Diffuse photon propagation in multilayered ... - Semantic Scholar

INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 51 (2006) 497–516

PHYSICS IN MEDICINE AND BIOLOGY

doi:10.1088/0031-9155/51/3/003

Diffuse photon propagation in multilayered geometries Jan Sikora1, Athanasios Zacharopoulos2, Abdel Douiri2, Martin Schweiger2, Lior Horesh3, Simon R Arridge2 and Jorge Ripoll4 1 Institute of the Theory of Electrical Engineering, Measurement and Information Systems, Warsaw University of Technology, Koszykowa 75, 00-661 Warsaw, Poland 2 Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK 3 Department of Medical Physics and BioEngineering, University College London, Gower Street, London, WC1E 6BT, UK 4 Institute of Electronic Structure and Laser, FORTH, PO Box 1527, Vassilika Vouton, 71110 Heraklion, Crete, Greece

Received 23 August 2005, in final form 15 November 2005 Published 11 January 2006 Online at stacks.iop.org/PMB/51/497 Abstract Diffuse optical tomography (DOT) is an emerging functional medical imaging modality which aims to recover the optical properties of biological tissue. The forward problem of the light propagation of DOT can be modelled in the frequency domain as a diffusion equation with Robin boundary conditions. In the case of multilayered geometries with piecewise constant parameters, the forward problem is equivalent to a set of coupled Helmholtz equations. In this paper, we present solutions for the multilayered diffuse light propagation for a three-layer concentric sphere model using a series expansion method and for a general layered geometry using the boundary element method (BEM). Results are presented comparing these solutions to an independent Monte Carlo model, and for an example three layered head model.

1. Introduction Over recent years, diffuse optical tomography (DOT) has attracted increasingly intense research interest around the world (Arridge et al 1991, Arridge and Hebden 1997, Yodh and Boas 2003, Gibson et al 2005, Hebden et al 2004, Kolehmainen et al 2000a, 2000b, Boas et al 2001, Chance et al 1998) due to advances both in measurement technology and in theoretical and practical understanding of the nature of the image reconstruction problem. Medical optical tomography aims to recover the optical properties of biological tissue from measurements of the transmitted light made at multiple points on the surface of the body. This boundary data measurement can be used to recover the spatial distribution of internal absorption and scattering coefficients. It is a non-invasive modality and can generate images of clinically relevant parameters, such as blood volume and oxygenation, with applications to 0031-9155/06/030497+20$30.00 © 2006 IOP Publishing Ltd Printed in the UK

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peripheral muscle, breast and the brain (Hebden et al 1999, Pogue et al 2000, Fantini et al 1998). An increasingly active topic within this field is the development of an efficient and accurate method for calculating the intensity of light transmitted or reflected from the subject under experimental investigation, sometimes referred to as the forward problem. A general model of light propagation can be developed using the radiative transfer equation (RTE), but a simpler model that can be derived from this equation in the case of sufficiently high scattering, and that has had a high degree of success, is the diffusion equation with Robin boundary conditions (Arridge 1999). Existing methods to solve this problem are either deterministic, based on the solutions to governing equations, or stochastic, based on simulations of the individual scattering and absorption events undertaken by each photon. The former include analytical expressions based on Green’s functions (Arridge et al 1992b), and numerical methods based on finite difference methods (FDM) or finite element methods (FEM) (Arridge et al 1993, Schweiger et al 1995, Schweiger and Arridge 1997, Paulsen and Jiang 1995, Model et al 1997, Takahashi et al 1997). In this paper we are concerned with the case where the object being studied can be considered as a set of disjoint simply connected regions with constant optical coefficients within each region, but that may differ between regions. In this case the diffusion equation can be replaced by a set of Helmholtz equations for each domain, together with interface conditions. For this problem, analytical solutions are less readily available, usually requiring particular symmetries of the geometry. Although volume based PDE solvers such as FDM or FEM can certainly be applied to this problem, there are often practical difficulties in constructing meshes for general geometries that respect the interfaces accurately. In contrast, the use of boundary integral methods involves only representation of the surface meshes and can be much easier to implement. In this paper we consider a boundary element method for the multilayer photon propagation problem. The boundary element method (BEM) has been introduced as a powerful tool for solving different types of engineering problems, ranging from linear to nonlinear and time-dependent problems (Brebbia 1978, Brebbia and Walker 1980, Brebbia et al 1984). The most important applications of BEM are in the fields of mechanical and aerospace engineering for the study of electrical and fluid flow problems, and the range of applications is extensive (Itagaki and Brebia 1988, de Munck 1992, Park and Kwon 1996, de Munck et al 2000, Frijns et al 2000, Lu and Yevick 2002). Nowadays, BEM are frequently used by the automotive and aerospace industries as they are simple to implement and can easily model complex 3D structures. BEM has been used in many biomedical applications problems, such as electroencephalography and magnetoencephalography EEG/MEG (Haueisen et al 1997, Leahy et al 1998, Mosher et al 1999, Fan et al 2001, Fuchs et al 2001, Kybic et al 2005), electrical impedance tomography (de Munck et al 2000, Duraiswami et al 1998) and potential and electromagnetic source imaging (Sarvas 1987, Ferguson and Stroink 1997, Gencer and Tanzer 1999, Bradley and Pullan 2002, Akahn-Acar and Gencer 2004) where it shows particular advantages in modelling of the layered structure of the head. In optical applications, BEM has been used in diffusing-wave spectroscopy for determining the correlation function for different boundary conditions and source properties in a cone-plate geometry (Vanel et al 2001). For the application to DOT the integral formulation was introduced by Ripoll (Ripoll and Nieto-Vesperinas 1999a, 1999b, 1999c, Ripoll et al 2000), and was based on the extinction theorem method. In computational mathematics this method was introduced by Kantorovich and Krylov (1964) and since then has been frequently referred to as the method of moments (MOM) (Harrington 1968, Strait 1980, Ney 1985). An example of the use of BEM was given in (Heino et al 2003) where it was used for a multilayer geometry in two dimensions including

