In an absorbing meium, under plane wave or weakly focused conditions, the radiation force at a given depth,

*z* and ultrasound intensity,

*I*(

*x*,

*y*) are related by [

21]

where

*F* is the spatially and temporally modulated body force in newtons per cubic meter,

*α* is the attenuation coefficient of the medium in nepers per meter,

*c* is the speed of sound in the medium in meters per second, and

*I* is the intensity of the beam in watts per square meter.

Assuming that the attenuation coefficient and the speed of sound in the medium are constant, we can achieve the desired forcing function by manipulating ultrasound beams to create a spatially varying intensity pattern. In SMURF imaging, the radiation force is shaped by varying the applied ultrasound intensity as a function of the lateral coordinate. The spatial frequency (*k*) of the intensity pattern, and thus force variation, is set equal to the desired spatial frequency of the shear wave to be generated (*k* = 2*π*/*λ*).

The ideal spatially modulated push force would be well localized in space, to provide spatial resolution, and have a well-defined spatial frequency, so that the shear wave frequency may be readily measured. In this work, a Gaussian-enveloped sinusoidal variation in intensity has been selected as the desired model for the lateral intensity pattern. Spatial resolution and purity of spatial frequency may be traded off by varying the width of the Gaussian envelope. Both methods proposed here will generate lateral beam profiles conforming to this model.

Field II [

22], an acoustic field simulator tool, is used to simulate the beams as generated by a linear array similar to a Siemens VF 7–3 ultrasound transducer (Siemens AG, Munich, Germany). Field II is used in this work to confirm that for a desired ROI pattern, the calculated input parameters used in the transducer apodization and delay profiles result in the expected pattern. It is also used in comparing the relative ultrasound intensity generated at the ROI by both methods for various combinations of the ROI focal depth and width. The simulator is linear, and scaling the impulse response results in an equivalent scaling of the measured amplitudes. Consequently, it is sufficient to normalize the apodizations and thus constrain the drive voltage before performing the ROI intensity comparison of the two methods. We have also shown in previous work [

23] that the tracked shear waves generated in phantoms using SMURF methods are in agreement with the corresponding patterns simulated with FIELD II. We explore further which method is more effective in terms of relative ultrasound intensity at the focus for different combinations of parameters such as the depth of the focal region, the width of the desired pushing region, and the transducer center frequency.

A. Focal Fraunhofer Method

The FF method relies on the fact that lateral pressure distribution within the focal zone (see ) is determined by the Fourier transform of the transducer’s apodization function and can thus be written as [

10]

and

*v*_{0} is the peak velocity amplitude of the transducer;

*k*_{l},

*c*, and

*λ*_{l} are the wave number, the sound speed, and the wavelength of the longitudinal wave, respectively.

Again, the ideal pushing force (and hence intensity pattern) should be well localized to maximize the energy deposition at the ROI. It should also have a well defined spatial frequency so that the resulting shear wave has a known wavelength. A sinusoidal pressure distribution limited in its lateral extent resembles such a pattern. Based on the Fourier transform relationship, the apodization that guarantees a sinusoidal pressure distribution with equally spaced peaks at the focal zone, in theory, consists of a pair of impulses. However, because it is not possible to create perfect impulses with the transducer elements and the applied force is to be confined laterally, the apodization function consists instead of a pair of narrow Gaussian functions. Our model pushing pulse is thus a Gaussian-weighted sinusoid of the desired spatial frequency.

The function describing the apodization and delay profiles to be applied to the transducer elements can be represented by the expression

where

*x*_{0} is the lateral distance at the transducer face (

*z* = 0),

*M*(

*x*_{0}) represents the magnitude of the apodization, and

(

*x*_{0}) contains the phase information.

where Δ is the distance from the center of the Gaussian function to the center of the transducer face and

*σ*_{a} is the spread of the Gaussian function. The phases of the elements are determined geometrically.

With the lateral pressure variation shown in (

2), the intensity at the region of interest is found to be (see [

10])

Δ is found to be

where

*c*_{l} is the longitudinal speed of sound,

*f*_{0} is the transducer center frequency, and

*λ*_{s} is the desired shear wavelength. As indicated by (

8), the spatial frequency of the pressure distribution can be controlled by the distance between the peaks of the two Gaussians in the apodization profile.

The widths of the beams at the aperture are represented by the spread of the Gaussians (*σ*_{a}) and the width of the ROI is represented by the spread of its Gaussian envelope (*σ*_{f}). The width of the pushing region is controlled by the widths of the two Gaussians in the apodization profile. The relationship between the width of the beams at the transducer (apodization profile) and that of the desired spatial pattern is found to be

For a given focal depth and limited drive voltage, the energy at the focus is proportional to the effective aperture width, represented by *σ*_{f}. If the desired focal width is large, the widths of the apodized beams at the transducer must be narrow because of the inverse Fourier relationship between the apodization and the lateral intensity variation in the far-field. Narrowing the width of the apodization beams at the transducer implies that the energy at the focus will decrease accordingly.

