Quantitative reconstruction of refractive index distributionand imaging of glucose concentration by usingdiffusing light

Xiaoping Liang, Qizhi Zhang, and Huabei Jiang

We show that a two-step reconstruction method can be adapted to improve the quantitative accuracy of therefractive index reconstruction in phase-contrast diffuse optical tomography (PCDOT). We also describe thepossibility of imaging tissue glucose concentration with PCDOT. In this two-step method, we first use ourexisting finite-element reconstruction algorithm to recover the position and shape of a target. We then usethe position and size of the target as a priori information to reconstruct a single value of the refractive indexwithin the target and background regions using a region reconstruction method. Due to the extremely lowcontrast available in the refractive index reconstruction, we incorporate a data normalization scheme intothe two-step reconstruction to combat the associated low signal-to-noise ratio. Through a series of phantomexperiments we find that this two-step reconstruction method can considerably improve the quantitativeaccuracy of the refractive index reconstruction. The results show that the relative error of the reconstructedrefractive index is reduced from 20% to within 1.5%. We also demonstrate the possibility of PCDOT forrecovering glucose concentration using these phantom experiments. © 2006 Optical Society of America

OCIS codes: 170.5280, 170.6960.

1. Introduction

It is well known that the refractive index in tissuedepends on the tissue’s physical and chemical prop-erties. Phase contrast due to the spatial variation ofthe tissue refractive index has been exploited for bi-ological imaging. To date there are several methodsto image phase contrast including the phase-contrastoptical microscope,1 optical coherent tomography,2x-ray phase-contrast computed tomography (CT),3 andultrasound tomography.4 These imaging methods havebrought significant contributions to the biomedicalfield.

We have previously described the possibility ofphase-contrast imaging of tissue using near-infrared(NIR) diffusing light.5 In that work, we developed afinite-element reconstruction algorithm based on aphoton diffusion model by considering a spatially vary-ing refractive index to extract the spatial map of the

tissue refractive index, a new recovery parameter rel-ative to tissue absorption, and scattering availablefrom conventional diffuse optical tomography (DOT).While we showed that refractive index images could bereconstructed using both phantom and clinical data,the quantitative accuracy of the reconstruction wasmoderate with a relative error of approximately 20%.The primary goal of this paper is to improve the quan-titative accuracy of refractive index reconstructionwith a two-step reconstruction method. After segment-ing the position and shape of a target from our existingfinite-element reconstruction procedure, we then as-sign the nodes in the target and background regionswith appropriate labels (e.g., background � 0, tar-get � 1) in this two-step method. We recover a singlevalue of the refractive index within each region forquantitative improvement of the reconstruction. Wealso consider a data normalization scheme in the two-step reconstruction in order to reduce the effect of mea-surement noise. The method is evaluated using aseries of phantom experiments with a target size assmall as 5 mm in diameter, and the results show thatthe relative error of the reconstructed refractive in-dex is significantly reduced from approximately 20%to within 1.5%.

We also show that we can recover the glucose con-centration using phantom data. As is known, glucoseis the primary food source for the brain, providing

The authors are with the J. Crayton Pruitt Family Departmentof Biomedical Engineering, University of Florida, Gainesville,Florida 32611. X. Liang’s e-mail address is xliang@bme.ufl.edu.H. Jiang’s e-mail address is hjiang@bme.ufl.edu.

Received 14 February 2006; revised 13 June 2006; accepted 7July 2006; posted 10 July 2006 (Doc. ID 67992).

0003-6935/06/328360-06$15.00/0© 2006 Optical Society of America

8360 APPLIED OPTICS � Vol. 45, No. 32 � 10 November 2006

approximately 90%–95% of its energy needs. Moni-toring of the level of glucose utilization or the changein glucose utilization is critical in many medical ap-plications, such as Alzheimer’s disease, and it cancurrently be measured only with a tracer or analog ofglucose using positron emission tomography (PET).6It would be clinically significant if phase-contrast dif-fuse optical tomography (PCDOT) could be used forimaging glucose metabolism in vivo.

