yoshihara 2014sds

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2014; 100:277–299Published online 28 July 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4735

A combined fluid–structure interaction and multi–field scalartransport model for simulating mass transport in biomechanics

L. Yoshihara*,†, M. Coroneo, A. Comerford, G. Bauer, T. Klöppel and W. A. Wall

Institute for Computational Mechanics, Technische Universität München, Boltzmannstr. 15, 85747 Garching b.,Munich, Germany

SUMMARY

Mass transport processes are known to play an important role in many fields of biomechanics such as res-piratory, cardiovascular, and biofilm mechanics. In this paper, we present a novel computational modelconsidering the effect of local solid deformation and fluid flow on mass transport. As the transport pro-cesses are assumed to influence neither structure deformation nor fluid flow, a sequential one-way couplingof a fluid–structure interaction (FSI) and a multi-field scalar transport model is realized. In each time step,first the non-linear monolithic FSI problem is solved to determine current local deformations and velocities.Using this information, the mass transport equations can then be formulated on the deformed fluid and soliddomains. At the interface, concentrations are related depending on the interfacial permeability. First numer-ical examples demonstrate that the proposed approach is suitable for simulating convective and diffusivescalar transport on coupled, deformable fluid and solid domains. Copyright © 2014 John Wiley & Sons, Ltd.

Received 13 June 2013; Revised 14 February 2014; Accepted 8 June 2014

KEY WORDS: mass transport; fluid–structure interaction; cardiovascular mechanics; respiratory mechan-ics; biofilm mechanics

1. INTRODUCTION

Transport of substances through biological fluids and solids plays an essential role in many fieldsof biomechanics. For instance, understanding particle transport in the lungs is essential for the opti-mization of targeted drug delivery. Another application in respiratory mechanics is the improvementof blood oxygenation during mechanical ventilation. In this case, the transport and exchange ofoxygen and carbon dioxide between air, tissue, and blood have to be studied. In biofilm mechan-ics, a proper understanding of nutrient transport and uptake is indispensable to enable the control ofbiofilm formation and development. While being mostly detrimental in the health sector, biofilmscan also be beneficial in a wide range of applications, such as waste water treatment. A prominentexample in cardiovascular mechanics is atherosclerosis, a prevalent disease of the large arteries thatinvolves an accumulation of lipoproteins in the arterial wall [1].

Computational models can fundamentally contribute to the understanding of involved phenom-ena and, as a consequence, to the improvement of therapeutical approaches. However, establishingreasonable numerical models is very challenging because the interaction of various physical fieldssuch as fluid, solid, and transported species has to be considered. In addition, most of the transportprocesses take place in a highly deforming environment. For instance, gas transport in the lungs isaffected by the periodic expansion and contraction of the organ during breathing.

Previous studies have been predominantly confined to passive transport processes in fluids. Cor-responding computational approaches are based on a one-way coupling of a computational fluid

*Correspondence to: L. Yoshihara, Institute for Computational Mechanics, Technische Universität München,Boltzmannstr. 15, 85747 Garching b., Munich, Germany.

†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

278 L. YOSHIHARA ET AL.

dynamics (CFD) and a scalar transport model. Hence, information on the flow field obtained fromthe CFD simulation is utilized to formulate the convection–diffusion equation for the mass transfer.Applications of this methodology include the transport of (macro-)molecules in both blood [2–5]and air [6, 7]. In addition, there have been a number of approaches considering also the coupling ofmacromolecule transport in the arterial lumen to transport within the wall [8–12]. To model inter-stitial flow within the immovable wall, either Darcy’s law or Brinkman’s equation has been utilizedin this context.

More complex models also include fluid–structure interaction (FSI) effects. For instance, it wasdemonstrated in [13] that wall compliance affects the transport of oxygen to the arterial wall in thecarotid bifurcation. Mass transport within the wall, though, was ignored in this study. By contrast,the approach proposed in [14] considers macro-molecule transport in both the lumen and the wall ofa coronary artery. The tissue was modeled as a poroelastic material, allowing the expression of theinterstitial velocity via Darcy’s equation. Subsequent to the FSI analysis, macro-molecule transportwas simulated based on the computed velocity and deformation field. However, transport processesin the wall and in the lumen were considered in a decoupled manner. After having calculated theconcentration field within the fluid, interfacial lumen concentrations were utilized as a boundarycondition to the transport simulations within the wall. As a consequence, mass transfer was onlypossible in one direction, that is, from fluid to tissue.

While the models mentioned so far have been concerned with rather basic transport processes,there have also been a number of studies looking specifically at drug release from stents into thearterial wall and lumen [15–17]. Again, a convection–diffusion equation was utilized to model thetransport of the drug through the different wall layers. In [16], the deformation of the wall followingstent deployment was considered simply by remeshing both the wall and the stent structure. Hence,wall deformation and mass transport were essentially uncoupled. In [17], a structural simulation ofstent deployment was utilized to calculate the interstitial water velocity from the deformation field.This was achieved by means of a decomposition of the total stress into the effective stress of thetissue matrix and the pore pressure. The latter component was then used to calculate the interstitialvelocity via Darcy’s equation. However, the mass transfer problem was again formulated on the finaldeformed structure. This methodology is suitable for drug eluting stent deployment simulations, butnot for more dynamically moving structures.

In addition to the 3D finite element (FE) based approaches discussed earlier, there have been anumber of studies utilizing a combination of 1D transport models and Fick’s law to describe thediffusion through the tissue (see, for instance, [18]). The major disadvantage of these reduced-dimensional models is that no information is available about local transport processes.

To overcome the drawbacks of previous approaches, we have developed a novel computationalapproach based on the sequential one-way coupling of a monolithic FSI model and a monolithicmulti-field transport model. The proposed methodology enables the consideration of (i) the effect oflocal tissue deformation and fluid flow on mass transport and (ii) the mutual interaction of transportprocesses in the lumen and the wall.

The remainder of this paper is organized as follows. After having summarized the governingequations of the FSI and the multi-field transport subproblem in Sections 2 and 3, the coupling ofsubproblems will be discussed in Section 4. The general validity and versatility of the proposedmodel will be illustrated by selected numerical examples in Section 5. Finally, concluding remarksand a brief outlook will be provided in Section 6.

2. FLUID–STRUCTURE INTERACTION

FSI problems can formally be described as three-field problems. To begin with, there are two phys-ical fields, fluid and structure, which share a common FSI interface � . Furthermore, to account fordeformations of the fluid domain, an arbitrary Lagrangian–Eulerian (ALE) approach is employed,constituting a third, non-physical mesh field.

In the following, the governing equations for each field will be summarized. A definition ofthe individual domains and the partitions of boundaries is given in Figure 1. In this context,

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2014; 100:277–299DOI: 10.1002/nme

MODEL FOR SIMULATING MASS TRANSPORT IN BIOMECHANICS 279

Figure 1. FSI subproblem: definition of domains�F; �S; �G, and their boundaries. The boundaries of solidand fluid domains are partitioned in the form @�K D �K

D [ �KN [ � with K 2 ¹F;Sº, whereas the boundary

of �G is given by @�G D �GE [ � .

fluid, structure, and mesh (or grid) quantities are denoted by the superscripts .�/F; .�/S, and .�/G,respectively.

