PAMM · Proc. Appl. Math. Mech. 8, 10167 – 10168 (2008) / DOI 10.1002/pamm.200810167

An efficient quantitative ultrasonic elastography based on variationalh-refinement

Alexander Arnold∗1, Jorn Mosler1, Stefan Reichling1, Walaa Khaled2, and Otto Timme Bruhns1

1 Institute of Mechanics, Ruhr-University Bochum, Universitatsstr. 150, D-44780 Bochum, Germany2 Institute of High Frequency Engineering, Ruhr-University Bochum, Universitatsstr. 150, D-44780 Bochum, Germany

Quantitative elastography is a method to visualise a stiffness distribution. It is motivated by the observation that changesin mechanical properties of soft tissue mostly include important diagnostic information. With an ultrasound-elastographysystem, the displacement field can be calculated from the pre- and post-deformation image. Using the assumption that thematerial is elastic, isotropic and nearly incompressible, the distribution of shear modulus is determined by solving an inverseproblem. While common approaches use a constant mesh, a variational h-refinement is in the focus of this work. In doingso, the efficiency and accuracy of the determination can be increased. The presented results are generated by using numericalsimulations.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

A method known as palpation is usually utilised in medicine to detect pathological changes in soft tissue such as breastcancers or prostate tumours, where the changes in stiffness define diagnostic information. With ultrasound elastography thestiffness can be measured in an objective way in regions deeper than those reachable by palpation. In doing so, sonogramsare taken before and after the defined compression. With these sonograms a displacement field can be calculated. Two waysto analyse this displacement field are the definition of a strain and a stiffness distribution. By using qualitative elastographythe strain distribution is treated as the measure for stiffness. However, this indirect way of stiffness characterisation impliessome difficulties. If the material distribution is more complex, the visible correlation between strain and stiffness is hard todefine. With quantitative elastography the stiffness distribution is directly determined. To reduce the numerical cost of thedetermination of the stiffness distribution in quantitative elastography and to improve the accuracy, a variational h-refinementis applied. This way, the discretisation can be adapted by indicating the areas with a high gradient in the stiffness distributionwhose positions are a priori unknown, which are then finer discretised.

2 Determination of the stiffness distribution

In order to define the mechanical behaviour of soft tissue many simplifications and assumptions are needed. Here soft tissueis assumed as elastic, isotropic and incompressible, and the Neo-Hooke material model is used. There is also a simplificationin the dimensions. The three dimensional problem is reduced to a two dimensional one; a plain strain problem. Sincethe calculated displacements in the axial directions show a good quality, only these displacements are used for the stiffnessreconstruction. The Tikhonov-Philips-Functional described in [1] is applied to determine the stiffness distribution. Therelative shear modulus distribution that minimises this functional is determined in an iterative way. The iteration is given bythe following algorithm:

Find µ = µ(x) :(

12‖T(u)−T(um)‖2

Ω +α

2‖µ − µ∗‖2

b

)→ min! , (1)

where um is the measured and u the calculated displacement field, while T denotes a projection tensor onto the axial direction.The parameter α is the Tikhonov parameter, µ the current and µ∗ the initial shear modulus distribution.

3 Variational h-refinement

To improve the ratio between the number of nodes and the accuracy, h-refinement is added to the quantitative elastography.This way, a higher resolution of the displacement field can be used, so that the interfaces between different materials canbe better localised. The h-refinement is included by a higher level iteration. By h-refining the mesh, the shear modulusdistribution is also refined, through the increase in the number of elements. If the shear modulus distribution is refined inthis manner, the minimum of the Tikhonov-Philips-Functional must be equal or smaller then in the previous iteration. The h-refinement error indicator is defined as a difference between the minimum of the functional before and after the refinement. To

∗ Corresponding author: e-mail: arnold@tm.bi.rub.de, Phone: + 49 (0)234 32 22036, Fax: + 49 (0)234 32 02036

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10168 Sessions of Short Communications 02: Biomechanics

identify the elements which improve the minimisation, the h-refinement error indicator must be calculated for each element.For each tested element, a global shear modulus distribution with a mesh that is refined on this element must be determined.This is computationally very expensive, and therefore the h-refinement error indicator is estimated, cf. [2]. By definition,the estimated error indicator is smaller or equal to the effective indicator. The idea to refine the elements that show thehighest estimated h-refinement error indicator corresponds to the gamma refinement criterion (4). Here, for each patch ofelements which are divided by refining an element, the gamma criterion must be calculated. The elements whose patches havethe highest gamma criterion will be refined. By using the alpha criterion (2) the elements whose patches show the highestTikhonov-Philips-Functional will be refined. The global functional as a sum of the functionals of all elements will decrease, ifthe highest functionals of elements also decrease. The decreasing of a functional of an element is done by refining this element.The beta criterion (3) is different from the alpha criterion (2) only in the initial shear modulus. The initial shear modulus staysalways on the lower level. By this way, elements with the highest shear modulus will be refined. The refinement criteria aregiven by:

α − criterion : ghα(µh) =

n∑e=1

⎛⎝1

2

∫Be

(x1

he − xg

1he

)2dVe +t

2

∫Be

(µh

e − µ∗he

)2 dVe

⎞⎠ (2)

β − criterion : ghβ(µh) =

n∑e=1

⎛⎝1

2

∫Be

(x1

he − xg

1he

)2dVe +t

2

∫Be

(µh

e − 1)2 dVe

⎞⎠ (3)

γ − criterion : ghγ (µh) = ‖gh

α(µh) − ghα ref (µh)‖2. (4)

4 Results

The variational h-refinement is demonstrated on a simulation of a shear modulus reconstruction. The displacement field isgenerated by solving the forward problem with an exact shear modulus distribution. The observed area consists of a big anda small circular inclusion and the surrounding material. The big inclusion is stiffer than the small one. The ratio of the shearmodulus is set to µinc1/µinc2/µmat = 5/3/1. By using all refinement criteria the differences between the displacement fieldwhich is generated by solving the forward problem with the exact and the reconstructed shear modulus distribution decreaseswith the increase in the number of the iteration steps, cf. Fig.(2). According to Fig.(2) the efficiency of the algorithm describedin [3] could be increased. An example for an adaption of a mesh according to the shear modulus distribution and the γ-criterionis shown in Fig.(1).

(a) (b) (c)

Fig. 1 Mesh and shear modulus distribution after 2(a), 7(b) and13(c) iteration steps of the h-refinement by using γ-criterion

γ-criterion

β-criterion

α-criterion

iteration step i

‖T(u

)−

T(u

m)‖

Ω

252015105

0.08

0.06

0.04

0.02

0

Fig. 2 Difference of the displacement fields

Acknowledgements The support of the DFG grant BR 580/32-1 is gratefully acknowledged.

References

[1] A. A. Oberai, N. H. Gokhale, M. M. Doyley, and J. C. Bamber. Evaluation of the adjoint equation based algorithm for elasticity imaging.Physics in Medicine and Biology, 49:2955–2974, 2004.

[2] J. Mosler. On the numerical modeling of localized material failure at finite strains by means of variational mesh adaption and cohesiveelements. Habilitation thesis, Ruhr-Universitat Bochum, 2007.

[3] S. Reichling. Das inverse Problem der quantitativen Ultraschallelastografie unter Bercksichtigung großer Deformationen. PhD thesis,Ruhr-Universitat Bochum, 2007.

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