rapid and accurate estimation of blood saturation, melanin …pilon/publications/ao2010.pdf ·...

Rapid and accurate estimation of blood saturation,melanin content, and epidermis thickness from

spectral diffuse reflectance

Dmitry Yudovsky and Laurent Pilon*University of California, Los Angeles, Henri Samueli School of Engineering and Applied Science,

Mechanical and Aerospace Engineering Department, Biomedical Inter-Department Program,Los Angeles, California 90095-1597, USA

*Corresponding author: [email protected]

Received 13 November 2009; revised 25 February 2010; accepted 1 March 2010;posted 4 March 2010 (Doc. ID 119962); published 22 March 2010

We present a method to determine chromophore concentrations, blood saturation, and epidermal thick-ness of human skin from diffuse reflectance spectra. Human skin was approximated as a plane-parallelslab of variable thickness supported by a semi-infinite layer corresponding to the epidermis and dermis,respectively. The absorption coefficient was modeled as a function of melanin content for the epidermisand blood content and oxygen saturation for the dermis. The scattering coefficient and refractive index ofeach layer were found in the literature. Diffuse reflectance spectra between 490 and 620nm were gen-erated using Monte Carlo simulations for a wide range of melanosome volume fraction, epidermal thick-ness, blood volume, and oxygen saturation. Then, an inverse method was developed to retrieve thesephysiologically meaningful parameters from the simulated diffuse reflectance spectra of skin. A pre-viously developed accurate and efficient semiempirical model for diffuse reflectance of two layered mediawas used instead of time-consuming Monte Carlo simulations. All parameters could be estimated withrelative root-mean-squared error of less than 5% for (i) melanosome volume fraction ranging from 1% to8%, (ii) epidermal thickness from 20 to 150 μm, (iii) oxygen saturation from 25% to 100%, (iv) bloodvolume from 1.2% to 10%, and (v) tissue scattering coefficient typical of human skin in the visible partof the spectrum. A similar approach could be extended to other two-layer absorbing and scatteringsystems. © 2010 Optical Society of America

OCIS codes: 100.3190, 170.1470, 170.1870, 170.3660, 170.3880, 170.6510.

1. Introduction

Diffuse reflectance spectroscopy has found manyapplications in noninvasive monitoring of biologicaltissues [1–7]. This technique investigates tissuestructure, chromophore concentration, and healthby measuring the tissue’s optical properties. Com-mercially available devices typically analyze experi-mental data using the modified Beer–Lambert law todetermine the relative concentrations of tissue chro-mophores, such as melanin, blood, water, or hemoglo-bin, in arbitrary units [6,8–13]. However, the tissue

scattering coefficient cannot be retrieved [10,14]. Al-ternatively, diffuse reflectance data processed withthe diffusion approximation [15] can yield absolutechromophore concentration and measure the tissue’sscattering coefficient [16], which is related to tissuemicrostructure [16–19]. However, this techniquerequires emitter–detector separation of up to 1 cm,thus limiting the spatial resolution of these devices[20,21].

Current spectroscopic techniques are based onthe assumption that tissue is homogeneous and thatproperties are independent of depth [6,7,9,11,22,23].In reality, most bodily organs, such as skin, the intes-tine, or the cervix, are protected by a thin liningcalled the epithelial layer [24]. While the organ is

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1 April 2010 / Vol. 49, No. 10 / APPLIED OPTICS 1707

typically composed of connective tissues and per-fused with blood vessels and nerves, the protectiveepithelial layer is bloodless and consists of struc-tured cell layers [24]. Differences in cellular struc-ture and chemical composition give rise to distinctoptical properties, making the assumption of tissuehomogeneity questionable [25]. In skin, for example,the outer epidermal layer is pigmented by melanin,which absorbs strongly in the UV, while the innerdermal layer is pigmented by blood, which absorbsin the visible and near-infrared parts of the spectrum[26]. Furthermore, the thickness of the epidermallayer may vary with anatomical location, gender, andage [27–29].Multilayer optical models of tissue have been

developed [13,30–32] to study the effects of tissuestructure and chromophore distribution on light pro-pagation. However, such models are computationallyintensive and cannot be used in real-time clinical ap-plications [33]. Semiempirical models of light trans-fer have been developed to accelerate computationwithout significant loss of accuracy [34–38]. Mantisand Zonios [34], for example, developed a semiempi-rical model for diffuse reflectance of two-layer media.They used their model in an inverse method to deter-mine the optical properties of two-layer tissue phan-toms of variable epithelial thickness. However, theirmethod requires that the bottom layer be scatteringbut nonabsorbing [34], which is not the case for mostorgans [24,26]. In addition, Tsumura et al. [36] devel-oped a semiempirical model for diffuse reflectance oftwo-layer scattering and absorbing media based onthe modified Beer–Lambert law. The authors as-sumed that the thickness of the top layer was equalto 70 μm [36]. Recently, Yudovsky and Pilon [35]developed a semiempirical model that predicts thediffuse reflectance of strongly scattering two-layermedia. The model accounted for (i) absorption andanisotropic scattering in both layers, (ii) variablethickness of the top layer, and (iii) internal reflectionat the medium/air interface. It was shown to accu-rately predict the diffuse reflectance of skin [35].The objective of this study is to develop an inverse

method based on our semiempirical model [35] and toassess its robustness in estimating the scatteringcoefficients, chromophore concentrations in both epi-dermis and dermis, and the epidermal thicknessfrom spectral diffuse reflectance of human skin.

