## Implementation of the equation of radiative transfer on block-structured grids for modeling light propagation in tissue |

Biomedical Optics Express, Vol. 1, Issue 3, pp. 861-878 (2010)

http://dx.doi.org/10.1364/BOE.1.000861

Acrobat PDF (3883 KB)

### Abstract

We present the first algorithm for solving the equation of radiative transfer (ERT) in the frequency domain (FD) on three-dimensional block-structured Cartesian grids (BSG). This algorithm allows for accurate modeling of light propagation in media of arbitrary shape with air-tissue refractive index mismatch at the boundary at increased speed compared to currently available structured grid algorithms. To accurately model arbitrarily shaped geometries the algorithm generates BSGs that are finely discretized only near physical boundaries and therefore less dense than fine grids. We discretize the FD-ERT using a combination of the upwind-step method and the discrete ordinates (*S _{N}
*) approximation. The source iteration technique is used to obtain the solution. We implement a first order interpolation scheme when traversing between coarse and fine grid regions. Effects of geometry and optical parameters on algorithm performance are evaluated using numerical phantoms (circular, cylindrical, and arbitrary shape) and varying the absorption and scattering coefficients, modulation frequency, and refractive index. The solution on a 3-level BSG is obtained up to 4.2 times faster than the solution on a single fine grid, with minimal increase in numerical error (less than 5%).

© 2010 OSA

## 1. Introduction

3. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. **47**(2), 131–146 (1995). [CrossRef] [PubMed]

11. M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume ﬁnite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B **48**(5), 477–497 (2005). [CrossRef]

12. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. **37**(7), 1531–1560 (1992). [CrossRef] [PubMed]

15. M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency domain fluorescence spectroscopy in biological tissues,” Appl. Opt. **33**(10), 1963–1974 (1994). [CrossRef]

7. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. **43**(5), 1285–1302 (1998). [CrossRef] [PubMed]

*a priori*information is central to the finite differences method, where any given grid point is assumed to have neighboring grid points that also connect to form cuboids. Conversely, the connectivity of unstructured grid elements must be explicitly provided because unstructured grid nodes connect to form elements that vary in size, shape, and orientation [16]. Thus, algorithms that solve the ERT on unstructured grids are more complex than algorithms on structured grids because node connectivity information must be explicitly provided and processed. Generating unstructured grids can in its-self be an arduous task, often requiring third party applications, while generating structured grids is a relatively simple task [16]. Numerical algorithms on both unstructured and structured grids have been developed for solving the ERT. For example, the unstructured finite-element method has been used by Arridge et al. [13

13. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. **20**(2), 299–309 (1993). [CrossRef] [PubMed]

11. M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume ﬁnite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B **48**(5), 477–497 (2005). [CrossRef]

17. J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. **33**(12), 4685–4700 (2006). [CrossRef] [PubMed]

9. H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. **25**(1), 015010 (2009). [CrossRef]

8. K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. **29**(6), 578–580 (2004). [CrossRef] [PubMed]

18. X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. **46**(10), 1624–1632 (2007). [CrossRef] [PubMed]

6. A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. **72**(5), 691–713 (2002). [CrossRef]

19. A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. **28**(12), 1019–1021 (2003). [CrossRef] [PubMed]

21. A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. **37**(1), 329–338 (2010). [CrossRef] [PubMed]

21. A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. **37**(1), 329–338 (2010). [CrossRef] [PubMed]

22. J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. **78**(8), 083706 (2007). [CrossRef] [PubMed]

23. B. W. Pogue and G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. **37**(31), 7429–7436 (1998). [CrossRef] [PubMed]

24. R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. **14**(11), 1665–1678 (1979). [CrossRef]

25. M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. **53**(3), 484–512 (1984). [CrossRef]

26. W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. **144**(3-4), 327–369 (1997). [CrossRef]

27. M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. **82**(1), 64–84 (1989). [CrossRef]

28. J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Reﬁnement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. **139**(2), 380–398 (1998). [CrossRef]

29. A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. **33**(5), 1299–1310 (2006). [CrossRef] [PubMed]

7. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. **43**(5), 1285–1302 (1998). [CrossRef] [PubMed]

18. X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. **46**(10), 1624–1632 (2007). [CrossRef] [PubMed]

30. A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. **16**(1), 79–88 (2005). [CrossRef] [PubMed]

31. A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. **64**(2), 239–245 (2005). [CrossRef] [PubMed]

## 2. Methods

### 2.1. Grid Generation

21. A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. **37**(1), 329–338 (2010). [CrossRef] [PubMed]

*x*, Δ

*y*, and Δ

*z*). It is necessary that the resolution be small enough to capture all physical effects. In our work a grid spacing of 1/10 μ

_{s}is generally sufficient. Next, the code finds all grid points within the discretized domain that lie within the volume enclosed by the boundary.

