aneurisk project: laboratory of biological structure mechanicsale/socci-aneurisk3-labs.pdf · 1...

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www.labsmech.polimi.it

ANEURISK PROJECT: LABORATORY OF BIOLOGICAL STRUCTURE MECHANICS

Numerical models of cerebral aneurysm biomechanics

Laura Socci, Dario Gastaldi, Giancarlo Pennati, Pasquale Vena

Gabriele Dubini, Francesco Migliavacca

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Realistic geometries: ANEURISK database

Internal Carotid Artery

Anterior Communicating

Artery

Basilar ArteryMiddle Cerebral

Artery

ANEURISK project aim: to evaluate aneurysm rupture risk

Different locations and geometries

Aim of the study

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Aim of the study




Internal Carotid Artery: U bend

Rupture risk ? Formation site ?

Collection of longitudinal patient-based information

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Aim of the study




Rupture risk ? Formation site ?

Adopted approach: investigation on aneurysm initiation


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Aim of the study

The investigation of the phenomena linked to aneurysms development from a biomechanical point of view. To achieve this target, the research has to be developed in different topics:structural aspects, adaptive phenomena and fluid dynamics

The project aim

The specific goal Development of theoretical models

able to simulate the aneurysm initiation and early development: from healthy vessel to bulge formation

and implementation of numerical tools

The path

To implement a constitutive model for cerebral vascular wallTo develop an adaptive law to mimic the development of aneurysmTo couple structural features with fluid dynamics boundary conditions

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CFD studies

Kyriacou and Humphery (1996); Ryan and Humphrey (1999); Humphrey and Canham (2000): non linear incompressible isotropic strain energy function on a membrane (cerebral aneurysms)

Wuyts et al. (1995), Humphrey and Rajagopal (2002), Zulliger et al (2004), Holzapfel et al. (2005), Gasser and Holzapfel (2006): non linear incompressible anisotropic material with matrix and fibers (healthy vessels)

Kroon and Holzapfel (2007): stress-mediated matrix turnover on saccular aneurysms

Watton et al. (2004): strain-based micro structural recruitment model

Steinmann et al (2003), Hoi et al (2004), Cebral et al. (2005), Hassan et al (2006): hemodynamic analysis of cerebral aneurysms on a patient-specific basis

Baek et al. (2005; 2006): stress-mediated matrix turnover on fusiform and saccular aneurysms

already developed aneurysms

Very few studies on the aneurysm initiation

Structural studies: elastic behavior

Structural studies: adaptive behavior

Biomechanical models of cerebral aneurysms: constitutive, adaptive and fluid dynamic approaches

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Kyriacou and Humphery (1996); Ryan and Humphrey (1999); Humphrey and Canham (2000): non linear incompressible isotropic strain energy function on a membrane (cerebral aneurysms)

Wuyts et al. (1995), Humphrey and Rajagopal (2002), Zulliger et al (2004), Holzapfel et al. (2005), Gasser and Holzapfel (2006): non linear incompressible anisotropic material with matrix and fibers (healthy vessels)

Kroon and Holzapfel (2007): stress-mediated matrix turnover on saccular aneurysms

Watton et al. (2004): strain-based micro structural recruitment model

Steinmann et al (2003), Hoi et al (2004), Cebral et al. (2005), Hassan et al (2006): hemodynamic analysis of cerebral aneurysms on a patient-specific basis

Baek et al. (2005; 2006): stress-mediated matrix turnover on fusiform and saccular aneurysms

Aneurysms initiation studiesWulandana and Robertson (2005): inelastic behaviour (not fluid dynamics and adaptive behaviours)

Feng et al. (2006), Chatziprodromou et al. (2006): fluid-structure approach (simple elastic behaviour and not adaptive behaviour)

CFD studies

Structural studies: elastic behavior

Structural studies: adaptive behavior

Work Novelty: Numerical model that couple passive, adaptive and fluid dynamics features of cerebral vascular wall on aneurysms genesis

Biomechanical models of cerebral aneurysms: constitutive, adaptive and fluid dynamic approaches

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Cerebral vessel wall behaviour:Nonlinearity (Scott et al, 1972; Steiger et al, 1989; Monson et al, 2003, 2005, 2006)

Anisotropy (Finlay and Canham 1998;Monson et al, 2006)

Large Deformations (Fung, 1983; Holzapfel et al, 2005)

Incompressibility constraint (Holzapfel et al, 2003, 2005)

