IEEE Signal Processing - March 2018 - 169
Parameter sensitivity
Based on the d -ESF model, it is easy to
see that Ddx i = dx (x i) - dx (x i - 1) =
Ts (b - a) di (x i) . In muscle tissue, for
example, b ~ 6.5 # 10 -4 and a ~ 1.23 #
10 -4 in the normal physiological body
temperature range. Therefore, the infinitesimal echo strain is ~5 # 10 -4 /cC or
0.05%/cC.
Signal processing considerations
Speckle tracking for echo-shift
estimation
The coherent nature of pulse-echo US
gives rise to speckle phenomenon, which
gives B-mode images their grainy appearance. In imaging, speckle reduces the softtissue contrast and is normally considered
as undesirable. For tissue displacement
estimation, however, speckle signal components provide an opportunity to estimate tissue motion and deformation with
very high resolution [8]. The displacement estimation is typically achieved by
performing localized cross correlation of
RF echo signals from consecutive frames,
s (x, z, Ti), s (x, z, Ti + 1) . The echo shift is
simply obtained by determining the peak
of the cross correlation.
The temperature imaging filters operate on the echo shifts induced by temperature change as well as other tissue
motions and deformations. Depending on
the frame rate, the frame-to-frame temperature-induced echo shifts are in the
range nanoseconds to low microseconds.
These shifts represent minute-to-small
fractions of the sample times for a typical
diagnostic US system (RF echo sampling
in the range of 20−40 MHz.) Robust estimation of echo-shift profiles is the key to
the success of UST and thermometry.
Several implementations of crosscorrelation techniques for time delay
estimation have been described in the
literature. In [2], [3], and [7], a complex
cross correlation with zero-phase projection was used. This has the advantage
of determining the true peak of the cross
correlation with subsample accuracy
without the need for interpolation. This
is due to the fact that, in the absence of
pulse distortion, the phase of the cross
correlation is linear and makes a zero
crossing at the true peak of the cross-
correlation function (See [9] and references therein for the 2-D case.)
Model limitations, imaging artifacts,
and mitigations
The echo-shift models and the corresponding temperature imaging equations
ignored tissue inhomogeneities and assumed linear dependence of the speed of
sound on temperature. For example, 2-D
temperature estimates obtained based on
the d-ESF model exhibited thermal lensing artifacts [2] due to the distortion of
the imaging beam as traversed the heated
region. Briefly, since the speed of sound
changes with temperature, the heated region behaves like a lens, which distorts the
imaging beam. The distortion can be especially severe where the imaging beam
traverses regions with sharp temperature
gradients. This distortion produces axial
and lateral oscillations in the echo shifts
distal to the heated region, which are accentuated by the axial gradient 2dx/2z
in (6). In [6], spatial compounding was
shown to be effective in reducing thermal lensing artifacts.
Another source of artifacts is tissue
heterogeneity (e.g., variation in fat content), which leads to variation in the values
of a, b, and c as well as their temperature
dependencies. These artifacts are inherent to
pulse-echo US imaging, where beamforming is performed assuming a constant speed
of sound, i.e., ignoring tissue inhomogeneity. However, modern scanners provide
research mode with access to raw channel
data. These software-defined US systems
allow users to investigate reconstructive
imaging accounting for tissue inhomogeneity. If successful, reconstructive imaging
could provide a basis for quantitative UST
or US thermometry. In addition, the effects
of temperature dependencies could be addressed by iterative reconstructive imaging solutions to the temperature estimation
problem. At a minimum, one could implement a postprocessing algorithm that enforces some of the known constrains about
temperature fields:
1) Nonnegativity: for example, in a heating experiment, the temperature field
cannot have values below the steadystate baseline temperature.
2) Spatiotemporal evolution: temperature fields in tissue media is governed
IEEE Signal Processing Magazine
|
March 2018
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by the bioheat-transfer equation
(BHTE), which is a modified version
of the parabolic heat conduction
equation. The Green's function of
the BHTE is a time-varying spatial
Gaussian kernel [9].
3) Boundary conditions: skin temperature or core temperature away from
the heated region.
These constraints can be combined to
regularize the 2-D temperature field using
the projection onto convex sets (POCS)
algorithm described by Youla in [10]
i k + 1 (x, z, Ti) = G (Ti) P (Ti)
# B (Ti) i k (x, z, Ti), (8)
where Ti is the ith frame (wall clock)
time and, G (Ti), P (Ti), and B (Ti) are
(convex) projection operators corresponding to the Gaussian spatial bandwidth, positivity, and boundary condition constraints, respectively.
From thermography to
thermometry
UST as described previously is more
like echography due to the assumption
of homogeneous medium in deriving the
imaging equations. Nonetheless, UST
has been shown to be useful in guiding
therapy [2] and even quantitative tissue
property measurements based on temperature transients [7]. The ultimate goal
is to perform thermometry by developing quantitative measurements based
on truly localized analysis of echo data.
For this purpose, spectral analysis of the
echo data could provide the answer [11].
Amini et al. compared the infinitesimal
echo-shift model with a frequency shift
model given by
dfm . fm (i 0) 6b (i 0) - a (i 0)@ di,
(9)
where fm is the mth harmonic of the
resonance frequency associated with
mean scatterer spacing in the region of
interest and a, b, and i are as in the
aforementioned RESF model. The frequency shift leads to a more localized
temperature change measurement. To
improve the spatial resolution of this
measurement, high-resolution spectral
estimation methods were employed (see
[11] and references therein). We note
169
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