2. Background

A. Human Skin

Skin is the largest organ of the human body, repre-senting a total surface area of approximately 1:8m2

and a total weight of approximately 11kg for adults[39]. The epidermis and dermis are the two mainlayers. They are separated by the basement mem-brane and rest on the subcutaneous fat layer [39].The topmost layer of the epidermis is called the stra-tum corneum and is composed of dead cells em-bedded in a lipid matrix. The rest of the epidermis

is mainly composed of keratinocytes, melanocytes,and langerhans [39]. Melanocytes synthesize mela-nin, the skin protein mainly responsible for skincolor. Melanin is contained in organelles know asmelanosomes, which are distributed throughoutthe epidermis [39]. Depending on genetic factorsand UV light exposure, melanosomes occupy 1% to43% of the epidermal volume, corresponding to lightor dark pigmented skin, respectively [40–42]. Epider-mal thickness varies with bodily location and rangesbetween 20 and 150 μm [26,42–44].

The dermis, located beneath the epidermis, is re-sponsible for the skin’s pliability, mechanical resis-tance, and temperature control. It contains touch,pressure, and temperature receptors, as well as se-baceous and sweat glands and hair follicles [39].The dermis is composed of collagen fibers perfusedby nerves, capillaries, and blood vessels [26,45,46].The thickness of the dermis ranges between 450and 650 μm [27,47]. Depending on body locationand tissue health, the volume of blood in the dermisranges between 0.2% and 7% [42,48,49]. Approxi-mately half of the blood volume is occupied by ery-throcytes (red blood cells), which are responsiblefor oxygen transfer from the lungs to the rest ofthe body [42,49]. Erythrocytes are composed mainlyof hemoglobin molecules, which reversibly bind tooxygen molecules in the lungs to form oxyhemoglo-bin. Hemoglobin is known as deoxyhemoglobin onceit has released its oxygen molecules. The ratio of oxy-hemoglobin molecules to the total number of hemo-globin molecules in the blood is the so-called oxygensaturation denoted by SO2. Hemoglobin absorptiondominates the total absorption of the dermis in thevisible range [26,39,46]. Furthermore, the spectralextinction coefficient of oxyhemoglobin differs signif-icantly from that of deoxyhemoglobin. Thus, the colorof the dermis depends on the average oxygen satura-tion of its blood content.

B. Skin Properties Measurement

Various techniques exist to measure chromophoreconcentrations and blood saturation of human skin.Commercially available noninvasive, optical devicestypically measure these quantities in a small regionof 1 to 2 cm in diameter and report a device specificmelanin (MI) and erythema (EI) index [40,50]. TheMI corresponds qualitatively to the darkness of skin,while EI corresponds to the redness or inflammationof skin. Such devices have been used to predict therisk of melanoma skin cancer [40] and as dosimetryfeedback during laser treatment of port-wine stains[51] and acne [52]. Recently, hyperspectral imagingin the visible and near-infrared parts of the spectrumhas been used to determine the spatial distributionof oxygen saturation in the human skin [12]. Thistechnique has been applied clinically to studydiabetic neuropathy [53] and predict the healing po-tential of diabetic foot ulcers [4,54]. Such devices ty-pically assume a homogeneous tissue structure anddo not model changes in the scattering coefficient

1708 APPLIED OPTICS / Vol. 49, No. 10 / 1 April 2010

with wavelength or biological state, or from patientto patient. Thus, only relative chromophore concen-tration in arbitrary units can be reported [10,12,13].Furthermore, epidermal thickness and blood volumecannot be determined [10].Epidermal thickness varies naturally with age,

gender, and body location [27–29,55]. It may also in-crease or decrease due to external stimuli. For exam-ple, UV exposure of human skin has been shown toincrease the thickness of the epidermis in addition toincreasing its melanin content [56,57]. On the otherhand, smoking has been shown to decrease epi-dermal thickness [55]. Epidermal thickness can bemeasured reliably with punch biopsy, whereby asample of the skin is removed and analyzed ex vivo[27,55,57]. This invasive technique can be painfuland destroys the sample. Alternatively, noninvasivemeasurements of epidermal thickness can be madewith techniques including optical coherent tomogra-phy or ultrasound [58,59]. However, these techniquesare primarily sensitive to the tissue’s scattering coef-ficient; therefore, simultaneous determination ofchromophore concentration is difficult [60,61].

C. Radiative Transfer Equation

Biological tissues, such as skin, are generally absorb-ing and strongly scattering media [33]. Light trans-fer through such turbid media is governed by theradiative transfer equation (RTE), written as [15]

s ·∇Iðr; s; λÞ ¼ −μaðλÞIðr; s; λÞ − μsðλÞIðr; s; λÞ

þ μsðλÞ4π

Z4πIðr; si; λÞΦðsi; s; λÞdΩi; ð1Þ

where Iðr; s; λÞ is the spectral intensity at location r ina unit solid angle dΩ around direction s expressed inW=cm2 · sr · nm. The linear spectral absorption andscattering coefficients are denoted by μaðλÞ andμsðλÞ, respectively, and are expressed in cm−1, whilethe scattering phase function is denoted by Φðs; si; λÞ.The Henyey–Greenstein scattering phase function isan approximate expression that accounts for the an-isotropic nature of scattering and is given by [62]

