Prior studies conducted by Bolch et al. (2001) and Bolch et al. (2003) systematically reviewed the human respiratory tract deposition and clearance model parameters within the ICRP Publication 66 lung model and investigated the influence of parameter uncertainties on particle deposition and clearance using Lung Dose Uncertainty Code (LUDUC). However, the source code for LUDUC (Huston 1995) is unavailable, and its computational framework is not compatible for integration into modern computational workflows.

This present study is part of a broader project aimed at developing a computational framework to propagate total uncertainty in inhalation dose coefficients governed by the ICRP Publication 66 HRTM. The effort, which includes application to radionuclides of interest to the US Federal Radiological Monitoring and Assessment Center (FRMAC), is discussed in a companion paper. The specific objective of this study is to characterize uncertainty in particle deposition fractions across particle sizes using an in-house Python-based computational module, Radiological Exposure Dose Calculator (REDCAL). This subtask represents a critical component of the overall framework, as deposition is a major source of uncertainty and variability in the HRTM pathway.

METHODOLOGY

The REDCAL module incorporates the ICRP deposition model and governing physics and interfaces with Sandia National Laboratories (SNL) Dakota toolkit, a robust mathematical toolkit optimized for parametric analysis. The Dakota toolkit permits sampling from many distribution types to conduct uncertainty quantification (Adams et al. 2012), including the use of Latin Hypercube Sampling (LHS). The deposition of radioactive material within the respiratory tract is modeled in REDCAL as a series of filtration processes occurring during both inhalation and exhalation. These filters, illustrated in ICRP Publication 66 (ICRP 1994), represent distinct regions of the respiratory tract where material is deposited via aerodynamic and thermodynamic mechanisms. The dominant deposition mechanism depends on particle size, with aerodynamic mechanisms governing particles larger than 1 μm and thermodynamic mechanisms dominating for particles smaller than 0.1 μm.

Between 0.1 and 1 μm, both mechanisms compete, and particle characteristics such as particle density and shape factor impact which mechanism controls particle behavior for depositing along the airways. Aerodynamic deposition encompasses impaction, sedimentation, and gravitational settling, while thermodynamic deposition is governed by particle diffusion via Brownian motion. The distributions for the uncertainty-modifying term applied to aerodynamic and thermodynamic regional deposition efficiency values are displayed in Table 1. The combined deposition efficiency for each region is determined as the geometric mean of these competing processes, as shown in eqn (1). The geometric mean computation ensures that the governing mechanism for particle deposition predominates within its particle size range, but the overlapping region has competition between the thermodynamic and aerodynamic terms:

$${\eta }_{Region}^2={\eta }_{ae}^2+{\eta }_{th}^2,$$ (1)

where ${\eta }_{Region}$ is the overall regional deposition efficiency, ${\eta }_{th}$ is the thermodynamic filtration efficiency, and ${\eta }_{ae}$ is the aerodynamic filtration efficiency. The regional deposition efficiency quantifies the filtration efficacy for each lung region during inhalation and exhalation. Both parts of the breathing cycle, inhalation and exhalation, yield computed deposition efficiencies. The regional deposition efficiency for each phase of the breathing cycle, across both aerodynamic and thermodynamic mechanisms, is calculated using the following equation, with the subscript indicating the corresponding deposition method (${\eta }_{ae}$):

$${\eta }_{ae}=1-\exp (a{R}^p),{\eta }_{th}=1-\exp (a{R}^p),$$ (2)

where $a$, $p$ are values defined uniquely for each region separately for aerodynamic and thermodynamic efficiency values for both inhalation and exhalation parts of the breathing cycle, and $R$ is a computed term that involves parameters relevant to the aerodynamic or thermodynamic mechanisms of deposition. The uncertainty modifying terms listed in Table 1 of this paper serve as multipliers of the $a$ term in eqn (2) according to the aerodynamic or thermodynamic efficiency value. The parameters defining the $R$ term in eqn (2) are discussed in detail for both aerodynamic and thermodynamic calculations. The uncertainty modifying terms for each region and each deposition efficiency mechanism are described in Table 1.

Table 1.

Sampled input parameter probability distributions for the ICRP human respiratory tract model (Bolch et al. 2001, 2003; James et al. 1994).

