## Analytical modelling techniques

Green's functions provide a method for modelling the diffusion equation or the RTE analytically. The Green's function is the solution when the source is a spatial and temporal δ-function, from which solutions for extended sources can be derived by convolution. Unfortunately, solutions only exist for simple homogeneous objects (Arridge et al. 1992) or media which include a single spherical perturbation (Boas et al. 1994), although the range of solutions is being extended to more complex geometries such as layered slabs (Martelli et al. 2002).

A number of workers have extended this type of approach. Kim (Kim and Ishimaru 1998, Kim 2004) has devised a method for calculating the Green's function as an expansion into plane-wave solutions, which allows analytical solutions for the diffusion equation or the RTE to be derived. Ripoll et al. (2001) has advocated the use of the Kirchhoff approximation, which models the Green's function between two points in a medium of arbitrary geometry as a sum of the infinite space Green's function plus other Green's functions calculated for diffusive waves which are multiply reflected off the boundary. It can be used alone, or to improve the accuracy of existing modelling techniques. Green's function techniques are now commonly used to solve the forward problem for image reconstruction, particularly for fast imaging techniques where the geometry can be approximated as a slab or an infinite halfspace, such as for optical topography (Culver et al. 2003b, Li et al. 2004).

Spinelli et al. (2003) have developed a perturbation approach first introduced by (Arridge 1995) in which the optical properties are modelled by a Green's function for a slab representing the homogeneous background, with an additional perturbation term representing a spherical insertion. It is well-suited to analysing transmission data measured across a compressed breast with a single, isolated lesion ((Torricelli et al. 2003).^{[1]}

## Statistical modelling techniques

Statistical (or stochastic) modelling techniques model individual photon trajectories and have the advantage that the Poisson error is incorporated into the model in a natural and elegant way. The most commonly used statistical technique in diffuse optics, and that which is often regarded as the gold standard to which other techniques are compared, is the Monte Carlo method. The geometry of the model is defined in terms of *µ _{a}*,

*µ*, and the refractive index, and the trajectories of photons, or packets of photons, are followed until they either escape from the object under study or are absorbed. By continuing until the required counting statistics are obtained, data with arbitrarily low statistical errors can be simulated. In optical imaging, Monte Carlo techniques are commonly used to calculate light propagation in non-diffusive domains where the diffusion approximation does not hold (Boas et al. 2002, Okada and Delpy 2003, Hayashi et al. 2003), or to validate results obtained using other, faster, methods (Schweiger et al. 1995, Chernomordik et al. 2002b, Dehghani et al. 2003a).

_{s}Random walk theory provides a distinct approach in which photon transport is modelled as a series of steps on a discrete cubic lattice. The time steps may be discrete or continuous (Weiss et al. 1998). Random walk theory is particularly suited to modelling time-domain measurements (Chernomordik et al. 2000) and has been used, for example, to determine the time spent by photons in a scattering inclusion (Chernomordik et al. 2002a) and to quantify the optical properties of a breast tumour (Chernomordik et al. 2002b). Recently, an extension to the technique has been developed for modelling media with anisotropic optical properties (see section 3.3.6), maintaining the cubic lattice but allowing the transition properties along different axes to differ (Dagdug et al. 2003). Similarly, Carminati et al. (2004) have developed a more general method based on random walk theory which allows the transition from single-scattering to diffusive regimes to be explored.

## Numerical modelling techniques

Numerical techniques are required if more complex geometries are to be modelled. The natural choice for representing the inhomogeneous distribution of optical properties in an arbitrary geometry is the finite element method (FEM) which was first introduced into optical tomography by Arridge et al. (1993). The method is explained in detail by Arridge and Schweiger (1995), Arridge (1999) and Arridge et al. (2000). FEM has become the method of choice for modelling complex inhomogeneous domains in optical imaging (Bluestone et al. 2001, Dehghani et al. 2003b), although the finite difference method (FDM) (Culver et al. 2003a, Hielscher et al. 2004), finite volume method (FVM) (Ren et al. 2004) and boundary element method (BEM) (Ripoll and Ntziachristos 2003, Heino et al. 2003) have been used in more specialised applications.

