Development of Analytical and Experimental Tools for Microwave Breast Imaging Research Fuqiang Gao Department of Electrical and Computer Engineering, Madison, WI [email protected]Introduction The incidence rate of breast cancer is the highest of any cancer among women in the United States, and its mortality rate is second only to lung cancer. It is well known that early detection is crucial to successful treatment of breast cancer. X-ray mammography has been the standard clinical tools for detecting and treating breast cancer for many years. However, the sensitivity and specificity of this method are significantly lower in women with extremely dense breasts, and it also involves breast compression and exposure to ionizing. Hence, there is a critical need of alternative non- ionizing molecular imaging technologies that are capable of 3D tomographic breast imaging and suitable for low-cost cancer screening particularly in high-risk patients. Microwave imaging, which utilizes the dielectric contrast between malignant and normal breast tissues, has shown great potential to meet these needs in recent studies. My master’s project involves development of both analytical and experimental tools for microwave breast imaging research. In the analytical part, the potential of incorporating both dielectric and elastic contrasts for improving breast tumor detection is studied. This work resulted in a comment that is published in IEEE Antennas and Wireless Propagation Letters. In the experimental part, a technique to fabricate 3D realistic breast phantom is successfully developed. The fabricated breast phantoms can be used for rigorous pre-clinical validation of microwave breast imaging systems. A manuscript based on this work has been submitted to a journal for peer-review. Hence, the main body of this report consists of the aforementioned two materials. The appendix includes the MATLAB codes used to generate the results in the IEEE AWPL comment.
Transcript
Development of Analytical and Experimental Tools for Microwave Breast Imaging Research
Fuqiang GaoDepartment of Electrical and Computer Engineering, Madison, WI
The incidence rate of breast cancer is the highest of any cancer among women in the United States, and its mortality rate is second only to lung cancer. It is well known that early detection is crucial to successful treatment of breast cancer. X-ray mammography has been the standard clinical tools for detecting and treating breast cancer for many years. However, the sensitivity and specificity of this method are significantly lower in women with extremely dense breasts, and it also involves breast compression and exposure to ionizing. Hence, there is a critical need of alternative non-ionizing molecular imaging technologies that are capable of 3D tomographic breast imaging and suitable for low-cost cancer screening particularly in high-risk patients. Microwave imaging, which utilizes the dielectric contrast between malignant and normal breast tissues, has shown great potential to meet these needs in recent studies.
My master’s project involves development of both analytical and experimental tools for microwave breast imaging research. In the analytical part, the potential of incorporating both dielectric and elastic contrasts for improving breast tumor detection is studied. This work resulted in a comment that is published in IEEE Antennas and Wireless Propagation Letters. In the experimental part, a technique to fabricate 3D realistic breast phantom is successfully developed. The fabricated breast phantoms can be used for rigorous pre-clinical validation of microwave breast imaging systems. A manuscript based on this work has been submitted to a journal for peer-review. Hence, the main body of this report consists of the aforementioned two materials. The appendix includes the MATLAB codes used to generate the results in the IEEE AWPL comment.
Copyright (c) 2010 IEEE. Personal use is permitted. For any other purposes, Permission must be obtained from the IEEE by emailing [email protected].
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 9, 2010 1
Comments and Replies
Comments on “Numerical Study of Microwave Scatteringin Breast Tissue via Coupled Dielectric
and Elastic Contrasts”
Fuqiang Gao, Student Member, IEEE,Susan C. Hagness, Fellow, IEEE, andBarry D. Van Veen, Fellow, IEEE
In [1], the dielectric properties in the 2-D ��� validation study ofFig. 1 were stated incorrectly as (�� � ��� �� � ����� S/m) for thecylinder and (�� � ��� �� � ��� S/m) for the background. Theactual properties for the results presented in Fig. 1 in [1] are (�� �
��� �� � � S/m) and (�� � ��� �� � � S/m), respectively. Also, therewas an inconsistency in the definition of � � �
�. The corrected Fig. 1is presented here.A validation for the lossy case (cylinder: �� � ��� �� � ����� S/m;
background: �� � ��� �� � ��� S/m) is presented in Fig. 2.The excellent agreement between the two curves in Fig. 2 supportsour conclusion in [1] regarding the accuracy of the FDTD-SBCimplementation.The peak applied strain of Fig. 3 in [1] was stated incorrectly as 10%.
The correct peak applied strain is 8%.The results shown in Fig. 4 in [1] were obtained for lossless ver-
sions of the materials stated; that is, �� � � S/m for all tissues. Theresults based on the lossy material properties stated in [1] are presentedin Fig. 3. These results support our conclusion in [1] regarding the en-hancement of the overall microwave scattering contrast using the hy-brid modality.
REFERENCES
[1] M. Zhao, J. D. Shea, S. C. Hagness, D. W. van der Weide, B. D. VanVeen, and T. Varghese, “Numerical study of microwave scattering inbreast tissue via coupled dielectric and elastic contrasts,” IEEE An-tennas Wireless Propag. Lett., vol. 7, pp. 247–250, 2008.