*i*,

*j*,

*k*) be

**=**

*n**n*

_{x}**ê**+

_{1}*n*

_{y}**+**

*ê*_{2}*n*

_{z}**with**

*ê*_{3}*n*<0. Then, nodes (

_{x}*i*-1,

*j*,

*k*) and (

*i*+ 1,

*j*,

*k*) are classified as “exterior” and “interior” points, respectively. Furthermore, all grid points to the right of (

*i*+ 1,

*j*,

*k*) are also “interior” points (i.e. (

*i*+ 2,

*j*,

*k*), (

*i*+ 3,

*j*,

*k*), etc.). This labeling scheme continues until a new boundary point is encountered, at which point the process is restarted. The result of this subroutine is a Cartesian grid where all nodes are classified as exterior, boundary, or interior points. The resulting grid is used as input to the BSG generating subroutine. An overview of the major aspects of the BOR method is presented in Fig. 3 .

*x*) as provided by the user. A subroutine then superimposes the next coarsest grid (Δ

_{f}*x*= 2Δ

*x*) over the fine grid. Next, the algorithm removes coarse grid points on the boundary of the computational domain. In the final step, all fine grid points within the coarse grid are removed from the computational domain. This process is repeated to generate higher-order BSGs. The output of the BSG generator contains the following information for each grid point: 1) location: exterior, main boundary, or interior, 2) grid level, and 3) type of fine/coarse boundary. The term “fine/coarse boundary” refers to points in the “active” domain at the union of coarse and fine grid segments (Fig. 4a ). The C + + pseudo code for the algorithm generating a two-dimensional

_{f}*n-level*grid is as follows (finest grid = level 0, coarsest grid = level

*n*):

- 1. Generate grid hierarchy: (for
*m = 0*;*m < = n*;*m + +*)- a. Determine all level
*m*grid points in exterior, main boundary, or interior. - b. If
*m > 0*, for all interior grid points level*m*i. Delete all points level*m*on the main numerical boundary.ii. Delete all points level*m-1*interior to a cell of 4 level*m*points.iii. Delete all points level*m-1*interior to a cell of 6 level*m*points.

- 2. Classify active grid points into four categories
- a. On main boundary
- b. On fine/coarse boundary (Fig. 4a)i. Missing west neighborii. Missing south neighboriii. Missing east neighboriv. Missing north neighborv. Coarse point (black dot)c. Interior point, in active domain, and not on fine/coarse boundary.d. Interior point not in active domain (black diamond, Fig. 4b).

### 2.2 Light Propagation Model

32. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. **32**(4), 607–616 (1993). [CrossRef] [PubMed]

^{−1}cm

^{−2}) [1,4]. A second important quantity in clinical applications is the partial current

22. J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. **78**(8), 083706 (2007). [CrossRef] [PubMed]

23. B. W. Pogue and G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. **37**(31), 7429–7436 (1998). [CrossRef] [PubMed]

33. U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. **79**(3), 034301 (2008). [CrossRef] [PubMed]

30. A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. **16**(1), 79–88 (2005). [CrossRef] [PubMed]

34. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. **8**(1), 1–33 (2006). [CrossRef] [PubMed]

37. O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods **48**(2), 139–145 (2009). [CrossRef] [PubMed]

*ERT 1*) is used to model the excitation field inside the medium due to a modulated boundary source. The fluorophore inside the medium is modeled as an internal source (

*Q*). Fluorophore emission is a function, among other things, of the excitation field. Thus, a second ERT is used to model the fluorophore emission field (

*ERT 2*).

*ERT 1*is obtained from Eqs. (1),2) by defining the boundary source as

*Q*) to zero, and defining the total attenuation coefficient as

*ERT 2*is obtained by setting the boundary source (

*S*) to zero and defining the fluorescent source as a function of the excitation field. This relationship is given by

*S*) to zero and defining the source of bioluminescence (

*Q*). The total attenuation coefficient is given by Eq. (8).

