Influence of fiber optic probe geometry on the applicability of inverse models of tissue reflectance...

Influence of fiber optic probe geometry on the applicabilityof inverse models of tissue reflectance spectroscopy:computational models and experimental measurements

Jiantang Sun, Kun Fu, Adrien Wang, Alex W. H. Lin, Urs Utzinger, and Rebekah Drezek

Accurate recovery of tissue optical properties from in vivo spectral measurements is crucial for improvingthe clinical utility of optical spectroscopic techniques. The performance of inversion algorithms can beoptimized for the specific fiber optic probe illumination–collection geometry. A diffusion-theory-basedinversion method has been developed for the extraction of tissue optical properties from the shape ofnormalized tissue diffusion reflectance spectra, specifically tuned for a fiber probe that comprises sevenhexagonally close-packed fibers. The central fiber of the probe goes to the spectrometer as the detectingfiber, and the surrounding six outer fibers are connected to the white-light source as illumination fibers.The accuracy of the diffusion-based inversion algorithm has been systematically assessed against MonteCarlo (MC) simulation as a function of probe geometry and tissue optical property combinations. By useof this algorithm, the spectral absorption and scattering coefficients of normal and cancerous tissue areefficiently retrieved. Although there are significant differences between the diffusion approximation andthe MC simulation at short source–detector (SD) separations, we show that with our algorithm the tissueoptical properties are well retrieved within the SD separation of 0.5–3 mm that is compatible withendoscopic specifications. The presented inversion method is computationally efficient for eventualreal-time in vivo tissue diagnostics application. © 2006 Optical Society of America

OCIS codes: 170.6510, 170.7050, 290.1990.

1. Introduction

Recent advances in fiber optics, light sources anddetectors, and computer-controlled instrumentationhave stimulated a period of unprecedented growth inthe development of photonic technologies for a widevariety of diagnostic and therapeutic clinical appli-cations.1–4 The diffusion approximation is an approachcommonly used to analyze optical data obtained by (1)in vivo reflectance spectroscopy,5–8 (2) in vivo fluores-

cence spectroscopy,1,9–13 and (3) noninvasive in vivomeasurement of tissue optical properties.14–19 Initialclinical studies using diffusion-based approximationhave, in fact, yielded promising results. For example,Zonios et al.5 used an analytical light diffusion model tostudy the diffuse reflectance spectra of adenomatouscolon polyps (cancer precursors) and normal colonicmucosa. The results suggested that diffuse reflectancecould be used to gather information about tissue struc-ture and composition in vivo. Georgakoudi et al.9 re-covered the intrinsic (undistorted) tissue fluorescenceby combining simultaneously measured fluorescenceand reflectance spectra. The extraction of the intrinsicfluorescence allowed the investigators to determinethe fluorescence spectra of NAD(P)H and collagen inan in vivo environment. Subsequently, they used ex-tracted NAD(P)H and collagen to describe diagnos-tically, and quantitatively, significant biochemicalchanges between normal and dysplastic tissues. Toobtain tissue fluorescence spectra free of distor-tion due to intrinsic scattering and absorption, theyused the same method described by Zonios et al.5 inanalyzing white-light reflectance spectra. Other re-searchers, for example, Stasic et al.,20 investigated the

J. Sun, K. Fu, A. Wang, A. W. H. Lin, and R. Drezek([email protected]) are with Rice University, 6100 Main Street,Houston, Texas 77005. J. Sun, A. Wang, A. W. H. Lin, and R.Drezek are with the Department of Bioengineering. J. Sun and K.Fu are also with the Rice Quantum Institute. K. Fu and R. Drezekare also with the Department of Electrical and Computer Engi-neering. U. Utzinger is with the Department of Biomedical Engi-neering, Obstetrics, and Gynecology and the Department ofElectrical and Computer Engineering, University of Arizona, Tuc-son, Arizona 85724.

Received 13 March 2006; revised 29 June 2006; accepted 7 July2006; posted 17 July 2006 (Doc. ID 68926).

