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A semi-analytical perturbation model for diffusion tomogram reconstruction from time-resolved optical projections

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This paper proposes a perturbation model for time-domain diffuse optical tomography in the flat layer transmission geometry. We derive an analytical representation of the weighting function that models the imaging operator by using the diffusion
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    A semi-analytical perturbation model for diffusion tomogram reconstruction from time-resolved optical projections   Alexander B Konovalov *a , Vitaly V. Vlasov a , Alexander S. Uglov a , Vladimir V. Lyubimov  b   a Russian Federal Nuclear Center – Zababakhin Institute of Applied Physics, PO Box 245, Shezhinsk, Chelyabinsk Region, 456770 Russia;  b Institute for Laser Physics of Vavilov State Optical Institute Corporation, 12 Birzhevaya Lin., St.-Petersburg, 199034 Russia ABSTRACT   This paper proposes a perturbation model for time-domain diffuse optical tomography in the flat layer transmission geometry. We derive an analytical representation of the weighting function that models the imaging operator by using the diffusion approximation of the radiative transfer equation and the perturbation theory by Born. To evaluate the weighing function for the flat layer geometry, the Green’s function of the diffusion equation for a semi-infinite scattering medium with the Robin boundary condition is used. For time-domain measurement data we use the time-resolved optical projections defined as relative disturbances in the photon fluxes, which are caused by optical inhomogeneities. To demonstrate the efficiency of the proposed model, a numerical experiment was conducted, wherein the rectangular scattering objects with two absorbing inhomogeneities and a randomly inhomogeneous component were reconstructed. Test tomograms are recovered by means of the multiplicative algebraic reconstruction technique modified by us. It is shown that nonstandard interpretation of the time-domain measurement data makes it possible to use different time-gating delays for regularization of the reconstruction procedure. To regularize the solution, we state the reconstruction problem for an augmented system of linear algebraic equations. At the recent stage of study the time-gating delays for regularization are selected empirically. Keywords:  Diffuse optical tomography, perturbation model, diffusion approximation, Robin boundary condition, time-resolved optical projection, multiplicative algebraic reconstruction technique   1.   INTRODUCTION The main problem of diffuse optical tomography (DOT) is low spatial resolution due to the multiple scattering of  photons which do not have regular trajectories and are distributed in the entire tissue being probed. The nonlinear reconstruction methods [1,2] based on multi-step linearization of the imaging operator allow gaining relatively high spatial resolution (3-5 mm) for diffusion tomograms, but they are not as fast as required for real-time medical diagnosis. A variety of approximate methods are investigated to find a reasonable trade-off between reconstruction accuracy and gain in computational time. The perturbation methods [1-8] are considered to be the most efficient of them. This paper develops the perturbation method for time-domain DOT in the flat layer transmission geometry. To evaluate the weighting function that models the imaging operator, we use the analytical expression for the Green’s function of the nonstationary diffusion equation for a semi-infinite medium. The novelties of the model developed here are as follows. Firstly, we consider the Robin boundary condition that is not simple as the zero or extrapolated boundary condition, but most appropriate in the view of the diffusion theory [9]. Secondly, nonstandard measurement data we call the time-resolved optical projections are used for diffusion tomogram reconstruction. The time-resolved optical projection is defined as relative disturbance in the