Pseudo-Spectrum Time Domain and Time Reversal Mirror technique using in Microwave-induced Thermo-Acoustic Tomography System

Published Online June 2011 in MECS

Recent research has suggested that microwave -induced thermo-acoustic tomography can be a powerful imaging technology with good spatial resolution. In a MITAT system, a narrow pulse modulated microwave irradiation is used to illuminate the biologic tissue. When the pulsed microwave irradiation is absorbed by a tissue, the heating and subsequent expansion of the tissue give

Manuscript received January 1, 2008; revised June 1, 2008; accepted

July 1, 2008.

Copyright credit, project number, corresponding author, etc.

rise to an instantaneous acoustic stress or pressure distribution inside the tissue. The induced pressure distribution prompts acoustic wave propaga- tion toward the surface of the tissue with various time dalays. The ultrasound transducers, which can convert mechanical stress into electrical signals, are placed around the tissue to record these outgoing acoustic waves, commonly referred to as thermo-acoustic signals. These thermo-acoustic signals carry information about the acoustic properties of the tissue. So using these detected acoustic waves can compute the distribution of the initial acoustic pressure or microwave absorption, which is related to the properties of the tissue. Microwave-induced thermo-acoustic tomography in biological tissues combines the advantages of pure microwave imaging and pure ultrasound imaging [1], [2]. As a microwave pulse is used to irradiate the tissue, the image of a MITAT system has higher contrast than a conventional ultrasound system. This is because that the high electrical (relative permittivity and electrical conductivity) contrasts due to the significantly different sodium concentrations, fluid contents and electrochemical properties in the different tissues, especially between the normal soft tissue and malignant tumors (For example, the ratios of relative permittivity and electrical conductivity between the normal soft tissue and malignant tumors are 1:3.75 and 1:6.75 respectively at the frequency of 800MHz) [3]. On the other hand, the MITAT image has higher resolution than the conventional microwave imaging. Because the velocity of acoustic wave in the soft tissue is 1.5mm/us, thermo-acoustic signals at megahertz can provide millimeter or better spatial resolution (Generally, the irradiated microwave in MITAT system works in the frequency range of 400~3000MHz, and the corresponding wavelength is in the range of 100~750mm, while the received ultrasound is in the frequency range of 200~2000KHz, and the corresponding wavelength is in the range of 0.75~7.5m m). Furthermore, a MITAT system utilizes non-ionizing microwave pulse radiation; therefore it causes less harm to human body compared with the X-ray imaging system [4].

However, due to the heterogeneous of the biologic tissue, the induced thermo-acoustic wave propagating in it is affected. The imaging algorithm, such as Back-Projection (BP), neglected this situation will decrease the contrast and resolution of the image. Time Reversal Mirror (TRM) technique [5] is a combination based on wave propagation and array signals process and can well employ the propagation information and achieve much better resolution especially in the complex environments. It has the ability to effectively suppress system noise due to the characteristics of its spatial-time matched filtering. Further more, the statistic stability due to the self-average character of TRM technique makes it has the capability to decrease the effects induced by random fluctuating of the media parameter. So this technique is a good choice for biomedical tomography.

In this paper, we present the Time Reversal Mirror technique based on Pseudo-Spectrum Time Domain method for the microwave-induced thermo-acoustic imaging of biological tissue. Numerical simulations are given to illustrate the validity of our methods.

II .Theoretical Background

The thermo-acoustic effect refers to the induction of acoustic radiation following temperature elevation due to absorption of microwave pulse energy. The magn- itude of thermo-acoustic response is proportional to th- e local absorbed power density and thermoelastic properties of the imaged tissue. The thermo-acoustic theory has been discussed in many literature reviews such as [6]. Here, we briefly review only the fundam- ental equations. The pressure p ( r ^v , t ) at position r ^v and time t in an acoustically homogeneous medium in respond to a heat source H ( r ^v ′ , t ′ ) obeys the following equation [7]:

∇ ² p ( r ^v , t ) - ^{1 ∂ 2} p ( r ^v , t ) = - ^{β ∂} H ( r ^v ′ , t ′ ) , (1) c ² ∂ t ² C p ∂ t