0003-6935/06/318152-11$15.00/0© 2006 Optical Society of America

8152 APPLIED OPTICS � Vol. 45, No. 31 � 1 November 2006

possibility of using spatially resolved fluorescence andreflectance to recover tissue optical properties, fluoro-phore concentration, and the upper layer thickness ofa two-layer tissue-simulating phantom. A diffusion-theory model was used in this study to fit reflectanceand fluorescence data generated from Monte Carlo(MC) simulations and data experimentally obtainedby using tissue-simulating phantoms. Finally, Choiet al.21 adopted near-infrared spectroscopy to noninva-sively determine the optical properties of the adulthuman brain by using a two-layer diffusion approxi-mation model. They concluded that adult humanheads can be reasonably approximated by their diffu-sion model and that the hemoglobin concentrationsand oxygen saturation of adult brains could be calcu-lated with good accuracy.21

The prediction of photon propagation in tissue canbe achieved with greater computational efficiency, andan inverse algorithm used to retrieve tissue opticalproperties from diagnosed signals can be more easilyderived from the diffusion model.5,22,23 These inversionalgorithms usually depend heavily on the geometry ofthe specific fiber probes used. In some limiting cases,which do not sufficiently satisfy the prerequisite of thediffusion approximation, solutions of the diffusionmodels may deviate from the true values as deter-mined by MC models.13,15,20 However, the diffusionequation is still an important tool for recovering tissueoptical properties as a consequence of its easily ob-tained inverse algorithms. To improve the clinical util-ity of optical spectroscopic methods, we are interestedin the development of inversion methods that will al-low recovery of tissue optical properties from in vivospectral measurements with a cylindrical optical fiberprobe that is composed of seven hexagonally close-packed fibers (Fig. 1). The central fiber of the probegoes to a spectrometer as the collection fiber, and thesix surrounding fibers are connected to a white-lightsource as illumination fibers. Although diffusion the-ory offers a good potential approach to the inversion,there are natural concerns over the applicability of thistheory for probe geometries that employ relatively

short source–detector (SD) separations.24–27 A SD dis-tance greater than 1 to 2 mm has been used to validatethe use of diffusion or random-walk models in studiesby Dudko et al.28 For their two-layered model and tis-sue optical properties, Alexandrakis et al.15 showedthat the differences between the diffusion and the MCresults were �8.5% at a 0.5 mm SD separation, in-creasing to �17% at a 1 mm SD separation andbecoming progressively smaller thereafter. Theseinvestigators indicated that the discrepancy was theresult of the breakdown of the diffusion solution atsmall SD separations and hence a limitation of themodel. Therefore, in consideration of both the potentialand the limitation of our diffusion-based inversionalgorithm for particular optical fiber probes andexpected tissue optical properties, we explored the ap-plicability of the presented inversion method. Two setsof different tissue optical properties and correspondinganisotropy factors adopted from previous publicationsby Drezek et al.29 and Qu et al.30 have been used in theinversions to assess the general applicability of thealgorithm. We have successfully obtained the spectralabsorption and scattering coefficients from a singleinput diffuse reflectance spectrum collected at the tis-sue surface. The recovery ability of our inversionmethod is systematically studied for different SD sep-arations, collection fiber geometries, and a wide rangeof similar tissue optical properties. As an importantstep toward a better understanding of the recoveryerror sources of the presented inversion model, wehave also accessed the accuracy of the diffusion ap-proximation against MC predictions of tissue reflec-tance for the specific geometries and tissue opticalproperties of interest. Although there are significantdifferences between the diffusion approximation andthe MC simulation for short SD separations, we haveshown that with our algorithm the tissue opticalproperties can still be recovered well under certainconditions. Additionally, the retrieved quantitativeinformation about hemoglobin oxygen saturation inthe preliminary experimental study will help thereal-time study of tissue physiological states. As itrequires only one input spectrum collected with anOcean Optics R400-7-SR fiber probe (Fig. 1) at thetissue surface, the inversion method we present ismore convenient to use than those with multiple SDseparations.

2. Methods

A diffusion-based inversion algorithm has been devel-oped to recover the wavelength-dependent absorptionand scattering coefficients from an input diffusion re-flectance spectrum. In this inversion method a bound-ary mismatched 2D semi-infinite homogeneous tissuemodel with a pencil incident light beam (N.A. of 0)normal to the tissue surface is used for the diffusionapproximation of tissue reflectance6,14,22,31 (Fig. 2). Tosimplify this model, the collecting fiber is assumed tohave a finite diameter and a N.A. of 1.32 For steady-state light propagation subjected to extrapolatedboundary conditions, when the light source is as-sumed to be a single scattering source with strength

Fig. 1. Cross section of the hexagonally close-packed fiber opticprobe. The central collection fiber goes to the spectrometer, and thesix outer illumination fibers go to the white-light source.