### 2.3 Discretization of Frequency Domain ERT

*S*) for the angular variable [4,20

_{N}20. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. **202**(1), 323–345 (2005). [CrossRef]

*2.3.1. Discretization on single grid:*The first step in discretizing the ERT is to use the discrete-ordinates method to replace the integral term with the

*extended trapezoidal rule*[5

5. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. **26**(8), 1698–1707 (1999). [CrossRef] [PubMed]

*k*is the ordinate number,

*ψ*is the radiance in the

_{k}*k*ordinate (where

^{th}*k*denotes the

*k*discrete angle), and

^{th}*ω*is a predetermined ordinate weight with full level symmetry [40,41]. The integral term does not require special treatment for implementation on BSGs.

_{k}*eight*possible numerical schemes, one for each octant in the three dimensional Cartesian coordinate system. For example, when all directional cosines are positive, the upwind-step method requires an Euler step in the negative

*x*-,

*y*-, and

*z*-axis. For this example, the discretization of Eqs. (1),2) is given by

*p*in Eq. (10) is the Henyey-Greenstein phase function and is given by

_{kk’}*g*is the anisotropy factor. The radiance,

*ψ*, can be solved from Eqs. (10),11) with any number of established algorithms. In this work we implement the source iteration technique (i.e. the matrix-free point-wise

*Gauss-Seidel Method*). The fluence is given by

*extended trapezoidal rule*, to Eq. (3) [20

20. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. **202**(1), 323–345 (2005). [CrossRef]

*2.3.2. Discretization on block-structured grid*: To solve the FD-ERT on BSGs, the Euler step used in the step-method, Eq. (10), must be changed to a step of variable size and become dependent on the local grid. For example, Δ

*x*becomes Δ

*x*where

_{ijl}*ijl*denotes the current grid point and its value is determined by the size of the local grid. Implementing a finite differences numerical scheme with a variable Euler step in a matrix-free formulation on BSGs is a complicated endeavor and requires great care. The main difficulty arises when solving for the FD-ERT on mesh points on a fine/coarse grid boundary (i.e. points that straddle both a coarse grid and a fine grid region).

*i*,

*j*,

*l*) to have four neighbors in two-dimensions and six neighbors in three-dimensions. However, mesh points on a fine/coarse boundary do not always have a full set of neighbors. This problem can be overcome by adjusting the numerical scheme for each individual fine/coarse boundary point, or by creating the missing point so that the normal scheme is applicable. In this work we consider the latter option. The missing point (virtual point) is constructed through interpolation using neighboring points. For illustration consider the two-dimensional example in Fig. 5a . Here, five cases must be considered independently for a given octant. For example, it is clear from Eq. (10) that when all directional cosines are positive, points

*ii*and

*iii*will require creating a virtual point interior to the coarse grid. However, points

*i*and

*iv*have all neighbors necessary to complete the stencil. The fifth case (black dots in Fig. 5a) can be treated as points on the fine grid or on the coarse grid. By treating these boundary points as coarse grid points we assure that they will always have a complete set of neighbors and no further special treatment is necessary. The set of points used to create the virtual point varies according to the type of boundary point. The different types of boundary points can be reduced to five in two dimensions (Fig. 5a) and nineteen in three dimensions.

*i*,

*j*,

*l*), represented by the black triangle. It is clear from Eq. (10) that the solution at grid point (

*i*-1,

*j*,

*l*), represented by a black dot, is necessary to solve the equation. However, that grid point does not exist and must be created by averaging the solution at the four neighboring grid points denoted by white dots.

### 2.4. Numerical Phantoms

*x*. With this setup we ensure the number of photons injected into the phantom is independent of the number of boundary points. The source density is 8 × 10

^{9}photons cm

^{−2}sr

^{−1}. The optical properties of the disk phantom are varied and they are summarized in Tables 2 and 3 .

^{9}photons cm

^{−2}sr

^{−1}. The specific optical properties used are summarized in Tables 2 and 3.

*x*= 2/256, 2/128, 2/64, 2/32 cm (where Δ

*x*= Δ

*y*). For each grid refinement, we determine the solution to the ERT on 1-, 2-, and 3-level BSGs and compare them to the benchmark solution. The benchmark solution for these simulations is the solution on the finest single grid (Δ

*x*= 2/256 cm). In addition, we vary the optical parameters (μ

_{a}, μ

_{s}), modulation frequency (ω), and refractive index (

*n*) of each phantom and analyze the performance of the algorithm for each case.