2. Reconstruction Methods

A. Finite-Element Reconstruction Algorithm

The new photon diffusion equation considering thespatially varying refractive index, which forms the the-oretical foundation for PCDOT, is stated as follows7,8:

� · D���r� �2Dn �n · � ��r� � �a��r� � �S0��r � r0�,

(1)

where � is the photon density; n is the refractiveindex; c is the speed of light in the medium; �a is theabsorption coefficient; and D is the diffusion coeffi-cient, which can be written as D � 1�3��s� � �a�,where �s� is the reduced scattering coefficient. [Since�s� �� �a in highly scattering tissues, D � 1�3��s��.]S0 is the source strength, and ��r � r0� is the Diracdelta function at position r0. In cases where the ab-sorption coefficient is of the same order as the scatter-ing coefficient, the first-order diffuse approximation[Eq. (1)] will not be applicable. A third-order diffusion-equation-based optical image reconstruction algorithmwill be needed to handle these cases.9

For a known optical property distribution, Eq. (1)becomes a standard boundary-value problem for thespatially varying photon density subject to appropriateboundary conditions. The following type III boundaryconditions (which are common with the diffusion equa-tion) are used for the solution to Eq. (1)10,11:

�D � � · n � ��, (2)

where n is the unit normal vector for the boundarysurface and � is a coefficient related to the internalreflection at the boundary.

By the use of finite-element discretization and re-alizing other derived matrix relations through differ-entiation, we obtain the following set of equationscapable of inverse problem solution:

�A���� � �b�, (3)

�A��

n�bn� �A

n����, (4)

��T� � I��n � �T���m� � ��c��, (5)

where the elements of matrix [A] are aij � �D��j ·��i � �a�j�i�, where � indicates integration over the

problem domain; �i and �j are locally spatially vary-ing Lagrangian basis functions at nodes i and j, re-spectively; � is the Jacobian matrix formed by ��nat the boundary measurement sites; �n � ��n1,�n2, . . . , �nN�T is the update vector for the refrac-tive index profile, where N is the total number ofnodes of the finite-element mesh used; and ��m�

� ��1�m�,�2

�m�, . . . , �M�m�� and ��c� � ��1

�c�, �2�c�, . . . ,

�M�c��, where �i

�m� and �i�c�, respectively, are the mea-

sured and calculated data for i � 1, 2, . . . , M bound-ary locations. To estimate n spatially, we need toexpand this quantity in a similar manner to � as afinite sum of unknown coefficient multiplied by thelocally defined Lagrangian basis function. In phase-contrast imaging, the goal is to iteratively update then distribution through the solution of Eqs. (3)–(5) sothat a weighted sum of the squared difference be-tween computed and measured optical data can beminimized. Since we are only interested in recon-structing the refractive index in phase-contrast im-aging, we have assumed that the diffusion andabsorption coefficients are constant during the recon-struction of the refractive index.

B. Data Normalization

To reduce the noise effect, we perform normalizationon the experimental data. In our imaging experi-ments, there are 16 sources and 16 detectors, whichprovide a set of measured data Di, j, where i is thesource number from 1 to 16 and j is the detectornumber from 1 to 16. For each source i, we first finda maximum value of the photon density for all 16detectors:

Di,max � max�Di, j�, j � 1, 2, . . . , 16. (6)

We then divide each data set Di, j by the above max-imum value Di,max:

Di, j � Di, j�Di,max. (7)

Thus the normalized data set Di, j is used for imagereconstructions.

C. Region Reconstruction

We adapt a region reconstruction method for PCDOT.While this region reconstruction method for conven-tional DOT has been detailed elsewhere,12,13 here webriefly describe it and adapt it for PCDOT. Since ourexisting algorithm can accurately recover the locationand size of a target as shown previously in Ref. 5, wecan segment the recovered image into different re-gions (i.e., target and background) by assigning thenodes in different regions with appropriate labels(background � 0, target � 1). In the reconstruction, weapply a matrix transformation to Jacobian matrix �,

� � �K, (8)

where � is an NM � NR matrix (number of mea-

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surements by number of regions) and K is thea priori matrix,

K � �R1 R2 · · · Rn

k1,1 k1,2 · · · k1,n

k2,1 k2,2 · · · k2,n

É É Ì É

kj,1 kj,2 · · · kj,n

�, where k ,� �1, � R�

0, � R�.

(9)

We solve the following equation to obtain an inter-mediate update of n:

��T���n � �T���m� � ��c��. (10)

The final updates from the region reconstructionbecome

�n � �nK�1. (11)

Note that because NR �� NM, the new Hessian�T� is a small, well-conditioned matrix. In the regionreconstruction, we use the same uniform mesh as inthe nonregion reconstruction. This is different fromthat in Ref. 11, where a locally refined mesh (aroundthe target) was used. We incorporate the above regionalgorithm into our regular reconstruction program.Now the inverse problem becomes reconstructing sin-gle values of the refractive index within each region.