Parameter	Geometric Mean/Mean	GSD/SD	Distribution Type
Aerodynamic deposition efficiency error terms
$C_{ae}(E{T}_1)$	1	1.82	Lognormal
$C_{ae}(E{T}_2)$	1	1.82	Lognormal
$C_{ae}(BB)$	1	1.58	Lognormal
$C_{ae}(bb)$	1	1.58	Lognormal
$C_{ae}(AI)$	1	1.3	Lognormal
Thermodynamic deposition efficiency error terms
$C_{th}(E{T}_1)$	1	1.18	Lognormal
$C_{th}(E{T}_2)$	1	1.18	Lognormal
$C_{th}(BB)$	1	1.23	Lognormal
$C_{th}(bb)$	1	1.23	Lognormal
$C_{th}(AI)$	1	1.23	Lognormal
Compartmental sequestration deposition fraction terms
$f_d(E{T}_{seq})$	0.0005	1.73	Lognormal
$f_d(B{B}_{seq})$	0.007	1.73	Lognormal
$f_d(b{b}_{seq})$	0.007	1.73	Lognormal
slow-clearing fraction error term
$E(F_s)$	0	0.1	Normal
Fast-clearing deposition fraction for alveolar interstitial term
$F_d(A{I}_1)$	0.3	1.1	Lognormal

For the regional deposition efficiency computation, the $R$ term is computed with parameters defining particle behavior relevant to the mechanisms governing deposition. For the aerodynamic calculations, particle aerodynamic diameter, particle velocity, regional residency time, volumetric flow rates, and respiratory regional scaling factors are the defining terms as depicted in Table 12 of ICRP Publication 66 (ICRP 1994). The extrathoracic regional scaling factor is a dominant term in the regional deposition equations, as it is cubed in the defining equations of the $R$ term used in eqn (2) given in Table 12 of ICRP Publication 66. This cubed dependence significantly influences deposition during the initial phase of inhalation and the final phase of exhalation in nasal breathing, due to the directional changes in airflow through the nasal passages. These deposition efficiency terms compute the likelihood for the particle to impact the airway walls via impaction and sedimentation or if the particle will gravitationally settle within a specific region of the respiratory tract. The aerodynamic diameter ($d_{ae}$) of the particle is computed by eqn (3):

$$d_{ae}=d_e{\left[\frac{\rho C(d_e)}{\chi {\rho }_{0}C(d_{ae})}\right]}^{0.5},$$ (3)

where $d_e$ is the equivalent volume diameter, $\chi$ is the particle shape factor, $\rho$ is the particle mass density, ${\rho }_0$ is unity mass density, and $C(d)$ is the Cunningham slip correction factor for the diameter specified.

For the thermodynamic calculations, key parameters include the diffusion coefficient, volumetric flow rates, regional scaling factors, empirical correction factor accounting for nonlaminar flow, and particle residence time as defined in Table 12 of ICRP Publication 66. These thermodynamic terms compute the likelihood of a particle having sufficient time to impact the walls of the airway while diffusing at a specific rate. The residence time is the time the particle is diffusing within a specific respiratory region defined by the void volume of the region divided by the velocity of the particle multiplied by a term defined by breathing parameters of tidal volume and functional residual capacity. The diffusion coefficient requires the computation of the Cunningham slip coefficient correction factor, a term defining the drag coefficient for small particles moving via free molecular flow and thus experiencing drag by individual air molecules striking the particle rather than by the viscosity of the airflow as a unified force (ICRP 1994). The diffusion coefficient describes the distance traversed per unit time by a particle by means of Brownian motion (ICRP 1994). The calculation of diffusion includes the slip correction term that is also used in eqn (3); the diffusion coefficient and slip correction factor calculations are defined in eqns (4) and (5):

$$\frac{\overline{x^2}}{t}=2D=\frac{2C(d_e)kT}{3\pi \mu d_e},$$ (4)

where $D$ is the diffusion coefficient; $\frac{\overline{x^2}}{t}$ is the mean square displacement per unit time of a particle by Brownian motion; $kT$ is the product of the Boltzmann constant and absolute temperature; and $\mu$ is the dynamic viscosity of air.

