The Gaussian Shear Wave in a Dispersive Medium

Abstract

Within the field of “imaging the biomechanical properties of tissues,” a number of approaches analyze shear wave propagation initiated by a short radiation force push. Unfortunately, it is experimentally observed that the displacement vs. time curves in lossy tissues are rapidly damped and distorted in ways that confound any simple tracking approach. This paper addresses the propagation, decay, and distortion of pulses in lossy and dispersive media, in order to derive closed form analytic expressions for the propagating pulses. The theory identifies key terms that drive the distortion and broadening of the pulse. Furthermore, the approach taken is not dependent on any particular viscoelastic model of tissue, but instead takes a general first order approach to dispersion. Examples with a Gaussian beam pattern and realistic dispersion parameters are given along with general guidelines for identifying the features of the distorting wave that are the most compact.

Introduction

There has been extensive development of techniques to estimate and image the elastic properties of tissues (Parker et al. 2011; Doyley 2012). These provide useful biomechanical and clinically relevant information not available from conventional radiology. A subset of techniques utilizes short duration pushing pulses of acoustic radiation force as an initial condition, which then results in a propagating shear wave. By tracking the propagating wave, the shear velocity can be estimated, and this yields the Young's modulus or stiffness of the material (Sarvazyan et al. 1998). An impressive set of approaches employing radiation force, with important clinical applications, has been developed (Fatemi and Greenleaf 1998; Nightingale et al. 1999; Nightingale et al. 2001; Konofagou and Hynynen 2003; McAleavey and Menon 2007; Parker et al. 2011; Hah et al. 2012; Hazard et al. 2012).

Unfortunately, in many lossy tissues a propagating shear wave produced by a focused ultrasound beam radiation force will rapidly devolve. Specifically, within a few millimeters the observed displacements will be significantly attenuated. Furthermore, the displacement wave has an extended “tail” and the shape becomes distorted. These effects complicate attempts to track the key features of the propagating pulse in order to estimate shear velocity. As an example, see Figure 1 where an approximately Gaussian axis-symmetric beam at 5 MHz is used to produce a short radiation force push in a gelatin phantom. The displacement vs. radial position at the focal depth is shown for regular intervals of time after the radiation force push. Even in this case where gelatin is a relatively elastic (very weakly attenuating and dispersive) medium, the loss of amplitude from cylindrical spreading is pronounced. The long tail of the displacement and softening of the leading edge are evident, even in this “best case” scenario of a low-loss medium. In more lossy tissues, these factors are more pronounced. Analytical and numerical models have been proposed to model the evolution and decay of pulses in viscoelastic media (Sarvazyan et al. 1998; Nightingale et al. 1999; Bercoff et al. 2004a; Fahey et al. 2005). However, there is still the need for a closed form analytical solution that clearly identifies the key terms responsible for the distortion and decay of the pulse. Furthermore, there are different models for wave propagation in lossy media (Szabo 1994; Chen and Holm 2003; Bercoff et al. 2004b; Chen et al. 2004; Chen and Holm 2004; Giannoula and Cobbold 2008; Giannoula and Cobbold 2009; Urban et al. 2009). Since the precise model and mechanism of loss for shear waves in tissue is still uncertain, it is useful to have analytical expressions that are independent of any particular model, but still valid over the operating range of shear wave frequencies.

Figure 1.

Experimental data demonstrating the displacements (vertical axis) in a gelatin phantom at different radial positions (horizontal axis) at 1, 2, 3…msec time intervals, following a short radiation force pulse at 5 MHz. The displacements were calculated from tracking scans taken after the radiation force push. An approximately Gaussian, axially symmetric beam pattern was produced at the focal depth with a standard deviation of approx. 1 mm. In a low-loss, low dispersion media such as gelatin, the amplitude loss is largely due to cylindrical spreading. Still, the long tail of the displacement curves and softening of the leading edge can be seen. Decay and distortion of the propagating wave are more pronounced in lossy tissues.

The Gaussian shape is of particular importance due to a number of fortunate properties. First, the Gaussian function is an eigenfunction of many linear operators including the Fourier transform. Related to this, a focused ultrasound transducer with Gaussian apodization will produce a focal region that has a Gaussian shape in the transverse direction. If this intensity distribution is used to create a radiation force excitation in an absorbing medium, the initial body forces will have a corresponding Gaussian distribution in the axial direction. If a shear wave with Gaussian characteristics can be launched by these means, this wave could be characterized by three parameters, the amplitude, the position of the centroid, and the spread. However, dispersion or frequency dependence of wave speed and attenuation will act on the Gaussian pulse to distort its shape, complicating any simple relations and estimators derived in the case of a purely elastic (lossless) medium. This paper first treats the idealized one-dimensional case of a Gaussian shear wave generated by radiation force, then moves to the more realistic but more intractable case of an axis symmetric beam in cylindrical coordinates. We make use of the unifying mathematical framework and fundamental theorems derived recently by Prof. Baddour (2011). In some cases, approximate solutions are derived where exact analytical solutions are not found. In all cases, the goal is to elucidate the key factors that re-shape a propagating Gaussian pulse, and reduce these to closed-form solutions that are valid in a dispersive media, even when the precise viscoelastic model is uncertain.

Theory

The 1d Solution

We begin with a one dimensional solution based on an experimental configuration shown in Figure 2. An ultrasound beam has a Gaussian intensity profile given by

Figure 2.

