Reconstruction for thermoacoustic tomography in an arbitrary detection geometry is proposed by time-reversing the measured field back to the time when the thermoacoustic sources are excited. Time reversal of the field can be implemented efficiently by applying the delay-and-sum algorithm. The theoretical conclusions are supported by a numerical simulation of three-dimensional thermoacoustic tomography

Wang, Lihong V.

Xu, Yuan

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Application of time reversal to
thermoacoustic tomography
Yuan  Xu, Lihong V. Wang
Yuan  Xu, Lihong V. Wang, "Application of time reversal to thermoacoustic
tomography," Proc. SPIE 5320, Photons Plus Ultrasound: Imaging and
Sensing,  (12 July 2004); doi: 10.1117/12.532395
Event: Biomedical Optics 2004, 2004, San Jose, CA, United States
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Application of time reversal to thermoacoustic tomography
Yuan Xu and Lihong V. Wang
Optical Imaging Laboratory, Department of Biomedical Engineering,Texas A&M University,
3120 TAMU, College Station, Texas 77843-3120;
ABSTRACT
Reconstruction for thermoacoustic tomography in an arbitrary detection geometry is proposed by time-reversing
the measured ﬁeld back to the time when the thermoacoustic sources are excited. Time reversal of the ﬁeld can
be implemented eﬃciently by applying the delay-and-sum algorithm. The theoretical conclusions are supported
by a numerical simulation of three-dimensional thermoacoustic tomography.
Keywords: time reversal, thermoacoustic tomography
1. INTRODUCTION
Developing reconstruction methods in thermoacoustic tomography (TAT) has been a craft work. The derivation
of an algorithm depends strongly on the speciﬁc detection conﬁguration, and exact reconstruction algorithms
are only available in a few special geometries. Although it has been noted that the energy deposition, the to-be-
reconstructed value in TAT, is equivalent to the initial pressure distribution, no method for deriving the initial
pressure distribution from the signals measured across the detection surface ( or to time-reverse the measured
signals) is available. In our recent work,1 we ﬁnd that when time reversal is considered in the time domain, exact
time-reversal methods that use only the ﬁeld on an arbitrary closed surface can be derived for a wide variety
of applications such as tomography with diﬀracting sources, inverse diﬀraction, and ultrasound therapy. In this
paper, we apply time-reversal methods to deriving exact and approximate reconstruction algorithms for TAT in
an arbitrary conﬁguration for the ﬁrst time.
Further author information: (Send correspondence to L. Wang)
L. Wang: E-mail: Lwang@tamu.edu, Telephone: 1 979 847 9040
Photons Plus Ultrasound: Imaging and Sensing, edited by
Alexander A. Oraevsky, Lihong V. Wang, Proceedings of SPIE Vol. 5320
(SPIE, Bellingham, WA, 2004) 1605-7422/04/$15 · doi: 10.1117/12.532395
257
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In the METHOD section, we ﬁrst show that the energy deposition in TAT is equivalent to the initial pressure
distribution. Then, we introduce some basic concepts of time reversal and an exact time-reversal formula as
well as an approximate one in the form of delay-and-sum algorithm. In the RESULTS section, we present a
reconstruction formula in TAT by time-reversing the acoustic ﬁeld to the time when the to-be-reconstructed
point is excited. The reconstruction algorithm is veriﬁed by a numerical simulation of 3-D TAT. Conclusions are
presented in the ﬁnal section.
2. METHOD
2.1. Reconstruction in TAT
In this subsection, we will show that the energy deposition is equivalent to the initial pressure distribution in
TAT. The thermoacoustic signal p0(r, t) is related to the energy deposition H(r, t) = I0η(t)ϕ(r) by the following
wave equation
∇2p0(r, t)− 1
v2s
∂p0(r, t)
∂t2
= − β
C
∂H(r, t)
∂t
, (1)
