Draft:Confocal Time-of-Flight Diffuse Optical Tomography

Confocal time-of-flight diffuse optical tomography (CToF-DOT) represents a significant advancement in the realm of diffuse optical tomography, tailored specifically to enhance both resolution and imaging speed. By harnessing the depth resolution capabilities of time-of-flight alongside the spatial precision afforded by confocal optical techniques and multiplexing, researchers have successfully engineered a novel DOT system capable of achieving remarkable spatial resolution down to 1 mm, coupled with an impressive >50 mean free path s.

Motivation
Diffuse optical tomography (DOT) serves as a valuable tool in biomedical imaging, offering a non-invasive means to probe tissue composition and function. Its ability to exploit the scattering and absorption properties of biological tissues to reconstruct spatial maps of parameters like hemoglobin concentration and oxygen saturation has found widespread use in fields ranging from cancer detection to brain imaging. However, despite its versatility, DOT faces inherent challenges, primarily stemming from the scattering of light as it traverses through tissue. This scattering leads to blurred images with limited spatial resolution and depth penetration, hindering its efficacy, particularly in applications requiring precise localization of structures or accurate characterization of deeper regions within the tissue.

To overcome these limitations, researchers have been actively exploring novel variants and enhancements to traditional DOT methodologies. One promising avenue involves integrating principles from confocal microscopy and time-of-flight measurements into DOT systems. The integration of the confocal time-of-flight variant into DOT serves a dual purpose. On one hand, confocal principles tackle the pressing need for improved spatial resolution, crucial for delineating small anatomical structures. On the other hand, time-of-flight measurements specifically address depth penetration, particularly from deeper tissue layers. This combined approach enhances the fidelity and accuracy of DOT reconstructions, making it more effective for clinical diagnosis and monitoring while opening up new avenues for biomedical research.

Time-of-Flight
Time-of-flight (ToF) in the context of optical imaging refers to the measurement of the time it takes for light to travel from a source to a detector. It is a fundamental principle used in various imaging techniques, including DOT. In ToF imaging, a short pulse of light is emitted from a source, and its time of arrival at a detector is recorded. By analyzing these time-of-flight measurements, valuable information about the properties of the imaged medium or objects within it can be obtained.

In the case of diffuse optical tomography, ToF is particularly useful for imaging through strongly diffusive media like biological tissue. When light propagates through such media, it follows a random path due to multiple scattering events. This leads to a broadening of the light pulse and a delay in its arrival time at the detector compared to a direct, unobstructed path. By measuring these time delays, ToF imaging can provide insights into the scattering and absorption properties of the medium, as well as the presence and characteristics of embedded objects.

ToF imaging offers several advantages. Firstly, it enables depth-resolved imaging, allowing researchers to distinguish between signals originating from different depths within the medium. This depth localization is crucial for applications such as medical imaging, where precise information about the location of anomalies or structures within tissue is required. Secondly, ToF measurements can help mitigate the effects of scattering, as direct photons that travel without scattering will arrive earlier than those undergoing multiple scattering events. By focusing on these early-arriving photons, ToF imaging can improve image contrast and resolution, particularly in highly scattering environments.

Confocal Optics
A technique also used in confocal microscopy, confocal optics is the recording of only in focus light during collection. To achieve this in DOT, the number of sources and detectors are simplified to be equal to one another. The source and detectors then grouped into pairs, and each source-detector is placed side by side with its pair (collocated). With this method, only one measurement is taken per source, and only light returning directly back is being measured. Furthermore. this configuration will lead to a much faster computational runtime, as explained later on.

Multiplexing
Illumination multiplexing has historically been used to increase the irradiance during imaging, thus improving the signal-to-noise ratio while not increasing exposure time. With DOT, however, multiplexing can be used to speed up the data capture time. When sources are separated by sufficient distance, there is nearly no cross-talk between the measurements, as the signal is attenuated exponentially with distance as according to Beer's Law. Thus, sources that are separated sufficiently can be on and taking measurements simultaneously. Given the source-detector pairs from to the confocal setup, the measurements at each detector are attributed to the nearest source. The speed-up in capture time is directly proportional to the number of sources multiplexed together.

Even given small amounts of cross-talk, multiplexing is still useful in this system. The SNR of the image is improved, and the multiplexed signals with cross-talk can be demultiplexed computationally. The gains from multiplexing sources with and without cross-talk can also be decoupled during reconstruction.

Forward Model
The equation for the forward model is a linearization of the radiative transfer equation:

$$f(\mu) = J\mu$$

Where $$J$$ is the Jacobian or sensitivity matrix that represents contribution of each voxel in the imaging volume for a given source-detector measurement and is calculated via Monte Carlo simulations. The forward model thus returns the predicted measurement given a set of optical parameters.