Manuscript received July 26, 2010; accepted July 26, 2010. Date of publica-tion August 05, 2010; date of current version nulldate.F. Gao, S. C. Hagness, and B. D. Van Veen are with the Department of Elec-
trical and Computer Engineering, University of Wisconsin-Madison, Madison,WI 53706 USA (e-mail: [email protected]; [email protected]; [email protected]).Digital Object Identifier 10.1109/LAWP.2010.2063450
Fig. 1. Comparison of the amplitudes of the normalized Doppler componentcomputed via the finite-difference time-domain method with sheet boundaryconditions (FDTD-SBC) (dashed line) and the normalized difference betweenstationary scattered field solutions computed via FDTD with repositionedboundaries (FDTD-REP) (solid line). the 2-D scattering object is a deforming4-mm-diameter lossless circular cylinder undergoing a peak radial displace-ment of 0.45 mm.
Fig. 2. Comparison of the amplitudes of the normalized Doppler componentcomputed via FDTD-SBC (dashed line) and the normalized difference betweenstationary scattered field solutions computed via FDTD-REP (solid line). The2-D scattering object is a deforming 4-mm-diameter lossy circular cylinder un-dergoing a peak radial displacement of 0.45 mm.
Fig. 3 Computed Doppler components in the backward (� � ��� ) and for-ward (� � � ) scattered fields under different peak applied strains, for malig-nant and normal fibroglandular scatterers.
grey). In addition, the CT images show the variations in fibroglandular tissue density across the
four different phantoms. The small dark circles in the images are air bubbles (typically less than
1 mm in diameter) that are introduced unintentionally during fabrication of the TM materials. The
dielectric properties of the various TM materials inside the phantom were verified by making a
coronal cross-sectional cut through each phantom and making measurements at approximately 1
cm intervals along a trans-sectional line across the exposed coronal plane. Figures 5, 6 (a),(b)
show the dielectric properties in the 1-6 GHz range at various trans-sectional positions for class I
and class IV phantoms. This data confirms that a large dielectric contrast (>5:1 in some cases)
between fatty and fibroglandular content is present, as desired. Parts (c) and (d) of Figures 5 and
6 show the dielectric properties of the phantoms at 3 GHz as a function of position. These graphs
provide a spatial snapshot of the interior heterogeneity as well as the increasing fibroglandular
density with phantom class number. For example, a comparison of Figures 5(c) and 5(d) and
Figures 6(c) and 6(d) shows that there is significantly more higher-water-content TM material in
the class IV phantom (extremely dense) than the class I phantom (mostly fat), as desired.
These phantoms are well suited for experimental imaging tests. The phantoms are relatively
sturdy and are not easily damaged during gentle handling. In addition, the TM materials used to
construct these phantoms have been shown to be stable over a nine-week period as long as they
are not exposed to air for long periods of time [1]. The TM materials are also stable in oil (1:1
mixture of kerosene and safflower oil); this is an important feature for experiments that require
the phantoms to be immersed in a coupling medium.
4. CONCLUSIONS
We have described the construction of four different realistic breast phantoms, each representing
a different volumetric breast density classification. The phantoms were constructed using TM
materials that accurately represent the dielectric properties of various breast tissues. The
phantoms were heterogeneous, each with a skin layer and a complex network of fibroglandular
and fatty tissues. We verified the integrity of the internal structure of the phantom by obtaining
CT images. The dielectric properties of the phantom constituents were verified by cutting the
phantoms in half and measuring the dielectric properties of the phantom cross-section. These
phantoms were constructed using simple techniques and have long term stability when not
exposed to air for long periods of time. The techniques described here may be easily modified to
construct phantoms with malignant lesions.
ACKNOWLEDGMENTS
The authors would like to thank Christine Jaskowiak (University of Wisconsin, School of
Medicine and Public Health) for her assistance in obtaining the CT images. This work was
supported by the Department of Defense Breast Cancer Research Program under grant
W81XWH-07-1-0629 and the National Institutes of Health under grant R01CA112398 awarded
by the National Cancer Institute.
