システム/制御/情報，Vol. 64, No. 3, pp. 87–91, 202087||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||解説
Multi-static Inverse Wave Scattering Theory
and Microwave Mammography
Kenjiro Kimura* and Noriaki Kimura†||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

1.

Introduction

molecular level, is ideal for visualizing breast cancer
tumours.
However, microwave mammography cannot be achieved if one important problem is not ﬁrst addressed:
the issue of inverse wave scattering. The challengeof visualizing breast cancer tumours when breasts areirradiated with microwave, referred to as the “scatterer,” lies in the fact the microwave are scatteredin various directions by breast cancer tumours withhigh relative permittivity. In other words, if we could
develop a mathematical theory and algorithm thatvisualizes the three-dimensional structures of a scatterer based on multi-static (multi-static) scattering
data obtained on a curved surface such as the surface
of a breast, the use of microwave mammography asa tool in early detection could be realized [5]. In ourresearch, we have achieved to establish the world’s
ﬁrst multi-static inverse scattering theory against theunresolved issue.

With the highest incidence occurring in women,
breast cancer causes about 520,000 deaths worldwide[1] and about 14,000 deaths in Japan alone [2]. If
discovered early, the 5-year survival rate is 93% [3].
Throughout the history of development of breast cancer screening technology, there is no doubt that Xray mammography has survived as the most superiorbreast-cancer-screening technique in comparison toother techniques, including ultrasound, nuclear magnetic resonance imaging, and positron emission tomography. Unfortunately, despite advances in theﬁeld, X-ray mammography is still deﬁcient when itcomes to early and accurate detection of tumours,
especially in cases where breast tissue densities arehigher than average. This is because when collagenﬁbres are compact and highly concentrated, a condition known as “dense breasts,” X-rays cannot produce the contrast of the breast tumor due to boththe large X-ray absorption and its large ﬂuctuationof the collagen density, thus smaller cancer tumours
are not easily detected. Since dense breasts are foundin about 79% of women under 50 in Asia (61% in
Caucasians, 57% in Black, and 51% in Hispanic) [4],a new breast-cancer-imaging technique is desperately
needed.

2.

3.

Four-dimensional Partial Diﬀerence Equation of Scattering Field

In this solution, it is assumed that the visualization target is a cone with a linear generatrix (Fig. 1);
thus, antenna elements are linearly arranged. Let us
consider the tangent plane of a cone, including its array antennas. On this tangent plane, we deﬁne thex axis on the reference plane, the axis of the antennas as the y axis, and the z axis is seen as being
perpendicular to these two axes. Thus, the x axis
rotates on the reference plane by θ degrees around
the Z axis from the X axis. Assuming that the hypothetical tangent plane rotates and the scattering dataobtained for all θ, we present the theory for reconstructing three-dimensional structures within a conefrom multi-static time-series data on the cone’s surface
If we deﬁne the multi-static coordinate y1 , y2 on
the y axis, a reconstruction theory can be establishedon the fact that data are only obtained at one point(x = 0,y1 ,y2 ) for the x coordinate along the tangent
plane.
Below, we describe a method for obtaining therelevant diﬀerential operator. Let us think of a situation in which one-dimensional array antenna (Fig. 2)
shifts to the x axis direction along the x−y plane. As
shown in Fig. 2, let us consider a situation whereina microwave emitted from the point P1 (x,y1 ,z) is reﬂected at the point P (ξ,η,ζ) and received at the point
P2 (x,y2 ,z). When Point P moves throughout Range

Electromagnetic Properties ofBreast Cancer

Breasts are mostly fat tissues by volume; the aforementioned collagen ﬁbres and mammary glands, which
serve to generate and transport breast milk, are foundin these fat tissues. Unfortunately, cancer tumours
can inﬁltrate the abovementioned mammary glands,
where they have easy access to the surrounding tissues
and are well supplied with blood capillaries. Interestingly, when this is viewed in light of the theory ofelectromagnetic properties, most breasts largely comprise insulating macromolecules with low relative permittivity, whereas breast-cancer tumours have a highrelative permittivity. If we consider such properties,it becomes clear that the microwave range, with relatively long wavelengths that do not absorb well at the∗Kobe University
Integral Geometry Science. IncKey Words : microwave mammography, analytical solution,
multi-static inverse scattering theory, dense breasts,
relative permittivity.
†–7–