Φðsi; s; λÞ ¼1 − gðλÞ

½1þ gðλÞ2 − 2gðλÞ cosΘ�3=2 ; ð2Þ

where Θ is the angle between s and si and gðλÞ is theHenyey–Greenstein asymmetry factor used exten-sively in tissue optics [33,63,64]. The values of gðλÞmeasured for the epidermis and dermis were ap-proximately the same and range between 0.73 and0.82 in the visible range [46]. To account for the mag-nitude and anisotropy of the scattering phenomenon,the transport single scattering albedo ωtrðλÞ is de-fined as [15]

ωtrðλÞ ¼μs;trðλÞ

μs;trðλÞ þ μaðλÞ¼ μsðλÞ½1 − gðλÞ�

μsðλÞ½1 − gðλÞ� þ μaðλÞ; ð3Þ

where μs;trðλÞ ¼ μsðλÞ½1 − gðλÞ� is the transport scat-tering coefficient.

Monte Carlo simulation is a stochastic method forsolving differential equations, such as the RTE[15,32], and has been applied to study light transferin skin [23,31,37,65]. To do so, a stochastic model isconstructed such that the expected value of a certainrandom variable is equivalent to the value of aphysical quantity that is determined by the exactdifferential equation [66]. The expected value isestimated by sampling the random variable multipletimes. In effect, by repeating the simulation, thevariance of the estimate diminishes. Thus, albeitat the cost of time, the solution may be found witharbitrary accuracy by increasing the number ofsimulations [32,66].

D. Semiempirical Model of Diffuse Reflectance

Recently, Yudovsky and Pilon [35] developed an ap-proximate expression for the diffuse reflectance ofskin treated as a two-layer media and given by

Re ¼ R�½R−ðn1;ωtr;epiÞ − R−ðn1;ωtr;dermÞ�þ R−ðn1;ωtr;dermÞ; ð4Þ

where ωtr;epi and ωtr;derm are the transport single scat-tering albedos of the epidermis and dermis, respec-tively, while n1 is the index of refraction of bothlayers. The reduced reflectance R� is a function ofa single semiempirical parameter α and is expressedas [35]

R� ¼ tanhðYepiÞ1=αþ ð1 − 1=αÞ tanhðYepiÞ

: ð5Þ

The parameter Yepi is the modified optical thickness,defined as Yepi ¼ ζðμa;epi þ μs;trÞLepi, where Lepi is thephysical thickness of the epidermis [67]. The param-eter ζ was approximated by a third-order polynomial[35]:

ζ2 ¼ 4752

þ 3149

ωtr;epi −4954

ω2tr;epi −

1727

ω3tr;epi: ð6Þ

Similarly, assuming the refractive index of tissue tobe n1 ¼ 1:44, the semiempirical parameter α wasfound to be [35]

1=α ¼ −0:569ω2tr;derm − 0:055ωtr;derm þ 0:993: ð7Þ

The function R−ðn1;ωtrÞ appearing in Eq. (4) is thediffuse reflectance of a semi-infinite homogeneouslayer, with transport single scattering albedo ωtrand index of refraction n1 given by [35]

R−ðn1;ωtrÞ ¼ ½1 − ρ01ðn1Þ�½1 − ρ10ðn1;ωtrÞ�

×RdðωtrÞ

1 − ρ10ðn1;ωtrÞRdðωtrÞ; ð8Þ


where ρ01ðn1Þ is the normal–normal reflectivity ofthe tissue/air interface, defined as,

ρ01ðn1Þ ¼�n1 − n0

n1 þ n0

�2: ð9Þ

Expressions for ρ10 and Rd were given in Eqs. (26)and (27) of Ref. [35], respectively. The relative errorbetween the semiempirical model and Monte Carlosimulations was typically around 3% and never morethan 8% for the optical properties of skin in the visi-ble [35].

3. Methods

A. Assumptions

Figure 1 shows the two-layer medium considered toapproximate skin structure. It consisted of a plane-parallel slab of thickness Lepi representing theepidermis characterized by μa;epiðλÞ and μs;trðλÞ andindex of refraction n1. This top layer was supportedby a semi-infinite sublayer representing the dermisand characterized by μa;dermðλÞ, μs;trðλÞ, and n2. Theindices of refraction of both layers were assumedto be identical and constant with wavelength anddepth (i.e., n1 ¼ n2) [63]. The physical distance fromthe surface was denoted z. The thickness of epider-mis Lepi was considered between 20 and 150 μm.The incident light source was modeled as a colli-mated, monochromatic, and normally incident beamof infinite radius and intensity I0ðλÞ ¼ q0ðλÞδðθÞ. Thequantity q0ðλÞ denotes the radiative flux of the colli-mated beam and δðθÞ is the Dirac delta function. Theair/epidermis interface and the interface between theslab and the semi-infinite sublayer were assumed tobe optically smooth and, therefore, specularly reflect-ing. Then, radiative transfer can be considered as onedimensional [15,68]. Thus, the local intensity de-pends only on depth z and angle θ, i.e., Iðr; s; λÞ ¼Iðz; θ; λÞ [15].

B. Closure Laws

To solve the RTE along with the associated boundaryconditions [35] using the Monte Carlo method [32],the radiation properties of both the epidermis anddermis must be specified on a spectral basis.