TABLE 1 Summary of Previously Published Work on Breast Phantoms for Microwave Imaging Applications
Materials used for construction
Breast shape
�r, �(S/m) of constituents Frequency (GHz)
Citation Skin Fat Fibroglandular Tumor
Corn syrup, NaCl, agar Cylinder
�r=23 �=0.2
�r=53.5 �=1.2 1.35 [5]
Corn syrup, agar Cylinder �r=42
�=0.67 �r=74 �=1.6 0.9 [6]
Silicone sheet, dough, alginate Cylinder
�r ~ 34.3 �=2-4
�r=4.5-4 �=0.06-0.2
�r ~ 43.7 �=5-8 1-7 [7]
Oil-in-gelatin Realistic �r=40-30
�=0.6-10.5 �r=8-7.1
�=0.05-1.1
1-11 [8]
Liquid, gelatin, polythene
Hemi-sphere
�r=35 � NA
�r=10 � NA
�r=27 � NA
�r=50 � NA 3 [9]
Dough, water Truncated
cone �r=80-64
�=0.1-15 �r=50-27 �=0.1-9 1-9 [10]
Isoparaffin, salt, sugar, water Realistic
�r=40.8 �=1.79
�r=5.3 �=0.13
2.5 [11]
Oil-in-gelatin Realistic �r=34-21 �=0.5-10
�r=7.5-5.5 �=0.01-1.8
�r=25-22.5 �=0.5-10
�r=47.5-30 �=0.5-015 0.05-13.51 [12]
Oil-in-gelatin Cylinder �r=46-35
�=12.5-18 �r=5-4.5
� ~ 1 �r=26-20 �=2.5-5
�r=55-40 �=15-21 0.5-8 [13]
Figure 1. Comparison of the dielectric properties of (a)-(b) fibroglandular, (c)-(d) heterogeneous
mix (both fibroglandular and fatty tissue), and (e)-(f) fatty tissue and the TM materials used to
construct the four different breast phantoms. The curves without vertical bars are the 75th (dashed
lines), 50th (solid lines), and 25th (dash-dotted lines) percentile breast tissue data reported in [2].
The solid curves with vertical bars are the average of 12 dielectric properties measurements made
on two TM material samples with the specified concentration of oil; the vertical bars span the
maximum and minimum property values at specific frequencies.
Figure 2. Comparison of the dielectric properties of wet and dry skin [15] and the 30%-oil TM
material. The solid curve with vertical bars is the average of 12 dielectric properties measurement
made on two 30%-oil TM material samples; the vertical bars span the maximum and minimum
property values at specific frequencies.
Figure 3. Photograph of a breast phantom after construction.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 4. CT images of sagittal (left) and coronal (right) cross-sections of four phantoms: (a)-(b)
class I (mostly fat), (c)-(d) class II (scattered fibroglandular), (e)-(f) class III (heterogeneously
dense), and (g)-(h) class IV (extremely dense). Features present in each phantom include a skin
layer and areola (medium grey), fat (black), heterogeneous mix (medium grey), and
fibroglandular (light grey) tissue.
Figure 5. Dielectric properties at nine positions along an interior trans-sectional line through the
class I phantom. (a)-(b) Dielectric properties across the 1-6 GHz frequency range. (c)-(d)
Dielectric properties at 3 GHz as a function of position.
Figure 6. Dielectric properties at nine positions along an interior trans-sectional line through the
class IV phantom. (a)-(b) Dielectric properties across the 1-6 GHz frequency range. (c)-(d)
Dielectric properties at 3 GHz as a function of position.
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2. M. Lazebnik, L. McCartney, D. Popovic, C. B. Watkins, M. J. Lindstrom, J. Harter, S. Sewall, A. Magliocco, J. H. Booske, M. Okoniewski and S. C. Hagness, A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries, Physics in Medicine and Biology 52 (2007), no. 10, 2637-2656.
3. K. Kerlikowske, L. Ichikawa, D. L. Miglioretti, D. S. M. Buist, P. M. Vacek, R. Smith-Bindman, B. Yankaskas, P. A. Carney and R. Ballard-Barbash, Longitudinal measurement of clinical mammographic breast density to improve estimation of breast cancer risk, Journal of the National Cancer Institute 99 (2007), no. 5, 386-395.
4. P. A. Carney, D. L. Miglioretti, B. C. Yankaskas, K. Kerlikowske, R. Rosenberg and C. M. Rutter, Individual and combined effects of age, breast density, and hormone replacement therapy use on the accuracy of screening mammography, Annals of Internal Medicine 138 (2003), no. 3, 168-175.
5. P. M. Meaney, N. K. Yagnamurthy and K. D. Paulsen, Pre-scaled two-parameter Gauss-Newton image reconstruction to reduce property recovery imbalance, Physics in Medicine and Biology 47 (2002), no. 7, 1101-1119.
6. D. Li, P. M. Meaney, T. D. Tosteson, S. Jiang, T. E. Kerner, T. O. McBride, B. W. Pogue, A. Hartov and K. D. Paulsen, Comparisons of three alternative breast modalities in a common phantom imaging experiment, Medical Physics 30 (2003), no. 8, 2194-2205.
7. J. M. Sill and E. C. Fear, Tissue sensing adaptive radar for breast cancer detection – experimental investigation of simple tumor models, IEEE Transactions on Microwave Theory and Techniques 53 (2005), no. 11, 3312-3319.
8. D. W. Winters, J. D. Shea, E. L. Madsen, G. R. Frank, B. D. Van Veen and S. C. Hagness, Estimating the breast surface using UWB microwave monostatic backscatter measurements, IEEE Transactions on Biomedical Engineering 55 (2008), no. 1, 247-256.
9. M. Klemm, J. A. Leendertz, D. Gibbins, I. J. Craddock, A. Preece and R. Benjamin, Microwave radar-based breast cancer detection: Imaging in inhomogeneous breast phantoms, IEEE Transactions Antennas and Wireless Propagation Letters 8 (2009), 1349-1352.
