A Multi-field Bidirectional Fluid Structure Interaction for Human Artery from Clinical Imaging

Rui Wang

Final Report

For the fulfillment of class ASEN5367 Instructor: Dr. Carlos Felippa

May 6, 2006

University of Colorado at Boulder Department of Mechanical Engineering

Abstract: The hemodynamic characteristics of blood flow in the human circulatory

system are significantly changed by the biomechanical properties of the distensible blood

vessel while the fluidic force introduced by the flow could also alter the nature of the

structure. A bio-directional fluid structure interaction model was developed for the

surgical patch after Fontan palliative procedure to accurately simulate the venous return

through vena cava into pulmonary artery under compliant wall boundary based on patient

specific imaging data. The effect of the compliance of the wall will be quantified with

respect to the energy efficiency and hemodynamics of the flow through such connection.

On the other hand, the pressure force and wave form incurred by the flow on the vessel

wall will be investigated, followed by the study how they will change its structure and

induce certain disease such as arterial/venous hypertension.

Introduction: For patients with single ventricle pathology, the Fontan operation by which the

two vena cava are connected directly to the pulmonary arteries excluding the diseased

right ventricle have provided the required palliation to maintain necessary

cardiopulmonary perfusion. Much research has been performed on examining alternative

connection geometries to improve hydrodynamic efficiency through the cavopulmonary

connection, but with a questionable rigid wall assumption. In addition, hypertension

commonly happens in the late stage of those after-Fontan patients which greatly

increased the mortality and morbidity rates and also trigger the development of

atherosclerosis. The difference between the deformation profiles of healthy and stiffer

arteries demonstrates significant and measurable differences particularly in the radial

direction. Furthermore, it has been well documented that there is a relationship between

the genesis and progression of this disease under abnormal flow situations. Fluid

dynamics largely influence the endothelial function of the vessel wall. 1 The most

important effect is the locally distributed mass transfer determined by the oscillating

shear stress, elongated particle resident time and flow recirculation zone, etc. 2 Finally the

pressure wave propagation within the blood vessel can be used as a clinical tool to

diagnose the stiffness of the wall, i.e. atherosclerosis.

Computational method is becoming more popular and powerful in understanding

complicated system involving multifield interaction such as Fluid-Structure, Structure-

Structure, Thermal-Structure, Electromagnetic-Structure etc, and has been used as a

useful tool in supporting the experimental studies. It often enables possible the

acquisition of variable values such as the wall shear loading, pressure wave propagation

etc which are unavailable otherwise. Blood flow through arteries represents a very

complex two way coupling system, the pressure exerted by the blood flow on the vessel

wall can result in large structural deformation which can in turn affect the flow. Fluid

structure interaction stands for a coupling between the 3D Navier-Stokes equations and a

structural field (interior or exterior) which is subjected to linear/large deformation.

Strong coupling between the blood and artery is due to the relatively low stiffness of the

artery compared to that of fluid. Such a highly involved system can raise some numerical

difficulties: (1) because of the nature of the problem – distensible wall and

incompressible flow, the densities of the artery walls and the blood have to be carefully

managed to avoid numerical instabilities; (2) nonlinear displacements are present in the

structure and the fluid problem have to be solved on a moving domain; (3) fake reflection

phenomena from the structural boundary should be taken good care of.

There is much research on how optimize the connection geometry can minimize

the energy loss, but only a few considered the effect of the wall compliance. The present

study will more accurately mimic the biological system from a physiological point of

view. A general idea on how to reconstruct the patient specific geometry, generate the

structural and fluidic mesh, formulate the FSI solution and special concerns will be given

to facilitate such type of study.

Formulation and Implementation: 1) Solid property and modeling:

All arteries and veins consist of three distinct tissue layers, though the latter is

thinner and softer. The innermost layer, the tunica intima, also called tunica interna, is

simple squamous epithelium surrounded by a connective tissue basement membrane with

elastic fibers. It is a single layer of endothelial cells. This part contributes little to the

mechanical properties of the blood vessel. However with age and disease, this is because

the layer the intima becomes thicker and stiffer and provides important mechanical

support. This is known as atherosclerosis. Basically atherosclerosis is the build-up along

the vessel wall of fatty substances, calcium, collagen fibers, cellular waste products and

fibrin, which is referred to as atherosclerotic plaque and make blood vessel have

significantly different mechanical behavior. The middle layer, the tunica media, is

primarily smooth muscle and is usually the thickest layer. It is made up of elastin, smooth

muscle cells, protoglycans, and collagen. Together these form a fibrous helix which is

oriented circumferentially. Such arrangement provides the blood vessel with high

strength, resilience and the ability to resist loads in both the longitudinal and

circumferential directions. To help keep blood flowing, the artery expands with pressure

by activating the smooth muscles in this layer. The outermost layer, which attaches the

vessel to the surrounding tissue, is the tunica externa or tunica adventitia. This part is

mainly consists of connective tissue with some elastic and collagenous fibers.