3. Experiments

Phantom experiments were conducted with a ten-wavelength DOT system, which is schematicallyshown in Fig. 1. This newly developed imaging sys-tem and its calibration have been described in detailelsewhere.14 In short, laser light from one of the tenlaser modules is transmitted to an optical switch,which sequentially passes it to 16 detection points atthe surface of the phantom for two-dimensional im-aging experiments. The 16 � 16 measured data are

then input into our reconstruction algorithm to gen-erate a two-dimensional cross-sectional image of thephantom. In this study, we were only interested inusing the laser at 775 nm.

Several tissuelike phantom measurements wereconducted with different contrasts of the refractiveindex between the target and the background. A lipidemulsion solution (Intralipid) was used as the back-ground to mimic tissue scattering ��s� � 1.0 mm�1�.Tissue absorption was simulated with India ink��a � 0.007 mm�1�. Agar powder (2%) was used tosolidify the mixed Intralipid and India ink solution.The cylindrical background phantom had a diameterof 50 mm and a refractive index of 1.33, close to thatof water. One 10 or 5 mm diameter off-center cylin-drical hole was drilled in the homogeneous backgroundphantom for inclusion of targets with various glucoseconcentrations and Intralipid–India ink solutions tomimic different refractive index contrasts. Figure 2gives the geometrical configuration for the test casesunder study.

We have used a refractometer to perform measure-ments of glucose and intralipid solution in order toobtain the following fitting formula that relates therefractive index value and glucose concentration forthe phantom materials used in our study:

n � 0.2015 � �C� � 1.3292, (12)

where n is the refractive index and [C] is the concen-tration of the glucose solution. Values of the refrac-tive index at several glucose concentrations used inthis study were calculated using the above formulaand are listed in Table 1.

4. Results and Discussion

A two-dimensional finite-element mesh with 717nodes and 1368 triangular elements was used for

Fig. 1. Schematic of the diffuse optical imaging system. Lightfrom ten laser modules with wavelengths from 673 to 965 nm wasdelivered to 16 excitation positions through a 1 � 10 optical switch.Diffused light is received by 16 detection optical fiber bundles andsent to 16 silicon photodiodes controlled by a homemade circuitboard, and data are recorded by a 16-bit analog-to-digital (A/D)board coupled with a personal computer (PC). Note that 16 sourceand 16 detector fibers are alternately attached to the surface of thephantom with equal spacing (filled circles, source fibers bundles;empty circles, detector fibers).

Fig. 2. Schematic of the phantom geometry under study. R1 �

50 mm; R2 � 10 or 5 mm; d � 14 or 18 mm.

Table 1. Values of the Refractive Index Corresponding toGlucose Concentration

Glucose Concentration 1% 2% 3% 5%

Refractive index 1.3312 1.3332 1.3353 1.3393

8362 APPLIED OPTICS � Vol. 45, No. 32 � 10 November 2006

reconstructions with both the existing and region re-construction algorithms. Figures 3 and 4 present thereconstructed refractive index images using the tworeconstruction methods for four cases where the tar-get contained 1%, 2%, 3%, and 5% glucose. It can beseen from Fig. 3 that the target is clearly detected foreach case. We note that these images are quantitativein terms of the location and size of the target. How-ever, the values of refractive index in the target areconsiderably larger than the ideal values shown inTable 1.

From Fig. 4, we immediately see that the recoveredvalues of the refractive index of the target are im-proved dramatically with the region reconstructionmethod for all cases. Note that the background andtarget in these images are homogeneous due to theuse of region reconstruction. To quantify these re-sults from the two-region reconstruction, we calcu-lated the relative errors of the recovered refractiveindex in the target for the four cases and listed themin Table 2. We see that the errors range from 0.7% to1.4%. It is interesting to note that the 1% glucose casehad the smallest error, while the 5% case had thelargest error.

