Coupled Problems in
Computational Modeling of the Respiratory System
Mechanical ventilation is a vital
supportive therapy for critical care patients suffering from acute lung diseases in view of oxygen supply. However, preexisting lung injuries predispose patients to a number of complications which are collectively termed ventilator associated lung injuries (VALI). VALI mainly occurs in the walls of the alveoli, i.e. the small lung compartments constituting the bloodgas barrier. Understanding the reason why alveoli still become damaged or inflamed despite recent developments towards more "protective" ventilation protocols is a key question sought by the medical community. Computational models of the respiratory system can provide essential insights into involved phenomena. In particular, computational models offer the possibility to predict data that cannot be measured in vivo such as local alveolar strains and stresses which are relevant for the development and progress of VALI. However, establishing reasonable models is difficult since the lung comprises more than 20 generations of bifurcating airways ending in approximately 500 million alveoli. This complexity inhibits a direct numerical simulation resolving all relevant structures from the onset. Therefore, as a first step, we have developed detailed computational models of distinct parts of the lung, i.e. the tracheobronchial and the alveolar region, which will later be combined to one overall "virtual lung" model [1].

Figure 1: Contours of phaseaveraged velocity in the
laryngeal and tracheal region. Contours of velocity are plotted at the locations L1L4 indicated along the
trachea (taken from [3]).

All simulations were performed with our multipurpose finite element (FE) software platform BACI [2]. BACI has been and is developed within an objectoriented C++ environment. Parallelization is based on domain decomposition methods using MPI. Stateoftheart solution techniques for nonlinear and linear systems of equations as well as for coupling of several physical fields are incorporated in BACI and are continuously
developed further in our group. For the implementation of efficient parallel
sparse linear algebra operations, BACI makes use of the opensource software framework "Trilinos" (Sandia National Laboratories).
There is an ongoing debate about how turbulence in the upper airways affects flow and pressure in lower airway generations. As opposed to previous simulations, we considered a pulsatile inflow condition, allowing the development of turbulence over a pulse cycle to be investigated, which is obviously more physiologically
realistic. Our results (see, e.g., Fig. 1)
suggest that turbulence effects in the bronchial airways are rather weak and can completely decay as early as the third generation, depending on both geometry and flow distribution.
Due to limitations on the number of airways visible on CT scans, only a part of the airway tree can be resolved
in 3D. Therefore, realistic boundary conditions need to be applied at the outlets of the 3D domain in order to consider the effect of the unresolved peripheral region appropriately. For this purpose, we have developed a
reduceddimensional model of the non
imageable vessels [4]. Briefly, the 3D
airway model is supplemented by simplified 1D trees attached to every
3D outlet. By considering the unresolved
peripheral impedances, reasonable outflow boundary conditions are
derived for the resolved 3D domain.

Figure 2: Comparison of pressure contours, for light
activity, at maximum inspiration. Top: 3D model with
free outflow. Bottom: 3D model considering peripheral
impedances (taken from [4]).

With these boundary conditions, we found that the maximum pressure in
the tracheabronchial region is
approximately 44% higher than in
previous models neglecting peripheral
impedances (see also Figure 2).

Figure 3: Tissue stress normalized by the
maximum stress.
Left: Airways without
surrounding lung tissue.
Right: Airways embedded in surrounding tissue.

As a next step, we also studied the influence of the surrounding tissue on both air flow and stresses in the airway walls. To solve the fully coupled nonlinear fluidstructure interaction problem, a monolithic approach with algebraic multigrid preconditioning was utilized (see, e.g., [5] and [6]). We found that essential features of the velocity distribution, such as locations of high velocity jets and recirculation zones, remain the same.
However, a fivefold reduction in airway
wall stresses was observed for airway models embedded in lung tissue.
In addition, we also found that the distribution of stresses slightly changed (see Figure 3).
Our detailed model of the tracheabronchial region enables us to quantify the distribution of pressure and flow into the peripheral regions. Since alveoli are the major site of VALI, we have also developed a comprehensive model of individual alveoli allowing us to determine local stresses and strains in the tissue. In contrast to former studies, our model is based on a realistic threedimensional geometry obtained from microCT imaging. We found that strains in individual alveolar walls are up to 400% higher than averaged global tissue strains (see Figure 4). Consequently, resolving the realistic alveolar morphology is crucial when investigating phenomena
of VALI.

Figure 4: Strain distribution in a cube of lung tissue. Red color indicates strain "hot spots" at risk of overdistension (taken from [7]).

Ongoing work is concerned with combining our detailed models of the
tracheobronchial and the alveolar region to one overall "virtual lung" model. For validation purposes, we plan to correlate our simulation results with medical data obtained for patientspecific disease conditions.
In the future, we hope to gain further insights into how different ventilation protocols affect local stresses and strains and, thereby, the development and progress of VALI.
References
[1] Wall, W. A., Wiechert, L.,
Comerford, A., Rausch, S.
Towards a comprehensive computational model for the respiratory system, International Journal for Numerical Methods in Biomedical Engineering 26, 807827, 2010
[2] Wall, W. A., Gee, M. W.
BACI: A parallel multiphysics simulation environment, Technical Report, Institute for Computational Mechanics, Technische Universität München, 2010
[3] Comerford, A., Gravemeier, V.,
Wall, W. A.
Turbulent pulsatile flow in the pulmonary airways, Journal of Fluid Mechanics,
submitted, 2012
[4] Comerford, A., Förster, C., Wall, W. A.
Structured tree impedance outflow
boundary conditions for 3D lung simulations, Journal of Biomechanical Engineering 132, 081002 110, 2010
[5] Gee, M. W., Küttler, U., Wall, W. A.
Truly monolithic algebraic multigrid for fluidstructure interaction, International Journal for Numerical Methods in Engineering, 85, 9871016, 2010
[6] Wiesner, T., Tuminaro, R. S., Wall, W. A.,
Gee, M. W.Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections. Numerical Linear
Algebra with Applications, submitted, 2011
[7] Rausch, S., Haberthür, D., Stampanoni,
M., Schittny, J. C., Wall, W. A.
Local strain distribution in real threedimensional alveolar geometries, Annals of Biomedical Engineering 39, 28352843, 2011
[8] http://www.lnm.mw.tum.de/research/
applications/biomedicalrespiratory
system/
• Wolfgang A. Wall Institute for
Computational
Mechanics,
Technische
Universität München
• Michael W. Gee Institute for
Computational
Mechanics,
Technische
Universität München
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