Statement of the problem - IEEE Xplore

Diffraction of Electromagnetic Wave on Skin Blood Vessel: Statement of the Problem Ivan Starkov and Zbynˇek Raida

Alexander Starkov

SIX Research Centre, Brno University of Technology, Brno, Czech Republic E-mail: {starkov|raida}@feec.vutbr.cz

Institute of Refrigeration and Biotechnology, National Research University of Information Technologies, Mechanics and Optics, St. Petersburg, Russia

Abstract—A theoretical model for the diffraction of electromagnetic wave on the skin blood vessel is developed. For this purpose, the Green’s function formalism is used. The calculations performed in the paper are based on real experimental data, which makes it more compelling. The proposed approach would eliminate fundamental uncertainties in the modeling of biological tissues. Index Terms—blood vessel, capillary, diffraction, scattering, Green’s function.

I. I NTRODUCTION Human skin is a prime example of a multicomponent biological structure which is very difficult to describe by mathematical models. Optical characteristics of such a complex environment depend on many factors. In order to correctly build the model of the human skin describing its optical properties, it is necessary to have a understanding on the biological features of this system. The thickness of the human skin strongly depends on the part of the body and is varying from 0.5 to 4mm [1] The upper layer of the skin is epidermis (see Fig. 1) which is multilayered epithelium. Let us denote thickness of epidermis as he . This value varies from 0.02-0.05mm on the neck and face to 0.5-2.4mm on palms and soles[2]. Radiation incident from air firstly goes through the epidermis where the largest dielectric constant corresponds to the melanin. By virtue of this, optical properties of the epidermis can be considered equal to those of the melanin with refractive index of 1.44 [3]. The next layer in the skin tissue is dermis. At the interface of the dermis some part of the radiation is reflected while the rest goes inside. The blood vessel is situated in dermis at a depth of h and is represented by the cylinder with center at point M0 and radius a. Cylinder axis is considered parallel to the interface. The typical value of a for blood caterpillars lays in the range 0.05-0.1mm. The wavelength of the incident electromagnetic field for medical purposes is around 1cm. Therefore, parameter ka with k as wavenumber can be considered small. As a consequence, in this paper the problem of diffraction on the capillary located in the layered media is investigated in the long-wave approximation. Below the dermis is the subcutaneous fat which possesses a high absorption coefficient and reflection from which is insignificant [4]. Thus, we can consider the thickness of dermis as infinite.

It is necessary to note that there exists a number of models (see e.g.[5]) of dermis as two sub-layers. We are not following this strategy but the employed long-wave approximation is applicable for an arbitrary number of layers. In-spite of the fact that the capillary has walls we will not take this issue into account. Moreover, it is assumed that the refraction index of the capillary wall coincides with index of the environment. In our formulation of the model, the consideration of capillary wall will result only in a small modification of the derived formulas. II. T HE STATEMENT OF

THE PROBLEM

As it well known, any plane electromagnetic wave can be represented by the superposition of two waves: the first one with vector Eint is perpendicular to the plane of incidence while the second lays in this plane. The approach in the both cases are similar and we restrict ourselves by the consideration of TE waves. That is, we assume that Eint = U int ey . The rectangular coordinate system is chosen that plane xy coincides with the skin interface. From the geometry of the system it follows that the total electric field E will be will be oriented in the same direction, i.e. E = U ey . For convenience, we denote quantities related to air, epidermis, dermis, and vessel by indexes {a, e, d, v}, respectively. The absence of index implies a general dependence. As a starting point, we consider the system of Maxwell’s equations iω iω µH, ∇ × H = − εE. (1) c c For electromagnetic waves of the same polarization, this system can be reduced to the Helmholtz’s equation [6] ∇×E =

ΔU + k 2 U = 0.