Where ( r ^v ′ , t ′ ) and ( r ^v , t ) mean that spatial and time coordinates of source and the ultrasound transducer respectively. H ( r ^v ′ , t ′ ) is the heating function defined as the thermal energy deposited by the energy source per time and volume, β is the isobaric volume expansion coefficient, C _p is the heat capacity. c is the velocity of ultrasound wave, in most biologic tissue, its value is about 1.5mm/us. The heating function H ( r ^v ′ , t ′ ) can be written as the product of a spatial absorption function and a temporal illumination function of the microwave source:

H ( r ^v ′ , t ′ ) = I 0 A ( r ^v ′ ) ⋅ η ( t ′ ) (2)

Where I ₀ is the amplitude of the microwave pulse,

η ( t ′ ) is the temporal profile of the microwave pulse. A ( r ^v ′ ) is the spatial distribution of the microwave energy absorbed in the tissues. If the microwave pulse duration is short enough so that both heat and temporal stress are confined to the size of the desired resolutionlimited voxel [8], one may assume the excitation to be a delta function in time

η ( t ′ ) = δ ( t ′ ) , (3)

So, the solution of equation (1) based on the free space Green’s function in the time domain can be expressed as:

p ( r ^v ) = G ( r ^v , r ^v ′ ) ⊗ p_a ( r ^v ) ,          (4)

Where G(rv, rv′) is Green’s function, and the integral volume is a sphere defined by|rv-rv′|=|t-t′|⋅c, ⊗ is the convolute operator, and pa (rv) = βI0 ∫∫∫ A(rv′)drv , (5) cρ |rv-rv′|=|t-t′|c

Where p_a ( r ^v ) depicts observed pressure induced by the A ( r ^v ′ ) enclosed in the sphere | r ^v - r ^v ′ | = | t - t ′ | ⋅ c . Obviously, p_a ( r ^v ) can be got by initial pressure defined in| r ^v - r ^v ′ | → 0 case:

p_a ( r ^v ) =    ∫∫∫    p _δ ( r ^v ′ ) dr ^v

|rv-rv′|=|t-t′|c pδ(rv′) = βI0 A(rv′) |rv→rv′ cρ

The purpose of MITAT is to reconstruct the spatial distribution of the microwave energy absorbed function A ( r ^v ′ ) based on the received ultrasound pressure wave p ( r ^v , t ) , and we can see from the equation (6) that the relationship between the initial pressure p _δ ( r ^v ′ ) and the microwave energy deposition A ( r ^v ′ ) distribution is proportional. So in MITAT system, the reconstruction of the distributions of the absorbed microwave energies A ( r ^v ′ ) in biologic tissue is equivalent to the reconstruction of the distributions of the microwave induced thermo-acoustic sources p _δ ( r ^v ′ ) .

Ш.Тше Reversal Mirror

MITAT is an imaging problem for targets in multiple physic field, the noise accompanying with measurement is inevitable. Therefore, the studies of MITAT focus on how to acquire signal with high SNR and on the image algorithms with high stability for random parameter. The TRM technique which combines the process of wave propagating and array signal processing has become a hot point in recent years. It has significant meaning for applying TRM in MITAT.

So there have two processes in a time reversal procedure includes the source propagates in the media, and the received signals are re-emitted into the same media. The first process can be depicted by equation (4), the re-emitting and refocus imaging process can be depicted by a convolution between the p(rv)and Green’s function again pT(rv)=G(rv,rv′)⊗pδ(rv′)⊗G(rv′,rv), (7)

For the spatial reciprocity of the Green’s function, equation (7) can be rewritten as

P^T ( r ^v ) = p _δ ( r ^v ′ ) ⊗Γ ( r ^v , r ^v ′ )

Γ ( r ^v , r ^v ′ ) = G ( r ^v , r ^v ′ ) ⊗ G ( r ^v ′ , r ^v )    ^,

Where Γ ( r ^v , r ^v ′ ) means the system function of the T-RM process.

