DOI 10.1007/s10812-016-0303-4

Journal of Applied Spectroscopy, Vol. 83, No. 3, July, 2016 (Russian Original Vol. 83, No. 3, May–June, 2016)

METHOD FOR CALCULATING THE OPTICAL DIFFUSE REFLECTION COEFFICIENT FOR THE OCULAR FUNDUS S. A. Lisenko* and M. M. Kugeiko

UDC 551.508

We have developed a method for calculating the optical diffuse reflection coefficient for the ocular fundus, taking into account multiple scattering of light in its layers (retina, epithelium, choroid) and multiple reflection of light between layers. The method is based on the formulas for optical “combination” of the layers of the medium, in which the optical parameters of the layers (absorption and scattering coefficients) are replaced by some effective values, different for cases of directional and diffuse illumination of the layer. Coefficients relating the effective optical parameters of the layers and the actual values were established based on the results of a Monte Carlo numerical simulation of radiation transport in the medium. We estimate the uncertainties in retrieval of the structural and morphological parameters for the fundus from its diffuse reflectance spectrum using our method. We show that the simulated spectra correspond to the experimental data and that the estimates of the fundus parameters obtained as a result of solving the inverse problem are reasonable. Keywords: ocular fundus, structural and morphological parameters, spectral measurements, diffuse reflection, multiple scattering, inverse problem. Introduction. The eye is closely connected with many organs and systems in the human body, and so often functional or morphological changes occur in the eye in general diseases. Thus, for example, changes in the blood vessels and tissue of the retina in most cases are due to cardiovascular diseases, endocrine disorders, blood diseases, inflammatory and degenerative processes in the body [1]. Accordingly, study of the condition of the eye allows us to diagnose not only eye diseases (retinal dystrophy, diabetic retinopathy, retinal microhemorrhage, etc.), but also general or systemic diseases. Currently, using a fundus camera to photograph the ocular fundus is considered as the most effective diagnostic tool in ophthalmology [2–6]. Modern fundus cameras can take multispectral photographs and correct for dynamic aberration of the optical system of the eye by using adaptive optics. However, visual study of the fundus by no means always can ensure objectivity of the diagnosis of disease, since due to many individual factors the photographic images of the fundus may be quite different in different patients with the same disease. The fundus consists of several layers with different structural and morphological characteristics, and also different optical properties: the white sclera, the dark-red choroid, the thin pigment epithelium that strongly absorbs light, and the weakly scattering retina with photoreceptors and a network of blood vessels. The resulting photographic image of the fundus is formed from light fluxes multiply scattered by all its layers, and consequently considerable differences are possible in interpretation of the results of the examination. In order to improve the objectivity of assessment of the condition of the fundus, we can draw on information about the content in its layers of blood vessels, effective scatterers and optically active chromophores (macular pigment and melanin), having a direct effect on the vision of the patient and quantitatively characterizing different eye diseases. Such information can be obtained by spectroscopic methods, such as those based on multispectral images of the fundus [5, 7–9] or the spectral dependences of light fluxes multiply scattered by the tissues of the eye [3, 10–12]. A large number of spectroscopic methods are known for determining the optical density of the macular pigment in the central zone of the retina [13–16] and the hemoglobin oxygenation level in its arteries and veins [5, 10, 12]. Some of these methods are realized as modules for existing fundus cameras, such as the VISUCAM Series 200 and 500 (Carl Zeiss, Germany), and also as separate instruments such as the QuantifEye (ZeaVision, USA). However, clinical trials of these methods show that the results obtained are subjective, _____________________ *

To whom correspondence should be addressed.

Belorussian State University, 4 Nezavisimost′ Ave., Minsk, 220030, Belarus; e-mail: [email protected]. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 83, No. 3, pp. 419–429, May–June, 2016. Original article submitted January 14, 2016. 412

