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Diffuse optical tomography (DOT) using near-infrared (NIR) light is a promising tool for noninvasive imaging of deep tissue. The approach is capable of reconstructing the quantitative optical parameters (absorption coefficient and scattering coefficient) of a soft tissue. The motivation for reconstructing the optical property variation is that it and, in particular, the absorption coefficient variation, can be used to diagnose different metabolic and disease states of tissue. In DOT, like any other medical imaging modality, the aim is to produce a reconstruction with good spatial resolution and in contrast with noisy measurements. The parameter recovery known as inverse problem in highly scattering biological tissues is a nonlinear and ill-posed problem and is generally solved through iterative methods. The algorithm uses a forward model to arrive at a prediction flux density at the tissue boundary. The forward model uses light transport models such as stochastic Monte Carlo simulation or deterministic methods such as radioactive transfer equation (RTE) or a simplified version of RTE namely the diffusion equation (DE). The finite element method (FEM) is used for discretizing the diffusion equation. The frequently used algorithm for solving the inverse problem is Newton-based Model based Iterative Image Reconstruction (N-MoBIIR). Many Variants of Gauss-Newton approaches are proposed for DOT reconstruction. The focuses of such developments are 1) to reduce the computational complexity; 2) to improve spatial recovery; and 3) to improve contrast recovery. These algorithms are 1) Hessian based MoBIIR; 2) Broyden-based MoBIIR; 3) adjoint Broyden-based MoBIIR; and 4) pseudo-dynamic approaches.

Diffuse Optical Tomography (DOT) provides an approach to probing highly scattering media such as tissue using low-energy near infra-red light (NIR) using the boundary measurements to reconstruct images of the optical parameter map of the media. Low power (1 - 10 milliwatt) NIR laser light, modulated by 100 MHz sinusoidal signal is passed through a tissue, and the existing light intensity and phase are measured on the boundary. The predominant effects are the optical absorption and scattering. The transport of photons through a biological tissue is well established through diffusion equation [1-6] which models the propagation of light through the highly scattering turbid media.

The forward model frequently uses light transport models such as stochastic Monte Carlo simulation [

A discretized version of diffusion equation is solved using finite element method (FEM) for providing the forward model for photon transport. The solution of the forward problem is used for computing the Jacobian and the simultaneous equation is solved using conjugate gradient search.

In this study, we look at many approaches used for solving the DOT problem. In DOT, the number of unknowns far exceeds the number of measurements. An accurate and reliable reconstruction procedure is essential to make DOT a practically relevant diagnostic tool. The iterative methods are often used for solving this type of both nonlinear and ill-posed problems based on nonlinear optimization in order to minimize a data-model misfit functional. The algorithm comprises a two-step procedure. The first step involves propagation of light to generate the so-called ‘forward data’ or prediction data and an update procedure that uses the difference between the prediction data and measurement data for the incremental parameter distribution. For the parameter update, one often uses a variation of Newton’s method in the hope of producing the parameter update in the right direction leading to the minimization of the error functional. This involves the computation of the Jacobian of the forward light propagation equation in each iteration. The approach is termed as model based iterative image reconstruction (MoBIIR).

The DOT involves an intense computational block that iteratively recovers unknown discretized parameter vectors from partial and noisy boundary measurement data. Being ill-posed, the reconstruction problem often requires regularization to yield meaningful results. To keep the dimension of the unknown parameters vector within reasonable limits and thus ensure the inversion procedure less ill-posed and tractable, the DOT usually attempts to solve only 2-D problems, recovering 2-D cross-sections with which 3-D images could be built-up by stacking multiple 2-D planes. The most formidable difficulty in crossing over a full-blown 3D problem is the disproportionate increase in the parameter vector dimension (a typical tenfold increase) compared to the data dimension where one cannot expect an increase beyond 2 - 3 folds. This makes the reconstruction problem more severely ill-posed to the extent that the iterations are rendered intractably owing to larger null-spaces for the (discretized) system matrices. As the iteration progresses, the mismatch (, the difference between the computed and measurement value) decreases.