Schematic of a Gaussian beam, high f number, applied in an absorbing medium to produce a radiation force push.

$$I(x)=(1/\sigma ){\mathbf{e}}^{-\left(\raisebox{1ex}{$x^2$}\!\left/ \!\raisebox{-1ex}{$2{\sigma }^2$}\right.\right)}$$ (1)

in the x -direction, and extends nearly uniformly in the axial (z) direction. This approximates a high f-number beam, propagating in a weakly attenuating medium. The beam is extended in the y dimension. The initial conditions are that velocity and displacement are everywhere zero, in an infinite, homogeneous medium. We will assume that, given the direction and extent of the radiation force, that the resulting displacements are polarized in the z -dimension and the derivatives with respect to y and z are zero. Thus, these conditions lead to an approximation of a one dimensional plane wave solution following a temporal impulse of radiation force. Using notation from Graff (1975) the governing equation in an elastic medium can be written as:

$${\nabla }^2{u}_z(x,t)-\frac{1}{c^2}\frac{{\partial }^2{u}_z(x,t)}{\partial t^2}=-F_z(x)T(t)$$ (2)

where : u_z = displacement of shear wave in z direction, F_z = the applied body force which is proportional to radiation force $\frac{2{\alpha }_{L}I(x)}{c_L}$, T (t) = temporal application, which we will take as an impulse, δ (t), and α_L and c_L are the longitudinal wave (ultrasound) attenuation and wave speed in the medium, respectively. The gradient reduces in this problem to a second derivative in x.

The version of the Fourier transform that we use is defined as follows. Under the suitability of integration of the function f (t), the temporal Fourier transform is defined as $F(\omega )=\underset{-\infty }{\overset{\infty }{\int }}f(t){\mathbf{e}}^{-i\omega t}\mathit{\text{dt}}$. The temporal inverse Fourier transform is given by $f(t)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F(\omega ){\mathbf{e}}^{+i\omega t}d\omega$. The Fourier transform can also be applied to the spatial variables. In 1D, the forward and inverse spatial Fourier transform are defined as for the temporal transform with the spatial variable x replacing the temporal variable t in the definition, and the spatial frequency ρ replacing the temporal frequency ω. It is noted that various other conventions are possible regarding the location of the positive and negative signs of the complex exponential kernel, as well as for the placement of the factors of 2π in the forward or inverse Fourier transform. This particular convention for the Fourier transform is the non-unitary version that uses angular frequency (as opposed to ordinary frequency).

Taking the spatial Fourier transform of eqn (2) followed by a temporal Fourier transform leads to

$$-{\rho }^2{\widehat{U}}_z(\rho ,\omega )+k^2{\widehat{U}}_z(\rho ,\omega )=-{\widehat{F}}_z(\rho )$$ (3)

where, on the right half side, the temporal Fourier transform of the impulse function yields unity, the overhat indicates a temporal Fourier transform, k is the wave number, $=\frac{\omega }{c}$ in a lossless medium, ρ is the spatial frequency corresponding to the spatial variable x, and −F̂_z (ρ) indicates the spatial Fourier transform of F_z (x). Rearranging yields the sinusoidal Green's function:

$${\widehat{u}}_z(x,\omega )={\Im }_{\rho }^{-1}\left\{\frac{{\widehat{F}}_z(\rho )}{{\rho }^2-k^2}\right\}$$ (4)

where ${\Im }_{\rho }^{-1}\{\cdot \}$ indicates the inverse Fourier transform of the spatial frequency variable, ρ. This inverse transform can be difficult to obtain. However, recently Prof. Baddour has demonstrated a set of theorems, applicable to 1, 2, and 3D wave sources, that provides a solution for this type of transform. Specifically, Prof. Baddour's Theorem 5 has shown that the following results are true (Baddour 2011):

$${\Im }_{\rho }^{-1}\left\{\frac{{\widehat{F}}_z(\rho )}{{\rho }^2-k^2}\right\}=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\frac{{\widehat{F}}_z(\rho )}{{\rho }^2-k^2}{\mathbf{e}}^{i\rho x}d\rho =\frac{-i}{2k}{\widehat{F}}_z(-k){\mathbf{e}}^{-\mathit{\text{ikx}}}\text{for}x>0$$ (5)

This new theorem effectively takes the spatial source transform F̂_z (ρ) and converts it to a temporal spectrum ${\widehat{F}}_z\left(-\frac{\omega }{c}\right)$; a spatial-temporal transformation involving the speed of the shear wave, c. Theorem 5 is applicable to analytic functions F̂_z (ρ) that remain bounded, meaning they have no poles. If F̂_z (ρ) has poles then a simple application of a partial fractions expansion is required prior to the application of Theorem 5 in order to separate the poles of F̂_z (ρ) from the (ρ² − k²)⁻¹ term.

Now, let us examine the velocity v_z (x, ω) :

$${\widehat{v}}_z(x,\omega )=i\omega \widehat{u}(x,\omega )=+\frac{c}{2}{\widehat{F}}_z(-k){\mathbf{e}}^{-\mathit{\text{ikx}}}$$ (6)

So if F_z (x) is a Gaussian, then its spatial Fourier transform, F̂_z (ρ) is also a Gaussian, and so is F̂_z (−k) by Prof. Baddour's theorem. Thus v̂_z (x, ω) is a Gaussian with phase shift proportional to $-\mathit{\text{kx}}=-\frac{x}{c}\omega$, which in turn corresponds to a shift in time from t to $t-\frac{x}{c}$. Note that, in accordance with D'Alembert's solution for an initial force, it is v and not u that propagates the form of F_z (x) (Graff 1975; Blackstock 2000).