1. Reduced Scattering Spectrum

The reduced scattering coefficients of the epidermisand dermis were assumed to be equal and are givenby the power law [33]:

μs;trðλÞ ¼ C

� λλ0

�−b; ð10Þ

where λ0 ¼ 1nm was introduced to ensure consis-tency in units. This approximate relationship isbased on analytical [19,69–72] and experimental[17,73,74] studies of scattering in biological mediaand has been used to model tissue scattering inthe visible and near-infrared parts of the spectrum.It has been shown experimentally that C and b de-pend on the average size of the microscopic features,such as cells or connective tissue, responsible forlight scattering in the skin [17,70].

2. Epidermis

Absorption in the epidermis is mainly due to melaninand flesh. Thus, the absorption coefficient in theepidermal layer μa;epiðλÞ was expressed as [42]

μa;epiðλÞ ¼ μa;melðλÞfmel þ μa;backðλÞð1 − fmelÞ; ð11Þ

where fmel is the volume fraction of melanosomesand μa;backðλÞ is the background absorption of humanflesh given by [42,75]

μa;backðλÞ ¼ 7:84 × 108λ−3:255: ð12Þ

The absorption coefficient of melanosomes as a func-tion of wavelength has been approximated as [76]

μa;melðλÞ ¼ 6:60 × 1011λ−3:33; ð13Þ

where λ and μa;melðλÞ are expressed in nanometersand cm−1, respectively. Figure 2(a) shows μa;backand μa;mel predicted by Eqs. (12) and (13) as a func-tion of wavelength between 450 and 700nm. It illus-trates that melanin absorption of UV light is muchstronger than that of near-infrared light. Indeed,the primary function of melanin is to protect the hu-man body from harmful UV radiation [40–42].

3. Dermis

The absorption coefficient of the dermis is deter-mined primarily by the absorption of blood[26,39,46] and can be written as [48,77]

Fig. 1. Simplified skin geometry, biological properties, and opti-cal characteristics of the epidermis and dermis considered in thisstudy.


μa;dermðλÞ ¼ f bloodμa;bloodðλÞ þ μa;backðλÞð1 − f bloodÞ;ð14Þ

where f blood is the volume fraction of the dermis oc-cupied by blood and μa;backðλÞ is given by Eq. (12). Inthe visible range, oxyhemoglobin and deoxyhemoglo-bin are mainly responsible for blood absorption, i.e.,μa;bloodðλÞ ¼ μa;oxyðλÞ þ μa;deoxyðλÞ. The absorption coef-ficient of oxyhemoglobin is given by [48,77]

μa;oxyðλÞ ¼ ϵoxyðλÞChemeSO2=66; 500; ð15Þ

where ϵoxyðλÞ is the molar extinction coefficient ofoxyhemoglobin in cm−1=ðmole=literÞ of molecularweight 66; 500 g=mol whileCheme is the concentrationratio of hemoglobin in blood (grams=liter), and SO2 isthe oxygen saturation. Similarly, the absorption coef-ficient of deoxyhemoglobin is given by [48,77]

μa;deoxyðλÞ ¼ ϵdeoxyðλÞChemeð1 − SO2Þ=66; 500; ð16Þ

where ϵdeoxyðλÞ is the molar extinction coefficient ofdeoxyhemoglobin. While the blood volume fractionf blood and oxygen saturation SO2 may vary with loca-tion and metabolic state, the average value of hemo-globin concentration Cheme is typically constant andequal to 150 g=l [44,77,78]. The spectral molar extinc-tion coefficients of oxyhemoglobin and deoxyhemo-globin are available in the literature for a widerange of wavelengths [77,79–81]. Figure 2(b) showsthe values of ϵoxyðλÞ and ϵdeoxyðλÞ used in this studyin the visible range from 450 to 700nm, as reportedin the literature [77]. Oxyhemoglobin exhibits twoabsorption peaks, at 542 and 578nm, while deoxyhe-moglobin exhibits a single peak, at 554nm. Further-more, oxyhemoglobin is nearly transparent forwavelengths above 600nm, giving oxygen rich bloodits red color.

C. Simulated Diffuse Reflectance

The biological properties required to estimate the ra-diative properties of skin in order to solve the RTEcan be represented by the input property vector:

~ai ¼ hfmel;Lepi; f blood;SO2;C; bi: ð17Þ

For a given vector ~ai, the absorption and scatteringcoefficients of the epidermis and dermis were deter-mined as a function of wavelength using Eqs. (10)–(16). Then, the RTE was solved on a spectral basisfor 40 evenly spaced wavelengths between 480 and650nm using the Monte Carlo simulation softwaredeveloped by Wang and Jacques [32]. A completeand detailed description of the implementationand theoretical underpinnings of this software is gi-ven in Ref. [32]. The number of simulated photonpackets per simulation was adjusted until the var-iance associated with the estimate of the diffuse re-flectance fell below 1%. EachMonte Carlo simulationwas allowed to run with 1,000,000 photon bundles,which effectively reduced the variance of the simu-lated diffuse reflectance spectra to zero. The com-puted diffuse reflectance is referred to as the“simulated diffuse reflectance spectrum,” denotedby Rið~ai; λjÞ, where λj is the jth wavelength and j isan integer between 1 and K .