10. S. M. Salvador and G. Vecchi, Experimental tests of microwave breast cancer detection on phantoms, IEEE Transactions on Antennas and Propagation 57 (2009), no. 6, 1705-1712.
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14. A. Mashal, B. Sitharaman, X. Li, P. K. Avti, A. V. Sahakian, J. H. Booske and S. C. Hagness, Toward carbon-nanotube-based theranostic agents for microwave detection and treatment of breast cancer: Enhanced dielectric and heating response of tissue-mimicking materials, IEEE Trans Biomed Eng 57 (2010), no. 8, 1831-1834.
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Appendix 1. FDTD_SBC.m %This function calls the corresponding subroutines to perform the FDTD-SBC %algorithm function FDTD_SBC(nmax) [fi1 E_i Ei ez_boundary ez_obs]=background_UPML(nmax); [ez_boundarys ez_obst]=scatter_UPML(nmax); eoz=ez_boundarys; [fi2 Ed ez_obss Jz rc]=sbc(nmax,eoz); Ed_m=Ed; plot(fi1,10*log10(abs(Ed).^2/abs(E_i)^2)); 2. background_UPML.m %********************************************************************** % 2D scatter problem, without scatter % a current line source is launched 7.5cm away from the center. The % electric field is recorded at the points 4.0cm away from the center. the % update coefficients for the main grid use the media pointer, and the upml % update equations are only used on the boundary. "On the fly" DFT is taken % for both observation points and the source, so that the final Ez_b is % nomalized %********************************************************************** % Program author: Fuqiang Gao % Department of Electrical and Computer Engineering % University of Wisconsin-Madison % 1415 Engineering Drive % Madison, WI 53706-1691 % [email protected] %********************************************************************** function [fi E_i Ez_b ez_boundary ez_obs]=background_UPML(nmax) %********************************************************************** % Fundamental constants %********************************************************************** cc=2.99792458e8; %speed of light in free space muz=4.0*pi*1.0e-7; %permeability of free space epsz=1.0/(cc*cc*muz); %permittivity of free space %********************************************************************** % Source Parameters %********************************************************************** epsrb=44.0; sigrb=0; epsrs=50.0; sigrs=0; %Parameter for lossy case % epsrb=44.0; % sigrb=4.162; % epsrs=50.0;
D3hx=D3x*D3hx; D4hx=D4x*D4hx; D5hx=D5x*D5hx; D6hx=D6x*D6hx; D1hy=D1y*D1hy; D2hy=D2y*D2hy; D3hy=D3y*D3hy; D4hy=D4y*D4hy; D5hy=D5y*D5hy; D6hy=D6y*D6hy; %Get index of observation points I=ones(1,2); J=ones(1,2); index=1; for i=1:ie_tot for j=1:je_tot if (((i-i_cen)^2+(j-j_cen)^2)>=(0.04/delta)^2 && ((i-i_cen)^2+(j-j_cen)^2)<(0.04/delta+1)^2) I(index)=i; J(index)=j; index=index+1; end; end end %Get index of source locations Id=ones(1,2); Jd=ones(1,2); index=1; for i=1:ie_tot for j=1:je_tot if (((i-i_cen)^2+(j-j_cen)^2)>(0.002/delta+0.4)^2 && ((i-i_cen)^2+(j-j_cen)^2)<(0.002/delta+1.4)^2) Id(index)=i; Jd(index)=j; index=index+1; end; end end source=[i_src-i_cen 0]; phi=ones(1,size(Id,2)); for i=1:size(Id,2) a=[Id(i)-i_cen Jd(i)-j_cen]; l=sqrt(source(1)^2+source(2)^2)*sqrt(a(1)^2+a(2)^2); cfi=dot(a,source)/l; phi(i)=acos(cfi);
if a(2)<0 phi(i)=2*pi-phi(i); end end for i=1:length(phi)-1 for j=i+1:length(phi) if phi(j)<phi(i) temp=phi(i); phi(i)=phi(j); phi(j)=temp; temp=Id(i); Id(i)=Id(j); Id(j)=temp; temp=Jd(i); Jd(i)=Jd(j); Jd(j)=temp; end end end %*********************************************************************** % compute observation points angle %*********************************************************************** source=[i_src-i_cen 0]; fi=ones(1,size(I,2)); for i=1:size(I,2) a=[I(i)-i_cen J(i)-j_cen]; l=sqrt(source(1)^2+source(2)^2)*sqrt(a(1)^2+a(2)^2); cfi=dot(a,source)/l; fi(i)=acos(cfi); if a(2)<0 fi(i)=2*pi-fi(i); end end for i=1:size(fi,2)-1 for j=i+1:size(fi,2) if fi(j)<fi(i) temp=fi(i); fi(i)=fi(j); fi(j)=temp; temp=I(i); I(i)=I(j); I(j)=temp; temp=J(i); J(i)=J(j); J(j)=temp; end end