signal caused by optical inhomogeneities. Here the signal is the photon flux measured at a chosen time-gating delay of receiver, i.e. the single count of the temporal point spread function (TPSF). Similar interpretation of the time-domain measurement data makes it possible to use different time-gating delays for regularization of the reconstruction procedure. We demonstrate the efficiency of our model by a numerical experiment on reconstruction of the rectangular scattering objects with two absorbing inhomogeneities and a randomly inhomogeneous component. The time-resolved optical projections are simulated by the finite element solution of the *  a_konov@mail.vega-int.ru; phone 73514654639; fax 73514652233; vniitf.ru Diffuse Optical Imaging III, edited by Andreas H. Hielscher, Paola Taroni, Proc. of SPIE-OSA Biomedical Optics,SPIE Vol. 8088, 80880T · © 2011 SPIE-OSA · CCC code: 1605-7422/11/$18 · doi: 10.1117/12.889755SPIE-OSA/ Vol. 8088 80880T-1 Downloaded from SPIE Digital Library on 30 Jun 2011 to 83.146.114.9. Terms of Use: http://spiedl.org/terms    nonstationary diffusion equation. And the tomograms are recovered with the help of the multiplicative algebraic reconstruction technique (MART) we modified to attain better convergence of the iterative procedure [10,11]. 2.   THE PERTURBATION MODEL 2.1   The fundamental equation Let us assume that the photon density (,) t  ϕ  r  in a volume V   of a scattering medium limited piecewise-closed smooth surface Σ  satisfies the nonstationary diffusion equation 2 (,)()(,)()(,)(,) a s s t cD t c t t t t  ϕ ϕ μ ϕ δ  ∂− ∇ + = − −∂ rrrrrrr  (1) with the Robin boundary condition [9] (,)(,)2()0 d  d d d  t t AD  ϕ ϕ η  ∈Σ ∂⎡ ⎤+ =⎢ ⎥∂⎣ ⎦ r rrr , (2) where c  is the velocity of light in the medium, ()  D r  is the diffusion coefficient, () a μ  r  is the absorption coefficient, (,)  s s t t  δ   − − rr  is an instantaneous point source,  A  is the boundary term that incorporates the refractive index mismatch at the tissue-air boundary, /  η  ∂ ∂  is the derivative in the direction of the outer normal to the surface Σ  at the point d   ∈Σ r . Let the optical inhomogeneities be defined by the local disturbances ()()  D D D δ   = − rr  and ()() a a a δμ μ μ  = − rr , where  D  and a μ   are the optical parameters for the homogeneous medium. Let the signal disturbance (,;,)  s s d d   I t t  δ  rr  caused by optical inhomogeneities be small in comparison with the value of the unperturbed signal 0 (,;,)  s s d d   I t t  rr  measured on the medium boundary at the point d  r  for time-gating delay d s t t  − . In this case the  perturbation theory can be used and the solution of equation (1) at the first Born approximation is defined as the sum 0 (,)(,)(,) t t t  ϕ ϕ δϕ  = + rrr  (3) where 0 (,) t  ϕ  r  is the solution of equation (1) for the homogeneous medium and (,) t  δϕ  r  is a solution disturbance induced by inhomogeneities. Let us rewrite (1) as follows 2 ()(,)1(,)(,)(,)()()() a s s c t t t c t t  D t D D μ δ ϕ ϕ ϕ   − −∂− ∇ + =∂ rrrrrrrrr . (4) Substituting solution (3) as well as the expressions ()() a a a μ μ δμ  = + rr  and 11()1()  D D D D δ  ⎡ ⎤≈ −⎢ ⎥⎣ ⎦ rr  into equation (4), opening the brackets, and rejecting terms of a higher order of smallness, we get the following equation for (,) t  δϕ  r   [ ] [ ] 200 (,)(,)(,)(,)(,),11(,)()ln(,)(). aa a t cD t c t cS t t t S t t D D c t  δϕ δϕ μ δϕ ϕ δμ μ ϕ δ  ∂− ∇ + =∂⎡ ⎤∂= − + +⎢ ⎥∂⎣ ⎦ rrrrrrrrr  (5) The solution of (5) at the spatiotemporal point (,) d d  t  r  of receiver can be written as follows [12] 30 (,)(,)(,)(,) d  s t d d d d t V  t cS t t G t t d rdt  δϕ ϕ  = − − ∫ ∫ rrrrr , (6) where (,) G t t  ′ ′− − rr  is the Green’s function of equation (1) for the homogeneous medium. Let us use expression (6) to derive the integral representation for the time-resolved optical projection 0 (,;,)(,;,)/(,;,)  s s d d s s d d s s d d   g t t I t t I t t  δ  = rrrrrr . According to the first Fick law SPIE-OSA/ Vol. 8088 80880T-2 Downloaded from SPIE Digital Library on 30 Jun 2011 to 83.146.114.9. Terms of Use: http://spiedl.org/terms    (,)(,;,)()  d d  