Fusing microwave radar and microwave-induced thermoacoustics for breast cancer detection

Abstract

Microwave-based techniques for breast tumour detection rely on the inherent electrical difference between malign and healthy tissue in the microwave range. Microwave-radar and microwave-induced thermoacoustic methods both struggle when the dielectric contrast between the tumorous and background tissues is relatively small. In this work, we propose a detection technique that uses a hypothesis testing framework to fuse the information provided by these two sensor modalities and hence provide more reliable detection results in low-contrast, high-clutter environments when compared to the results that either of the techniques would provide alone.

Full text

Page 1

FUSING MICROWAVE RADAR AND MICROWAVE-INDUCED THERMOACOUSTICS FOR
BREAST CANCER DETECTION
Evgeny Kirshin Borislav OreshkinKevin Guangran ZhuMilica Popovi´ cMark Coates
Department of Electrical and Computer Engineering, McGill University, H3A 2A7, Montreal, QC, Canada
ABSTRACT
Microwave-based techniques for breast tumour detection rely on
the inherent electrical difference between malign and healthy tissue
in the microwave range. Microwave-radar and microwave-induced
thermoacoustic methods both struggle when the dielectric contrast
between the tumorous and background tissues is relatively small. In
this work, we propose a detection technique that uses a hypothesis
testing framework to fuse the information provided by these two
sensor modalities and hence provide more reliable detection results
in low-contrast, high-clutter environments when compared to the
results that either of the techniques would provide alone.
Index Terms— Microwave imaging, UWB, breast cancer de-
tection, hypotheses testing, sensor fusion
1. INTRODUCTION
Earlydetection of breast cancer leadsto amuch higher recovery rate.
The X-ray mammography and Magnetic Resonance Imaging (MRI)
screening techniques require expensive equipment and involve sig-
nificant patient discomfort; X-ray mammography involves undesir-
able exposure to ionizing radiation. Microwave-based techniques
have some promise as complementary modalities [1]. Microwave-
radar (MR) methods measure and analyze the backscatter signal
when the breast is illuminated by microwaves. Microwave-induced
thermoacoustic (MIT) methods measure and process the acoustic
signals induced by differential microwave heating. Both techniques
rely on the difference in the dielectric properties of malign and
healthy tissue and their individual performance can suffer when the
dielectric contrast is low. Since the techniques rely on different
physical processes (wave reflection versus heating) and measure
different types of signals (microwave versus acoustic), the fusion of
the information provided by the two modalities has the potential to
provide significantly better detection performance.
The primary contribution of this paper is the proposal of a
new microwave-based method for breast cancer detection based
on jointly processing MR and MIT signals in a hypothesis-testing
framework. We identify a test statistic and explain how to derive
its null-hypothesis distribution for the purpose of setting a thresh-
old. We conduct numerical simulations using structurally-realistic
breast models (derived from MRI scans). These simulations explore
multiple settings of the dielectric parameters and tissue properties to
examine the impact of low contrast and high clutter. The simulations
indicate that the proposed fusion approach can provide important
improvement.
2. PROBLEM STATEMENT
Weaddress thedetectionproblem of atumour inside abreast consist-
ing of heterogeneous tissue (glandular, fatty, etc.). The measurement
Fig. 1: A 2-D system model for the joint MR and MIT simulations.
system, depicted fora2-D settinginFigure1, operatesintwomodes.
In the MR mode, a wideband microwave pulse is radiated and the
respective backscattered signal is recorded sequentially at M1 dif-
ferent locations (channels). In the MIT mode, modulated pulses are
radiated from M2 locations, and transducers on the opposite side of
the breast measure the induced acoustic signals.
After pre-processing (see Section 4), the signals from the
MR and MIT modalities form a set of column-vectors xi
[xi,1,...,xi,N1]Tfor i = 1...M1 and yj = [yj,1,...,yj,N2]T
for j = 1...M2 of length N1 and N2 respectively (we focus on
the case N1 = N2 = N and M1 = M2 = M). Taking into
account the inherent limitations of the antenna array resolution, we
model the location of the tumour as taking one of a finite set of
locations {rℓ},ℓ = 1,...,L. The parameters θℓcapture the tissue
and geometry properties that affect the propagation and shape of
the received signals (see Section 2.1 for a concrete example for the
2-dimensional setting). The index ℓ0 denotes the “true” location, if
the tumour exists. Signals xiand yican then be modeled as:
=
xi = βℓ0si(θℓ0) + ξi,
yj = ηℓ0dj(θℓ0) + ζj,
i = 1,...,M
j = 1,...,M
(1)
(2)
These expressions model xi and yj as a combination of scaled
signals si(θℓ0) (scattered microwave signal) and dj(θℓ0) (induced
pressure signal) and the corresponding noise components ξiand ζj.
These latter components are comprised of clutter (microwave scat-
tering or acoustic signals generated by the normal tissue variation in
the breast), as well as receiver noise and digital quantization noise.
We model the random noise vectors ξi and ζj as jointly Gaus-

