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Simultaneous Estimation of Material Parametersand Neumann Boundary Conditions in a LinearElastic Model by PDE-Constrained Optimization
Tom Seidl, Bart van Bloemen Waanders, Tim Wildey
Optimization and Uncertainty Quantification Department, Sandia National Laboratories
SIAM Computational Science and Engineering
March 1 2017Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly ownedsubsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 1 / 13
SAND2017-1689C
Motivation and Formulation
Biomechanical Imaging 1
Biomechanical imaging (BMI) is a technique used tonon-invasively and quantitatively infer tissue mechanicalproperties [Barbone and Oberai, 2010] [Doyley, 2012]An ultrasound scanner can apply and measure a quasi-staticdisplacement um [Jiang and Hall, 2011]
1Data courtesy of T.J. Hall (U. Wisconsin)Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 2 / 13
Motivation and Formulation
Biomechanical Imaging 1
An estimate of µ is obtained by solving an inverse problem
1Data courtesy of T.J. Hall (U. Wisconsin)Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 2 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
Inverse Problem Formulation
PDE-constrained optimization:
minµ
12kW (u � um)k2
| {z }weighted data match
+ ↵R(µ)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0
| {z }equilibrium PDE + BCs
L µ U| {z }bound constraints
Algorithmic details:Finite element discretizationTotal variation regularizationL-BFGS approximation of HessianAdjoint-based gradient computation
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 3 / 13
Motivation and Formulation
The BCs Problem
The forward problem requires sufficient traction and/ordisplacement BCs to be well-posedPoorly characterized BCs must be assumed, which consequentlybiases the inverse problem solution [Richards et al., 2009]
measured displacement
measured displacement
measured displacement
measured displacement
dis
pla
cem
ent =
?
measured displacement
measured displacement
stre
ss =
?
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 4 / 13
Motivation and Formulation
Simultaneous Inversion
Treat tx as an optimization variable and use H1 regularization:
minµ,tx
12kW (u � um)k2
| {z }weighted data match
+↵R(µ) + �R(tx)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = tx| {z }
equilibrium PDE + BCs
L µ U| {z }bound constraints
R(tx) =12
Z
�rStx ·rStx d�
rS = (I � n ⌦ n)r
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 5 / 13
Motivation and Formulation
Simultaneous Inversion
Treat tx as an optimization variable and use H1 regularization:
minµ,tx
12kW (u � um)k2
| {z }weighted data match
+↵R(µ) + �R(tx)| {z }regularization
s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = tx| {z }
equilibrium PDE + BCs
L µ U| {z }bound constraints
R(tx) =12
Z
�rStx ·rStx d�
rS = (I � n ⌦ n)r
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 5 / 13
Phantom Experiment and Results
Phantom Experiment 2
Ultrasound phantoms are used to validate BMI techniques
2Data courtesy of T.J. Hall (U. Wisconsin)Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 6 / 13
Phantom Experiment and Results
Phantom Experiment 2
The phantom contains a stiff spherical inclusion embedded withina softer homogeneous background [Pavan et al., 2012]Size of the computational domain is approximately 40 mm (y) by25 mm (x)
2Data courtesy of T.J. Hall (U. Wisconsin)Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 6 / 13
Phantom Experiment and Results
Traction Reconstructions
For a fixed ↵, the magnitude of � has a major effect on the tractionreconstructions
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 7 / 13
Phantom Experiment and Results
Traction Reconstructions
For a fixed ↵, the magnitude of � has a major effect on the tractionreconstructions
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 7 / 13
Phantom Experiment and Results
Modulus Reconstructions
For a fixed ↵, the magnitude of � has a minor effect on themodulus reconstructionsIndependent mechanical testing of phantom materials measuredan inclusion to background µ contrast of 3.54
(a) � = 1e-0 (b) � = 1e-1 (c) � = 1e-2
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 8 / 13
Phantom Experiment and Results
Comparison to Traction-Free Reconstruction
The greatest differences occur around the edge of the inclusionand at the top of the domain
(d) Simultaneous (e) Traction-free (f) Difference
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 9 / 13
Phantom Experiment and Results
Comparison to Traction-Free Reconstruction
The greatest differences occur around the edge of the inclusionand at the top of the domain
(g) Simultaneous (h) Traction-free (i) Line plot
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 9 / 13
Conclusions
Summary
BMI is a PDE-constrained optimization problem with incompleteinterior dataUnknown traction boundary conditions can be estimatedconcurrently with mechanical propertiesAn additional regularization constant � must be determined,although it does not appear to have a significant influence onmodulus reconstructions
Thank you!
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 10 / 13
Conclusions
Summary
BMI is a PDE-constrained optimization problem with incompleteinterior dataUnknown traction boundary conditions can be estimatedconcurrently with mechanical propertiesAn additional regularization constant � must be determined,although it does not appear to have a significant influence onmodulus reconstructions
Thank you!
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 10 / 13
Conclusions
References I
Barbone, P. E. and Oberai, A. A. (2010).A review of the mathematical and computational foundations ofbiomechanical imaging.In Computational Modeling in Biomechanics, pages 375–408.Springer.
Doyley, M. (2012).Model-based elastography: a survey of approaches to the inverseelasticity problem.Physics in medicine and biology, 57(3):R35.
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 11 / 13
Conclusions
References II
Jiang, J. and Hall, T. J. (2011).A fast hybrid algorithm combining regularized motion tracking andpredictive search for reducing the occurrence of largedisplacement errors.IEEE transactions on ultrasonics, ferroelectrics, and frequencycontrol, 58(4):730–736.
Pavan, T. Z., Madsen, E. L., Frank, G. R., Jiang, J., Carneiro, A. A.,and Hall, T. J. (2012).A nonlinear elasticity phantom containing spherical inclusions.Physics in medicine and biology, 57(15):4787.
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 12 / 13
Conclusions
References III
Richards, M. S., Barbone, P. E., and Oberai, A. A. (2009).Quantitative three-dimensional elasticity imaging from quasi-staticdeformation: a phantom study.Physics in medicine and biology, 54(3):757.
Tom Seidl (SNL) Simultaneous Elastic Inversion SIAM CSE 2017 13 / 13
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