Biomedical Engineering Reference
In-Depth Information
mation states can be observed. Although imposed multi-scale boundary conditions
are still very simple, alveolar deformation can be simulated more realistically than
in the comparative simulation neglecting the influence of the surrounding tissue
completely.
28.2.3 Coupling of 3D Parenchyma and Airway Models
Although VALI is known to occur primarily in the alveolar region, the conducting
part of the lung also has to be included in the model. After all, local parenchyma de-
formations are determined by the distribution of airflow into the peripheral domains.
Hence, we have been recently focusing on completing our 'bridging of scales' by
combining our multi-scale parenchyma model with 3D airway models (cf., e.g.,
Wall and Rabczuk,
2008
; Comerford et al.,
2010
). Again, due to limited computa-
tional resources and the insufficient resolution of CT imaging techniques (minimum
voxel size of 0
.
5
0
.
5 mm), a detailed modeling of all relevant airway struc-
tures from the trachea—where the endotracheal tube is situated during mechanical
ventilation—down to the alveoli is not possible. Therefore, airway models are usu-
ally restricted to the first generations of the tracheo-bronchial tree.
To compensate for the gap between resolvable airways and the alveolar region,
we have proposed a general concept for the homogenization of unresolvable struc-
tures (Yoshihara and Wall,
2012
). Briefly, our approach considers two different types
of interactions of airway and tissue models. Firstly, lung parenchyma surrounds the
main part of the airway tree, thereby affecting airflow and inducing an interdepen-
dence of neighboring airways not present in the isolated airway tree. This effect
can be considered by means of fluid-structure interaction (FSI) procedures (see,
e.g., Gee et al.,
2010
; Küttler et al.,
2010
). Secondly, the parenchyma is inflated
by the air transported in the conducting part. To consider this coupling, we divide
the parenchyma model into subdomains associated with the outlets of the resolved
three-dimensional airway tree. Each subdomain can be thought of as a homogenized
continuum consisting of smaller airways and alveoli that is provided with air by the
associated 3D airway. Hence, the volume of air passing through each outlet has to
equal the change in volume of the corresponding tissue subdomain. To enforce this
constraint within the framework of FSI problems, we utilize a Lagrange multiplier
technique. For the parallel and iterative solution of the resulting linear systems, a
specific preconditioning algorithm has been introduced.
The simple numerical example shown in Fig.
28.3
illustrates the novel volume-
coupled FSI approach. A cuboidal parenchyma model is split into two parts consis-
tent with the two outlets of the embedded deformable cylindrical airways. If both
parenchyma blocks exhibit the same material properties, a perfectly symmetric dis-
tribution of parenchyma deformations and airflow develops for a prescribed inflow.
However, if the Young's modulus of the left block is doubled, most of the inflow-
ing air is transported into the softer right parenchyma block resulting in a hetero-
geneous deformation state. In combination with the multi-scale approach of lung
×
0
.
5
×