B. Intersecting Plane Wave Method

A spatially modulated function can also be generated by the interference pattern of two intersecting planar waves. The pressure caused by the interference of the two plane waves propagating at angles ±*θ* to the *z*-axis, can be expressed by

where

*f*_{l} and

*k*_{l} are the frequency and wave number of the ultrasound wave. By using the appropriate trigonometric identities, the expression for the lateral pressure distribution can be rewritten as

where the spatial frequency of the lateral pressure variation is given by

*k*,

The wavenumber of the induced shear wave, *k*_{s}, is thus

Consequently, the angle, *θ*, required to result in the desired spatial variation must then be

Given that we have a finite aperture and the ROI is to be limited in its lateral extent, two unfocused Gaussian beams are used to approximate the planar wave fronts.

The function describing the apodization and delay profiles to be applied to the transducer elements can be represented by the expression:

where

*x*_{0} is the lateral distance at the transducer face (

*z* = 0), Δ is the distance from the center of the Gaussian function to the center of the transducer face, and

*σ*_{a} is the spread of the Gaussian function. The complex function

*A*(

*x*_{0}) is a Gaussian-modulated version of the equation of two forward-traveling plane waves at an angle.

Gaussian beams are of particular interest because they maintain a Gaussian profile everywhere in the field of propagation [

24]. The shape of a propagating Gaussian beam is represented by the hyperbolic profile shown in .

Let *W*(*z*) represent the width of the beam, where the pressure amplitude is reduced by 4.3 dB (i.e., *e*^{−1}), from the on-axis value. It is represented mathematically as

and the radius of curvature

*R*(

*z*) of a wave front in the beam is given by

where

*z*_{0} is the Rayleigh distance and

*z*_{f} is the focal distance. The width of the waist is

*W*_{0}. By observing (

17) and (

18), we can see that, first, the Rayleigh distance is directly proportional to the square of the beam width at the waist. Second, near the waist, i.e., for |

*z* −

*z*_{f}| <

*z*_{0},

*W*(

*z*) can be approximated by

*W*_{0}. In other words, in this region, the beam diverges slowly. Further from the waist (|

*z* −

*z*_{f}|

*z*_{0}), the beam width increases proportionally with distance and can be approximated as

*W*(

*z*) ≈ (

*λ*/(&

*pi;W*_{0}))(

*z* −

*z*_{f}). Note that the larger the waist, the slower the divergence of the beam profile, and the greater the region over which the plane wave approximation holds.

Also, from (

19), for |

*z* −

*z*_{f}|

*z*_{0} (far from the beam waist), the wave front radius asymptotically approaches

*z* and is approximated as a spherical wave. Near the waist however, the wave front radius is very large compared with the beam width

*W*(

*z*) and the beam can thus be approximated as a plane wave (infinite radius) with a Gaussian lateral profile. Based on these relationships, one can determine the Gaussian apodization that would be required for a desired ROI width.

The spatial frequency at the focus is determined by the angle at which the beams intersect. Knowing the longitudinal wavelength and the desired shear wavelength,

*θ* is first determined by using (

15).

From ,

The distance from the center of the beam profiles to the center of the transducer, Δ, depends on the focal depth, because *θ* is kept constant. Recall that the apodization consists of two Gaussians separated by a distance of 2Δ. The delay profile is such that the waves propagate at an angle so that the pattern is generated by a constructive and destructive interference at the focus. The resulting pattern also has a Gaussian profile. The widths of the beams at the aperture are thus represented by the spread, *σ*_{a}, of the Gaussians. The width of the ROI is also represented by the spread of its Gaussian envelope, *σ*_{f}. The relationship between the width of the beams at the transducer (apodization profile) and the desired spatial pattern is found to be

where

*p* = (2

*zλ*_{0})/

*π*,

*λ*_{0} is the transducer wavelength, and

*z* is the focal depth.

Eq. (21) indicates that there are two possible apodization widths (

*σ*_{a}) that would result in an ROI of a desired width (

*σ*_{f}). However, the solution that is more consistent with the plane wave assumption, and is likely to produce a greater acoustic intensity at the ROI, is the one that requires a wider Gaussian apodization (see ). That is,

Note that there will be combinations of focal depths (*z*) and ROI beam widths (*σ*_{f}) that will result in a non-real number for the spread of the Gaussian apodization (*σ*_{a}). For these cases, the focal Fraunhofer will automatically be the method of choice (see ).