II. METHODS

Active microwave imaging approaches typically pose an inverse scattering problem, where multiple microwave transmitters illuminate an object and scattered fields in numerous locations are measured [15]. The shape of the object and spatial distribution of the complex permittivity are obtained from the transmitted (incident) and scattered (received) fields [7]. The solutions to most inverse scattering problems are difficult because of the relationship between object dimensions, discontinuity, separation, and contrast in properties of inhomogeneities compared to the wavelength, the wave undergoes multiple scattering within the object that is to be reconstructed. This makes the relationship between the measured scattered fields and object function nonlinear. Altogether, the inverse scattering approaches generally suffer from non uniqueness and multiple wrappings of the scattered field phases [16]; however, with smaller geometries, such as the breasts, these concerns are minimal.

MICROWAVE TOMOGRAPHIC IMAGING

Moving on to specific methods, we first explore tomographic image reconstruction. In tomographic systems, the object to be imaged is immersed in water, weak saline solutions, or some type of coupling medium that possess minimal electrical contrasts with that of the areas of interest; these areas are the breasts for our case. Antennas are scanned over planar or cylindrical surfaces and waves are transmitted and received by numerous antennas to reconstruct the object function (spatial distributions of the dielectric constants and or conductivity). From there, the image reconstruction involves iteratively matching the measured data and computed data. Computed data is compared to measured data and nonlinear reconstruction procedures are applied to obtain material updates of the model. Repetition of this procedure is performed until the measured data and computed data obtains convergence

One exhibition of microwave tomographic imaging experimental studies is performed in Kochi, India. Samples of normal and malignant breast tissues of four patients are collected from the Department of Surgery, Lourde Hospital, Kochi and are subjected to study within 30 minutes of mastectomy [13].

System

The 2D microwave imaging system is shown in Figure 4. Breast samples are supported on a PVC holder that is mounted on a circular platform capable of circular motion in a horizontal plane. The platform and samples are kept inside a tomographic chamber of radius 12 cm and height 30 cm that is coated inside with a suitable absorbing material. The chamber is also filled with a coupling medium.

Bowtie antennas are suspended and used for both transmission and reception of microwave energy. Measurements are done using HP 8510C network analyzer; interfaced with a Compaq work station SP 750 using a GPIB bus.

Figure 1. Experimental setup used in the Department of Electronics, Microwave Tomography and Materials Research Laboratory at the Cochin University of Science and Technology, Kochi, India [13].

The coupling medium is an essential part of acquiring a better resolution of reconstructed images. In this study the coupling medium that is used in corn syrup. Dielectric parameters of this material in the frequency range of 2000-4000 MHz are done using cavity perturbation technique [18], [19]. The results are compared with that of the breast tissue data [17] and good agreement is observed in [18]. The frequency range is of 2000-4000 MHz is adopted and the resonant frequency of the antenna used is 3000MHz. The corn syrup also conveniently includes the Industrial Scientific and Medical applications band of 2450 MHz. The dielectric permittivity and conductivity of corn syrup is displayed in Figure 5.

Figure 2. Dielectric permittivity and conductivity for corn syrup in frequency ranges 2000-4000 MHz [13].

The medium’s complex permittivity can be written as:

εr=εr'−jεr} {}εr=εr'−jεr} {} size 12{ε rSub { size 8{r} } =ε rSub { size 8{r} } ' - jε rSub { size 8{r} } "} {};(1)

εr'εr' size 12{ε rSub { size 8{r} } '} {} is the dielectric permittivity and

εr} {}εr} {} size 12{ε rSub { size 8{r} } "} {} is the dielectric loss of the medium.

The loss tangent is :

tanδ=εr/ε rSub { size 8{r} } '} {}tanδ=εr/ε rSub { size 8{r} } '} {} size 12{"tan"δ=ε rSub { size 8{r} } ""/"ε rSub { size 8{r} } '} {}(2)

The propagation constant is:

γ=jωμ0(σ+jωε)=α+jβγ=jωμ0(σ+jωε)=α+jβ size 12{γ= sqrt {j ital "ωμ" rSub { size 8{0} } \( σ+j ital "ωε" \) } =α+jβ} {}(3)

Where α is the attenuation factor and β is the phase factor.