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anisotropy, and preliminary results in 3D can be found in Sikora and Arridge (2002), Sikora et al (2003). Recently BEM has been applied to the related problem of fluorescence diffuse propagation (Fedele et al 2005). In this paper we focus our application on the problem of optical imaging of the brain using 3D methods. This paper is organized as follows. In section 2 we give the mathematical model in terms of coupled Helmholtz equations with piecewise constant coefficients and interface conditions. In section 3 we derive an example analytic solution for a multi concentric layered sphere geometry. In section 4 we derive the integral formulation of the problem, and in section 5 the discrete version using BEM. In section 6 we present results comparing the analytical threelayer model for concentric spheres and the equivalent BEM formulation; we also present results for a realistic three-layer head model derived from MRI data. In section 7 we give some conclusions. Some technical details of the numerical integration rules employed are given in the appendix. 2. Formulation of the problem The problem of optical tomography in a highly diffusive body
with boundary  can be modelled by the use of the diffusion equation in the frequency domain form: iω (1) −∇ · κ(r)∇(r; ω) + µa (r)(r; ω) + (r, ω) = q(r; ω) c with Robin boundary conditions (m; ω) + 2ακ(m)

∂(m; ω) = h− (m; ω), ∂ν

m on 

(2)

where ω ∈ R+ is the frequency modulation,  is the radiance, c is the velocity of light, q is an internal source of light in the medium, h− is an incoming flux, α is a boundary term which incorporates the refractive index mismatch at the tissue–air boundary, ν is the outward normal at the boundary , κ and µa are the diffusion and absorption coefficients, respectively. We define, κ = 3(µa1+µ ) , where µs is the reduced scattering coefficient (Arridge 1999, Schweiger s et al 1995). We use the notation r for a position vector in
and m for a position vector restricted to a surface. For the scope of our model, we assume that the body
is divided into L nested subregions
 (1

L) shown in figure 1. Each region is defined by the smooth boundaries  (1

L) and is characterized by constant material parameters, diffusion κ and absorption µa, coefficients for the respective sub-domain
 ,  = 1, . . . , L. In a piecewise homogeneous medium the diffusion equation is equivalent to the Helmholtz equation, since the sub-domains have constant optical parameters. If we denote by  the restriction of  in the sub-domain
 , we get ∇ 2  (r; ω) − 2  (r; ω) = − −1 | =  | ,

q (r; ω) κ (r)

2

L

κ−1 ∂−1 −1 | = κ ∂  | , 1 |1 + 2ακ1 ∂1 1 |1 = h−

2

L

in
 ,

(3) (4) (5) (6)

where  is the complex wave number associated with the Helmholtz equation in each region

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Γ1

ΓL Γ2

ΩL (µa,L, µ
s,L)

Ω2 (µa,2, µ
s,2)

Ω1 (µa,1, µ
s,1)

Figure 1. The domain
, divided into disjoint regions
i with constant optical parameters (µa,l , µs,l ), separated by interfaces l with l = 1, . . . , L.


 given by
 =

µa, + iω/c κ

 12 (7)

and we used the abbreviated notation ∂ := ν · ∇ for the normal derivatives at the interfaces  . 3. Analytical model for concentric spheres Analytical models for concentric spheres could be a plausible model for diffuse photon propagation in the head, in analogy to its usage in MEG. In addition it provides a valuable comparison to the more general BEM. In this section, we derive this model using a series expansion method. The analytic solution for a homogeneous sphere with Dirichlet boundary conditions was previously given in (Arridge et al 1992a), and the multilayered case in 2D was introduced in (Arridge 2001). Here we give the 3D version. Consider a model of L concentric spheres, with radii {ρ ,  = 1, . . . , L}, and regions {
 ,  = 1, . . . , L − 1} representing the spherical shells bounded by spheres with radii ρ and ρ+1 , with
L the innermost sphere. As defined in (7), we let  be the complex wave number in the th layer. We use a spherical polar coordinate system and write the solution in the th layer as  (r, ϑ, ϕ) =

∞  n 

Yn,m (ϑ, ϕ)[a,n in (  r) + b,n kn (  r)]

(8)

n=0 m=−n

 2 1/2  2 1/2 In+1/2 (x), kn (x) = πx Kn+1/2 (x) are the modified spherical Bessel where in (x) = πx functions of the first and second kinds respectively, and Yn,m (ϑ, ϕ) is the spherical harmonic of order n and degree m. If we represent h− (m) in spherical polars we have h− (m) = h− (ρ1 , ϑ, ϕ) =

∞  n  n=0 m=−n

h− n,m Yn,m (ϑ, ϕ).

(9)

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Without loss of generality, let h− be azimuthally symmetric around the ϑ = 0 direction; then the ϕ dependence can be neglected and each spherical harmonic term is decoupled. At the internal boundaries the interface conditions (4), (5) lead to the following two conditions on the coefficients: a−1,n in ( −1 ρ ) + b−1,n kn ( −1 ρ ) = a,n in (  ρ ) + b,n kn (  ρ ),

(10)

a−1,n κ−1 −1 in ( −1 ρ ) + b−1,n κ−1 −1 kn ( −1 ρ ) = a,n κ  in (  ρ ) + b,n κ  kn (  ρ )

1 <  < L.

(11)

In the innermost layer, bL,n = 0, ∀n in order that the solution is finite at the origin, which leads to the condition for the innermost interface aL−1,n in ( L−1 ρL ) + bL−1,n kn ( L−1 ρL ) = aL,n in ( L ρL ),

(12)

aL−1,n κL−1 L−1 in ( L−1 ρL ) + bL−1,n κL−1 L−1 kn ( L−1 ρL ) = aL,n κL L in ( L ρL ).