0021-9037/16/8303-0412 ©2016 Springer Science+Business Media New York

support the influence of the physiological characteristics of the patient [17], and poorly correlate with each other [15, 18]. The reason for this is crude assumptions about the structure of the fundus and the nature of the interaction of light with it, making it possible to rather simply carry out a theoretical calculation of the recorded light signals (most often based on the Lambert–Beer approximation), but reducing the accuracy of the assessments of the characteristics of the object. The multilayer structure of the fundus and multiple scattering of light in each of its layers can be properly taken into account in the interpretation of the recorded light signals only by using models based on numerical solution of the radiation transport equation [11, 19]. An advantage of such models is the possibility of using them as the basis for determining the set of structural and morphological parameters of the fundus, comprehensively characterizing its condition in normal health and when various pathologies are present. An obvious disadvantage is the considerable computing resources required, especially in pixel-by-pixel processing of multispectral images of the fundus, making on-the-fly diagnosis impossible. In order to shorten machine time in solving the inverse problem while maintaining the accuracy of the solution inherent to numerical methods in radiation transport theory, we obtained stable regression relations between the structural and morphological parameters of the fundus and the linearly independent quantities comprised of its spectral diffuse reflection coefficients [20]. On this basis, we can process the multispectral images of the fundus on-the-fly and retrieve two-dimensional distributions of the hemoglobin and macular pigment content in the retina, the melanin content in the pigment epithelium and the choroid, the blood oxygenation level and the structural parameter of the retina, characterizing the volume concentration of its effective scatterers. In this case, it is suggested that the real values of the structural and morphological parameters of the studied object should not be outside the ranges for which the regressions used are valid. Unfortunately, the fundus is one of the least studied tissues in the human body, and literature data on the scattering properties of the layers in the fundus and the content in the layers of optically active chromophores are insufficient to obtain reliable regression solutions to the spectroscopic inverse problems. An alternative to the regression approach to solving the inverse problem with the same speed is iterative solution of the inverse problem using a fast method for calculating the optical diffuse reflectance spectrum for biological tissue, taking into account anatomical features of the studied object, which is comparable in accuracy to numerical methods in radiation transport theory. Ultrafast methods were previously developed for calculating the spectral and spatial characteristics of a radiation field formed by backscattering, diffuse reflection, and transmission by biological objects (human skin, mucous membranes, and blood) [21–25]. These methods are based on analytical approximations of the dependences of the characteristics of multiple scattering of light by the object on its optical and structural parameters, and ensure accuracy in the calculations comparable with the Monte Carlo method [26]. In this paper, we propose an analytical method for calculating the optical diffuse reflection coefficient for the fundus. The method is based on formulas for optical “combination” of the layers of a medium from [27], in which the optical parameters of the layers are replaced by some effective values depending on the optical thicknesses and albedo of single scattering off the layers, which ensures a correspondence between the results of the analytical calculations of the diffuse reflection coefficients and the numerical Monte Carlo calculations. We studied the effectiveness of using our method to solve the inverse problem by retrieval of the structural and morphological parameters of the fundus from its diffuse reflectance spectrum. Diffuse Reflectance Spectroscopy of the Fundus. Optical diffuse reflectance spectra from the fundus generally are measured based on commercial fundus cameras integrated with a device for spectral selection of optical radiation (a monochromator tunable by an optical filter, a mapping spectrometer, etc.) [3–7]. In order to eliminate glare spots in the images of the patient’s fundus, the fundus camera illuminates the surface of the pupil of the eye under study in the form of a peripheral ring. The image of this ring completely covers the receiving channel of the fundus camera, and only the light flux diffusely reflected from the fundus enters the photographic recorder and the entrance aperture of the spectrometric device. The field of view of the camera is guided to the section of the fundus of interest by having the patient fix his or her gaze on an illuminated mark, which can be moved over the patient’s field of view either arbitrarily (by the operator) or according to a programmed sequence. Figure 1 shows a possible schematic of a device for measuring the optical reflectance spectra from the fundus. It includes light source 1; receiving/transmitting lens components 2, 3, 4; plane mirror 5 with a circular hole in the center; fiber holder 6; optical fiber bundle 7; fiber-optic spectrometer 8; USB cable 9 and personal computer 10. Light from a lamp or white light-emitting diode 1 is transmitted by optical system 2, 3, 5 to the pupil of the eye 13 as a hollow cone of rays, converging at the eyeball and illuminating the section of the fundus opposite the pupil. Some of the light flux incident on the eye is reflected from the surface of the optical system of the eye, consisting of the cornea 11, the aqueous humor 12,

