technique for handling wave propagation speciﬁc...

Technique for handling wavepropagation specific effects in biologicaltissue: Mapping of the photon transport

equation to Maxwell’s equations

Chintha C Handapangoda1, Malin Premaratne1, David M Paganin2

and Priyantha R D S Hendahewa3

1Advanced Computing and Simulation Laboratory (AXL), Department of Electrical andComputer Systems Engineering, Monash University, Clayton 3800, Victoria, Australia

2School of Physics, Faculty of Science, Monash University, Clayton 3800, Victoria, Australia3Department of Mechanical Engineering, Monash University, Clayton 3800, Victoria,

[email protected].

Abstract: A novel algorithm for mapping the photon transport equation(PTE) to Maxwell’s equations is presented. Owing to its accuracy, wavepropagation through biological tissue is modeled using the PTE. Themapping of the PTE to Maxwell’s equations is required to model wavepropagation through foreign structures implanted in biological tissue forsensing and characterization of tissue properties. The PTE solves foronly the magnitude of the intensity but Maxwell’s equations require thephase information as well. However, it is possible to construct the phaseinformation approximately by solving the transport of intensity equation(TIE) using the full multigrid algorithm.

© 2008 Optical Society of America

OCIS codes: (170.1470) Blood or tissue constituent monitoring; (170.4580) Optical diagnos-tics for medicine; (170.6930) Tissue.

References and links1. S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scat-

tering media,” Appl. Opt. 35, 3372–3378 (1996).2. M. Premaratne, E. Premaratne, A. J. Lowery, “The photon transport equation for turbid biological media with

spatially varying isotropic refractive index,” Opt. Express 13, 389–399 (2005).3. C. C. Handapangoda, M. Premaratne, L. Yeo, J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating

laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).4. F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between

two different media,” Appl. Sci. Res. 28, 73–88 (1973).5. S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, O. Samek, “Enhanced transmission versus localization of

a light pulse by a subwavelength metal slit,” Phys. Rev. B. 70, 195428 (2004).6. S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).7. S. C. Chapra, R. P. Canale, Numerical Methods For Engineers, (4th ed., McGraw-Hill, New York, 2002).8. D. M. Paganin, Coherent X-Ray Optics, (Oxford University Press, New York, 2006).9. M. Born, E. Wolf, Principles of Optics, (7th ed., Cambridge University Press, Cambridge, 1999).

10. M. R. Spiegel, Vector Analysis And An Introduction To Tensor Analysis, (McGraw-Hill, 1959).11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441

(1983).12. T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric

phase-contrast microscopy,” J. Phys. D: Appl. Phys. 32, 563–567 (1999).

#101155 - $15.00 USD Received 4 Sep 2008; revised 28 Sep 2008; accepted 10 Oct 2008; published 17 Oct 2008

(C) 2008 OSA 27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 17792

13. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun.199, 65–75 (2001).

14. T. E. Gureyev, K.A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solu-tion for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).

15. T. E. Gureyev, K.A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt.Commun. 133, 339–346 (1997).

16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes In C: The Art Of ScientificComputing, (2nd ed., Cambridge University Press, Cambridge, 1992).

17. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phaseuniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).

18. G. E. Thomas, K. Stamnes, Radiative Transfer In The Atmosphere And Ocean, (Cambridge University Press,1999).

19. R. Ramamoorthi, P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumina-tion from images of a convex Lambertian object,” J. Opt. Soc. Am. A 18, 2448–2459 (2001).

20. D. Paganin, K. A. Nugent, “Non-interferometric phase determination,” in Peter Hawkes (editor), Advances inImaging and Electron Physics (volume 118, 85–127, Harcourt Publishers, Kent, 2001).

21. D. Paganin, K. A. Nugent, “Non-interferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).

22. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819(1998).

