## Implicitly causality enforced solution of multidimensional transient photon transport equation

Optics Express, Vol. 17, Issue 26, pp. 23423-23442 (2009)

http://dx.doi.org/10.1364/OE.17.023423

Acrobat PDF (947 KB)

### Abstract

A novel method for solving the multidimensional transient photon transport equation for laser pulse propagation in biological tissue is presented. A Laguerre expansion is used to represent the time dependency of the incident short pulse. Owing to the intrinsic causal nature of Laguerre functions, our technique automatically always preserve the causality constrains of the transient signal. This expansion of the radiance using a Laguerre basis transforms the transient photon transport equation to the steady state version. The resulting equations are solved using the discrete ordinates method, using a finite volume approach. Therefore, our method enables one to handle general anisotropic, inhomogeneous media using a single formulation but with an added degree of flexibility owing to the ability to invoke higher-order approximations of discrete ordinate quadrature sets. Therefore, compared with existing strategies, this method offers the advantage of representing the intensity with a high accuracy thus minimizing numerical dispersion and false propagation errors. The application of the method to one, two and three dimensional geometries is provided.

© 2009 Optical Society of America

## 1. Introduction

1. Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer **16**, 289–296 (2002). [CrossRef]

2. K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. **55**, 429–446 (1998). [CrossRef]

3. G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. part I: A general method of solution,” -J. Atmos. Sci. **45**, 1818–1836 (1988). [CrossRef]

1. Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer **16**, 289–296 (2002). [CrossRef]

4. Z. M. Tan and P. F. Hsu, “Transient radiative transfer in three-dimensional homogeneous and non-homogeneous participating media,” J. Quant. Spectrosc. Radiat. Transfer **73**, 181–194 (2002). [CrossRef]

5. C. C. Handapangoda, M. Premaratne, D. M. Paganin, and P. R. D. S. Hendahewa, “Technique for handling wave propagation specific effects in biological tissue: Mapping of the photon transport equation to Maxwell’s equations,” Opt. Express **16**, 17792–17807 (2008). [CrossRef] [PubMed]

4. Z. M. Tan and P. F. Hsu, “Transient radiative transfer in three-dimensional homogeneous and non-homogeneous participating media,” J. Quant. Spectrosc. Radiat. Transfer **73**, 181–194 (2002). [CrossRef]

4. Z. M. Tan and P. F. Hsu, “Transient radiative transfer in three-dimensional homogeneous and non-homogeneous participating media,” J. Quant. Spectrosc. Radiat. Transfer **73**, 181–194 (2002). [CrossRef]

*et al*. [4

**73**, 181–194 (2002). [CrossRef]

*et al*. [1

1. Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer **16**, 289–296 (2002). [CrossRef]

*et al*. [11

11. Z. Guo and K. Kim, “Ultrafast-laser-radiation transfer in heterogeneous tissues with the discrete-ordinates method,” Appl. Opt. **42**, 2897–2905 (2003). [CrossRef] [PubMed]

11. Z. Guo and K. Kim, “Ultrafast-laser-radiation transfer in heterogeneous tissues with the discrete-ordinates method,” Appl. Opt. **42**, 2897–2905 (2003). [CrossRef] [PubMed]

*et al*. [12

12. J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional trnasient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transfer **86**, 299–313 (2004). [CrossRef]

*et al*. [13

13. A. Sawetprawichkul, P. F. Hsu, and K. Mitra, “Parallel computing of three-dimensional monte carlo simulation of transient radiative transfer in participating media,” in *Proceedings of the 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conferece*, (American Institute of Aeronautics and Astronautics, St. Louse, Missouri, 2002), pp. 1–10.

*et al*. [14

14. J. V. P. de Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer prolems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer **73**, 55–62 (2002). [CrossRef]

15. C. C. Handapangoda, M. Premaratne, L. Yeo, and 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). [CrossRef]

**16**, 289–296 (2002). [CrossRef]

*et al*. in [1

**16**, 289–296 (2002). [CrossRef]

## 2. Formulation

### 2.1. For three-dimensional PTE

**16**, 289–296 (2002). [CrossRef]

*I*(

*x,y, z, s, t*) is the light intensity (radiance) with units

*W.m*

^{-2}.

*sr*

^{-1}.

