Principal component analysis of dynamic
Principal component analysis of dynamic fluorescence diffuse optical tomography images Xin Liu, Daifa Wang, Fei Liu, and Jing Bai* Department of Biomedical Engineering, School of Medicine, Tsinghua University, Beijing 100084, China *[email protected] Abstract: Challenges remain in resolving drug distributions within small animals utilizing fluorescence diffuse optical tomography (FDOT). In this paper, we present a new method for detecting and visualizing organs with different kinetics utilizing principal component analysis (PCA). Indocynaine green (ICG) metabolic processes are simulated and imaged using FDOT. When applied to the time series of generated FDOT images, PCA provides a set of the principal components (PCs) which can represent spatial patterns associated with different kinetic behavior. Simulation and experiment studies are both performed to validate the performance of the proposed algorithm. The results suggest that we are able to extract and illustrate changes in ICG kinetic behavior between the heart and the lungs. ©2010 Optical Society of America OCIS codes: (170.6960) Tomography; (170.3010) Image reconstruction techniques; (170.6280) Spectroscopy, fluorescence and luminescence; (170.3880) Medical and biological imaging. References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005). V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8(7), 757–761 (2002). J. Haller, D. Hyde, N. Deliolanis, R. de Kleine, M. Niedre, and V. Ntziachristos, “Visualization of pulmonary inflammation using noninvasive fluorescence molecular imaging,” J. Appl. Physiol. 104(3), 795–802 (2008). V. Ntziachristos, A. G. 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Biomed. 13(5), 766–773 (2009). 1. Introduction Fluorescence diffuse optical tomography (FDOT) is an optical imaging method that allows non-invasive, quantitative, three-dimensional (3-D) imaging of biological and biochemical processes in living small animals. At present, FDOT has been successfully applied in enzymes [1,2], inflammation [3], oncology [4–6], and therapy [7], etc. With the advances in optical probes [8,9], imaging systems [8,10,11], and reconstruction algorithms [12–16], tomographic imaging of dynamic biological activities in small animal body in vivo is now possible. Compared with static FDOT image reconstructed at a specific time point, the time series of tomographic imaging impart the ability to capture the complete dynamic course of absorption, distribution, and elimination of the fluorophores, contrast agents, or drug, by adding time as a new dimension. We call this time series dynamic fluorescence diffuse optical tomography (D-FDOT). By allowing longitudinal time-lapse visualization of metabolic processes of contrast agents, D-FDOT provides an attractive approach in studying drug delivery, tumor detection, and treatment monitoring in small animals in vivo. However, challenges remain in resolving organs and functional structures with different kinetics in small animal body in vivo. The main problem is that the delivery route of drug (non-specific contrast agents) is complex. They may distribute all over the body through circulatory system, instead of accumulating at some specific place, which will result in poor imaging contrast and specificity. Together, the highly diffusive nature of photon migration in tissues and ill-posed characteristic of the inverse problem are responsible for the poor image quality and low spatial resolution. Hence, it is difficult to detect and visualize changes in kinetic behavior between major organs of small animals directly from FDOT images obtained at a specific time point. Here, instead of focusing on static tomographic image, we look at the dynamics of drug during FDOT imaging. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6301 Each organ in the body plays a different role in the course of drug circulating, accumulating or metabolizing, and so each organ will exhibit a distinctive time course in its optical signal [17]. An exposition of resolving different functional structures based on DFDOT images reduces to an examination of its correlation structure. Here, the correlation feature refers to the correlative kinetic behavior obtained over time in the same organ. Principal component analysis (PCA) is very suitable for this examination. PCA provides the ability of extracting the important features of the correlation matrix in terms of principal components or eigenvectors [18]. Each principal component (PC) represents a major trend of kinetic behavior within which there are high