where I0 is a scaling factor proportional to the incident radiation intensity; ϕ(r) describes the microwave absorp-
tion properties of the medium at r; η(t) describes the shape of the irradiating pulse and is a nonnegative function
whose integration over time equals the pulse energy; C is the speciﬁc heat; β is the coeﬃcient of volume thermal
expansion; and vs is the acoustic speed in the background. We will consider a delta pulse (the eﬀect of ﬁnite
pulse length has been discussed elsewhere2). Substituting a delta pulse into (1), and applying the integration of
t over [0−, 0+] twice, where 0+ (0-) is an inﬁnite small, positive (negative) real number, we have
p0(r, 0+) = εϕ(r), (2)
where ε = βv2sI0/C. It is assumed that the thermoacoustic pressure is ﬁnite. Therefore, the reconstruction in
TAT is equivalent to reverting the pressure ﬁeld to time zero. This concept can also be adopted to derive the
TAT reconstruction algorithm in a known acoustically heterogeneous medium.
2.2. Time reversal
The time reversal of an acoustic or electromagnetic wave is based on the invariance of the wave equation in
a lossless medium under the transform t → −t (t represents the time). The time reversal of a wave can be
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understood as the generation of the back-propagation ﬁeld from the measured forward-propagation ﬁeld and/or
its normal derivative after removing the initial sources. The concept of time reversal has been implemented
experimentally and applied to a wide range of studies such as inverse scattering,3–5 wavefront distortion correc-
tion,6, 7 and multiple scattering phenomena.8 Here we will introduce some basic concepts of time reversal and
an exact time-reversal formula as well as an approximate one in the form of delay-and-sum. We start from the
wave equation for pressure p0(r, t) in a non-absorbing and non-dispersive medium9
∇2p0(r, t)− 1
v2s
∂2p0(r, t)
∂t2
= −q(r, t) , (3)
where q(r, t) is the source term, which is nonzero only in R [the space enclosed by the detection surface Σ] and
within the time period [0, Ts]. We have
p0(r, t) =
∫ Ts
0
dt0
∫
R
dr0q(r0, t0)g+(r, t |r0, t0) , (4)
where g±(r, t |r0, t0) = δ(t− t0 ∓ |r− r0| /vs) /(4π |r− r0|) is a diverging (g+) or converging (g−) Green’s func-
tion.
Time reversal of p0(r, t) at time T0, is deﬁned as
pr(r, T0) = p0(r, T0), p′r(r, T0) = −p′0(r, T0) , (5)
where prime represents the temporal derivative in this paper; pr(r, t) is the time-reversed ﬁeld of p0(r, t); and
T0 is chosen to be large enough so that p0(r, t) = 0 for r ∈ R, t > T0. Then, pr(r, t) can be uniquely determined
by the initial conditions at T0. The above deﬁnition of time reversal is analogous to the fact that a particle will
move back along its trajectory if its velocity is reversed and position unchanged. According to this deﬁnition,
we mean pr(r, tˆ) when we say time-reversing a ﬁeld back to time t, where a hat over a time variable t represents
2T0 − t.
In the case of a point source δ(r− r0)δ(t− t0) in R, (5) becomes
gr(r, T0 |r0, t0 ) = g+(r, T0 |r0, t0 ), g′r(r, T0 |r0, t0 ) = −g′+(r, T0 |r0, t0 ). (6)
It can be easily veriﬁed that the time-reversed ﬁeld
gr(r, t |r0, t0 ) = g−(r, t
∣∣r0, tˆ0 )− g+(r, t ∣∣r0, tˆ0 ), t > T0 , (7)
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because it satisﬁes both the homogeneous wave equation and the initial-value conditions shown above. This
result is also obtained by Cassereau7 and Porter.10 (7) means that if we time-reverse, at T0, the ﬁeld of a point
source which is located at r0 and excited at t0, the time-reversed ﬁeld converges to r0 at time tˆ0 and then diverges
with an opposite amplitude. The diverging wave g+(r, t
∣∣r0, tˆ0) , however, does not have a counterpart in the
forward propagation. It exists because, unlike the forward propagation, there is no source inside Σ for gr(r, t).
The diverging wave with an opposite amplitude exactly cancels the source term related to the converging one.
