Breast Cancer Detection by Scattering of Electromagnetic Waves

Transcription

1 57 MSAS'2006 Breast Cancer Detection by Scattering of Electromagnetic Waves F. Seydou & T. Seppänen University of Oulu Department of Electrical and Information Engineering P.O. Box 3000, 9040 Oulu Finland [seydou, tapio.seppanen]ee.oulu.fi Abstract. In this paper we discuss the problem of finding the dielectric parameters for the tumor region in microwave breast cancer imaging. We use fixed frequencies and boundary integral equations in the optimization method. Different realistic breast models are considered and discussed. Introduction Although the incidence of breast cancer in under-developed regions is small at 20 per compared to 90 per people in the West, the mortality rate seen among women in those regions is disproportionately high compared to the incident rate []. Their tumors tend to be very aggressive with short periods of time between the onset of symptoms and diagnosis []. Despite improvements due to progress in early detection, still breast cancer is by far the second most frequent cause of cancer deaths in women today (after lung cancer) and is the most common cancer among women, excluding non-melanoma skin cancers [2]. According to the World Health Organization, more than.2 million people will be diagnosed with breast cancer each year worldwide and over will die from the disease [2]. By looking at these numbers it is clear that any research in this area is strongly motivated. Therefore, a cheap and powerful solution must be found to remedy the problem of global mortality from breast tumors. In particular, complementary and/or alternative techniques are highly desirable. An increasing interest of research in this field has been observed during the last decades and medical imaging is considered as the most effective way of diagnostic breast tumors. Among current clinical and experimental breast-cancer imaging technologies, X-ray mammography, which exposes women to ionizing radiation, is widely used and, in general, considered as an accepted method. However, a part from the ionizing radiation and painful breast compression, mammography suffers from several limitations. It has high false-positive and false-negative rates. One major reason of this high rate is the difficulty to image dense breasts and the problem in detecting breast tumors at their earlier stages [3]. Other non-ionizing imaging methods exist today. Modalities like Magnetic Resonance Imaging with contrast enhancement and Ultrasound Imaging are used, however those techniques are not suitable for large scale screening program [4]. As an alternative and complementary modality for breast imaging, microwave (MW) techniques have been proposed over the last few years [5] and is considered to be one of the needed imaging modalities in the future. The physical basis of microwave imaging in this application is the high contrast between the dielectric properties of the healthy breast tissue and the malignant tumors at microwave frequencies, and that it is a non-ionizing method which probably will be rather inexpensive. Moreover, after major research efforts, it was observed that microwave imaging has a strong potential to detecting tumors at their earlier stages [5]. The goal in MW imaging is therefore to detect, localize and characterize hidden tumors in the breast using electromagnetic waves at microwave frequencies. Mathematically, we have a nonlinear inverse scattering problem in which a given cost functional is minimized via an iterative algorithm. The challenge is that the inverse problem is difficult to solve, due to two unpleasant facts: it is nonlinear and, more seriously, it falls in a group of problems called ill-posed, that is the solution-if it exists at all-does not depend on the data. Previous research into the use of microwave imaging for breast tumors has mostly

2 MSAS' focused on time-domain techniques. However, there exist frequency-domain techniques which attempt to solve the full non-linear inverse problem and have shown successful reconstructions of subsurface biological materials [5], [3]. The goal with this work is to develop a microwave imaging system, based on the dual space method [6], able to detect breast tumor for fixed frequencies, using boundary integral equations. We utilize realistic dielectric and conductivity values for skin, internal tissue and tumors, and generate synthetic data to test the suitability of the inversion algorithm. 2 The Breast model and problem formulation Skin Breast Tissue Tumor Figure : The breast model used in the simulations A breast model for nonmagnetic isotropic media is considered. The breast tissue is embedded in a 2 mm layer of skin, and contains tumor inside (Fig. ), with complex medium permittivity parameter ɛ = ɛ r + iσ/ɛ 0 ω, where ɛ 0 is the permittivity of free space, ɛ r is the relative permittivity, σ is the conductivity, and ω is the angular frequency. The original three-dimensional Maxwell equations could be converted into an equivalent two-dimensional formulation in the (x, y) plane. The electric field is either perpendicular to the plane (TM mode) or lies in the plane (TE mode). The total field can be characterized by a single scalar function u, which represents either the E z or the H z component for the case of TM or TE polarization, respectively. The whole space is separated into four homogeneous regions Ω out (the outer region), Ω skin (the skin region), Ω breast (the breast tissue region) and Ω tumor (the tumor region), characterized by the complex permittivity parameter ɛ given as piecewise constant function in each region as follows: ɛ out x Ω out ɛ skin x Ω skin ɛ(x) =. ɛ breast x Ω breat ɛ tumor x Ω tumor 2