1 November 2006 � Vol. 45, No. 31 � APPLIED OPTICS 8153

S�z� � a��z � 1��t�� located at z0 � 1��t�, a corre-sponding image source of the same strength shouldbe placed at ��z0 � 2zb� (the z-axis origin is at thetissue surface, and the z axis penetrates the tissue);the diffuse reflectance Rd�r� at the tissue surface canthen be obtained from the diffusion approximation asderived by Farrell et al.14:

Rd�r� �a�

4� � 1�t�

��eff �1r1� exp��effr1�

r12 � � 1

�t�� 2zb�

� ��eff �1r2� exp��effr2�

r22 �, (1)

where a� � �s��a � �s�� is the transport albedo,�eff � 3�a��a � �s��1�2 is the effective attenuationcoefficient, �s� � �1 � g��s is the transport scatteringcoefficient, and g is the anisotropic parameter (the av-erage cosine of the scattering angle). The total inter-action coefficient is �t� � �a � �s�. The extrapolatedboundary position is given by zb � 2AD, where thediffusion constant is D � �3�a � �1 � g��s��1 and Ais the internal-reflection-related parameter. A is givenby A � �1 � rd��1 � rd� with rd � �1.44 nrel

�2

� 0.71 nrel�1 � 0.668 � 0.0636 nrel. The relative re-

fractive index is nrel � ninside�noutside. For a matchedboundary, A is equal to 1. The distance from thecenter of the source fiber to the center of the detectionfiber, i.e., the SD separation, is expressed as r. Thedistance from the single scattering source to the col-lection point is r1 � �z � z0�2 � r21�2, and the corre-sponding distance from the image source to that pointis r2 � �z � z0 � 2zb�2 � r21�2.13,14,22,33

A normalized input diffuse reflectance spectrumis generated from MC simulation with the tissueoptical properties and corresponding anisotropyfactors previously published by Drezek et al.29 or Quet al.30 MCML34 is used as the MC simula-tion program, and a packet of 500,000 photons islaunched per simulation.34–37 To recover the tissueoptical properties with the inversion algorithm, theleast-squares fitting method has been used to matchthe calculated normalized spectrum, which is ob-tained from Eq. (1), with the input normalizedspectrum. During the inversion, the wavelength-dependent scattering coefficients are approximated

by �s�� c1�f, and the absorption coefficients are

given by �a�� c2HbO2�� 1 � �Hb��. Heref � �1.1 (Refs. 38 and 39); c1, c2, and � are fittingparameters; and HbO2�� and Hb�� are absorptioncoefficients of oxyhemoglobin and deoxyhemoglobin,respectively.27,38,40–43 The best combination of c1, c2,and �, corresponding to the best match of input andcalculated normalized diffuse reflectance spectra, ispicked up from a large range of initial input fittingcoefficients, giving the recovered absorption and scat-tering coefficients. For all the simulations in this pa-per, the wavelength-dependent anisotropy factor g issimplified to be a constant associated with a specifictissue site. Simulations show that small variations inthe g value do not change the results significantly,which is in agreement with previous publications.29

The approximation of reduced scattering coefficient�s� � c��0.4 can also be used in Eq. (1), instead of g and�s, to avoid concerns over constant g. By integratingthe results of the 2D model (Fig. 2) over different fibergeometries and SD separations and applying themodel to different ranges of tissue optical properties,the recovery ability of our inversion algorithm hasbeen systematically studied.13,27,44

The spatially resolved diffuse reflectance at thetissue surface, Rd�r�, contains valuable diagnosticinformation about the tissue site.5,18,31,45 Somepublications have pointed out that the diffusion ap-proximation is invalid for analyzing the diffuse re-flectance when the detector is too close to the lightsource owing to the collapse of the isotropic scatter-ing assumption.5 Notwithstanding this complaint,the SD separation is an important parameter forclinical measurement of Rd�r� at the tissue surface.32

A sufficient separation is, in fact, demanded in orderto justify using the inverse diffusion algorithm toanalyze the tissue optical properties. On the otherhand, with the exponentially decreasing Rd�r� signalintensity with increasing SD,6,13,14,18,46 the SD sepa-ration values are usually limited. Therefore an opti-mal SD separation range for our specific sets of tissueoptical parameter combinations is desirable to re-solve the dilemma.22,32,46 To fully implement ourdiffusion-based inversion algorithm, we have charac-terized the performance of the proposed diffusion ap-proximation in Eq. (1) against MC simulations. Thesame 2D tissue model and probe geometries shown inFig. 2 have been used with a matched boundary con-dition. The tissue optical coefficients used in thesecomputations are as follows: �s � 4–10 mm�1, �a �0.01–0.25 mm�1, and g � 0.84, the typical range ofoptical coefficients in human epithelial tissue.29,31,47

A preliminary inversion experiment has beenconducted with a HR2000CG-UV-NIR spectrometer(Ocean Optics, Dunedin, Florida), which has a spec-trum resolution of about 0.66 nm. Fresh normal andcancerous excised human breast tissue samples ofabout 1 cm3 each are obtained from the CooperativeHuman Tissue Network. Diffuse reflectance spectra ofthese tissue samples are collected at the tissue surfacewith an Ocean Optics R400-7-SR fiber probe (Fig. 1).