23. D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy III: The effectsof noise,” J. Microscopy 214, 51–61 (2004).

24. W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. QuantumElectron. 26, 2166–2185 (1990).

25. M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, (Springer, Germany, 2004).26. S. A. Prahl, J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the

adding-doubling method,” Appl. Opt. 32, 559–568 (1993).27. C. Y. Wu, S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering

medium,” Int. J. Heat Mass Trans. 43, 2009–2020 (2000).28. A. E. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216–2222 (1989).29. M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue 1. Models of radiation

transport and their application,” Lasers in Medical Science 6, 155–168, (1991).30. G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert. “Development and application of three-dimensional

light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. 23, 1721–1733 (1987).31. W. D. Burnett, “Evaluation of laser hazards to the eye and the skin,” Amer. Ind. Hyg. Assoc. J. 30, 582–587

(1969).

1. Introduction

Optical techniques in biomedical applications such as optical tomography and light-aided sens-ing of substances have been receiving tremendous interest recently [1]. These techniques ofsensing substances in tissue or blood require foreign structures to be embedded/implanted intissue in order to condition optical signals. Therefore, having a detailed understanding of howlight interacts with tissue is required. Owing to its ability to accurately represent light propa-gation through tissue, wave propagation through biological tissue is modeled using the photontransport equation (PTE) [2, 3]. However, the interaction of electromagnetic energy with em-bedded objects can be best studied using Maxwell’s equations. Therefore, in order to modelwave propagation through tissue with implanted foreign structures, a mapping of the PTE toMaxwell’s equations is required.

In this paper we develop a technique to map the photon transport equation to Maxwell’sequations using phase-retrieval techniques. To illustrate the applicability and accuracy of ourmethod we analyze laser pulse propagation through a metal slit in biological tissue. Light prop-agation in scattering and absorbing media such as biological tissue is modeled by the photontransport equation (PTE) [2, 3] which is written in terms of the magnitude of the intensity butnot the phase. Light propagation through a slit in a metal screen is described by Maxwell’sequations [4, 5], which take into consideration both the magnitude as well as the phase of theelectric and magnetic fields together with the vectorial character of these fields. However, to



the authors’ knowledge no work has been reported which addresses the problem of couplingthese two sets of equations. Therefore, we introduce a novel strategy to couple these two setsof equations at the tissue - metal screen interface by retrieving the phase information from theintensity profile.

This paper is organized in five sections. In section 2 we introduce the formulation of theproposed technique for coupling the PTE to Maxwell’s equations using a phase-retrieval tech-nique. In section 3 we address the problem of modeling wave propagation through a slit in ametal screen implanted in tissue. Section 4 provides the simulation results for a composite slabof tissue layer and a metal screen with a slit, a discussion on those results and possible exten-sions of the proposed technique. We conclude in section 5 summarizing the key features andadvantages of the proposed technique.

2. Formulation: construction of phase information from a radiance profile

This section presents how we map the PTE to Maxwell’s equations by constructing the phaseinformation using the radiance profile obtained by solving the PTE. In section 2.1 we discusshow to obtain the radiance profile by solving the PTE. In section 2.2 we present the derivationof the transport-of-intensity equation which is used for phase retrieval. In section 2.3 we showhow to construct the phase profile from the radiance profile obtained by solving the PTE.

2.1. Modeling light propagation in tissue

For modeling light propagation through biological tissue, we use the photon transport equation(PTE) [3]:

1v

∂∂ t

IPTE (z,u,φ ,t)+u∂∂ z

IPTE (z,u,φ ,t)+ σt IPTE (z,u,φ ,t)

− σs

4π

∫ 2π

0

∫ 1

−1P

(u′,φ ′;u,φ

)IPTE

(z,u′,φ ′,t

)du′dφ ′ = F (z,u,φ ,t) , (1)

where IPTE (z,u,φ ,t) is the radiance (units: W.m−2.sr−1.Hz−1), (z,θ ,φ) are the standard spher-ical coordinates, u = cosθ , t is the time variable, σt and σs are attenuation and scattering co-efficients, respectively, and σt = σs + σa, where σa is the absorption coefficient. The speed oflight in the medium is denoted by v , P(u ′,φ ′;u,φ) is the phase function and F (z,u,φ ,t) refersto the source term.

The Laguerre Runge-Kutta-Fehlberg (LRKF) method [3] can be used to solve Eq.(1) forIPTE . The LRKF method reduces the transient PTE, given by Eq.(1), to an ordinary differentialequation of only one independent variable, z, and solves it numerically. In the LRKF algorithmthe discrete ordinate method [6] is used to discretize the azimuthal and zenith angles (φ andθ ), a Laguerre expansion [3] is used to represent the time dependence and the Runge-Kutta-Fehlberg (RKF) method [7] is used to solve the one-variable ordinary differential equation inz.

