Statistical analysis of nonlinearly reconstructed near-infrared tomographic images: Part I--Theory and simulations

Periodic-net: an end-to-end data driven framework for diffuse optical imaging of breast cancer from noisy boundary data

SIGNIFICANCE

The machine learning (ML) approach plays a critical role in assessing biomedical imaging processes especially optical imaging (OI) including segmentation, classification, and reconstruction, intending to achieve higher accuracy efficiently.

AIM

This research aims to develop an end-to-end deep learning framework for diffuse optical imaging (DOI) with multiple datasets to detect breast cancer and reconstruct its optical properties in the early stages.

APPROACH

The proposed Periodic-net is a nondestructive deep learning (DL) algorithm for the reconstruction and evaluation of inhomogeneities in an inverse model with high accuracy, while boundary measurements are calculated by solving a forward problem with sources/detectors arranged uniformly around a circular domain in various combinations, including 16 × 15 , 20 × 19 , and 36 × 35 boundary measurement setups.

RESULTS

The results of image reconstruction on numerical and phantom datasets demonstrate that the proposed network provides higher-quality images with a greater amount of small details, superior immunity to noise, and sharper edges with a reduction in image artifacts than other state-of-the-art competitors.

CONCLUSIONS

The network is highly effective at the simultaneous reconstruction of optical properties, i.e., absorption and reduced scattering coefficients, by optimizing the imaging time without degrading inclusions localization and image quality.

Selfrec-Net: self-supervised deep learning approach for the reconstruction of Cherenkov-excited luminescence scanned tomography

As an emerging imaging technique, Cherenkov-excited luminescence scanned tomography (CELST) can recover a high-resolution 3D distribution of quantum emission fields within tissue using X-ray excitation for deep penetrance. However, its reconstruction is an ill-posed and under-conditioned inverse problem because of the diffuse optical emission signal. Deep learning based image reconstruction has shown very good potential for solving these types of problems, however they suffer from a lack of ground-truth image data to confirm when used with experimental data. To overcome this, a self-supervised network cascaded by a 3D reconstruction network and the forward model, termed Selfrec-Net, was proposed to perform CELST reconstruction. Under this framework, the boundary measurements are input to the network to reconstruct the distribution of the quantum field and the predicted measurements are subsequently obtained by feeding the reconstructed result to the forward model. The network was trained by minimizing the loss between the input measurements and the predicted measurements rather than the reconstructed distributions and the corresponding ground truths. Comparative experiments were carried out on both numerical simulations and physical phantoms. For singular luminescent targets, the results demonstrate the effectiveness and robustness of the proposed network, and comparable performance can be attained to a state-of-the-art deep supervised learning algorithm, where the accuracy of the emission yield and localization of the objects was far superior to iterative reconstruction methods. Reconstruction of multiple objects is still reasonable with high localization accuracy, although with limits to the emission yield accuracy as the distribution becomes more complex. Overall though the reconstruction of Selfrec-Net provides a self-supervised way to recover the location and emission yield of molecular distributions in murine model tissues.

Deep-learning based image reconstruction for MRI-guided near-infrared spectral tomography

Non-invasive near-infrared spectral tomography (NIRST) can incorporate the structural information provided by simultaneous magnetic resonance imaging (MRI), and this has significantly improved the images obtained of tissue function. However, the process of MRI guidance in NIRST has been time consuming because of the needs for tissue-type segmentation and forward diffuse modeling of light propagation. To overcome these problems, a reconstruction algorithm for MRI-guided NIRST based on deep learning is proposed and validated by simulation and real patient imaging data for breast cancer characterization. In this approach, diffused optical signals and MRI images were both used as the input to the neural network, and simultaneously recovered the concentrations of oxy-hemoglobin, deoxy-hemoglobin, and water via end-to-end training by using 20,000 sets of computer-generated simulation phantoms. The simulation phantom studies showed that the quality of the reconstructed images was improved, compared to that obtained by other existing reconstruction methods. Reconstructed patient images show that the well-trained neural network with only simulation data sets can be directly used for differentiating malignant from benign breast tumors.

