Introduction Finite Element Model Updating Approach ...

Identification of the Anisotropic Hyperelastic Properties of Normal and Diseased Aortic Walls from 4D Ultrasound Strain Imaging Andreas Wittek 1,2, Wojciech Derwich 3, Thomas Schmitz-Rixen 3 , Claus-Peter Fritzen2, Christopher Blase 1 1 Johann

Wolfgang Goethe University, Institute for Cell Biology and Neuroscience, Frankfurt am Main, Germany 2 University Siegen, Institute of Mechanics and Control Engineering-Mechatronics, Siegen, Germany 3Johann Wolfgang Goethe University, Department of Vascular and Endovascular Surgery, Frankfurt am Main, Germany

Introduction  

 Finite Element Analysis of AAA wall stresses as rupture

risk predictor. Reliable computational models have to be patient-individual with regard to: geometry, loads, and material properties. Nonlinear, orthotropic material of the individual AAA wall is unknown in vivo.

Verification Numerically generated input data for the FEMU Systolic deformation field

Diastolic geometry

Finite Element Model Updating Approach Blood pressure

4D ultrasound Artida (Toshiba) with speckle traking

Identified constitutive parameter set Master Best Fit Deviation [%] Strain computation

µ [MPa] 0.071 0.072 1.4

k1[MPa] 0.567 0.593 4.6

k2 353.64 331.92 -6.1

κ 0.26 0.26 0.0

φ [°] 68.8 68.5 -0.4

Computational model

3D geometry

3D geometry

Individual elastic properties in vivo Patients 3D displacements / mm

HV (volunteer, M4) 22y / m / 22.4

Constitutive equation  C, a a  1, 2  k1  C1  3  2k 2

PAOD (aged, KGU012) 80y / m / 28.4 Peripheral arterial occlusive disease

--

AAA (KGU079) 58y / m / 26.3 AAA (dmax = 32 mm)

 e k 2 [ C1 3 13 C4   1]2  1      1 2



Guess of constitutive parameters μ, k1 , k2 , φ, κ

Iterative optimal design algorithm Heterogeneous strain fields

Estimated load free geometry

inverse FEA Identified nonlinear orthotropic properties axial CAUCHY stress

circumferential CAUCHY stress HV

AAA

µ [MPa] 0.078 0.155 0.228 k1[MPa] 0.001 0.609 4.736 k2 169.0 874.9 1960.7 κ 0.28 0.00 0.21 φ [°] 19.5 85.6 4.6

direct FEA Circumferential strain

PAOD

Computationally estimated strain fields

Mean deviation of strain per element 4

strain [%]

3 2 1 0

Error function minimum?

no

HV

PAOD NE11

NE22

AAA NE12

measurement

model

yes

Constitutive parameters

μ, k1, k2, κ, φ References Derwich W, Wittek A, Pfister K, Nelson K, Bereiter-Hahn J, Fritzen C-P, Blase C, Schmitz-Rixen T (2015): High Resolution Strain Analysis Comparing Aorta and Abdominal Aortic Aneurysm with Real Time Three Dimensional Speckle Tracking Ultrasound. Eur. J. Vasc. Endovasc. Surg. doi:10.1016/j.ejvs.2015.07.042. Wittek A, Derwich W, Karatolios K, Vogt S, Schmitz-Rixen T, Blase C (2015): A finite element updating approach for identification of the anisotropic hyperelastic properties of normal and diseased aortic walls from 4D ultrasound strain imaging. JMBBM, doi:10.1016/j.jmbbm.2015.09.022

Conclusions Using heterogeneous strain fields, it is feasible to identify the parameters of an anisotropic, nonlinear elastic constitutive equation based on the observation of just two load cases, even though the load free geometry is unknown. These data are available from in vivo observation of deformed configurations of the aorta by the use of non-invasive 4D-US imaging and measurement of diastolic and systolic blood pressure.