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Adaptive finite element based tomography for fluorescence optical imaging in tissue

Amit Joshi, Wolfgang Bangerth, and Eva M. Sevick-Muraca

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Abstract

A three-dimensional fluorescence-enhanced optical tomography scheme based upon an adaptive finite element formulation is developed and employed to reconstruct fluorescent targets in turbid media from frequency-domain measurements made in reflectance geometry using area excitation illumination. The algorithm is derived within a Lagrangian framework by treating the photon diffusion model as a constraint to the optimization problem. Adaptively refined meshes are used to separately discretize maps of the forward/adjoint variables and the unknown parameter of fluorescent yield. A truncated Gauss-Newton method with simple bounds is used as the optimization method. Fluorescence yield reconstructions from simulated measurement data with added Gaussian noise are demonstrated for one and two fluorescent targets embedded within a 512ml cubical tissue phantom. We determine the achievable resolution for the area-illumination/area-detection reflectance measurement geometry by reconstructing two 0.4cm diameter spherical targets placed at at a series of decreasing lateral spacings. The results show that adaptive techniques enable the computationally efficient and stable solution of the inverse imaging problem while providing the resolution necessary for imaging the signals from molecularly targeting agents.

©2004 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Adaptive tomography algorithm. GN stands for Gauss-Newton; see Section 2.4 for a description of symbols.

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Fig. 2.
Fig. 2. Area illumination and area detection geometry employed by Thompson et al. [23]

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Fig. 3.
Fig. 3. Single target reconstruction: A black wire-frame depicts the actual target and colored blocks represent the reconstruction. Top 10% of the contour levels of µaxf are shown.

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Fig. 4.
Fig. 4. Adaptive mesh evolution for state/adjoint (left) and parameter discretization (right). Meshes are shown at 1st, 11th and 22nd Gauss-Newton iterations.

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Fig. 5.
Fig. 5. Dual target reconstructions: A black wire-frame depicts the actual targets and colored blocks represent the reconstruction. Top 10% of the contour levels of µaxf are shown. Edge to edge spacing: (a) 1.0142cm, (b) 0.6607cm, (c) 0.3071cm, and (d) 0.1657cm.

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Fig. 6.
Fig. 6. Dual target reconstruction for 0.1cm target separation.

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Tables (1)

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Table 1. Summary of results for dual fluorescent target reconstructions. d is the edge to edge target separation in cm; Iter. is the Gauss-Newton iteration for which the other results are reported; ‖q-q true‖2 is the error in reconstructed parameter; 1 2 ∥ v − z ∥ Σ 2 is the meausurement error; Nq is the number of elements (unknowns) in the parameter mesh.

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Equations (34)