end %delete some observation points, avoid 'noisy' data i=1; while (i<length(fi)) if abs(fi(i)-fi(i+1))<0.02 fi(i+1)=[]; I(i+1)=[]; J(i+1)=[]; else i=i+1; end end eps=[epsrb epsrs]; sig=[sigrb sigrs]; mu=[1.0 1.0]; sigstar=[0.0 0.0]; %*********************************************************************** % Main Grid Coefficients %*********************************************************************** ca=(1.0-(dt*sig)./(2.0*epsz*eps))./(1.0+(dt*sig)./(2.0*epsz*eps)); cb=(dt./epsz./eps./delta)./(1.0+(dt*sig)./(2.0*epsz*eps)); da=(1.0-(sigstar*dt)./(2.0*muz*mu))./(1.0+(dt*sigstar)./(2.0*muz*mu)); db=(dt./muz./mu./delta)./(1.0+(dt*sigstar)./(2.0*muz*mu)); %*********************************************************************** % UPML Coefficients %*********************************************************************** %Update coefficients in PML region R0=exp(-16); %reflection error m=4; %order of polynomial grading d=upml*delta; %thickness of pml slabs; eta=sqrt(muz/epsrb/epsz); %x-direction sigxmax=-(m+1)*log(R0/100)/2/d/eta; kxmax=1.0; sigfactor=sigxmax/(m+1)/(d^m)/delta; kfactor=(kxmax-1)/(m+1)/(d^m)/delta; for i=1:upml %coefficients on the boundary x2=(upml-i+1.5)*delta; x1=(upml-i+0.5)*delta; sigx=(x2^(m+1)-x1^(m+1))*sigfactor; kx=1+(x2^(m+1)-x1^(m+1))*kfactor; C1=(2*epsrb*epsz*kx-sigx*dt)/(2*epsrb*epsz*kx+sigx*dt); C2=(2*epsrb*epsz*dt)/(2*epsrb*epsz*kx+sigx*dt);
D5=(2*epsrb*epsz*kx+sigx*dt); D6=(2*epsrb*epsz*kx-sigx*dt); C1ez(i,:)=C1; C1ez(ie_tot-i+1,:)=C1; C2ez(i,:)=C2; C2ez(ie_tot-i+1,:)=C2; D5hx(i,:)=D5; D5hx(ie_tot-i+1,:)=D5; D6hx(i,:)=D6; D6hx(ie_tot-i+1,:)=D6; %coefficients in the center x2=(upml-i+1)*delta; x1=(upml-i)*delta; sigx=(x2^(m+1)-x1^(m+1))*sigfactor; kx=1+(x2^(m+1)-x1^(m+1))*kfactor; D3=(2*epsrb*epsz*kx-sigx*dt)/(2*epsrb*epsz*kx+sigx*dt); D4=1.0/(2*epsrb*epsz*kx+sigx*dt)/muz; D3hy(i,:)=D3; D3hy(ih_tot-i+1,:)=D3; D4hy(i,:)=D4; D4hy(ih_tot-i+1,:)=D4; end % Model PEC walls C1ez(1,:)=-1.0; C1ez(ie_tot,:)=-1.0; C2ez(1,:)=0.0; C2ez(ie_tot,:)=0.0; %y-direction sigymax=-(m+1)*log(R0/100)/2/d/eta; kymax=1.0; sigfactor=sigymax/(m+1)/(d^m)/delta; kfactor=(kymax-1)/(m+1)/(d^m)/delta; for j=1:upml %coefficients on the boundary y2=(upml-j+1.5)*delta; y1=(upml-j+0.5)*delta; sigy=(y2^(m+1)-y1^(m+1))*sigfactor; ky=1+(y2^(m+1)-y1^(m+1))*kfactor; C3=(2*epsrb*epsz*ky-sigy*dt)/(2*epsrb*epsz*ky+sigy*dt); C4=1.0/(2*epsrb*epsz*ky+sigy*dt)/epsrb/epsz; D5=(2*epsrb*epsz*ky+sigy*dt); D6=(2*epsrb*epsz*ky-sigy*dt); C3ez(:,j)=C3;
C3ez(:,je_tot-j+1)=C3; C4ez(:,j)=C4; C4ez(:,je_tot-j+1)=C4; D5hy(:,j)=D5; D5hy(:,je_tot-j+1)=D5; D6hy(:,j)=D6; D6hy(:,je_tot-j+1)=D6; %coefficients in the center y2=(upml-j+1)*delta; y1=(upml-j)*delta; sigy=(y2^(m+1)-y1^(m+1))*sigfactor; ky=1+(y2^(m+1)-y1^(m+1))*kfactor; D1=(2*epsrb*epsz*ky-sigy*dt)/(2*epsrb*epsz*ky+sigy*dt); D2=2*epsrb*epsz*dt/(2*epsrb*epsz*ky+sigy*dt); D1hx(:,j)=D1; D1hx(:,jh_tot-j+1)=D1; D2hx(:,j)=D2; D2hx(:,jh_tot-j+1)=D2; end % Model PEC walls C3ez(:,1)=-1.0; C3ez(:,je_tot)=-1.0; C4ez(:,1)=0.0; C4ez(:,je_tot)=0.0; %*********************************************************************** % BEGIN TIME-STEPPING LOOP %*********************************************************************** ez_obs(nmax,size(I,2))=0.0; ez_boundary=zeros(nmax,length(Id)); %store ez field on boundary at every time step hx_boundary=zeros(nmax,length(Id)); %store ez field on boundary at every time step hy_boundary=zeros(nmax,length(Id)); %store ez field on boundary at every time step ftr(1,size(I,2))=0.0; fti(1,size(I,2))=0.0; E_i=0; for n=1:nmax %*********************************************************************** % Update electric fields %*********************************************************************** es=ez; ez(upml+1:ie_tot-upml,upml+1:je_tot-upml)=ca(MEDIAez(upml+1:ie_tot-upml,upml+1:je_tot-upml)).*ez(upml+1:ie_tot-upml,upml+1:je_tot-upml)+... cb(MEDIAez(upml+1:ie_tot-upml,upml+1:je_tot-upml)).*... (hy(upml+1:ie_tot-upml,upml+1:je_tot-upml)-hy(upml:ih_tot-upml,upml+1:je_tot-upml)+...