In order to enable the numerical solution of the field equations, discretization in space and time isnecessary. In this work, we utilize FEs and implicit time integration schemes for all fields, leadingto a set of non-linear algebraic equations, which is solved using a Newton-type method. It has tobe noted, though, that the algorithms discussed here are not restricted to any specific discretizationscheme.

After having reviewed the individual field equations, the coupling of fields will be discussed indetail at the close of this section.

2.1. Structure field

In this work, the structure field is modeled by the non-linear elastodynamics equation

�S d2dS

dt2D r � .F � S/C �SbS in �S � .0; T /; (1)

where dS is the unknown structural displacement, �S is the structural density, �S denotes the unde-formed domain, and bS represents the external body forces. In Equation (1), the internal forces areexpressed in terms of the second Piola–Kirchhoff stress tensor S and the deformation gradient F. Torelate stresses and strains, a suitable constitutive model needs to be formulated.

At the Dirichlet and Neumann boundaries, �SD and �S

N, boundary conditions have to be defined,that is,

dS D NdS on �SD � .0; T / (2)

.F � S/ � nS D NhS on �SN � .0; T /; (3)

where nS refers to the outward-pointing normal vector. To complete the initial boundary valueproblem, the following initial conditions need to be specified:

dS.x; 0/ D dS0.x/ for x 2 �S; (4)

ddS

dt.x; 0/ D PdS

0.x/ for x 2 �S: (5)

The weak form of the structure problem stated earlier is the starting point for the spatial dis-cretization. Details about the FE method for structure fields can be found, for example, in [19–21].Although mixed or hybrid FE are equally suitable, a displacement-based FE formulation is consid-ered exclusively in this work for the sake of simplicity. In this case, the weak form is obtained bymultiplication of Equation (1) with the virtual displacements ıdS followed by integration by parts,that is,

0 D

�ıdS; �S d2dS

dt2

��S

C�rıdS;F � S

��S �

�ıdS; �SbS

��S �

�ıdS; NhS

��S

N: (6)



Here, .�; �/�S and .�; �/�SN

denote the L2 inner products on �S and �SN, respectively. Equation (6) can

now be discretized in space and time. In the following, the residual of the discrete weak form isdenoted by rS D rS.dS/, where dS is from now on the vector of discretized nodal displacements.After consistent linearization, the system to be solved in every iteration step i of the non-linearalgorithm reads

"SII SI�

S�I S��

#inC1

"�dS

I

�dS�

#iC1nC1

D �

"rS

I

rS�

#inC1

(7)

for time step n C 1. In preparation of the coupling of fields introduced in Section 2.4, the setof equations is already split into interior quantities identified by .�/I and those defined at the FSIinterface denoted by .�/� , that is,

S˛ˇ D@rS˛

@dSˇ

; (8)

with ˛; ˇ 2 ¹I; �º.

2.2. Fluid field

The considered flow problem is assumed to be governed by the instationary, incompressible Navier–Stokes equations for a Newtonian fluid on a deformable fluid domain �F. The deformation of thefluid domain dG is defined by a unique, arbitrary mapping ' given by

dG.x; t / D '�dG� ; x; t

�for .x; t / 2 �F � .0; T /; (9)

where dG� represents the mesh interface displacement that will later be related to the structure

interface displacement dS� . The grid velocity uG is then defined by

uG D@'

@tin �F � .0; T /: (10)

Equation (10) allows for the definition of the ALE convective velocity

cF D uF � uG (11)

representing the fluid velocity uF relative to the arbitrarily moving fluid domain. Hence, theconvective formulation of the ALE form of the Navier–Stokes equations reads

�F @uF

@tC �F

�cF � r

�uF � 2�r � "

�uF�C rpF D �FbF; (12)

r � uF D 0; (13)

where the fluid velocity uF and the fluid pressure pF are the unknown physical fields. In the momen-tum Equation (12), bF denotes a prescribed body force, ".uF/ D 1

2

�ruF C .ruF/T

�the strain rate

tensor of the Newtonian fluid, and � its dynamic viscosity. Equation (13) states the conservation ofmass given that the fluid density �F is constant.

At the Dirichlet boundary �FD and the Neumann boundary �F

N, the following conditions are set:

uF D Nu on �FD � .0; T /; (14)

� F � nF D Nh on �FN � .0; T /; (15)



where nF denotes the outward-pointing normal vector. The Cauchy stress tensor � F for a Newtonianfluid is given by

� F D �pFIC 2�"�uF�; (16)

with I denoting the second-order identity tensor. To complete the initial boundary value problem, adivergence-free initial velocity field uF.x; 0/ D uF

0.x/ has to be specified for all x 2 �F.The weak form of the incompressible Navier–Stokes Equations (12) and (13) is obtained by

multiplying these equations with test functions ıuF for velocity and ıpF for pressure and integratingby parts, that is,

0 D ��ıuF; �FbF

��F C

�ıuF; �F @uF

@t

��F

C�ıuF; �F

�cF � r

�uF��F

C�rıuF; 2�".uF/

��F �

�r � ıuF; pF

��F �

�ıpF;r � uF

��F C

�ıuF; Nh

��F

N:

(17)

In this work, an implicit time integration method is utilized in combination with stabilized FEs todiscretize the fluid problem (17). Stabilization terms are applied to account for instabilities arisingfrom equal-order discretization of fluid and pressure fields as well as for convection-dominatedproblems. For details on the FE discretization of the fluid field and stabilization methods we refer,for example, to [22–24].

In the following, the residual of the discrete weak form is denoted by rF D rF.uF; pF;dG/.Again, uF;pF, and dG now represent the vectors of discretized nodal unknowns. The total residualdifferential then reads

drF D@rF

@uFduF C

@rF

@pFdpF C

@rF

@dGddG: (18)

The resulting set of linear equations can again be split into interior quantities and those defined atthe FSI interface, yielding the following system of equations

"F II F I� FG

II FGI�

F�I F�� FG�I FG

��

#inC1

266664�uF

I

�uF�

�dGI

�dG�

377775

iC1

nC1

D �

"rF

I

rF�

#inC1

(19)

with F˛ˇ ;FG˛ˇ .˛; ˇ 2 ¹I; �º/ denoting the matrix representations of the partial derivatives

introduced in Equation (18), that is,

F˛ˇ D@rF˛

@uFˇ

; FG˛ˇ D

@rF˛

@dGˇ

: (20)

For ease of notation, the vector pF of nodal pressure values has been merged with the vector of theinterior fluid unknowns uF

I . A detailed description of the so-called shape derivatives FG˛ˇ can be

found, for instance, in [25, 26].

