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A Robust Monte Carlo Model for the Extraction of Biological Absorption and Scattering In Vivo

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A Robust Monte Carlo Model for the Extraction of Biological Absorption and Scattering In Vivo
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  960 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 56, NO. 4, APRIL 2009 A Robust Monte Carlo Model for the Extractionof Biological Absorption and Scattering  In Vivo Janelle E. Bender ∗ , Karthik Vishwanath, Laura K. Moore, J. Quincy Brown  , Member, IEEE  ,Vivide Chang  , Student Member, IEEE  , Gregory M. Palmer, and Nirmala Ramanujam  Abstract —We have a toolbox to quantify tissue optical prop-erties that is composed of specialized fiberoptic probes for UV-visible diffuse reflectance spectroscopy and a fast, scalable inverseMonte Carlo (MC) model. In this paper, we assess the robustnessof the toolbox for quantifying physiologically relevant parametersfromturbidtissue-likemedia.Inparticular,weconsidertheeffectsof using different instruments, fiberoptic probes, and instrument-specific settings for a wide range of optical properties. Addition-ally, we test the quantitative accuracy of the inverse MC model forextracting the biologically relevant parameters of hemoglobin sat-uration and total hemoglobin concentration. We also test the effectof double-absorber phantoms (hemoglobin and crocin to modelthe absorption of hemoglobin and beta carotene, respectively, inthe breast) for a range of absorption and scattering properties. Weinclude an assessment on which reference phantom serves as thebest calibration standard to enable accurate extraction of the ab-sorption and scattering properties of the target sample. We foundthe best reference–target phantom combinations to be ones withsimilar scattering levels. The results from these phantom studiesprovide a set of guidelines for extracting optical parameters fromclinical studies.  Index Terms —Biomedical optical spectroscopy, diffuse re-flectance, Monte Carlo (MC) methods, tissue diagnostics, turbidmedia. I. I NTRODUCTION D IFFUSE reflectance spectroscopy in the UV–visible(UV-VIS) range can be used to quantitatively and non-invasively measure a tissue’s physiological (hemoglobin satu-ration, total hemoglobin content, and absorber concentrations)and morphological parameters (nuclear size and cellular den-sity)  in vivo . Diffuse reflectance spectroscopy has a broad rangeofapplicationsincancer-relatedfields,assystematicandsignifi-cantdifferencesappearintheopticalspectroscopicpropertiesof malignant and nonmalignant tissues [1]–[8]. These differencesenable diffuse reflectance spectroscopy to be used in applica-tions such as margin assessment during core needle biopsy [9]and tissue diagnostics [7], [8], [10]. Similar spectroscopic tech-niques are also being explored for monitoring tumor response Manuscript received May 8, 2008; revised July 24, 2008. First publishedOctober 3, 2008; current version published May 6, 2009. This work wassupported in part by the National Institutes of Health (NIH) under Grant5R01-CA-100559-05.  Asterisk indicates corresponding author  . ∗ J. E. Bender is with the Department of Biomedical Engineering, Duke Uni-versity, Durham, NC 27708 USA (e-mail: janelle.bender@duke.edu).K. Vishwanath, L. K. Moore, J. Q. Brown, V. Chang, and N. Ramanujam arewith the Department of Biomedical Engineering, Duke University, Durham,NC 27708 USA (e-mail: karthik.v@duke.edu; laura.k.moore@duke.edu;quincy.brown@duke.edu; vivide.chang@duke.edu; nimmi@duke.edu).G. M. Palmer is with the Department of Radiation Oncology, Duke Univer-sity, Durham, NC 27710 USA (e-mail: greg.palmer@duke.edu).Digital Object Identifier 10.1109/TBME.2008.2005994 to therapy [11]–[14], as changes in the tumor vasculature andoxygenation, and thus, optical properties are expected duringthe course of cancer therapy. Another area in which diffuse re-flectancespectroscopycouldbeappliedisinmonitoringchangesin tissue hemoglobin levels due to blood loss or fluid replace-ment [15], in order to help guide transfusions during surgery.Opticalspectroscopicmethodshavehighchemicalspecificitydue to the large number of molecules, including hemoglobin,that interact with light. Fiberoptic technology can be employedtomeasurespectraremotelyandnoninvasivelyfromseveralmil-limeters deep within intact human tissue. In diffuse reflectancespectroscopy, the sample is