D6hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*bs(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)); %*********************************************************************** % Get obseverations, compute phasor quantities(Ei) %*********************************************************************** for i=1:size(I,2) ez_obs(n,i)=ez(I(i),J(i)); ftr(i)=ftr(i)+ez_obs(n,i)*cos(omega*n*dt); fti(i)=fti(i)+ez_obs(n,i)*sin(omega*n*dt); end E_i=E_i+ez(i_cen,j_cen)*(cos(omega*n*dt)-1j*sin(omega*n*dt)); for i=1:length(Id) ez_boundary(n,i)=ez(Id(i),Jd(i)); hx_boundary(n,i)=hx(Id(i),Jd(i)); hy_boundary(n,i)=hy(Id(i),Jd(i)); end end Ez_b=(ftr-1j*fti); 3. scatter_UPML.m %*********************************************************************** % 2D scatter problem, with scatter. the scatter is put in the center of % the grid. and it's diameter is 4mm. %*********************************************************************** % % Program author: Fuqiang Gao % Department of Electrical and Computer Engineering % University of Wisconsin-Madison % 1415 Engineering Drive % Madison, WI 53706-1691 % [email protected] % %*********************************************************************** function [ez_boundarys ez_obs]=scatter_UPML(nmax) %*********************************************************************** % Fundamental constants %*********************************************************************** cc=2.99792458e8; %speed of light in free space muz=4.0*pi*1.0e-7; %permeability of free space epsz=1.0/(cc*cc*muz); %permittivity of free space % nmax=1000; %***********************************************************************
D6y=2.0*epsrb*epsz-sigrb*dt; %Assign main domain parameters to whole region, then modify the PML region C1ez=C1*C1ez; C2ez=C2*C2ez; C3ez=C3*C3ez; C4ez=C4*C4ez; C5ez=C5*C5ez; C6ez=C6*C6ez; D1hx=D1x*D1hx; D2hx=D2x*D2hx; D3hx=D3x*D3hx; D4hx=D4x*D4hx; D5hx=D5x*D5hx; D6hx=D6x*D6hx; D1hy=D1y*D1hy; D2hy=D2y*D2hy; D3hy=D3y*D3hy; D4hy=D4y*D4hy; D5hy=D5y*D5hy; D6hy=D6y*D6hy; %assing media pointers to scatter for i=1:ie_tot for j=1:je_tot if ((i-i_cen)^2+(j-j_cen)^2)<=(0.002/delta)^2 %scatter MEDIAez(i,j)=2; MEDIAhx(i,j)=2; MEDIAhy(i,j)=2; end end end %Get index of observation points I=ones(1,2); J=ones(1,2); index=1; for i=1:ie_tot for j=1:je_tot if (((i-i_cen)^2+(j-j_cen)^2)>=(0.04/delta)^2 && ((i-i_cen)^2+(j-j_cen)^2)<(0.04/delta+1)^2) I(index)=i; J(index)=j; index=index+1; end; end end
%Get index of source locations Id=ones(1,2); Jd=ones(1,2); index=1; for i=1:ie_tot for j=1:je_tot if (((i-i_cen)^2+(j-j_cen)^2)>(0.002/delta+0.4)^2 && ((i-i_cen)^2+(j-j_cen)^2)<(0.002/delta+1.4)^2) Id(index)=i; Jd(index)=j; index=index+1; end; end end source=[i_src-i_cen 0]; phi=ones(1,size(Id,2)); for i=1:size(Id,2) a=[Id(i)-i_cen Jd(i)-j_cen]; l=sqrt(source(1)^2+source(2)^2)*sqrt(a(1)^2+a(2)^2); cfi=dot(a,source)/l; phi(i)=acos(cfi); if a(2)<0 phi(i)=2*pi-phi(i); end end for i=1:length(phi)-1 for j=i+1:length(phi) if phi(j)<phi(i) temp=phi(i); phi(i)=phi(j); phi(j)=temp; temp=Id(i); Id(i)=Id(j); Id(j)=temp; temp=Jd(i); Jd(i)=Jd(j); Jd(j)=temp; end end end eps=[epsrb epsrs 1/2*(epsrb+epsrs)]; sig=[sigrb sigrs 1/2*(sigrs+sigrb)]; mu=[1.0 1.0 1.0]; sigstar=[0.0 0.0 0.0];
