Additional topics

Phase aberration

The spatial and contrast resolution of medical ultrasound can be severely limited by a phenomenon known as phase aberration. The steering and focusing of an ultrasound beam using a phased array relies on an approximation of the speed of sound in tissue, usually 1540 m/s. In fact, the speed of sound through different tissues can vary greatly. As an acoustic wavefront passes through inhomogeneous tissues, it can become distorted as portions of its surface are advanced or retarded. On transmit this phenomenon affects the focusing and steering of the system point spread function (PSF). The returning echoes incident on the elements of the transducer array are also misaligned such that when these signals are summed to form a single echo line they no longer sum coherently, resulting in focal and steering errors, and consequently, diminished echo amplitude.

The discussion is separated into two separate sections, on ``first'' and ``second'' order aberrators. In this terminology, first order refers to geometric focal errors due to the difference between the actual mean sound speed in the tissue and that assumed by the ultrasound scanner beamformer. Second order aberration refers to the effects of the inhomogeneity of the tissue that still remain after the first order focal error is removed. Both types of aberration result in blurring of the point spread function and consequently the reduction of spatial and contrast resolution.

A first order aberrator arises from a gross sound speed error in the ultrasound system beamformer relative to the actual tissue, and is usually caused by the application of a single assumed tissue sound speed to all patients. For example, the first order aberration resulting from a 5% overestimation of sound speed is shown in Fig. 9.1. The impact of this aberrator on the system point spread function is shown in Fig. 9.3. In this example, the aberrator was applied to the element delays of a typical array at the f/1 focal range (7.5 MHz, 60% relative bandwidth, 128 elements of width equal to the 1540 m/s wavelength, f/8 in elevation) in an acoustic field simulation program[3]. The aberration leads to an 85% reduction in the PSF energy and a 75% reduction in its peak amplitude. The lateral resolution has been degraded by a factor of 3.9.

Figure 9.1: The first order phase aberration corresponding to a 5% overestimation in sound speed at the f/1 focal range.

An example of a second order aberrator is shown in Fig. 9.2. The impact of this aberrator on the system point spread function is shown in in Fig. 9.4. Once again, this aberrator was applied to the element delays of a typical array (7.5 MHz, 60% relative bandwidth, 128 elements of width equal to the wavelength, f/8 in elevation) in an acoustic field simulation program[3]. The aberration produces an 95% reduction in the PSF energy and a 91% reduction in its peak amplitude. The lateral resolution in this case has been degraded by a factor of 2.3.

A variety of techniques have been proposed to compensate for the effects of phase aberration[35,36,37,38,39,40,41,42,43,44,45]. Ideally, the estimated aberrator should be applied on both transmit and receive, as the phase screen affects wave propagation on both transmit and receive.

Figure 9.2: One realization of a simulated 50 ns RMS second order aberrator with a spatial autocovariance FWHM of 3 mm. This particular aberrator creates 41.5 ns RMS phase error across the array, and has a spatial autocovariance FWHM of 2.7 mm.

Figure 9.3: The impact of the first order aberrator (sound speed error) shown in Fig. 9.1 on the system point spread function. The aberrated case (bottom) is compared to the unaberrated control (top). The ``sharpness'' of the PSF is markedly reduced, reflecting losses of spatial and contrast resolution. The images are not displayed with the same gray-scale, so they do not show the 75% reduction in PSF amplitude that the aberration causes.

Figure 9.4: The impact of the second order aberrator shown in Fig. 9.2 on the system point spread function. The aberrated case (bottom) is compared to the unaberrated control (top). As in the case of first order aberration, the ``sharpness'' of the PSF is reduced, reflecting losses of spatial and contrast resolution. Once again, the images are not displayed with the same gray-scale, so they do not show the 91% reduction in PSF amplitude caused by the aberration.

The Van Cittert-Zernicke theorem

Speckle has a finite correlation length due to aperture-scatterer path length differences that distinguish one aperture position from another. A point target as infinite correlation length due to the lack of such path length differences.

The Van Cittert-Zernicke theorem states that for an incoherent, diffuse source, the receive spatial correlation function is equal to the Fourier transform of the intensity pattern of the incoherent source[46]. Speckle is a truly incoherent source. The location of scatterers on one line is uncorrelated with that of those in the next.

Figure 9.5: The cross-correlation between speckle signals from the focus received on two elements of an array obeys the predictions of the Van Cittert-Zernicke theorem (solid line). Here this curve is compared to the cross-correlation between apertures given lateral translation on receive only (dashed) and on transmit and receive (dotted).

Cross-correlation methods[47] are affected by VCZ. The VCZ curve falls off even faster in the presence of a transmit phase aberrator due to be broadening of the beam, which scales in k-space to a narrower intensity pattern, thereby producing a steeper correlation curve.