The finite element method requires that the reconstruction domain be divided into a finite element mesh. In principle, this is a completely general technique which can be applied to any geometry. In practise, however, it is difficult to create a finite element mesh of irregular objects with complex internal structure, and the development of robust, efficient 3D meshing techniques is still the subject of active research. A particular challenge is meshing the head, while respecting the convoluted internal boundaries of the scalp, skull, cerebrospinal fluid (CSF), grey matter and white matter derived from a segmented MRI image. Inclusion of such anatomical details has been shown to improve the quality of reconstructed images in EIT (Bagshaw et al. 2003), although generating the 3D finite element mesh was difficult and time-consuming (Bayford et al. 2001, Lionheart 2004). The use of realistic finite element meshes in optical imaging has progressed from 2D models of internal structure taken from segmented MR images of the head and breast (Schweiger and Arridge 1999, Brooksby et al. 2003), through 3D finite element meshes with complex surface shape but no internal structure (Bluestone et al. 2001, Gibson et al. 2003, Dehghani et al. 2004), to a recent report by (Schweiger et al. 2003) of simulations from 3D finite element meshes of the breast and head with anatomically realistic internal structure. A further complication is the consideration that optimal computational efficiency requires a finite element mesh which can represent the internal field adequately whilst using the smallest possible number of elements. One approach is to adaptively refine the mesh, placing more elements where the field changes most rapidly (Molinari et al. 2002, Joshi et al. 2004). Another approach, which is suited to modelling a segmented volume taken from MRI, is to resample the pixels in the MRI image onto a regular grid which can then be solved with FDM (Barnett et al. 2003)

## Use of prior information

Perhaps the most significant recent trend in optical imaging has been the inclusion of prior anatomical information into the forward problem, most commonly in the form of an anatomical MRI image. Prior information was first used in optical tomography reconstruction by Pogue and Paulsen (1998), and by Schweiger and Arridge (1999). The latter used a more sophisticated reconstruction procedure, which began by building a finite element mesh with four regions segmented from an MR image. They smoothed the model to account for gradual transitions between regions and generated simulated data, to which noise was added. They then used a two-step reconstruction process in which the optical properties of the regions were reconstructed first, followed by a full reconstruction onto the finite element mesh basis. The use of prior anatomical information was shown to improve the image quality in all cases. Ideally, the anatomical prior and the optical image should be acquired simultaneously so the two images are correctly registered. This impacts on the design of the image acquisition system and the experimental procedure and so these approaches are reviewed in the appropriate section on clinical applications below.

## Non-diffusive regions

Light transport can be modelled adequately using the diffusion approximation in most biological tissues. Among the exceptions to this are tissue volumes which are smaller than a few scattering lengths, and regions where *µ _{a}* is comparable to or greater than

*µ'*, such as the CSF which surrounds the brain and fills the central ventricles. These regimes are generally modelled using higher order approximations to the RTE.

_{s}It is generally too computationally expensive to solve the RTE fully in a practical reconstruction scheme. Instead, approximations are made such as the method of discrete ordinates, which solves the full RTE on a regular grid using FDM or FVM by quantising the allowed directions of travel, . This makes the problem tractable but because only preferred angles of travel are allowed, there may be regions where light cannot reach. This is known as the Ray Effect and restricts the application of the method. The method of discrete ordinates was applied to optical imaging by Dorn (1998) and has been used successfully (Klose and Hielscher 1999, Ren et al. 2004) for imaging small objects such as the finger (Hielscher et al. 2004), and in fluorescence and molecular imaging of small animals (Klose et al. 2004). Alternatively, the RTE can be solved using the *P _{N}* approximation (Aydin et al. 2002) or by expansion into a rotated spherical harmonic basis (Markel 2004).

A different approach can be taken for modelling light transport in the head, where the healthy CSF is non-scattering (although brain injury may cause proteins and blood products to leak into the CSF, making it diffusive (Seehusen et al. 2003)). If an anatomical MRI is available, then it is possible to segment the head into diffusive and non-diffusive regions. A coupled radiosity-diffusion model has been developed at UCL which models light transport in scattering regions using the diffusion model, and in clear regions using a visibility model which couples points on the void boundary which are mutually visible. The model has been applied to 2D (see Figure 2) and some 3D geometries (Riley et al. 2000), but it has not yet been applied to a 3D head-like model. A similar approach has been adopted by Schulz et al. (2003, 2004) who use a coupled model for non-contact molecular imaging where the animal is modelled diffusively and the space between the optical fibres and the animal is modelled using a free-space propagation model. One disadvantage of the current implementation of the radiosity-diffusion model is that the boundary of the clear region must be known.