illuminated and the intensity of backscattered light is measured as a function of wavelength,typically with a fiberoptic probe. The reflected intensity relatesto the attenuation of light as it propagates through the sample,undergoing elastic scattering and absorption interactions. Theabsorption and scattering coefficients of the tissue can be ex-tracted from the intensity of the reflected light via analyticalapproximation to the transport equation [3], [16], [17], empir-ical methods [5], [18]–[21], or with Monte Carlo (MC) mod-eling [22], [23]. Of these techniques, the MC method is con-sidered to provide a “gold standard” for accurate calculationsof absorbed, reflected, or transmitted light in a turbid medium.However, the MC technique operates in a forward fashion, bycalculating the diffuse reflectance from a medium given its op-tical properties. This procedure is very time consuming, andtherefore, impractical to use for interpreting experimental mea-surements.As a solution to this problem, we have developed an in-verse scalable MC model of light transport [22] to operate inan inverse fashion by using a scaling technique to speed up theforward calculations [23], [24]. The model is flexible in that itcan model a wide range of optical properties, any well-definedprobe geometry, and is computationally efficient via the useof scaling techniques. In this paper, we present the quantita-tive accuracy of extracting a wide range of optical propertiesfrom tissue-mimicking phantoms with our inverse MC modelof reflectance under a variety of different experimental condi-tions and with different instruments and fiberoptic probes toassess the robustness and clinical utility of the algorithm. Thequantitative accuracy of the extracted absorption and scatter-ing coefficients was tested with single-absorber phantoms com-posedofhemoglobinorcrocin.Thesesingle-absorberphantomswere also used for an assessment on which reference phantomserves as the best calibration standard to extract optical prop-erties from target samples. Hemoglobin phantoms were used totesttheclinicalutilityofthealgorithminextractinghemoglobin 0018-9294/$25.00 © 2009 IEEE  BENDER  et al. : ROBUST MONTE CARLO MODEL FOR EXTRACTION OF BIOLOGICAL ABSORPTION 961 Fig. 1. (a) Generalized schematic of the instruments used. (b) Schematic of the probes used. saturation and total hemoglobin for a large range of concen-trations. To model breast tissue, double-absorber phantoms(hemoglobin and crocin) were used to test the accuracy inextracting information about multiple absorbers using single-absorber reference phantoms.II. M ATERIALS AND  M ETHODS  A. Instruments and Probes Two different instruments designed to make diffuse re-flectance measurements through a fiberoptic probe were tested.Fig. 1(a) shows a general representation of the instruments, andFig. 1(b) shows a schematic representation of the common arm(in contact with the sample) for the two probes employed inthe experiments described here.  Instrument A  used a 450-W Xearc lamp (JY Horiba) filtered via a scanning double-excitationmonochromator (Gemini 180, JY Horiba) as the source, whilethe remitted light was coupled through an imaging spectro-graph (Triax 320, JY Horiba) and detected by a Peltier-cooledopen-electrode charge-coupled device (CCD) (Symphony, JYHoriba).  Instrument B  (SkinSkan, JY Horiba) consisted of a150-W Xe arc lamp and double-grating excitation monochro-mator as the source, while the reflected light was collected viaan emission monochromator, and delivered to an extended redphotomultiplier tube (PMT). For both instruments, the illumi-nation and collection light was coupled to the tissue phantomvia a bifurcated fiberoptic probe bundle. Probe A  (RoMack, Inc., Williamsburg, VA) used a core of 19fibers for illumination and detected the reflected signal throughan 18-fiber collection ring. Each individual optical fiber in thebundle had a core diameter of 200  µ m with numerical aperture(NA) of 0.22.  Probe B  (RoMack, Inc.) consisted of 29 illumina-tionfibersarrangedaround29collectionfibers.Theilluminationfibers had NA of 0.125, while the collection fibers had NA of 0.12. The core/cladding diameter of each individual fiber inprobe B was 200/245  µ m. Both probes were custom-designedin-house. Probe geometry in each case was accounted for in theMC model via convolution over all illumination and collection TABLE II LLUMINATION AND  C OLLECTION  P ARAMETERS fibers. The centers of all illumination and collection fibers weredetermined by imaging the common end of the fiber bundleand calculating the coordinates of the illumination and collec-tion centers using ImageJ [25]. The probe geometry was thenintegrated for every illumination–collection pair to determinecollection probability [22].  