Page 2

sian with zero means and covariances σ2
unit-norm matrices Rξ and Rζ capture the structural properties of
the covariance, and the scalars σ2
ers. We model the noise signals as independent among the channels
and between the MR and MIT methods. We assume that Rξ and
Rζ can be estimated using a training data set containing responses
from a number of healthy breasts. The signal amplitudes βℓand ηℓ
are treated as unknown deterministic parameters and signals si(θℓ),
dj(θℓ) as deterministic and dependent on a set of parameters θℓ.
In addition, tissue parameters, included in θ, can be also estimated.
Thus, for any given location rℓwe consider s and d as known and
refer to them as “signal templates”.
We consider the binary hypothesis testing framework for the
joint MR/MIT breast tumour detection formulated on the finite set
of scan-locations {rℓ}L
ξRξ and σ2
ζRζ, where the
ξand σ2
ζspecify the noise pow-
ℓ=1:
H(0)
ℓ
: βℓ,ηℓ= 0 vs. H(1)
ℓ
: βℓ,ηℓ?= 0.
(3)
The hypothesis H(1)
H(0)
ℓ
is the null hypothesis. To discern between these hypotheses
and to address the issue of unknown parameters in (1) and (2) we
adopt a generalized likelihood ratio test (GLRT) approach [2].
ℓ
denotes the scenario with tumour present while
2.1. 2D Signal Templates
For the 2-dimensional experiments described in this paper, we con-
struct signal templates as follows. In the discussed model the set of
parameters θℓincludes average Debye tissue properties [3] (relative
permittivity at the infinite frequency ǫ∞, difference in the relative
permittivity at DC and the infinite frequency ∆ǫ, static conductivity
σsand relaxation time constant τ) and tumour location rℓ. The 2-D
signal templatefor theMR process models theeffect of thedelay and
attenuation in the matching medium, skin and tissue. Its frequency
domain representation for channel i is:
Si(θℓ) = G(jω)e−j2kb|ri−rb,i|e−j2ks|rb,i−rs,i|
· e−j2kn(θℓ)|rs,i−rℓ|,
(4)
where rℓdenotes the location of the tumour, ridenotes the location
of the i-th antenna; rb,iand rs,ilie on the line between riand rℓat
the transition points of the background-skin interface and the skin-
tissue interface respectively. kband ksdenote the known wave num-
bers of the matching medium and skin. kn(θℓ) is the wave number
from the estimated Debye parameters. G(jω) denotes the Gaussian
modulated pulse reflected at the interface between the tissue and the
tumor.
The 2-D signal template for the acoustic wave generated by the
heated tumour is [4]
Di(θℓ) = I(jω)
?
Ω
e−jka|ri−rℓ|
?|ri− rℓ|
dΩ
(5)
where ka denotes the acoustic wave number. I(jω) denotes the
