A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging

δ-SPN approximation for numerical modeling of directional sources and scattering

We propose the δ-SP_N approximation for the frequency domain coupled radiative transfer equations modeling fluorescence with collimated incident beams and present its numerical implementation using the finite element method. The performance of the proposed model is investigated with respect to Monte Carlo simulations and the standard SP_N approximation over sub-centimeter domains for various optical properties. We find that the δ-SP_N approximation is more accurate than the SP_N in the near-source region, and provides improved estimates of phase and partial currents, at both excitation and emission wavelengths, over a wider range of optical properties. The accuracy of the δ-SP_N model improves with increase in approximation order for normally incident beams.

Development of tomographic reconstruction for three-dimensional optical imaging: From the inversion of light propagation to artificial intelligence

Optical molecular tomography (OMT) is an imaging modality which uses an optical signal, especially near-infrared light, to reconstruct the three-dimensional information of the light source in biological tissue. With the advantages of being low-cost, noninvasive and having high sensitivity, OMT has been applied in preclinical and clinical research. However, due to its serious ill-posedness and ill-condition, the solution of OMT requires heavy data analysis and the reconstruction quality is limited. Recently, the artificial intelligence (commonly known as AI)-based methods have been proposed to provide a different tool to solve the OMT problem. In this paper, we review the progress on OMT algorithms, from conventional methods to AI-based methods, and we also give a prospective towards future developments in this domain.

Incorporating reflection boundary conditions in the Neumann series radiative transport equation: application to photon propagation and reconstruction in diffuse optical imaging

We propose a formalism to incorporate boundary conditions in a Neumann-series-based radiative transport equation. The formalism accurately models the reflection of photons at the tissue-external medium interface using Fresnel's equations. The formalism was used to develop a gradient descent-based image reconstruction technique. The proposed methods were implemented for 3D diffuse optical imaging. In computational studies, it was observed that the average root-mean-square error (RMSE) for the output images and the estimated absorption coefficients reduced by 38% and 84%, respectively, when the reflection boundary conditions were incorporated. These results demonstrate the importance of incorporating boundary conditions that model the reflection of photons at the tissue-external medium interface.

Frequency-domain optical tomographic image reconstruction algorithm with the simplified spherical harmonics (SP3) light propagation model

We introduce here the finite volume formulation of the frequency-domain simplified spherical harmonics model with n-th order absorption coefficients (FD-SP_N) that approximates the frequency-domain equation of radiative transfer (FD-ERT). We then present the FD-SP_N based reconstruction algorithm that recovers absorption and scattering coefficients in biological tissue. The FD-SP_N model with 3^rd order absorption coefficient (i.e., FD-SP₃) is used as a forward model to solve the inverse problem. The FD-SP₃ is discretized with a node-centered finite volume scheme and solved with a restarted generalized minimum residual (GMRES) algorithm. The absorption and scattering coefficients are retrieved using a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Finally, the forward and inverse algorithms are evaluated using numerical phantoms with optical properties and size that mimic small-volume tissue such as finger joints and small animals. The forward results show that the FD-SP₃ model approximates the FD-ERT (S₁₂) solution within relatively high accuracy; the average error in the phase (<3.7%) and the amplitude (<7.1%) of the partial current at the boundary are reported. From the inverse results we find that the absorption and scattering coefficient maps are more accurately reconstructed with the SP₃ model than those with the SP₁ model. Therefore, this work shows that the FD-SP₃ is an efficient model for optical tomographic imaging of small-volume media with non-diffuse properties both in terms of computational time and accuracy as it requires significantly lower CPU time than the FD-ERT (S₁₂) and also it is more accurate than the FD-SP₁.

Near-Infrared Fluorescence-Enhanced Optical Tomography

Fluorescence-enhanced optical imaging using near-infrared (NIR) light developed for in vivo molecular targeting and reporting of cancer provides promising opportunities for diagnostic imaging. The current state of the art of NIR fluorescence-enhanced optical tomography is reviewed in the context of the principle of fluorescence, the different measurement schemes employed, and the mathematical tools established to tomographically reconstruct the fluorescence optical properties in various tissue domains. Finally, we discuss the recent advances in forward modeling and distributed memory parallel computation to provide robust, accurate, and fast fluorescence-enhanced optical tomography.