Mechanically, the adventitia gives structural reinforcement to the vessel and prevent

overstretch and rupture at high levels of mechanical force.

For major blood vessel, the thickness is typically considered to be 5 ~ 10% of its

radius. Therefore, it is appropriate to use thin shell element, defined by two curved

surface coordinates, to model such structure. Since the maximum structural deformation

can approach 15%, a nonlinear displacement needs to be handled in present study. In

shell theories, the bending and membrane strain are merged in the energy expression, and

the coupled deformations including the stretching and change of the middle surface

curvature are required to predict the stress/strain of the shell element.

The governing equation for the structure model is:

ug SBijij &&ρτ =−,

Where τ is the stress, g is the body force, Sρ is the material density, and u&& is the

acceleration.

The governing equation for non-linear dynamic system is:

∑

∑

∑

=

=

=

−∆+=+∆++∆+

iii

TiiV

iii

TiiiV

iii

TiiiV

dVCBBK

dVHHkC

dVHHMtFttRutKuttCuttM

)(

)(

)(

)()()()()(ρ

&

Where M is the mass matrix, C is the damping matrix, K is the master stiffness matrix,

R is the external load, F is the element stress, u is the displacement, iK is the element

damping factor, iH is the element displacement matrix, iC is the elasticity matrix, iB is

the strain displacement matrix and iV is the element volume.

In the present study, we chose low-order general 4-node QUAD shell element

instead of the TRI shell element due to the nature of the problem and some special

numerical concerns. For the geometrically exact TRI element, although it gives excellent

accuracy for some FSI problem and was tested for some structural problems, it is

inappropriate for FSI problem with blood flow because the bending and membrane

phenomena are both present. In contrast, for QUAD shell element, it gives excellent

results for both bending and membrane dominated problem like the coupled structure and

blood flow interaction problem. ANSYS was used in this study to solve the structure part

of the problem.

2) Fluid Property and modeling:

CFX, also from ANSYS Inc, was used to solve the fluid part of the problem.

With the assumption that the blood is homogenous, incompressible and the flow is

laminar, the governing equations are as follows:

)(21

*2

0

T

B

vve

epI

fvvtvv

∇+∇=

+−=

=∇−∇⋅+∂∂=⋅∇

µτ

τρρ

Where v is velocity, Bf is the volume body force, I is the identity matrix and µ is the

dynamic viscosity.

However there is some numerical difficulty regarding the FSI solution with

incompressible fluid. This is caused applying unphysical boundary conditions that lead

to severe convergence problems. Initially the fluid domain is unaware of the constraint

of the structural domain, and vice versa. If the iteration converges this discrepancy will

be settled, but sometimes the initial phase is so ill posed that convergence is practically

impossible to obtain. For example, considering an elastic tube with inlet only, in the start

of the iteration the wall is at rest, if the fluid is incompressible it is impossible to maintain

the continuity since there is no net flux out the domain. Therefore, we applied a small

compressibility to the fluid to deal with this numerical difficulty with the following

formula:

]/1[0 ff kp+= ρρ

Where 0ρ has the value of 1060 kg/m^3 and fk is the bulk modulus of blood.

This bulk modulus parameter is used in the transient algorithm for incompressible

flows, yet it accounts for mild compressibility effects, i.e., non-infinite speeds of sound.

The speed of sound, or the acoustic speed, is the speed at which a sound wave or small

pressure disturbance propagates in a fluid medium. The artificial compressibility has a

natural physical explanation in FSI simulation: the compressibility is defined so that it

makes the fluid follow the elastic response of the structure. Then we have the continuity

as follows:

0)( =⋅∇+∇⋅+∂∂ vpv

tp

k ff

ρρ

Geometry Acquisition:

Two orthonormal images are obtained from bi-plane X-ray angiograms performed

during routine catheterization. A series of points representative of the vessel centerline

for a single branch is identified. A spline is then fit to the initial points and a deformation

model is used to drive the spline to the actual vessel centerline (same procedure is

repeated for each branch). Then bifurcation points, directional vectors and cross

sectional diameters are calculated. Finally a 3D skeleton is created based on those points

and the corresponding vessel diameter. Based on this skeleton, a NURBS based patient

specific geometry is constructed by CAD package Solidworks.

Mesh Generation:

The mesh generator ANSYS ICEMCFD is used to generate both shell and

hexahedron for the structural and fluidic model respectively. Careful attention was given

to the parts of the model whenever there was some sharp geometric change and where

multiple surfaces met. Staggered (non-uniform) mesh algorithm was employed along the

wall boundary of the fluid model to more accurately catch the boundary layer effect in

such regions. In order to maintain a good mesh quality, targeted mesh vortices, edges

and faces were projected particular points, curves and surfaces of the model respectively.

At the end, the mesh quality was checked quantitatively by displaying the value of

determinant, angle and warping ratio etc for each element. In the present study, they had

the minimum values of 0.25, 25○ and 2.06 for the above parameters respectively, which

will pave the way for an accurate, smooth and fast converging FSI solution procedure.