システム/制御/情報

88

第 64 巻

第3号(2020)

Then, we obtain the following equation about thekernel function ϕ:[1{∆4 − 2(ik)2 }2 − ∂y1 2 ∂y2 2 +4]2224(ik) (∂y1 + ∂y2 ) − (ik) ϕ = 0(4)Though this equation is derived under the assumption of steady state, it can be easily expanded to include an unsteady state by substituting a variable asseen below:
1(5)−ik → ∂tc
Where c is the speed of light. Ultimately, we obtainthe following equation:
4{∆4 2 − 2 (∂t 2 ∂x 2 + ∂t 2 ∂z 2 ) − 4∂y21 ∂y2 2 }ϕ = 0cFig. 1

Linear antenna array and a cone(6)

∆4 = ∂ x 2 + ∂ y 1 2 + ∂ y 2 2 + ∂ z 2
By applying the derivative to the integral kernel, φsatisﬁes the above-described partial diﬀerential equation. This equation is a four-dimensional psuedo-wave
equation consisting of ﬁve variables (t,x,y1 ,y2 ,z).

4.

Analytical Solution andThree-dimensional Structure Image of ScatterLet us now solve this equation using Fourier transform. First, we perform multiple Fourier transform of
φ with respect to t,x,y1 ,y2Fig. 2

One-dimensional array antennaD, the signal received at P2 can be expressed as follows:∫∫ ikρ1 ikρ2eeε(ξ,η,ζ)dξdηdζ
φ(x,y1 ,y2 ,z) =ρ
ρ21D√
(1)
ρ1 = (x − ξ)2 + (y1 − η)2 + (z − ζ)2ρ2 =√φ̃(kx ,ky1 ,ky2 ,z,ω) =∫∞−∞∫∞∫∞∫∞eiωt dt −∞ eiky1 y1 dy1 −∞ eiky2 y2 dy2 −∞ eikx φ(x,y1 ,y2 ,z,t)dx(7)

If we use Dz for the diﬀerentiation with respect toz, we obtain the following formula:2{(Dz 2 − kx 2 − ky1 2 − ky2 2 ) +4k 2 (Dz 2 − kx 2 ) − 4ky1 2 ky2 2 }φ̃ = 0(x − ξ)2 + (y2 − η)2 + (z − ζ)2where k is the wave number of the microwave, and thefactor of time is assumed to be relative to exp(−iωt).
Let the kernel function in the integration term of theabove Equation ϕ, then:(8)
where ω = c k is used, the four fundamental solutions
for this equation are√ √√2i{ ( k2 −ky1 2 + k2 −ky2 2 ) −kx 2 }z
E1 = eeikρ1 eikρ2(2)ρ1 ρ2Thus, a partial diﬀerential equation is obtained in
ϕ=E2 = ewhich this equation or the diﬀerentiation and integration of this equation with respect to ξ,η,ζ gives
an asymptotic solution. To that end, higher orderterms for 1/ρ created by the diﬀerentiation should be
ignored in the calculation; we then deﬁne the abbreviation for diﬀerentiation as−i{

E3 = e√ √√2( k2 −ky1 2 + k2 −ky2 2 ) −kx 2 }z(9)√ √√2i{ ( k2 −ky1 2 − k2 −ky2 2 ) −kx 2 }z

E4 = e−i{√ √√2( k2 −ky1 2 − k2 −ky2 2 ) −kx 2 }zConsidering that the temporal factor is e−iωt , phaseis added by the microwave path and that the microwave reﬂected by an object bounces back to the∂∂∂∂∂→ ∂t ,→ ∂x ,→ ∂z (3)→ ∂ y1 ,→ ∂ y2 ,∂t
∂x
∂y1
∂y2
∂z–8–

89

Kimura and Kimura: Multi-static Inverse Wave Scattering Theory and Microwave Mammography