inter-correlations. At present, PCA has been widely applied in analyzing dynamic positron emission tomography (PET) [19–21] and functional magnetic resonance imaging (fMRI) data [22,23]. In 2007, Hillman and Moore [17] firstly demonstrated the capability of applying PCA in analyzing dynamic optical projections. Using a CCD camera to image the dynamics metabolic processes of optical contrast agents injected through tail veins, they mainly focused on decomposing the 2-D fluorescence reflectance imaging and extracted anatomical information of various internal organs utilizing PCA. Despite of these successful applications of PCA, no trial has been reported for PCA in 3-D D-FDOT images. As PCA is a data-driven method and 3-D D-FDOT has its own characteristics of poor spatial resolution and diffusive nature, investigation on the applying PCA-based methods in 3-D D-FDOT is very important. In this paper, we propose to apply a PCA-based method on a time series of 3-D FDOT images to resolve organs and functional structures with different kinetic patterns. The contrast agent we employed, indocyanine green (ICG), is a widely used optical fluorophores in the near-infrared (NIR) range for small animal imaging [11], as well as for human breast cancer imaging [4]. The proposed method was validated by simulation and experiment studies on a non-contact, full angle FDOT imaging system [10]. In simulation studies, we simulated a case where ICG distributed through the heart and the lungs of mouse after tail vein injection, based on a 3-D digital mouse model [24,25]. In experiment studies, we repeated the simulated conditions by two tubes filled with different concentrations of ICG at different time points. The results suggest that we are able to detect and visualize regions with different kinetics based on D-FDOT images reconstructed much better. The outline of this paper is as follows. In section 2, the methods used are detailed. In section 3, the simulation and experiment are shown. In section 4, the results are described. Finally, we discuss and conclude the major findings of this study in section 5. 2. Methods 2.1. Forward and inverse problems In highly scattering tissue medium, the photon migration in biological tissues can be modeled using the coupled diffusion equation with the Robin-type boundary condition [16]. In FDOT, the Green's function G (rs , r ) due to a continuous wave (CW) point excitation source δ (r − rs ) , describing the photon density at position r, can be obtained as follows, ∇ ⋅ [ D(r )∇G (rs , r )] − µa (r )G (rs , r ) = −δ (r − rs ) ∂G (rs , r ) 2qD(r ) ∂n + G (rs , r ) = 0 r ∈Ω r ∈ ∂Ω , (1) where Ω is the domain of the imaged object and ∂Ω is the boundary; µa (r ) is the absorption coefficient and D(r ) = 1 / (3µ s' (r )) is the diffusion coefficient of the tissue with the reduced scattering coefficient µ s' (r ) at position r; n denotes the outward normal vector to the boundary and q is a constant depending upon the optical reflective index mismatch at the boundary. A collimated point source spot can be modeled as an isotropic source δ (r − rs ) , #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6302 where rs is the point one transport mean free path ltr = 1 / µ s' (r ) into the medium from the illumination spot [16]. The diffusion equation can be solved using the Galerkin finite element method to obtain the Green's functions [14,15]. After that, the inverse problem is generated based on Normalized Born approximation [13], which reduces the influence of mouse tissue heterogeneity. The normalized ratio between the measured fluorescence signal Φ m (rd ) and the corresponding excitation signal Φ x (rd ) at detector point rd is given as follows, G (rd , rp )G (rs , rp )n(rp ) Φ m (rd ) drp , = Θ ∫V G (rs , rd ) Φ x (rd ) (2) where rp is the point inside the volume V considered for reconstruction; n(rp ) denotes the fluorescence yield at point rp , which is directly proportional to the fluorophores's concentration; G (rd , rp ) denotes the Green's function value at rp due to a point source δ (r − rd ) at rd ; Strictly speaking, G (rs , rp ) describes the light propagation at the excitation wavelength and G (rd , rp ) corresponds to light propagation at the emission wavelength. The excitation and emission wavelength are close to each other. In addition, the Normalized Born method can largely reduce the effects of incorrect assumption of optical properties. For the moment, we assume that the Green's functions are similar at these two wavelengths for simplification. Θ is a calibration factor which accounts for the unknown gain and attenuation