The anisotropic constituent-based computational model for cerebral vascular wall

Mechanical properties of cerebral vessels

Constitutive relationship ij

ij EWS∂∂

=

Incompressibility constraint ( ) ( )421isovol0 I,I,IWJWW +=

Anisotropy: matrix + families of fiber (α)

α

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Mechanical properties of cerebral vessels

Constitutive relationship

Incompressibility constraint ( ) ( )421isovol0 I,I,IWJWW +=

α

Anisotropy ( )( ) ( )( )( )( )1131exp)3( 222

2

1 −−+−−+−= 411 III ρρμ KfKKW n

ijij E

WS∂∂

=

Cerebral vessel wall behaviour:Nonlinearity (Scott et al, 1972; Steiger et al, 1989; Monson et al, 2003, 2005, 2006)

Anisotropy (Finlay and Canham 1998;Monson et al, 2006)

Large Deformations (Fung, 1983; Holzapfel et al, 2005)

Incompressibility constraint (Holzapfel et al, 2003, 2005)


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α

Constitutive parameters identification

Scarce and not complete literature on cerebral vessels (Scott et al, 1972; Steiger et al, 1989; Monson et al, 2003)

A recent single study on biaxial tensile tests (Monson et al, 2006)

( )( ) ( )( )( )( )1131exp)3( 222

2

1 −−+−−+−= 411 III ρρμ KfKKW n

Several histological studies (Finlay and Canham 1998, Canham et al, 2003)

α : circumferential and longitudinal directions

MPa045.01 =KLongitudinal fibers

75.23MPa11.0

2

1

==

KK

Circumferential fibers

MPa0756.0=μ

Matrix

~ K1

~ K2

5.472 =Kμ


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Important results A constituent-based computational model regarding non linearity, anisotropyand large deformations was implemented in a displacement-based commercial code (ABAQUS, 3DS, USA)

An identification of proper constitutive parameters allow us to reproduce the elastic behaviour of cerebral vessels

Future Developments

To consider a multi-layer structure to better reproduce vascular histology

Realistic constitutive modeling is a prerequisite to quantifying changes in the structure and function of tissue in response to altered mechanical stimulus

To identify constitutive parameters based on more accurate experimental tests

Critical Issues


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en

pn FFF =

inelastic elastic

Finite element approach for the tissue growth and remodeling in aneurysms development

The adaptive law implementation

Adaptive phenomena are implemented as inelastic mechanisms: multiplicative decomposition of deformation gradient tensor (Skalak, 1989; Rodriguez et al, 1994; Taber et al, 1998; Humphrey et al, 1999)

Undeformed reference configuration

Grown configuration

Grown deformed configuration

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Perturbation Stimulus adaptiveeffect

Definition of the suitable stimulus: strain energy density of fibers (Watton et al, 2004; Baek et al, 2006)

Definition of the adaptive effects: fiber lengthening, change of fiber density (Watton et al, 2004, Baek et al, 2006)

growth remodelling

Adaptive law

Definition of the adaptive law: linear relationship (Rodriguez et al, 1994; Watton et al, 2004; Baek et al, 2006; Kroon and Holzapfel, 2007)

Perturbation: increase of pressure (Kroon and Holzapfel, 2007)

The adaptive law implementation


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Saccular simplifiedmodel

Axial displacement

Stimulus

Fiber density

Fiber lengthening

Stableresponse for

all kf /kp

Time


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Saccular simplifiedmodel

Stimulus

Stableresponse only

for higherkf /kp values

Time

Saccular model

Stimulus


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Fusiformmodel

Radial displacement

Stimulus

Fiber density

Fiber lengthening

Stableresponse forhigher kf /kp

Time


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Radial displacement

Different adaptive behaviour according to different constitutive models

Fusiformmodel

Time


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The mechanisms of growth (change in reference configuration) and remodeling (change of volumetric fiber density) are competing mechanisms

Important results

Future DevelopmentsTo investigate patient-based geometriesTo consider more realistic perturbationTo consider more realistic boundary conditionsTo implement more sophisticated adaptive mechanisms

The proposed computational model for growth/remodeling is capable of simulating the time evolution of stable and unstable aneurysms

The boundary conditions imposed could influence the adaptive behaviour

To identify proper adaptive constants based on clinical observations: it is difficult to collect longitudinal patient-based information

Critical Issues


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Biomechanical analysis of cerebral aneurysms formation in realistic geometries

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Choice of realisticidealized U-bend in

terms of diameters and curvature

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Realistic idealized geometries and realistic perturbations

CFD model (Fluent, Ansys)

+

Structural and adaptive model (ABAQUS, 3DS)