The Fourier transform of $F_z(x)=\frac{1}{\sigma }{\mathbf{e}}^{\raisebox{1ex}{$-x^2$}\!\left/ \!\raisebox{-1ex}{$2{\sigma }^2$}\right.}$ is given by $F_z(\rho )={\sqrt{2\pi \mathbf{e}}}^{\raisebox{1ex}{$-{\sigma }^2{\rho }^2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. Therefore, taking the inverse temporal transform of eqn (6) assuming F̂_z is a Gaussian of magnitude P yields:

$$v_z(x,t)=+\frac{P{c}^2}{2\sigma }{\mathbf{e}}^{\raisebox{1ex}{$-{(x-\mathit{\text{ct}})}^2$}\!\left/ \!\raisebox{-1ex}{$2{\sigma }^2$}\right.}$$ (7)

which agrees with the D'Alembert's solution for the right traveling wave. The displacement wave u_z (x, t), being the time integration of velocity, is an Erf $[(x-\mathit{\text{ct}})/\sqrt{2\sigma }]$ function which never returns to zero in theory, consistent with the discussions in Blackstock (2000) and Graff (1975). However, in practice, displacements will return to zero by a combination of reflections from boundaries and/or restorative forces. These are low frequency or long time phenomena that are difficult to know a priori. Background vibration and baseline drift add to the uncertainties. For these reasons, the tracking of displacement is seen to be inferior to the tracking of velocity.

In fact, we can consider that in a very general sense the inverse Fourier transform of eqn (6), where the inverse transform takes the temporal frequency variable ω and transforms to time, t. We recall that $k=\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.$ and make use of the Fourier shift and scaling theorems in taking the inverse Fourier transform of eqn (6) to obtain:

$$v_z(x,t)=+\frac{c^2}{2}{F}_z\left(-c\left(t-\frac{x}{2}\right)\right)=\frac{c^2}{2}{F}_z(x-\mathit{\text{ct}})\text{for}x>0$$ (8)

The previous equation states very simply and concisely that the original shape of the forcing function is propagated outwards (towards increasing x, to the right in this case) and is unchanged by the velocity wave. As previously stated, the displacement wave u (x, t), is a time-integration of velocity and thus the displacement wave would not have the same shape as the original forcing function. Furthermore, the temporal Fourier transform of v_z (x, t), or its bandwidth, is related by eqn (5) - (8) to the beam pattern shape that produces F_z (x). That is, narrow beam patterns produce sharper v_z (x, t), comprised of higher bandwidths. Experimental work with focused ultrasound typically produces bandwidths under 1000 Hz, peaking below 500 Hz (McAleavey and Menon 2007; Hah et al. 2012).

Now consider the more relevant case of a medium with attenuation and dispersion. As shown in Blackstock Chapter 9 (2000), a variety of loss mechanisms exist in materials, each with their own particular frequency dependent solution. However even if the precise mechanism is unknown, the harmonic plane wave solution can be generalized by a complex wave number such that the general solution is,

$$u_z(x,t)=u_0{\mathbf{e}}^{-\mathit{\text{ikx}}}{\mathbf{e}}^{i\omega t}$$ (9)

where the wave number k is defined such that its imaginary part is negative. Whatever the underlying mechanisms, these parameters will be frequency dependent and the particular form of frequency dependence will depend strictly on the underlying mechanism expressed in the constitutive equations. Of course it must be kept in mind that the dispersion of velocity, and attenuation, are linked by Kramers-Kronig relations (Nachman et al. 1990; Szabo 1994; Blackstock 2000).

In this treatment, we are agnostic as to the precise mechanism and, therefore, the extended frequency dependence of the attenuation and phase velocity. Furthermore, we assume that only a limited shear wave frequency bandwidth is available so that any particular frequency dependence can be expressed using a Taylor series expansion over the restricted frequency range.

We therefore begin with a complex wave number, setting $k=\frac{\omega }{c}-i\alpha$ where α is the absorption coefficient. The corresponding statement to eqn (5) for complex wave numbers comes from the application of Theorem 6 (complex wave number with negative imaginary part) from (Baddour 2011), which states:

$$\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\frac{{\widehat{F}}_z(\rho )}{{\rho }^2-k^2}{\mathbf{e}}^{i\rho x}d\rho =\frac{-i}{2k}{\widehat{F}}_z(-k){\mathbf{e}}^{-\mathit{\text{ikx}}}\text{for}x>0$$ (10)

With our choice of negative real part for the complex k, Prof. Baddour's Theorem 6 (Baddour 2011), ensures that eqn (4) and (5) are still valid for complex k. Hence, for F_z a Gaussian of amplitude P = 1, eqn (4) and (5), and v̂_z (x, ω) = iωû (x, ω) yield:

$$v_z(x,\omega )=\frac{\omega }{2k}{\mathbf{e}}^{-\frac{1}{2}{\sigma }^2{k}^2}{\mathbf{e}}^{-\mathit{\text{ikx}}}=\frac{\omega c}{2(\omega -i\alpha c)}{\mathbf{e}}^{-\frac{1}{2}{\sigma }^2\left({\left(\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)}^2+{\alpha }^2\right)}{\mathbf{e}}^{-i\left(\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)x}{\mathbf{e}}^{-\alpha x}$$ (11)

where we have taken k ² as the product of its conjugate roots.