Experimental Comparison of Continuous-Wave and Frequency-Domain Fluorescence Tomography in a Commercial Multi-Modal Scanner

The performance evaluation of a variety of small animal tomography measurement approaches and algorithms for recovery of fluorescent absorption cross section has not been conducted. Herein, we employed an intensified CCD system installed in a commercial small animal CT (Computed Tomography) scanner to compare image reconstructions from time-independent, continuous wave (CW) measurements and from time-dependent, frequency domain (FD) measurements in a series of physical phantoms specifically designed for evaluation. Comparisons were performed as a function of (1) number of projections, (2) the level of preprocessing filters used to improve the signal-to-noise ratio (SNR), (3) endogenous heterogeneity of optical properties, as well as in the cases of (4) two fluorescent targets and (5) a mouse-shaped phantom. Assessment of quantitative recovery of fluorescence absorption cross section was performed using a fully parallel, regularization-free, linear reconstruction algorithm with diffusion approximation (DA) and high order simplified spherical harmonics ( SPN) approximation to the radiative transport equation (RTE). The results show that while FD measurements may result in superior image reconstructions over CW measurements, data acquisition times are significantly longer, necessitating further development of multiple detector/source configurations, improved data read-out rates, and detector technology. FD measurements with SP3 reconstructions enabled better quantitative recovery of fluorescent target strength, but required increased computational expense. Despite the developed parallel reconstruction framework being able to achieve more than 60 times speed increase over sequential implementation, further development in faster parallel acceleration strategies for near-real time and real-time image recovery and more precise forward solution is necessary.

Analytical Green's function for the fluorescence simplified spherical harmonics equations in turbid medium

It is more complicated to write the analytical expression for the fluorescence simplified spherical harmonics ( SPN) equations in a turbid medium, since both the processes of the excitation and emission light and the composite moments of the fluence rate are described by coupled equations. Based on an eigen-decomposition strategy and the well-developed analytical methods of diffusion approximation (DA), we derive the analytical solutions to the fluorescence SPN equations for regular geometries using the Green's function approach. By means of comparisons with the results of fluorescence DA and Monte Carlo simulations, we have shown the effectiveness of our proposed method and the expected advantages of the SPN equations in the case of small source-detector separation and high absorption.

Far-red fluorescence gene reporter tomography for determination of placement and viability of cell-based gene therapies

Non-invasive injectable cellular therapeutic strategies based on sustained delivery of physiological levels of BMP-2 for spinal fusion are emerging as promising alternatives, which could provide sufficient fusion without the associated surgical risks. However, these injectable therapies are dependent on bone formation occurring only at the specific target region. In this study, we developed and deployed fluorescence gene reporter tomography (FGRT) to provide information on in vivo cell localization and viability. This information is sought to confirm the ideal placement of the materials with respect to the area where early bone reaction is required, ultimately providing three dimensional data about the future fusion. However, because almost all conventional fluorescence gene reporters require visible excitation wavelengths, current in vivo imaging of fluorescent proteins is limited by high tissue absorption and confounding autofluorescence. We previously administered fibroblasts engineered to produce BMP-2, but is difficult to determine 3-D information of placement prior to bone formation. Herein we used the far-red fluorescence gene reporter, IFP1.4 to report the position and viability of fibroblasts and developed 3-D tomography to provide placement information. A custom small animal, far-red fluorescence tomography system integrated into a commercial CT scanner was used to assess IFP1.4 fluorescence and to demark 3-D placement of encapsulated fibroblasts with respect to the vertebrae and early bone formation as assessed from CT. The results from three experiments showed that the placement of the materials within the spine could be detected. This work shows that in vivo fluorescence gene reporter tomography of cell-based gene therapy is feasible and could help guide cell-based therapies in preclinical models.

