Solution of Radiative Transfer Equation with a Continuous and Stochastic Varying Refractive Index by Legendre Transform Method

Abstract

The present paper gives a new computational framework within which radiative transfer in a varying refractive index biological tissue can be studied. In our previous works, Legendre transform was used as an innovative view to handle the angular derivative terms in the case of uniform refractive index spherical medium. In biomedical optics, our analysis can be considered as a forward problem solution in a diffuse optical tomography imaging scheme. We consider a rectangular biological tissue-like domain with spatially varying refractive index submitted to a near infrared continuous light source. Interaction of radiation with the biological material into the medium is handled by a radiative transfer model. In the studied situation, the model displays two angular redistribution terms that are treated with Legendre integral transform. The model is used to study a possible detection of abnormalities in a general biological tissue. The effect of the embedded nonhomogeneous objects on the transmitted signal is studied. Particularly, detection of targets of localized heterogeneous inclusions within the tissue is discussed. Results show that models accounting for variation of refractive index can yield useful predictions about the target and the location of abnormal inclusions within the tissue.

1. Introduction

A special attention in diffuse optical tomography is focused on the development of methods for detection of photons providing the information concerning optical parameters of the explored medium. This gives the targets of localized nonhomogeneous inclusion arising in tissues due to various pathologies, like tumor formation, local increase in blood volume, and other abnormalities [1–4]. In radiative transfer theory, the most used parameters in modeling laser radiation interaction with biological tissue are absorption and scattering [5–7]. However some other studies evoked a significant variation of refractive index of abnormal biological tissues especially in the near infrared range. More precisely, experimental results [8, 9] showed that the tissue of malignant tumors could manifest an increase of the refractive index which can attain until 10% of that of a normal tissue which encircles them. So, medical imaging by diffuse optical tomography should take advantage from the emergence of a third contrast parameter which is the refractive index. This led to the appearance of a big number of numerical and fundamental works in the field of radiative transfer in a varying refractive index biological medium. While the conventional radiative transfer equation (RTE) has been widely used to study interaction of near infrared radiation with biological media, there exist a number of works dealing with a modified radiative transfer equation in spatially varying refractive index media [10, 11]. Some of these papers are interested in varying refractive index biological tissues [12–14]. In the present paper, our first concern is to contribute to the usability of the radiative transfer theory in a potential optical tomography setting in medical imaging. At this level, studying the effect of refractive index on the transmitted light through a biological rectangular layer should be crucial. This could improve detectability of heterogeneous objects in a typical tomography scheme. However, it is important to note that in a varying refractive index medium, the rays are not straight lines but curves. So even in a rectangular geometry, the varying index radiative transfer equation displays the classical form of the angular derivative terms commonly appearing when dealing with spherical and cylindrical geometries with uniform refractive index [15–17]. This finding gives rise to the use of Legendre transform as a manner for modeling these terms. Although this technique was used by Sghaier et al. [17] in a uniform refractive index spherical domain as an innovative view to handle these terms, it prevails as useful in this contemporary problem. This fact is our second concern in this paper. So, we present a computational RTE-based model suitable for basic diffuse optical tomography forward problem with spatially varying refractive index biological medium. We treat angular derivative terms by using the Legendre integral transform technique. We investigate cases concerning optical tomography applications. Results concerning the effect of the refractive index variation on the detected signal are shown.

2. Mathematical Model

In this work, the radiative transfer equation in a human biological tissue is described by using a stationary varying refractive index RTE [18, 19] Ω⋅∇Ψ𝑟,⃗Ω+𝜇_𝑎(𝑟)⃗+𝜇_𝑠(𝑟⃗Ψ⃗Ω+1)𝑟,𝑛∇𝑛⋅∇_ΩΨ⃗Ω−2𝑟,𝑛Ψ⃗Ω=𝜇Ω⋅∇𝑛𝑟,_𝑠(𝑟)4𝜋₀^4𝜋𝑃Ω,Ω^′Ψ𝑟,Ω^′𝑑Ω^′+𝑆𝑟,⃗Ω, (1) where(i)Ψ(𝑟,⃗Ω) is the directional energetic radiance at the spatial position vector 𝑟⃗=(𝑥,𝑦,𝑧),(ii)𝑛(𝑟)⃗ is the refractive index distribution,(iii)𝜇_𝑎(𝑟)⃗ and 𝜇_𝑠(𝑟)⃗ are the absorption and scattering coefficients, respectively,(iv)𝑐=𝑐_vac/𝑛_𝑟 is the ratio of speed of light in a vacuum, 𝑐_vac, and the refractive index 𝑛_𝑟,(v)the source term 𝑆(𝑟,⃗Ω) is an injected radiance at the medium’s boundary,(vi)the phase function 𝑃(Ω,Ω^′) describes the probability that, during a scattering event, a photon with direction Ω^′ is scattered in the direction Ω.