2.3. ALE field

The ALE formulation of the incompressible Navier–Stokes equations necessitates the definition ofthe mapping' introduced in Equation (9). In case of the FSI problems considered here, the boundaryof the ALE mesh is coupled to the Lagrangian mesh of the flanking structures (cf. also Section 2.4)and an Eulerian mesh at the inflow and outflow portions denoted by �E, that is,

dG D 0 on �GE � .0; T /; (21)

dG D dS on � � .0; T /: (22)



Within the domain, the ALE mesh is allowed to deform arbitrarily. In this study, the ALE field istreated as a quasi-elastostatic pseudo-structure following [27]. In this case, the ALE equation ofmotion reads

r � �G D 0 in �G � .0; T / (23)

with

�G D �Gtr��G�

IC 2�G�G; �G D1

2

�rdG C

�rdG

�T�: (24)

In this context, tr.�/ represents the trace operator and �G; �G denote the Lamé constants of thepseudo-structure, which can be chosen arbitrarily.

The procedure for setting up the linearized discrete weak form associated with Equation (23) iscomparable to those presented in the previous sections. As the mesh movement is not allowed toinfluence the interface, the resulting set of linear equations reduces to

�GII GI�

inC1

"�dG

I

�dG�

#iC1nC1

D 0: (25)

2.4. FSI coupling

At the fluid–structure interface � , different kinematic and dynamic constraints have to be fulfilled.Equilibrium of forces requires the surface tractions of fluid and structure to be equal, yielding

hS� D �hF

� on � � .0; T /: (26)

In addition, the grid velocity uG� has to match the fluid velocity uF

� at the interface:

uF� D uG

� on � � .0; T /: (27)

Usually, both a mass flow across and a relative tangential movement of fluid and structure at � areprohibited, that is,

@dS�

@tD uF

� on � � .0; T /: (28)

In combination with Equation (27), this condition is equivalent to

dS� D dG

� on � � .0; T /; (29)

at least in the continuous setting. Hence, structural deformation and fluid movement (represented bythe ALE-based fluid domain deformation dG

� ) must match at � .The fully coupled non-linear FSI problem is solved within one global Newton loop. Consequently,

the individual field variables, that is, solid displacements, ALE mesh displacements, fluid velocities,and pressure fields, are determined simultaneously. For complex biological problems involving thecoupling of incompressible flows and soft tissue, these so-called monolithic schemes were foundto be the best (and sometimes even the only feasible) alternative [28, 29]. As a matter of fact, thesimilar densities of fluid and structure and the incompressible flow constitute a very challengingscenario for coupling schemes, and partitioned schemes were found to be less robust and inefficientin these conditions. In the following, it is assumed that fluid and solid FE discretizations coincideat the interface, although the extension to non-conforming meshes is straightforward (cf., e.g., [30]for a novel coupling algorithm based on a dual mortar method). In this case, there is only oneindependent set of degrees of freedom at the interface as a consequence of the kinematic constraints(27)–(29). Thus, either interface displacements or velocities are retained in the monolithic system ofequations. In the following, the monolithic FSI problem will be derived for the latter case, although



both alternatives are in general equivalent. For the coupling of discrete mesh displacements and fluidvelocities at the interface, the following relation was derived in [31]

dG�InC1 D dG

�In C�t

2

�uF�InC1 C uF

�In

�: (30)

This trapezoidal rule correctly preserves the size of the fluid domain and is exact for every fluid timeintegration scheme that assumes constant accelerations within the time step �t . Rearrangement ofEquation (30) yields the subsequent expression for the increments of interface displacements

�dGIiC1�InC1 D

�t

2�uFIiC1

�InC1 C �.i/�t uF�In; (31)

where i again denotes the iteration step, and we have introduced the mapping

�.i/ D

²1; i D 00; i > 0

: (32)

Based on the linearized problems of the individual fields and the constraint conditions discussedearlier, the monolithic FSI problem to be solved at time tnC1, and iteration step i C 1 finally reads2

666664

SII�t2SI� 0 0

S�I F�� C�t2

�S�� CFG

��

�F�I FG

�I

0 F I� C�t2FG

I� F II FGII

0 �t2GI� 0 GII

3777775

i

nC1

266664�dS

I

�uF�

�uFI

�dGI

377775

iC1

nC1

D

2666664

fSI

fSF�

fFI

fGI

3777775

i

nC1

(33)

with 2666664

fSI

fSF�

fFI

fGI

3777775

i

nC1

D �

2666664

rSI

rS� C rF

�

rFI

rGI

3777775

i

nC1

C �.i/�t

2666664

SI� 0 0 0

0 S�� CFG�� 0 0

0 0 FGI� 0

0 0 0 GI�

3777775

i

nC1

266664

uF�

uF�

uF�

uF�

377775n

:

(34)

After solution of Equation (33), the vector of unknowns is updated as follows:266664

dSI

uF�

uFI

dGI

377775

iC1

nC1

D

266664

dSI

uF�

uFI

dGI

377775

i

nC1

C

266664�dS

I

�uF�

�uFI

�dGI

377775

iC1

nC1

: (35)

The iterative procedure is aborted when a convergence criterion is met, for example,

�rFSI�inC1

D

2666664

fSI

fSF�

fFI

fGI

3777775

i

nC1

< FSI; (36)

where FSI is a problem-specific tolerance.



Figure 2. Multi-field scalar transport subproblem: Definition of domains �K and boundary partitions@�ÎK D �ÎKD [ �ÎKN [ � with K 2 ¹F;Sº.

3. MULTI-FIELD SCALAR TRANSPORT

To study passive transport phenomena in biological systems, a multi-field scalar transport modelis established. On each domain �K (with K 2 ¹F;Sº, cf. also Figure 2), the temporal andspatial variation of the scalar concentration field ˆK is governed by the following partialdifferential equation

@ˆK

@tC cK � rˆK CˆK

�r � uK

�� r �

�DKrˆK

�CRK D 0: (37)

Here, DK denotes the domain-specific diffusion coefficient, uK the velocity, RK the reactionterm, and cK the convective velocity. The latter has already been introduced for the fluid field inEquation (11). On the solid domain, the velocity uS D @dS

@tand the grid velocity are identical, and,

hence, the convective velocity vanishes. In contrast to the momentum equation for the incompress-ible fluid (12), a conservative formulation needs to be chosen for the mass transport because, ingeneral, r � uS ¤ 0.

Essential (or Dirichlet) boundary conditions are defined by

ˆK D N̂ K on �ÎKD � .0; T /: (38)

A typical example is an inflow boundary where the mass concentration entering the domain isknown. At the Neumann boundary �ÎKN , the negative normal flux is prescribed as follows:

DKrˆK � nK D NhK on �ÎKN � .0; T /: (39)

At the interface � between the domains �F and �S, the mass fluxes are given by

hS D P�ˆS �ˆF

�; hF D P

�ˆF �ˆS

�on � � .0; T /; (40)

where P denotes the interface permeability. The initial boundary value problem is completed by thedefinition of appropriate initial concentration fields reading

ˆK.x; 0/ D ˆK0 .x/ for x 2 �K: (41)

For each domain �K, the weak form is obtained by multiplying Equation (37) with the virtualconcentrations ıˆK and integrating by parts, yielding

0 D

�ıˆK;

@ˆK

@tC cK � rˆK CˆK

�r � uK

��K

C�rıˆK;DrˆK

��K

C�ıˆK; RK

��K �

�ıˆK; NhK

��ÎKN��ıˆK; hK

��:

(42)