1= lim [y1 →y (2π)3measurement surface, we obtain E1 as the only meaningful solution. Therefore,φ̃(kx ,ky1 ,ky2 ,z,k) =a(kx ,ky1 ,ky2 ,k)e

√ √√2( k2 −ky1 2 + k2 −ky2 2 ) −kx 2 }zi{(10)dkx dky1 dky2

If we apply this to the above equation with z = 0,we then obtainφR (y1 ,y2 ,k)δ(x) =−∞ −∞ −∞=(13)

Applying Fourier transform by (x,y1 ,y2 ) on bothsides of the equation, we obtain the following:

(14)

∫ ∞

lim [φ(x,y1 ,y,z,k)]dk−∞ y1 →yei{∫ ∞∫ ∞∫ ∞−∞ −∞ −∞√ √√2( k2 −ky1 2 + k2 −ky2 2 ) −kx 2 }z

∫ ∞1=lim [3y→y(2π)
1−∞y1 →ydkx dky1 dky2 ]dk∫ ∞∫ ∞∫ ∞−∞ −∞ −∞e−i(kx x+ky1 y1 +ky2 y) eikz z φ̃R (ky1 ,ky2 ,k,θ))(dkdkx dky1 dky2 dkz ]dkz−∞ −∞ −∞e−i(kx x+ky1 y1 +ky2 y) a(kx ,ky1 ,ky2 ,k)e(16)e−i(kx x+ky1 y1 +ky2 y) a(kx ,ky1 ,ky2 ,k)∫ ∞∫ ∞∫ ∞}{√ √√2( k2 −ky1 2 + k2 −ky2 2 ) −kx 2k(kx 2 + kz 2 )1=lim [
3y→y(2π)−∞ 1φ(x,y,y,z,k) = lim [φ(x,y1 ,y,z,k)]iz

2(ky1 2 − ky2 2 )+ 2(ky1 2 + ky2 2 )kx 2 + kz 2√
√k z k 2 − k y1 2 k 2 − k y2 2
k x 2 + kz 2 +∫ ∞Thus, we obtain the solution φ(x,y1 ,y2 ,z,k) of thepartial diﬀerential equation. We can then take Tracewith respect to y1 ,y2 of the function φ(x,y1 ,y2 ,z,k),speciﬁcally, if the visual function is deﬁned as y1 →y,y2 → y, where the term k,z is ﬁxed:

1= lim [y1 →y (2π)312√−∞

e−i(kx x+ky1 y1 +ky2 y2 )a(kx ,ky1 ,ky2 ,k) = φ̃R (ky1 ,ky2 ,k)φ̃R (ky1 ,ky2 ,k)dkx dky1 dky2 ]Next, we perform Fourier transform on Equation(15) with respect to k, and apply t = 0. Then, we obtain the visualization function ρ(r,θ) in the localized
coordinate system at the angle θ in Equation (17). We
describe θ dependency of φ̃R (ky1 ,ky2 ,k) positively asφ̃R (ky1 ,ky2 ,k,θ)∫ ∞ρ(x,y,z,θ) =φ(x,y,y,z,k)dk(12)a(kx ,ky1 ,ky2 ,k)dkx dky1 dky2−∞ −∞ −∞edk=dkz

Let us now consider that measured data only exists for x = 0; thus, the following equation stands:∫ ∞∫ ∞∫ ∞∫ ∞∫ ∞∫ ∞−i(kx x+ky1 y1 +ky2 y) ikz z

k=(11)

1(2π)3dkx dky1 dky2 ]We can now convert the variables to integrate withrespect to k:√()2√√k 2 − k y1 2 + k 2 − k y2 2 − k x 2kz =−∞ −∞ −∞φ(x,y1 ,y2 ,0,k) = φR (y1 ,y2 ,k)δ(x)φ̃R (ky1 ,ky2 ,k)(15)∫ ∞∫ ∞∫ ∞√ √√2( k2 −ky1 2 + k2 −ky2 2 ) −kx 2 }ze}
{√ √√
2iz
( k2 −ky1 2 + k2 −ky2 2 ) −kx 2ee−i(kx x+ky1 y1 +ky2 y2 ) a(kx ,ky1 ,ky2 ,k)ei{

e1y1 →y (2π)3φ(x,y1 ,y2 ,z,k) =1(2π)3−∞ −∞ −∞−i(kx x+ky1 y1 +ky2 y)= lim [

By applying Fourier transform to the above equation, we obtain the solution φ for the wave Equation(6):∫ ∞∫ ∞∫ ∞(17)By integrating results for each θ, we obtain imagesof three-dimensional structures.

dkx dky1 dky2 ]–9–システム/制御/情報90

5.

