in vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations

In vivo breast imaging with diffuse optical tomographybased on higher-order diffusion equations

Yong Xu, Xuejun Gu, Laurie L. Fajardo, and Huabei Jiang

We report on in vivo absorption and scattering imaging of a human breast cyst and implant, using areconstruction algorithm based on our third-order diffusion equations. To validate these in vivo images,a series of phantom experiments were conducted, in which we used low-absorbing and low-scatteringheterogeneities to mimic a breast cyst or implant. These heterogeneities or targets were composed ofpure water or a mixture of water and very dilute Intralipid �0.05% and 0.1%�. The phantom experimentsconfirmed the quantitative imaging capability of our improved algorithm for reconstructing heterogene-ities where the conventional diffusion approximation is inadequate. Pilot clinical results from femalevolunteers indicate that enhanced diffuse optical tomography can quantitatively image findings such asbreast cysts or implants in which the absorption and scattering coefficients are usually low. © 2003Optical Society of America

OCIS codes: 170.0170, 170.6960, 170.3010.

1. Introduction

In diffuse optical tomography �DOT�,1 light–tissueinteraction is generally described by the diffusionequation with the lowest-order approximation to theBoltzmann transport equation �BTE�.2 Althoughthe diffusion approximation is capable of providingaccurate results in many practical situations, it doesnot hold when the domain under consideration isnonscattering or low scattering. This occurs whenDOT is performed for structures such as a cyst in thebreast, the cerebrospinal fluid �CSF� in the brain, orthe synovial fluid in a joint. To overcome this prob-lem, several advanced light–tissue models have beendeveloped. Klose and Heilscher3 implemented aBTE-based reconstruction algorithm, Dehghani etal.4,5 studied a radiosity-diffusion model, and Jiang6

developed a reconstruction scheme based on thethird-order diffusion equations �TODE�.

Thus far, image reconstructions that use these ad-vanced light–tissue models have been largely limited

Y. Xu, X. Gu, and X. Jiang are with the Department of Physicsand Astronomy, Clemson University, Clemson, South Carolina29634. L. L. Fajardo is with the Department of Radiology, Uni-versity of Iowa, Iowa City, Iowa 52242. H. Jiang’s e-mail addressis [email protected].

Received 2 September 2002; revised manuscript received 2 De-cember 2002.

0003-6935�03�163163-07$15.00�0© 2003 Optical Society of America

to numerical simulations. In addition, the numeri-cal configurations of nonscattering or low-scatteringheterogeneities used have largely been aimed atmimicking the CSF-filled regions within the brain.3–5

Recently, Hielscher et al.7 presented an experimentalstudy on finger joints, using their BTE-based recon-struction algorithm. In this paper we experimen-tally evaluate our TODE-based reconstructionalgorithm with phantom configurations mimicking aclear-fluid–filled cyst or a saline-filled implant in thebreast. We also present what we believe is the firstin vivo absorption and scattering images of a breastwith a cyst and a breast with an implant, using ourenhanced reconstruction approach. This studyshows that quantitative optical images can be ob-tained from phantoms that have low-absorbing andlow-scattering heterogeneities and from a breast thathas cysts or implants with the TODE-based recon-struction algorithm.

2. Reconstruction Algorithm

Our reconstruction algorithm is based on the follow-ing TODE6:

� � D�r��1��r� � �a�r��1��r� � � � D�r��2��r�

� 6� � D�r��1��3��r� � 6� � D�r��2�

�4��r� � �S�r�,(1)

1 June 2003 � Vol. 42, No. 16 � APPLIED OPTICS 3163

�� D�r��1��r� �257

� � D�r��2��r�

� 5��t�r��2��r� �607

� � D�r��1��3��r�

�607

� � D�r��2��4��r� � 0, (2)

� � D�r��1��1��r� �

107

� � D�r��1��2��r�

�907

� � D�r��3��r� � 10��t�r��3��r� � 0, (3)

12

� � D�r��2��1��r� �

57

� � D�r��2��2��r�

�457

� � D�r��4��r� � 5��t�r��4��r� � 0, (4)

where

� � x�

� x� y

�

� y,

�1 � x�

� x� y

�

� y,

�2 � x�

� y� y

�

� x.

