Corrections to “The Role of Viscosity in the Impulse Diffraction Field of Elastic Waves Induced by the Acoustic Radiation Force” and “Supersonic Shear Imaging: A New Technique for Soft Tissue Elasticity Mapping”

Abstract

This paper presents corrections to, ”The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force,” (Bercoff et al.), IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 51, no. 11, pp. 1523–1536, Nov. 2004, and ”Supersonic shear imaging: A new technique for soft tissue elasticity mapping,” (Bercoff, J., Tanter, M., and Fink, M.), IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 51, no. 4, pp. 396–409. April 2004.

I. Introduction

Reference [1] presents a Green’s function approach to model the displacement field generated by a force density distribution in an unbounded, homogeneous, and isotropic viscoelastic medium. This approach extended the Green’s function method presented in reference [2] by accounting for viscosity using the Voigt model. The theory in [1] formed the basis of a novel ultrasound shear wave elastography approach [3] that has been very useful and widely used. The equations and derivation for the viscoelastic Green’s function method in references [1] and [3] contain several typographical errors. In this note we provide corrections for these equations in order to clarify the derivation and help enable others to correctly implement and use this novel Green’s function method.

II. Corrections to Green’s Function Derivation

Equation (8) in [1] is the wave equation for the Lamé potentials. This equation can be simplified by taking into account the spherical symmetry of the medium—the solution is only a function of the radial distance r. Equation (9) in [1] was obtained by using this symmetry condition and by writing the Laplacian in spherical polar coordinates. However, Equation (9) in [1] contains typographical errors. Its corrected form is

$$\frac{{\partial }^2{g}_i}{\partial t^2}-\left(c_i^2+v_i\frac{\partial }{\partial t}\right)\left(\frac{{\partial }^2{g}_i}{\partial r^2}+\frac{2}{r}\frac{\partial g_i}{\partial r}\right)=\frac{\delta (r)}{4\pi r^2}\delta (t)i=p,s.$$ (1)

Equation (10) in [1] now directly follows from Equation (1).

The Fourier transform pair is defined by Equation (11) in [1]. Given this definition, the inverse Fourier transform of the Gaussian exponential shown by Equation (16) in [1] is incorrect. Its corrected form is

$$F^{-1}\left\{e^{-a{k}^2}\right\}=\frac{1}{2\pi }\sqrt{\frac{\pi }{a}}{e}^{\frac{-r^2}{4a}}.$$ (2)

Equation (17) in [1] follows from the application of Equation (2).

For Equation (21) in [1] the v_p should be replaced with a v_s. The equation’s corrected form is

$$\begin{gathered} \overrightarrow{{\psi }_u}=\frac{a{c}_s}{4\pi \rho }\frac{1}{\sqrt{2\pi v_{s}t}}(0,\frac{\partial }{\partial x_3}\left(\frac{1}{|\overrightarrow{r}|}\right), \\ -\frac{\partial }{\partial x_2}\left(\frac{1}{|\overrightarrow{r}|}\right)){\int }_0^{\frac{r}{c_s}}\tau e^{-\frac{{(t-\tau )}^2{c}_s^2}{2v_{s}t}}d\tau . \end{gathered}$$ (3)

III. Corrections to Viscoelastic Green’s Function

Equations (22), (23), (25), and (26) in [1] are the final equations for the viscoelastic Green’s function. However, these equations have sign and/or typographical errors. The corrected form of these equations is presented in the following.

Equation (22) in [1] writes the Green’s function tensor $g_{ij}(\overrightarrow{r},t)$ as a sum of $g_{ij}^p(\overrightarrow{r},t)$, $g_{ij}^s(\overrightarrow{r},t)$ and $g_{ij}^{ps}(\overrightarrow{r},t)$ which represent the P-wave, S-wave, and near-field terms, respectively. However, a_i, the amplitude of the forcing function, should not be part of this Green’s function expression. The correct form of Equation (22) in [1] is

$$g_{ij}(\overrightarrow{r},t)=g_{ij}^p(\overrightarrow{r},t)+g_{ij}^s(\overrightarrow{r},t)+g_{ij}^{ps}(\overrightarrow{r},t).$$ (4)

The P-wave and S-wave terms from Equation (23) in [1] have typographical errors in several variable subscripts. The near-field term from Equation (23) in [1] contains a sign error. Equation (5) provides the corrected Green’s function terms.