*Hz*

^{-1}, (

*x,y, z*) are the cartesian coordinates, s is the local solid angle,

*µ*,

*ξ*and

*η*are direction cosines such that

*µ*=cos

*θ, ξ*=sin

*θ*cos

*ϕ, η*=sin

*θ*sin

*ϕ*and

*t*is time;

*σ*(

_{t}*x,y,z*) and

*σ*(

_{s}*x,y,z*) represent the position dependent attenuation and scattering coefficients, respectively. The speed of light in the medium is

*v*,

*P*(s′; s) is the phase function and

*F*(

*x,y, z, s, t*) is the source term. Since we consider a constant refractive index, the speed of light in the medium is a constant. In this paper we assume that there are no internal sources inside the medium and hence

*F*=0. Figure 1 shows the incident direction,

**s**, the scattered direction, s′, and the scattering angle. Figure 2 shows the relationship between the Cartesian and spherical coordinates, showing the zenith and azimuthal angles.

*I*(

*x,y, z, s,τ*) in a summation.

*δ*is the Kronecker delta [18]. Laguerre polynomials are causal [19

_{nm}19. C. J. Rivero-Moreno and S. Bres, “Video spatio-temporal signatures using polynomial transforms,” Lect. Notes Compt. Sci. **3736**, 50–59 (2005). [CrossRef]

*I*, as a summation of

*N*Laguerre polynomials;

^{∞}

_{0}

*L*(

_{n}*τ*)

*L*(

_{m}*τ*)

*e*

^{-τ}

*dτ*=

*δ*. Using Eq. (5) in Eq. (4) and taking moments (i.e. multiplying by

_{mn}*L*(

_{n}*τ*)

*e*and integrating over [0,∞)), the time dependence can be removed resulting in

^{-τ}*N*uncoupled equations, one for each Laguerre coefficient,

*B*.

_{n}*n*=1, …,

*N*. Thus, the transient photon transport equation has been reduced to a set of uncoupled steady state photon transport equations. Equation (6) can be solved using the discrete ordinates method as detailed in [16], and outlined below.

*L*coupled equations for each Laguerre coefficient.

*w*are quadrature weights,

_{j}*i*=1, …,

*L*and

*n*=1, …,

*N*. We divide the tissue specimen into infinitesimally small control volumes and apply the discrete ordinates method. We integrate Eq. (7) over a control volume:

*V*=Δ

*x*Δ

*y*Δ

*z*is the volume of the control volume, (

*B*)

_{n}*and (*

_{xu}*B*)

_{n}*are average values of*

_{xd}*B*over the faces

_{n}*A*and

_{xu}*A*of the control volume, respectively. Similarly,

_{xd}*γ*,

_{x}*γ*and

_{y}*γ*can be determined using the scheme proposed by Lathrop [1

_{z}**16**, 289–296 (2002). [CrossRef]

22. K. D. Lathrop, “Spatial differnecing of the transport equation: positive vs accuracy,” J. Comput. Phys. **4**, 475–498 (1968). [CrossRef]

*B*)

^{i}_{n}*, (*

_{xd}*B*)

^{i}_{n}*and (*

_{yd}*B*)

^{i}_{n}*,*

_{zd}*V*, we get

23. W. A. Fiveland, “Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer **2**, 309–316 (1988). [CrossRef]

*x*, Δ

*y*and Δ

*z*. According to this scheme

*B*)

^{i}_{n}*, (*

_{xu}*B*)

^{i}_{n}*and (*

_{yu}*B*)

^{i}_{n}*values for each corner control volume are given by the Laguerre coefficients of the boundary condition. Hence, the centre values, (*

_{zu}*B*)

^{i}_{n}*p*, can be determined from Eq. (17). Once (

*B*)

^{i}_{n}*is calculated (*

_{p}*B*)

^{i}_{n}*, (*

_{xd}*B*)

^{i}_{n}*and (*

_{yd}*B*)

^{i}_{n}*can be determined using Eq. (12) to Eq. (14). Also, (*

_{zd}*S*)

^{i}*, which is initially assumed to be equal to zero, is updated using the calculated (*

_{p}*B*)

^{i}_{n}*value from Eq. (16). (*

_{p}*B*)

^{i}_{n}*, (*

_{xd}*B*)

^{i}_{n}*and (*

_{yd}*B*)

^{i}_{n}*of the current control volume are equal to (*

_{zd}*B*)

^{i}_{n}*, (*

_{xu}*B*)

^{i}_{n}*and (*

_{yu}*B*)

^{i}_{n}*of the adjacent control volumes, respectively. A more detailed description on this method can be found in [16].*

_{zu}*I*(

*x,y, z, s, t*), using a Laguerre basis.