��1�, ��2�, ��3�, and ��4� are the first four componentsin the spherical harmonic expansion of the photonradiance where the first component, ��1�, is the aver-age diffused photon density. �a is the absorptioncoefficient. ��t �a ��s, where ��s is the reduced-scattering coefficient. D 1�3��t is the diffusion co-efficient. S is the light-source term. In thealgorithm we apply type III boundary conditions tothe first component, ��1�, i.e., �Dn � ��1� ��1�,where n is the unit normal vector on the boundarysurface and � is a coefficient related to the internalreflection at the boundary, and we employ type Iboundary conditions to the remaining components,i.e., ��2� ��3� ��4� 0 on the boundary. Theboundary coefficient, �, can be calculated through adata-preprocessing scheme.7

For recovering spatial maps of absorption and scat-tering coefficients we use a regularized Newton’smethod to update an initial optical property distribu-tion iteratively, which minimizes an object functioncomposed of a weighted sum of the squared differencebetween computed and measured optical data at themedium surface. We obtain the computed opticaldata by solving the above TODE with the finite-element method. The mathematical details of ouralgorithm have been described previously.6 A two-dimensional finite-element mesh with 717 nodes and1368 triangular elements was used in the reconstruc-tion for both forward and inverse solutions. Twentyiterations were used in the reconstruction, after

which no noticeable improvement was observed.The computations were conducted in a 600-MHz Pen-tium III personal computer.

3. Experiments

A. Phantom Experiments

The phantom experiments were performed with anautomated multichannel frequency-domain system�we just needed dc measurements to reconstruct theabsorption and scattering images reported here�.This system has been described in detail and evalu-ated extensively in prior phantom studies.8,9

Briefly, an intensity-modulated light from a 785-nm50-mW diode laser is sequentially sent to the phan-tom by sixteen 3-mm fiber-optic bundles. For eachsource position, the diffused light is received at 16detector positions along the surface of the phantomand sequentially delivered to a photomultiplier tube.A second photomultiplier tube is used to record thereference signal, and dc, ac, and phase-shift signalsare obtained by use of the standard heterodyne tech-nique controlled by fast-Fourier-transform Labviewroutines. A complete set of tomographic data �16 �16 measurements� can be obtained within 8 min.

The experimental phantoms comprised Intralipidas a scatterer and India ink as an absorber. Agarpowder �1–2%� was used to solidify the Intralipid andIndia ink solutions.10 The background phantomwas a 50-mm-diameter solid cylinder with �a 0.007�mm and ��s 1.0�mm. One or two 14-mm-diameter cylindrical holes were drilled into thehomogeneous-background phantom for inclusion oftargets with various optical contrasts. To mimic abreast cyst and saline implant to validate our in vivoimages, we used one or two targets consisting of puredistilled water or a mixture of water and very diluteIntralipid �0.05% and 0.1%�. Figure 1 depicts thegeometrical configurations for the test cases understudy.

B. Clinical Experiments

The clinical experiments were conducted with ournewly developed diffuse optical mammography sys-tem, which has been reported in detail elsewhere.11

The system primarily consists of three diode lasers at785, 808, and 830 nm; 64 � 64 channel source–detector optic fibers for light delivery and receiving;and 16 computer gain-controlled photomultipliertubes for parallel detection. The fiber-optic probeused in this system provides data sets for both two-and three-dimensional imaging. The system is com-puter controlled with Labview software. A full set oftomographic data �64 � 64� at one wavelength can beobtained in 4 min.

Two female volunteers were examined with thisnew imaging system. The protocol was approved bythe Clemson institutional review board, and writteninformed consent was obtained from both partici-pants. The first patient �43 years old� had a 2-cmcyst in her left breast that was detected by a sono-gram. The second volunteer �41 years old� had a

3164 APPLIED OPTICS � Vol. 42, No. 16 � 1 June 2003

known saline implant �4 cm in diameter� in her rightbreast.

