We propose a multi-scale lung model to investigate spatio-temporal distributions of ventilation variables. Lung envelope and large airway geometries are derived from CT scans; smaller airways are generated using a physiologically consistent algorithm. Tissue mechanics is modeled using nonlinear elasticity under small deformations, coupled with local air pressure from fluid dynamics within the bronchial tree. Airflow accounts for inertia and static airway compliance. Simulations employ finite elements. Using this model, we explore spatio-temporal airflows and shear stresses distributions.
The bronchial mucus protects the lung from external aggression by trapping and removing inhaled particulate matter. Mucus is continuously produced and transported toward the esophagus through two primary mechanisms: mucociliary clearance and cough. In pathological conditions where these clearance processes are impaired, mucus stagnation occurs, increasing susceptibility to infections. Chest physiotherapy (CP) is often prescribed to compensate for this dysfunction and facilitate mucus transport. Many common CP techniques involve generating high airflow rates, based on the rationale that such flows interact with mucus and promote its mobilization. However, analyses of this interaction and the resulting mucus dynamics throughout the bronchial tree remain largely empirical.
Mathematical and computational modeling is particularly well suited to studying the lung’s internal dynamics, which are challenging to observe in vivo. Recent studies employing Weibel-like models of the bronchial tree have demonstrated that air-mucus interactions can indeed mobilize mucus during chest physiotherapy. Key drivers of these interactions have been identified and characterized, including air wall shear stress, mucus rheology, and airway geometry and structure [1,2]. However, these idealized models assume that all pathways from the trachea to the acini are geometrically and biophysically identical, and thus cannot capture spatial inhomogeneities. A significant paradigmatic gap remains in modeling the three-dimensional spatial distributions of the underlying physical processes.
In this abstract, we propose a prototype for a new synthetic multi-scale lung model that addresses this gap for resting ventilation conditions. Using this model, we explore the spatio-temporal distributions of air wall shear stresses and airflows within our integrative lung model at rest ventilation. Geometrical Model. The model geometry consists in 3D models of the left (L) and right (R) lung envelopes, whose walls are denoted W p , and from a 3D model of the upper bronchial tree B. B comprises the airway walls W a , the tracheal opening T , and n o terminal openings (O i ) i=1..no , each located in either the left (i ∈ I L ) or right (i ∈ I R ) lung. From each terminal opening O i of the 3D mesh, we decompose its host lung (L or R) into subdomains (A i )i = 1..n o via Voronoi tessellation based on the barycenters of the O i .
The thorax is modeled as the convex hull C of both lungs dilated by 10%. The pleural cavity is simplified by assuming an elastic medium fills the space between the lungs and thoracic wall.
Geometries are represented using surface and volumetric meshes. L, R and B are reconstructed from patient CT scans [3]. Here, segmentations from the LUNA16 challenge were used, with surface meshes generated from 3D masks via CGal and volumetric meshes obtained using MeshLab. Mesh resolution is controlled throughout the process.
Fluid Mechanics. Airflow in B is governed by the nonlinear incompressible Navier-Stokes equations, solved using 3D finite elements and the method of characteristics. Atmospheric pressure is the reference. No-slip conditions are imposed on W a . The extrathoracic airways are modeled via a resistance R T = 50 000 Pa•m -3 •s, imposing -p T,mean = R T f T , where p T,mean is the mean pressure at T and f T the tracheal flow. The associated Lagrange multiplier corresponds to a uniform pressure applied over T . Similarly, for each terminal opening O i , flow is constrained to equal the volume change rate of its associated lung region A i , yielding uniform pressures p Oi as Lagrange multipliers.
Tissue Mechanics. Lung tissue mechanics is modeled using small-strain elasticity with Poisson’s ratio 0.3 and a nonlinear local deformation-dependent Young’s modulus, calibrated from literature’s static compliance data. Lamé parameters µ and λ are deduced from these quantities. Lung density accounts for its 90 % air, 10 % tissue composition, ρ = 100 kg.m -3 . The stress-strain relation incorporates both tissue deformation and local air pressure [4]:
where u is displacement and p is local air pressure. Air pressure at a point x within region A i combines the pressure at its feeding terminal opening O i and the pressure drop through the distal airway subtree. Modeling these pressures requires generating physiologically consistent subtrees peripheral to B, see section II B.
Boundary conditions. At the intersections of B with the walls of L and R, and on the spine region at the back of C, we assume no displacements (u = 0). A time-dependant pressure is applied to the lower part W D of C to mimic the diaphragm-induced stress. Half that pressure is applied on the rest of C wall to mimic coastal muscles-induced stress. The pressure amplitude is calibrated to obtain a physiologic rest lung tidal volume. The pressure time dependance is a smoothed Heaviside, with a 1.3 s active inspiration and a 2.6 s passive expiration, see [4].
Small Airways
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