Equation (1) takes into account the fact that the rays are not straight lines but curves. It involves terms that illustrate the expansion or the contraction of the cross section of the tube of light rays in the medium.

For a two-dimensional problem and in Cartesian coordinate system of the 𝑥-𝑦 plane, the terms due to the refractive index variation can be expressed as 1𝑛∇𝑛⋅∇_Ω2Ψ−𝑛Ψ1Ω⋅∇𝑛=−𝑛𝜕𝑛𝜕𝑥sin𝜑𝜕Ψ+1𝜕𝜑+2cos𝜑Ψ𝑛𝜕𝑛𝜕𝑦cos𝜑𝜕Ψ,𝜕𝜑−2sin𝜑Ψ (2) where cos𝜑 and sin𝜑 are the Cartesian coordinates of the unit direction vector in the 𝑥-𝑦 plane. In fact we assume that the radiance of out of plane directions is negligible. By using notations 𝜉=cos𝜑 and 𝜂=sin𝜑, (2) displays the classical form of the angular redistribution term commonly appearing when dealing with spherical and cylindrical geometries with uniform refractive index [20] 1𝑛∇𝑛⋅∇_Ω2Ψ−𝑛Ψ=1Ω⋅∇𝑛2𝑛²𝜕𝑛²𝜕𝜕𝑥𝜕𝜉1−𝜉²Ψ+𝜕𝑛²𝜕𝜕𝑦𝜕𝜂1−𝜂²Ψ. (3)

The angular redistribution terms will be noted 𝐷_𝑥=𝜕𝜕𝜉1−𝜉²Ψ,𝐷_𝑦=𝜕𝜕𝜂1−𝜂²Ψ (4) so (3) becomes 1𝑛∇𝑛⋅∇_Ω2Ψ−𝑛1Ω⋅∇𝑛Ψ=2𝑛²𝜕𝑛²𝐷𝜕𝑥^𝑥+𝜕𝑛²𝐷𝜕𝑦^𝑦. (5)

3. Numerical Method

In our numerical implementation, we use a rectangular domain which is divided into a set of 𝐼×𝐽 elementary uniform volumes Δ𝑉=Δ𝑥Δ𝑦Δ𝑧 with a uniform unitary depth (Δ𝑧=1). The angular discretization is obtained through a discrete ordinate technique. This yields a set of 𝑀 discrete directions, 𝜑_𝑚,𝑚=1⋯𝑀 giving a set of angular discrete direction cosines (𝜉_𝑚,𝜂_𝑚),𝑚=1⋯𝑀. An orientation depending on the incident ray direction is adopted for each cell [7]. Calculations are done by using integration of (1) over an elementary volume Δ𝑉 for each discrete direction. This gives Δ𝑦𝜉_𝑚Ψ_𝑚,𝐸−Ψ_𝑚,𝑊+Δ𝑥𝜂_𝑚Ψ_𝑚,𝑁−Ψ_𝑚,𝑆+Δ𝑥Δ𝑦𝜇_𝑎+𝜇_𝑠Ψ_𝑚,𝑃+1𝑛𝜕𝑛𝐷𝜕𝑥^𝑚,𝑥+𝜕𝑛𝐷𝜕𝑦^𝑚,𝑦=Δ𝑥Δ𝑦𝑆_𝑚,𝑃+𝜇_𝑠^𝑀_{𝑚′=1,𝑚′≠𝑚}𝑤_𝑚′𝑝_𝑚𝑚′Ψ_𝑚′,𝑃, (6) where 𝐷_𝑚,𝑥 and 𝐷_𝑚,𝑦 are the discrete angular derivative terms at the angular ordinates 𝜉_𝑚 and 𝜂_𝑚, respectively, and 𝑤_𝑚 is a weighting factor. The discrete term of Henyey-Greenstein phase function is written as 𝑝_𝑚𝑚′=1−𝑔²4𝜋1+𝑔²−2𝑔𝜉_𝑚𝜉_𝑚′+𝜂_𝑚𝜂_𝑚′^3/2, (7) where 𝑔 is the anisotropy factor. If the direction cosines are positive, the directional radiance is known on the faces 𝑊 and 𝑆 and they are unknown on the faces 𝐸 and 𝑁 of the (𝑖,𝑗)-cell and also in the centre 𝑃. Therefore, we need two complementary relations to eliminate Ψ_𝑚,𝐸 and Ψ_𝑚,𝑁; this can be obtained by using interpolation formula Ψ_𝑚,𝑃=𝛼Ψ_𝑚,𝐸+(1−𝛼)Ψ_𝑚,𝑊,Ψ_𝑚,𝑃=𝛼Ψ_𝑚,𝑁+(1−𝛼)Ψ_𝑚,𝑆, (8) where 𝛼 is an interpolation parameter. Using these relations, (6) becomes Δ𝑦𝜉_𝑚𝛼Ψ_𝑚,𝑃−Ψ_𝑚,𝑊+Δ𝑥𝜂_𝑚𝛼Ψ_𝑚,𝑃−Ψ_𝑚,𝑆+Δ𝑥Δ𝑦𝜇_𝑎+𝜇_𝑠Ψ_𝑚,𝑃+12𝑛²𝜕𝑛²𝐷𝜕𝑥^𝑚,𝑥+𝜕𝑛²𝐷𝜕𝑦^𝑚,𝑦=Δ𝑥Δ𝑦𝑆_𝑚,𝑃+𝜇_𝑠^𝑀_{𝑚′=1,𝑚′≠𝑚}𝑤_𝑚′𝑝_𝑚𝑚′Ψ_𝑚′,𝑃. (9)

