Effect of Lesion Boundary Conditions on Axial Strain Elastograms: A Parametric Study

Abstract

Ultrasound elastography produces strain images of compliant tissues under quasi-static compression. When a material is compressed, there are several parameters that affect the stress-distribution and, hence, the strain distribution in the material. The state of bonding of an inclusion to the background material is a critical parameter. Heretofore, in the field of elastography, the inclusion was considered to be firmly bonded to the background material and analytical solutions were derived for the elasticity problem involving simple geometries like circular inclusion (for two dimensional [2D]) and spherical inclusion (three dimensional [3D]). Under these conditions, simple analytical expressions relating the strain contrast to the modulus contrast were derived. However, it is known that the state of bonding of some tumors to their surrounding tissues depends on the type of the lesion. For example, benign lesions of the breast are known to be loosely bonded to the surrounding tissue, while malignant breast lesions are firmly bonded. In this study, we perform a parametric study using finite element modeling (FEM) to investigate the validity of the analytical expression relating the strain contrast to the modulus contrast, when the state of bonding at the inclusion/background interface spans a large dynamic range. The results suggest that estimated modulus contrast using the analytical expression is sensitive to the region-of-interest within the inclusion that is considered in the computation of the strain contrast. By considering the inclusion region lying along the axis of lateral symmetry instead of whole region of the inclusion, the estimated modulus contrast (obtained using the analytical expression present in the literature) can be computed to within a systematic error of 10% of the actual modulus contrast. Additional estimation errors are expected to accrue in experimental and in vivo conditions. (E-mail: [email protected])

Introduction

Elastography is a technique that produces images (elastograms) that map the strain experienced by tissue elements subjected to a quasi-static compression (Ophir et al. 1991). Ultrasound elastography typically produces high resolution axial strain elastograms due to high sampling possible in that direction and the ability to use the ultrasound transducer as a tissue compression device. It has been recently shown that modulus contrast approximates the inverse of strain contrast under certain conditions (Srinivasan et al. 2004). This relationship justifies the use of an inverse strain image as a first approximation for a modulus image under certain conditions.

Prior literature reports have investigated the relationship between the strain contrast and the modulus contrast (Ponnekanti et al 1995, Kallel et al 1996, Bilgen and Insana 1998). Ponnekanti et al. (1995) described this relationship in terms of contrast transfer efficiency (CTE), defined as the ratio of estimated modulus contrast from elastogram (as inverse of strain contrast) to actual modulus contrast. Later, Kallel et al. (1996) reported an analytic study (2D) on the fundamental limitations on the CTE in elastography. They reported a closed form expression to estimate the modulus contrast from the observed strain contrast viz.

$$\frac{1}{C_s}=[\frac{(1-2\upsilon )}{C_m+(1-2\upsilon )}+\frac{2}{1+C_m(3-4\upsilon )}]$$

where, C_s is the strain contrast, C_m is the corresponding Young’s modulus contrast, and υ is the Poisson’s ratio of both the inclusion and background. For incompressible materials (υ = 0.5), eqn 1 reduces to

$$\frac{1}{C_s}=\frac{2}{1+C_m}$$

Bilgen and Insana (1998) reported a similar study, but for a 3D situation. In all these studies, it was assumed that the inclusion was firmly bonded to the background material. However, this assumption may not always hold. In fact, in the case of breast tumors, the literature suggests the existence of differences in the way that benign tumors and malignant tumors are bonded to the surrounding tissues (Chen et al. 1995). Breast fibroadenomas are loosely bonded to their surrounding tissue and possess strong mobility and slip upon palpation (Fry 1954). Breast carcinomas are thought to be firmly bonded to the background due to the formation of stellate boundaries, whereas fibroadenomas are thought to be loosely bonded to the background due to their smooth boundaries (Fry 1954, Chen et al 1995, Garra et al 1997, Bamber et al 1988, Ueno et al 1988). The nature of the bonding of the lesion to the background has been related to differences in lesion mobility (Bamber et al 1988, Ueno et al 1988, Konofagou et al 2000).

In a recent report (Thitaikumar et al. 2006), we recognized the need to obtain an estimate of the modulus contrast for an inclusion that is loosely bonded to the inclusion. The analytic expression derived in Kallel et al. (1996) and presented in eqn 2 seemed attractive due to its simple form. This expression allows estimating the modulus contrast using the strain contrast measured from axial elastograms. However, as mentioned earlier, it was derived under the assumption that the inclusion was firmly bonded to the surrounding. Therefore, the present objective was to study the validity of this expression when the assumption of firm bonding at the inclusion/background interface does not strictly hold. This was investigated using finite element modeling (FEM) based parametric study by changing one model parameter at a time, as explained in the following section.