B. Optical Measurements TheinstrumentsettingsforinstrumentsAandBaretabulatedin Table I.The fixed parameters for instrument A were the CCD analog-to-digital conversion (ADC) gain (which was set to 6.7 × ) andits ADC speed (which was set to 20 kHz). CCD gain is de-fined as the number of electrons generated in the CCD requiredto generate a single ADC count reported by the detector sys-tem. A 1200-grooves/mm grating and 1-mm slit widths wereused for the illumination spectrometer. The spectral bandpassof the imaging spectrograph was fixed at 1.9 nm using a 600-grooves/mm grating and 0.6-mm slit width unless otherwisenoted. The other spectral bandpass that was tested was fixed at10 nm using a 300-grooves/mm grating and 1-mm slit width.Bandpass was calculated by measuring the emission spectrumfrom a krypton gas emission lamp (90-0014-01, UVP, Upland,CA) under the same experimental parameters used for the phan-toms. The full-width at half-maximum of the lamp spectrumwas determined from the 645.6 nm spectral line. Zero-order il-lumination was used, and the diffuse reflectance was collectedover the wavelength range in 0.13 or 0.26 nm increments, de-pending on the grating used. For the 1.9-nm spectral bandpasssetting,a600-grooves/mmgratingblazedat400nmwasusedtocollect two scans with an approximately 10 nm overlap to coverthe entire wavelength range. The first scan was from 348.5 to479.9 nm, and the second scan was from 470.3 to 600.1 nm.The two scans were combined in postprocessing in MATLAB(TheMathWorks,Natick,MA).Tocombinethespectra,thetwoscanswereaveragedintheregionofoverlap,beginningwiththepixel of closest overlap. For the 10-nm spectral bandpass set-ting, a 300-grooves/mm grating blazed at 500 nm was used tocollect a single scan that covered 334.9–602.1 nm.ThefixedparametersforinstrumentBweretheexcitationandemission bandpasses, which were fixed at 5 nm and the PMT  962 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 56, NO. 4, APRIL 2009 high voltage, which was set at 350 V. The spectral bandpassof the illumination and collection ends of instrument B wasfixed at 2.1 and 3.5 nm, respectively, using 1200-grooves/mmexcitation and emission monochromator gratings, and 0.5 mmslit widths. The bandpass of the collection end was calculatedby measuring the emission spectrum from an He–Ne laser inthe same manner as described earlier. The diffuse reflectancewas measured using synchronous scanning over the wavelengthrange in 5 nm increments. A 1200-grooves/mm grating blazedat 350 nm was used to collect a single scan that covered 350–600 nm. Cubic splines were used to resample the spectra to350–600 nm in increments reflective of the bandpass of theinstrument. Dark subtract was enabled on both instruments toaccount for the noise from the dark signal. The reflectance fromeach phantom was measured once with the room lights off,and diffuse reflectance was collected. The exposure time of theinitial phantom in a set was adjusted to reach a maximum SNR(at 500 nm) of greater than 100 for instrument A and instrumentB. Phantom preparation and optical properties are described inSection II-D. C. MC Model of Reflectance The inverse MC model is based on a scaling approach de-scribed previously [24]. The diffuse reflectance from a single-baseline MC simulation for a given set of absorption and scat-tering coefficients (optical properties) is scaled using simpleanalytical expressions to predict the diffuse reflectance for anycombinationofopticalabsorptionandscattering;thesearecom-piled to create a lookup table for a wide range of absorptionand scattering [22]. The inverse MC model we employed usesa flexible convolution scheme that accurately accounts for thefiber-probegeometryusedintheexperimentalmeasurementsbyconvolving the photon collection probability over each source–detector separation [22]. 