PAMM · Proc. Appl. Math. Mech. 10, 71 – 72 (2010) / DOI 10.1002/pamm.201010028

A novel algorithm for the inverse problem in elasticity imaging by meansof variational r-adaption

Alexander Arnold1,∗, Otto T. Bruhns1, and Jörn Mosler2,∗∗

1 Mechanics - Continuum Mechanics, Ruhr-University Bochum, 44780 Bochum, Germany2 Institute of Materials Research, Materials Mechanics, GKSS-Forschungzentrum Geesthacht, 21502 Geesthacht, Germany

Elasticity imaging or elastography is a powerful technique in medicinal imaging for visualizing the stiffness distribution insoft tissue in vivo. It is motivated by the observation that the stiffness in soft tissue is affected by pathologies in many cases.More precisely, diseased tissue tends to be stiffer than the healthy surrounding tissue. The two steps involved in elasticityimaging are: first deforming the tissue and measuring the displacement field in the region of interest using ultrasound or MRIsignals; second calculating the underlying stiffness distribution using an inverse analysis. While in common approaches thisinverse analysis is based on minimizing the distance between the measured and the computed deformation field dependingonly on the unknown stiffness distribution, an additional variation of the underlying finite element discretization is the focusof the present work. In doing so, the triangulation is optimized improving the accuracy of the results and increasing theefficiency of the computational framework.

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

It is well known that pathologies, e.g. breast or prostate tumors, affect the mechanical properties of soft tissue. Manydiagnostic methods rely on that observation. By using ultrasound techniques for example, the difference in speed of sound isutilized to visualize the material distribution and therefore, to identify diseased tissue. However, the contrast between healthyand diseased tissue in speed of sound is less than the contrast in the stiffness modulus. Therefore, elasticity imaging, a methodto visualize the stiffness distribution in soft tissue, has been developed. Using elasticity imaging, the region of interest isimaged first by ultrasound or MRI before and after a quasi-static deformation. With such images, the displacement field iscalculated subsequently by using either correlation-based algorithms or by minimizing a suitable objective function. Basedon this displacement field, the stiffness distribution can be determined by solving the inverse problem of elasticity imaging.Usually, the region of interest is discretized by means of the finite element method, whereas the stiffness inside each elementis assumed as constant. Consequently, the accuracy of the solution depends strongly on the approximation of the stiffnessdistribution. While common approaches use a uniform mesh, a variational mesh adaption is discussed in the present work.Using a combination of a variational h-adaption, a clustering technique and a variational r-adaption the accuracy of the solutionis increased by keeping the resulting numerical cost relatively low.

2 The inverse problem of elasticity imaging

The inverse problem of elasticity imaging is defined by computing the shear modulus distribution µ(X) inside the region ofinterest based on boundary data and a measured deformation field. For that purpose and in line with [1], the functional

g(µ) =12‖P(ϕ−ϕm)‖2 +

α

2‖µ− µ∗‖2 (1)

depending on the unknown shear modulus distribution µ(X) has to be minimized. Here, P is a projection of the deformationmapping ϕ, i.e., only the axial components of the measured deformation ϕm and the predicted deformation ϕ (FE-analysis)are considered. α represents a regularization parameter and µ∗ denotes a referential shear modulus distribution (i.e., onlya relative distribution can be computed). The minimization problem is solved by applying an iterative procedure, e.g., anL-BFGS algorithm.

3 Variational mesh-adaption in elasticity imaging

Clearly, the numerically computed solution of the inverse problem of elasticity imaging depends strongly on the underlyingfinite element triangulation. For improving the quality of the solution, the space of admissible shear moduli distributions canbe adapted. For a variational h-adaption and a clustering technique, the reader is referred to [4], see also [2, 3]. In the present

∗ E-mail: arnold@lkm.rub.de, Phone: +00 49 (0) 234 32-26956, Fax: +00 49 (0) 234 32-14229∗∗ E-mail: joern.mosler@gkss.de

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

72 Section 2: Biomechanics

work, the quality of the solution is further improved by combining the aforementioned techniques with a variational r-adaption(reallocation of the nodal positions denoted asXh). For that purpose, the functional

h(Xh) =12‖P(ϕ−ϕm)‖2 + α̃ hgeo(Xh) (2)

is introduced. The additional term hgeo(Xh) represents a penalty function guaranteeing small aspect ratios of the finiteelements and α̃ is an additional regularization parameter, cf. [5]. Using an L-BFGS algorithm, the functional h(Xh) and itsgradient with respect to the nodal positions Xh are required. Denoting the linearization of an approximated field ah withrespect toXh as ∆Xh

ah, the aforementioned gradient reads (for an element e)

∆Xh

hhe =

∫B0e

(x2

he − xm

2he

)(∆Xh

x2he −∆Xh

xm2

he

)dVe +

12

∫Bξ

(x2

he − xm

2he

)2

DXh

(detF Φ̂e

)·∆Xh

e dVξ

+α̃[DXh

(ggeoe

)·∆Xh

e

].

(3)

Here, xm2 are the measured and x2 are the predicted nodal deformations and F Φ̂ = ∂Φ̂/∂ξ, where Φ̂ is a mapping connecting

the reference configuration to the material configuration.

4 Results and conclusionsThe performance of the proposed variational mesh adaption is demonstrated in this section. For that purpose, a displacementfield is generated first by solving a forward problem. According to Figs. 1, that problem is characterized by one inclusionembedded within a softer matrix (the ratio of shear moduli is µinc/µmat = 5/1). The results of the inverse problem ofelasticity imaging using a relatively coarse and uniform mesh without employing any adaption scheme are shown in Fig. 1(a).Although the inclusion is detected, the interface between both materials is not captured in a sharp manner. Based on this initialresult, variational mesh refinement is applied. More precisely, two h-refinement steps combined with a clustering techniquein line with the algorithm advocated in [4] are considered first. This way, the quality of the solution is significantly improved,see Fig. 1(b). A further improvement can be achieved by using the variational r-adaption as discussed within the presentpaper, see Fig. 1(c). The interface between the two different materials is now very sharp and in excellent agreement withthe forward problem. The difference between the predicted and measured displacements is illustrated in Fig. 2. As evident,the displacement error decreases with each variational mesh refinement step. The same holds for the error of the stiffnessdistribution, see Fig. (3).

(a) (b) (c)

Fig. 1 Finite element meshes and computed shearmodulus distribution: (a) no mesh adaption; (b) h-adaption; (c) h- and r-adaption

0

0.01

0.02

0.03

0.04

0 1 2;0 10 20 30 40

h- r-adaption

‖u−

um‖ B

iteration step i

Fig. 2 L2-norm of the difference be-tween the displacement field of the for-ward and the inverse problem

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2;0 10 20 30 40

h- r-adaption

‖µ−µ

m‖ B/‖µ

m‖ B

iteration step i

Fig. 3 L2-norm of the difference (rela-tive) between the shear modulus field ofthe forward and the inverse problem

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, Physics in Medicine and Biology 49, 2955-2974 (2004).[2] J. Mosler and M. Ortiz, International Journal for Numerical Methods in Engineering 72, 505-523 (2007).[3] J. Mosler, International Journal for Numerical Methods in Engineering 77, 437-450 (2009).[4] A. Arnold, S. Reichling, O. T. Bruhns and J. Mosler, Physics in Medicine and Biology 55, 2035-2056 (2010).[5] P. M. Knupp, International Journal for Numerical Methods in Engineering 48, 401-420 (2000).

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