The standard way of imaging the stiffness or elasticity of living tissue is through display of the axial strain. Tissue-specific lateral-strain,^{1} Poisson's ratio,^{2} and shear-strain^{3} elastograms are usually also visualized. The so-called inverse-problem method makes it possible to image other related tissue parameters, such as Young's modulus and the shear modulus.^{4} All of these techniques aim at distinguishing among different types of elastic-tissue behavior using parameters related to the strain or elastic-modulus tensor. Being able to display the full strain tensor in a single image has already led to promising results, triggering new approaches in other medical-imaging applications, such as in diffusion-tensor magnetic-imaging resonance (DT-MRI).^{5} Our goal is to provide clinicians with new tools enabling the extraction of more information from their diagnosis or from pathologies related to changes in the elastic-tissue properties. One of the remaining unsolved issues in the field relates to the discrimination between benign and malign lesions in different cancers without the use of biopsies. To achieve this, we have proposed the introduction of new scalar strain parameters, including the ‘vorticity’ and ‘strain index’ (SI), as well as a tensorial representation of the strain.

Mathematical basis

We define u = (*u*_{x}, u_{y}) to be a 2D displacement-vector field. It is well known from mechanical engineering that the ‘displacement-gradient matrix,’ J, also called the Jacobian or unit-relative-displacement matrix, can be decomposed into the strain tensor, which is the symmetric component, and the (antisymmetric) vorticity tensor,

The strain tensor, E, with the elongational and shearing-strain elements located on and off the diagonal of the matrix, respectively, measures the changes of shape locally (i.e., stretching or shortening), whereas the vorticity matrix, Ω, contains information about rotational changes.

Vorticity

Although elastography can detect breast tumors, in vivo biopsies are still needed to assess their malignancy. Malignant tumors form ramified boundaries which become firmly bound to the surrounding tissue, unlike benign lesions, which have smooth borders and are only loosely attached to the tissue medium.

We have developed^{6} the theory and procedures for assessing the rotation of a tumor by visualizing the vorticity image, which leads to an improved determination of tumor infiltration. Preliminary but promising results (see Figure 1), supported by a phantom study, show the potential of this technique for the diagnosis and prognosis of tumors, through noninvasive detection of their infiltrating nature.

**Figure 1.** Vorticity images, both with symmetric boundary conditions. (left) Bounded, malign case. (right) Unbounded, benign case. For the bounded case the vorticity is homogeneous (for both the inclusion and the background). For the unbounded case one may observe the ring, which also appears in the axial or the shear-strain images, but now isolated from the other parameters.

Vorticity isolates information about tumor rotation in the deformation tensor and may also contribute to the diagnosis and prognosis of in vivo cancers (i.e., breast and prostate tumors). As far as we know, these have not yet been assessed in other elastography studies.

**Figure 2.** Synthetic phantom with a circular inclusion three times harder than the background. (left) Strain elastogram. (right) SI elastogram.

Strain index

We will now perform the eigenvalue decomposition of the strain tensor, E, to obtain its eigenvalues, *d*_{i}. From DT-MRI analysis it has been established that the sum of the eigenvalues of the diffusion tensor contains information about the total amount of diffusion in the tissue. Similarly, the sum of the eigenvalues of the strain tensor contains information about the total tissue strain. However, unlike the diffusion tensor, the strain tensor is not always a symmetric-positive-definite matrix. Negative eigenvalues can occur, depending on the choice of the coordinate system. This problem can be overcome easily by taking the absolute eigenvalues.

We propose that the elastic information of tissues in elastography is represented by the SI, which is defined as

where ( |ε_{ij}| ) and ( |*J*_{ij}| ) are the matrices containing the absolute values of the elements in E and J, respectively. The sum of the eigenvalues is equivalent to the trace of the matrix and, as the strain and vorticity tensors are the symmetric and antisymmetric parts of ∇ u, respectively, the SI can be computed either from the strain tensor or directly from the displacement-gradient matrix. The SI thus provides a more global insight into the relevant tissue's elasticity, as it takes information from both the axial and the lateral strain. In Figure 2 a synthetic phantom is shown with a circular inclusion three times harder than the background. The inclusion is clearly visible in both elastograms. However, the shape is more accurately displayed using the SI representation, where the boundaries are also more clearly visible. On the strain elastogram (see the left-hand panel of Figure 2) an undesired vertical pattern is observed, which does not appear in the novel SI representation.