Theoretically, if we know the solution in the (𝑖,𝑗)-cell, we can do calculus over the cells (𝑖+1,𝑗) and (𝑖,𝑗+1) using the boundary conditions and the following relations: Ψ_𝑚,𝑊𝑖+1,𝑗=Ψ_𝑚,𝐸Ψ𝑖,𝑗;𝑖=1⋯𝐼−1,_𝑚,𝐸𝑖,𝑗+1=Ψ_𝑚,𝑁𝑖,𝑗;𝑗=1⋯𝐽−1. (10)

If the direction cosines are both positive, we get the following equation: Δ𝑦𝜉_𝑚𝛼Ψ_{𝑚,𝑖,𝑗}−Ψ_{𝑚,𝑖−1,𝑗}+Δ𝑥𝜂_𝑚𝛼Ψ_{𝑚,𝑖,𝑗}−Ψ_{𝑚,𝑖,𝑗−1}+Δ𝑥Δ𝑦𝜇_𝑎+𝜇_𝑠Ψ_{𝑚,𝑖,𝑗}+12𝑛²𝜕𝑛²𝐷𝜕𝑥^𝑚,𝑥+𝜕𝑛²𝐷𝜕𝑦^𝑚,𝑦=Δ𝑥Δ𝑦𝑆_{𝑚,𝑖,𝑗}+𝜇_𝑠^𝑀_𝑚′=1𝑤_𝑚′𝑝_𝑚𝑚′Ψ_{𝑚′,𝑖,𝑗}; (11) this gives Ψ_{𝑚,𝑖,𝑗}=Ψ_{𝑚,𝑖,𝑗}−𝑛_𝑖,𝑗Ψ2𝑐Δ𝑡^{𝑚,𝑖,𝑗}+Δ𝑦𝜉_𝑚𝛼Ψ_{𝑚,𝑖−1,𝑗}+Δ𝑥𝜂_𝑚𝛼Ψ_{𝑚,𝑖,𝑗−1}+12𝑛²_𝑖,𝑗Δ𝑦𝑛²_𝑖,𝑗−𝑛²_{𝑖−1,𝑗}𝐷_{𝑚,𝑥,𝑖,𝑗}+Δ𝑦𝑛²_𝑖,𝑗−𝑛²_{𝑖,𝑗−1}𝐷_{𝑚,𝑥,𝑖,𝑗}+Δ𝑥Δ𝑦𝑆_{𝑚,𝑖,𝑗}+𝜇^𝑀_{𝑠,𝑖,𝑗𝑚′=1}𝑤_𝑚′𝑝_𝑚𝑚′Ψ_{𝑚′,𝑖,𝑗}×Δ𝑦𝜉_𝑚𝛼+Δ𝑥𝜂_𝑚𝛼+Δ𝑥Δ𝑦𝜇_{𝑎,𝑖,𝑗}+𝜇_{𝑠,𝑖,𝑗}⁻¹. (12)