Back-propagation neural network-based reconstruction algorithm for diffuse optical tomography

Diffuse optical tomography (DOT) is a promising noninvasive imaging modality and is capable of providing functional characteristics of biological tissue by quantifying optical parameters. The DOT image reconstruction is ill-posed and ill-conditioned, due to the highly diffusive nature of light propagation in biological tissues and limited boundary measurements. The widely used regularization technique for DOT image reconstruction is Tikhonov regularization, which tends to yield oversmoothed and low-quality images containing severe artifacts. It is necessary to accurately choose a regularization parameter for Tikhonov regularization. To overcome these limitations, we develop a noniterative reconstruction method, whereby optical properties are recovered based on a back-propagation neural network (BPNN). We train the parameters of BPNN before DOT image reconstruction based on a set of training data. DOT image reconstruction is achieved by implementing a single evaluation of the trained network. To demonstrate the performance of the proposed algorithm, we compare with the conventional Tikhonov regularization-based reconstruction method. The experimental results demonstrate that image quality and quantitative accuracy of reconstructed optical properties are significantly improved with the proposed algorithm.

Weighting function effects in a direct regularization method for image-guided near-infrared spectral tomography of breast cancer

Structural image-guided near-infrared spectral tomography (NIRST) has been developed as a way to use diffuse NIR spectroscopy within the context of image-guided quantification of tissue spectral features. A direct regularization imaging (DRI) method for NIRST has the value of not requiring any image segmentation. Here, we present a comprehensive investigational study to analyze the impact of the weighting function implied when weighting the recovery of optical coefficients in DRI based NIRST. This was done using simulations, phantom and clinical patient exam data. Simulations where the true object is known indicate that changes to this weighting function can vary the contrast by 10%, the contrast to noise ratio by 20% and the full width half maximum (FWHM) by 30%. The results from phantoms and human images show that a linear inverse distance weighting function appears optimal, and that incorporation of this function can generally improve the recovered total hemoglobin contrast of the tumor to the normal surrounding tissue by more than 15% in human cases.

Optical property uncertainty estimates for spatial frequency domain imaging

Spatial frequency domain imaging (SFDI) is a wide-field diffuse optical imaging modality that has attracted considerable interest in recent years. Typically, diffuse reflectance measurements of spatially modulated light are used to quantify the optical absorption and reduced scattering coefficients of tissue, and with these, chromophore concentrations are extracted. However, uncertainties in estimated absorption and reduced scattering coefficients are rarely reported, and we know of no method capable of providing these when look-up table (LUT) algorithms are used to recover the optical properties. We present a method to generate optical property uncertainty estimates from knowledge of diffuse reflectance measurement errors. By employing the Cramér-Rao bound, we can quickly and efficiently explore theoretical SFDI performance as a function of spatial frequencies and sample optical properties, allowing us to optimize spatial frequency selection for a given application. In practice, we can also obtain useful uncertainty estimates for optical properties recovered with a two-frequency LUT algorithm, as we demonstrate with tissue-simulating phantom and in vivo experiments. Finally, we illustrate how absorption coefficient uncertainties can be propagated forward to yield uncertainties for chromophore concentrations, which could significantly impact the interpretation of experimental results.

Direct regularization from co-registered anatomical images for MRI-guided near-infrared spectral tomographic image reconstruction

Combining anatomical information from high resolution imaging modalities to guide near-infrared spectral tomography (NIRST) is an efficient strategy for improving the quality of the reconstructed spectral images. A new approach for incorporating image information directly into the inversion matrix regularization was examined using Direct Regularization from Images (DRI), which encodes the gray-scale data into the NIRST image reconstruction problem. This process has the benefit of eliminating user intervention such as image segmentation of distinct regions. Specifically, the Dynamic Contrast Enhanced Magnetic Resonance (DCE-MR) image intensity value differences within the anatomical image were used to implement an exponentially-weighted regularization function between the image pixels. The algorithm was validated using simulated reconstructions with noise, and the results showed that spatial resolution and robustness of the reconstructed images were significantly improved by appropriate choice of the regularization weight parameters. The proposed approach was also tested on in vivo breast data acquired in a recent clinical trial combining NIRST / MRI for cancer tumor characterization. Relative to the standard "no priors" diffuse recovery, the contrast of the tumor to the normal surrounding tissue increased from 2.4 to 3.6, and the difference between the tumor size segmented from DCE-MR images and reconstructed optical images decreased from 18% to 6%, while there was an overall decrease in surface artifacts.