*B*)

^{i}_{n}*, (*

_{xu}*B*)

^{i}_{n}*and (*

_{yu}*B*)

^{i}_{n}*for this control volume are known from the boundary condition.*

_{zu}*S*)

^{i}*p*is assumed to be zero initially for all control volumes.

*B*)

^{i}_{n}*, (*

_{xu}*B*)

^{i}_{n}*and (*

_{yu}*B*)

^{i}_{n}*for adjacent control volumes.*

_{zu}*S*)

^{i}*p*is converged.

*I*.

### 2.2. For two-dimensional PTE

*I*, is expanded using a Laguerre basis using Eq. (5) and take moments as done for the three-dimensional case. This results in the two-dimensional steady state PTE:

### 2.3. For one-dimensional PTE

*n*=1, …,

*N*. Then we discretize the solid angle, s, of Eq. (23) using the discrete ordinates method, which results in

*i*=1, …,

*L*and

*n*=1, …,

*N*. Thus, there are

*L*coupled equations for each Laguerre coefficient,

*B*and Eq. (24) corresponds to the

_{n}*i*

^{th}discrete ordinate of

*s*. We can write this set of equations in matrix form as;

**B**

*=[*

_{n}*B*(

_{n}*z*,

*s*]

^{i}_{L,1},

**P**=[

*P*(

**s**

*;*

^{j}**s**

*)]*

^{i}*, and,*

_{L,L}**A**is a

*L*by

*L*diagonal matrix with diagonal elements

*µ*. The matrix

^{i}**W**is also a

*L*by

*L*diagonal matrix with diagonal elements

*w*.

_{j}^{th}order Runge-Kutta-Fehlberg(RKF) method [24]. A more detailed description about solving the one-dimensional PTE using this technique (the Laguerre Runge-Kutta-Fehlberg method) can be found in our paper [15

15. C. C. Handapangoda, M. Premaratne, L. Yeo, and 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). [CrossRef]

## 3. Numerical results and discussion

^{™}. First, we validate the proposed technique for special cases, for which the results are intuitive and readily computable using the one-dimensional PTE. Then we provide a comparison for all the three cases; one, two and three dimensions, with results obtained from the technique proposed by Guo

*et al*. [1

**16**, 289–296 (2002). [CrossRef]

**16**, 289–296 (2002). [CrossRef]

**16**, 289–296 (2002). [CrossRef]

### 3.1. Normalization of units

*I*, and thus arbitrary units for intensity can be used. Also, it is a first order, linear differential equation in time and thus time units can also be selected to enhance the numerical accuracy. It is very clear that input pulse parameters influence the numerical accuracy and efficiency of the algorithm. The input pulse was taken to be a Gaussian pulse described mathematically by

*T*=

_{s}*T*/1.5. We choose this normalization factor due to the fact that the Laguerre approximation of the Gaussian pulse is very accurate for pulses with

*T*=1.5 or greater. This effect is shown in Fig. 5 and Fig. 6. Figure 5 presents a comparison of direct Laguerre approximation (i.e. without scaling with

*T*) with the exact plot of the Gaussian pulse, for four different pulse widths. This shows that Laguerre approximation for pulses with

_{s}*T*=1 and

*T*=1.5 are very accurate. However, for the

*T*=1 case, a very small bump appears for 6.5<

*x*<7.5 (see Fig. 5b). Such inaccuracies can be always eliminated by choosing larger number of Laguerre coefficients. However, if the computational efficiency is also of concern, then better scaling can be used to reduce the required number of Laguerre coefficients. Therefore, we have chosen the

*T*=1.5 and

*t*

_{0}=4 approximation. A pulse with different

*T*and

*t*

_{0}values should be scaled and shifted accordingly to obtain

*T*=1.5 and

*t*

_{0}=4 before the Laguerre fit is carried out. The results obtained by the simulations should be scaled and shifted back (reversing this process). This idea is illustrated in Fig. 6. Four different Gaussian pulses with different

*T*and

*t*

_{0}values are scaled and shifted (up or down) to obtain the Gaussian pulse with

*T*=1.5 and

*t*

_{0}=4 and a Laguerre fit is carried out. Then the scaling and shifting is reversed and the resulting fit is plotted with the exact plot of the corresponding pulse. It is clearly shown in Fig. 6 that this scaled Laguerre fit produces very accurate approximations.