3.1. Numerical Treatment of Angular Derivative Terms with Finite Legendre Transform

As is explained in [17], we consider the following Legendre transforms: ℒ𝜕𝜕𝜉1−𝜉²Ψ,𝑠=¹₋₁𝜕𝜕𝜉1−𝜉²Ψ𝑃_𝑠(𝜉)ℒ𝜕𝑑𝜉,𝜕𝜂1−𝜂²Ψ,𝑠=¹₋₁𝜕𝜕𝜂1−𝜂²Ψ𝑃_𝑠𝜂𝑑𝜂, (13) where 𝑃_𝑚 is the 𝑚 degree Legendre polynomial. According to Sturn-Liouville equation defining Legendre Polynomials, we can write ¹₋₁𝜕𝜕𝜂1−𝜉²𝜓𝑃_𝑠(𝜉)=𝑠()𝑑𝜉𝑠+12𝑠+1¹₋₁Ψ𝑃_𝑠+1(𝜉)𝑑𝜉−¹₋₁Ψ𝑃_𝑠−1(𝜉)𝑑𝜉₁₋₁,𝜕𝜕𝜂1−𝜉²𝜓𝑃_𝑠𝜂=𝑠()𝑑𝜂𝑠+12𝑠+1¹₋₁Ψ𝑃_𝑠+1𝜂𝑑𝜂−¹₋₁Ψ𝑃_𝑠−1𝜂𝑑𝜂. (14)

To obtain derivative terms, we make use of numerical quadrature: ^𝑀_𝑚=1𝑤_𝑚𝐷_𝜉,𝑚𝑃_𝑠𝜉_𝑚=𝑠(𝑠+1)2𝑠+1^𝑀_𝑚=1𝑤_𝑚𝑃_𝑠+1𝜉_𝑚−^𝑀_𝑚=1𝑤_𝑚𝑃_𝑠−1𝜉_𝑚,𝑠=1,…,𝑀−1,_𝑀𝑚=1𝑤_𝑚𝐷_𝜂,𝑚𝑃_𝑠𝜂_𝑚=𝑠(𝑠+1)2𝑠+1^𝑀_𝑚=1𝑤_𝑚𝑃_𝑠+1𝜂_𝑚−^𝑀_𝑚=1𝑤_𝑚𝑃_𝑠−1𝜂_𝑚,𝑠=1,…,𝑀−1. (15) The above systems of equations are closed by the obvious relations: ^𝑀_𝑚=1𝑤_𝑚𝐷_𝜉,𝑚=¹₋₁𝜕𝜕𝜂1−𝜉²𝜓𝑑(𝜉),_𝑀𝑚=1𝑤_𝑚𝐷_𝜂,𝑚=¹₋₁𝜕𝜕𝜂1−𝜂²𝜓𝑑𝜂. (16) So discrete derivative terms in the (𝑖,𝑗)-cell into the medium can be obtained by solving the linear algebraic equations 𝐴_𝜉𝐷_{𝜉,𝑖,𝑗}=𝐵_{𝜉,𝑖,𝑗},𝐴_𝜂𝐷_{𝜂,𝑖,𝑗}=𝐵_{𝜂,𝑖,𝑗}, (17) where 𝐀_𝜉=𝑤₁𝑤₂…𝑤_𝑀𝑤₁𝑃₁𝜉₁𝑤₂𝑃₁𝜉₂…𝑤_𝑀𝑃₁𝜉_𝑀𝑤⋮⋮⋮⋮₁𝑃_𝑀−1𝜉₁𝑤₂𝑃_𝑀−1𝜉₂…𝑤_𝑀𝑃_𝑀−1𝜉_𝑀,𝐃_{𝜂,𝑖,𝑗}=𝐷_{𝜉,1,𝑖,𝑗}𝐷_{𝜉,2,𝑖,𝑗}⋮𝐷_{𝜉,𝑀,𝑖,𝑗},𝐁_{𝜂,𝑖,𝑗}=023^𝑀_𝑚′=1𝑤_𝑚′𝜓_{𝑚′,𝑖,𝑗}𝑃₂𝜉_𝑚′−𝑃₀𝜉_𝑚′⋮𝑀(𝑀−1)2𝑀−1^𝑀_𝑚′=1𝑤_𝑚′𝜓_{𝑚′,𝑖,𝑗}𝑃_𝑀𝜉_𝑚′−𝑃_𝑀−2𝜉_𝑚′ (18) and 𝐴_𝜂,𝐷_{𝜂,𝑖,𝑗},𝐵_{𝜂,𝑖,𝑗} are straightforward by replacing 𝜉 by 𝜂.

