Simultaneous Estimation of Material Parameters …Simultaneous Estimation of Material Parameters and Neumann Boundary Conditions in a Linear Elastic Model by PDE-Constrained Optimization

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


Biomechanical Imaging 1

An estimate of µ is obtained by solving an inverse problem



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





minµ

12kW (u � um)k2



s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0








minµ

12kW (u � um)k2



s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0








minµ

12kW (u � um)k2



s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0








minµ

12kW (u � um)k2



s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0








minµ

12kW (u � um)k2



s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = 0






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




dis

pla

cem

ent =

?



stre

ss =

?



Simultaneous Inversion

Treat tx as an optimization variable and use H1 regularization:

minµ,tx

12kW (u � um)k2


+↵R(µ) + �R(tx)| {z }regularization

s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = tx| {z }

equilibrium PDE + BCs


R(tx) =12

Z

�rStx ·rStx d�

rS = (I � n ⌦ n)r



Simultaneous Inversion

Treat tx as an optimization variable and use H1 regularization:

minµ,tx

12kW (u � um)k2


+↵R(µ) + �R(tx)| {z }regularization

s.t. r · �(µ,u) = 0, uy |� = umy , (� · n)x |� = tx| {z }

equilibrium PDE + BCs


R(tx) =12

Z

�rStx ·rStx d�

rS = (I � n ⌦ n)r


Phantom Experiment and Results

Phantom Experiment 2

Ultrasound phantoms are used to validate BMI techniques



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)



Traction Reconstructions

For a fixed ↵, the magnitude of � has a major effect on the tractionreconstructions



Traction Reconstructions

For a fixed ↵, the magnitude of � has a major effect on the tractionreconstructions



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



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



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


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!


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!


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.


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.


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.


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Simultaneous Estimation of Material Parameters …Simultaneous Estimation of Material Parameters and Neumann Boundary Conditions in a Linear Elastic Model by PDE-Constrained Optimization