4. Results and Discussion

Four phantom experiments were performed to eval-uate our TODE-based algorithm for the cases withlow-absorbing and low-scattering targets. For di-rect comparison on raw boundary data, Fig. 2 showsmeasured boundary data from a representativephantom case and boundary data computed fromTODE and first-order diffusion equation �FODE�.As can be seen, small but significant differences ex-ists between TODE and FODE, especially in the po-sitions farthest from the source position where thesignal-to-noise ratio is the smallest. Both recon-structed gray-scale images and quantitative plots ofone-dimensional �1D� profiles of the exact and recov-ered property distributions contained in these imagesare presented �Figs. 3–5�. For comparison, 1D pro-files of optical properties obtained from the FODE-

Fig. 1. Phantom geometry for �a� one target cases, and �b� two targets cases.

Fig. 2. Comparison of boundary data for a representative phan-tom case �long-dashed curve, measured; short-dashed curve, com-puted from FODE; solid curve, computed from TODE�. Note thatdata in four detector points closest to the source are not shown.

Fig. 3. Reconstructed absorption and scattering images: �a� ab-sorption image for one pure-water target, �b� scattering image forone pure-water target, �c� absorption image for one water plus 0.05%Intralipid target, �d� scattering image for one water plus 0.05%Intralipid target, �e� absorption image for one water plus 0.1% In-tralipid target, �f � scattering image for one water plus 0.1% In-tralipid target, �g� absorption image for two targets �water plus 0.1%Intralipid�, �h� scattering image for two targets �water plus 0.1%Intralipid�.


based algorithm are also shown �Figs. 4 and 5�. Thein vivo results from the two volunteers are given inFig. 6 and Table 1.

Figure 3 shows the simultaneous reconstruction ofboth absorption and scattering images from the fourphantom experiments. Figures 3�a� and 3�b� are,respectively, the absorption and scattering imagesrecovered for the first case with one pure-water tar-get. In this case the expected optical properties of

the target are �a 0.002�mm and ��s � 0.05�mm.10

Figures 3�c� and 3�d� present the reconstructed im-ages for the second case with a water plus 0.05%Intralipid target where the expected optical proper-ties of the target are �a 0.002�mm and ��s � 0.1�mm. Figures 3�e� and 3�f � show the recoveredimages for the third case with a water plus 0.1%Intralipid target where the expected target opticalproperties are �a 0.002�mm and ��s � 0.15�mm.

Fig. 4. Comparison of exact �dashed–dotted curve� and reconstructed �solid and dotted curves� absorption distributions along transectCD, EF, or C*D* shown in Fig. 1 for the images appearing in Fig. 3, where the solid curves are the recovered results from the TODEalgorithm and the dotted curves give the recovered results from the FODE algorithm: �a� absorption profiles along transect CD for theimage shown in Fig. 3�a�, �b� absorption profiles along transect EF for the image shown in Fig. 3�a�, �c� absorption profiles along transectCD for the image shown in Fig. 3�c�, �d� absorption profiles along transect EF for the image shown in Fig. 3�c�, �e� absorption profiles alongtransect CD for the image shown in Fig. 3�e�, �f � absorption profiles along transect EF for the image shown in Fig. 3�e�, �g� absorptionprofiles along transect C*D* for the image shown in Fig. 3�g�.


The recovered �a and ��s images for the fourth casewith two targets are shown in Figs. 3�g� and 3�h�where each target contained a mixture of water and0.1% Intralipid. From the images presented in Fig.3 we can see that good-quality absorption and scat-tering images can be simultaneously obtained fromdc data by use of our TODE-based reconstructionalgorithm when the scattering coefficient of the tar-get is low.