Now assuming weak attenuation, $\alpha \ll \frac{\omega }{c}$ over most of the bandwidth, we neglect the imaginary term in the denominator and then to first order we have

$$v_z(x,\omega )\cong \frac{c}{2}{\mathbf{e}}^{-\frac{1}{2}{\sigma }^2\left[{\left(\frac{\omega }{c}\right)}^2+{\alpha }^2\right]}{\mathbf{e}}^{-i\left(\frac{\omega }{c}\right)x}{\mathbf{e}}^{-\alpha x}$$ (12)

Using the fact that the Fourier transform of $F_z(t)=\frac{1}{\sigma \sqrt{2\pi }}{\mathbf{e}}^{-\frac{{\mathbf{t}}^2}{2{\mathbf{\sigma }}^2}}$ is given by ${\widehat{F}}_z(\omega )={\mathbf{e}}^{-\frac{{\sigma }^2{\omega }^2}{2}}$ and taking the temporal inverse Fourier transform yields

$$v(x,t)=\frac{c^2}{2\sigma \sqrt{2\pi }}{\mathbf{e}}^{\frac{-{(x-\mathit{\text{ct}})}^2}{2{\sigma }^2}}{\mathbf{e}}^{\frac{-{\sigma }^2{\alpha }^2}{2}}{\mathbf{e}}^{-\alpha x}$$ (13)

an attenuated D'Alembert's solution valid for a lossy media with relatively constant attenuation over the frequency range of the pulse.

We next introduce first order dispersion terms as a Taylor series approximation over a limited bandwidth, so that c ≡ c₀ + c₁ ·|ω|; α ≡ α₀ + α₁ ·|ω| where c₀ ≫ c₁ω and α₀ ≫ α₁ω. However, for lowpass functions like the Gaussian, the behavior of c and k near zero frequency is particularly important. Under most conventional loss mechanisms (Blackstock 2000), as ω → 0, c → c₀, and α → 0. Thus α₀ = 0 for lowpass functions in conventional lossy media. Substituting these into eqn (12) and again assuming weak attenuation, $\alpha \ll \frac{\omega }{c}$, then retaining only the most significant terms, we find:

$$v_z(x,\omega )\cong \frac{c_0}{2}{\mathbf{e}}^{-\frac{1}{2}{\sigma }^2{\left(\frac{\omega }{c_0}\right)}^2}{\mathbf{e}}^{-i\left(\frac{\omega }{c_0+c_1\left|\omega \right|}\right)x}{\mathbf{e}}^{-({\alpha }_1|\omega |)x}\cong \frac{c_0}{2}{\mathbf{e}}^{-\frac{1}{2}{\sigma }^2{\left(\frac{\omega }{c_0}\right)}^2}{\mathbf{e}}^{-\mathit{\text{ix}}\left[\frac{\omega }{c_0}-\frac{c_1{\omega }^2\text{sign}[\omega ]}{c_0^2}\right]}{\mathbf{e}}^{-{\alpha }_1\left|\omega \right|x}$$ (14)

where a first order series expansion $\frac{1}{1+\frac{c_1}{c_0}\left|\omega \right|}\cong 1-\frac{c_1}{c_0}\left|\omega \right|$ is used in the phase term.

Taking the inverse Fourier transform with respect to temporal frequency and rearranging some terms we obtain

$$v_z(x,t)=\frac{c_0^2}{2\sigma \sqrt{2\pi }}\left[{\mathbf{e}}^{\frac{-{(c_{0}t-x)}^2}{2{\sigma }^2}}\right]\ast \left(\frac{1}{\pi }\frac{{\alpha }_{1}x}{t^2+{({\alpha }_{1}x)}^2}\right)\ast \text{Quad}(t)$$ (15)

where the asterisks represent convolution, and the term $\left(\frac{1}{\pi }\frac{{\alpha }_{1}x}{t^2+{({\alpha }_{1}x)}^2}\right)$ is the inverse Fourier transform (IFT) of e^{−α₁|ω|x}, essentially a lowpass filter. Quad(t) is the IFT of the quadratic phase term ${\mathbf{e}}^{+\mathit{\text{ix}}{c}_1\frac{{\omega }^2\text{sign}[\omega ]}{c_0^2}}$. The quadratic phase all-pass filter has significant distortion effects. Its time domain impulse response, Quad(t), is a type of right-sided chirp. When convolved with a Gaussian, a distortion and warping is produced, with the degree of distortion depending on the magnitude of the quadratic phase term $\left[x\cdot c_1\cdot {\left(\frac{\omega }{c_0}\right)}^2\right]$ over the Gaussian bandwidth σ. Its exact form from Mathematica (Wolfram Research, Champaign, IL, USA) for $x\frac{c_1}{c_0^2}=1$ is:

$$\text{Quad}(t)=\frac{1}{2}\left[cos\left(\frac{t^2}{4}\right)\left(1+2\mathit{C}\left(\frac{t}{\sqrt{2\pi }}\right)\right)+sin\left(\frac{t^2}{4}\right)\left(1+2\mathit{S}\left(\frac{t}{2\pi }\right)\right)\right]$$ (16)

where C (·) and S (·) are the Fresnel cosine and sine integrals, respectively. The result of these convolutions is a waveform that is diminished in amplitude and broadened, as it propagates.

The cylindrical coordinate solution: Green's function in an elastic medium

In most cases the beam in Figure 2 will not be extended in the y -axis, and in the case of an axisymmetric beam the solution can be found as a function of cylindrical radius, $r=\sqrt{x^2+y^2}$. The Laplacian operator in eqn (2) for cylindrical coordinates leads naturally to the Hankel transform (Graff 1975). Unfortunately, the singularity and peculiarities of Bessel functions in Hankel transforms make it very difficult to obtain closed form solutions involving Gaussian functions. Nonetheless, approximate solutions can be derived and compared with the 1D solution. The problem begins with a quiescent infinite homogeneous medium and an extended line source of force, along the z -axis at the origin. The resulting solution will have only a non-zero polarized shear wave with displacement and velocity in the z -axis, and non-zero derivatives in the radial dimension.