2.2. Derivation of the transport-of-intensity equation for phase construction

As described in detail in [8], for a static (i.e. the electrical permittivity and the magnetic perme-ability are independent of time), non-magnetic (i.e. with constant permeability) medium with-out any current or charge densities inside, and with scatterers which slowly vary over lengthscales comparable to the wavelength of the incident radiation, the Maxwell equations can be



reduced to [8][

ε(x,y,z)μ0∂ 2

∂ t2 −∇2]

E(x,y,z,t) = 0, (2)

[ε(x,y,z)μ0

∂ 2

∂ t2 −∇2]

H(x,y,z,t) = 0, (3)

where ε is the electric permittivity of the medium, μ0 is the permeability of free space,

∇ =(

∂∂x i+ ∂

∂y j+ ∂∂ zk

), t represents time and E and H denote the electric field and the mag-

netic field, respectively. From Eq.(2) and Eq.(3), since there is no mixing between any of thecomponents of the electric and the magnetic field vectors, we can move on to a scalar theory[8]. Thus,

[ε(x,y,z)μ0

∂ 2

∂ t2 −∇2]

Ψ(x,y,z,t) = 0. (4)

In Eq.(4) Ψ(x,y,z,t) describes the electromagnetic field and it is complex. Using the Fourierintegral Ψ(x,y,z,t) can be expressed as [8]

Ψ(x,y,z,t) =1√2π

∫ ∞

0ψω (x,y,z)e− jωtdω , (5)

where ω is the angular frequency. Using Eq.(5) in Eq.(4) we get[∇2 + εω(x,y,z)μ0c2k2

0

]ψω (x,y,z) = 0, (6)

where ∇2 =(

∂ 2

∂x2 + ∂ 2

∂y2 + ∂ 2

∂ z2

), c is the speed of light in free space and k0 = ω/c is the wave

number in free space. Then, identifying εω (x,y,z)μ0c2 as the square of the position-dependentrefractive index, nω(x,y,z), of the medium, we can re-write Eq.(6) as

[∇2 + k0

2n2ω(x,y,z)

]ψω (x,y,z) = 0, (7)

which is called the homogeneous Helmholtz equation [8].In order to incorporate scattering we express ψω (x,y,z) in Eq.(7) as a perturbed plane wave

[8]:ψω (x,y,z) = ψω (x,y,z)e jkz, (8)

where e jkz represents the unscattered plane wave and ψn (x,y,z) represents the complex en-velope [9]. That is, we have considered the paraxial condition where the rays are not exactlyparallel to each other; or in other words, a field with perturbed wave fronts. Using Eq.(8) inEq.(7), we obtain

∇2(

ψω (x,y,z)e jkz)

+ k20n2

ω (x,y,z) ψω (x,y,z)e jkz = 0. (9)

Using the identity [10]

∇2 [A(x,y,z)B(x,y,z)] ≡ A(x,y,z)∇2B(x,y,z)+B(x,y,z)∇2A(x,y,z)+ 2∇A(x,y,z).∇B(x,y,z), (10)

in Eq.(9) and simplifying we get [8][∇2

xy+∂ 2

∂ z2 +2 jk0∂∂ z

+k20

(n2

ω(x,y,z)−1)]

ψω(x,y,z) = 0, (11)



where ∇2xy =

(∂ 2

∂x2 + ∂ 2

∂y2

). With the paraxial approximation we consider the envelope,

ψω (x,y,z), to be “beam-like” so that its second derivative in the z direction is much smaller

in magnitude than its second derivative in the x and y directions. Therefore, the ∂ 2

∂ z2 ψω(x,y,z)term in Eq.(11) can be dropped. Then Eq.(11) reduces to

[∇2

xy+2 jk0∂∂ z

+k20

(n2

ω(x,y,z)−1)]

ψω(x,y,z) = 0. (12)

Letψω (x,y,z) =

√Ie jφ . (13)

Here, I represents the irradiance (units: W.m−2.Hz−1) and φ represents the phase. Using Eq.(13)in Eq.(12) and separating the imaginary part we get the following relationship [8, 11]:

∇xy · (I (x,y,z)∇xyφ (x,y,z)) = −k∂ I (x,y,z)

∂ z. (14)

Equation (14) is called the transport-of-intensity equation (TIE). It shows how the intensity andthe phase are related, and this forms the basis of the phase construction. In the next subsectionwe show how to retrieve the phase information from the intensity profile, by solving Eq.(14).