acoustic pulse generated by a point in the tumour region Ω. Note
that this signal template assumes the tissue and matching medium
have the same acoustic properties.
3. METHODOLOGY
Prior to the application of the GLRT framework, the input signals x
and y are whitened using the estimated clutter covariance matrices:
˜ x =˜R−1/2
ξ
x; ˜ y =˜R−1/2
ζ
y
(6)
The clutter covariance matrices are estimated from a training set of
tumour-free breasts, using the procedure described in [5]. We set
the regularization parameter to 1e-7, a value that provides good per-
formance in practice. In order to compensate for the distortion in-
troduced by whitening, we expose the signal templates to the same
whitening procedure, which produces ˜ s and˜d vectors.
The GLRT performs the comparison of the generalized likeli-
hood ratio LG(˜ x, ˜ y) against a threshold γ:
LG(˜ x, ˜ y)
H(1)
ℓ
≷
H(0)
ℓ
γ
(7)
The generalized likelihood ratio can be factorized under the
noise independency assumption:
LG(˜ x, ˜ y) =p(˜ x|ˆβℓ1, ˆ σ2
ξ,ℓ1,H1) · p(˜ y|ˆ ηℓ1, ˆ σ2
p(˜ x|ˆ σ2
ζ,ℓ1,H1)
ζ,ℓ0,H0)
ξ,ℓ0,H0) · p(˜ y|ˆ σ2
(8)
Maximum likelihood estimates for the unknown variables are
given by [2, Appendix 9A]:
ˆβℓ1=
?
NM˜ xT(I − P˜ s) ˜ x; ˆ σ2
1
NM˜ yT(I − P˜d) ˜ y; ˆ σ2
˜ sT˜ s
?−1˜ sT˜ x; ˆ ηℓ1=
?˜dT˜d
ξ,ℓ0=
?−1˜dT˜ y;
1
NM˜ xT˜ x;
1
NM˜ yT˜ y.
(9)
ˆ σ2
ξ,ℓ1=
1
(10)
ˆ σ2
ζ,ℓ1=
ζ,ℓ0=
(11)
Intheexpressionsabove P˜ s= ˜ s?˜ sT˜ s?−1˜ sTandP˜d=˜d
and noise subspaces respectively.
Making use of the MLE expressions in the ratio of Gaussian
PDFs (8) results in:
?˜dT˜d
?−1˜dT
represent projection matrices that project a vector onto the signal
LG(˜ x, ˜ y) =
?
ˆ σ2
ˆ σ2
ξ,ℓ0
ξ,ℓ1
? NM
2
·
?
ˆ σ2
ˆ σ2
ζ,ℓ0
ζ,ℓ1
? NM
2
=
?
˜ xT˜ x
˜ xT(I − P˜ s) ˜ x
?NM
NM
2
2
·
?
˜ yT˜ y
˜ yT(I − P˜d) ˜ y
?NM
2
= (Λ˜ x(˜ x) · Λ˜ y(˜ y))
(12)
Combining this expression with (7) and noting that (·)
monotone increasing function, results in the following decision rule:
NM
2
is a
Λ′(˜ x, ˜ y) = Λ˜ x(˜ x) · Λ˜ y(˜ y)
H(1)
ℓ
≷
H(0)
ℓ
γ′,
(13)
For a fixed probability of false alarm, the threshold γ′can be de-
termined from the inverse cumulative distribution function (CDF) of
thetest statisticΛ′(˜ x, ˜ y), which can be acquired by means of numer-
ical simulations or empirically estimated from training data obtained
from simulations with healthy cases (distribution only under H(0)
required to perform the test).
In our work we also consider an empirical test statistic which is
the product of two alternative, zero shifted, test statistics:
ℓ
is
T(˜ x, ˜ y) = T˜ x(˜ x) · T˜ y(˜ y) =
˜ xTP˜ s˜ x
˜ xTP⊥
˜ s˜ x·
˜ yTP˜d˜ y
˜ yTP⊥
˜d˜ y.
(14)