%*********************************************************************** % Main Grid Coefficients %*********************************************************************** ca=(1.0-(dt*sig)./(2.0*epsz*eps))./(1.0+(dt*sig)./(2.0*epsz*eps)); cb=(dt./epsz./eps./delta)./(1.0+(dt*sig)./(2.0*epsz*eps)); da=(1.0-(sigstar*dt)./(2.0*muz*mu))./(1.0+(dt*sigstar)./(2.0*muz*mu)); db=(dt./muz./mu./delta)./(1.0+(dt*sigstar)./(2.0*muz*mu)); %*********************************************************************** % UPML Coefficients %*********************************************************************** %Update coefficients in PML region R0=exp(-16); %reflection error m=4; %order of polynomial grading d=upml*delta; %thickness of pml slabs; eta=sqrt(muz/epsrb/epsz); %x-direction sigxmax=-(m+1)*log(R0/100)/2/d/eta; kxmax=1.0; sigfactor=sigxmax/(m+1)/(d^m)/delta; kfactor=(kxmax-1)/(m+1)/(d^m)/delta; for i=1:upml %coefficients on the boundary x2=(upml-i+1.5)*delta; x1=(upml-i+0.5)*delta; sigx=(x2^(m+1)-x1^(m+1))*sigfactor; kx=1+(x2^(m+1)-x1^(m+1))*kfactor; C1=(2*epsrb*epsz*kx-sigx*dt)/(2*epsrb*epsz*kx+sigx*dt); C2=(2*epsrb*epsz*dt)/(2*epsrb*epsz*kx+sigx*dt); D5=(2*epsrb*epsz*kx+sigx*dt); D6=(2*epsrb*epsz*kx-sigx*dt); C1ez(i,:)=C1; C1ez(ie_tot-i+1,:)=C1; C2ez(i,:)=C2; C2ez(ie_tot-i+1,:)=C2; D5hx(i,:)=D5; D5hx(ie_tot-i+1,:)=D5; D6hx(i,:)=D6; D6hx(ie_tot-i+1,:)=D6; %coefficients in the center x2=(upml-i+1)*delta; x1=(upml-i)*delta; sigx=(x2^(m+1)-x1^(m+1))*sigfactor; kx=1+(x2^(m+1)-x1^(m+1))*kfactor;
D3=(2*epsrb*epsz*kx-sigx*dt)/(2*epsrb*epsz*kx+sigx*dt); D4=1.0/(2*epsrb*epsz*kx+sigx*dt)/muz; D3hy(i,:)=D3; D3hy(ih_tot-i+1,:)=D3; D4hy(i,:)=D4; D4hy(ih_tot-i+1,:)=D4; end % Model PEC walls C1ez(1,:)=-1.0; C1ez(ie_tot,:)=-1.0; C2ez(1,:)=0.0; C2ez(ie_tot,:)=0.0; %y-direction sigymax=-(m+1)*log(R0/100)/2/d/eta; kymax=1.0; sigfactor=sigymax/(m+1)/(d^m)/delta; kfactor=(kymax-1)/(m+1)/(d^m)/delta; for j=1:upml %coefficients on the boundary y2=(upml-j+1.5)*delta; y1=(upml-j+0.5)*delta; sigy=(y2^(m+1)-y1^(m+1))*sigfactor; ky=1+(y2^(m+1)-y1^(m+1))*kfactor; C3=(2*epsrb*epsz*ky-sigy*dt)/(2*epsrb*epsz*ky+sigy*dt); C4=1.0/(2*epsrb*epsz*ky+sigy*dt)/epsrb/epsz; D5=(2*epsrb*epsz*ky+sigy*dt); D6=(2*epsrb*epsz*ky-sigy*dt); C3ez(:,j)=C3; C3ez(:,je_tot-j+1)=C3; C4ez(:,j)=C4; C4ez(:,je_tot-j+1)=C4; D5hy(:,j)=D5; D5hy(:,je_tot-j+1)=D5; D6hy(:,j)=D6; D6hy(:,je_tot-j+1)=D6; %coefficients in the center y2=(upml-j+1)*delta; y1=(upml-j)*delta; sigy=(y2^(m+1)-y1^(m+1))*sigfactor; ky=1+(y2^(m+1)-y1^(m+1))*kfactor; D1=(2*epsrb*epsz*ky-sigy*dt)/(2*epsrb*epsz*ky+sigy*dt); D2=2*epsrb*epsz*dt/(2*epsrb*epsz*ky+sigy*dt); D1hx(:,j)=D1; D1hx(:,jh_tot-j+1)=D1; D2hx(:,j)=D2;
D2hx(:,jh_tot-j+1)=D2; end %*********************************************************************** % BEGIN TIME-STEPPING LOOP %*********************************************************************** ez_obs(nmax,size(I,2))=0.0; ez_boundarys=zeros(nmax,length(Id)); %store ez field on boundary at every time step hx_boundarys=zeros(nmax,length(Id)); %store ez field on boundary at every time step hy_boundarys=zeros(nmax,length(Id)); %store ez field on boundary at every time step Je=0.0; for n=1:nmax %*********************************************************************** % Update electric fields %*********************************************************************** es=ez; ez(upml+1:ie_tot-upml,upml+1:je_tot-upml)=ca(MEDIAez(upml+1:ie_tot-upml,upml+1:je_tot-upml)).