2.3. Construction of phase information from the irradiance profile

To construct the phase we re-write the TIE in Eq.(14) as

I (x,y,z)∇2xyφ (x,y,z)+

∂ I (x,y,z)∂x

∂φ (x,y,z)∂x

+∂ I (x,y,z)

∂y∂φ (x,y,z)

∂y= −k

∂ I (x,y,z)∂ z

. (15)

Equation (15) can be solved for φ (x,y,z) numerically using a suitable technique such as the fullmultigrid algorithm [12, 13], a Green-function method [11] or a fast-Fourier-transform-basedmethod [14, 15].

Out of these techniques for solving the TIE, we adopted the full multigrid algorithm [12, 13,16]. Equation (15) is a linear, elliptic partial differential equation of the second order and has aunique solution if I (x,y,z) > 0 over a simply-connected planar region [17]. We adopt the fullmultigrid algorithm which solves the TIE exactly [12, 13, 16].

The full multigrid algorithm we used is briefly described below, as explained in [16]. Referto [16] for further details.

Equation (15) can be expressed asΓu = f , (16)

where Γ =(

I (x,y,z)∇2xy + ∂ I(x,y,z)

∂x∂∂x + ∂ I(x,y,z)

∂y∂∂y

), u = φ (x,y,z) and f = −k ∂ I(x,y,z)

∂ z .

In multigrid methods, we discretize the original equation on a uniform grid. We can discretizeEq.(16) as follows:

(Ii+1, j − Ii−1, j

2Δ

)(φi+1, j −φi−1, j

2Δ

)+

(Ii, j+1− Ii, j−1

2Δ

)(φi, j+1 −φi, j−1

2Δ

)

+Ii, j

(φi−1, j + φi, j−1 + φi+1, j + φi, j+1−4φi, j

Δ2

)= fi, j, (17)

where i = 1, . . . ,M, j = 1, . . . ,M for M×M grid points. Also, Ii, j = I (xi,y j,z), φi, j = φ (xi,y j,z),

Δ = xi+1 − xi = y j+1 − y j and fi, j = −k∂ I(xi,y j ,z)

∂ z . By solving the PTE on two closely separated



planes z = z and z = z + δ z, we obtain two intensity profiles. Thus, we can use the followingapproximations in Eq.(17).

I (x,y,z) ≈ I (x,y,z+ δ z)+ I (x,y,z)2

, (18)

and∂ I (x,y,z)

∂ z≈ I (x,y,z+ δ z)− I (x,y,z)

δ z. (19)

Since in Eq.(18) and Eq.(19) I represents the irradiance, I and I PTE are related by

I =∫

2πIPTE cosθdω, (20)

where θ is the zenith angle used in Eq.(1) and d ω is an infinitesimal solid angle [18, 19]. Thisconversion of the intensity is required because in the PTE we use the ray model of optics, butin Maxwell’s equations we use the wave model; and these two models deals with two differentdefinitions of intensity, radiance and irradiance, respectively.

Since the intensity and its partial derivatives with respect to x, y and z can be approximatelycalculated from the two intensity profiles, as shown in Eq.(17), Eq.(18) and Eq.(19), the onlyunknown in Eq.(17) is φ i, j . Hence, we can use the full multigrid algorithm to solve Eq.(17) forφi, j and thus the phase can be retrieved on each grid point.