Fig. 2. 2D Semi-infinite homogeneous tissue model.


The probe is composed of seven fibers that each have a400 �m core, 490 �m outer diameter, and N.A. of0.22. They are hexagonally close packed, with thecentral fiber going to the spectrometer and the sixouter fibers going to a tungsten-halogen broadbandlight source (DH2000, Ocean Optics). The diffuse re-flectance spectra recorded are percent reflectancewith respect to a diffuse reflectance standard (WS-1,Ocean Optics) as the 100% reflectance reference.

3. Results

A. Recovery Ability of Diffusion-Based InversionAlgorithm

One limitation of previously published diffusion-based inversion algorithms is that they usually dealwith one set of tissue optical properties at one specificwavelength at a time. To improve the clinical utilityof optical spectroscopic methods, we are interested inthe extraction of wavelength-dependent tissue opticalproperties in one run.4,8 An input tissue diffusionreflectance spectrum is generated with MC simula-tion and normalized at a selected wavelength, whichwill be obtained by in vivo measurement in a clin-ical environment. A set of wavelength-dependentstromal tissue optical properties, previously pub-lished by Drezek et al.,29 has been adopted in thestudies because we are particularly interested in theovarian tissue where the epithelium is just a thinlayer of cells and the stromal tissue is dominant.Here, where nrel � 1.37, simulating a noncontactmode, reflectance signals are all collected by a 1 mmdiameter fiber, and the anisotropy factor g is heldconstant at 0.9 as given in the literature.4,8,29 Thediffusion-based inversion algorithm is used to recoverthe wavelength-dependent tissue optical properties.Figure 3 shows the original and the best-fit spectra,as well as the true and predicted absorption and scat-tering coefficients. For this and the following figures,�a, �s are represented by u_a, u_s, respectively, inunits of inverse millimeters. “M” indicates the resultof the MC simulation, and “D” indicates that of thediffusion approximation. The results (Fig. 3) indicatethat this diffusion-based inversion method workswell at relatively short center-to-center separationsof the source and detection fibers, i.e., SD separationsof r � 1.19 mm and r � 3.01 mm. The influence of theSD separation on recovery ability is shown in Fig. 3.At both lower and upper bounds of SD separations,the method presented can recover the tissue opticalproperties well. To test the general applicability ofthe diffusion-approximation-based inversion method,another set of tumor tissue optical properties, pub-lished by Qu et al.,30 has been chosen. This time,where nrel � 1.37, simulating a noncontact mode,reflectance signals are collected by a 1 mm diameterfiber, at SD separation r � 1.19 mm, while the an-isotropy factor g is held constant at 0.94 as given inthe literature.30 All the recovered coefficients andspectra agree well with their corresponding real val-ues, as shown in Fig. 4. For most of the inversionexamples given in this paper, the anisotropy factor g

is usually held constant at 0.9, since we find theinversion algorithm is not very sensitive to it.29 Therecovery ability of this inversion method is also testedfor the extreme condition of SD equal to 0 for thestromal tissue case; i.e., the same 1 mm fiber is usedas the incident and the collection fiber. It is not sur-prising that the inversion algorithm fails to give rea-sonable recoveries. This is because, in a region tooclose to the incident fiber, the critical assumptions ofthe diffusion theory are breaking down. However, forshort SD separations, the diffusion-approximation-based inversion method can still be used to extracttissue optical properties, and at the extreme ofSD � 0 the largest errors are expected.14,15,33,48