## Anisotropy

Another regime in which the diffusion approximation does not apply is that of anisotropic scattering. Certain tissues such as the skin, nervous tissue and muscle cells are anatomically anisotropic at the cellular level, and this leads to anisotropic scattering properties (Nickell et al. 2000).

Heino and Somersalo (2002) derived a modified form of the diffusion equation in which the diffusion coefficient, , is replaced by a diffusion tensor , where S is (in 3D) a 3×3 matrix whose diagonal contains the anisotropic scattering coefficients. They make the anatomically realistic assumption that *µ _{a}* is isotropic everywhere and distinguish between knowing a priori the direction but not the strength of the scatter anisotropy (as may be obtained from MR diffusion tensor imaging (Le Bihan et al. 2001)), and having full knowledge of both the direction and the strength of the anisotropy. Such prior information is required if the solution is to be unique. Statistical reconstructions of simulated data demonstrated that artefacts in

*µ*will result if anisotropic

_{a}*µ'*is not fully considered. They validated the results by comparison with a Monte Carlo model (Heino et al. 2003).

_{s}Dagdug et al. (2003) studied the same problem using a random walk formulation in which the transition probability is allowed to vary with direction. To date, this method only allows the direction of anisotropy to be parallel to the co-ordinate axes. The model has been shown agree with time-resolved measurements on anisotropic phantoms (Hebden et al. 2004).