Page 3

where P⊥= I − P. The statistic T˜ x(˜ x) is derived in [5] for the
microwave radar modality. We are motivated to explore the per-
formance of the statistic T(˜ x, ˜ y) because in our experiments the
Λ′(˜ x, ˜ y) statistic is completely dominated by the T˜ x(˜ x) term. This
occurs because our signal model siis much more accurate than dj.
It can be shown that Λ′(˜ x, ˜ y) can be represented in terms of
T(˜ x, ˜ y), T˜ x(˜ x) and T˜ y(˜ y) as:
Λ′(˜ x, ˜ y) = 1 + T(˜ x, ˜ y) + T˜ x(˜ x) + T˜ y(˜ y).
(15)
This expression provides better form of comparison Λ′(˜ x, ˜ y) with
T(˜ x, ˜ y). After incorporating the constant “1” into the threshold, it
can be seen that Λ′(˜ x, ˜ y) consists of the sum of the two separate test
statistics for MR and MIT and their cross-product term. If T˜ x(˜ x)
and T˜ y(˜ y) are of approximately equal power, such a combination
should lead to better performance, in comparison to T(˜ x, ˜ y), when
one of the two methods fails to detect the tumour or gives false pos-
itives in substantially wrong locations.
4. NUMERICAL SIMULATIONS
The electromagnetic and acoustic signals were simulated using the
finite-difference time-domain (FDTD) method and two-dimensional
numerical breast phantoms derived from MRI images. The FDTD
grid (650 X 650 cells) resolves the geometry depicted in Figure 1 to
0.4mm. For the MR technique, we use a pulsed excitation. For the
MIT simulation, we use a 3-GHz continuous waveform source. The
specific absorption rate is computed and used in the acoustic signal
simulation [6].
Nine series of breast tissue models are considered, whose Debye
properties are represented in Table 1. ∆ǫ, and τ have been fixed to
constant values (see the Table) while ǫ∞, and σshave been assigned
withvalues by linearly mapping the variation of the MRI pixel inten-
sities into the variation of ǫ∞ in the range of ǫ∞,b(1 ± 0.01Var/2)
and of σs in the range of σs,b(1 ± 0.01Var/2) with the param-
eters ǫ∞,b,σs,b taken from Table 1.
max(ǫr,m/ǫr,b), min(σm/σb), max(σm/σb) represent the mini-
mum and maximum ratios of the resulting dielectric properties of
malignant to benign tissues after mapping. The dielectric properties
are evaluated using the Debye model at central frequency 6.85 GHz.
For each series, Nr = 10 different realizations of tissue struc-
ture are generated, nine of which are used as the training set for the
whitening procedure. On the basis of the tenth one we build two tu-
morous breast phantoms by placing a circular tumour (R = 3mm)
into two positions relative to the centre of the breast: “pos.1” at
[-20mm;8mm]; “pos.2” at [15mm;-15mm]. Location “pos.2” has
higher density of tissue, which decreases the dielectric contrast and
makes the detection task more complicated.
In order to model measurement noise, white Gaussian noise is
added to the signal. Prior to the whitening procedure, the signals are
pre-processed. A simple calibration procedure is used to remove the
incident pulses from the MR mode signals. Artifacts in the signals
due to reflections at the skin-breast boundary are removed using the
minimum mean-squared-error algorithm described in [7].
Columns min(ǫr,m/ǫr,b),
5. RESULTS AND DISCUSSION
For each series 1,...,9 three cases are considered (two different
tumour locations and the healthy scenario). For each of these cases
images of four test statistics are computed: two (’MR’ and ’MIT’)
corresponding to the individual MR and the MIT decision rules, and
two representing the outputs from the data fusion rules (the derived
123456789
0
20
40
60
80
Series #
SINR, dB


MR
MIT
F
FP
Fig. 2: SINR for “pos.1” tests. As the dielectric contrast between
malignant and benign tissues drops (see Table 1), the detection per-
formance (SINR) decreases.
test statisticin (13) denoted by “F” and theproduct-rule (14) denoted
by “FP”). Each pixel of the images represents the value of a test
statistic at locations {rℓ}L
of 1mm.
For each image we compute the signal to interference-plus-
noise ratio (SINR) as the log-ratio between the maximums of
the tumorous image and the corresponding healthy one: SINR =
10log10max(TH1)/max(TH0).
Fig. 2 shows the SINR values for the “pos.1” tests. The SINR
level drops as the contrast in dielectric properties decreases and the
heterogeneity level goes up (see Table 1). The FP test statistic pro-
vides a significant improvement in the SINR along the whole range
of contrast levels while the performance of F is biased to the MR.
ℓ=1on the the grid with spatial resolution
0
4x 10−3
b)
0.005
0.01
T
0
2
T
0
1x 10−5
d)
0.005
0.01
123456789
0
0.5
Series #


T, H1
T, H0
a)
c)
Fig. 3: Test statistic maximum values (“pos.1” case) under H1 and
H0 hypotheses for each series: a) MR; b) MIT; c) F; and d) FP.
Thresholds γmin and γmax are shown as the red and green lines.
The image shows improvement in performance for FP test statistic
(d) versus MR (a) and MIT(b). The cases where improvement is
achieved are surrounded by black boxes.
We assess the detection performance for two different thresh-
olds: γmin, the largest threshold such that all tumours are detected;
and γmax, the smallest threshold such that there are no false alarms.
Figure 3 depicts the maximum values of the test statistics obtained
for the “pos.1” scenario. Two cases are represented: the tumorous
(i.e., under hypothesis H1) and the healthy (H0). An improved de-

Page 4

Table 1: Tissue properties for data series. Tumour is characterized by: ǫ∞ = 6.75, σe = 0.79 S/m, ∆ǫ = 48.35, τ = 10.47 ps.
Series #
Debye model parameters
ǫ∞,b
σs,b
3.10.05
4.00.08
4.00.08
13.0 0.4
13.00.4
13.00.4
13.80.7
13.8 0.7
14.20.8
Var, %
min(ǫr,m/ǫr,b)
10.6
6.4
6.3
1.4
1.3
1.3
1.1
1.1
1.0
max(ǫr,m/ǫr,b)
11.1
7.7
8.0
1.6
1.7
1.7
1.2
1.2
1.1
min(σm/σb)
24.8
11.5
11.5
1.7
1.7
1.7
1.2
1.2
1.0
max(σm/σb)
25.1
11.9
12.0
1.8
1.8
1.8
1.2
1.2
1.0
∆ǫ
1.6
3.5
3.5
24.4
24.4
24.4
35.6
35.6
40.5
τ, ps
13
13
13
13
13
13
13
13
13
1
2
3
4
5
6
7
8
9
7
30
40
30
50
70
10
30
10
z, mm