D. Inverse Method

The goal of the inverse problem was to estimate thevector ~ai from the simulated diffuse reflectanceRið~ai; λjÞ. This was achieved by finding an estimatevector ~ae that minimizes the sum of the squaredresiduals δ expressed as

δ ¼XKi¼1

½Rið~ai; λjÞ − Reð~ae; λjÞ�2W2j ; ð18Þ

where Reð~ae; λjÞ is the estimated spectral diffuse re-flectance predicted by Eqs. (4) and (5), Wj is the

Fig. 2. Absorption properties of endogenous skin chromophoresin the visible range (480 to 700nm). (a) Absorption coefficient ofmelanosomes [76]. (b) Spectral molar extinction coefficient of hu-man oxyhemoglobin and deoxyhemoglobin [77].


weight associated with the jth residual, and K is thenumber of wavelengths considered. The input diffusereflectance spectra Rið~ai; λjÞ were calculated byMonte Carlo simulations for a given input parametervector~ai. Diffuse reflectance was predicted at K ¼ 40evenly space wavelengths between 490 and 650nm.This wavelength range was chosen such thatmelanin, oxyhemoglobin, and deoxyhemoglobin ex-hibit distinct and significant absorption [Figs. 2(a)and 2(b)].Note that the number of wavelengths considered

(K ¼ 40) was chosen arbitrarily. Theoretically, sixwavelengths are required to retrieve the six un-knowns fmel, Lepi, f blood, SO2, C, and b. In practice,more measurements can be made to reduce theeffects of experimental uncertainty. Furthermore,wavelengths can be strategically chosen to coincidewith the absorption peaks of oxyhemoglobin anddeoxyhemoglobin to increase the inverse method’ssensitivity to, for example, SO2 [13]. The goal of thisstudy, however, was to assess the use of the semiem-pirical model [Eq. (4)] in predicting ~ai. Therefore, Kwas chosen to be relatively large, yet practically im-plementable so as to reduce the effects of numericalerror and wavelength selection.The values of Wj were chosen to be 2.0 for λj <

600nm and 1.0 for λj ≥ 600nm to increase the meth-od’s sensitivity to changes in SO2 observed mainly forλ < 600nm. Equation (18) was solved iterativelywith the constrained Levenberg–Marquardt algo-rithm [82]. The values of the estimated propertyvector ~ae were constrained between physiologicallyrealistic upper and lower bounds reported in the lit-erature and summarized in Table 1. The minimiza-tion was stopped once successive iterations of thealgorithm no longer reduced δ by more than 10−9.Furthermore, multiple random initial guesses for~ae were attempted to prevent convergence to a localminimum.

4. Results and Discussion

A. Accuracy of Semiempirical Model for Skin

Figure 3 compares the diffuse reflectance spectrumRiðλÞ calculated by Monte Carlo simulations to thatpredicted by Eq. (4) for Lepi ¼ 30 μm and fmel ¼ 5:0%.Two different values for both f blood and SO2 were ex-plored, namely, f blood ¼ 0:41% and 7% and SO2 ¼ 0%and 100%. The effects of oxygen saturation on theshape of the diffuse reflectance spectrum was mostapparent when f blood ¼ 7:0%. Skin with oxygen de-

pleted blood, corresponding to SO2 ¼ 0%, exhibitedthe single absorption peak of deoxyhemoglobin at554nm, while reflectance for SO2 ¼ 100% exhibitedboth oxyhemoglobin absorption peaks at 542 and578nm. Furthermore, fully oxygenated blood re-sulted in 1.5 times more reflective skin for λ >600nm than when blood was deoxygenated. Figure 3also shows that decreasing the blood volume fractionto f blood ¼ 0:41% greatly diminished the effects ofoxygen saturation on the reflectance spectrum. Then,the oxyhemoglobin and deoxyhemoglobin peaks andchanges in reflectance were far more subtle since theabsorption coefficient of the dermis predicted byEq. (14) was dominated by μa;back. Note that it typi-cally took 2 min on a 2:66GHz processor to computethe diffuse reflectance spectrum for 40 discrete wave-lengths using Monte Carlo simulations [32]. Onthe other hand, the diffuse reflectance spectrumcould be estimated nearly instantaneously by usingEqs. (4)–(9).

The accuracy of the semiempirical model with re-spect to predictions by Monte Carlo simulations canalso be assessed from Fig. 3. For f blood ¼ 7:0%, predic-tions of the diffuse reflectance by the semiempiricalmodel fell within 3% of that calculated by MonteCarlo simulations for allwavelengths considered. Theerrorwas less that 0.5% for λbetween530and600nm,the region associated with the oxyhemoglobin and

Table 1. Lower and Upper Bound Values for Biological Properties of Human Skin Estimated by the Inverse Method

Biological Property Symbol Lower Bound Upper Bound Reference

Volume fraction of melanosomes fmel 1% 10% [40–42]Epidermal thickness Lepi 20 μm 150 μm [26,42–44]Blood volume fraction f blood 0.2% 7% [42,49]Oxygen saturation SO2 0% 100%Scattering constant C 4:0 × 105 cm−1 5:5 × 105 cm−1 [42]Scattering power constant b 1.10 1.50 [16,17,42]

Fig. 3. Diffuse reflectance of human skin as a function of wave-length calculated by Monte Carlo simulates and predicted by thesemiempirical model for Lepi ¼ 30 μmand fmel ¼ 50% and differentvalues of blood volume fraction f blood (0.41% and 7%) and oxygensaturation SO2 (0 and 100%).


deoxyhemoglobin peaks. However, for f blood ¼ 0:41%,the semiempirical model underpredicted the diffusereflectance by approximately 4% for all wavelengthsand for SO2 equal to 0% and 100%. As discussed byYudovsky and Pilon [35], the prediction error asso-ciated with the semiempirical model used in thisstudy increased when the transport single scatteringalbedo of either layer approached unity. Thus, as f blooddecreased, the absorption coefficient of the dermisμa;derm decreased and its single scattering albedo ap-proached 1.0, resulting in larger error in the predic-tion of the diffuse reflectance.

