PAMM · Proc. Appl. Math. Mech. 9, 153 – 154 (2009) / DOI 10.1002/pamm.200910051

An efficient quantitative ultrasonic elastography based on variationalmesh refinement

Alexander Arnold∗1, Otto T. Bruhns1, and Jörn Mosler2∗∗

1 Institute of Mechanics, Ruhr-University Bochum, Universitätsstr. 150, D-44780 Bochum, Germany2 Institute of Materials Research, Materials Mechanics GKSS Research Centre Geesthacht, Geesthacht, Germany

Palpation is often utilised in medicine to detect pathological changes in soft tissue such as those associated with breast canceror prostate tumor. Ultrasound elastography can be used to measure the stiffness in regions deeper than those reachable bypalpation. Conceptually, ultrasound echo signals are measured before and after a defined compression. A displacement fieldcan be calculated with these signals and subsequently, the stiffness distribution can be determined using quantitative elastog-raphy. In this regard, the difference between measured und estimated displacement fields is minimised (inverse problem).The region of interest is discretised using the finite element method. In order to reduce the resulting numerical costs and toimprove the accuracy of the method, a variational mesh refinement combined with a clustering technique known from digitalimage compression is elaborated and presented.

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

It is well known that many different pathologies affect the stiffness of soft tissue. This is evident in breast and prostate cancerand in other tumors being recognised as hard lumps. Ultrasonic elastography is an emerging medical imaging technique inwhich images of the spatial distribution of the stiffness of soft tissue are visualised. The region of interest is imaged byultrasound before and during a quasi-static deformation. With such images, the displacement field is calculated subsequentlyby using either correlation-based algorithms or by minimising a suitable objective function. Based on this displacement field,the stiffness distribution can be determined by solving an inverse problem. Usually, the region of interest is discretised bymeans of the finite element method, whereas the stiffness inside an element is assumed as constant. While common approachesuse a constant mesh, a variational refinement method is presented in this work. In doing so, the accurancy of the numericalformulation can be increased by keeping the resulting numerical cost relatively low. The efficiency of the proposed algorithmis further increased by applying a clustering technique known from digital image compression.

2 Determination of the stiffness distribution

The inverse problem of quantitative ultrasonic elastography is solved by utilising a Tikhonov regularisation, cf. [1]. Conse-quently, the functional

g(µ) =12‖P(ϕ − ϕg)‖2 +

α

2‖µ − µ∗‖2 (1)

depending on the unknown shear modulus distribution µ(X) is to be minimised. Here, P is a projection of the deformationmapping ϕ, i.e., only the axial components of the measured deformation ϕg and the predicted deformation ϕ (FE-analysis)are considered. α represents a regularisation parameter and µ∗ denotes a referential shear modulus distribution (i.e., only arelative distribution can be computed). The minimisation problem is solved by applying an L-BFGS-B algorithm. Thus, ineach step of this iterative algorithm, the functional g(µ) and its gradient with respect to the shear modulus [Dµg] have to becomputed.

3 Variational h-refinement and clustering

Within the proposed quantitative elastography, the region of interest is discretised by a finite element triangulation T0. Con-sequently the approximated shear modulus distribution is represented by a finite-dimensional subspace V

hµ

0 , with Vhµ

0 ⊂V µ (V µ is the solution space containing all possible and physically attainable shear modulus distributions). Evidently,V

hµ

1 ≥ Vhµ

0 , if triangulation T1 corresponding to Vhµ

1 can be obtained form triangulation T0 by a sequence of edge bisections

∗ Corresponding author: e-mail: arnold@tm.bi.rub.de, Phone: + 49 (0)234 32-22036, Fax: + 49 (0)234 32-02036∗∗ Corresponding author: e-mail: joern.mosler@gkss.de, Phone: +49 (0)4152 87-2679, Fax: +49 (0)4152 87-2595

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

154 Short Communications 2: Biomechanics

(nested sequence of triangulations). In order to identify edges e of E(Ti), whose bisection improves the minimisation of g(µ)significantly, the estimate

∆grefj = inf g(Ti) − inf g(Ti+1,e=j) (2)

can be applied (Ti+1,e=j is obtained from triangulation Ti by edge bisection of edge j). Hence the indicator ∆grefj determines

the influence of a local mesh adaption on the solution. Clearly, this indicator has to be calculated for each edge. Thus the shearmodulus distribution with a modified triangulation must be calculated for each edge. This is computationally very expensiveand therefore the indicator ∆g

refj is estimated, cf. [2] and [3]. More precisely, only the shear modulus of those elements

which are refined is varied. The shear modulus of the other elements is kept constant. Accordingly, this estimated indicatorreads

∆grefj = inf g(Ti) − inf g(Ti+1,e=j)

∣∣∣ϕ=const,∀ϕ∈Vi∩Vi+1,e=j

≤ ∆grefj . (3)

In order to reduce the number of degrees of freedom resulting from h-refinement further, elements are grouped according totheir shear moduli. A positive side effect of such a clustering technique is the reduction of data noise (averaging technique).Roughly speaking, neighbouring elements with similar shear moduli are collected in groups. Each group shows only onedegree of freedom (shear modulus).

4 Results

The performance of the quantitative elastography combined with variational h-refinement and the proposed clustering tech-nique is demonstrated in this section. First, a displacement field is generated by solving a forward problem. Accord-ing to Fig. 1(a), the problem is characterized by a large and a small inclusion. The ratio of the shear modulus is set toµinc1/µinc2/µmat = 5/3/1 (the first inclusion is the large one). Subsequently, the proposed inverse method, together withthe h-refinement, is applied (a homogenuous initial shear modulus distribution is assumed within the algorithm). The results(the error) with and without employing the clustering technique are summarized in Fig. 2. Accordingly, the clustering tech-nique shows the same accurancy as the reference solution. However, the number of degrees of freedom is reduced siginficantly,cf. 3. The predicted finite element trinagulations and the associated distribution of the shear modulus are shown in Fig.(1).

(a) (b) (c)

Fig. 1 Mesh and shear modulus distribution af-ter 5(a), 10(b) and 15(c) iteration steps of the h-refinement combined with the proposed clusteringtechnique

0

0.02

0.04

0.06

0.08

0.1

0.12

0 2 4 6 8 10 12 14 16

|u−

um|

iteration step i

I.IndicatorI.Indicator-clustering

Fig. 2 Difference of the displacementfields

0

500

1000

1500

0 2 4 6 8 10 12 14 16

degr

ees

offr

eedo

mn

iteration step i

I.IndicatorI.Indicator-clustering

Fig. 3 Number of degrees offreedom

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 and M. Ortiz. Variational h-adaption in finite deformation elasticity and plasticity. International Journal for NumericalMethods in Engineering, 72:505–523, 2007.

[3] J. Mosler. An error-estimate-free and remapping-free variational mesh refinement and coarsening method for dissipative solids at finitestrains. International Journal for Numerical Methods in Engineering, 77:437–450, 2009.

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