The geometry of the collection fiber does affect theinversion ability of this algorithm, but not signifi-cantly. Both a 0.4 mm diameter collection fiber and a1 mm one have been used to get the spectra for thestromal tissue case at a SD separation of 1.19 mm.The recovered coefficients are shown in Fig. 5. The0.4 mm fiber works better for recovery than the 1 mmone in this specific case. The influence of the geom-etry of the incident fiber has also been studied. Forstromal tissue with a 1 mm collection fiber andSD � 1.19 mm, tissue optical properties have beenextracted from a spectrum that is generated with afinite incident beam34,36 (1 J, 1 mm diameter flatbeam), and they are similar to those optical proper-ties extracted from a spectrum that is generated froma pencil incident beam under the same conditions.Although good recovery efficiency and inversion re-sults are obtained with the pencil incident beam sim-plification, the pencil source beam is still one of thelimitations of the current method and will be im-proved in the future. Another influence on the recov-ery ability comes from the ratio of the scatteringcoefficients to absorption coefficients. To illustrate,for stromal tissue29 with a 1 mm collection fiber andSD � 1.19 mm, the absorption coefficients are keptthe same while the scattering coefficients are manip-ulated to be from 1�10 to 10 times the original value.Then diffusion reflectance spectra are generated fromthese new coefficients, and the corresponding tissueoptical properties are recovered. The average percenterror (ARPE) of the recovered absorption (scattering)coefficients relative to that of the real values over thewhole spectrum is calculated. Here, the recovery per-cent error is expressed in the following equation,where �a�s�

r is the recovered absorption (scattering)coefficient:

RPE ��a�s�

r � �a�s�

�a�s�� 100%. (2)

A positive ARPE indicates that the coefficient is over-estimated, and a negative value indicates that it isunderestimated. When �s is manipulated to be from1�2 to 1 times the original value, the �a ARPE variesfrom �14.6% to 4.5%, and the corresponding �s

ARPE varies from �41.2% to �70.6%. When �s ismanipulated to be from 1 to 5 times the originalvalue, the �a ARPE varies from 4.5% to 33.8%, and


the corresponding �s ARPE varies from �70.6% to�85.3%. No systematical ARPE trend is observedwith the variation of the scattering coefficients toabsorption coefficients ratio, since the errors are av-eraged over the whole spectrum. The errors shownhere are within a range similar to those in previouspublications; errors of �50% or greater have beenreported.13,15,20,40,49 In general, all these scatteringand absorption coefficients can be extracted with theinversion algorithm, but the recovery ability varieswith the scattering-to-absorption ratio.33 Moreover, it

is worthwhile to stress that only a fixed SD separa-tion is employed in the fiber probe geometry, andthe wavelength-dependent tissue optical coefficientspectra are all recovered together in one run; thus, intheory, this inversion method is considered more ef-ficient.

B. Applicability of Diffusion Approximation for theInversion Algorithm

The performance of the diffusion-based inversion algo-rithm for tissue optical properties recovery may be

Fig. 3. Extraction results of stromal tissue recovered at SD separations of (a1)–(a3) 1.19 mm and (b1)–(b3) 3.01 mm with a 1 mmdiameter collection fiber: (a1), (b1) input diffuse reflectance spectra (Rd_M) and best-fit spectra; (a2), (b2) real absorption coefficients(u_a stroma) and the best-fit results; (a3), (b3) real scattering coefficients (u_s stroma) and the best-fit results.


influenced by many external and internal factors.8 Theexternal factors include the illumination–collection ge-ometry4 and presence of various tissue chromophoresand scatterers. The internal influence is dominatedprimarily by the limitations of the diffusion theory.To fully implement our diffusion-based inversion algo-rithm, it is necessary to characterize the performanceand limitation of the proposed diffusion approxima-tion. The characterization of the diffusion algorithmwill serve as the base for optimization of the inversion

algorithm coupling to specific tissue type and fiber ge-ometry. We have chosen the MC simulation as thestandard against which the diffusion algorithm is com-pared at realistic ranges of tissue optical propertiesand illumination–collection fiber geometries. Thetissue optical coefficients used in this computationare the following: �s � 4–10 mm�1, �a � 0.01–0.25mm�1, g � 0.84, the typical range of optical coeffi-cients in human epithelial tissue,29,31,46 and SD sep-aration ranges from 0 to 3 mm. To quantitatively

Fig. 4. Extraction results of tumor recovered at a SD separation of 1.19 mm with a 1 mm diameter collection fiber: (a) normalized inputdiffuse reflectance spectrum (Rd_M) and best-fit spectrum; (b) absolute input diffuse reflectance spectrum (MC) and best-fit spectrum;(c) real absorption coefficients (u_a tumor) and the best-fit results; (d) real scattering coefficients (u_s tumor) and the best-fit results.