We can discretize Eq.(16) on a uniform grid with mesh size h as

Γhuh = fh. (21)

If uh denotes an approximate solution to Eq.(21), then the error in u h is

vh = uh − uh, (22)

and the residual or the defect isdh = Γuh − fh. (23)

Since Γh is a linear operator, the error satisfies

Γhvh = −dh. (24)

In order to find the next approximate solution, we need to make an approximation to Γ h in orderto find vh. Classical iteration methods, such as Jacobi or Gauss-Seidel can be used to do this.The next approximation is generated by

unewh = uh + vh. (25)

Next, we form an appropriate approximation Γ H of Γh on a coarser grid with mesh size H. Thenthe residual equation, Eq.(24), is approximated by

ΓHvH = −dH . (26)

Since ΓH has smaller dimension, Eq.(26) is easier to solve than Eq.(24). In the full multigridalgorithm, the first approximation is obtained by interpolating from a coarse-grid solution andat the coarsest level we start with the exact solution [16]. Using the full multigrid algorithm asdetailed above we can solve Eq.(16) for u. Thus, the phase at each grid point, φ i, j is retrieved.

Before proceeding with an example that makes use of the TIE and PTE in the context of mod-eling wave-propagation specific effects for electromagnetic pulses traversing foreign structures



implanted in biological tissue, we briefly discuss the validity and accuracy of both the TIE andthe PTE in this context.

Regarding the validity and accuracy of deterministic phase retrieval using the transport-of-intensity equation, Eq.(14), we make the following three remarks: (a) The TIE has been widelyemployed for quantitative phase retrieval using monochromatic and polychromatic electromag-netic fields in both the visible-light and X-ray region, given a series of defocused intensityimages. The TIE has also been successfully solved for the phase using matter waves such aselectrons and neutrons. For a review of this work, see reference [20]. (b) Since the TIE is thecontinuity equation associated with the paraxial equation [11], an exact solution to which isfurnished by the Fresnel diffraction integral [8], its regime of validity is restricted to paraxialbeam-like fields. Interestingly, it may be used with both coherent and partially-coherent fields,a point which has been studied from both a theoretical [21] and an experimental [22] perspec-tive. (c) Errors, in the phase retrieved using a TIE analysis, are primarily due to two sources:the finite-difference approximation to the right-hand-side of the TIE (Eq.(14)) that is given inEq.(19), together with the presence of noise in the detected images. While the latter factor isirrelevant in the context of the analysis in the present paper, as shall become clear in the follow-ing sections, errors in the retrieved phase due to the former effect need to be considered. For ananalysis of both factors, see reference [23]. The upshot of this analysis is that the error in theTIE-retrieved phase, due to a non-infinitesimal spacing δ z (cf. Eq.(19)), leads to a blurring ofthe retrieved phase which becomes negligibly small if δ z tends to zero from above. Reference[23] develops an expression for the optimal δ z in the presence of a given level of noise, thisoptimal defocus distance being proportional to the cube root of the standard deviation of thenoise. Typical reconstruction accuracies from experiments involving TIE-based phase retrievalare on the order of 1-5% [8, 22].

In this paper, we use the transport theory to model light propagation through tissue. Mostof the recent advances in describing the transfer of laser energy in tissue are based on trans-port theory [24]. This theory is preferred in tissue optics instead of analytic approaches usingthe Maxwell equations because of inhomogeneities in biological tissue [24]. Several differentmodels for numerically solving the photon transport equation have been reported in the litera-ture and comparison of these models with the Monte-Carlo method [25] or measured data havebeen reported [24, 26, 27]. Thus, the application of the PTE for light propagation in biologicaltissue is established to be valid and accurate. The LRKF method we use for solving the PTEin this paper uses a combination of the discrete ordinates method [28] and functional expan-sion methods [29]. The validity and applicability of these methods in solving the PTE had beenpreviously established [28, 29].

3. Example: wave propagation through a slit in a metal screen implanted in tissue

In this section, we apply the previously constructed theory to an object buried in tissue.Figure 1 shows a composite object composed of a layer of biological tissue and a layer of a

metal screen with a slit; and Figure 2 shows the end elevation of Figure 1. A short laser pulseis incident on the tissue layer as shown. In general, due to the index mismatch at the interface,radiation gets reflected. Even though the proposed method can easily handle such reflectionsat interfaces, due to the increased mathematical complexity in formulation which masks themain points of our algorithm, we limit our analysis to an index-matched surrounding at the leftboundary of the tissue layer. However, the reflections at the tissue-metal screen interface andmetal screen-tissue interface will be taken into consideration with a detailed formulation.