factors of the system, such as excitation light power. To solve the inverse problem, Eq. (2) is discretized on a 3-D grid of size N x , N y , N z . For each single source-detector pair (rs1 , rd 1 ) and each point (rp ) p =1... N xN y Nz on the grid, Eq. (2) is described as follows, n(r1 ) . Φ m (rd 1 ) Θ∆V = [G (rd 1 , r1 )G (r1 , rs1 ),..., G (rdN , rN )G (rN , rsN )] ⋅ . , Φ x (rd 1 ) G (rs1 , rd 1 ) . n( r ) N (3) where N is the total number of the descretized voxels. For M source-detector pairs, a linear system is generated as, Φ m (rd 1 ) / Φ x (rd 1 ) W11 . . = . . . . Φ m (rdM ) / Φ x (rdM ) WM 1 . . . . . . W1N n(r1 ) . . . ⋅ . . . . . . WMN n(rN ) . (4) where Wij is an element of the weight matrix W connecting the measurement to the unknown fluorescence yield. The unknown n(rp ) in each voxel is obtained by solving Eq. (4) using algebraic reconstruction technique (ART) [26] with non-negative constraints. 2.2. Principal component analysis In kinetic analysis, the drug distribution within biological body is sequentially imaged at different time points. These sequential images can be regarded as multivariate images from which physiological, biochemical and functional information can be derived. Principal #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6303 component analysis (PCA), as a common multivariate image analysis method, provides a graceful strategy for resolving different kinetic behavior of the signal in each pixel by remapping the data into a new coordinate system. In this study, the input data can be denoted as X = { X 1 , X 2 ,..., X t ,..., X N } , where N is the number of frames and X t is the frame taken at a given time t. Here, image frame denotes the reconstructed tomographic image from the same slice but from different time points. To apply PCA to the spatial and temporal FDOT images at the same slice, we assume that the variables are the frames taken in a time sequence, and a pixel's temporal distribution is an observation of these variables. If the reconstructed image has M = R × C pixels, then we can represent one frame data as X tT = ( x1,t , x2,t ,..., xk ,t ,..., xM ,t ) , where xk ,t is the density of the pixel k at the frame t (T means transposition operation), and the size of the image sequence X is of M × N matrix. Then, we define the matrix, S , as follows, S= 1 XT X, M −1 (5) where X is a data set with zero mean obtained from the original one by subtracting its mean value from each frame. Mathematically, principal component expansion is obtained by the diagonalization of matrix S , which produces a set of eigenvalues, λi , and eigenvectors, Ei . If we define the set of eigenvectors as E = {E1 , E2 ,..., Ei ,..., EN } , where the i column contains the elements of the vector EiT = {e1,i , e2,i ,..., e j ,i ,..., EN ,i } , the principal components of the image sequence can be obtained as a M × N matrix, P , as follows, P = XE, (6) and each principal components (PCs) is given below, N Pi = ∑ et ,i X t . (7) t The condition of Cov ( Pj , Pk ) = 0, j ≠ k is required, meaning that each principal component is functionally uncorrelated to each other. Further each element of principal component is utilized as a weight factor for creating image, termed PC image [20,27] that illustrates organs and functional structures with different kinetic behavior. To sum up, the corresponding flowchart of the proposed algorithm is shown in Fig. 1. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6304 Fig. 1. The flowchart of the proposed algorithm. 3. Materials In validating the performance of PCA on D-FDOT images, we performed simulation and experiment studies based on a non-contact, full angle FDOT system. The sketch of the system is shown in Fig. 2, which is similar to that in [10]. Fig. 2. The schematic of the imaging system. 3.1. Simulation studies 3.1.1. Setup for simulation studies In simulation studies, a virtual mouse atlas was employed to provide 3-D anatomical information [24,25]. To simplify the experimental design, we only focused on imaging kinetic #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6305 behavior inside mouse chest region instead of whole body. Therefore, only the mouse torso from the neck to the base of the lungs was simulated, with a length of 1.6 cm. The imaged mouse was suspended on a rotating stage, as shown in Fig. 2. The rotational axis of the mouse was defined as the Z axis with the bottom plane set as z = 0 cm. According to [28], optical parameters (absorption, scattering) were assigned to the heart and lungs to simulate photons propagation in biological tissues. The optical properties outside heart and lungs were regarded as homogeneous. Detailed information about optical properties of biological tissues in mouse is presented in Table 1. Based on [29], in this simulations, fluorescence tomography of 360o full view was performed with 18 projections at every 20o with the excitation source at height 0.8 cm. Table 1. Optical parameters of biological tissues in mouse at 700-800 nma µ a (cm -1 ) Tissue type Heart Lungs 0.156 0.516 µ 's (cm -1 ) 9.0 21.2 a From Phys. Med. Biol. 50, 4225 (2005). 