Applying the Hankel transform ℋ in space and the Fourier transform in time to eqn (2) in cylindrical coordinates yields:

$$\widehat{U}(\epsilon ,\omega )=\frac{\widehat{F}(\epsilon )}{{\epsilon }^2-\frac{{\omega }^2}{c^2}}$$ (17)

where F̂ (ε) = ℋ [F (r)], ε is the spatial frequency, and T̂ (ω) = ℑ[T (t)] = 1 assuming an impulse, and

$$\widehat{u}(r,\omega )=\underset{0}{\overset{\infty }{\int }}\frac{\widehat{F}(\epsilon )}{{\epsilon }^2-\frac{{\omega }^2}{c^2}}{J}_0(\epsilon r)\epsilon d\epsilon$$ (18)

Now applying Prof. Baddour's Theorem 3 (Baddour 2011), the following results hold true:

$$\underset{0}{\overset{\infty }{\int }}\frac{\widehat{F}(\epsilon )}{{\epsilon }^2-k^2}{J}_n(\epsilon r)\epsilon d\epsilon =\{\begin{array}{c}-\frac{\pi }{2}{Y}_n(\mathit{\text{kr}})\widehat{F}(k)\\ \pi i\frac{1}{2}{H}_n^{(1)}(\mathit{\text{kr}})\widehat{F}(k)\\ -\pi i\frac{1}{2}{H}_n^{(2)}(\mathit{\text{kr}})\widehat{F}(k)\end{array}$$ (19)

Where $k=\frac{\omega }{c}$ is a real wave-number, Y_n (x) is a Bessel function of the second kind, and $H_n^{(1,2)}(x)$ are Hankel functions of the first and second kind, respectively. In applying this theorem, the Sommerfield radiation condition must be applied to select the appropriate solution that yields out-wards propagating waves, the correct application of which depends on the chosen time-dependence of the solution, which in turn is reflected in the chosen definition of the Fourier transform. In our case, we have stated the chosen definition of the Fourier transform above and this corresponds to an implied time-dependence of e^iωt, which in turn implies that the correct application of the theorem yields

$$\widehat{u}(r,\omega )=\frac{-\pi i}{2}{H}_0^{(2)}\left(\frac{\omega }{c}\cdot r\right)\widehat{F}\left(\frac{\omega }{c}\right)$$ (20)

where

$$u(r,t)={\Im }_{\omega }^{-1}\{\widehat{u}(r,\omega )\}$$ (21)

Now, for the specific solutions we first examine the Green's function for a force that is impulsive in space and time. Let F_z (r)T (t) = A_oδ (t) δ (r)/r, where A₀ is a constant of proportionality, then

$$\widehat{u}(r,\omega )=A_0\frac{-\pi i}{2}{H}_0^{(2)}\left(\frac{\omega }{c}\cdot r\right)$$ (22)

Note that the general requirement for a causal function f (t) is: f (t) = 0 for t < 0 (Papoulis 1987), in which case

$$f(t)=\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}R(\omega )cos(\omega t)d\omega =-\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}X(\omega )sin(\omega t)d\omega$$ (23)

where R (ω) ≡ real part of ℑ{f (t)} and X (ω) ≡ imaginary part of ℑ{f (t)}.

Now, for our case

$$\widehat{u}(r,\omega )=A_0\frac{-\pi }{2}i{H}_0^{(2)}\left(\frac{\omega }{c}\cdot r\right)=A_0\frac{-\pi }{2}\left[Y_0\left(\omega \cdot \frac{r}{c}\right)+i{J}_0\left(\omega \cdot \frac{r}{c}\right)\right]$$ (24)

So for a causal u (r, t) :

$$u(r,t)=A_0\frac{-\pi }{2}\left(\frac{-2}{\pi }\right)\underset{0}{\overset{\infty }{\int }}{J}_0\left(\omega \cdot \frac{r}{c}\right)sin(\omega t)d\omega$$ (25)

and from Gradshteyn and Ryzhik p. 73 (1965):

$$u(r,t)=A_0\frac{1}{\sqrt{t^2-{\left(\frac{r}{c}\right)}^2}}\text{for}0<\frac{r}{c}<t$$ (26)

and this agrees with Graffs solution (Graff 1975) by Laplace transform.

The corresponding solution for velocity, v, is given by $\frac{d}{\mathit{\text{dt}}}u(r,t)$:

$$v(r,t)=\frac{A_0\delta \left(t-\frac{r}{c}\right)}{{\left(t^2-\frac{r^2}{c^2}\right)}^{\frac{1}{2}}}-\frac{A_{0}t\cdot H\left(t-\frac{r}{c}\right)}{{\left(t^2-\frac{r^2}{c^2}\right)}^{\frac{3}{2}}}$$ (27)

where H (·) is the Heaviside step function.

Comparing u and v, we see features that were also evident in the 1D case. The Green's function for u is persistent in time with a $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.$ decay, which in experiments is prone to contamination by ambient vibration and baseline drift, and reflections from distant boundaries. The leading edge is a singularity which survives to infinite distances, a result which is experimentally unrealizable. The Green's function for v (r,t) is more promising: it contains a delta function which will be useful in convolving with and reproducing a beam pattern. Unlike the 1D case it has a negative “recoil” term (the second term of eqn (27)), which decays asymptotically as $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t^2$}\right.$, so is more localized spatially than u.

