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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.
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.
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 treeparenchyma
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.
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.
Tree-parenchyma coupled model
In this section we investigate the tree-parenchyma coupled model described in subsection 9.1. We simulate spontaneous tidal breathing driven by Neumann homogeneous pressure boundary conditions, first in a healthy configuration, then in the case of a bronchoconstriction.
Linear (3) and non-linear (4) resistance models are compared. There is no consensus on the spatiotemporal pleural pressure profile during the breathing cycle [89]. In this section we impose an academic piecewise constant pressure pattern with a physiological plateau of at inspiration [89] and at expiration so as to simulate passive recoil. This approach isolates the effect of the tree resistance distribution by ensuring that simulated ventilation heterogeneities are not due to spatial heterogeneities in the boundary conditions.
In Figure 29 left lung lobar volumes are plotted. Branch resistances are computed with the Poiseuille resistance model. The time step is .
Dirichlet boundary conditions registered from medical images
In this section we reconstruct the lung parenchyma surface evolution based on HRCT data provided
by [91]. It can be applied as a boundary condition to our finite element model.
Impact of boundary conditions on lung regional expansion
In addition to MLV data, [91] provides TLC HRCT scans of the lung envelope along with lobar segmentation. Following subsection 10.3 we perform non-linear surface registration from the MLV to the TLC state. Physiological landmarks and surface lobe fissures (see subsection 13.6) can be included In what follows we assume MLV to be the reference state. Here imaging measurements are static.
Thus we do not consider any dynamics in the transition from MLV to TLC, and airway resistances are set to zero. The registered displacement field is used as Dirichlet boundary conditions. In Figure 40 we compare experimental lobar ventilation ratios (issued from image segmentation) with results obtained from the model in three cases: surface displacement field registered without landmarks or lobe fissures, with landmarks only (10 on the left lung surface, 16 on the right one), and with both landmarks and lobar fissures.
segmentations on CT images at the two inflation states. Simulations are carried out with Dirichlet boundary conditions issued from image registration. Three registered surface displacement fields are used: crude registration performed without landmarks or lobar fissures, registration performed with landmarks, and registration performed with both landmarks and lobar fissures.
As we add physiological information to the registration process, results get more accurate. Whether landmarks or fissures contribute more to the improvement depends on the number of landmarks used and on their relevance. Here adding landmarks to the process improves the result by 7% on average. Adding fissures brings a further 2% improvement. The residual error can have several sources: here we neglect the lobe sliding, though it may affect the parenchyma displacement field [100]. The volume increase from MLV to TLC is about 70%: intermediary states images between the two configurations would ensure a better registration and hence more accuracy in ventilation prediction. Linear elasticity is a rough approximation for large displacements. With this constitutive relation, recoil effort is increasingly underestimated as the parenchyma expands. Displacement may be well predicted but a proper effort computation requires an appropriate mechanical law for the parenchyma. Despite these strong assumptions, results are encouraging. This shows that surface parenchyma displacement field is an appropriate boundary condition when it comes to lung ventilation modeling. This points out also how crucial it is for the registration to be precise. If 4D-CT[80] or 4D-MRI [81] data along with segmented upper airways were available in a pathological case where tree resistance is increased, it would be of great interest to run the model with dynamic Dirichlet boundary conditions and to compare the resulting simulation to the corresponding dynamic ventilation acquisition; depending on the tree resistance distribution the tree-parenchyma coupling could then be emphasized.
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