第 64 巻Conversion from Tangent Space
to the Whole Coordinate

Then the reconstruction function for angle θ becomes∫ ∞∫ ∞∫ ∞∫ ∞1
ρ(x,y,z,θ) =lim
2(2π)3 yo →0 −∞ −∞ −∞ −∞ei(ξX+ηY +kyo yo +ςZ) φ̃R (ky1 ,ky2 ,k,θ)(18)

Where α is the angle between the coordinate axis zof the tangent plane and the Z axis of the reference
plane. The equations converting from (x,y ′ ) to (X,Y) are
x = X cosθ + Y sinθy ′ = −X sinθ + Y cosθ

(20)

12kz

√

k 2 − k y1 2√
k 2 − k y2 2k(kx 2 + kz 2 )√2k x 2 + kz 2 +(ky1 2 − ky2 2 )+ 2(ky1 2 + ky2 2 )kx 2 + kz 2(26)(21)

Since kx ,ky1 ,ky2 , and kz are functions with respect to ξ,η,ς, and kyo as in Equations (22) and (25),these can be converted by performing Fourier transform on data localized in the coordinate of angle θon the whole coordinate. Finally, the reconstruction
image is obtained by integrating with respect to theangle θ:∫ 2π
P(X,Y,Z) =ρ(x,y,z,θ)dθ(22)The above equation can be written as∫ ∞∫ ∞∫ ∞∫ ∞1lim
ρ(x,y,z,θ) =2(2π)3 yo →0 −∞ −∞ −∞ −∞0(23)=We perform the following variable conversions inthe spectral range:∫ 2π ∫ ∞ ∫ ∞ ∫ ∞ ∫ ∞1lim
2(2π)3 yo →0 0−∞ −∞ −∞ −∞ei(ξX+ηY +kyo yo +ςZ) φ̃R (ky1 ,ky2 ,k,θ))(dkdξdηdkyo dςdθdkz

ξ = −kx cosθ + (kye cosα + kz sinα)sinθ
(24)

ς = −kye sinα + kz cosαThis inverse transformation gives the following equations:kx = −ξ cosθ − ηsinθkye = (ξ sinθ − ηcosθ)cosα − ς sinα)dkdξdηdkyo dςdkz

k=We introduce the following variables:η = −kx sinθ − (kye cosα + kz sinα)cosθ∫ ∞∫ ∞∫ ∞∫ ∞1lim2(2π)3 yo →0 −∞ −∞ −∞ −∞dk=dkzWe apply these conversion formulas to the following equation:∫ ∞∫ ∞∫ ∞∫ ∞1
ρ(x,y,z,θ) =(2π)3 −∞ −∞ −∞ −∞x+kye y+kyo yo −kz z)e−i(kx)φ̃R (ky1 ,ky2 ,k,θ)(dkdkx dkye dkyo dkz
dkz=(z = X sinαsinθ − Y sinαcosθ + Z cosαkye = ky1 + ky2
kyo = ky1 − ky2)
dkdkx dkye dkyo dkz
dkz

ei(ξX+ηY +kyo yo +ςZ) φ̃R (ky1 ,ky2 ,k,θ)x = X cosθ + Y sinθx+ky1 y+ky2 y) ikz ze−i(kx)eφ̃R (ky1 ,ky2 ,k,θ)(dkdkx dky1 dky2 dkz
dkz

((19)

Summarizing the above gives

y = −X cosαsinθ + Y cosαcosθ + Z sinα(2020)kz = (ξ sinθ − ηcosθ)sinα + ς cosα

We then convert the results calculated in the tangent space to the whole coordinate (X, Y, Z). If weconsider the projection of y axis on the (X, Y) plane
as y ′ , the following equations are established:y = y ′ cosα + Z sinαz = −y ′ sinα + Z cosα

