A tree-parenchyma coupled model for lung ventilation simulation.

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Zero-dimensional tree coupled to a three-dimensional parenchyma

In [41], the authors describe the parenchyma as a porous medium irrigated by a space-filling resistive tree. Energy dissipation within airways makes it harder to pull flow into the parenchyma. The tree and the parenchyma are coupled through the pressure exerted by the former on the latter and by ensuring a volume conservation constraint: under the gas incompressibility hypothesis, the gas volume entering or leaving a parenchyma region induces a corresponding volume variation of this region. The lung geometry is segmented from CT images. An arbitrary surface-displacement is imposed as boundary condition. With this model, the flow distribution within the tree can be studied along with the impact of resistance increase in a tree region or local elasticity change within the parenchyma.
Instead of imposing a volume constraint, the action of the tree on the parenchyma may be integrated into the parenchyma constitutive relation. Following [72], [73], this is the approach followed in this thesis (see CHAPTER II and [74]). Both pressure and surface displacement boundary conditions can be applied to the parenchyma. As the pleural pressure is not easily experimentally measurable, we propose to impose realistic Dirichlet boundary conditions extracted from lung images at different inflation states. The parenchyma model could be further coupled to a full flow model in the proximal part of the airway tree.

Full three-dimensional model

To our knowledge, [75] is the only study in which a description of the airways is coupled to a parenchyma model (see Figure 16). Upper airways geometry is segmented from a CT scan, and the flow within the tree is obtained from the resolution of Navier-Stokes equations. Unlike studies mentioned in subsection 4.3, airways are not assumed to be rigid, and fluid-structure interaction between the gas and the airway wall is taken into account. The parenchyma is described as a Neo-Hookean material. The coupling between the respiratory tract and the parenchyma is ensured by imposing volume conservation with Lagrange multipliers. This framework is used to mimic mechanical ventilation so as to investigate ventilator-associated lung injuries (see subsection 3). This fully integrated model provides relevant ventilation and strain calculation results, though such a precise description is done at the price of time-consuming heavy computations.
Figure 16: Lung ventilation model coupling upper airways and the parenchyma. Geometries are segmented from CT scans. Surface displacement boundary conditions are applied to the parenchyma, this generates flow boundary conditions at the tree outlets for CFD calculations. Taken from [75].

A perspective on boundary conditions for lung ventilation modeling

As presented above, the various models require the knowledge of either the pleural pressure or the dynamic of the surface displacement of the parenchyma envelope. They correspond to the applied forces appearing as the right hand side of the 0D models for instance or as boundary conditions for the 3D parenchyma equations. Nevertheless pleural pressure cannot be measured in time and space around the lung, and assessing the parenchyma surface displacement requires heavy imaging techniques.
In [76], the authors propose an interesting method to recover the surface displacement from the knowledge of the skin motion only. The assessment of the parenchyma surface dynamics is based on the modeling of the rib cage motion and diaphragm action. More precisely, the thoracic cage outer skin displacement is first measured with captors and a kinematic model enables to deduce ribs motions. The action of the diaphragm is modeled based on the measured tracheal flow and sliding conditions are applied within the pleural cavity. Finally, the lung surface evolution is deduced from a mechanical model. This work has been further completed with models for the diaphragm [77] and for ribs kinematic [78].
As a conclusion to this section, we note that a large hierarchy of models has been proposed to describe lung ventilation. Depending on the needs, available data and numerical resources, one would rather use a fully reduced description, a tree-parenchyma coupled model or a model in between. In the next section, contributions to lung ventilation modeling brought by this thesis are presented.

A tree-parenchyma coupled model

We assume that the lung parenchyma is an isotropic elastic media occupying a domain denoted by . Here, since our aim is to describe the tree-parenchyma coupling, we choose to consider a linearized behavior law and to neglect tissue viscosity. This assumption can be justified when considering normal breathing (see section 12). Let us denote the displacement of the parenchyma by defined in , taken as the reference state of the lung. The linearized stress tensor of the media is given by0.
where and are the Lamé parameters of the effective material, and the strain tensor defined by Note that these effective macroscopic coefficients and may be obtained using a homogenization process as described in [73] and [88]. The latter takes into account the alveoli microstructure so that we may refer to the homogenized media when considering the lung tissue. In particular, when the alveoli are assumed to have a hexagonal shape and to be uniform in size then the isotropic behavior of the homogenized law can be derived [73].
As stated above, the lung parenchyma is supplied with gas that flows through a dyadic resistive tree. If an airway of the tracheo-bronchial tree is nearly closed (due to a stenosis for example) the related supplied region will require more effort to stretch, even if its elastic properties are not affected.

From the tree-parenchyma coupled model to the exit-compartment model

In this section we compute the average pressure endured by terminal region in the tree-parenchyma coupled model. This resulting pressure naturally includes the mechanical connection between alveolar regions and its evolution can be driven by “realistic” Dirichlet boundary conditions (see subsection 8.2). Then is applied as a driving force to the balloon of the exit compartment model. Doing so accounts for mechanical interactions between compartments and avoids the need to know the pleural pressure distribution around the lung.
The pressure applied to the terminal balloon of the exit-compartment model is where is the parenchyma stress tensor and . As tangential forces are not included in , energy spent to induce motion might be different in both models although ventilation is the same. We have: The coupling term is easily computed following (17).