We next seek a solution to the Gaussian beam pattern. Let $F_z(r)T(t)=\left(\frac{1}{2{\sigma }^2}\right){\mathbf{e}}^{-r^2/4{\sigma }^2}\delta (t)$. Then, from eqn (20), and taking the Hankel Transform of F_z (r), we have

$$\widehat{u}(r,\omega )=-\pi i\frac{1}{2}{H}_0^{(2)}\left(\frac{\omega }{c}\cdot r\right)\left[{\mathbf{e}}^{-{\sigma }^2{\left(\frac{\omega }{c}\right)}^2}\right]$$ (28)

and the causality constraint, eqn (23) yields:

$$u(r,t)=\underset{0}{\overset{\infty }{\int }}{J}_0\left(\omega \cdot \frac{r}{c}\right){\mathbf{e}}^{-{\sigma }^2{\left(\frac{\omega }{c}\right)}^2}sin(\omega t)d\omega$$ (29)

or

$$u(r,t)=-\underset{0}{\overset{\infty }{\int }}{Y}_0\left(\omega \cdot \frac{r}{c}\right){\mathbf{e}}^{-{\sigma }^2{\left(\frac{\omega }{c}\right)}^2}cos(\omega t)d\omega$$ (30)

or, since v̂ (r, ω) = iωû (r, ω) then:

$$v(r,t)=\underset{0}{\overset{\infty }{\int }}\omega \cdot J_0\left(\omega \cdot \frac{r}{c}\right){\mathbf{e}}^{-{\sigma }^2{\left(\frac{\omega }{c}\right)}^2}cos(\omega t)d\omega$$ (31)

or

$$v(r,t)=\underset{0}{\overset{\infty }{\int }}\omega Y_0\left(\omega \cdot \frac{r}{c}\right){\mathbf{e}}^{-{\sigma }^2{\left(\frac{\omega }{c}\right)}^2}sin(\omega t)d\omega$$ (32)

No closed form solutions to these have been found. Alternatively, we could convolve the Green's functions for u or v (eqn (26) and (27)) with a Gaussian in the time domain, however no closed form solution has been found for this approach either. We therefore turn to an approximate solution. If as an approximation to the Gaussian we use a similar shape:

$$F_z(r)T(t)=\frac{A_0}{{(r^2+a^2)}^{\frac{1}{2}}}\delta (t)$$ (33)

where a is a beamwidth parameter, then

$$\mathcal{H}[F_z(r)]=\frac{A_0{\mathbf{e}}^{-a\epsilon }}{\epsilon }$$ (34)

Superficially, this F_z (r) resembles the “bell shape” of the Gaussian beam. An important difference is the relatively slow $\frac{1}{r}$ drop off, as depicted in Figure 3. This slowly decaying “tail” of the function leads to a singularity in ℋ[F_z (r)] at zero frequency, and to long “tails” in the vsolutions. Nonetheless, the approximation provides a useful vehicle for examining the effects of dispersion. Substituting eqn (34) into eqn (20) and invoking causality yields

Figure 3.

The comparison of a Gaussian function of σ = 2 units and a substitute function $\frac{1}{{(r^2+1)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$, both normalized to unit amplitude at r = 0. The substitute function and its Hankel transform lead to an analytical solution which is useful for illustrating the effects of dispersion. However, it has a slow asymptotic decay of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.$, and therefore much wider “tails” compared with the Gaussian.

$$u(r,t)=A_0\underset{0}{\overset{\infty }{\int }}{J}_0\left(\omega \cdot \frac{r}{c}\right)\frac{{\mathbf{e}}^{-a\left(\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)}}{\left(\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)}sin(\omega t)d\omega$$ (35)

which can be interpreted as the Fourier sine transform of the function $\frac{c}{\omega }{\mathbf{e}}^{-a\left(\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)}{J}_0\left(\omega \cdot \frac{r}{c}\right)$, Which is known (Erdélyi, Theorem 17) (Erdélyi and Bateman 1954). Thus,

$$u(r,t)=A_{0}c\cdot {sin}^{-1}\left(\frac{2t}{r_1+r_2}\right)\text{for}t\ge 0\text{and}r\ge o$$ (36)

where $r_1^2={\left(\frac{a}{c}\right)}^2+{\left(t+\frac{r}{c}\right)}^2$ and $r_2^2={\left(\frac{a}{c}\right)}^2+{\left(t-\frac{r}{c}\right)}^2$, and

$$v_{\mathit{\text{approx}}}(r,t)=\frac{d}{\mathit{\text{dt}}}u(r,t)$$ (37)

where the subscript designates an approximation, meaning the solution is for a substitute beam pattern, not an exact Gaussian. Due to the long “tails” of the substitute beam pattern, and the nature of the cylindrical Green's function, the velocity is extended in time and space.

The cylindrical coordinate solution: Cylindrical coordinates and a dispersive medium

To introduce dispersion, we substitute a complex k and seek a solution. Prof. Baddour's Theorem 4 for Bessel Functions holds for complex wave number (Baddour 2011). This theorem states

$$\underset{0}{\overset{\infty }{\int }}\frac{\widehat{F}(\epsilon )}{{\epsilon }^2-k^2}{J}_n(\epsilon r)\epsilon d\epsilon =\{\begin{array}{cc}\pi i\frac{1}{2}{H}_n^{(1)}(\mathit{\text{kr}})\widehat{F}(k)& Im(k)>0\\ -\pi i\frac{1}{2}{H}_n^{(2)}(\mathit{\text{kr}})\widehat{F}(k)& Im(k)<0\end{array}$$ (38)

Using this theorem along with eqn (34), and complex wave number:

$$u(r,\omega )=-A_0\frac{\pi i}{2}{H}_0^{(2)}\left(\frac{\omega }{c}r-i\alpha r\right)\frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c}-i\alpha \right)}}{\left(\frac{\omega }{c}-i\alpha \right)}$$ (39)

