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

System configuration optimization for mesoscopic fluorescence molecular tomography

Tissue engineering applications demand 3D, non-invasive, and longitudinal assessment of bioprinted constructs. Current emphasis is on developing tissue constructs mimicking in vivo conditions; however, these are increasingly challenging to image as they are typically a few millimeters thick and turbid, limiting the usefulness of classical fluorescence microscopic techniques. For such applications, we developed a Mesoscopic Fluorescence Molecular Tomography methodology that collects high information content data to enable high-resolution tomographic reconstruction of fluorescence biomarkers at millimeters depths. This imaging approach is based on an inverse problem; hence, its imaging performances are dependent on critical technical considerations including optode sampling, forward model design and inverse solver parameters. Herein, we investigate the impact of the optical system configuration parameters, including detector layout, number of detectors, combination of detector and source numbers, and scanning mode with uncoupled or coupled source and detector array, on the 3D imaging performances. Our results establish that an MFMT system with a 2D detection chain implemented in a de-scanned mode provides the optimal imaging reconstruction performances.

Sensitivity Uniformity Ratio as a New Index to Optimize the Scanning Geometry for Fluorescent Molecular Tomography

Background:

Molecular fluorescence imaging is widely used as a noninvasive method to study the cellular and molecular mechanisms. In the optical imaging system, the sensitivity is the change of the intensity received by the detector for the changed optical characteristics (fluorescence) in each sample point. Sensitivity could be considered as a function of imaging geometry. A favor imaging system has a uniform and high-sensitivity coefficient for each point of the sample. In this study, a new parameter was proposed which the optimal angle between the source and detector could be determined based on this parameter.

Methods:

For evaluation of the new method, a two-dimensional mesh with a radius of 20 mm and 5133 nodes was built. In each reconstruction, 0.5-mm fluorescence heterogeneity with a contrast-to-purpose ratio of fluorescence yield of 10 was randomly added at different points of the sample. The source and the detector were simulated in different geometric conditions. The calculations were performed using the NIRFAST and MATLAB software. The relationship between mean squared error (MSE) and sensitivity uniformity ratio (SUR) was evaluated using the correlation coefficient. Finally, based on the new index, an optimal geometrical strategy was introduced.

Results:

There was a negative correlation coefficient (R = −0.78) with the inverse relationship between the SUR and MSE indices. The reconstructed images showed that the better image quality achieved using the optimal geometry for all scanning depths. For the conventional geometry, there is an artifact in the opposite side of the inhomogeneity at the shallow depths, which has been eliminated in the reconstructed images achieved using the optimal geometry.

Conclusion:

The SUR is a powerful computational tool which could be used to determine the optimal angles between the source and detector for each scanning depth.

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.

Sampling scheme optimization for diffuse optical tomography based on data and image space rankings

We present a methodology for the optimization of sampling schemes in diffuse optical tomography (DOT). The proposed method exploits singular value decomposition (SVD) of the sensitivity matrix, or weight matrix, in DOT. Two mathematical metrics are introduced to assess and determine the optimum source–detector measurement configuration in terms of data correlation and image space resolution. The key idea of the work is to weight each data measurement, or rows in the sensitivity matrix, and similarly to weight each unknown image basis, or columns in the sensitivity matrix, according to their contribution to the rank of the sensitivity matrix, respectively. The proposed metrics offer a perspective on the data sampling and provide an efficient way of optimizing the sampling schemes in DOT. We evaluated various acquisition geometries often used in DOT by use of the proposed metrics. By iteratively selecting an optimal sparse set of data measurements, we showed that one can design a DOT scanning protocol that provides essentially the same image quality at a much reduced sampling.

Multiplexed fluorescence tomography with spectral and temporal data: demixing with intrinsic regularization

We consider the joint use of spectral and temporal data for multiplexed fluorescence molecular tomography to enable high-throughput imaging of multiple fluorescent targets in bulk tissue. This is a challenging problem due to the narrow near-infrared diagnostic window and relatively broad emission spectra of common fluorophores, and the distortion ("redshift") that the fluorophore signals undergo as they propagate through tissue. We show through a Cramér-Rao lower bound analysis that demixing with spectral-temporal data could result in an order of magnitude improvement in performance over either modality alone. To cope with the resulting large data set, we propose a novel two-stage algorithm that decouples the demixing and tomographic reconstruction operations. In this work we concentrate on the demixing stage. We introduce an approach which incorporates ideas from sparse subspace clustering and compressed sensing and does not require a regularization parameter. We report on simulations in which we simultaneously demixed four fluorophores with closely overlapping spectral and temporal profiles in a 25 mm diameter cross-sectional area with a root-mean-square error of less than 3% per fluorophore, as well as on studies of sensitivity of the method to model mismatch.