(13)

At the outermost sphere with radius ρ1 we impose the Robin condition (2) which takes the form (in ( 1 ρ1 ) + 2ακ1 1 in ( 1 ρ1 ))a1,n + (kn ( 1 ρ1 ) + 2ακ1 1 kn ( 1 ρ1 ))b1,n = h− n.

(14)

Equations (10)–(14) constitute a system of 2L − 1 equations for each spherical harmonic order n, which can be solved to determine the L coefficients {a,n ,  = 1, . . . , L} and the L − 1 coefficients {b,n ,  = 1, . . . , L − 1}. In matrix form we need to solve the following,  n n C1,1 + 2αD1,1 n  C1,2  n  D1,2   ..  .   0 0

n n E1,1 + 2αF1,1 n E1,2 n F1,2 .. .

0 n −C2,2 n −D2,2 .. .

0 n −E2,2 n −F2,2 .. .

... ... ... .. .

0 0 0 .. .

0 0 0 .. .

0 0 0 .. .

0 0 

0 0

0 0

... ...

n CL−1,L n DL−1,L

n EL−1,L n FL−1,L

n −CL,L n −DL,L



a1  − hn  b   1  0   a     2    0     b2    ..    ×  ..  =  .    .     0   aL−1     0  bL−1  0 aL

        

(15)

n n n n where Cj,k = in ( j ρk ), Dj,k = κj j in ( j ρk ), Ej,k = kn ( j ρk ), Fj,k = κj j kn ( j ρk ). Once these coefficients have been determined the field at any point in the interior can be reconstructed using (8). Taking into account the azimuthal symmetry this becomes

 (r, ϑ) =

∞  n=0

(2n + 1)Pn (cos ϑ)[a,n in (  r) + b,n kn (  r)].

(16)

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4. Integral formulation The boundary integral formulation makes use of the Green’s function of the Helmholtz equation, which in the absence of boundary conditions is also referred to as the fundamental solution. We define a Green’s function of equation (3) in each sub-domain
 as solutions of the equations ∇ 2 G (r, r  ; ω) − 2 G (r, r  ; ω) = −δ(r − r  )

(17)

with the asymptotic G (r, r  ; ω)|r→∞ = 0

(18)

G (r, r  ; ω) is the response of the infinite media to a single source q = δ at position r = r  and takes the form of a spherical wave (Arridge et al 1992a) G (r, r  ) =

1  e−  |r−r | . 4π |r − r  |

The normal derivative of the fundamental solution can then be written as
 −1 r − r   ∂ G = ν · e−  |r−r | . − |r − r  | 4π |r − r  |2 4π |r − r  |

(19)

(20)

From equations (17) and (3), by multiplying (17) with  (r  ; ω) and (3) with G (r, r  ; ω), and subtracting we get  (r  ; ω)∇ 2 G (r, r  ; ω) − G (r, r  ; ω)∇ 2  (r  ; ω) q (r  ; ω) = −δ(r − r  ) (r  ; ω) + G (r, r  ; ω). κ

(21)

Integrating both parts of the equation over the whole subregion
 with respect to r  , applying the second Green’s theorem and simplifying by the Dirac filter δ(r − r  ) (r  ; ω) dn r  (22)  (r; ω) =


and using the notations U (m; ω) :=  | = −1 | V (m; ω) := κ ∂  | = κ−1 ∂−1 −1 | we get the following integral formula for the internal interfaces: 
G (r, m ; ω) ∂ G (r, m ; ω)U (m ; ω) − V (m ; ω) dS(m )  (r; ω) + κ  
G (r, m ; ω) ∂ G (r, m ; ω)U+1 (m ; ω) − V+1 (m ; ω) dS(m ) − κ+1 +1 = Q (r; ω) 2

L−1 (23) where L+1 = ∅ and the source term Q is G (r, r  ; ω) q (r  ) dn r  . Q (r; ω) = κl


(24)

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For 1 , we use the boundary conditions (6) to eliminate V1 h− (m; ω) − U1 (m; ω) V1 (m; ω) = (25) 2α with the result 
G (r, m ; ω) ∂1 G1 (r, m ; ω) + U1 (m ; ω) dS(m ) 1 (r; ω) + 2ακ1 1 
G1 (r, m ; ω) ∂1 G1 (r, m ; ω)U2 (m ; ω) − V2 (m ; ω) dS(m ) − κ2 2 = Q1 (r; ω) + H (r; ω) (26) where the term H is G1 (r, m ; ω) −  h (m ) dS(m ). (27) H (r; ω) = 2ακ 1 1 Considering (23), if we let r approach the outer boundary  , we note that the first integral is singular whilst the second has a continuous kernel. A limiting process, described in Becker (1992) and Bonnet (1999) must be invoked, designed to account for the singularity of the fundamental solutions and (23) will take the form
+ ∂ G (m, m ; ω)U (m ; ω) C (m)U (m; ω) + ( −σε+ )


G (m, m ; ω)   ∂ G (m, m ; ω)U+1 (m ; ω) V (m ; ω) dS(m ) − − κ +1  G (m, m ; ω)  V+1 (m ; ω) dS(m ) = Q (m; ω) 2

L − 1. − κ+1 (28) Similarly, if the variable m approaches the inner interface +1 , we get the equation
− ∂ G (m, m ; ω)U+1 (m ; ω) C (m)U+1 (m; ω) − (+1 −σε− ) 


G (m, m ; ω) ∂ G (m, m ; ω)U (m ; ω) V+1 (m ; ω) dS(m ) + κ+1   G (m, m ; ω) V (m ; ω) dS(m ) = Q (m; ω) 2