B. Parameter Estimation

Diffuse reflectance spectra were calculated withMonte Carlo simulations while varying the compo-nents of ~ai between the upper and lower bounds re-ported in Table 1. Six values of oxygen saturationSO2, epidermal thickness Lepi, melanosome volumefraction fmel, and blood volume fraction f blood, andthree values of scattering constants C and b, wereconsidered for a total of 11,664 simulated spectra.For each spectrum, the estimate vector ~ae was foundby minimizing δ defined in Eq. (17). To explore theeffects of each parameter separately, two scenarioswere considered. First, the inverse method’s abilityto estimate Lepi, fmel, f blood, and SO2 was assessed,assuming that the scattering constants C and b wereknown. Second, the scattering constants C and bwere assumed unknown and were retrieved alongwith the other four input parameters. To assessthe accuracy of the inverse method, the root-mean-squared (rms) error in the retrieval of a given param-eter was computed over the entire range of biologicalproperties considered for an arbitrary value of thatparameter.

1. Estimation with Known C and b

In this section, C and b were assumed to be knownand equal to C ¼ 5:0 × 105 cm−1 and b ¼ 1:30. Thesevalues were measured ex vivo for healthy humanskin [42]. Here, diffuse reflectance spectra were si-mulated only for input melanin concentrationfmel ¼ 5%. Then, values of fmel, Lepi, SO2, and f bloodwere retrieved by the inverse method by using thesimulated diffuse reflectance spectrum obtained byusing Monte Carlo simulations for the range offmel, Lepi, SO2, and f blood specified in Table 1. Therms relative error between the estimated and inputvalue of fmel ¼ 5%was less than 0.5% for all values ofLepi, SO2, and f blood considered.Figs. 4(a) and 4(b) show the estimated versus input

oxygen saturation SO2 and 10% absolute errorbounds for Lepi ranging from 20 to 150 μm and forf blood equal to 0.20% and 7.0%, respectively. Figure 4(a) indicates that SO2 was overestimated for lowvalues of f blood. This was primarily due to the under-prediction of diffuse reflectance by the semiempiricalmodel for λ ≥ 600nm (Fig. 3). Indeed, highly oxyge-nated blood exhibited stronger reflectance in thisspectral range. Thus, an overestimate of SO2 com-

pensated for the inaccuracy of the semiempiricalmodel. Figure 4(a) also shows the effect of epidermalthickness on the prediction of SO2. A thicker epider-mis resulted in poorer estimates of SO2 because,then, the epidermis optically shielded the dermis.The effect of epidermal thickness on the predictionof SO2 was best illustrated for input SO2 ¼ 60%.In this case, when Lepi ¼ 20 μm, the absolute estima-tion error was less than 5%. However, for Lepi largerthan 100 μm, the absolute error in SO2 increasedbeyond 10%.

Figure 4(b) indicates that, for blood volume equalto 7.0%, epidermal thickness has little effect on theprediction error associated with SO2. The rms andthe maximum absolute errors between the estimatedand input values of SO2 increased from 2.9% and9.2% to 13.5% and 47.9%, respectively, as Lepiincreased from 20 to 150 μm. In contrast, as f blood in-creased from 0.20% to 7.0%, the rms and maximum

Fig. 4. Estimated versus input oxygen saturation SO2 and 10%absolute error bounds for input values fmel ¼ 5:0%, 20 ≤ Lepi ≤

150 μm, and (a) f blood ¼ 0:20% and (b) 7%, respectively. The scat-tering constants were taken as C ¼ 5:0 × 105 cm−1 and b ¼ 1:30.


absolute errors between the estimated and inputvalues of SO2 decreased from 13.8% to 3% and from47.%9 to 9.4%, respectively. The largest maximum er-ror of 47.9% occurred for the lowest values of f bloodand the largest values of Lepi. However, it was nottypical of this inverse method’s performance, as sug-gested by the rms error. Nouvong et al. [54] recentlymeasured the oxygen saturation near healing andnonhealing diabetic foot ulcers using hyperspectralimaging [12]. Their data suggests that skin near non-healing ulcers exhibit a 10% lower oxygen saturationon average when compared with skin near healingulcers. Furthermore, the values of SO2 reported werenear 55%. The rms error of the present algorithm forSO2 ¼ 55% was found to be 3.6%. Thus, the presentmethod is accurate enough to assess wound healingon the diabetic foot.Figures 5(a) and 5(b) show the estimated versus

input values of f blood and 0.5% absolute error boundsfor 20 ≤ Lepi ≤ 150 μm and SO2 ¼ 0% and 100%, re-spectively. Unlike the estimate of SO2 previously dis-cussed, the estimate of f blood was almost unaffectedby epidermal thickness Lepi because the effect of in-creasing or decreasing f blood was to shift the entirereflectance spectrum intensity down or up, respec-tively. Changes in SO2, on the other hand, affectedthe spectral shape of the diffuse reflectance spectrumnear the absorption peaks of oxyhemoglobin anddeoxyhemoglobin (530 <≤ λ <≤ 600nm). However,SO2 had an effect only outside of this wavelengthrange for larger values of f blood. Thus, the proposedinverse method predicted changes in f blood more ac-curately than changes in SO2. Furthermore, as Lepiincreased from 20 to 150 μm, the rms and the max-imum absolute errors between the input and esti-mated values of f blood increased from 0.074% to0.21% andfrom 0.30% to 0.71%, respectively. How-ever, these errors remained small and acceptablefor all values of Lepi.Figures 6(a) and 6(b) show the estimated versus