1) FixedandFreeParametersofFit:  Thediffusereflectancespectrum is a function of the wavelength-dependent absorp-tion and scattering coefficients that, in turn, are determinedby specified absorber concentrations, scatter size, and scatterdensity [22]. The free parameters that are iteratively updatedduring the reflectance fitting include the absorber concentra-tion and the scatterer size and volume density. The extinctioncoefficients for the absorber and the wavelength-dependent re-fractive indexes of the scatterer and surrounding medium arefixed parameters [22]. The average refractive indexes for thescatterer (polystyrene spheres) and surrounding medium (wa-ter) over 350–600 nm were 1.60 and 1.34, respectively. Therefractive index of polystyrene spheres has been reported to beconstant within approximately 1% of this value over the wave-length range used [26]. The extinction coefficients for oxy-Hb(HbO 2 ),deoxy-Hb(HbH),andcrocin(Cr)weremeasuredusinga standard UV-VIS spectrophotometer (Cary 300, Varian, Inc.,Palo Alto, CA). 2) Reference Phantom:  In order to calibrate for systemthroughput and wavelength dependence, the measured diffusereflectance spectrum from the sample is calibrated to the diffusereflectance spectrum from a reference phantom with knownoptical properties. The reflectance spectrum of the referencephantom can be modeled by the MC algorithm using the scalingmethod. A ratio of the measured reference phantom reflectanceto the modeled reference phantom reflectance gives a calibra-tion factor that enables a direct comparison between measuredand predicted reflectance spectra during the inversion process.When a reference phantom measurement from one day is usedto calibrate for a sample measurement from a different day, anadditional calibration step is carried out. The diffuse reflectancespectrum is measured from a reflectance standard, such as aspectralon puck (SRS-99-010, Labsphere, Inc., North Sutton,NH)oranintegratingsphereimmediatelyafterthemeasurementto calibrate for day-to-day variations in system throughput. Theuse of different day reference–target inversions enabled us tomimic what is done in a clinical situation, where it is not prac-tical to measure diffuse reflectance from a reference phantom. 3) Inversion Process:  Given an experimentally measureddiffuse reflectance spectrum, the inverse MC model minimizesthe sum of squares error between the predicted diffuse re-flectance and the measured diffuse reflectance by iterativelyupdating the optical properties. When the sum of squares differ-ence between the modeled and measured diffuse reflectance isminimized, the concentrations of absorber(s) and the scatterersize and volume density that best predict the measured diffusereflectance spectrum are extracted. We used 100 fits to ensure astability of convergence.  D. Phantom Preparations The first portion of phantom experiments tested the quantita-tive accuracy of the inverse MC model for extracting scatteringand absorption coefficients using three sets of single-absorberphantoms. The accuracy in optical properties for the single-absorber phantom inversions was assessed using all referenceand target phantom combinations within the same experiment.From the single-absorber phantom results, the best referencephantoms were selected. These reference phantoms were usedto test the effect of spectral bandpass with instrument A andthe effect of different instruments and probes. These referencephantomswerealsousedtotestthealgorithm’sabilitytoextractabsorber concentrations from two sets of biologically relevantphantoms: Hb saturation phantoms and phantoms with a largerange of Hb concentrations. We then tested the robustness of thealgorithm in extracting the concentrations of multiple absorbersinadouble-absorberphantomset,usingthebestreferencephan-toms, which were measured on a different day.Liquid tissue-simulating phantoms were prepared by mix-ing predetermined volumes of absorber with scatterer. Sus-pensions of 1- µ m-diameter monodisperse polystyrene micro-spheres(07310,Polysciences,Inc.,Warrington,PA)withknownvolume density were used as the scatterer. Powdered forms of human hemoglobin  A 0  (H0267 ferrous stabilized, Sigma Co.,St. Louis, MO) and/or crocin (17304 standard Fluka, Fluka,Allentown, PA) were used as the absorbers. Concentrated stock solutionsofHb(114 µ M)and/orCr(12.3mM)werepreparedbydissolving a known weight of the dry powders in deionized (DI)water and its absorption spectrum determined by measuring a  BENDER  et al. : ROBUST MONTE CARLO MODEL FOR EXTRACTION OF BIOLOGICAL ABSORPTION 963 TABLE IIA BSORBER  (HB)  AND  S CATTERER  L EVELS  U SED IN  SA_HB_A  AND  SA_HB_B dilutedsampleinaspectrophotometer.WeusedHbandCrastheabsorbers since they simulate blood and beta-carotene, two of the absorbing species commonly present in breast tissues [27],whichourgrouphasstudiedextensively.Thereducedscatteringcoefficient of the stock scatterer was determined from the Mietheory using freely available software [28], given the knownsize, density, and refractive index of the polystyrene spheres,as well as the refractive index of the surrounding medium. Thephantoms were prepared and measured from 2.5-cm-diametercylindrical plastic containers by diluting a small volume of theabsorption stock solution with the scatterer and DI water to atotal initial volume of 8 mL. The ratios of stock solution andscatterer volume to the total phantom volume enabled us to cal-culate the expected absorption ( µ a ) and reduced scattering ( µ ′ s )coefficientsofthepreparedphantom.Forallpreparedphantoms,the expected optical properties were comparable to the knownabsorption and scattering coefficients reported for human breasttissuesintheUV-VISwavelengthrange[29].Reportedaveragesin µ a  and µ ′ s  weretakenoverthe350–600nmwavelengthrange. 1) Single-Absorber Phantoms:  Three sets of single-absorber phantoms were used to test the accuracy of the in-verse MC algorithm in extracting  µ a  and  µ ′ s . Each phantom set“SA_Hb_a” and “SA_Hb_b” had Hb as the absorber, whereeither Hb or polystyrene spheres was added incrementally(SA_Hb_a and SA_Hb_b, respectively), while “SA_Cr” had Cras the absorber. Each of SA_Hb_a and SA_Hb_b consisted of ten phantoms with absorber (Hb) and scatterer levels shownin Table II. Two containers were used for SA_Hb_a, eachcorresponding to an initial absorber level (A1) for a set scat-tering level (S2 and S4). The stock Hb solution was incre-mentally added to each container to increase the absorptioncoefficient from A1 to A5. Similarly, two containers were usedfor SA_Hb_b, each corresponding to an initial scattering level(S1) for a set absorber level (A2 and A4). Polystyrene sphereswere incrementally added to each container to increase the scat-tering coefficient from S1–S5. Between each addition of ab-sorber in SA_Hb_a or scatterer in SA_Hb_b, the reflectancespectrum was measured. The incremental additions of absorberin SA_Hb_a caused the scattering coefficients to decrease dueto dilution by up to 12%, while the additions of scatterer inSA_Hb_b caused a dilution of the absorption coefficient up to23%.Thesechangesintheabsorptionandscatteringcoefficientswere appropriately accounted for in subsequent calculations of the expected absorption and scattering coefficients.SA_Cr consisted of 12 phantoms with absorber (Cr) levelsshown in Table III. Three containers were used for SA_Cr, each TABLE IIIA BSORBER  (CR) L EVELS  U SED IN  SA_CR correspondingtoaninitialscatteringlevel(S2–S4).ThestockCrsolution was incrementally added to each container to increasethe absorption coefficient. The magnitude of the absorption co-efficients in SA_Cr was selected to be fractions of the A2 level(see Table II) in phantom sets SA_Hb_a and SA_Hb_b. The ad-dition of absorber caused the scattering coefficient to decreaseby 2% for each scattering level. 2) Hb Saturation Phantoms:  In order to evaluate the extrac-tion accuracy of Hb saturation, an experiment was conductedwhere concurrent optical and percent oxygen measurementswere taken, while a phantom was deoxygenated over a 1-h timeperiod. The gradual deoxygenation was accomplished via theaddition of a small amount of Baker’s yeast. A subset of phan-tomsfromSA_Hb_aconsistingofthefiveabsorberlevelsfortheS2 scattering level was used as reference for the deoxygenatedphantom. The phantom used for deoxygenation had absorptionand scattering coefficients corresponding to levels A5 and levelS2 (see Table II), respectively. Phantoms in this set, “Sat_Hb,”were buffered in 10 ×  PBS to keep the pH constant (pH  =  6.94).Optical measurements were taken approximately every minute.The temperature in the room was regulated at 24 ◦ C. The phan-tom was continuously stirred to ensure uniform deoxygenationby the yeast. Hemoglobin saturation (100[HbO 2 ]/([HbO 2 ]  + [HbH])) was calculated from the extracted concentrations of HbO 2  and HbH using reference phantoms measured on thesame day.The percent oxygen over the course of the optical mea-surements was independently and continuously monitored byan oxygen-sensitive electrode (MI-730, Microelectrodes, Inc.,Bedford, NH). Prior to taking measurements, the electrode wascalibrated in an air-saturated water sample, and a water samplecompletely deoxygenated by the addition of the reducing agentsodium dithionite (Na 2 S 2 O 4 ). The percent oxygen measured bythe electrode was recorded at the beginning and end of eachoptical measurement and averaged for that time point. Oxygenpartial pressure was determined from the percent oxygen mea-sured by the electrode assuming a linear relationship. Expectedsaturation measurements were derived using the subroutine re-portedbyKelman[30].ThepartialpressureofCO 2  wasderivedfrom the percent CO 2  in ambient air. For quantifiable compari-son, values for the Hill’s coefficient ( n ) and the partial pressureof oxygen at which Hb is 50% saturated (p 50 ) were calculated. 