L − 1. − κ (29) ± The term C (m) arises due to the singularities on the boundary, and the integration of the fundamental solution over the interface. C± (m) can be calculated by surrounding the point m, which lays on the boundary, by a small hemisphere σ(ε) of radius ε and taking each term in (28)–(29) in the limit when ε → 0, as shown in figure 2. We note that in this form both integrals are now continuous. For the outer surface we have
 G1 (r, m ; ω) +  ∂1 G1 (r, m ; ω) + U1 (m ; ω) dS(m ) C1 (m)U1 (m; ω) + 2ακ1 (1 −σε+ ) 
G1 (r, m ; ω) ∂1 G1 (r, m ; ω)U2 (m ; ω) − − V2 (m ; ω) dS(m ) κ2 2 = Q1 (m; ω) + H (m; ω). (30) −

In Becker (1992), Bonnet (1999) it is shown that the term C± does not need to be calculated explicitly, and can be obtained indirectly by physical considerations. In the case of

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Γ−1 Γ

rs

Γ+1

r ε

Ω−1

σε Ω+1

Ω

Figure 2. Limiting process for the small hemisphere σ(ε) .

the observation point upon a smooth surface, which is the case considered here C+ (m) = C− (m) = 12 . Using (28)–(30) we can construct a system of 2n − 1 equations with 2n − 1 unknowns, {f} = {U1 , U2 , . . . , UL , V2 , . . . , VL }.

(31)

Solving for these functions, the integral representation (23)–(26) yields the field  at internal points. 5. Numerical implementation 5.1. Boundary element method The surface interfaces  are discretized in P surface elements τ,k , k = 1, . . . , P , with N vertices N ,k , k  = 1, . . . , N , after which we can approximate the functions U and V by the use of nodal basis functions φk , restricted to  . U (m; ω) 

N 

V (m; ω) 

U,k (ω)φ,k (m),

k  =1

N 

V,k (ω)φ,k (m).

(32)

k  =1

Representation (32) expresses both U and V in terms of the complex coefficients U,k , V,k interpolated by the nodal basis functions and thus enforces V to be at least C 0 continuous. The integrals occurring in the boundary integral equations (28) and (29) take the form u (m; ω) =

N 

U,k (ω)

v (m; ω) =

N  k  =1

∂ G (m, m ; ω)φ,k (m ) dS(m )

(33)

G (m, m ; ω)φ,k (m ) dS(m ).

(34)



k  =1

V,k (ω) 

The function v (m; ω), which is obtained by convolution with a Green’s function, is known as a single layer potential, and the function u (m; ω), which is obtained by convolution with the normal derivative of a Green’s function, is known as a double layer potential. The convolution kernels contain a weak and a strong singularity respectively. To obtain the

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discrete representation, we formally consider integrating the single and double layer potentials in equations (33) and (34) with the test function ψk (m): u,k (ω) = ψk (m)u (m; ω) dS(m) 

=



N 

ψk (m)∂ G (m, m ; ω)φ,k (m ) dS(m ) dS(m)

U,k (ω) 

k  =1

(35)



v,k (ω) =

ψk (m)v (m; ω) dS(m) 

=

N 



ψk (m)G (m, m ; ω)φ,k (m ) dS(m ) dS(m).

V,k (ω) 

k  =1

(36)



In this paper we consider a Collocation BEM approach, where the integral equation is enforced exactly at the nodal points N ,k and therefore: ψk (m) = δ(m − N ,k ) =: δ,k (m).

(37)

5.2. Matrix assembly For the solution of the system of the 2L−1 equations that we formed we introduce the following notation. Let A , B be N × N matrices corresponding to surface  of subdomain
 with entries:   A (k, k ) = δ ,k (m)∂ G (m, m ; ω)φ ,k (m ) dS(m ) dS(m)   = ∂ G (N  ,k , m ; ω)φ ,k (m ) dS(m ) (38) 

B (k, k  ) =





G (m, m ; ω) φ ,k (m ) dS(m ) dS(m) κ  G (N  ,k , m ; ω) φ ,k (m ) dS(m ). κ δ ,k (m)



=



Equations (28)–(30) give rise to the discrete form

 1 1 I + A11 + B11 U 1 − A12 U 2 + B12 V 2 = Q1 + H 2 2α .. . 

1 I + A U  − B V  − A+1 U +1 + B+1 V +1 = 0 2 

1 A U  − B V  + I − A+1 U +1 + B+1 V +1 = 0 2 .. .

 1 I + ALL U L − BLL V L = 0. 2

(39)

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We express this as the matrix system  1 C11 + 2α B11 −A12 B12 0 ···  · · · · · ·   C −B −A+1 B+1 0   ..  A −B D+1 B+1 .   0 0 0 C − B+1+1 +1+1   0 0 · · · A − B+1+1 +1+1   .. ..  . . ··· ··· 0 0 ···     Q1 + H U1 U   Q   2    2 V   0   2     .    . ..  ..        U   0           ×  V = 0      U +1   0      V +1   0   .    ..  .     .    .      UL   0  VL 0

0

0

−A+1+2

B+1+2 B+1+2

D+1+2

CLL

0

             

−BLL

(40)

where C = 12 I + A , D = 12 I − A ,

and we have assumed that the source term q is located only in
1 . Equation (40) can be summarized as the linear matrix equation Kf = b.

(41)

Here, f is the discrete version of {f} and contains the coefficients of the unknowns when using the approximations in (32) for the functions in (31), K is the system matrix, and b is the vector of known coefficients calculated from the light sources in the problem. The matrix K is of dense un-symmetric block form. The generalized minimum residuals method (GMRES) is used to solve (41) (Saad and Schultz 1986). 6. Results We present results for a three-layer concentric spherical shell model and for a head model using surfaces derived from MRI. The BEM meshes used are illustrated in figure 3. 6.1. Comparison to analytic model We consider a three-sphere model with radii 20, 15 and 10 mm. Two sets of optical parameters are given in table 1 in units of mm−1 . The number of elements and nodes for the BEM case are also given in table 1. The modulation frequency used was 100 MHz.