3.1.2. Dynamic modeling in simulation studies To optimally evaluate the performance of the proposed method, we simulated the metabolic processes of ICG within small animals. Firstly, we set a series of fluorescent yields (concentrations of ICG) to the heart and the lungs at six time points (5 min, 10 min, 15 min, 30 min, 60 min, and 120 min) according to [30]. The anatomical information of the heart and the lungs is described in Fig. 3(a) and the concentration time course of ICG is plotted in Fig. 3(b). Secondly, six tomographic images were reconstructed to depict ICG distribution within the body at corresponding time points. Finally, we assembled these tomographic images to form D-FDOT images which were used to simulate dynamic course of ICG kinetics. Fig. 3. Setup for simulation studies. (a) The mouse geometry model used in simulation studies. The gray part in (a) depicts the mouse surface with a length of 1.6 cm from the neck to the base of the lungs. The red part in (a) depicts anatomical information of the heart and the green part in (a) depicts anatomical information of the lungs. In order to reduce the boundary artifacts which interfere with the finite element computation, the model is 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 torso surface. (b) ICG concentration time course in the heart and the lungs after tail vein injection. The circles in (b) depict actual concentration value at corresponding time points according to [30]. Different colors correspond to different time course curves (red:heart; green:lungs). It should be noted that, the uptake of ICG in organ was assumed to be approximately uniformly distributed rather than accumulated in limited regions of organ. Herein, we #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6306 performed simulations assuming that the whole region of organ was tagged with ICG. In addition, considering that ICG was a kind of non-specific fluorescence contrast agent, the heart and the lungs were labeled simultaneously in this study. 3.1.3. Reconstructions of synthetic data Reconstructions were performed to obtain tomographic images at different time points. The excitation and emission data were synthesized using finite element method based on Eq. (2). The geometrical model in Fig. 3(a) was discretized into 10,987 nodes and 55,398 tetrahedral elements. The detectors were located on the boundary finite element nodes which were between 1.4 cm height range and inside 150o FOV corresponding to each point source. The volume considered for reconstruction was 2.0 cm × 2.5 cm × 1.6 cm and sampled to 32 × 40 × 32 voxels. Only the 16,518 voxels inside the imaged object were considered for reconstruction and the reconstruction was terminated after 100 ART iterations. 3.1.4. Principal component analysis for synthetic D-FDOT images To extract ICG kinetic properties in the heart and the lungs from D-FDOT images, PCA was performed. Here, the input data was six frames reconstructed tomographic images, having zero mean, from D-FDOT images at the same slice. Each frame with a spatial size of 32 × 40 pixels was reshaped as a column vector containing 1,280 elements. By the transformation, the set of input data was placed in a matrix with 1,280 rows and 6 columns. After applying Eq. (6), we obtained six principal components which were functionally uncorrelated to each other. Each positive and negative element of principal component was respectively utilized as a weight factor for creating image that illustrated regions with different kinetic behavior using different colors. Red color depicted negative PC images and green color depicted positive PC images. 