Fig. 5. Extraction results of stromal tissue recovered at a SD separation of 1.19 mm, with 0.4 and 1 mm diameter collection fibers;(a) real absorption coefficients (u_a stroma) and the best-fit results; (b) real scattering coefficients (u_s stroma) and the best-fit results.


study the agreement of these two methods, we definePE(r) as

PE�r� �RdD�r� � RdM�r�

RdD�r�� 100%, (3)

the percent error of the two reflectance spectra as afunction of SD separation distance r. Here, RdD�r� isthe spatially resolved diffuse reflectance of the diffu-sion approximation, and RdM�r� is the correspondingresult of the MC simulation. We also define absolutepercent error (APE) as the absolute value of the cor-responding PE. The agreement among the results ofdiffusion approximation and those of MC simulationvaries with the combinations of tissue optical prop-erties.14,15,33 Since we are interested in the influenceof various �a and �s combinations on that agreement,g is held constant at g � 0.84. In the separationdistance of 0.5–3 mm, the average APE (AAPE)between the diffusion approximation and the MCsimulation ranges from 6.5% (for �a � 0.01 mm�1,�s � 10 mm�1) to 19.2% (for �a � 0.25 mm�1, �s �4 mm�1); this is shown in Table 1.

Table 1 also shows that, if �a (or �s) is held con-stant, the increasing ratios of tissue scattering toabsorption ��s��a� lead to better congruence betweenthe diffusion and the MC computations. The increas-ing �s��a ratio enhances the validity of the diffusionassumption and consequently results in decreaseddiscrepancies between diffusion and MC simula-tions.14,33 Figure 6 compares the APE(r) of g � 0.84,�a � 0.25 mm�1, �s � 0.1 mm�1, and �s � 4 mm�1.From the results of Fig. 6, it is clear that, when �s isof the same order as �a, the steady-state diffusionapproximation is not a good choice for inversion.

The diffuse reflectance signal Rd at the tissue sur-face decreases exponentially as a function of separa-tion radius r. Therefore, for our specific inversionapplication and instrumentation, we are primarilyinterested in the region of 0.5–3 mm. In real clinicalmeasurement, the collection fiber is of a finite size;i.e., diameter d � 1 mm, and the measured signal isan integration of the point signals over that fibercollection region.25,26,36 Figure 7 gives the integratedpercent errors of the diffusion approximation and theMC simulation with g � 0.84 and �a � 0.01 mm�1,�s � 10 mm�1; �a � 0.1 mm�1, �s � 7 mm�1; and�a � 0.25 mm�1, �s � 4 mm�1. Since we have scannedonly the separation region of 0.5–3 mm and thecollection fiber is 1 mm wide, photon detection effec-tively begins at r � 1 mm and terminates at

r � 2.5 mm. For our tissue optical properties, theintegrated percent errors of the diffusion approxima-tion and the MC simulation range between the upperand the lower curves of Fig. 7.25,26,36 The integratedpercent errors of the diffusion approximation and theMC simulation might also depend on the size of thefibers used in the system, assuming that the incidentand collection fibers are of the same size. Figure 8gives the integrated percent errors of the diffusionapproximation and the MC simulation for differentfiber diameters (d), where d � 0.2, 0.4, 1 mm, with�a � 0.25 mm�1, �s � 4 mm�1, and g � 0.84. Thenegative integrated percent errors indicate thatthe diffusion approximation signals are less than inthe MC results. From Fig. 8 we can tell, for thesespecific tissue optical properties, that the fiber diame-ters have little influence on the integrated percent er-rors over most of their overlapped SD separationregion. The diffusion reflectance spectra of stromal tis-

Fig. 6. APE(r) of the diffusion approximation and the MCsimulation with g � 0.84, �a � 0.25 mm�1, �s � 0.1 mm�1, and�s � 4 mm�1.

Fig. 7. Integrated percent errors of the diffusion approxima-tion and the MC simulation with g � 0.84, �a � 0.01 mm�1

�s � 10 mm�1, �a � 0.1 mm�1 �s � 7 mm�1, and �a � 0.25 mm�1

�s � 4 mm�1.

Table 1. AAPE of the Diffusion Approximation and the MC Simulationfor Different �a and �s Combinations in the SD Separation Region of

0.5–3 mm with g � 0.84

�a (mm�1)

�s (mm�1)

4 7 10

0.01 10.7% 8.0% 6.5%0.1 14.6% 10.2% 7.3%0.25 19.2% 10.9% 7.9%


sue obtained by MC simulation and diffusion approx-imation, at SD separations of 1.19 and 2.29 mm, arealso shown in Fig. 9. A 1 mm diameter fiber is used tocollect the reflectance signals. The same stromal tissueoptical properties as adopted in a previous inversionpart29 have been used here, with the mismatchedboundary condition, nrel � 1.37, and the anisotropyfactor at g � 0.9. The results show that at both SDseparations the AAPE of the spectra obtained from thediffusion approximation and those from the MC sim-ulation are around 10%. These closely agreeing spec-tra obtained from direct calculation with the specifiedconditions justify our use of the diffusion approxima-tion in the presented inversion algorithm.

