Quantification of bioluminescence images of point source objects using diffusion theory models

Algorithm for localized adaptive diffuse optical tomography and its application in bioluminescence tomography

A reconstruction algorithm for diffuse optical tomography based on diffusion theory and finite element method is described. The algorithm reconstructs the optical properties in a permissible domain or region-of-interest to reduce the number of unknowns. The algorithm can be used to reconstruct optical properties for a segmented object (where a CT-scan or MRI is available) or a non-segmented object. For the latter, an adaptive segmentation algorithm merges contiguous regions with similar optical properties thereby reducing the number of unknowns. In calculating the Jacobian matrix the algorithm uses an efficient direct method so the required time is comparable to that needed for a single forward calculation. The reconstructed optical properties using segmented, non-segmented, and adaptively segmented 3D mouse anatomy (MOBY) are used to perform bioluminescence tomography (BLT) for two simulated internal sources. The BLT results suggest that the accuracy of reconstruction of total source power obtained without the segmentation provided by an auxiliary imaging method such as x-ray CT is comparable to that obtained when using perfect segmentation.

Maximum likelihood tomographic reconstruction of extremely sparse solutions in diffuse fluorescence flow cytometry

We apply reparameterization and the maximum likelihood method to a specific fluorescence-mediated tomography problem where the solution is known a priori to be extremely sparse (i.e., all image values are zero except for one). Our algorithm performs significantly better than a standard image reconstruction method, particularly for deep-seated targets, and achieves close to 150 μm accuracy in a 3 mm diameter cross-sectional area with only 12 measurements. Moreover, results do not depend on the selection of a regularization parameter or other ad hoc values, and since reconstructions can be computed very quickly, the algorithm is also suitable for real-time implementation.

Self-calibrated algorithms for diffuse optical tomography and bioluminescence tomography using relative transmission images

Reconstruction algorithms for diffuse optical tomography (DOT) and bioluminescence tomography (BLT) have been developed based on diffusion theory. The algorithms numerically solve the diffusion equation using the finite element method. The direct measurements of the uncalibrated light fluence rates by a camera are used for the reconstructions. The DOT is self-calibrated by using all possible pairs of transmission images obtained with external sources along with the relative values of the simulated data and the calculated Jacobian. The reconstruction is done in the relative domain with the cancelation of any geometrical or optical factors. The transmission measurements for the DOT are used for calibrating the bioluminescence measurements at each wavelength and then a normalized system of equations is built up which is self-calibrated for the BLT. The algorithms have been applied to a three dimensional model of the mouse (MOBY) segmented into tissue regions which are assumed to have uniform optical properties. The DOT uses the direct method for calculating the Jacobian. The BLT uses a reduced space of eigenvectors of the Green's function with iterative shrinking of the permissible source region. The reconstruction results of the DOT and BLT algorithms show good agreement with the actual values when using either absolute or relative data. Even a small calibration error causes significant degradation of the reconstructions based on absolute data.

Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region

A reconstruction algorithm for bioluminescence tomography (BLT) has been developed. The algorithm numerically calculates the Green's function at different wavelengths using the diffusion equation and finite element method. The optical properties used in calculating the Green's function are reconstructed using diffuse optical tomography (DOT) and assuming anatomical information is provided by x-ray computed tomography or other methods. A symmetric system of equations is formed using the Green's function and the measured light fluence rate and the resulting eigenvalue problem is solved to get the eigenvectors of this symmetric system of equations. A space can be formed from the eigenvectors obtained and the reconstructed source is written as an expansion of the eigenvectors corresponding to non-zero eigenvalues. The coefficients of the expansion are found to obtain the reconstructed BL source distribution. The problem is solved iteratively by using a permissible source region that is shrunk by removing nodes with low probability to contribute to the source. Throughout this process the permissible region shrinks from the entire object to just a few nodes. The best estimate of the reconstructed source is chosen that which minimizes the difference between the calculated and measured light fluence rates. 3D simulations presented here show that the reconstructed source is in good agreement with the actual source in terms of locations, magnitudes, sizes, and total powers for both localized multiple sources and large inhomogeneous source distributions.

Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region

Reconstruction algorithms are presented for two-step solutions of the bioluminescence tomography (BLT) and the fluorescence tomography (FT) problems. In the first step, a continuous wave (cw) diffuse optical tomography (DOT) algorithm is used to reconstruct the tissue optical properties assuming known anatomical information provided by x-ray computed tomography or other methods. Minimization problems are formed based on L1 norm objective functions, where normalized values for the light fluence rates and the corresponding Green's functions are used. Then an iterative minimization solution shrinks the permissible regions where the sources are allowed by selecting points with higher probability to contribute to the source distribution. Throughout this process the permissible region shrinks from the entire object to just a few points. The optimum reconstructed bioluminescence and fluorescence distributions are chosen to be the results of the iteration corresponding to the permissible region where the objective function has its global minimum This provides efficient BLT and FT reconstruction algorithms without the need for a priori information about the bioluminescence sources or the fluorophore concentration. Multiple small sources and large distributed sources can be reconstructed with good accuracy for the location and the total source power for BLT and the total number of fluorophore molecules for the FT. For non-uniform distributed sources, the size and magnitude become degenerate due to the degrees of freedom available for possible solutions. However, increasing the number of data points by increasing the number of excitation sources can improve the accuracy of reconstruction for non-uniform fluorophore distributions.

Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties

Reconstruction algorithms are presented for a two-step solution of the bioluminescence tomography (BLT) problem. In the first step, a priori anatomical information provided by x-ray computed tomography or by other methods is used to solve the continuous wave (cw) diffuse optical tomography (DOT) problem. A Taylor series expansion approximates the light fluence rate dependence on the optical properties of each region where first and second order direct derivatives of the light fluence rate with respect to scattering and absorption coefficients are obtained and used for the reconstruction. In the second step, the reconstructed optical properties at different wavelengths are used to calculate the Green's function of the system. Then an iterative minimization solution based on the L1 norm shrinks the permissible regions where the sources are allowed by selecting points with higher probability to contribute to the source distribution. This provides an efficient BLT reconstruction algorithm with the ability to determine relative source magnitudes and positions in the presence of noise.

Improving the accuracy of a solid spherical source radius and depth estimation using the diffusion equation in fluorescence reflectance mode

Background

Non-invasive planar fluorescence reflectance imaging (FRI) is used for accessing physiological and molecular processes in biological tissue. This method is efficiently used to detect superficial fluorescent inclusions. FRI is based on recording the spatial radiance distribution (SRD) at the surface of a sample. SRD provides information for measuring structural parameters of a fluorescent source (such as radius and depth). The aim of this article is to estimate the depth and radius of the source distribution from SRD, measured at the sample surface. For this reason, a theoretical expression for the SRD at the surface of a turbid sample arising from a spherical light source embedded in the sample, was derived using a steady-state solution of the diffusion equation with an appropriate boundary condition.

Methods

The SRD was approximated by solving the diffusion equation in an infinite homogeneous medium with solid spherical sources in cylindrical geometry. Theoretical predications were verified by experiments with fluorescent sources of radius 2-6 mm embedded at depths of 2-4 mm in a tissue-like phantom.

Results

The experimental data were compared with the theoretical values which shows that the root mean square (RMS) error in depth measurement for nominal depth values d = 2, 2.5, 3, 3.5, 4 mm amounted to 17%, 5%, 2%, 1% and 5% respectively. Therefore, the average error in depth estimation was ≤ 4% for depths larger than the photon mean free path.

Conclusions

An algorithm is proposed that allows estimation of the location and radius of a spherical source in a homogeneous tissue-like phantom by accounting for anisotropic light scattering effect using FRI modality. Surface SRD measurement enabled accurate estimates of fluorescent depth and radius in FRI modality, and can be used as an element of a more general tomography reconstruction algorithm.

Particle size, surface coating, and PEGylation influence the biodistribution of quantum dots in living mice

This study evaluates the influence of particle size, PEGylation, and surface coating on the quantitative biodistribution of near-infrared-emitting quantum dots (QDs) in mice. Polymer- or peptide-coated 64Cu-labeled QDs 2 or 12 nm in diameter, with or without polyethylene glycol (PEG) of molecular weight 2000, are studied by serial micropositron emission tomography imaging and region-of-interest analysis, as well as transmission electron microscopy and inductively coupled plasma mass spectrometry. PEGylation and peptide coating slow QD uptake into the organs of the reticuloendothelial system (RES), liver and spleen, by a factor of 6-9 and 2-3, respectively. Small particles are in part renally excreted. Peptide-coated particles are cleared from liver faster than physical decay alone would suggest. Renal excretion of small QDs and slowing of RES clearance by PEGylation or peptide surface coating are encouraging steps toward the use of modified QDs for imaging living subjects.

Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography

We investigate fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography for applications in small animal imaging. Our forward model uses a diffusion approximation for optically inhomogeneous tissue, which we solve using a finite element method (FEM). We examine two approaches to incorporating the forward model into the solution of the inverse problem. In a conventional direct calculation approach one computes the full forward model by repeated solution of the FEM problem, once for each potential source location. We describe an alternative on-the-fly approach where one does not explicitly solve for the full forward model. Instead, the solution to the forward problem is included implicitly in the formulation of the inverse problem, and the FEM problem is solved at each iteration for the current image estimate. We evaluate the convergence speeds of several representative iterative algorithms. We compare the computation cost of those two approaches, concluding that the on-the-fly approach can lead to substantial reductions in total cost when combined with a rapidly converging iterative algorithm.