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Figure 3. BEM meshes used for two problems considered. Left the concentric spherical shell model, right the head model. The mesh details are given in tables 1 and 2. Table 1. Optical parameters and mesh discretizations for the three-sphere example.

Outer shell Middle shell Inner shell

Case 1: µa

Case 1: µs

Case 2: µa

Case 2: µs

Nodes

Elements

0.005 0.005 0.01

1 1 1.5

0.005 0.005 0.01

1 0.75 1.5

2658 1410 646

1328 704 322

Use of a δ-function source would require an infinite number of terms in the representation (9). In the results presented here, we assume that h− (m) is a narrow Gaussian profile input beam with coefficients in (9) given by

2 2 n s . (42) = exp − h− n 2 The analytical expressions were solved using Mathematica for 15 370 Legendre coefficients. The Gaussian decay parameter was s = 1/1500. Results are shown for case 1 optical parameters in figure 4, and for case 2 in figure 5, for amplitude and phase as a function of angular separation from the source position. In addition, we show the results of a Monte Carlo simulation for the flux on the outermost surface. Every node on the BEM meshes is displayed, demonstrating the azimuthal symmetry. In the Monte Carlo code, the exiting photons are binned together for the same azimuthal angle, to provide more robust statistics. For the comparison of amplitudes a scale factor is applied to match the values on the outer layer at the 45◦ position. No scale factor is required for the phase. These figures show that all three techniques agree very well. The amplitudes decrease away from the source and the phases increase, with the range of values reducing with the depth of the layer. When comparing parameter set one with parameter set two, the amplitudes are consistently increased and the phases consistently reduced, with the change in phases being more noticeable. This is consistent with the physical interpretation of photons exhibiting shorter path lengths through the middle layer as its scattering is lowered, and therefore undergoing less attenuation. 6.2. Three-layer head model In this section we present results for a three-layer head model. The surfaces modelled were the outer skin, the skull and the brain, and were generated from an MRI scan following tissue

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−2

0.8

−4

0.7

0.6

phase (radians)

log intensity

−6

−8

−10

0.5

0.4 0.3

−12 0.2 −14

−16

0.1

0

20

40

60

80 100 120 Angle from source

140

160

0

180

0

20

40

60

80 100 120 Angle from source

140

160

180

Figure 4. Spheres case 1. Curves are the solutions on the outer middle and inner spheres plotted against angular distance from the source. Diamonds are the analytical solution. Solid lines are the BEM solution. Open circles are Monte Carlo on the outer sphere. The left figure is amplitude on logarithmic scale, with the three solutions having decreasing values at the ϑ = 0 point. The right figure is phase in units of radians, having increasing values at the ϑ = 0 point. −2

0.7

−4

0.6

0.5 phase (radians)

log intensity

−6

−8

0.4

0.3

−10 0.2 −12

−14

0.1

0

20

40

60

80 100 120 Angle from source

140

160

180

0

0

20

40

60

80 100 120 Angle from source

140

160

180

Figure 5. Spheres case 2. Notation for the curves is the same as in figure 4. Table 2. Optical parameters and mesh discretizations for the three-layer head model.

Outer shell Middle shell Inner shell

Case 1: µa

Case 1: µs

Case 2: µa

Case 2: µs

Nodes

Elements

0.0149 0.01 0.0178

0.8 1 1.25

0.0149 0.01 0.0178

0.8 0.5 1.25

2806 3294 2098

1402 1646 1048

segmentation. The complexity of these meshes is such that a volume mesh such as that used for a finite element method (FEM) would be very difficult to construct. The optical parameters and mesh sizes are given in table 2. Two different parameter sets were considered with the first set having twice the scattering coefficient in the middle layer, with respect to the first set. A source was placed at the back of the head at position r = (0, 45.8345, 0). The modulation frequency used was 100 MHz. Results are shown for case 1 optical parameters in figure 6, and for case 2 in figure 7. These figures show that the BEM is producing smoothly varying amplitude and phase on the

Diffuse photon propagation in multilayered geometries

509

Figure 6. Three-layer head model, parameter case 1. Left: amplitude, right: phase.

three surfaces with values consistent with the three-sphere model. As in the three-sphere model the amplitudes decrease away from the source and the phases increase, with the range of values reducing with the depth of the layer. Also as in the three-sphere model, the lower scattering coefficient in the middle layer leads to a reduction in phase and increase in amplitude on all nodes. The change in phase is much more noticeable than the change in amplitude.

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Figure 7. Three-layer head model, parameter case 2. Left: amplitude, right: phase.

7. Conclusion In this paper we have developed methods for the solution of the diffuse photon propagation problem in multilayered geometries, as relevant to optical tomography. Both an analytic model for concentric spheres and a 3D boundary element method (BEM) have been presented.

Diffuse photon propagation in multilayered geometries

511 (x5, y5, z5)

z

(x6, y6, z6) (x4, y4, z4) y

x

(x2, y2, z2) (x1, y1, z1)

(x3, y3, z3)

Figure 8. Quadratic surface triangle.