A set of weights are judiciously chosen with an equally spaced distribution of angular ordinates (𝜉,𝜂). The matrices 𝐴_𝜉 and 𝐴_𝜂 must be regular and they are inversed each only once in all the calculation process.

To solve (12), we use successive iterations to actualise the implicit internal source term in the right member. So, this gives Ψ^𝑘+1_{𝑚,𝑖,𝑗}=Δ𝑦𝜉_𝑚𝛼Ψ^𝑘+1_{𝑚,𝑖−1,𝑗}+Δ𝑥𝜂_𝑚𝛼Ψ^𝑘+1_{𝑚,𝑖,𝑗−1}+12𝑛²_𝑖,𝑗Δ𝑦𝑛²_𝑖,𝑗−𝑛²_{𝑖−1,𝑗}𝐷_{𝑚,𝑥,𝑖,𝑗}+Δ𝑦𝑛²_𝑖,𝑗−𝑛²_{𝑖,𝑗−1}𝐷_{𝑚,𝑥,𝑖,𝑗}+Δ𝑥Δ𝑦𝑆^𝑘+1_{𝑚,𝑖,𝑗}+𝜇^𝑀_{𝑠,𝑖,𝑗𝑚′=1}𝑤_𝑚′𝑝_𝑚𝑚′Ψ^𝑘_{𝑚′,𝑖,𝑗}×Δ𝑦𝜉_𝑚𝛼+Δ𝑥𝜂_𝑚𝛼+Δ𝑥Δ𝑦𝜇_{𝑎,𝑖,𝑗}+𝜇_{𝑠,𝑖,𝑗}⁻¹. (19)

The iteration process is repeated until a convergence criterion is attempted. To improve convergence speed, we use a successive overrelaxation method. So the updated value Ψ^𝑘+1_{𝑚,𝑖,𝑗} is a linear combination of the iterated value Ψ^𝑘_{𝑚,𝑖,𝑗} and the previously computed value Ψ^𝑘+1_{𝑚,𝑖,𝑗updated}=𝜌Ψ^𝑘_{𝑚,𝑖,𝑗}+1−𝜌Ψ^𝑘+1_{𝑚,𝑖,𝑗}, (20) where 𝜌 is a relaxation parameter whose value is usually between 1 and 2. The solution is obtained when the relative discrepancy value Ψ𝜀=^{𝑘+1𝑚,𝑖,𝑗}−Ψ^𝑘_{𝑚,𝑖,𝑗}Ψ^𝑘_{𝑚,𝑖,𝑗} (21) is smaller than a tolerance value. In all our calculus, we have taken 10⁻⁸ as a tolerance value. As initial condition, we take a field of null radiance. Also, all our calculations are done in the case of interpolation diamond scheme (𝛼=0.5).

If the direction cosines are not both positive, the precedent equations are valid provided that the orientation WESN of cells is done according to the direction of propagation [7]. In all our investigations, the injected power source is assumed to be equivalent to a forward collimated monochromatic intensity placed at a source point on the middle of the bottom side of the boundary. Results shown below are obtained by using a continuous wave source with a uniform equivalent intensity value of 50 mW·cm⁻¹.

3.2. Boundary Conditions

On the boundary, the radiance is the sum of the external source contribution and the partly reflected radiance due to the refractive index mismatch at the boundary(i)Ψ(𝑟,⃗Ω)=𝑆(𝑟_𝑏,Ω)+𝑅⋅Ψ(𝑟,⃗Ω_ref);(ii)𝑢_𝑏⋅Ω<0 and 𝑢_𝑏⋅Ω_ref=−𝑢_𝑏⋅Ω,where 𝑟_𝑏 is a position on the boundary and 𝑢_𝑏 is an outer normal unit vector. The reflectivity 𝑅 can be calculated for each direction using Fresnel’s relations.

