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Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization

Scott C. Davis, Hamid Dehghani, Jia Wang, Shudong Jiang, Brian W. Pogue, and Keith D. Paulsen

Open Access Open Access

Abstract

A promising method to incorporate tissue structural information into the reconstruction of diffusion-based fluorescence imaging is introduced. The method regularizes the inversion problem with a Laplacian-type matrix, which inherently smoothes pre-defined tissue, but allows discontinuities between adjacent regions. The technique is most appropriately used when fluorescence tomography is combined with structural imaging systems. Phantom and simulation studies were used to illustrate significant improvements in quantitative imaging and linearity of response with the new algorithm. Images of an inclusion containing the fluorophore Lutetium Texaphyrin (Lutex) embedded in a cylindrical phantom are more accurate than in situations where no structural information is available, and edge artifacts which are normally prevalent were almost entirely suppressed. Most importantly, spatial priors provided a higher degree of sensitivity and accuracy to fluorophore concentration, though both techniques suffer from image bias caused by excitation signal leakage. The use of spatial priors becomes essential for accurate recovery of fluorophore distributions in complex tissue volumes. Simulation studies revealed an inability of the “no-priors” imaging algorithm to recover Lutex fluorescence yield in domains derived from T1 weighted images of a human breast. The same domains were reconstructed accurately to within 75% of the true values using prior knowledge of the internal tissue structure. This algorithmic approach will be implemented in an MR-coupled fluorescence spectroscopic tomography system, using the MR images for the structural template and the fluorescence data for region quantification.

©2007 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. An axial T1 weighted MR slice of a human breast is shown in (a), from which the test domain for simulation studies was derived. Darker regions indicate fibro-glandular tissue imbedded in adipose tissue indicated by lighter values. The image was acquired during a clinical exam with one of our MR-coupled NIR tomography systems and shows the indentations caused by the fiber optic probes. The test domain (b) was a discretization of the MR image thresholded into regions. The yellow anomaly was added to simulate a targeted cancerous tumor.

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Fig. 2.
Fig. 2. The normalized absorbance and fluorescence emission spectra of Lutetium Texaphyrin are shown.

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Fig. 3.
Fig. 3. The experimental spectrometer-based system depicted at left couples directly into the MR via 13 meter fiber optic bundles. Sixteen spectrometers are computer controlled for rapid image acquisition (left photograph). An animal interface (right photograph) is composed of a rodent MR coil custom built by Philips Research Hamburg to accommodate the optical fiber array for simultaneous MR and NIR fluorescence imaging.

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Fig. 4.
Fig. 4. An example of a pair of basis spectra for the excitation and fluorescence light (in counts/s as a function of CCD pixel number) is shown in (b). These spectra are recorded for each detector prior to imaging. In practice, the basis spectra are used to perform a least squares fit (a) to the spectrum measured for each source-detector pair to determine the relative contribution of the fluorescence and excitation light to the measured response.

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Fig. 5.
Fig. 5. Target and recovered values of μa,x, μ ′ s,x, μa,m, μ ′ s,m, and fluorescence yield, ημaf, for reconstruction implementations using no prior information and with spatial prior information. In this case, the simulated cancer region is near the edge, which is known to be easier to recover without spatial priors. The image scales are at right.

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Fig. 6.
Fig. 6. Target and recovered values of μa,x, μ ′ s,x, μa,m, μ ′ s,m, and fluorescence yield, ημaf, for reconstruction implementations using no priors and spatial priors. In this case, the simulated tumor region is near the center of the imaging domain, which is known to be more difficult to recover accurately. Image scales are at right.

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Fig. 7.
Fig. 7. Cross sectional plots of fluorescence yield are shown for the simulated imaging domains in (a) the case with an object near the edge and (b) the case with an object near the center. In both cases, the cross section is in the y-direction just off center from x = 0. The solid line represents the target value, the small dotted line the recovered value using a no-priors based algorithm, and the dashed line the recovered value using spatially guided reconstruction.

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Fig. 8.
Fig. 8. Recovered images of fluorescence yield are shown for varying concentrations of Lutex. The 14 mm diameter fluorescent inclusion was embedded in a 55 mm diameter solid epoxy tissue simulating phantom. Images were generated from the same data using algorithms based on no-priors and spatial soft prior implementations.

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Fig. 9.
Fig. 9. A narrower colorbar-scale version of the 0.3125 μM Lutex phantom images shown in Fig. 8 further illustrates the improvement in image accuracy for the spatially guided algorithm.

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Tables (2)

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Table 1. Chromophore concentrations and scattering parameter values assigned to the mesh regions in the simulation studies

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Table 2. Target and recovered fluorescence yield regional contrasts for the images in Figs. 4 and 5.

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Equations (16)

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− ∇ ∙ κ x ( r ) ∇ Φ x r ω + ( μ a x ( r ) + i ω c ( r ) ) Φ x r ω = q 0 r ω
− ∇ ∙ κ m ( r ) ∇ Φ m r ω + ( μ a m ( r ) + i ω c ( r ) ) Φ m r ω = Φ x r ω η μ a f ( r ) 1 − i ω τ ( r ) 1 + [ ω τ ( r ) ] 2
Φ x , m ξ ω + 2 A n ̂ ∙ κ x , m ( ξ ) ∇ Φ x , m ξ ω = 0
A = 2 ( 1 − R 0 ) − 1 + ∣ cos θ c ∣ 3 1 − ∣ cos θ c ∣ 2
( K x ( κ ) + C x ( μ a x + i ω c ( r ) ) + 1 2 A F x ) Φ x = Q 0
( K m ( κ ) + C m ( μ a m + i ω c ( r ) ) + 1 2 A F m ) Φ m = − Q m
K x , m ij = ∫ Ω κ x , m ( r ) ∇ u i ( r ) . ∇ u j ( r ) d n r
C x , m ij = ∫ Ω ( μ a x , m ( r ) + i ω c ( r ) ) u i ( r ) u j ( r ) d n r
F x , m ij = ∮ ∂ Ω u i ( r ) u j ( r ) d n − 1 r
Q 0 i = ∫ Ω u i ( r ) q 0 ( r ) d n r
Q m i = ∫ Ω u i ( r ) [ Φ x r ω η μ a f ( r ) 1 − j ω τ ( r ) 1 + [ ω τ ( r ) ] 2 ] d n r
χ 2 = ∑ i = 1 N M ( Φ x i Meas − Φ x i C ) 2 + λ ∑ j = 1 N N I ( μ x j − μ x 0 ) 2
Δ μ x = [ J T J + λ I ] − 1 J T ( Φ x Meas − Φ x C )
χ 2 = ∑ i = 1 N M ( Φ x i Meas − Φ x i C ) 2 + β ∑ j = 1 N N ( L ( μ x j − μ x 0 ) ) 2
Δ μ x = [ J T J + β L T L ] − 1 J T ( Φ x Meas − Φ x C )
S = ∑ i = 1 N [ y i − ( a F ( λ i ) + b G ( λ i ) ) ] 2
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