The results show a good agreement between both methods and an independent Monte Carlo model. For future work we want to apply these models to the problem of optical imaging in the brain. One advantage is that it becomes easy to restrict the inverse solution to the surface of the brain. Alternatively using a coupled FEM-BEM model, the volume solution inside the brain could be derived relatively easily, assuming piecewise constant values for the outer layers. The BEM solution method is a natural choice for a shape-based reconstruction method along the lines of the 2D method presented in Kolehmainen et al (2000a), where it was shown that variable shape regions can be reconstructed even in the presence of tissue heterogeneity. An obvious drawback of the simple three-layer model presented here is that is does not take account of the non-scattering cerebral spinal fluid (CSF) layer surrounding the brain. One possibility for involving this layer is to use the radiosity-diffusion model introduced in Arridge et al (2000), which utilized a hybrid FEM-BEM model in 2D; the extension to a three-layer sphere was presented in Riley et al (2000). A purely integral equation based implementation of this model was presented in 2D in Ripoll et al (2000). The extension to a 3D BEM solution of the radiosity-diffusion model is currently under development. Acknowledgments We thank David Holder, Richard Bayford and Andrew Tizzard for assistance with obtaining the meshes used in section 6. This work was supported by EPSRC grant GR/R86201/01 and the Integrated Technologies for In-Vivo Molecular Imaging project funded by FP6 EU contract LSHG-CT-2003-503259. Appendix. Numerical integration on elements The evaluation of the matrix elements (38), (39) is carried out element-wise with summation over contributions where the same node pairs occur in multiple elements. In contrast to the finite element method (FEM) where integrals of products of shape functions can be evaluated analytically, in the BEM we require the product of a shape function and a kernel function that can contain singularities (single and double layer potentials). There are many choices of element type possible. In the following explanation of the numerical implementation we discuss the isoparametric quadratic triangle elements that have been used in our approach, as illustrated in figure 8. The isoparametric triangular element shown in figure 8 is mapped from global coordinates m = (x, y, z) system to local coordinates ξ = (ξ1 , ξ2 ) by m(ξ) =

6  i=1

φi (ξ)mi

(A.1)

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η2

ξ2 1

1

z

η1 y

0

1

ξ

1

-1 -1

x

1

Figure 9. Quadratic surface triangle mapped onto the flat triangle with local coordinates (ξ1 , ξ2 ), and the secondary mapping on the square (η1 , η2 ). Table 3. Gaussian points and weights for a flat triangular element. m

ξ1,m

ξ2,m

wm

1 2 3 4 5 6 7

1/3 0.059 72 0.470 14 0.470 14 0.797 43 0.101 29 0.101 29

1/3 0.470 14 0.059 72 0.470 14 0.101 29 0.797 43 0.101 29

0.112 5 0.066 20 0.066 20 0.066 20 0.062 97 0.062 97 0.062 97

where φi (ξ) are the shape functions given by φ1 (ξ) = (1 − ξ1 − ξ2 )(1 − 2ξ1 − 2ξ2 )

φ2 (ξ) = 4ξ1 (1 − ξ1 − ξ2 )

φ3 (ξ) = ξ1 (2ξ1 − 1)

φ4 (ξ) = 4ξ1 ξ2

φ5 (ξ) = ξ2 (2ξ2 − 1)

φ6 (ξ) = 4ξ2 (1 − ξ1 − ξ2 )

(A.2)

and mi are the global coordinates of the element nodes (see figure 9). With this mapping integrals of functions on an element are transformed into integrals over a regular right angled triangle 1 1−ξ1 f (m) dS(m) → f (m(ξ))J (ξ) dξ1 dξ2 (A.3) 0

τ

0

where J (ξ) is the Jacobian of the mapping (A.1) given by  ∂x ∂x     ∂ξ1 ∂ξ2 νx      ∂y ∂y νy  J (ξ) = mξ1 × mξ2  =  ∂ξ1 ∂ξ 2  ∂z  ∂z  νz  ∂ξ1

(A.4)

∂ξ2

where mξ1 and mξ2 are vectors in the local tangent plane of the element and ν = (νx , νy , νz ) is the local unit normal vector. For evaluation of the integral on the right-hand side of (A.3) we use a Gaussian quadrature scheme for flat triangular elements taken from the literature (Zienkiewicz and Taylor 1989), using the points and weights given in table 3. 1 1−ξ1 gm  f (m(ξ))J (ξ) dξ1 dξ2  f (m(ξ m ))J (ξ m )wm . (A.5) 0

0

m=1

Diffuse photon propagation in multilayered geometries

z

1

513

ξ2

η2

T4

y

T1

x T1

T3

T4

J(ξ(η)) T2 0 1

T1

J (η)

T3

ξ1

T2 1

ξ2

η2 T1

T4 T1

T4

T3 T2 0

T1

η1

η1

η2 T3

ξ1

T2

T2

η2 η1

T3

η1

1

Figure 10. Two cases of iso-parametric triangle subdivision for different positions of the singular point.

For the matrix elements that involve nodes that are close to each other the kernels of the integral vary sharply, with a singularity occurring when they are coincident. There are several approaches for singular integration, including integration by regularization and by subtraction and series expansion. In this work, we use a regularization method introducing a division of the singular elements into several triangles and then a secondary mapping of the singular triangles to squares. Gaussian quadrature is then applied to the squares for the integration. The secondary mapping involves two different approaches depending on the position of the singular node in the triangle. In both cases the triangle is first divided into four subtriangles T1 , T2 , T3 and T4 . Singularity at vertex node. Taking node 1 as representative, as shown in the top row of figure 10, the singularity is contained in subtriangle T1 and the integral over T2 , T3 and T4 can be calculated by the use of the non-singular method. Subtriangle T1 is mapped onto a square by introducing a new space of local coordinates η = (η1 , η2 ) as shown in the top right of figure 10. The mapping is defined by
 1 (1 + η1 )(1 − η2 ) ξ(η) = . (A.6) 8 (1 + η1 )(1 + η2 ) Singularity at mid edge node. Taking node 2 as an example, as shown in figure 10 on the bottom row, the singular node participates in the three subtriangles T1 , T2 , T3 , with integration over the triangle T4 calculated as in the non-singular case. Subtriangles T1 , T2 , T3 are individually mapped to squares with the transformations • For sub-triangle T1 : ξ(η) =


 1 2(1 − η1 ) . 8 (1 + η1 )(1 − η2 )

(A.7)

• For sub-triangle T2 :


 1 (1 + η1 )(3 − η2 ) + 2(1 − η1 ) ξ(η) = . (1 + η1 )(1 + η2 ) 8

(A.8)

