Level Set Contouring for Breast Tumor in Sonography

The echogenicity, echotexture, shape, and contour of a lesion are revealed to be effective sonographic features for physicians to identify a tumor as either benign or malignant. Automatic contouring for breast tumors in sonography may assist physicians without relevant experience, in making correct diagnoses. This study develops an efficient method for automatically detecting contours of breast tumors in sonography. First, a sophisticated preprocessing filter reduces the noise, but preserves the shape and contrast of the breast tumor. An adaptive initial contouring method is then performed to obtain an approximate circular contour of the tumor. Finally, the deformation-based level set segmentation automatically extracts the precise contours of breast tumors from ultrasound (US) images. The proposed contouring method evaluates US images from 118 patients with breast tumors. The contouring results, obtained with computer simulation, reveal that the proposed method always identifies similar contours to those obtained with manual sketching. The proposed method provides robust and fast automatic contouring for breast US images. The potential role of this approach might save much of the time required to sketch a precise contour with very high stability.

Appendices

Appendix 1

Let I(x, y) denote an input image. The MCDE equation is given as

$$ f_{t} = {\left| {\nabla f} \right|}\nabla \cdot c{\left( {{\left| {\nabla f} \right|}} \right)}\frac{{\nabla f}} {{{\left| {\nabla f} \right|}}}, $$

(4)

where f = f(x,y,t) and $f{\left( {x,y,0} \right)} = I{\left( {x,y} \right)}$. Progressively smoothed versions of the image are obtained by choosing progressively greater values of t from the solution. The conductance function c(·) is monotonically decreasing and $c{\left( {{\left| {\nabla f} \right|}} \right)} = {k^{2} } \mathord{\left/ {\vphantom {{k^{2} } {{\left( {k^{2} + {\left| {\nabla f} \right|}^{2} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {k^{2} + {\left| {\nabla f} \right|}^{2} } \right)}}$, where k is a constant parameter used to determine the contrast of edges.

Appendix 2

The basic concept of level set approach is to express a closed curve as a set of 2-dimension planar curve function ΓΓ(t), which consists of the zero level set point at time t.18 Let Γ(t = 0) be a closed initial planar curve; instead of propagating the curve directly, it embeds the curve as the zero level set of a higher order function ϕ called the level set function. The function ϕ is defined by

$$ \phi {\left( {x,t = 0} \right)} = \pm d, $$

(5)

where x is a point in Euclidean plane R ² and d is the distance from x to the initial contour Γ(0). The sign is chosen if the point x is outside (+) or inside (−) the initial contour. Thus, the initial function ϕ(x,t = 0) has the property of

$$ \Gamma {\left( 0 \right)} = {\left( {x\left| {\phi {\left( {x,t = 0} \right)} = 0} \right.} \right)}. $$

(6)

The objective is to produce an equation for the evolving function ϕ(x,t), which contains the embedded motion of Γ(t) as the level set {ϕ = 0}. Because the evolving function ϕ is always zero on the propagating hypersurface, no matter how much time t is, ϕ(x(t),t) is zero. Now, Γ(t) can be expressed as ϕ(x,t) = 0, by the chain rule, the equation can be expressed as

$$ \frac{{\partial \phi }} {{\partial t}} + \nabla \phi {\left( {x{\left( t \right)},t} \right)} \cdot x\prime {\left( t \right)} = 0. $$

(7)

Suppose F is a speed function in direction of normal vector in planar curve, then $x\prime {\left( t \right)} \cdot \frac{{\nabla \phi }} {{{\left| {\nabla \phi } \right|}}} = F,$ where ${\left| {\nabla \phi } \right|}$ represents the absolute gradient value of level set function. Thus, the level set evolution equation can be defined as

$$ \frac{{\partial \phi }} {{\partial t}} + F{\left| {\nabla \phi } \right|} = 0, $$

(8)

with a given value of the initial function ϕ(x,t = 0). The speed function F at any one point is based solely on the input intensity u ₀ at that point:29

$$ F = - {\left( {\alpha D{\left( {u_{0} } \right)} + {\left( {1 - \alpha } \right)}\nabla \cdot \frac{{\nabla \phi }} {{{\left| {\nabla \phi } \right|}}}} \right)} $$

(9)

and

$$ D{\left( {u_{0} } \right)} = \frac{{{\left( {U - L} \right)}}} {2} - {\left( {u_{0} - \frac{{{\left( {U + L} \right)}}} {2}} \right)}, $$

(10)

where α is a free curvature parameter that controls the degree of smoothness, and U and L denote the adaptive maximal and minimal intensities in the identified ROI subimage, respectively. Figure 10 shows the surface propagation of an initial curve and the accompany movement of the level set function ϕ.

Fig 10

Level set curve propagation: (a) the initial curve and the corresponding surface, (b) the curve and the corresponding surface at time t.