Approximations of the measurable quantity in diffuse optical problems: theoretical analysis of model deviations

Photon density and photon flux are widely used to model the measurable quantity in diffuse optical tomography problems. However, it is not these two quantities that are actually measured, but rather the radiance accepted by the detection system. We provide a theoretical analysis of the model deviations related to the choice of the measurable quantity-either photon density or flux. By using the diffusion approximation to the radiative transfer equation and its solution with extrapolated boundary conditions, an exact analytical expression of the measurable quantity has been obtained. This expression has been employed as a reference to assess model deviation when considering the photon density or the photon flux as the measurable quantity. For the case of semi-infinite geometry and for both continuous wave and time domains, we show that the photon density approximates the measurable quantity better than the photon flux. We also demonstrate that the validity of this approximation strongly depends on the optical parameters.

Bioluminescence imaging of point sources implanted in small animals post mortem: evaluation of a method for estimating source strength and depth

The performance of a simple approach for the in vivo reconstruction of bioluminescent point sources in small animals was evaluated. The method uses the diffusion approximation as a forward model of light propagation from a point source in a homogeneous tissue to find the source depth and power. The optical properties of the tissue are estimated from reflectance images obtained at the same location on the animal. It was possible to localize point sources implanted in mice, 2-8 mm deep, to within 1 mm. The same performance was achieved for sources implanted in rat abdomens when the effects of tissue surface curvature were eliminated. The source power was reconstructed within a factor of 2 of the true power for the given range of depths, even though the apparent brightness of the source varied by several orders of magnitude. The study also showed that reconstructions using optical properties measured in situ were superior to those based on data in the literature.

microPET-based biodistribution of quantum dots in living mice

This study evaluates the quantitative biodistribution of commercially available CdSe quantum dots (QD) in mice.

METHODS

(64)Cu-Labeled 800- or 525-nm emission wavelength QD (21- or 12-nm diameter), with or without 2,000 MW (molecular weight) polyethylene glycol (PEG), were injected intravenously into mice (5.55 MBq/25 pmol QD) and studied using well counting or by serial microPET and region-of-interest analysis.

RESULTS

Both methods show rapid uptake by the liver (27.4-38.9 %ID/g) (%ID/g is percentage injected dose per gram tissue) and spleen (8.0-12.4 %ID/g). Size has no influence on biodistribution within the range tested here. Pegylated QD have slightly slower uptake into liver and spleen (6 vs. 2 min) and show additional low-level bone uptake (6.5-6.9 %ID/g). No evidence of clearance from these organs was observed.

CONCLUSION

Rapid reticuloendothelial system clearance of QD will require modification of QD for optimal utility in imaging living subjects. Formal quantitative biodistribution/imaging studies will be helpful in studying many types of nanoparticles, including quantum dots.

Validation of luminescent source reconstruction using single-view spectrally resolved bioluminescence images

We characterize the capabilities and limitations of the Living Image Software 3D Analysis package (Xenogen, Alameda, California) in the reconstruction of calibrated light sources. Sources shallower than the mean free path of light propagation suffered reconstruction inaccuracy. For sources deeper than the mean free path, the average error in depth and intensity reconstruction was less than 4% and 12%, respectively, for homogeneous tissue. The reconstruction of luminescent beads implanted within an optically heterogeneous mouse abdomen proved less accurate. The ability to distinguish multiple sources decreased with increasing source depth. A number of factors influence the accuracy of light source reconstruction.

All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast

Optical molecular imaging in small animals harnesses the power of highly specific and biocompatible contrast agents for drug development and disease research1-7. However, the widespread adoption of in vivo optical imaging has been inhibited by its inability to clearly resolve and identify targeted internal organs. Optical tomography8-11 and combined X-ray and micro-computed tomography (micro-CT)12 approaches developed to address this problem are generally expensive, complex or incapable of true anatomical co-registration. Here, we present a remarkably simple all-optical method that can generate co-registered anatomical maps of a mouse's internal organs, while also acquiring in vivo molecular imaging data. The technique uses a time series of images acquired after injection of an inert dye. Differences in the dye's in vivo biodistribution dynamics allow precise delineation and identification of major organs. Such co-registered anatomical maps permit longitudinal organ identification irrespective of repositioning or weight gain, thereby promising greatly improved accuracy and versatility for studies of orthotopic disease, diagnostics and therapies.