input values of Lepi and 10% relative error boundsfor SO2 between 0% and 100% and for f blood ¼0:20% and 7.0%, respectively. The relative error inLepi was larger than 10% for f blood less than 1%and Lepi between 40 and 100 μm. However, for Lepismaller than 40 μm or larger than 100 μm, the epider-mal thickness was dominant in determining theshape of the diffuse reflectance spectrum. In bothcases, Lepi was estimated accurately for all valuesof f blood. Indeed, for large values of f blood, ωtr;derm de-creased but remained larger than 0.5, resulting inbetter agreement between the semiempirical modeland the Monte Carlo simulations. Thus, the epider-mal thickness Lepi was better estimated for larger va-lues of f blood. In fact, as f blood increased from 0.20% to7.0%, the rms relative error and the maximum rela-tive error between the input and estimated values ofLepi decreased from 8.1% to 6.6% and from 20.1% to16.7%, respectively. Note that, for all cases, the max-imum absolute error in estimating Lepi was less than

14 μm, which is approximately the diameter of akeratinocyte cell [83].

2. Estimation of Melanin Concentration

Here also, C and b were assumed to be known andequal to C ¼ 5:0 × 105 cm−1 and b ¼ 1:30. The valueof fmel was varied between 1% and 10% and was re-trieved by the inverse method along with Lepi, SO2,and f blood. While melanosome volume fractions of upto 43% have been reported [42], the range consideredin this study was abridged to 10%. Indeed, beyond10%, the transport single scattering albedo ωtr;epi be-came less than 0.50 and the accuracy of the semiem-pirical model in predicting the diffuse reflectancegreatly diminished [35], making accurate inversiondifficult or even impossible.

Fig. 5. Estimated versus input blood volume fraction f blood and0.5% absolute error bounds for input values fmel ¼ 5:0%, 20 ≤

Lepi ≤ 150 μm, and (a) SO2 ¼ 0% and (b) 100%, respectively. Thescattering constantswere takenasC ¼ 5:0 × 105 cm−1 andb ¼ 1:30.


Figure 7 shows the estimated versus input valuesof fmel and 0.5% absolute error bounds for variousinput values of Lepi and f blood, and for SO2 ¼ 50%.The estimate of fmel was well within 0.5% for all va-lues of Lepi. As with f blood, shown in Fig. 5, the abso-lute error was small and acceptable. Similar resultsas those discussed in Subsection 4.B.1 for fmel ¼ 5%were found for the retrieved values of SO2, Lepi, andf blood and need not be repeated. In other words, theability of the inverse method to estimate Lepi, SO2,and f blood was not altered by the melanosome volumefraction fmel.

3. Estimation of Tissue Scattering

So far, the values of C and b were considered to beconstant and known. In reality, they depend on theaverage diameter and concentration of collagen fi-bers in the skin, for example. They may vary from

4:0 × 105 to 5:5 × 105 cm−1 and from 1.1 to 1.5, respec-tively [17,18,70]. To test the ability of the proposedinverse method to estimate C and b from diffuse re-flectance measurements, Monte Carlo simulationswere performed while varying C and b between thesebounds. Then, δ given by Eq. (18) was minimized tofind optimal estimates of C and b for each diffuse re-flectance spectrum, in addition to the four param-eters fmel, Lepi, f blood, and SO2. It was found thatfmel, Lepi, f blood, SO2, and b or C could be estimatedaccurately only if C or b was assumed to be known,respectively. However, if both b and C were retrievedsimultaneously, only SO2, fmel, and Lepi could be es-timated reliably, while f blood, b, and C could not beaccurately estimated. This was despite the fact thatthe estimated diffuse reflectance by the semiempiri-cal model using the retrieved parameters matchedthe input diffuse reflectance closely for all casesconsidered.

Therefore, values of b measured in vitro andreported in the literature [16,17,42] were used inconjunction with the present inverse method.Figures 8(a) and 8(b) show the estimated versus in-put scattering constant C and 5% relative errorbounds for 20 <≤ Lepi <≤ 150 μm, 0:20% ≤ f blood ≤

7%, fmel ¼ 5%, and for SO2 ¼ 0% and 100%, respec-tively. The parameter b was assumed to be constantand equal to 1.30 for both simulated and estimateddiffuse reflectance. The blood volume fraction f bloodhad the strongest effect on the relative error in theretrieved C. Greater error was observed for smallvalues of f blood for reasons previously discussed (Sub-section 4.B.1). As f blood increased from 0.20% to 7.0%,the rms relative error and the maximum relativeerror between the input and estimated values of Cdecreased from 5.9% to 1.4% and from 23.4% to3.4%, respectively. These errors remained smalland acceptable.