413

Fig. 1. Schematic drawing of device for recording optical diffuse reflectance spectra from the ocular fundus. the crystalline lens 13, and the vitreous body 14. All the components of the optical system of the eye have similar refractive indices, approximately corresponding to the refractive index of water, and so the light is mainly reflected from the surface of the cornea and from the interface between the vitreous body 14 and the retina 15 (the inner limiting membrane). Within the optical system of the eye, attenuation of the light flux is mainly due to its absorption by the crystalline lens 13, the optical transmission of which has a monotonic spectral dependence and diminishes with the age of the patient [28]. Absorption of light by the cornea, the aqueous humor, and the vitreous body is negligibly small in the visible region of the spectrum [29]. After the light flux passes through the optical system of the eye, it is incident on the fundus, consisting of the retina 15, the pigment epithelium 16, the choroid 17, and the sclera 18. Here the light is partially absorbed by blood and macular pigment in the retina, melanin in the epithelium, and blood in the choroid, and is multiply scattered and multiply reflected between layers of the fundus. As a result, some diffuse light flux reaches the surface of the retina. Most of it is distributed over the inner surface of the eyeball and does not enter the aperture of the receiving device. The scattered light from the center of the illuminated section of the fundus, emerging from the pupil of the eye 13 and incident on the objective lens 3, through the receiving optics 4 enters the fiber-optic probe 6 and through fibers 7 is delivered to spectrometer 8, controlled via a USB interface 9 by computer 10. A small fraction of the diffuse light flux, emerging from the retina, is reflected from the inner limiting membrane, is once again scattered and absorbed in the layers of the fundus, and escapes. This process is repeated many times. The resulting light flux, entering the receiving device, is formed from diffuse components, rapidly diminishing in magnitude and originating from multiple reflections of light between the retina and the inner limiting membrane. Analytical Method for Calculating the Diffuse Reflection Coefficient for Reflection of Light from the Fundus. Due to the small thickness of the fundus layers (the total thickness of the retina, epithelium, and choroid is ~0.5 mm) compared with the diameter of the eyeball (~24 mm) and the strong absorption of light by the epithelium, the curvature of the eyeball has practically no effect on the light conditions in the layers of the fundus [19]. This lets us use the model of a plane-parallel layered medium to describe diffuse reflection of light from the fundus. Since the angle of divergence for the light beam incident on the pupil of the eye is generally small (~5 to 10o ), for simplicity we assume that the fundus is illuminated by a collimated light beam, and backscattered light is collected in the receiving device. The ratio of the total light flux incident on the receiver to the light flux S incident on the pupil is calculated as the sum of terms in an infinitely decreasing geometric progression, formed by the diffuse fluxes multiply reflected between the retina and the boundary of the vitreous body adjacent to it (the inner limiting membrane, ILM): 2 S = A (1 − rcornea ) Tlens R, 2

R =

∗ (1 − rILM )(1 − rILM ) R1− 4 ∗ 1 − rILM R1∗− 4

(1)

,

(2)

where A is the fraction of scattered light flux incident on the entrance aperture of the receiving device; rcornea and r ILM are the Fresnel coefficients for reflection of light from the cornea and the inner limiting membrane for normal incidence of ∗ the light; rILM is the coefficient for reflection of light from the inner limiting membrane, illuminated by the diffuse flux from the retina; Tlens = exp [ln (10)Dlens] is the optical transmission of the crystalline lens; Dlens is the optical density 414

of the crystalline lens, depending on the age of the patient [28]; R is the diffuse reflection coefficient for the fundus; R1–4 is the diffuse reflection coefficient for a four-layer medium (retina (1), epithelium (2), choroid (3), and sclera (4)) with refractive index equal to the refractive index of the substance surrounding it (the vitreous body), illuminated along the normal to the surface; R1∗− 4 is the same but for diffuse illumination of the medium (below, all the quantities relating to diffuse light are labeled with an asterisk (*)). Assuming that the optical system of the eye and the retina have refractive indices relative to air of 1.336 and 1.470 [19], from the Fresnel formulas we obtain the reflection coefficients appearing in formulas (1) and (2): ∗ rcornea = 0.0207, r ILM = 0.0023, rILM = 0.1953. The retina weakly scatters light, and so after passing through the retina, a significant fraction of the initially collimated light flux retains its direction and the subsequent layers of the fundus are illuminated by both collimated and scattered flux. As a result of multiple scattering, the radiation becomes practically diffuse. Neglecting the differences in refractive indices for the layers of the fundus, let us rewrite the expression for its diffuse reflection coefficient, obtained using the familiar rules for combining diffuse reflection coefficients of individual layers of a medium [27]: R1− 4 = R1 +

T1rT1∗ R2 − 4 1 − R1∗ R2*− 4

R2 − 4 = R2 +

+

(T1 − T1r )T1∗ R2*− 4 1 − R1∗ R2*− 4

T2T2∗ R3 − 4 1 − R2∗ R3*− 4

R3*− 4 = R3∗ +

(T3∗ ) 2 R4∗ 1 − R3∗ R4*

,

,

(3)

(4)

,

(5)

where Ri, Ri* and Ti, Ti* are the reflection and transmission coefficients for the individual layer i; Ri–j and Ri∗− j are the reflection coefficients for the "cake” made from layers i–j; T1r = exp (–μex,1L1) is the collimated transmission coefficient for the retina; L1 and μex,1 are the thickness of the retina and its extinction coefficient. The coefficients R2*− 4 and R1*− 4 are also calculated from formulas (3)–(5), replacing all the quantities appearing in them by the analogous quantities with asterisks (*) and T1r = 0. Thus the diffuse reflection coefficient of the eye measured in the experiment depends on the transmission and reflection coefficients for directional and diffuse light transmitted and reflected by the layers of the fundus. At the moment, there is no analytical method for calculating these coefficients for layers of finite thickness with arbitrary optical parameters, and the familiar formulas relate to semi-infinite media [30, 31] or to special cases of weak absorption [32, 33] and scattering [34] of light in a homogeneous layer. None of the familiar formulas are applicable to layers of the fundus, the optical characteristics of which, depending on the anatomical section of the fundus and the wavelength of the light, vary from weak absorption and scattering (typical for the retina beyond the macula) to very strong absorption and scattering (typical for the thin epithelium) [35]. Accordingly, in order to calculate the diffuse reflection coefficients of the eye, we start from the formulas for the reflection coefficients R and the transmission coefficients T for a one-dimensional medium of optical thickness τ [27]: R = R∞