3.2. Experiment studies 3.2.1. Setup for experiment studies A physical experiment was performed to further validate the performance of the proposed method. As shown in Fig. 4(a), two transparent glass tubes (0.3 cm diameter, 0.6 cm length) filled with different concentrations of ICG were immerged in a cylinder phantom. The phantom was made of a glass cylinder (3.0 cm diameter) filled with 1% intralipid ( µ s' = 10.0 cm -1 ,µa =0.02 cm -1 ). The two tubes were separated with an edge-to-edge distance of 0.2 cm along Y axis. In the experiments, the imaged phantom was placed on a rotating stage which allowed rotation and shift of the target around Z axis for collecting projections evenly distributed over 360o , as shown in Fig. 2. The small light spot from a 250 W Halogen lamp traveled through a 775 ± 23 nm band-pass filter and focused on the surface of imaged object. A 512 × 512 element, -70o C cooled EMCCD array coupled with a Nikkor 60 mm f/2.8D lens was placed on the opposite side of imaged object, collecting the photons propagating through the imaged object. When collecting the fluorescence images, a 840 ± 6 nm band-pass filter was placed in front of EMCCD camera. When collecting the excitation light images, a neutral density filter of 1% transmittance was used. When collecting the white light images, a white light bulb was employed to replace the excitation light. The white light images at different projection angles were acquired for recovering the 3-D geometry of the imaged object, which was necessary for modeling diffuse light transportation. 3.2.2. Dynamic modeling in experiment studies To simulate the time active curve of ICG in the heart and the lungs, the concentrations of ICG in tube 1 and tube 2 at different time points were set according to Fig. 4(b). Tube 1 depicted ICG metabolic processes in the heart and tube 2 depicted ICG metabolic processes in the #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6307 lungs. For each experiment at specific point (frame), 24 excitation and emission images were collected at every 15o using EMCCD with the excitation source at height 2.9 cm. The total power of the point source was about 20 mW. 72 white light images acquired in 5o steps were back-projected to form a 3-D geometry of phantom [31]. Fig. 4. Setup for experiment studies. (a) Experiment setup. Two glass tubes (diameter of 0.3 cm and height of 0.6 cm) filled with different concentrations of ICG were placed inside a cylinder phantom (a glass cylinder of 3.0 cm diameter filled with 1% intralipid). The edge to edge distance along Y axis was 0.2 cm. (b) ICG concentration time course in tube 1 and tube 2. Tube 1 simulated ICG metabolic processes in the heart and tube 2 simulated ICG metabolic processes in the lungs. The circles in (b) depicted actual concentration value at corresponding time points. Different colors corresponded to different time course curves (red:tube 1; green:tube 2). 3.2.3. Reconstructions of experimental data The cylinder phantom in Fig. 4(a) was discretized into 17,558 nodes and 78,782 tetrahedral elements. The detectors were located on the boundary finite element nodes which were between 3.2 cm height range and inside 150o FOV corresponding to each point source. The volume considered for reconstruction was 3.0 cm × 3.0 cm × 5.0 cm and sampled to 30 × 30 × 50 voxels. The 27,178 voxels inside the imaged object were used for reconstruction. Reconstructions were terminated after 100 ART iterations. 3.2.4. Principal component analysis for experimental D-FDOT images The main description was the same as in section 3.1.4. Briefly, the input data was 6 frames reconstructed tomographic imaging with a spatial size of 30 × 30 pixels and the input matrix had 900 rows and 6 columns. 4. Results 4.1. Reconstructions of synthetic data Figure 5 shows a slice through the chest, including the heart and the lungs, in the mouse after ICG injection. It is a study of ICG metabolic processes in the heart and the lungs. The six reconstructed images (5 min, 10 min, 15 min, 30 min, 60 min, and 120 min) from D-FDOT images at the same height (z = 0.8 cm) are described in Figs. 5(a) to 5(f). The results suggest that it is quite difficult to resolve the uptake of ICG in the heart and the lungs directly from static tomographic image. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6308 Fig. 5. Reconstruction of synthetic data from a dynamic study of ICG metabolic processes in the heart and the lungs. The images are at z = 0.8 cm. (a)-(f) The reconstructed results at 5 min, 10 min, 15 min, 30 min, 60 min, and 120 min. Different colors correspond to actual boundary of different organs (red:heart; green:lungs; yellow:surface). 