The solution should resemble the previous case except for the effects of the imaginary term in the argument. Assuming $\alpha \ll \raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$c$}\right.$, and taking only leading terms of $\frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c}-i\alpha \right)}}{\left(\frac{\omega }{c}-i\alpha \right)}$, leaves us with a real approximation for $\widehat{F}(k)=\frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c}\right)}}{\left(\frac{\omega }{c}\right)}$ where we also substituted Re[e^+iαa] = cos[aα] ≅ 1 by assuming aα ≪ 1. Thus

$$u(r,\omega )=-A_0\frac{\pi i}{2}{H}_0^{(2)}\left(\frac{\omega }{c}r-i\alpha r\right)\cdot \frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c}\right)}}{\left(\frac{\omega }{c}\right)}$$ (40)

Now, adding dispersion to eqn (40), c = c₀ + c₁ |ω|, α = α₁|ω| and retaining the most significant terms we have

$$u(r,\omega )=-A_0\frac{\pi i}{2}\left[H_0^{(2)}\left(\frac{\omega }{c_0}r-\left(\frac{{\omega }^2{c}_1\text{sign}(\omega )}{c_0^2}\right)r-i({\alpha }_1\left|\omega \right|)r\right)\right]\frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c_0}\right)}}{\left(\frac{\omega }{c_0}\right)}$$ (41)

Consider the asymptotic form for $H_0^{(2)}$ (Abramowitz and Stegun 1964) for complex z :

$$H_0^{(2)}(z)\cong \sqrt{\frac{2}{\pi z}}{\mathbf{e}}^{-i\left(z-\frac{\pi }{4}\right)}$$ (42)

Now if z = a − ib and 0 < b ≪ a, then eqn (42) suggests a useful approximation: $H_0^{(2)}(a-\mathit{\text{ib}})\cong {\mathbf{e}}^{-b}{H}_0^{(2)}(a)$, and it can be verified that this approximation works well over all real, positive a, as long as b ≪ a. Substituting this into eqn (41) yields

$$u(r,\omega )\cong -A_0\frac{\pi i}{2}{\mathbf{e}}^{-({\alpha }_1|\omega |)r}\cdot H_0^{(2)}\left(\left[\frac{\omega }{c_0}-{\left(\frac{\omega }{c_0}\right)}^2{c}_1\text{sign}(\omega )\right]r\right)\cdot \frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c_0}\right)}}{\left(\frac{\omega }{c_0}\right)}$$ (43)

Similarly, by substituting z = (ax − bx²) into eqn (42), we have found that $H_0^{(2)}(\mathit{\text{ax}}-b{x}^2)$ is well approximated by a chirp of the form

$$H_0^{(2)}(\mathit{\text{ax}}-b{x}^2)\cong H_0^{(2)}(\mathit{\text{ax}}){\left[\frac{1}{1-\frac{\mathit{\text{bx}}}{a}}\right]}^{\frac{1}{2}}{\mathbf{e}}^{+\mathit{\text{ib}}{x}^2}\text{for}x>0\text{and}a>b>0$$ (44)

Thus:

$$u(r,\omega )\cong -A_0\frac{\pi i}{2}{\mathbf{e}}^{-{\alpha }_1\left|\omega \right|r}\cdot H_0^{(2)}\left(\frac{\omega }{c_0}r\right)\cdot {\left[\frac{c_0}{c_0-c_1\left|\omega \right|}\right]}^{\frac{1}{2}}\cdot {\mathbf{e}}^{+i{\left(\frac{\omega }{c_0}\right)}^2\text{sign}(\omega )c_{1}r}\cdot \frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c_0}\right)}}{\left(\frac{\omega }{c_0}\right)}\text{for}r>0$$ (45)

Note that the kernel of this expression $H_0^{(2)}\left(\frac{\omega }{c}r\right)\cdot \frac{{\mathbf{e}}^{-a\left(\frac{\omega }{c_0}\right)}}{\left(\frac{\omega }{c_0}\right)}$ is that which was obtained for the lossless, elastic case (eqn (20) and (34)). The other terms resemble those observed in the 1D case (eqn (14)), an exponential decay with frequency and a quadratic phase distortion. There is an additional amplitude term associated with cylindrical dispersive spreading proportional to ${\left[\frac{c_0}{c_0-c_1\left|\omega \right|}\right]}^{\frac{1}{2}}$; however assuming ωc₁ ≪ c₀, this term is ≅ 1 over the bandwidth of the pulse. Thus, in the time domain we conclude that

$$v(r,t)\cong [v_{\text{approx}}(r,t)]\ast \frac{1}{\pi }\frac{{\alpha }_{1}r}{(t^2+{({\alpha }_{1}r)}^2)}\ast \text{Quad}(t)$$ (46)

for a dispersive medium where v_approx (r, t) is defined by eqn (36) and (37). Note that eqn (46) has the same structure as eqn (15) for the 1D case, and suggests the possibility of a common convolution term for modeling dispersion.