The conductivity is given by:

σ=ωε0εr} {}σ=ωε0εr} {} size 12{σ= ital "ωε" rSub { size 8{0} } ε rSub { size 8{r} } "} {}(4)

When you substitute equations (1), (2) and (4) into (3) and simplify, you get:

α=2πfμ0ε0εr'[1+tan2δ−1]α=2πfμ0ε0εr'[1+tan2δ−1] size 12{α=2πf sqrt {μ rSub { size 8{0} } ε rSub { size 8{0} } ε rSub { size 8{r} } ' \[ sqrt {1+"tan" rSup { size 8{2} } δ} - 1 \] } } {}(5)

and

β=2πfμ0ε0εr'[1+tan2δ+1]β=2πfμ0ε0εr'[1+tan2δ+1] size 12{β=2πf sqrt {μ rSub { size 8{0} } ε rSub { size 8{0} } ε rSub { size 8{r} } ' \[ sqrt {1+"tan" rSup { size 8{2} } δ} +1 \] } } {}(6)

The total loss of a wave over a distance consists of dissipation loss LdissLdiss size 12{L rSub { size 8{ ital "diss"} } } {} due to conduction currents being excited in the medium and diffusion loss LdiffLdiff size 12{L rSub { size 8{ ital "diff"} } } {} due to the spherical spreading of energy [20]. They are given by:

Ldiss=20log10eαzLdiss=20log10eαz size 12{L rSub { size 8{ ital "diss"} } ="20""log" rSub { size 8{"10"} } e rSup { size 8{αz} } } {}(7)

Ldiff=20log10(βz)−29.14(dB)Ldiff=20log10(βz)−29.14(dB) size 12{L rSub { size 8{ ital "diff"} } ="20""log" rSub { size 8{"10"} } \( βz \) - "29" "." "14" \( ital "dB" \) } {}(8)

So the total loss is

Ltotal=Ldiss+LdiffLtotal=Ldiss+Ldiff size 12{L rSub { size 8{ ital "total"} } =L rSub { size 8{ ital "diss"} } +L rSub { size 8{ ital "diff"} } } {}(9)

The losses of corn syrup increase with frequency due to the increase of conductivity. Table 1 compares the loss parameters of distilled water and saline [20] with corn syrup at 3000 MHz, at a distance of 12 cm from the transmitter. The loss values are acceptable when compared to the loss parameters of conventional coupling medium like distilled water and saline [20].

Table 1. Propagation loss parameters of water, corn syrup, and saline at 3000 MHz 12 cm. from the transmitter [13].

The antenna design is a coplanar stripline fed bowtie antenna that generates TM01 mode that is designed for both transmission and reception of the microwave signals. The bandwidth is a major deciding factor in the antenna design because of the time domain approaches that are used. The antenna, in air, exhibits enhanced 2:1 VSWR bandwidth of approximately 46% in the operational band of 1850-3425 MHz with a return loss of approximately -53 dB. With corn syrup, the bandwidth is enhanced to 91% in the range of 1215-3810 MHz with resonant frequency at 2855 MHz and a return loss of -41 dB [13]. Figure 6 shows the radiation characteristics of the antenna.

Figure 3. Return loss versus frequency of the bowtie antenna [13].

As stated earlier, the sample tissues are acquired within 30 minutes of mastectomy. Cancerous tissues of approximately 0.5 cm in radius are inserted in normal tissues of approximately 1 cm radius of patient 1, sample 1. Patient 2 and 3, samples 2 and 3, consists of 4 tumorous inclusions of approximately 0.25 cm in radius and is inserted in normal tissues of approximately 1 cm radius. Patient 4, sample 4, contains scattered inclusions of cancerous tissues of radius 0.1 cm inserted in normal tissue of approximately 1 cm radius.

For the data acquisition, the breast samples are illuminated by a bowtie antenna at 3000 MHz. As shown in Figure 4, the transmit antenna is fixed at a radius of 6 cm on the circular rail while the receive antenna is rotated around the object at a radius of 6 cm. The platform of the mounted object then rotates from 0° to 360° in increments of 10° and the receive antenna is rotated from 30° to 330° in increments of 10°. For every 10° increment rotation of the platform, the receive antenna also makes an increment in 10°.

After acquisition of the data, the data is then analyzed and interpreted. The contrast in dielectrical properties of the object creates multiple scattering of the wave inside the object. A nonlinear inverse scattering problem formulated in terms of Fredholm integral equation of the second kind is then posed [21]. For an incident TM wave, the total electric field at the receiver is given by,

φ(r)=φinc,b(r)+ω2μ∫SdSgb(r,r')δε(r')φ(r')φ(r)=φinc,b(r)+ω2μ∫SdSgb(r,r')δε(r')φ(r') size 12{φ \( r \) =φ rSub { size 8{ ital "inc",b} } \( r \) +ω rSup { size 8{2} } μ Int cSub { size 8{S} } { ital "dSg" rSub { size 8{b} } \( r,r' \) ital "δε" \( r' \) φ \( r' \) } } {}(10)

Where r stands for a point in the measurement domain and r’ for the object domain. φinc,b(r)φinc,b(r) size 12{φ rSub { size 8{ ital "inc",b} } \( r \) } {} is the incident field in the presence of the background inhomogeneity and the integral term is the scattered field due to the dielectric contrast between the scatterer and the background medium [22].