4.2. Principal component analysis for synthetic D- FDOT images As shown in Fig. 6, when PCA was applied to D-FDOT images at specific slice (z = 0.8 cm) shown in Fig. 5, we obtained six PC images reflecting different kinetic behavior. In Figs. 6(a) to 6(f), the obtained positive PC images are depicted using green color. In Figs. 6(g) to 6(l), the obtained negative PC images are depicted using red color. Based on the anatomical information of the heart and the lungs, we find that a very high uptake of ICG in the lungs is nicely demonstrated using the positive PC2 image, as shown in Fig. 6(b). Similarly, in Fig. 6(h), a very high uptake of ICG in the heart is demonstrated using the negative PC2 image. Higher PC images (PC4 to PC6) are similar to the PC3 image, containing no structure. The results suggest that PCA is a useful tool to detect and visualize organs and functional structures with different kinetics. In addition, it is possible to give a simple interpretation that PCA is also seen as data reduction method since only the first two PC images contain structure. Fig. 6. The PC images obtained when PCA was applied to D-FDOT images shown in Fig. 5. (a)-(f) The six positive PC images. (g)-(l) The six negative PC images. In the PC2 image, the uptake of ICG in the heart (negative) and the lungs (positive) was indicated. Different colors corresponded to actual boundary of different organs (red:heart; green:lungs; yellow:surface). To demonstrate the performance of PCA on D-FDOT images at all slices, we describe the 3-D visualization results of the negative and the positive PC2 images, by extracting isosurfaces from all obtained PC2 images, as shown in Figs. 7(a) and 7(d). Figure 7(a) #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6309 indicates the 3-D region of ICG distribution in the heart. Figure 7(d) indicates the 3-D region of ICG distribution in the lungs. In the 3-D visualization of the PC2 images, reconstructed fluorescence signals, which was less than 2% of the maximum signal intensity, were not considered. These fluorescence signals did not contain information but induced artifacts into the reconstructed images. A visual comparison of the PC2 images in Figs. 7(a) and 7(d) to anatomical information in Figs. 7(b) and 7(e) are shown in Figs. 7(c) and 7(f). The red parts in Figs. 7(c) and 7(f) depict the 3-D anatomical information of the heart and the lungs. The green parts in Figs. 7(c) and 7(f) depict the 3-D visualization results of the negative and the positive PC2 images. Fig. 7. The 3-D visualization results of the PC2 images. (a) and (d) The 3-D visualization results of the negative and the positive PC2 images obtained when PCA was applied to the DFDOT images at all slices. (a) indicated the uptake of ICG in the heart. (d) indicated the uptake of ICG in the lungs. (b) and (e) The 3-D anatomical information of the heart and the lungs in the mouse. (c) and (f) A visual comparison of the PC2 images in (a) and (d) to anatomical information in (b) and (e). The red parts in (c) and (f) indicated the 3-D anatomical information of the heart and the lungs. The green parts in (c) and (f) indicated the 3-D visualization results of the negative and the positive PC2 images. Figure 8 shows a pictogram of how principal component analysis works with D-FDOT images shown in Fig. 5. Each frame in Fig. 5, after subtraction of its mean value, was arranged as a columns vector of input matrix X . After applying Eq. (6), six functionally uncorrelated principal components were generated and arranged in the matrix P . Subsequently, each positive and negative element of principal component was respectively utilized as a weight factor for creating image that illustrated functional structures with different kinetic behavior. The relation between matrices X and P was given by the matrix E. Comparison between 3-D reconstructed results at a specific time point (5 min) and the merged results of the negative and positive PC2 images, we further demonstrated that organs and functional structures with different kinetic patterns can be resolved when PCA is applied to a time series of FDOT images. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6310 Fig. 8. Pictogram of how principal component expansion works with D-FDOT images. In this plot, we show how the reconstructed results ( X ,X ,...,X ) in Fig. 5 (after subtraction of 1 2 6 the mean value) were transformed into six principal components (PC1,PC2,...,PC6) that were presented as images through the matrix E . The red part in the PC2 image indicated the uptake of ICG in the heart and the green part in the PC2 image indicated the uptake of ICG in the lungs. The black curves in 3-D view depicted the height of selected dynamic tomographic images. The 3-D view (left) depicted the 3-D reconstructed results of frame 1 (5 min). The 3-D view (right) depicted the merged results of the positive and the negative PC2 images. The red part in 3-D view (right) depicted the 3-D visualization results of the negative PC2 images in Fig. 7(a). The green part in 3-D view (right) depicted the 3-D visualization results of the positive PC2 images in Fig. 7(d). 