On the use of the Cramér-Rao lower bound for diffuse optical imaging system design

We evaluated the potential of the Cramér-Rao lower bound (CRLB) to serve as a design metric for diffuse optical imaging systems. The CRLB defines the best achievable precision of any estimator for a given data model; it is often used in the statistical signal processing community for feasibility studies and system design. Computing the CRLB requires inverting the Fisher information matrix (FIM), however, which is usually ill-conditioned (and often underdetermined) in the case of diffuse optical tomography (DOT). We regularized the FIM by assuming that the inhomogeneity to be imaged was a point target and assessed the ability of point-target CRLBs to predict system performance in a typical DOT setting in silico. Our reconstructions, obtained with a common iterative algebraic technique, revealed that these bounds are not good predictors of imaging performance across different system configurations, even in a relative sense. This study demonstrates that agreement between the trends predicted by the CRLBs and imaging performance obtained with reconstruction algorithms that rely on a different regularization approach cannot be assumed a priori. Moreover, it underscores the importance of taking into account the intended regularization method when attempting to optimize source-detector configurations.

Implicit and explicit prior information in near-infrared spectral imaging: accuracy, quantification and diagnostic value

Near-infrared spectroscopy (NIRS) of tissue provides quantification of absorbers, scattering and luminescent agents in bulk tissue through the use of measurement data and assumptions. Prior knowledge can be critical about things such as (i) the tissue shape and/or structure, (ii) spectral constituents, (iii) limits on parameters, (iv) demographic or biomarker data, and (v) biophysical models of the temporal signal shapes. A general framework of NIRS imaging with prior information is presented, showing that prior information datasets could be incorporated at any step in the NIRS process, with the general workflow being: (i) data acquisition, (ii) pre-processing, (iii) forward model, (iv) inversion/reconstruction, (v) post-processing, and (vi) interpretation/diagnosis. Most of the development in NIRS has used ad hoc or empirical implementations of prior information such as pre-measured absorber or fluorophore spectra, or tissue shapes as estimated by additional imaging tools. A comprehensive analysis would examine what prior information maximizes the accuracy in recovery and value for medical diagnosis, when implemented at separate stages of the NIRS sequence. Individual applications of prior information can show increases in accuracy or improved ability to estimate biochemical features of tissue, while other approaches may not. Most beneficial inclusion of prior information has been in the inversion/reconstruction process, because it solves the mathematical intractability. However, it is not clear that this is always the most beneficial stage.

Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography

PURPOSE

A computationally efficient algorithm (linear iterative type) based on singular value decomposition (SVD) of the Jacobian has been developed that can be used in rapid dynamic near-infrared (NIR) diffuse optical tomography.

METHODS

Numerical and experimental studies have been conducted to prove the computational efficacy of this SVD-based algorithm over conventional optical image reconstruction algorithms.

RESULTS

These studies indicate that the performance of linear iterative algorithms in terms of contrast recovery (quantitation of optical images) is better compared to nonlinear iterative (conventional) algorithms, provided the initial guess is close to the actual solution. The nonlinear algorithms can provide better quality images compared to the linear iterative type algorithms. Moreover, the analytical and numerical equivalence of the SVD-based algorithm to linear iterative algorithms was also established as a part of this work. It is also demonstrated that the SVD-based image reconstruction typically requires O(NN2) operations per iteration, as contrasted with linear and nonlinear iterative methods that, respectively, require O(NN3) and O(NN6) operations, with "NN" being the number of unknown parameters in the optical image reconstruction procedure.

CONCLUSIONS

This SVD-based computationally efficient algorithm can make the integration of image reconstruction procedure with the data acquisition feasible, in turn making the rapid dynamic NIR tomography viable in the clinic to continuously monitor hemodynamic changes in the tissue pathophysiology.