In vivo imaging of orthotopic prostate cancer with far-red gene reporter fluorescence tomography and in vivo and ex vivo validation

Fluorescence gene reporters have recently become available for excitation at far-red wavelengths, enabling opportunities for small animal in vivo gene reporter fluorescence tomography (GRFT). We employed multiple projections of the far-red fluorescence gene reporters IFP1.4 and iRFP, excited by a point source in transillumination geometry in order to reconstruct the location of orthotopically implanted human prostate cancer (PC3), which stably expresses the reporter. Reconstruction was performed using a linear radiative-transfer-based regularization-free tomographic method. Positron emission tomography (PET) imaging of a radiolabeled antibody-based agent that targeted epithelial cell adhesion molecule overexpressed on PC3 cells was used to confirm in vivo GRFT results. Validation of GRFT results was also conducted from ex vivo fluorescence imaging of resected prostate tumor. In addition, in mice with large primary prostate tumors, a combination of GRFT and PET showed that the radiolabeled antibody did not penetrate the tumor, consistent with known tumor transport limitations of large (∼150  kDa) molecules. These results represent the first tomography of a living animal using far-red gene reporters.

Time-domain fluorescence-guided diffuse optical tomography based on the third-order simplified harmonics approximation

Extensive efforts have been made to integrate diffuse optical tomography (DOT) with other imaging modalities, such as magnetic-resonance imaging and x-ray computerized tomography, for its performance improvement. However, the experimental apparatus is in general intricate and costly due to adoption of the physically distinct radiation regimes. In this study, a time-domain fluorescence-guided DOT methodology that incorporates a priori localization information provided by diffuse fluorescence tomography (DFT) is investigated in an attempt to optimize recovery of the optical property distributions. The methodology is based on a specifically designed multichannel time-correlated single-photon-counting DOT/DFT system as well as a featured-data image reconstruction scheme that is developed within the framework of the generalized pulse spectrum technique and employs the third-order simplified harmonics approximation to the radiative transfer equation as the forward model. We have validated the methodology using phantom experiments and demonstrated that, with the guidance of fluorescence a priori, the quantitativeness and spatial resolution of the recovered optical target can be considerably improved in terms of the absorption and scattering images.

Extended finite element method with simplified spherical harmonics approximation for the forward model of optical molecular imaging

An extended finite element method (XFEM) for the forward model of 3D optical molecular imaging is developed with simplified spherical harmonics approximation (SP(N)). In XFEM scheme of SP(N) equations, the signed distance function is employed to accurately represent the internal tissue boundary, and then it is used to construct the enriched basis function of the finite element scheme. Therefore, the finite element calculation can be carried out without the time-consuming internal boundary mesh generation. Moreover, the required overly fine mesh conforming to the complex tissue boundary which leads to excess time cost can be avoided. XFEM conveniences its application to tissues with complex internal structure and improves the computational efficiency. Phantom and digital mouse experiments were carried out to validate the efficiency of the proposed method. Compared with standard finite element method and classical Monte Carlo (MC) method, the validation results show the merits and potential of the XFEM for optical imaging.

A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography

The work presented herein describes the system design and performance evaluation of a miniaturized near-infrared fluorescence (NIRF) frequency-domain photon migration (FDPM) system with non-contact excitation and homodyne detection capability for small animal fluorescence tomography. The FDPM system was developed specifically for incorporation into a Siemens micro positron emission tomography/computed tomography (microPET/CT) commercial scanner for hybrid small animal imaging, but could be adapted to other systems. Operating at 100 MHz, the system noise was minimized and the associated amplitude and phase errors were characterized to be ±0.7% and ±0.3°, respectively. To demonstrate the tomographic ability, a commercial mouse-shaped phantom with 50 µM IRDye800CW and ⁶⁸Ga containing inclusion was used to associate PET and NIRF tomography. Three-dimensional mesh generation and anatomical referencing was accomplished through CT. A third-order simplified spherical harmonics approximation (SP₃) algorithm, for efficient prediction of light propagation in small animals, was tailored to incorporate the FDPM approach. Finally, the PET-NIRF target co-localization accuracy was analyzed in vivo with a dual-labeled imaging agent targeting orthotopic growth of human prostate cancer. The obtained results validate the integration of time-dependent fluorescence tomography system within a commercial microPET/CT scanner for multimodality small animal imaging.