## References

- Arridge S R (1995), "Photon measurement density functions. Part I: Analytical forms" Applied Optics 34 7395-7409.
- Arridge S R (1999), "Optical tomography in medical imaging" Inverse Problems 15 R41-R93.
- Arridge S R, M Cope, and D T Delpy (1992), "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis" Phys. Med. Biol. 37 1531-1559.
- Arridge S R, H Dehghani, M Schweiger, and E Okada (2000), "The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions" Med. Phys. 27(1) 252-264.
- Arridge S R and M Schweiger (1995), "Photon measurement density functions. Part II: Finite element method calculations" Applied Optics 34 8026-8037.
- Arridge S R, M Schweiger, M Hiraoka, and D T Delpy (1993), "Finite element approach for modelling photon transport in tissue" Med. Phys. 20(2) 299-309.
- Aydin E D, C R E de Oliveira, and A J H Goddard (2002), "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method" Med. Phys. 29(9) 2013-2023.
- Bagshaw A P, A D Liston, R H Bayford, A Tizzard, A P Gibson, A T Tidswell, M K Sparkes, H Dehghani, C D Binnie, and D S Holder (2003), "Electrical impedance tomography of human brain function using reconstruction algorithms based on the finite element method" Neuroimage 20 752-763.
- Barnett A H, J P Culver, A G Sorensen, A Dale, and D A Boas (2003), "Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging" Appl. Opt. 42(16) 3095-3108.
- Bayford R H, A P Gibson, A Tizzard, T Tidswell, and D S Holder (2001), "Solving the forward problem in electrical impedance tomography for the human head using IDEAS (integrated design engineering analysis software), a finite element modelling tool" Physiol Meas. 22(1) 55-64.
- Bluestone A Y, G Abdouleav, C H Schmitz, R L Barbour, and A H Hielscher (2001), "Three-dimensional optical tomography of hemodynamics in the human head" Optics Express 9(6) 272-286.
- Boas D A, J P Culver, J J Stott, and A K Dunn (2002), "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head" Optics Express 10(3) 159-169.
- Boas D A, M A O'Leary, B Chance, and A G Yodh (1994), "Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications" Proc. Natl. Acad. Sci. USA 91 4887-4891.
- Brooksby B, H Dehghani, B W Pogue, and K D Paulsen (2003), "Near infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities" IEEE Quantum Electronics 9(2) 199-209.
- Carminati R, R Elaloufi, and J-J Greffet (2004), "Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light" Phys. Rev. Lett. 92(21) 213904-1-213904-4.
- Chernomordik V, D W Hattery, A H Gandjbakhche, A Pifferi, P Taroni, A Torricelli, G Valentini, and R Cubeddu (2000), "Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms" Opt. Lett. 25(13) 951-953.
- Chernomordik V, D W Hattery, I Gannot, G Zaccanti, and A H Gandjbakhche (2002a), "Analytical calculation of the mean time spent by photons inside an absorptive inclusion embedded in a highly scattering medium" J. Biomed. Opt. 7(3) 486-492.
- Chernomordik V, D W Hattery, D Grosenick, H Wabnitz, H Rinneberg, K T Moesta, P M Schlag, and A H Gandjbakhche (2002b), "Quantification of optical properties of a breast tumour using random walk theory" J. Biomed. Opt 7(1) 80-87.
- Culver J P, R Choe, M J Holboke, L Zubkov, T Durduran, A Slemp, V Ntziachristos, B Chance, and A G Yodh (2003a), "Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging" Med. Phys. 30(2) 235-247.
- Culver J P, A M Siegel, J J Stott, and D A Boas (2003b), "Volumetric diffuse optical tomography of brain activity" Opt. Lett. 28(21) 2061-2063.
- Dagdug L, G H Weiss, and A H Gandjbakhche (2003), "Effects of anisotropic optical properties on photon migration in structured tissues" Phys. Med. Biol. 48 1361-1370.
- Dehghani H, B Brooksby, K Vishwanath, B W Pogue, and K D Paulsen (2003a), "The effect of internal refractive index variation on near-infrared optical tomography: a finite element modelling approach" Phys. Med. Biol. 48 2713-2727.
- Dehghani H, M M Doyley, B W Pogue, S Jiang, J Geng, and K D Paulsen (2004), "Breast deformation modelling for image reconstruction in near infrared optical tomography" Phys. Med. Biol. 49 1131-1145.
- Dehghani H, B W Pogue, S P Poplack, and K D Paulsen (2003b), "Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom and clinical results" Appl. Opt. 42(1) 135-145.
- Dorn O (1998), "A transport-backtransport method for optical tomography" Inverse Problems 14(5) 1107-1130.
- Gibson A P, J Riley, M Schweiger, J C Hebden, S R Arridge, and D T Delpy (2003), "A method for generating patient-specific finite element meshes for head modelling" Phys. Med. Biol. 48 481-495.
- Hayashi T, Y Kashio, and E Okada (2003), "Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region" Applied Optics 42(16) 2888-2896.
- Hebden J C, J J Garcia Guerrero, V Chernomordik, and A H Gandjbakhche (2004), "Experimental evaluation of an anisotropic scattering model of a slab geometry" Optics Letters 29 2518-2520.