s s d d d  t  I t t cD  ϕ η  ∂= −∂ rrrr . (7) Expression (7), combined with the boundary condition (2), gives the simpler form ),(~),;,( d d d d  s s  t t t  I  rrr  ϕ  . (8) Thus for the time-resolved optical projection we can write 3000 (,)(,)(,)(,;,)(,)(,)(,) d  s t d d d d  s s d d t V d d d d  t t G t t  g t t cS t d rdt t t  δϕ ϕ ϕ ϕ  − −= = ∫ ∫ rrrrrrrrr . (9) Taking into account that for the instantaneous point source 0 (,)(,)  s s t G t t  ϕ   = − − rrr , we get the following expression 3 (,;,)(,;,;)()(,;,;)() a  s s d d s s d d a D s s d d V   g t t W t t W t t D d r  μ   δμ δ  ⎡ ⎤= +⎣ ⎦ ∫ rrrrrrrrrr , (10) where (,;,;) a  s s d d  W t t  μ  rrr  and (,;,;)  D s s d d  W t t  rrr  are the weighting functions (,)(,)(,;,;)(,) d a s t  s s d d  s s d d t d s d s G t t G t t W t t c dt G t t  μ  − − − −=− − ∫ rrrrrrrrr , and (11) (,)(,)1(,;,;)ln(,)(,) d  s t  s s d d  D s s d d a s st d s d s G t t G t t W t t c G t t dt  D G t t t  μ  − − − − ∂⎡ ⎤= + − −⎢ ⎥− − ∂⎣ ⎦ ∫ rrrrrrrrrrr . (12) Expression (10) is the fundamental equation that describes the reconstruction model we propose. 2.2   Evaluation of the weighting function The aim of this paper is to create the reconstruction model for the flat layer transmission geometry frequently applied in diffuse optical mammotomography [13]. In practice, the distance between two glass plates compressing the breast, on which radiation sources and receivers are located, is no less than 5cm. It does mean that the influence of the boundary on the diffusion process of photon migration in breast tissue near the opposite boundary is negligibly small and the analytical expressions for the semi-infinite medium can be used for its description. The Green’s function of equation (1) with boundary condition (2) for the semi-infinite homogeneous medium can be written as [12] [ ] [ ] 3/2222222 (,)4()exp()()()()()()()expexp,4()4() a G t t Dc t t c t t  x x y y z z x x y y z z J  Dc t t Dc t t AD π μ  − ′ ′ ′ ′− − = − − −⎧ ⎫′ ′ ′ ′ ′ ′⎡ ⎤ ⎡ ⎤− + − + − − + − + +⎪ ⎪× − + − −⎨ ⎬⎢ ⎥ ⎢ ⎥′ ′− −⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ rr  (13) where  J   has the integral representation 2220 ()()()expexp24()  x x y y z z  J d  AD Dc t t  ξ ξ ξ  ∞  ′ ′ ′⎡ ⎤− + − + + +⎛ ⎞= − −⎢ ⎥⎜ ⎟′−⎝ ⎠ ⎣ ⎦ ∫ . (14) Integral (14) is expressed via the complementary error function 2 2efrc()exp() d  α  α ξ ξ π  ∞ = − ∫  as [14] 2222 ()()()()()2()()efrcexp4()44() c t t A z z x x y y c t t A z z  J Dc t t  Dc t t A D A Dc t t  π  ⎡ ⎤′ ′ ′ ′ ′ ′⎡ ⎤− + + − + − − + +′ ⎢ ⎥= − − +⎢ ⎥′−⎢ ⎥′− ⎣ ⎦⎣ ⎦ . (15) Then we can   rewrite the Green’s function in the most suitable form for analytical derivations as well as for numerical calculations SPIE-OSA/ Vol. 8088 80880T-3 Downloaded from SPIE Digital Library on 30 Jun 2011 to 83.146.114.9. Terms of Use: http://spiedl.org/terms    [ ] [ ] 2223/222 ()()()(,)4()exp()expexp4()4()()()()()()2(expefrcexp4()4() a  x x y y z z G t t Dc t t c t t  Dc t t Dc t t  Dc t t  z z c t t A z z c t t A z z  Dc t t AD A Dc t t  π μ π  −  ⎧′ ′ ′⎡ ⎤ ⎡ ⎤− + − −⎪′ ′ ′ ′− − = − − − − −⎨⎢ ⎥ ⎢ ⎥′ ′− −⎪⎣ ⎦ ⎣ ⎦⎩⎡ ⎤′′ − ′ ′ ′ ′⎡ ⎤+ − + + − + +⎢ ⎥+ − −⎢ ⎥′− ⎢ ⎥′−⎣ ⎦ ⎣ ⎦ rr 2 ).4  A D ⎫⎡ ⎤⎬⎢ ⎥⎣ ⎦⎭  (16) By substituting expressions (16) into (11) and (12), we can obtain the analytical representations of the weighting functions. But they seem to be bulky for practical use. To test the model developed, we restrict ourselves here to the case of absorbing inhomogeneities and calculate numerically function (11) that is responsible for their reconstruction. Then we consider a 2D case described by coordinates (,)  x z  * . The result of calculations for parameters: 0.0214cm/ps c  = , 0.034cm  D  = , 0  s t   = , 3000 = d  t   ps, and 3.25  A  =  is presented in figure 1(a) as the 2D "banana-like" distribution. To gain a better understanding, we inscribed this distribution in the rectangular scattering object of size 10×8 cm 2  chosen for test reconstruction. The source is located inside the object at a point (3.0,7.4cm)  s r  and the receiver is located on the  boundary 0  z   =  at a point (7.0cm,0) d  r . (a) (b) Figure 1. The "banana-like" distributions as the results of weighting function evaluation for the semi-infinite medium (a) and the flat layer transmission geometry (b). The coordinate axes are graduated in centimeters. To obtain a similar distribution for the case of the flat layer transmission geometry, we use the central symmetry method described in [10,11]. The basic assumption here is that the character of photon migration near the boundaries of the source and the receiver must be the same. The result of reflection is presented in figure 1(b). After reflection the source appears on the boundary 8 =  z  cm at the point (3.0,8.0cm)  s r . The "banana-like" distribution shown in figure 1(b) defines the weights for the single source-receiver pair. By computing similar distributions for different connections  between sources and receivers, we can construct the weighting matrix for reconstruction. 3.   RECONSTRUCTION FROM THE TIME-RESOLVED OPTICAL PROJECTIONS 3.1   The reconstruction algorithm The discrete model of 2D reconstruction is formulated traditionally [7]. The reconstruction problem is reduced to the solution of the system of linear algebraic equations g=W×f  , where {} i  g  = g  is the set of time-resolved optical  projections, {} ij W  = W  is the weighing matrix, and {}  j  f  = f   is a vector that defines the sought distribution of absorbing inhomogeneities. For reconstruction, we use the MART modified by us [10,11] to attain better convergence of the iterative reconstruction. The main idea is to take into account the distribution of the weight sums over picture elements. This distribution is usually non-uniform. As a result, the MART often converges to a wrong solution if data are incomplete. Because of the incorrect redistribution of intensity, tomograms exhibit distinct artifacts which are often *  In the 2D case formula (16) will contain [ ] 1 4()  Dc t t  π   − ′−  instead of [ ] 3/2 4()  Dc t t  π   − ′− . SPIE-OSA/ Vol. 8088 80880T-4 Downloaded from SPIE Digital Library on 30 Jun 2011 to 83.146.114.9. Terms of Use: http://spiedl.org/terms     present in the regions where the structures are actually absent. The solution correction formula of the modified MART has the following form [10,11] /(1)()() ij j W W  s s s j j i ij j j  f f g W f  λ  +  ⎛ ⎞= ⋅⎜ ⎟⎝ ⎠ ∑ % , (17) where ij Li W W N  = ∑ %  is the reduced weight sum for the  j - picture element,  L  N   is the total number of source-receiver connections used in reconstruction. Figure 2 demonstrates the efficiency of the proposed approach. Figure 2(a) shows an example of the distribution of the weight sums. Figures 2(b) and 2(c) present results of the reconstruction of the rectangular scattering object with a randomly inhomogeneous component (RIC). The results are obtained for a low number of iterations (10) with the help of the non-modified and modified MART, respectively. We can see that the result obtained with the use of formula (17) is much better. (a)   (b) (c) Figure 2. An example of the distribution of the weight sums over picture elements (a) and the results of reconstruction with the help of the non-modified (b) and modified (c) MART. 3.2   The regularization approach The feature of the reconstruction model proposed is that the time-resolved optical projections we use for reconstruction are defined only for the single time-gating delay of receivers. In other words, to reconstruct a tomogram, we need not to know full TPSF, but the single count of the TPSF only. As for the different time-gating delays, we try to use them for regularization. To regularize the reconstruction procedure, we state the reconstruction problem for an augmented system which is constructed with additional time-resolved optical projections taken for different time-gating delays of receivers. At the recent stage of study the time-gating delays for regularization are selected empirically. The basic value of time- SPIE-OSA/ Vol. 8088 80880T-5 Downloaded from SPIE Digital Library on 30 Jun 2011 to 83.146.114.9. Terms of Use: http://spiedl.org/terms
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