3 59 MSAS'2006 In each region the function u is written as u(x) = u s + u i u skin u breast u tumor x Ω out x Ω skin x Ω breat x Ω tumor where u s is the scattered field and u i represents the incident field generated by exterior sources. The total field must satisfy the following conditions: () the Helmholtz equation u + k 2 ɛu = 0, in R 2 ; (2) a set of continuity conditions on the interfaces between regions: ( ) u u s +u i s = u skin and ρ ν + ui = u skin on the interface between the outer and the skin regions (Γ ), ν ν u skin = u breat and and ρ 2 u skin ν = u breast ν on the interface between the skin and the breast regions (Γ 2 ), u breast = u tumor and ρ 3 u breast ν = u tumor ν and (3) the Sommerfeld radiation condition for the scattered field on the interface between the breast and the tumor regions (Γ 3 ); lim r( r r ikɛ out)u s = 0, r = x. Here k = ω/c (c is the light velocity in the vacuum); u/ ν is the normal derivative; ν is the outward normal unit vector; and the coefficient ρ j, j =, 2, 3, are given as ρ j = for TM polarization case, whereas for the TE case we have ρ = ɛ skin /ɛ out, ρ 2 = ɛ breast /ɛ skin and ρ 3 = ɛ tumor /ɛ breast. Time dependence is adopted as exp( iωt) and omitted throughout the paper In the sequel we use as incident field u i (x) = exp(ikx d) with incident direction d, d =. The direct problem is to find u for given the breast geometry, the permittivity ɛ, the frequency ω and the incident direction d. We are interested in the inverse problem, which is to determine ɛ and/or the tumor region Ω tumor from the knowledge of the scattered field u s at long distances from inhomogeneities, i.e. the far field pattern u. It can be shown (cf. [7]) that the scattered field u s has the asymptotic behavior u s (x) = exp(ik x ) { u (ˆx; d) + O( } x x ), x, ˆx = x x. () 3 The dual space method for solving the inverse scattering problem The dual space method is to solve the inverse problem as follows (cf. [6], [7]). We assume that we have approximate measurements of the far field pattern for given incident waves and wave directions. In addition, we assume that a domain B, with boundary B, is a priori known such that the breast and skin 3

4 MSAS' regions are contained in its interior. First, from the measured data (u ), we need to find g p (θ) solutions to the first kind integral equation π π [u (θ; α) iγ exp(ika sin(θ α))] g(θ) dθ = γ exp(ipα i pπ 2 ), (2) for P p P, P N, where γ = 2/(πk) exp( iπ/4), the constant a is a positive real number assumed to satisfy the following conditions: the domain B is contained in the disc of radius a, and J l (ka) 0 for every l, with J l being the Bessel function of order l. Now, for u being the far field pattern corresponding to the direct problem, setting h p (θ) := g p (θ π), equation (2) is satisfied if and only if there exists a total field w p = w i p + w s p that satisfies where and w i p(x) = π π v p (x) = i a w p + k 2 ɛw p = 0 in B (3) w p w i p v p u p = 0 and ν (w p w i p v p u p ) = 0 on B, (4) u i (x; α)h p (α) dα, w s p(x) = y =a π π u s (x; α)h p (α) dα, u p (r; θ) = H p () (kr) exp(ipθ), ( ) H () 0 (k x y )(Rh p) a y ds(y), (Rh)(φ) := i l h l exp(ilφ), (5) where h l is the lth Fourier coefficient of h and H () p is the Hankel function of first kind and order p. The method can be summarized in the following two steps: Step. Solve the integral equation (2) for given u. This equation is ill-posed and is solved via Tikhonov regularization method. Step 2. Given the approximated solution of (2), we solve the inverse problem by minimizing the functional J(q, Ω) = w p w i p v p u p ν (w p w i p v p u p ) on B, where q and Ω represent the sought permittivity and tumor region, respectively. 4 Numerical simulation and results Table : Parameter values for different frequencies [8], [9], [0] Example Frequency ɛ out ɛ skin ɛ breast ɛ tumor 800 MHz (air) i i i GHz i (water) i i i 3 6 GHz (air) i i i In this section we present our numerical results. We only consider the TE-polarization case, the TM mode being similar. We use synthetic data, i.e., for a given wave number k, permittivity ɛ, incident 4