$$C(d)=1+\frac{\lambda }{d}\left[2.514+0.800exp\left(-0.55\frac{d}{\lambda }\right)\right],$$ (5)

where $C(d)$ is the slip correction factor for a particle of a given diameter, and $\lambda$ is the mean free path of air molecules, equaling 0.0712 μm at body temperature, 37 °C, atmospheric pressure, and 100% relative humidity (ICRP 1994). The slip correction factor equation, along with particle density and dynamic shape factor, converts thermodynamic diameters to aerodynamic diameters by the recursive solving of Equation D.13 of ICRP Publication 66 Annex D, with the initial definition of D.13 set to $d_{th}=d_{ae}\sqrt{\chi ∕\rho }$ yielding Equation D.13, reproduced in eqn (6) as:

$$d_{th}=d_{ae}\sqrt{\frac{\chi {\rho }_0}{\rho }\times \frac{C(d_{ae})}{C(d_{th})}},$$ (6)

where $d$ is diameter, specified to be either aerodynamic or thermodynamic; $\rho$ is particle density; $\chi$ is the dynamic shape factor; and $C(d)$ is the slip correction factor equation at the specified particle diameter, being either aerodynamic or thermodynamic.

The deposition module in REDCAL computes the regional deposition efficiency values in accordance with Table 12 of ICRP Publication 66, with the addition of the uncertainty modifying terms shown in Table 1 in the scope of this study. However, the appropriate modifications recommended in the updated HRTM in the ICRP Publication 130 (ICRP 2015) were also incorporated. For each particle size run, REDCAL defines the particle size distribution as a lognormal distribution with the activity median aerodynamic diameter (AMAD) defining the median value and the geometric standard deviation defined by the following equations (eqn 7):

$${\sigma }_g=1+1.5\left[1-{(100AMT{D}^{1.5}+1)}^{-1}\right],$$ (7)

where ${\sigma }_g$ is the geometric standard deviation (GSD) of particle sizes for an activity median thermodynamic diameter (AMTD). REDCAL uses the lognormal distribution defined by the computed GSD to define the probability term for the particle sizes within the distribution for a given AMAD or AMTD. The deposition efficiency terms are computed for all particle sizes within the distribution range, and the deposition value is calculated for each region. The deposition values for activity for each region of the respiratory tract for the distribution are computed by eqn (8) and eqn (9), with AMAD for particles above 1 μm and AMTD for particles below 0.1 μm. In the overlap region of 0.1–1 μm, the diameters are converted to equivalent diameter, then to aerodynamic or thermodynamic diameter, respectively, to match the majority term of the particle size distribution:

$$D{E}_A(AMTD,{\sigma }_g)=\sum _0^{\infty }DE[D(d_{th})]{\rho }_A[D(d_{th})]\delta d_{th}$$ (8)

$$D{E}_A(AMAD,{\sigma }_g)=\sum _0^{\infty }DE(d_{ae}){\rho }_A(d_{ae})\delta d_{ae},$$ (9)

where $D{E}_A$ is the fractional deposition of inhaled activity, ${\rho }_A\delta d$ is share of aerosol activity in the particle size interval defined by either aerodynamic diameter or the diffusion coefficient as a function of thermodynamic diameter, and $DE(d)$ is the fractional deposition of activity at the specified particle size as defined by either aerodynamic diameter or the diffusion coefficient as a function of thermodynamic diameter. Both eqn (8) and eqn (9) require the computation of the regional deposition fraction for each particle size spanned by the distribution range. The regional fractional deposition of activity is computed by applying the deposition efficiencies, both aerodynamic and thermodynamic, to the regions of the respiratory tract, treating each region as a filter that receives a fractional portion of the inhaled activity.