This SI visualization can potentially assist physicians in the detection of different types of lesions. The expected improvement in elastogram quality will allow the introduction of segmentation methods in the analysis of elastography images aimed at detecting the borders of the inclusions. This task is still very difficult to carry out with the elastogram quality thus far obtained.

Strain-tensor elastography

The strain-tensor field visualizes in a single image the standard scalar parameters that are usually represented separately. Using this technique, physicians will have access to complementary information on the tissue properties. In addition, this novel visual image offers the possibility of extracting new parameters related to elastic-tissue behavior. Visualization of tensor fields significantly improves the understanding and interpretation of tensor data.

**Figure 3.** Tensorial images overlaid on the axial-strain field. Comparison of the bounded and unbounded case for a virtual phantom. (left) Bounded, malign case. (right) Unbounded, benign case.

Most representations in elasticity imaging exhibit scalar tissue parameters, i.e., components of the strain tensor. Tensor formulations are not widely employed in signal-processing or related fields, but some biomedical applications, such as diffusion-tensor imaging (DTI) and cardiac strain-rate imaging, use tensor visualizations. Although the DTI-visualization techniques are quite well developed, their application to strain-tensor fields is not obvious. The strain tensor is symmetric, but does not satisfy the positive-semi-definite condition. However, the sign of the strain-tensor eigenvalues represents material stretching or shortening in the direction of the corresponding eigenvector for positive and negative eigenvalues, respectively. We represent the axes of the strain-tensor ellipse with the absolute eigenvalues, thus appreciating how much deformation from the total has been absorbed by the different tissues.

We developed^{7} a technique similar to that used for the visualization of myocardial strain-rate tensors.^{5} We propose to visualize tensors as ellipsoids colored according to the sign of the largest eigenvalue, representing stretch or compression in the principal directions. We use blue and red coloring for shortening and stretching, respectively, when we display the tensorial image on its own or overlaid on a gray-scale image. Where the tensorial image is displayed superposed on a color image, the ellipses are drawn in black for shortening, to distinguish them from the background colors.

In Figure 3 we show a virtual phantom with a circular inclusion for the two main cases, i.e., where the inclusion is, respectively, firmly and loosely bounded to the background.

**Figure 4.** Tensorial image overlaid on the axial-strain field (mapped to a color scale and expressed as the fractional strain) for a commercial phantom, CIRS 059 (tissue-mimicking commercial breast), manually compressed. Acquisition was done using the Ultrasonix RP500 US ultrasound research scanner.

Future work

A more in-depth comparison of the visualization parameters is needed, as well as clinical validation using in vivo experiments. Once this validation has been achieved, the final goal is to provide physicians with different visualization methods built into a standard scanner, allowing application of the most suitable technique in each unique situation, or to obtain complementary information from different visualization methods. We continue to work on the development of novel parameters to visualize the elastic tissue properties with a better shape performance than the axial elastograms used to date.

*Part of this work has been supported by a Spanish government USIMAG grant (TEC2004-06647C03-02) and the European Network of Excellence SIMILAR (FP6-507609 WP10).*

Darío Sosa-Cabrera

Signals and Communications Laboratory

Center for Technology in Medicine

University of Las Palmas de Gran Canaria

Canary Islands, Spain

http://www.ctm.ulpgc.es Darío Sosa-Cabrera obtained his first degree in mechanical engineering from the University of Las Palmas de Gran Canaria in 2001. He is currently associated with a number of European and Spanish research projects. He will shortly submit his PhD thesis in the field of telecommunications engineering, focusing on ultrasound elastography. He also studied at the French Institute for Advanced Mechanics. After briefly working in Latvia, he completed an MSc in management of technology at the University of Texas (UT) at San Antonio and subsequently joined the Department of Radiology at UT Houston for one semester.

Juan Ruiz-Alzola

Technological Institute of the Canary Islands

Las Palmas de Gran Canaria, Spain