Evaluation of local changes in radio-frequency signal waveform and brightness caused by vessel dilatation for ascertaining reliability of elasticity estimation inside heterogeneous plaque: a preliminary study

Abstract

Purpose

To diagnose plaque characteristics, we previously developed an ultrasonic method to estimate the local elastic modulus from the ratio of the pulse pressure to the strain of the arterial wall due to dilatation in systole by transcutaneously measuring the minute thinning in thickness during one cardiac cycle. For plaques, however, some target regions became thicker as the vessel dilates, resulting in false elasticity. Therefore, a method to identify a reliable target for the elastic modulus estimation is indispensable. As a candidate for an identification index of plaques that become thicker during one cardiac cycle, the correlation of the radio-frequency (RF) signals remains high and it is not sufficient to obtain the elasticity. In this study, we thoroughly observed the target with a high correlation but positive strain in the plaque and characterized it by the property of the surrounding area.

Methods

For the plaque formed in the right carotid sinus of a patient with hyperlipidemia and the wall of the right common carotid artery of a young healthy male, (1) the correlation value as the similarity between the RF signals, (2) change in brightness obtained from the log-compressed envelope signals, and (3) strain obtained between the time of the R-wave and that of the maximum vessel dilatation were observed to characterize the region in the plaque.

Results

In the plaque, it was found that the region with high correlation and positive strain and its surrounding area could be classified into one of the three typical patterns.

Conclusion

As a preliminary study, this study provides a clue to assert the reliability of elasticity estimates for a region with high correlation and positive strain in the plaque based on measurable properties.

Appendix: Derivation of the local elasticity of the arterial wall and the assumptions in Eq. (4)

In the estimation of the local elastic modulus ${E}_{\uptheta }\left({d}_{0}\right)$, it is assumed that the vessel wall is elastically incompressible (Poisson ratio $\nu =0.5$) and isotropic; the strain of the vessel wall in the axial direction, ${\varepsilon }_{z}$, can be negligible (${\varepsilon }_{z}=0)$ because the artery is strongly restricted in the axial direction in vivo; and the change of inner radius $\Delta r$ and the change of target thickness $\Delta {h}_{d}$ are sufficiently small compared to the initial radius ${r}_{0}$ and the target thickness $w$ at the time of R-wave, respectively. It is also assumed that the pressure in the vessel wall changes linearly from the pressure on the inner surface of the vessel wall (i.e., internal pressure of lumen) to the pressure outside the vessel (i.e., atmospheric pressure).

From the assumption of elastic isotropy, the local elastic modulus ${E}_{\uptheta }\left({d}_{0}\right)$ in the circumferential direction on the local target set around the center depth ${d}_{0}$ with width $w$ along the ultrasonic beam inside the wall is defined as [30]

$$\begin{array}{*{20}c} {E_{{\uptheta }} \left( {d_{0} } \right) = \left( {{\Delta }\sigma_{r} \left( {d_{0} } \right) - \nu {\Delta }\sigma_{{\uptheta }} \left( {d_{0} } \right) - \nu {\Delta }\sigma_{z} \left( {d_{0} } \right)} \right)\frac{1}{{\varepsilon \left( {d_{0} } \right)}}. \left[ {{\text{Pa}}} \right]} \\ \end{array}$$

(A1)

The first term $\Delta {\sigma }_{r}\left({d}_{0}\right)$ in Eq. (A1) is the time change of stress in the radial direction at a depth ${d}_{0}$. The second and third terms are the contribution of time changes of stresses $\Delta {\sigma }_{\uptheta }\left({d}_{0}\right)$ in the circumferential direction and $\Delta {\sigma }_{z}\left({d}_{0}\right)$ in the axial direction to the radial strain $\varepsilon \left({d}_{0}\right)$ at depth ${d}_{0}$.