Treating the regions of the respiratory tract as filters, the regional deposition efficiency values calculated from eqns (1) and (2) represent the fraction of inhaled particles that deposit within each respective region. Since the breathing cycle has air entering the respiratory tract during inhalation and then exiting during exhalation, the total filter series is extrathoracic, bronchial, bronchiolar, alveolar-interstitial, bronchiolar, bronchial, and extrathoracic. In series, the filtration efficiency expresses the complement of the filter output divided by its input in terms of particle activity (ICRP 1994). Therefore, the activity deposited within each region can be solved for recursively by eqn (10):

$$D{E}_j=D{E}_{j-1}{\eta }_j{\xi }_j\left(\frac{1}{{\eta }_{j-1}}-1\right),\text{for}\mathrm{j}=1,\mathrm{N},$$ (10)

where $D{E}_j$ is the deposition fraction of activity for region $j$; ${\eta }_j$ is the regional deposition efficiency for region $j$, shown earlier in eqn (1); and ${\xi }_j$ is the quotient of the volumetric fraction of region $j$ and its preceding region. The yield from this equation is the fraction of the inhaled activity deposited within each region of the lungs during one complete breathing cycle. The deposition in the first extrathoracic region requires the computation of inhalability, which is treated as the prefiltration of inhaled activity based on particle size as defined by equation D.4 in ICRP Publication 66 Annex D. The inhalability, therefore, defines the upper cap for the total deposition of inhaled activity. This ceiling value is defined by $D{E}_1={\eta }_1(1-(1-{\eta }_I))$, with ${\eta }_I=inhalability$ to start the recursive solving of eqn (10).

The regional deposition fraction is partitioned to the compartments of each region to account for various absorption and clearance rates. Different compartments within the same region will have unique absorption, movement of radioactive material into the bloodstream, and clearance, movement of radioactive material upwards in the respiratory tract, and eventually to the gastrointestinal tract. REDCAL uses sampled parameters from Table 1 to introduce uncertainty into the compartmental activity allocation; the default fractions of activity deposition within each respiratory compartment are defined in Table 17B of ICRP Publication 66. REDCAL uses a sampled term to define the fraction of activity deposited to the sequestered compartments of ET₂, BB, and bb, and for the AI₁ compartment. REDCAL has an error term that includes uncertainty in the computed slow-clearing fractions of the bronchial and bronchiolar regions. The error term is added to the $f_s$ term calculated (eqn 11 and eqn 12) in accordance with particle size classification:

$$f_s=0.5\text{for}{d}_{ae}\le 2.5\sqrt{\rho ∕\chi }[\mu m]$$ (11)

$$f_s=0.5e^{-0.63\left(d_{ae}\sqrt{\chi ∕{\rho }^{-2.5}}\right)}\text{for}{d}_{ae}>2.5\sqrt{\rho ∕\chi }[\mu m],$$ (12)

where $f_s$ is the slow-clearing fraction, $\rho$ is particle density, $\chi$ is the shape factor, and $d_{ae}$ is the aerodynamic diameter, which can be computed from the thermodynamic diameter by Eq. 6 discussed earlier. For partitioning the activity to compartments within a region, REDCAL calculates the slow-clearing fraction first, then the sequestered fraction, and finally the nominal fraction as the remaining activity. The summation of the compartmental fractional activity equals the regional deposition of activity for said region of the respiratory tract. The sampled error term for $f_s$ can either increase or decrease the fraction of activity partitioned for the slow clearing fraction, as seen in Fig. 2, which displays the sampled distribution. The compartmental deposition fractions establish the initial activity within the body for biokinetic modeling of inhaled radioactive material. A companion publication (Mate-Kole et al. 2025) discusses the impact of uncertainty in activity deposition for systemic biokinetic models of various radionuclides on committed dose, harnessing analysis in this deposition study presented here.

Fig. 2.

The sampled distribution of the slow clearing fraction used by REDCAL, with the default value shown as the red vertical line and the blue vertical lines denoting 2 standard deviations from the mean.

Beyond uncertainty propagation, for a selected particle size distribution, the characteristic probability distribution functions (PDFs) were inferred. This involved fitting a distribution to the deposition fraction results associated with each particle size. Candidate PDFs were chosen based on subject-matter expertise, prior work, and the prior distributions of the uncertain parameters. In this study, the considered PDFs included the lognormal, normal, gamma, Johnson Special Bounded (johnsonsb), and generalized normal (gennorm) distributions. The fits of the empirical distribution (derived from 10,000 realizations of the deposition fractions) to the theoretical distributions were assessed visually using quantile-quantile (Q-Q) plots accompanied by the probability density plots.