Mesh generation from HRCT images

To generate a mesh for the finite element solver, we use HRCT DICOM data from [91]. DICOM images are first treated with Matlab to generate a surface triangle mesh. This surface mesh is then processed with Meshlab software [93]. First we perform decimation to adjust the mesh size as desired, then smoothing through a Taubin filter [94] to improve mesh quality. We use Gmsh [95] to generate from the surface mesh a tetrahedric volume mesh. Once the mesh is built, each tetrahedron is assigned to one of the tree exits according to the algorithm described in Figure 24. We denote by m a mother branch, m a region of the parenchyma fed by m, d1 and d2 the daughter branches of m, 1 and 2 the regions of the parenchyma fed respectively by d1 and d2, T1 and T 2 the subsets of the tracheo-bronchial tree fed respectively by d1 and d2, R and L the daughters of the trachea and L the list of the chosen tree exit branches: L={exi}. So-called nodes are the bifurcation junctions’ positions.

Surface displacement registration

To impose Dirichlet boundary conditions, we need to have the displacement field of the surface over time. To that end we used Deformetrica software [96] that performs non-linear surface registration on surface meshes via the Currents method [97]. This computed displacement is not physical in the sense that no mechanics is included in the underlying process. There is no uniqueness in surface points correspondence from one inflation state to another, this may lead to non-physiological evolution. Yet Deformetrica allows the addition of landmarks and curves correspondence to the process so that physiological patterns can be tracked along the breathing cycle. Doing so provides registration with a more physical basis.
In the following section results of simulations performed on the tree-parenchyma coupling and the exit compartment models are presented.

Table of contents :

CHAPTER I- Introduction
1. Lung ventilation physiology
1.1. Role of the lung
1.2. Human lung architecture
1.3. Ventilation mechanism
1.4. Pathophysiology
2. Lung properties, measured by in vivo and in vitro experiments
3. Motivations for lung ventilation modeling
4. State of the art on mathametical lung ventilation modeling
4.1. Mono-compartment zero-dimensional model
4.2. Multi-compartment zero-dimensional model
4.3. Three-dimensional upper airways
4.4. Zero-dimensional tree coupled to a three-dimensional parenchyma
4.5. Full three-dimensional model
4.6. A perspective on boundary conditions for lung ventilation modeling
5. Thesis organization and contributions
6. List of publications
CHAPTER II- A tree-parenchyma coupled model for lung ventilation simulation..
7. Introduction
8. Model
8.1. Tracheo-bronchial tree
8.2. A tree-parenchyma coupled model
8.3. An exit-compartment model
8.4. From the tree-parenchyma coupled model to the exit-compartment model
9. Numerical methods
9.1. Tree-parenchyma coupled model
9.2. Exit-compartment model
10. Patient specific structural elements
10.1. Tree geometry
10.2. Mesh generation from HRCT images
10.3. Surface displacement registration
11. Simulations and results
11.1. Tree-parenchyma coupled model
11.2. Mechanical independence in the exit compartment model
11.3. From the tree-parenchyma coupled model to boundary conditions for the exitcompartment
model
11.4. Dirichlet boundary conditions registered from medical images
11.4.1. Impact of boundary conditions on lung regional expansion
11.4.2. Tree-parenchyma coupled model with Dirichlet boundary conditions
11.5. Pressure controlled mechanical ventilation
11.6. Computation time
12. Limitations and conclusion
13. Appendix
13.1. On the evolution of pressure drops along the tree
13.2. On the link between exit tree pressures and flows
13.3. On assuming flow into the mother branch equals the sum of the flows entering the daughters
13.4. Robustness of the resolution scheme
13.5. Mesh and time convergence analysis
13.6. Lobar surface fissure detection
13.7. Dynamical surface displacement built based on two static images
CHAPTER III- Calculated ventilation and effort distribution as a measure of respiratory disease and Heliox effectiveness
14. Introduction
15. Model and methods
15.1. Tree-parenchyma coupling and measure of the effort
15.2. Modeling disease of the tracheo-bronchial tree
16. Results and discussion
16.1. Impact of a bronchoconstriction
16.2. Impact of the regional position
16.3. Impact of the generational position
16.4. Asthma attack simulation
17. Conclusion
CHAPTER IV- From 4D lung dynamical maps to plugs distribution in asthma – a diagnosis tool
18. Introduction
19. Direct ventilation model and identification problem approaches
19.1. Global methodology
19.2. Direct ventilation problem
19.3. Identification problem
19.4. A priori prediction process
19.4.1. Ventilation information extraction
19.4.2. Machine learning process
19.5. Global approach summary
20. Numerical methods and patient-specific data exploitation
20.1. Input data construction
20.1.1. Lung geometry
20.1.2. Temporal and spatial resolutions
20.1.3. Tree structure
20.2. Numerical methods
21. Results
21.1. Results for the direct problem
21.2. Results for the machine learning step
21.3. Results for the identification problem step
22. Limits and conclusions
23. Appendix
23.1. On the dimension of
23.2. On the reason why if is not invertible, it has some collinear lines.
CHAPTER V- Perspectives
24. On spirometry modeling
24.1. Introduction
24.2. Exit-compartment model
24.2.1. Model and methods for compartments mechanical behavior
24.2.2. Results
24.2.3. Towards a more sophisticated resistance model
24.3. Tree-parenchyma coupled model
24.3.1. Model and methods
24.3.2. Results
24.4. Appendix
24.4.1. Gravity consideration in the exit-compartment model
24.4.2. Effect of geometrical perturbations on the pressure drops at forced expiratory flows
25. Towards a statistical lung model
25.1. Introduction
25.2. Methods and results
26. Experimental validation of the tree-parenchyma coupled model
Conclusion
References

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