第3号

(25)
– 10 –(27)

In this paper, we describe the multi-static inverse
scattering theory that we have developed successfully
as the core application for microwave mammography[5–7]. In clinical trials using a 1–14 GHz ultra-wideband
microwave mammography system [6,7] with over 350
breast cancer patients and healthy individuals, according to the physics, high detection sensitivity was
shown in dense breasts. Fig. 3 shows the demonstration our microwave mammography image. In the fu-

Kimura and Kimura: Multi-static Inverse Wave Scattering Theory and Microwave Mammography

Fig. 391

Microwave mammography image[5]ture, we hope that this system will become widespread,thus helping many breast-cancer patients.
(2019 年 5 月 30 日受付)

ing field tomography for next-generation breast cancer screening; 31th International Microprocesses andNanotechnology Conference, Physics T. J. S. o. A.,
Ed, The Japan Society of Applied Physics (2018)

ReferencesAuthors[1] World Health Organization(WHO): Global Cancer
Statistics, International Agency for Research on Cancer (2012）[2] National Cancer Center Japan: Cancer registrationand statistics, Cancer information service.
[3] Joint investigation of survival rates of the National
Cancer Center for Adult Diseases Association（2019）[4] J. A. Tice, S. R. Cummings, R. S. Bindman, L.
Ichikawa, W. E. Barlow and K. Kerlikowske: Using
clinical factors and mammographic breast density toestimate breast cancer risk: Development and validation of a new predictive model; Ann Intern Med,
Vol. 148, No. 5, pp. 337–347 (2008)[5] K. Kimura and N. Kimura: Research and development of real-time microwave mammography;
ICT Innovation Forum 2016, Strategic Information
and Communications R&D Promotion Programme(SCOPE), Makuhari Messe International ConferenceCenter, pp. 26–27 (2016)[6] K. Kimura, A. Inagaki, S. Kono, H. Tanino, T.
Hashimoto, T. Sakuma, M. Miki, S. Takao, N.
Watanabe, Y. Konishi, K. Okamoto, H. Matsumoto,K. Yamagami, Y. Mima, K. Doi and N. Kimura: Development of microwave scattered field tomographic
imaging system and clinical trial results; the 26th
annual meeting of the Japanese Breast Cancer Society ＠ Kyoto International Conference Center, p.
259 (2018)[7] K. Kimura, A. Inagaki, Y. Mima, K. Doi, N. Kimura,N. Yamamoto, T. Tsuruhara, N. Watanabe, Y. Konishi, K. Okamoto, Y. Sachiko, H. Matsumoto andK. Yamagami: Development of microwave scatter-– 11 –Kenjiro Kimura
1978 年 11 月 28 日生．2006 年 3 月京都
大学大学院工学研究科電子工学専攻博士課程修了（工学博士）．2008 年神戸大学
大学院理学研究科化学専攻講師，2012 年同大学准教授，2018 年より同大学数理・データサイエンスセンター教授．場の逆解析理論と非破壊画像計測システムに関する研究に従事．世
界で初めて応用数学史上の未解決問題である波動散乱逆問
題の解析解の導出に成功．2017 年第一回日本医療研究開発大賞受賞，次世代乳がん検査技術としてマイクロ波マンモグ
ラフィを開発．ほか，表面科学会，応用物理学会，分子科学会，全国発明コンテストなど受賞多数．

Noriaki Kimura
1946 年 12 月 16 日生．1981 年東京大学
大学院工学研究科工学博士取得．2003 年
より 2012 年まで三井造船株式会社技術本部，2008 年蛍光フローサイトメータ装置の技術開発に関し計測自動学会にて論文賞受賞，2007 年より大阪大学大学院工学研究科三井造船共同研究講座招聘教授，2012 年より合同会
社 Integral Geometry Instruments（現，株式会社 IntegralGeometry Science）代表社員，2015 年株式会社 Integral
Geometry Science 代表取締役就任（現任），2016 年 5 月合同会社木村財団代表社員就任（現任）．