This example focuses on obtaining the magnitude and the phase of the field at z = z A andthen converting these to the corresponding electric and magnetic fields, so that the field due tothe slit in the metal screen can be modeled. Then, at the exit of the metal screen, the electric



Tissue layer

Incident pulse

φ

θz

x

y

Metal screen witha slit

E0ej(kz-wt)

E0

Tissue layer

Fig. 1. (Color online) Metal screen implanted in biological tissue.

zz = 0 z = d

z = zA-

z = zA- - z

z = zA

Pulse

TissueMetal screen

Metal screen

slit

Fig. 2. End elevation of the tissue-metal screen model.

and magnetic fields can be converted back to the intensity profile so that the tissue layer beyondthis plane can be modeled by solving the PTE.

For modeling light propagation through biological tissue, (i.e. up to the tissue-metal screeninterface), we use the PTE given by Eq.(1). In this paper, without loss of generality, we considerthat there is no source contained inside the medium which results in F (z,u,φ ,t) = 0 in Eq. (1).

The Laguerre Runge-Kutta-Fehlberg (LRKF) method [3] can be used to solve the PTE fromz = 0 to z = zA− . Thus, we can obtain the radiance profile at the plane just before the tissue-metalscreen interface (i.e. at z = zA−). However, in order to model the propagation of the laser pulsebeyond this plane, Maxwell’s equations should be used. Maxwell’s equations require the phaseof the field in addition to the magnitude. Thus, the phase information of the field at z = z A−should be retrieved in order to model the light propagation through the slit in the metal screen.

In order to apply the phase retrieval technique detailed above, first the radiance profile, I PTE



obtained by solving Eq.(1) should be converted to an irradiance profile I, using the relationshipin Eq.(20). Thus, the irradiance at z = zA− , IA− , and at z = zA−−δ z, IA−−δ z, can be obtained bysolving the PTE for radiance and integrating over the hemisphere. Then, we use the approxi-mations in Eq.(18) and Eq.(19) in order to solve the TIE. That is,

I ≈ IA− + IA−−δ z

2, (27)

and∂ I∂ z

≈ IA− − IA−−δ z

δ z. (28)

The full multigrid algorithm [12, 13] is then used to solve the TIE, given by Eq.(14), for thephase, φ (x,y,z). Thus, the phase at the tissue-metal screen interface is retrieved using the in-tensity values at two infinitesimally separated planes.

Once the phase is retrieved, if the incident electric field is known, the field at the tissue-metalscreen interface can be obtained. If the incident polarization vector is E 0, as shown in Figure 1,the electric field at z = z+

A can thus be written as

EA =√

IA−e jφAe j(kz−ωt)E0. (29)

Then, the corresponding magnetic field at z = z+A can be obtained from

HA = j1ω

∇×EA. (30)

Thus, we have obtained the incident electric and magnetic fields at the interface. In order tocalculate the field distribution just after the metal screen with the slit, we have adopted thetechnique introduced by Neerhoff and Mur [4, 5]. We consider a time-dependent incident profileas opposed to the time-independent profile used by Neerhoff and Mur. However, since the timevariation is very slow, their technique [4, 5] can be applied for our case as outlined below. We

x

z

Diffractedwave

region 3region 2

region 1

Incident wave

Met

al s

cree

nM

etal

scr

een

k

E

H 0b

-a

a

y

Fig. 3. Propagation of an incident wave through a slit in a thick metal screen.

consider TM polarization and the magnetic field is assumed to be approximately time harmonicand constant in y direction as shown in figure 3. Thus

Hy (x,y,z,t) = U (x,z,t)e− jωtey, (31)



where ey is the unit vector in y direction. Since the time variation of U (x,z,t) is very slow, itapproximately satisfies the Helmholtz equation. Hence,

(∇2 + k2

j

)Uj (x,z,t) = 0, (32)

where j = 1,2,3 and k j is the wave number in region j. The field in region 1 can be decomposedinto three components:

U1 (x,z,t) = Ui (x,z,t)+Ur (x,z,t)+Ud (x,z,t) , (33)

where Ui represents the incident field, U r represents the field that would be reflected if therewere no slit in the screen and U d represents the diffracted field in region 1 due to the presenceof the slit [5]. Each term on the right hand side of Eq.(33) approximately satisfies the Helmholtzequation. Also it can be shown [4, 5],

Ui (x,z,t) = e− jk1z, (34)

Ur (x,z,t) = Ui (x,2b− z,t). (35)

With the above set of equations and standard boundary conditions for a perfectly conductingscreen, there exists a unique solution for the diffraction problem [5]. Thus, the field in region 3,close to the metal screen can be obtained using the 2-dimensional Green’s theorem as discussedin [4] and [5].

Once the electric field(Ed2

)just after the metal screen is obtained, these can be combined

to obtain the intensity (i.e. the irradiance) using the relationship

I =12

vε|E|2, (36)

where v and ε are the propagation speed and the permittivity in the medium, respectively. Once

z

x

y

R

P

Irradiance at point P

Local directionof propagation

Forward hemisphere centred on the propagation direction

Fig. 4. An illustration of the strategy used for mapping radiance to irradiance.

the irradiance profile on the plane z = d2 is obtained, it should be converted back to a radianceprofile so that the PTE can be used to model the light propagation beyond this plane. Figure 4shows a strategy that can be used for mapping the irradiance profile to the radiance profile, asneeded for solving the PTE. In Fig. 4 , the axes (x,y,z) represent the global coordinate systemused in solving the PTE; also shown is the ray-centred spherical coordinate system used fordescribing the irradiance-to-radiance mapping forward hemisphere.



Based on the work of Ramamoorthi et al. [19], this strategy uses a hemisphere positionedcentrally at the ray propagation direction and uses the relationship between radiance and irra-diance given by Eq.(20), and spherical harmonic representation to achieve this task. Once theirradiance profile on the plane z = d2 is converted back to a radiance profile at each point, inthe forward hemisphere, on the interface, the LRKF method can be used to model the lightpropagation through the remaining layers of tissue.

4. Numerical results and discussion

We have simulated the proposed technique using Matlab. In the simulations, all the units werenormalized. The PTE, Eq.(1), is linear in intensity, IPTE , and thus representing IPTE as IPTE/I0

does not change the equation. Therefore, we use an arbitrary scale for I PTE throughout thispaper. The time units are normalized by Ts, spatial units by Zs and scattering and absorptioncoefficients by 1/Zs.

The input pulse was taken to be a Gaussian pulse (see Fig.5) described mathematically by

f (t) = I0e−

((t−t0)

T

)2

, (37)

where T is the factor determining the width of the input pulse while t 0 determines the time atwhich pulse attains its peak value.

In this paper, we have set Ts = T . We choose this normalization factor due to the fact thatthe Laguerre approximation of the Gaussian pulse is very accurate for pulses with T = 1 orgreater. Therefore, with this scaling it is possible to obtain very accurate results even for verynarrow pulses, which are used in many biomedical applications. For pulses with other shapes,it is recommended that a least square fit is used to obtain a Gaussian approximation, subse-quently setting Ts to be the width of that Gaussian pulse. We have set Zs = v×T . Here, T canbe chosen to suit the particular application. However, these scaling factors should be chosencarefully so that the matrices that are used in the LRKF method remain well-conditioned. Forthe simulations presented in this paper, without loss of generality, we have chosen T/T = 1 sothat Zs = v×T .

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

11

Time

Irrad

ianc

e

Fig. 5. The incident radiance profile on the tissue layer (with arbitrary units).

Figure 5 shows the irradiance profile, in a particular direction, incident on the centre of thetissue layer from the left hand side on figure 1. The proposed technique does not depend onthe type of the input source. Therefore, in order to minimize the additional mathematical com-plexity which might mask the main idea of the proposed technique, without loss of generality,



we have assumed an input source with the same radiance profile in all directions in the forwardhemisphere, as depicted in Fig. 5. However, the proposed technique can be applied to otherkinds of input sources; for example, one may use the afore-mentioned technique proposed byRamamoorthi et al. [19], to construct an input radiance profile with non-uniform profile.

Fig. 6. (Color online) The irradiance profile on a plane just before the tissue-metal screeninterface (with arbitrary units).