To present our results, we use the detected fluence rate which is given in a (𝑖_𝑑,𝑗_𝑑)-detector point on the boundary as Φ_𝑑=^𝑀_𝑚=11−𝑅_{𝑚,𝑖𝑑,𝑗𝑑}𝑤_𝑚Ψ_{𝑚,𝑖𝑑,𝑗𝑑}, (22) with 𝑅_{𝑚,𝑖𝑑,𝑗𝑑}=1if𝜑_𝑚𝑛>arcsin^air𝑛_{𝑖𝑑,𝑗𝑑}12𝑛_{𝑖𝑑,𝑗𝑑}cos𝜑_𝑚−𝑛_aircos𝜑_𝑡𝑛_{𝑖𝑑,𝑗𝑑}cos𝜑_𝑚+𝑛_aircos𝜑_𝑡²+𝑛_{𝑖𝑑,𝑗𝑑}cos𝜑_𝑡−𝑛_aircos𝜑_𝑚𝑛_{𝑖𝑑,𝑗𝑑}cos𝜑_𝑡+𝑛_aircos𝜑_𝑚²,𝜑else,_𝑡𝑛=arcsin^airsin𝜑_𝑚𝑛_{𝑖𝑑,𝑗𝑑},𝑛_air≈1. (23)

Also, we make use of a normalized detected fluence rate defined as Φ_𝑁=Φ_𝑑(1/𝐷)∑^𝐷_𝑑=1𝑤^′_𝑑Φ_𝑑, (24) where 𝐷 is the number of the detector points on one side of the boundary and 𝑤^′_𝑑 is a weighting factor from the generalized trapezoidal integration rule.

In all calculations, we have used 28 detector points on each side. Also, all calculus is carried out by using 16 uniformly distributed discrete directions and a space grid of 121 × 121 cells.

4. Results and Discussion

4.1. Continuous Varying Refractive Index Medium

In this investigation, we study near infrared radiation transport in a rectangular medium exposed to a continuous collimated source which is placed on the bottom side of the boundary. Figure 1 shows the considered medium; it is assumed to be 2 × 2 cm sized with varying refractive index. Within the medium, we consider a 𝑥-axis linear refractive index variation with different gradient values. To show the effect of the refractive index on detected fluence rate, we have used a weakly absorbing and forward scattering background medium whose optical parameters are shown in Figure 1.

Figure 1 

A test-medium with background properties and detector points.

Figures 2 and 3 represent the response of the medium through the detected signal on the top and right side of the boundary in the case of linear variation of refractive index. We note that the response of the medium varies linearly according to the gradient of refractive index. The maximum transmission moves to regions of height index (Figure 2). Indeed, according to boundary conditions of the medium (Descartes Laws) the transmission window increases proportionally of refractive index between the medium and outside medium (air).

Figure 2 

Response of a continuous varying refractive index medium detected fluence rate on the top side: gradient effect.

Figure 3 

Response of a continuous varying refractive index medium detected fluence rate on the right side: gradient effect.

On the other hand, the right side (Figure 3) shows the response of the medium detected on the right side of the boundary in the case of linear and parabolic variation, respectively. The detected fluence rate curves presents distinguished effect of refractive index gradient in linear case. The transmission zone on the boundary increases with decreasing gradient values. Even though most detected transmission is obtained near the source, a weak gradient can augment transmitted radiation relatively far from the source while high values of gradient can block transmitted radiation within the medium.

4.2. Effect of Stochastic Varying Refractive Index

In this investigation, we keep the same medium precedent, but we will study the stochastic variation refractive index. We control this variation by uniform random fixing of a given interval. The other optical properties are the same as in the precedent investigation. In such cases, there is an obvious effect of the heterogeneity on the detected signal especially when the refractive index is increased or decreased by 10% in a stochastic way. The detected signal on different sides of the medium is shown in Figures 4 and 5. There is a significant effect of this variation on the detected signal on the top side and right side, because there is strong perturbation in medium. These findings highlight the potential of refractive index as a possible detection parameter of a tumor in a surrounding safe tissue. These findings open prospects for further study of the effect of local refractive variation in domain and frequency-domain schemes in diffuse optical tomography.

Figure 4 

Response of stochastic varying refractive index medium on the top side.

Figure 5 

Response of stochastic varying refractive index medium on the right side.

5. Conclusion

This study attempted to develop a computational way helping in detection of abnormalities in a biological tissue. This should enable predictions of eventual tumor existence when using a diffuse optical tomography scheme. The used model is based on stationary radiative transfer equation including a possible continuous and stochastic variation of refractive index. In particular, computational technique of Legendre transform is extended to handle angular derivative terms arising by the varying refractive index consideration. The obtained computational model is implemented to investigate some practical situations in diffuse optical tomography (DOT) setting. Obtained results showed that variation of refractive index can yield useful predictions about the target and the location of abnormal inclusions within the tissue. These findings open prospects for further study of the effect of local refractive variation in time-domain and frequency-domain schemes in diffuse optical tomography.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.