RESULTS

To propagate uncertainty and conduct sensitivity analyses in the ICRP deposition model, the ICRP Publication 66 HRTM was reconstructed as a Python computational module. Key uncertain parameters were prioritized and sampled via LHS using SNL’s Dakota, generating deposition fractions for 10,000 realizations. For a given deterministic particle deposition fraction, the relative differences between total deposition fractions computed using the REDCAL deposition module and those reported by the ICRP were within ±1%. REDCAL’s uncertainty propagation simulations produced multiple deposition fraction sets for each particle size distribution. Fig. 3 presents the mean total deposition ±1 standard deviation based on the 10,000 samples.

Fig. 3.

Total deposition, mean ± 1 SD, as a function of particle size, based on 10,000 samples derived from all input distributions.

The distribution of the deposition fractions of each particle size distribution set was evaluated using Q-Q plots with accompanying probability density plots. Figs. 4 through 10 illustrate the best fit distribution of the dataset based on visual inspection of the Q-Q plots for the selected particle size distributions, along with the corresponding density function plots. Table 2 lists the average breakdown of compartmental activity fractions for each particle size distribution.

Fig. 4.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 0.01 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Fig. 10.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 5 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Table 2.

Average compartmental deposition fractions for each particle size distribution.

AMAD (μm)	AI_1	AI_2	AI_3	BB_gel	BB_sol	BB_seq	bb_gel	bb_sol	bb_seq	ET2_seq	ET2_sur	ET1	Total
0.01	7.73 × 10−02	1.54 × 10−01	2.57 × 10−02	3.87 × 10−02	3.93 × 10−02	6.16 × 10−04	1.68 × 10−01	1.71 × 10−01	2.68 × 10−03	9.59 × 10−05	1.71 × 10−01	1.51 × 10−01	8.78 × 10−01
0.03	1.84 × 10−01	3.66 × 10−01	6.11 × 10−02	1.55 × 10−02	1.57 × 10−02	2.46 × 10−04	1.07 × 10−01	1.09 × 10−01	1.71 × 10−03	4.09 × 10−05	7.30 × 10−02	6.61 × 10−02	7.24 × 10−01
0.1	1.93 × 10−01	3.84 × 10−01	6.41 × 10−02	1.39 × 10−02	1.41 × 10−02	2.21 × 10−04	9.82 × 10−02	9.98 × 10−02	1.56 × 10−03	3.68 × 10−05	6.56 × 10−02	6.52 × 10−02	2.94 × 10−01
0.3	1.73 × 10−01	3.43 × 10−01	5.73 × 10−02	1.23 × 10−02	1.25 × 10−02	1.95 × 10−04	7.31 × 10−02	7.43 × 10−02	1.16 × 10−03	6.99 × 10−05	1.25 × 10−01	1.28 × 10−01	2.08 × 10−01
1	1.09 × 10−01	2.16 × 10−01	3.60 × 10−02	9.67 × 10−03	9.82 × 10−03	1.54 × 10−04	3.29 × 10−02	3.34 × 10−02	5.24 × 10−04	1.70 × 10−04	3.03 × 10−01	2.50 × 10−01	4.17 × 10−01
3	5.86 × 10−02	1.17 × 10−01	1.95 × 10−02	1.09 × 10−02	1.11 × 10−02	1.74 × 10−04	2.08 × 10−02	2.11 × 10−02	3.31 × 10−04	2.29 × 10−04	4.09 × 10−01	3.32 × 10−01	7.40 × 10−01
5	3.88 × 10−02	7.72 × 10−02	1.29 × 10−02	1.60 × 10−02	5.80 × 10−03	1.72 × 10−04	2.37 × 10−02	8.62 × 10−03	2.55 × 10−04	2.48 × 10−04	4.44 × 10−01	3.73 × 10−01	8.36 × 10−01

For completeness, Table 3 presents the mean and standard deviation of the compartmental deposition fractions, summarizing the statistical range of deposition based on 10,000 realizations and the analyses conducted.

Table 3.

Statistical summary for compartmental deposition fractions for particle sizes analyzed for the 10,000 realizations.