4.3. Reconstructions of experimental data Figure 9 shows reconstructed results of experimental data from the same slice but from different frames. The six reconstructed images at the same height (z = 3.0 cm) are shown in Figs. 9(a) to 9(f). Similar as in section 4.1, it is difficult to resolve the uptake of ICG in two tubes based on each static tomographic image. Fig. 9. Reconstruction of experimental data at different frames. The images are at z = 3.0 cm. (a)-(f) Reconstruction results at frames 1 to 6. The red curve on the cross images depicts the phantom boundary, and the black circles depict the actual tubes. All images are displayed at the same range. 4.4. Principal component analysis for experimental D-FDOT images The results in section 4.2 suggest that function structures with different dynamic behavior can be resolved utilizing PCA in simulation studies. As expected, in experiment studies, when PCA was applied to D-FDOT images shown in Fig. 9, we detected and visualized the uptake of ICG in two tubes with an edge-to-edge distance of 0.2 cm. As shown in Figs. 10(b) and 10(c), the uptakes of ICG in tube 1 and tube 2 are illustrated using the negative and the positive PC3 image, respectively. The two parts indicate the kinetics of ICG in the heart and the lungs. When PCA was applied to D-FDOT images at all slices, we generated the 3-D visualization results of the positive and the negative PC3 images, as shown in Figs. 10(e) and 10(f). In the 3-D visualization, reconstructed fluorescence signals, which was less than 2% of #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6311 the maximum signal intensity, were not considered. Comparing the reconstructed result of frame 3 in Figs. 10(a) with the PC3 images in Figs. 10(b) and 10(c), we validated the capability of PCA in illustrating changes in kinetic behavior in experiment studies. Fig. 10. Comparison of reconstructed result to PCA-based result. (a) The 2-D reconstructed result of frame 3. The cross section image is at z = 3.0 cm, which is depicted by the red curve in (d). The red circle in (a) depicts the phantom boundary, and the black circles depict the actual tubes. (b) and (c) The positive and the negative PC3 images obtained when PCA was applied to D-FDOT images shown in Fig. 9. The red circles in (b) and (c) depict the phantom boundary, and the cyan circles in (b) and (c) depict the actual tubes. (d) The 3-D reconstructed result of frame 3. (e) and (f) The 3-D visualization results of the positive and the negative PC3 images obtained when PCA was applied to the D-FDOT images at all slices. Figure 11 further shows differences in the uptake of ICG in two tubes using the merged results of the positive and the negative PC3 image. Figure 11(a) depicts the 2-D merged result of the positive PC3 image in Fig. 10(b) and the negative PC3 image in Fig. 10(c). Figures 11(b) and 11(c) depict the 3-D merged results of the positive PC3 images in Fig. 10(e) and the negative PC3 images in Fig. 10(f), using different views. The red part in Fig. 11 indicates the uptake of ICG in tube 1 that is used to simulate ICG metabolic processes in the heart. The green part in Fig. 11 indicates the uptake of ICG in tube 2 that is used to simulate ICG metabolic processes in the lungs. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6312 Fig. 11. The merged results of the positive and the negative PC3 images. (a) The 2-D merged results of the positive PC3 image in Fig. 10(b) and the negative PC3 image in Fig. 10(c). The cross section image is at z = 3.0 cm, which is depicted by the red curves in (b) and (c). The red circle in (a) depicts the phantom boundary, and the cyan circles depict the actual tubes. (b) and (c) The 3-D merged results of the positive PC3 images in Fig. 10(e) and the negative PC3 images in Fig. 10(f) using different views. The red parts indicate the uptake of ICG in tube 1 that is used to simulate ICG metabolic processes in the heart. The green parts indicate the uptake of ICG in tube 2 that is used to simulate ICG metabolic processes in the lungs. 