−40
−20
0
20
40
0
0.005
0.01
0.015
0.02


0
x 10
0.5
1
1.5
2
x 10
−3
y, mm
z, mm


−40 −200 20 40
−40
−20
0
20
40
0
0.005
0.01
0.015
0.02
y, mm


−40 −2002040
0
1
2
3
4
−5
a)
c)
d)
b)
Fig. 4: Images of the test statistics for a breast model slice: a) MR;
b) MIT; c) F; d) FP. True tumour location is shown by circle.
tection performance can be observed for FP with respect to MR (the
H0statistics for series 4, 6 and 8 are shifted below the γminthresh-
old) and MIT (the H0statistics for series 2 and 3 go below γminand
the H1statisticsfor series 7 and 8 increase dramatically). The detec-
tion capability for other series is preserved. The γmin thresholding
generates four false-alarms for the MR case and two false-alarms for
the MIT case, but only one false-alarm for the FP rule.
Table 2 summarizes the detection performance for both “pos.1”
and “pos.2” tests. The product rule FP provides the best perfor-
mance. Fig. 4 illustrates example images for the case where the FP
test statistic leads to significant improvement (series #8, “pos.1”).
One may notice the difference in the spatial distribution of clutter
between the MR and the MIT test statistics, which is the principle
property that the fusion benefits from. Considerable suppression of
the clutter may be observed for the FP test statistic. The F test statis-
tic is dominated by the MR signal, i.e. the T˜ xterm in (15). We con-
jecture that this is because the MR signal template is a much better
match to the simulated signal than the acoustic MIT signal template.
6. CONCLUSIONS
The conducted study shows that the fusion of data from microwave
radar and microwave thermoacoustic measurements has the potential
to lead to improved breast cancer detection performance. In the pre-
sented simulations, the performance of the derived GLRT test statis-
Table 2: Detection performance. False alarms are reported for the
case when threshold γmin is used. Misses are reported for the case
when γmax is used. Note that MR, F and FP can all achieve 1 miss
and 0 false-alarms when γmaxis used.
pos.1pos.2
False alarms
4
2
4
1
Misses
1
3
1
1
False alarms
7
9
7
6
Misses
1
6
1
1
MR
MIT
F
FP
tic is dominated by the MR signal and the heuristic test statistic per-
forms better. We believe this can be addressed by employing a more
accurate MIT signal template. Although the theoretically derived
test statistic does not show significant improvement in performance
over the separate MR and MIT methods, our study demonstrates that
thedatafusionusing simpleempiricallyderivedproduct rulecansig-
nificantly improve the overall detection performance.
7. REFERENCES
[1] E.C. Fear, P.M. Meaney, and M.A. Stuchly, “Microwaves for
breast cancer detection?,” IEEE potentials, vol. 22, no. 1, pp.
12–18, 2003.
[2] S.M. Kay, Fundamentals of Statistical Signal Processing, Vol-
ume 2: Detection Theory, Prentice Hall PTR, 1998.
[3] Ari Sihvola, Electromagnetic Mixing Formulas and Applica-
tions, Institution of Electrical Engineers, 1999.
[4] Yao Xie, Bin Guo, Jian Li, Geng Ku, and Lihong V. Wang,
“Adaptive and robust methods of reconstruction for thermoa-
coustic tomography,” IEEE Trans. Biomed. Eng., vol. 55, pp.
2741–2842, 2008.
[5] S.K.Davis, H.Tandradinata, S.C.Hagness, andB.D.vanVeen,
“Ultrawideband microwave breast cancer detection: adetection-
theoretic approach using the generalized likelihood ratio test,”
IEEE Trans. Biomed. Eng., vol. 52, no. 7, pp. 1237–1250, 2005.
[6] GuangranZhu,MilicaPopovi´ c,
“Microwave-induced thermoacoustics:
tomography,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1654 –
1657, 2009.
[7] E. J. Bond, Xu Li, S. C. Hagness, and B. D. van Veen, “Mi-
crowave imaging via space-time beamforming for early detec-
tion of breast cancer,” IEEE Trans. Antennas Propag., vol. 51,
pp. 1690–1705, 2003.
and
Assisting microwave
Qianqian Fang,