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Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm

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Abstract

With the development of in-vivo free-space fluorescence molecular imaging and multi-modality imaging for small animals, there is a need for new reconstruction methods for real animal-shape models with a large dataset. In this paper we are reporting a novel hybrid adaptive finite element algorithm for fluorescence tomography reconstruction, based on a linear scheme. Two different inversion strategies (Conjugate Gradient and Landweber iterations) are separately applied to the first mesh level and the succeeding levels. The new algorithm was validated by numerical simulations of a 3-D mouse atlas, based on the latest free-space setup of fluorescence tomography with 360° geometry projections. The reconstructed results suggest that we are able to achieve high computational efficiency and spatial resolution for models with irregular shape and inhomogeneous optical properties.

©2007 Optical Society of America

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Figures (9)

Fig. 1.
Fig. 1. Flow diagram for the hybrid adaptive finite element reconstruction algorithm
Fig. 2.
Fig. 2. Mesh refinement method. (a) is the original tetrahedral element. (b) is the second generation elements after refinement.
Fig. 3.
Fig. 3. Experimental setup and the geometrical model for a mouse abdomen part. The sketch of the free-space FMT system is illustrated in (a), which is similar to that in Ref. [6]. The plane of excitation sources is shown in (b) where the black points represent positions of the isotropic point sources. The points 01–y015 are the first excitation sources for the 15 projections and the points 016–030 are the second ones. The field of view (FOV) of 150° for detection with respect to the excitation source point 01 is also shown. (c) and (d) are different views of the mouse geometry model used in the reconstruction. In order to reduce the boundary artifacts, which interfere with the finite element computation, the model was generated by sampling the original atlas data (intersections of the vertical and horizontal lines) and then approximating the curves using spline function to form the kidney surface and the body surface.
Fig. 4.
Fig. 4. The forward geometry model for generating surface measurements is shown in (a) and (b) with a fluorescent probe in the left kidney, where the red part represents the fluorescent probe. Different views for the mesh of tetrahedral elements are shown in (c) and (d).
Fig. 5.
Fig. 5. (a). and (b). show different surface views of the reconstruction mesh in the initial mesh level.(c) plots of the Residual Error in the three mesh levels. See text for details.
Fig. 6.
Fig. 6. Reconstruction results for a single fluorescence target embedded in the left kidney using the proposed algorithm. (a) the 3D reconstructed results obtained from the initial uniformly coarse mesh, with a threshold of 70% of the maximum value. (The red cylinders denote the real target). (b) the transverse view of the reconstruction at z=77 in the same mesh, where the white circle indicates the real fluorescence target. (c) and (d) the corresponding reconstruction using the refined, second mesh level. (e) and (f) the final reconstruction utilizing our algorithm, which is based on the third mesh level.
Fig. 7.
Fig. 7. Reconstruction results for a single fluorescence target outside the kidneys using the proposed algorithm. (a) the 3D reconstructed results obtained from the initial uniformly coarse mesh, with a threshold of 70% of the maximum value. (The red cylinders denote the real target). (b) the transverse view of the reconstruction at z=77 in the same mesh, where the white circle indicates the real fluorescence target. (c) and (d) the corresponding reconstruction using the refined, second mesh level. (e) and (f) the final reconstruction utilizing our algorithm, which is based on the third mesh level.
Fig. 8.
Fig. 8. Reconstruction results of double fluorescent targets in different discretized levels using the proposed algorithm. (a) the 3D reconstructed results obtained from the initial uniformly coarse mesh, with a threshold of 70% of the maximum value. (The red cylinders denote the real target). (b) the transverse view of the reconstruction at z=77 in the same mesh, where the white circle indicates the real fluorescence target. (c) and (d) the corresponding reconstruction using the refined, second mesh level. (e) and (f) the final reconstruction utilizing our algorithm, which is based on the third mesh level.
Fig. 9.
Fig. 9. The transverse view of the reconstructed results at z=77, based on a fixed fine mesh with 7339 nodes and 34147 elements. For a single fluorescence target with the same size and location as in Fig. 6, the reconstruction results using CG is shown in (a), with a threshold of 70% of the maximum value. The corresponding results for two fluorescence targets with the same sizes and locations as in Fig. 8, with the same threshold as (a) is shown in (b),

Tables (3)

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Table 1. Summary of the reconstruction results for a single target

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Table 2. Summary of the reconstruction results for double targets

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Table 3. Comparison of reconstruction in a fixed mesh and using the proposed algorithm

Equations (14)

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{ D x ( r ) Φ x ( r ) μ ax ( r ) Φ x ( r ) = Θ s δ ( r r sl ) ( 1.1 ) D m ( r ) Φ m ( r ) μ am ( r ) Φ m ( r ) = Φ x ( r ) η μ af ( r ) ( 1.2 ) ,
2 D x , m Φ x , m n + q Φ x , m = 0 ,
{ Ω ( D x ( r ) Φ x ( r ) · ψ ( r ) + μ ax ( r ) Φ x ( r ) ψ ( r ) ) dr + Ω 1 2 q Φ x ( r ) ψ ( r ) dr = Ω Θ s δ ( r r sl ) ψ ( r ) dr ( 3.1 ) Ω ( D m ( r ) Φ m ( r ) · ψ ( r ) + μ am ( r ) Φ m ( r ) ψ ( r ) ) dr + Ω 1 2 q Φ m ( r ) ψ ( r ) dr = Ω Φ x ( r ) η μ af ( r ) ψ ( r ) dr ( 3.2 ) ,
Φ x , m ( r ) = i = 1 N p Φ xi , mi ψ i ( r ) ,
x ( r ) = η μ af ( r ) = i = l N p ( η μ af ) i γ i ( r ) = i = 1 N p x i · γ i ( r ) ,
[ K x ] { Φ x } = { L x }
[ K m ] { Φ m } = [ F ] { X }
{ Φ m , sl } = [ K m 1 ] · [ F sl ] · { X } = [ B sl ] · { X } ,
{ Φ m , sl Meas } = [ A sl ] · { X } .
{ Φ m , s 1 Meas Φ m , s 2 Meas Φ m , sL Meas } = [ A s 1 A s 2 A sL ] { η μ a f ( 1 ) η μ a f ( 2 ) η μ a f ( N p ) } = [ A ] { X } .
A k X k = Φ k mea
min x L x k x h E k ( X k ) = A k X k Φ k mea 2 2 + λ k η k ( X k ) .
X k + 1 0 = F k k + 1 ( X k )
X k + 1 n + 1 = X k + 1 n + α ( A k + 1 per ) T ( Φ k + 1 mea A k + 1 per X k + 1 n ) ,
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