δε(r')=ε(r')−εb(r')δε(r')=ε(r')−εb(r') size 12{ ital "δε" \( r' \) =ε \( r' \) - ε rSub { size 8{b} } \( r' \) } {}(11)

Equation (11) is called the object function, the Green’s function and gb(r,r’) the total electric field inside the scatterer. Equation (10) is used for both the forward and inverse solutions. In the forward problem, both the medium properties and the domain of inhomogeneity are known and the equation is solved to obtain the total electric field. In the inverse problem, scattered fields are measured at discrete points and the medium properties are the unknowns to be determined.

The problem is then linearized using distorted Born approximations [22] by replacing φ(r)φ(r) size 12{φ \( r \) } {} with φinc,b(r)φinc,b(r) size 12{φ rSub { size 8{ ital "inc",b} } \( r \) } {}. As the background medium is inhomogeneous, Green’s function is solved numerically [23]. Discretization of the integral equation in the inverse problem yields a vector representation of the scattered field and the object profile. An optimization technique is then adopted to minimize the error by minimizing a cost functional. The instability of this problem is thus avoided and a sufficient solution is provided. The obtained δεδε size 12{ ital "δε"} {} is used to improve εb(r)εb(r) size 12{ε rSub { size 8{b} } \( r \) } {} which in turn is used to update the parameters in Equation (10). The iteration is continued until convergence is reached. The imaging area is restricted to 16 x 16 pixels due to computational complexity. The sampling rate considered is 0.1λ.

CONFOCAL MICROWAVE IMAGING (CMI)

A more recent active microwave breast imaging approach is confocal microwave imaging (CMI) [14], [26-29]. This approach avoids complex image reconstruction algorithms by synthetically focusing reflections from the breast. As the illuminating signal is ultrawideband, this translates to simply time-shifting and summing signals. This imaging method also differs from tomographic microwave imaging in terms of the overall goal. This goal of this method is to identify the presence and general location of the significant scatterers in the breast, rather than to recover the complete dielectric properties of the breast [7].

Confocal microwave imaging focuses on backscattered signals to create images that indicate regions of significant scattering. When compared with normal tissues, the malignant tissues have larger microwave scattering cross-sections of comparable size at the frequencies of interest. Spatial focusing overcomes challenges of breast heterogeneity, thereby permitting millimeter-sized tumors to be detected and localized by CMI [14].

The example of confocal microwave imaging (CMI) is also performed by the same group who performed the microwave tomographical imaging experiments we described earlier. The same four samples are used during this method. The same antenna is used for both the transmission and reception of microwave energy. The data is acquired and collected by illuminating the tissue sample with the same wide band bowtie antenna. The antenna is rotated at a radius of 6 cm and measurements are taken for every 10° increments of the antenna.

A time-shift-and-add algorithm is then applied to the set of recorded pulses to enhance the returns from high contrast regions to reduce clutter. This involves computing the time delay for the roundtrip between each antenna position to a point in the domain of interest, then adding the corresponding portions of the time signals recorded at each antenna position [13].

To validate the experimental investigation performing CMI, the theoretical analysis is done using the finite difference time domain (FDTD) method. The nature of the dielectric medium is incorporated in the constitutive FDTD equations using first order Debye dispersion relation [24], [25].

The geometry under consideration consists of an infinitely long multilayered cylinder of dispersive dielectric nature with its axis in the z direction. In the z direction the scatterer geometry is assumed to be uniform, and hence the field variations are zero. When this assumption is incorporated in the Maxwell’s curl equations, the variations of the magnetic and electric fields only exist with respect to the y and x spatial coordinate variables and with respect to the time parameter. The source excitation is a constant current source confined in the xz plane polarized in the negative z direction. So the x and y components of the electric current density do not exist. Thus the problem is treated as 2D with only electric field in the z direction and magnetic fields in the x and y directions present. The electric fields in a non magnetic medium are given by:

Hx(i,j,t+1)=Hx(i,j,t)−dtμ0dy(Ez(i,j,t)−Ez(i,j−1,t))Hx(i,j,t+1)=Hx(i,j,t)−dtμ0dy(Ez(i,j,t)−Ez(i,j−1,t)) size 12{H rSub { size 8{x} } \( i,j,t+1 \) =H rSub { size 8{x} } \( i,j,t \) - { { ital "dt"} over {μ rSub { size 8{0} } ital "dy"} } \( E rSub { size 8{z} } \( i,j,t \) - E rSub { size 8{z} } \( i,j - 1,t \) \) } {} (12)