Interaction with CFD simulations


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Q inlet

P outlet

Perturbation: increased flow rate (Feng et al, 2006)

Output: pressure and wall shear stresses

P WSS


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Q inlet

P outlet

Perturbation: increased flow rate (Feng et al, 2006)

Output: pressure and wall shear stresses

Degradation function according to highest wall shear stresses (Wulandanaand Robertson 2005, Feng et al 2006)

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

h

hn Kf

τττ

degWSS


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Q inlet

P outlet

+


Result of simulations Stimulus

TimeQ inlet

P outlet


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Q inlet

P outlet




Q inlet

P outlet

+



Time


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Q inlet

P outlet




Q inlet

P outlet

+



Time


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Realistic idealized geometries and realistic perturbationsWorks in progress


2Qph62R6D4

6

6

6

Rcurv[mm]

1.5Qph1.5R6D3Q

2Qph1R6D2

2Qph1.5R6D3

QpertR

[mm]

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Realistic idealized geometries and realistic perturbationsWorks in progress


55.762R6D4

6

6

6

Rcurv[mm]

64.31.5R6D3Q

78.71R6D2

64.31.5R6D3

Dean/Q [s/ml]

R [mm]

0

0.001

0.002

0.003

0.004

3 8 13 18 23

R6D3

R6D2

Stimulus

Time

Stimulus

Time

R6D3Q0

0.001

0.002

0.003

3 8 13 18 23

Stimulus

Time

0

0.0005

0.001

0.0015

0.002

0.0025

3 8 13 18 23

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12⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅=curvRR

QDeanπμρ

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Conclusions and Future Developments

The developed “fluid-structure” interaction allow us to couple realistic fluid dynamic boundary conditions to a proper structural and adaptive model for cerebral vascular wallThe implemented model represents a novel approach to study the aneurysms initiationThe application to realistic geometry is a potentially useful tool to predict different aneurysms formation site according to geometrical and fluid dynamics features

Conclusions

Future developments

Adaptive perturbation (Pressure and WSS) changing during aneurysm formation and haemodynamic evolution;To apply the model to more realistic anatomy from clinical data;

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Publications

1. P. Vena, D.Gastaldi, L.Socci, G. Pennati, Contro R. A constituent-based computational model for vascular remodelling of a growing cerebral aneurysm. Computational Plasticity 2007 . (vol. 1, pp. 220-223). Proceeding of IX Computational Plasticity, Barcelona, September 5th – 7th.

2. Vena P., Gastaldi G., Socci L., Pennati G. An anisotropic model for tissue growthand remodeling during early development of cerebral aneurysms. ComputationalMaterials Science (2008). doi:10.1016/j.commatsci.2007.12.023

3. L. Socci Numerical models of cerebral aneurysm biomechanics PhD Thesis on Bioengineering

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International congresses

2. Computational Plasticity 2007 . IX Computational Plasticity, Barcelona, September 5 – 7, 2007. (Invited)P. Vena, D.Gastaldi, L.Socci, G. Pennati, Contro R. A constituent-based computational model for

vascular remodelling of a growing cerebral aneurysm. Computational Plasticity 2007 . (vol. 1, pp. 220-223).

1. 5th World Conference of Biomechanics, Munich, Germany, July 29- August 4, 2006. Socci L., Pennati G., Migliavacca F., Dubini G. Computational haemodynamics in cerebral aneurysm custom models based on different reconstructive methodologies J. Biomechanics 2006;39(Supp.1):S603(4859).

3. 8th International Symposyum on Computer Methods in Biomechanics and Biomedical Engineering. February 27 – March 1, 2008. L. Socci, D. Gastaldi,P. Vena, G.Pennati. Finite Element implementation of an anisotropic remodeling law for cerebral aneurysm development.

4. 16th Congress of European Society of Biomechanics. July 6 – 9, 2008. (Accepted)L. Socci, D. Gastaldi,G. Pennati, P. Vena. Biomechanical analyses of cerebral aneurysms development: a

numerical model

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Acknowledgments

LaBS Staff involved:Dario GastaldiGiancarlo PennatiPasquale VenaGabriele DubiniFrancesco Migliavacca

financial support of Fondazione Politecnico within the researchproject ANEURYSK is gratefully acknowledged.

MOX Staff involved:Tiziano PasseriniSimone VantiniPiercesare SecchiAlessandro Veneziani

Mario Negri Institute Staff involved:Marina PiccinelliLuca Antiga

Clinical Staff involved:Edoardo BoccardiSusanna Bacigaluppi

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