After discretization in space and time, the discrete form of Equation (42) reads

0 D ıˆK �tÎK C hÎK�; (43)

which has to hold for arbitrary virtual concentrations ıˆK. The discrete vectors tÎK and hÎK

arise from the first four summands and the last term of the right hand side of (42), respectively. Incase of a linear reaction term, the transport problem is completely linear. Hence, the system to besolved in every time step tnC1 can be formulated in terms of either absolute concentrationsˆK

nC1 orconcentration increments �ˆK

nC1. If, however, the reaction term is nonlinear, Equation (43) has tobe solved iteratively. Hence, we restrict ourselves to the more general incremental formulation here.In this case, the monolithic system of equations reads

2666664

T FII T F

I� 0 0

T F�I T F

�� CH 0 �H

0 0 T SII T S

I�

0 �H T S�I T S

�� CH

3777775

i

nC1

2666664

�ˆFI

�ˆF�

�ˆSI

�ˆS�

3777775

iC1

nC1

D �

26666664

tÎFI

tÎF� C hÎF

tÎSI

tÎS� C hÎS

37777775

i

nC1

;

(44)

where T K˛ˇ is the matrix representation of the partial derivatives of tÎK˛ with respect to the nodal

concentrations ˆKˇ .˛; ˇ 2 ¹I; �º/. The coupling matrix H arises from the discrete flux condition

hÎK and has a mass matrix like structure. For the sake of clarity, the set of Equations (44) hasagain been split into interior and boundary quantities. After solution of Equation (44), the vector ofunknowns is updated as follows:

2666664

ˆFI

ˆF�

ˆSI

ˆS�

3777775

iC1

nC1

D

2666664

ˆFI

ˆF�

ˆSI

ˆS�

3777775

i

nC1

C

2666664

�ˆFI

�ˆF�

�ˆSI

�ˆS�

3777775

iC1

nC1

: (45)

The iterative procedure is aborted when a convergence criterion is met, for example,

.rmass/inC1

D

26666664

tÎFI

tÎF� C hÎF

tÎSI

tÎS� C hÎS

37777775

i

nC1

< mass; (46)

where mass is a problem-specific tolerance.In the limiting case of an infinite permeability, the interfacial concentrations are equal, that is,

ˆS� D ˆ

F� : (47)



As a consequence, the concentration increments are also identical, and, hence, the discrete multi-field scalar transport problem simplifies to

2664

T FII T F

I� 0

T F�I T F

�� C T S�� T S

�I

0 T SI� T S

II

3775i

nC1

2664�ˆF

I

�ˆF�

�ˆSI

3775iC1

nC1

D �

2664

tÎFI

tÎF� C tÎS�

tÎSI

3775i

nC1

: (48)

As it is well-known, for convection–diffusion–reaction problems in convection-dominatedregimes, the standard Galerkin FE method is potentially unstable. It is possible to avoid such insta-bilities through the use of very fine spatial resolutions. This strategy, however, usually becomesprohibitively expensive in terms of computational costs for most examples. As a consequence,stabilized FE formulations for the numerical solution of convection–diffusion–reaction equationswere introduced. The proposed stabilization technique, which basically is a streamline upwindPetrov/Galerkin method [32], accounts for spurious oscillations of the standard Galerkin FE methodwhen convection dominates. Details of the stabilized FE formulation implemented in the presentapproach are provided in [33].

4. COUPLING OF FSI AND MASS TRANSPORT SUBPROBLEMS

In most cases, it can be assumed that fluid flow and structural deformation are not influenced bymass transport processes. In line with this supposition, a one-way coupling of CFD and transportmodels has been proposed in the past, for example, to study multi-ion transport in electrochemicalsystems [34] or nanoparticle transport in the lung [7]. A sequential procedure also seems suitable forthe coupling of the FSI and multi-field scalar transport models discussed in Sections 2 and 3. Hence,in each time step, the non-linear FSI problem is solved first. In the present study, the same spatialdiscretization for the FSI and the mass transport equations is utilized. Hence, local deformations andvelocities obtained from the FSI calculation can be transferred directly (i.e., without necessitatingany interpolation) to the mass transport subproblem. Using this information, the multi-field scalartransport equations can then be solved on the deforming fluid and solid domains. The proposedprocedure is summarized in Algorithm 1 and Figure 3. It is, however, important to highlight thatthere are some specific applications in which the scalar transport has an influence on the fluid flow.In these cases, a two-way coupling between the FSI and the scalar transport subproblems has to beemployed [34, 35].

Algorithm 1 Coupling of FSI and mass transport subproblemswhile t < tmax do

whilerFSI

> FSI doSet up and solve (33)Update dS;uF, and dG (35)Check convergence (36)

end whileTransfer dS;dG; uS;uF, and cF to mass transport subproblemwhile krmassk > mass do

Set up and solve (44) or (48)Update ˆF and ˆS (45)Check convergence (46)

end whileUpdate t

end while



Figure 3. Schematic of the one-way coupling of FSI and mass transport subproblems.

5. NUMERICAL EXAMPLES

To demonstrate that the presented approach is suitable for simulating multi-field scalar transport ondeformable domains, selected numerical examples will be provided in the following. All simulationsare based on a parallel implementation of the algorithms discussed earlier in our in-house multi-physics research code BACI [36]. For time discretization of fluid, solid, and transport equations, aone-step- time integration scheme is utilized. For the purpose of increasing the numerical robust-ness, a slight shift of towards higher values as compared to the trapezoidal rule is chosen, and inall the examples, D 0:66 is always used. For the first four examples, trilinear hexahedral FE areused. Hence, they are based on three-dimensional models and discretizations, albeit a pseudo-two-dimensional deformation and flow state are enforced by prescribing specific boundary conditionsin most simulations considered. However, for the last example, a two-dimensional model is cho-sen to reduce computational costs. As quantitative results are of minor importance for the first fourexamples, details on material parameters and others are provided only where they seem to be help-ful for a better understanding. Besides, instead of computing absolute concentrations (i.e., the totalmass of a substance), we confine ourselves to calculate the spatial and temporal development ofconcentrations relative to an arbitrarily chosen prescribed value. Hence, ˆK is dimensionless inthese cases. By contrast, the last example accurately reproduces a real application. Therefore, alldetails regarding material parameters and operating conditions are reported, and ˆK is given inabsolute values.

In Sections 5.1 and 5.2, conservation of mass will be proven for structural and FSI problems,respectively. The influence of the permeability on the coupling of interfacial concentrations will beaddressed in Section 5.3. Afterwards, a possible application of the proposed methodology to oxygentransport in an idealized alveolus will be discussed in Section 5.4. Finally, the importance of theproposed model is highlighted in Section 5.5 using the example of a real biofilm structure.