T =

1 − exp (−2k τ) 1 − R∞2 exp (−2k τ)

(1 − R∞2 ) exp (−2k τ) 1 − R∞2 exp (−2k τ)

,

,

(6)

(7)

where R∞ is the reflection coefficient for a medium of infinitely large optical thickness; k is the depth (or asymptotic) attenuation coefficient. In a one-dimensional medium, the light scattering process is characterized by the probability of survival of a photon ω and the probabilities of forward scattering x and backscattering 1 – x. In order to go to a three-dimensional medium, we need to specify the scattering indicatrix of the medium. Let us represent it in the form of the sum of the two Dirac delta functions δ(μ – 1) and δ(μ + 1), describing scattering into the forward and backward hemispheres: p(μ) = [xδ(μ – 1) + (1 – x)δ(μ + 1)]/(2π), where μ is the cosine of the scattering angle. The average cosine of the scattering indicatrix p(μ) is unambiguously related to the forward scattering probability as g = 2x – 1, which allows us, from the formulas for R∞ and k obtained in [27] for a onedimensional medium, to go to the corresponding formulas for a semi-infinite three-dimensional medium: 415

TABLE 1. Variation Ranges for Optical Parameters in Layers of the Fundus i

μs,i, mm–1

μa,i, mm–1

βi (1 – gi)/ki

1

1–50

0.001–100

0.005–500

2

5–300

0.5–500

0.01–10

3

5–300

0.005–300

0.01–200

R∞ =

ω (1 + g ) ⎧⎪ 2 − ⎨1 − ω(1 − g ) ⎪⎩ 2

⎫

(1 − ω) (1 − ωg ) ⎪⎬ ,

k = ω(1 – g)(1 – R∞2)/4R∞ .

⎪⎭

(8) (9)

Let us apply formulas (6)–(9) to calculate the coefficients Ri, Ri* and Ti, Ti* for three layers of the fundus: the retina (i = 1), the epithelium (i = 2), and the choroid (i = 3). In this case, for each layer we introduce effective optical characteristics ∗ of its elementary volume: the absorption coefficient ( μ a , i , μ ∗a , i ) and the scattering coefficient ( μ s , i , μ s , i ), different for collimated and diffuse light fluxes. We select the functional relationship between the effective optical parameters of the layer and the actual values (μa,i, μs,i ) in such a way that for a relatively small number of its independent parameters, we ensure that the results of the numerical and analytical calculations match for the diffuse reflection coefficients of the medium under consideration. Taking into account the broad ranges of variation in the optical parameters of layers of the medium (several orders of magnitude), let us relate their effective and actual values by power-law functions with coefficients depending on the albedo for single scattering from the layers:  a , i Li = ai (μ a , i Li ) (bi ,1 + bi , 2ωi ) (μ s , i Li ) (ci ,1 + ci , 2ω i ) , μ

(10)

 s , i Li = d i (μ a , i Li ) ( fi ,1 + fi , 2ωi ) (μ s , i Li ) ( hi ,1 + hi , 2ωi ) , μ

(11)

where Li and ωi are the geometric thickness and albedo for single scattering by the ith layer; ai, bi,1, bi,2, ci,1, ci,2, di, fi,1, fi,2, hi,1, hi,2 are some constants corresponding to illumination of the layer along the normal to the surface by a parallel beam of rays. We determine the effective parameters of the layer μ a , i and μ ∗s , i , corresponding to its diffuse illumination, with analogous expressions. Expressing the coefficients Ri and Ti using formulas (6)–(9) in terms of the effective optical thicknesses τi = (μ a , i + μ s , i ) Li and the albedo for single scattering ωi = μ s , i (μ a , i + μ s , i ) by the layers of the medium and substituting them into (2)–(5), we obtain an analytical model for the diffuse reflection coefficients of the fundus, including a set of unknown constants. In order to identify the constants of the model, we used the results of numerical simulation of radiation transport in the fundus by the Monte Carlo method [26]. The calibration data were obtained by independent variations of the optical parameters for the layers in the fundus (μa,i, μs,i ) in the ranges indicated in Table 1 and simulation of a large number of photon trajectories in the medium under study, taking into account their absorption, scattering, and reflection from the surfaces of the optical system of the eye and the sclera. The ranges of variation in the parameters μa,i and μs,i were selected based on the experimental results in [35, 36] and calculations based on data on the content of optically active chromophores in the fundus layers [11]. The thicknesses of the fundus layers were assumed to be equal to L1 = 200 μm, L2 = 10 μm, L3 = 250 μm; the average cosines of the scattering indicatrix for the layers were g1 = 0.97, g2 = 0.84, g3 = 0.94 [11, 35]. Our study [20] of the information content in the diffuse reflectance spectra for the fundus shows that as a result of solution of the inverse problem taking into account variations in the geometric thicknesses of the fundus layers, we can only determine the structural and morphological parameters of the layers reduced to some fixed thickness. Accordingly, in this paper, we initially select fixed values for the thicknesses of the fundus layers. Variations in the parameters of the scattering indicatrix for the fundus layers gi are neglected, first of all due to the lack of appropriate experimental data, and secondly because of the preferential dependence of the characteristics for multiple scattering by biological tissues on the product of the quantities μs,i and gi rather than on each of them individually. In the Monte Carlo calculations, the sclera was considered as an orthotropically reflecting medium with an albedo varying in the range R4∗ = 0.1–1.0. The refractive indices of the fundus layers and the substance surrounding them were assumed to be equal to 1.470 and 1.336 [19]. 416