The radial stress ${\sigma }_{r}({d}_{0})$ is obtained by the average of stresses on the lumen and adventitia sides of the target. Under the assumptions mentioned above, $\Delta {\sigma }_{r}\left({d}_{0}\right)$ is obtained as [31]

$$\begin{array}{*{20}c} {\Delta \sigma_{r} \left( {d_{0} } \right) = - \frac{1}{2}\left\{ {{\Delta }p^{\prime}\left( {d_{0} - \frac{w}{2}} \right) + {\Delta }p^{\prime}\left( {d_{0} + \frac{w}{2}} \right)} \right\}, \left[ {{\text{Pa}}} \right]} \\ \end{array}$$

(A2)

where ${h}_{0}\ge w$ is the initial thickness of the vessel wall at the time of R-wave and ${\Delta }p^{\prime}\left( d \right)$ is the time change of the pressure on depth $d$ inside the wall. Under the assumption of linearity of pressure change in the wall [31], ${\Delta }p^{\prime}\left( d \right)$ can be expressed using the pulse pressure $\Delta p$ (time change of internal pressure of lumen minus atmospheric pressure) measured by a sphygmomanometer as

$$\begin{array}{*{20}c} {\Delta p^{\prime}\left( d \right) = \frac{{h_{0} - \left( {d - d_{w} } \right)}}{{h_{0} }}\Delta p. \left[ {{\text{Pa}}} \right]} \\ \end{array}$$

(A3)

From the conditions for the equilibrium of forces in the radial direction on the target and the assumptions mentioned above, $\Delta {\sigma }_{\uptheta }({d}_{0})$ is obtained as [31]

$$\begin{array}{*{20}c} {\Delta \sigma_{{\uptheta }} \left( {d_{0} } \right) = \frac{1}{w}\left( {r_{0} + d_{0} - \frac{w}{2} - d_{w} } \right)\Delta p^{\prime}\left( {d_{0} - \frac{w}{2}} \right)} \\ { - \frac{1}{w}\left( {r_{0} + d_{0} + \frac{w}{2} - d_{w} } \right)\Delta p^{\prime}\left( {d_{0} + \frac{w}{2}} \right), \left[ {{\text{Pa}}} \right]} \\ \end{array}$$

(A4)

where ${d}_{\mathrm{w}}<{d}_{0}$ is the depth of the interface between the lumen and the inner surface of the vessel wall at the time of R-wave. The first and second terms of Eq. (A4) are the contributions of pressures on the lumen side surface and the adventitia side surface of the target, respectively. $\Delta {\sigma }_{z}({d}_{0})$ is obtained from the assumption of ${\varepsilon }_{z}=0$ as [30]

$$\begin{array}{*{20}c} {\Delta \sigma_{{\text{z}}} \left( {d_{0} } \right) = \nu \left( {{\Delta }\sigma_{r} \left( {d_{0} } \right) + {\Delta }\sigma_{{\uptheta }} \left( {d_{0} } \right)} \right). \left[ {{\text{Pa}}} \right]} \\ \end{array}$$

(A5)

By substituting Eqs. (A2, A3, A4, A5) into Eq. (A1), the local elastic modulus $E_{{\uptheta }} \left( {d_{0} } \right)$ on the target is obtained as

$$\begin{gathered} E_{{\uptheta }} \left( {d_{0} } \right) = \left\{ {\left( {1 - {\nu }^{2} } \right){\Delta }\sigma_{r} \left( {d_{0} } \right) - \nu \left( {1 + \nu } \right){\Delta }\sigma_{{\uptheta }} \left( {d_{0} } \right)} \right\}\frac{1}{{\varepsilon \left( {d_{0} } \right)}} \hfill \\ = \frac{3}{4}\left( { - {\Delta }\sigma_{r} \left( {d_{0} } \right) + {\Delta }\sigma_{{\uptheta }} \left( {d_{0} } \right)} \right)\frac{1}{{ - \varepsilon \left( {d_{0} } \right)}} \hfill \\ = \frac{3}{4}\left( {\frac{1}{2} + \frac{{r_{0} + d_{0} - \frac{w}{2} - d_{{\text{w}}} }}{w}} \right)\frac{{\frac{w}{{h_{0} }}{\Delta }p}}{{ - \varepsilon \left( {d_{0} } \right)}}. \left[ {{\text{Pa}}} \right] \hfill \\ \end{gathered}$$

(A6)

Equation (A6) is the expansion of the elasticity definition derived by Hasegawa et al. [30] and equals the elasticity for the entire wall [30] by substituting $d_{0} = d_{{\text{w}}} + h_{0} /2$ and $w = h_{0}$ into Eq. (A6). Thus, Eq. (4) is derived.