C. Preliminary Experimental Study

The inversion parameters were calibrated during theprocedures for retrieving the known tissue opticalproperties in Subsection 3.A, ensuring optimum scat-tering and absorption spectra output. The tissue re-fractive index used is n � 1.4, the SD separation here

is 0.49 mm, and the anisotropic factor is g � 0.9. Thecollected diffuse reflectance spectra are smoothedwith a moving window average of 4 nm and thennormalized at 600 nm. The input and best-fit diffusereflectance and the tissue optical properties extractedfrom normal and cancerous breast tissues are shownin Fig. 10. Differences in both the scattering and theabsorption spectra of the normal and cancerous tis-sue samples are observed,5,30 and the inverted spec-tra are in the range of optical properties typical oftissue.4,29,30,47,50 It should be noted that a wavelengthrange different from all previous spectra is chosen inthis experimental study; this is because we are moreinterested in the visible–near-infrared range for fu-ture inclusion of exogenous scattering and absorbingagents.

4. Discussion

We present a method that employs the shape infor-mation of a diffuse reflectance spectrum, collected atthe tissue surface, to extract the tissue optical prop-erties. As shown in the figures in Section 3, the in-verted absorption and scattering spectra agree wellwith their real spectra, especially the value and rel-ative shape of the absorption spectra. These goodagreements are expected. It is noticed that the shapeof the normalized tissue diffuse reflectance spectrumis predominantly determined by its absorption spec-trum and that the tissue scattering coefficientsspectrum has little structure.5 During the recoveryprocedure, the wavelength-dependent absorption co-efficient is expressed as the weighed combination ofbase oxyhemoglobin and deoxyhemoglobin absorp-tion spectra. The ultimate goal of the algorithm pre-sented is to perform the best tissue optical propertyextractions from an input diffuse reflectance spec-trum by minimizing the difference between the cal-culated and the input diffuse reflectance. Thereforethe recovered fitting parameters of the absorptionspectra are determined by the best fit of the entirereflectance spectra. Although the recovered and theinput scattering spectra in Fig. 4(d) agree well witheach other, there might be concerns over the rela-tively big errors in the scattering spectra in Figs. 3and 5, which are of a similar order of magnitude asin some previous publications.13,15,20,40,49 As statedabove, the recovered optical properties are extractedfrom the best shape fit of the normalized spectra. Thisextraction method is believed to be one of the sourcesthat introduce scattering coefficient errors, sincethere is little structure in the tissue scattering spec-tra [Figs. 4(d) and 5(b)]. Information from the abso-lute values of the diffuse reflectance spectra can helpsolve this problem. The percent difference PD��0�of the recovered diffuse reflectance from the inputabsolute diffuse reflectance spectra at the selectednormalization wavelength ��0� can be calculated.Then the average percent difference (APD) at �0 andits adjacent wavelengths ��0 �� are obtained, i.e.,APD��0, ��. Subsequently an adjusted intensity,with PD��0� of the absolute input spectra set to be �times APD��0, ��, is calculated. For example, if

Fig. 8. Integrated percent errors of the diffusion approximationand the MC simulation for various diameter (d) fibers with�a � 0.25 mm�1, �s � 4 mm�1, and g � 0.84. (The incident andcollection fibers are assumed to have the same diameters.)

Fig. 9. Diffusion reflectance spectra of stromal tissue obtained byMC simulation (Rd_M) and diffusion approximation (Rd_D) at SDseparations of 1.19 and 2.29 mm, with nrel � 1.37 and g � 0.9. A 1mm diameter fiber was used to collect the reflectance signals.


APD��0, �� 30% and � � 50%, PD��0� betweenthe adjusted intensity and the absolute input spectrais 15%. Here the probe-associated adjusting constant� is given by system calibration according to calibra-tion standards. Next the adjusted reflectance in-tensity value and the extracted best absorptioncoefficient at �0 are used to calculate the adjustedscattering coefficient. Then the ratio of the adjustedto the unadjusted scattering coefficient at �0 is usedas a scale factor to get the final adjusted scatteringcoefficient spectrum. The results shown in Fig. 11have good agreement with the real values.29 The rel-ative values of the recovered scattering spectra, alongwith the accurately recovered absorption spectra, canstill provide important physiological informationsuch as relative size of the scatterers and the hemo-globin oxygen saturation. The quantitative informa-tion about the Hb oxygen saturation can be obtainedfrom extracted � values.