5. Discussion and conclusion Fluorescence diffuse optical tomography plays an important role in drug research for allowing non-invasive, quantitative, 3-D imaging of biological and biochemical processes in living subjects. However, the complex dynamic behavior of drug often makes it difficult to resolve organs with different kinetics in small animal body. In this paper, we proposed a PCA-based method for kinetic studies, to aid in the identification of functional structures with different kinetic patterns, and then evaluated its performance using simulation and experiment studies. It was clearly seen from the reconstructed images at different time points, ICG kinetic behavior in the heart and the lungs (simulation studies) and in the two tubes with an edge-toedge distance of 0.2 cm (experiment studies) could not be resolved. This was probably caused by the diffusive nature of photon migration in biological tissues and the ill-posed characteristic of the inverse problem. In contrast, when PCA was applied to the time series of tomographic images, we generated a set of PC images that illustrate organs and functional structures with different kinetic patterns. For example, in simulation studies, the uptake of ICG in the heart and the lungs were illustrated using the negative and the positive PC2 images. Similar in experiment studies, we resolved the spatial structure of two tubes associated with different kinetic behavior using the negative and the positive PC3 images. Based on the satisfying results, we believe that PCA-based method provides an attractive approach in kinetic study. Firstly, this method allows to non-invasive measure changes in the function of organs. The detecting capability is critical to evaluate the effects of a drug on the organs in vivo. Secondly, this method has the potential in enhancing the resolution and specificity of FDOT by emphasizing differences in kinetic behavior. Thirdly, utilizing the technique, additional anatomical information of organs may be achieved without resorting to other imaging equipment. Finally, the proposed method is totally independent of any kinetic model and thus not subject to model-based restrictions. In this article, the ICG concentration was kept constant while collecting one frame of fluorescence projection images. It is mainly for simplification. However, it is reasonable. In experiments, the full angle imaging time of collecting one frame typically takes approximately 1.3 minutes to 2.0 minutes, depending on the sum of rotation time and total exposure time of images. During this imaging course, the change of ICG concentration can be seen as linear approximately. Then, the fluorescence projections of the same frame can be corrected using linear model, which will lead to the corrected projection images similar to the #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6313 projection images used in this article. Of course, in dynamic FDOT, good data correction methods for fluorescence projections can improve the reconstructed image quality a lot. Systematic studies in this field will also be investigated in our future work. All the obtained results are based on a basic assumption that accurate optical properties are known. Tissue optical properties are important for FDOT reconstruction and wrong optical parameters may lead to significant errors to the reconstructed results [32]. It will directly affect the performance of PCA identifying regions with different kinetics. Moreover, it should be noted that PCA has difficulties in separating the signal from the noise when the magnitude of the noise is relatively high [19,23]. Ultimately, it is also difficult to illustrate how to choose the optimal PC images reflecting different kinetics since the obtained PC images may be influenced by the reconstructed errors, noise, and time courses of drug, etc. Hence, in in vivo experiments, many practical factors need further consideration. It is out of the scope of this article, and will be analyzed in our future work. In conclusion, using simulation and experiment studies, we have demonstrated the capabilities of PCA in enhancing FDOT performances in imaging kinetics. This analysis can be used as a starting point for dynamic studies, e.g. tumor detection, disease progression, and drug delivery, etc. Future works will be focused applying the method in resolving ICG distribution within small animal in vivo. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No. 30670577, 60831003, 30930092, 30872633; the Tsinghua-Yue-Yuen Medical Science Foundation; the National Basic Research Program of China (973) under Grant No. 2006CB705700; the National High-Tech Research and Development Program of China (863) under Grant No. 2006AA020803. #121176 - $15.00 USD (C) 2010 OSA Received 10 Dec 2009; revised 6 Feb 2010; accepted 10 Mar 2010; published 12 Mar 2010 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 6314
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