Hy(i,j,t+1)=Hy(i,j,t)−dtμ0dx(Ez(i,j,t)−Ez(i,j−1,t))Hy(i,j,t+1)=Hy(i,j,t)−dtμ0dx(Ez(i,j,t)−Ez(i,j−1,t)) size 12{H rSub { size 8{y} } \( i,j,t+1 \) =H rSub { size 8{y} } \( i,j,t \) - { { ital "dt"} over {μ rSub { size 8{0} } ital "dx"} } \( E rSub { size 8{z} } \( i,j,t \) - E rSub { size 8{z} } \( i,j - 1,t \) \) } {} (13)

E z ( i , j , t + 1 ) = ε ∞ ε ∞ + χ 0 ( i , j ) E z ( i , j , t ) E z ( i , j , t + 1 ) = ε ∞ ε ∞ + χ 0 ( i , j ) E z ( i , j , t ) size 12{E rSub { size 8{z} } \( i,j,t+1 \) = { {ε rSub { size 8{ infinity } } } over {ε rSub { size 8{ infinity } } +χ rSub { size 8{0} } \( i,j \) } } E rSub { size 8{z} } \( i,j,t \) } {}

+ 1 ε ∞ + χ 0 ( i , j ) ∑ m = 0 t − 1 E z ( i , j , t − m ) Δχ m ( i , j ) + dt ε ∞ + χ 0 ( i , j ) ε 0 dx ( H y ( i + 1, j , t ) − H y ( i , j , t ) ) − dt ε ∞ + χ 0 ( i , j ) ε 0 dy ( H x ( i , j + 1, t ) − H x ( i , j , t ) ) + 1 ε ∞ + χ 0 ( i , j ) ∑ m = 0 t − 1 E z ( i , j , t − m ) Δχ m ( i , j ) + dt ε ∞ + χ 0 ( i , j ) ε 0 dx ( H y ( i + 1, j , t ) − H y ( i , j , t ) ) − dt ε ∞ + χ 0 ( i , j ) ε 0 dy ( H x ( i , j + 1, t ) − H x ( i , j , t ) ) alignl { stack { size 12{+ { {1} over {ε rSub { size 8{ infinity } } +χ rSub { size 8{0} } \( i,j \) } } Sum cSub { size 8{m=0} } cSup { size 8{t - 1} } {E rSub { size 8{z} } \( i,j,t - m \) } Δχ rSub { size 8{m} } \( i,j \) } {} # + { { ital "dt"} over {ε rSub { size 8{ infinity } } +χ rSub { size 8{0} } \( i,j \) ε rSub { size 8{0} } ital "dx"} } \( H rSub { size 8{y} } \( i+1,j,t \) - H rSub { size 8{y} } \( i,j,t \) \) {} # - { { ital "dt"} over {ε rSub { size 8{ infinity } } +χ rSub { size 8{0} } \( i,j \) ε rSub { size 8{0} } ital "dy"} } \( H rSub { size 8{x} } \( i,j+1,t \) - H rSub { size 8{x} } \( i,j,t \) \) {} } } {}

(14)

Where

χ0(i,j)=(εs−ε∞)(1−exp(−dt/t0))χ0(i,j)=(εs−ε∞)(1−exp(−dt/t0)) size 12{χ rSub { size 8{0} } \( i,j \) = \( ε rSub { size 8{s} } - ε rSub { size 8{ infinity } } \) \( 1 - "exp" \( - ital "dt"/t rSub { size 8{0} } \) \) } {} (15)

is the susceptibility function.

Δχm(i,j)=(εs−ε∞)(exp(−mdt/t0)(1−exp(−dt/t0))2Δχm(i,j)=(εs−ε∞)(exp(−mdt/t0)(1−exp(−dt/t0))2 size 12{Δχ rSub { size 8{m} } \( i,j \) = \( ε rSub { size 8{s} } - ε rSub { size 8{ infinity } } \) \( "exp" \( - ital "mdt"/t rSub { size 8{0} } \) \( 1 - "exp" \( - ital "dt"/t rSub { size 8{0} } \) \) rSup { size 8{2} } } {} (16)

Where εsεs size 12{ε rSub { size 8{s} } } {} is the static permittivity, ε∞ε∞ size 12{ε rSub { size 8{ infinity } } } {} is optical permittivity, and t0t0 size 12{t rSub { size 8{0} } } {} is the dielectric relaxation time. A Gaussian pulse of half width T as 18 ps with time delay of 54 ps is selected as the source of excitation. The computational domain is discretized as 120 x 120 Yee cells. Space step of 1 mm and time stop of 6.05 ps are chosen to ensure propagation of the waves in the entire domain. Mur’s second order absorbing boundary conditions are applied to terminate the TDTD grid [13]