### 2.5. Quantification of computation time and solution accuracy

*x*= 2/256 cm for all phantoms introduced in Section 2.4. The relative speed up (RSU) achieved with the BSG algorithm is used to compare the computational cost of a solution obtained on a BSG (

*T*) and on a single fine grid (

_{BSG}*T*), where the solutions we compare are partial current measurements at the boundary as defined by Eq. (4). All simulations were performed on a computer with a 2.93 GHz Intel Core 2 Duo processor.

_{F}5. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. **26**(8), 1698–1707 (1999). [CrossRef] [PubMed]

9. H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. **25**(1), 015010 (2009). [CrossRef]

*f*and

_{i}^{a}*f*are the approximate and benchmark solutions, respectively. Only partial currents are compared and the MPE is reported. This is done because cross-sectional fluence solutions on coarse grids cannot, in general, be directly compared to solutions computed on fine grids since the numerical boundary of each grid is unique. This point is perhaps best illustrated by referring to Fig. 2b-d.

_{i}^{b}## 3. Results

*x*) always refers to the spacing of the finest grid (i.e. the grid near the boundaries). As an example, a 2-level BSG with grid spacing Δ

*x*has an embedded section of coarse grid points whose spacing is 2Δ

*x*. The terms 1L, 2L, and 3L in all results tables refer to the number of grid levels in the BSGs.

### 3.1 Block Structured Grids

### 3.2 Disk Phantom

*x*= 2/256, 2/128, and 2/64 cm. Solutions were computed on 1-, 2-, and 3- level BSGs with Δ

*x*= 2/256 cm. It is clear that, compared to solutions on single coarse grids, the solution computed on BSGs better approximates the true solution to fluorescence excitation and emission. The solution to fluorescence emission is less accurate than the solution to the excitation problem. This occurs because the numerical error from the excitation solution propagates into the emission solution. Thus, the solution to emission computed on BSGs is more accurate than the solution to emission computed on single coarse grids because the error in the solution to excitation on BSGs is lower.

*x*= 2/64 cm, the MPE of the solution obtained from a single grid is 3.3%. The solution on a 2-level BSG has MPE of 7.4% and is obtained 2.1 times faster than the solution computed on a single grid. As was explained before, the interior of the 2-level BSG has Δ

*x*= 2/32 cm. The MPE of the solution on a single grid with Δ

*x*= 2/32 cm is 18.9%. Therefore, the solution computed on a 2-level BSG is twice as accurate. Results from the other cases are similar.

*x*= 2/32 cm) in cases 1 and 4 are similar, i.e. the refractive index mismatch at the boundary of the circular phantom does not have a significant impact on the error in the solution. This is unexpected.

### 3.3 Small Animal Phantom

*x*= 2/32 cm. In these cases the solution on 2-level BSGs is obtained only 1 time faster than the single grid solution and the increase in error is very large. This occurs because the interior of the BSG is very coarse (Δ

*x*= 2/16 cm) causing the numerical error to be large.

### 3.4 Three dimensional phantoms

*x*= 2/128 cm) is only 0.28%, while the solution computed on a single coarse grid (Δ

*x*= 2/64 cm) 2.94%. Thus, solving the FD-ERT on a 2-level grid instead of a coarse grid reduced the error by 2.66%. Similarly, the solution computed on a 3-level grid (Δ

*x*= 2/128 cm) is 30.21%, while the error in the solution computed on the coarsest grid (Δ

*x*= 2/32 cm) is 78.25%. The error in the solution is reduced by 48.29%. In addition, using BSGs reduces computation time. Solutions on 2- and 3- level grids are obtained 1.5 and 3.0 times faster than the solution on the fine grid, respectively.

## 4. Discussions and Conclusion

*x*). In addition, there is numerical error due to the

*S*approximation to the integral terms (it decreases with increasing order of the

_{N}*S*method). Error due to poorly resolved boundaries arises when the single Cartesian grid does not accurately approximate the physical boundary. This error is particularly large when a coarse grid is used to approximate curved geometries.