(2)

Here ω is frequency, c speed of light in vacuum, ε,µ dielectric √ and magnetic permittivities, k = ωc εµ wave number. We believe that media listed in the introduction have a constant wave number k = k(z): ka ,ke ,kd ,kv . At the interfaces we demand the fulfillment of continuous conditions for the tangential component of electric and magnetic field. These requirements can be written in the following form Ui = Uj ,

mi

∂Uj ∂Ui = mj , ∂n ∂n

(3)

The main role in the computational procedure plays the transmission (and reflection) coefficients: from air to dermis Tad (Rad ) and from dermis to air Tda (Rda ). These quantities are determined as follows [7] rae + red eiβ , 1 + rae red e2iβ tae ted eiβ Tad (θa ) = , 1 + rae red e2iβ

Rad (θa ) =

(9)

where mi sin θi − mj sin θj 2mi sin θi , tij = mi sin θi + mj sin θj mi sin θi + mj sin θj (10) are the coefficients of reflection and transmition from the boundary with indexes i and j. The coefficient Tda is defined similarly to (9). rij =

Fig. 1. Simplified three-layer skin model takes into account the location of a blood vessel in the subcutaneous tissues.

� where n is normal to the interface and m = µ/ε is the wave resistance of the medium. There are 3 possible combinations for the pair of indexes {i, j}: {a, e} at the air/epidermis interface, {e, d} at the epidermis/dermis interface, and {d, v} at the dermis/capillary interface. Here it is necessary to mention that in the case of TM waves the coefficient m should replaced by m′ = 1/m. The incident field can can be expressed as U int = eika (x cos θa −z sin θa ) ,

(4)

where θa is the angle of incidence (see Fig. 1). Let us denote through U 1 the wave field in the absence of the capillary. Obviously, such a field is a superposition of the plane waves. The presence of the capillary changes the field on the scattering additive. In other words, the total field is given by U = U 1 + U SC .

B. Green’s function For further calculations it is convenient to define Green’s function, i.e. the function of the point source. Due to the details of the experiment we consider the point of observation M (x, z) located in the air and the source is place at the M ′ (x′ , z ′ ) point in dermis. Then, the Green’s function G(M, M ′ ) can be calculated via the Fourier transformation 1 G(M, M ) = 2π ′

A. Plane wave formalism For the solution of the formulated problem (2)-(6), we can seek U 1 in the form of plane waves � � Ua1 = eika x cos θa e−ika z sin θa + Rad eika z sin θa , � � Ue1 = eike x cos θe Ae−ike z sin θe + Beike z sin θe , (7) Ud1

= Tad e

ikd {x cos θd −(z−he ) sin θd }

The angles {θa , θe , θd } are linked by the Snell’s law ka cos θa = ke cos θe = kd cos θd .

.

(8)

(11)

where the one-dimensional Green’s function g(z, z ′ , ζ) is given by g + (z)g − (z ′ ) g(z, z ′ , ζ) = . (12) W (g + , g − ) The quantities g ± (z) are the solution of the ordinary differential equation d2 g + (k 2 (z) − ζ 2 )g = 0, dz 2

SC

with (r, ϕ) as a polar coordinate system with center at the M0 point.

′

eiζ(x−x ) g(z, z ′ , ζ)dζ,

−∞

(5)

In turn, the scattered field U must satisfy the Sommerfeld radiation condition � � √ ∂U SC SC = 0, (6) − ikU lim r r→∞ ∂r

�∞

(13)

and satisfy the boundary conditions (3) together with the decay conditions at z → ±∞ for Imk > 0. The Wronskian of the above solutions is denoted by W (g + , g − ). It is easy to check that solution g − after the substitution ζ = ka cos θ coincides with a set of plane waves U 1 up to the factor eika cos θa . The solution g + is built analogously and can be represented by the set of exponents as well. Asymptotic of the Green’s function for r → ∞ can be found in a standard manner [8], [9] Aei(ka r) Tda (ϕa ), G(M, M ′ ) = √ 2πka r with A = e−i(he

√

ke2 −cos2 ϕe −h

√

2 −cos2 ϕ −3π/4) kd d

(14)

.