*ez(upml+1:ie_tot-upml,upml+1:je_tot-upml)+... cb(MEDIAez(upml+1:ie_tot-upml,upml+1:je_tot-upml)).*... (hy(upml+1:ie_tot-upml,upml+1:je_tot-upml)-hy(upml:ih_tot-upml,upml+1:je_tot-upml)+... hx(upml+1:ie_tot-upml,upml:jh_tot-upml)-hx(upml+1:ie_tot-upml,upml+1:je_tot-upml)); ds=dz; dz(2:upml,2:je_tot-1)=C1ez(2:upml,2:je_tot-1).*dz(2:upml,2:je_tot-1)+... C2ez(2:upml,2:je_tot-1)./delta.*((hy(2:upml,2:je_tot-1)-hy(1:upml-1,2:je_tot-1))-... (hx(2:upml,2:je_tot-1)-hx(2:upml,1:je_tot-2))); ez(2:upml,2:je_tot-1)=C3ez(2:upml,2:je_tot-1).*es(2:upml,2:je_tot-1)+... C4ez(2:upml,2:je_tot-1).*(C5ez(2:upml,2:je_tot-1).*dz(2:upml,2:je_tot-1)-C6ez(2:upml,2:je_tot-1).*ds(2:upml,2:je_tot-1)); dz(upml+1:ie_tot-1,2:upml)=C1ez(upml+1:ie_tot-1,2:upml).*ds(upml+1:ie_tot-1,2:upml)+... C2ez(upml+1:ie_tot-1,2:upml)./delta.*((hy(upml+1:ie_tot-1,2:upml)-hy(upml:ie_tot-2,2:upml))-... (hx(upml+1:ie_tot-1,2:upml)-hx(upml+1:ie_tot-1,1:upml-1))); ez(upml+1:ie_tot-1,2:upml)=C3ez(upml+1:ie_tot-1,2:upml).*es(upml+1:ie_tot-1,2:upml)+... C4ez(upml+1:ie_tot-1,2:upml).*(C5ez(upml+1:ie_tot-1,2:upml).*dz(upml+1:ie_tot-1,2:upml)-... C6ez(upml+1:ie_tot-1,2:upml).*ds(upml+1:ie_tot-1,2:upml)); dz(upml+1:ie_tot-1,je_tot-upml+1:je_tot-1)=C1ez(upml+1:ie_tot-1,je_tot-upml+1:je_tot-1).*ds(upml+1:ie_tot-1,je_tot-upml+1:je_tot-1)+... C2ez(upml+1:ie_tot-1,je_tot-upml+1:je_tot-1)./delta.*...
(D5hy(upml+1:ih_tot,1:upml).*by(upml+1:ih_tot,1:upml)-D6hy(upml+1:ih_tot,1:upml).*bs(upml+1:ih_tot,1:upml)); by(ih_tot-upml+1:ih_tot,upml+1:je_tot)=D1hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*bs(ih_tot-upml+1:ih_tot,upml+1:je_tot)-... D2hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*(-(ez(ih_tot-upml+2:ih_tot+1,upml+1:je_tot)-ez(ih_tot-upml+1:ih_tot,upml+1:je_tot)))./delta; hy(ih_tot-upml+1:ih_tot,upml+1:je_tot)=D3hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*hs(ih_tot-upml+1:ih_tot,upml+1:je_tot)+... D4hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*(D5hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*by(ih_tot-upml+1:ih_tot,upml+1:je_tot)-... D6hy(ih_tot-upml+1:ih_tot,upml+1:je_tot).*bs(ih_tot-upml+1:ih_tot,upml+1:je_tot)); by(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)=D1hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*bs(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)-... D2hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*... (-(ez(upml+2:ih_tot-upml+1,je_tot-upml+1:je_tot)-ez(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)))./delta; hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)=D3hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*hs(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)+... D4hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*... (D5hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*by(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)-... D6hy(upml+1:ih_tot-upml,je_tot-upml+1:je_tot).*bs(upml+1:ih_tot-upml,je_tot-upml+1:je_tot)); for i=1:size(I,2) ez_obs(n,i)=ez(I(i),J(i)); end for i=1:length(Id) ez_boundarys(n,i)=ez(Id(i),Jd(i)); hx_boundarys(n,i)=hx(Id(i),Jd(i)); hy_boundarys(n,i)=hy(Id(i),Jd(i)); end end