Automatic exposure control and estimation of effective system noise in diffuse fluorescence tomography

A diffuse fluorescence tomography system, based upon time-correlated single photon counting, is presented with an automated algorithm to allow dynamic range variation through exposure control. This automated exposure control allows the upper and lower detection levels of fluorophore to be extended by an order of magnitude beyond the previously published performance and benefits in a slight decrease in system effective noise. The effective noise level is used as a metric to characterize the system performance, integrating both model-mismatch and calibration bias errors into a single parameter. This effective error is near 7% of the reconstructed fluorescent yield value, when imaging in just few minutes. Quantifying protoporphyrin IX concentrations down to 50 ng/ml is possible, for tumor-sized regions. This fluorophore has very low fluorescence yield, but high biological relevance for tumor imaging, given that it is produced in the mitochondria, and upregulated in many tumor types.

Improving image quality of diffuse optical tomography with a projection-error-based adaptive regularization method

Diffuse optical tomography (DOT) reconstructs the images of internal optical parameter distribution using noninvasive boundary measurements. The image reconstruction procedure is known to be an ill-posed problem. In order to solve such a problem, a regularization technique is needed to constrain the solution space. In this study, a projection-error-based adaptive regularization (PAR) technique is proposed to improve the reconstructed image quality. Simulations are performed using a diffusion approximation model and the simulated results demonstrate that the PAR technique can improve reconstruction precision of object more effectively. The method is demonstrated to have low sensitivity to noise at various noise levels. Moreover, with the PAR method, the detectability of an object located both at the center and near the peripheral regions has been increased largely.

A simulation study of the variability of indocyanine green kinetics and using structural a priori information in dynamic contrast enhanced diffuse optical tomography (DCE-DOT)

We investigated (1) the variability of indocyanine green kinetics (ICG) between different cases in the existence of random noise, changing the size of the imaging region, the location and the size of the inclusion, (2) the use of structural a priori information to reduce the variability. We performed two-dimensional simulation studies for this purpose. In the simulations, we used a two-compartmental model to describe the ICG transport and obtained pharmacokinetic parameters. The transfer constant and the rate constant showed a wide variation, i.e. 60% and 95%, respectively, when random Gaussian noise with a standard deviation of 1% in amplitude and 0.4 degrees in phase was added to data. Moreover, recovered peak ICG concentration and time to reach the peak concentration was different within different cases. When structural a priori information was used in the reconstructions, the variations in the transfer and the rate constant were reduced to 29%, 15%, respectively. As a result, although the recovered peak concentration was still case dependent, the variability of the shape of the kinetic curve was reduced.

Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography

Imaging of oxygen saturation provides a spatial map of the tissue metabolic activity and has potential in diagnosis and treatment monitoring of breast cancer. Oxygen-saturation imaging is possible through near-infrared (NIR) tomography, but has low signal-to-noise ratio (SNR). This can be augmented by using NIR tomography as an add-on to MRI. Presented are results from a free-standing NIR system and a hybrid MR-guided system for breast imaging. In results from imaging 60 healthy volunteers in the initial NIR system, oxygen saturation was a significant discriminator between the BIRADS classifications of adipose tissue, heterogeneously dense, and extremely dense tissue. By using the MR-guided NIR system, more accurate tissue-specific data were obtained on adipose and fibroglandular volumes, with 11 healthy volunteers. In these data, oxygen saturation in the adipose tissue correlated with percentage of adipose tissue. In two case studies of infiltrating ductal carcinomas, oxygen saturation was reduced at the site of the tumor, as compared with the surrounding healthy tissue, agreeing with conventional thought that hypoxia exists in larger solid tumors. The MRI-guided NIR images of oxygen saturation provide higher resolution and superior SNR and will likely be used in the future to study and characterize specific tissue volumes.

Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography

Diffuse optical tomography (DOT) involves estimation of tissue optical properties using noninvasive boundary measurements. The image reconstruction procedure is a nonlinear, ill-posed, and ill-determined problem, so overcoming these difficulties requires regularization of the solution. While the methods developed for solving the DOT image reconstruction procedure have a long history, there is less direct evidence on the optimal regularization methods, or exploring a common theoretical framework for techniques which uses least-squares (LS) minimization. A generalized least-squares (GLS) method is discussed here, which takes into account the variances and covariances among the individual data points and optical properties in the image into a structured weight matrix. It is shown that most of the least-squares techniques applied in DOT can be considered as special cases of this more generalized LS approach. The performance of three minimization techniques using the same implementation scheme is compared using test problems with increasing noise level and increasing complexity within the imaging field. Techniques that use spatial-prior information as constraints can be also incorporated into the GLS formalism. It is also illustrated that inclusion of spatial priors reduces the image error by at least a factor of 2. The improvement of GLS minimization is even more apparent when the noise level in the data is high (as high as 10%), indicating that the benefits of this approach are important for reconstruction of data in a routine setting where the data variance can be known based upon the signal to noise properties of the instruments.

Experimental fluorescence tomography of tissues with noncontact measurements

Noncontact optical measurements from diffuse media could facilitate the use of large detector arrays at multiple angles that are well suited for diffuse optical tomography applications. Such imaging strategy could eliminate the need for individual fibers in contact with tissue, restricted geometries, and matching fluids. Thus, it could significantly improve experimental procedures and enhance our ability to visualize functional and molecular processes in vivo. In this paper, we describe the experimental implementation of this novel concept and demonstrate capacity to perform small animal imaging.

Automated region detection based on the contrast-to-noise ratio in near-infrared tomography

The contrast-to-noise ratio (CNR) was used to determine the detectability of objects within reconstructed images from diffuse near-infrared tomography. It was concluded that there was a maximal value of CNR near the location of an object within the image and that the size of the true region could be estimated from the CNR. Experimental and simulation studies led to the conclusion that objects can be automatically detected with CNR analysis and that our current system has a spatial resolution limit near 4 mm and a contrast resolution limit near 1.4. A new linear convolution method of CNR calculation was developed for automated region of interest (ROI) detection.

Three-dimensional optical tomography: resolution in small-object imaging

Near-infrared (NIR) optical tomography provide estimates of the internal distribution of optical absorption and transport scattering from boundary measurement of light propagation within biological tissue. Although this is a truly three-dimensional (3D) imaging problem, most research to date has concentrated on two-dimensional modeling and image reconstruction. More recently, 3D imaging algorithms are demonstrating better estimation of the light propagation within the imaging region and are providing the basis of more accurate image construction algorithms. As 3D methods emerge, it will become increasingly important to evaluate their resolution, contrast, and localization of optical property heterogeneity. We present a concise study of 3D reconstructed resolution of a small, low-contrast, absorbing and scattering anomaly as it is placed in different locations within a cylindrical phantom. The object is an 8-mm-diameter cylinder, which represents a typical small target that needs to be resolved in NIR mammographic imaging. The best resolution and contrast is observed when the object is located near the periphery of the imaging region (12-22 mm from the edge) and is also positioned within the multiple measurement planes, with the most accurate results seen for the scatter image when the anomaly is at 17 mm from the edge. Furthermore, the accuracy of quantitative imaging is increased to almost 100% of the target values when a priori information regarding the internal structure of imaging domain is utilized.

Statistical analysis of nonlinearly reconstructed near-infrared tomographic images: Part II--Experimental interpretation

Image error analysis of a diffuse near-infrared tomography (NIR) system has been carried out on simulated data using a statistical approach described in Part I of this paper (Pogue et al., 2002). The methodology is used here with experimental data acquired on phantoms with a prototype imaging system intended for characterizing breast tissue. Results show that imaging performance is not limited by random measurement error, but rather by calibration issues. The image error over the entire field of view is generally not minimized when an accurate homogeneous estimate of the phantom properties is available; however, local image error over a target region of interest (ROI) is reduced. The image reconstruction process which includes a Levenberg-Marquardt style regularization provides good minimization of the objective function, yet its reduction is not always correlated with an overall image error decrease. Minimization of the bias in an ROI which contains localized changes in the optical properties can be achieved through five to nine iterations of the algorithm. Precalibration of the algorithm through statistical evaluation of phantom studies may provide a better measure of the image accuracy than that implied by minimization of the standard objective function.