• A.    Spatial-temporal filtering of TRM

Generally, the measured signals have noise in an actual MITAT system, so the received signals in forward procedure should be modified as p's (v) = Ps (v) + n ,              (9)

Where n is an additive noise function, equation (8) should be modified in terms of the received signals model defined in equation (9), pT (v) = ps (v) ®r(v, v)       , (10)

= p _δ ( r ^v ′ ) ⊗ R ( r ^v , r ^v ′ ) + n ⊗ G ( r ^v , r ^v ′ )

Where R ( r ^v , r ^v ′ ) is the self correlation function of Γ ( r ^v , r ^v ′ ) . For the ultrasonic wave p _δ ( r ^v ′ ) propagating through the system defined by G ( x_f , x_p ) function and the received signals propagating again this system in reversal time sequence, the refocused point is enhanced N times due to the in-phase accumulation by time reversal procedure, while the noise has not this character contrarily. In fact, the maximum of the refocu- sed function p^T ( r ^v ′ ) can be got at r ^v = r ^v ′ due to the f-unction R ( r ^v , r ^v ′ ) . This character of the TRM technique is usually called spatial-temporal filtering.

• B.    Self-average character of TRM

Self-average is another character of TRM, which means that time reversal procedure has good statistical stability for a heterogeneous media. Given the maximum of the ultrasonic wavelength is far less than the distance between source and the receiver, self-average is determined by the independent of the function Γ ⁾ ( r ^v , r ^v ′ , ω ) for different frequency « 1 , m 2 [9] :

Where E { ⋅ } is the operator of expectation. And there has ,

E { Γ ( r ^v , r ^v ′ , t )²} =

E { ^+∞ d ω ₁ ^+∞ d ω ₂ e ^{- i ( ω 1 + ω 2) t} Γ ⁾ ( r ^v , r ^v ′ , ω ₁ ) ⋅Γ ⁾ ( r ^v , r ^v ′ , ω ₂ )} -∞      -∞

≈ ^+∞ d ω ₁ ^+∞ d ω ₂ e ^{- i ( ω 1 + ω 2) t}E { Γ ⁾ ( r ^v , r ^v ′ , ω ₁ )} ⋅ -∞      -∞

E { Γ ⁾ ( r ^v , r ^v ′ , ω 2 )} = E ²{ Γ ( r ^v , r ^v ′ , t )}

For an arbitrary small plus value γ >  0 , the relationship between the function Γ ( r , r ′ , t ) and its expectation can be deduced by Chebyshev’s inequality:

Prob {| Γ ( r ^v , r ^v ′ , t ) - E { Γ ( r ^v , r ^v ′ , t )} | > γ } ≤

E {[ Г ( r , V , t ) - E { Г ( r , V ,t )}]²} « 0 ^,

γ ²

The equation (12) depicts that there has the time and frequency domain equivalent relationship for the expectation of function Γ . And Γ ( r ^v , r ^v ′ , t ) ≈ E { Γ ( r ^v , r ^v ′ , t )} can be deduced by the equation (13), that is, Γ ( r ^v , r ^v ′ , t ) has the self-average character in statistics.

W. Pseudo-Spectrum Time Domain Method

The core of the TRM is the system function Γ ( r ^v , r ^v ′ , t ) . The exact Green’s function is difficult to present by analytical method in terms of the complexity in an actual media. However the numerical method is feasible for its flexibility in modeling and computer efficiency. In this paper, the numeric method Pseudo-Spectrum Time Domain (PSTD) is employed to complete the forward core in a TRM procedure, and the PSTD can also numerical presents the Green’s function of the complex biologic tissue easily [10], [11]. The Pseudo-S pectrum time domain method is a higher order approxi- mation method in the whole domain which can use the FFT to calculate the spatial derivatives.

For the finite difference time domain (FDTD) method uses one order partial differential equation, the equation (1) should be rewritten as:

∂ v ^v      1

= - ∇ p

^{∂ t      ρ} ,              (14)

^{∂ p} = - K ∇⋅ v ^v ∂ t

Where K = ρ c ² , v ^v is the velocity of elastic media cub defined in Fig. 1, which equals to ∂ u_i / ∂ t . The num-uerical system function can get by discretizing equatio- n (14) into N=Nx×Ny ( Nx, Ny are total nodes along x and y directions, respectively) orthogonal coordinates.

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