In the case of an arbitrary source, similar results can be obtained after considering the linearity of the wave
equation with respect to the source
pr(r, t) =
∫ Ts
0
dt0
∫
R
dr0q(r0, t0)gr(r, t |r0, t0) . (8)
After substituting (7) into (8), using g−(r, t |r0, t0) = g+(r,−t |r0,−t0) , and a variable transform, we obtain for
rd ∈ Σ ( although it is valid for any r)
pr(rd, t) = p0(rd, 2T0 − t) + pdiv(rd, t) , (9)
pdiv(rd, t) = −
∫ 2T0
2T0−Ts
dt0
∫
R
dr0q(r0, tˆ0)g+(rd, t |r0, t0) .
(10)
As in the case of a point source, the diverging component pdiv(rd, t) has no counterpart in the forward propagation
and is, in general, not available from the experimental measurements of p0 except in special cases. Nevertheless,
it is shown that pr(r, t) before a speciﬁed time can be derived using only p0(rd, t)1
pr(r, t) = −
∫ t+
T0
dt0
∮
Σ
dSdp0(rd, tˆ0)
∂g1(r, t |rd , t0)
∂n
, (11)
where ∂/∂n is the derivative along the normal of Σ at rd pointing away from the volume R; g1(r, t |rd, t0) with
r, rd ∈ (R ∪ Σ) is Green’s function subject to the homogeneous Dirichlet boundary condition on Σ; and t+ is
inﬁnitesimally greater than t. An explicit expression of g1, in the form of a series of eigenfunctions, can be
found for the boundaries that ﬁt with separable coordinates.11 In TAT, after assuming that the object is not
close to the detection surface and following the method shown in our recent work,1 we arrive at the following
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approximate algorithm in the form of delay-and-sum
pr(r, t) ≈ 12π
∮
Σ
dSd
n · (rd − r)
|r− rd|2
[
p0(rd, trd)
|r− rd|
−p′0(rd, trd)/vs] , (12)
where trd = 2T0 − t + |r− rd| /vs.
3. RESULTS
Combining (2) and (11) or (12), we have the following reconstruction algorithm in TAT
ϕ(r) =
1
ε
p0(r, 0) =
1
ε
pr(r, 2T0). (13)
A three-dimensional TAT is numerically simulated (Fig. 1). The objects are two spheres with a radius of 8
mm. ϕ(r) is set to be 1 in the sphere and zero otherwise. 2048 detection positions are randomly distributed over
a sphere with a radius of 80 mm and the center at the origin. The imaging space is a cube with a side length
of 112 mm, centered at the origin, and mapped with a 64 by 64 by 64 mesh. In the forward problem, p(r, t) is
computed by the integration of the object value along a series of spheres. The signals are within [0 1] MHz. The
acoustic speed is 1500 m/s. In the inverse problem, we use (12) to time-reverse ﬁelds to save computation time.
Figures 1(a)-(b) show that the shape and position of the object is reconstructed correctly. The line graphs in
Fig. 1(c) show that the object is reconstructed quantitatively.
Figure 1. (a) Reconstructed image section in DT along the y = 0 and (b) x = 0 planes. (c) The line graphs along the
z = 0 (dotted), y = 0 (dashed) lines in (b) and the corresponding real value (solid).
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4. CONCLUSIONS
Reconstruction for thermoacoustic tomography in an arbitrary detection geometry is proposed by time-reversing
the measured ﬁeld back to the time when the thermoacoustic sources are excited. Time reversal of the ﬁeld can
be implemented eﬃciently by applying the delay-and-sum algorithm. The theoretical conclusions are supported
by the numerical simulation of three-dimensional thermoacoustic tomography.
ACKNOWLEDGMENTS
We are grateful to Minhua Xu for useful discussion about this study. This project was sponsored in part by U.S.
Army Medical Research and Materiel Command Grant No. DAMD17-00-1-0455, National Institutes of Health
Grant No. R01 EB000712, and Texas Higher Education Coordinating Board Grant No. ARP 000512-0063-2001.
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Application of time reversal to thermoacoustic tomography

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Application of time reversal to thermoacoustic tomography

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