5.1. Mass conservation for solid problems

In the first numerical example, a cube of lung tissue (edge length L D 100�m) compressed to halfits original volume. The chosen initial distribution of oxygen concentrations in the cube is given by

ˆS.x; y; z/ D 2:0C cos.0:02�x/ cos.0:02�y/ cos.0:02�z/

with �50�m 6 x; y; z 6 50�m (cf. also Figure 4). In line with [37], the diffusivity of oxygenin lung tissue is assumed to be DS D 1:0�m2=ms. At the boundary @�S of the cube, zero-fluxconditions are prescribed. In each time step tnC1, the transport equation

T SnC1�ˆ

SnC1 D �tˆISnC1 (49)

is solved on the deforming solid domain �S.In Figure 5, simulated concentration profiles are plotted at different points in time. Owing to the

diffusion processes within the cube, the initial gradient of the concentration field gradually reducesto zero. At the same time, the concentration increases owing to the prescribed compression ofthe cube.



Figure 4. Initial concentration field.

Figure 5. Simulated concentration profiles along the x-axis at different points in time.

Figure 6 shows the time-dependent course of the cube’s volume, the mean oxygen concentration,and the overall mass of oxygen. Clearly, the mean concentration increases to the same extent as thevolume of the cube decreases. Hence, the overall mass of oxygen remains constant at all times.

5.2. Mass conservation for FSI problems

The previous example demonstrated the validity of the proposed approach for mass transport insolids. Next, a simple FSI example will be considered. On the left hand side of Figure 7, the generalset-up of the problem is shown. The fluid domain �F (dimensions 100�m � 80�m � 5�m) isbound by a rigid wall at the bottom (indicated by �F

D) and a movable, nearly rigid solid cuboid(dimensions 100�m � 20�m � 5�m) at the top. The interface between the cuboid and the fluiddomain is again denoted by � . At �F

N, zero-traction boundary conditions are defined for the fluidfield. By prescribing the following Dirichlet condition

dSy.t/ D 15�m

�1:0C sin

��.t � 250:0ms/

500:0ms

��on �S

D � .0; T D 500ms/; (50)



Figure 6. Time-dependent course of the cube’s volume�R

d�S�, the mean oxygen concentration

�RˆSd�SR

d�S

�,

and the overall mass of oxygenRˆSd�S during compression of the cube. All quantities are referred to their

respective initial values.

Figure 7. Left: general set-up of the FSI validation example. Right: fluid flow at t D 250 ms, that is, at thetime of maximum flow. Colors and vector lengths indicate local velocity magnitudes.

Figure 8. Distribution of concentrations ˆK at t D 0 ms, t D 250 ms, and t D 500 ms (from left to right).

the solid cuboid is forced to move downwards, thereby decreasing the volume of the fluid domain.Consequently, flow through �F

N is induced (cf. also the right hand side of Figure 7).The initial configuration of the mass transport problem is defined by a linear concentration gra-

dient in vertical direction. The permeability of the interface is assumed to be infinite, and, thus,



concentrations at � are identical in both fields. The diffusivities are chosen to beDF D 1:6�m2=msandDS D 1:0�m2=ms, corresponding to the diffusivity of oxygen in blood and lung tissue, respec-tively. In Figure 8, the distribution of concentrations is shown at different points in time. As aconsequence of the convective and diffusive transport processes, the concentrations in both fieldsagain gradually equalize.

At the boundaries �FN, mass is transported out of the computational domain. Hence, in contrast

to the example discussed in Section 5.1, the overall mass within the domain changes with time. Ascan be clearly seen in Figure 9, this change in mass equals the mass flux out of the domain given by�ˆFc �DFrˆF

�� nF. Hence, conservation of mass also holds in case of FSI problems.

5.3. Coupling of concentrations at the interface

So far, the permeability of the interface between fluid and solid domains was assumed to be infinite.In practice, however, this presumption is unlikely to hold. The following example serves to providean insight into how permeability affects the distribution of concentrations within both domains.

For this purpose, mass transport is studied in a channel of fluid .100�m � 20�m � 2�m/ andan adjacent solid structure (dimensions 100�m � 6�m � 2�m/, which is fixed at �S

D. A detailedoverview on the respective domains and boundaries is given in Figure 10. The fluid is loaded by abody force in the x-direction with

bFx D

´100:0�m=ms2

�1:0C sin

�� t�50ms100ms

��for t < 100ms

200:0�m=ms2 for 100ms 6 t 6 300ms;(51)

yielding a maximum velocity of uFxImax D 0:64�m=ms. The diffusivities of the substance under

consideration are chosen to be DF D 1:6�m2=ms and DS D 0:5�m2=ms on the fluid and solid

Figure 9. Time-dependent course of the change in overall mass within the domainP

K�RˆKd�K

�t, the mass

flux out of the domain�ˆFc �DFrˆF

�� nF computed at �F

N, and the sum of both.

Figure 10. Coupling of concentrations at the interface: definition of domains and boundaries. For the sakeof simplicity, Neumann boundaries are not explicitly marked.



Figure 11. Concentration fields at t D 300ms for P D 0:0ms�1; P D 0:05ms�1; P D 0:2ms�1; P D1:0ms�1, and the limiting case P ! 1 (from top to bottom). Fluid flows from left to right. For the sakeof a better perceptibility of the concentration differences, the interface � between fluid and solid domains is

not explicitly marked.

domain, respectively. In the beginning, the concentration is zero throughout both domains except

for the boundaries �ˆIFD and �ˆISD , where ˆF D ˆS D 1:0 is prescribed. Depending on whether afinite or infinite permeability of the interface is assumed, either the weakly coupled problem (44) orthe simplified, strongly coupled system (48) is solved.

In Figure 11, the computed concentration fields at t D 300:0 ms can be compared for differ-ent permeabilities of the interface. Owing to the pronounced convective transport in the fluid phasein combination with the different diffusivities in both domains, an uneven distribution of concen-trations develops in all cases. However, the actual dispersion strongly depends on the interfacialpermeability. If P D 0:0ms�1 (cf. the top of Figure 11), concentrations at the interface are com-pletely decoupled. With increasing permeability, interfacial concentrations gradually equalize, and,hence, transport processes in both domains affect each other more and more. In the limiting case ofan infinite permeability (cf. the bottom of Figure 11), concentrations on both sides of the interfaceare identical.

Exemplarily, the difference in interfacial concentrations at the outlet of the fluid and solid domain,respectively, is plotted in Figure 12. Starting from �ˆ D 0:14, the difference drops quickly withincreasing permeability, thereby asymptotically approaching the limiting case �ˆ D 0:0.



Figure 12. Magnitude of the difference in interfacial concentrations �ˆ at the outlet of the fluid and soliddomain, respectively, for different surface permeabilities P .

5.4. Oxygen transport in an idealized terminal lung unit

After the general validity of the combined FSI and multi-field scalar transport model hasbeen demonstrated, a potential application in the field of respiratory mechanics is discussed inthe following.

The primary function of the lung is to enable the transport of oxygen and carbon dioxide betweenenvironment and blood. Efficient gas exchange requires that the blood–gas barrier possesses anextremely large surface area combined with a small thickness, such that the passage of gas moleculesis impeded as slightly as possible. For serving this purpose, multiple thin interior walls are formed,thereby subdividing the lung into a large number of small air chambers (also known as alveoli),which are connected with the outside air through the airway tree. The pulmonary arteries and veinsform similar trees that finally converge in a dense capillary network wrapped around the alveoli.