Results

The solutions are evaluated with parameters that approximate soft bio-materials (Zhang et al. 2007). For the lossless cases, c₀ = 2 m/sec and a 1 mm beamwidth Gaussian parameter isused. For lossy and dispersive media, we additionally include c₁ = 0.1 mm/sec/Hz (in other words, the speed rises linearly from 2.00 m/sec at low frequencies to 2.01 m/sec at 100Hz), and α₁=1x10⁻⁴ Np/mm/Hz (in other words the attenuation coefficient is 0.1 Np/cm at 100 Hz). These values are somewhat less dispersive than those found in liver tissue (Zhang et al. 2007; Barry et al. 2012; Hah et al. 2012), however these values of dispersion are sufficient to show marked changes in the propagating waveforms, as compared with the lossless reference cases. Figure 4a shows the time history of velocity at regularly spaced observation points out to 8 mm, for the 1D solution of the right-travelling shear wave. Since this is essentially D'Alembert's solution, there is no change in the characteristics as the shear wave propagates. Peak tracking would provide an accurate estimation of shear velocity in this case. Figure 4b shows the solution in the dispersive medium, at the same observation points, derived from numerical integration of the inverse Fourier transform of eqn (14). The loss of amplitude and spreading of the shear wave are evident, and this is progressive as the dispersive effects accumulate. The effects of attenuation, which acts as a low pass filter, are pronounced. For the 8 mm observation point (the right-most curve) the peak has shifted from 4 msec in the lossless case to approximately 3.7 msec, a drop of 10% due to relatively modest loss parameters. Figure 5 provides the results for a cylindrically symmetric beam. The time history is shown at identical observation points as in Figure 4, at regular spacings out to 8 mm. Solutions are obtained by numerical integration of the inverse Fourier transform of eqn (45). In the lossless case, Figure 5a, the rapid decrease of amplitude followed by the asymptotic approach to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{r}$}\right.$ cylindrical spreading is demonstrated. With dispersion, Figure 5b, the low-pass smoothing of the velocity waveforms is evident. These waveforms are more extended than in the 1D case because of the cylindrical Green's function and because the approximation to a Gaussian function, with slower convergence, was used in the closed form analytic solution to eqn (28) - (36). In this example, the 8 mm observation point (right-most curve) has a peak near 3.8 msec in the lossless case, dropping to 3.1 msec in the lossy case (Figure 5b). Neither case has the simple value of 4 msec expected from consideration of a plane wave in a lossless medium. The combined effects of dispersion and cylindrical spreading in eqn (46) are complicated; for example higher temporal frequency components are traveling faster but are also more highly attenuated. It should also be noted that the experimental task of finding a “peak” becomes quite difficult for dispersive examples like Figure 5b, even more so in the presence of noise.

Figure 4.

Theoretical values of velocity (arbitrary units) vs. time (sec.) profiles for the right-traveling shear wave of the one dimensional case, at observation points 0, 2, 4, 6, and 8 mm from the center of the Gaussian beam. In 4a is the simple case of the lossless medium, reducing to D'Alembert's solution. In 4b is the same initial beam and observation points but with modest loss and dispersion characteristic of some soft tissues.

Figure 5.

Theoretical values of velocity (arbitrary units) vs. time (sec.) profiles for the shear wave of the cylindrical case, at observation points 0, 2, 4, 6, and 8 mm radially from the center of an approximately Gaussian beam. Rapid initial loss of amplitude followed by cylindrical spreading is seen even in the lossless case, Figure 5a. With dispersion, Figure 5b, the smoothing of the waveforms and amplitude losses are pronounced.

Discussion and Conclusions

Closed form analytic expressions have been derived for the case of a Gaussian shear wave in a lossy, dispersive medium. The framework for solving inverse transforms employs the recent theorems published by Prof. Baddour. In all cases a general first order approximation to dispersion is employed, so the results are not tied to any particular model of tissue loss. In the resulting solutions, one key term that drives the distortion and diminution of the wave is a lowpass filter term resulting from the progressively higher attenuation of higher frequencies as a result of dispersion. Another key term is the quadratic phase term resulting from the progressively higher wave speed as a result of dispersion. Some specific results of interest emerge from the analysis:

• –By extending the beam in the out-of-plane dimension, an approximate 1D configuration is generated. This leads to the simplest solution, D'Alembert's solution where the shape of the beam pattern is carried by the velocity wave, not the displacement wave.
• –The displacement wave has a long “tail” that is not well confined in space and time, and is dependent on low frequency terms that are difficult to know a-priori, thus the velocity wave is better suited for tracking.
• –In cylindrical coordinates, a dramatic initial drop in the velocity wave is followed by the $\frac{1}{\sqrt{r}}$ cylindrical decay, plus additional terms due to dispersion.
• –The cylindrical Green's function contains a negative “recoil” term (eqn (27)) however after convolution with a beam pattern term, the solution may or may not contain a negative velocity region. This depends on the particular shape of the beampattern and its compactness.
• –In all cases the unwanted quadratic phase term can be eliminated from consideration by taking the power spectrum of the velocity pulse
• –Some approximations for Hankel functions of complex arguments and quadratic arguments are proposed and used within solutions.

These factors help to characterize the rapid diminuation and distortion of shear wave pulses in lossy tissue, and suggest approaches to more robust tracking.

Some key limitations of this study should be highlighted for further research. First, dispersion relations assume only a first-order approximation. Next, the approximate solution to the Gaussian shear wave in cylindrical coordinates, eqn (37), is from a “similar” beam pattern. It is useful in demonstrating the extra terms that must be included in a dispersive media, as opposed to a simple elastic media. The dispersive terms can be generalized to apply to other beam patterns. However, finding an exact closed form analytical solution to eqn (29), (30), (31), or (32) with exact Gaussians is still desirable. Another important limitation of this study is the assumed uniformity of the beam and radiation force in (y, z) or (z) axes for the one dimensional and cylindrical cases, respectively. In reality, the beam will have finite extent in the y -direction and a diffraction pattern in the z -axis with progressive attenuation loss. These spatial factors will add additional decay terms to the solutions, further diminishing the propagating shear wave as time increases.