- Heino J, S R Arridge, J Sikora, and E Somersalo (2003), "Anisotropic effects in highly scattering media" Phys. Rev. E 68(031908)
- Heino J and E Somersalo (2002), "Estimation of optical absorption in anisotropic background" Inverse Problems 18 559-573.
- Hielscher A H, A D Klose, A K Scheel, B Moa-Anderson, M Backhaus, U Netz, and J Beuthan (2004), "Sagittal laser optical tomography for imaging of rheumatoid finger joints" Phys. Med. Biol. 49 1147-1163.
- Joshi A, A B Thompson, E M Sevick-Muraca, and W Bangerth (2004), "Adaptive finite element methods for forward modelling in fluorescence enhanced frequency domain optical tomography" OSA Biomedical Topical Meetings, Miami WB7.
- Kim A (2004), "Transport theory for light propagation in biological tissue" J. Opt. Soc. Am. A 21(5) 820-827.
- Kim A D and A Ishimaru (1998), "Optical diffusion of continuous-wave, pulsed and density waves in scattering media and comparisons with radiative transfer" Appl. Opt. 37(22) 5313-5319.
- Klose A D and A H Hielscher (1999), "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer" Medical Physics 28(8) 1698-1707.
- Klose A D, V Ntziachristos, and A H Hielscher (2004), "Experimental validation of a fluorescence tomography algorithm based on the equation of radiative transfer" OSA Biomedical Topical Meetings, Miami SA6.
- Le Bihan D, J-F Mangin, C Poupon, C A Clark, S Pappata, N Molko, and H Chabriet (2001), "Diffusion tensor imaging: Concepts and applications" J. Magn. Res. Imaging 13 534-546.
- Li A, Q Zhang, J P Culver, E L Miller, and D A Boas (2004), "Reconstructing chromophore concentration images directly by continuous wave diffuse optical tomography" Opt. Lett. 29 (3) 256-258.
- Lionheart W R B (2004), "EIT reconstruction algorithms: pitfalls, challenges and recent developments" Physiol. Meas. 25 125-142.
- Markel V A (2004), "Modified spherical harmonics method for solving the radiative transport equation" Waves in Random Media 14 L13-19.
- Martelli F, A Sassaroli, Y Yamada, and G Zaccanti (2002), "Analytical approximate solutions of the time-domain diffusion equation in layered slabs" J. Opt. Soc. Am. A 19 71-80.
- Molinari M, B H Blott, S J Cox, and G J Daniell (2002), "Optimal imaging with adaptive mesh refinement in electrical impedance tomography" Physiol. Meas. 23 121-128.
- Nickell S, M Hermann, M Essenpreis, T J Farrell, U Krämer, and M S Patterson (2000), "Anisotropy of light propagation in human skin" Phys. Med. Biol. 45 2873-2886.
- Okada E and D T Delpy (2003), "Near-infrared light propagation in an adult head model. I. Modeling of low-level scattering in the cerebrospinal fluid layer" Applied Optics 42(16) 2906-2914.
- Pogue B W and K D Paulsen (1998), "High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information" Opt. Lett. 23(21) 1716-1718.
- Ren K, G S Abdoulaev, G Bal, and A H Hielscher (2004), "Algorithm for solving the equation of radiative transfer in the frequency domain" Opt. Lett. 29(6) 578-580.
- Riley J, H Dehghani, M Schweiger, S R Arridge, J Ripoll, and M Nieto-Vesperinas (2000), "3D optical tomography in the presence of void regions" Optics Express 7 462-467.
- Ripoll J and V Ntziachristos (2003), "Iterative boundary method for diffuse optical tomography" J. Opt Soc. Am. A 20(6) 1103-1110.
- Ripoll J, V Ntziachristos, R Carminati, and M Nieto-Vesperinas (2001), "Kirchhoff approximation for diffusive waves" Phys. Rev. E 64051917.
- Schulz R B, J Ripoll, and V Ntziachristos (2003), "Noncontact optical tomography of turbid media" Optics Letters 28(18) 1701-1703.
- Schulz R B, J Ripoll, and V Ntziachristos (2004), "Experimental fluorescence tomography of tissues with noncontact measurements" IEEE Trans. Med. Imag. 23(4) 492-500.
- Schweiger M, A P Gibson, and S R Arridge (2003), "Computational aspects of diffuse optical tomography" IEEE Computing in Science and Engineering Nov/Dec 2003 33-41.
- Schweiger M and S R Arridge (1999), "Optical tomographic reconstruction in a complex head model using a priori region boundary information" Phys. Med. Biol. 44(11) 2703-2721.
- Schweiger M, S R Arridge, M Hiraoka, and D T Delpy (1995), "The finite element method for the propagation of light in scattering media: boundary and source conditions" Med. Phys. 22(11 Pt 1) 1779-1792.
- Seehusen D A, M M Reeves, and D A Fomin (2003), "Cerebrospinal fluid analysis" American Family Physician 68(6) 1103-1108.
- Spinelli L, A Torricelli, A Pifferi, P Taroni, and R Cubeddu (2003), "Experimental test of a perturbation model for time-resolved imaging in diffusive media" Appl. Opt. 42(16) 3145-3153.
- Torricelli A, L Spinelli, A Pifferi, P Taroni, R Cubeddu, and G M Danesini (2003), "Use of a nonlinear perturbation approach for in vivo breast lesion characterization by multi-wavelength time-resolved optical mammography" Optics Express 11(8) 853-867.
- Weiss G H, J M Porrà, and J Masoliver (1998), "The continuous-time random walk description of photon migration on an isotropic medium" Opt. Comm. 146 268-276.