An example of a three-dimensional reconstructed image is shown in . The phantom, depicted in , consisted of a single absorbing tube with an absorption contrast of 50:1. This is the absorption contrast in the near-infrared that we expect between superficial veins and the surrounding tissue. The center of the tube was located at a depth of 3 mm. is a volume rendering of the three-dimensional reconstructed absorption image. The tube surface in the volume rendering represents an isosurface of 60% of the maximum reconstructed value in the image. Individual slices through the three-dimension image at depths of 1–5 mm are shown in . At the surface the tube is not visible. The maximum amount of absorption occurs in the 1 mm slice. Absorption then decreases with depth until at a depth of 5 mm no absorption contrast is present.
Fig. 2 Example of a reconstructed image of a single absorbing tube located 3 mm below the surface of a tissue simulating phantom. (a) Schematic of the tissue simulating phantom. (b) Volume rendering of the three-dimensional reconstructed image. (c) Slices through (more ...)
Image reconstruction leads to a significant increase in resolution compared to images created by fitting for optical properties on pixel by pixel basis while assuming a homogeneous model. Profiles of tomographic images are shown in . The profiles are from experiments with a single absorbing tube at depths of 2 mm () and 3 mm (). In both cases the contrast in 50:1. The curves are normalized to a maximum value of one to facilitate a comparison of the two methods. The solid lines correspond to the lateral profiles of the reconstructed images. The full-width-half-maximum (FWHM) for the plots shown are 1.8 mm and 2.2 mm at depths of 2 mm and 3 mm respectively. We also measured the FWHM for tubes with contrasts ranging from 3:1 to 100:1. The FWHM ranged from 1.8 to 2.3 mm at a depth of 2 mm and from 2.2 to 3.2 mm at a depth of 3 mm, with the low contrast images having slightly lower resolution. For comparison, profiles of the images obtained by fitting pixel by pixel to a homogeneous model are shown with dotted lines. The FWHM is approximately three times larger. For the different concentrations, the FWHM determined from homogeneous model fits ranged from 4.4 to 6.8 mm at 2 mm depth, and from 5.5 to 9.5 mm at 3 mm depth.
Fig. 3 Line profiles of the reconstructed images of a single absorbing tube at depths of (a) 2mm and (b) 3 mm. Solid line corresponds to tomographic reconstruction, and dotted lines correspond to pixel by pixel fitting to a homogeneous model. Curves are normalized (more ...)
The improved resolution not only allows one to distinguish between nearby absorbers, but also decreases errors in quantification due to the blurring of nearby objects in the image. As a demonstration, we imaged four absorbing tubes at a depth of 2 mm. Each tube contained the same concentration of nigrosin, with the expected contrast being 50:1. The lateral spacing between the tubes was 3, 4, and 5 mm (see ). In the reconstructed image and line profile () the tubes are clearly resolved and all have approximately the same reconstructed value of absorption. In contrast, for the image obtained by fitting pixel by pixel (), the center two tubes appear to contain much more contrast than the outer tubes. This is because the image the absorption from nearby tubes, which appears broadened, is added to absorption of any given tube such that the central tubes falsely appear to have more absorption. Note, the maximum intensity also differs greatly, as images created using the homogeneous model underestimate the actual contrast by about a factor ten.
Fig. 4 (a) Schematic of tissue simulating phantom with four absorbing tubes located at a depth of z0 = 2 mm, and with lateral separations of d1 = 5 mm, d2 = 4 mm, and d3 = 3 mm. (b) Image and line profile of the reconstructed image using the tomographic method. (more ...)
In order to quantify the ability of both imaging methods to accurately recover the amount of absorption in a small sub-surface heterogeneity, we performed titration experiments in which the expected contrast in a single tube was varied from 3:1 to 100:1. shows the measured contrast versus the expected contrast using both the tomographic () and pixel by pixel () methods. The depth of the absorbing tube was 2 mm for , and 3 mm for . The contrast was determined by selecting a region of interest (ROI) 1 mm wide along the length of the tube and dividing the mean absorption value for the voxels in the ROI by the background value. For the tomographic images we selected all voxels from the depth at which the contrast was greatest. Error bars represent the standard deviation of voxels in the ROI.
Fig. 5 Plot of measured (reconstructed) vs. expected (known) contrast for absorbing tubes whose contrast ranged from 3:1 to 100:1 at depths of (a,b) 2 mm and (c,d) 3 mm. The tomographic method was used in (a) and (c), whereas (b) and (d) were calculated by fitting (more ...)
Regardless of depth or imaging method, all curves are linear up to a contrast of 30:1, and then begin to saturate as expected due to the non-linearity of the inverse problem for diffuse light. We note that for this experiment, the linear range for tomographic reconstruction was larger than in previous work by a factor of about three [14
]. However, non-linearity is expected to be more severe as the size of the inclusion becomes larger. Here the absorbing tube has a thickness of 1.5
*, where as in the previous work the thickness of the absorber was 13
*. Within the linear range both methods accurately give the relative change in contrast between the absorber and the background. However, the tomographic method comes closer to the actual values. Within the linear range, the percent error in measured absorption (i.e. 100 × (measured – expected)/expected) had a mean value of −2% and −28% at depths 2 mm and 3 mm respectively using the tomographic method.
The slight decrease in resolution at 3 mm suggests that the decrease in contrast measured at 3 mm is in part due to a partial volume effect. That is, the 3 mm tube appears broader and less intense due to the fact that the sensitivity of each detector is sharply peaked at the surface, and becomes both broader and smaller in magnitude with depth. While the reconstruction algorithm accounts for this effect, it does not remove it completely. Unlike the maximum absorption value, the total amount of reconstructed absorption does remain relatively constant with the depth of the tube. By integrating the reconstructed absorption of the entire tube, the total amount of reconstructed contrast is equal for the 2 mm and 3 mm depths to within 10%. In contrast, when fitting pixel by pixel to a homogeneous model the absolute values of absorption are underestimated by up to a factor of about ten. This was expected, since the trajectories of the detected photons are primarily located in background medium, and not in the absorbing tube.
We were able to determine the relative depths of absorbers, but not their absolute depths. show the depth profiles of the tomographic images of the single tubes at both depths. For clarity, only depth profiles for the 5×, 10×, 20×, 30×, and 50× concentrations are shown. The grey area denotes the range of depths at which the maximum absorption was reconstructed. For both the 2 and 3 mm depths, the reconstructed images underestimate the depth of the tube. The maximum absorption occurred from 0.6 to 0.7 mm and 0.9–1 mm for actual depths of 2 mm and 3 mm respectively. We attempted to use spatially varying regularization [24
] to overcome this effect. However, for these particular experiments we found that the depth at which the tubes appeared in the image was more determined by our choice of regularization parameter, than by the actual depth of the absorber.
Fig. 6 Depth profiles of the three-dimensional images. For clarity only contrasts of 5× (dashed line), 10× (dash-dot line), 20× (dotted line), 30× (solid line with dots), and 50× (solid line) are shown. The grey area corresponds (more ...)
In these experiments the absorbers were located at 2–3
* from the surface of the scattering medium, where the diffusion approximation to the radiative transport equation (RTE) is known to break down. It may be possible to more accurately determine the depths of buried absorbers by modeling photon transport using the RTE. Reconstruction algorithms such as the one used in this manuscript can be modified to incorporate radiative transport provided that the Fourier components of the Green’s functions (see Eq. (7)
) can be calculated for the RTE [25
]. We are currently exploring the use of both analytic [27
] and Monte Carlo methods to solve for these Fourier components. Use of the RTE may also result in more accurate quantification and improved resolution.