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− ∇ . [ D x ( r ) ∇ u ( r , ω ) ] + k x u ( r , ω ) = 0 ,
− ∇ ∙ [ D m ( r ) ∇ v ( r , ω ) ] + k m v ( r , ω ) = β xm u ( r , ω ) ,
2 D x ∂ u ∂ n + γ u + S ( r ) = 0 , 2 D m ∂ v ∂ n + γ v = 0 ,
A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = 0 ∀ ζ , ξ ∈ H 1 ,
A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = ( D x ∇ u , ∇ ζ ) Ω + ( k x u , ζ ) ∂ Ω + γ 2 ( u , ζ ) ∂ Ω + 1 2 ( S , ζ ) ∂ Ω
+ ( D m ∇ v , ∇ ξ ) Ω + ( k m v , ξ ) Ω + γ 2 ( v , ξ ) ∂ Ω − ( β xm u , ξ ) Ω .
J ( q , v ) = 1 2 ∥ v − z ∥ Σ 2 + β r ( q ) ,
min q , u , v J ( q , v ) subject to A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = 0 .
L ( [ u , v ] , [ λ ex , λ em ] , q ) = J ( q , v ) + A ( q ; [ u , v ] ) ( [ λ ex , λ em ] ) .
L x ( x ) ( y ) = 0 ∀ y = { φ ex , φ em , ψ ex , ψ em , χ } ,
L u ( x ) ( φ ex ) = A u ( q ; [ u , v ] ) ( φ ex ) ( [ λ ex , λ em ] ) = 0 ,
L v ( x ) ( φ em ) = J v ( q , v ) ( φ em ) + A v ( q ; [ u , v ] ) ( φ em ) ( [ λ ex , λ em ] ) = 0 ,
L λ ex ( x ) ( ψ ex ) = A ( q ; [ u , v ] ) ( [ ψ ex , 0 ] ) = 0 ,
L λ em ( x ) ( ψ em ) = A ( q ; [ u , v ] ) ( [ 0 , ψ em ] ) = 0 ,
L q ( x ) ( χ ) = J q ( q , v ) ( χ ) + A q ( q ; [ u , v ] ) ( χ ) ( [ λ ex , λ em ] ) = 0 .
L xx ( x k ) ( δ x k , y ) = − L x ( x k ) ( y ) ∀ y ,
A u ( q k ; [ u k , v k ] ) ( ϕ e x ) ( [ δ λ k e x , 0 ] ) + A u ( q k ; [ u k , v k ] ) ( ϕ e x ) ( [ 0 , δ λ k e m ] ) + A u q ( q k ; [ u k , v k ] ) ( ϕ e x , δ q k ) ( [ λ e x , λ e m ] ) = − L u ( x k ) ( ϕ e x ) ,
J v v ( q k , v k ) ( δ v k , ϕ e m ) + A v ( q k ; [ u k , v k ] ) ( ϕ e m ) ( [ δ λ k e x , 0 ] ) + A v ( q k ; [ u k , v k ] ) ( ϕ e m ) ( [ 0 , δ λ k e m ] ) + J v q ( q k , v k ) ( δ q k , ϕ e m ) + A v q ( q k ; [ u k , v k ] ) ( ϕ e m , δ q k ) ( [ λ e x , λ e m ] ) = − L v ( x k ) ( ϕ e m ) ,
A u ( q k ; [ u k , v k ] ) ( δ u k ) ( [ ψ e x , 0 ] ) + A v ( q k ; [ u k , v k ] ) ( δ v k ) ( [ ψ e x , 0 ] ) + A q ( q k ; [ u k , v k ] ) ( δ q k ) ( [ ψ e x , 0 ] ) = − L λ e x ( x k ) ( ψ e x ) ,
A u ( q k ; [ u k , v k ] ) ( δ u k ) ( [ 0 , ψ e m ] ) + A v ( q k ; [ u k , v k ] ) ( δ v k ) ( [ 0 , ψ e m ] ) + A q ( q k ; [ u k , v k ] ) ( δ q k ) ( [ 0 , ψ e x ] ) = − L λ e m ( x k ) ( ψ e m ) ,
A q u ( q k ; [ u k , v k ] ) ( δ u k , χ ) ( [ λ e x , λ e m ] ) + A q v ( q k ; [ u k , v k ] ) ( δ v k , χ ) ( [ λ e x , λ e m ] ) + J q v ( q k , v k ) ( δ v k , χ ) + J q q ( q k , v k ) ( δ q k , χ ) + A q ( q k ; [ u k , v k ] ) ( χ ) ( [ δ λ e x , 0 ] ) + A q ( q k ; [ u k , v k ] ) ( χ ) ( [ 0 , δ λ e m ] ) = − L q ( x k ) ( χ ) .
x k + 1 = x k + α k δ x k .
[ M 0 P T 0 R C T P C 0 ] [ δ p k δ q k δ d k ] = [ F 1 F 2 F 3 ] ,
M = [ 0 0 0 ( φ i , φ j ) Σ ] i j , R = [ β r " ( q k , χ i , χ j ) ] ij , P T = [ A global ex − B global ex → em 0 A global em ] .
C 1 = ( ∂ D x ( q k ) ∂ q ∇ u k ∙ ∇ ψ i , χ j ) ij + ( ∂ k x ( q k ) ∂ q u k ψ i , χ j ) ij ,
C 2 = ( ∂ D m ( q k ) ∂ q ∇ v k ∙ ∇ ψ i , χ j ) ij + ( ∂ k m ( q k ) ∂ q v k ψ i , χ j ) ij − ( ∂ β xm ( q k ) ∂ q u k ψ i , χ j ) ij .
F 1 = [ − ( D x ( q k ) ∇ λ k ex , ∇ φ i ) − ( k x ( q k ) λ k ex , φ i ) − γ 2 ( λ k ex , φ i ) ∂ Ω + ( β xm ( q k ) λ k em , φ i ) − ( v k − z , φ i ) Σ − ( D m ( q k ) ∇ λ k em , ∇ φ i ) − ( k m ( q k ) λ k em , φ i ) − γ 2 ( λ k em , φ i ) ∂ Ω ] i ,
F 2 = [ − β r ′ ( q k , χ i ) − ( ∂ D x ( q k ) ∂ q ∇ u k ∙ ∇ λ k ex , χ i ) − ( ∂ k x ( q k ) ∂ q u k λ k em , χ i ) − ( ∂ D m ( q k ) ∂ q ∇ v k . ∇ λ k em , χ i ) − ( ∂ k m ( q k ) ∂ q v k λ k em , χ i ) + ( ∂ β xm ( q k ) ∂ q u k λ k em , χ i ) ] i ,
F 3 = [ − D x ( q k ) ∇ ψ i , ∇ u k ) − ( k x ( q k ) ψ i , u k ) − 1 2 ( S x , ψ i ) ∂ Ω − γ 2 ( ψ i , u k ) ∂ Ω − D m ( q k ) ∇ ψ i , ∇ v k ) − ( k m ( q k ) ψ i , v k ) − γ 2 ( ψ i , v k ) ∂ Ω + ( β xm ( q k ) ψ i , u k ) ] i .
{ R + C T P − T M P − 1 C } δ q k = F 2 − C T P − T F 1 + C T P − T M P − 1 F 3 ,
P δ p k = F 3 − C δ q k ,
P T δ d k = F 1 − M δ p k .
∥ u − u h ∥ ≤ C ( u ) h 2 ,
η K u = h 24 ∥ ∂ n u h ∥ ∂ K 2 , η K v = h 24 ∥ ∂ n v h ∥ ∂ K 2 , η K = α η K u + ( 1 − α ) η K v ,
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