*T*to the width of that Gaussian pulse divided by 1.5. With the Laguerre fit it is not possible to approximate any pulse shape to an arbitrary accuracy because of numerical considerations. Therefore, in practice, there is a finite domain in which this approximation is valid. However, as it can be seen in Fig. 6 this domain expands as

_{s}*T*is increased. However, since in this algorithm the Laguerre polynomials are propagated with the pulse, the approximation will always be accurate in the domain we are interested in (i.e. the domain in which the Laguerre fit is accurate extends few time units beyond the required observation point). A discussion on the observation window in which the Laguerre approximation is accurate can be found in [15

15. C. C. Handapangoda, M. Premaratne, L. Yeo, and 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). [CrossRef]

*Z*=

_{s}*v*×

*T*̄. Here,

*T*̄ can be chosen to suit the particular application. However, these scaling factors should be chosen carefully so that the matrices that are used remain well-conditioned. For the simulations presented in this paper, without loss of generality, we have chosen

*T/T*̄=1.5 so that

### 3.2. Computational complexity

*x*̄<+1,-1<

*y*̄<+1 and 0<

*z*̄<2. However, we have assumed refractive index-matched boundaries in all the three dimensions. Therefore, in order to minimize effects from internal reflections at the boundaries the simulation volume was taken to be larger than the observation volume in all the three dimensions (-5<

*x*̄<+5,-5<

*y*̄<+5 and 0<

*z*̄<5). The total number of voxels used in the finite volume computations was 108000. The number of voxels considered in the observation volume was 1728. The number of discrete ordinates taken for the simulations was 80. We have used the discrete ordinates for

*S*

_{8}approximation given in Table 16.1 of reference [16]. The two main drawbacks of the discrete ordinates method are false scattering and the ray effect [4

**73**, 181–194 (2002). [CrossRef]

*x*̄=0,

*y*̄=0,

*z*̄=0). This is the model used for the simulations presented in this paper. For all the simulations, we have used

*T*̄=1.5 and

*v*̄=1 and the normalized thickness of the tissue layer, at which the results were obtained, to be

*z*̄=2. The isotropic phase function (

*P*(

**s**′;

**s**)=1) was used in the simulations. However, the proposed method is able to handle any other phase function.

*σ*and

_{s}*σ*in the PTE are set to zero. With this setup, there should be neither a decay in the intensity, nor scattering to other directions. Figure 7 shows the variation of radiance with time, along the direction of incidence (ie.

_{t}*µ*=1), at

*z*̄=2. As expected, at

*z*̄=2, we have obtained the same Gaussian pulse, without any loss, but with the corresponding time delay.

*µ*, at

*z*=2. As expected, a decayed radiance profile exists along the incident direction,

*µ*=0.5, but since there was no scattering no radiance values are seen along other directions.

**16**, 289–296 (2002). [CrossRef]

*µ*=1.

*z*̄=2, for the one-dimensional PTE using the Laguerre DOM and the Transient DOM.

*z*̄=2, for the two-dimensional PTE using the two methods. In Fig. 11 one can observe that the time shift of the peak value of the irradiance profile increases as

*x*̄ moves away from

*x*̄=0. The reason for this is that light takes the shortest time to reach (

*x*̄=0,

*z*̄=2) point and the time increases as the point moves away from

*x*̄=0 central axis. However, in the simulation obtained using Transient DOM shown in Fig. 12 this physical phenomenon is not clearly visible.