The main drawback of a Newton based MoBIIR algorithm (N-MoBIIR) is the computational complexity of the algorithm. The Jacobian computation in each iteration is the root cause of the high computation time. The Broyden approach is proposed to reduce the computation time by an order of magnitude. In the Broyden-based approach, Jacobian is calculated only once with uniform distribution of optical parameters to start with and then in each iteration. It is updated over the region of interest (ROI) only using a rank-1 update procedure.. The idea behind the Jacobian (J) update is the step gradient of adjoint operator at ROI that localizes the inhomogeneities. The other difficulty with MoBIIR is that it requires regularization to ease the ill-posedness of the problem. However, the selection of a regularization parameter is arbitrary. An alternative route to the above iterative solution is through introducing an artificial dynamics in the system and treating the steady-state response of the artificially evolving dynamical system as a solution. This alternative also avoids an explicit inversion of the linearized operator as in the Gauss-Newton update equation and thus helps to get away with the regularization.

The light diffusion equation in frequency domain is,

where is the photon flux, is the diffusion coefficient and is given by

and are absorption coefficient and reduced scattering coefficient respectively. The input photon is from a source of constant intensity located at. The transmitted output optical signal measured by a photomultiplier tube.

The DOT problem is represented by a non-linear operator given by where gives model predicted data over the domain and M is the computed measurement vector obtained from and.

The image reconstruction problem seeks to find a solution such that the difference between the model predicted and the experimental measurement is minimum. To minimize the error, the cost functional is minimized and the cost functional is defined as [

where is norm. Through Gauss-Newton and Levenberg-Marquardt [1,15,16] algorithms, the minimized form of the above equation is given as,

where is the difference between model predicted data and experimental measurement data

, and J is the Jacobian matrix which has been evaluated at each iteration of MoBIIR algorithm (

The iterative reconstruction algorithm recovers an approximation to from the set of boundary measurements. By directly Taylor expanding Equation 3, and using a quadratic term involving Hessian, the perturbation equation can be written as,

where is the Hessian corresponding to the measurement. For d number of detectors, the above equation can be rewritten as,

The Equation 7 is the update equation obtained from the quadratic perturbation equation. The term is often observed to be diagonally dominant and can be denoted by, neglecting the off diagonal terms. Thus, through the incorporation of the second derivative term, the update equation has a generalized form with a physically consistent regularization term.

The major constraint of Newton’s method is the computationally expensive Jacobian evaluation [17,18]. The quasi-Newton methods provide an approximate Jacobian [

The Broyden’s Jacobian update equation becomes

Equation 9 is referred to as Broyden’s update equation. In Broyden’s method there is no clue about the initial Jacobian estimate [

Least change secant based Adjoint Broyden [

The direct and adjoint tangent conditions are

and

respectively. With respect to the Frobenius norm, the least change update of a matrix to a matrix

satisfies the direct secant condition and the adjoint secant condition, for normalized directions and, and is given as [

where,. The rank-1 update for Jacobian update based on adjoint method is given as [

The Adjoint Broyden update is categorized based on the choice of. For simplicity, we consider only secant direction [

where is the step size and is estimated through line search method. The above equation has been solved in Adjoint Broyden based MoBIIR image reconstruction.

The image reconstruction flowchart using Broyden based MoBIIR is shown in

Diffuse optical tomographic imaging is an ill-posed problem, and a regularization term is used in image reconstruction to overcome this limitation. Several regularization schemes have been proposed in the literature [

For the DOT problem, the pseudo-time linearized ODE-s for the Gauss-Newton’s normal equation for is given by:

where, ,

and

when we use Equation 5. For the case where the quadratic perturbation is used (Equation 7), then S is replaced by

We first consider the linear case wherein Equation 5 is used to arrive at the pseudo-dynamic system. While initiating the pseudo-time recursion for, the initial parameter vector may be taken corresponding to the background value which is assumed to be known. Equation 13 may be integrated in closed-form leading to the following pseudo-time evolution,

where and. In the ideal case, when the measured data is noise-free, the sequence has a limit point, which yields the desired reconstruction. In a practical scenario where the measured data is noisy, i.e, with being a measure of the noise ‘strength’. In this case, a stopping rule may have to be imposed so that the sequence is stopped at (

is the stopping time) with.