Unfortunately, ‘in vivo’ imaging techniques such as magnetic resonance tomography or computedtomography are inapplicable to alveoli owing to their small characteristic size in combination withthe high water content of the tissue. Hence, realistic representations are only available for excisedsmall animal lungs (see, for instance, [38] for an imaging-based model of rat lung tissue). However,the geometry of alveoli in the intact human lung is still unknown. Besides, many quantities such asexact flow velocities or material parameters of the tissue are currently investigated (see, e.g., [39]).Hence, at the moment, gas transport in the alveolus can only be simulated qualitatively at best. How-ever, owing to the complex interplay of tissue deformation, air flow, blood flow, and gas transport,this application is suited to demonstrate the general capabilities of the proposed methodology.

Details on the simplified configuration considered here are given in Figure 13. The lumen of asingle alveolus is approximated by a cuboid �FIa (dimensions 100�m� 30�m� 1�m) filled withair. On both sides of the cuboid �FIa

DI1, no-slip conditions are prescribed. Vis-á-vis of the inlet �FIaDI2,

the alveolar wall�S is situated. The wall itself consists of two layers of tissue (dimensions 100�m�5�m � 1�m) surrounding a pulmonary capillary �FIb (dimensions 100�m � 5�m � 1�m). Infact, the upper tissue layer is of minor interest here and is introduced only to enable a reasonablemovement of the capillary by providing an FSI interface. At �S

D, deformations in the x- and z-directions are prohibited, whereas the wall is allowed to move freely in the y-direction. To describethe material behavior of the tissue, a neo-Hookean constitutive model with Young’s modulus ES D

105 pN=�m2, Poisson’s ratio �S D 0:49, and density �S D 0:001 ng=�m3 is utilized. The materialparameters for air and blood are given by �FIa D 0:01983 ng=.�m ms/; �FIa D 1:2 � 10�6 ng=�m3

and �FIb D 3:0 ng=.�m ms/; �FIb D 1:06 � 10�3 ng=�m3. In line with [37, 40], the diffusivities ofoxygen in tissue, air, and blood are chosen to beDS D 1:0�m2=ms;DFIa D 2:0 �104 �m2=ms, andDFIb D 1:6�m2=ms, respectively.



Figure 13. Oxygen transport in an idealized terminal lung unit: definition of domains and boundaries. Forthe sake of simplicity, Neumann boundaries are not explicitly marked.

Figure 14. Distribution of flow velocities in the capillary at t D 150 ms (left) and in the lumen at t D 650ms (right). Owing to the different magnitudes of velocity, blood and air flow are displayed separately. Blood

flows from left to right.

In the first 150 ms of the simulation, a body force in the x-direction bFx D 1100:0�m=ms2 is grad-

ually applied to the capillary, yielding a maximum blood velocity of 1:19�m=ms. Subsequently, airflow and relative oxygen concentration at the inlet of the alveolus are prescribed as follows:

uFIay D 0:1�m=ms � C.x; t / on �

FIaDI2; ˆFIa D 1:0 � C.x; t / on �

ˆIFIaD (52)

with

C.x; t / D�1:0 � 0:0004x2

�sin

��t � 150:0ms

1000:0ms

�: (53)

To model the deoxygenated state of the blood entering the alveolus, ˆFIb D 0:0 is chosen at theinlet of the capillary �ˆIFIbD . As transport of oxygen from the blood to the upper tissue layer is notassumed to occur, concentrations are decoupled at the interface �3. At both �1 and �2, the limitingcase of an infinite permeability is assumed.

In Figures 14 and 15, snapshots of the distribution of flow and relative oxygen concentration areshown. As a consequence of the inflow, the volume of the air domain increases, thereby causingan upward movement of the alveolar wall. Oxygen is transported through both the lumen of thealveolus and the lower tissue layer into the capillary, where it is removed from the alveolus.

Although the presented example is still very simple, it clearly illustrates the suitability ofthe presented approach to model complex multi-field transport processes on deforming domains.It is moreover important to highlight that with a rigid model, there would not be any flow,and consequently, mass transfer would not be properly calculated because only diffusion wouldbe present.



Figure 15. Distribution of concentrations in all three fields at t D 350 ms, t D 450 ms, t D 650 ms (i.e., atmaximum inflow), t D 800 ms, t D 950 ms, and t D 1100 ms (from left top to right bottom).

Figure 16. Computational domain (a) and particular of the streamer (b).

5.5. Nutrient transport in a biofilm structure

The importance of coupling mass transfer with FSI can be particularly appreciated when modelingthe interaction of biofilms with the surrounding fluid. In this case, the coupled simulation high-lights the beneficial effect of the structure motion on food supply, which plays an important rolefor the formation and growth of biofilms. Among the several structures a biofilm can form, a fila-mentous shape in flow direction, usually called streamer, is selected for this example. This type ofbiofilm architecture was experimentally studied by Stoodley et al. [41], who grew streamers in aflow cell and observed their lateral oscillation, highlighting that their formation is due both to highshear rates and to a limited nutrient availability [42]. Hence, to reproduce the macro-scale dynam-ics of this type of biofilm structures, it is essential to consider fluid flow, structure deformation, andsubstrate transport.



In the present example, a streamer located in a two-dimensional rectangular flow channel is con-sidered as sketched in Figure 16. Streamer dimensions, flow conditions, and material properties areassumed similar to those found experimentally in [41] and utilized in [43, 44] for biofilm simulations(cf. Tables I and II, respectively).

The streamer consists of a head attached to the substratum and a tail free to move in the flow.Hence, only the interface between the streamer tail and the fluid, � , is considered as the FSI surface,while the streamer head boundary, �head , is fixed. At the inlet of the channel, a uniform horizontalvelocity profile and a constant substrate concentration are prescribed. At the upper and lower domainboundaries, a slip condition is imposed on the fluid to avoid the development of any boundary layers.

For the scalar transport problem, an infinite permeability is assumed along the entire interfacebetween the fluid and the structure domain, that is, in both the tail and the head region. In the fluiddomain, substrate transport is modeled using a convection–diffusion equation, whereas a diffusion–reaction equation is utilized to consider substrate transport and consumption within the streamer.The corresponding reaction term is based on the non-linear Monod kinetic (cf. [44] for a comparisonof different reaction models for biofilm applications). In this case, the reaction term R is given by

R D kˆS

K CˆS(54)

where k denotes the reaction rate constant and K represents the half saturation constant.Computations are run on a 2D mesh of 165.000 quad cells refined in the interface region. A

fixed time step of 0.01 ms is utilized. At first, a simulation with a completely fixed streamer, thatis, neglecting FSI effects, is run. Corresponding results are then used as initial condition for thecoupled simulation considering the combined FSI and multi-field mass transport. Simulation resultsat different time steps for both the fixed and moving tail configuration are given in Figure 17.

By considering the coupling between FSI and mass transport, the effect of both the local biofilmdeformation and the fluid flow on the transport and reaction of nutrients can be analyzed. The scalar

Table I. Geometrical details.

Parameter Symbol Value (mm)

Tail length Ltail 1.499Head diameter Dhead 3:33 � 10�1

Tip diameter Dtip 6:7 � 10�2

Distance from inlet Linlet 1.5Domain length L 12Domain height H 3

Table II. Material parameters and operating conditions.