5 6 MSAS'2006 directions d j, j =, 2,, 64, and the geometry of the breast model, we solve the direct problem to obtain an approximation of the far field pattern u,j, j =, 2,, 64,. The approximated far field is used to try and recover the permittivity or the tumor region using the algorithm presented in the previous section. We have used boundary integral method (cf. []) for solving the direct problem and for the approximation of the functional J in Step 2. It is interesting to analyze the functional J, which contains the necessary information in the three different frequencies (cf. Table ). In the numerical computation we have used the parameter values in Table, and the tumor region is a disc of radius 2 mm for the computation of the far field pattern. In Fig. 2, Fig. 4 and Fig. 6 we plot the functional J against the radius R of the tumor region for Examples, 2 and 3, respectively. In all cases the global minimum is clearly seen at R = 2 mm. We observe however that the higher the frequency the more we have local minima. When trying to plot the functional against the real and imaginary parts of the permittivity in the tumor region (Fig. 3, Fig. 5 and Fig. 7) we see that we have clear minimum at the values of the permittivity that were used in the forward solver. But in this case we do not have local minima. 5 x J(q) R Figure 2: The functional J(R) against the radius of the tumor region for Example 5

6 MSAS' Figure 3: The functional J(q) against the real and imaginary parts of the permittivity in the tumor region for Example 6 x J(q) R Figure 4: The functional J(R) against the radius of the tumor region for Example 2 6

7 63 MSAS'2006 Figure 5: The functional J(q) against the real and imaginary parts of the permittivity in the tumor region for Example 2.5 x 0 7 J(q) R Figure 6: The functional J(R) against the radius of the tumor region for Example 3 7

8 MSAS' Figure 7: The functional J(q) against the real and imaginary parts of the permittivity in the tumor region for Example 3 References [] A. Fregene, L.A. Newman, Breast cancer in cub-saharan Africa: now does it relate to breast cancer in African-American women?, Cancer 03, (2005) [2] [3] Q. H. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, Active microwave imaging I-2-D forward and inverse scattering methods, IEEE Trans. Microwave Theory Tech., 50, 23-33, (2002). [4] M. Stabel and H. Aichinger, Recent Developments in Breast Imaging, Phys.Med. Biol., 4, , (996). [5] E.C. Fear, S.C. Hagness, P.M. Meaney, M. Okoniewski, M.A. Stuchly Enhancing breast tumor detection with near-field imaging, IEEE Microwave Magazine, 3, (2002) [6] F. Seydou, Profile inversion in scattering theory: the TE case, J. Comput. Appl. Math (200). [7] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second ed., Springer, Berlin/Heidelberg/New York, 998. [8] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack and K.D. Paulsen, A clinical prototype for active microwave imaging of the breast, IEEE Trans. Microwave Theory Tech., 48, , (2000). [9] X. Li, S. K. Davis, S. C. Hagness, D. W. van der Weide and B. D. Van Veen, Microwave imaging via spacetime beamforming: experimental investigation of tumor detection in multilayer breast phantoms, IEEE Trans. Microwave Theory Tech., 52, , (2004). [0] X. Li, P.M. Meaney and K.D. Paulsen, Conformal microwave imaging for breast cancer detection, IEEE Trans. Microwave Theory Tech., 5, 79-86, (2003). [] F. Seydou, R. Duraiswami, N.A. Gumerov& T. Seppänen, TM electromagnetic scattering from multilayered dielectric bodies - numerical solution, ACES Journal (2004). 8