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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References

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Acad. Radiol. (1)

M. G. Pomper, �??Molecular Imaging: An Overview,�?? Acad. Radiol. 8, 1141�??1153 (2001).
[CrossRef] [PubMed]

Acta Numerica (1)

R. Becker and R. Rannacher, �??An optimal control approach to error estimation and mesh adaptation in finite element methods,�?? Acta Numerica 10, 1�??102 (2001).
[CrossRef]

Appl. Opt. (6)

Biomedical Photonics Handbook (1)

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

ECMOR VIII 2002 (1)

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Naevdal, and H. Urkedal, �??Adaptive selection of parameterization for reservoir history matching,�?? in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E�??46 (European Association of Geoscientists and Engineers (EAGE), 2002).

IEEE J. Sel. Top. Quantum Electron. (1)

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, �??Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,�?? IEEE J. Sel. Top. Quantum Electron. 54, 930�??935 (1999).

Int. J. Num. Meth. Engrg. (1)

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, �??A posteriori error analysis and adaptive processes in the finite element method: Part I�??Error Analysis,�?? Int. J. Num. Meth. Engrg. 19, 1593�??1619 (1983).
[CrossRef]

Intl. Symp. Biomedical Imaging 2004 (1)

A. Joshi, W. Bangerth, and E. Sevick-Muraca, �??Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,�?? in International Symposium on Biomedical Imaging, pp. 1103�??1106 (IEEE, 2004).

Inverse Problems (2)

H. Ben Ameur, G. Chavent, and J. Jaffrè, �??Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,�?? Inverse Problems 18, 775�??794 (2002).
[CrossRef]

R. Luce and S. Perez, �??Parameter identification for an elliptic partial differential equation with distributed noisy data,�?? Inverse Problems 15, 291�??307 (1999).
[CrossRef]

J. Biomed. Opt. (2)

A. B. Thompson and E. M. Sevick-Muraca, �??NIR fluorescence contrast enhanced imaging with ICCD homodyne detection: measurement precision and accuracy,�?? J. Biomed. Opt. 8, 111�??120 (2002).
[CrossRef]

J. P. Houston, S. Ke,W.Wang, C. Li, and E. M. Sevick-Muraca, �??Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,�?? J. Biomed. Opt. (submitted) (2004).

J. Luminescence (1)

M. A. O�??Leary, D. A. Boas, B. Chance, and A. Yodh, �??Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,�?? J. Luminescence 60, 281�??286 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

Med. Phys. (2)

X. Gu, Y. Xu, and H. Jiang, �??Mesh-based enhancement schemes in diffuse optical tomography,�?? Med. Phys. 30(5), 861�??869 (2003).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, �??A submillimeter resolution fluorescence molecular imaging system for small animal imaging,�?? Med. Phys. 30, 901�??911 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

Phys. Med. Biol. (1)

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, �??Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,�?? Phys. Med. Biol. 48, 1701�??1720 (2003).
[CrossRef] [PubMed]

Physiological Measurement (2)

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, �??Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,�?? Physiological Measurement 22, 91�??96 (2001).
[CrossRef] [PubMed]

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, �??Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,�?? Physiological Measurement 23, 121�??128 (2002).
[CrossRef] [PubMed]

Proc. Nat. Acad. Sci. (1)

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, �??Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,�?? Proc. Nat. Acad. Sci. 99, 9619�??9624 (2002).
[CrossRef] [PubMed]

SIAM J. Contr. Optim. (1)

R. Becker, H. Kapp, and R. Rannacher, �??Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept,�?? SIAM J. Contr. Optim. 39, 113�??132 (2000).
[CrossRef]

SIAM J. Control Optim. (1)

R. Li, W. Liu, H. Ma, and T. Tang, �??Adaptive finite element approximation for distributed elliptic optimal control problems,�?? SIAM J. Control Optim. 41, 1321�??1349 (2002).
[CrossRef]

Other (12)

W. Bangerth, R. Hartmann, and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference (2004). <a href= "http://www.dealii.org/">http://www.dealii.org/</a>

A. Thompson, �??Development of a new optical imaging modality for detection of fluorescence enhanced disease,�?? Ph.D. thesis, Texas A & M University (2003).

D. G. Luenberger, Optimization by Vector Space Methods (John Wiley, 1969).

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, New York, 1999).
[CrossRef]

S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Elements (Springer, Berlin-Heidelberg-New York, 1994).

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (Wiley/Teubner, New York, Stuttgart, 1996).

W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations (Birkhäuser Verlag, 2003).

W. Bangerth, �??Adaptive Finite Element Methods for the Identification of Distributed Coefficients in Partial Differential Equations,�?? Ph.D. thesis, University of Heidelberg (2002).

W. Bangerth, �??A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques,�?? Tech. Rep. 04-39, ICES, University of Texas at Austin (2004).

R. A. Adams, Sobolev Spaces (Academic Press, 1975).

A. N. Tikhonov and V. Y. Arsenin, eds., Solution of Ill-Posed Problems (Winston, Washington, DC, 1977).

L. Beilina, �??Adaptive Hybrid FEM/FDM Methods for Inverse Scattering Problems,�?? Ph.D. thesis, Chalmers University of Technology (2002).

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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.

Fig. 2.
Fig. 2.

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

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.

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.

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.

Fig. 6.
Fig. 6.

Dual target reconstruction for 0.1cm target separation.

Tables (1)

Tables Icon

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.

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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