Stochastic Lung Model

Model - Theoretical basis

Introduction

Deterministic models traditionally used in lung exposure calculations fail to capture the inherent variabilities in biological systems. These models rely on average values for morphometric parameters, overlooking both the variability of the airway geometry in one person’s lungs and the variability between the different members of the population. To get more realistic results, stochastic models are proposed, which treat some key parameters - like the length and diameter of the airways and the gravitational and branching angles - as random variables described by probability distributions based on real measured data. This provides a more realistic approach to determine the airway deposition of the inhaled particles.

Stochastic Lung Model

The aim of a joint Austrian-Hungarian project was to integrate anatomical variability into lung exposure calculations. The stochastic approach analyzed detailed anatomical data from sources such as the Raabe database [1] for tracheobronchial airways [2] and the data of Haefeli and Bleuer [3] for pulmonary airways, supplemented by additional information from the literature [4], to establish probability distributions for the data which describe the structure of the human airways. These distributions were integrated into the Stochastic Lung Model (SLM) which was initially developed by Koblinger and Hofmann [4] and Hofmann and Koblinger [5,6]. Since then, it has been continually further developed and extended to a variety of specific applications over the past three decades [7-18]. The main specific feature of this model is, that it uses a Monte Carlo airway geometry selection. Unlike in deterministic models, which use fixed lengths and diameters in each airway generation, in the SLM, the simulations of 10 000-100 000 particle pathways in a stochastically generated airway structure provide information on both the intrasubject and intersubject variability of the airway geometry. The effect of individual breathing patterns (volume of inhaled air, breathing frequency, etc) on the airway deposition distribution can be determined by specified input data.

Monte Carlo deposition calculations

In the Stochastic Lung Model, the branching airway structure is modelled by a sequence of Y-shaped bifurcation units, consisting of a parent tube and two randomly and asymmetrically dividing daughter airways (note: the use of airway bifurcation numbers instead of generation numbers is only a mere question of bookkeeping and does not affect particle deposition calculations).

In the Stochastic Lung Model, Monte Carlo methods are employed to construct the asymmetric branching structure of the human airways. Particles are inhaled at random times during the inspiration phase, either through nasal or oral airways. Upon entering the lungs, the random walk of inhaled particles through a stochastic airway structure is simulated by randomly selecting a sequence of airways for each individual particle, resulting in a variable number of airway generations along each path. At each asymmetric airway bifurcation, the major and minor daughter parameters are randomly selected from the related frequency distributions (airway lengths, diameters, branching and gravity angles) and their various correlations are selected by Monte Carlo methods along the path of an inhaled particle in an iterative manner, interpreting the morphometric frequency distributions as probability density functions.

Upon inhalation, the distribution of airflow among the five lobes, and hence the probability of a particle entering a given lobe, depends on the respective lobar volume, based on the assumption that the number of alveoli in a given lobe is proportional to the volume of that lobe. In subsequent asymmetric bronchial airway bifurcations, the entrance probability to an airway at a given airway bifurcation is assumed to be proportional to the cross-sectional area of that daughter branch. In the acinar airways, entrance probabilities are based on the ratio of air volume entering the associated alveoli.

Particles are tracked from inhalation until they are deposited or being exhaled. In case of a deposition event, which would terminate the path of an inhaled particle, deposition in a given airway is simulated by decreasing its statistical weight in subsequent airways, i.e. the particle continues its path with a smaller statistical weight instead of starting again at the entrance to the lungs. The contribution of a deposition event to the deposition fraction in a given airway is then determined by the product of the actual statistical weight times the deposition efficiency in that airway.

The computation of deposition in extrathoracic nasal or oral airways upon inhalation and exhalation is based on empirical deposition equations [19, 20]. Throughout their paths through the random airway structure, particles can deposit by impaction, sedimentation, or diffusion. Deposition probabilities for each mechanism are calculated using the equations provided by for cylindrical tubes (bronchi and acinar ducts) and spherical spaces (alveoli) [4, 21, 22, 23, 24, 25, 26].

The model computes the average fraction of inhaled spherical, inert particles deposited in extrathoracic, bronchial and alveolar airway generations. Because of the underlying structure of the simulation code, the generations denote the individual tubes in the bifurcation tree. E.g. the 1st generation is the trachea, the 2^nd generation is the main bronchi towards the left and right lung, etc. These number or mass deposition fractions are calculated for multiple breathing cycles (inhalation, exhalation, and two breath-holds), assuming nasal or oral breathing. Through the multiple cycles the same particle uses the same path of tubes (but a different path is selected for every inhaled particle). The simulation of a particle ends either with exiting to the environment (exhaling it), or by reaching a limit (1e-5) in the statistical weight. When a simulated particle enters an alveolus, it is actually split into two parts, according to the probability of entering the alveoli.

Data requirements

The main input data for the Stochastic Lung Deposition Model include spirometric data (e.g., functional residual capacity) and inhalation parameters (e.g., tidal volume, inhalation time, breath-hold duration, exhalation time). Particle properties (e.g. size or size distribution, and density) are also needed for the calculations. These data can be added as input parameters for each run. The detailed description of the input and output data can be found under the "Description of input parameters" and "Description of output parameters" menu, respectively. (link)

Model validation

The stochastic lung deposition model was validated by comparison with the experimental data of Heyder at al. [27] for a wide range of particles sizes and of Schiller et al. [28] for ultrafine particles, for several flow rates, and the fitted deposition fractions provided by the International Commission on Radiological Protection (ICRP) [20,29]. The excellent agreement with these experimental data thus provides a reliable basis for the simulation of inhaled particle deposition patterns.

References