(15)

Here ϕa is the angle of observation and the angles ϕd , e connected with ϕa by Snell’s law.

U

SC

mv − md (M ) = ma

� �

∂S

� �� � � SC md ∂U SC ∂U 1 G(M, M ′ )dS + +(kd2 − kv2 ) + + U 1 G(M, M ′ )dS. U ∂n ∂n ma S

approximate expression for the field of scattered wave � � mv ka2 2 SC − kv U 1 (M0 )G(M, M0 )S. U (r, ϕ) = ma

Fig. 2. The normalized radiation pattern F (θa , ϕa ) of a blood vessel illuminated by a plane wave.

C. Determination of the scattering additive By using the obtained Green’s function, we may reduce the solution of the diffraction problem to the solution of the integral equation. For this purpose we apply the second Green formula to U SC /M and to the Green’s function in all media. The integrals over two interfaces vanish and only integrals over the capillary area S and its border ∂S are left and we obtain equation (16). In (16) dS is differential of the arc length at ∂S, dS is differential of the area. During the integration the point M ′ runs either through the area of capillary S (for the double integral) or its border ∂S (for the single integral). Unfortunately, it is not possible to obtain an exact analytical solution for the rigorous integral equation (16). In the longwave approximation, when dimensions of the scatterer much smaller than the wavelength (ka ≪ 1), an approximate solution can be derived. In order to do this, we replace in (16) Green’s function G(M, M ′ ) on G(M, M0 ). That is, we neglect the dependence of G(M, M ′ ) on the variables of integration. Such a simplification is feasible because the deference between G(M, M ′ ) and G(M, M0 ) is small due to smallness of the distance |M0 − M ′ |. Finally, the quantities left in (16) that can be significantly changed in the integration area are normal derivatives. To get rid of them, we may use the first Green formula and transform aforementioned terms as follows �� �� � ∂U 1 1 2 = ΔU dS = −kv U 1 dS. (17) ∂n ∂S

S

(16)

S

Furthermore, in (17) we neglected U and replaced U 1 by its value in the M0 point. As a result, we obtain the SC

(18)

Note that S = πa2 . From the above expression we can see that U SC (µ) is really small because it contains a small √ parameter k 2 S. At long distances U SC (µ) decreases as 1/ ka r. It is necessary to emphasize that (18) is universal and is independent either on the geometry of capillary or on the number of layers in the system. Moreover, it can be easily generalized on the three-dimensional case when we consider the diffraction of a small object. For this purpose it is enough to change the area S of blood vessel on its volume V . In addition, (18) can be generalized for the case when capillary has more complicated form. For example, when the capillary walls are taken into account. Further, by virtue of the Green’s function asymptotic at large distances (14), let us rewrite (18) as � � mv ka2 ei(ka r−3π/4) SC 2 U (θa , ϕa ) = − kv S √ F (θa , ϕa ). ma 2πka r (19) where the radiation pattern F (θa , ϕ) is defined by √ 2 √ 2 2 2 F (θ, ϕ) = e−i(he ke +cos φe +h kd −cos φd ) Tad (θ)Tda (ϕ), (20) The results of the calculations for F (θ, ϕ) are demonstrated in Fig. 2. III. C ONCLUSION In this work, we have formulated and solved the problem of the electromagnetic wave diffraction on skin blood vessel. The developed approach yields important results. The amplitude of the scattered field is proportional to the square of the capillary. The comparison of the simulation results for TE and TM waves allows to determine the dielectric constant of the vessel. For a case of long waves, the reflected field is linearly dependent on the depth position of the capillary. The findings of this work have significant implications and contribute valuable insights and for the modeling of biological tissues. ACKNOWLEDGMENT The research leading to the results exposed has received funding from the SIX project CZ.1.05/2.1.00/03.0072 and from the project CZ.1.07/2.3.00/30.0039 of Brno University of Technology. Additionally, this work was partially financially supported by Government of Russian Federation, Grant 074U01.

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