Technical Notes
An analysis of elastographic contrast-to-noise ratio

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Abstract

We present a theoretical formalism and simulation results that allow the incorporation of the elastic contrast properties of tissues with simple geometries into the elastographic noise models developed previously. This analysis results in the computation of the elastographic contrast-to–noise ratio (CNRe). The CNRe in elastography is an important quantity that is related to the detectability of a lesion or inhomogeneity. In this paper, the upper bound on the elastographic CNRe is derived for both a one-dimensional (1-D) and 2-D analytic plane-strain tissue model. The CNRe in the elastogram depends on the contrast-transfer efficiency (CTE) for both the 1-D and 2-D geometries discussed in this paper. The 1-D model is used to characterize layered structures and the 2-D model is derived for circular inclusion within a background of uniform elasticity. A previously derived classical analytic solution of the elasticity equations, for a circular inclusion embedded in an infinite medium and subjected to a uniaxial compression, is used to compute the upper bound of the CNRe. Monte Carlo simulations illustrate the close correspondence between the theoretical and simulation results.

Introduction

Elastography is a method for imaging the elastic behavior of soft tissues, producing grey-scale strain images referred to as elastograms Ophir et al 1991, Ophir et al 1996, Ophir et al 1997, Ophir et al 1998. The process of generating an elastogram can be viewed as a two-step process, as illustrated in Fig. 1. First, the elastic medium is compressed to generate a strain field. The nature of the tissue compression, the elastic moduli of the medium, and the boundary conditions determine how the medium is strained in 3-D. Second, ultrasonic echo correlation techniques are used to estimate the axial strain field from the pre- and postcompression RF echo signals. Only the axial strain is estimated because the sampling of the RF signals along the beam axis is significantly finer than the lateral sampling interval.

In this paper, we combine the properties of the ultrasound imaging system and signal processing algorithms with the elastic behavior of the tissue, as illustrated in Fig. 1. This combined theoretical model enables prediction of the elastographic contrast-to–noise ratio (CNRe) for layered or circular lesions embedded in a uniformly elastic background. The approach discussed in this paper may be modified in the future to include varying lesion sizes and shapes. However, the analysis in this paper is limited to sufficiently large diameters of hard and soft lesions.

We propose the use of analytical models to characterize tissue elastic properties. For layered structures, a 1-D model is proposed, where the strains in the tissue are directly proportional to their elastic moduli (Céspedes and Ophir 1993). For embedded circular lesions, a 2-D analytic solution of the elasticity equations Honein and Herman 1933, Muskhelishvili 1963, Kallel et al 1996 is used. In this case, the elastographic strains are not proportional to the elastic moduli, giving rise to fundamental limitations in displaying the modulus contrast. These limitations have been quantified in terms of the contrast-transfer efficiency (CTE) by Ponnekanti et al. (1995) and Kallel et al. (1996).

The CTE was defined as the ratio of the strain contrast as measured from the elastogram to the actual modulus contrast. The CTE for 1-D models equals 1 (maximum value) because the strain contrast and the modulus contrast are equal. However, for 2-D geometries, the CTE is less than or equal to 1 (0 dB), and is highly dependent on the modulus contrast as illustrated by Ponnekanti et al. (1995) using finite element analysis, and by Kallel et al. (1996) using an analytic solution for the elasticity equations.

The noise performance of the strain estimator in elastography may be characterized using the strain filter (SF) developed by Varghese and Ophir (1997a). The novel concept in this paper is the incorporation of the analytic tissue CTE model into the SF formulation (see Fig. 1) for simple 1-D and 2-D tissue models. This results in a description of the combined effect of the elastic tissue parameters and the signal processing parameters; specifically, it provides a means of predicting the upper bound of the CNRe between layered tissues of various modulus contrasts, or between a lesion and its background.

Elastographic detectability of lesions depends on their size and contrast, the elastographic imaging system, display system and observer or detector performance (Belaid et al. 1994). The strain noise in the elastogram and the elastographic resolution are the other factors that limit the detector performance. Contrast detail curves for circular elastographic lesions with varying diameters and contrasts were analyzed by Belaid et al. (1994). They found that, for identical object contrasts, elastography had significantly higher detectability of all lesion diameters when compared to sonography.