3) Hb Concentration Phantoms:  A set of phantoms with Hbconcentrations ranging from 1 to 35 µ M was constructed to testthe sensitivity and accuracy of our inversion model in extractingtotal Hb over a large range of concentrations, to model the vari-ability in absorber concentrations seen in biological systems.One container was used for this phantom set, “Conc_Hb,” in  964 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 56, NO. 4, APRIL 2009 TABLE IVA BSORBER  (HB)  AND  S CATTERER  L EVELS  U SED IN  CONC_HBTABLE VA BSORBER  (CR  AND  HB) L EVELS  U SED IN  DA_HBCR which the initial scattering level was approximately equal toS4 (see Table II). The phantoms in Conc_Hb spanned a largerrange of absorption coefficients (H1–H17) than the phantomsin SA_Hb_a and SA_Hb_b, as shown in Table IV. Sequentialadditionoftheabsorbercauseddilutionsinthescatteringcoeffi-cientbyupto30%.Phantomsmeasuredonthesameday,withinthe same set, were used as references for this set of phantoms. 4) Double-AbsorberPhantoms:  Asetof20double-absorber(Hb and Cr) phantoms was used to test the algorithm’s ability toextractconcentrationsofmultipleabsorbers,whichmorecloselymimicsbreasttissueabsorptionproperties.Eachofthefourcon-tainers used in this phantom set, “DA_HbCr,” corresponded toan initial scattering and Hb absorption level. The first phantomin each container contained Hb as the only absorber. There werefour combinations of scattering levels and initial Hb levels:A2–S2, A2–S4, A4–S2, and A4–S4 (see Table II). For eachof these combinations, the stock Cr solution was added incre-mentally to yield four Cr absorber levels. The Cr levels weredesigned as fractions of the initial Hb level. The ranges of theabsorbers are shown in Table V. As before, changes in the ex-pected values of the scattering and absorption coefficients wereaccounted for in the prepared phantoms.III. R ESULTS  A. Single-Absorber Phantoms We first tested the quantitative accuracy for extracting µ a  and µ ′ s  with phantom sets SA_Hb_a, SA_Hb_b, and SA_Cr using Fig. 2. Extraction accuracy for all reference–target combinations within setsSA_Hb_a, SA_Hb_b, and SA_Cr. Averages and standard deviation are calcu-lated over all reference phantoms for each target within a phantom set. Theblack line is the line of perfect agreement.TABLE VIE XTRACTION  A CCURACY  (A VERAGE  P ERCENT  E RROR  ±  S TANDARD D EVIATION )  FOR ALL  R EFERENCE –T ARGET  C OMBINATIONS FOR S INGLE -A BSORBER  P HANTOMS instrument A with the 1.9-nm bandpass setting and probe A.The quantitative accuracy for wavelength-averaged extracted µ a  and  µ ′ s  was determined for all reference–target phantomcombinations and then averaged over all reference phantoms toproduce the average percent error  ±  standard deviation within aphantomset.Thepercenterrorsarepresentedasabsolutevalues.Fig.2showstheextractedversusexpectedwavelength-averaged µ a  and  µ ′ s  for all phantoms in the three sets. Table VI showsthe extraction accuracy for all reference–target combinations.The similarity in extracted optical parameters, regardless of theabsorber, indicates the MC model’s excellent adaptability todifferent absorbing species. 1) Choice of Reference Phantom:  Fig. 3 shows errorgrids depicting the wavelength-averaged percent errors in ex-tracted  µ a  and  µ ′ s  for all reference–target combinations withinSA_Hb_a, SA_Hb_b, and SA_Cr. Each square represents a sin-gle reference–target combination; light gray squares are combi-nations in which the error is under 10%, medium gray squaresare between 10% and 20% error, and black squares are greaterthan 20% error. White squares are combinations for which nophantom measurements were taken. For each scattering level,there are five possible absorber levels, which are indicated forS1. The dark black lines separate each scattering level. It is
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