514

J Sikora et al Table 4. Gauss points and weights for the flat square figure 9. m

η1,m −0.932 47 −0.661 21 −0.238 62 0.238 62 0.661 21 0.932 47

1 2 3 4 5 6

η2,m −0.932 47 −0.661 21 −0.238 62 0.238 62 0.661 21 0.932 47

wm 0.171 32 0.360 76 0.467 91 0.467 91 0.360 76 0.171 32

• For sub-triangle T3 :


 1 (1 + η1 )(1 − η2 ) + 2(1 − η1 ) ξ(η) = . 2(1 + η1 ) 8

(A.9)

Singularities at nodes 3 and 5 are handled as those at node 1, and those at nodes 4 and 6 are handled as at node 2. In all cases, the integral (A.3) then becomes 1 1 f (m(ξ(η)))J (ξ(η))J
(η) dη1 dη2 (A.10) −1

−1

where each of the mappings (A.6)–(A.9) has the same Jacobian J
(η) = (1 + η1 )/32.

(A.11)

The Gaussian quadrature scheme for calculation of integrals on the square is taken from (Aliabadi 2002), using the 6 by 6 combinations of the points and weights in table 4. 6  6 

f (m(ξ(η m,n )))J (ξ(η m,n ))J
(η m,n )wm wn .

(A.12)

m=1 n=1

With this scheme, if the Gauss points from the square are mapped back to the flat triangle, they are concentrated around the node with the singularity. References Akahn-Acar Z and Gencer N G 2004 An advanced boundary element method (BEM) implementation for the forward problem of electromagnetic source imaging Phys. Med. Biol. 49 5011–28 Aliabadi M H 2002 The Boundary Element Method (New York: Wiley) Arridge S R 1999 Optical tomography in medical imaging Inverse Problems 15 R41–93 Arridge S R 2001 Scattering chapter 2.4.2: Diffusive Tomography in Dense Media (New York: Academic) pp 920–36 Arridge S R, Cope M and Delpy D T 1992a Theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis Phys. Med. Biol. 37 1531–60 Arridge S R, Dehghani H, Schweiger M and Okada E 2000 The finite element model for the propagation of light in scattering media: a direct method for domains with non-scattering regions Med. Phys. 27 252–64 Arridge S R and Hebden J C 1997 Optical imaging in medicine: II. Modelling and reconstruction Phys. Med. Biol. 42 841–53 Arridge S R, Schweiger M and Delpy D T 1992b Iterative reconstruction of near-infrared absorption images Inverse Problems in Scattering and Imaging (Proc. SPIE) vol 1767 ed M A Fiddy pp 372–83 Arridge S R, Schweiger M, Hiraoka M and Delpy D T 1993 A finite element approach for modeling photon transport in tissue Med. Phys. 20 299–309 Arridge S R, van der Zee P, Delpy D T and Cope M 1991 Reconstruction methods for infra-red absorption imaging Time-Resolved Spectroscopy and Imaging of Tissues (Proc. SPIE) vol 1431 ed B Chance and A Katzir pp 204–15 Becker A A 1992 The Boundary Element Method in Engineering (Maidenhead: McGraw-Hill)

Diffuse photon propagation in multilayered geometries

515

Boas D A, Brooks D H, Miller E L, DiMarzio C A, Kilmer M, Gaudette R J and Zhang Q 2001 Imaging the body with diffuse optical tomography IEEE Sig. Proc. Mag. 18 57–75 Bonnet M 1999 Boundary Integral Equation Methods for Fluids and Solids (New York: Wiley) Bradley C and Pullan A 2002 Application of the BEM in biopotential problems Eng. Anal. Bound. Elem. 26 391–403 Brebbia C A 1978 The Boundary Element Method for Engineers (New York and London: Pentech Press and Halstead Press) Brebbia C A, Telles J C F and Wrobel L C 1984 Boundary Element Techniques. Theory and Applications in Engineering (Berlin: Springer) Brebbia C A and Walker S 1980 The Boundary Element Method for Engineers (London: Newnes-Butterworths) Chance B et al 1998 A novel method for fast imaging of brain function non-invasively with light Opt. Express 2 411–23 de Munck J C 1992 A linear discretization of the volume conductor boundary integral equation using analytically integrated elements IEEE Trans. Biomed. Eng. 39 986–90 de Munck J C, Faes T J C and Heethaar R M 2000 The boundary element method in the forward and inverse problem of electrical impedance tomography IEEE Trans. Biomed. Eng. 47 792–800 Duraiswami R, Sarkar K and Chahine G L 1998 Efficient 2D and 3D electrical impedance tomography using dual reciprocity boundary element techniques Eng. Anal. Bound. Elem. 22 13–31 Fan Y, Jiang T Z and Evans D J 2001 Parallel algorithm for BEM based EEG/MEG forward problem Neuroimage 13 S1301 Fantini S, Walker S A, Franceschini M A, Kaschke M, Schlag P M and Moesta K T 1998 Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods Appl. Opt. 37 1982–9 Fedele F, Eppstein M J, Laible J P, Godavarty A and Sevick-Muraca E M 2005 Fluorescence photon migration by the boundary element method J. Comp. Phys. 210 109–32 Ferguson A S and Stroink G 1997 Factors affecting the accuracy of the boundary element method in the forward problem: 1. Calculating surface potentials IEEE Trans. Biomed. Eng. 44 1139–55 Frijns J H M, de Snoo S L and Schoonhoven R 2000 Improving the accuracy of the boundary element method by the use of second-order interpolation functions IEEE Trans. Biomed. Eng. 47 1336–46 Fuchs M, Wagner M and Kastner J 2001 A standard BEM head model for EEG source reconstruction Neuroimage 13 S1303 Gencer N G and Tanzer I O 1999 Forward problem solution of electromagnetic source imaging using a new BEM formulation with high-order elements Phys. Med. Biol. 44 2275–87 Gibson A, Hebden J C and Arridge S R 2005 Recent advances in diffuse optical tomography Phys. Med. Biol. 50 R1–R43 Harrington R F 1968 Field Computation by Moment Methods (Malabar, FL: Krieger) Haueisen J, Bottner A, Funke M, Brauer H and Nowak H 1997 The influence of boundary element discretization on the forward and inverse problem in electroencephalography and magnetoencephalography Biomed. Tech. 42 240–8 Hebden J C, Gibson A P, Austin T, Yusof R Md, Everdell N, Delpy D T, Arridge S R, Meek J H and Wyatt J S 2004 Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography Phys. Med. Biol. 49 1117–30 Hebden J C, Schmidt F E W, Fry M E, Schweiger M, Hillman E M C, Delpy D T and Arridge S R 1999 Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data Opt. Lett. 24 534–6 Heino J, Arridge S R, Sikora J and Somersalo E 2003 Anisotropic effects in highly scattering media Phys. Rev. E 68 31908 Itagaki M and Brebia C A 1988 Boundary element method applied to neutron diffusion problems Proc. 10th Int. Conf. BEM (Southampton University) (Berlin: Springer) Kantorovich L V and Krylov V I 1964 Approximate Methods of Higher Analysis (New York: Wiley) (translated from Russian by C D Benster) Kolehmainen V, Arridge S R, Vauhkonen M and Kaipio J P 2000a Simultaneous reconstruction of internal tissue region boundaries and coefficients in optical diffusion tomography Phys. Med. Biol. 45 3267–83 Kolehmainen V, Prince S, Arridge S R and Kaipio J P 2000b A state estimation approach to non-stationary optical tomography problem J. Opt. Soc. Am. A 20 876–84 Kybic J, Clerc M, Abboud T, Faugeras O, Keriven R and Papadopoulo T 2005 A common formalism for the integral formulations of the forward EEG problem IEEE Trans. Med. Imaging 24 12–28 Leahy R M, Mosher J C, Spencer M E, Huang M X and Lewine J D 1998 A study of dipole localization accuracy for MEG and EEG using a human skull phantom Electroencephalogr. Clin. Neurophysiol. 107 159–73 Lu T and Yevick D O 2002 Boundary element analysis of dielectric waveguides J. Opt. Soc. 19 1197–206