TABLE 2. Constants in Formulas (10), (11) a1

0.3438

a1*

0.23784

a2

0.42543

a2*

0.23080

a3*

0.43487

0.16102

* b3,1

1.72017

b1,1

2.5697

* b1,1

–0.08933

b2,1

–0.45023

* b2,1

b1,2

–1.8954

* b1,2

0.09522

b2,2

0.31071

* b2,2

–0.25937

* b3,2

–1.7329

c1,1

–8.34265

* c1,1

1.10800

c2,1

1.34589

* c2,1

0.93512

* c3,1

–0.75554

c1,2

9.17747

* c1,2

0.33733

c2,2

–0.12610

* c2,2

0.68127

* c3,2

2.00560

d1

1.14292

d1∗

1.26579

d2

1.38360

d 2∗

0.97370

d3*

1.45288

f1,1

0.81853

* f1,1

0.99444

f2,1

0.87901

* f 2,1

1.09937

* f 3,1

2.49012

f1,2

0.08941

* f1,2

0.09276

f2,2

0.06310

* f 2,2

0.10345

* f 3,2

–1.41808

h1,1

–0.02890

* h1,1

0.01797

h2,1

0.02330

* h2,1

0.01922

* h3,1

–1.33113

h1,2

0.09086

* h1,2

0.21533

h2,2

0.09923

* h2,2

0.48703

* h3,2

1.47406

Fig. 2. Comparison of the diffuse reflection coefficients for the eye calculated by the Monte Carlo method (RMC) and using formulas (2)–(11) (R) for 3000 random combinations of the optical parameters for the ocular fundus. The line represents R = RMC. The calibration data set includes 3000 random realizations of the parameters μa,i, μs,i, R4∗ and the corresponding diffuse reflection coefficients R for the fundus. It does not include realizations with R < 10–3 and R > 0.2, since such values are not encountered in practice [3]. The constants in the analytical method for calculating the diffuse reflection coefficients for the fundus that appear in expressions (10), (11) were determined by minimization of its error relative to the Monte Carlo method: N R (n) − R (n) MC 1 δR = , (12) ∑ (n) N n =1 RMC (n) where RMC and R(n) are the diffuse reflection coefficients for the fundus, calculated by the Monte Carlo method and by using formulas (2)–(11); n is the sequence number for the realization of the optical parameters of the fundus; N is the total number of realizations. The constants for the analytical method, corresponding to minimum error (12), are given in Table 2. The effectiveness (n) of the method can be evaluated from Fig. 2, in which we compare the coefficients RMC and R(n) for the calibration data set. We see that the proposed method for calculating the diffuse reflectance coefficients for the fundus after the calibration reproduces the results of the numerical calculations rather well. The discrepancy from the Monte Carlo method is ~1.9%, which can be considered as a quite good result if we take into account the complex structure of the object under study and the broad ranges of variations in its optical parameters, covering the analogous ranges for other biological tissues [37], and also if we take into account the random noise in the Monte Carlo method.

417

Solution of the Inverse Problem. Let us study the feasibility of using our method for quantitative interpretation of the optical diffuse reflectance spectra from the fundus. For this purpose, let us turn to the Monte Carlo method, but we will simulate the spectra R(λ) for random combinations of structural and morphological parameters of the fundus, among which are considered [11, 20] the volume concentrations of blood vessels in the retina f V,1 and in the choroid f V,3, the hemoglobin oxygenation level in the blood S, the optical density of the macular pigment in the retina Dmac at the wavelength of its maximum absorption 460 nm, the volume concentration of melanin in the epithelium fm, the structural parameters of the layers αi, characterizing the volume concentrations of their effective scatterers. We give the ranges of variations in the structural and morphological parameters as: f V,1 = 0.002–0.1, f V,3 = 0.02–0.5, S = 0.10–0.96, Dmac = 0.001–0.5, fm = 0.1–0.9, α1 = 0.22–1.0, α2 = 0.45–1.0, α3 = 0.5–3.0 [11]. The presence of a small amount of melanin in the choroid is neglected since first of all, its contribution to the resulting diffuse reflection coefficients of the fundus is insignificant (light flux backscattered by the choroid passes twice through the epithelium and is strongly attenuated) and secondly, taking it into account along with melanin in the epithelium leads to nonuniqueness of the solution to the inverse problem [20]. For the specified values of the model parameters, the spectral dependences of the absorption and scattering coefficients for the fundus layers are calculated using the formulas: μ s, i (λ ) = α i μ s,expi (λ ) ,