The ability to make a decision on accepting or rejecting the presence of a lesion or layer is quantified by the contrast-to–noise ratio (CNR) which, in elastography, is computed from the means and variances of the strains in the lesion and the background, respectively (Bilgen and Insana 1997). The CNR has been used as a basis for measuring the separability of two different data clusters. The CNR is not rigorously related to the ideal observer performance; however, it is correlated with the visual impressions of the human observer when the lesions are larger than the noise correlations and the noise is roughly constant throughout the entire sonographic image (Insana and Hall 1994). Because the probability density function (pdf) for an homogeneous elastogram is characterized by Gaussian statistics (Belaid et al. 1994), similar results can be obtained for elastograms. This note does not discuss the performance of the ideal observer; rather, the strain dependence of the CNRe, and the range of strains over which the CNRe is enhanced, are analyzed for some special cases.

The upper bound on the CNRe in elastography is of interest to determine the ability of elastograms to differentiate between different regions in tissue. Contrast contour maps (changes in the CNRe with modulus contrast) presented in this paper illustrate the strain dependence of the CNRe. This upper bound on the CNRe may, in turn, be derated by various realistic effects, such as tissue attenuation (Varghese and Ophir 1997b), lateral and elevational signal decorrelation (Kallel et al. 1997), quantization noise, bias errors due to interpolation, etc., by accounting for their effects on the SF.

The range of tissue modulus contrasts considered in this note was obtained from stiffness measurements in breast tissue in vitro (Krouskop et al. 1997). The stiffness measurements for normal breast tissue range from 20 ± 8 kPa for normal fat tissue (n = 40), 48 ± 15 kPa for normal glandular tissue (n = 31), to 220 ± 88 kPa for fibrous tissue (n = 21) using a 20% precompression and a 2%/s strain rate (Krouskop et al. 1997). For abnormal breast tissue, the stiffness measurements were 291 ± 67 kPa (n = 23) for ductal tumors and 558 ± 180 kPa (n = 32) for infiltrating ductal carcinomas in breast tissue (Ophir et al. 1998) under similar loading conditions. The above measurements indicate that the stiffness ratio between the softest tissue in breast (normal fat) and hard lesions (carcinomas) is about 29 dB, and the ratio between ductal tumors and carcinomas is about 5.65 dB. Generally, lesions with a contrast of 6 dB and below may be defined as low-contrast lesions. The CNRe values that correspond to such low-contrast elastographic lesions are of primary interest in this note.

The analysis in this note does not account for the increase in the contrast in the elastograms due to the mechanical stress concentration artifacts Cespedes 1993, Ponnekanti et al 1995, Kallel and Ophir 1998. Contrast is defined as the ratio of the strain within the lesion to the strain in the homogeneous region in the absence of the lesion or far away from the lesion (Kallel et al. 1996). Defining the contrast in this manner underestimates the contrast observed in the elastogram. It has been noted that the presence of the mechanical stress concentration artifacts in elastography may improve the detectability of the lesion because it enhances the contrast between the lesion and the background region in its immediate vicinity. It has in fact been shown that the CTE improves for both hard and soft lesions when the peak artifacts are taken into account (Kallel and Ophir 1998). The upper bound computed in this note ignores these effects.

The following section presents the analytic tissue models used in this note. Monte Carlo simulations are used to corroborate the theoretically predicted CNRe results. Finally, the contributions of this note are summarized.

Section snippets

Theory

The expression for the CNRe is derived by Bilgen and Insana (1997), and is given by: CNRe=2(s1−s2)2s12s22), where s1 = sI or sL1 and s2 = sB or sL2 represent the mean value of strain in the lesion (sI) or the first layer (sL1) and the background (sB) or second layer (sL2), and σs12, σs22 denote the strain variances respectively. Note from eqn (1) that the CNRe in the elastogram can be estimated from knowledge of the means and the variances of the strain estimates within the lesion and the

Simulation results

Monte–Carlo simulations in MATLAB are used in this section to corroborate the theoretical results presented in the previous section for the 1-D and 2-D tissue models. The B-scans for the 1-D and 2-D analytic models were obtained as follows: First, the analytic model was used to generate displacement information with applied strain. An acoustic Rayleigh scattering model was added to the displacement information to simulate tissue scattering profiles. Pre- and postcompression RF signals were

Discussion and conclusions

In this note, we derived the upper bound on the CNRe for specific 1-D and 2-D analytical models. Simulation results illustrate the close correspondence between the theoretical and simulation results. Knowledge of the theoretical upper bound on the CNRe in elastography is crucial for determining the ability to discriminate between different regions in the elastograms. Trade-offs among the various elastographic parameters, namely the elastographic SNRe, sensitivity, dynamic range, CNRe and

Acknowledgements

The authors thank Dr. Faouzi Kallel for providing the analytic solution for the elasticity equations and Dr. S. Kaiser Alam for his helpful comments.

This work was supported in part by NIH grants R01-CA60520 and P01-CA64597.

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