	AMAD (μm)	AI_1	AI_2	AI_3	BB_gel	BB_sol	BB_seq	bb_gel	bb_sol	bb_seq	ET2_seq	ET2_sur	ET1_sur
Mean	0.003	0.0019	0.0038	0.0006	0.0612	0.0622	0.0010	0.0769	0.0782	0.0012	0.0002	0.3461	0.3278
Std		0.0011	0.0022	0.0004	0.0154	0.0156	0.0005	0.0178	0.0181	0.0006	0.0001	0.0364	0.0262
Mean	0.01	0.0678	0.1350	0.0225	0.0340	0.0345	0.0005	0.1477	0.1501	0.0023	0.0001	0.1504	0.1330
Std		0.0115	0.0206	0.0033	0.0085	0.0086	0.0003	0.0318	0.0324	0.0012	0.0000	0.0199	0.0163
Mean	0.03	0.1332	0.2650	0.0442	0.0112	0.0114	0.0002	0.0777	0.0790	0.0012	0.0000	0.0528	0.0478
Std		0.0157	0.0249	0.0037	0.0027	0.0027	0.0001	0.0187	0.0191	0.0006	0.0000	0.0076	0.0067
Mean	0.1	0.0567	0.1129	0.0188	0.0041	0.0041	0.0001	0.0288	0.0293	0.0005	0.0000	0.0193	0.0191
Std		0.0097	0.0176	0.0028	0.0009	0.0009	0.0000	0.0072	0.0073	0.0002	0.0000	0.0028	0.0027
Mean	0.3	0.0358	0.0713	0.0119	0.0026	0.0026	0.0000	0.0152	0.0154	0.0002	0.0000	0.0260	0.0267
Std		0.0055	0.0098	0.0015	0.0006	0.0006	0.0000	0.0037	0.0037	0.0001	0.0000	0.0087	0.0063
Mean	1	0.0452	0.0900	0.0150	0.0040	0.0041	0.0001	0.0137	0.0139	0.0002	0.0001	0.1262	0.1041
Std		0.0087	0.0160	0.0026	0.0016	0.0015	0.0000	0.0045	0.0045	0.0001	0.0000	0.0429	0.0258
Mean	3	0.0434	0.0863	0.0144	0.0081	0.0082	0.0001	0.0154	0.0156	0.0002	0.0002	0.3024	0.2459
Std		0.0101	0.0191	0.0031	0.0041	0.0040	0.0001	0.0069	0.0069	0.0002	0.0001	0.0655	0.0404
Mean	5	0.0324	0.0645	0.0108	0.0090	0.0091	0.0001	0.0134	0.0136	0.0002	0.0002	0.3711	0.3115
Std		0.0083	0.0158	0.0026	0.0047	0.0047	0.0001	0.0064	0.0065	0.0001	0.0001	0.0593	0.0378

DISCUSSION

The results demonstrate several key characteristics of particle deposition behavior in the respiratory tract. The AMAD, which defines the GSD of particle sizes, is a key variable in determining particle distribution. When the GSD exceeds 2.5, the lognormal distribution becomes excessively broad and impractical for sampling. Larger particles that cannot be inhaled are excluded from the sampling distribution. As the GSD increases, the distribution deviates further from a lognormal pattern, which is inherently a function of particle diameter. Larger particle diameters exhibit less dispersed total deposition due to reduced inhalability, and this trimming effect is evident in the results. Notably, peak deposition occurs at 5 μm AMAD, attributed to a reduction in the number of particles exhaled, resulting in a higher overall deposition. However, particles exceeding a certain size fail to reach the lungs, instead colliding with the nasal passages. Conversely, minimum deposition occurs at 0.3 μm, where the number of particles exhaled surpasses those remaining, thereby reducing total deposition.