(13)

μ a,1(λ) = (ln10)Dmac Λmac (λ)/L1 + f V,1μ a,bl( λ) ,

(14)

μ a,2( λ) = fm μ a,m(λ) ,

(15)

μ a,3( λ) = f V,3 μ a,bl(λ) ,

(16)

μ a,bl ( λ) = CHb[SεHbO2(λ) + (1 – S)εHb(λ)] ,

(17)

exp

where the μ s, i are the experimental scattering coefficients for the fundus layers [35]; Λmac(λ) is the optical density spectrum for the macular pigment, normalized to its value at λ = 460 nm [28]; L1 is the thickness of the retina, μa,bl is the absorption coefficient of the blood; CHb = 2.3 mmol/L is the average molar hemoglobin concentration in human blood; ε HbO 2 and εHHb are the molar absorption coefficients for oxidized and reduced hemoglobin [38]; μa,m is the volume absorption coefficient for retinal melanin [39]. Since the sclera has an insignificant effect on the diffuse reflection coefficients of the eye [11], in order to calculate its diffuse reflectance spectrum it is sufficient to use the simple empirical formula [3]: R4∗ ( λ) = 0.5 exp (–0.00261(λ – 675)) ,

(18)

providing good accuracy of the approximation of the experimental spectra for λ ≥ 675 nm [40]. Light with shorter wavelengths is practically completely absorbed by the epithelium and does not reach the sclera. The test data set includes 250 random realizations of the diffuse reflectance spectra of the fundus, calculated by the Monte Carlo method in the range 450–800 nm in 5 nm steps. When inverting the test spectra, it is assumed that they are known to accuracy up to some multiplicative constant, depending on the measurement geometry. The model parameters are retrieved by minimization of the residuals between the diffuse reflectance spectra of the fundus calculated using formulas (2)–(11), (13)–(18) and the corresponding test data. The actual and the retrieved values of the parameters for the medium modeling the fundus are compared in Fig. 3. Without absolute calibration of the diffuse reflectance spectra of the fundus, we cannot use them to determine the structural parameters of the epithelium and the choroid (α2 and α3), which is consistent with the previous analysis of the information content of the data under consideration [20]. On the whole, as we can tell from the correlation coefficients ρ between the actual and retrieved values of the structural and morphological parameters of the fundus in Fig. 3, our method for calculating the diffuse reflection coefficients for the fundus provides accuracy in solution of the inverse problem which is sufficient in practice. The mean-square deviations of the retrieved values of the optical density for the macular pigment Dmac and the blood oxygenation level S from their actual values are 0.021 and 0.024, which are significantly less than the analogous errors for the familiar diagnostic methods for Dmac and S [9, 13, 15, 18]. Furthermore, the proposed method allows us to determine other parameters of the fundus that are useful for identifying a wide range of pathologies. Let us compare the results of simulating the diffuse reflection coefficients of the fundus with experiment. The experimental data are taken from [11] and represent the optical diffuse reflectance spectrum from the macular region of the 2 fundus. Since the age of the patient is not indicated in [11], we considered the product KR(λ) Tlens (λ) as the initial data, where R( λ) is the optical diffuse reflectance spectrum from the fundus, Tlens( λ) is the transmission of light by the crystalline lens, 418

Fig. 3. Results of retrieval of volume concentrations of blood vessels in the retina (a) and in the choroid (b), hemoglobin oxygenation level in the blood (c), optical density of macular pigment in the retina (d), volume concentration of melanin in the epithelium (e), structural parameter of the retina (f) from diffuse reflectance spectra of the eye, simulated by the Monte Carlo method. The lines correspond to equal actual and retrieved (*) parameters of the fundus. K is the calibration constant. The age of the patient and the constant K along with the unknown parameters of the fundus, using successive iteration approximations of the calculated diffuse reflectance spectrum of the eye, approached the corresponding experimental data. In Fig. 4, we see that the differences between the theoretical and experimental spectra are at the level of measurement error. The retrieved structural and morphological parameters of the fundus have the following values: f V,1 = 2·10–4, f V,3 = 0.41, S = 0.97, Dmac = 0.41, fm = 0.68, α1 = 0.19. The small value of f V,1 corresponds to sections of the retina not containing large veins and arteries (in the absence of pathologies and traumas in the patient that would lead to retinal microhemorrhage) [36]. The hemoglobin oxygenation level in the choroid S corresponds well to the experimental data for people [41] and animals [42], according to which the difference in the S values for venous (96–98% oxygenated) and arterial blood in the choroid of the fundus are statistically insignificant. The value of Dmac is typical for the macular zone of the retina [11, 28]. The parameters f V,3 and fm, when recalculated as equivalent molar concentrations of hemoglobin and melanin, also correspond to the results of independent studies [11]. The retrieved value of the parameter α1 = 0.21 is close to the lower limit 0.22 of its range, indicated in [11], which (considering the uncertainty in the age of the test subject) is quite acceptable. 419