516

J Sikora et al

Model R, Orlt M, Walzel M and H¨unlich R 1997 Reconstruction algorithm for near-infrared imaging in turbid media by means of time-domain data J. Opt. Soc. Am. A 14 313–24 Mosher J C, Leahy R M and Lewis P S 1999 EEG and MEG: forward solutions for inverse methods IEEE Trans. Biomed. Eng. 46 245–59 Ney M M 1985 Methods of moments as applied to electromagnetics problems IEEE Trans. Microw. Theory Tech. 33 972–80 Park S J and Kwon T H 1996 Sensitivity analysis formulation for 3D conduction heat transfer with complex geometries using a boundary element method Int. J. Numer. Meth. Eng. 39 2837–62 Paulsen K D and Jiang H 1995 Spatially-varying optical property reconstruction using a finite element diffusion equation approximation Med. Phys. 22 691–701 Pogue B W, Paulsen K D, Abele C and Kaufman H 2000 Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms J. Biomed. Opt. 5 185–93 Riley J, Dehghani H, Schweiger M, Arridge S R, Ripoll J and Nieto-Vesperinas M 2000 3D optical tomography in the presence of void regions Opt. Express 7 462–7 Ripoll J, Arridge S R, Dehghani H and Nieto-Vesperinas M 2000 Boundary conditions for light propagation in diffusive media with nonscattering regions J. Opt. Soc. Am. A 17 1671–81 Ripoll J and Nieto-Vesperinas M 1999a Scattering integral equations for diffusive waves. detection of objects buried in diffusive media in the presence of rough interfaces J. Opt. Soc. Am. A 16 1453–65 Ripoll J and Nieto-Vesperinas M 1999b Index mismatch for diffusive photon density waves both at flat and rough diffuse-diffuse interfaces J. Opt. Soc. Am. A 16 1947–57 Ripoll J and Nieto-Vesperinas M 1999c Reflection and transmission coefficients for diffusive photon density waves Opt. Lett. 24 796–8 Saad Y and Schultz M H 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–69 Sarvas J 1987 Basic mathematical and electromagnetic concepts of the biomagnetic inverse problems Phys. Med. Biol. 32 1122 Schweiger M and Arridge S R 1997 The finite element model for the propagation of light in scattering media: frequency domain case Med. Phys. 24 895–902 Schweiger M, Arridge S R, Hiraoka M and Delpy D T 1995 The finite element method for the propagation of light in scattering media: boundary and source conditions Med. Phys. 22 1779–92 Sikora J and Arridge S R 2002 Some numerical aspects of 3D BEM application to optical tomography IVth International Workshop Computational Problems of Electrical Engineering pp 59–62 Sikora J, Riley J, Arridge S R, Zacharopoulos A D and Ripoll J 2003 Analysis of light propagation in diffusive media with non-scattering regions using 3D BEM XIIth International Symposium on Theoretical Electrical Engineering ISTET03 (Warsaw, Poland) pp 511–4 Strait B J 1980 Applications of the Method of Moments to Electromagnetics (St. Cloud, FL: SCEEE) Takahashi S, Imai D and Yamada Y 1997 Fundamental 3D FEM analysis of light propagation in head model towards 3D optical tomography Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II vol 2979 (Proc. SPIE) pp 130–8 Vanel L, Lemieux P A and Durian D J 2001 Diffusing wave spectroscopy for arbitrary geometries: numerical analysis by boundary element method Appl. Opt. 40 1336–46 Yodh A G and Boas D A 2003 Biomedical Photonics Handbook (Boca Raton, FL: CRC Press) Zienkiewicz O C and Taylor R L 1989 The Finite Element Method vol 1, 4th edn (New York: McGraw-Hill)