*z*̄=2, for the three-dimensional PTE using Laguerre DOM. In these figures the value of the normalized irradiance (ranging from 0 to 16×10

^{-4}) is represented by the color scale. Figures 13 to 16 were obtained using the same data set. Figure 13 shows a slice plane at

*y*̄=0. Thus, in Fig. 13 the variation of irradiance with time along the x-axis on (

*y*̄=0,

*z*̄=2) plane is shown. As expected, the light reaches (

*x*̄=0,

*y*̄=0) point first as rays traveling at an angle to the central axis (

*x*̄=0,

*y*̄=0) take longer to reach

*z*̄=2 plane (because the diagonal distance is more than the central axis distance and the speed is constant). For the same reason the irradiance profile decays first at (

*x*̄=0,

*y*̄=0) and later at other points. The more we move away from the central axis, the longer it takes the irradiance profile to decay.

*x*and

*y*axes on

*z*̄=2 plane at

*t*̄=6. With isotropic scattering and normal incidence we have obtained a circular spatial distribution on the

*xy*plane. From Fig. 13 it can be seen that at

*t*̄=6 points around (

*x*̄=0,

*y*̄=0) are receiving the second half of the gaussian pulse (i.e. the maximum irradiance has been reached earlier) and points further away from the central axis are receiving light corresponding to the maximum or the first half of the incident pulse. This phenomenon is reflected in Fig. 14. However, in the simulation result shown in Fig. 13 and Fig. 14, an uncharacteristically low number of photons can be seen along the central axis and the majority of diffuse photons appear off axis resulting in a well defined shape. The reason for this phenomenon is not very clear yet, and we intend to investigate this issue using Monte-Carlo techniques in the future. Figures 15 and 16 show that the irradiance profile at

*z*̄=2 is symmetrical along

*x*and

*y*axes due to isotropic scattering and normal incidence.

## 4. Conclusion

## References and links

1. | Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer |

2. | K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. |

3. | G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. part I: A general method of solution,” -J. Atmos. Sci. |

4. | Z. M. Tan and P. F. Hsu, “Transient radiative transfer in three-dimensional homogeneous and non-homogeneous participating media,” J. Quant. Spectrosc. Radiat. Transfer |

5. | C. C. Handapangoda, M. Premaratne, D. M. Paganin, and P. R. D. S. Hendahewa, “Technique for handling wave propagation specific effects in biological tissue: Mapping of the photon transport equation to Maxwell’s equations,” Opt. Express |

6. | K. W. Johnson, J. J. Mastrototaro, D. C. Howey, R. L. Brunelle, P. L. Burden-Brady, N. A. Bryan, C. C. Andrew, H. M. Rowe, D. J. Allen, B. W. Noffke, W. C. McMahan, R. J. Morff, D. Lipson, and R. S. Nevin, “In vivo evaluation of an electroenzymatic glucose sensor implanted in subcutaneous tissue,” Biosens. Bioelectron. |

7. | D. Moatti-Sirat, F. Capron, V. Poitout, G. Reach, D. S. Bindra, Y. Zhang, G. S. Wilson, and D. R. Thevenot, “Towards continuous glucose monitoring: in vivo evaluation of a miniaturized glucose sensor implanted for several days in rat subcutaneous tissue,” Diabetologia |

8. | V. Poitout, D. Moatti-Sirat, G. Reach, Y. Zhang, G. S. Wilson, F. Lemonnier, and J. C. Klein, “A glucose monitoring system for on line estimation in man of blood glucose concentration using a miniaturized glucose sensor implanted in the subcutaneous tissue and a wearable control unit,” Diabetologia |

9. | C. C. Handapangoda, M. Premaratne, and D. M. Paganin, “Simulation of a device concept for noninvasive sensing of blood glucose levels,” in |

10. | C. C. Handapangoda, M. Premaratne, and D. M. Paganin, “Simulation of embedded photonic crystal structures for blood glucose measurement using Raman spectroscopy,” |

11. | Z. Guo and K. Kim, “Ultrafast-laser-radiation transfer in heterogeneous tissues with the discrete-ordinates method,” Appl. Opt. |

12. | J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional trnasient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transfer |

13. | A. Sawetprawichkul, P. F. Hsu, and K. Mitra, “Parallel computing of three-dimensional monte carlo simulation of transient radiative transfer in participating media,” in |

14. | J. V. P. de Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer prolems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer |

15. | C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. |

16. | M. F. Modest, “The method of discrete ordinates” in |

17. | M. Abramomitz and I. A. Stegun, “Orthogonal polynomials” in |

18. | G. A. Korn and T. M. Korn, “Tensor analysis” in |

19. | C. J. Rivero-Moreno and S. Bres, “Video spatio-temporal signatures using polynomial transforms,” Lect. Notes Compt. Sci. |

20. | G. A. Korn and T. M. Korn, “Special functions” in Mathematical handbook for scientists and engineers: definitions, theorems and formulas for reference and review, 2nd ed., (Dover, New York, 2000), pp. 848–856. |

21. | S. Chandrasekhar, “Quadrature formulae” in |

22. | K. D. Lathrop, “Spatial differnecing of the transport equation: positive vs accuracy,” J. Comput. Phys. |

23. | W. A. Fiveland, “Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer |

24. | S. C. Chapra and R. P. Canale, “Runge-Kutta methods” in |

**OCIS Codes**

(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine

(170.6930) Medical optics and biotechnology : Tissue

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: September 30, 2009

Revised Manuscript: October 29, 2009

Manuscript Accepted: December 4, 2009

Published: December 7, 2009

**Virtual Issues**

Vol. 5, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Chintha C. Handapangoda and Malin Premaratne, "Implicitly causality enforced solution of multidimensional transient photon transport equation," Opt. Express **17**, 23423-23442 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23423

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### References

- Z. Guo, and S. Kumar, "Three-dimensional discrete ordinates method in transient radiative transfer," J. Thermophys. Heat Transfer 16, 289-296 (2002). [CrossRef]
- K. F. Evans, "The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998). [CrossRef]
- G. L. Stephens, "Radiative transfer through arbitrarily shaped optical media. part I: A general method of solution," J. Atmos. Sci. 45, 1818-1836 (1988). [CrossRef]
- Z. M. Tan, and P. F. Hsu, "Transient radiative transfer in three-dimensional homogeneous and non-homogeneous participating media," J. Quantum Spectrosc. Radiat. Transfer 73, 181-194 (2002). [CrossRef]
- C. C. Handapangoda, M. Premaratne, D. M. Paganin, and P. R. D. S. Hendahewa, "Technique for handling wave propagation specific effects in biological tissue: Mapping of the photon transport equation to Maxwell’s equations," Opt. Express 16, 17792-17807 (2008). [CrossRef] [PubMed]
- K. W. Johnson, J. J. Mastrototaro, D. C. Howey, R. L. Brunelle, P. L. Burden-Brady, N. A. Bryan, C. C. Andrew, H. M. Rowe, D. J. Allen, B. W. Noffke, W. C. McMahan, R. J. Morff, D. Lipson, and R. S. Nevin, "In vivo evaluation of an electroenzymatic glucose sensor implanted in subcutaneous tissue," Biosens. Bioelectron. 7, 709-714 (1992). [CrossRef] [PubMed]
- D. Moatti-Sirat, F. Capron, V. Poitout, G. Reach, D. S. Bindra, Y. Zhang, G. S. Wilson, and D. R. Thevenot, "Towards continuous glucose monitoring: in vivo evaluation of a miniaturized glucose sensor implanted for several days in rat subcutaneous tissue," Diabetologia 35, 224-230 (1992). [CrossRef] [PubMed]
- V. Poitout, D. Moatti-Sirat, G. Reach, Y. Zhang, G. S. Wilson, F. Lemonnier, and J. C. Klein, "A glucose monitoring system for on line estimation in man of blood glucose concentration using a miniaturized glucose sensor implanted in the subcutaneous tissue and a wearable control unit," Diabetologia 36, 658-663 (1993). [CrossRef] [PubMed]
- C. C. Handapangoda, M. Premaratne, and D. M. Paganin, "Simulation of a device concept for noninvasive sensing of blood glucose levels," in Information and Automation for Sustainability, 2007. ICIAFS 2007. Third International Conference, (Melbourne, Australia, December 2007), pp. 31-34.
- C. C. Handapangoda, M. Premaratne, and D. M. Paganin, "Simulation of embedded photonic crystal structures for blood glucose measurement using Raman spectroscopy," 2008 International conference on Nanoscience and Nanotechnology, (Melbourne, Australia, February 2008).
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