$$\begin{gathered} g_{ij}^p(\overrightarrow{r},t)=\frac{1}{4\pi \rho c_p}\frac{1}{\sqrt{2\pi v_{p}t}}{\gamma }_i{\gamma }_j\frac{1}{r}{e}^{-\frac{{\left(t-\frac{r}{c_p}\right)}^2{c}_p^2}{2v_{p}t}} \\ {g}_{ij}^s(\overrightarrow{r},t)=\frac{1}{4\pi \rho c_s}\frac{1}{\sqrt{2\pi v_{s}t}}\frac{{\delta }_{ij}-{\gamma }_i{\gamma }_j}{r}{e}^{-\frac{{\left(t-\frac{r}{c_s}\right)}^2{c}_s^2}{2v_{s}t}} \\ {g}_{ij}^{ps}(\overrightarrow{r},t)=\frac{1}{4\pi \rho }\left({\delta }_{ij}-3{\gamma }_i{\gamma }_j\right)\frac{1}{r^3}\left[\frac{c_p}{\sqrt{2\pi v_{p}t}}{{\int }_0^{\frac{r}{c_p}}\tau e}^{-\frac{{(t-\tau )}^2{c}_p^2}{2v_{p}t}}d\tau -\frac{c_s}{\sqrt{2\pi v_{s}t}}{{\int }_0^{\frac{r}{c_s}}\tau e}^{-\frac{{(t-\tau )}^2{c}_s^2}{2v_{s}t}}d\tau \right]. \end{gathered}$$ (5)

Equation (25) in [1] has a sign error. The correct form is

$$g_{ij}^{ps}(\overrightarrow{r},t)=\frac{1}{4\pi \rho }\left({\delta }_{ij}-3{\gamma }_i{\gamma }_j\right)\frac{1}{r^3}\left[I_p(r,t)-I_s(r,t)\right].$$ (6)

To derive Equation (6) we followed the method outlined in [1]. Others may find it instructive to compare Equation (6) with the first term of Equation (4.23) in [2].

Equation (26) in [1] has variable subscript errors. The corrected form is shown in Equation (7).

$$\begin{gathered} I_p(r,t)=\frac{\sqrt{v_{p}t}}{c_p\sqrt{2\pi }}\left[e^{-\frac{t^2{c}_p^2}{2v_{p}t}}-e^{-\frac{{\left(t-\frac{r}{c_p}\right)}^2{c}_p^2}{2v_{p}t}}\right]+\frac{t}{2}\left[\text{Erf}\left(\frac{c_{p}t}{\sqrt{2v_{p}t}}\right)-\text{Erf}\left(\frac{c_p\left(t-\frac{r}{c_p}\right)}{\sqrt{2v_{p}t}}\right)\right] \\ {I}_s(r,t)=\frac{\sqrt{v_{s}t}}{c_s\sqrt{2\pi }}\left[e^{-\frac{t^2{c}_s^2}{2v_{s}t}}-e^{-\frac{{\left(t-\frac{r}{c_s}\right)}^2{c}_s^2}{2v_{s}t}}\right]+\frac{t}{2}\left[\text{Erf}\left(\frac{c_{s}t}{\sqrt{2v_{s}t}}\right)-\text{Erf}\left(\frac{c_s\left(t-\frac{r}{c_s}\right)}{\sqrt{2v_{s}t}}\right)\right]. \end{gathered}$$ (7)

Reference [3] provides a simplified form of the viscoelastic Green’s function terms. The bulk viscosity is assumed to be negligible and the shear viscosity is the only viscous term accounted for. Equation (3) in [3] is the Green’s function for this simplified model and contains typographical errors in the variable subscripts, an extra c_p in the denominator, a $\sqrt{c_s}$ that should be c_s, multiplication instead of subtraction in one of the near-field term exponentials, and missing negative signs in several of the exponentials. Equation (8) presents the corrected Green’s function terms for this less general viscoelastic model.