A simple hexagonally close-packed fiber probe isused to collect the diffuse reflectance signal. Thesteady-state diffuse reflectance equation derived byFarrell et al.14 is adopted in this study. Although ithas been reported that this equation may not be pre-cise enough in certain cases,5,22 most of our spectracalculated from this equation are still in good agree-ment with the MC results: the AAPE differs byaround 10%.14,40,41 Zonios et al.5 used this steady-state diffuse reflectance model as the starting point in

their study for the simplicity of this equation andproved that it could work well. For the same reason,we directly adopted the diffusion equation, Eq. (1), inour inversion algorithm. For our specific fiber probegeometry, the central fiber and each of the six sur-rounding illumination fibers form a set of incidentand collection pairs that satisfies the simple model.This specific probe geometry gives improved illumi-nation, since it employs multiple incident fibers andhence raises the collected signal-to-noise ratio. Fur-

Fig. 10. Extraction results for normal and cancerous human breast tissue: input and recovered diffuse reflectance (normalized) of (a)normal tissue and (b) tumor; (c) extracted absorption coefficients spectra; (d) extracted scattering coefficients spectra.

Fig. 11. Real scattering coefficients of stromal tissue (u_s stroma)and the adjusted best-fit scattering results for different probe ge-ometries.


thermore, the fixed probe geometry facilitates consis-tent data collection and easy handling.5,25,40

While the preliminary experimental inversionstudy has only one case, the information extracted isin accordance with previous publications.5,10,30,49 Theretrieved tumor scattering coefficients are higherthan those of normal breast tissue, in agreement withcancerous tissue’s tending to have bigger scatterers.5The extracted Hb oxygen saturation of normal tissueis 100%, and that of a tumor is only 75%. Althoughthe Hb saturation extraction may not be reliable forex vivo samples, these extracted values are still in thereasonable range, because the Hb oxygen saturationdrops5 within the capillaries from approximately 97%(arterial blood) to approximately 40% (venous blood).Furthermore, our results agree with the general ideathat tumors are often characterized by abnormallylow Hb oxygenation. There are absorbers other thanoxyhemoglobin and deoxyhemoglobin in our tissuesamples, but the hypothesis that Hb absorption dom-inates gives a good starting point for �a��, and theextracted results are reasonable. Previous publicationson tissue optical property studies also support thishypothesis.5,42,50 Since the tissue optical absorptionand scattering properties are extracted simulta-neously and are correlated, the retrieved quantitativecoefficients can also be further used to extract tissueintrinsic spectra.5,8,9 Further and detailed experimen-tal study of the presented inversion method is cur-rently under way.

5. Conclusions

In conclusion, we have reported a diffusion-theory-based inversion method for recovering tissue opticalproperties from diffuse reflectance spectra collectedat the tissue surface with a cylindrical optical fiberprobe that is composed of seven hexagonally close-packed fibers. The central fiber of the probe is thedetecting fiber, and the six outer source fibers ensurebetter illumination and a lower signal-to-noise ratio.Although there are significant differences betweenthe diffusion approximation and the MC simulationfor short SD separations, the tissue scattering andabsorption coefficients can still be extracted well un-der certain conditions. The recovery ability of thismethod is affected by the optical fiber probe geome-tries used and by the tissue optical properties. For thehexagonally close-packed fiber probe geometry, opti-mized SD separation, fiber diameter, normalizedwavelength, and computing parameters can be foundto yield accurate recoveries for a specific range oftissue optical properties. Since the only input of theinversion algorithm is a diffusion reflectance spec-trum collected with a fixed-SD-separation opticalfiber probe at the tissue surface, our method iscomputationally more efficient than some of the pre-viously published methods. Additionally, the quanti-tative information about Hb oxygen saturation andthe absorption and scattering spectra obtained couldenhance the clinical utility of various spectroscopictechniques.

The authors thank Maureen J. Mueller for herhelp in proofreading this paper. We also gratefullyacknowledge support from the National Institutes ofHealth, grant R01 CA 098341-01A1, the WhitakerFoundation, grant RG-02-0125, and a Coulter-Foundation Early Career Award.

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