Parameter Symbol Value Unit

LiquidDynamic viscosity �F 10�3 kg m�1 s�1

Density �F 103 kg m�3

Inlet velocity uFIN 3 � 10�1 m s�1

BiofilmDensity �S 103 kg m�3

Young’s modulus ES 4 � 103 kg m�1 s�2

Poisson ratio �S 4 � 10�1

SubstrateDiffusion coefficient D 2:5 � 10�9 m2 s�1

Uptake rate coefficient k 3 � 10�2 mol m�3 s�1

Saturation coefficient K 3 � 10�3 mol m�3

Inlet concentration ˆFIN 2:5 � 10�2 mol m�3



Figure 17. Velocity (left) and concentration (right) maps for a fixed tail (top) and for a moving tail at threedifferent time steps (u in m/s; ˆ in mol/m3).

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1

UncoupledCoupled

Figure 18. Molar flux along the tail boundary for both the fixed and moving tail configuration.

field maps highlight an important concentration boundary layer on the tail surface. Hence, the sub-strate concentration at the interface is significantly lower than in the bulk of the fluid. The streamermovement permits the tip to reach zones with higher substrate concentration, thereby causing anincrease of the mass transfer compared to the fixed tail configuration. In Figure 18, the molar flux



along the interface between fluid and tail, � , is reported for both the fixed and moving tail con-figuration. In the latter case, results are averaged in time because the flapping tail produces a timedependent flux. In particular, reported results refer to the system condition 150 ms after the tail flap-ping starts. Without considering the tail movement, the positions of local maxima and minima ofmass flux cannot be predicted, which is a very important information when the local growth of thebiofilm structure is of interest. Besides, an overall higher mass flux can be observed for the movingtail configuration, reaching a percentage increase of 39% at the tail tip. Hence, to enable a realis-tic determination of substrate uptake and consumption within the biofilm streamer, the presentedapproach considering both FSI and multi-field mass transport effects is indispensable.

6. CONCLUSION AND OUTLOOK

To enable the investigation of transport processes in biomechanical applications, we have developedan advanced computational model considering the effect of local structure deformation and fluidflow on passive mass transport. For this purpose, a sequential one-way coupling of an FSI and amulti-field scalar transport model has been realized. In each time step, the non-linear monolithic FSIproblem is solved first to determine local deformations and velocities. Using this information, themonolithic multi-field transport problem can then be solved on the deforming domains. Dependingon the surface permeability P , concentrations at the interface between fluid and solid fields arecompletely decoupled .P D 0:0/, coupled by means of a flux condition .P > 0:0/ or identical.P !1/ in both domains.

Although the effect of structure deformation on mass transport has already been consideredbefore, previous approaches suffered from a number of limitations. For instance, in [14], the trans-port of macromolecules was considered separately on both domains. As a consequence, masstransfer was only possible in one direction, that is, from fluid to tissue. In [17], the structural sim-ulation was essentially decoupled. Hence, instead of solving the solid field equations in each timestep, the mass transfer problem was formulated on the final deformed structure. Although this sim-plification seems to be suitable in the realm of drug eluting stent modeling, it may be too restrictivefor other applications. All mentioned limitations are overcome by the new methodology presentedin this paper. The proposed combination of FSI and multi-field scalar transport models enables themutual coupling of (i) flow and solid deformation and (ii) transport processes on deforming fluidand solid domains. Hence, our approach can provide insights into how local deformations influencetransport processes in both fluid and solid fields, which is of utmost importance in highly dynamicstructures such as alveoli during ventilation and biofilm structures under specific flow conditions.Another major advantage of the proposed approach is the generality of its implementation. Asalready mentioned before, the algorithms discussed in Sections 2 and 3 are by no means restrictedto any specific discretization scheme or solution technique. Furthermore, arbitrary constitutive lawsincluding poroelastic models (cf., for instance, [45]) and multi-scale approaches (see, e.g., [46])may be utilized to describe the behavior of the tissue under consideration as accurately as possible.

The general validity and versatility of the proposed methodology was illustrated by selectednumerical examples. Conservation of mass was proven for both structural and FSI problems.Besides, it was shown that a reasonable coupling of concentrations at the field interface has beenachieved. Finally, the application of the novel approach to oxygen transport in an idealized alveolusand in a real biofilm application was discussed. It was successfully demonstrated that our approachis in general suitable for modeling convective and diffusive scalar transport as well as reactionprocesses on deformable, coupled fluid and solid domains for the first time.

Future work will be concerned with employing the proposed model to new realistic scenariosfrom different fields of biomechanics. Planned applications include the optimization of targeteddrug delivery, the investigation of atherosclerosis, and the growth of biofilms. In the latter case,flow and deformation states are affected by transport-mediated growth processes. Hence, ongoingwork focuses on extending the presented approach towards a mutual coupling of FSI, transport, andbiochemical models.



REFERENCES

1. Weinberg PD. Rate-limiting steps in the development of atherosclerosis: the response-to-influx theory. Journal ofVascular Research 2004; 41:1–17.

2. Rappitsch G, Perktold K. Pulsatile albumin transport in large arteries: a numerical simulation study. Journal ofBiomechanical Engineering 1996; 118:511–519.

3. Wada S, Koujiya M, Karino T. Theoretical study of the effect of local flow disturbances on the concentration of low-density lipoproteins at the luminal surface of end-to-end anastomosed vessels. Medical and Biological Engineeringand Computing 2002; 40:576–587.

4. Kaazempur-Mofrad MR, Ethier CR. Mass transport in an anatomically realistic human right coronary artery. Annalsof Biomedical Engineering 2001; 29:121–127.

5. Comerford A, David T. Computer model of nucleotide transport in a realistic porcine aortic trifurcation. Annals ofBiomedical Engineering 2008; 36:1175–1187.

6. Zhang Z, Kleinstreuer C, Kim CS. Airflow and nanoparticle deposition in a 16-generation tracheobronchial airwaymodel. Annals of Biomedical Engineering 2008; 36:2095–2110.

7. Comerford A, Bauer G, Wall WA. Nanoparticle transport in a realistic model of the tracheobronchial region.International Journal for Numerical Methods in Biomedical Engineering 2010; 26:904–914.

8. Quarteroni A, Veneziani A, Zunino P. Mathematical and numerical modeling of solute dynamics in blood flow andarterial walls. SIAM Journal on Numerical Analysis 2002; 39:1488–1511.

9. Stangeby DK, Ethier CR. Computational analysis of coupled blood-wall arterial LDL transport. Journal ofBiomechanical Engineering 2002; 124:1–8.

10. Prosi M, Zunino P, Perktold K, Quarteroni A. Mathematical and numerical models for transfer of low-density lipopro-teins through the arterial walls: a new methodology for the model set up with applications to the study of disturbedlumenal flow. Journal of Biomechanics 2005; 38:903–917.

11. Sun N, Torii R, Wood NB, Hughes AD, Thom SAM, Xu XY. Computational modeling of LDL and albumin transportin an in vivo CT image-based human right coronary artery. Journal of Biomechanical Engineering 2009; 131:021003.