_{N}## Acknowledgments

## References and links

1. | A. J. Welch, and M. J. C. Van Gemert, |

2. | S.A. Prahl, M. Keijzer, S.L. Jacques, A.J. Welch, “A Monte Carlo model of light propagation in tissue,” in |

3. | L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. |

4. | A. D. Klose, “Radiative Transfer of Luminescence in Biological Tissue”, in |

5. | A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. |

6. | A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. |

7. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. |

8. | K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. |

9. | H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. |

10. | O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. |

11. | M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume ﬁnite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B |

12. | S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. |

13. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. |

14. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

15. | M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency domain fluorescence spectroscopy in biological tissues,” Appl. Opt. |

16. | V. D. Liseikin, |

17. | J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. |

18. | X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. |

19. | A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. |

20. | A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. |

21. | A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. |

22. | J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. |

23. | B. W. Pogue and G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. |

24. | R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. |

25. | M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. |

26. | W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. |

27. | M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. |

28. | J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Reﬁnement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. |

29. | A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. |

30. | A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. |

31. | A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. |

32. | B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. |

33. | U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. |

34. | V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. |

35. | M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. |

36. | A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging |

37. | O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods |

38. | K. M. Case, and P. F. Zweifel, |

39. | J. J. Duderstadt, and W. R. Martin, |

40. | E. E. Lewis, and W. F. Miller, |

41. | B. G. Carlson, and K. D. Lathrop, “Transport theory - the method of discrete ordinates”, in |

42. | K. E. Atkinson, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(170.3660) Medical optics and biotechnology : Light propagation in tissues

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: June 8, 2010

Revised Manuscript: September 12, 2010

Manuscript Accepted: September 13, 2010

Published: September 13, 2010

**Virtual Issues**

Optical Imaging and Spectroscopy (2010) *Biomedical Optics Express*

**Citation**

Ludguier D. Montejo, Alexander D. Klose, and Andreas H. Hielscher, "Implementation of the equation of radiative transfer on block-structured grids for modeling light propagation in tissue," Biomed. Opt. Express **1**, 861-878 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-3-861

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### References

- A. J. Welch, and M. J. C. Van Gemert, Optical-thermal response of laser-irradiated tissue, (Plenum Press, New York, NY, 1995).
- S.A. Prahl, M. Keijzer, S.L. Jacques, A.J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, SPIE Institute Series IS (5) 102–111 (1989).
- L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef] [PubMed]
- A. D. Klose, “Radiative Transfer of Luminescence in Biological Tissue”, in Light Scattering Reviews, Volume 4, A.A. Kokhanovsky (Ed.), 293–345 (Springer, Berlin, 2009).
- A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26(8), 1698–1707 (1999). [CrossRef] [PubMed]
- A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002). [CrossRef]
- A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998). [CrossRef] [PubMed]
- K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29(6), 578–580 (2004). [CrossRef] [PubMed]
- H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. 25(1), 015010 (2009). [CrossRef]
- O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998). [CrossRef]
- M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005). [CrossRef]
- S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992). [CrossRef] [PubMed]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993). [CrossRef] [PubMed]
- M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989). [CrossRef] [PubMed]
- M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33(10), 1963–1974 (1994). [CrossRef]
- V. D. Liseikin, Grid generation methods, Second Edition (Springer, Netherlands, 2010).
- J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006). [CrossRef] [PubMed]
- X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. 46(10), 1624–1632 (2007). [CrossRef] [PubMed]
- A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28(12), 1019–1021 (2003). [CrossRef] [PubMed]
- A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005). [CrossRef]
- A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010). [CrossRef] [PubMed]
- J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007). [CrossRef] [PubMed]
- B. W. Pogue and G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. 37(31), 7429–7436 (1998). [CrossRef] [PubMed]
- R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. 14(11), 1665–1678 (1979). [CrossRef]
- M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. 53(3), 484–512 (1984). [CrossRef]
- W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997). [CrossRef]
- M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. 82(1), 64–84 (1989). [CrossRef]
- J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998). [CrossRef]
- A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006). [CrossRef] [PubMed]
- A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005). [CrossRef] [PubMed]
- A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005). [CrossRef] [PubMed]
- B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32(4), 607–616 (1993). [CrossRef] [PubMed]
- U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]
- V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006). [CrossRef] [PubMed]
- M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008). [CrossRef] [PubMed]
- A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005). [CrossRef] [PubMed]
- O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods 48(2), 139–145 (2009). [CrossRef] [PubMed]
- K. M. Case, and P. F. Zweifel, Linear transport theory, (Addison-Wesley, Reading, 1967).
- J. J. Duderstadt, and W. R. Martin, Transport theory, (John Wiley, New York, 1979).
- E. E. Lewis, and W. F. Miller, Computational Methods of Neutron Transport, (Wiley, New York, 1984).
- B. G. Carlson, and K. D. Lathrop, “Transport theory - the method of discrete ordinates”, in Computing Methods in Reactor Physics, H. Greenspan et al, eds. (Gordon and Breach, New York, 1968, pp. 166–266).
- K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., (John Wiley & Sons, Canada, 1989).

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