IEEE Signal Processing - March 2018 - 168
proposed a model for the speed of sound
in tissue given by
1 = 1 + 1 + 1 , (1)
c (i)
c w (i) c f (i) c r (i)
where i is the temperature; c is the
speed of sound, and the subscripts
w, f, and r refer to water, fat, and residual tissue (protein, etc.), respectively.
Empirical formulas for the three components were also given
c w (i) =
c f (i) =
5
/ k iw i k,
i=0
k 0f -
k 1f i,
c r (i) = k 0r - k 1r i.
(2)
Based on (1) and specific values of parameters in (2), in most aqueous tissues,
the temperature dependence can be described as an inverted quadratic curve,
which peaks in the 45−55 °C range.
In addition to the speed of sound, the
change in backscatter energy (CBE) was
also proposed as a method for the estimation of temperature change using pulseecho US [5]. The backscatter coefficient
from a random distribution of subwavelength scatterers is given by
h (i) =
^ t m c m (i) - t s c s (i) h2
^ t s c s (i) h2
3t s - 3 t m 2
m,
+ 1c
3 2t s + t m
(3)
where t m, c m (t s, c s) are the density
and the speed of sound for the medium
(scatterer), respectively. For a plane wave
propagating in the z direction, the backscatter energy is given by
E (i) =
h (i)
^1 - e -2ar (i)z h,
ar (i)
(4)
where ar (i) is the attenuation coefficient
of the medium. In [5], a temperature imaging relation was proposed based on the
ratio E (i) /E (i 0) for some reference temperature, i 0 . The basis for this approach
is the empirical result showing that the
ratio in decibels increases or decreases
monotonically in the 37−45 °C range [5]
for fat and aqueous tissues.
The echo-shift model and
imaging equations
A survey of the literature reveals that the
echo-shift model is the most widely used
168
in the medical US imaging community.
Temperature change in tissue induces
changes in the speed of sound as well as
thermal expansion of the affected tissue.
The former produces apparent echo shifts
due to the change in the acoustical path
length. The latter produces physical shifts
due to the actual change in the location of
the scattering elements within the heated
volume. The details of the derivation can
be found in [2] for what is now called the
infinitesimal echo-strain filter ( d-ESF).
More recently, [3] introduced a modified
version that resulted in a recursive echostrain filter (RESF), which includes the
d-ESF as a special case.
To derive the RESF, assume a simple model of pulse-echo response from
a single point scatterer at a distance z
from the transducer face. Assume further that the medium is homogeneous
at a constant baseline temperature, i 0 .
An emitted pulse, p (t), will reach the
scatterer after a delay of z/c 0 , and the
scattered echo will be detected by the
transducer after an additional delay of
z/c 0 for a total delay of 2z/c 0 . For the
inhomogeneous case, c (z, i) and assuming infinitesimal temperature
c h a n g e , i (z) = i 0 + di (z), |di| % i 0,
we can express the echo shift from two
transmissions at T0 and T1:
dx (z) = x (z) - x 0 (z) . 2
#;
#0
z (i)(1 - adi)
1 + a (p) di (p)
1 E dp,
c (p, i (p))
c (p, i 0)
(5)
where a is the linear coefficient of thermal expansion. The second term in the
integrand accounts for the propagation
delay at time T0 .
The RESF
Si nc e | di | % i 0, one m ay a ssu me
c (z; i (z)) = c 0 (z) [1+ b (z) di (z)], which
is justifiable near normal physiological
temperatures. Substituting and differentiating (5) with respect to z, we get
- az 2di + (a - b) di =
2z
c (z, i) 2dx (z)
.
2
2z
(6)
This is a differential equation with spatially varying coefficients, which can
be solved numerically using a variety of
IEEE Signal Processing Magazine
|
March 2018
|
methods. However, we seek a filtering
approach for real-time implementation
on a variety of platforms, including fieldprogrammable gate arrays (FPGAs). A
considerable insight can be obtained by
treating the coefficients as space invariant, including the coefficient of 2di/2z.
Setting z = z m and assuming a, b, and c
are constants, we transform (6) into a
constant-coefficient differential equation.
Using x = 2z/c & 2x = 22z/c allows us
to discretize the equation on the uniform
RF sampling grid " x i = iTs ,iN=s 1 to arrive
at the recursive equation [3]:
di (x i) = a $ di (x i - 1)
+ b $ ^dx (x i) - dx (x i - 1)h, (7)
with the filter coefficients given by
a=
ax m
ax m - Ts (a - b)
b =-
and
1
ax m - Ts (a - b)
where x m = 2z m /c 0 . The higher the value of z m, the narrower the bandwidth of
the lowpass filter defined by the recursive
term. The recursive term determines the
extent of spatial memory, which is controlled by z m .
It should be noted that (6) is an
approximation of an integrodifferential
equation with higher-order spatial derivatives of di and even nonlinear terms. The
low-order approximation is shown here
due to the insight provided by the coefficients a and b in (7). These coefficients
produce well calibrated temperature
estimates in terms of the computed echo
strain differentials ^dx (x i) - dx (x i - 1)h .
The d-ESF
Setting z m = 0 reduces (7) to the discrete-space differentiator originally proposed in [2] or the d-ESF. The echo strain
equation described in [4] was derived under the same assumptions used to derive
the d-ESF while ignoring the thermal
expansion term. Several other authors
have used this model, including [6] and
[7], to demonstrate temperature imaging (thermography) in tissue-mimicking
phantoms and tissue media. It was shown
to be valid when the range of di is such
that i 0 + di 1 50 °C when appropriate
values of a, b, and c are used.
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
Contents
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