FSI Formulation:

After the numerical models were created for both domains, a FSI simulation was

performed to solve such coupled systems by using staggering partition scheme by using a

tied ANSYS & CFX solver. The fluid solver CFX will start first because it is the driving

force of the whole problem and provides the structure ANSYS solver with pressure load,

then the solid solver starts and feeds the fluid solver with the displacement of the wall

boundary. Since there is a two way feedback between them, a bidirectional FSI should

be performed.

To obtain a solution on the FSI interface, the compatibility condition should be

satisfied all the time which has the following formula:

fS uu =

Form the no slip wall boundary condition, we have:

fS

Sf

nn

uv

ττ =

= &

Then the fluidic force acting on the solid elements on the interface is:

∫= dShF fdτ

where dh represents the virtual displacement of the solid element.

Then the governing equation for the FSI problem can be written as:

=

)](,[

)](,[][

ffSS

SSff

XXF

XdXFXF

τ

where fS XX & are the solution vectors.

A partitioned staggering scheme was used to solve the above problem serially

instead of simultaneously. Convergence was maintained for both fields and the interface

as well.

Boundary conditions and material properties:

The four outer surfaces of the structure were assumed to be stress fee and all

nodes on them were fully constrained, which was due to the reasons of computational

reasons and also is also physiologically reasonable. A single shell layer with a thickness

value of 1 mm was used globally. Elastic modulus of 1E08, Poisson ratio of 0.48 and

material density of 1200 kg/m^3 were assumed for the blood vessel.

The working fluid was assumed to be homogeneous and incompressible with a

dynamic viscosity of 0.003. A normal velocity of 0.25 m/sec was used at both inlet IVC

& SVC while a normal velocity of 0.3 m/sec was used at one outlet RPA and a fixed

pressure value of 0 Pa was used at the other. While these values were in the range of

physiological flow conditions, there were some simplifications used in this study. For

example, the pulsatility of the real blood flow was not considered here, an impedance

outflow boundary condition was more appropriate in a physiological point of view and

only linear wall deformation was used etc. All these will be investigated in the future

study.

Results and Conclusion: The largest displacement of the vessel wall was found to be 7.1E-07 m which was

corresponding to approximately 2% of maximum diameter of the blood vessel diameter.

The areas at the branch junctions have relatively large shear stress values which has a

peak value of 14111 Pa where the IVC, SVC and LPA meets. It was also observed that

the structure has a relatively small rotation in current study. On the other hand, it was

also found larger wall deformation was associated with larger flow rate. The big IVC

branch has a more profound displacement and stress values than the rest three relatively

small model branches. Moreover, the area with significant curvature change also

incurred large wall displacement and stress force like the elbow area of the SVC branch.

Form the results of fluid field solution, a strong wave propagation was seen which is

critical factor influencing the pattern of blood flow and the wall dynamics as well, and it

will be addressed in more details in the future research with more realistic boundary

condition. Strong vortical flow was found at the two outflow boundaries with a maximum

value of 7486 1/s at LPA, which also has a great potential to incur some property changes

on the wall structure such as its stiffness. The shear stress on the wall from the blow was

found to be less than 35 Pa globally with high values happening at the two outflow

boundaries and the regions with big sharp geometry changes, the corresponding shear

strain rate was found to have a peak value of 7126 1/s at the opening of LPA.

In conclusion, by using the advanced bidirectional FSI simulation tool, an

insightful view could be obtained regarding the involved responses of the exterior blood

vessel and the interior blood flow. The problematic regions of the structural wall with

large deformation and stress could be identified, followed by its potential property

changes most importantly the stiffness in present study. In the meanwhile, a detailed

flow pattern characteristics analysis under moving boundary could be performed and

derive the more accurate energy efficiency, the shear force distribution, pressure wave

propagation and finally, the prediction, diagnosis and understanding of the disease

associated with blood vessel.

Legend: Figure1: CAD Model from Solidworks indicating the four branches of the blood

vessel

Figure 2: Structural mesh (Shell) for the vessel wall

Figure 3: Close view of the Shell element at the junction section of the structure

Figure 4: Hexa mesh for the fluid domain (Brick element)

Figure 5: Cross-section view of the Hexa mesh

Figure 6: Deformation of the wall from FEA results

Figure 7: Von Mises stress results from FEA results

Figure 8: Vorticity value from fluid solution

Figure 9: Wall shear stress (WSS) contour from fluid solution

Figure1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure8

Figure 9

References

1. Perktold K, Rappitsch G: Computer-Simulation of Local Blood-Flow and Vessel Mechanics in a Compliant Carotid-Artery Bifurcation Model. Journal of Biomechanics 28:845-856, 1995 2. Caro CG, Fitzgera.Jm, Schroter RC: Proposal of a Shear Dependent Mass Transfer Mechanism for Atherogenesis. Clinical Science 40:P5-&, 1971