In Table 2, we also list the values and errors ofrecovered glucose concentration in the target from thetwo-region reconstruction. It is clear that the errorsare as large as 400% (the 1% glucose case). However,we observe that the trend of the glucose concentrationfrom 1% to 5% is correctly reconstructed as seen in

Fig. 5(a). If we subtract the recovered glucose concen-tration for each case from a common baseline, we canobtain an interesting change in the glucose concen-tration relative to the baseline as we show in Fig.5(b). Now the errors of these relative glucose concen-trations are 53%, 8%, 38%, and 25% for 1%, 2%, 3%,and 5% glucose concentration cases, respectively. Al-though the recovery of glucose concentration is notabsolutely quantitative, the ability of the current semi-quantitative glucose concentration imaging may stillprove to be useful in certain clinical applications, aswe pointed out earlier that imaging the change ofglucose utilization is important in monitoring Alzhei-mer’s disease.

To test the ability to detect a small target, we con-ducted two more experiments with a 5 mm diametertarget containing 3% and 5% glucose concentrations,respectively. The results are shown in Fig. 6. As canbe seen, while the shape of the recovered target isdistorted, we note that the target is clearly detectedand that the value of the reconstructed refractiveindex is quantitative after the region reconstructionfor both cases. Here the distortion is due to the lowersignal-to-noise ratio available for these cases. Thenumeric results of the refractive index images for thesmaller targets can be seen from the color bar shownin Fig. 6 (e.g., for the 3% case, the refractive index inthe target is 1.3453, and for the 5% case, the refrac-tive index of the target is 1.3464). Using the samebaseline as before, the accuracy of calculated glucose

Fig. 3. (Color online) Refractive index images reconstructed from the measurements on tissue-mimicking phantoms where the scatteringand absorption coefficients were �a � 0.007 mm�1 and �s� � 1.0 mm�1 for both the background and the target (10 mm in diameter)containing different glucose concentrations: (a) 1% glucose concentration �n � 1.3312�, (b) 2% glucose concentration �n � 1.3332�, (c) 3%glucose concentration �n � 1.3353�, (d) 5% glucose concentration �n � 1.3393�.

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concentration is similar to the previous cases with alarger target size. Our baseline calibration is effectivefor phantom experiments. Modifications are neces-sary for more complex heterogeneous media such asreal tissues. This warrants further study.

In summary, we have presented a two-step phase-contrast DOT imaging method that can image therefractive index of heterogeneous media quantita-tively. The method has been confirmed with phantomexperiments. The ability of imaging the tissue refrac-tive index as well as the glucose concentration mayoffer DOT a new opportunity for biomedical applica-tions. We expect to evaluate this method with in vivodata in the near future. However, we anticipate thatdifficulties may be encountered for refractive indeximaging within more complex media, especially as itrelates to solution uniqueness and parameter crosstalk (from D and �a). We plan to minimize thesedifficulties by, for example, combining PCDOT withthe conventional DOT from which the D and �a dis-tributions can be obtained first. An alternative way to

Fig. 4. (Color online) Refractive index images recovered with the region reconstruction method. The optical properties of the phantomwere the same as those for the images shown in Fig. 2. (a) 1% glucose concentration �n � 1.3312�. (b) 2% glucose concentration�n � 1.3332�. (c) 3% glucose concentration �n � 1.3353�. (d) 5% glucose concentration �n � 1.3393�. Note that the target shape is not roundbecause of the limited number of finite-element nodes used.

Fig. 5. (Color online) (a) Comparison of ideal and calculated absolute values of glucose concentration for all four cases. (b) Comparisonof ideal and calculated relative values of glucose concentration for all four cases.

Table 2. Comparison of Ideal and Reconstructed Values of theRefractive Index and Glucose Concentration

Ideal(C )(%)

Idealn

Calculatedn

Calculated(C )(%)

RelativeError (n)

RelativeError (C )

1 1.3312 1.3393 5.01 0.0069 4.00642 1.3332 1.3406 5.64 0.0079 1.82013 1.3353 1.3445 7.60 0.0108 1.53425 1.3393 1.3488 9.71 0.0139 0.9412

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improve the reconstruction is to use a priori geomet-rical information obtained from another modalitysuch as magnetic resonance imaging or x ray.

This work was supported in part by a grant from theNational Institutes of Health (NIH) (R01 CA090533).

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Fig. 6. (Color online) Refractive index images reconstructed (a), (b) without and (c), (d) with the region reconstruction method for the5 mm diameter target experiments. The absorption and scattering coefficients of the background and target were �a � 0.007 mm�1

and �s� � 1.0 mm�1. The target contained (a), (c) 3% and (b), (d) 5% glucose concentration.

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