Fig. 6. Estimated versus input epidermal thickness Lepi and 10%relative error bounds for input values fmel ¼ 5:0%, 0% ≤ SO2 ≤

100%, and (a) f blood ¼ 0:20% and (b) 7.0%, respectively. The scat-tering constants were taken as C ¼ 5:0 × 105 cm−1 and b ¼ 1:30.

Fig. 7. Estimated versus input melanosome volume fraction fmeland 0.5% absolute error bounds for input values 20 ≤ Lepi ≤ 150 μmand 0:20 ≤ f blood ≤ 7:0%, and SO2 ¼ 50%. The scattering constantswere taken as C ¼ 5:0 × 105 cm−1 and b ¼ 1:30.


To illustrate the accuracy of the inverse methodusing the semiempirical model, simulated diffuse re-flectance spectra were produced for Lepi ¼ 50 and100 μm and fmel equal to 1%, 5%, and 10%, represent-ing weakly, moderately, and strongly pigmented skin,respectively. The input values of the other param-

eters SO2, f blood, and C were chosen in the rangesgiven in Table 1. Then, all five parameters were es-timated, while b was imposed to be 1.30. Table 2shows the relative rms error between the inputand estimated values of fmel, Lepi, f blood, SO2, andC for the three skin complexions and two epidermalthicknesses considered. Note that input values ofSO2 smaller than 25%were ignored while calculatingthe relative rms error associated with estimatingSO2 because these small values are unrealistic forliving tissues.

Finally, the present inverse algorithmwas recentlyused to detect callus and ulcer formation on the footof a diabetic patient [84]. Hyperspectral tissue reflec-tance images of the foot of a diabetic patient were col-lected at regular interval for 18 months [54]. Duringthis period, a foot ulcer developed on the plantar sur-face of the left sole [54]. Hyperspectral data of theulceration site was available 30 days before the ulcerformed. A diabetic foot ulcer is often preceded by cal-lus formation (i.e., epidermal thickening) around thepreulcerative site [85,86]. As the ulcer formation pro-gresses, the epidermis immediately above the necro-tic tissue diminishes in thickness [85,86]. Estimationof the epidermal thickness was performed with thepresent algorithm to confirm that such changes couldbe detected optically. In fact, the preulcerative siteexhibited an epidermal thickness of 85 μm and wassurrounded by much thicker epidermis of approxi-mately 130 μm. These results show promise thatthe present algorithm could be used to assess tissuehealth in vivo and in a clinical setting. However, com-plete discussion of this topic falls outside the scope ofthis paper.

5. Conclusion

In this paper, human skin was modeled as a slab ofvariable thickness, corresponding to the epidermis,supported by a semi-infinite layer, correspondingto the dermis. Absorption in the epidermis wasdue to melanin and varied depending on the melano-some volume fraction. Absorption in the epidermiswas due to oxyhemoglobin and deoxyhemoglobinand varied with blood volume fraction and oxygen sa-turation. The index of refraction and the scatteringcoefficient in both layers were assumed to be identi-cal. The radiative transfer equation was solved on aspectral basis between 490 and 650nm by MonteCarlo simulations, to produce simulated diffuse

Fig. 8. Estimated versus input scattering constant C and 5% re-lative error bounds for input values 20 ≤ Lepi ≤ 150 μm, 0:20 ≤

f blood ≤ 7:0%, fmel ¼ 5:0%, and b ¼ 1:30 for (a) SO2 ¼ 0% and(b) SO2 ¼ 100%. The scattering constant b is assumed to be known.

Table 2. Relative RMS Error between Retrieved and Input Values of fmel, Lepi, SO2, f blood, and C for Lepi ¼ 50 and 100 μm and fmel Equal to 1%, 5%,and 10%, Representing Weakly, Moderately, and Strongly Pigmented Skin, Respectivelya

Lepi ¼ 50 μm Lepi ¼ 100 μm

fmel fmel Lepi SO2 f blood C fmel Lepi SO2 f blood C

1% 9.3% 16.1% 1.3% 2.5% 2.4% 10.1% 15.4% 4.8% 5.2% 2.2%5% 9.9% 9.6% 4.6% 10.2% 2.4% 3.7% 5.7% 4.9% 6.4% 2.5%10% 3.5% 1.8% 3.2% 11.8% 3.1% 0.5% 3.8% 15.8% 11.3% 2.2%

aThe input values of SO2, f blood, and C were chosen in the ranges given in Table 1. The rms error in estimating SO2 was calculatedfor SO2 > 25%.


reflectance spectra for a wide range of biologicalproperties. Then, an inverse method was used to re-trieve physiologically meaningful parameters fromthe simulated diffuse reflectance spectra. A quickersemiempirical model [35] was used instead of MonteCarlo simulations in the iterative inversion proce-dure. The accuracy of the inverse method in estimat-ing fmel, Lepi, SO2, f blood, and C was explored for arange of physiologically meaningful values specifiedin Table 1. In summary, all parameters could be es-timated with relative rms error of less than 5% forfmel between 1% and 8%, Lepi ranging from 20 to150 μm, SO2 from 25% to 100%, f blood from 1.2% to10%, and C from 4:0 × 105 to 5:5 × 105 cm−1. Themethodology presented can be applied to any two-layer optical system where scattering dominatesover absorption.

This research was funded in part through a grantfrom the National Institute of Diabetes and Diges-tive and Kidney Diseases (R42-DK069871) throughHyperMed, Inc. The authors would like to thankA. Nouvong, K. Schomacker, and R. Lifsitz for usefuldiscussion and exchange of information.

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