L1-regularized Cerenkov luminescence tomography with a SP3 method and CT fusion

Imaging modality of radionuclides has been enriched by an optical approach, Cerenkov luminescence tomography (CLT). Referred to the traditional radionuclide imaging, such as positron emission tomography (PET) or single photon emission computed tomography (SPECT), any incremental improvement of CLT imaging is consistent with the application to information needs. In this contribution, the paper presents an l(1)-regularized imaging method for CLT problem. After utilizing the Vavilov-Cerenkov effect via third-order simplified spherical harmonics (SP(3)) approximation, we establish the large-scale linear equations in the CLT framework. The derived linear problem is seriously ill-posed, and transformed into an l(1)-regularized least squares program. The inverse solution to these equations is the three-dimensional radioisotope recovery data by an interior-point method. In the physical phantom and the in vivo mouse experiment, results demonstrate that the proposed technique produces better imaging quality and improves the reconstruction efficacy, compared with those from diffusion approximation with the Tikhonov regularization.

Total variation regularization for bioluminescence tomography with the split Bregman method

Regularization methods have been broadly applied to bioluminescence tomography (BLT) to obtain stable solutions, including l2 and l1 regularizations. However, l2 regularization can oversmooth reconstructed images and l1 regularization may sparsify the source distribution, which degrades image quality. In this paper, the use of total variation (TV) regularization in BLT is investigated. Since a nonnegativity constraint can lead to improved image quality, the nonnegative constraint should be considered in BLT. However, TV regularization with a nonnegativity constraint is extremely difficult to solve due to its nondifferentiability and nonlinearity. The aim of this work is to validate the split Bregman method to minimize the TV regularization problem with a nonnegativity constraint for BLT. The performance of split Bregman-resolved TV (SBRTV) based BLT reconstruction algorithm was verified with numerical and in vivo experiments. Experimental results demonstrate that the SBRTV regularization can provide better regularization quality over l2 and l1 regularizations.

Multilevel, hybrid regularization method for reconstruction of fluorescent molecular tomography

In this paper, a multilevel, hybrid regularization method is presented for fluorescent molecular tomography (FMT) based on the hp-finite element method (hp-FEM) with a continuous wave. The hybrid regularization method combines sparsity regularization and Landweber iterative regularization to improve the stability of the solution of the ill-posed inverse problem. In the first coarse mesh level, considering the fact that the fluorescent probes are sparsely distributed in the entire reconstruction region in most FMT applications, the sparse regularization method is employed to take full advantage of this sparsity. In the subsequent refined mesh levels, since the reconstruction region is reduced and the initial value of the unknown parameters is provided from the previous mesh, these mesh levels seem to be different from the first level. As a result, the Landweber iterative regularization method is applied for reconstruction. Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are conducted to evaluate the performance of our method. The reconstructed results show the potential and feasibility of the proposed approach.

Perturbative forward solver software for small localized fluorophores in tissue

In this paper a forward solver software for the time domain and the CW domain based on the Born approximation for simulating the effect of small localized fluorophores embedded in a non-fluorescent biological tissue is proposed. The fluorescence emission is treated with a mathematical model that describes the migration of photons from the source to the fluorophore and of emitted fluorescent photons from the fluorophore to the detector for all those geometries for which Green's functions are available. Subroutines written in FORTRAN that can be used for calculating the fluorescent signal for the infinite medium and for the slab are provided with a linked file. With these subroutines, quantities such as reflectance, transmittance, and fluence rate can be calculated.

Reconstruction of sectional images in frequency-domain based photoacoustic imaging

Photoacoustic (PA) imaging is based upon the generation of an ultrasound pulse arising from subsurface tissue absorption due to pulsed laser excitation, and measurement of its surface time-of-arrival. Expensive and bulky pulsed lasers with high peak fluence powers may provide shortcomings for applications of PA imaging in medicine and biology. These limitations may be overcome with the frequency-domain PA measurements, which employ modulated rather than pulsed light to generate the acoustic wave. In this contribution, we model the single modulation frequency based PA pressures on the measurement plane through the diffraction approximation and then employ a convolution approach to reconstruct the sectional image slices. The results demonstrate that the proposed method with appropriate data post-processing is capable of recovering sectional images while suppressing the defocused noise resulting from the other sections.

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.

A study on tetrahedron-based inhomogeneous Monte Carlo optical simulation

Monte Carlo (MC) simulation is widely recognized as a gold standard in biophotonics for its high accuracy. Here we analyze several issues associated with tetrahedron-based optical Monte Carlo simulation in the context of TIM-OS, MMCM, MCML, and CUDAMCML in terms of accuracy and efficiency. Our results show that TIM-OS has significant better performance in the complex geometry cases and has comparable performance with CUDAMCML in the multi-layered tissue model.