Fig. 7. (Color online) The electric field distribution on a plane just before the tissue-metalscreen interface (with arbitrary units).

Figure 6 shows the irradiance profile at z = 2, on a plane just before the tissue-metal screen



interface, obtained by solving the PTE using the afore-mentioned LRKF method. Here, we havemodeled the tissue layer with a normalized scattering coefficient of 0.3, normalized absorptioncoefficient of 0.5 and the Henyey-Greenstein phase function [18] with an asymmetry factor of0.7. The normalized velocity was taken to be 1 while the refractive index of the tissue layer wasassumed to be 1.37. Figure 6 shows how the irradiance profile on the (x,y) grid at z = 2 varieswith time.

Fig. 8. (Color online) The magnetic field distribution on a plane just before the tissue-metalscreen interface (with arbitrary units).

Fig. 9. (Color online) The magnetic field distribution on a plane just after the tissue-metalscreen interface (with arbitrary units).



Using the same technique, the irradiance profile at z = 1.95 was obtained, and these two pro-files were used to retrieve the phase of the field at z = 2. For phase retrieval, we first translatedthe code given in [16] for the full multigrid algorithm to Matlab scripting, and then modified itto solve the TIE, which involved slight modifications to a few subroutines. Then, the irradiancevalues and the phase values were combined according to Eq.(29) to construct the electric fieldat z = 2, which is shown in figure 7.

Fig. 10. (Color online) The electric field component in the x-direction on a plane just afterthe tissue-metal screen interface (with arbitrary units).

Fig. 11. (Color online) The electric field component in the z-direction on a plane just afterthe tissue-metal screen interface (with arbitrary units).



Equation (30) was used to calculate the magnetic field distribution on a plane just before themetal screen, using the electric field distribution. This result is shown in figure 8. Then, thefield distribution on a plane just after the screen was obtained using the technique introducedby Neerhoff and Mur [4], as discussed in the previous section. The magnetic and electric fieldsthus obtained are shown in figures 9, 10 and 11. The irradiance profile constructed according toEq.(36) is shown in figure 12.

The proposed technique can be extended to model more complicated structures, such asphotonic crystals, implanted in biological tissue.

Fig. 12. (Color online) The irradiance profile on a plane just after the tissue-metal screeninterface (with arbitrary units).

5. Conclusion

This paper introduced a novel strategy by which Maxwell’s equations and the photon transportequation can be seamlessly integrated to analyze electromagnetic radiation in tissue-like me-dia. The need for such analysis stems from the perceived biomedical applications in sensingand characterizing tissue properties. By using the present technique it is possible to analyzediffraction effects within the frame work of radiative transfer theory.

The proposed technique can be used to assist the development of biomedical instrumentsthat can be used for noninvasive diagnosis of diseases. Moreover, it enables us to calculate thelight energy distribution within tissue structures with surgically implanted foreign structures.Estimating the light energy distribution in tissue is essential for dosimetry for either photody-namic or for thermal coagulation therapy [30]. Laser treatment is very popular at present andtechniques developed for analyzing reflected or transmitted light from tissue can be used as adiagnostic tool or for evaluation of the progress of laser treatment [30].

The maximum safe exposure of laser light for the skin is 0.1 Joule/cm 2 per pulse or 1.0Watt/cm2 for continuous exposure [31]. However, structures such as photonic crystals can beimplanted to obtain enhanced signals by properly engineering the photon density of states. Theproposed technique can be used to model such foreign structures implanted in tissue. Sinceour main focus is on how to map the photon transport equation to Maxwell equations, for oursimulations we have used a simple foreign structure, a metal screen with a slit, in order to



introduce the proposed technique without the additional mathematical complexity of modelingmore complicated structures.

In modeling wave propagation through biological tissue with foreign structures implanted,the PTE models wave propagation through the tissue layer. At the interface, the phase is re-trieved from the irradiance profile and thus the electromagnetic field is determined. The wavepropagation through the foreign structure is modeled using Maxwell’s equations. Then, theelectromagnetic field is converted back to an irradiance profile at the exit of the foreign struc-ture.



technique for handling wave propagation speciﬁc...

Documents