Deconvolution and super resolution methods

Deconvolution or ``super-resolution'' methods rely mainly on two approximations. The first of these is known as the Born approximation, which disregards echoes reduced by multiple scattering from targets. The second is that the point spread function is accurately known throughout the region of interest.

Under ideal conditions, deconvolution can work very well. This implies that the noise is very low and that the number of targets is relatively small. Deconvolution does not work well for larger numbers of targets under conditions of low signal-to-noise ratio, especially away from the focus and/or in the presence of phase aberration or attenuation. There performance is also degraded when physical conditions reduce the stationarity of the system's point spread function.

Detection:

Detection refers to a variety of methods of estimating or approximating the RF echo envelope. Ideally, the detection process shifts the signal band to base band, removing the carrier frequency, as shown in Figure 9.6.

Figure 9.6: Through ``ideal'' demodulation, the relationships of signal frequency components to the center frequency are translated to the baseband, reestablishing these component relationships relative to DC.

The simplest form of detection is AM demodulation using a full wave rectifier followed by a low pass filter. This is shown schematically in Figure 9.7, using the squaring operation to represent rectification.

Figure 9.7: If we model rectification with the squaring operation of the echo signal $e(t)$, the frequency domain representation of this detection method is shown above. The original RF signal (left) is convolved with itself to produce base-band and double-frequency components (center). The application of a low pass filter (LPF) removes the double-frequency components, leaving the desired baseband signal (right).

A more sophisticated demodulation technique is to mix the signal with that from a local oscillator. This produces base-band and double-frequency components, the latter of which are typically filtered out using a low pass filter. An extension of this concept, known as quadrature demodulation, is to mix the original signal with two local oscillators having the same frequency but shifted by 90$^\circ$ in phase relative to one another. The two baseband signals that remain after low-pass filtering are often referred to as the I (for ``in-phase'') and Q (for ``quadrature'') signals. Generating both I and Q signals preserves the phase information in the original signal.

Given the trigonometric identities,

$$\begin{split}\cos\alpha \sin\beta &= \frac{1}{2}[\cos(\alpha-... ... \text{and }&=-\frac{1}{2}\sin(\Delta\omega_{1})\ \end{split}$$ (9.1)

Neither of these techniques achieve ``ideal'' demodulation. While the level of energy at zero Hertz (DC) is preserved, the base-band signal away from DC contains the integration across $f$ of the product of frequencies at shift $\Delta f$:

$$\Bbb{F}(\Delta f) = \int^{BW/2}_{-BW/2} f_{axial}(f) f_{axial}(f+\Delta f) df$$ (9.2)

The energy at $\Delta f$ contains the integral of cross products of all frequencies at this shift. The original signal is irreversibly lost. This process mixes spatial frequency components across the bandwidth.

The most accurate demodulation technique is to use the Hilbert operator or the Hilbert transform to generate the quadrature signal at the RF center frequency. This operation does not assume any particular center frequency, and is thus immune to discrepancies between the local oscillator's frequency and that of the echo signal. The echo envelope is then found by calculating the magnitude of the complex signal formed by combining the original RF signal with its synthesized quadrature signal. With this method, no filtering is necessary.

Elastography imaging

Elastography refers to the imaging of the mechanical elastic properties of tissue[48,49]. These methods are based on vibrating or compressing the tissue while imaging the tissue with ultrasound. This approach grew from the dynamic palpation of the body during imaging.

Various methods for creating elastic waves or motion in a tissue are under investigation, including an internally or externally applied vibrator, an externally-applied compression paddle, or pumping waves using low frequency sound. Motion under vibration is imaged using Doppler techniques or speckle-tracking techniques. Standing waves are a problem, especially with strong vibration necessary for the targets.

Compression based techniques typically used some kind of correlation search before and after compression. Some error is produced in these techniques by the decorrelation of speckle during compression. Decorrelation during compression results in part from scatterers moving into or out of the resolution volume, and partly from changes in the geometric relationships among particles within the resolution volume[50,51].

Limitations of k-space

In closing, it is important to reiterate the various limitations of k-space. The most obvious limitation is that it is a linear systems approach that follows from the Fraunhofer approximation, and thus can only be applied under conditions of linear propagation and within the constraints of that approximation. Thus k-space cannot be applied in analyzing non-linear imaging modalities or the near field. Also, modeling the interaction between an imaging system and scatterers is only straightforward for targets that can be represented as point targets or collections of point targets, and under conditions justifying the Born approximation. There is no obvious means to represent non-Rayleigh scatterers, or to consider conditions in which the Born approximation is unjustified, such as in blood with physiological levels of hematocrit, or a bolus of a microbubble ultrasonic contrast agent. Finally, some k-space operations cannot be applied in the analysis of a spatially shift-variant system, such as in a complex flow field in which shear and turbulence are present that are warping the point-spread function as a function of location.

Despite these limitations, we hope this text has demonstrated the utility of a k-space approach in understanding and analyzing many aspects of ultrasound imaging systems and beamforming techniques.