Fig. 4. Experimental diffuse reflectance spectrum of the ocular fundus [11] (solid line) and theoretical spectrum calculated using formulas (2)–(11), (13)–(18) (dotted line). Conclusions. We propose a simple analytical method for calculating the optical diffuse reflection coefficient for the ocular fundus. The method takes into account the anatomical structure of the object and multiple scattering of light in it, does not require considerable computer resources, and provides an accuracy in the calculations that is sufficient for solving the spectroscopic inverse problems. By numerical simulation of the optical diffuse reflectance spectra from the fundus, we have shown that our method can retrieve a set of its structural and morphological parameters: the volume concentrations of blood vessels in the retina and the choroid, the hemoglobin oxygenation level in the blood, the optical density of the macular pigment in the retina, the volume concentration of melanin in the epithelium, the structural parameter of the retina characterizing the content of scattering centers in it. Calculations of the spectral diffuse reflection coefficients for the fundus using the formulas obtained allow us to reproduce the corresponding experimental data within measurement error limits, and to obtain in real time reasonable estimates of diagnostically important parameters of the fundus. We should point out that the model for the optical characteristics of the fundus layers, represented by formulas (13)–(17), is certainly not complete. In order to refine it, we need experimental optical diffuse reflectance spectra from the fundus in normal health and for different pathologies. Improvement of the model may be related to taking into account minor optically active chromophores in the fundus (bilirubin, neuroglobin etc.), and also separation of the gas composition of hemoglobin in the retina and the choroid. Considering that all hemoglobin in the choroid is actually oxygenated [41, 42], we can use a fixed value (0.96–0.98) for its oxygenation level. At the same time, the hemoglobin oxygenation level in the retina varies over a broad range [5, 7], and so should be an independent parameter of the model. In this case, in order to determine the hemoglobin oxygenation level in large blood vessels in the retina, in the model we need to take into account localized light absorption (the "sieve" effect) [43].

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

420

L. A. Katsnel′son, V. S. Lysenko, and T. I. Balishanskoi, Clinical Atlas of Pathologies of the Ocular Fundus, 4th edn., Reprint [in Russian], GEOTAP-Media, Moscow (2013). T. V. Anaf′yanova, A. A. Volkov, and L. A. Karamchakova, Fundamental’nye Issledovaniya, 9, No. 3, 382–384 (2011). F. C. Delori and K. P. Pflibsen, Appl. Opt., 28, No. 6, 1061–1077 (1989). V. Nourrit, J. Denniss, M. M. K. Muqit, I. Schiessl, C. Fenerty, P. E. Stanga, and D. B. Henson, J. Fran. Ophtalmol., 33, No. 10, 686–692 (2010). D. J. Mordant, I. Al-Abboud, G. Muyo, A. Gorman, A. Sallam, P. Ritchie, A. R. Harvey, and A. I. McNaught, Eye (Lond.), 25, No. 3, 309–320 (2011). L. Gao, R. T. Smith, and T. S. Tkaczyk, Biomed. Opt. Express, 3, No. 1, 48–54 (2012). S. H. Hardarson, A. Harris, R. A. Karlsson ,G. H. Halldorsson, L. Kagemann, E. Rechtman, G. M. Zoega, T. Eysteinsson, J. A. Benediktsson, A. Thorsteinsson, P. C. Jensen, J. Beach, and E. Stefansson, IOVS, 47, No. 11, 5011–5016 (2006). J. C. Ramella-Roman, S. A. Mathews, H. Kandimalla, A. Nabili, D. D. Duncan, S. A. D’Anna, S. M. Shah, and Q. D. Nguyen, Opt. Express, 16, No. 9, 6170–6182 (2008). D. J. Mordant, I. Al-Abboud, G. Muyo, A. Gorman, A. Sallam, P. Rodmell, J. Crowe, S. Morgan, P. Ritchie, A. R. Harvey, and A. I. McNaught, IOVS, 52, No. 5, 2851–2859 (2011).

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27. 28. 29. 30. 31. 32.