$$\begin{gathered} g_{ij}^p(\overrightarrow{r},t)=\frac{1}{4\pi \rho c_p}{\gamma }_i{\gamma }_j\frac{1}{r}\delta \left(t-\frac{r}{c_p}\right) \\ {g}_{ij}^s(\overrightarrow{r},t)=\frac{1}{4\pi \rho c_s}\frac{1}{\sqrt{2\pi v_{s}t}}\frac{{\delta }_{ij}-{\gamma }_i{\gamma }_j}{r}{e}^{-\frac{{\left(t-\frac{r}{c_s}\right)}^2{c}_s^2}{2v_{s}t}} \\ {g}_{ij}^{ps}(\overrightarrow{r},t)=\frac{1}{4\pi \rho }\left(3{\gamma }_i{\gamma }_j-{\delta }_{ij}\right)\frac{1}{r^3}\{\frac{\sqrt{v_{s}t}}{c_s\sqrt{2\pi }}\left[e^{-\frac{t^2{c}_s^2}{2v_{s}t}}-e^{-\frac{{\left(t-\frac{r}{c_s}\right)}^2{c}_s^2}{2v_{s}t}}\right] \\ +\frac{t}{2}\left[\text{Erf}\left(\frac{c_{s}t}{\sqrt{2v_{s}t}}\right)-\text{Erf}\left(\frac{c_s\left(t-\frac{r}{c_s}\right)}{\sqrt{2v_{s}t}}\right)\right]-tH(t)H\left(\frac{r}{c_p}-t\right)\}. \end{gathered}$$ (8)

IV. Alternate Notation

Equation (27) in [1] highlights the utility of the viscoelastic Green’s function approach. The displacement field $\overrightarrow{u}(\overrightarrow{r},t)$ for an arbitrary forcing distribution can be numerically computed by convolving (in space and time) the Green’s function with the force density distribution. However, the notation used by Equation (27) in [1] doesn’t explicitly define how the displacement field is calculated from the various components of the $g_{ij}(\overrightarrow{r},t)$ tensor. Here we provide an alternate form of Equation (27) in [1] to clarify how the displacements along the three Cartesian axes are calculated from the nine components of the Green’s function tensor $g_{ij}(\overrightarrow{r},t)$. The modified form of Equation (27) is

$$u_i(\overrightarrow{r},t)=\sum _{j=1}^3\left({\int }_{\tau }d\tau {\iiint }_V{f}_j(\overrightarrow{\xi },\tau )g_{ij}(\overrightarrow{r}-\overrightarrow{\xi },t-\tau )d\overrightarrow{\xi }\right)$$ (9)

where $u_i(\overrightarrow{r},t)$ is the displacement component along the ${\widehat{x}}_i$ axis and $f_j(\overrightarrow{r},t)$ is the component of the force density along the ${\widehat{x}}_j$ axis.

V. Discussion

Many typographical errors in References [1] and [3] are due to subscript and exponent sign errors that can be readily identified by an engaged reader. However, the near-field term sign error, Equation (23) in [1], is more subtle and can result in erroneous simulated displacements if not identified and corrected. Figure 1 compares the incorrect and corrected $g_{zz}(\overrightarrow{r},t)$ Green’s function term. The value of $g_{zz}(\overrightarrow{r},t)$ was plotted at 4 different positions, all having the same source-observer distance. Figure 1 demonstrates that the effect of the sign error in the near-field term can be significant and depends on the angle of the observer relative to the source. (See next page for Figure 1.)

Fig. 1.

The incorrect and correted g_zz Green’s function is plotted as a function of time at four different spatial positions in (a)–(d). For all plots, the source-observer distance is 1 cm and the angle speficied by θ denotes the polar angle from the direction of propagation of the observer position in spherical coordinates. For example, the θ = 90° plot in (a) corresponds to g_zz measured 1 cm from the origin in the z = 0 plane. For all plots, the following medium parameters were used: c_s = 1.5 m/s, c_p = 1500 m/s, ρ = 1000 kg/m³, η_p = 0, and η_s = 2 Pa · s, where c_s is the shear velocity, c_p is the bulk velocity, ρ is the density, η_p is the bulk viscosity, and η_s is the shear viscosity. Conceptually, g_zz can be interpreted as the displacement along the direction z generated by a impulse forcing function along z applied at the origin at time t = 0.

VI. Conclusion

The novel Green’s function approach presented in [1] and [3] is extremely useful for modeling the 3D displacement field in a viscoelastic medium. The method has been widely cited and is particularly useful for modeling the displacement field generated by the acoustic radiation force from an ultrasound transducer. In this note, we provide a set of corrected equations for this important viscoelastic Green’s function method.