12. Badia S, Quaini A, Quarteroni A. Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic mediainteraction. Journal of Computional Physics 2009; 228:7986–8014.

13. Tada S, Tarbell JM. Oxygen mass transport in a compliant carotid bifurcation model. Annals of BiomedicalEngineering 2006; 34:1389–1399.

14. Koshiba N, Ando J, Chen X, Hisada T. Multiphysics simulation of blood flow and LDL transport in a porohyperelasticarterial wall model. Journal of Biomechanical Engineering 2007; 129:374–385.

15. Zunino P. Multidimensional pharmacokinetic models applied to the design of drug-eluting stents. CardiovascularEngineering 2004; 4:181–191.

16. Migliavacca F, Gervaso F, Prosi M, Zunino P, Minisini S, Formaggia L, Dubini G. Expansion and drug elution modelof a coronary stent. Computer Methods in Biomechanics and Biomedical Engineering 2007; 10:63–73.

17. Feenstra PH, Taylor CA. Drug transport in artery walls: A sequential porohyperelastic-transport approach. ComputerMethods in Biomechanics and Biomedical Engineering 2009; 12:263–276.

18. Swan AJ, Tawhai MH. Evidence for minimal oxygen heterogeneity in the healthy human pulmonary acinus. Journalof Applied Physiology 2011; 110:528–537.

19. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley: Chichester,New York, Weinheim, 2005.

20. Bischoff M, Wall WA, Bletzinger KU, Ramm E. Encyclopedia of Computational Mechanics, Volume 2: Solidsand Structures. In Models and Finite Elements for Thin-Walled Structures, Stein E, de Borst R, Hughes TJR (eds).John Wiley & Sons, Ltd: Chichester, 2004; 59–137.

21. Hughes TJR. The Finite Element Method - Linear Static and Dynamic Finite Element Analysis. Dover Publications:Mineola, 1981.

22. Donea J, Huerta A. Finite Element Methods for Flow Problems. Wiley: New York, 2003.23. Förster C, Wall WA, Ramm E. Stabilized finite element formulation for incompressible flow on distorted meshes.

International Journal of Numerical Methods in Fluids 2009; 60:1103–1126.24. Hughes TJR, Scovazzi G, Franca LP. Encyclopedia of Computational Mechanics, Volume 3: Fluids. In Multiscale

and Stabilized Methods, Stein E, de Borst R, Hughes TJR (eds). John Wiley & Sons, Ltd: Chichester, 2004; 5–59.25. Braess H, Wriggers P. Arbitrary Lagrangian Eulerian finite element analysis of free surface flow. Computer Methods

in Applied Mechanics and Engineering 2000; 190:95–109.26. Fernandez MA, Moubachir M. A Newton method using exact Jacobians for solving fluid-structure coupling.

Computers & Structures 2005; 83:127–142.27. Wall WA, Ramm E. Fluid-structure interaction based upon a stabilized (ALE) finite element method. In Computa-

tional Mechanics – New Trends and Applications, Idelsohn SR, Onate E, Dvorkin EN (eds). CIMNE: Barcelona,1998; 1–20.

28. Küttler U, Gee M, Förster C, Comerford A, Wall WA. Coupling strategies for biomedical fluid-structure interactionproblems. International Journal for Numerical Methods in Biomedical Engineering 2010; 26:305–321.

29. Klöppel T, Wall WA. A novel two-layer, coupled finite element approach for the nonlinear elastic and viscoelasticbehavior of human erythrocytes. Biomechanics and Modeling in Mechanobiology 2010; 10:445–459.

30. Klöppel T, Popp A, Küttler U, Wall WA. Fluid-structure interaction for non-conforming interfaces based on a dualmortar formulation. Computer Methods in Applied Mechanics and Engineering 2011; 200:3111–3126.



31. Förster C, Wall WA, Ramm E. On the geometric conservation law in transient flow calculations on deformingdomains. International Journal for Numerical Methods in Fluids 2006; 50:1369–1379.

32. Brooks A, Hughes T. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with par-ticular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics andEngineering 1982; 32:199–259.

33. Gravemeier V, Wall W. Residual-based variational multiscale methods for laminar transitional and turbulent variable-density flow at low Mach number. International Journal for Numerical Methods in Fluids 2011; 65:1260–1278.

34. Bauer G, Gravemeier V, Wall WA. A 3D finite element approach for the coupled numerical simulation ofelectrochemical systems and fluid flow. International Journal for Numerical Methods in Engineering 2011; 86:1339–1359.

35. Ehrl A, Bauer G, Gravemeier V, Wall WA. Simulating natural convection in electrochemical cells by a stabilizedfinite element approach. Journal of Computational Physics 2013; 235:764–785.

36. Wall WA, Gee MW. BACI - a multiphysics simulation environment. Technical Report, Technische UniversitätMünchen: Munich, 2012.

37. Weibel ER. The Pathway for Oxygen. Harvard University Press: Cambridge, 1984.38. Rausch S, Haberthuer D, Stampanoni M, Schittny JC, Wall WA. Local strain distribution in real three-dimensional

alveolar geometries. Annals of Biomedical Engineering 2011; 39:2835–2843.39. Wall WA, Wiechert L, Comerford A, Rausch S. Towards a comprehensive computational model for the respiratory

system. International Journal for Numerical Methods in Biomedical Engineering 2010; 26:807–827.40. Spaeth EE, Friedlander SK. The diffusion of oxygen, carbon dioxide, and inert gas in flowing blood. Biophysical

Journal 1967; 7:827–851.41. Stoodley P, Lewandowski Z, Boyle J, Lappin-Scott HM. Oscillation characteristics of biofilm streamers in turbulent

flowing water as related to drag and pressure drop. Biotechnology and Bioengineering 1998; 57:536–544.42. Stoodley P, Dodds I, Boyle J, Lappin-Scott H. Influence of hydrodynamics and nutrients on biofilm structure. Journal

of Applied Microbiology 1998; 85:19S–28S.43. Taherzadeh D, Picioreanu C, Küttler U, Simone A, Wall WA, Horn H. Computational study of the drag and oscillatory

movement of biofilm streamers in fast flows. Biotechnology and Bioengineering 2010; 105:600–610.44. Coroneo M, Yoshihara L, Bauer G, Wall WA. A finite element approach for the coupled numerical simulation of fluid-

structure interaction and mass transfer of moving biofilm structures. In Proceedings of the 6th European Congress onComputational Methods in Applied Sciences and Engineering, Eberhardsteiner J, Böhm HJ, Rammerstorfer FG (eds).Vienna University of Technology: Austria, 2012; 1–14.

45. Chapelle D, Gerbeau JF, Sainte-Marie J, Vignon-Clementel IE. A poroelastic model valid in large strains withapplications to perfusion in cardiac modeling. Computational Mechanics 2010; 46:91–101.

46. Wiechert L, Wall WA. A nested dynamic multi-scale approach for 3D problems accounting for micro-scale multi-physics. Computer Methods in Applied Mechanics and Engineering 2010; 199:1342–1351.


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