To obtain further quantitative information about

the reconstructed images, we calculated the recon-structed optical property distribution along twotransects through the center of the target �transectsCD�EF or C*D*; see Fig. 1�. The calculated 1D �aand ��s distributions �solid curves� are shown in Figs.4 and 5, respectively, where we plotted the exactoptical property profiles �dashed–dotted curves� andthe �a and ��s distributions �dotted curves� recoveredfrom the FODE-based algorithm for each figure.From these 1D �a and ��s profiles, we note that the

Fig. 5. Comparison of exact �dashed–dotted curve� and reconstructed �solid and dotted curves� scattering distributions along transect CD,EF, or C*D* shown in Fig. 1 for the images appearing in Fig. 3, where the solid curves are the recovered results from the TODE algorithmand the dotted curves present the recovered results from the FODE algorithm: �a� scattering profiles along transect CD for the imageshown in Fig. 3b, �b� scattering profiles along transect EF for the image shown in Fig. 3�b�, �c� scattering profiles along transect CD for theimage shown in Fig. 3�d�, �d� scattering profiles along transect EF for the image shown in Fig. 3�d�, �e� scattering profiles along transectCD for the image shown in Fig. 3�f �, �f � scattering profiles along transect EF for the image shown in Fig. 3�f �; �g� scattering profiles alongtransect C*D* for the image shown in Fig. 3�h�.


recovered images are quantitative in terms of thetarget location, size, and shape. We found that theoptical properties of the targets can be recoveredwithin 10% of the expected peak values for both �aand ��s images. When we examine Fig. 4, it is obvi-ous that the TODE algorithm offers consistently bet-ter and more-accurate �a reconstruction in the low-scattering target region than the FODE algorithm,whereas the FODE-based reconstruction alwaysgives an unphysical �negative� �a value in the targetregion. From Fig. 5 we note that the TODE andFODE algorithms provide almost the same quality of��s reconstruction in the target region, but the TODEalgorithm shows better overall background recon-struction than the FODE algorithm. These compar-isons clearly indicate that the TODE provide more-accurate description of light propagation innonscattering or low-scattering regions than the con-ventional FODE. It is unclear at the present time

why the recovered scattering images from the TODEand FODE algorithms are almost the same quanti-tatively, whereas the reconstructed absorption im-ages from both algorithms are quantitativelydifferent, which warrants further study. We sus-pect that the imbalance existing between the recon-structions of these two parameters may play animportant role here.

In vivo breast images with simultaneous recon-struction of absorption and scattering by use of theTODE algorithm are displayed in Fig. 6 for the twofemale volunteers. Figures 6�a� and 6�b� are, respec-tively, the reconstructed �a and ��s images in thelesion plane for the cyst-bearing patient. The recov-ered in vivo absorption and scattering images for theimplant-bearing patient are presented in Figs. 6�c�and 6�d�, respectively. These in vivo images clearlyshow a marked decrease in both scattering and ab-sorption coefficients in the cyst and implant regions.The sizes of cyst and implant can be estimated fromthe FWHM of the reconstructed optical properties ofthe cyst and implant.12 With this threshold, thesizes of cyst and implant were calculated �Table 1�.From sonography and mammography we note thatthe sizes of the cyst and implant resolved by opticalimaging are comparable with expected values. Theaverage �a and ��s values of the cyst or implant andthe surrounding normal tissues are also presented inTable 1. From the previously described phantomvalidation results, we believe that the error of theabsorption and scattering coefficients obtained hereshould be within 10%.

In summary, we have experimentally validated thecapabilities of our TODE-based reconstruction algo-rithm for imaging tissue-like phantoms with low-

Fig. 6. Reconstructed absorption and scattering images for two female volunteers: �a� absorption image and �b� scattering image for thevolunteer with a 2-cm cyst in the left breast; �c� absorption image and �d� scattering image for the volunteer with a 4-cm saline implantin the right breast.

Table 1. Recovered Geometric Information and Optical Properties ofClinical Results

Object Size�mm�a

OpticalProperties

�mm�1�

Lx Ly ObjectNormalTissue

Cyst case �a Image 25.0 23.0 0.0030 0.011��s Image 16.0 18.0 0.55 0.70

Implant case �a Image 39.0 35.0 0.0034 0.0062��s Image 38.0 38.0 0.38 0.49

aLx and Ly are the FWHM of the object along the x and the ydirections, respectively.


scattering heterogeneities, using dc data. Theclinical results from two female volunteers show thatenhanced diffuse optical tomography �DOT� canquantitatively image a breast cyst and implant wherethe absorption and scattering coefficients are usuallylow.

This research was supported in part by a grantfrom the National Institutes of Health �NIH� �R01 CA90533�.

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in vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations

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