For extremely small particle sizes, the uncertainty in particle deposition increases significantly due to the acute influence of thermodynamic phenomena, such as heightened diffusion and dispersion. This behavior is governed by the distinct deposition mechanisms in the extrathoracic region of the respiratory tract, including the nose, naso-oropharyngeal passages, and larynx. Particles with an aerodynamic diameter larger than about 0.5 μm are primarily deposited through aerodynamic transport processes, such as inertial motion (impaction) and gravitational settling (sedimentation). For very large particles and fibers, interception with airway surfaces further contributes to deposition. However, particles with an equivalent physical diameter smaller than a few tenths of a micrometer are predominantly deposited by thermodynamic transport processes, such as Brownian diffusion. These processes, specific to small particle sizes, result in increased deposition as these particles exhibit greater diffusion and dispersion, reducing their likelihood of remaining suspended in the airflow and being exhaled. Consequently, small particles are more prone to deposition within the respiratory tract, contributing to the observed variability in deposition and dose distributions.

Furthermore, significant deposition is observed in the extrathoracic regions (ET1 and ET2), particularly for smaller particles, where over 85% of inhaled activity is deposited. The isolation of ET1 in the ICRP Publication 66 model, with no systemic connections, leads to a high concentration of material that exits the respiratory tract over time. This isolation introduces challenges in calculating effective dose coefficients, as material deposited in ET1 is excluded from the systemic dosimetry model. Conversely, ICRP Publication 130 (ICRP 2015) offers a more realistic representation by linking ET1 to ET2 transit of material into deeper respiratory or systemic compartments, which is a modeling approach currently under investigation.

To complement the analysis, Q-Q plots were employed to interpret the quality of the fits for the distributions of particle deposition fractions across a range of particle size distributions relevant to internal contamination (0.01 μm, 0.03 μm, 0.1 μm, 0.3 μm, 1 μm, 3 μm, and 5 μm). In a Q–Q plot, theoretical quantiles derived from a candidate PDF are plotted against the empirical quantiles of the observed data. If the observed data conform to the theoretical distribution, the points in the Q–Q plot align along the reference (45°) line. Deviations from this line suggest discrepancies between the observed data and the assumed distribution, particularly in the distribution tails. Among the candidate distributions assessed, the Johnson Special Bounded (johnsonsb) distribution consistently provided the best fit to the observed deposition fractions for all considered particle sizes. As shown in the Q-Q plots of Figs. 4, 5, 6, 7, 8, 9, and 10, the data closely follow the theoretical quantile line, indicating excellent agreement with the johnsonsb distribution.

Fig. 5.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 0.03 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Fig. 6.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 0.1 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Fig. 7.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 0.3 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Fig. 8.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 1 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

Fig. 9.

Q–Q plot (left) and probability density plot (right) for the best-fit distribution (Johnson Special Bounded – johnsonsb) of total deposition fractions for particle size 3 μm AMAD. The Q–Q plot shows the agreement between the empirical data and the fitted distribution, while the probability density plot illustrates the full range of computed deposition fractions and the corresponding fitted density.

CONCLUSION

For this study, the Python code, REDCAL, was adapted to propagate uncertainty in the deposition of radioactivity within the human respiratory tract, following the HRTM defined in ICRP Publication 66. To enable stochastic sampling of model input parameters, REDCAL was integrated with the Dakota software suite from Sandia National Laboratories, applying LHS across input distributions informed by both published ICRP Annex data and updated literature sources. The resulting simulation outputs were used to estimate deposition fractions for particles across a range of aerodynamic diameters, enabling the construction of PDFs characterizing total activity deposition as a function of size. Beyond total deposition, regional deposition patterns were examined across the extrathoracic, bronchial, bronchiolar, and alveolar-interstitial compartments to evaluate spatial variability. These results enable quantification of the propagation of input parameter uncertainties into regional deposition fractions, which represent a key source of variability in internal dose estimates for inhalation exposure scenarios. The modeling framework developed here improves upon earlier tools, such as LUDUC, by providing a modern, extensible platform compatible with contemporary workflows. This study forms a foundational component of a broader effort to develop a full probabilistic dose assessment capability aligned with the ICRP Publication 66 HRTM, applicable to a wide range of inhaled radionuclides of interest to radiological emergency response and regulatory assessment. The REDCAL module serves as the deposition component of this framework, and future work will incorporate subsequent stages of the HRTM to propagate uncertainty through clearance, absorption, and systemic biokinetics. By enabling quantification of uncertainty in respiratory tract deposition, this work strengthens the basis for more reliable internal dose assessment in regulatory, occupational, and emergency response contexts.