Convolutional Approximation
As explained previously, ToF DOT can be modeled as a linear system. However, when utilizing confocal geometries, the size of the Jacobian matrix is greatly reduced. Furthermore, the previous linear system also becomes shift invariant. This is because when source-detectors are collocated, there is greater chance that photons will pass through. As the Jacobian matrix represents these probabilities, these measurements allow for a well-conditioned matrix, leading to a shift invariant system. This allows for a convolutional approach to the forward model to be utilized:

$$m(x, y, t) = \rho(x, y, t)\circledast\mu(x, y)$$

With a blur kernel matrix taking the place of a typical Jacobian matrix.

Inverse Model
The reconstruction goal, otherwise known as the inverse problem, in DOT is to produce a spatial distribution of the tissue optical parameters (given as $$\mu$$) from a measured dataset. Given collected measurements, $$m$$, the inverse model can then be represented as:

$$\mu = J^{-1}m$$

With $$\mu$$ being solved via a matrix inversion of the sensitivity matrix. However, for anything beyond simple geometries, the weighing matrix condition number becomes very large and the problem is both ill-defined and ill-conditioned. Thus, an iterative approach is necessary. The inverse problem is formulated as an optimization instead, given as:

$$\mu = \underset{\mu}{argmin} ||m - f(\mu)||_2 + \lambda||\mu||_1$$

Where $$\lambda||\mu||_1$$ is a regularization term to ensure the variation in $$\mu$$ is small and the linearized forward model is used. The above equation is convex and can be solved with a variety of optimization algorithms, such as the fast iterative shrinkage thresholding algorithm (FISTA). An additional modification

3D Reconstruction
The reconstruction method outlined above is readily extended to 3D image reconstruction. The sensitivity matrix $$J$$ is a 3D mapping of optical parameters to measurements and is again calculated using the Monte Carlo method. A separate regularization term is used for each depth to account for the lower sensitivity at larger depths. The overall optimization is given as:

$$\mu(x,y,z) = \underset{\mu}{argmin} ||m - f(\mu(x,y,z))||_2 + \lambda||\mu(x,y,z)||_1$$

Extending to the convolutional approximation, the forward model can be given as:

$$m(x,y,t) = \sum_{z}\rho_z(x,y,t)\circledast \mu_z(x,y)$$

The above equation can then be substituted into the optimization.

Runtime Analysis
Reconstruction for DOT can take on the order of an hour if many iterations are necessary. The iterative optimization involves repeated forward model computations, which requires matrix multiplication using the sensitivity matrix, which can have hundreds of billions of elements. The Big-O runtime for the standard forward model is given by $$O(N_sN_dN_tN_{voxels})$$, where $$N_s$$ is the number of sources, $$N_d$$ is the number of detectors, $$N_t$$ is the number of time bins, and $$N_{voxels}$$ is the number of voxels.

The convolutional approach due to the confocal setup both decreases the number of measurements from $$N_sN_dN_t$$ to $$N_sN_t$$ allows for the use of the Fast Fourier Transform. Using a blur kernel of size $$K \times K$$, the runtime complexity is given by $$O(N_tK^2\log(K))$$, which is significantly smaller than the standard runtime. Using this, the reconstruction time can be reduced to the order of milliseconds.

Spatial Resolution
As part of ToF, a technique known as time-gating is utilized. This method refers to rejecting photons that return too quickly to the detector, as they are most likely excitation light that contains no information about the target. When utilized, the spatial resolution is improved to 1 mm. The system has been shown to resolve two 0.5 mm-thick lines separated by 0.5 mm both in simulation and experimentally.

Imaging Depth
2D imaging using CToF-DOT has been shown to image at 1 mm spatial resolution at 60 mean free pathlengths (~6.5 mm). Other optical imaging techniques that rely on ballistic photons, such as two-photon microscopy or OCT, are limited to a depth of ~1 mm. The imaging depth of CToF-DOT establishes it clearly as a deep tissue imaging technique.

Applications
CToF-DOT has the potential to be used in various domains such as biomedical research and clinical imaging. In brain imaging, DOT has been instrumental in advancing neuroimaging studies by employing both linearization and nonlinear iterative approaches. With the use of continuous wave instruments, DOT facilitates tasks such as retinotopic mapping and somatosensory processing, offering insights into brain function. Notably, DOT's ability to discriminate between cerebral tissue and scalp signals has enhanced studies of functional connectivity, particularly in infants.

In breast cancer imaging, DOT emerges as a promising alternative to traditional x-ray mammography, showing potential for improved detection and treatment assessment. Despite facing challenges related to image resolution, ongoing efforts are focused on enhancing image quality through structural-prior guided approaches. Additionally, DOT extends its utility to muscle physiology, peripheral artery disease assessment, and rheumatoid arthritis imaging. By offering non-invasive assessment of skeletal muscle functions and direct visualization of peripheral circulation, DOT contributes to our understanding of physiological processes and disease diagnosis. While still in its early stages in thyroid imaging, DOT holds promise for addressing diagnostic challenges related to thyroid lesions. Its potential to overcome limitations of existing modalities, coupled with advancements in imaging techniques and algorithms, highlight its significance in modern medical practice and research.