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

M. Hammer, E. Thamm, and D. Schweitzer, Phys. Med. Biol., 47, No. 17, N233–N238 (2002). M. Hammer and D. Schweitzer, Phys. Med. Biol., 47, No. 2, 179–191 (2002). V. Diaconu, Appl. Opt., 48, No. 10, D52–D61 (2009). F. C. Delori, D. G. Goger, B. R. Hammond, D. M. Snodderly, and S. A. Burns, J. Opt. Soc. Am. A, 18, No. 6, 1212–1230 (2001). J. van de Kraats, T. T. J. M. Berendschot, S. Valen, and D. van Norren, J. Biomed. Opt., 11, No. 6, 064031-1–064031-7 (2006). R. A. Bone, B. Brener, and J. C. Gibert, Vision Res., 47, No. 26, 3259–3268 (2007). M. Sharifzadeh, P. S. Bernstein, and W. Gellermann, J. Biomed. Opt., 18, No. 11, 116001-1–116001-9 (2013). E. N. Khomyakova, S. G. Sergushev, A. A. Ryabtseva, and O. M. Andryukhina, Al’manakh Klin. Meditsiny, No. 29, 14–22 (2013). C. Creuzot-Garcher, P. Koehrer, C. Picot, S. Aho, and A. M. Bron, IOVS, 55, No. 5, 2941–2946 (2014). Y. Guo, G. Yao, B. Lei, and J. Tan, J. Opt. Soc. Am. A, 25, No. 2, 304–311 (2008). S. A. Lisenko, M. M. Kugeiko, V. A. Firago, and A. I. Kubarko, Opt. Spektrosk., 117, No. 3, 157–162 (2014). S. A. Lisenko and M. M. Kugeiko, Zh. Prikl. Spektrosk., 80, No. 3, 432–441 (2013) [S. A. Lisenko and M. M. Kugeiko, J. Appl. Spectrosc., 80, 419–428 (2013) (English translation)]. S. A. Lisenko and M. M. Kugeiko, Izmerit. Tekhnika, No. 11, 68–73 (2013). S. A. Lisenko, M. M. Kugeiko, V. A. Firago, and A. N. Sobchuk, Zh. Prikl. Spektrosk., 81, No. 1, 120–126 (2014) [S. A. Lisenko, M. M. Kugeiko, V. A. Firago, and A. N. Sobchuk, J. Appl. Spectrosc., 81, 118–126 (2014) (English translation)]. S. A. Lisenko, M. M. Kugeiko, V. A. Firago, and A. N. Sobchuk, Kvant. Élektron., 44, No. 1, 69–75 (2014). S. A. Lisenko and M. M. Kugeiko, Kvant. Élektron., 44, No. 3, 252–258 (2014). L. Wang, S. L. Jacques, and L. Zheng, Comput. Method. Program. Biomed., 47, 131–146 (1995). V. V. Sobolev, Radiant Energy Transport in Atmospheres of Stars and Planets [in Russian], Gostekhizdat, Moscow (1959). N. P. A. Zagers and D. van Norren, J. Opt. Soc. Am. A, 21, No. 12, 2257–2268 (2004). E. A. Boettner and R. Wolter, IOVS, 1, No. 6, 776–783 (1962). T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys., 19, No. 4, 879–888 (1992). G. Zonios and A. Dimou, Opt. Express, 14, No. 19, 8661–8674 (2006). É. P. Zege, O. V. Bushmakova, I. L. Katsev, and N. V. Konovalov, Zh. Prikl. Spektrosk., 30, No. 5, 900–907 (1979) [É. P. Zege, O. V. Bushmakova, I. L. Katsev, and N. V. Konovalov, J. Appl. Spectrosc., 30, 646–652 (1979) (English translation)]. W. G. Egan and T. W. Hilgerman, Optical Properties of Inhomogeneous Materials, Academic Press, New York (1979). L. G. Sokoletsky, A. A. Kokhanovsky, and F. Shen, Appl. Opt., 52, No. 35, 8471–8483 (2013). M. Hammer, A. Roggan, D. Schweitzer, and G. Müller, Phys. Med. Biol., 40, No. 6, 963–978 (1995). B. G. Yust, L. C. Mimun, and D. K. Sardar, Lasers Med. Sci., 27, No. 2, 413–422 (2012). A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, J. Innov. Opt. Health Sci., 4, No. 1, 9–38 (2011). S. A. Prahl, http://omlc.ogi.edu/spectra/hemoglobin/index.html. S. Jacques, http:/omlc.org/spectra/melanin/mua.html. A. N. Bashkatov, É. A. Genina, V. I. Kochubei, and V. V. Tuchin, Opt. Spektrosk., 109, No. 2, 226–234 (2010). J. V. Kristjansdottir, S. H. Hardarson, A. R. Harvey, O. B. Olafsdottir, T. S. Eliasdottir, and E. Stefansson, IOVS, 54, No. 5, 3234–3239 (2013). A. Alm and A. Bill, Acta Physiol. Scand., 80, No. 1, 19–28 (1970). A. Talsma, B. Chance, and R. Graaff, J. Opt. Soc. Am. A, 18, No. 4, 932–939 (2001).

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