TOWARDS THE DETECTION OF BREAST CANCER VIA EIT

OHIN KWON, JIN KEUN SEO, AND JEONG-ROCK YOON

The inverse problem by EIT(Electrical Impedance Tomography) is imaging the conductivity distribution inside a body from current-voltage relations on the outer surface. EIT uses a chain of electrodes attached to the surface where different current patterns are applied and their corresponding voltage potentials are measured.

Difficulties of this problem are due to its inherent ill-posedness and nonlinearity. Many authors proposed various reconstruction algorithms, most of which are based on laborious least square algorithms and Newton-type iteration schemes. In these methods, one has to take a good initial guess a priori. Without taking a good initial guess, to identify the conductivity distribution seems to be pessimistic: One needs tremendous computational costs and time to get a close image to the true solution, since Newton-type iteration schemes may not converge to an approximate solution unless the initial guess is close to the true solution. Evidently, success or failure of Newton-type procedures heavily depend on how one can take a good initial guess. In spite of its importance, both the mathematical theory and the numerical algorithm for giving a good initial guess are at an early stage(almost a beginning) as far as the authors know.

The purpose of this work is to develop a real time algorithm of giving a good initial guess which is reasonably close to the true solution. If we are able to extract the core informations such as the location or the size of the anomaly, it will be very useful to select an appropriate initial guess. In this paper, we initiate a real time method of searching for the location of anomalies which is immediately determined from the pattern of a simple weighted combination of the input current and the output voltage. Combing it with the suggested algorithm in the earlier paper[?] for finding the size of anomalies, we can give a good initial guess for the inverse conductivity problem for piecewise constant conductivity distributions.

Let us set up our problem mathematically. We consider an electrically conducting body occupied in the region Ω in ℝ^{n}(n = 2, 3), of the form

γ(x) = 1 + μχ_{D}(x) (−1 < μ ≠ 0 < ∞),

where D is a simply connected subdomain of Ω and χ_{D} denotes the characteristic function of D. Then the relationship between the applied current g across the boundary ∂Ω and the corresponding measured voltage potential f := u|_{∂Ω} on ∂Ω is determined by solving the Neumann problem:

div ((1 + μχ_{D})∇u) = 0 in Ω,

\frac{∂u}{∂ν} = g on ∂Ω, and ∫_{∂Ω} u dσ = 0.

(0.1)

Here ν denotes the unit outward normal vector on ∂Ω and g is chosen so that the compatibility condition ∫_{∂Ω} g dσ = 0 is satisfied. The inverse problem with one measurement is to determine D from one pair of data (g, f) and our goal is to develop a real time algorithm for finding a reasonable initial guess for D.

Figure 1. A numerical experiment of the location search and the size estimation algorithm in the case of μ = 5 and with 5% random noise in measured voltage potential f, when the current pattern is given by g = e⃗_{2} · ν.

Our location search algorithm is based on simple aspects of the function H(·; g, f) which is computed directly from the data g and f,

H(x; g, f) := ∫_{∂Ω} \frac{∂Φ(x − y)}{∂ν(y)}f(y) dσ_{y} − ∫_{∂Ω} Φ(x − y)g(y) dσ_{y}, x ∈ ℝ^{n} \ ∂Ω,  (0.2)

where Φ is the fundamental solution of Laplacian given by Φ(x − y) := 1/(2π) log | x − y | for n = 2 and Φ(x − y) := −1/(4π)| x − y |^{−1} for n = 3, respectively. Now we are in a position to present our location search algorithm.

Location search algorithm. As an applied current pattern, we choose g(x) = a⃗ · ν(x) for some fixed constant vector a⃗. Take two observation regions Σ_{1} and Σ_{2} contained in ℝ^{3} \ Ω̄, given by

Σ_{1} := a line parallel to a⃗ and Σ_{2} := a plane normal to a⃗.  (0.3)

Next we find two points P_{i} ∈ Σ_{i}(i = 1, 2) so that H(P_{1}; g, f) = 0 and

H(P_{2}; g, f) = \begin{cases}min_{x∈Σ_{2}} H(x; g, f) & if μ > 0, \\ max_{x∈Σ_{2}} H(x; g, f) & if μ < 0. \end{cases}

Finally, we draw the corresponding plane Π_{1}(P_{1}) and the corresponding line Π_{2}(P_{2}) given by

Π_{1}(P_{1}) := {x | a⃗ · (x − P_{1}) = 0} and Π_{2}(P_{2}) := {x | (x − P_{2}) is parallel to a⃗}.  (0.4)

Then the intersecting point P of Π_{1}(P_{1}) ∩ Π_{2}(P_{2}) is close to the domain D. For two dimensional case, the same argument holds if one adjust Σ_{2} and Π_{1}(P_{1}) to be simply lines in ℝ^{2}.

We give the rigorous justification for this algorithm in two cases: Either when D is a small ball, or when the conductivity μ is close to zero without geometric restrictions on D. Although the basic idea of our algorithm is simple, several technical arguments are needed for its proof. Combining this location search algorithm with the size estimation algorithm proposed in [?], we are able to select an appropriate initial guess. Figure 1 demonstrates the remarkable achievement of our algorithm.

Though our numerical algorithm works satisfactorily for quite general cases of D and μ, the theoretical justifications are given for the restricted cases. Thus we expect these restrictions to be relaxed in the theoretical sense. And moreover it is worthwhile extending our method to the case when D has many components.

   E-mail address: oikwon@math.snu.ac.kr
   E-mail address: seoj@bubble.yonsei.ac.kr
   E-mail address: jryoon@numer.kaist.ac.kr